{"header":{"message":"add standard-library","description":null,"timestamp":"2026-02-13T12:59:03.602578522Z","authors":[{}]},"hash":"GP7YEIGGU3IPEBURKAKH5SIXYRVFYZILAQRSUDU3FDFI2G66HOQQC","authors":[{"login":"tzemanovic","key":"3FpVTsmZthvnKhEGTe7aUW47SMW2aCxSdqSHT4iuQREo"}],"deps":{"hashes":[["GP7YEIGGU3IPEBURKAKH5SIXYRVFYZILAQRSUDU3FDFI2G66HOQQC","add standard-library"],["AA",null],["SHHLAOUJR4YNATSPEACPP56ITB6353COYJGUDEN47747EFPYJNQAC","Bump sass from 1.89.2 to 1.90.0 (#1124)"]],"deps":["SHHLAOUJR4YNATSPEACPP56ITB6353COYJGUDEN47747EFPYJNQAC"],"known":[]},"hunks":[{"FileAdd":{"meta":{"basename":"standard-library","metadata":576},"context":[{"change":2,"pos":2}],"contents":[]}},{"FileAdd":{"meta":{"basename":"tests","metadata":576},"context":[{"change":0,"pos":1}],"contents":[]}},{"FileAdd":{"meta":{"basename":"text","metadata":576},"context":[{"change":0,"pos":31}],"contents":[]}},{"FileAdd":{"meta":{"basename":"tabular","metadata":576},"context":[{"change":0,"pos":50}],"contents":[]}},{"FileAdd":{"meta":{"basename":"tabular.agda-lib","metadata":0},"context":[{"change":0,"pos":68}],"contents":[{"Add":"name: tabular"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":68}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":68}],"contents":[{"Add":"┌───┬───┬───┐"},{"Add":"│foo│bar│baz│"},{"Add":"├───┼───┼───┤"},{"Add":"│ 1│2 │ 3 │"},{"Add":"├───┼───┼───┤"},{"Add":"│ 6│5 │ 4 │"},{"Add":"└───┴───┴───┘"},{"Add":"┌───┬───┐"},{"Add":"│foo│bar│"},{"Add":"├───┼───┤"},{"Add":"│ 1│2 │"},{"Add":"├───┼───┤"},{"Add":"│ 4│3 │"},{"Add":"└───┴───┘"},{"Add":"+-------+"},{"Add":"|foo|bar|"},{"Add":"|---+---|"},{"Add":"| 1|2 |"},{"Add":"|---+---|"},{"Add":"| 4|3 |"},{"Add":"+-------+"},{"Add":"foo bar"},{"Add":" 1 2 "},{"Add":" 4 3 "},{"Add":"┌───┬───┐"},{"Add":"│foo│bar│"},{"Add":"│ 1│2 │"},{"Add":"│ 4│3 │"},{"Add":"└───┴───┘"},{"Add":"foo│bar"},{"Add":"───┼───"},{"Add":" 1│2 "},{"Add":"───┼───"},{"Add":" 4│3 "},{"Add":"┌─────┬─────┐"},{"Add":"│ foo │ bar │"},{"Add":"├─────┼─────┤"},{"Add":"│ 1 │ 2 │"},{"Add":"├─────┼─────┤"},{"Add":"│ 4 │ 3 │"},{"Add":"└─────┴─────┘"},{"Add":"┌─────┬─────┐"},{"Add":"│ foo │ bar │"},{"Add":"│ 1 │ 2 │"},{"Add":"│ 4 │ 3 │"},{"Add":"└─────┴─────┘"},{"Add":"┌───────┬────┬───┬───┬─────────┐"},{"Add":"│ foo │ bar│ │ │ │"},{"Add":"├───────┼────┼───┼───┼─────────┤"},{"Add":"│partial│rows│are│ok │ │"},{"Add":"├───────┼────┼───┼───┼─────────┤"},{"Add":"│ 3 │ 2│1 │...│surprise!│"},{"Add":"└───────┴────┴───┴───┴─────────┘"},{"Add":"┌───┬───┐"},{"Add":"│foo│bar│"},{"Add":"│ 1│ 2│"},{"Add":"│ 4│ 3│"},{"Add":"└───┴───┘"}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":68}],"contents":[{"Add":"-- This is taken from README.Text.Tabular"},{"Add":""},{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Function.Base"},{"Add":""},{"Add":"open import Data.List.Base"},{"Add":"open import Data.String.Base"},{"Add":"open import Data.Vec.Base"},{"Add":""},{"Add":"open import IO.Base"},{"Add":"open import IO.Finite"},{"Add":""},{"Add":"open import Text.Tabular.Base"},{"Add":"import Text.Tabular.List as Tabularˡ"},{"Add":"import Text.Tabular.Vec as Tabularᵛ"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" putStrLn $"},{"Add":" unlines (Tabularᵛ.display unicode"},{"Add":" (Right ∷ Left ∷ Center ∷ [])"},{"Add":" ( (\"foo\" ∷ \"bar\" ∷ \"baz\" ∷ [])"},{"Add":" ∷ (\"1\" ∷ \"2\" ∷ \"3\" ∷ [])"},{"Add":" ∷ (\"6\" ∷ \"5\" ∷ \"4\" ∷ [])"},{"Add":" ∷ []))"},{"Add":""},{"Add":" let foobar = (\"foo\" ∷ \"bar\" ∷ [])"},{"Add":" ∷ (\"1\" ∷ \"2\" ∷ [])"},{"Add":" ∷ (\"4\" ∷ \"3\" ∷ [])"},{"Add":" ∷ []"},{"Add":""},{"Add":" putStrLn $"},{"Add":" unlines (Tabularᵛ.display unicode"},{"Add":" (Right ∷ Left ∷ [])"},{"Add":" foobar)"},{"Add":""},{"Add":" putStrLn $"},{"Add":" unlines (Tabularᵛ.display ascii"},{"Add":" (Right ∷ Left ∷ [])"},{"Add":" foobar)"},{"Add":""},{"Add":" putStrLn $"},{"Add":" unlines (Tabularᵛ.display whitespace"},{"Add":" (Right ∷ Left ∷ [])"},{"Add":" foobar)"},{"Add":""},{"Add":" putStrLn $"},{"Add":" unlines (Tabularᵛ.display (compact unicode)"},{"Add":" (Right ∷ Left ∷ [])"},{"Add":" foobar)"},{"Add":""},{"Add":" putStrLn $"},{"Add":" unlines (Tabularᵛ.display (noBorder unicode)"},{"Add":" (Right ∷ Left ∷ [])"},{"Add":" foobar)"},{"Add":""},{"Add":" putStrLn $"},{"Add":" unlines (Tabularᵛ.display (addSpace unicode)"},{"Add":" (Right ∷ Left ∷ [])"},{"Add":" foobar)"},{"Add":""},{"Add":" putStrLn $"},{"Add":" unlines (Tabularᵛ.display (compact (addSpace unicode))"},{"Add":" (Right ∷ Left ∷ [])"},{"Add":" foobar)"},{"Add":""},{"Add":" putStrLn $"},{"Add":" unlines (Tabularˡ.display unicode"},{"Add":" (Center ∷ Right ∷ [])"},{"Add":" ( (\"foo\" ∷ \"bar\" ∷ [])"},{"Add":" ∷ (\"partial\" ∷ \"rows\" ∷ \"are\" ∷ \"ok\" ∷ [])"},{"Add":" ∷ (\"3\" ∷ \"2\" ∷ \"1\" ∷ \"...\" ∷ \"surprise!\" ∷ [])"},{"Add":" ∷ []))"},{"Add":""},{"Add":" putStrLn $"},{"Add":" unlines (unsafeDisplay (compact unicode)"},{"Add":" ( (\"foo\" ∷ \"bar\" ∷ [])"},{"Add":" ∷ (\" 1\" ∷ \" 2\" ∷ [])"},{"Add":" ∷ (\" 4\" ∷ \" 3\" ∷ [])"},{"Add":" ∷ []))"}]}},{"FileAdd":{"meta":{"basename":"regex","metadata":576},"context":[{"change":0,"pos":50}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":4146}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"regex.agda-lib","metadata":0},"context":[{"change":0,"pos":4146}],"contents":[{"Add":"name: regex"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":4146}],"contents":[{"Add":"Match found: $1 --compile-dir=../../_build -c [Main.agda] > log"},{"Add":"Match found: ./../../[_build/]Main > output"},{"Add":"No match found"},{"Add":"Match found: rm ../../[_build/]Main"},{"Add":"Match found: rm ../../[_build/MAlonzo/Code/]Main*"}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":4146}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Data.Bool.Base using (true; false)"},{"Add":"open import Data.List.Base using (_∷_; [])"},{"Add":"open import Data.List.Relation.Binary.Infix.Heterogeneous using (toView; MkView)"},{"Add":"open import Data.String using (String; toList; fromList; lines; concat)"},{"Add":"open import IO"},{"Add":"open import Function.Base using (_$_; case_of_)"},{"Add":"open import Relation.Binary.PropositionalEquality using (_≡_)"},{"Add":"open import Relation.Nullary"},{"Add":"open import Text.Regex.String"},{"Add":""},{"Add":"show : ∀ e {xs} → Dec (Match (Span e _≡_) xs (Regex.expression e)) → String"},{"Add":"show _ (no _) = \"No match found\""},{"Add":"show e (yes match) = case toView (toInfix e (match .Match.related)) of λ where"},{"Add":" (MkView pref x suff) → concat"},{"Add":" $ \"Match found: \""},{"Add":" ∷ fromList pref"},{"Add":" ∷ \"[\""},{"Add":" ∷ fromList (match .Match.list)"},{"Add":" ∷ \"]\""},{"Add":" ∷ fromList suff"},{"Add":" ∷ []"},{"Add":""},{"Add":"agdaFile : Exp"},{"Add":"agdaFile = [ 'a' ─ 'z' ∷ 'A' ─ 'Z' ∷ [] ] +"},{"Add":" ∙ singleton '.'"},{"Add":" ∙ singleton 'a'"},{"Add":" ∙ singleton 'g'"},{"Add":" ∙ singleton 'd'"},{"Add":" ∙ singleton 'a'"},{"Add":""},{"Add":"buildDir : Exp"},{"Add":"buildDir = singleton '_'"},{"Add":" ∙ ([ 'a' ─ 'z' ∷ 'A' ─ 'Z' ∷ [] ] + ∙ singleton '/') +"},{"Add":""},{"Add":"regex : Regex"},{"Add":"regex .Regex.fromStart = false"},{"Add":"regex .Regex.tillEnd = false"},{"Add":"regex .Regex.expression"},{"Add":" = agdaFile ∣ buildDir"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" text ← readFiniteFile \"run\""},{"Add":" List.forM′ (lines text) $ λ l →"},{"Add":" putStrLn (show regex $ search (toList l) regex)"}]}},{"FileAdd":{"meta":{"basename":"printf","metadata":576},"context":[{"change":0,"pos":50}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":6143}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"printf.agda-lib","metadata":0},"context":[{"change":0,"pos":6143}],"contents":[{"Add":"name: printf"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":6143}],"contents":[{"Add":"example: 3 + 2 ≡ 5"},{"Add":"example: 3 + -3 ≡ 0"},{"Add":"gcd(-9,15) ≡ 3"},{"Add":"lcm(-9,15) ≡ 45"},{"Add":"A char: c"},{"Add":"A float: 3.14"}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":6143}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Data.Integer.Base as ℤ using (+_)"},{"Add":"open import Data.Integer.Literals"},{"Add":"open import Data.Integer.GCD"},{"Add":"open import Data.Integer.LCM"},{"Add":"open import Data.Nat.Base"},{"Add":"open import Data.String.Base"},{"Add":"open import Data.Unit.Base"},{"Add":"open import Function.Base using (_$_)"},{"Add":"open import IO.Base"},{"Add":"open import IO.Finite"},{"Add":""},{"Add":"open import Text.Printf"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" let instance _ = negative"},{"Add":" putStrLn $ printf \"%s: %u + %u ≡ %u\" \"example\" 3 2 5"},{"Add":" putStrLn $ printf \"%s: %u + %i ≡ %d\" \"example\" 3 -3 (+ 3 ℤ.+ -3)"},{"Add":" putStrLn $ printf \"gcd(%d,%i) ≡ %d\" -9 (+ 15) (gcd -9 (+ 15))"},{"Add":" putStrLn $ printf \"lcm(%d,%i) ≡ %d\" -9 (+ 15) (lcm -9 (+ 15))"},{"Add":" putStrLn $ printf \"A %s: %c\" \"char\" 'c'"},{"Add":" putStrLn $ printf \"A %s: %f\" \"float\" 3.14"}]}},{"FileAdd":{"meta":{"basename":"pretty","metadata":576},"context":[{"change":0,"pos":50}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":7364}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"pretty.agda-lib","metadata":0},"context":[{"change":0,"pos":7364}],"contents":[{"Add":"name: pretty"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":7364}],"contents":[{"Add":"(setq column-number-mode t)"},{"Add":"(setq-default show-trailing-whitespace"},{"Add":" t)"},{"Add":"(add-hook 'write-file-hooks"},{"Add":" 'delete-trailing-whitespace)"},{"Add":"(setq column-number-mode t)"},{"Add":"(setq-default show-trailing-whitespace t)"},{"Add":"(add-hook 'write-file-hooks 'delete-trailing-whitespace)"}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":7364}],"contents":[{"Add":"{-# OPTIONS --guardedness --sized-types #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Size"},{"Add":""},{"Add":"open import Data.List.Base"},{"Add":"open import Data.Maybe.Base using (Maybe; just; nothing)"},{"Add":"open import Data.Nat.Base using (ℕ)"},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Data.Tree.Rose using (Rose; node)"},{"Add":""},{"Add":"open import IO.Base"},{"Add":"open import IO.Finite"},{"Add":""},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"open import Text.Pretty using (Doc; render)"},{"Add":"open module Pretty {w} = Text.Pretty w hiding (Doc; render)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" i : Size"},{"Add":" w : ℕ"},{"Add":""},{"Add":"pretty : Rose (Maybe String) i → Doc w"},{"Add":"pretty (node nothing ts) = vcat (map pretty ts)"},{"Add":"pretty (node (just a) []) = text a"},{"Add":"pretty (node (just a) ts) = parens $ text a <+> sep (map pretty ts)"},{"Add":""},{"Add":"SEXP = Rose (Maybe String) _"},{"Add":""},{"Add":"atom : String → SEXP"},{"Add":"atom a = node (just a) []"},{"Add":""},{"Add":"list : List SEXP → SEXP"},{"Add":"list = node nothing"},{"Add":""},{"Add":"colMode : SEXP"},{"Add":"colMode = node (just \"setq\") (atom \"column-number-mode\" ∷ atom \"t\" ∷ [])"},{"Add":""},{"Add":"showTrailing : SEXP"},{"Add":"showTrailing = node (just \"setq-default\")"},{"Add":" $ atom \"show-trailing-whitespace\" ∷ atom \"t\" ∷ []"},{"Add":""},{"Add":"deleteTrailing : SEXP"},{"Add":"deleteTrailing = node (just \"add-hook\")"},{"Add":" $ atom \"'write-file-hooks\""},{"Add":" ∷ atom \"'delete-trailing-whitespace\""},{"Add":" ∷ []"},{"Add":""},{"Add":"dotEmacs : SEXP"},{"Add":"dotEmacs = node nothing"},{"Add":" $ colMode ∷ showTrailing ∷ deleteTrailing ∷ []"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" let doc : Doc w; doc = pretty dotEmacs"},{"Add":" putStrLn $ render 40 doc"},{"Add":" putStrLn $ render 80 doc"}]}},{"FileAdd":{"meta":{"basename":"system","metadata":576},"context":[{"change":0,"pos":31}],"contents":[]}},{"FileAdd":{"meta":{"basename":"random","metadata":576},"context":[{"change":0,"pos":9460}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":9480}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > random_values"},{"Add":"touch output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"random.agda-lib","metadata":0},"context":[{"change":0,"pos":9480}],"contents":[{"Add":"name: ansi"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":9480}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":9480}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Level using (0ℓ)"},{"Add":"open import Data.Unit.Polymorphic.Base using (⊤)"},{"Add":"import Data.Char.Base as Char"},{"Add":"import Data.Fin.Base as Fin"},{"Add":"import Data.Fin.Show as Fin"},{"Add":"import Data.Float.Base as Float"},{"Add":"open import Data.Integer.Base using (-[1+_]; +_; -≤+)"},{"Add":"import Data.Integer.Show as ℤ"},{"Add":"open import Data.List.Base as List using ([]; _∷_)"},{"Add":"import Data.List.Show as List"},{"Add":"import Data.Nat.Base"},{"Add":"import Data.Nat.Properties as ℕ"},{"Add":"import Data.Nat.Show as ℕ"},{"Add":"open import Data.String.Base as String"},{"Add":"import Data.Vec.Show as Vec"},{"Add":"import Data.Word.Base as Word64"},{"Add":"open import IO.Base"},{"Add":"open import IO.Finite"},{"Add":"open import Function.Base using (_$_; _∘_)"},{"Add":"open import System.Random"},{"Add":"import Data.Vec.Bounded.Show as Vec≤"},{"Add":""},{"Add":"open import Relation.Binary.Construct.Closure.Reflexive"},{"Add":""},{"Add":"asA : String → String → IO {0ℓ} ⊤"},{"Add":"asA ty str = putStrLn (ty ++ \": \" ++ str)"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" asA \"Char\" ∘ Char.show =<< Char.randomIO"},{"Add":" asA \"Char\" ∘ Char.show ∘ InBounds.value =<< Char.randomRIO ' ' '~' [ ℕ.≤ᵇ⇒≤ _ _ _ ]"},{"Add":" asA \"Float\" ∘ Float.show =<< Float.randomIO"},{"Add":" asA \"Float\" ∘ Float.show ∘ InBounds.value =<< Float.randomRIO 0.0 1.0 _"},{"Add":" asA \"Integer\" ∘ ℤ.show =<< ℤ.randomIO"},{"Add":" asA \"Integer\" ∘ ℤ.show ∘ InBounds.value =<< ℤ.randomRIO -[1+ 2 ] (+ 5) -≤+"},{"Add":" asA \"Nat\" ∘ ℕ.show =<< ℕ.randomIO"},{"Add":" asA \"Nat\" ∘ ℕ.show ∘ InBounds.value =<< ℕ.randomRIO 1 10 (ℕ.≤ᵇ⇒≤ _ _ _)"},{"Add":" asA \"Word\" ∘ ℕ.show ∘ Word64.toℕ =<< Word64.randomIO"},{"Add":" asA \"Word\" ∘ ℕ.show ∘ Word64.toℕ ∘ InBounds.value =<<"},{"Add":" Word64.randomRIO (Word64.fromℕ 10) (Word64.fromℕ 20) (ℕ.≤ᵇ⇒≤ _ _ _)"},{"Add":" asA \"Fin 10\" ∘ Fin.show =<< Fin.randomIO {n = 10}"},{"Add":" asA \"Fin 10\" ∘ Fin.show ∘ InBounds.value =<<"},{"Add":" Fin.randomRIO {n = 10}"},{"Add":" (Fin.fromℕ< {m = 3} (ℕ.≤ᵇ⇒≤ _ _ _))"},{"Add":" (Fin.fromℕ< {m = 8} (ℕ.≤ᵇ⇒≤ _ _ _))"},{"Add":" (ℕ.≤ᵇ⇒≤ _ _ _)"},{"Add":" asA \"Vec≤ Integer 10\" ∘ Vec≤.show (ℤ.show ∘ InBounds.value) =<< Vec≤.randomIO (ℤ.randomRIO -[1+ 10 ] (+ 11) -≤+) 10"},{"Add":" asA \"Vec Float 5\" ∘ Vec.show (Float.show ∘ InBounds.value) =<< Vec.randomIO (Float.randomRIO -20.0 20.0 _) 5"},{"Add":" asA \"String≤ 20\" ∘ String.show =<< RangedString≤.randomIO ' ' '~' [ ℕ.≤ᵇ⇒≤ _ _ _ ] 20"}]}},{"FileAdd":{"meta":{"basename":".gitignore","metadata":0},"context":[{"change":0,"pos":9480}],"contents":[{"Add":"random_values"}]}},{"FileAdd":{"meta":{"basename":"io","metadata":576},"context":[{"change":0,"pos":9460}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":12192}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main hello world < input > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"io.agda-lib","metadata":0},"context":[{"change":0,"pos":12192}],"contents":[{"Add":"name: io"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"input","metadata":0},"context":[{"change":0,"pos":12192}],"contents":[{"Add":"hello"},{"Add":"my"},{"Add":"name"},{"Add":"is"},{"Add":"Agda"},{"Add":"exit"}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":12192}],"contents":[{"Add":"echo< echo> hello"},{"Add":"echo< echo> my"},{"Add":"echo< echo> name"},{"Add":"echo< echo> is"},{"Add":"echo< echo> Agda"},{"Add":"echo< "}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":12192}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Data.String.Base"},{"Add":"open import Function.Base using (_$_; case_of_)"},{"Add":"open import IO"},{"Add":"open import System.Exit using (exitSuccess)"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" -- Ensure no buffering so that the prompt \"echo< \""},{"Add":" -- gets outputed immediately"},{"Add":" hSetBuffering stdout noBuffering"},{"Add":" forever $ do"},{"Add":" putStr \"echo< \""},{"Add":" -- Get a line from the user and immediately inspect it"},{"Add":" -- If it's a magic \"exit\" keyword then we exit, otherwise"},{"Add":" -- we print the echo'd message and let the `forever` action"},{"Add":" -- continue"},{"Add":" str ← getLine"},{"Add":" case str of λ where"},{"Add":" \"exit\" → exitSuccess"},{"Add":" _ → putStrLn (\"echo> \" ++ str)"}]}},{"FileAdd":{"meta":{"basename":"environment","metadata":576},"context":[{"change":0,"pos":9460}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":13369}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main hello world > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":13369}],"contents":[{"Add":"Main"},{"Add":"hello world"},{"Add":"hello back!"}]}},{"FileAdd":{"meta":{"basename":"environment.agda-lib","metadata":0},"context":[{"change":0,"pos":13369}],"contents":[{"Add":"name: environment"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":13369}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Data.Maybe.Base using (fromMaybe)"},{"Add":"open import Data.String.Base using (unwords)"},{"Add":"open import IO"},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"open import System.Environment"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" prog ← getProgName"},{"Add":" putStrLn prog"},{"Add":" args ← getArgs"},{"Add":" putStrLn $ unwords args"},{"Add":" let var = \"AGDA_STDLIB_ENVIRONMENT_TEST\""},{"Add":" setEnv var \"hello back!\""},{"Add":" msg ← lookupEnv var"},{"Add":" unsetEnv var"},{"Add":" putStrLn $ fromMaybe \":(\" msg"}]}},{"FileAdd":{"meta":{"basename":"directory","metadata":576},"context":[{"change":0,"pos":9460}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":14244}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":14244}],"contents":[{"Add":"Creating tmp1"},{"Add":"Creating tmp2"},{"Add":"Saw _build"},{"Add":"Saw tmp1"},{"Add":"Saw tmp2"},{"Add":"Removing tmp1"},{"Add":"Removing tmp2"}]}},{"FileAdd":{"meta":{"basename":"directory.agda-lib","metadata":0},"context":[{"change":0,"pos":14244}],"contents":[{"Add":"name: directory"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":14244}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Data.Bool.Base using (true; false)"},{"Add":"open import Data.List.Base as List using (_∷_; [])"},{"Add":"import Data.List.Sort as Sort"},{"Add":"open import Data.String.Base using (_++_; unwords)"},{"Add":"import Data.String.Properties as Stringₚ"},{"Add":"open import IO"},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (DecTotalOrder)"},{"Add":"import Relation.Binary.Construct.On as On"},{"Add":""},{"Add":"open import System.Directory"},{"Add":"open import System.FilePath.Posix"},{"Add":""},{"Add":"decTotalOrder : DecTotalOrder _ _ _"},{"Add":"decTotalOrder ="},{"Add":" On.decTotalOrder Stringₚ.≤-decTotalOrder-≈"},{"Add":" (getFilePath {n = Nature.relative})"},{"Add":""},{"Add":"open Sort decTotalOrder using (sort)"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" let dirs = \"tmp1\" ∷ \"tmp2\" ∷ []"},{"Add":" List.forM′ dirs $ λ d → do"},{"Add":" putStrLn $ \"Creating \" ++ d"},{"Add":" createDirectory (mkFilePath d)"},{"Add":" ds ← listDirectory (mkFilePath \".\")"},{"Add":" List.forM′ (sort ds) $ λ d → do"},{"Add":" true ← doesDirectoryExist d"},{"Add":" where false → pure _"},{"Add":" let str = getFilePath d"},{"Add":" putStrLn $ \"Saw \" ++ str"},{"Add":" List.forM′ dirs $ λ d → do"},{"Add":" putStrLn $ unwords $ \"Removing\" ∷ d ∷ []"},{"Add":" removeDirectory (mkFilePath d)"}]}},{"FileAdd":{"meta":{"basename":"ansi","metadata":576},"context":[{"change":0,"pos":9460}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":15851}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main hello world > 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Function.Base using (_$_)"},{"Add":"open import System.Console.ANSI"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" putStrLn $ withCommands (setColour foreground classic blue"},{"Add":" ∷ setColour background classic white"},{"Add":" ∷ [])"},{"Add":" \"Classic blue on white\""},{"Add":" putStrLn $ withCommands (setColour foreground classic white"},{"Add":" ∷ setColour background bright blue"},{"Add":" ∷ [])"},{"Add":" \"White on bright blue\""},{"Add":" putStrLn $ withCommands (setWeight bold"},{"Add":" ∷ setColour foreground bright yellow"},{"Add":" ∷ setColour background bright blue"},{"Add":" ∷ [])"},{"Add":" \"Bold bright yellow on bright blue\""},{"Add":" putStrLn $ withCommands (setStyle italic"},{"Add":" ∷ setColour foreground bright red"},{"Add":" ∷ setColour background classic yellow"},{"Add":" ∷ [])"},{"Add":" \"Italic bright red on yellow\""},{"Add":" putStrLn $ withCommands (setBlinking rapid"},{"Add":" ∷ setUnderline double"},{"Add":" ∷ setWeight 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../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":17962}],"contents":[{"Add":"standard-library"},{"Add":" ├ Agda"},{"Add":" │ ├ Haskell"},{"Add":" │ │ ├ Haskell (bootstrap)"},{"Add":" │ │ └ C"},{"Add":" │ ├ cabal"},{"Add":" │ │ └ Haskell"},{"Add":" │ │ ├ Haskell (bootstrap)"},{"Add":" │ │ └ C"},{"Add":" │ ├ alex"},{"Add":" │ │ └ Haskell"},{"Add":" │ │ ├ Haskell (bootstrap)"},{"Add":" │ │ └ C"},{"Add":" │ └ happy"},{"Add":" │ └ Haskell"},{"Add":" │ ├ Haskell (bootstrap)"},{"Add":" │ └ C"},{"Add":" ├ cabal"},{"Add":" │ └ Haskell"},{"Add":" │ ├ Haskell (bootstrap)"},{"Add":" │ └ C"},{"Add":" └ Haskell"},{"Add":" ├ Haskell (bootstrap)"},{"Add":" └ C"}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":17962}],"contents":[{"Add":"{-# OPTIONS --guardedness --sized-types #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Data.List.Base using (_∷_; [])"},{"Add":"open import Data.String.Base using (String; unlines)"},{"Add":"open import Data.Tree.Rose"},{"Add":"open import Data.Tree.Rose.Show"},{"Add":"open import IO.Base"},{"Add":"open import IO.Finite"},{"Add":"open import Function.Base using (_$_; id)"},{"Add":""},{"Add":"dependencies : Rose String _"},{"Add":"dependencies = node \"standard-library\""},{"Add":" $ agda"},{"Add":" ∷ cabal"},{"Add":" ∷ haskell"},{"Add":" ∷ [] where"},{"Add":""},{"Add":" haskell : Rose String _"},{"Add":" haskell = node \"Haskell\""},{"Add":" $ node \"Haskell (bootstrap)\" []"},{"Add":" ∷ node \"C\" []"},{"Add":" ∷ []"},{"Add":""},{"Add":" cabal : Rose String _"},{"Add":" cabal = node \"cabal\" (haskell ∷ [])"},{"Add":""},{"Add":" agda : Rose String _"},{"Add":" agda = node \"Agda\""},{"Add":" $ haskell"},{"Add":" ∷ cabal"},{"Add":" ∷ node \"alex\" (haskell ∷ [])"},{"Add":" ∷ node \"happy\" (haskell ∷ [])"},{"Add":" ∷ []"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" putStrLn $ unlines $ showSimple id dependencies"}]}},{"FileAdd":{"meta":{"basename":"reflection","metadata":576},"context":[{"change":0,"pos":17944}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":19652}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"reflection.agda-lib","metadata":0},"context":[{"change":0,"pos":19652}],"contents":[{"Add":"name: reflection"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":19652}],"contents":[{"Add":"Data.List.Base.map : Π ({A.a} : Agda.Primitive.Level) (Π ({A} : (Set (var 0))) (Π ({B.b} : Agda.Primitive.Level) (Π ({B} : (Set (var 0))) (Π (_ : (Π (_ : (var 2)) (var 1))) (Π (_ : (Agda.Builtin.List.List {var 4} (var 3))) (Agda.Builtin.List.List {var 3} (var 2)))))))"},{"Add":"Data.List.Base.map = function {[ {A.a : Agda.Primitive.Level}{A : Set (var 0)}{B.b : Agda.Primitive.Level}{B : Set (var 0)}(f : Π (_ : (var 2)) (var 1)) ] {pat-var 4} {pat-var 3} {pat-var 2} {pat-var 1} pat-var 0 Agda.Builtin.List.List.[] → Agda.Builtin.List.List.[] {unknown} {unknown} ; [ {A.a : Agda.Primitive.Level}{A : Set (var 0)}{B.b : Agda.Primitive.Level}{B : Set (var 0)}(f : Π (_ : (var 2)) (var 1))(x : var 3)(xs : Agda.Builtin.List.List {var 5} (var 4)) ] {pat-var 6} {pat-var 5} {pat-var 4} {pat-var 3} pat-var 2 (Agda.Builtin.List.List._∷_ pat-var 1 pat-var 0) → Agda.Builtin.List.List._∷_ {unknown} {unknown} (var 2 (var 1)) (Data.List.Base.map {var 6} {var 5} {var 4} {var 3} (var 2) (var 0)) ;}"}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":19652}],"contents":[{"Add":"{-# OPTIONS --guardedness --sized-types #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Data.List.Base as List using (_∷_; [])"},{"Add":"open import Data.String.Base as String using (String)"},{"Add":"open import Data.Unit.Base using (⊤)"},{"Add":"open import Function.Base using (_$_; id)"},{"Add":""},{"Add":"open import Reflection using (TC; Term; getType; getDefinition)"},{"Add":"open import Reflection.AST.Show"},{"Add":""},{"Add":"macro"},{"Add":" runTC : TC String → Term → TC ⊤"},{"Add":" runTC tc t = let open Reflection in do"},{"Add":" u ← tc"},{"Add":" unify t (lit (string u))"},{"Add":""},{"Add":"open import IO.Base hiding (_<$>_)"},{"Add":"open import IO.Finite"},{"Add":""},{"Add":"open import Reflection.TCM.Syntax using (_<$>_)"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" let name = quote List.map"},{"Add":" putStr $ showName name String.++ \" : \""},{"Add":" putStrLn $ runTC (showTerm <$> getType name)"},{"Add":" putStr $ showName name String.++ \" = \""},{"Add":" putStrLn $ runTC (showDefinition <$> getDefinition name)"}]}},{"FileAdd":{"meta":{"basename":"num","metadata":576},"context":[{"change":0,"pos":17944}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":21837}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"num.agda-lib","metadata":0},"context":[{"change":0,"pos":21837}],"contents":[{"Add":"name: num"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":21837}],"contents":[{"Add":"0"},{"Add":"0"},{"Add":"13"},{"Add":"1101"},{"Add":"42"},{"Add":"101010"},{"Add":"1024"},{"Add":"10000000000"},{"Add":"0"},{"Add":"0"},{"Add":"13"},{"Add":"-13"},{"Add":"42"},{"Add":"-42"},{"Add":"1024"},{"Add":"-1024"},{"Add":"0/1 * 1/14 ≡ 0/14"},{"Add":"0/1 * 1/43 ≡ 0/43"},{"Add":"0/1 * 1/1025 ≡ 0/1025"},{"Add":"13/1 * 1/1 ≡ 13/1"},{"Add":"13/1 * 1/43 ≡ 13/43"},{"Add":"13/1 * 1/1025 ≡ 13/1025"},{"Add":"42/1 * 1/1 ≡ 42/1"},{"Add":"42/1 * 1/14 ≡ 42/14"},{"Add":"42/1 * 1/1025 ≡ 42/1025"},{"Add":"1024/1 * 1/1 ≡ 1024/1"},{"Add":"1024/1 * 1/14 ≡ 1024/14"},{"Add":"1024/1 * 1/43 ≡ 1024/43"},{"Add":"0/1 * 1/14 ≡ 0/1"},{"Add":"0/1 * 1/43 ≡ 0/1"},{"Add":"0/1 * 1/1025 ≡ 0/1"},{"Add":"13/1 * 1/1 ≡ 13/1"},{"Add":"13/1 * 1/43 ≡ 13/43"},{"Add":"13/1 * 1/1025 ≡ 13/1025"},{"Add":"42/1 * 1/1 ≡ 42/1"},{"Add":"42/1 * 1/14 ≡ 3/1"},{"Add":"42/1 * 1/1025 ≡ 42/1025"},{"Add":"1024/1 * 1/1 ≡ 1024/1"},{"Add":"1024/1 * 1/14 ≡ 512/7"},{"Add":"1024/1 * 1/43 ≡ 1024/43"}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":21837}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Data.List.Base using (List; _∷_; [])"},{"Add":"open import Data.String using (unwords)"},{"Add":"open import IO"},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"import Data.Nat.Base as Nat"},{"Add":"import Data.Nat.Show as ShowNat"},{"Add":""},{"Add":"tests : List Nat.ℕ"},{"Add":"tests = 0 ∷ 13 ∷ 42 ∷ 2 Nat.^ 10 ∷ []"},{"Add":""},{"Add":"nats : IO {0ℓ} _"},{"Add":"nats = let open ShowNat in"},{"Add":" List.forM′ tests $ λ n → do"},{"Add":" putStrLn (show n)"},{"Add":" putStrLn (showInBase 2 n)"},{"Add":""},{"Add":"import Data.Integer.Base as Int"},{"Add":"import Data.Integer.Show as ShowInt"},{"Add":""},{"Add":"ints : IO {0ℓ} _"},{"Add":"ints = let open Int; open ShowInt in"},{"Add":" List.forM′ tests $ λ n → do"},{"Add":" putStrLn (show (+ n))"},{"Add":" putStrLn (show (- + n))"},{"Add":""},{"Add":"import Data.Rational.Unnormalised.Base as URat"},{"Add":"import Data.Rational.Unnormalised.Show as ShowURat"},{"Add":""},{"Add":"urats : IO {0ℓ} _"},{"Add":"urats = let open URat; open ShowURat in"},{"Add":" List.forM′ tests $ λ num →"},{"Add":" List.forM′ tests $ λ denum →"},{"Add":" unless (num Nat.≡ᵇ denum) $ do"},{"Add":" putStrLn $ unwords"},{"Add":" $ show (mkℚᵘ (Int.+ num) 0)"},{"Add":" ∷ \"*\""},{"Add":" ∷ show (mkℚᵘ (Int.+ 1) denum)"},{"Add":" ∷ \"≡\""},{"Add":" ∷ show (mkℚᵘ (Int.+ num) denum)"},{"Add":" ∷ []"},{"Add":""},{"Add":"import Data.Rational.Base as Rat"},{"Add":"import Data.Rational.Show as ShowRat"},{"Add":"open import Data.Nat.Coprimality"},{"Add":""},{"Add":"rats : IO {0ℓ} _"},{"Add":"rats = let open Rat; open ShowRat in"},{"Add":" List.forM′ tests $ λ num →"},{"Add":" List.forM′ tests $ λ denum →"},{"Add":" unless (num Nat.≡ᵇ denum) $ do"},{"Add":" putStrLn $ unwords"},{"Add":" $ show (normalize num 1)"},{"Add":" ∷ \"*\""},{"Add":" ∷ show (normalize 1 (Nat.suc denum))"},{"Add":" ∷ \"≡\""},{"Add":" ∷ show (normalize num (Nat.suc denum))"},{"Add":" ∷ []"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" nats"},{"Add":" ints"},{"Add":" urats"},{"Add":" rats"}]}},{"FileAdd":{"meta":{"basename":"runtests.agda","metadata":0},"context":[{"change":0,"pos":31}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module runtests where"},{"Add":""},{"Add":"open import Data.List.Base as List using (_∷_; [])"},{"Add":"open import Data.String.Base using (String; _++_)"},{"Add":"open import IO.Base"},{"Add":"open import Function.Base"},{"Add":"open import Test.Golden"},{"Add":""},{"Add":"dataTests : TestPool"},{"Add":"dataTests = mkTestPool \"Data structures\""},{"Add":" $ \"appending\""},{"Add":" ∷ \"colist\""},{"Add":" ∷ \"list\""},{"Add":" ∷ \"rational\""},{"Add":" ∷ \"rational-unnormalised\""},{"Add":" ∷ \"trie\""},{"Add":" ∷ \"bytestring\""},{"Add":" ∷ []"},{"Add":""},{"Add":"systemTests : TestPool"},{"Add":"systemTests = mkTestPool \"System modules\""},{"Add":" $ \"ansi\""},{"Add":" ∷ \"directory\""},{"Add":" ∷ \"environment\""},{"Add":" ∷ \"io\""},{"Add":" ∷ \"random\""},{"Add":" ∷ []"},{"Add":""},{"Add":"showTests : TestPool"},{"Add":"showTests = mkTestPool \"Show instances\""},{"Add":" $ \"num\""},{"Add":" ∷ \"reflection\""},{"Add":" ∷ \"tree\""},{"Add":" ∷ []"},{"Add":""},{"Add":"textTests : TestPool"},{"Add":"textTests = mkTestPool \"Text libraries\""},{"Add":" $ \"pretty\""},{"Add":" ∷ \"printf\""},{"Add":" ∷ \"regex\""},{"Add":" ∷ \"tabular\""},{"Add":" ∷ []"},{"Add":""},{"Add":"monadTests : TestPool"},{"Add":"monadTests = mkTestPool \"Monad transformers\""},{"Add":" $ \"counting\""},{"Add":" ∷ \"fibonacci\""},{"Add":" ∷ \"pythagorean\""},{"Add":" ∷ \"tcm\""},{"Add":" ∷ []"},{"Add":""},{"Add":"reflectionTests : TestPool"},{"Add":"reflectionTests = mkTestPool \"Reflection machinery\""},{"Add":" $ \"assumption\""},{"Add":" ∷ []"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ ignore $ runner"},{"Add":" $ testPaths \"data\" dataTests"},{"Add":" ∷ testPaths \"monad\" monadTests"},{"Add":" ∷ testPaths \"reflection\" reflectionTests"},{"Add":" ∷ testPaths \"show\" showTests"},{"Add":" ∷ testPaths \"system\" systemTests"},{"Add":" ∷ testPaths \"text\" textTests"},{"Add":" ∷ [] where"},{"Add":""},{"Add":" testPaths : String → TestPool → TestPool"},{"Add":" testPaths dir pool ="},{"Add":" let testCases = List.map ((dir ++ \"/\") ++_) (pool .TestPool.testCases)"},{"Add":" in record pool { testCases = testCases }"}]}},{"FileAdd":{"meta":{"basename":"reflection","metadata":576},"context":[{"change":0,"pos":31}],"contents":[]}},{"FileAdd":{"meta":{"basename":"assumption","metadata":576},"context":[{"change":0,"pos":26118}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":26142}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":26142}],"contents":[{"Add":"function {[ {A.a : Agda.Primitive.Level}{A : Set (var 0)}{B.b : Agda.Primitive.Level}{B : Set (var 0)}(x : var 2)(y : var 1) ] {pat-var 5} {pat-var 4} {pat-var 3} {pat-var 2} pat-var 1 pat-var 0 → var 0 ;}"},{"Add":"function {[ {A.a : Agda.Primitive.Level}{A : Set (var 0)}{B.b : Agda.Primitive.Level}{B : Set (var 0)}(x : var 2)(y : var 1) ] {pat-var 5} {pat-var 4} {pat-var 3} {pat-var 2} pat-var 1 pat-var 0 → var 1 ;}"},{"Add":"function {[ {A.a : Agda.Primitive.Level}{A : Set (var 0)}{B.b : Agda.Primitive.Level}{B : Set (var 0)}(x : var 2)(y : var 1)(z : var 2) ] {pat-var 6} {pat-var 5} {pat-var 4} {pat-var 3} pat-var 2 pat-var 1 pat-var 0 → var 2 ;}"},{"Add":"function {[ {A.a : Agda.Primitive.Level}{A : Set (var 0)}{B.b : Agda.Primitive.Level}{B : Set (var 0)}(x : Agda.Builtin.List.List {Agda.Primitive._⊔_ (var 3) (var 1)} (Π (_ : (var 2)) (var 1)))(y : var 3)(z : var 2)(a : var 3) ] {pat-var 7} {pat-var 6} {pat-var 5} {pat-var 4} pat-var 3 pat-var 2 pat-var 1 pat-var 0 → var 3 ;}"},{"Add":"function {[ {A.a : Agda.Primitive.Level}{A : Set (var 0)}{B.b : Agda.Primitive.Level}{B : Set (var 0)}(x : Π (_ : (var 2)) (Agda.Builtin.List.List {var 2} (var 1)))(y : var 3)(z : var 2)(a : Agda.Builtin.List.List {var 4} (var 3)) ] {pat-var 7} {pat-var 6} {pat-var 5} {pat-var 4} pat-var 3 pat-var 2 pat-var 1 pat-var 0 → var 3 ;}"}]}},{"FileAdd":{"meta":{"basename":"assumption.agda-lib","metadata":0},"context":[{"change":0,"pos":26142}],"contents":[{"Add":"name: assumption"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":26142}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Data.Bool.Base using (if_then_else_)"},{"Add":"open import Data.List.Base using (List; []; _∷_; concatMap; map)"},{"Add":"open import Data.Maybe.Base using (Maybe; nothing; just)"},{"Add":"open import Data.Nat.Base using (ℕ; suc)"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Level using (Level)"},{"Add":"open import Data.Unit.Base using (⊤)"},{"Add":"open import Function.Base using (case_of_; _$_)"},{"Add":"open import Reflection.TCM hiding (pure)"},{"Add":"open import Reflection.AST.Term"},{"Add":"open import Reflection.AST.Literal hiding (_≟_)"},{"Add":"open import Reflection.AST.Argument using (Arg; unArg; arg)"},{"Add":"open import Reflection.AST.DeBruijn"},{"Add":"open import Reflection.AST.Show"},{"Add":"open import Relation.Nullary.Decidable using (does)"},{"Add":""},{"Add":"open import Effect.Monad"},{"Add":"open RawMonad {{...}}"},{"Add":""},{"Add":"open import Data.Maybe.Instances"},{"Add":"open import Reflection.TCM.Instances"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":""},{"Add":"-- As the doc states (cf. https://agda.readthedocs.io/en/latest/language/reflection.html#type-checking-computations)"},{"Add":"-- Note that the types in the context are valid in the rest of the context."},{"Add":"-- To use in the current context they need to be weakened by 1 + their position"},{"Add":"-- in the list."},{"Add":""},{"Add":"-- That is to say that the type of the current goal needs to be strengthened"},{"Add":"-- before being compared to the type of the most local variable. The goal"},{"Add":"-- indeed lives below that variable's binding site!"},{"Add":""},{"Add":"searchEntry : ℕ → Type → List (String × Arg Type) → Maybe ℕ"},{"Add":"searchEntry n ty [] = nothing"},{"Add":"searchEntry n ty ((_ , e) ∷ es) = do"},{"Add":" ty ← strengthen ty"},{"Add":" if does (ty ≟ unArg e)"},{"Add":" then just n"},{"Add":" else searchEntry (suc n) ty es"},{"Add":""},{"Add":"macro"},{"Add":" assumption : Term → TC ⊤"},{"Add":" assumption hole = do"},{"Add":" asss ← getContext"},{"Add":" goal ← inferType hole"},{"Add":" debugPrint \"\" 10"},{"Add":" (strErr \"Context : \""},{"Add":" ∷ concatMap (λ where (_ , arg info ty) → strErr \"\\n \" ∷ termErr ty ∷ []) asss)"},{"Add":" let res = searchEntry 0 goal asss"},{"Add":" case res of λ where"},{"Add":" nothing → typeError (strErr \"Couldn't find an assumption of type: \" ∷ termErr goal ∷ [])"},{"Add":" (just idx) → unify hole (var idx [])"},{"Add":""},{"Add":"test₀ : A → B → B"},{"Add":"test₀ x y = assumption"},{"Add":""},{"Add":"test₁ : A → B → A"},{"Add":"test₁ x y = assumption"},{"Add":""},{"Add":"test₂ : A → B → B → A"},{"Add":"test₂ x y z = assumption"},{"Add":""},{"Add":"test₃ : List (A → B) → A → B → B → List (A → B)"},{"Add":"test₃ x y z a = assumption"},{"Add":""},{"Add":"test₄ : (A → List B) → A → B → List B → A → List B"},{"Add":"test₄ x y z a = assumption"},{"Add":""},{"Add":"open import IO.Base using (Main; run)"},{"Add":"open import IO.Finite"},{"Add":"open import IO.Instances"},{"Add":""},{"Add":"macro"},{"Add":" runTC : TC String → Term → TC ⊤"},{"Add":" runTC tc t = do"},{"Add":" u ← tc"},{"Add":" unify t (lit (string u))"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" putStrLn $ runTC (showDefinition <$> getDefinition (quote test₀))"},{"Add":" putStrLn $ runTC (showDefinition <$> getDefinition (quote test₁))"},{"Add":" putStrLn $ runTC (showDefinition <$> getDefinition (quote test₂))"},{"Add":" putStrLn $ runTC (showDefinition <$> getDefinition (quote test₃))"},{"Add":" putStrLn $ runTC (showDefinition <$> getDefinition (quote test₄))"}]}},{"FileAdd":{"meta":{"basename":"monad","metadata":576},"context":[{"change":0,"pos":31}],"contents":[]}},{"FileAdd":{"meta":{"basename":"tcm","metadata":576},"context":[{"change":0,"pos":30888}],"contents":[]}},{"FileAdd":{"meta":{"basename":"tcm.agda-lib","metadata":0},"context":[{"change":0,"pos":30907}],"contents":[{"Add":"name: tcm"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":30907}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":30907}],"contents":[{"Add":"IO.Base.Main"},{"Add":"Π ({a} : Agda.Primitive.Level) (Π (_ : Agda.Builtin.String.String) (IO.Base.IO {var 1} (Data.Unit.Polymorphic.Base.⊤ {var 1})))"}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":30907}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"-- Taken from #1530"},{"Add":""},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Data.Maybe.Base using (Maybe; nothing; just)"},{"Add":"open import Data.Unit using (⊤; tt)"},{"Add":""},{"Add":"open import Reflection.AST.Literal"},{"Add":"open import Reflection.AST.Term"},{"Add":"open import Reflection.AST.Show using (showTerm)"},{"Add":"open import Reflection.TCM using (TC; inferType; unify)"},{"Add":""},{"Add":"open import Effect.Monad"},{"Add":"open RawMonad {{...}} public using (pure; _>>=_; _>>_)"},{"Add":""},{"Add":"open import Data.Maybe.Instances"},{"Add":"open import Reflection.TCM.Instances"},{"Add":""},{"Add":"macro"},{"Add":" goalErr : Term → Term → TC ⊤"},{"Add":" goalErr t goal = do"},{"Add":" goalType ← inferType t"},{"Add":" unify goal (lit (string (showTerm goalType)))"},{"Add":""},{"Add":""},{"Add":"open import IO.Base hiding (_>>=_; _>>_)"},{"Add":"open import IO.Finite"},{"Add":"open import IO.Instances"},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" putStrLn (goalErr main)"},{"Add":" putStrLn (goalErr putStrLn)"}]}},{"FileAdd":{"meta":{"basename":"pythagorean","metadata":576},"context":[{"change":0,"pos":30888}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":32279}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"pythagorean.agda-lib","metadata":0},"context":[{"change":0,"pos":32279}],"contents":[{"Add":"name: pythagorean"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":32279}],"contents":[{"Add":"3² + 4² = 5²"},{"Add":"4² + 3² = 5²"},{"Add":"5² + 12² = 13²"},{"Add":"6² + 8² = 10²"},{"Add":"7² + 24² = 25²"},{"Add":"8² + 6² = 10²"},{"Add":"8² + 15² = 17²"},{"Add":"9² + 12² = 15²"},{"Add":"9² + 40² = 41²"},{"Add":"10² + 24² = 26²"},{"Add":"11² + 60² = 61²"},{"Add":"12² + 5² = 13²"},{"Add":"12² + 9² = 15²"},{"Add":"12² + 16² = 20²"},{"Add":"12² + 35² = 37²"},{"Add":"13² + 84² = 85²"},{"Add":"14² + 48² = 50²"},{"Add":"15² + 8² = 17²"},{"Add":"15² + 20² = 25²"},{"Add":"15² + 36² = 39²"},{"Add":"16² + 12² = 20²"},{"Add":"16² + 30² = 34²"},{"Add":"16² + 63² = 65²"},{"Add":"18² + 24² = 30²"},{"Add":"18² + 80² = 82²"},{"Add":"20² + 15² = 25²"},{"Add":"20² + 21² = 29²"},{"Add":"20² + 48² = 52²"},{"Add":"21² + 20² = 29²"},{"Add":"21² + 28² = 35²"},{"Add":"21² + 72² = 75²"},{"Add":"24² + 7² = 25²"},{"Add":"24² + 10² = 26²"},{"Add":"24² + 18² = 30²"},{"Add":"24² + 32² = 40²"},{"Add":"24² + 45² = 51²"},{"Add":"24² + 70² = 74²"},{"Add":"25² + 60² = 65²"},{"Add":"27² + 36² = 45²"},{"Add":"28² + 21² = 35²"},{"Add":"28² + 45² = 53²"},{"Add":"28² + 96² = 100²"},{"Add":"30² + 16² = 34²"},{"Add":"30² + 40² = 50²"},{"Add":"30² + 72² = 78²"},{"Add":"32² + 24² = 40²"},{"Add":"32² + 60² = 68²"},{"Add":"33² + 44² = 55²"},{"Add":"33² + 56² = 65²"},{"Add":"35² + 12² = 37²"},{"Add":"35² + 84² = 91²"},{"Add":"36² + 15² = 39²"},{"Add":"36² + 27² = 45²"},{"Add":"36² + 48² = 60²"},{"Add":"36² + 77² = 85²"},{"Add":"39² + 52² = 65²"},{"Add":"39² + 80² = 89²"},{"Add":"40² + 9² = 41²"},{"Add":"40² + 30² = 50²"},{"Add":"40² + 42² = 58²"},{"Add":"40² + 75² = 85²"},{"Add":"42² + 40² = 58²"},{"Add":"42² + 56² = 70²"},{"Add":"44² + 33² = 55²"},{"Add":"45² + 24² = 51²"},{"Add":"45² + 28² = 53²"},{"Add":"45² + 60² = 75²"},{"Add":"48² + 14² = 50²"},{"Add":"48² + 20² = 52²"},{"Add":"48² + 36² = 60²"},{"Add":"48² + 55² = 73²"},{"Add":"48² + 64² = 80²"},{"Add":"51² + 68² = 85²"},{"Add":"52² + 39² = 65²"},{"Add":"54² + 72² = 90²"},{"Add":"55² + 48² = 73²"},{"Add":"56² + 33² = 65²"},{"Add":"56² + 42² = 70²"},{"Add":"57² + 76² = 95²"},{"Add":"60² + 11² = 61²"},{"Add":"60² + 25² = 65²"},{"Add":"60² + 32² = 68²"},{"Add":"60² + 45² = 75²"},{"Add":"60² + 63² = 87²"},{"Add":"60² + 80² = 100²"},{"Add":"63² + 16² = 65²"},{"Add":"63² + 60² = 87²"},{"Add":"64² + 48² = 80²"},{"Add":"65² + 72² = 97²"},{"Add":"68² + 51² = 85²"},{"Add":"70² + 24² = 74²"},{"Add":"72² + 21² = 75²"},{"Add":"72² + 30² = 78²"},{"Add":"72² + 54² = 90²"},{"Add":"72² + 65² = 97²"},{"Add":"75² + 40² = 85²"},{"Add":"76² + 57² = 95²"},{"Add":"77² + 36² = 85²"},{"Add":"80² + 18² = 82²"},{"Add":"80² + 39² = 89²"},{"Add":"80² + 60² = 100²"},{"Add":"84² + 13² = 85²"},{"Add":"84² + 35² = 91²"},{"Add":"96² + 28² = 100²"}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":32279}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Data.Bool.Base using (if_then_else_)"},{"Add":"open import Data.List.Base using (List; []; _∷_; catMaybes)"},{"Add":"open import Data.List.Effectful.Transformer"},{"Add":"open import Data.List.Effectful"},{"Add":"open import Data.Maybe.Base using (just; nothing)"},{"Add":"open import Data.Maybe.Effectful.Transformer"},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc; _+_; _^_; _≡ᵇ_)"},{"Add":"open import Data.Nat.Show using (show)"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Data.String.Base using (_++_)"},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"open import Effect.Applicative"},{"Add":"open import Effect.Monad"},{"Add":"open import Effect.Monad.Identity"},{"Add":""},{"Add":"open import IO.Base using (Main; run)"},{"Add":"open import IO.Finite using (putStrLn)"},{"Add":""},{"Add":"open import Data.List.Instances"},{"Add":"open import Data.Maybe.Instances"},{"Add":"open import Effect.Monad.Identity.Instances"},{"Add":"open import IO.Instances"},{"Add":""},{"Add":"open RawMonad {{...}}"},{"Add":"open RawApplicativeZero {{...}} using (guard)"},{"Add":"open TraversableA {{...}}"},{"Add":""},{"Add":"1⋯100 : List ℕ"},{"Add":"1⋯100 = rangeFrom 1 100 where"},{"Add":""},{"Add":" rangeFrom : (start steps : ℕ) → List ℕ"},{"Add":" rangeFrom start zero = []"},{"Add":" rangeFrom start (suc n) = start ∷ rangeFrom (suc start) n"},{"Add":""},{"Add":"triples : MaybeT (ListT Identity) (ℕ × ℕ × ℕ)"},{"Add":"triples = do"},{"Add":" let range = mkMaybeT (mkListT (pure (pure <$> 1⋯100)))"},{"Add":" p ← range"},{"Add":" q ← range"},{"Add":" r ← range"},{"Add":" mkMaybeT (mkListT (pure (pure (guard (p ^ 2 + q ^ 2 ≡ᵇ r ^ 2)))))"},{"Add":" pure (p , q , r)"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run"},{"Add":" $ ignore $ forA (catMaybes $ runIdentity $ runListT $ runMaybeT triples)"},{"Add":" $ λ (p , q , r) → do let sqStr = λ x → show x ++ \"²\""},{"Add":" putStrLn (sqStr p ++ \" + \" ++ sqStr q ++ \" = \" ++ sqStr r)"}]}},{"FileAdd":{"meta":{"basename":"fibonacci","metadata":576},"context":[{"change":0,"pos":30888}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":36265}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"fibonacci.agda-lib","metadata":0},"context":[{"change":0,"pos":36265}],"contents":[{"Add":"name: fibonacci"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":36265}],"contents":[{"Add":"fib 0 = 0"},{"Add":"fib 1 = 1"},{"Add":"fib 2 = 1"},{"Add":"fib 3 = 2"},{"Add":"fib 5 = 5"},{"Add":"fib 6 = 8"},{"Add":"fib 7 = 13"},{"Add":"fib 8 = 21"},{"Add":"fib 9 = 34"},{"Add":"fib 10 = 55"}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":36265}],"contents":[{"Add":"{-# OPTIONS --guardedness --rewriting #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Data.Bool.Base using (Bool; true; false)"},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc; _+_)"},{"Add":"open import Data.Nat.Show using (show)"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Data.String.Base using (String; _++_)"},{"Add":""},{"Add":"open import Data.Default"},{"Add":"open import Debug.Trace using (trace)"},{"Add":""},{"Add":"open import IO.Base using (Main; run)"},{"Add":"open import IO.Finite using (putStrLn)"},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"open import Effect.Monad"},{"Add":"open import Effect.Monad.State"},{"Add":"open import Effect.Monad.State.Instances"},{"Add":"open import Effect.Monad.Identity.Instances"},{"Add":"open import IO.Instances"},{"Add":""},{"Add":"open RawMonad {{...}}"},{"Add":"open RawMonadState {{...}}"},{"Add":""},{"Add":"record Fib : Set where"},{"Add":" field tracing : Bool"},{"Add":" index : ℕ"},{"Add":" current : ℕ"},{"Add":" next : ℕ"},{"Add":"open Fib"},{"Add":""},{"Add":"initFib : Bool → Fib"},{"Add":"initFib b = record"},{"Add":" { tracing = b"},{"Add":" ; index = 0"},{"Add":" ; current = 0"},{"Add":" ; next = 1"},{"Add":" }"},{"Add":""},{"Add":"displayFib : ℕ → ℕ → String"},{"Add":"displayFib idx fibn = \"fib \" ++ show idx ++ \" = \" ++ show fibn"},{"Add":""},{"Add":"fibM : ℕ → State Fib ℕ"},{"Add":"fibM 0 = gets current"},{"Add":"fibM (suc n) = do"},{"Add":" b ← gets tracing"},{"Add":" when b $ do"},{"Add":" idx ← gets index"},{"Add":" fibn ← gets current"},{"Add":" trace (displayFib idx fibn) (pure _)"},{"Add":" modify (λ r → record r { index = suc (index r) ; current = next r ; next = next r + current r })"},{"Add":" fibM n"},{"Add":""},{"Add":"fib : ℕ → Bool → ℕ"},{"Add":"fib n b = evalState (fibM n) (initFib b)"},{"Add":""},{"Add":"fibStr : ℕ → {{WithDefault false}} → String"},{"Add":"fibStr n {{b}} = displayFib n (fib n (value b))"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" putStrLn $ fibStr 0"},{"Add":" putStrLn $ fibStr 1"},{"Add":" putStrLn $ fibStr 2"},{"Add":" putStrLn $ fibStr 3"},{"Add":" putStrLn $ fibStr 5"},{"Add":" putStrLn $ fibStr 6"},{"Add":" putStrLn $ fibStr 7"},{"Add":" putStrLn $ fibStr 8"},{"Add":" putStrLn $ fibStr 9"},{"Add":" putStrLn $ fibStr 10"}]}},{"FileAdd":{"meta":{"basename":"counting","metadata":576},"context":[{"change":0,"pos":30888}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":38467}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":38467}],"contents":[{"Add":"First: 1"},{"Add":"Second: 3"},{"Add":"Third: 8"}]}},{"FileAdd":{"meta":{"basename":"counting.agda-lib","metadata":0},"context":[{"change":0,"pos":38467}],"contents":[{"Add":"name: counting"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":38467}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Data.Nat.Base"},{"Add":"open import Data.Nat.Show using (show)"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Data.String.Base using (String; _++_)"},{"Add":"open import IO.Base as IO using (IO; Main; run)"},{"Add":"open import IO.Finite using (putStrLn)"},{"Add":"open import IO.Effectful as IO"},{"Add":"open import Function.Base using (_∘′_; _$_; const)"},{"Add":""},{"Add":"open import Level using (Lift; 0ℓ; lift; lower)"},{"Add":""},{"Add":"open import Effect.Monad"},{"Add":"open import Effect.Monad.Reader.Transformer as Reader"},{"Add":"open import Effect.Monad.State.Transformer as State"},{"Add":"open import Effect.Monad.IO"},{"Add":""},{"Add":"open import IO.Instances"},{"Add":""},{"Add":"open import Effect.Monad.IO.Instances"},{"Add":"open import Effect.Monad.State.Instances"},{"Add":"open import Effect.Monad.Reader.Instances"},{"Add":""},{"Add":"open RawMonad {{...}}"},{"Add":"open RawMonadReader {{...}}"},{"Add":"open RawMonadState {{...}}"},{"Add":"open RawMonadIO {{...}}"},{"Add":""},{"Add":"step : ∀ {M : Set Level.zero → Set (Level.suc Level.zero)} →"},{"Add":" {{RawMonad M}} →"},{"Add":" {{RawMonadReader String M}} →"},{"Add":" {{RawMonadState ℕ M}} →"},{"Add":" {{RawMonadIO M}} →"},{"Add":" (ℕ → ℕ) →"},{"Add":" M _"},{"Add":"step f = do"},{"Add":" modify f"},{"Add":" str ← ask"},{"Add":" n ← get"},{"Add":" let msg = str ++ show n"},{"Add":" liftIO (putStrLn msg)"},{"Add":""},{"Add":"script : ∀ {M} →"},{"Add":" {{RawMonad M}} →"},{"Add":" {{RawMonadReader String M}} →"},{"Add":" {{RawMonadState ℕ M}} →"},{"Add":" {{RawMonadIO M}} →"},{"Add":" M _"},{"Add":"script = do"},{"Add":" step suc"},{"Add":" local (const \"Second: \") $ step (3 *_)"},{"Add":" local (const \"Third: \") $ step (2 ^_)"},{"Add":""},{"Add":"it : ∀ {a} {A : Set a} → {{A}} → A"},{"Add":"it {{x}} = x"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ evalStateT it (runReaderT script \"First: \") 0"}]}},{"FileAdd":{"meta":{"basename":"data","metadata":576},"context":[{"change":0,"pos":31}],"contents":[]}},{"FileAdd":{"meta":{"basename":"trie","metadata":576},"context":[{"change":0,"pos":40349}],"contents":[]}},{"FileAdd":{"meta":{"basename":"trie.agda-lib","metadata":0},"context":[{"change":0,"pos":40367}],"contents":[{"Add":"name: trie"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":40367}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":40367}],"contents":[{"Add":"ID \"fix\""},{"Add":"ID \"f\""},{"Add":"ID \"x\""},{"Add":"EQ"},{"Add":"LET"},{"Add":"ID \"b\""},{"Add":"EQ"},{"Add":"ID \"fix\""},{"Add":"ID \"f\""},{"Add":"IN"},{"Add":"LPAR"},{"Add":"ID \"f\""},{"Add":"ID \"b\""},{"Add":"RPAR"},{"Add":"ID \"x\""}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":40367}],"contents":[{"Add":"-- Taken from README.Data.Trie.NonDependent"},{"Add":""},{"Add":"{-# OPTIONS --guardedness --sized-types #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Data.Unit"},{"Add":"open import Data.Bool"},{"Add":"open import Data.Char as Char hiding (show)"},{"Add":"import Data.Char.Properties as Char"},{"Add":"open import Data.List.Base as List using (List; []; _∷_)"},{"Add":"open import Data.List.Fresh as List# using (List#; []; _∷#_)"},{"Add":"open import Data.Maybe as Maybe"},{"Add":"open import Data.Product as Prod"},{"Add":"open import Data.String as String using (String; unlines; _++_)"},{"Add":"open import Data.These as These"},{"Add":""},{"Add":"open import Function.Base using (case_of_; _$_; _∘′_; id; _on_)"},{"Add":"open import Relation.Nary"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Nullary.Decidable using (¬?)"},{"Add":""},{"Add":"open import Data.Trie Char.<-strictTotalOrder"},{"Add":"open import Data.Tree.AVL.Value"},{"Add":""},{"Add":"open import IO.Base"},{"Add":"open import IO.Finite"},{"Add":""},{"Add":"record Lexer t : Set (suc t) where"},{"Add":" field"},{"Add":" Tok : Set t"},{"Add":""},{"Add":" Keyword : Set t"},{"Add":" Keyword = String × Tok"},{"Add":""},{"Add":" Distinct : Rel Keyword 0ℓ"},{"Add":" Distinct a b = ⌊ ¬? ((proj₁ a) String.≟ (proj₁ b)) ⌋"},{"Add":""},{"Add":" field"},{"Add":" keywords : List# Keyword Distinct"},{"Add":" breaking : Char → ∃ λ b → if b then Maybe Tok else Lift _ ⊤"},{"Add":" default : String → Tok"},{"Add":""},{"Add":""},{"Add":"module _ {t} (L : Lexer t) where"},{"Add":" open Lexer L"},{"Add":""},{"Add":" tokenize : String → List Tok"},{"Add":" tokenize = start ∘′ String.toList where"},{"Add":""},{"Add":" mutual"},{"Add":""},{"Add":" Keywords : Set _"},{"Add":" Keywords = Trie (const _ Tok) _"},{"Add":""},{"Add":" init : Keywords"},{"Add":" init = fromList $ List.map (Prod.map₁ String.toList) $ proj₁ $ List#.toList keywords"},{"Add":""},{"Add":" start : List Char → List Tok"},{"Add":" start = loop [] init"},{"Add":""},{"Add":" loop : (acc : List Char) → -- chars read so far in this token"},{"Add":" (toks : Keywords) → -- keyword candidates left at this point"},{"Add":" (input : List Char) → -- list of chars to tokenize"},{"Add":" List Tok"},{"Add":" loop acc toks [] = push acc []"},{"Add":" loop acc toks (c ∷ cs) = case breaking c of λ where"},{"Add":" (true , m) → push acc $ maybe′ _∷_ id m $ start cs"},{"Add":" (false , _) → case lookupValue toks (c ∷ []) of λ where"},{"Add":" (just tok) → tok ∷ start cs"},{"Add":" nothing → loop (c ∷ acc) (lookupTrie toks c) cs"},{"Add":""},{"Add":" push : List Char → List Tok → List Tok"},{"Add":" push [] ts = ts"},{"Add":" push cs ts = default (String.fromList (List.reverse cs)) ∷ ts"},{"Add":""},{"Add":"module LetIn where"},{"Add":""},{"Add":" data TOK : Set where"},{"Add":" LET EQ IN : TOK"},{"Add":" LPAR RPAR : TOK"},{"Add":" ID : String → TOK"},{"Add":""},{"Add":" show : TOK → String"},{"Add":" show LET = \"LET\""},{"Add":" show EQ = \"EQ\""},{"Add":" show IN = \"IN\""},{"Add":" show LPAR = \"LPAR\""},{"Add":" show RPAR = \"RPAR\""},{"Add":" show (ID x) = \"ID \\\"\" ++ x ++ \"\\\"\""},{"Add":""},{"Add":" keywords : List# (String × TOK) (λ a b → ⌊ ¬? ((proj₁ a) String.≟ (proj₁ b)) ⌋)"},{"Add":" keywords = (\"let\" , LET)"},{"Add":" ∷# (\"=\" , EQ)"},{"Add":" ∷# (\"in\" , IN)"},{"Add":" ∷# []"},{"Add":""},{"Add":" breaking : Char → ∃ (λ b → if b then Maybe TOK else Lift 0ℓ ⊤)"},{"Add":" breaking c = if isSpace c then true , nothing else parens c where"},{"Add":""},{"Add":" parens : Char → _"},{"Add":" parens '(' = true , just LPAR"},{"Add":" parens ')' = true , just RPAR"},{"Add":" parens _ = false , _"},{"Add":""},{"Add":" default : String → TOK"},{"Add":" default = ID"},{"Add":""},{"Add":"letIn : Lexer 0ℓ"},{"Add":"letIn = record { LetIn }"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" let open LetIn"},{"Add":" putStrLn $ unlines $ List.map show $"},{"Add":" tokenize letIn \"fix f x = let b = fix f in (f b) x\""}]}},{"FileAdd":{"meta":{"basename":"rational-unnormalised","metadata":576},"context":[{"change":0,"pos":40349}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":44085}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"rational-unnormalised.agda-lib","metadata":0},"context":[{"change":0,"pos":44085}],"contents":[{"Add":"name: rational"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":44085}],"contents":[{"Add":"[-4, -3, -2, -1, 0, 1, 2, 3]"},{"Add":"[-3, -2, -1, 0, 1, 2, 3, 4]"},{"Add":"[-3, -2, -1, 0, 0, 1, 2, 3]"},{"Add":"[-4, -3, -2, 0, 0, 2, 3, 4]"},{"Add":"[5/10, 7/10, 5/10, 3/10, 3/10, 5/10, 7/10, 5/10]"}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":44085}],"contents":[{"Add":"{-# OPTIONS --guardedness --sized-types #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Data.Integer.Base using (ℤ; +_)"},{"Add":"import Data.Integer.Show as ℤ"},{"Add":"open import Data.List.Base as List using (List; _∷_; [])"},{"Add":"import Data.Nat.Base"},{"Add":"open import Data.Rational.Unnormalised.Base"},{"Add":" using (ℚᵘ; -_; _/_; floor; ceiling; truncate; round; fracPart)"},{"Add":"import Data.Rational.Unnormalised.Show as ℚᵘ"},{"Add":"open import Data.String.Base as String using (String)"},{"Add":""},{"Add":"open import IO.Base"},{"Add":"open import IO.Finite"},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"testList : List ℚᵘ"},{"Add":"testList = - (+ 35 / 10) ∷ - (+ 27 / 10) ∷ - (+ 15 / 10) ∷ - (+ 3 / 10)"},{"Add":" ∷ + 3 / 10 ∷ + 15 / 10 ∷ + 27 / 10 ∷ + 35 / 10 ∷ []"},{"Add":""},{"Add":"showInts : List ℤ → String"},{"Add":"showInts is = String.concat"},{"Add":" $ \"[\""},{"Add":" ∷ String.intersperse \", \" (List.map ℤ.show is)"},{"Add":" ∷ \"]\" ∷ []"},{"Add":""},{"Add":"showRats : List ℚᵘ → String"},{"Add":"showRats ps = String.concat"},{"Add":" $ \"[\""},{"Add":" ∷ String.intersperse \", \" (List.map ℚᵘ.show ps)"},{"Add":" ∷ \"]\" ∷ []"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" putStrLn $ showInts (List.map floor testList)"},{"Add":" putStrLn $ showInts (List.map ceiling testList)"},{"Add":" putStrLn $ showInts (List.map truncate testList)"},{"Add":" putStrLn $ showInts (List.map round testList)"},{"Add":" putStrLn $ showRats (List.map fracPart testList)"}]}},{"FileAdd":{"meta":{"basename":"rational","metadata":576},"context":[{"change":0,"pos":40349}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":45929}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > 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Data.Nat.Base"},{"Add":"open import Data.Rational.Base"},{"Add":" using (ℚ; -_; _/_; floor; ceiling; truncate; round; fracPart)"},{"Add":"import Data.Rational.Show as ℚ"},{"Add":"open import Data.String.Base as String using (String)"},{"Add":""},{"Add":"open import IO.Base"},{"Add":"open import IO.Finite"},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"testList : List ℚ"},{"Add":"testList = - (+ 35 / 10) ∷ - (+ 27 / 10) ∷ - (+ 15 / 10) ∷ - (+ 3 / 10)"},{"Add":" ∷ + 3 / 10 ∷ + 15 / 10 ∷ + 27 / 10 ∷ + 35 / 10 ∷ []"},{"Add":""},{"Add":"showInts : List ℤ → String"},{"Add":"showInts is = String.concat"},{"Add":" $ \"[\""},{"Add":" ∷ String.intersperse \", \" (List.map ℤ.show is)"},{"Add":" ∷ \"]\" ∷ []"},{"Add":""},{"Add":"showRats : List ℚ → String"},{"Add":"showRats ps = String.concat"},{"Add":" $ \"[\""},{"Add":" ∷ String.intersperse \", \" (List.map ℚ.show ps)"},{"Add":" ∷ \"]\" ∷ []"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" putStrLn $ showInts (List.map floor testList)"},{"Add":" putStrLn $ showInts (List.map ceiling testList)"},{"Add":" putStrLn $ showInts (List.map truncate testList)"},{"Add":" putStrLn $ showInts (List.map round testList)"},{"Add":" putStrLn $ showRats (List.map fracPart testList)"}]}},{"FileAdd":{"meta":{"basename":"list","metadata":576},"context":[{"change":0,"pos":40349}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":47702}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"list.agda-lib","metadata":0},"context":[{"change":0,"pos":47702}],"contents":[{"Add":"name: list"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":47702}],"contents":[{"Add":"[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]"},{"Add":"[14, 36, 20, 16, 15, 196, 13, 12, 11, 20, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]"},{"Add":"[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 20, 20, 36, 196]"},{"Add":"[14, 18, 17, 16, 15, 392, 13, 12, 11, 20, 9, 11, 7, 6, 5, 4, 3, 2, 1, 0]"},{"Add":"[0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 11, 12, 13, 14, 15, 16, 17, 18, 20, 392]"}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":47702}],"contents":[{"Add":"{-# OPTIONS --guardedness --sized-types #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Data.List.Base as List using (List; _∷_; []; _++_; reverse)"},{"Add":"open import Data.List.Zipper"},{"Add":"import Data.List.Sort as Sort"},{"Add":"open import Data.Maybe.Base"},{"Add":"open import Data.Nat.Base"},{"Add":"open import Data.Nat.Show using (show)"},{"Add":"import Data.Nat.Properties as ℕₚ"},{"Add":"open import Data.String.Base as String using (String)"},{"Add":"import Data.Vec.Base as Vec"},{"Add":""},{"Add":"open import Codata.Sized.Stream using (nats; take)"},{"Add":""},{"Add":"open Sort ℕₚ.≤-decTotalOrder"},{"Add":""},{"Add":"open import IO.Base"},{"Add":"open import IO.Finite"},{"Add":"open import Function.Base using (_$_; _∘_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"data Direction : Set where Left Right : Direction"},{"Add":""},{"Add":"turn : Direction → Zipper A → Zipper A"},{"Add":"turn Left zip = fromMaybe zip (left zip)"},{"Add":"turn Right zip = fromMaybe zip (right zip)"},{"Add":""},{"Add":"follow : List Direction → Zipper A → Zipper A"},{"Add":"follow dirs init = go dirs init where"},{"Add":""},{"Add":" go : List Direction → Zipper A → Zipper A"},{"Add":" go [] zip = zip"},{"Add":" go (d ∷ dirs) zip = go dirs (turn d zip)"},{"Add":""},{"Add":"updateFocus : (A → A) → Zipper A → Zipper A"},{"Add":"updateFocus f (mkZipper ctx (a ∷ val)) = mkZipper ctx (f a ∷ val)"},{"Add":"updateFocus f zip = zip"},{"Add":""},{"Add":"updateAt : List Direction → (A → A) → Zipper A → Zipper A"},{"Add":"updateAt dirs f = updateFocus f ∘ follow dirs"},{"Add":""},{"Add":"applyAt : List Direction → (A → A) → List A → List A"},{"Add":"applyAt dirs f xs = toList"},{"Add":" $ updateFocus f"},{"Add":" $ follow dirs"},{"Add":" $ fromList xs"},{"Add":""},{"Add":"someNats : List ℕ"},{"Add":"someNats = Vec.toList $ take 20 $ nats"},{"Add":""},{"Add":"otherNats : List ℕ"},{"Add":"otherNats"},{"Add":" = applyAt (Right ∷ Right ∷ []) (3 +_)"},{"Add":" $ applyAt (List.replicate 10 Right ++ Left ∷ []) (10 +_)"},{"Add":" $ applyAt (List.replicate 10 Left) (_∸ 5)"},{"Add":" $ applyAt (Left ∷ Right ∷ Right ∷ Left ∷ []) (2 *_)"},{"Add":" $ applyAt (List.replicate 5 Right) (_^ 2)"},{"Add":" $ List.reverse someNats"},{"Add":""},{"Add":"chaoticNats : List ℕ"},{"Add":"chaoticNats"},{"Add":" = toList"},{"Add":" $ updateAt (Right ∷ Right ∷ []) (3 +_)"},{"Add":" $ updateAt (List.replicate 10 Right ++ Left ∷ []) (10 +_)"},{"Add":" $ updateAt (List.replicate 10 Left) (_∸ 5)"},{"Add":" $ updateAt (Left ∷ Right ∷ Right ∷ Left ∷ []) (2 *_)"},{"Add":" $ updateAt (List.replicate 5 Right) (_^ 2)"},{"Add":" $ fromList"},{"Add":" $ List.reverse someNats"},{"Add":""},{"Add":"showNats : List ℕ → String"},{"Add":"showNats ns = String.concat"},{"Add":" $ \"[\""},{"Add":" ∷ String.intersperse \", \" (List.map show ns)"},{"Add":" ∷ \"]\" ∷ []"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" putStrLn $ showNats someNats"},{"Add":" putStrLn $ showNats otherNats"},{"Add":" putStrLn $ showNats $ sort otherNats"},{"Add":" putStrLn $ showNats chaoticNats"},{"Add":" putStrLn $ showNats $ sort chaoticNats"}]}},{"FileAdd":{"meta":{"basename":"colist","metadata":576},"context":[{"change":0,"pos":40349}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":51076}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":51076}],"contents":[{"Add":"λ 0"},{"Add":"(λ 0) ((λ 0) (λ 0) ((λ 0) (λ 0)))"},{"Add":"(λ 0) (λ 0) ((λ 0) (λ 0))"},{"Add":"(λ 0) ((λ 0) (λ 0))"},{"Add":"(λ 0) (λ 0)"},{"Add":"λ 0"}]}},{"FileAdd":{"meta":{"basename":"colist.agda-lib","metadata":0},"context":[{"change":0,"pos":51076}],"contents":[{"Add":"name: colist"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":51076}],"contents":[{"Add":"{-# OPTIONS --guardedness --sized-types #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Level using (0ℓ)"},{"Add":"open import Size"},{"Add":"open import Data.Bool.Base using (Bool; true; false; if_then_else_)"},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc)"},{"Add":"open import Data.Fin"},{"Add":"import Data.Fin.Show as Fin"},{"Add":"open import Data.String.Base using (String; _++_; parens)"},{"Add":"open import Data.Unit.Polymorphic.Base using (⊤)"},{"Add":"open import Codata.Sized.Thunk"},{"Add":"open import Function.Base using (_$_; _∘_; id)"},{"Add":"open import Relation.Nullary"},{"Add":""},{"Add":"open import Codata.Musical.Notation"},{"Add":"open import Codata.Sized.Colist using (Colist; _∷_; [])"},{"Add":"open import Codata.Musical.Colist renaming (Colist to Colist♩) using (_∷_; [])"},{"Add":"open import Codata.Musical.Conversion"},{"Add":""},{"Add":"variable"},{"Add":" i : Size"},{"Add":" m n : ℕ"},{"Add":" A : Set"},{"Add":""},{"Add":"data Lam (n : ℕ) : Set where"},{"Add":" var : Fin n → Lam n"},{"Add":" app : Lam n → Lam n → Lam n"},{"Add":" lam : Lam (suc n) → Lam n"},{"Add":""},{"Add":"data Loc : Set where appL appR lam : Loc"},{"Add":""},{"Add":"appParens : Loc → String → String"},{"Add":"appParens appR str = parens str"},{"Add":"appParens _ str = str"},{"Add":""},{"Add":"lamParens : Loc → String → String"},{"Add":"lamParens lam str = str"},{"Add":"lamParens _ str = parens str"},{"Add":""},{"Add":"show : Loc → Lam n → String"},{"Add":"show i (var v) = Fin.show v"},{"Add":"show i (app f t) = appParens i $ show appL f ++ \" \" ++ show appR t"},{"Add":"show i (lam b) = lamParens i $ \"λ \" ++ show lam b"},{"Add":""},{"Add":"variable"},{"Add":" b f t : Lam n"},{"Add":""},{"Add":"_∙_ : (Fin m → A) → A → Fin (suc m) → A"},{"Add":"(ρ ∙ v) zero = v"},{"Add":"(ρ ∙ v) (suc k) = ρ k"},{"Add":""},{"Add":"rename : (Fin m → Fin n) → Lam m → Lam n"},{"Add":"rename ρ (var v) = var (ρ v)"},{"Add":"rename ρ (app f t) = app (rename ρ f) (rename ρ t)"},{"Add":"rename ρ (lam b) = lam (rename ((suc ∘ ρ) ∙ zero) b)"},{"Add":""},{"Add":"subst : (Fin m → Lam n) → Lam m → Lam n"},{"Add":"subst ρ (var v) = ρ v"},{"Add":"subst ρ (app f t) = app (subst ρ f) (subst ρ t)"},{"Add":"subst ρ (lam b) = lam (subst ((rename suc ∘ ρ) ∙ var zero) b)"},{"Add":""},{"Add":"data Redex {n} : Lam n → Set where"},{"Add":" here : ∀ b t → Redex (app (lam b) t)"},{"Add":" lam : Redex b → Redex (lam b)"},{"Add":" appL : Redex f → ∀ t → Redex (app f t)"},{"Add":" appR : ∀ f → Redex t → Redex (app f t)"},{"Add":""},{"Add":"redex : ∀ {n} (t : Lam n) → Dec (Redex t)"},{"Add":"redex (var v) = no (λ ())"},{"Add":"redex (app (lam b) t) = yes (here b t)"},{"Add":"redex (app f@(var _) t) with redex t"},{"Add":"... | yes rt = yes (appR f rt)"},{"Add":"... | no nrt = no λ where (appR _ rt) → nrt rt"},{"Add":"redex (app f@(app _ _) t) with redex f | redex t"},{"Add":"... | yes rf | _ = yes (appL rf t)"},{"Add":"... | _ | yes rt = yes (appR f rt)"},{"Add":"... | no nrf | no nrt = no λ where"},{"Add":" (appL rf _) → nrf rf"},{"Add":" (appR _ rt) → nrt rt"},{"Add":"redex (lam b) with redex b"},{"Add":"... | yes rb = yes (lam rb)"},{"Add":"... | no nrb = no λ where (lam rb) → nrb rb"},{"Add":""},{"Add":"fire : Redex {n} t → Lam n"},{"Add":"fire (here b t) = subst (var ∙ t) b"},{"Add":"fire (lam rt) = lam (fire rt)"},{"Add":"fire (appL rf t) = app (fire rf) t"},{"Add":"fire (appR f rt) = app f (fire rt)"},{"Add":""},{"Add":"eval : Lam n → Colist (Lam n) i"},{"Add":"eval t with redex t"},{"Add":"... | yes rt = t ∷ λ where .force → eval (fire rt)"},{"Add":"... | no nrt = t ∷ λ where .force → []"},{"Add":""},{"Add":"open import IO.Base"},{"Add":"open import IO.Finite"},{"Add":""},{"Add":"trace : Colist♩ (Lam n) → IO {0ℓ} ⊤"},{"Add":"trace [] = pure _"},{"Add":"trace (t ∷ ts) = seq (♯ putStrLn (show lam t))"},{"Add":" (♯ trace (♭ ts))"},{"Add":""},{"Add":"`id : Lam 0"},{"Add":"`id = lam (var (# 0))"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" trace (Colist.toMusical $ eval `id)"},{"Add":" trace (Colist.toMusical $ eval (app `id (app (app `id `id) (app `id `id))))"}]}},{"FileAdd":{"meta":{"basename":"bytestring","metadata":576},"context":[{"change":0,"pos":40349}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":54810}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":54810}],"contents":[{"Add":"\"\\SOH\\STX\\NUL\\NUL\\NUL\\NUL\\NUL\\NUL\\NUL\\NUL\\NUL\\NUL\\NUL\\NUL\\NUL\\NUL\\ETX\""},{"Add":"------------------------------------------------------------------------"},{"Add":"1 = 0b00000001"},{"Add":"1 = 0x01"},{"Add":"1 = 1"},{"Add":"144 = 0b10010000"},{"Add":"144 = 0x90"},{"Add":"144 = 144"},{"Add":"------------------------------------------------------------------------"},{"Add":"2 = 0b0000000000000000000000000000000000000000000000000000000000000010"},{"Add":"2 = 0x00000002"},{"Add":"2 = 2"},{"Add":"------------------------------------------------------------------------"},{"Add":"3 = 0b0000000000000000000000000000000000000000000000000000000000000011"},{"Add":"3 = 0x00000003"},{"Add":"3 = 3"},{"Add":"------------------------------------------------------------------------"},{"Add":"2024 = 0b0000000000000000000000000000000000000000000000000000011111101000"},{"Add":"2024 = 0x000007e8"},{"Add":"2024 = 2024"}]}},{"FileAdd":{"meta":{"basename":"bytestring.agda-lib","metadata":0},"context":[{"change":0,"pos":54810}],"contents":[{"Add":"name: bytestring"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":54810}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":"open import Agda.Builtin.FromNat"},{"Add":""},{"Add":"open import Data.Bytestring.Base as Bytestring"},{"Add":"open import Data.Bytestring.Builder.Base"},{"Add":"open import Data.List.Base using ([]; _∷_)"},{"Add":"import Data.Nat.Literals; instance numberNat = Data.Nat.Literals.number"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Data.String using (String; _++_; fromVec)"},{"Add":"open import Data.Unit.Base using (⊤; tt)"},{"Add":"import Data.Vec.Base as Vec"},{"Add":"open import Data.Word8.Base as Word8"},{"Add":"import Data.Word8.Show as Word8"},{"Add":"import Data.Word8.Literals; instance numberWord8 = Data.Word8.Literals.number"},{"Add":"open import Data.Word64.Base as Word64 using (Word64)"},{"Add":"import Data.Word64.Unsafe as Word64"},{"Add":"import Data.Word64.Show as Word64"},{"Add":"import Data.Word64.Literals; instance numberWord64 = Data.Word64.Literals.number"},{"Add":""},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"open import IO.Base"},{"Add":"open import IO.Finite"},{"Add":""},{"Add":"1⋯3 : Bytestring"},{"Add":"1⋯3 = toBytestring"},{"Add":" $ List.concat"},{"Add":" $ word8 1"},{"Add":" ∷ word64LE 2"},{"Add":" ∷ word64BE 3"},{"Add":" ∷ []"},{"Add":""},{"Add":"1,⋯,3 : Word8 × Word64 × Word64"},{"Add":"1,⋯,3 = getWord8 1⋯3 0"},{"Add":" , getWord64LE 1⋯3 1"},{"Add":" , getWord64BE 1⋯3 9"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ do"},{"Add":" let separation = fromVec (Vec.replicate 72 '-')"},{"Add":" putStrLn (Bytestring.show 1⋯3)"},{"Add":" putStrLn separation"},{"Add":" let (one , two , three) = 1,⋯,3"},{"Add":""},{"Add":" let word8test : Word8 → IO _"},{"Add":" word8test w = do"},{"Add":" putStrLn (Word8.show w ++ \" = \" ++ Word8.showBits w)"},{"Add":" putStrLn (Word8.show w ++ \" = \" ++ Word8.showHexa w)"},{"Add":" putStrLn (Word8.show w ++ \" = \" ++ Word8.show (Word8.fromBits (Word8.toBits w)))"},{"Add":""},{"Add":""},{"Add":" let word64test : Word64 → IO _"},{"Add":" word64test w = do"},{"Add":" putStrLn separation"},{"Add":" putStrLn (Word64.show w ++ \" = \" ++ Word64.showBits w)"},{"Add":" putStrLn (Word64.show w ++ \" = \" ++ Word64.showHexa w)"},{"Add":" putStrLn (Word64.show w ++ \" = \" ++ Word64.show (Word64.fromBits (Word64.toBits w)))"},{"Add":""},{"Add":" word8test one"},{"Add":" word8test (Word8.fromℕ 144)"},{"Add":" word64test two"},{"Add":" word64test three"},{"Add":" word64test (Word64.fromℕ 2024)"}]}},{"FileAdd":{"meta":{"basename":"appending","metadata":576},"context":[{"change":0,"pos":40349}],"contents":[]}},{"FileAdd":{"meta":{"basename":"run","metadata":0},"context":[{"change":0,"pos":57886}],"contents":[{"Add":"$1 --compile-dir=../../_build -c Main.agda > log"},{"Add":"./../../_build/Main < input > output"},{"Add":""},{"Add":"rm ../../_build/Main"},{"Add":"rm ../../_build/MAlonzo/Code/Main*"}]}},{"FileAdd":{"meta":{"basename":"input","metadata":0},"context":[{"change":0,"pos":57886}],"contents":[{"Add":"helLo"},{"Add":"teST"},{"Add":"parcequecestnotrePROJEEET"}]}},{"FileAdd":{"meta":{"basename":"expected","metadata":0},"context":[{"change":0,"pos":57886}],"contents":[{"Add":"hel"},{"Add":"te"},{"Add":"parcequecestnotre"}]}},{"FileAdd":{"meta":{"basename":"appending.agda-lib","metadata":0},"context":[{"change":0,"pos":57886}],"contents":[{"Add":"name: list"},{"Add":"include: ../../../src/ ."}]}},{"FileAdd":{"meta":{"basename":"TakeWhile.agda","metadata":0},"context":[{"change":0,"pos":57886}],"contents":[{"Add":"{-# OPTIONS --safe --cubical-compatible #-}"},{"Add":""},{"Add":"module TakeWhile where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Data.List.Base hiding (takeWhile)"},{"Add":"open import Data.List.Relation.Unary.All as List using ([]; _∷_)"},{"Add":"open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_; refl)"},{"Add":"open import Data.List.Relation.Ternary.Appending.Propositional"},{"Add":"open import Data.Product.Base using (_×_; proj₁)"},{"Add":"open import Data.Maybe.Relation.Unary.All as Maybe using (nothing; just)"},{"Add":"import Data.Nat"},{"Add":"open import Relation.Unary"},{"Add":"open import Relation.Binary.PropositionalEquality as P using (_≡_)"},{"Add":"open import Relation.Nullary"},{"Add":"open import Relation.Nullary.Product"},{"Add":""},{"Add":"-- Original bug reported in #1765 by James Wood"},{"Add":"_ : Appending (3 ∷ []) (2 ∷ []) (3 ∷ 2 ∷ [])"},{"Add":"_ = P.refl ∷ []++ P.refl ∷ []"},{"Add":""},{"Add":""},{"Add":"variable"},{"Add":" a p : Level"},{"Add":" A : Set a"},{"Add":" P : A → Set p"},{"Add":""},{"Add":"infix 1 _,_,_"},{"Add":"record TakeWhile {A : Set a} (P : A → Set p) xs : Set (a ⊔ p) where"},{"Add":" constructor _,_,_"},{"Add":" field"},{"Add":" {prefix rest} : List A"},{"Add":" goodPrefix : List.All P prefix"},{"Add":" isPrefix : Appending prefix rest xs"},{"Add":" isBiggest : Maybe.All (proj₁ ⊢ ∁ P) (uncons rest)"},{"Add":"open TakeWhile public"},{"Add":""},{"Add":"takeWhile : Decidable P → Π[ TakeWhile P ]"},{"Add":"takeWhile P? [] = [] , []++ [] , nothing"},{"Add":"takeWhile P? (x ∷ xs) with P? x"},{"Add":"... | yes px = let (pxs' , prf , biggest) = takeWhile P? xs in"},{"Add":" px ∷ pxs' , P.refl ∷ prf , biggest"},{"Add":"... | no ¬px = [] , []++ refl P.refl , just ¬px"},{"Add":""},{"Add":"open import Data.Char"},{"Add":"open import Data.String using (toList)"},{"Add":""},{"Add":"lower? : (c : Char) → Dec ('a' ≤ c × c ≤ 'z')"},{"Add":"lower? c = 'a' ≤? c ×-dec c ≤? 'z'"},{"Add":""},{"Add":"_ : takeWhile lower? (toList \"helLo\")"},{"Add":" ≡ record { prefix = toList \"hel\""},{"Add":" ; isPrefix = P.refl ∷ P.refl ∷ P.refl ∷ []++ P.refl ∷ P.refl ∷ []"},{"Add":" }"},{"Add":"_ = P.refl"}]}},{"FileAdd":{"meta":{"basename":"Main.agda","metadata":0},"context":[{"change":0,"pos":57886}],"contents":[{"Add":"{-# OPTIONS --guardedness #-}"},{"Add":""},{"Add":"module Main where"},{"Add":""},{"Add":""},{"Add":"open import Data.List.Base using (replicate)"},{"Add":"open import Data.String using (toList; fromList)"},{"Add":""},{"Add":"open import IO"},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"open import TakeWhile"},{"Add":""},{"Add":"main : Main"},{"Add":"main = run $ List.sequence′ $ replicate 3 $ do"},{"Add":" str ← getLine"},{"Add":" let taken = takeWhile lower? (toList str)"},{"Add":" putStrLn $ fromList (taken .prefix)"}]}},{"FileAdd":{"meta":{"basename":"Makefile","metadata":0},"context":[{"change":0,"pos":31}],"contents":[{"Add":"INTERACTIVE ?= --interactive"},{"Add":""},{"Add":"runtests: runtests.agda"},{"Add":"\trm -f _build/runtests"},{"Add":"\trm -f _build/MAlonzo/Code/Qruntests*"},{"Add":"\t$(AGDA) --compile-dir=_build/ -c runtests.agda"},{"Add":""},{"Add":"test: runtests"},{"Add":"\t./_build/runtests $(AGDA_EXEC) $(INTERACTIVE) --timing --failure-file failures --only $(only)"},{"Add":""},{"Add":""},{"Add":"retest: runtests"},{"Add":"\t./_build/runtests $(AGDA_EXEC) $(INTERACTIVE) --timing --failure-file failures --only-file failures --only $(only)"}]}},{"FileAdd":{"meta":{"basename":"standard-library.agda-lib","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"name: standard-library-2.1"},{"Add":"include: src"},{"Add":"flags:"},{"Add":" --warning=noUnsupportedIndexedMatch"}]}},{"FileAdd":{"meta":{"basename":"stack-9.6.2.yaml","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"resolver: nightly-2023-08-27"},{"Add":"compiler: ghc-9.6.2"},{"Add":"compiler-check: match-exact"},{"Add":""},{"Add":"packages:"},{"Add":"- '.'"}]}},{"FileAdd":{"meta":{"basename":"stack-9.4.5.yaml","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"resolver: lts-21.7"},{"Add":"compiler: ghc-9.4.5"},{"Add":"compiler-check: match-exact"},{"Add":""},{"Add":"packages:"},{"Add":"- '.'"}]}},{"FileAdd":{"meta":{"basename":"stack-9.2.8.yaml","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"resolver: lts-20.26"},{"Add":"compiler: ghc-9.2.8"},{"Add":"compiler-check: match-exact"},{"Add":""},{"Add":"packages:"},{"Add":"- '.'"}]}},{"FileAdd":{"meta":{"basename":"stack-9.0.2.yaml","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"resolver: lts-19.33"},{"Add":"compiler: ghc-9.0.2"},{"Add":"compiler-check: match-exact"},{"Add":""},{"Add":"packages:"},{"Add":"- '.'"}]}},{"FileAdd":{"meta":{"basename":"stack-8.8.4.yaml","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"resolver: lts-16.31"},{"Add":"compiler: ghc-8.8.4"},{"Add":"compiler-check: match-exact"},{"Add":""},{"Add":"packages:"},{"Add":"- '.'"}]}},{"FileAdd":{"meta":{"basename":"stack-8.8.3.yaml","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"resolver: lts-16.5"},{"Add":"compiler: ghc-8.8.3"},{"Add":"compiler-check: match-exact"},{"Add":""},{"Add":"packages:"},{"Add":"- '.'"}]}},{"FileAdd":{"meta":{"basename":"stack-8.8.2.yaml","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"resolver: lts-15.3"},{"Add":"compiler: ghc-8.8.2"},{"Add":"compiler-check: match-exact"},{"Add":""},{"Add":"packages:"},{"Add":"- '.'"}]}},{"FileAdd":{"meta":{"basename":"stack-8.6.5.yaml","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"resolver: lts-14.27"},{"Add":"compiler: ghc-8.6.5"},{"Add":"compiler-check: match-exact"},{"Add":""},{"Add":"packages:"},{"Add":"- '.'"}]}},{"FileAdd":{"meta":{"basename":"stack-8.4.4.yaml","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"resolver: lts-12.26"},{"Add":"compiler: ghc-8.4.4"},{"Add":"compiler-check: match-exact"},{"Add":""},{"Add":"packages:"},{"Add":"- '.'"}]}},{"FileAdd":{"meta":{"basename":"stack-8.2.2.yaml","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"resolver: lts-11.22"},{"Add":"compiler: ghc-8.2.2"},{"Add":"compiler-check: match-exact"},{"Add":""},{"Add":"packages:"},{"Add":"- '.'"}]}},{"FileAdd":{"meta":{"basename":"stack-8.10.7.yaml","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"resolver: lts-18.23"},{"Add":"compiler: ghc-8.10.7"},{"Add":"compiler-check: match-exact"},{"Add":""},{"Add":"# extra-deps:"},{"Add":"# - filemanip-0.3.6.3"},{"Add":"# - unix-compat-0.5.2"},{"Add":""},{"Add":"packages:"},{"Add":"- '.'"}]}},{"FileAdd":{"meta":{"basename":"stack-8.10.5.yaml","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"resolver: lts-18.0"},{"Add":"compiler: ghc-8.10.5"},{"Add":"compiler-check: match-exact"},{"Add":""},{"Add":"# extra-deps:"},{"Add":"# - filemanip-0.3.6.3"},{"Add":"# - unix-compat-0.5.2"},{"Add":""},{"Add":"packages:"},{"Add":"- '.'"}]}},{"FileAdd":{"meta":{"basename":"stack-8.0.2.yaml","metadata":0},"context":[{"change":0,"pos":1}],"contents":[{"Add":"resolver: lts-9.21"},{"Add":"compiler: ghc-8.0.2"},{"Add":"compiler-check: match-exact"},{"Add":""},{"Add":"packages:"},{"Add":"- '.'"}]}},{"FileAdd":{"meta":{"basename":"src","metadata":576},"context":[{"change":0,"pos":1}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Text","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Tree","metadata":576},"context":[{"change":0,"pos":62939}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Linear.agda","metadata":0},"context":[{"change":0,"pos":62957}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use the Data.Tree.Rose.Show module"},{"Add":"-- directly"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --sized-types #-}"},{"Add":""},{"Add":"module Text.Tree.Linear where"},{"Add":""},{"Add":"open import Data.Tree.Rose.Show public using (display)"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Text.Tree.Linear was deprecated in v1.6. Use Data.Tree.Rose.Show instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Tabular","metadata":576},"context":[{"change":0,"pos":62939}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Vec.agda","metadata":0},"context":[{"change":0,"pos":63518}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Fancy display functions for Vec-based tables"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --safe --cubical-compatible #-}"},{"Add":""},{"Add":"module Text.Tabular.Vec where"},{"Add":""},{"Add":"open import Data.List.Base using (List)"},{"Add":"open import Data.Product.Base as Prod using (uncurry)"},{"Add":"open import Data.String using (String; rectangle; fromAlignment)"},{"Add":"open import Data.Vec.Base"},{"Add":"open import Function.Base"},{"Add":""},{"Add":"open import Text.Tabular.Base"},{"Add":""},{"Add":"display : ∀ {m n} → TabularConfig → Vec Alignment n → Vec (Vec String n) m →"},{"Add":" List String"},{"Add":"display c a = unsafeDisplay c"},{"Add":" ∘ toList"},{"Add":" ∘ map toList"},{"Add":" ∘ transpose"},{"Add":" ∘ map (uncurry rectangle ∘ unzip)"},{"Add":" ∘ transpose"},{"Add":" ∘ map (zip (map fromAlignment a))"}]}},{"FileAdd":{"meta":{"basename":"List.agda","metadata":0},"context":[{"change":0,"pos":63518}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Fancy display functions for List-based tables"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --safe --cubical-compatible #-}"},{"Add":""},{"Add":"module Text.Tabular.List where"},{"Add":""},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Data.List.Base"},{"Add":"import Data.Nat.Properties as ℕ"},{"Add":"open import Data.Product.Base using (-,_; proj₂)"},{"Add":"open import Data.Vec.Base as Vec using (Vec)"},{"Add":"open import Data.Vec.Bounded.Base as Vec≤ using (Vec≤)"},{"Add":"open import Function.Base"},{"Add":""},{"Add":"open import Text.Tabular.Base"},{"Add":"import Text.Tabular.Vec as Show"},{"Add":""},{"Add":"display : TabularConfig → List Alignment → List (List String) → List String"},{"Add":"display c a rows = Show.display c alignment rectangle"},{"Add":" where"},{"Add":" alignment : Vec Alignment _"},{"Add":" alignment = Vec≤.padRight Left"},{"Add":" $ Vec≤.≤-cast (ℕ.m⊓n≤m _ _)"},{"Add":" $ Vec≤.take _ (Vec≤.fromList a)"},{"Add":""},{"Add":" rectangle : Vec (Vec String _) _"},{"Add":" rectangle = Vec.fromList"},{"Add":" $ map (Vec≤.padRight \"\")"},{"Add":" $ proj₂"},{"Add":" $ Vec≤.rectangle"},{"Add":" $ map (λ row → -, Vec≤.fromList row) rows"}]}},{"FileAdd":{"meta":{"basename":"Base.agda","metadata":0},"context":[{"change":0,"pos":63518}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Fancy display functions for List-based tables"},{"Add":"--"},{"Add":"-- The functions in this module assume some (unenforced) invariants."},{"Add":"-- If you cannot guarantee that your data respects these invariants,"},{"Add":"-- you should instead use Text.Tabular.List."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --safe --cubical-compatible #-}"},{"Add":""},{"Add":"module Text.Tabular.Base where"},{"Add":""},{"Add":"open import Data.Bool.Base using (if_then_else_)"},{"Add":"open import Data.Char.Base using (Char)"},{"Add":"open import Data.List.Base as List"},{"Add":" using (List; []; _∷_; _?∷_; _++_; _∷ʳ?_; null; map; intersperse)"},{"Add":"open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; maybe)"},{"Add":"open import Data.Nat.Base"},{"Add":"open import Data.String.Base as String"},{"Add":" using (String; fromChar; unlines; replicate; length)"},{"Add":"open import Function.Base"},{"Add":"open import Agda.Builtin.Equality"},{"Add":""},{"Add":"open String"},{"Add":" using ( Alignment"},{"Add":" ; Left"},{"Add":" ; Center"},{"Add":" ; Right"},{"Add":" ) public"},{"Add":""},{"Add":"record TabularLine : Set where"},{"Add":" field"},{"Add":" left : Maybe String"},{"Add":" cont : Maybe Char"},{"Add":" sep : String"},{"Add":" right : Maybe String"},{"Add":"open TabularLine"},{"Add":""},{"Add":"record TabularConfig : Set where"},{"Add":" field"},{"Add":" top : Maybe TabularLine"},{"Add":" sep : Maybe TabularLine"},{"Add":" row : TabularLine"},{"Add":" bot : Maybe TabularLine"},{"Add":"open TabularConfig"},{"Add":""},{"Add":"unicode : TabularConfig"},{"Add":"unicode .top = just λ where"},{"Add":" .left → just \"┌\""},{"Add":" .cont → just '─'"},{"Add":" .sep → \"┬\""},{"Add":" .right → just \"┐\""},{"Add":"unicode .sep = just λ where"},{"Add":" .left → just \"├\""},{"Add":" .cont → just '─'"},{"Add":" .sep → \"┼\""},{"Add":" .right → just \"┤\""},{"Add":"unicode .row = λ where"},{"Add":" .left → just \"│\""},{"Add":" .cont → nothing"},{"Add":" .sep → \"│\""},{"Add":" .right → just \"│\""},{"Add":"unicode .bot = just λ where"},{"Add":" .left → just \"└\""},{"Add":" .cont → just '─'"},{"Add":" .sep → \"┴\""},{"Add":" .right → just \"┘\""},{"Add":""},{"Add":"ascii : TabularConfig"},{"Add":"ascii .top = just λ where"},{"Add":" .left → just \"+\""},{"Add":" .cont → just '-'"},{"Add":" .sep → \"-\""},{"Add":" .right → just \"+\""},{"Add":"ascii .sep = just λ where"},{"Add":" .left → just \"|\""},{"Add":" .cont → just '-'"},{"Add":" .sep → \"+\""},{"Add":" .right → just \"|\""},{"Add":"ascii .row = λ where"},{"Add":" .left → just \"|\""},{"Add":" .cont → nothing"},{"Add":" .sep → \"|\""},{"Add":" .right → just \"|\""},{"Add":"ascii .bot = just λ where"},{"Add":" .left → just \"+\""},{"Add":" .cont → just '-'"},{"Add":" .sep → \"-\""},{"Add":" .right → just \"+\""},{"Add":""},{"Add":"compact : TabularConfig → TabularConfig"},{"Add":"compact c = record c { sep = nothing }"},{"Add":""},{"Add":"private"},{"Add":" dropBorder : TabularLine → TabularLine"},{"Add":" dropBorder l = record l { left = nothing; right = nothing }"},{"Add":""},{"Add":"noBorder : TabularConfig → TabularConfig"},{"Add":"noBorder c .top = nothing"},{"Add":"noBorder c .sep = Maybe.map dropBorder (c .sep)"},{"Add":"noBorder c .row = dropBorder (c .row)"},{"Add":"noBorder c .bot = nothing"},{"Add":""},{"Add":""},{"Add":"private"},{"Add":" space : TabularLine → TabularLine"},{"Add":" space l = let pad = maybe fromChar \" \" (l .cont) in λ where"},{"Add":" .left → Maybe.map (String._++ pad) (l .left)"},{"Add":" .cont → l .cont"},{"Add":" .sep → pad String.++ l .sep String.++ pad"},{"Add":" .right → Maybe.map (pad String.++_) (l .right)"},{"Add":""},{"Add":"addSpace : TabularConfig → TabularConfig"},{"Add":"addSpace c .top = Maybe.map space (c .top)"},{"Add":"addSpace c .sep = Maybe.map space (c .sep)"},{"Add":"addSpace c .row = space (c .row)"},{"Add":"addSpace c .bot = Maybe.map space (c .bot)"},{"Add":""},{"Add":"whitespace : TabularConfig"},{"Add":"whitespace .top = nothing"},{"Add":"whitespace .sep = nothing"},{"Add":"whitespace .row = λ where"},{"Add":" .left → nothing"},{"Add":" .cont → nothing"},{"Add":" .sep → \" \""},{"Add":" .right → nothing"},{"Add":"whitespace .bot = nothing"},{"Add":""},{"Add":"-- /!\\ Invariants:"},{"Add":"-- * the table is presented as a list of rows"},{"Add":"-- * header has the same length as each one of the rows"},{"Add":"-- i.e. we have a rectangular table"},{"Add":"-- * all of the strings in a given column have the same length"},{"Add":""},{"Add":"unsafeDisplay : TabularConfig → List (List String) → List String"},{"Add":"unsafeDisplay _ [] = []"},{"Add":"unsafeDisplay c (header ∷ rows) ="},{"Add":" map String.concat $ th ++ (trs ∷ʳ? lbot)"},{"Add":" where"},{"Add":""},{"Add":" cellsOf : Maybe Char → List String → List String"},{"Add":" cellsOf nothing = id"},{"Add":" cellsOf (just c) = map (λ cell → replicate (length cell) c)"},{"Add":""},{"Add":" lineOf : TabularLine → List String → List String"},{"Add":" lineOf l xs = l .left"},{"Add":" ?∷ intersperse (l .sep) (cellsOf (l .cont) xs)"},{"Add":" ∷ʳ? l .right"},{"Add":""},{"Add":" mlineOf : Maybe TabularLine → List String → Maybe (List String)"},{"Add":" mlineOf l xs = Maybe.map (λ l → lineOf l xs) l"},{"Add":""},{"Add":" ltop : Maybe (List String)"},{"Add":" lsep : Maybe (List String)"},{"Add":" tr : List String → List String"},{"Add":" lbot : Maybe (List String)"},{"Add":""},{"Add":" ltop = mlineOf (c. top) header"},{"Add":" lsep = mlineOf (c. sep) header"},{"Add":" tr = lineOf (c. row)"},{"Add":" lbot = mlineOf (c. bot) header"},{"Add":""},{"Add":" th = ltop ?∷ tr header ∷ []"},{"Add":" trs = if null rows then id else (maybe _∷_ id lsep)"},{"Add":" $ maybe intersperse id lsep"},{"Add":" $ map tr rows"}]}},{"FileAdd":{"meta":{"basename":"Regex.agda","metadata":0},"context":[{"change":0,"pos":62939}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Regular expressions"},{"Add":"--"},{"Add":"-- The content of this module (and others in the Regex subdirectory) is"},{"Add":"-- based on Alexandre Agular and Bassel Mannaa's 2009 technical report:"},{"Add":"-- Regular Expressions in Agda"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (DecPoset)"},{"Add":""},{"Add":"module Text.Regex {a e r} (decPoset : DecPoset a e r) where"},{"Add":""},{"Add":""},{"Add":"private"},{"Add":" preorder = DecPoset.preorder decPoset"},{"Add":""},{"Add":"import Text.Regex.Base preorder as Regex"},{"Add":"import Text.Regex.SmartConstructors preorder as Smart"},{"Add":"import Text.Regex.Derivative.Brzozowski decPoset as Eat"},{"Add":"import Text.Regex.Search decPoset as Search"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting basic definition and semantics"},{"Add":""},{"Add":"open Regex public"},{"Add":" using ( Range; module Range"},{"Add":" ; [_]; _─_"},{"Add":" ; Regex; module Regex; Exp; module Exp"},{"Add":" ; ε; [^_]; ∅; ·; singleton"},{"Add":" ; _∈ᴿ_; _∉ᴿ_"},{"Add":" ; _∈_; _∉_; sum; prod; star"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting smart constructors"},{"Add":""},{"Add":"open Smart public"},{"Add":" using (_∣_; _∙_; _⋆; _+; _⁇)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting semantics decidability"},{"Add":""},{"Add":"open Eat public"},{"Add":" using (_∈?_; _∉?_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting search algorithms"},{"Add":""},{"Add":"open Search public"},{"Add":" using (Span; toInfix; Match; mkMatch; search)"}]}},{"FileAdd":{"meta":{"basename":"Regex","metadata":576},"context":[{"change":0,"pos":62939}],"contents":[]}},{"FileAdd":{"meta":{"basename":"String.agda","metadata":0},"context":[{"change":0,"pos":72073}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Regular expressions acting on strings"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Text.Regex.String where"},{"Add":""},{"Add":"import Data.Char.Properties as Char"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting definitions"},{"Add":""},{"Add":"open import Text.Regex Char.≤-decPoset public"}]}},{"FileAdd":{"meta":{"basename":"String","metadata":576},"context":[{"change":0,"pos":72073}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Unsafe.agda","metadata":0},"context":[{"change":0,"pos":72615}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Regular expressions acting on strings, using unsafe features"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --with-K #-}"},{"Add":""},{"Add":"module Text.Regex.String.Unsafe where"},{"Add":""},{"Add":"open import Data.String.Base using (String; toList; fromList)"},{"Add":"import Data.String.Unsafe as Stringₚ"},{"Add":"open import Function.Base using (_on_; id)"},{"Add":"open import Level using (0ℓ)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Definitions using (Decidable)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; sym; subst)"},{"Add":"open import Relation.Nullary.Decidable using (map′)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting safe definitions"},{"Add":""},{"Add":"open import Text.Regex.String as Regex public"},{"Add":" hiding (_∈_; _∉_; _∈?_; _∉?_; Span; Match; search)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Specialised definitions"},{"Add":""},{"Add":"infix 4 _∈_ _∉_ _∈?_ _∉?_"},{"Add":""},{"Add":"_∈_ : String → Exp → Set"},{"Add":"str ∈ e = toList str Regex.∈ e"},{"Add":""},{"Add":"_∉_ : String → Exp → Set"},{"Add":"str ∉ e = toList str Regex.∉ e"},{"Add":""},{"Add":"_∈?_ : Decidable _∈_"},{"Add":"str ∈? e = toList str Regex.∈? e"},{"Add":""},{"Add":"_∉?_ : Decidable _∉_"},{"Add":"str ∉? e = toList str Regex.∉? e"},{"Add":""},{"Add":"Span : Regex → Rel String 0ℓ"},{"Add":"Span e = Regex.Span e _≡_ on toList"},{"Add":""},{"Add":"-- A match is a string, a proof it matches the regular expression,"},{"Add":"-- and a proof it appears as the right sort of substring."},{"Add":"record Match (str : String) (e : Regex) : Set where"},{"Add":" constructor mkMatch"},{"Add":" field"},{"Add":" string : String"},{"Add":" match : string ∈ Regex.expression e"},{"Add":" related : Span e string str"},{"Add":"open Match public"},{"Add":""},{"Add":"search : Decidable Match"},{"Add":"search str e = map′ from to (Regex.search input e) where"},{"Add":""},{"Add":" input = toList str"},{"Add":" exp = Regex.expression e"},{"Add":""},{"Add":" from : Regex.Match (Regex.Span e _≡_) input exp → Match str e"},{"Add":" from (Regex.mkMatch list match related) ="},{"Add":" let eq = sym (Stringₚ.toList∘fromList list) in"},{"Add":" mkMatch (fromList list)"},{"Add":" (subst (Regex._∈ exp) eq match)"},{"Add":" (subst (λ str → Regex.Span e _≡_ str input) eq related)"},{"Add":""},{"Add":" to : Match str e → Regex.Match (Regex.Span e _≡_) input exp"},{"Add":" to (mkMatch string match related) ="},{"Add":" Regex.mkMatch (toList string) match related"}]}},{"FileAdd":{"meta":{"basename":"SmartConstructors.agda","metadata":0},"context":[{"change":0,"pos":72073}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Regular expressions: smart constructors"},{"Add":"-- Computing the Brzozowski derivative of a regular expression may lead"},{"Add":"-- to a blow-up in the size of the expression. To keep it tractable it"},{"Add":"-- is crucial to use smart constructors."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (Preorder)"},{"Add":""},{"Add":"module Text.Regex.SmartConstructors {a e r} (P : Preorder a e r) where"},{"Add":""},{"Add":"open import Data.List.Base using ([])"},{"Add":"open import Data.List.Relation.Ternary.Appending.Propositional"},{"Add":"open import Data.Sum.Base using (inj₁; inj₂; fromInj₁; fromInj₂)"},{"Add":""},{"Add":"open import Relation.Nullary.Decidable using (yes; no)"},{"Add":"open import Relation.Nullary.Negation using (contradiction)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (refl)"},{"Add":""},{"Add":"open import Text.Regex.Base P as R hiding (_∣_; _∙_; _⋆)"},{"Add":"open import Text.Regex.Properties.Core P"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Sum"},{"Add":""},{"Add":"infixr 5 _∣_"},{"Add":""},{"Add":"_∣_ : (e f : Exp) → Exp"},{"Add":"e ∣ f with is-∅ e | is-∅ f"},{"Add":"... | yes _ | _ = f"},{"Add":"... | _ | yes _ = e"},{"Add":"... | _ | _ = e R.∣ f"},{"Add":""},{"Add":"∣-sound : ∀ {w} e f → w ∈ (e ∣ f) → w ∈ (e R.∣ f)"},{"Add":"∣-sound e f p with is-∅ e | is-∅ f"},{"Add":"... | yes _ | _ = sum (inj₂ p)"},{"Add":"... | no _ | yes _ = sum (inj₁ p)"},{"Add":"... | no _ | no _ = p"},{"Add":""},{"Add":"∣-complete : ∀ {w} e f → w ∈ (e R.∣ f) → w ∈ (e ∣ f)"},{"Add":"∣-complete e f pr@(sum p) with is-∅ e | is-∅ f"},{"Add":"... | yes refl | _ = fromInj₂ (λ p → contradiction p ∉∅) p"},{"Add":"... | no _ | yes refl = fromInj₁ (λ p → contradiction p ∉∅) p"},{"Add":"... | no _ | no _ = pr"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Product"},{"Add":""},{"Add":"infixr 6 _∙_"},{"Add":""},{"Add":"_∙_ : (e f : Exp) → Exp"},{"Add":"e ∙ f with is-∅ e | is-ε e | is-∅ f | is-ε f"},{"Add":"... | yes _ | _ | _ | _ = R.∅"},{"Add":"... | _ | yes _ | _ | _ = f"},{"Add":"... | _ | _ | yes _ | _ = R.∅"},{"Add":"... | _ | _ | _ | yes _ = e"},{"Add":"... | _ | _ | _ | _ = e R.∙ f"},{"Add":""},{"Add":"∙-sound : ∀ {w} e f → w ∈ (e ∙ f) → w ∈ (e R.∙ f)"},{"Add":"∙-sound e f p with is-∅ e | is-ε e | is-∅ f | is-ε f"},{"Add":"... | yes refl | _ | _ | _ = contradiction p ∉∅"},{"Add":"... | no _ | yes refl | _ | _ = prod ([] ++ _) ε p"},{"Add":"... | no _ | no _ | yes refl | _ = contradiction p ∉∅"},{"Add":"... | no _ | no _ | no _ | yes refl = prod (_ ++[]) p ε"},{"Add":"... | no _ | no _ | no _ | no _ = p"},{"Add":""},{"Add":"∙-complete : ∀ {w} e f → w ∈ (e R.∙ f) → w ∈ (e ∙ f)"},{"Add":"∙-complete e f pr@(prod eq p q) with is-∅ e | is-ε e | is-∅ f | is-ε f"},{"Add":"... | yes refl | _ | _ | _ = contradiction p ∉∅"},{"Add":"... | no _ | yes refl | _ | _ = ∈ε∙e-inv pr"},{"Add":"... | no _ | no _ | yes refl | _ = contradiction q ∉∅"},{"Add":"... | no _ | no _ | no _ | yes refl = ∈e∙ε-inv pr"},{"Add":"... | no _ | no _ | no _ | no _ = pr"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Kleene star"},{"Add":""},{"Add":"infix 7 _⋆"},{"Add":""},{"Add":"_⋆ : Exp → Exp"},{"Add":"e ⋆ with is-∅ e | is-ε e"},{"Add":"... | yes _ | _ = R.ε"},{"Add":"... | _ | yes _ = R.ε"},{"Add":"... | _ | _ = e R.⋆"},{"Add":""},{"Add":"⋆-sound : ∀ {w} e → w ∈ (e ⋆) → w ∈ (e R.⋆)"},{"Add":"⋆-sound e p with is-∅ e | is-ε e"},{"Add":"... | yes refl | _ = star (sum (inj₁ p))"},{"Add":"... | no _ | yes refl = star (sum (inj₁ p))"},{"Add":"... | no _ | no _ = p"},{"Add":""},{"Add":"⋆-complete : ∀ {w} e → w ∈ (e R.⋆) → w ∈ (e ⋆)"},{"Add":"⋆-complete e pr with is-∅ e | is-ε e"},{"Add":"... | yes refl | no _ = ∈∅⋆-inv pr"},{"Add":"... | no _ | yes refl = ∈ε⋆-inv pr"},{"Add":"... | no _ | no _ = pr"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Derived notions: at least one and maybe one"},{"Add":""},{"Add":"infixl 7 _+ _⁇"},{"Add":"_+ : Exp → Exp"},{"Add":"e + = e ∙ e ⋆"},{"Add":""},{"Add":"_⁇ : Exp → Exp"},{"Add":"∅ ⁇ = ε"},{"Add":"e ⁇ = ε ∣ e"}]}},{"FileAdd":{"meta":{"basename":"Search.agda","metadata":0},"context":[{"change":0,"pos":72073}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Regular expressions: search algorithms"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (DecPoset)"},{"Add":""},{"Add":"module Text.Regex.Search {a e r} (P? : DecPoset a e r) where"},{"Add":""},{"Add":"open import Level using (_⊔_)"},{"Add":"open import Data.Bool.Base using (if_then_else_; true; false)"},{"Add":"open import Data.List.Base using (List; []; _∷_)"},{"Add":"open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)"},{"Add":"open import Function.Base using (id; _∘′_; _∘_)"},{"Add":""},{"Add":"open import Data.List.Relation.Binary.Prefix.Heterogeneous"},{"Add":" using (Prefix; []; _∷_) hiding (module Prefix)"},{"Add":"open import Data.List.Relation.Binary.Infix.Heterogeneous"},{"Add":" using (Infix; here; there) hiding (module Infix)"},{"Add":"import Data.List.Relation.Binary.Infix.Heterogeneous.Properties as Infixₚ"},{"Add":"open import Data.List.Relation.Binary.Pointwise"},{"Add":" as Pointwise using (Pointwise; []; _∷_)"},{"Add":"open import Data.List.Relation.Binary.Suffix.Heterogeneous"},{"Add":" using (Suffix; here; there) hiding (module Suffix)"},{"Add":""},{"Add":"open import Relation.Nullary using (Dec; ¬_; yes; no)"},{"Add":"open import Relation.Nullary.Decidable using (map′)"},{"Add":"open import Relation.Nullary.Negation using (contradiction)"},{"Add":"open import Relation.Binary.Core using (Rel; _⇒_)"},{"Add":"open import Relation.Binary.Definitions using (Decidable)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core"},{"Add":""},{"Add":"open DecPoset P? using (preorder) renaming (Carrier to A)"},{"Add":"open import Text.Regex.Base preorder"},{"Add":"open import Text.Regex.Properties P?"},{"Add":"open import Text.Regex.Derivative.Brzozowski P?"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Type corresponding to a match"},{"Add":""},{"Add":"-- Users have control over whether the match should start at the"},{"Add":"-- beginning or stop at the end. So we have a precise type of spans"},{"Add":"-- ensuring their demands are respected"},{"Add":"Span : ∀ {r} → Regex → Rel A r → Rel (List A) (a ⊔ r)"},{"Add":"Span regex ="},{"Add":" if Regex.fromStart regex"},{"Add":" then if Regex.tillEnd regex"},{"Add":" then Pointwise"},{"Add":" else Prefix"},{"Add":" else if Regex.tillEnd regex"},{"Add":" then Suffix"},{"Add":" else Infix"},{"Add":""},{"Add":"-- All matches are selecting an infix sublist"},{"Add":"toInfix : ∀ {r} {R : Rel A r} e → Span e R ⇒ Infix R"},{"Add":"toInfix e with Regex.fromStart e | Regex.tillEnd e"},{"Add":"... | true | true = Infixₚ.fromPointwise"},{"Add":"... | true | false = here"},{"Add":"... | false | true = Infixₚ.fromSuffix"},{"Add":"... | false | false = id"},{"Add":""},{"Add":"-- A match is a list, a proof it matches the regular expression,"},{"Add":"-- and a proof it is the right sort of sublist."},{"Add":"record Match {s} (R : Rel (List A) s) (xs : List A) (exp : Exp)"},{"Add":" : Set (a ⊔ e ⊔ r ⊔ s) where"},{"Add":" constructor mkMatch"},{"Add":" field"},{"Add":" list : List A"},{"Add":" match : list ∈ exp"},{"Add":" related : R list xs"},{"Add":"open Match public"},{"Add":""},{"Add":"map : ∀ {r s} {R : Rel (List A) r} {S : Rel (List A) s} {xs ys e} →"},{"Add":" (∀ {a} → R a xs → S a ys) → Match R xs e → Match S ys e"},{"Add":"map f (mkMatch ys ys∈e pys) = mkMatch ys ys∈e (f pys)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Search algorithms"},{"Add":""},{"Add":"module Prefix where"},{"Add":""},{"Add":" []ᴹ : ∀ {xs e} → [] ∈ e → Match (Prefix _≡_) xs e"},{"Add":" []ᴹ p = mkMatch [] p []"},{"Add":""},{"Add":" []⁻¹ᴹ : ∀ {e} → Match (Prefix _≡_) [] e → [] ∈ e"},{"Add":" []⁻¹ᴹ (mkMatch .[] p []) = p"},{"Add":""},{"Add":" infixr 5 _∷ᴹ_ _∷⁻¹ᴹ_"},{"Add":""},{"Add":" _∷ᴹ_ : ∀ {xs e} x → Match (Prefix _≡_) xs (eat x e) → Match (Prefix _≡_) (x ∷ xs) e"},{"Add":" x ∷ᴹ (mkMatch ys ys∈e\\x ys≤xs) = mkMatch (x ∷ ys) (eat-sound x _ ys∈e\\x) (refl ∷ ys≤xs)"},{"Add":""},{"Add":" _∷⁻¹ᴹ_ : ∀ {xs x e} → [] ∉ e →"},{"Add":" Match (Prefix _≡_) (x ∷ xs) e → Match (Prefix _≡_) xs (eat x e)"},{"Add":" []∉e ∷⁻¹ᴹ (mkMatch .[] []∈e []) = contradiction []∈e []∉e"},{"Add":" []∉e ∷⁻¹ᴹ (mkMatch (._ ∷ ys) ys∈e (refl ∷ ys≤xs)) = mkMatch ys (eat-complete _ _ ys∈e) ys≤xs"},{"Add":""},{"Add":" shortest : Decidable (Match (Prefix _≡_))"},{"Add":" shortest xs ∅ = no (∉∅ ∘ match)"},{"Add":" shortest xs e with []∈? e"},{"Add":" ... | yes []∈e = yes ([]ᴹ []∈e)"},{"Add":" shortest [] e | no []∉e = no ([]∉e ∘′ []⁻¹ᴹ)"},{"Add":" shortest (x ∷ xs) e | no []∉e with shortest xs (eat x e)"},{"Add":" ... | yes p = yes (x ∷ᴹ p)"},{"Add":" ... | no ¬p = no (¬p ∘ ([]∉e ∷⁻¹ᴹ_))"},{"Add":""},{"Add":" longest : Decidable (Match (Prefix _≡_))"},{"Add":" longest [] e = map′ []ᴹ []⁻¹ᴹ ([]∈? e)"},{"Add":" longest xs ∅ = no (∉∅ ∘ match)"},{"Add":" longest (x ∷ xs) e with longest xs (eat x e)"},{"Add":" ... | yes p = yes (x ∷ᴹ p)"},{"Add":" ... | no ¬p with []∈? e"},{"Add":" ... | yes []∈e = yes ([]ᴹ []∈e)"},{"Add":" ... | no []∉e = no (¬p ∘ ([]∉e ∷⁻¹ᴹ_))"},{"Add":""},{"Add":"module Infix where"},{"Add":""},{"Add":" []⁻¹ᴹ : ∀ {e acc} → Match (Infix _≡_) [] e ⊎ Match (Prefix _≡_) [] acc → [] ∈ e ⊎ [] ∈ acc"},{"Add":" []⁻¹ᴹ (inj₁ (mkMatch .[] []∈e (here []))) = inj₁ []∈e"},{"Add":" []⁻¹ᴹ (inj₂ (mkMatch .[] []∈acc [])) = inj₂ []∈acc"},{"Add":""},{"Add":" step : ∀ {e acc} x {xs} → Match (Infix _≡_) xs e ⊎ Match (Prefix _≡_) xs (eat x (acc ∣ e)) →"},{"Add":" Match (Infix _≡_) (x ∷ xs) e ⊎ Match (Prefix _≡_) (x ∷ xs) acc"},{"Add":" step x (inj₁ (mkMatch ys ys∈e p)) = inj₁ (mkMatch ys ys∈e (there p))"},{"Add":" step {e} {acc} x (inj₂ (mkMatch ys ys∈e p)) with eat-sound x (acc ∣ e) ys∈e"},{"Add":" ... | sum (inj₂ xys∈e) = inj₁ (mkMatch (x ∷ ys) xys∈e (here (refl ∷ p)))"},{"Add":" ... | sum (inj₁ xys∈e) = inj₂ (mkMatch (x ∷ ys) xys∈e (refl ∷ p))"},{"Add":""},{"Add":" step⁻¹ : ∀ {e acc} x {xs} →"},{"Add":" [] ∉ e → [] ∉ acc →"},{"Add":" Match (Infix _≡_) (x ∷ xs) e ⊎ Match (Prefix _≡_) (x ∷ xs) acc →"},{"Add":" Match (Infix _≡_) xs e ⊎ Match (Prefix _≡_) xs (eat x (acc ∣ e))"},{"Add":" -- can't possibly be the empty match"},{"Add":" step⁻¹ x []∉e []∉acc (inj₁ (mkMatch .[] ys∈e (here []))) = contradiction ys∈e []∉e"},{"Add":" step⁻¹ x []∉e []∉acc (inj₂ (mkMatch .[] ys∈e [])) = contradiction ys∈e []∉acc"},{"Add":" -- if it starts 'there', it's an infix solution"},{"Add":" step⁻¹ x []∉e []∉acc (inj₁ (mkMatch ys ys∈e (there p))) = inj₁ (mkMatch ys ys∈e p)"},{"Add":" -- if it starts 'here' we're in prefix territory"},{"Add":" step⁻¹ {e} {acc} x []∉e []∉acc (inj₁ (mkMatch (.x ∷ ys) ys∈e (here (refl ∷ p))))"},{"Add":" = inj₂ (mkMatch ys (eat-complete x (acc ∣ e) (sum (inj₂ ys∈e))) p)"},{"Add":" step⁻¹ {e} {acc} x []∉e []∉acc (inj₂ (mkMatch (.x ∷ ys) ys∈e (refl ∷ p)))"},{"Add":" = inj₂ (mkMatch ys (eat-complete x (acc ∣ e) (sum (inj₁ ys∈e))) p)"},{"Add":""},{"Add":" -- search non-deterministically: at each step, the `acc` regex is"},{"Add":" -- changed to accomodate the fact the match may be starting just now"},{"Add":" searchND : ∀ xs e acc → [] ∉ e → Dec (Match (Infix _≡_) xs e ⊎ Match (Prefix _≡_) xs acc)"},{"Add":" searchND xs e acc []∉e with []∈? acc"},{"Add":" ... | yes []∈acc with Prefix.longest xs acc -- get the best match possible"},{"Add":" ... | yes longer = yes (inj₂ longer)"},{"Add":" ... | no noMatch = contradiction (mkMatch [] []∈acc []) noMatch"},{"Add":" searchND [] e acc []∉e | no []∉acc = no ([ []∉e , []∉acc ]′ ∘′ []⁻¹ᴹ)"},{"Add":" searchND (x ∷ xs) e acc []∉e | no []∉acc"},{"Add":" = map′ (step x) (step⁻¹ x []∉e []∉acc) (searchND xs e (eat x (acc ∣ e)) []∉e)"},{"Add":""},{"Add":" search : Decidable (Match (Infix _≡_))"},{"Add":" search xs e with []∈? e"},{"Add":" ... | yes []∈e = yes (mkMatch [] []∈e (here []))"},{"Add":" ... | no []∉e with searchND xs e ∅ []∉e"},{"Add":" ... | no ¬p = no (¬p ∘′ inj₁)"},{"Add":" ... | yes (inj₁ p) = yes p"},{"Add":" ... | yes (inj₂ p) = contradiction (match p) ∉∅"},{"Add":""},{"Add":"module Whole where"},{"Add":""},{"Add":" whole : ∀ xs e → xs ∈ e → Match (Pointwise _≡_) xs e"},{"Add":" whole xs e p = mkMatch xs p (Pointwise.refl refl)"},{"Add":""},{"Add":" whole⁻¹ : ∀ xs e → Match (Pointwise _≡_) xs e → xs ∈ e"},{"Add":" whole⁻¹ xs e (mkMatch ys ys∈e p) with Pointwise.Pointwise-≡⇒≡ p"},{"Add":" whole⁻¹ xs e (mkMatch .xs xs∈e p) | refl = xs∈e"},{"Add":""},{"Add":" search : Decidable (Match (Pointwise _≡_))"},{"Add":" search xs e = map′ (whole xs e) (whole⁻¹ xs e) (xs ∈? e)"},{"Add":""},{"Add":"module Suffix where"},{"Add":""},{"Add":" []⁻¹ᴹ : ∀ {e acc} → Match (Suffix _≡_) [] e ⊎ Match (Pointwise _≡_) [] acc → [] ∈ e ⊎ [] ∈ acc"},{"Add":" []⁻¹ᴹ (inj₁ (mkMatch .[] ys∈e (here []))) = inj₁ ys∈e"},{"Add":" []⁻¹ᴹ (inj₂ (mkMatch .[] ys∈acc [])) = inj₂ ys∈acc"},{"Add":""},{"Add":" step : ∀ {e acc} x {xs} →"},{"Add":" Match (Suffix _≡_) xs e ⊎ Match (Pointwise _≡_) xs (eat x (e ∣ acc)) →"},{"Add":" Match (Suffix _≡_) (x ∷ xs) e ⊎ Match (Pointwise _≡_) (x ∷ xs) acc"},{"Add":" step x (inj₁ (mkMatch ys ys∈e p)) = inj₁ (mkMatch ys ys∈e (there p))"},{"Add":" step {e} {acc} x (inj₂ (mkMatch ys ys∈e p)) with eat-sound x (e ∣ acc) ys∈e"},{"Add":" ... | sum (inj₁ xys∈e) = inj₁ (mkMatch (x ∷ ys) xys∈e (here (refl ∷ p)))"},{"Add":" ... | sum (inj₂ xys∈acc) = inj₂ (mkMatch (x ∷ ys) xys∈acc (refl ∷ p))"},{"Add":""},{"Add":" step⁻¹ : ∀ {e acc} x {xs} →"},{"Add":" Match (Suffix _≡_) (x ∷ xs) e ⊎ Match (Pointwise _≡_) (x ∷ xs) acc →"},{"Add":" Match (Suffix _≡_) xs e ⊎ Match (Pointwise _≡_) xs (eat x (e ∣ acc))"},{"Add":" -- match starts later"},{"Add":" step⁻¹ x (inj₁ (mkMatch ys ys∈e (there p))) = inj₁ (mkMatch ys ys∈e p)"},{"Add":" -- match starts here!"},{"Add":" step⁻¹ {e} {acc} x (inj₁ (mkMatch (.x ∷ ys) ys∈e (here (refl ∷ p))))"},{"Add":" = inj₂ (mkMatch ys (eat-complete x (e ∣ acc) (sum (inj₁ ys∈e))) p)"},{"Add":" step⁻¹ {e} {acc} x (inj₂ (mkMatch (.x ∷ ys) ys∈e (refl ∷ p)))"},{"Add":" = inj₂ (mkMatch ys (eat-complete x (e ∣ acc) (sum (inj₂ ys∈e))) p)"},{"Add":""},{"Add":" searchND : ∀ xs e acc → Dec (Match (Suffix _≡_) xs e ⊎ Match (Pointwise _≡_) xs acc)"},{"Add":" searchND [] e acc with []∈? e | []∈? acc"},{"Add":" ... | yes []∈e | _ = yes (inj₁ (mkMatch [] []∈e (here [])))"},{"Add":" ... | _ | yes []∈acc = yes (inj₂ (mkMatch [] []∈acc []))"},{"Add":" ... | no []∉e | no []∉acc = no ([ []∉e , []∉acc ]′ ∘′ []⁻¹ᴹ)"},{"Add":" searchND (x ∷ xs) e acc"},{"Add":" = map′ (step x) (step⁻¹ x) (searchND xs e (eat x (e ∣ acc)))"},{"Add":""},{"Add":" search : Decidable (Match (Suffix _≡_))"},{"Add":" search xs e with searchND xs e ∅"},{"Add":" ... | no ¬p = no (¬p ∘′ inj₁)"},{"Add":" ... | yes (inj₁ p) = yes p"},{"Add":" ... | yes (inj₂ p) = contradiction (match p) ∉∅"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Search for the user-specified span"},{"Add":""},{"Add":"search : ∀ xs e → Dec (Match (Span e _≡_) xs (Regex.expression e))"},{"Add":"search xs e with Regex.fromStart e | Regex.tillEnd e"},{"Add":"... | true | true = Whole.search xs (Regex.expression e)"},{"Add":"... | true | false = Prefix.shortest xs (Regex.expression e)"},{"Add":"... | false | true = Suffix.search xs (Regex.expression e)"},{"Add":"... | false | false = Infix.search xs (Regex.expression e)"}]}},{"FileAdd":{"meta":{"basename":"Properties.agda","metadata":0},"context":[{"change":0,"pos":72073}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of regular expressions and their semantics"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (DecPoset)"},{"Add":""},{"Add":"module Text.Regex.Properties {a e r} (P? : DecPoset a e r) where"},{"Add":""},{"Add":"open import Data.List.Base using (List; []; _∷_)"},{"Add":"open import Data.List.Relation.Unary.All using (all?)"},{"Add":"open import Data.List.Relation.Unary.Any using (any?)"},{"Add":"open import Data.Product.Base using (_×_; _,_; uncurry)"},{"Add":"open import Data.Sum.Base using (_⊎_; inj₁; inj₂)"},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"open import Relation.Nullary.Decidable"},{"Add":" using (Dec; yes; no; map′; ¬?; _×-dec_; _⊎-dec_)"},{"Add":"open import Relation.Nullary.Negation"},{"Add":" using (¬_; contradiction)"},{"Add":""},{"Add":"import Relation.Unary as U"},{"Add":"open import Relation.Binary.Definitions using (Decidable)"},{"Add":""},{"Add":"open DecPoset P? renaming (Carrier to A)"},{"Add":"open import Text.Regex.Base preorder"},{"Add":"open import Data.List.Relation.Binary.Pointwise.Base using ([])"},{"Add":"open import Data.List.Relation.Ternary.Appending.Propositional {A = A}"},{"Add":"open import Data.List.Relation.Ternary.Appending.Propositional.Properties {A = A}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Publicly re-export core properties"},{"Add":""},{"Add":"open import Text.Regex.Properties.Core preorder public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Decidability results"},{"Add":""},{"Add":"[]∈?_ : U.Decidable ([] ∈_)"},{"Add":"[]∈? ε = yes ε"},{"Add":"[]∈? [ rs ] = no (λ ())"},{"Add":"[]∈? [^ rs ] = no (λ ())"},{"Add":"[]∈? (e ∣ f) = map′ sum (λ where (sum pr) → pr)"},{"Add":" $ ([]∈? e) ⊎-dec ([]∈? f)"},{"Add":"[]∈? (e ∙ f) = map′ (uncurry (prod ([]++ []))) []∈e∙f-inv"},{"Add":" $ ([]∈? e) ×-dec ([]∈? f)"},{"Add":"[]∈? (e ⋆) = yes (star (sum (inj₁ ε)))"},{"Add":""},{"Add":"infix 4 _∈ᴿ?_ _∉ᴿ?_ _∈?ε _∈?[_] _∈?[^_]"},{"Add":""},{"Add":"_∈ᴿ?_ : Decidable _∈ᴿ_"},{"Add":"c ∈ᴿ? [ a ] = map′ [_] (λ where [ eq ] → eq) (c ≟ a)"},{"Add":"c ∈ᴿ? (lb ─ ub) = map′ (uncurry _─_) (λ where (ge ─ le) → ge , le)"},{"Add":" $ (lb ≤? c) ×-dec (c ≤? ub)"},{"Add":""},{"Add":"_∉ᴿ?_ : Decidable _∉ᴿ_"},{"Add":"a ∉ᴿ? r = ¬? (a ∈ᴿ? r)"},{"Add":""},{"Add":"_∈?ε : U.Decidable (_∈ ε)"},{"Add":"[] ∈?ε = yes ε"},{"Add":"(a ∷ _) ∈?ε = no (λ ())"},{"Add":""},{"Add":"_∈?[_] : ∀ w rs → Dec (w ∈ [ rs ])"},{"Add":"[] ∈?[ rs ] = no (λ ())"},{"Add":"(a ∷ b ∷ _) ∈?[ rs ] = no (λ ())"},{"Add":"(a ∷ []) ∈?[ rs ] = map′ [_] (λ where [ p ] → p)"},{"Add":" $ any? (a ∈ᴿ?_) rs"},{"Add":""},{"Add":"_∈?[^_] : ∀ w rs → Dec (w ∈ [^ rs ])"},{"Add":"[] ∈?[^ rs ] = no (λ ())"},{"Add":"(a ∷ []) ∈?[^ rs ] = map′ [^_] (λ where [^ p ] → p) $ all? (a ∉ᴿ?_) rs"},{"Add":"(a ∷ b ∷ _) ∈?[^ rs ] = no (λ ())"}]}},{"FileAdd":{"meta":{"basename":"Properties","metadata":576},"context":[{"change":0,"pos":72073}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Core.agda","metadata":0},"context":[{"change":0,"pos":92574}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Regular expressions: core properties (only require a Preorder)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (Preorder)"},{"Add":""},{"Add":"module Text.Regex.Properties.Core {a e r} (P : Preorder a e r) where"},{"Add":""},{"Add":"open import Level using (_⊔_)"},{"Add":""},{"Add":"open import Data.Bool.Base using (Bool)"},{"Add":"open import Data.List.Base as List using (List; []; _∷_; _++_)"},{"Add":""},{"Add":"open import Data.List.Relation.Unary.Any using (Any; here; there)"},{"Add":"open import Data.List.Relation.Unary.All using (All; []; _∷_)"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Data.Sum.Base using (_⊎_; inj₁; inj₂)"},{"Add":""},{"Add":"open import Relation.Nullary using (¬_; Dec; yes; no)"},{"Add":"open import Relation.Nullary.Negation using (contradiction)"},{"Add":"open import Relation.Unary using (Pred)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)"},{"Add":""},{"Add":"open Preorder P using (_≈_) renaming (Carrier to A; _∼_ to _≤_)"},{"Add":"open import Text.Regex.Base P"},{"Add":""},{"Add":"open import Data.List.Relation.Ternary.Appending.Propositional.Properties {A = A}"},{"Add":" using (++[]⁻¹; []++⁻¹; conicalˡ; conicalʳ)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Views"},{"Add":""},{"Add":"is-∅ : ∀ (e : Exp) → Dec (e ≡ ∅)"},{"Add":"is-∅ ε = no (λ ())"},{"Add":"is-∅ [ [] ] = yes refl"},{"Add":"is-∅ [ r ∷ rs ] = no (λ ())"},{"Add":"is-∅ [^ rs ] = no (λ ())"},{"Add":"is-∅ (e ∣ f) = no (λ ())"},{"Add":"is-∅ (e ∙ f) = no (λ ())"},{"Add":"is-∅ (e ⋆) = no (λ ())"},{"Add":""},{"Add":"is-ε : ∀ (e : Exp) → Dec (e ≡ ε)"},{"Add":"is-ε ε = yes refl"},{"Add":"is-ε [ rs ] = no (λ ())"},{"Add":"is-ε [^ rs ] = no (λ ())"},{"Add":"is-ε (e ∣ f) = no (λ ())"},{"Add":"is-ε (e ∙ f) = no (λ ())"},{"Add":"is-ε (e ⋆) = no (λ ())"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Inversion lemmas"},{"Add":""},{"Add":"∉∅ : ∀ {xs} → xs ∉ ∅"},{"Add":"∉∅ [ () ]"},{"Add":""},{"Add":"∈ε⋆-inv : ∀ {w} → w ∈ (ε ⋆) → w ∈ ε"},{"Add":"∈ε⋆-inv (star (sum (inj₁ ε))) = ε"},{"Add":"∈ε⋆-inv (star (sum (inj₂ (prod eq ε p)))) rewrite []++⁻¹ eq = ∈ε⋆-inv p"},{"Add":""},{"Add":"∈∅⋆-inv : ∀ {w} → w ∈ (∅ ⋆) → w ∈ ε"},{"Add":"∈∅⋆-inv (star (sum (inj₁ ε))) = ε"},{"Add":"∈∅⋆-inv (star (sum (inj₂ (prod eq p q)))) = contradiction p ∉∅"},{"Add":""},{"Add":"∈ε∙e-inv : ∀ {w e} → w ∈ (ε ∙ e) → w ∈ e"},{"Add":"∈ε∙e-inv (prod eq ε p) rewrite []++⁻¹ eq = p"},{"Add":""},{"Add":"∈e∙ε-inv : ∀ {w e} → w ∈ (e ∙ ε) → w ∈ e"},{"Add":"∈e∙ε-inv (prod eq p ε) rewrite ++[]⁻¹ eq = p"},{"Add":""},{"Add":"[]∈e∙f-inv : ∀ {e f} → [] ∈ (e ∙ f) → [] ∈ e × [] ∈ f"},{"Add":"[]∈e∙f-inv (prod eq p q) rewrite conicalˡ eq | conicalʳ eq = p , q"}]}},{"FileAdd":{"meta":{"basename":"Derivative","metadata":576},"context":[{"change":0,"pos":72073}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Brzozowski.agda","metadata":0},"context":[{"change":0,"pos":95331}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Regular expressions: Brzozowski derivative"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (DecPoset)"},{"Add":""},{"Add":"module Text.Regex.Derivative.Brzozowski {a e r} (P? : DecPoset a e r) where"},{"Add":""},{"Add":"open import Data.List.Base using (List; []; _∷_)"},{"Add":"open import Data.List.Relation.Binary.Equality.Propositional"},{"Add":"open import Data.Sum.Base as Sum using (inj₁; inj₂)"},{"Add":""},{"Add":"open import Function.Base using (_$_; _∘′_; case_of_)"},{"Add":"open import Relation.Nullary.Decidable using (Dec; yes; no)"},{"Add":"open import Relation.Nullary.Negation using (contradiction)"},{"Add":"open import Relation.Nullary.Decidable using (map′; ¬?)"},{"Add":"open import Relation.Binary.Definitions using (Decidable)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)"},{"Add":""},{"Add":"open DecPoset P? using (preorder) renaming (Carrier to A)"},{"Add":"open import Text.Regex.Base preorder as R hiding (_∣_; _∙_; _⋆)"},{"Add":"open import Text.Regex.Properties P?"},{"Add":"open import Text.Regex.SmartConstructors preorder"},{"Add":""},{"Add":"open import Data.List.Relation.Ternary.Appending.Propositional {A = A}"},{"Add":"open import Data.List.Relation.Ternary.Appending.Propositional.Properties {A = A}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Action of characters on regular expressions"},{"Add":""},{"Add":"private"},{"Add":" decToExp : ∀ {p} {P : Set p} → Dec P → Exp"},{"Add":" decToExp (yes _) = ε"},{"Add":" decToExp (no _) = ∅"},{"Add":""},{"Add":"eat : A → Exp → Exp"},{"Add":"eat a ε = ∅"},{"Add":"eat a [ rs ] = decToExp ((a ∷ []) ∈?[ rs ])"},{"Add":"eat a [^ rs ] = decToExp ((a ∷ []) ∈?[^ rs ])"},{"Add":"eat a (e R.∣ f) = eat a e ∣ eat a f"},{"Add":"eat a (e R.∙ f) = case []∈? e of λ where"},{"Add":" (yes _) → (eat a e ∙ f) ∣ (eat a f)"},{"Add":" (no ¬p) → eat a e ∙ f"},{"Add":"eat a (e R.⋆) = eat a e ∙ (e ⋆)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- This action is sound and complete with respect to matching"},{"Add":""},{"Add":"eat-sound : ∀ x {xs} e → xs ∈ eat x e → (x ∷ xs) ∈ e"},{"Add":"eat-sound x ε pr = contradiction pr ∉∅"},{"Add":"eat-sound x [ rs ] pr with (x ∷ []) ∈?[ rs ]"},{"Add":"... | yes p = case pr of λ where ε → p"},{"Add":"... | no _ = contradiction pr ∉∅"},{"Add":"eat-sound x [^ rs ] pr with (x ∷ []) ∈?[^ rs ]"},{"Add":"... | yes p = case pr of λ where ε → p"},{"Add":"... | no _ = contradiction pr ∉∅"},{"Add":"eat-sound x (e R.∣ f) pr with ∣-sound (eat x e) (eat x f) pr"},{"Add":"... | sum pr′ = sum $ Sum.map (eat-sound x e) (eat-sound x f) pr′"},{"Add":"eat-sound x (e R.∙ f) pr with []∈? e"},{"Add":"... | yes []∈e with ∣-sound (eat x e ∙ f) (eat x f) pr"},{"Add":"... | sum (inj₂ pr') = prod ([] ++ _) []∈e (eat-sound x f pr')"},{"Add":"... | sum (inj₁ pr') with ∙-sound (eat x e) f pr'"},{"Add":"... | prod eq p q = prod (refl ∷ eq) (eat-sound x e p) q"},{"Add":"eat-sound x (e R.∙ f) pr | no ¬p with ∙-sound (eat x e) f pr"},{"Add":"... | prod eq p q = prod (refl ∷ eq) (eat-sound x e p) q"},{"Add":"eat-sound x (e R.⋆) pr with ∙-sound (eat x e) (e ⋆) pr"},{"Add":"... | prod eq p q ="},{"Add":" star (sum (inj₂ (prod (refl ∷ eq) (eat-sound x e p) (⋆-sound e q))))"},{"Add":""},{"Add":"eat-complete′ : ∀ x {xs w} e → w ≋ (x ∷ xs) → w ∈ e → xs ∈ eat x e"},{"Add":"eat-complete : ∀ x {xs} e → (x ∷ xs) ∈ e → xs ∈ eat x e"},{"Add":""},{"Add":"eat-complete x e = eat-complete′ x e ≋-refl"},{"Add":""},{"Add":"eat-complete′ x [ rs ] (refl ∷ []) [ p ]"},{"Add":" with (x ∷ []) ∈?[ rs ]"},{"Add":"... | yes _ = ε"},{"Add":"... | no ¬p = contradiction [ p ] ¬p"},{"Add":"eat-complete′ x [^ rs ] (refl ∷ []) [^ p ]"},{"Add":" with (x ∷ []) ∈?[^ rs ]"},{"Add":"... | yes _ = ε"},{"Add":"... | no ¬p = contradiction [^ p ] ¬p"},{"Add":"eat-complete′ x (e R.∣ f) eq (sum p) ="},{"Add":" ∣-complete (eat x e) (eat x f) $ sum $"},{"Add":" Sum.map (eat-complete′ x e eq) (eat-complete′ x f eq) p"},{"Add":"eat-complete′ x (e R.∙ f) eq p with []∈? e"},{"Add":"eat-complete′ x (e R.∙ f) (refl ∷ eq) (prod ([]++ _) p q) | no []∉e"},{"Add":" = contradiction p []∉e"},{"Add":"eat-complete′ x (e R.∙ f) (refl ∷ eq) (prod (refl ∷ app) p q) | no []∉e"},{"Add":" = ∙-complete (eat x e) f (prod (respʳ-≋ app eq) (eat-complete x e p) q)"},{"Add":"eat-complete′ x (e R.∙ f) eq (prod ([]++ eq′) p q) | yes []∈e"},{"Add":" = ∣-complete (eat x e ∙ f) (eat x f) $ sum $ inj₂"},{"Add":" $ eat-complete′ x f (≋-trans eq′ eq) q"},{"Add":"eat-complete′ x (e R.∙ f) (refl ∷ eq) (prod (refl ∷ app) p q) | yes []∈e"},{"Add":" = ∣-complete (eat x e ∙ f) (eat x f) $ sum $ inj₁"},{"Add":" $ ∙-complete (eat x e) f (prod (respʳ-≋ app eq) (eat-complete x e p) q)"},{"Add":"eat-complete′ x (e R.⋆) eq (star (sum (inj₂ (prod ([]++ app) p q))))"},{"Add":" = eat-complete′ x (e R.⋆) (≋-trans app eq) q"},{"Add":"eat-complete′ x (e R.⋆) (refl ∷ eq) (star (sum (inj₂ (prod (refl ∷ app) p q))))"},{"Add":" = ∙-complete (eat x e) (e ⋆) $ prod (respʳ-≋ app eq) (eat-complete x e p)"},{"Add":" $ ⋆-complete e q"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Consequence: matching is decidable"},{"Add":""},{"Add":"infix 4 _∈?_ _∉?_"},{"Add":""},{"Add":"_∈?_ : Decidable _∈_"},{"Add":"[] ∈? e = []∈? e"},{"Add":"(x ∷ xs) ∈? e = map′ (eat-sound x e) (eat-complete x e) (xs ∈? eat x e)"},{"Add":""},{"Add":"_∉?_ : Decidable _∉_"},{"Add":"xs ∉? e = ¬? (xs ∈? e)"}]}},{"FileAdd":{"meta":{"basename":"Base.agda","metadata":0},"context":[{"change":0,"pos":72073}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Regular expressions: basic types and semantics"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (Preorder)"},{"Add":""},{"Add":"module Text.Regex.Base {a e r} (P : Preorder a e r) where"},{"Add":""},{"Add":"open import Level using (_⊔_)"},{"Add":""},{"Add":"open import Data.Bool.Base using (Bool)"},{"Add":"open import Data.List.Base as L using (List; []; _∷_)"},{"Add":"open import Data.List.Relation.Unary.Any using (Any)"},{"Add":"open import Data.List.Relation.Unary.All using (All)"},{"Add":"open import Data.Sum.Base using (_⊎_)"},{"Add":""},{"Add":"open import Relation.Nullary.Negation.Core using (¬_)"},{"Add":""},{"Add":"open Preorder P using (_≈_) renaming (Carrier to A; _∼_ to _≤_)"},{"Add":"open import Data.List.Relation.Ternary.Appending.Propositional {A = A}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Regular expressions on the alphabet A"},{"Add":""},{"Add":"infix 10 [_] _─_"},{"Add":"data Range : Set a where"},{"Add":" [_] : (a : A) → Range"},{"Add":" _─_ : (lb ub : A) → Range"},{"Add":""},{"Add":"infixr 5 _∣_"},{"Add":"infixr 6 _∙_"},{"Add":"infixl 7 _⋆"},{"Add":"infix 10 [^_]"},{"Add":""},{"Add":"data Exp : Set a where"},{"Add":" ε : Exp"},{"Add":" [_] : (rs : List Range) → Exp"},{"Add":" [^_] : (rs : List Range) → Exp"},{"Add":" _∣_ : (e f : Exp) → Exp"},{"Add":" _∙_ : (e f : Exp) → Exp"},{"Add":" _⋆ : (e : Exp) → Exp"},{"Add":""},{"Add":"-- A regular expression has additional parameters:"},{"Add":"-- * should the match begin at the very start of the input?"},{"Add":"-- * should it span until the very end?"},{"Add":""},{"Add":"record Regex : Set a where"},{"Add":" field"},{"Add":" fromStart : Bool"},{"Add":" tillEnd : Bool"},{"Add":" expression : Exp"},{"Add":""},{"Add":"updateExp : (Exp → Exp) → Regex → Regex"},{"Add":"updateExp f r = record r { expression = f (Regex.expression r) }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Derived notions: nothing, anything, and singleton"},{"Add":""},{"Add":"pattern ∅ = [ List.[] ]"},{"Add":"pattern · = [^ List.[] ]"},{"Add":""},{"Add":"pattern singleton a = [ Range.[ a ] ∷ [] ]"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Semantics: matching words"},{"Add":""},{"Add":"infix 4 _∈ᴿ_ _∉ᴿ_"},{"Add":"data _∈ᴿ_ (c : A) : Range → Set (a ⊔ r ⊔ e) where"},{"Add":" [_] : ∀ {val} → c ≈ val → c ∈ᴿ [ val ]"},{"Add":" _─_ : ∀ {lb ub} → lb ≤ c → c ≤ ub → c ∈ᴿ (lb ─ ub)"},{"Add":""},{"Add":"_∉ᴿ_ : A → Range → Set (a ⊔ r ⊔ e)"},{"Add":"a ∉ᴿ r = ¬ (a ∈ᴿ r)"},{"Add":""},{"Add":"infix 4 _∈_ _∉_"},{"Add":"data _∈_ : List A → Exp → Set (a ⊔ r ⊔ e) where"},{"Add":" ε : [] ∈ ε"},{"Add":" [_] : ∀ {a rs} → Any (a ∈ᴿ_) rs → L.[ a ] ∈ [ rs ]"},{"Add":" [^_] : ∀ {a rs} → All (a ∉ᴿ_) rs → L.[ a ] ∈ [^ rs ]"},{"Add":" sum : ∀ {w e f} → (w ∈ e) ⊎ (w ∈ f) → w ∈ (e ∣ f)"},{"Add":" prod : ∀ {v w vw e f} → Appending v w vw → v ∈ e → w ∈ f → vw ∈ (e ∙ f)"},{"Add":" star : ∀ {v e} → v ∈ (ε ∣ e ∙ (e ⋆)) → v ∈ (e ⋆)"},{"Add":""},{"Add":"_∉_ : List A → Exp → Set (a ⊔ r ⊔ e)"},{"Add":"w ∉ e = ¬ (w ∈ e)"}]}},{"FileAdd":{"meta":{"basename":"Printf.agda","metadata":0},"context":[{"change":0,"pos":62939}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Printf"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Text.Printf where"},{"Add":""},{"Add":"open import Data.String.Base using (String; fromChar; concat)"},{"Add":"open import Function.Base using (id)"},{"Add":""},{"Add":"import Data.Integer.Show as ℤ"},{"Add":"import Data.Float.Base as Float"},{"Add":"import Data.Nat.Show as ℕ"},{"Add":""},{"Add":"open import Text.Format as Format hiding (Error)"},{"Add":"open import Text.Printf.Generic"},{"Add":""},{"Add":"printfSpec : PrintfSpec formatSpec String"},{"Add":"printfSpec .PrintfSpec.renderArg ℕArg = ℕ.show"},{"Add":"printfSpec .PrintfSpec.renderArg ℤArg = ℤ.show"},{"Add":"printfSpec .PrintfSpec.renderArg FloatArg = Float.show"},{"Add":"printfSpec .PrintfSpec.renderArg CharArg = fromChar"},{"Add":"printfSpec .PrintfSpec.renderArg StringArg = id"},{"Add":"printfSpec .PrintfSpec.renderStr = id"},{"Add":""},{"Add":"module Printf = Type formatSpec"},{"Add":"open Printf public hiding (map)"},{"Add":"open Render printfSpec public renaming (printf to gprintf)"},{"Add":""},{"Add":"printf : (fmt : String) → Printf (lexer fmt) String"},{"Add":"printf fmt = Printf.map (lexer fmt) concat (gprintf fmt)"}]}},{"FileAdd":{"meta":{"basename":"Printf","metadata":576},"context":[{"change":0,"pos":62939}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Generic.agda","metadata":0},"context":[{"change":0,"pos":104739}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Generic printf function."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Text.Printf.Generic where"},{"Add":""},{"Add":"open import Level using (Level; 0ℓ; _⊔_; Lift)"},{"Add":"open import Data.List.Base as List using (List; []; _∷_)"},{"Add":"open import Data.Char.Base"},{"Add":"open import Data.Maybe.Base hiding (map)"},{"Add":"open import Data.Nat.Base using (ℕ)"},{"Add":"open import Data.Product.Base hiding (map)"},{"Add":"open import Data.Product.Nary.NonDependent"},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Data.Sum.Base using (_⊎_; inj₁; inj₂)"},{"Add":"open import Data.Unit.Base using (⊤)"},{"Add":"open import Function.Nary.NonDependent"},{"Add":"open import Function.Base"},{"Add":""},{"Add":"open import Text.Format.Generic"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ : Level"},{"Add":" A B : Set ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Printf argument specifications."},{"Add":"-- Defines the rendering of chunks."},{"Add":""},{"Add":"record PrintfSpec {ℓ} (spec : FormatSpec) (A : Set ℓ) : Set (Level.suc 0ℓ ⊔ ℓ) where"},{"Add":""},{"Add":" open FormatSpec spec public"},{"Add":""},{"Add":" field"},{"Add":" renderArg : ∀ arg → ArgType arg → A"},{"Add":" renderStr : String → A"},{"Add":""},{"Add":"module Type (spec : FormatSpec) where"},{"Add":""},{"Add":" open Format spec renaming (Error to FormatError)"},{"Add":""},{"Add":" record Error (e : FormatError) : Set ℓ where"},{"Add":""},{"Add":" private"},{"Add":""},{"Add":" Size : FormatError ⊎ Format → ℕ"},{"Add":" Size (inj₁ err) = 0"},{"Add":" Size (inj₂ fmt) = size fmt"},{"Add":""},{"Add":" Printf : ∀ pr → Set ℓ → Set (ℓ ⊔ ⨆ (Size pr) 0ℓs)"},{"Add":" Printf (inj₁ err) _ = Error err"},{"Add":" Printf (inj₂ fmt) B = Arrows _ ⟦ fmt ⟧ B"},{"Add":""},{"Add":" map : ∀ pr → (A → B) → Printf pr A → Printf pr B"},{"Add":" map (inj₁ err) f p = _"},{"Add":" map (inj₂ fmt) f p = mapₙ _ f p"},{"Add":""},{"Add":"module Render {spec : FormatSpec} (render : PrintfSpec spec A) where"},{"Add":""},{"Add":" open PrintfSpec render"},{"Add":" open Type spec"},{"Add":""},{"Add":" open Format spec renaming (Error to FormatError)"},{"Add":""},{"Add":" assemble : ∀ fmt → Product⊤ _ ⟦ fmt ⟧ → List A"},{"Add":" assemble [] vs = []"},{"Add":" assemble (Arg a ∷ fmt) (x , vs) = renderArg a x ∷ assemble fmt vs"},{"Add":" assemble (Raw str ∷ fmt) vs = renderStr str ∷ assemble fmt vs"},{"Add":""},{"Add":" private"},{"Add":""},{"Add":" printf′ : ∀ pr → Printf pr (List A)"},{"Add":" printf′ (inj₁ err) = _"},{"Add":" printf′ (inj₂ fmt) = curry⊤ₙ _ (assemble fmt)"},{"Add":""},{"Add":" printf : (input : String) → Printf (lexer input) (List A)"},{"Add":" printf input = printf′ (lexer input)"}]}},{"FileAdd":{"meta":{"basename":"Pretty.agda","metadata":0},"context":[{"change":0,"pos":62939}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Pretty Printing"},{"Add":"-- This module is based on Jean-Philippe Bernardy's functional pearl"},{"Add":"-- \"A Pretty But Not Greedy Printer\""},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --with-K #-}"},{"Add":""},{"Add":"open import Data.Nat.Base using (ℕ)"},{"Add":""},{"Add":"module Text.Pretty (width : ℕ) where"},{"Add":""},{"Add":"import Level"},{"Add":"open import Data.Char.Base using (Char)"},{"Add":"open import Data.List.Base"},{"Add":" using (List; _∷_; []; [_]; uncons; _++_; map; filter)"},{"Add":"open import Data.List.NonEmpty as List⁺ using (foldr₁)"},{"Add":"open import Data.Maybe.Base using (maybe′)"},{"Add":"open import Data.Product.Base using (uncurry)"},{"Add":"open import Data.String.Base using (String; fromList; replicate)"},{"Add":"open import Function.Base using (_∘_; _∘′_; _$_)"},{"Add":""},{"Add":"open import Effect.Monad using (RawMonad)"},{"Add":"import Data.List.Effectful as List"},{"Add":"open RawMonad (List.monad {Level.zero})"},{"Add":""},{"Add":"import Data.Nat.Properties as ℕ"},{"Add":"open import Data.List.Extrema.Core ℕ.≤-totalOrder using (⊓ᴸ)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Internal representation of documents and rendering function"},{"Add":""},{"Add":"import Text.Pretty.Core as Core"},{"Add":""},{"Add":"record Doc : Set where"},{"Add":" constructor mkDoc"},{"Add":" field runDoc : List Core.Block"},{"Add":"open Doc public"},{"Add":""},{"Add":"render : Doc → String"},{"Add":"render = Core.render"},{"Add":" ∘ maybe′ (foldr₁ (⊓ᴸ Core.Block.height) ∘′ uncurry List⁺._∷_) Core.empty"},{"Add":" ∘ uncons"},{"Add":" ∘′ runDoc"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Basic building blocks"},{"Add":""},{"Add":"fail : Doc"},{"Add":"fail = mkDoc []"},{"Add":""},{"Add":"text : String → Doc"},{"Add":"text = mkDoc ∘′ filter (Core.valid width) ∘ pure ∘ Core.text"},{"Add":""},{"Add":"empty : Doc"},{"Add":"empty = text \"\""},{"Add":""},{"Add":"char : Char → Doc"},{"Add":"char c = text (fromList (c ∷ []))"},{"Add":""},{"Add":"spaces : ℕ → Doc"},{"Add":"spaces n = text (replicate n ' ')"},{"Add":""},{"Add":"semi colon comma space dot : Doc"},{"Add":"semi = char ';'; colon = char ':'"},{"Add":"comma = char ','; space = char ' '; dot = char '.'"},{"Add":""},{"Add":"backslash forwardslash equal : Doc"},{"Add":"backslash = char '\\\\'; forwardslash = char '/'; equal = char '='"},{"Add":""},{"Add":"squote dquote : Doc"},{"Add":"squote = char '\\''; dquote = char '\"'"},{"Add":""},{"Add":"lparen rparen langle rangle : Doc"},{"Add":"lparen = char '('; rparen = char ')'"},{"Add":"langle = char '<'; rangle = char '>'"},{"Add":""},{"Add":"lbrace rbrace llbrace rrbrace : Doc"},{"Add":"lbrace = char '{'; rbrace = char '}'"},{"Add":"llbrace = char '⦃'; rrbrace = char '⦄'"},{"Add":""},{"Add":"lbracket rbracket llbracket rrbracket : Doc"},{"Add":"lbracket = char '['; rbracket = char ']'"},{"Add":"llbracket = char '⟦'; rrbracket = char '⟧'"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Combining two documents"},{"Add":""},{"Add":"infixr 5 _<>_"},{"Add":"_<>_ : Doc → Doc → Doc"},{"Add":"xs <> ys = mkDoc $"},{"Add":" let candidates = Core._<>_ <$> runDoc xs ⊛ runDoc ys in"},{"Add":" filter (Core.valid width) candidates"},{"Add":""},{"Add":""},{"Add":"flush : Doc → Doc"},{"Add":"flush = mkDoc ∘′ map Core.flush ∘′ runDoc"},{"Add":""},{"Add":"infixr 5 _<+>_"},{"Add":"_<+>_ : Doc → Doc → Doc"},{"Add":"x <+> y = x <> space <> y"},{"Add":""},{"Add":"infixr 5 _$$_"},{"Add":"_$$_ : Doc → Doc → Doc"},{"Add":"x $$ y = flush x <> y"},{"Add":""},{"Add":"infixr 4 _<|>_"},{"Add":"_<|>_ : Doc → Doc → Doc"},{"Add":"x <|> y = mkDoc (runDoc x ++ runDoc y)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Combining lists of documents"},{"Add":""},{"Add":"foldDoc : (Doc → Doc → Doc) → List Doc → Doc"},{"Add":"foldDoc _ [] = empty"},{"Add":"foldDoc _ (x ∷ []) = x"},{"Add":"foldDoc f (x ∷ xs) = f x (foldDoc f xs)"},{"Add":""},{"Add":"hsep vcat : List Doc → Doc"},{"Add":"hsep = foldDoc _<+>_"},{"Add":"vcat = foldDoc _$$_"},{"Add":""},{"Add":"sep : List Doc → Doc"},{"Add":"sep [] = empty"},{"Add":"sep xs = hsep xs <|> vcat xs"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Defined combinators"},{"Add":""},{"Add":"parens : Doc → Doc"},{"Add":"parens d = lparen <> d <> rparen"},{"Add":""},{"Add":"commaSep : List Doc → Doc"},{"Add":"commaSep = foldDoc (λ d e → d <> comma <+> e)"},{"Add":""},{"Add":"newline : Doc"},{"Add":"newline = flush empty"}]}},{"FileAdd":{"meta":{"basename":"Pretty","metadata":576},"context":[{"change":0,"pos":62939}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Core.agda","metadata":0},"context":[{"change":0,"pos":110969}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Pretty Printing"},{"Add":"-- This module is based on Jean-Philippe Bernardy's functional pearl"},{"Add":"-- \"A Pretty But Not Greedy Printer\""},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --with-K #-}"},{"Add":""},{"Add":"module Text.Pretty.Core where"},{"Add":""},{"Add":"import Level"},{"Add":""},{"Add":"open import Data.Bool.Base using (Bool)"},{"Add":"open import Data.Irrelevant as Irrelevant using (Irrelevant) hiding (module Irrelevant)"},{"Add":"open import Data.List.Base as List using (List; []; _∷_)"},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc; _+_; _⊔_; _≤_; z≤n)"},{"Add":"open import Data.Nat.Properties using (≤-refl; ≤-trans; +-identityʳ;"},{"Add":" module ≤-Reasoning; m≤n⊔m; +-monoʳ-≤; m≤m⊔n; +-comm; _≤?_)"},{"Add":"open import Data.Product.Base as Prod using (_×_; _,_; uncurry; proj₁; proj₂)"},{"Add":"import Data.Product.Relation.Unary.All as Allᴾ"},{"Add":""},{"Add":"open import Data.Tree.Binary as Tree using (Tree; leaf; node; #nodes; mapₙ)"},{"Add":"open import Data.Tree.Binary.Relation.Unary.All as Allᵀ using (leaf; node)"},{"Add":"open import Data.Unit.Base using (⊤; tt)"},{"Add":"import Data.Tree.Binary.Relation.Unary.All.Properties as Allᵀ"},{"Add":"import Data.Tree.Binary.Properties as Tree"},{"Add":""},{"Add":"open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; maybe′)"},{"Add":"open import Data.Maybe.Relation.Unary.All as Allᴹ using (nothing; just)"},{"Add":""},{"Add":"open import Data.String.Base as String"},{"Add":" using (String; length; replicate; _++_; unlines)"},{"Add":"import Data.String.Unsafe as String"},{"Add":"open import Function.Base using (_∘_; flip; _on_; id; _∘′_; _$_)"},{"Add":"open import Relation.Nullary.Decidable.Core using (Dec)"},{"Add":"open import Relation.Unary using (IUniversal; _⇒_; U)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core"},{"Add":" using (_≡_; refl; sym; cong; cong₂; subst)"},{"Add":"open import Relation.Binary.PropositionalEquality.Properties"},{"Add":" using (module ≡-Reasoning)"},{"Add":""},{"Add":"open import Data.Refinement using (Refinement-syntax; _,_; value; proof)"},{"Add":"import Data.Refinement.Relation.Unary.All as Allᴿ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Block of text"},{"Add":""},{"Add":"-- Content is a representation of the first line and the middle of the"},{"Add":"-- block. We use a tree rather than a list for the middle of the block"},{"Add":"-- so that we can extend it with lines on the left and on the line for"},{"Add":"-- free. We will ultimately render the block by traversing the tree left"},{"Add":"-- to right in a depth-first manner."},{"Add":""},{"Add":"Content : Set"},{"Add":"Content = Maybe (String × Tree String ⊤)"},{"Add":""},{"Add":"size : Content → ℕ"},{"Add":"size = maybe′ (suc ∘ #nodes ∘ proj₂) 0"},{"Add":""},{"Add":"All : ∀ {p} (P : String → Set p) → (Content → Set p)"},{"Add":"All P = Allᴹ.All (Allᴾ.All P (Allᵀ.All P U))"},{"Add":""},{"Add":"All≤ : ℕ → Content → Set"},{"Add":"All≤ n = All (λ s → length s ≤ n)"},{"Add":""},{"Add":"record Block : Set where"},{"Add":" field"},{"Add":" height : ℕ"},{"Add":" block : [ xs ∈ Content ∣ size xs ≡ height ]"},{"Add":" -- last line"},{"Add":" lastWidth : ℕ"},{"Add":" last : [ s ∈ String ∣ length s ≡ lastWidth ]"},{"Add":" -- max of all the widths"},{"Add":" maxWidth : [ n ∈ ℕ ∣ lastWidth ≤ n × All≤ n (block .value) ]"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Raw string"},{"Add":""},{"Add":"text : String → Block"},{"Add":"text s = record"},{"Add":" { height = 0"},{"Add":" ; block = nothing , ⦇ refl ⦈"},{"Add":" ; lastWidth = width"},{"Add":" ; last = s , ⦇ refl ⦈"},{"Add":" ; maxWidth = width , ⦇ (≤-refl , nothing) ⦈"},{"Add":" } where width = length s; open Irrelevant"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Empty"},{"Add":""},{"Add":"empty : Block"},{"Add":"empty = text \"\""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Helper functions"},{"Add":""},{"Add":"node? : Content → String → Tree String ⊤ → Content"},{"Add":"node? (just (x , xs)) y ys = just (x , node xs y ys)"},{"Add":"node? nothing y ys = just (y , ys)"},{"Add":""},{"Add":"∣node?∣ : ∀ b y ys → size (node? b y ys)"},{"Add":" ≡ size b + suc (#nodes ys)"},{"Add":"∣node?∣ (just (x , xs)) y ys = refl"},{"Add":"∣node?∣ nothing y ys = refl"},{"Add":""},{"Add":"≤-Content : ∀ {m n} {b : Content} → m ≤ n → All≤ m b → All≤ n b"},{"Add":"≤-Content {m} {n} m≤n = Allᴹ.map (Prod.map step (Allᵀ.mapₙ step))"},{"Add":""},{"Add":" where"},{"Add":""},{"Add":" step : ∀ {p} → p ≤ m → p ≤ n"},{"Add":" step = flip ≤-trans m≤n"},{"Add":""},{"Add":"All≤-node? : ∀ {l m r n} →"},{"Add":" All≤ n l → length m ≤ n → Allᵀ.All (λ s → length s ≤ n) U r →"},{"Add":" All≤ n (node? l m r)"},{"Add":"All≤-node? nothing py pys = just (py , pys)"},{"Add":"All≤-node? (just (px , pxs)) py pys = just (px , node pxs py pys)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Appending two documents"},{"Add":""},{"Add":"private"},{"Add":" module append (x y : Block) where"},{"Add":""},{"Add":" module x = Block x"},{"Add":" module y = Block y"},{"Add":""},{"Add":" blockx = x.block .value"},{"Add":" blocky = y.block .value"},{"Add":""},{"Add":" widthx = x.maxWidth .value"},{"Add":" widthy = y.maxWidth .value"},{"Add":""},{"Add":" lastx = x.last .value"},{"Add":" lasty = y.last .value"},{"Add":""},{"Add":" height : ℕ"},{"Add":" height = (_+_ on Block.height) x y"},{"Add":""},{"Add":" lastWidth : ℕ"},{"Add":" lastWidth = (_+_ on Block.lastWidth) x y"},{"Add":""},{"Add":" pad : Maybe String"},{"Add":" pad with x.lastWidth"},{"Add":" ... | 0 = nothing"},{"Add":" ... | l@(suc _) = just (replicate l ' ')"},{"Add":""},{"Add":" size-pad : maybe′ length 0 pad ≡ x.lastWidth"},{"Add":" size-pad with x.lastWidth"},{"Add":" ... | 0 = refl"},{"Add":" ... | l@(suc _) = String.length-replicate l"},{"Add":""},{"Add":" indent : Maybe String → String → String"},{"Add":" indent = maybe′ _++_ id"},{"Add":""},{"Add":" size-indent : ∀ ma str → length (indent ma str)"},{"Add":" ≡ maybe′ length 0 ma + length str"},{"Add":" size-indent nothing str = refl"},{"Add":" size-indent (just pad) str = String.length-++ pad str"},{"Add":""},{"Add":" indents : Maybe String → Tree String ⊤ → Tree String ⊤"},{"Add":" indents = maybe′ (mapₙ ∘ _++_) id"},{"Add":""},{"Add":" size-indents : ∀ ma t → #nodes (indents ma t) ≡ #nodes t"},{"Add":" size-indents nothing t = refl"},{"Add":" size-indents (just pad) t = Tree.#nodes-mapₙ (pad ++_) t"},{"Add":""},{"Add":" unfold-indents : ∀ ma t → indents ma t ≡ mapₙ (indent ma) t"},{"Add":" unfold-indents nothing t = sym (Tree.map-id t)"},{"Add":" unfold-indents (just pad) t = refl"},{"Add":""},{"Add":" vContent : Content × String"},{"Add":" vContent with blocky"},{"Add":" ... | nothing = blockx"},{"Add":" , lastx ++ lasty"},{"Add":" ... | just (hd , tl) = node?"},{"Add":" {-,--------------,-}"},{"Add":" {-|-} blockx {-|-}"},{"Add":" {-|-} {-'---,-} {-,------------------,-}"},{"Add":" {-|-} (lastx {-|-} ++ {-|-} hd) {-|-}"},{"Add":" {-'------------------'-} {-|-} {-|-}"},{"Add":" (indents pad {-|-} tl) {-,----'-}"},{"Add":" , indent pad {-|-} lasty {-|-}"},{"Add":" {-'-------------'-}"},{"Add":""},{"Add":" vBlock = proj₁ vContent"},{"Add":" vLast = proj₂ vContent"},{"Add":""},{"Add":" isBlock : size blockx ≡ x.height → size blocky ≡ y.height →"},{"Add":" size vBlock ≡ height"},{"Add":" isBlock ∣x∣ ∣y∣ with blocky"},{"Add":" ... | nothing = begin"},{"Add":" size blockx ≡⟨ ∣x∣ ⟩"},{"Add":" x.height ≡⟨ +-identityʳ x.height ⟨"},{"Add":" x.height + 0 ≡⟨ cong (_ +_) ∣y∣ ⟩"},{"Add":" x.height + y.height ∎ where open ≡-Reasoning"},{"Add":" ... | just (hd , tl) = begin"},{"Add":" ∣node∣ ≡⟨ ∣node?∣ blockx middle rest ⟩"},{"Add":" ∣blockx∣ + suc (#nodes rest) ≡⟨ cong ((size blockx +_) ∘′ suc) ∣rest∣ ⟩"},{"Add":" ∣blockx∣ + suc (#nodes tl) ≡⟨ cong₂ _+_ ∣x∣ ∣y∣ ⟩"},{"Add":" x.height + y.height ∎ where"},{"Add":""},{"Add":" open ≡-Reasoning"},{"Add":" ∣blockx∣ = size blockx"},{"Add":" middle = lastx ++ hd"},{"Add":" rest = indents pad tl"},{"Add":" ∣rest∣ = size-indents pad tl"},{"Add":" ∣node∣ = size (node? blockx middle rest)"},{"Add":""},{"Add":" block : [ xs ∈ Content ∣ size xs ≡ height ]"},{"Add":" block .value = vBlock"},{"Add":" block .proof = ⦇ isBlock (Block.block x .proof) (Block.block y .proof) ⦈"},{"Add":" where open Irrelevant"},{"Add":""},{"Add":" isLastLine : length lastx ≡ x.lastWidth →"},{"Add":" length lasty ≡ y.lastWidth →"},{"Add":" length vLast ≡ lastWidth"},{"Add":" isLastLine ∣x∣ ∣y∣ with blocky"},{"Add":" ... | nothing = begin"},{"Add":" length (lastx ++ lasty) ≡⟨ String.length-++ lastx lasty ⟩"},{"Add":" length lastx + length lasty ≡⟨ cong₂ _+_ ∣x∣ ∣y∣ ⟩"},{"Add":" x.lastWidth + y.lastWidth ∎ where open ≡-Reasoning"},{"Add":" ... | just (hd , tl) = begin"},{"Add":" length (indent pad lasty) ≡⟨ size-indent pad lasty ⟩"},{"Add":" maybe′ length 0 pad + length lasty ≡⟨ cong₂ _+_ size-pad ∣y∣ ⟩"},{"Add":" x.lastWidth + y.lastWidth ∎ where open ≡-Reasoning"},{"Add":""},{"Add":" last : [ s ∈ String ∣ length s ≡ lastWidth ]"},{"Add":" last .value = vLast"},{"Add":" last .proof = ⦇ isLastLine (Block.last x .proof) (Block.last y .proof) ⦈"},{"Add":" where open Irrelevant"},{"Add":""},{"Add":" vMaxWidth : ℕ"},{"Add":" vMaxWidth = widthx ⊔ (x.lastWidth + widthy)"},{"Add":""},{"Add":" isMaxWidth₁ : y.lastWidth ≤ widthy → lastWidth ≤ vMaxWidth"},{"Add":" isMaxWidth₁ p = begin"},{"Add":" lastWidth ≤⟨ +-monoʳ-≤ x.lastWidth p ⟩"},{"Add":" x.lastWidth + widthy ≤⟨ m≤n⊔m _ _ ⟩"},{"Add":" vMaxWidth ∎ where open ≤-Reasoning"},{"Add":""},{"Add":" isMaxWidth₂ : length lastx ≡ x.lastWidth →"},{"Add":" x.lastWidth ≤ widthx →"},{"Add":" All≤ widthx blockx →"},{"Add":" All≤ widthy blocky →"},{"Add":" All≤ vMaxWidth vBlock"},{"Add":" isMaxWidth₂ ∣x∣≡ ∣x∣≤ ∣xs∣ ∣ys∣ with blocky"},{"Add":" ... | nothing = ≤-Content (m≤m⊔n _ _) ∣xs∣"},{"Add":" isMaxWidth₂ ∣x∣≡ ∣x∣≤ ∣xs∣ (just (∣hd∣ , ∣tl∣))"},{"Add":" | just (hd , tl) ="},{"Add":" All≤-node? (≤-Content (m≤m⊔n _ _) ∣xs∣)"},{"Add":" middle"},{"Add":" (subst (Allᵀ.All _ U) (sym $ unfold-indents pad tl)"},{"Add":" $ Allᵀ.mapₙ⁺ (indent pad) (Allᵀ.mapₙ (indented _) ∣tl∣))"},{"Add":" where"},{"Add":""},{"Add":" middle : length (lastx ++ hd) ≤ vMaxWidth"},{"Add":" middle = begin"},{"Add":" length (lastx ++ hd) ≡⟨ String.length-++ lastx hd ⟩"},{"Add":" length lastx + length hd ≡⟨ cong (_+ _) ∣x∣≡ ⟩"},{"Add":" x.lastWidth + length hd ≤⟨ +-monoʳ-≤ x.lastWidth ∣hd∣ ⟩"},{"Add":" x.lastWidth + widthy ≤⟨ m≤n⊔m _ _ ⟩"},{"Add":" vMaxWidth ∎ where open ≤-Reasoning"},{"Add":""},{"Add":" indented : ∀ s → length s ≤ widthy →"},{"Add":" length (indent pad s) ≤ vMaxWidth"},{"Add":" indented s ∣s∣ = begin"},{"Add":" length (indent pad s) ≡⟨ size-indent pad s ⟩"},{"Add":" maybe′ length 0 pad + length s ≡⟨ cong (_+ _) size-pad ⟩"},{"Add":" x.lastWidth + length s ≤⟨ +-monoʳ-≤ x.lastWidth ∣s∣ ⟩"},{"Add":" x.lastWidth + widthy ≤⟨ m≤n⊔m (widthx) _ ⟩"},{"Add":" vMaxWidth ∎ where open ≤-Reasoning"},{"Add":""},{"Add":" maxWidth : [ n ∈ ℕ ∣ lastWidth ≤ n × All≤ n vBlock ]"},{"Add":" maxWidth .value = vMaxWidth"},{"Add":" maxWidth .proof ="},{"Add":" ⦇ _,_ ⦇ isMaxWidth₁ (map proj₁ (Block.maxWidth y .proof)) ⦈"},{"Add":" ⦇ isMaxWidth₂ (Block.last x .proof)"},{"Add":" (map proj₁ (Block.maxWidth x .proof))"},{"Add":" (map proj₂ (Block.maxWidth x .proof))"},{"Add":" (map proj₂ (Block.maxWidth y .proof))"},{"Add":" ⦈"},{"Add":" ⦈ where open Irrelevant"},{"Add":""},{"Add":"infixl 4 _<>_"},{"Add":"_<>_ : Block → Block → Block"},{"Add":"x <> y = record { append x y }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Flush (introduces a new line)"},{"Add":""},{"Add":"private"},{"Add":" module flush (x : Block) where"},{"Add":""},{"Add":" module x = Block x"},{"Add":" blockx = x.block .value"},{"Add":" lastx = x.last .value"},{"Add":" widthx = x.maxWidth .value"},{"Add":" heightx = x.height"},{"Add":""},{"Add":" height = suc heightx"},{"Add":" lastWidth = 0"},{"Add":" vMaxWidth = widthx"},{"Add":""},{"Add":" last : [ s ∈ String ∣ length s ≡ lastWidth ]"},{"Add":" last = \"\" , ⦇ refl ⦈ where open Irrelevant"},{"Add":""},{"Add":" vContent = node? blockx lastx (leaf tt)"},{"Add":""},{"Add":" isBlock : size blockx ≡ heightx → size vContent ≡ height"},{"Add":" isBlock ∣x∣ = begin"},{"Add":" size vContent ≡⟨ ∣node?∣ blockx lastx (leaf tt) ⟩"},{"Add":" size blockx + 1 ≡⟨ cong (_+ 1) ∣x∣ ⟩"},{"Add":" heightx + 1 ≡⟨ +-comm heightx 1 ⟩"},{"Add":" height ∎ where open ≡-Reasoning"},{"Add":""},{"Add":" block : [ xs ∈ Content ∣ size xs ≡ height ]"},{"Add":" block .value = vContent"},{"Add":" block .proof = Irrelevant.map isBlock $ Block.block x .proof"},{"Add":""},{"Add":" maxWidth : [ n ∈ ℕ ∣ lastWidth ≤ n × All≤ n vContent ]"},{"Add":" maxWidth .value = widthx"},{"Add":" maxWidth .proof = map (z≤n ,_)"},{"Add":" ⦇ All≤-node? ⦇ proj₂ (Block.maxWidth x .proof) ⦈"},{"Add":" ⦇ middle (Block.last x .proof) ⦇ proj₁ (Block.maxWidth x .proof) ⦈ ⦈"},{"Add":" (pure (leaf tt))"},{"Add":" ⦈ where"},{"Add":""},{"Add":" open Irrelevant"},{"Add":""},{"Add":" middle : length lastx ≡ x.lastWidth → x.lastWidth ≤ vMaxWidth →"},{"Add":" length lastx ≤ vMaxWidth"},{"Add":" middle p q = begin"},{"Add":" length lastx ≡⟨ p ⟩"},{"Add":" x.lastWidth ≤⟨ q ⟩"},{"Add":" vMaxWidth ∎ where open ≤-Reasoning"},{"Add":""},{"Add":"flush : Block → Block"},{"Add":"flush x = record { flush x }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Other functions"},{"Add":""},{"Add":"render : Block → String"},{"Add":"render x = unlines"},{"Add":" $ maybe′ (uncurry (λ hd tl → hd ∷ Tree.Infix.toList tl)) []"},{"Add":" $ node? (Block.block x .value) (Block.last x .value) (leaf tt)"},{"Add":""},{"Add":"valid : (width : ℕ) (b : Block) → Dec (Block.maxWidth b .value ≤ width)"},{"Add":"valid width b = Block.maxWidth b .value ≤? width"}]}},{"FileAdd":{"meta":{"basename":"Format.agda","metadata":0},"context":[{"change":0,"pos":62939}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Format strings for Printf and Scanf"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Text.Format where"},{"Add":""},{"Add":"open import Data.Maybe.Base"},{"Add":"open import Text.Format.Generic"},{"Add":""},{"Add":"-- Formatted types"},{"Add":"open import Data.Char.Base using (Char)"},{"Add":"open import Data.Integer.Base using (ℤ)"},{"Add":"open import Data.Float.Base using (Float)"},{"Add":"open import Data.Nat.Base using (ℕ)"},{"Add":"open import Data.String.Base using (String)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Basic types"},{"Add":""},{"Add":"data ArgChunk : Set where"},{"Add":" ℕArg ℤArg FloatArg CharArg StringArg : ArgChunk"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Semantics"},{"Add":""},{"Add":"ArgType : (fmt : ArgChunk) → Set"},{"Add":"ArgType ℕArg = ℕ"},{"Add":"ArgType ℤArg = ℤ"},{"Add":"ArgType FloatArg = Float"},{"Add":"ArgType CharArg = Char"},{"Add":"ArgType StringArg = String"},{"Add":""},{"Add":"lexArg : Char → Maybe ArgChunk"},{"Add":"lexArg 'd' = just ℤArg"},{"Add":"lexArg 'i' = just ℤArg"},{"Add":"lexArg 'u' = just ℕArg"},{"Add":"lexArg 'f' = just FloatArg"},{"Add":"lexArg 'c' = just CharArg"},{"Add":"lexArg 's' = just StringArg"},{"Add":"lexArg _ = nothing"},{"Add":""},{"Add":"formatSpec : FormatSpec"},{"Add":"formatSpec .FormatSpec.ArgChunk = ArgChunk"},{"Add":"formatSpec .FormatSpec.ArgType = ArgType"},{"Add":"formatSpec .FormatSpec.lexArg = lexArg"},{"Add":""},{"Add":"open Format formatSpec public"},{"Add":""},{"Add":"pattern `ℕ = Arg ℕArg"},{"Add":"pattern `ℤ = Arg ℤArg"},{"Add":"pattern `Float = Arg FloatArg"},{"Add":"pattern `Char = Arg CharArg"},{"Add":"pattern `String = Arg StringArg"}]}},{"FileAdd":{"meta":{"basename":"Format","metadata":576},"context":[{"change":0,"pos":62939}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Generic.agda","metadata":0},"context":[{"change":0,"pos":125865}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Format strings for Printf and Scanf"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Text.Format.Generic where"},{"Add":""},{"Add":"open import Level using (0ℓ)"},{"Add":"open import Effect.Applicative"},{"Add":"open import Data.Char.Base using (Char)"},{"Add":"open import Data.List.Base as List"},{"Add":"open import Data.Maybe.Base as Maybe"},{"Add":"open import Data.Nat.Base"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Data.Product.Nary.NonDependent"},{"Add":"open import Data.Sum.Base"},{"Add":"open import Data.String.Base"},{"Add":"import Data.Sum.Effectful.Left as Sumₗ"},{"Add":"open import Function.Base"},{"Add":"open import Function.Nary.NonDependent using (0ℓs; Sets)"},{"Add":"open import Function.Strict"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Format specifications."},{"Add":"-- Defines the supported %-codes."},{"Add":""},{"Add":"record FormatSpec : Set₁ where"},{"Add":" field"},{"Add":" ArgChunk : Set"},{"Add":" ArgType : ArgChunk → Set"},{"Add":" lexArg : Char → Maybe ArgChunk"},{"Add":""},{"Add":"module _ where"},{"Add":" open FormatSpec"},{"Add":""},{"Add":" -- Left-biased union of format specs."},{"Add":" unionSpec : FormatSpec → FormatSpec → FormatSpec"},{"Add":" unionSpec spec₁ spec₂ .ArgChunk = spec₁ .ArgChunk ⊎ spec₂ .ArgChunk"},{"Add":" unionSpec spec₁ spec₂ .ArgType (inj₁ a) = spec₁ .ArgType a"},{"Add":" unionSpec spec₁ spec₂ .ArgType (inj₂ a) = spec₂ .ArgType a"},{"Add":" unionSpec spec₁ spec₂ .lexArg c ="},{"Add":" Maybe.map inj₁ (spec₁ .lexArg c) <∣>"},{"Add":" Maybe.map inj₂ (spec₂ .lexArg c)"},{"Add":""},{"Add":"module Format (spec : FormatSpec) where"},{"Add":""},{"Add":" open FormatSpec spec"},{"Add":""},{"Add":" ----------------------------------------------------------------------"},{"Add":" -- Basic types"},{"Add":""},{"Add":" data Chunk : Set where"},{"Add":" Arg : ArgChunk → Chunk"},{"Add":" Raw : String → Chunk"},{"Add":""},{"Add":" Format : Set"},{"Add":" Format = List Chunk"},{"Add":""},{"Add":" ----------------------------------------------------------------------"},{"Add":" -- Semantics"},{"Add":""},{"Add":" size : Format → ℕ"},{"Add":" size = List.sum ∘′ List.map λ { (Raw _) → 0; _ → 1 }"},{"Add":""},{"Add":" -- Meaning of a format as a list of value types"},{"Add":""},{"Add":" ⟦_⟧ : (fmt : Format) → Sets (size fmt) 0ℓs"},{"Add":" ⟦ [] ⟧ = _"},{"Add":" ⟦ Arg a ∷ cs ⟧ = ArgType a , ⟦ cs ⟧"},{"Add":" ⟦ Raw _ ∷ cs ⟧ = ⟦ cs ⟧"},{"Add":""},{"Add":" ----------------------------------------------------------------------"},{"Add":" -- Lexer: from Strings to Formats"},{"Add":""},{"Add":" -- Lexing may fail. To have a useful error message, we defined the"},{"Add":" -- following enumerated type"},{"Add":""},{"Add":" data Error : Set where"},{"Add":" UnexpectedEndOfString : String → Error"},{"Add":" -- ^ expected a type declaration; found an empty string"},{"Add":" InvalidType : String → Char → String → Error"},{"Add":" -- ^ invalid type declaration"},{"Add":" -- return a focus: prefix processed, character causing failure, rest"},{"Add":""},{"Add":" lexer : String → Error ⊎ List Chunk"},{"Add":" lexer input = loop [] [] (toList input) where"},{"Add":""},{"Add":" open RawApplicative (Sumₗ.applicative Error 0ℓ)"},{"Add":""},{"Add":" -- Type synonyms used locally to document the code"},{"Add":" RevWord = List Char -- Mere characters accumulated so far"},{"Add":" Prefix = RevWord -- Prefix of the input String already read"},{"Add":""},{"Add":" toRevString : RevWord → String"},{"Add":" toRevString = fromList ∘′ reverse"},{"Add":""},{"Add":" -- Push a Raw token if we have accumulated some mere characters"},{"Add":" push : RevWord → List Chunk → List Chunk"},{"Add":" push [] ks = ks"},{"Add":" push cs ks = Raw (toRevString cs) ∷ ks"},{"Add":""},{"Add":" -- Main loop"},{"Add":" loop : RevWord → Prefix → List Char → Error ⊎ List Chunk"},{"Add":" type : Prefix → List Char → Error ⊎ List Chunk"},{"Add":""},{"Add":" loop acc bef [] = pure (push acc [])"},{"Add":" -- escaped '%' character: treat like a mere character"},{"Add":" loop acc bef ('%' ∷ '%' ∷ cs) = loop ('%' ∷ acc) ('%' ∷ '%' ∷ bef) cs"},{"Add":" -- non escaped '%': type declaration following"},{"Add":" loop acc bef ('%' ∷ cs) = push acc <$> type ('%' ∷ bef) cs"},{"Add":" -- mere character: push onto the accumulator"},{"Add":" loop acc bef (c ∷ cs) = loop (c ∷ acc) (c ∷ bef) cs"},{"Add":""},{"Add":" type bef [] = inj₁ (UnexpectedEndOfString input)"},{"Add":" type bef (c ∷ cs) = _∷_ <$> chunk c ⊛ loop [] (c ∷ bef) cs where"},{"Add":""},{"Add":" chunk : Char → Error ⊎ Chunk"},{"Add":" chunk c ="},{"Add":" case lexArg c of λ where"},{"Add":" (just ch) → pure (Arg ch)"},{"Add":" nothing →"},{"Add":" force′ (toRevString bef) λ prefix →"},{"Add":" force′ (fromList cs) λ suffix →"},{"Add":" inj₁ (InvalidType prefix c suffix)"}]}},{"FileAdd":{"meta":{"basename":"Test","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Golden.agda","metadata":0},"context":[{"change":0,"pos":130286}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Golden testing framework"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- This is a port of the golden testing framework used by the Idris2"},{"Add":"-- compiler and various Idris2 libraries."},{"Add":"-- It provides the core features required to perform golden file testing."},{"Add":"--"},{"Add":"-- We provide the core functionality to run a *single* golden file test,"},{"Add":"-- or a whole test tree. This allows the developer freedom to use as is"},{"Add":"-- or design the rest of the test harness to their liking."},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Test Structure"},{"Add":"--"},{"Add":"-- This harness works from the assumption that each individual golden"},{"Add":"-- test comprises of a directory with the following structure:"},{"Add":"--"},{"Add":"-- + `run` a *shell* script that runs the test. We expect it to:"},{"Add":"-- * Use `$1` as the variable standing for the Agda executable to be tested"},{"Add":"-- * Clean up after itself (e.g. by running `rm -rf build/`)"},{"Add":"--"},{"Add":"-- + `expected` a file containting the expected output of `run`"},{"Add":"--"},{"Add":"-- During testing, the test harness will generate an `output` file."},{"Add":"-- It will be compared to the `expected` golden file provided by the user."},{"Add":"-- In case there is a mismatch, the framework will:"},{"Add":""},{"Add":"-- + either display output & expected if the session is not interactive"},{"Add":"-- + or use the following command line to produce a diff and ask the user"},{"Add":"-- whether they want to overwrite the currently `expected` value:"},{"Add":"--"},{"Add":"-- git diff --no-index --exit-code --word-diff=color expected output"},{"Add":"--"},{"Add":"-- If `git` fails then the runner will simply present the expected and"},{"Add":"-- 'given' files side-by-side."},{"Add":"--"},{"Add":"-- Of note, it is helpful to add `output` to a local `.gitignore` instance"},{"Add":"-- to ensure that it is not mistakenly versioned."},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Options"},{"Add":"--"},{"Add":"-- The test harness has several options that may be set:"},{"Add":"--"},{"Add":"-- + `exeUnderTest` The path of the executable we are testing (typically `agda`)"},{"Add":"-- + `onlyNames` The tests to run relative to the generated executable."},{"Add":"-- + `onlyFile` The file listing the tests to run relative to the generated executable."},{"Add":"-- + `interactive` Whether to offer to update the expected file or not."},{"Add":"-- + `timing` Whether to display time taken for each test."},{"Add":"-- + `failureFile` The file in which to write the list of failing tests."},{"Add":"-- + `colour` The output should use ANSI escape codes"},{"Add":"--"},{"Add":"-- We provide an options parser (`options`) that takes the list of command line"},{"Add":"-- arguments and constructs this for you."},{"Add":"--"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Usage"},{"Add":"--"},{"Add":"-- When compiled to an executable the expected usage is:"},{"Add":"--"},{"Add":"--```sh"},{"Add":"--runtests "},{"Add":"-- [--timing]"},{"Add":"-- [--interactive]"},{"Add":"-- [--only-file PATH]"},{"Add":"-- [--failure-file PATH]"},{"Add":"-- [--no-colour]"},{"Add":"-- [--only [NAMES]]"},{"Add":"--```"},{"Add":"--"},{"Add":"-- assuming that the test runner is compiled to an executable named `runtests`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module Test.Golden where"},{"Add":""},{"Add":"open import Data.Bool.Base using (Bool; true; false; if_then_else_)"},{"Add":"import Data.Char as Char"},{"Add":"open import Data.Fin using (#_)"},{"Add":"import Data.Integer.Base as ℤ"},{"Add":"open import Data.List.Base as List using (List; []; _∷_; _++_; filter; partitionSums)"},{"Add":"open import Data.List.Relation.Binary.Infix.Heterogeneous.Properties using (infix?)"},{"Add":"open import Data.List.Relation.Unary.Any using (any?)"},{"Add":"open import Data.Maybe.Base using (Maybe; just; nothing; fromMaybe)"},{"Add":"open import Data.Nat.Base using (ℕ; _≡ᵇ_; _<ᵇ_; _+_; _∸_)"},{"Add":"import Data.Nat.Show as ℕ using (show)"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Data.String.Base as String using (String; lines; unlines; unwords; concat)"},{"Add":"open import Data.String.Properties as String using (_≟_)"},{"Add":"open import Data.Sum.Base using (_⊎_; inj₁; inj₂)"},{"Add":"open import Data.Unit.Base using (⊤)"},{"Add":""},{"Add":"open import Function.Base using (id; _$_; case_of_)"},{"Add":""},{"Add":"open import Relation.Nullary.Decidable.Core using (does)"},{"Add":""},{"Add":"open import Codata.Musical.Notation using (♯_)"},{"Add":"open import IO"},{"Add":""},{"Add":"open import System.Clock as Clock using (time′; Time; seconds)"},{"Add":"open import System.Console.ANSI"},{"Add":"open import System.Directory using (doesFileExist; doesDirectoryExist)"},{"Add":"open import System.Environment using (getArgs; lookupEnv)"},{"Add":"open import System.Exit"},{"Add":"open import System.FilePath.Posix using (mkFilePath)"},{"Add":"open import System.Process using (callCommand; system)"},{"Add":""},{"Add":"record Options : Set where"},{"Add":" field"},{"Add":" -- What is the name of the Agda executable?"},{"Add":" exeUnderTest : String"},{"Add":" -- Should we only run some specific cases?"},{"Add":" onlyNames : List String"},{"Add":" -- Should we run the test suite interactively?"},{"Add":" interactive : Bool"},{"Add":" -- Should we time and display the tests?"},{"Add":" timing : Bool"},{"Add":" -- Should we write the list of failing cases to a file?"},{"Add":" failureFile : Maybe String"},{"Add":" -- Should we use ANSI escape codes to colour the output?"},{"Add":" colour : Bool"},{"Add":"open Options"},{"Add":""},{"Add":"initOptions : String → Options"},{"Add":"initOptions exe = record"},{"Add":" { exeUnderTest = exe"},{"Add":" ; onlyNames = []"},{"Add":" ; interactive = false"},{"Add":" ; timing = false"},{"Add":" ; failureFile = nothing"},{"Add":" ; colour = true"},{"Add":" }"},{"Add":""},{"Add":"usage : String"},{"Add":"usage = unwords"},{"Add":" $ \"Usage:\""},{"Add":" ∷ \"runtests \""},{"Add":" ∷ \"[--timing]\""},{"Add":" ∷ \"[--interactive]\""},{"Add":" ∷ \"[--failure-file PATH]\""},{"Add":" ∷ \"[--only-file PATH]\""},{"Add":" ∷ \"[--no-colour]\""},{"Add":" ∷ \"[--only [NAMES]]\""},{"Add":" ∷ []"},{"Add":""},{"Add":"data Error : Set where"},{"Add":" MissingExecutable : Error"},{"Add":" InvalidArgument : String → Error"},{"Add":""},{"Add":"show : Error → String"},{"Add":"show MissingExecutable = \"Expected a path to Agda, got nothing\""},{"Add":"show (InvalidArgument arg) = \"Invalid argument: \" String.++ arg"},{"Add":""},{"Add":"-- Process the command line options"},{"Add":"options : List String → IO (Error ⊎ Options)"},{"Add":"options(exe ∷ rest) = mkOptions exe rest where"},{"Add":""},{"Add":" go : List String → Maybe String → Options → String ⊎ (Maybe String × Options)"},{"Add":" go [] mfp opts = inj₂ (mfp , opts)"},{"Add":" go (\"--timing\" ∷ args) mfp opts ="},{"Add":" go args mfp (record opts { timing = true })"},{"Add":" go (\"--interactive\" ∷ args) mfp opts ="},{"Add":" go args mfp (record opts { interactive = true })"},{"Add":" go (\"--failure-file\" ∷ fp ∷ args) mfp opts ="},{"Add":" go args mfp (record opts { failureFile = just fp })"},{"Add":" go (\"--only\" ∷ args) mfp opts ="},{"Add":" inj₂ (mfp , (record opts { onlyNames = args }))"},{"Add":" go (\"--only-file\" ∷ fp ∷ args) mfp opts ="},{"Add":" go args (just fp) (record opts { onlyNames = args })"},{"Add":" go (\"--no-colour\" ∷ args) mfp opts ="},{"Add":" go args mfp (record opts { colour = false })"},{"Add":" go (arg ∷ _) _ _ = inj₁ arg"},{"Add":""},{"Add":" mkOptions : String → List String → IO (Error ⊎ Options)"},{"Add":" mkOptions exe rest = do"},{"Add":" inj₂ (mfp , opts) ← pure $ go rest nothing (initOptions exe)"},{"Add":" where inj₁ arg → pure (inj₁ (InvalidArgument arg))"},{"Add":" term ← fromMaybe \"\" <$> lookupEnv \"TERM\""},{"Add":" let opts = if does (term ≟ \"DUMB\")"},{"Add":" then record opts { colour = false }"},{"Add":" else opts"},{"Add":" just fp ← pure mfp"},{"Add":" where _ → pure (inj₂ opts)"},{"Add":" only ← readFiniteFile fp"},{"Add":" pure $ inj₂ $ record opts { onlyNames = lines only ++ onlyNames opts }"},{"Add":""},{"Add":"options [] = pure (inj₁ MissingExecutable)"},{"Add":""},{"Add":""},{"Add":"-- The result of a test run"},{"Add":"-- `Left` corresponds to a failure and `Right` to a success"},{"Add":"Result : Set"},{"Add":"Result = String ⊎ String"},{"Add":""},{"Add":"-- Run the specified golden test with the supplied options."},{"Add":"runTest : Options → String → IO Result"},{"Add":"runTest opts testPath = do"},{"Add":""},{"Add":" true ← doesDirectoryExist (mkFilePath testPath)"},{"Add":" where false → fail directoryNotFound"},{"Add":""},{"Add":" time ← time′ $ callCommand $ unwords"},{"Add":" $ \"cd\" ∷ testPath"},{"Add":" ∷ \"&&\" ∷ \"sh ./run\" ∷ opts .exeUnderTest"},{"Add":" ∷ \"| tr -d '\\\\r' > output\""},{"Add":" ∷ []"},{"Add":""},{"Add":" just out ← readLocalFile \"output\""},{"Add":" where nothing → fail (fileNotFound \"output\")"},{"Add":" just exp ← readLocalFile \"expected\""},{"Add":" where nothing → do if opts .interactive"},{"Add":" then mayOverwrite nothing out"},{"Add":" else putStrLn (fileNotFound \"expected\")"},{"Add":" pure (inj₁ testPath)"},{"Add":""},{"Add":" let result = does (out String.≟ exp)"},{"Add":""},{"Add":" if result"},{"Add":" then printTiming (opts .timing) time"},{"Add":" $ if opts .colour"},{"Add":" then withCommand (setColour foreground classic green)"},{"Add":" else id"},{"Add":" $ \"success\""},{"Add":" else do printTiming (opts .timing) time"},{"Add":" $ if opts .colour"},{"Add":" then withCommand (setColour foreground bright red)"},{"Add":" else id"},{"Add":" $ \"FAILURE\""},{"Add":" if opts .interactive"},{"Add":" then mayOverwrite (just exp) out"},{"Add":" else putStrLn (unlines (expVsOut exp out))"},{"Add":""},{"Add":" pure $ if result then inj₂ testPath else inj₁ testPath"},{"Add":""},{"Add":" where"},{"Add":""},{"Add":" directoryNotFound : String"},{"Add":" directoryNotFound = unwords"},{"Add":" $ \"Directory\""},{"Add":" ∷ testPath"},{"Add":" ∷ \"does not exist\" ∷ []"},{"Add":""},{"Add":" fileNotFound : String → String"},{"Add":" fileNotFound name = unwords"},{"Add":" $ \"File\""},{"Add":" ∷ (testPath String.++ \"/output\")"},{"Add":" ∷ \"does not exist\" ∷ []"},{"Add":""},{"Add":" fail : String → IO Result"},{"Add":" fail msg = do putStrLn msg"},{"Add":" pure (inj₁ testPath)"},{"Add":""},{"Add":" readLocalFile : String → IO (Maybe String)"},{"Add":" readLocalFile name = do"},{"Add":" let fp = concat (testPath ∷ \"/\" ∷ name ∷ [])"},{"Add":" true ← doesFileExist (mkFilePath fp)"},{"Add":" where false → pure nothing"},{"Add":" just <$> readFiniteFile fp"},{"Add":""},{"Add":" getAnswer : IO Bool"},{"Add":" getAnswer = untilJust $ do"},{"Add":" str ← getLine"},{"Add":" case str of λ where"},{"Add":" \"y\" → pure $ just true"},{"Add":" \"n\" → pure $ just false"},{"Add":" \"\" → pure $ just false -- default answer is no"},{"Add":" _ → do putStrLn \"Invalid answer.\""},{"Add":" pure nothing"},{"Add":""},{"Add":" expVsOut : String → String → List String"},{"Add":" expVsOut exp out = \"Expected:\" ∷ exp ∷ \"Given:\" ∷ out ∷ []"},{"Add":""},{"Add":" hasFailed : ExitCode → Bool"},{"Add":" hasFailed ExitSuccess = false"},{"Add":" hasFailed (ExitFailure code) = code ℤ.≤ᵇ ℤ.+ 0"},{"Add":""},{"Add":" mayOverwrite : Maybe String → String → IO _"},{"Add":" mayOverwrite mexp out = do"},{"Add":" case mexp of λ where"},{"Add":" nothing → putStrLn $ unlines"},{"Add":" $ \"Golden value missing. I computed the following result:\""},{"Add":" ∷ out"},{"Add":" ∷ \"Accept new golden value? [y/N]\""},{"Add":" ∷ []"},{"Add":" (just exp) → do"},{"Add":" code ← system $ concat"},{"Add":" $ \"git diff --no-index --exit-code --word-diff=color \""},{"Add":" ∷ testPath ∷ \"/expected \""},{"Add":" ∷ testPath ∷ \"/output\""},{"Add":" ∷ []"},{"Add":" putStrLn $ unlines"},{"Add":" $ \"Golden value differs from actual value.\""},{"Add":" ∷ (if hasFailed code then expVsOut exp out else [])"},{"Add":" ++ \"Accept actual value as new golden value? [y/N]\""},{"Add":" ∷ []"},{"Add":""},{"Add":" b ← getAnswer"},{"Add":" when b $ writeFile (testPath String.++ \"/expected\") out"},{"Add":""},{"Add":" printTiming : Bool → Time → String → IO _"},{"Add":" printTiming false _ msg = putStrLn $ concat (testPath ∷ \": \" ∷ msg ∷ [])"},{"Add":" printTiming true time msg ="},{"Add":" let time = ℕ.show (time .seconds) String.++ \"s\""},{"Add":" spent = 9 + List.sum (List.map String.length (testPath ∷ time ∷ []))"},{"Add":" -- ^ hack: both \"success\" and \"FAILURE\" have the same length"},{"Add":" -- can't use `String.length msg` because the msg contains escape codes"},{"Add":" pad = String.replicate (72 ∸ spent) ' '"},{"Add":" in putStrLn (concat (testPath ∷ \": \" ∷ msg ∷ pad ∷ time ∷ []))"},{"Add":""},{"Add":"-- A test pool is characterised by"},{"Add":"-- + a name"},{"Add":"-- + and a list of directory paths"},{"Add":"record TestPool : Set where"},{"Add":" constructor mkTestPool"},{"Add":" field"},{"Add":" poolName : String"},{"Add":" testCases : List String"},{"Add":"open TestPool"},{"Add":""},{"Add":"module Summary where"},{"Add":""},{"Add":" -- The summary of a test pool run"},{"Add":" record Summary : Set where"},{"Add":" constructor mkSummary"},{"Add":" field"},{"Add":" success : List String"},{"Add":" failure : List String"},{"Add":" open Summary public"},{"Add":""},{"Add":" init : Summary"},{"Add":" init = mkSummary [] []"},{"Add":""},{"Add":" merge : Summary → Summary → Summary"},{"Add":" merge (mkSummary ws1 ls1) (mkSummary ws2 ls2) ="},{"Add":" mkSummary (ws2 ++ ws1) (ls2 ++ ls1)"},{"Add":""},{"Add":" update : List Result → Summary → Summary"},{"Add":" update res sum ="},{"Add":" let (ls2 , ws2) = partitionSums res in"},{"Add":" merge sum (mkSummary ws2 ls2)"},{"Add":""},{"Add":"open Summary using (Summary) hiding (module Summary)"},{"Add":""},{"Add":"-- Only keep the tests that have been asked for"},{"Add":"filterTests : Options → List String → List String"},{"Add":"filterTests opts = case onlyNames opts of λ where"},{"Add":" [] → id"},{"Add":" xs → let names = List.map String.toList xs in"},{"Add":" filter (λ n → any? (λ m → infix? Char._≟_ m (String.toList n)) names)"},{"Add":""},{"Add":"poolRunner : Options → TestPool → IO Summary"},{"Add":"poolRunner opts pool = do"},{"Add":" -- check there is a non-empty list of tests we want to run"},{"Add":" let tests = filterTests opts (pool .testCases)"},{"Add":" (_ ∷ _) ← pure tests"},{"Add":" where [] → pure Summary.init"},{"Add":""},{"Add":" -- announce the test pool and run them"},{"Add":" putStrLn banner"},{"Add":" loop Summary.init tests"},{"Add":""},{"Add":" where"},{"Add":""},{"Add":" separator : String"},{"Add":" separator = String.replicate 72 '-'"},{"Add":""},{"Add":" banner : String"},{"Add":" banner = unlines"},{"Add":" $ \"\" ∷ separator"},{"Add":" ∷ pool .poolName"},{"Add":" ∷ separator ∷ \"\""},{"Add":" ∷ []"},{"Add":""},{"Add":" loop : Summary → List String → IO Summary"},{"Add":" loop acc [] = pure acc"},{"Add":" loop acc (test ∷ tests) = do"},{"Add":" res <- runTest opts test"},{"Add":" loop (Summary.update (res ∷ []) acc) tests"},{"Add":""},{"Add":"runner : List TestPool → IO ⊤"},{"Add":"runner tests = do"},{"Add":" -- figure out the options"},{"Add":" args ← getArgs"},{"Add":" inj₂ opts ← options args"},{"Add":" where inj₁ err → die (unlines (show err ∷ usage ∷ []))"},{"Add":""},{"Add":" -- run the tests"},{"Add":" ress ← List.mapM (poolRunner opts) tests"},{"Add":" let open Summary"},{"Add":" let res = List.foldl merge init ress"},{"Add":""},{"Add":" -- report the result"},{"Add":" let nsucc = List.length (res .success)"},{"Add":" let nfail = List.length (res .failure)"},{"Add":" let ntotal = nsucc + nfail"},{"Add":" putStrLn $ concat $ ℕ.show nsucc ∷ \"/\" ∷ ℕ.show ntotal ∷ \" tests successful\" ∷ []"},{"Add":""},{"Add":" -- deal with failures"},{"Add":" let list = unlines (res .failure)"},{"Add":" when (0 <ᵇ nfail) $ do"},{"Add":" putStrLn \"Failing tests:\""},{"Add":" putStrLn list"},{"Add":" whenJust (opts .failureFile) $ λ fp → writeFile fp list"},{"Add":""},{"Add":" -- exit"},{"Add":" if 0 ≡ᵇ nfail"},{"Add":" then exitSuccess"},{"Add":" else exitFailure"}]}},{"FileAdd":{"meta":{"basename":"Tactic","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"RingSolver.agda","metadata":0},"context":[{"change":0,"pos":144494}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- A solver that uses reflection to automatically obtain and solve"},{"Add":"-- equations over rings."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Tactic.RingSolver where"},{"Add":""},{"Add":"open import Algebra"},{"Add":"open import Data.Fin.Base as Fin using (Fin)"},{"Add":"open import Data.Vec.Base as Vec using (Vec; _∷_; [])"},{"Add":"open import Data.List.Base as List using (List; _∷_; [])"},{"Add":"open import Data.Maybe.Base as Maybe using (Maybe; just; nothing; fromMaybe)"},{"Add":"open import Data.Nat.Base using (ℕ; suc; zero; _<ᵇ_)"},{"Add":"open import Data.Bool.Base using (Bool; if_then_else_; true; false)"},{"Add":"open import Data.Unit.Base using (⊤)"},{"Add":"open import Data.String.Base as String using (String; _++_; parens)"},{"Add":"open import Data.Product.Base using (_,_; proj₁)"},{"Add":"open import Function.Base"},{"Add":"open import Relation.Nullary.Decidable"},{"Add":""},{"Add":"open import Reflection"},{"Add":"open import Reflection.AST.Argument"},{"Add":"open import Reflection.AST.Term as Term"},{"Add":"open import Reflection.AST.AlphaEquality"},{"Add":"open import Reflection.AST.Name as Name"},{"Add":"open import Reflection.TCM.Syntax"},{"Add":"open import Data.Nat.Reflection"},{"Add":"open import Data.List.Reflection"},{"Add":"import Data.Vec.Reflection as Vec"},{"Add":""},{"Add":"open import Tactic.RingSolver.NonReflective renaming (solve to solver)"},{"Add":"open import Tactic.RingSolver.Core.AlmostCommutativeRing"},{"Add":"open import Tactic.RingSolver.Core.NatSet as NatSet"},{"Add":""},{"Add":"open AlmostCommutativeRing"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Utilities"},{"Add":""},{"Add":"private"},{"Add":" VarMap : Set"},{"Add":" VarMap = ℕ → Maybe Term"},{"Add":""},{"Add":" getVisible : Arg Term → Maybe Term"},{"Add":" getVisible (arg (arg-info visible _) x) = just x"},{"Add":" getVisible _ = nothing"},{"Add":""},{"Add":" getVisibleArgs : ∀ n → Term → Maybe (Vec Term n)"},{"Add":" getVisibleArgs n (def _ xs) = Maybe.map Vec.reverse"},{"Add":" (List.foldl f c (List.mapMaybe getVisible xs) n)"},{"Add":" where"},{"Add":" f : (∀ n → Maybe (Vec Term n)) → Term → ∀ n → Maybe (Vec Term n)"},{"Add":" f xs x zero = just []"},{"Add":" f xs x (suc n) = Maybe.map (x ∷_) (xs n)"},{"Add":""},{"Add":" c : ∀ n → Maybe (Vec Term n)"},{"Add":" c zero = just []"},{"Add":" c (suc _ ) = nothing"},{"Add":" getVisibleArgs _ _ = nothing"},{"Add":""},{"Add":" curriedTerm : NatSet → Term"},{"Add":" curriedTerm = List.foldr go Vec.`[] ∘ NatSet.toList"},{"Add":" where"},{"Add":" go : ℕ → Term → Term"},{"Add":" go x xs = var x [] Vec.`∷ xs"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Reflection utilities for rings"},{"Add":""},{"Add":"`AlmostCommutativeRing : Term"},{"Add":"`AlmostCommutativeRing = def (quote AlmostCommutativeRing) (2 ⋯⟨∷⟩ [])"},{"Add":""},{"Add":"record RingOperatorTerms : Set where"},{"Add":" constructor add⇒_mul⇒_pow⇒_neg⇒_sub⇒_"},{"Add":" field"},{"Add":" add mul pow neg sub : Term"},{"Add":""},{"Add":"checkIsRing : Term → TC Term"},{"Add":"checkIsRing ring = checkType ring `AlmostCommutativeRing"},{"Add":""},{"Add":"module RingReflection (`ring : Term) where"},{"Add":""},{"Add":" -- Takes the name of a function that takes the ring as it's first"},{"Add":" -- explicit argument and the terms of it's arguments and inserts"},{"Add":" -- the required ring arguments"},{"Add":" -- e.g. \"_+_\" $ʳ xs = \"_+_ {_} {_} ring xs\""},{"Add":" infixr 6 _$ʳ_"},{"Add":" _$ʳ_ : Name → Args Term → Term"},{"Add":" nm $ʳ args = def nm (2 ⋯⟅∷⟆ `ring ⟨∷⟩ args)"},{"Add":""},{"Add":" `Carrier : Term"},{"Add":" `Carrier = quote Carrier $ʳ []"},{"Add":""},{"Add":" `refl : Term"},{"Add":" `refl = quote refl $ʳ (1 ⋯⟅∷⟆ [])"},{"Add":""},{"Add":" `sym : Term → Term"},{"Add":" `sym x≈y = quote sym $ʳ (2 ⋯⟅∷⟆ x≈y ⟨∷⟩ [])"},{"Add":""},{"Add":" `trans : Term → Term → Term"},{"Add":" `trans x≈y y≈z = quote trans $ʳ (3 ⋯⟅∷⟆ x≈y ⟨∷⟩ y≈z ⟨∷⟩ [])"},{"Add":""},{"Add":" -- Normalises each of the fields of the ring operator so we can"},{"Add":" -- compare the result against the normalised definitions we come"},{"Add":" -- across when converting the term passed to the macro."},{"Add":" getRingOperatorTerms : TC RingOperatorTerms"},{"Add":" getRingOperatorTerms = ⦇"},{"Add":" add⇒ normalise (quote _+_ $ʳ [])"},{"Add":" mul⇒ normalise (quote _*_ $ʳ [])"},{"Add":" pow⇒ normalise (quote _^_ $ʳ [])"},{"Add":" neg⇒ normalise (quote (-_) $ʳ [])"},{"Add":" sub⇒ normalise (quote _-_ $ʳ [])"},{"Add":" ⦈"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Reflection utilities for ring solver"},{"Add":""},{"Add":"module RingSolverReflection (ring : Term) (numberOfVariables : ℕ) where"},{"Add":" open RingReflection ring"},{"Add":""},{"Add":" `numberOfVariables : Term"},{"Add":" `numberOfVariables = toTerm numberOfVariables"},{"Add":""},{"Add":" -- This function applies the hidden arguments that the constructors"},{"Add":" -- that Expr needs. The first is the universe level, the second is the"},{"Add":" -- type it contains, and the third is the number of variables it's"},{"Add":" -- indexed by. All three of these could likely be inferred, but to"},{"Add":" -- make things easier we supply the third because we know it."},{"Add":" infix -1 _$ᵉ_"},{"Add":" _$ᵉ_ : Name → List (Arg Term) → Term"},{"Add":" e $ᵉ xs = con e (1 ⋯⟅∷⟆ `Carrier ⟅∷⟆ `numberOfVariables ⟅∷⟆ xs)"},{"Add":""},{"Add":" -- A constant expression."},{"Add":" `Κ : Term → Term"},{"Add":" `Κ x = quote Κ $ᵉ (x ⟨∷⟩ [])"},{"Add":""},{"Add":" `I : Term → Term"},{"Add":" `I x = quote Ι $ᵉ (x ⟨∷⟩ [])"},{"Add":""},{"Add":" infixl 6 _`⊜_"},{"Add":""},{"Add":" _`⊜_ : Term → Term → Term"},{"Add":" x `⊜ y = quote _⊜_ $ʳ (`numberOfVariables ⟅∷⟆ x ⟨∷⟩ y ⟨∷⟩ [])"},{"Add":""},{"Add":" `correct : Term → Term → Term"},{"Add":" `correct x ρ = quote Ops.correct $ʳ (1 ⋯⟅∷⟆ x ⟨∷⟩ ρ ⟨∷⟩ [])"},{"Add":""},{"Add":" `solver : Term → Term → Term"},{"Add":" `solver `f `eq = quote solver $ʳ (`numberOfVariables ⟨∷⟩ `f ⟨∷⟩ `eq ⟨∷⟩ [])"},{"Add":""},{"Add":" -- Converts the raw terms provided by the macro into the `Expr`s"},{"Add":" -- used internally by the solver."},{"Add":" --"},{"Add":" -- When trying to figure out the shape of an expression, one of"},{"Add":" -- the difficult tasks is recognizing where constants in the"},{"Add":" -- underlying ring are used. If we were only dealing with ℕ, we"},{"Add":" -- might look for its constructors: however, we want to deal with"},{"Add":" -- arbitrary types which implement AlmostCommutativeRing. If the"},{"Add":" -- Term type contained type information we might be able to"},{"Add":" -- recognize it there, but it doesn't."},{"Add":" --"},{"Add":" -- We're in luck, though, because all other cases in the following"},{"Add":" -- function *are* recognizable. As a result, the \"catch-all\" case"},{"Add":" -- will just assume that it has a constant expression."},{"Add":" convertTerm : RingOperatorTerms → VarMap → Term → TC Term"},{"Add":" convertTerm operatorTerms varMap = convert"},{"Add":" where"},{"Add":" open RingOperatorTerms operatorTerms"},{"Add":""},{"Add":" mutual"},{"Add":" convert : Term → TC Term"},{"Add":" -- First try and match directly against the fields"},{"Add":" convert (def (quote _+_) xs) = convertOp₂ (quote _⊕_) xs"},{"Add":" convert (def (quote _*_) xs) = convertOp₂ (quote _⊗_) xs"},{"Add":" convert (def (quote -_) xs) = convertOp₁ (quote ⊝_) xs"},{"Add":" convert (def (quote _^_) xs) = convertExp xs"},{"Add":" convert (def (quote _-_) xs) = convertSub xs"},{"Add":" -- Other definitions the underlying implementation of the ring's fields"},{"Add":" convert (def nm xs) = convertUnknownName nm xs"},{"Add":" -- Variables"},{"Add":" convert v@(var x _) = pure $ fromMaybe (`Κ v) (varMap x)"},{"Add":" -- Special case to recognise \"suc\" for naturals"},{"Add":" convert (`suc x) = convertSuc x"},{"Add":" -- Otherwise we're forced to treat it as a constant"},{"Add":" convert t = pure $ `Κ t"},{"Add":""},{"Add":" -- Application of a ring operator often doesn't have a type as"},{"Add":" -- simple as \"Carrier → Carrier → Carrier\": there may be hidden"},{"Add":" -- arguments, etc. Here, we do our best to handle those cases,"},{"Add":" -- by just taking the last two explicit arguments."},{"Add":" convertOp₂ : Name → Args Term → TC Term"},{"Add":" convertOp₂ nm (x ⟨∷⟩ y ⟨∷⟩ []) = do"},{"Add":" x' ← convert x"},{"Add":" y' ← convert y"},{"Add":" pure (nm $ᵉ (x' ⟨∷⟩ y' ⟨∷⟩ []))"},{"Add":" convertOp₂ nm (x ∷ xs) = convertOp₂ nm xs"},{"Add":" convertOp₂ _ _ = pure unknown"},{"Add":""},{"Add":" convertOp₁ : Name → Args Term → TC Term"},{"Add":" convertOp₁ nm (x ⟨∷⟩ []) = do"},{"Add":" x' ← convert x"},{"Add":" pure (nm $ᵉ (x' ⟨∷⟩ []))"},{"Add":" convertOp₁ nm (x ∷ xs) = convertOp₁ nm xs"},{"Add":" convertOp₁ _ _ = pure unknown"},{"Add":""},{"Add":" convertExp : Args Term → TC Term"},{"Add":" convertExp (x ⟨∷⟩ y ⟨∷⟩ []) = do"},{"Add":" x' ← convert x"},{"Add":" pure (quote _⊛_ $ᵉ (x' ⟨∷⟩ y ⟨∷⟩ []))"},{"Add":" convertExp (x ∷ xs) = convertExp xs"},{"Add":" convertExp _ = pure unknown"},{"Add":""},{"Add":" convertSub : Args Term → TC Term"},{"Add":" convertSub (x ⟨∷⟩ y ⟨∷⟩ []) = do"},{"Add":" x' ← convert x"},{"Add":" -y' ← convertOp₁ (quote (⊝_)) (y ⟨∷⟩ [])"},{"Add":" pure (quote _⊕_ $ᵉ x' ⟨∷⟩ -y' ⟨∷⟩ [])"},{"Add":" convertSub (x ∷ xs) = convertSub xs"},{"Add":" convertSub _ = pure unknown"},{"Add":""},{"Add":" convertUnknownName : Name → Args Term → TC Term"},{"Add":" convertUnknownName nm xs = do"},{"Add":" nameTerm ← normalise (def nm [])"},{"Add":" if (nameTerm =α= add) then convertOp₂ (quote _⊕_) xs else"},{"Add":" if (nameTerm =α= mul) then convertOp₂ (quote _⊗_) xs else"},{"Add":" if (nameTerm =α= neg) then convertOp₁ (quote ⊝_) xs else"},{"Add":" if (nameTerm =α= pow) then convertExp xs else"},{"Add":" if (nameTerm =α= sub) then convertSub xs else"},{"Add":" pure (`Κ (def nm xs))"},{"Add":""},{"Add":" convertSuc : Term → TC Term"},{"Add":" convertSuc x = do x' ← convert x; pure (quote _⊕_ $ᵉ (`Κ (toTerm 1) ⟨∷⟩ x' ⟨∷⟩ []))"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Macros"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Quantified macro"},{"Add":""},{"Add":"open RingReflection"},{"Add":"open RingSolverReflection"},{"Add":""},{"Add":"malformedForallTypeError : ∀ {a} {A : Set a} → Term → TC A"},{"Add":"malformedForallTypeError found = typeError"},{"Add":" ( strErr \"Malformed call to solve.\""},{"Add":" ∷ strErr \"Expected target type to be like: ∀ x y → x + y ≈ y + x.\""},{"Add":" ∷ strErr \"Instead: \""},{"Add":" ∷ termErr found"},{"Add":" ∷ [])"},{"Add":""},{"Add":"quantifiedVarMap : ℕ → VarMap"},{"Add":"quantifiedVarMap numVars i ="},{"Add":" if i <ᵇ numVars"},{"Add":" then just (var i [])"},{"Add":" else nothing"},{"Add":""},{"Add":"constructCallToSolver : Term → RingOperatorTerms → List String → Term → Term → TC Term"},{"Add":"constructCallToSolver `ring opNames variables `lhs `rhs = do"},{"Add":" `lhsExpr ← conv `lhs"},{"Add":" `rhsExpr ← conv `rhs"},{"Add":""},{"Add":" pure $ `solver `ring numVars"},{"Add":" (prependVLams variables (_`⊜_ `ring numVars `lhsExpr `rhsExpr))"},{"Add":" (prependHLams variables (`refl `ring))"},{"Add":" where"},{"Add":" numVars : ℕ"},{"Add":" numVars = List.length variables"},{"Add":""},{"Add":" conv : Term → TC Term"},{"Add":" conv = convertTerm `ring numVars opNames (quantifiedVarMap numVars)"},{"Add":""},{"Add":"-- This is the main macro which solves for equations in which the"},{"Add":"-- variables are universally quantified over:"},{"Add":"--"},{"Add":"-- lemma : ∀ x y → x + y ≈ y + x"},{"Add":"-- lemma = solve-∀ ring"},{"Add":"--"},{"Add":"-- where ring is your implementation of AlmostCommutativeRing."},{"Add":"-- (Find some example implementations in"},{"Add":"-- Polynomial.Solver.Ring.AlmostCommutativeRing.Instances)."},{"Add":"solve-∀-macro : Name → Term → TC ⊤"},{"Add":"solve-∀-macro ring hole = do"},{"Add":" `ring ← checkIsRing (def ring [])"},{"Add":" commitTC"},{"Add":" operatorTerms ← getRingOperatorTerms `ring"},{"Add":""},{"Add":" -- Obtain and sanitise the goal type"},{"Add":" `hole ← inferType hole >>= reduce"},{"Add":" let variablesAndTypes , equation = stripPis `hole"},{"Add":""},{"Add":" let variables = List.map proj₁ variablesAndTypes"},{"Add":" just (lhs ∷ rhs ∷ []) ← pure (getVisibleArgs 2 equation)"},{"Add":" where nothing → malformedForallTypeError `hole"},{"Add":""},{"Add":" solverCall ← constructCallToSolver `ring operatorTerms variables lhs rhs"},{"Add":" unify hole solverCall"},{"Add":""},{"Add":"macro"},{"Add":" solve-∀ : Name → Term → TC ⊤"},{"Add":" solve-∀ = solve-∀-macro"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Unquantified macro"},{"Add":""},{"Add":"malformedArgumentListError : ∀ {a} {A : Set a} → Term → TC A"},{"Add":"malformedArgumentListError found = typeError"},{"Add":" ( strErr \"Malformed call to solve.\""},{"Add":" ∷ strErr \"First argument should be a list of free variables.\""},{"Add":" ∷ strErr \"Instead: \""},{"Add":" ∷ termErr found"},{"Add":" ∷ [])"},{"Add":""},{"Add":"malformedGoalError : ∀ {a} {A : Set a} → Term → TC A"},{"Add":"malformedGoalError found = typeError"},{"Add":" ( strErr \"Malformed call to solve.\""},{"Add":" ∷ strErr \"Goal type should be of the form: LHS ≈ RHS\""},{"Add":" ∷ strErr \"Instead: \""},{"Add":" ∷ termErr found"},{"Add":" ∷ [])"},{"Add":""},{"Add":"checkIsListOfVariables : Term → Term → TC Term"},{"Add":"checkIsListOfVariables `ring `xs = checkType `xs (`List (`Carrier `ring)) >>= normalise"},{"Add":""},{"Add":"-- Extracts the deBruijn indices from a list of variables"},{"Add":"getVariableIndices : Term → Maybe NatSet"},{"Add":"getVariableIndices = go []"},{"Add":" where"},{"Add":" go : NatSet → Term → Maybe NatSet"},{"Add":" go t (var i [] `∷` xs) = go (insert i t) xs"},{"Add":" go t `[]` = just t"},{"Add":" go _ _ = nothing"},{"Add":""},{"Add":"constructSolution : Term → RingOperatorTerms → NatSet → Term → Term → TC Term"},{"Add":"constructSolution `ring opTerms variables `lhs `rhs = do"},{"Add":" `lhsExpr ← conv `lhs"},{"Add":" `rhsExpr ← conv `rhs"},{"Add":" pure $ `trans `ring (`sym `ring `lhsExpr) `rhsExpr"},{"Add":" where"},{"Add":" numVars = List.length variables"},{"Add":""},{"Add":" varMap : VarMap"},{"Add":" varMap i = Maybe.map (λ x → `I `ring numVars (toFinTerm x)) (lookup variables i)"},{"Add":""},{"Add":" ρ : Term"},{"Add":" ρ = curriedTerm variables"},{"Add":""},{"Add":" conv = λ t → do"},{"Add":" t' ← convertTerm `ring numVars opTerms varMap t"},{"Add":" pure $ `correct `ring numVars t' ρ"},{"Add":""},{"Add":"-- Use this macro when you want to solve something *under* a lambda."},{"Add":"-- For example: say you have a long proof, and you just want the solver"},{"Add":"-- to deal with an intermediate step. Call it like so:"},{"Add":"--"},{"Add":"-- lemma₃ : ∀ x y → x + y * 1 + 3 ≈ 2 + 1 + y + x"},{"Add":"-- lemma₃ x y = begin"},{"Add":"-- x + y * 1 + 3 ≈⟨ +-comm x (y * 1) ⟨ +-cong ⟩ refl ⟩"},{"Add":"-- y * 1 + x + 3 ≈⟨ solve (x ∷ y ∷ []) Int.ring ⟩"},{"Add":"-- 3 + y + x ≡⟨ refl ⟩"},{"Add":"-- 2 + 1 + y + x ∎"},{"Add":"--"},{"Add":"-- The first argument is the free variables, and the second is the"},{"Add":"-- ring implementation (as before)."},{"Add":"solve-macro : Term → Name → Term → TC ⊤"},{"Add":"solve-macro variables ring hole = do"},{"Add":" `ring ← checkIsRing (def ring [])"},{"Add":" commitTC"},{"Add":" operatorTerms ← getRingOperatorTerms `ring"},{"Add":""},{"Add":" -- Obtain and sanitise the list of variables"},{"Add":" listOfVariables′ ← checkIsListOfVariables `ring variables"},{"Add":" commitTC"},{"Add":" just variableIndices ← pure (getVariableIndices listOfVariables′)"},{"Add":" where nothing → malformedArgumentListError listOfVariables′"},{"Add":""},{"Add":" -- Obtain and santise the goal type"},{"Add":" hole′ ← inferType hole >>= reduce"},{"Add":" just (lhs ∷ rhs ∷ []) ← pure (getVisibleArgs 2 hole′)"},{"Add":" where nothing → malformedGoalError hole′"},{"Add":""},{"Add":" solution ← constructSolution `ring operatorTerms variableIndices lhs rhs"},{"Add":" unify hole solution"},{"Add":""},{"Add":"macro"},{"Add":" solve : Term → Name → Term → TC ⊤"},{"Add":" solve = solve-macro"}]}},{"FileAdd":{"meta":{"basename":"RingSolver","metadata":576},"context":[{"change":0,"pos":144494}],"contents":[]}},{"FileAdd":{"meta":{"basename":"NonReflective.agda","metadata":0},"context":[{"change":0,"pos":159067}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- An implementation of the ring solver that requires you to manually"},{"Add":"-- pass the equation you wish to solve."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- You'll probably want to use `Tactic.RingSolver` instead which uses"},{"Add":"-- reflection to automatically extract the equation."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Tactic.RingSolver.Core.AlmostCommutativeRing"},{"Add":""},{"Add":"module Tactic.RingSolver.NonReflective"},{"Add":" {ℓ₁ ℓ₂} (ring : AlmostCommutativeRing ℓ₁ ℓ₂)"},{"Add":" (let open AlmostCommutativeRing ring)"},{"Add":" where"},{"Add":""},{"Add":"open import Algebra.Morphism"},{"Add":"open import Function.Base using (id; _⟨_⟩_)"},{"Add":"open import Data.Bool.Base using (Bool; true; false; T; if_then_else_)"},{"Add":"open import Data.Maybe.Base"},{"Add":"open import Data.Empty using (⊥-elim)"},{"Add":"open import Data.Nat.Base using (ℕ)"},{"Add":"open import Data.Product.Base using (_×_; proj₁; proj₂; _,_)"},{"Add":"open import Data.Vec.Base using (Vec)"},{"Add":"open import Data.Vec.N-ary"},{"Add":""},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Parameters"},{"Add":"open import Tactic.RingSolver.Core.Expression public"},{"Add":"open import Algebra.Properties.Semiring.Exp.TCOptimised semiring"},{"Add":""},{"Add":"module Ops where"},{"Add":" zero-homo : ∀ x → T (is-just (0≟ x)) → 0# ≈ x"},{"Add":" zero-homo x _ with 0≟ x"},{"Add":" zero-homo x _ | just p = p"},{"Add":" zero-homo x () | nothing"},{"Add":""},{"Add":" homo : Homomorphism ℓ₁ ℓ₂ ℓ₁ ℓ₂"},{"Add":" homo = record"},{"Add":" { from = record"},{"Add":" { rawRing = AlmostCommutativeRing.rawRing ring"},{"Add":" ; isZero = λ x → is-just (0≟ x)"},{"Add":" }"},{"Add":" ; to = record"},{"Add":" { isAlmostCommutativeRing = record"},{"Add":" { isCommutativeSemiring = isCommutativeSemiring"},{"Add":" ; -‿cong = -‿cong"},{"Add":" ; -‿*-distribˡ = -‿*-distribˡ"},{"Add":" ; -‿+-comm = -‿+-comm"},{"Add":" }"},{"Add":" }"},{"Add":" ; morphism = -raw-almostCommutative⟶ ring"},{"Add":" ; Zero-C⟶Zero-R = zero-homo"},{"Add":" }"},{"Add":" open Eval rawRing id public"},{"Add":""},{"Add":" open import Tactic.RingSolver.Core.Polynomial.Base (Homomorphism.from homo)"},{"Add":""},{"Add":" norm : ∀ {n} → Expr Carrier n → Poly n"},{"Add":" norm (Κ x) = κ x"},{"Add":" norm (Ι x) = ι x"},{"Add":" norm (x ⊕ y) = norm x ⊞ norm y"},{"Add":" norm (x ⊗ y) = norm x ⊠ norm y"},{"Add":" norm (⊝ x) = ⊟ norm x"},{"Add":" norm (x ⊛ i) = norm x ⊡ i"},{"Add":""},{"Add":" ⟦_⇓⟧ : ∀ {n} → Expr Carrier n → Vec Carrier n → Carrier"},{"Add":" ⟦ expr ⇓⟧ = ⟦ norm expr ⟧ₚ where"},{"Add":""},{"Add":" open import Tactic.RingSolver.Core.Polynomial.Semantics homo"},{"Add":" renaming (⟦_⟧ to ⟦_⟧ₚ)"},{"Add":""},{"Add":" correct : ∀ {n} (expr : Expr Carrier n) ρ → ⟦ expr ⇓⟧ ρ ≈ ⟦ expr ⟧ ρ"},{"Add":" correct {n = n} = go"},{"Add":" where"},{"Add":" open import Tactic.RingSolver.Core.Polynomial.Homomorphism homo"},{"Add":""},{"Add":" go : ∀ (expr : Expr Carrier n) ρ → ⟦ expr ⇓⟧ ρ ≈ ⟦ expr ⟧ ρ"},{"Add":" go (Κ x) ρ = κ-hom x ρ"},{"Add":" go (Ι x) ρ = ι-hom x ρ"},{"Add":" go (x ⊕ y) ρ = ⊞-hom (norm x) (norm y) ρ ⟨ trans ⟩ (go x ρ ⟨ +-cong ⟩ go y ρ)"},{"Add":" go (x ⊗ y) ρ = ⊠-hom (norm x) (norm y) ρ ⟨ trans ⟩ (go x ρ ⟨ *-cong ⟩ go y ρ)"},{"Add":" go (⊝ x) ρ = ⊟-hom (norm x) ρ ⟨ trans ⟩ -‿cong (go x ρ)"},{"Add":" go (x ⊛ i) ρ = ⊡-hom (norm x) i ρ ⟨ trans ⟩ ^-congˡ i (go x ρ)"},{"Add":""},{"Add":" open import Relation.Binary.Reflection setoid Ι ⟦_⟧ ⟦_⇓⟧ correct public"},{"Add":""},{"Add":"solve : ∀ (n : ℕ) →"},{"Add":" (f : N-ary n (Expr Carrier n) (Expr Carrier n × Expr Carrier n)) →"},{"Add":" Eqʰ n _≈_ (curryⁿ (Ops.⟦_⇓⟧ (proj₁ (Ops.close n f)))) (curryⁿ (Ops.⟦_⇓⟧ (proj₂ (Ops.close n f)))) →"},{"Add":" Eq n _≈_ (curryⁿ (Ops.⟦_⟧ (proj₁ (Ops.close n f)))) (curryⁿ (Ops.⟦_⟧ (proj₂ (Ops.close n f))))"},{"Add":"solve = Ops.solve"},{"Add":"{-# INLINE solve #-}"},{"Add":""},{"Add":"infixl 6 _⊜_"},{"Add":"_⊜_ : ∀ {n : ℕ} →"},{"Add":" Expr Carrier n →"},{"Add":" Expr Carrier n →"},{"Add":" Expr Carrier n × Expr Carrier n"},{"Add":"_⊜_ = _,_"},{"Add":"{-# INLINE _⊜_ #-}"}]}},{"FileAdd":{"meta":{"basename":"Core","metadata":576},"context":[{"change":0,"pos":159067}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Polynomial","metadata":576},"context":[{"change":0,"pos":162981}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Semantics.agda","metadata":0},"context":[{"change":0,"pos":162999}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- \"Evaluating\" a polynomial, using Horner's method."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Parameters"},{"Add":""},{"Add":"module Tactic.RingSolver.Core.Polynomial.Semantics"},{"Add":" {r₁ r₂ r₃ r₄}"},{"Add":" (homo : Homomorphism r₁ r₂ r₃ r₄)"},{"Add":" where"},{"Add":""},{"Add":"open import Data.Nat.Base using (ℕ; suc; zero; _≤′_; ≤′-step; ≤′-refl)"},{"Add":"open import Data.Vec.Base using (Vec; []; _∷_; uncons)"},{"Add":"open import Data.List.Base using ([]; _∷_)"},{"Add":"open import Data.Product.Base using (_,_; _×_)"},{"Add":"open import Data.List.Kleene using (_+; _*; ∹_; _&_; [])"},{"Add":""},{"Add":"open Homomorphism homo hiding (_^_)"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Base from"},{"Add":"open import Algebra.Properties.Semiring.Exp.TCOptimised semiring"},{"Add":""},{"Add":"drop : ∀ {i n} → i ≤′ n → Vec Carrier n → Vec Carrier i"},{"Add":"drop ≤′-refl xs = xs"},{"Add":"drop (≤′-step i+1≤n) (_ ∷ xs) = drop i+1≤n xs"},{"Add":""},{"Add":"drop-1 : ∀ {i n} → suc i ≤′ n → Vec Carrier n → Carrier × Vec Carrier i"},{"Add":"drop-1 si≤n xs = uncons (drop si≤n xs)"},{"Add":"{-# INLINE drop-1 #-}"},{"Add":""},{"Add":"_*⟨_⟩^_ : Carrier → Carrier → ℕ → Carrier"},{"Add":"x *⟨ ρ ⟩^ zero = x"},{"Add":"x *⟨ ρ ⟩^ suc i = ρ ^ (suc i) * x"},{"Add":"{-# INLINE _*⟨_⟩^_ #-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Evaluation"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Why do we have three functions here? Why are they so weird looking?"},{"Add":"--"},{"Add":"-- These three functions are the main bottleneck for all of the proofs:"},{"Add":"-- as such, slight changes can dramatically affect the length of proof"},{"Add":"-- code."},{"Add":""},{"Add":"mutual"},{"Add":""},{"Add":" _⟦∷⟧_ : ∀ {n} → Poly n × Coeff n * → Carrier × Vec Carrier n → Carrier"},{"Add":" (x , []) ⟦∷⟧ (ρ , ρs) = ⟦ x ⟧ ρs"},{"Add":" (x , (∹ xs)) ⟦∷⟧ (ρ , ρs) = ρ * ⅀⟦ xs ⟧ (ρ , ρs) + ⟦ x ⟧ ρs"},{"Add":""},{"Add":" ⅀⟦_⟧ : ∀ {n} → Coeff n + → (Carrier × Vec Carrier n) → Carrier"},{"Add":" ⅀⟦ x ≠0 Δ i & xs ⟧ (ρ , ρs) = ((x , xs) ⟦∷⟧ (ρ , ρs)) *⟨ ρ ⟩^ i"},{"Add":" {-# INLINE ⅀⟦_⟧ #-}"},{"Add":""},{"Add":" ⟦_⟧ : ∀ {n} → Poly n → Vec Carrier n → Carrier"},{"Add":" ⟦ Κ x ⊐ i≤n ⟧ _ = ⟦ x ⟧ᵣ"},{"Add":" ⟦ ⅀ xs ⊐ i≤n ⟧ Ρ = ⅀⟦ xs ⟧ (drop-1 i≤n Ρ)"},{"Add":" {-# INLINE ⟦_⟧ #-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Performance"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- As you might imagine, the implementation of the functions above"},{"Add":"-- seriously affect performance. What you might not realise, though,"},{"Add":"-- is that the most important component is the *order of the arguments*."},{"Add":"-- For instance, if we change:"},{"Add":"--"},{"Add":"-- (x , xs) ⟦∷⟧ (ρ , ρs) = ρ * ⅀⟦ xs ⟧ (ρ , ρs) + ⟦ x ⟧ ρs"},{"Add":"--"},{"Add":"-- To:"},{"Add":"--"},{"Add":"-- (x , xs) ⟦∷⟧ (ρ , ρs) = ⟦ x ⟧ ρs + ⅀⟦ xs ⟧ (ρ , ρs) * ρ"},{"Add":"--"},{"Add":"-- We get a function that's several orders of magnitude slower!"}]}},{"FileAdd":{"meta":{"basename":"Reasoning.agda","metadata":0},"context":[{"change":0,"pos":162999}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Polynomial reasoning"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Tactic.RingSolver.Core.AlmostCommutativeRing"},{"Add":""},{"Add":"-- Some specialised tools for equational reasoning."},{"Add":"module Tactic.RingSolver.Core.Polynomial.Reasoning"},{"Add":" {a ℓ} (ring : AlmostCommutativeRing a ℓ)"},{"Add":" where"},{"Add":""},{"Add":"open AlmostCommutativeRing ring"},{"Add":"open import Relation.Binary.Reasoning.Setoid setoid public"},{"Add":""},{"Add":"infixr 1 ≪+_ +≫_ ≪*_ *≫_"},{"Add":"infixr 0 _⊙_"},{"Add":""},{"Add":"≪+_ : ∀ {x₁ x₂ y} → x₁ ≈ x₂ → x₁ + y ≈ x₂ + y"},{"Add":"≪+ prf = +-cong prf refl"},{"Add":"{-# INLINE ≪+_ #-}"},{"Add":""},{"Add":"+≫_ : ∀ {x y₁ y₂} → y₁ ≈ y₂ → x + y₁ ≈ x + y₂"},{"Add":"+≫_ = +-cong refl"},{"Add":"{-# INLINE +≫_ #-}"},{"Add":""},{"Add":"≪*_ : ∀ {x₁ x₂ y} → x₁ ≈ x₂ → x₁ * y ≈ x₂ * y"},{"Add":"≪* prf = *-cong prf refl"},{"Add":"{-# INLINE ≪*_ #-}"},{"Add":""},{"Add":"*≫_ : ∀ {x y₁ y₂} → y₁ ≈ y₂ → x * y₁ ≈ x * y₂"},{"Add":"*≫_ = *-cong refl"},{"Add":"{-# INLINE *≫_ #-}"},{"Add":""},{"Add":"-- transitivity as an operator"},{"Add":"_⊙_ : ∀ {x y z} → x ≈ y → y ≈ z → x ≈ z"},{"Add":"_⊙_ = trans"},{"Add":"{-# INLINE _⊙_ #-}"}]}},{"FileAdd":{"meta":{"basename":"Parameters.agda","metadata":0},"context":[{"change":0,"pos":162999}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Bundles of parameters for passing to the Ring Solver"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"-- This module packages up all the stuff that's passed to the other"},{"Add":"-- modules in a convenient form."},{"Add":""},{"Add":"module Tactic.RingSolver.Core.Polynomial.Parameters where"},{"Add":""},{"Add":"open import Algebra.Bundles using (RawRing)"},{"Add":"open import Data.Bool.Base using (Bool; T)"},{"Add":"open import Level"},{"Add":"open import Relation.Unary"},{"Add":"open import Tactic.RingSolver.Core.AlmostCommutativeRing"},{"Add":""},{"Add":"-- This record stores all the stuff we need for the coefficients:"},{"Add":"--"},{"Add":"-- * A raw ring"},{"Add":"-- * A (decidable) predicate on \"zeroeness\""},{"Add":"--"},{"Add":"-- It's used for defining the operations on the Horner normal form."},{"Add":"record RawCoeff ℓ₁ ℓ₂ : Set (suc (ℓ₁ ⊔ ℓ₂)) where"},{"Add":" field"},{"Add":" rawRing : RawRing ℓ₁ ℓ₂"},{"Add":" isZero : RawRing.Carrier rawRing → Bool"},{"Add":""},{"Add":" open RawRing rawRing public"},{"Add":""},{"Add":"-- This record stores the full information we need for converting"},{"Add":"-- to the final ring."},{"Add":"record Homomorphism ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Set (suc (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄)) where"},{"Add":" field"},{"Add":" from : RawCoeff ℓ₁ ℓ₂"},{"Add":" to : AlmostCommutativeRing ℓ₃ ℓ₄"},{"Add":""},{"Add":" module Raw = RawCoeff from"},{"Add":" open AlmostCommutativeRing to public"},{"Add":""},{"Add":" field"},{"Add":" morphism : Raw.rawRing -Raw-AlmostCommutative⟶ to"},{"Add":" open _-Raw-AlmostCommutative⟶_ morphism renaming (⟦_⟧ to ⟦_⟧ᵣ) public"},{"Add":" field"},{"Add":" Zero-C⟶Zero-R : ∀ x → T (Raw.isZero x) → 0# ≈ ⟦ x ⟧ᵣ"}]}},{"FileAdd":{"meta":{"basename":"Homomorphism.agda","metadata":0},"context":[{"change":0,"pos":162999}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Some specialised instances of the ring solver"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Parameters"},{"Add":""},{"Add":"-- Here, we provide proofs of homomorphism between the operations"},{"Add":"-- defined on polynomials and those on the underlying ring."},{"Add":""},{"Add":"module Tactic.RingSolver.Core.Polynomial.Homomorphism"},{"Add":" {r₁ r₂ r₃ r₄}"},{"Add":" (homo : Homomorphism r₁ r₂ r₃ r₄)"},{"Add":" where"},{"Add":""},{"Add":"-- Proofs for each component of the polynomial"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Addition homo using (⊞-hom) public"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Multiplication homo using (⊠-hom) public"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Negation homo using (⊟-hom) public"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Exponentiation homo using (⊡-hom) public"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Constants homo using (κ-hom) public"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Variables homo using (ι-hom) public"}]}},{"FileAdd":{"meta":{"basename":"Homomorphism","metadata":576},"context":[{"change":0,"pos":162999}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Variables.agda","metadata":0},"context":[{"change":0,"pos":170296}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Homomorphism proofs for variables and constants over polynomials"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Parameters"},{"Add":""},{"Add":"module Tactic.RingSolver.Core.Polynomial.Homomorphism.Variables"},{"Add":" {r₁ r₂ r₃ r₄}"},{"Add":" (homo : Homomorphism r₁ r₂ r₃ r₄)"},{"Add":" where"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Data.Vec.Base as Vec using (Vec)"},{"Add":"open import Data.Fin.Base using (Fin)"},{"Add":"open import Data.List.Kleene"},{"Add":""},{"Add":"open Homomorphism homo"},{"Add":""},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Lemmas homo"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Base (Homomorphism.from homo)"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Reasoning (Homomorphism.to homo)"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Semantics homo"},{"Add":""},{"Add":"ι-hom : ∀ {n} (i : Fin n) (Ρ : Vec Carrier n) → ⟦ ι i ⟧ Ρ ≈ Vec.lookup Ρ i"},{"Add":"ι-hom i Ρ′ = let (ρ , Ρ) = drop-1 (space≤′n i) Ρ′ in begin"},{"Add":" ⟦ (κ Raw.1# Δ 1 ∷↓ []) ⊐↓ space≤′n i ⟧ Ρ′ ≈⟨ ⊐↓-hom (κ Raw.1# Δ 1 ∷↓ []) (space≤′n i) Ρ′ ⟩"},{"Add":" ⅀?⟦ κ Raw.1# Δ 1 ∷↓ [] ⟧ (ρ , Ρ) ≈⟨ ∷↓-hom-s (κ Raw.1#) 0 [] ρ Ρ ⟩"},{"Add":" ρ * ⟦ κ Raw.1# ⟧ Ρ ≈⟨ *≫ 1-homo ⟩"},{"Add":" ρ * 1# ≈⟨ *-identityʳ ρ ⟩"},{"Add":" ρ ≡⟨ drop-1⇒lookup i Ρ′ ⟩"},{"Add":" Vec.lookup Ρ′ i ∎"}]}},{"FileAdd":{"meta":{"basename":"Negation.agda","metadata":0},"context":[{"change":0,"pos":170296}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Homomorphism proofs for negation over polynomials"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Parameters"},{"Add":""},{"Add":"module Tactic.RingSolver.Core.Polynomial.Homomorphism.Negation"},{"Add":" {r₁ r₂ r₃ r₄}"},{"Add":" (homo : Homomorphism r₁ r₂ r₃ r₄)"},{"Add":" where"},{"Add":""},{"Add":"open import Data.Vec.Base using (Vec)"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Data.Nat.Base using (_<′_)"},{"Add":"open import Data.Nat.Induction"},{"Add":""},{"Add":"open import Function.Base using (_⟨_⟩_; flip)"},{"Add":""},{"Add":"open Homomorphism homo"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Lemmas homo"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Reasoning to"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Base from"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Semantics homo"},{"Add":""},{"Add":"⊟-step-hom : ∀ {n} (a : Acc _<′_ n) → (xs : Poly n) → ∀ ρ → ⟦ ⊟-step a xs ⟧ ρ ≈ - (⟦ xs ⟧ ρ)"},{"Add":"⊟-step-hom (acc _ ) (Κ x ⊐ i≤n) ρ = -‿homo x"},{"Add":"⊟-step-hom (acc wf) (⅀ xs ⊐ i≤n) ρ′ ="},{"Add":" let (ρ , ρs) = drop-1 i≤n ρ′"},{"Add":" neg-zero ="},{"Add":" begin"},{"Add":" 0#"},{"Add":" ≈⟨ sym (zeroʳ _) ⟩"},{"Add":" - 0# * 0#"},{"Add":" ≈⟨ -‿*-distribˡ 0# 0# ⟩"},{"Add":" - (0# * 0#)"},{"Add":" ≈⟨ -‿cong (zeroˡ 0#) ⟩"},{"Add":" - 0#"},{"Add":" ∎"},{"Add":" in"},{"Add":" begin"},{"Add":" ⟦ poly-map (⊟-step (wf i≤n)) xs ⊐↓ i≤n ⟧ ρ′"},{"Add":" ≈⟨ ⊐↓-hom (poly-map (⊟-step (wf i≤n)) xs) i≤n ρ′ ⟩"},{"Add":" ⅀?⟦ poly-map (⊟-step (wf i≤n)) xs ⟧ (ρ , ρs)"},{"Add":" ≈⟨ poly-mapR ρ ρs (⊟-step (wf i≤n)) -_ (-‿cong) (λ x y → *-comm x (- y) ⟨ trans ⟩ -‿*-distribˡ y x ⟨ trans ⟩ -‿cong (*-comm _ _)) (λ x y → sym (-‿+-comm x y)) (flip (⊟-step-hom (wf i≤n)) ρs) (sym neg-zero ) xs ⟩"},{"Add":" - ⅀⟦ xs ⟧ (ρ , ρs)"},{"Add":" ∎"},{"Add":""},{"Add":"⊟-hom : ∀ {n}"},{"Add":" → (xs : Poly n)"},{"Add":" → (Ρ : Vec Carrier n)"},{"Add":" → ⟦ ⊟ xs ⟧ Ρ ≈ - ⟦ xs ⟧ Ρ"},{"Add":"⊟-hom = ⊟-step-hom (<′-wellFounded _)"}]}},{"FileAdd":{"meta":{"basename":"Multiplication.agda","metadata":0},"context":[{"change":0,"pos":170296}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Homomorphism proofs for multiplication over polynomials"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Parameters"},{"Add":""},{"Add":"module Tactic.RingSolver.Core.Polynomial.Homomorphism.Multiplication"},{"Add":" {r₁ r₂ r₃ r₄}"},{"Add":" (homo : Homomorphism r₁ r₂ r₃ r₄)"},{"Add":" where"},{"Add":""},{"Add":"open import Data.Nat.Base as ℕ using (ℕ; suc; zero; _<′_; _≤′_; ≤′-step; ≤′-refl)"},{"Add":"open import Data.Nat.Properties using (≤′-trans)"},{"Add":"open import Data.Nat.Induction"},{"Add":"open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; map₁)"},{"Add":"open import Data.List.Kleene"},{"Add":"open import Data.Vec.Base using (Vec)"},{"Add":"open import Function.Base using (_⟨_⟩_; flip)"},{"Add":"open import Induction.WellFounded"},{"Add":"open import Relation.Unary"},{"Add":""},{"Add":"open Homomorphism homo hiding (_^_)"},{"Add":""},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Lemmas homo"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Addition homo"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Base from"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Reasoning to"},{"Add":"open import Tactic.RingSolver.Core.Polynomial.Semantics homo"},{"Add":"open import Algebra.Definitions.RawSemiring rawSemiring"},{"Add":" using () renaming (_^′_ to _^_)"},{"Add":""},{"Add":"reassoc : ∀ {y} x z → x * (y * z) ≈ y * (x * z)"},{"Add":"reassoc {y} x z = sym (*-assoc x y z) ⟨ trans ⟩ ((≪* *-comm x y) ⟨ trans ⟩ *-assoc y x z)"},{"Add":""},{"Add":"mutual"},{"Add":" ⊠-step′-hom : ∀ {n} → (a : Acc _<′_ n) → (xs ys : Poly n) → ∀ ρ → ⟦ ⊠-step′ a xs ys ⟧ ρ ≈ ⟦ xs ⟧ ρ * ⟦ ys ⟧ ρ"},{"Add":" ⊠-step′-hom a (x ⊐ p) = ⊠-step-hom a x p"},{"Add":""},{"Add":" ⊠-step-hom : ∀ {i n}"},{"Add":" → (a : Acc _<′_ n)"},{"Add":" → (xs : FlatPoly i)"},{"Add":" → (i≤n : i ≤′ n)"},{"Add":" → (ys : Poly n)"},{"Add":" → ∀ ρ → ⟦ ⊠-step a xs i≤n ys ⟧ ρ ≈ ⟦ xs ⊐ i≤n ⟧ ρ * ⟦ ys ⟧ ρ"},{"Add":" ⊠-step-hom a (Κ x) i≤n = ⊠-Κ-hom a x"},{"Add":" ⊠-step-hom a (⅀ xs) i≤n = ⊠-⅀-hom a xs i≤n"},{"Add":""},{"Add":" ⊠-Κ-hom : ∀ {n}"},{"Add":" → (a : Acc _<′_ n)"},{"Add":" → ∀ x"},{"Add":" → (ys : Poly n)"},{"Add":" → ∀ ρ"},{"Add":" → ⟦ ⊠-Κ a x ys ⟧ ρ ≈ ⟦ x ⟧ᵣ * ⟦ ys ⟧ ρ"},{"Add":" ⊠-Κ-hom (acc _) x (Κ y ⊐ i≤n) ρ = *-homo x y"},{"Add":" ⊠-Κ-hom (acc wf) x (⅀ xs ⊐ i≤n) ρ ="},{"Add":" begin"},{"Add":" ⟦ ⊠-Κ-inj (wf i≤n) x xs ⊐↓ i≤n ⟧ ρ"},{"Add":" ≈⟨ ⊐↓-hom (⊠-Κ-inj (wf i≤n) x xs) i≤n ρ ⟩"},{"Add":" ⅀?⟦ ⊠-Κ-inj (wf i≤n) x xs ⟧ (drop-1 i≤n ρ)"},{"Add":" ≈⟨ ⊠-Κ-inj-hom (wf i≤n) x xs (drop-1 i≤n ρ) ⟩"},{"Add":" ⟦ x ⟧ᵣ * ⅀⟦ xs ⟧ (drop-1 i≤n ρ)"},{"Add":" ∎"},{"Add":""},{"Add":" ⊠-Κ-inj-hom : ∀ {n}"},{"Add":" → (a : Acc _<′_ n)"},{"Add":" → (x : Raw.Carrier)"},{"Add":" → (xs : Coeff n +)"},{"Add":" → ∀ ρ"},{"Add":" → ⅀?⟦ ⊠-Κ-inj a x xs ⟧ ρ ≈ ⟦ x ⟧ᵣ * ⅀⟦ xs ⟧ ρ"},{"Add":" ⊠-Κ-inj-hom {n} a x xs (ρ , Ρ) ="},{"Add":" poly-mapR"},{"Add":" ρ"},{"Add":" Ρ"},{"Add":" (⊠-Κ a x)"},{"Add":" (⟦ x ⟧ᵣ *_)"},{"Add":" (*-cong refl)"},{"Add":" reassoc"},{"Add":" (distribˡ ⟦ x ⟧ᵣ)"},{"Add":" (λ ys → ⊠-Κ-hom a x ys Ρ)"},{"Add":" (zeroʳ _)"},{"Add":" xs"},{"Add":""},{"Add":" ⊠-⅀-hom : ∀ {i n}"},{"Add":" → (a : Acc _<′_ n)"},{"Add":" → (xs : Coeff i +)"},{"Add":" → (i>= normalise"},{"Add":" names ← findMonoidNames mon"},{"Add":" just (lhs , rhs) ← pure (getArgs hole′)"},{"Add":" where nothing → typeError (termErr hole′ ∷ [])"},{"Add":" let soln = constructSoln mon names lhs rhs"},{"Add":" unify hole soln"},{"Add":""},{"Add":"macro"},{"Add":" solve : Term → Term → TC _"},{"Add":" solve = solve-macro"}]}},{"FileAdd":{"meta":{"basename":"Cong.agda","metadata":0},"context":[{"change":0,"pos":144494}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- A simple tactic for used to automatically compute the function"},{"Add":"-- argument to cong."},{"Add":"--"},{"Add":"-- The main use for this tactic is getting a similar experience to"},{"Add":"-- 'rewrite' during equational reasoning. This allows us to write very"},{"Add":"-- succinct proofs:"},{"Add":"--"},{"Add":"-- example : ∀ m n → m ≡ n → suc (suc (m + 0)) + m ≡ suc (suc n) + (n + 0)"},{"Add":"-- example m n eq = begin"},{"Add":"-- suc (suc (m + 0)) + m"},{"Add":"-- ≡⟨ cong! (+-identityʳ m) ⟩"},{"Add":"-- suc (suc m) + m"},{"Add":"-- ≡⟨ cong! eq ⟩"},{"Add":"-- suc (suc n) + n"},{"Add":"-- ≡⟨ cong! (+-identityʳ n) ⟨"},{"Add":"-- suc (suc n) + (n + 0)"},{"Add":"-- ∎"},{"Add":"--"},{"Add":"-- Please see README.Tactic.Cong for more details."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Tactic.Cong where"},{"Add":""},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"open import Data.Bool.Base using (true; false; if_then_else_; _∧_)"},{"Add":"open import Data.Char.Base as Char using (toℕ)"},{"Add":"open import Data.Float.Base as Float using (_≡ᵇ_)"},{"Add":"open import Data.List.Base as List using ([]; _∷_)"},{"Add":"open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)"},{"Add":"open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _≡ᵇ_; _+_)"},{"Add":"open import Data.Unit.Base using (⊤)"},{"Add":"open import Data.Word64.Base as Word64 using (toℕ)"},{"Add":"open import Data.Product.Base using (_×_; map₁; _,_)"},{"Add":"open import Function using (flip; case_of_)"},{"Add":""},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)"},{"Add":"open import Relation.Nullary.Decidable.Core using (yes; no)"},{"Add":""},{"Add":"-- 'Data.String.Properties' defines this via 'Dec', so let's use the"},{"Add":"-- builtin for maximum speed."},{"Add":"import Agda.Builtin.String as String renaming (primStringEquality to _≡ᵇ_)"},{"Add":""},{"Add":"open import Reflection"},{"Add":"open import Reflection.AST.Abstraction"},{"Add":"open import Reflection.AST.AlphaEquality as Alpha"},{"Add":"open import Reflection.AST.Argument as Arg"},{"Add":"open import Reflection.AST.Argument.Information as ArgInfo"},{"Add":"open import Reflection.AST.Argument.Visibility as Visibility"},{"Add":"open import Reflection.AST.Literal as Literal"},{"Add":"open import Reflection.AST.Meta as Meta"},{"Add":"open import Reflection.AST.Name as Name"},{"Add":"open import Reflection.AST.Term as Term"},{"Add":"import Reflection.AST.Traversal as Traversal"},{"Add":""},{"Add":"open import Reflection.TCM.Syntax"},{"Add":"open import Reflection.TCM.Utilities"},{"Add":""},{"Add":"-- Marker to keep anti-unification from descending into the wrapped"},{"Add":"-- subterm."},{"Add":"--"},{"Add":"-- For instance, anti-unification of ⌞ a + b ⌟ + c and b + a + c"},{"Add":"-- yields λ ϕ → ϕ + c, as opposed to λ ϕ → ϕ + ϕ + c without ⌞_⌟."},{"Add":"--"},{"Add":"-- The marker is only visible to the cong! tactic, which inhibits"},{"Add":"-- normalisation. Anywhere else, ⌞ a + b ⌟ reduces to a + b."},{"Add":"--"},{"Add":"-- Thus, proving ⌞ a + b ⌟ + c ≡ b + a + c via cong! (+-comm a b)"},{"Add":"-- also proves a + b + c ≡ b + a + c."},{"Add":"⌞_⌟ : ∀ {a} {A : Set a} → A → A"},{"Add":"⌞_⌟ x = x"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Utilities"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"private"},{"Add":" -- Descend past a variable."},{"Add":" varDescend : ℕ → ℕ → ℕ"},{"Add":" varDescend ϕ x = if ϕ ℕ.≤ᵇ x then suc x else x"},{"Add":""},{"Add":" -- Descend a variable underneath pattern variables."},{"Add":" patternDescend : ℕ → Pattern → Pattern × ℕ"},{"Add":" patternsDescend : ℕ → Args Pattern → Args Pattern × ℕ"},{"Add":""},{"Add":" patternDescend ϕ (con c ps) = map₁ (con c) (patternsDescend ϕ ps)"},{"Add":" patternDescend ϕ (dot t) = dot t , ϕ"},{"Add":" patternDescend ϕ (var x) = var (varDescend ϕ x) , suc ϕ"},{"Add":" patternDescend ϕ (lit l) = lit l , ϕ"},{"Add":" patternDescend ϕ (proj f) = proj f , ϕ"},{"Add":" patternDescend ϕ (absurd x) = absurd (varDescend ϕ x) , suc ϕ"},{"Add":""},{"Add":" patternsDescend ϕ ((arg i p) ∷ ps) ="},{"Add":" let (p' , ϕ') = patternDescend ϕ p"},{"Add":" (ps' , ϕ'') = patternsDescend ϕ' ps"},{"Add":" in (arg i p ∷ ps' , ϕ'')"},{"Add":" patternsDescend ϕ [] ="},{"Add":" [] , ϕ"},{"Add":""},{"Add":" -- Construct an error when the goal is not 'x ≡ y' for some 'x' and 'y'."},{"Add":" notEqualityError : ∀ {A : Set} Term → TC A"},{"Add":" notEqualityError goal = typeError (strErr \"Cannot rewrite a goal that is not equality: \" ∷ termErr goal ∷ [])"},{"Add":""},{"Add":" unificationError : ∀ {A : Set} → TC Term → TC Term → TC A"},{"Add":" unificationError term symTerm = do"},{"Add":" term' ← term"},{"Add":" symTerm' ← symTerm"},{"Add":" -- Don't show the same term twice."},{"Add":" let symErr = case term' Term.≟ symTerm' of λ where"},{"Add":" (yes _) → []"},{"Add":" (no _) → strErr \"\\n\" ∷ termErr symTerm' ∷ []"},{"Add":" typeError (strErr \"cong! failed, tried:\\n\" ∷ termErr term' ∷ symErr)"},{"Add":""},{"Add":" record EqualityGoal : Set where"},{"Add":" constructor equals"},{"Add":" field"},{"Add":" level : Term"},{"Add":" type : Term"},{"Add":" lhs : Term"},{"Add":" rhs : Term"},{"Add":""},{"Add":" destructEqualityGoal : Term → TC EqualityGoal"},{"Add":" destructEqualityGoal goal@(def (quote _≡_) (lvl ∷ tp ∷ lhs ∷ rhs ∷ [])) ="},{"Add":" pure $ equals (unArg lvl) (unArg tp) (unArg lhs) (unArg rhs)"},{"Add":" destructEqualityGoal (meta m args) ="},{"Add":" blockOnMeta m"},{"Add":" destructEqualityGoal goal ="},{"Add":" notEqualityError goal"},{"Add":""},{"Add":" -- Helper for constructing applications of 'cong'"},{"Add":" `cong : ∀ {a} {A : Set a} {x y : A} → EqualityGoal → Term → x ≡ y → TC Term"},{"Add":" `cong {a = a} {A = A} {x = x} {y = y} eqGoal f x≡y = do"},{"Add":" -- NOTE: We apply all implicit arguments here to ensure that using"},{"Add":" -- equality proofs with implicits don't lead to unsolved metavariable"},{"Add":" -- errors."},{"Add":" let open EqualityGoal eqGoal"},{"Add":" eq ← quoteTC x≡y"},{"Add":" `a ← quoteTC a"},{"Add":" `A ← quoteTC A"},{"Add":" `x ← quoteTC x"},{"Add":" `y ← quoteTC y"},{"Add":" pure $ def (quote cong) $ `a ⟅∷⟆ `A ⟅∷⟆ level ⟅∷⟆ type ⟅∷⟆ vLam \"ϕ\" f ⟨∷⟩ `x ⟅∷⟆ `y ⟅∷⟆ eq ⟨∷⟩ []"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Anti-Unification"},{"Add":"--"},{"Add":"-- The core idea of the tactic is that we can compute the input"},{"Add":"-- to 'cong' by syntactically anti-unifying both sides of the"},{"Add":"-- equality, and then using that to construct a lambda"},{"Add":"-- where all the differences are replaced by the lambda-abstracted"},{"Add":"-- variable."},{"Add":"--"},{"Add":"-- For instance, the two terms 'suc (m + (m + 0)) + (m + 0)' and"},{"Add":"-- 'suc (m + m) + (m + 0)' would anti unify to 'suc (m + _) + (m + 0)'"},{"Add":"-- which we can then use to construct the lambda 'λ ϕ → suc (m + ϕ) + (m + 0)'."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"private"},{"Add":" antiUnify : ℕ → Term → Term → Term"},{"Add":" antiUnifyArgs : ℕ → Args Term → Args Term → Maybe (Args Term)"},{"Add":" antiUnifyClauses : ℕ → Clauses → Clauses → Maybe Clauses"},{"Add":" antiUnifyClause : ℕ → Clause → Clause → Maybe Clause"},{"Add":""},{"Add":" pattern apply-⌞⌟ t = (def (quote ⌞_⌟) (_ ∷ _ ∷ arg _ t ∷ []))"},{"Add":""},{"Add":" antiUnify ϕ (var x args) (var y args') with x ℕ.≡ᵇ y | antiUnifyArgs ϕ args args'"},{"Add":" ... | _ | nothing = var ϕ []"},{"Add":" ... | false | just uargs = var ϕ uargs"},{"Add":" ... | true | just uargs = var (varDescend ϕ x) uargs"},{"Add":" antiUnify ϕ (con c args) (con c' args') with c Name.≡ᵇ c' | antiUnifyArgs ϕ args args'"},{"Add":" ... | _ | nothing = var ϕ []"},{"Add":" ... | false | just uargs = var ϕ []"},{"Add":" ... | true | just uargs = con c uargs"},{"Add":" antiUnify ϕ (def f args) (apply-⌞⌟ t) = antiUnify ϕ (def f args) t"},{"Add":" antiUnify ϕ (def f args) (def f' args') with f Name.≡ᵇ f' | antiUnifyArgs ϕ args args'"},{"Add":" ... | _ | nothing = var ϕ []"},{"Add":" ... | false | just uargs = var ϕ []"},{"Add":" ... | true | just uargs = def f uargs"},{"Add":" antiUnify ϕ (lam v (abs s t)) (lam _ (abs _ t')) ="},{"Add":" lam v (abs s (antiUnify (suc ϕ) t t'))"},{"Add":" antiUnify ϕ (pat-lam cs args) (pat-lam cs' args') with antiUnifyClauses ϕ cs cs' | antiUnifyArgs ϕ args args'"},{"Add":" ... | nothing | _ = var ϕ []"},{"Add":" ... | _ | nothing = var ϕ []"},{"Add":" ... | just ucs | just uargs = pat-lam ucs uargs"},{"Add":" antiUnify ϕ (Π[ s ∶ arg i a ] b) (Π[ _ ∶ arg _ a' ] b') ="},{"Add":" Π[ s ∶ arg i (antiUnify ϕ a a') ] antiUnify (suc ϕ) b b'"},{"Add":" antiUnify ϕ (sort (set t)) (sort (set t')) ="},{"Add":" sort (set (antiUnify ϕ t t'))"},{"Add":" antiUnify ϕ (sort (lit n)) (sort (lit n')) with n ℕ.≡ᵇ n'"},{"Add":" ... | true = sort (lit n)"},{"Add":" ... | false = var ϕ []"},{"Add":" antiUnify ϕ (sort (propLit n)) (sort (propLit n')) with n ℕ.≡ᵇ n'"},{"Add":" ... | true = sort (propLit n)"},{"Add":" ... | false = var ϕ []"},{"Add":" antiUnify ϕ (sort (inf n)) (sort (inf n')) with n ℕ.≡ᵇ n'"},{"Add":" ... | true = sort (inf n)"},{"Add":" ... | false = var ϕ []"},{"Add":" antiUnify ϕ (sort unknown) (sort unknown) ="},{"Add":" sort unknown"},{"Add":" antiUnify ϕ (lit l) (lit l') with l Literal.≡ᵇ l'"},{"Add":" ... | true = lit l"},{"Add":" ... | false = var ϕ []"},{"Add":" antiUnify ϕ (meta x args) (meta x' args') with x Meta.≡ᵇ x' | antiUnifyArgs ϕ args args'"},{"Add":" ... | _ | nothing = var ϕ []"},{"Add":" ... | false | _ = var ϕ []"},{"Add":" ... | true | just uargs = meta x uargs"},{"Add":" antiUnify ϕ unknown unknown = unknown"},{"Add":" antiUnify ϕ _ _ = var ϕ []"},{"Add":""},{"Add":" antiUnifyArgs ϕ (arg i t ∷ args) (arg _ t' ∷ args') ="},{"Add":" Maybe.map (arg i (antiUnify ϕ t t') ∷_) (antiUnifyArgs ϕ args args')"},{"Add":" antiUnifyArgs ϕ [] [] ="},{"Add":" just []"},{"Add":" antiUnifyArgs ϕ _ _ ="},{"Add":" nothing"},{"Add":""},{"Add":" antiUnifyClause ϕ (clause Γ pats t) (clause Δ pats' t') ="},{"Add":" Maybe.when ((Γ =α= Δ) ∧ (pats =α= pats'))"},{"Add":" let (upats , ϕ') = patternsDescend ϕ pats"},{"Add":" in clause Γ upats (antiUnify ϕ' t t')"},{"Add":" antiUnifyClause ϕ (absurd-clause Γ pats) (absurd-clause Δ pats') ="},{"Add":" Maybe.when ((Γ =α= Δ) ∧ (pats =α= pats')) $"},{"Add":" absurd-clause Γ pats"},{"Add":" antiUnifyClause ϕ _ _ ="},{"Add":" nothing"},{"Add":""},{"Add":" antiUnifyClauses ϕ (c ∷ cs) (c' ∷ cs') ="},{"Add":" Maybe.ap (Maybe.map _∷_ (antiUnifyClause ϕ c c')) (antiUnifyClauses ϕ cs cs')"},{"Add":" antiUnifyClauses ϕ _ _ ="},{"Add":" just []"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Rewriting"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"macro"},{"Add":" cong! : ∀ {a} {A : Set a} {x y : A} → x ≡ y → Term → TC ⊤"},{"Add":" cong! x≡y hole ="},{"Add":" -- NOTE: We avoid doing normalisation here as this tactic"},{"Add":" -- is mainly meant for equational reasoning. In that context,"},{"Add":" -- the endpoints of the equality are already specified in"},{"Add":" -- the form that the -- programmer expects them to be in,"},{"Add":" -- so normalising buys us nothing."},{"Add":" withNormalisation false $ do"},{"Add":" goal ← inferType hole"},{"Add":" eqGoal ← destructEqualityGoal goal"},{"Add":" let makeTerm = λ lhs rhs → `cong eqGoal (antiUnify 0 lhs rhs) x≡y"},{"Add":" let lhs = EqualityGoal.lhs eqGoal"},{"Add":" let rhs = EqualityGoal.rhs eqGoal"},{"Add":" let term = makeTerm lhs rhs"},{"Add":" let symTerm = makeTerm rhs lhs"},{"Add":" let uni = _>>= flip unify hole"},{"Add":" -- When using ⌞_⌟ with ≡⟨ ... ⟨, (uni term) fails and"},{"Add":" -- (uni symTerm) succeeds."},{"Add":" catchTC (uni term) $"},{"Add":" catchTC (uni symTerm) $ do"},{"Add":" -- If we failed because of unresolved metas, restart."},{"Add":" blockOnMetas goal"},{"Add":" -- If we failed for a different reason, show an error."},{"Add":" unificationError term symTerm"}]}},{"FileAdd":{"meta":{"basename":"System","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Random.agda","metadata":0},"context":[{"change":0,"pos":266416}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- *Pseudo-random* number generation"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module System.Random where"},{"Add":""},{"Add":"import System.Random.Primitive as Prim"},{"Add":""},{"Add":"open import Data.Bool.Base using (T)"},{"Add":"open import Data.Nat.Base using (ℕ; z≤n) hiding (module ℕ)"},{"Add":"open import Foreign.Haskell.Pair using (_,_)"},{"Add":"open import Function.Base using (_$_; _∘_)"},{"Add":"open import IO.Base using (IO; lift; lift!; _<$>_; _>>=_; pure)"},{"Add":"import IO.Effectful as IO"},{"Add":"open import Level using (0ℓ; suc; _⊔_; lift)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Ranged generation shall return proofs"},{"Add":""},{"Add":"record InBounds {a r} {A : Set a} (_≤_ : Rel A r) (lo hi : A) : Set (a ⊔ r) where"},{"Add":" constructor _∈[_,_]"},{"Add":" field"},{"Add":" value : A"},{"Add":" .isLowerBound : lo ≤ value"},{"Add":" .isUpperBound : value ≤ hi"},{"Add":""},{"Add":"RandomRIO : ∀ {a r} {A : Set a} (_≤_ : Rel A r) → Set (suc (a ⊔ r))"},{"Add":"RandomRIO {A = A} _≤_ = (lo hi : A) → .(lo ≤ hi) → IO (InBounds _≤_ lo hi)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Instances"},{"Add":""},{"Add":"module Char where"},{"Add":""},{"Add":" open import Data.Char.Base using (Char; _≤_)"},{"Add":""},{"Add":" randomIO : IO Char"},{"Add":" randomIO = lift Prim.randomIO-Char"},{"Add":""},{"Add":" randomRIO : RandomRIO _≤_"},{"Add":" randomRIO lo hi _ = do"},{"Add":" value ← lift (Prim.randomRIO-Char (lo , hi))"},{"Add":" pure (value ∈[ trustMe , trustMe ])"},{"Add":" where postulate trustMe : ∀ {A} → A"},{"Add":""},{"Add":"module Float where"},{"Add":""},{"Add":" open import Data.Float.Base using (Float; _≤_)"},{"Add":""},{"Add":" randomIO : IO Float"},{"Add":" randomIO = lift Prim.randomIO-Float"},{"Add":""},{"Add":" randomRIO : RandomRIO _≤_"},{"Add":" randomRIO lo hi _ = do"},{"Add":" value ← lift (Prim.randomRIO-Float (lo , hi))"},{"Add":" pure (value ∈[ trustMe , trustMe ])"},{"Add":" where postulate trustMe : ∀ {A} → A"},{"Add":""},{"Add":"module ℤ where"},{"Add":""},{"Add":" open import Data.Integer.Base using (ℤ; _≤_)"},{"Add":""},{"Add":" randomIO : IO ℤ"},{"Add":" randomIO = lift Prim.randomIO-Int"},{"Add":""},{"Add":" randomRIO : RandomRIO _≤_"},{"Add":" randomRIO lo hi _ = do"},{"Add":" value ← lift (Prim.randomRIO-Int (lo , hi))"},{"Add":" pure (value ∈[ trustMe , trustMe ])"},{"Add":" where postulate trustMe : ∀ {A} → A"},{"Add":""},{"Add":"module ℕ where"},{"Add":""},{"Add":" open import Data.Nat.Base using (ℕ; _≤_)"},{"Add":""},{"Add":" randomIO : IO ℕ"},{"Add":" randomIO = lift Prim.randomIO-Nat"},{"Add":""},{"Add":" randomRIO : RandomRIO _≤_"},{"Add":" randomRIO lo hi _ = do"},{"Add":" value ← lift (Prim.randomRIO-Nat (lo , hi))"},{"Add":" pure (value ∈[ trustMe , trustMe ])"},{"Add":" where postulate trustMe : ∀ {A} → A"},{"Add":""},{"Add":"module Word64 where"},{"Add":""},{"Add":" open import Data.Word64.Base using (Word64; _≤_)"},{"Add":""},{"Add":" randomIO : IO Word64"},{"Add":" randomIO = lift Prim.randomIO-Word64"},{"Add":""},{"Add":" randomRIO : RandomRIO _≤_"},{"Add":" randomRIO lo hi _ = do"},{"Add":" value ← lift (Prim.randomRIO-Word64 (lo , hi))"},{"Add":" pure (value ∈[ trustMe , trustMe ])"},{"Add":" where postulate trustMe : ∀ {A} → A"},{"Add":""},{"Add":"module Fin where"},{"Add":""},{"Add":" open import Data.Nat.Base as ℕ using (suc; NonZero; z≤n; s≤s)"},{"Add":" import Data.Nat.Properties as ℕ"},{"Add":" open import Data.Fin.Base using (Fin; _≤_; fromℕ<; toℕ)"},{"Add":" import Data.Fin.Properties as Fin"},{"Add":""},{"Add":" randomIO : ∀ {n} → .{{NonZero n}} → IO (Fin n)"},{"Add":" randomIO {n = n@(suc _)} = do"},{"Add":" suc k ∈[ lo≤k , k≤hi ] ← ℕ.randomRIO 1 n (s≤s z≤n)"},{"Add":" pure (fromℕ< k≤hi)"},{"Add":""},{"Add":" toℕ-cancel-InBounds : ∀ {n} {lo hi : Fin n} →"},{"Add":" InBounds ℕ._≤_ (toℕ lo) (toℕ hi) →"},{"Add":" InBounds _≤_ lo hi"},{"Add":" toℕ-cancel-InBounds {n} {lo} {hi} (k ∈[ toℕlo≤k , k≤toℕhi ]) ="},{"Add":" let"},{"Add":" .k List.randomIO Char.randomIO"},{"Add":""},{"Add":"module String≤ (n : ℕ) where"},{"Add":""},{"Add":" import Data.Vec.Bounded.Base as Vec≤"},{"Add":" open import Data.String.Base using (String; fromList)"},{"Add":""},{"Add":" randomIO : IO String"},{"Add":" randomIO = fromList ∘ Vec≤.toList <$> Vec≤.randomIO Char.randomIO n"},{"Add":""},{"Add":"open import Data.Char.Base using (Char; _≤_)"},{"Add":""},{"Add":"module RangedString≤ (a b : Char) .(a≤b : a ≤ b) (n : ℕ) where"},{"Add":""},{"Add":" import Data.Vec.Bounded.Base as Vec≤"},{"Add":" open import Data.String.Base using (String; fromList)"},{"Add":""},{"Add":" randomIO : IO String"},{"Add":" randomIO ="},{"Add":" fromList ∘ Vec≤.toList ∘ Vec≤.map InBounds.value"},{"Add":" <$> Vec≤.randomIO (Char.randomRIO a b a≤b) n"}]}},{"FileAdd":{"meta":{"basename":"Random","metadata":576},"context":[{"change":0,"pos":266416}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Primitive.agda","metadata":0},"context":[{"change":0,"pos":272647}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Primitive System.Random simple bindings to Haskell functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module System.Random.Primitive where"},{"Add":""},{"Add":"open import Agda.Builtin.IO using (IO)"},{"Add":"open import Agda.Builtin.Char using (Char)"},{"Add":"open import Agda.Builtin.Float using (Float)"},{"Add":"open import Agda.Builtin.Int using (Int)"},{"Add":"open import Agda.Builtin.Nat using (Nat)"},{"Add":"open import Agda.Builtin.Word using (Word64)"},{"Add":"open import Foreign.Haskell.Pair using (Pair)"},{"Add":""},{"Add":"postulate"},{"Add":" randomIO-Char : IO Char"},{"Add":" randomRIO-Char : Pair Char Char → IO Char"},{"Add":" randomIO-Int : IO Int"},{"Add":" randomRIO-Int : Pair Int Int → IO Int"},{"Add":" randomIO-Float : IO Float"},{"Add":" randomRIO-Float : Pair Float Float → IO Float"},{"Add":" randomIO-Nat : IO Nat"},{"Add":" randomRIO-Nat : Pair Nat Nat → IO Nat"},{"Add":" randomIO-Word64 : IO Word64"},{"Add":" randomRIO-Word64 : Pair Word64 Word64 → IO Word64"},{"Add":""},{"Add":"{-# FOREIGN GHC import System.Random #-}"},{"Add":""},{"Add":"{-# COMPILE GHC randomIO-Char = randomIO #-}"},{"Add":"{-# COMPILE GHC randomRIO-Char = randomRIO #-}"},{"Add":"{-# COMPILE GHC randomIO-Int = randomIO #-}"},{"Add":"{-# COMPILE GHC randomRIO-Int = randomRIO #-}"},{"Add":"{-# COMPILE GHC randomIO-Float = randomIO #-}"},{"Add":"{-# COMPILE GHC randomRIO-Float = randomRIO #-}"},{"Add":"{-# COMPILE GHC randomIO-Nat = abs <$> randomIO #-}"},{"Add":"{-# COMPILE GHC randomRIO-Nat = randomRIO #-}"},{"Add":"{-# COMPILE GHC randomIO-Word64 = randomIO #-}"},{"Add":"{-# COMPILE GHC randomRIO-Word64 = randomRIO #-}"}]}},{"FileAdd":{"meta":{"basename":"Process.agda","metadata":0},"context":[{"change":0,"pos":266416}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Calling external processes"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module System.Process where"},{"Add":""},{"Add":"open import Level using (Level)"},{"Add":"open import Data.List.Base using (List)"},{"Add":"open import Data.Product.Base using (_×_; proj₁)"},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Data.Unit.Polymorphic using (⊤)"},{"Add":"open import Foreign.Haskell.Coerce"},{"Add":"open import IO.Base"},{"Add":""},{"Add":"open import System.Exit"},{"Add":"import System.Process.Primitive as Prim"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ : Level"},{"Add":""},{"Add":"callCommand : String → IO {ℓ} ⊤"},{"Add":"callCommand cmd = lift′ (Prim.callCommand cmd)"},{"Add":""},{"Add":"system : String → IO ExitCode"},{"Add":"system cmd = lift (Prim.system cmd)"},{"Add":""},{"Add":"callProcess : String → List String → IO {ℓ} ⊤"},{"Add":"callProcess exe args = lift′ (Prim.callProcess exe args)"},{"Add":""},{"Add":"readProcess"},{"Add":" : String -- Filename of the executable"},{"Add":" → List String -- any arguments"},{"Add":" → String -- standard input"},{"Add":" → IO String -- stdout"},{"Add":"readProcess exe args stdin = lift (Prim.readProcess exe args stdin)"},{"Add":""},{"Add":"readProcessWithExitCode"},{"Add":" : String -- Filename of the executable"},{"Add":" → List String -- any arguments"},{"Add":" → String -- standard input"},{"Add":" → IO (ExitCode × String × String) -- exitcode, stdout, stderr"},{"Add":"readProcessWithExitCode exe args stdin ="},{"Add":" lift (coerce Prim.readProcessWithExitCode exe args stdin)"},{"Add":""},{"Add":"callProcessWithExitCode : String → List String → IO ExitCode"},{"Add":"callProcessWithExitCode exe args = proj₁ <$> readProcessWithExitCode exe args \"\""}]}},{"FileAdd":{"meta":{"basename":"Process","metadata":576},"context":[{"change":0,"pos":266416}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Primitive.agda","metadata":0},"context":[{"change":0,"pos":275959}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Primitive System.Process simple bindings to Haskell functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module System.Process.Primitive where"},{"Add":""},{"Add":"open import Level using (Level)"},{"Add":"open import Agda.Builtin.List"},{"Add":"open import Agda.Builtin.String"},{"Add":"open import Agda.Builtin.Unit"},{"Add":"open import Foreign.Haskell using (Pair)"},{"Add":"open import IO.Primitive.Core using (IO)"},{"Add":"open import System.Exit.Primitive using (ExitCode)"},{"Add":""},{"Add":"postulate"},{"Add":""},{"Add":" callCommand : String → IO ⊤"},{"Add":" system : String → IO ExitCode"},{"Add":" callProcess : String → List String → IO ⊤"},{"Add":""},{"Add":" readProcess"},{"Add":" : String -- Filename of the executable"},{"Add":" → List String -- any arguments"},{"Add":" → String -- standard input"},{"Add":" → IO String -- stdout"},{"Add":""},{"Add":" readProcessWithExitCode"},{"Add":" : String -- Filename of the executable"},{"Add":" → List String -- any arguments"},{"Add":" → String -- standard input"},{"Add":" → IO (Pair ExitCode (Pair String String)) -- exitcode, stdout, stderr"},{"Add":""},{"Add":"{-# FOREIGN GHC import System.Process #-}"},{"Add":"{-# FOREIGN GHC import qualified Data.Text as T #-}"},{"Add":"{-# FOREIGN GHC import MAlonzo.Code.System.Exit.Primitive #-}"},{"Add":""},{"Add":"{-# COMPILE GHC callCommand = \\ cmd -> callCommand (T.unpack cmd) #-}"},{"Add":"{-# COMPILE GHC system = \\ cmd -> fmap fromExitCode (system (T.unpack cmd)) #-}"},{"Add":"{-# COMPILE GHC callProcess = \\ exe -> callProcess (T.unpack exe) . map T.unpack #-}"},{"Add":"{-# COMPILE GHC readProcess = \\ exe args -> fmap T.pack . readProcess (T.unpack exe) (map T.unpack args) . T.unpack #-}"},{"Add":"{-# COMPILE GHC readProcessWithExitCode = \\ exe args stdin ->"},{"Add":" do { (ex, out, err) <- readProcessWithExitCode (T.unpack exe) (map T.unpack args) (T.unpack stdin)"},{"Add":" ; pure (fromExitCode ex, (T.pack out, T.pack err))"},{"Add":" }"},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"FilePath","metadata":576},"context":[{"change":0,"pos":266416}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Posix.agda","metadata":0},"context":[{"change":0,"pos":277989}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Posix filepaths"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module System.FilePath.Posix where"},{"Add":""},{"Add":"open import Agda.Builtin.Bool using (Bool)"},{"Add":"open import Agda.Builtin.List using (List)"},{"Add":"open import Agda.Builtin.String using (String)"},{"Add":"open import IO.Base using (IO; lift)"},{"Add":"open import Data.Maybe.Base using (Maybe)"},{"Add":"open import Data.Product.Base using (_×_)"},{"Add":"open import Data.Sum.Base using (_⊎_)"},{"Add":""},{"Add":"open import Foreign.Haskell.Coerce"},{"Add":""},{"Add":"open import System.FilePath.Posix.Primitive as Prim"},{"Add":" public"},{"Add":" -- Some of these functions are not directly re-exported because their"},{"Add":" -- respective types mentions FFI-friendly versions of the stdlib's types"},{"Add":" -- e.g. `Pair` instead of `_×_`."},{"Add":" -- A wrapper is systematically defined below using Foreign.Haskell.Coerce's"},{"Add":" -- zero-cost coercion to expose a more useful function to users."},{"Add":" using ( module Nature"},{"Add":" ; Nature"},{"Add":" ; FilePath"},{"Add":" ; mkFilePath"},{"Add":" ; getFilePath"},{"Add":" ; AbsolutePath"},{"Add":" ; RelativePath"},{"Add":" ; SomePath"},{"Add":" ; Extension"},{"Add":" ; mkExtension"},{"Add":" ; getExtension"},{"Add":" -- Separator predicates"},{"Add":" ; pathSeparator"},{"Add":" ; pathSeparators"},{"Add":" ; isPathSeparator"},{"Add":" ; searchPathSeparator"},{"Add":" ; isSearchPathSeparator"},{"Add":" ; extSeparator"},{"Add":" ; isExtSeparator"},{"Add":" -- $PATH methods"},{"Add":" ; splitSearchPath"},{"Add":" -- ; getSearchPath see below: lift needed"},{"Add":" -- Extension functions"},{"Add":" -- ; splitExtension see below: coerce needed"},{"Add":" ; takeExtension"},{"Add":" ; replaceExtension"},{"Add":" ; dropExtension"},{"Add":" ; addExtension"},{"Add":" ; hasExtension"},{"Add":" ; takeExtensions"},{"Add":" ; replaceExtensions"},{"Add":" ; dropExtensions"},{"Add":" ; isExtensionOf"},{"Add":" -- ; stripExtension see below: coerce needed"},{"Add":" -- Filename/directory functions"},{"Add":" -- ; splitFileName see below: coerce needed"},{"Add":" ; takeFileName"},{"Add":" ; replaceFileName"},{"Add":" ; dropFileName"},{"Add":" ; takeBaseName"},{"Add":" ; replaceBaseName"},{"Add":" ; takeDirectory"},{"Add":" ; replaceDirectory"},{"Add":" ; combine"},{"Add":" ; splitPath"},{"Add":" ; joinPath"},{"Add":" ; splitDirectories"},{"Add":" -- Trailing slash functions"},{"Add":" ; hasTrailingPathSeparator"},{"Add":" ; addTrailingPathSeparator"},{"Add":" ; dropTrailingPathSeparator"},{"Add":" -- File name manipulations"},{"Add":" ; normalise"},{"Add":" ; equalFilePath"},{"Add":" ; makeRelative"},{"Add":" -- ; checkFilePath see below: coerce needed"},{"Add":" ; isRelative"},{"Add":" ; isAbsolute"},{"Add":" ; isValid"},{"Add":" ; makeValid"},{"Add":" )"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" m n : Nature"},{"Add":""},{"Add":"-- singleton type for Nature"},{"Add":"data KnownNature : Nature → Set where"},{"Add":" instance"},{"Add":" absolute : KnownNature Nature.absolute"},{"Add":" relative : KnownNature Nature.relative"},{"Add":""},{"Add":"currentDirectory : SomePath"},{"Add":"currentDirectory = mkFilePath \".\""},{"Add":""},{"Add":"splitExtension : FilePath n → FilePath n × Extension"},{"Add":"splitExtension = coerce Prim.splitExtension"},{"Add":""},{"Add":"splitExtensions : FilePath n → FilePath n × Extension"},{"Add":"splitExtensions = coerce Prim.splitExtensions"},{"Add":""},{"Add":"stripExtension : Extension → FilePath n → Maybe (FilePath n)"},{"Add":"stripExtension = coerce Prim.stripExtension"},{"Add":""},{"Add":"getSearchPath : IO (List SomePath)"},{"Add":"getSearchPath = lift Prim.getSearchPath"},{"Add":""},{"Add":"splitFileName : FilePath n → FilePath n × RelativePath"},{"Add":"splitFileName = coerce Prim.splitFileName"},{"Add":""},{"Add":"checkFilePath : FilePath n → RelativePath ⊎ AbsolutePath"},{"Add":"checkFilePath = coerce Prim.checkFilePath"}]}},{"FileAdd":{"meta":{"basename":"Posix","metadata":576},"context":[{"change":0,"pos":277989}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Primitive.agda","metadata":0},"context":[{"change":0,"pos":281533}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Primitive System.FilePath.Posix simple bindings to Haskell functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module System.FilePath.Posix.Primitive where"},{"Add":""},{"Add":"open import Agda.Builtin.Bool using (Bool)"},{"Add":"open import Agda.Builtin.Char using (Char)"},{"Add":"open import Agda.Builtin.List using (List)"},{"Add":"open import Agda.Builtin.Maybe using (Maybe)"},{"Add":"open import Agda.Builtin.String using (String)"},{"Add":"open import Foreign.Haskell as FFI using (Pair; Either)"},{"Add":"open import IO.Primitive.Core using (IO)"},{"Add":""},{"Add":"-- A filepath has a nature: it can be either relative or absolute."},{"Add":"-- We postulate this nature rather than defining it as an inductive"},{"Add":"-- type so that the user cannot inspect it. The only way to cast an"},{"Add":"-- arbitrary filepath nature @n@ to either @relative@ or @absolute@"},{"Add":"-- is to use @checkFilePath@."},{"Add":""},{"Add":"module Nature where"},{"Add":""},{"Add":" postulate"},{"Add":" Nature : Set"},{"Add":" relative absolute unknown : Nature"},{"Add":""},{"Add":"-- In the Haskell backend these @natures@ are simply erased as the"},{"Add":"-- libraries represent all filepaths in the same way."},{"Add":""},{"Add":" {-# FOREIGN GHC import Data.Kind #-}"},{"Add":" {-# COMPILE GHC Nature = type Type #-}"},{"Add":" {-# COMPILE GHC relative = type () #-}"},{"Add":" {-# COMPILE GHC absolute = type () #-}"},{"Add":" {-# COMPILE GHC unknown = type () #-}"},{"Add":""},{"Add":"open Nature using (Nature) public"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" m n : Nature"},{"Add":""},{"Add":"postulate"},{"Add":""},{"Add":" FilePath : Nature → Set"},{"Add":" getFilePath : FilePath n → String"},{"Add":""},{"Add":" Extension : Set"},{"Add":" mkExtension : String → Extension"},{"Add":" getExtension : Extension → String"},{"Add":""},{"Add":"{-# FOREIGN GHC import Data.Text #-}"},{"Add":"{-# FOREIGN GHC import System.FilePath.Posix #-}"},{"Add":"{-# FOREIGN GHC type AgdaFilePath n = FilePath #-}"},{"Add":"{-# COMPILE GHC FilePath = type AgdaFilePath #-}"},{"Add":"{-# COMPILE GHC getFilePath = const pack #-}"},{"Add":"{-# COMPILE GHC Extension = type String #-}"},{"Add":"{-# COMPILE GHC mkExtension = unpack #-}"},{"Add":"{-# COMPILE GHC getExtension = pack #-}"},{"Add":""},{"Add":"-- We provide convenient short names for the two types of filepaths"},{"Add":""},{"Add":"AbsolutePath = FilePath Nature.absolute"},{"Add":"RelativePath = FilePath Nature.relative"},{"Add":"SomePath = FilePath Nature.unknown"},{"Add":""},{"Add":"-- In order to prevent users from picking whether a string gets"},{"Add":"-- converted to a @relative@ or an @absolute@ path we have:"},{"Add":"-- * a postulated @unknown@ nature"},{"Add":"-- * a function @mkFilePath@ producing filepaths of this postulated nature"},{"Add":""},{"Add":"postulate"},{"Add":" mkFilePath : String → SomePath"},{"Add":""},{"Add":"{-# COMPILE GHC mkFilePath = unpack #-}"},{"Add":""},{"Add":"postulate"},{"Add":""},{"Add":" -- Separator predicates"},{"Add":""},{"Add":" pathSeparator : Char"},{"Add":" pathSeparators : List Char"},{"Add":" isPathSeparator : Char → Bool"},{"Add":" searchPathSeparator : Char"},{"Add":" isSearchPathSeparator : Char → Bool"},{"Add":" extSeparator : Char"},{"Add":" isExtSeparator : Char → Bool"},{"Add":""},{"Add":" -- $PATH methods"},{"Add":""},{"Add":" splitSearchPath : String → List SomePath"},{"Add":" getSearchPath : IO (List SomePath)"},{"Add":""},{"Add":" -- Extension functions"},{"Add":""},{"Add":" splitExtension : FilePath n → Pair (FilePath n) Extension"},{"Add":" takeExtension : FilePath n → Extension"},{"Add":" replaceExtension : FilePath n → Extension → FilePath n"},{"Add":" dropExtension : FilePath n → FilePath n"},{"Add":" addExtension : FilePath n → Extension → FilePath n"},{"Add":" hasExtension : FilePath n → Bool"},{"Add":" splitExtensions : FilePath n → Pair (FilePath n) Extension"},{"Add":" takeExtensions : FilePath n → Extension"},{"Add":" replaceExtensions : FilePath n → Extension → FilePath n"},{"Add":" dropExtensions : FilePath n → FilePath n"},{"Add":" isExtensionOf : Extension → FilePath n → Bool"},{"Add":" stripExtension : Extension → FilePath n → Maybe (FilePath n)"},{"Add":""},{"Add":" -- Filename/directory functions"},{"Add":""},{"Add":" splitFileName : FilePath n → Pair (FilePath n) RelativePath"},{"Add":" takeFileName : FilePath n → String"},{"Add":" replaceFileName : FilePath n → String → FilePath n"},{"Add":" dropFileName : FilePath n → FilePath n"},{"Add":" takeBaseName : FilePath n → String"},{"Add":" replaceBaseName : FilePath n → String → FilePath n"},{"Add":" takeDirectory : FilePath n → FilePath n"},{"Add":" replaceDirectory : FilePath m → FilePath n → FilePath n"},{"Add":" combine : FilePath n → RelativePath → FilePath n"},{"Add":" splitPath : FilePath n → List RelativePath"},{"Add":" joinPath : List RelativePath → RelativePath"},{"Add":" splitDirectories : FilePath n → List RelativePath"},{"Add":""},{"Add":" -- Drive functions"},{"Add":""},{"Add":" splitDrive : FilePath n → Pair (FilePath n) RelativePath"},{"Add":" joinDrive : FilePath n → RelativePath → FilePath n"},{"Add":" takeDrive : FilePath n → FilePath n"},{"Add":" hasDrive : FilePath n → Bool"},{"Add":" dropDrive : FilePath n → RelativePath"},{"Add":" isDrive : FilePath n → Bool"},{"Add":""},{"Add":" -- Trailing slash functions"},{"Add":""},{"Add":" hasTrailingPathSeparator : FilePath n → Bool"},{"Add":" addTrailingPathSeparator : FilePath n → FilePath n"},{"Add":" dropTrailingPathSeparator : FilePath n → FilePath n"},{"Add":""},{"Add":" -- File name manipulations"},{"Add":""},{"Add":" normalise : FilePath n → FilePath n"},{"Add":" equalFilePath : FilePath m → FilePath n → Bool"},{"Add":" makeRelative : FilePath m → FilePath n → RelativePath"},{"Add":" checkFilePath : FilePath n → Either RelativePath AbsolutePath"},{"Add":" isRelative : FilePath n → Bool"},{"Add":" isAbsolute : FilePath n → Bool"},{"Add":" isValid : FilePath n → Bool"},{"Add":" makeValid : FilePath n → FilePath n"},{"Add":""},{"Add":""},{"Add":"{-# FOREIGN GHC"},{"Add":"checkFilePath fp"},{"Add":" | isRelative fp = Left fp"},{"Add":" | otherwise = Right fp"},{"Add":"#-}"},{"Add":""},{"Add":"{-# COMPILE GHC pathSeparator = pathSeparator #-}"},{"Add":"{-# COMPILE GHC pathSeparators = pathSeparators #-}"},{"Add":"{-# COMPILE GHC isPathSeparator = isPathSeparator #-}"},{"Add":"{-# COMPILE GHC searchPathSeparator = searchPathSeparator #-}"},{"Add":"{-# COMPILE GHC isSearchPathSeparator = isSearchPathSeparator #-}"},{"Add":"{-# COMPILE GHC extSeparator = extSeparator #-}"},{"Add":"{-# COMPILE GHC isExtSeparator = isExtSeparator #-}"},{"Add":""},{"Add":"{-# COMPILE GHC splitSearchPath = splitSearchPath . unpack #-}"},{"Add":"{-# COMPILE GHC getSearchPath = getSearchPath #-}"},{"Add":""},{"Add":"{-# COMPILE GHC splitExtension = const splitExtension #-}"},{"Add":"{-# COMPILE GHC takeExtension = const takeExtension #-}"},{"Add":"{-# COMPILE GHC replaceExtension = const replaceExtension #-}"},{"Add":"{-# COMPILE GHC dropExtension = const dropExtension #-}"},{"Add":"{-# COMPILE GHC addExtension = const addExtension #-}"},{"Add":"{-# COMPILE GHC hasExtension = const hasExtension #-}"},{"Add":"{-# COMPILE GHC splitExtensions = const splitExtensions #-}"},{"Add":"{-# COMPILE GHC takeExtensions = const takeExtensions #-}"},{"Add":"{-# COMPILE GHC replaceExtensions = const replaceExtensions #-}"},{"Add":"{-# COMPILE GHC dropExtensions = const dropExtensions #-}"},{"Add":"{-# COMPILE GHC isExtensionOf = const isExtensionOf #-}"},{"Add":"{-# COMPILE GHC stripExtension = const stripExtension #-}"},{"Add":""},{"Add":"{-# COMPILE GHC splitFileName = const splitFileName #-}"},{"Add":"{-# COMPILE GHC takeFileName = const $ pack . takeFileName #-}"},{"Add":"{-# COMPILE GHC replaceFileName = const $ fmap (. unpack) replaceFileName #-}"},{"Add":"{-# COMPILE GHC dropFileName = const dropFileName #-}"},{"Add":"{-# COMPILE GHC takeBaseName = const $ pack . takeBaseName #-}"},{"Add":"{-# COMPILE GHC replaceBaseName = const $ fmap (. unpack) replaceBaseName #-}"},{"Add":"{-# COMPILE GHC takeDirectory = const takeDirectory #-}"},{"Add":"{-# COMPILE GHC replaceDirectory = \\ _ _ -> replaceDirectory #-}"},{"Add":"{-# COMPILE GHC combine = const combine #-}"},{"Add":"{-# COMPILE GHC splitPath = const splitPath #-}"},{"Add":"{-# COMPILE GHC joinPath = joinPath #-}"},{"Add":"{-# COMPILE GHC splitDirectories = const splitDirectories #-}"},{"Add":""},{"Add":"{-# COMPILE GHC splitDrive = const splitDrive #-}"},{"Add":"{-# COMPILE GHC joinDrive = const joinDrive #-}"},{"Add":"{-# COMPILE GHC takeDrive = const takeDrive #-}"},{"Add":"{-# COMPILE GHC hasDrive = const hasDrive #-}"},{"Add":"{-# COMPILE GHC dropDrive = const dropDrive #-}"},{"Add":"{-# COMPILE GHC isDrive = const isDrive #-}"},{"Add":""},{"Add":"{-# COMPILE GHC hasTrailingPathSeparator = const hasTrailingPathSeparator #-}"},{"Add":"{-# COMPILE GHC addTrailingPathSeparator = const addTrailingPathSeparator #-}"},{"Add":"{-# COMPILE GHC dropTrailingPathSeparator = const dropTrailingPathSeparator #-}"},{"Add":""},{"Add":"{-# COMPILE GHC normalise = const normalise #-}"},{"Add":"{-# COMPILE GHC equalFilePath = \\ _ _ -> equalFilePath #-}"},{"Add":"{-# COMPILE GHC makeRelative = \\ _ _ -> makeRelative #-}"},{"Add":"{-# COMPILE GHC isRelative = const isRelative #-}"},{"Add":"{-# COMPILE GHC isAbsolute = const isAbsolute #-}"},{"Add":"{-# COMPILE GHC checkFilePath = const checkFilePath #-}"},{"Add":"{-# COMPILE GHC isValid = const isValid #-}"},{"Add":"{-# COMPILE GHC makeValid = const makeValid #-}"}]}},{"FileAdd":{"meta":{"basename":"Exit.agda","metadata":0},"context":[{"change":0,"pos":266416}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Exiting the program."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module System.Exit where"},{"Add":""},{"Add":"open import Level using (Level)"},{"Add":"open import Data.Bool.Base using (Bool; true; false; not)"},{"Add":"open import Data.Integer.Base using (ℤ; +_)"},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Function.Base using (_$_; _∘′_)"},{"Add":"open import IO using (IO; lift; _>>_; putStrLn)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting the ExitCode data structure"},{"Add":""},{"Add":"open import System.Exit.Primitive as Prim"},{"Add":" public"},{"Add":" using ( ExitCode"},{"Add":" ; ExitSuccess"},{"Add":" ; ExitFailure"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Tests"},{"Add":""},{"Add":"isSuccess : ExitCode → Bool"},{"Add":"isSuccess ExitSuccess = true"},{"Add":"isSuccess (ExitFailure _) = false"},{"Add":""},{"Add":"isFailure : ExitCode → Bool"},{"Add":"isFailure = not ∘′ isSuccess"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Various exiting function"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"exitWith : ExitCode → IO A"},{"Add":"exitWith c = lift (Prim.exitWith c)"},{"Add":""},{"Add":"exitFailure : IO A"},{"Add":"exitFailure = exitWith (ExitFailure (+ 1))"},{"Add":""},{"Add":"exitSuccess : IO A"},{"Add":"exitSuccess = exitWith ExitSuccess"},{"Add":""},{"Add":"die : String → IO A"},{"Add":"die str = do"},{"Add":" putStrLn str"},{"Add":" exitFailure"}]}},{"FileAdd":{"meta":{"basename":"Exit","metadata":576},"context":[{"change":0,"pos":266416}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Primitive.agda","metadata":0},"context":[{"change":0,"pos":291823}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Primitive System.Exit simple bindings to Haskell functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module System.Exit.Primitive where"},{"Add":""},{"Add":"open import Agda.Builtin.Int using (Int)"},{"Add":"open import Agda.Builtin.IO using (IO)"},{"Add":""},{"Add":"data ExitCode : Set where"},{"Add":" ExitSuccess : ExitCode"},{"Add":" ExitFailure : Int → ExitCode"},{"Add":""},{"Add":"{-# FOREIGN GHC data AgdaExitCode = AgdaExitSuccess | AgdaExitFailure Integer #-}"},{"Add":"{-# COMPILE GHC ExitCode = data AgdaExitCode (AgdaExitSuccess | AgdaExitFailure) #-}"},{"Add":""},{"Add":"{-# FOREIGN GHC import qualified System.Exit as SE #-}"},{"Add":""},{"Add":"{-# FOREIGN GHC"},{"Add":"toExitCode :: AgdaExitCode -> SE.ExitCode"},{"Add":"toExitCode AgdaExitSuccess = SE.ExitSuccess"},{"Add":"toExitCode (AgdaExitFailure n) = SE.ExitFailure (fromIntegral n)"},{"Add":""},{"Add":"fromExitCode :: SE.ExitCode -> AgdaExitCode"},{"Add":"fromExitCode SE.ExitSuccess = AgdaExitSuccess"},{"Add":"fromExitCode (SE.ExitFailure n) = AgdaExitFailure (fromIntegral n)"},{"Add":"#-}"},{"Add":""},{"Add":"postulate"},{"Add":" exitWith : ∀ {a} {A : Set a} → ExitCode → IO A"},{"Add":""},{"Add":"{-# COMPILE GHC exitWith = \\ _ _ -> SE.exitWith . toExitCode #-}"}]}},{"FileAdd":{"meta":{"basename":"Environment.agda","metadata":0},"context":[{"change":0,"pos":266416}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Miscellanous information about the system environment"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module System.Environment where"},{"Add":""},{"Add":"open import IO using (IO; lift; run; ignore)"},{"Add":"open import Data.List.Base using (List)"},{"Add":"open import Data.Maybe.Base using (Maybe)"},{"Add":"open import Data.Product.Base using (_×_)"},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Data.Unit.Polymorphic using (⊤)"},{"Add":"open import Foreign.Haskell.Coerce"},{"Add":""},{"Add":"import System.Environment.Primitive as Prim"},{"Add":""},{"Add":"getArgs : IO (List String)"},{"Add":"getArgs = lift Prim.getArgs"},{"Add":""},{"Add":"getProgName : IO String"},{"Add":"getProgName = lift Prim.getProgName"},{"Add":""},{"Add":"lookupEnv : String → IO (Maybe String)"},{"Add":"lookupEnv var = lift (coerce (Prim.lookupEnv var))"},{"Add":""},{"Add":"setEnv : String → String → IO ⊤"},{"Add":"setEnv var val = ignore (lift (Prim.setEnv var val))"},{"Add":""},{"Add":"unsetEnv : String → IO ⊤"},{"Add":"unsetEnv var = ignore (lift (Prim.unsetEnv var))"},{"Add":""},{"Add":"withArgs : ∀ {a} {A : Set a} → List String → IO A → IO A"},{"Add":"withArgs args io = lift (Prim.withArgs args (run io))"},{"Add":""},{"Add":"withProgName : ∀ {a} {A : Set a} → String → IO A → IO A"},{"Add":"withProgName name io = lift (Prim.withProgName name (run io))"},{"Add":""},{"Add":"getEnvironment : IO (List (String × String))"},{"Add":"getEnvironment = lift (coerce Prim.getEnvironment)"}]}},{"FileAdd":{"meta":{"basename":"Environment","metadata":576},"context":[{"change":0,"pos":266416}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Primitive.agda","metadata":0},"context":[{"change":0,"pos":294480}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Primitive System.Environment: simple bindings to Haskell functions"},{"Add":"--"},{"Add":"-- Note that we currently leave out:"},{"Add":"-- * filepath-related functions (until we have a good representation of"},{"Add":"-- absolute vs. relative & directory vs. file)"},{"Add":"-- * functions that may fail with an exception"},{"Add":"-- e.g. we provide `lookupEnv` but not `getEnv`"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module System.Environment.Primitive where"},{"Add":""},{"Add":"open import IO.Primitive.Core using (IO)"},{"Add":"open import Data.List.Base using (List)"},{"Add":"open import Data.Maybe.Base using (Maybe)"},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Data.Unit.Base using (⊤)"},{"Add":""},{"Add":"open import Foreign.Haskell.Pair using (Pair)"},{"Add":""},{"Add":"{-# FOREIGN GHC import qualified System.Environment as SE #-}"},{"Add":"{-# FOREIGN GHC import qualified Data.Text as T #-}"},{"Add":"{-# FOREIGN GHC import Data.Bifunctor (bimap) #-}"},{"Add":"{-# FOREIGN GHC import Data.Function (on) #-}"},{"Add":""},{"Add":"postulate"},{"Add":" getArgs : IO (List String)"},{"Add":" getProgName : IO String"},{"Add":" lookupEnv : String → IO (Maybe String)"},{"Add":" setEnv : String → String → IO ⊤"},{"Add":" unsetEnv : String → IO ⊤"},{"Add":" withArgs : ∀ {a} {A : Set a} → List String → IO A → IO A"},{"Add":" withProgName : ∀ {a} {A : Set a} → String → IO A → IO A"},{"Add":" getEnvironment : IO (List (Pair String String))"},{"Add":""},{"Add":"{-# COMPILE GHC getArgs = fmap (fmap T.pack) SE.getArgs #-}"},{"Add":"{-# COMPILE GHC getProgName = fmap T.pack SE.getProgName #-}"},{"Add":"{-# COMPILE GHC lookupEnv = fmap (fmap T.pack) . SE.lookupEnv . T.unpack #-}"},{"Add":"{-# COMPILE GHC setEnv = SE.setEnv `on` T.unpack #-}"},{"Add":"{-# COMPILE GHC unsetEnv = SE.unsetEnv . T.unpack #-}"},{"Add":"{-# COMPILE GHC withArgs = \\ _ _ -> SE.withArgs . fmap T.unpack #-}"},{"Add":"{-# COMPILE GHC withProgName = \\ _ _ -> SE.withProgName . T.unpack #-}"},{"Add":"{-# COMPILE GHC getEnvironment = fmap (fmap (bimap T.pack T.pack)) SE.getEnvironment #-}"}]}},{"FileAdd":{"meta":{"basename":"Directory.agda","metadata":0},"context":[{"change":0,"pos":266416}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Directory manipulation"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module System.Directory where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import IO.Base"},{"Add":"open import Data.Unit.Polymorphic.Base using (⊤)"},{"Add":"open import Data.Bool.Base using (Bool)"},{"Add":"open import Data.List.Base using (List)"},{"Add":"open import Data.Maybe.Base using (Maybe)"},{"Add":"open import Data.Nat.Base using (ℕ)"},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Foreign.Haskell.Coerce using (coerce)"},{"Add":"open import Function.Base using (_∘′_)"},{"Add":"open import System.FilePath.Posix hiding (makeRelative)"},{"Add":""},{"Add":"open import System.Directory.Primitive as Prim"},{"Add":" public"},{"Add":" using ( XdgDirectory"},{"Add":" ; XdgData"},{"Add":" ; XdgConfig"},{"Add":" ; XdgCache"},{"Add":" ; XdgDirectoryList"},{"Add":" ; XdgDataDirs"},{"Add":" ; XdgConfigDirs"},{"Add":" ; exeExtension"},{"Add":" )"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a : Level"},{"Add":" m n : Nature"},{"Add":""},{"Add":"-- Actions on directories"},{"Add":""},{"Add":"createDirectory : FilePath n → IO {a} ⊤"},{"Add":"createDirectoryIfMissing : Bool → FilePath n → IO {a} ⊤"},{"Add":"removeDirectory : FilePath n → IO {a} ⊤"},{"Add":"removeDirectoryRecursive : FilePath n → IO {a} ⊤"},{"Add":"removePathForcibly : FilePath n → IO {a} ⊤"},{"Add":"renameDirectory : FilePath n → FilePath n → IO {a} ⊤"},{"Add":"listDirectory : FilePath n → IO (List RelativePath)"},{"Add":""},{"Add":"-- Current working directory"},{"Add":""},{"Add":"getDirectoryContents : FilePath n → IO (List RelativePath)"},{"Add":"getCurrentDirectory : IO AbsolutePath"},{"Add":"setCurrentDirectory : FilePath n → IO {a} ⊤"},{"Add":"withCurrentDirectory : {A : Set a} → FilePath n → IO A → IO A"},{"Add":""},{"Add":"-- Pre-defined directories"},{"Add":""},{"Add":"getHomeDirectory : IO AbsolutePath"},{"Add":"getXdgDirectory : XdgDirectory → RelativePath → IO AbsolutePath"},{"Add":"getXdgDirectoryList : XdgDirectoryList → IO (List AbsolutePath)"},{"Add":"getAppUserDataDirectory : RelativePath → IO AbsolutePath"},{"Add":"getUserDocumentsDirectory : IO AbsolutePath"},{"Add":"getTemporaryDirectory : IO AbsolutePath"},{"Add":""},{"Add":"-- Action on files"},{"Add":"removeFile : FilePath m → IO {a} ⊤"},{"Add":"renameFile : FilePath m → FilePath n → IO {a} ⊤"},{"Add":"renamePath : FilePath m → FilePath n → IO {a} ⊤"},{"Add":"copyFile : FilePath m → FilePath n → IO {a} ⊤"},{"Add":"copyFileWithMetadata : FilePath m → FilePath n → IO {a} ⊤"},{"Add":"getFileSize : FilePath n → IO ℕ"},{"Add":"canonicalizePath : FilePath n → IO AbsolutePath"},{"Add":"makeAbsolute : FilePath n → IO AbsolutePath"},{"Add":"makeRelative : FilePath n → IO RelativePath"},{"Add":""},{"Add":"toKnownNature : KnownNature m → FilePath n → IO (FilePath m)"},{"Add":"toKnownNature absolute = makeAbsolute"},{"Add":"toKnownNature relative = makeRelative"},{"Add":""},{"Add":"relativeToKnownNature : KnownNature n → RelativePath → IO (FilePath n)"},{"Add":"absoluteToKnownNature : KnownNature n → AbsolutePath → IO (FilePath n)"},{"Add":""},{"Add":"relativeToKnownNature absolute = makeAbsolute"},{"Add":"relativeToKnownNature relative = pure"},{"Add":""},{"Add":"absoluteToKnownNature absolute = pure"},{"Add":"absoluteToKnownNature relative = makeRelative"},{"Add":""},{"Add":"-- Existence tests"},{"Add":""},{"Add":"doesPathExist : FilePath n → IO Bool"},{"Add":"doesFileExist : FilePath n → IO Bool"},{"Add":"doesDirectoryExist : FilePath n → IO Bool"},{"Add":"findExecutable : String → IO (Maybe AbsolutePath)"},{"Add":"findExecutables : String → IO (List AbsolutePath)"},{"Add":"findExecutablesInDirectories : List (FilePath n) → String → IO (List (FilePath n))"},{"Add":"findFile : List (FilePath n) → String → IO (Maybe (FilePath n))"},{"Add":"findFiles : List (FilePath n) → String → IO (List (FilePath n))"},{"Add":"findFileWith : (FilePath n → IO Bool) → List (FilePath n) → String → IO (Maybe (FilePath n))"},{"Add":"findFilesWith : (FilePath n → IO Bool) → List (FilePath n) → String → IO (List (FilePath n))"},{"Add":""},{"Add":"-- Symbolic links"},{"Add":""},{"Add":"createFileLink : FilePath m → FilePath n → IO {a} ⊤"},{"Add":"createDirectoryLink : FilePath m → FilePath n → IO {a} ⊤"},{"Add":"removeDirectoryLink : FilePath n → IO {a} ⊤"},{"Add":"pathIsSymbolicLink : FilePath n → IO Bool"},{"Add":"getSymbolicLinkTarget : FilePath n → IO SomePath"},{"Add":""},{"Add":"createDirectory = lift! ∘′ lift ∘′ Prim.createDirectory"},{"Add":"createDirectoryIfMissing = λ b → lift! ∘′ lift ∘′ Prim.createDirectoryIfMissing b"},{"Add":"removeDirectory = lift! ∘′ lift ∘′ Prim.removeDirectory"},{"Add":"removeDirectoryRecursive = lift! ∘′ lift ∘′ Prim.removeDirectoryRecursive"},{"Add":"removePathForcibly = lift! ∘′ lift ∘′ Prim.removePathForcibly"},{"Add":"renameDirectory = λ fp → lift! ∘′ lift ∘′ Prim.renameDirectory fp"},{"Add":"listDirectory = lift ∘′ Prim.listDirectory"},{"Add":"getDirectoryContents = lift ∘′ Prim.getDirectoryContents"},{"Add":""},{"Add":"getCurrentDirectory = lift Prim.getCurrentDirectory"},{"Add":"setCurrentDirectory = lift! ∘′ lift ∘′ Prim.setCurrentDirectory"},{"Add":"withCurrentDirectory = λ fp ma → lift (Prim.withCurrentDirectory fp (run ma))"},{"Add":""},{"Add":"getHomeDirectory = lift Prim.getHomeDirectory"},{"Add":"getXdgDirectory = λ d fp → lift (Prim.getXdgDirectory d fp)"},{"Add":"getXdgDirectoryList = lift ∘′ Prim.getXdgDirectoryList"},{"Add":"getAppUserDataDirectory = lift ∘′ Prim.getAppUserDataDirectory"},{"Add":"getUserDocumentsDirectory = lift Prim.getUserDocumentsDirectory"},{"Add":"getTemporaryDirectory = lift Prim.getTemporaryDirectory"},{"Add":""},{"Add":"removeFile = lift! ∘′ lift ∘′ Prim.removeFile"},{"Add":"renameFile = λ a b → lift! (lift (Prim.renameFile a b))"},{"Add":"renamePath = λ a b → lift! (lift (Prim.renamePath a b))"},{"Add":"copyFile = λ a b → lift! (lift (Prim.copyFile a b))"},{"Add":"copyFileWithMetadata = λ a b → lift! (lift (Prim.copyFileWithMetadata a b))"},{"Add":"getFileSize = lift ∘′ Prim.getFileSize"},{"Add":"canonicalizePath = lift ∘′ Prim.canonicalizePath"},{"Add":"makeAbsolute = lift ∘′ Prim.makeAbsolute"},{"Add":"makeRelative = lift ∘′ Prim.makeRelativeToCurrentDirectory"},{"Add":""},{"Add":"doesPathExist = lift ∘′ Prim.doesPathExist"},{"Add":"doesFileExist = lift ∘′ Prim.doesFileExist"},{"Add":"doesDirectoryExist = lift ∘′ Prim.doesDirectoryExist"},{"Add":"findExecutable = lift ∘′ coerce ∘′ Prim.findExecutable"},{"Add":"findExecutables = lift ∘′ Prim.findExecutables"},{"Add":"findExecutablesInDirectories = λ fps str → lift (Prim.findExecutablesInDirectories fps str)"},{"Add":"findFile = λ fps str → lift (coerce Prim.findFile fps str)"},{"Add":"findFiles = λ fps str → lift (Prim.findFiles fps str)"},{"Add":"findFileWith = λ p fps str → lift (coerce Prim.findFileWith (run ∘′ p) fps str)"},{"Add":"findFilesWith = λ p fps str → lift (Prim.findFilesWith (run ∘′ p) fps str)"},{"Add":""},{"Add":"createFileLink = λ fp → lift! ∘′ lift ∘′ Prim.createFileLink fp"},{"Add":"createDirectoryLink = λ fp → lift! ∘′ lift ∘′ Prim.createDirectoryLink fp"},{"Add":"removeDirectoryLink = lift! ∘′ lift ∘′ Prim.removeDirectoryLink"},{"Add":"pathIsSymbolicLink = lift ∘′ Prim.pathIsSymbolicLink"},{"Add":"getSymbolicLinkTarget = lift ∘′ Prim.getSymbolicLinkTarget"}]}},{"FileAdd":{"meta":{"basename":"Directory","metadata":576},"context":[{"change":0,"pos":266416}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Primitive.agda","metadata":0},"context":[{"change":0,"pos":303797}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Primitive System.Direcotry simple bindings to Haskell functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module System.Directory.Primitive where"},{"Add":""},{"Add":"open import Agda.Builtin.Unit using (⊤)"},{"Add":"open import Agda.Builtin.Bool using (Bool)"},{"Add":"open import Agda.Builtin.IO using (IO)"},{"Add":"open import Agda.Builtin.List using (List)"},{"Add":"open import Agda.Builtin.Maybe using (Maybe)"},{"Add":"open import Agda.Builtin.Nat using (Nat)"},{"Add":"open import Agda.Builtin.String using (String)"},{"Add":"open import System.FilePath.Posix.Primitive"},{"Add":""},{"Add":"{-# FOREIGN GHC import System.Directory #-}"},{"Add":"{-# FOREIGN GHC import Data.Text #-}"},{"Add":""},{"Add":"data XdgDirectory : Set where"},{"Add":" XdgData XdgConfig XdgCache XdgState : XdgDirectory"},{"Add":""},{"Add":"data XdgDirectoryList : Set where"},{"Add":" XdgDataDirs XdgConfigDirs : XdgDirectoryList"},{"Add":""},{"Add":"{-# COMPILE GHC XdgDirectory = data XdgDirectory (XdgData|XdgConfig|XdgCache|XdgState) #-}"},{"Add":"{-# COMPILE GHC XdgDirectoryList = data XdgDirectoryList (XdgDataDirs|XdgConfigDirs) #-}"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" m n : Nature"},{"Add":""},{"Add":"postulate"},{"Add":""},{"Add":" -- Actions on directories"},{"Add":""},{"Add":" createDirectory : FilePath n → IO ⊤"},{"Add":" createDirectoryIfMissing : Bool → FilePath n → IO ⊤"},{"Add":" removeDirectory : FilePath n → IO ⊤"},{"Add":" removeDirectoryRecursive : FilePath n → IO ⊤"},{"Add":" removePathForcibly : FilePath n → IO ⊤"},{"Add":" renameDirectory : FilePath m → FilePath n → IO ⊤"},{"Add":" listDirectory : FilePath n → IO (List RelativePath)"},{"Add":" getDirectoryContents : FilePath n → IO (List RelativePath)"},{"Add":""},{"Add":" -- Current working directory"},{"Add":""},{"Add":" getCurrentDirectory : IO AbsolutePath"},{"Add":" setCurrentDirectory : FilePath n → IO ⊤"},{"Add":" withCurrentDirectory : ∀ {a} {A : Set a} → FilePath n → IO A → IO A"},{"Add":""},{"Add":" -- Pre-defined directories"},{"Add":""},{"Add":" getHomeDirectory : IO AbsolutePath"},{"Add":" getXdgDirectory : XdgDirectory → RelativePath → IO AbsolutePath"},{"Add":" getXdgDirectoryList : XdgDirectoryList → IO (List AbsolutePath)"},{"Add":" getAppUserDataDirectory : RelativePath → IO AbsolutePath"},{"Add":" getUserDocumentsDirectory : IO AbsolutePath"},{"Add":" getTemporaryDirectory : IO AbsolutePath"},{"Add":""},{"Add":" -- Actions on files"},{"Add":""},{"Add":" removeFile : FilePath m → IO ⊤"},{"Add":" renameFile : FilePath m → FilePath n → IO ⊤"},{"Add":" renamePath : FilePath m → FilePath n → IO ⊤"},{"Add":" copyFile : FilePath m → FilePath n → IO ⊤"},{"Add":" copyFileWithMetadata : FilePath m → FilePath n → IO ⊤"},{"Add":" getFileSize : FilePath n → IO Nat"},{"Add":" canonicalizePath : FilePath n → IO AbsolutePath"},{"Add":" makeAbsolute : FilePath n → IO AbsolutePath"},{"Add":" makeRelativeToCurrentDirectory : FilePath n → IO RelativePath"},{"Add":""},{"Add":" -- Existence tests"},{"Add":""},{"Add":" doesPathExist : FilePath n → IO Bool"},{"Add":" doesFileExist : FilePath n → IO Bool"},{"Add":" doesDirectoryExist : FilePath n → IO Bool"},{"Add":" findExecutable : String → IO (Maybe AbsolutePath)"},{"Add":" findExecutables : String → IO (List AbsolutePath)"},{"Add":" findExecutablesInDirectories : List (FilePath n) → String → IO (List (FilePath n))"},{"Add":" findFile : List (FilePath n) → String → IO (Maybe (FilePath n))"},{"Add":" findFiles : List (FilePath n) → String → IO (List (FilePath n))"},{"Add":" findFileWith : (FilePath n → IO Bool) → List (FilePath n) → String → IO (Maybe (FilePath n))"},{"Add":" findFilesWith : (FilePath n → IO Bool) → List (FilePath n) → String → IO (List (FilePath n))"},{"Add":" exeExtension : String"},{"Add":""},{"Add":" -- Symbolic links"},{"Add":""},{"Add":" createFileLink : FilePath m → FilePath n → IO ⊤"},{"Add":" createDirectoryLink : FilePath m → FilePath n → IO ⊤"},{"Add":" removeDirectoryLink : FilePath n → IO ⊤"},{"Add":" pathIsSymbolicLink : FilePath n → IO Bool"},{"Add":" getSymbolicLinkTarget : FilePath n → IO SomePath"},{"Add":""},{"Add":""},{"Add":"{-# COMPILE GHC createDirectory = const createDirectory #-}"},{"Add":"{-# COMPILE GHC createDirectoryIfMissing = const createDirectoryIfMissing #-}"},{"Add":"{-# COMPILE GHC removeDirectory = const removeDirectory #-}"},{"Add":"{-# COMPILE GHC removeDirectoryRecursive = const removeDirectoryRecursive #-}"},{"Add":"{-# COMPILE GHC removePathForcibly = const removePathForcibly #-}"},{"Add":"{-# COMPILE GHC renameDirectory = \\ _ _ -> renameDirectory #-}"},{"Add":"{-# COMPILE GHC listDirectory = const listDirectory #-}"},{"Add":"{-# COMPILE GHC getDirectoryContents = const getDirectoryContents #-}"},{"Add":""},{"Add":"{-# COMPILE GHC getCurrentDirectory = getCurrentDirectory #-}"},{"Add":"{-# COMPILE GHC setCurrentDirectory = const setCurrentDirectory #-}"},{"Add":"{-# COMPILE GHC withCurrentDirectory = \\ _ _ _ -> withCurrentDirectory #-}"},{"Add":""},{"Add":"{-# COMPILE GHC getHomeDirectory = getHomeDirectory #-}"},{"Add":"{-# COMPILE GHC getXdgDirectory = getXdgDirectory #-}"},{"Add":"{-# COMPILE GHC getXdgDirectoryList = getXdgDirectoryList #-}"},{"Add":"{-# COMPILE GHC getAppUserDataDirectory = getAppUserDataDirectory #-}"},{"Add":"{-# COMPILE GHC getUserDocumentsDirectory = getUserDocumentsDirectory #-}"},{"Add":"{-# COMPILE GHC getTemporaryDirectory = getTemporaryDirectory #-}"},{"Add":""},{"Add":"{-# COMPILE GHC removeFile = const removeFile #-}"},{"Add":"{-# COMPILE GHC renameFile = \\ _ _ -> renameFile #-}"},{"Add":"{-# COMPILE GHC renamePath = \\ _ _ -> renamePath #-}"},{"Add":"{-# COMPILE GHC copyFile = \\ _ _ -> copyFile #-}"},{"Add":"{-# COMPILE GHC copyFileWithMetadata = \\ _ _ -> copyFileWithMetadata #-}"},{"Add":"{-# COMPILE GHC getFileSize = const getFileSize #-}"},{"Add":"{-# COMPILE GHC canonicalizePath = const canonicalizePath #-}"},{"Add":"{-# COMPILE GHC makeAbsolute = const makeAbsolute #-}"},{"Add":"{-# COMPILE GHC makeRelativeToCurrentDirectory = const makeRelativeToCurrentDirectory #-}"},{"Add":""},{"Add":"{-# COMPILE GHC doesPathExist = const doesPathExist #-}"},{"Add":"{-# COMPILE GHC doesFileExist = const doesFileExist #-}"},{"Add":"{-# COMPILE GHC doesDirectoryExist = const doesDirectoryExist #-}"},{"Add":"{-# COMPILE GHC findExecutable = findExecutable . unpack #-}"},{"Add":"{-# COMPILE GHC findExecutables = findExecutables . unpack #-}"},{"Add":"{-# COMPILE GHC findExecutablesInDirectories = \\ _ fps -> findExecutablesInDirectories fps . unpack #-}"},{"Add":"{-# COMPILE GHC findFile = \\ _ fps -> findFile fps . unpack #-}"},{"Add":"{-# COMPILE GHC findFiles = \\ _ fps -> findFiles fps . unpack #-}"},{"Add":"{-# COMPILE GHC findFileWith = \\ _ p fps -> findFileWith p fps . unpack #-}"},{"Add":"{-# COMPILE GHC findFilesWith = \\ _ p fps -> findFilesWith p fps . unpack #-}"},{"Add":"{-# COMPILE GHC exeExtension = pack exeExtension #-}"},{"Add":""},{"Add":"{-# COMPILE GHC createFileLink = \\ _ _ -> createFileLink #-}"},{"Add":"{-# COMPILE GHC createDirectoryLink = \\ _ _ -> createDirectoryLink #-}"},{"Add":"{-# COMPILE GHC removeDirectoryLink = const removeDirectoryLink #-}"},{"Add":"{-# COMPILE GHC pathIsSymbolicLink = const pathIsSymbolicLink #-}"},{"Add":"{-# COMPILE GHC getSymbolicLinkTarget = const getSymbolicLinkTarget #-}"}]}},{"FileAdd":{"meta":{"basename":"Console","metadata":576},"context":[{"change":0,"pos":266416}],"contents":[]}},{"FileAdd":{"meta":{"basename":"ANSI.agda","metadata":0},"context":[{"change":0,"pos":311524}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- ANSI escape codes"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module System.Console.ANSI where"},{"Add":""},{"Add":"open import Data.List.Base as List using (List; []; _∷_)"},{"Add":"open import Data.Nat.Base using (ℕ; _+_)"},{"Add":"import Data.Nat.Show as ℕ using (show)"},{"Add":"open import Data.String.Base using (String; concat; intersperse)"},{"Add":"open import Function.Base using (_$_; case_of_)"},{"Add":""},{"Add":"data Layer : Set where"},{"Add":" foreground background : Layer"},{"Add":""},{"Add":"data Colour : Set where"},{"Add":" black red green yellow blue magenta cyan white : Colour"},{"Add":""},{"Add":"data Intensity : Set where"},{"Add":" classic bright : Intensity"},{"Add":""},{"Add":"data Weight : Set where"},{"Add":" bold faint normal : Weight"},{"Add":""},{"Add":"data Underlining : Set where"},{"Add":" single double : Underlining"},{"Add":""},{"Add":"data Style : Set where"},{"Add":" italic straight : Style"},{"Add":""},{"Add":"data Blinking : Set where"},{"Add":" slow rapid : Blinking"},{"Add":""},{"Add":"data Command : Set where"},{"Add":" reset : Command"},{"Add":" setColour : Layer → Intensity → Colour → Command"},{"Add":" setWeight : Weight → Command"},{"Add":" setStyle : Style → Command"},{"Add":" setUnderline : Underlining → Command"},{"Add":" setBlinking : Blinking → Command"},{"Add":""},{"Add":"private"},{"Add":""},{"Add":" escape : String → String"},{"Add":" escape txt = concat $ \"\\x1b[\" ∷ txt ∷ \"m\" ∷ []"},{"Add":""},{"Add":"encode : Command → String"},{"Add":"encode reset = \"0\""},{"Add":"encode (setColour l i c) = ℕ.show (layer l + colour c + intensity i)"},{"Add":""},{"Add":" where"},{"Add":""},{"Add":" layer : Layer → ℕ"},{"Add":" layer foreground = 30"},{"Add":" layer background = 40"},{"Add":""},{"Add":" colour : Colour → ℕ"},{"Add":" colour black = 0"},{"Add":" colour red = 1"},{"Add":" colour green = 2"},{"Add":" colour yellow = 3"},{"Add":" colour blue = 4"},{"Add":" colour magenta = 5"},{"Add":" colour cyan = 6"},{"Add":" colour white = 7"},{"Add":""},{"Add":" intensity : Intensity → ℕ"},{"Add":" intensity classic = 0"},{"Add":" intensity bright = 60"},{"Add":""},{"Add":"encode (setWeight w) = case w of λ where"},{"Add":" bold → \"1\""},{"Add":" faint → \"2\""},{"Add":" normal → \"22\""},{"Add":""},{"Add":"encode (setStyle s) = case s of λ where"},{"Add":" italic → \"3\""},{"Add":" straight → \"23\""},{"Add":""},{"Add":"encode (setUnderline i) = case i of λ where"},{"Add":" single → \"4\""},{"Add":" double → \"21\""},{"Add":""},{"Add":"encode (setBlinking b) = case b of λ where"},{"Add":" slow → \"5\""},{"Add":" rapid → \"6\""},{"Add":""},{"Add":"command : Command → String"},{"Add":"command c = escape (encode c)"},{"Add":""},{"Add":"withCommand : Command → String → String"},{"Add":"withCommand c txt = concat (command c ∷ txt ∷ command reset ∷ [])"},{"Add":""},{"Add":"commands : List Command → String"},{"Add":"commands [] = command reset"},{"Add":"commands cs = escape (intersperse \";\" $ List.map encode cs)"},{"Add":""},{"Add":"withCommands : List Command → String → String"},{"Add":"withCommands cs txt = concat (commands cs ∷ txt ∷ command reset ∷ [])"}]}},{"FileAdd":{"meta":{"basename":"Clock.agda","metadata":0},"context":[{"change":0,"pos":266416}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Measuring time"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module System.Clock where"},{"Add":""},{"Add":"open import Level using (Level; 0ℓ; Lift; lower)"},{"Add":"open import Data.Bool.Base using (if_then_else_)"},{"Add":"open import Data.Fin.Base using (Fin; toℕ)"},{"Add":"open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _+_; _∸_; _^_; _<ᵇ_)"},{"Add":"import Data.Nat.Show as ℕ using (show)"},{"Add":"open import Data.Nat.DivMod using (_/_)"},{"Add":"import Data.Nat.Properties as ℕ"},{"Add":"open import Data.String.Base using (String; _++_; padLeft)"},{"Add":""},{"Add":"open import IO.Base"},{"Add":"open import Function.Base using (_$_; _∘′_)"},{"Add":"open import Foreign.Haskell using (_,_)"},{"Add":""},{"Add":"open import Relation.Nullary.Decidable using (False; fromWitnessFalse; toWitnessFalse)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting the Clock data structure"},{"Add":""},{"Add":"open import System.Clock.Primitive as Prim"},{"Add":" public"},{"Add":" using ( Clock"},{"Add":" -- System-wide monotonic time since an arbitrary point in the past"},{"Add":" ; monotonic"},{"Add":" -- System-wide real time since the Epoch"},{"Add":" ; realTime"},{"Add":" -- Amount of execution time of the current process"},{"Add":" ; processCPUTime"},{"Add":" -- Amount of execution time of the current OS thread"},{"Add":" ; threadCPUTime"},{"Add":" -- A raw hardware version of Monotonic ignoring adjustments"},{"Add":" ; monotonicRaw"},{"Add":" -- Linux-specific clocks"},{"Add":" -- Similar to Monotonic, includes time spent suspended"},{"Add":" ; bootTime"},{"Add":" -- Faster but less precise alternative to Monotonic"},{"Add":" ; monotonicCoarse"},{"Add":" -- Faster but less precise alternative to RealTime"},{"Add":" ; realTimeCoarse"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Defining a more convenient representation"},{"Add":""},{"Add":"record Time : Set where"},{"Add":" constructor mkTime"},{"Add":" field seconds : ℕ"},{"Add":" nanoseconds : ℕ"},{"Add":"open Time public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Reading the clock"},{"Add":""},{"Add":"getTime : Clock → IO Time"},{"Add":"getTime c = do"},{"Add":" (a , b) ← lift (Prim.getTime c)"},{"Add":" pure $ mkTime a b"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Measuring time periods"},{"Add":""},{"Add":"diff : Time → Time → Time"},{"Add":"diff (mkTime ss sns) (mkTime es ens) ="},{"Add":" if ens <ᵇ sns"},{"Add":" then mkTime (es ∸ suc ss) ((1000000000 + ens) ∸ sns)"},{"Add":" else mkTime (es ∸ ss) (ens ∸ sns)"},{"Add":""},{"Add":"record Timed (A : Set a) : Set a where"},{"Add":" constructor mkTimed"},{"Add":" field value : A"},{"Add":" time : Time"},{"Add":""},{"Add":"time : IO A → IO (Timed A)"},{"Add":"time io = do"},{"Add":" start ← lift! $ getTime realTime"},{"Add":" a ← io"},{"Add":" end ← lift! $ getTime realTime"},{"Add":" pure $ mkTimed a $ diff (lower start) (lower end)"},{"Add":""},{"Add":"time′ : IO {0ℓ} A → IO Time"},{"Add":"time′ io = Timed.time <$> time io"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Showing time"},{"Add":""},{"Add":"show : Time → -- Time in seconds and nanoseconds"},{"Add":" Fin 10 → -- Number of decimals to show"},{"Add":" -- (in [0,9] because we are using nanoseconds)"},{"Add":" String"},{"Add":"show (mkTime s ns) prec = secs ++ \"s\" ++ padLeft '0' decimals nsecs where"},{"Add":" decimals = toℕ prec"},{"Add":" secs = ℕ.show s"},{"Add":" prf = ℕ.m^n≢0 10 (9 ∸ decimals)"},{"Add":" nsecs = ℕ.show ((ns / (10 ^ (9 ∸ decimals))) {{prf}})"}]}},{"FileAdd":{"meta":{"basename":"Clock","metadata":576},"context":[{"change":0,"pos":266416}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Primitive.agda","metadata":0},"context":[{"change":0,"pos":317534}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Primitive System.Clock simple bindings to Haskell functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module System.Clock.Primitive where"},{"Add":""},{"Add":"open import Agda.Builtin.Nat using (Nat)"},{"Add":"open import IO.Primitive.Core using (IO)"},{"Add":"open import Foreign.Haskell using (Pair)"},{"Add":""},{"Add":"data Clock : Set where"},{"Add":" monotonic realTime processCPUTime : Clock"},{"Add":" threadCPUTime monotonicRaw bootTime : Clock"},{"Add":" monotonicCoarse realTimeCoarse : Clock"},{"Add":""},{"Add":"{-# COMPILE GHC Clock = data Clock (Monotonic | Realtime | ProcessCPUTime"},{"Add":" | ThreadCPUTime | MonotonicRaw | Boottime"},{"Add":" | MonotonicCoarse | RealtimeCoarse )"},{"Add":"#-}"},{"Add":""},{"Add":"postulate getTime : Clock → IO (Pair Nat Nat)"},{"Add":""},{"Add":"{-# FOREIGN GHC import System.Clock #-}"},{"Add":"{-# FOREIGN GHC import Data.Function #-}"},{"Add":"{-# COMPILE GHC getTime = fmap (\\ (TimeSpec a b) -> ((,) `on` fromIntegral) a b) . getTime #-}"}]}},{"FileAdd":{"meta":{"basename":"Strict.agda","metadata":0},"context":[{"change":0,"pos":62922}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED, please use `Function.Strict` directly."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Strict where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Strict was deprecated in v1.8."},{"Add":"Use `Function.Strict instead (also re-exported by `Function`).\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Function.Strict public"},{"Add":" using"},{"Add":" ( force ; force-≡ ; force′ ; force′-≡"},{"Add":" ; seq ; seq-≡"},{"Add":" )"}]}},{"FileAdd":{"meta":{"basename":"Size.agda","metadata":0},"context":[{"change":0,"pos":62922}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Sizes for Agda's sized types"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --sized-types #-}"},{"Add":""},{"Add":"module Size where"},{"Add":""},{"Add":"open import Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export builtins"},{"Add":""},{"Add":"open import Agda.Builtin.Size public using"},{"Add":" ( SizeUniv -- sort SizeUniv"},{"Add":" ; Size -- : SizeUniv"},{"Add":" ; Size<_ -- : Size → SizeUniv"},{"Add":" ; ↑_ -- : Size → Size"},{"Add":" ; _⊔ˢ_ -- : Size → Size → Size"},{"Add":" ; ∞ -- : Size"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Concept of sized type"},{"Add":""},{"Add":"SizedSet : (ℓ : Level) → Set (suc ℓ)"},{"Add":"SizedSet ℓ = Size → Set ℓ"}]}},{"FileAdd":{"meta":{"basename":"Relation","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Unary.agda","metadata":0},"context":[{"change":0,"pos":320100}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Unary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Unary where"},{"Add":""},{"Add":"open import Data.Empty using (⊥)"},{"Add":"open import Data.Unit.Base using (⊤)"},{"Add":"open import Data.Product.Base using (_×_; _,_; Σ-syntax; ∃; uncurry; swap)"},{"Add":"open import Data.Sum.Base using (_⊎_; [_,_])"},{"Add":"open import Function.Base using (_∘_; _|>_)"},{"Add":"open import Level using (Level; _⊔_; 0ℓ; suc; Lift)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_)"},{"Add":"open import Relation.Nullary as Nullary using (¬_; Dec; True)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c ℓ ℓ₁ ℓ₂ : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":" C : Set c"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"-- Unary relations are known as predicates and `Pred A ℓ` can be viewed"},{"Add":"-- as some property that elements of type A might satisfy."},{"Add":""},{"Add":"-- Consequently `P : Pred A ℓ` can also be seen as a subset of A"},{"Add":"-- containing all the elements of A that satisfy property P. This view"},{"Add":"-- informs much of the notation used below."},{"Add":""},{"Add":"Pred : ∀ {a} → Set a → (ℓ : Level) → Set (a ⊔ suc ℓ)"},{"Add":"Pred A ℓ = A → Set ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Special sets"},{"Add":""},{"Add":"-- The empty set."},{"Add":"-- Explicitly not level polymorphic as this often causes unsolved metas;"},{"Add":"-- see `Relation.Unary.Polymorphic` for a level-polymorphic version."},{"Add":""},{"Add":"∅ : Pred A 0ℓ"},{"Add":"∅ = λ _ → ⊥"},{"Add":""},{"Add":"-- The singleton set."},{"Add":""},{"Add":"｛_｝ : A → Pred A _"},{"Add":"｛ x ｝ = x ≡_"},{"Add":""},{"Add":"-- The universal set."},{"Add":"-- Explicitly not level polymorphic (see comments for `∅` for more details)"},{"Add":""},{"Add":"U : Pred A 0ℓ"},{"Add":"U = λ _ → ⊤"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Membership"},{"Add":""},{"Add":"infix 4 _∈_ _∉_"},{"Add":""},{"Add":"_∈_ : A → Pred A ℓ → Set _"},{"Add":"x ∈ P = P x"},{"Add":""},{"Add":"_∉_ : A → Pred A ℓ → Set _"},{"Add":"x ∉ P = ¬ x ∈ P"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Subset relations"},{"Add":""},{"Add":"infix 4 _⊆_ _⊇_ _⊈_ _⊉_ _⊂_ _⊃_ _⊄_ _⊅_ _≐_ _≐′_"},{"Add":""},{"Add":"_⊆_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊆ Q = ∀ {x} → x ∈ P → x ∈ Q"},{"Add":""},{"Add":"_⊇_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊇ Q = Q ⊆ P"},{"Add":""},{"Add":"_⊈_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊈ Q = ¬ (P ⊆ Q)"},{"Add":""},{"Add":"_⊉_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊉ Q = ¬ (P ⊇ Q)"},{"Add":""},{"Add":"_⊂_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊂ Q = P ⊆ Q × Q ⊈ P"},{"Add":""},{"Add":"_⊃_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊃ Q = Q ⊂ P"},{"Add":""},{"Add":"_⊄_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊄ Q = ¬ (P ⊂ Q)"},{"Add":""},{"Add":"_⊅_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊅ Q = ¬ (P ⊃ Q)"},{"Add":""},{"Add":"_≐_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ≐ Q = (P ⊆ Q) × (Q ⊆ P)"},{"Add":""},{"Add":"-- The following primed variants of _⊆_ can be used when 'x' can't"},{"Add":"-- be inferred from 'x ∈ P'."},{"Add":""},{"Add":"infix 4 _⊆′_ _⊇′_ _⊈′_ _⊉′_ _⊂′_ _⊃′_ _⊄′_ _⊅′_"},{"Add":""},{"Add":"_⊆′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊆′ Q = ∀ x → x ∈ P → x ∈ Q"},{"Add":""},{"Add":"_⊇′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"Q ⊇′ P = P ⊆′ Q"},{"Add":""},{"Add":"_⊈′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊈′ Q = ¬ (P ⊆′ Q)"},{"Add":""},{"Add":"_⊉′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊉′ Q = ¬ (P ⊇′ Q)"},{"Add":""},{"Add":"_⊂′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊂′ Q = P ⊆′ Q × Q ⊈′ P"},{"Add":""},{"Add":"_⊃′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊃′ Q = Q ⊂′ P"},{"Add":""},{"Add":"_⊄′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊄′ Q = ¬ (P ⊂′ Q)"},{"Add":""},{"Add":"_⊅′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ⊅′ Q = ¬ (P ⊃′ Q)"},{"Add":""},{"Add":"_≐′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ≐′ Q = (P ⊆′ Q) × (Q ⊆′ P)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of sets"},{"Add":""},{"Add":"infix 10 Satisfiable Universal IUniversal"},{"Add":""},{"Add":"-- Emptiness - no element satisfies P."},{"Add":""},{"Add":"Empty : Pred A ℓ → Set _"},{"Add":"Empty P = ∀ x → x ∉ P"},{"Add":""},{"Add":"-- Satisfiable - at least one element satisfies P."},{"Add":""},{"Add":"Satisfiable : Pred A ℓ → Set _"},{"Add":"Satisfiable P = ∃ λ x → x ∈ P"},{"Add":""},{"Add":"syntax Satisfiable P = ∃⟨ P ⟩"},{"Add":""},{"Add":"-- Universality - all elements satisfy P."},{"Add":""},{"Add":"Universal : Pred A ℓ → Set _"},{"Add":"Universal P = ∀ x → x ∈ P"},{"Add":""},{"Add":"syntax Universal P = Π[ P ]"},{"Add":""},{"Add":"-- Implicit universality - all elements satisfy P."},{"Add":""},{"Add":"IUniversal : Pred A ℓ → Set _"},{"Add":"IUniversal P = ∀ {x} → x ∈ P"},{"Add":""},{"Add":"syntax IUniversal P = ∀[ P ]"},{"Add":""},{"Add":"-- Irrelevance - any two proofs that an element satifies P are"},{"Add":"-- indistinguishable."},{"Add":""},{"Add":"Irrelevant : Pred A ℓ → Set _"},{"Add":"Irrelevant P = ∀ {x} → Nullary.Irrelevant (P x)"},{"Add":""},{"Add":"-- Recomputability - we can rebuild a relevant proof given an"},{"Add":"-- irrelevant one."},{"Add":""},{"Add":"Recomputable : Pred A ℓ → Set _"},{"Add":"Recomputable P = ∀ {x} → Nullary.Recomputable (P x)"},{"Add":""},{"Add":"-- Stability - instances of P are stable wrt double negation"},{"Add":""},{"Add":"Stable : Pred A ℓ → Set _"},{"Add":"Stable P = ∀ x → Nullary.Stable (P x)"},{"Add":""},{"Add":"-- Weak Decidability"},{"Add":""},{"Add":"WeaklyDecidable : Pred A ℓ → Set _"},{"Add":"WeaklyDecidable P = ∀ x → Nullary.WeaklyDecidable (P x)"},{"Add":""},{"Add":"-- Decidability - it is possible to determine if an arbitrary element"},{"Add":"-- satisfies P."},{"Add":""},{"Add":"Decidable : Pred A ℓ → Set _"},{"Add":"Decidable P = ∀ x → Dec (P x)"},{"Add":""},{"Add":"-- Erasure: A decidable predicate gives rise to another one, more"},{"Add":"-- amenable to η-expansion"},{"Add":""},{"Add":"⌊_⌋ : {P : Pred A ℓ} → Decidable P → Pred A ℓ"},{"Add":"⌊ P? ⌋ a = Lift _ (True (P? a))"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Operations on sets"},{"Add":""},{"Add":"infix 10 ⋃ ⋂"},{"Add":"infixr 9 _⊢_"},{"Add":"infixr 8 _⇒_"},{"Add":"infixr 7 _∩_"},{"Add":"infixr 6 _∪_"},{"Add":"infixr 6 _∖_"},{"Add":"infix 4 _≬_"},{"Add":""},{"Add":"-- Complement."},{"Add":""},{"Add":"∁ : Pred A ℓ → Pred A ℓ"},{"Add":"∁ P = λ x → x ∉ P"},{"Add":""},{"Add":"-- Implication."},{"Add":""},{"Add":"_⇒_ : Pred A ℓ₁ → Pred A ℓ₂ → Pred A _"},{"Add":"P ⇒ Q = λ x → x ∈ P → x ∈ Q"},{"Add":""},{"Add":"-- Union."},{"Add":""},{"Add":"_∪_ : Pred A ℓ₁ → Pred A ℓ₂ → Pred A _"},{"Add":"P ∪ Q = λ x → x ∈ P ⊎ x ∈ Q"},{"Add":""},{"Add":"-- Intersection."},{"Add":""},{"Add":"_∩_ : Pred A ℓ₁ → Pred A ℓ₂ → Pred A _"},{"Add":"P ∩ Q = λ x → x ∈ P × x ∈ Q"},{"Add":""},{"Add":"-- Difference."},{"Add":""},{"Add":"_∖_ : Pred A ℓ₁ → Pred A ℓ₂ → Pred A _"},{"Add":"P ∖ Q = λ x → x ∈ P × x ∉ Q"},{"Add":""},{"Add":"-- Infinitary union."},{"Add":""},{"Add":"⋃ : ∀ {i} (I : Set i) → (I → Pred A ℓ) → Pred A _"},{"Add":"⋃ I P = λ x → Σ[ i ∈ I ] P i x"},{"Add":""},{"Add":"syntax ⋃ I (λ i → P) = ⋃[ i ∶ I ] P"},{"Add":""},{"Add":"-- Infinitary intersection."},{"Add":""},{"Add":"⋂ : ∀ {i} (I : Set i) → (I → Pred A ℓ) → Pred A _"},{"Add":"⋂ I P = λ x → (i : I) → P i x"},{"Add":""},{"Add":"syntax ⋂ I (λ i → P) = ⋂[ i ∶ I ] P"},{"Add":""},{"Add":"-- Positive version of non-disjointness, dual to inclusion."},{"Add":""},{"Add":"_≬_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _"},{"Add":"P ≬ Q = ∃ λ x → x ∈ P × x ∈ Q"},{"Add":""},{"Add":"-- Update."},{"Add":""},{"Add":"_⊢_ : (A → B) → Pred B ℓ → Pred A ℓ"},{"Add":"f ⊢ P = λ x → P (f x)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Predicate combinators"},{"Add":""},{"Add":"-- These differ from the set operations above, as the carrier set of the"},{"Add":"-- resulting predicates are not the same as the carrier set of the"},{"Add":"-- component predicates."},{"Add":""},{"Add":"infixr 2 _⟨×⟩_"},{"Add":"infixr 2 _⟨⊙⟩_"},{"Add":"infixr 1 _⟨⊎⟩_"},{"Add":"infixr 0 _⟨→⟩_"},{"Add":"infixl 9 _⟨·⟩_"},{"Add":"infix 10 _~"},{"Add":"infixr 9 _⟨∘⟩_"},{"Add":"infixr 2 _//_ _\\\\_"},{"Add":""},{"Add":"-- Product."},{"Add":""},{"Add":"_⟨×⟩_ : Pred A ℓ₁ → Pred B ℓ₂ → Pred (A × B) _"},{"Add":"(P ⟨×⟩ Q) (x , y) = x ∈ P × y ∈ Q"},{"Add":""},{"Add":"-- Sum over one element."},{"Add":""},{"Add":"_⟨⊎⟩_ : Pred A ℓ → Pred B ℓ → Pred (A ⊎ B) _"},{"Add":"P ⟨⊎⟩ Q = [ P , Q ]"},{"Add":""},{"Add":"-- Sum over two elements."},{"Add":""},{"Add":"_⟨⊙⟩_ : Pred A ℓ₁ → Pred B ℓ₂ → Pred (A × B) _"},{"Add":"(P ⟨⊙⟩ Q) (x , y) = x ∈ P ⊎ y ∈ Q"},{"Add":""},{"Add":"-- Implication."},{"Add":""},{"Add":"_⟨→⟩_ : Pred A ℓ₁ → Pred B ℓ₂ → Pred (A → B) _"},{"Add":"(P ⟨→⟩ Q) f = P ⊆ Q ∘ f"},{"Add":""},{"Add":"-- Product."},{"Add":""},{"Add":"_⟨·⟩_ : (P : Pred A ℓ₁) (Q : Pred B ℓ₂) →"},{"Add":" (P ⟨×⟩ (P ⟨→⟩ Q)) ⊆ Q ∘ uncurry _|>_"},{"Add":"(P ⟨·⟩ Q) (p , f) = f p"},{"Add":""},{"Add":"-- Converse."},{"Add":""},{"Add":"_~ : Pred (A × B) ℓ → Pred (B × A) ℓ"},{"Add":"P ~ = P ∘ swap"},{"Add":""},{"Add":"-- Composition."},{"Add":""},{"Add":"_⟨∘⟩_ : Pred (A × B) ℓ₁ → Pred (B × C) ℓ₂ → Pred (A × C) _"},{"Add":"(P ⟨∘⟩ Q) (x , z) = ∃ λ y → (x , y) ∈ P × (y , z) ∈ Q"},{"Add":""},{"Add":"-- Post-division."},{"Add":""},{"Add":"_//_ : Pred (A × C) ℓ₁ → Pred (B × C) ℓ₂ → Pred (A × B) _"},{"Add":"(P // Q) (x , y) = Q ∘ (y ,_) ⊆ P ∘ (x ,_)"},{"Add":""},{"Add":"-- Pre-division."},{"Add":""},{"Add":"_\\\\_ : Pred (A × C) ℓ₁ → Pred (A × B) ℓ₂ → Pred (B × C) _"},{"Add":"P \\\\ Q = (P ~ // Q ~) ~"}]}},{"FileAdd":{"meta":{"basename":"Unary","metadata":576},"context":[{"change":0,"pos":320100}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Sized.agda","metadata":0},"context":[{"change":0,"pos":328511}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Indexed unary relations over sized types"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Sized types live in the special sort `SizeUniv` and therefore are no"},{"Add":"-- longer compatible with the ordinary combinators defined in"},{"Add":"-- `Relation.Unary`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --sized-types #-}"},{"Add":""},{"Add":"module Relation.Unary.Sized where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Size"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ ℓ₁ ℓ₂ : Level"},{"Add":""},{"Add":"infixr 8 _⇒_"},{"Add":"_⇒_ : SizedSet ℓ₁ → SizedSet ℓ₂ → SizedSet (ℓ₁ ⊔ ℓ₂)"},{"Add":"F ⇒ G = λ i → F i → G i"},{"Add":""},{"Add":"∀[_] : SizedSet ℓ → Set ℓ"},{"Add":"∀[ F ] = ∀{i} → F i"}]}},{"FileAdd":{"meta":{"basename":"Relation","metadata":576},"context":[{"change":0,"pos":328511}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Binary","metadata":576},"context":[{"change":0,"pos":329305}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Subset.agda","metadata":0},"context":[{"change":0,"pos":329327}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Order properties of the subset relations _⊆_ and _⊂_"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Unary.Relation.Binary.Subset where"},{"Add":""},{"Add":"open import Level using (Level)"},{"Add":"import Relation.Binary.Structures as BinaryStructures"},{"Add":"open import Relation.Binary.Bundles"},{"Add":"open import Relation.Unary"},{"Add":"open import Relation.Unary.Properties"},{"Add":"open import Relation.Unary.Relation.Binary.Equality using (≐-isEquivalence; ≐′-isEquivalence)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"module _ {a : Level} {A : Set a} {ℓ : Level} where"},{"Add":""},{"Add":" open BinaryStructures {A = Pred A ℓ} _≐_"},{"Add":""},{"Add":" ⊆-isPreorder : IsPreorder _⊆_"},{"Add":" ⊆-isPreorder = record"},{"Add":" { isEquivalence = ≐-isEquivalence"},{"Add":" ; reflexive = ⊆-reflexive"},{"Add":" ; trans = ⊆-trans"},{"Add":" }"},{"Add":""},{"Add":" ⊆-isPartialOrder : IsPartialOrder _⊆_"},{"Add":" ⊆-isPartialOrder = record"},{"Add":" { isPreorder = ⊆-isPreorder"},{"Add":" ; antisym = ⊆-antisym"},{"Add":" }"},{"Add":""},{"Add":" ⊂-isStrictPartialOrder : IsStrictPartialOrder _⊂_"},{"Add":" ⊂-isStrictPartialOrder = record"},{"Add":" { isEquivalence = ≐-isEquivalence"},{"Add":" ; irrefl = ⊂-irrefl"},{"Add":" ; trans = ⊂-trans"},{"Add":" ; <-resp-≈ = ⊂-resp-≐"},{"Add":" }"},{"Add":""},{"Add":"module _ {a : Level} {A : Set a} {ℓ : Level} where"},{"Add":""},{"Add":" open BinaryStructures {A = Pred A ℓ} _≐′_"},{"Add":""},{"Add":" ⊆′-isPreorder : IsPreorder _⊆′_"},{"Add":" ⊆′-isPreorder = record"},{"Add":" { isEquivalence = ≐′-isEquivalence"},{"Add":" ; reflexive = ⊆′-reflexive"},{"Add":" ; trans = ⊆′-trans"},{"Add":" }"},{"Add":""},{"Add":" ⊆′-isPartialOrder : IsPartialOrder _⊆′_"},{"Add":" ⊆′-isPartialOrder = record"},{"Add":" { isPreorder = ⊆′-isPreorder"},{"Add":" ; antisym = ⊆′-antisym"},{"Add":" }"},{"Add":""},{"Add":" ⊂′-isStrictPartialOrder : IsStrictPartialOrder _⊂′_"},{"Add":" ⊂′-isStrictPartialOrder = record"},{"Add":" { isEquivalence = ≐′-isEquivalence"},{"Add":" ; irrefl = ⊂′-irrefl"},{"Add":" ; trans = ⊂′-trans"},{"Add":" ; <-resp-≈ = ⊂′-resp-≐′"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles"},{"Add":""},{"Add":"module _ {a : Level} (A : Set a) (ℓ : Level) where"},{"Add":""},{"Add":" ⊆-preorder : Preorder _ _ _"},{"Add":" ⊆-preorder = record"},{"Add":" { isPreorder = ⊆-isPreorder {A = A} {ℓ}"},{"Add":" }"},{"Add":""},{"Add":" ⊆-poset : Poset _ _ _"},{"Add":" ⊆-poset = record"},{"Add":" { isPartialOrder = ⊆-isPartialOrder {A = A} {ℓ}"},{"Add":" }"},{"Add":""},{"Add":" ⊂-strictPartialOrder : StrictPartialOrder _ _ _"},{"Add":" ⊂-strictPartialOrder = record"},{"Add":" { isStrictPartialOrder = ⊂-isStrictPartialOrder {A = A} {ℓ}"},{"Add":" }"},{"Add":""},{"Add":" ⊆′-preorder : Preorder _ _ _"},{"Add":" ⊆′-preorder = record"},{"Add":" { isPreorder = ⊆′-isPreorder {A = A} {ℓ}"},{"Add":" }"},{"Add":""},{"Add":" ⊆′-poset : Poset _ _ _"},{"Add":" ⊆′-poset = record"},{"Add":" { isPartialOrder = ⊆′-isPartialOrder {A = A} {ℓ}"},{"Add":" }"},{"Add":""},{"Add":" ⊂′-strictPartialOrder : StrictPartialOrder _ _ _"},{"Add":" ⊂′-strictPartialOrder = record"},{"Add":" { isStrictPartialOrder = ⊂′-isStrictPartialOrder {A = A} {ℓ}"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Equality.agda","metadata":0},"context":[{"change":0,"pos":329327}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Equality of unary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Unary.Relation.Binary.Equality where"},{"Add":""},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence)"},{"Add":"open import Relation.Unary using (Pred; _≐_; _≐′_)"},{"Add":"open import Relation.Unary.Properties"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a ℓ : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"≐-isEquivalence : IsEquivalence {A = Pred A ℓ} _≐_"},{"Add":"≐-isEquivalence = record"},{"Add":" { refl = ≐-refl"},{"Add":" ; sym = ≐-sym"},{"Add":" ; trans = ≐-trans"},{"Add":" }"},{"Add":""},{"Add":"≐′-isEquivalence : IsEquivalence {A = Pred A ℓ} _≐′_"},{"Add":"≐′-isEquivalence = record"},{"Add":" { refl = ≐′-refl"},{"Add":" ; sym = ≐′-sym"},{"Add":" ; trans = ≐′-trans"},{"Add":" }"},{"Add":""},{"Add":"≐-setoid : ∀ {a} (A : Set a) ℓ → Setoid _ _"},{"Add":"≐-setoid A ℓ = record"},{"Add":" { isEquivalence = ≐-isEquivalence {A = A} {ℓ}"},{"Add":" }"},{"Add":""},{"Add":"≐′-setoid : ∀ {a} (A : Set a) ℓ → Setoid _ _"},{"Add":"≐′-setoid A ℓ = record"},{"Add":" { isEquivalence = ≐′-isEquivalence {A = A} {ℓ}"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Properties.agda","metadata":0},"context":[{"change":0,"pos":328511}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of constructions over unary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Unary.Properties where"},{"Add":""},{"Add":"open import Data.Product.Base as Product using (_×_; _,_; swap; proj₁; zip′)"},{"Add":"open import Data.Sum.Base using (inj₁; inj₂)"},{"Add":"open import Data.Unit.Base using (tt)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core as Binary"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" hiding (Decidable; Universal; Irrelevant; Empty)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (refl)"},{"Add":"open import Relation.Unary"},{"Add":"open import Relation.Nullary.Decidable using (yes; no; _⊎-dec_; _×-dec_; ¬?)"},{"Add":"open import Function.Base using (id; _$_; _∘_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b ℓ ℓ₁ ℓ₂ ℓ₃ : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The empty set"},{"Add":""},{"Add":"∅? : Decidable {A = A} ∅"},{"Add":"∅? _ = no λ()"},{"Add":""},{"Add":"∅-Empty : Empty {A = A} ∅"},{"Add":"∅-Empty x ()"},{"Add":""},{"Add":"∁∅-Universal : Universal {A = A} (∁ ∅)"},{"Add":"∁∅-Universal = λ x x∈∅ → x∈∅"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The universe"},{"Add":""},{"Add":"U? : Decidable {A = A} U"},{"Add":"U? _ = yes tt"},{"Add":""},{"Add":"U-Universal : Universal {A = A} U"},{"Add":"U-Universal = λ _ → _"},{"Add":""},{"Add":"∁U-Empty : Empty {A = A} (∁ U)"},{"Add":"∁U-Empty = λ x x∈∁U → x∈∁U _"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Subset properties"},{"Add":""},{"Add":"∅-⊆ : (P : Pred A ℓ) → ∅ ⊆ P"},{"Add":"∅-⊆ P ()"},{"Add":""},{"Add":"⊆-U : (P : Pred A ℓ) → P ⊆ U"},{"Add":"⊆-U P _ = _"},{"Add":""},{"Add":"⊆-refl : Reflexive {A = Pred A ℓ} _⊆_"},{"Add":"⊆-refl x∈P = x∈P"},{"Add":""},{"Add":"⊆-reflexive : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐_ _⊆_"},{"Add":"⊆-reflexive (P⊆Q , Q⊆P) = P⊆Q"},{"Add":""},{"Add":"⊆-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊆_ _⊆_ _⊆_"},{"Add":"⊆-trans P⊆Q Q⊆R x∈P = Q⊆R (P⊆Q x∈P)"},{"Add":""},{"Add":"⊆-antisym : Antisymmetric {A = Pred A ℓ} _≐_ _⊆_"},{"Add":"⊆-antisym = _,_"},{"Add":""},{"Add":"⊆-min : Min {B = Pred A ℓ} _⊆_ ∅"},{"Add":"⊆-min = ∅-⊆"},{"Add":""},{"Add":"⊆-max : Max {A = Pred A ℓ} _⊆_ U"},{"Add":"⊆-max = ⊆-U"},{"Add":""},{"Add":"⊂⇒⊆ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂_ _⊆_"},{"Add":"⊂⇒⊆ = proj₁"},{"Add":""},{"Add":"⊂-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊂_ _⊂_ _⊂_"},{"Add":"⊂-trans (P⊆Q , Q⊈P) (Q⊆R , R⊈Q) = (λ x∈P → Q⊆R (P⊆Q x∈P)) , (λ R⊆P → R⊈Q (λ x∈R → P⊆Q (R⊆P x∈R)))"},{"Add":""},{"Add":"⊂-⊆-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊂_ _⊆_ _⊂_"},{"Add":"⊂-⊆-trans (P⊆Q , Q⊈P) Q⊆R = (λ x∈P → Q⊆R (P⊆Q x∈P)) , (λ R⊆P → Q⊈P (λ x∈Q → R⊆P (Q⊆R x∈Q)))"},{"Add":""},{"Add":"⊆-⊂-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊆_ _⊂_ _⊂_"},{"Add":"⊆-⊂-trans P⊆Q (Q⊆R , R⊈Q) = (λ x∈P → Q⊆R (P⊆Q x∈P)) , (λ R⊆P → R⊈Q (λ R⊆Q → P⊆Q (R⊆P R⊆Q)))"},{"Add":""},{"Add":"⊂-respʳ-≐ : _Respectsʳ_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂_ _≐_"},{"Add":"⊂-respʳ-≐ (Q⊆R , _) P⊂Q = ⊂-⊆-trans P⊂Q Q⊆R"},{"Add":""},{"Add":"⊂-respˡ-≐ : _Respectsˡ_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂_ _≐_"},{"Add":"⊂-respˡ-≐ (_ , R⊆Q) P⊂Q = ⊆-⊂-trans R⊆Q P⊂Q"},{"Add":""},{"Add":"⊂-resp-≐ : _Respects₂_ {A = Pred A ℓ} _⊂_ _≐_"},{"Add":"⊂-resp-≐ = ⊂-respʳ-≐ , ⊂-respˡ-≐"},{"Add":""},{"Add":"⊂-irrefl : Irreflexive {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐_ _⊂_"},{"Add":"⊂-irrefl (_ , Q⊆P) (_ , Q⊈P) = Q⊈P Q⊆P"},{"Add":""},{"Add":"⊂-antisym : Antisymmetric {A = Pred A ℓ} _≐_ _⊂_"},{"Add":"⊂-antisym (P⊆Q , _) (Q⊆P , _) = ⊆-antisym P⊆Q Q⊆P"},{"Add":""},{"Add":"⊂-asym : Asymmetric {A = Pred A ℓ} _⊂_"},{"Add":"⊂-asym (_ , Q⊈P) = Q⊈P ∘ proj₁"},{"Add":""},{"Add":"∅-⊆′ : (P : Pred A ℓ) → ∅ ⊆′ P"},{"Add":"∅-⊆′ _ _ = λ ()"},{"Add":""},{"Add":"⊆′-U : (P : Pred A ℓ) → P ⊆′ U"},{"Add":"⊆′-U _ _ _ = _"},{"Add":""},{"Add":"⊆′-refl : Reflexive {A = Pred A ℓ} _⊆′_"},{"Add":"⊆′-refl x x∈P = x∈P"},{"Add":""},{"Add":"⊆′-reflexive : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐′_ _⊆′_"},{"Add":"⊆′-reflexive (P⊆Q , Q⊆P) = P⊆Q"},{"Add":""},{"Add":"⊆′-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊆′_ _⊆′_ _⊆′_"},{"Add":"⊆′-trans P⊆Q Q⊆R x x∈P = Q⊆R x (P⊆Q x x∈P)"},{"Add":""},{"Add":"⊆′-antisym : Antisymmetric {A = Pred A ℓ} _≐′_ _⊆′_"},{"Add":"⊆′-antisym = _,_"},{"Add":""},{"Add":"⊆′-min : Min {B = Pred A ℓ} _⊆′_ ∅"},{"Add":"⊆′-min = ∅-⊆′"},{"Add":""},{"Add":"⊆′-max : Max {A = Pred A ℓ} _⊆′_ U"},{"Add":"⊆′-max = ⊆′-U"},{"Add":""},{"Add":"⊂′⇒⊆′ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂′_ _⊆′_"},{"Add":"⊂′⇒⊆′ = proj₁"},{"Add":""},{"Add":"⊂′-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊂′_ _⊂′_ _⊂′_"},{"Add":"⊂′-trans (P⊆Q , Q⊈P) (Q⊆R , R⊈Q) = ⊆′-trans P⊆Q Q⊆R , λ R⊆P → R⊈Q (⊆′-trans R⊆P P⊆Q)"},{"Add":""},{"Add":"⊂′-⊆′-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊂′_ _⊆′_ _⊂′_"},{"Add":"⊂′-⊆′-trans (P⊆Q , Q⊈P) Q⊆R = ⊆′-trans P⊆Q Q⊆R , λ R⊆P → Q⊈P (⊆′-trans Q⊆R R⊆P)"},{"Add":""},{"Add":"⊆′-⊂′-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊆′_ _⊂′_ _⊂′_"},{"Add":"⊆′-⊂′-trans P⊆Q (Q⊆R , R⊈Q) = ⊆′-trans P⊆Q Q⊆R , λ R⊆P → R⊈Q (⊆′-trans R⊆P P⊆Q)"},{"Add":""},{"Add":"⊂′-respʳ-≐′ : _Respectsʳ_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂′_ _≐′_"},{"Add":"⊂′-respʳ-≐′ (Q⊆R , _) P⊂Q = ⊂′-⊆′-trans P⊂Q Q⊆R"},{"Add":""},{"Add":"⊂′-respˡ-≐′ : _Respectsˡ_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂′_ _≐′_"},{"Add":"⊂′-respˡ-≐′ (_ , R⊆Q) P⊂Q = ⊆′-⊂′-trans R⊆Q P⊂Q"},{"Add":""},{"Add":"⊂′-resp-≐′ : _Respects₂_ {A = Pred A ℓ₁} _⊂′_ _≐′_"},{"Add":"⊂′-resp-≐′ = ⊂′-respʳ-≐′ , ⊂′-respˡ-≐′"},{"Add":""},{"Add":"⊂′-irrefl : Irreflexive {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐′_ _⊂′_"},{"Add":"⊂′-irrefl (_ , Q⊆P) (_ , Q⊈P) = Q⊈P Q⊆P"},{"Add":""},{"Add":"⊂′-antisym : Antisymmetric {A = Pred A ℓ} _≐′_ _⊂′_"},{"Add":"⊂′-antisym (P⊆Q , _) (Q⊆P , _) = ⊆′-antisym P⊆Q Q⊆P"},{"Add":""},{"Add":"⊆⇒⊆′ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊆_ _⊆′_"},{"Add":"⊆⇒⊆′ P⊆Q _ x∈P = P⊆Q x∈P"},{"Add":""},{"Add":"⊆′⇒⊆ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊆′_ _⊆_"},{"Add":"⊆′⇒⊆ P⊆Q x∈P = P⊆Q _ x∈P"},{"Add":""},{"Add":"⊂⇒⊂′ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂_ _⊂′_"},{"Add":"⊂⇒⊂′ = Product.map ⊆⇒⊆′ (_∘ ⊆′⇒⊆)"},{"Add":""},{"Add":"⊂′⇒⊂ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂′_ _⊂_"},{"Add":"⊂′⇒⊂ = Product.map ⊆′⇒⊆ (_∘ ⊆⇒⊆′)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Equality properties"},{"Add":""},{"Add":"≐-refl : Reflexive {A = Pred A ℓ} _≐_"},{"Add":"≐-refl = id , id"},{"Add":""},{"Add":"≐-sym : Sym {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐_ _≐_"},{"Add":"≐-sym = swap"},{"Add":""},{"Add":"≐-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _≐_ _≐_ _≐_"},{"Add":"≐-trans = zip′ (λ P⊆Q Q⊆R x∈P → Q⊆R (P⊆Q x∈P)) (λ Q⊆P R⊆Q x∈R → Q⊆P (R⊆Q x∈R))"},{"Add":""},{"Add":"≐′-refl : Reflexive {A = Pred A ℓ} _≐′_"},{"Add":"≐′-refl = (λ _ → id) , (λ _ → id)"},{"Add":""},{"Add":"≐′-sym : Sym {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐′_ _≐′_"},{"Add":"≐′-sym = swap"},{"Add":""},{"Add":"≐′-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _≐′_ _≐′_ _≐′_"},{"Add":"≐′-trans = zip′ (λ P⊆Q Q⊆R x x∈P → Q⊆R x (P⊆Q x x∈P)) λ Q⊆P R⊆Q x x∈R → Q⊆P x (R⊆Q x x∈R)"},{"Add":""},{"Add":"≐⇒≐′ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐_ _≐′_"},{"Add":"≐⇒≐′ = Product.map ⊆⇒⊆′ ⊆⇒⊆′"},{"Add":""},{"Add":"≐′⇒≐ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐′_ _≐_"},{"Add":"≐′⇒≐ = Product.map ⊆′⇒⊆ ⊆′⇒⊆"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Decidability properties"},{"Add":""},{"Add":"∁? : {P : Pred A ℓ} → Decidable P → Decidable (∁ P)"},{"Add":"∁? P? x = ¬? (P? x)"},{"Add":""},{"Add":"infix 2 _×?_ _⊙?_"},{"Add":"infix 10 _~?"},{"Add":"infixr 1 _⊎?_"},{"Add":"infixr 7 _∩?_"},{"Add":"infixr 6 _∪?_"},{"Add":""},{"Add":"_∪?_ : {P : Pred A ℓ₁} {Q : Pred A ℓ₂} →"},{"Add":" Decidable P → Decidable Q → Decidable (P ∪ Q)"},{"Add":"_∪?_ P? Q? x = (P? x) ⊎-dec (Q? x)"},{"Add":""},{"Add":"_∩?_ : {P : Pred A ℓ₁} {Q : Pred A ℓ₂} →"},{"Add":" Decidable P → Decidable Q → Decidable (P ∩ Q)"},{"Add":"_∩?_ P? Q? x = (P? x) ×-dec (Q? x)"},{"Add":""},{"Add":"_×?_ : {P : Pred A ℓ₁} {Q : Pred B ℓ₂} →"},{"Add":" Decidable P → Decidable Q → Decidable (P ⟨×⟩ Q)"},{"Add":"_×?_ P? Q? (a , b) = (P? a) ×-dec (Q? b)"},{"Add":""},{"Add":"_⊙?_ : {P : Pred A ℓ₁} {Q : Pred B ℓ₂} →"},{"Add":" Decidable P → Decidable Q → Decidable (P ⟨⊙⟩ Q)"},{"Add":"_⊙?_ P? Q? (a , b) = (P? a) ⊎-dec (Q? b)"},{"Add":""},{"Add":"_⊎?_ : {P : Pred A ℓ} {Q : Pred B ℓ} →"},{"Add":" Decidable P → Decidable Q → Decidable (P ⟨⊎⟩ Q)"},{"Add":"_⊎?_ P? Q? (inj₁ a) = P? a"},{"Add":"_⊎?_ P? Q? (inj₂ b) = Q? b"},{"Add":""},{"Add":"_~? : {P : Pred (A × B) ℓ} → Decidable P → Decidable (P ~)"},{"Add":"_~? P? = P? ∘ swap"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Irrelevant properties"},{"Add":""},{"Add":"U-irrelevant : Irrelevant {A = A} U"},{"Add":"U-irrelevant a b = refl"},{"Add":""},{"Add":"∁-irrelevant : (P : Pred A ℓ) → Irrelevant (∁ P)"},{"Add":"∁-irrelevant P a b = refl"}]}},{"FileAdd":{"meta":{"basename":"PredicateTransformer.agda","metadata":0},"context":[{"change":0,"pos":328511}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Predicate transformers"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Unary.PredicateTransformer where"},{"Add":""},{"Add":"open import Data.Product.Base using (∃)"},{"Add":"open import Function.Base using (_∘_)"},{"Add":"open import Level hiding (_⊔_)"},{"Add":"open import Relation.Nullary"},{"Add":"open import Relation.Unary"},{"Add":"open import Relation.Binary.Core using (REL)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c i ℓ ℓ₁ ℓ₂ ℓ₃ : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":" C : Set c"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Heterogeneous and homogeneous predicate transformers"},{"Add":""},{"Add":"PT : Set a → Set b → (ℓ₁ ℓ₂ : Level) → Set _"},{"Add":"PT A B ℓ₁ ℓ₂ = Pred A ℓ₁ → Pred B ℓ₂"},{"Add":""},{"Add":"Pt : Set a → (ℓ : Level) → Set _"},{"Add":"Pt A ℓ = PT A A ℓ ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Composition and identity"},{"Add":""},{"Add":"infixr 9 _⍮_"},{"Add":""},{"Add":"_⍮_ : PT B C ℓ₂ ℓ₃ → PT A B ℓ₁ ℓ₂ → PT A C ℓ₁ _"},{"Add":"S ⍮ T = S ∘ T"},{"Add":""},{"Add":"skip : PT A A ℓ ℓ"},{"Add":"skip P = P"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Operations on predicates extend pointwise to predicate transformers"},{"Add":""},{"Add":"-- The bottom and the top of the predicate transformer lattice."},{"Add":""},{"Add":"abort : PT A B 0ℓ 0ℓ"},{"Add":"abort = λ _ → ∅"},{"Add":""},{"Add":"magic : PT A B 0ℓ 0ℓ"},{"Add":"magic = λ _ → U"},{"Add":""},{"Add":"-- Negation."},{"Add":""},{"Add":"infix 8 ∼_"},{"Add":""},{"Add":"∼_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂"},{"Add":"∼ T = ∁ ∘ T"},{"Add":""},{"Add":"-- Refinement."},{"Add":""},{"Add":"infix 4 _⊑_ _⊒_ _⊑′_ _⊒′_"},{"Add":""},{"Add":"_⊑_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → Set _"},{"Add":"S ⊑ T = ∀ {X} → S X ⊆ T X"},{"Add":""},{"Add":"_⊑′_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → Set _"},{"Add":"S ⊑′ T = ∀ X → S X ⊆ T X"},{"Add":""},{"Add":"_⊒_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → Set _"},{"Add":"T ⊒ S = T ⊑ S"},{"Add":""},{"Add":"_⊒′_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → Set _"},{"Add":"T ⊒′ S = S ⊑′ T"},{"Add":""},{"Add":"-- The dual of refinement."},{"Add":""},{"Add":"infix 4 _⋢_"},{"Add":""},{"Add":"_⋢_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → Set _"},{"Add":"S ⋢ T = ∃ λ X → S X ≬ T X"},{"Add":""},{"Add":"-- Union."},{"Add":""},{"Add":"infixl 6 _⊓_"},{"Add":""},{"Add":"_⊓_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂"},{"Add":"S ⊓ T = λ X → S X ∪ T X"},{"Add":""},{"Add":"-- Intersection."},{"Add":""},{"Add":"infixl 7 _⊔_"},{"Add":""},{"Add":"_⊔_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂"},{"Add":"S ⊔ T = λ X → S X ∩ T X"},{"Add":""},{"Add":"-- Implication."},{"Add":""},{"Add":"infixl 8 _⇛_"},{"Add":""},{"Add":"_⇛_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂"},{"Add":"S ⇛ T = λ X → S X ⇒ T X"},{"Add":""},{"Add":"-- Infinitary union and intersection."},{"Add":""},{"Add":"infix 9 ⨆ ⨅"},{"Add":""},{"Add":"⨆ : ∀ (I : Set i) → (I → PT A B ℓ₁ ℓ₂) → PT A B ℓ₁ _"},{"Add":"⨆ I T = λ X → ⋃[ i ∶ I ] T i X"},{"Add":""},{"Add":"syntax ⨆ I (λ i → T) = ⨆[ i ∶ I ] T"},{"Add":""},{"Add":"⨅ : ∀ (I : Set i) → (I → PT A B ℓ₁ ℓ₂) → PT A B ℓ₁ _"},{"Add":"⨅ I T = λ X → ⋂[ i ∶ I ] T i X"},{"Add":""},{"Add":"syntax ⨅ I (λ i → T) = ⨅[ i ∶ I ] T"},{"Add":""},{"Add":"-- Angelic and demonic update."},{"Add":""},{"Add":"⟨_⟩ : REL A B ℓ → PT B A ℓ _"},{"Add":"⟨ R ⟩ P = λ x → R x ≬ P"},{"Add":""},{"Add":"[_] : REL A B ℓ → PT B A ℓ _"},{"Add":"[ R ] P = λ x → R x ⊆ P"}]}},{"FileAdd":{"meta":{"basename":"Polymorphic.agda","metadata":0},"context":[{"change":0,"pos":328511}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Polymorphic versions of standard definitions in Relation.Unary"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Unary.Polymorphic where"},{"Add":""},{"Add":"open import Data.Empty.Polymorphic using (⊥)"},{"Add":"open import Data.Unit.Polymorphic using (⊤)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Unary using (Pred)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a ℓ : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Special sets"},{"Add":""},{"Add":"-- The empty set."},{"Add":""},{"Add":"∅ : Pred A ℓ"},{"Add":"∅ = λ _ → ⊥"},{"Add":""},{"Add":"-- The universal set."},{"Add":""},{"Add":"U : Pred A ℓ"},{"Add":"U = λ _ → ⊤"}]}},{"FileAdd":{"meta":{"basename":"Polymorphic","metadata":576},"context":[{"change":0,"pos":328511}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Properties.agda","metadata":0},"context":[{"change":0,"pos":346766}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of polymorphic versions of standard definitions in"},{"Add":"-- Relation.Unary"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Unary.Polymorphic.Properties where"},{"Add":""},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Definitions hiding (Decidable; Universal; Empty)"},{"Add":"open import Relation.Nullary.Decidable using (yes; no)"},{"Add":"open import Relation.Unary hiding (∅; U)"},{"Add":"open import Relation.Unary.Polymorphic"},{"Add":"open import Data.Unit.Polymorphic.Base using (tt)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a ℓ ℓ₁ ℓ₂ : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The empty set"},{"Add":""},{"Add":"∅? : Decidable {A = A} {ℓ} ∅"},{"Add":"∅? _ = no λ()"},{"Add":""},{"Add":"∅-Empty : Empty {A = A} {ℓ} ∅"},{"Add":"∅-Empty _ ()"},{"Add":""},{"Add":"∁∅-Universal : Universal {A = A} {ℓ} (∁ ∅)"},{"Add":"∁∅-Universal _ ()"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The universe"},{"Add":""},{"Add":"U? : Decidable {A = A} {ℓ} U"},{"Add":"U? _ = yes tt"},{"Add":""},{"Add":"U-Universal : Universal {A = A} {ℓ} U"},{"Add":"U-Universal _ = _"},{"Add":""},{"Add":"∁U-Empty : Empty {A = A} {ℓ} (∁ U)"},{"Add":"∁U-Empty _ x∈∁U = x∈∁U _"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Subset properties"},{"Add":""},{"Add":"∅-⊆ : (P : Pred A ℓ₁) → ∅ {ℓ = ℓ₂} ⊆ P"},{"Add":"∅-⊆ _ ()"},{"Add":""},{"Add":"⊆-U : (P : Pred A ℓ₁) → P ⊆ U {ℓ = ℓ₂}"},{"Add":"⊆-U _ _ = _"},{"Add":""},{"Add":"⊆-min : Min {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊆_ ∅"},{"Add":"⊆-min = ∅-⊆"},{"Add":""},{"Add":"⊆-max : Max {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊆_ U"},{"Add":"⊆-max = ⊆-U"},{"Add":""},{"Add":"∅-⊆′ : (P : Pred A ℓ₁) → ∅ {ℓ = ℓ₂} ⊆′ P"},{"Add":"∅-⊆′ _ _ = λ ()"},{"Add":""},{"Add":"⊆′-U : (P : Pred A ℓ₁) → P ⊆′ U {ℓ = ℓ₂}"},{"Add":"⊆′-U _ _ _ = _"},{"Add":""},{"Add":"⊆′-min : Min {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊆′_ ∅"},{"Add":"⊆′-min = ∅-⊆′"},{"Add":""},{"Add":"⊆′-max : Max {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊆′_ U"},{"Add":"⊆′-max = ⊆′-U"}]}},{"FileAdd":{"meta":{"basename":"Indexed.agda","metadata":0},"context":[{"change":0,"pos":328511}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Indexed unary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Unary.Indexed where"},{"Add":""},{"Add":"open import Data.Product.Base using (∃; _×_)"},{"Add":"open import Level"},{"Add":"open import Relation.Nullary.Negation using (¬_)"},{"Add":""},{"Add":"IPred : ∀ {i a} {I : Set i} → (I → Set a) → (ℓ : Level) → Set _"},{"Add":"IPred A ℓ = ∀ {i} → A i → Set ℓ"},{"Add":""},{"Add":"module _ {i a} {I : Set i} {A : I → Set a} where"},{"Add":""},{"Add":" infix 4 _∈_ _∉_"},{"Add":""},{"Add":" _∈_ : ∀ {ℓ} → (∀ i → A i) → IPred A ℓ → Set _"},{"Add":" x ∈ P = ∀ i → P (x i)"},{"Add":""},{"Add":" _∉_ : ∀ {ℓ} → (∀ i → A i) → IPred A ℓ → Set _"},{"Add":" t ∉ P = ¬ (t ∈ P)"}]}},{"FileAdd":{"meta":{"basename":"Consequences.agda","metadata":0},"context":[{"change":0,"pos":328511}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Some properties imply others"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Unary.Consequences where"},{"Add":""},{"Add":"open import Relation.Unary"},{"Add":"open import Relation.Nullary using (recompute)"},{"Add":""},{"Add":"dec⇒recomputable : {a ℓ : _} {A : Set a} {P : Pred A ℓ} → Decidable P → Recomputable P"},{"Add":"dec⇒recomputable P-dec = recompute (P-dec _)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- DEPRECATED NAMES"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Please use the new names as continuing support for the old names is"},{"Add":"-- not guaranteed."},{"Add":""},{"Add":"-- Version 2.0"},{"Add":""},{"Add":"dec⟶recomputable = dec⇒recomputable"},{"Add":"{-# WARNING_ON_USAGE dec⟶recomputable"},{"Add":"\"Warning: dec⟶recomputable was deprecated in v2.0."},{"Add":"Please use dec⇒recomputable instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Closure","metadata":576},"context":[{"change":0,"pos":328511}],"contents":[]}},{"FileAdd":{"meta":{"basename":"StrictPartialOrder.agda","metadata":0},"context":[{"change":0,"pos":350696}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Closures of a unary relation with respect to a strict partial order"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (StrictPartialOrder)"},{"Add":""},{"Add":"module Relation.Unary.Closure.StrictPartialOrder"},{"Add":" {a r e} (P : StrictPartialOrder a e r) where"},{"Add":""},{"Add":"open StrictPartialOrder P renaming (_<_ to _∼_)"},{"Add":"open import Relation.Unary using (Pred)"},{"Add":""},{"Add":"-- Specialising the results proven generically in `Base`."},{"Add":"import Relation.Unary.Closure.Base _∼_ as Base"},{"Add":"open Base public"},{"Add":" using (□; ◇; Closed; curry; uncurry)"},{"Add":" hiding (module □; module ◇)"},{"Add":""},{"Add":"module □ {t} {T : Pred Carrier t} where"},{"Add":""},{"Add":" reindex : ∀ {x y} → x ∼ y → □ T x → □ T y"},{"Add":" reindex = Base.□.reindex trans"},{"Add":""},{"Add":" duplicate : ∀ {x} → □ T x → □ (□ T) x"},{"Add":" duplicate = Base.□.duplicate trans"},{"Add":""},{"Add":" closed : ∀ {t} {T : Pred Carrier t} → Closed (□ T)"},{"Add":" closed = Base.□.closed trans"},{"Add":""},{"Add":" open Base.□ public using (map)"},{"Add":""},{"Add":"module ◇ {t} {T : Pred Carrier t} where"},{"Add":""},{"Add":" reindex : ∀ {x y} → x ∼ y → ◇ T x → ◇ T y"},{"Add":" reindex = Base.◇.reindex trans"},{"Add":""},{"Add":" join : ∀ {x} → ◇ (◇ T) x → ◇ T x"},{"Add":" join = Base.◇.join trans"},{"Add":""},{"Add":" closed : Closed (◇ T)"},{"Add":" closed = Base.◇.closed trans"},{"Add":""},{"Add":" open Base.◇ public using (map; run)"}]}},{"FileAdd":{"meta":{"basename":"Preorder.agda","metadata":0},"context":[{"change":0,"pos":350696}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Closure of a unary relation with respect to a preorder"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (Preorder)"},{"Add":""},{"Add":"module Relation.Unary.Closure.Preorder {a r e} (P : Preorder a e r) where"},{"Add":""},{"Add":"open Preorder P"},{"Add":"open import Relation.Unary using (Pred)"},{"Add":""},{"Add":"-- Specialising the results proven generically in `Base`."},{"Add":"import Relation.Unary.Closure.Base _∼_ as Base"},{"Add":"open Base public"},{"Add":" using (□; ◇; Closed; curry; uncurry)"},{"Add":" hiding (module □; module ◇)"},{"Add":""},{"Add":"module □ {t} {T : Pred Carrier t} where"},{"Add":""},{"Add":" reindex : ∀ {x y} → x ∼ y → □ T x → □ T y"},{"Add":" reindex = Base.□.reindex trans"},{"Add":""},{"Add":" extract : ∀ {x} → □ T x → T x"},{"Add":" extract = Base.□.extract refl"},{"Add":""},{"Add":" duplicate : ∀ {x} → □ T x → □ (□ T) x"},{"Add":" duplicate = Base.□.duplicate trans"},{"Add":""},{"Add":" closed : ∀ {t} {T : Pred Carrier t} → Closed (□ T)"},{"Add":" closed = Base.□.closed trans"},{"Add":""},{"Add":" open Base.□ public using (map)"},{"Add":""},{"Add":"module ◇ {t} {T : Pred Carrier t} where"},{"Add":""},{"Add":" reindex : ∀ {x y} → x ∼ y → ◇ T x → ◇ T y"},{"Add":" reindex = Base.◇.reindex trans"},{"Add":""},{"Add":" pure : ∀ {x} → T x → ◇ T x"},{"Add":" pure = Base.◇.pure refl"},{"Add":""},{"Add":" join : ∀ {x} → ◇ (◇ T) x → ◇ T x"},{"Add":" join = Base.◇.join trans"},{"Add":""},{"Add":" closed : Closed (◇ T)"},{"Add":" closed = Base.◇.closed trans"},{"Add":""},{"Add":" open Base.◇ public using (map; run)"}]}},{"FileAdd":{"meta":{"basename":"Base.agda","metadata":0},"context":[{"change":0,"pos":350696}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Closures of a unary relation with respect to a binary one."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Definitions using (Transitive; Reflexive)"},{"Add":""},{"Add":"module Relation.Unary.Closure.Base {a b} {A : Set a} (R : Rel A b) where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Data.Product.Base using (Σ-syntax; _×_; _,_; -,_)"},{"Add":"open import Function.Base using (flip)"},{"Add":"open import Relation.Unary using (Pred)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definitions"},{"Add":""},{"Add":"-- Box"},{"Add":""},{"Add":"-- We start with the definition of □ (\"box\") which is named after the"},{"Add":"-- box modality in modal logic. `□ T x` states that all the elements one"},{"Add":"-- step away from `x` with respect to the relation R satisfy `T`."},{"Add":""},{"Add":"□ : ∀ {t} → Pred A t → Pred A (a ⊔ b ⊔ t)"},{"Add":"□ T x = ∀ {y} → R x y → T y"},{"Add":""},{"Add":"-- Use cases of □ include:"},{"Add":"-- * The definition of the accessibility predicate corresponding to R:"},{"Add":"-- data Acc (x : A) : Set (a ⊔ b) where"},{"Add":"-- step : □ Acc x → Acc x"},{"Add":""},{"Add":"-- * The characterization of stability under weakening: picking R to be"},{"Add":"-- `Data.List.Relation.Sublist.Inductive`, `∀ {Γ} → Tm Γ → □ T Γ`"},{"Add":"-- corresponds to the fact that we have a notion of weakening for `Tm`."},{"Add":""},{"Add":"-- Diamond"},{"Add":""},{"Add":"-- We then have the definition of ◇ (\"diamond\") which is named after the"},{"Add":"-- diamond modality in modal logic. In modal logic, `◇ T x` states that"},{"Add":"-- there exists an element one step away from x with respect to the"},{"Add":"-- relation R that satisfies T. It is worth noting that the modal logic"},{"Add":"-- metaphor breaks down here: this only is a closure operator if the"},{"Add":"-- step we take is *backwards* with respect to R."},{"Add":""},{"Add":"◇ : ∀ {t} → Pred A t → Pred A (a ⊔ b ⊔ t)"},{"Add":"◇ T x = Σ[ support ∈ A ] (R support x × T support)"},{"Add":""},{"Add":"-- Use cases of ◇ include:"},{"Add":"-- * The characterization of strengthening: picking R to be"},{"Add":"-- `Data.List.Relation.Sublist.Inductive`, `∀ {Γ} → Tm Γ → ◇ Tm Γ`"},{"Add":"-- is the type of a function strengthening a term to its support:"},{"Add":"-- all the unused variables are discarded early on by the `related`"},{"Add":"-- proof."},{"Add":"-- Cf. Conor McBride's \"Everybody's got to be somewhere\" for a more"},{"Add":"-- detailed treatment of such an example."},{"Add":""},{"Add":"-- Closed"},{"Add":""},{"Add":"-- Whenever we have a value in one context, we can get one in any"},{"Add":"-- related context."},{"Add":""},{"Add":"record Closed {t} (T : Pred A t) : Set (a ⊔ b ⊔ t) where"},{"Add":" field next : ∀ {x} → T x → □ T x"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Basic functions relating □ and ◇"},{"Add":""},{"Add":"module _ {t p} {T : Pred A t} {P : Pred A p} where"},{"Add":""},{"Add":" curry : (∀ {x} → ◇ T x → P x) → (∀ {x} → T x → □ P x)"},{"Add":" curry f tx x∼y = f (-, x∼y , tx)"},{"Add":""},{"Add":" uncurry : (∀ {x} → T x → □ P x) → (∀ {x} → ◇ T x → P x)"},{"Add":" uncurry f (_ , y∼x , ty) = f ty y∼x"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"module □ {t} {T : Pred A t} where"},{"Add":""},{"Add":" reindex : Transitive R → ∀ {x y} → R x y → □ T x → □ T y"},{"Add":" reindex trans x∼y □Tx y∼z = □Tx (trans x∼y y∼z)"},{"Add":""},{"Add":" -- Provided that R is reflexive and Transitive, □ is a comonad"},{"Add":" map : ∀ {u} {U : Pred A u} {x} → (∀ {x} → T x → U x) → □ T x → □ U x"},{"Add":" map f □Tx x~y = f (□Tx x~y)"},{"Add":""},{"Add":" extract : Reflexive R → ∀ {x} → □ T x → T x"},{"Add":" extract refl □Tx = □Tx refl"},{"Add":""},{"Add":" duplicate : Transitive R → ∀ {x} → □ T x → □ (□ T) x"},{"Add":" duplicate trans □Tx x∼y y∼z = □Tx (trans x∼y y∼z)"},{"Add":""},{"Add":" -- Provided that R is transitive, □ is a closure operator"},{"Add":" -- i.e. for any `T`, `□ T` is closed."},{"Add":" closed : Transitive R → Closed (□ T)"},{"Add":" closed trans = record { next = duplicate trans }"},{"Add":""},{"Add":"module ◇ {t} {T : Pred A t} where"},{"Add":""},{"Add":" reindex : Transitive R → ∀ {x y} → R x y → ◇ T x → ◇ T y"},{"Add":" reindex trans x∼y (z , z∼x , tz) = z , trans z∼x x∼y , tz"},{"Add":""},{"Add":" -- Provided that R is reflexive and Transitive, ◇ is a monad"},{"Add":" map : ∀ {u} {U : Pred A u} {x} → (∀ {x} → T x → U x) → ◇ T x → ◇ U x"},{"Add":" map f (y , y∼x , ty) = y , y∼x , f ty"},{"Add":""},{"Add":" pure : Reflexive R → ∀ {x} → T x → ◇ T x"},{"Add":" pure refl tx = -, refl , tx"},{"Add":""},{"Add":" join : Transitive R → ∀ {x} → ◇ (◇ T) x → ◇ T x"},{"Add":" join trans (_ , y∼x , _ , z∼y , tz) = _ , trans z∼y y∼x , tz"},{"Add":""},{"Add":" -- Provided that R is transitive, ◇ is a closure operator"},{"Add":" -- i.e. for any `T`, `◇ T` is closed."},{"Add":" closed : Transitive R → Closed (◇ T)"},{"Add":" closed trans = record { next = λ ◇Tx x∼y → reindex trans x∼y ◇Tx }"},{"Add":""},{"Add":" run : Closed T → ∀ {x} → ◇ T x → T x"},{"Add":" run closed (_ , y∼x , ty) = Closed.next closed ty y∼x"}]}},{"FileAdd":{"meta":{"basename":"Algebra.agda","metadata":0},"context":[{"change":0,"pos":328511}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Algebraic properties of constructions over unary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Unary.Algebra where"},{"Add":""},{"Add":"open import Algebra.Bundles"},{"Add":"import Algebra.Definitions as AlgebraicDefinitions"},{"Add":"open import Algebra.Lattice.Bundles"},{"Add":"import Algebra.Lattice.Structures as AlgebraicLatticeStructures"},{"Add":"import Algebra.Structures as AlgebraicStructures"},{"Add":"open import Data.Empty.Polymorphic using (⊥-elim)"},{"Add":"open import Data.Product.Base as Product using (_,_; proj₁; proj₂; <_,_>; curry; uncurry)"},{"Add":"open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_])"},{"Add":"open import Data.Unit.Polymorphic using (tt)"},{"Add":"open import Function.Base using (id; const; _∘_)"},{"Add":"open import Level"},{"Add":"open import Relation.Unary hiding (∅; U)"},{"Add":"open import Relation.Unary.Polymorphic using (∅; U)"},{"Add":"open import Relation.Unary.Relation.Binary.Equality using (≐-isEquivalence)"},{"Add":""},{"Add":"module _ {a ℓ : Level} {A : Set a} where"},{"Add":""},{"Add":" open AlgebraicDefinitions {A = Pred A ℓ} _≐_"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of _∩_"},{"Add":""},{"Add":" ∩-cong : Congruent₂ _∩_"},{"Add":" ∩-cong (P⊆Q , Q⊆P) (R⊆S , S⊆R) = Product.map P⊆Q R⊆S , Product.map Q⊆P S⊆R"},{"Add":""},{"Add":" ∩-comm : Commutative _∩_"},{"Add":" ∩-comm _ _ = Product.swap , Product.swap"},{"Add":""},{"Add":" ∩-assoc : Associative _∩_"},{"Add":" ∩-assoc _ _ _ = Product.assocʳ′ , Product.assocˡ′"},{"Add":""},{"Add":" ∩-idem : Idempotent _∩_"},{"Add":" ∩-idem _ = proj₁ , < id , id >"},{"Add":""},{"Add":" ∩-identityˡ : LeftIdentity U _∩_"},{"Add":" ∩-identityˡ _ = proj₂ , < const tt , id >"},{"Add":""},{"Add":" ∩-identityʳ : RightIdentity U _∩_"},{"Add":" ∩-identityʳ _ = proj₁ , < id , const tt >"},{"Add":""},{"Add":" ∩-identity : Identity U _∩_"},{"Add":" ∩-identity = ∩-identityˡ , ∩-identityʳ"},{"Add":""},{"Add":" ∩-zeroˡ : LeftZero ∅ _∩_"},{"Add":" ∩-zeroˡ _ = proj₁ , ⊥-elim"},{"Add":""},{"Add":" ∩-zeroʳ : RightZero ∅ _∩_"},{"Add":" ∩-zeroʳ _ = proj₂ , ⊥-elim"},{"Add":""},{"Add":" ∩-zero : Zero ∅ _∩_"},{"Add":" ∩-zero = ∩-zeroˡ , ∩-zeroʳ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of _∪_"},{"Add":""},{"Add":" ∪-cong : Congruent₂ _∪_"},{"Add":" ∪-cong (P⊆Q , Q⊆P) (R⊆S , S⊆R) = Sum.map P⊆Q R⊆S , Sum.map Q⊆P S⊆R"},{"Add":""},{"Add":" ∪-comm : Commutative _∪_"},{"Add":" ∪-comm _ _ = Sum.swap , Sum.swap"},{"Add":""},{"Add":" ∪-assoc : Associative _∪_"},{"Add":" ∪-assoc _ _ _ = Sum.assocʳ , Sum.assocˡ"},{"Add":""},{"Add":" ∪-idem : Idempotent _∪_"},{"Add":" ∪-idem _ = [ id , id ] , inj₁"},{"Add":""},{"Add":" ∪-identityˡ : LeftIdentity ∅ _∪_"},{"Add":" ∪-identityˡ _ = [ ⊥-elim , id ] , inj₂"},{"Add":""},{"Add":" ∪-identityʳ : RightIdentity ∅ _∪_"},{"Add":" ∪-identityʳ _ = [ id , ⊥-elim ] , inj₁"},{"Add":""},{"Add":" ∪-identity : Identity ∅ _∪_"},{"Add":" ∪-identity = ∪-identityˡ , ∪-identityʳ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of _∩_ and _∪_"},{"Add":""},{"Add":" ∩-distribˡ-∪ : _∩_ DistributesOverˡ _∪_"},{"Add":" ∩-distribˡ-∪ _ _ _ ="},{"Add":" ( uncurry (λ x∈P → [ inj₁ ∘ (x∈P ,_) , inj₂ ∘ (x∈P ,_) ])"},{"Add":" , [ Product.map₂ inj₁ , Product.map₂ inj₂ ]"},{"Add":" )"},{"Add":""},{"Add":" ∩-distribʳ-∪ : _∩_ DistributesOverʳ _∪_"},{"Add":" ∩-distribʳ-∪ _ _ _ ="},{"Add":" ( uncurry [ curry inj₁ , curry inj₂ ]"},{"Add":" , [ Product.map₁ inj₁ , Product.map₁ inj₂ ]"},{"Add":" )"},{"Add":""},{"Add":" ∩-distrib-∪ : _∩_ DistributesOver _∪_"},{"Add":" ∩-distrib-∪ = ∩-distribˡ-∪ , ∩-distribʳ-∪"},{"Add":""},{"Add":" ∪-distribˡ-∩ : _∪_ DistributesOverˡ _∩_"},{"Add":" ∪-distribˡ-∩ _ _ _ ="},{"Add":" ( [ < inj₁ , inj₁ > , Product.map inj₂ inj₂ ]"},{"Add":" , uncurry [ const ∘ inj₁ , (λ x∈Q → [ inj₁ , inj₂ ∘ (x∈Q ,_) ]) ]"},{"Add":" )"},{"Add":""},{"Add":" ∪-distribʳ-∩ : _∪_ DistributesOverʳ _∩_"},{"Add":" ∪-distribʳ-∩ _ _ _ ="},{"Add":" ( [ Product.map inj₁ inj₁ , < inj₂ , inj₂ > ]"},{"Add":" , uncurry [ (λ x∈Q → [ inj₁ ∘ (x∈Q ,_) , inj₂ ]) , const ∘ inj₂ ]"},{"Add":" )"},{"Add":""},{"Add":" ∪-distrib-∩ : _∪_ DistributesOver _∩_"},{"Add":" ∪-distrib-∩ = ∪-distribˡ-∩ , ∪-distribʳ-∩"},{"Add":""},{"Add":" ∩-abs-∪ : _∩_ Absorbs _∪_"},{"Add":" ∩-abs-∪ _ _ = proj₁ , < id , inj₁ >"},{"Add":""},{"Add":" ∪-abs-∩ : _∪_ Absorbs _∩_"},{"Add":" ∪-abs-∩ _ _ = [ id , proj₁ ] , inj₁"},{"Add":""},{"Add":" ∪-∩-absorptive : Absorptive _∪_ _∩_"},{"Add":" ∪-∩-absorptive = ∪-abs-∩ , ∩-abs-∪"},{"Add":""},{"Add":" ∩-∪-absorptive : Absorptive _∩_ _∪_"},{"Add":" ∩-∪-absorptive = ∩-abs-∪ , ∪-abs-∩"},{"Add":""},{"Add":"module _ {a : Level} (A : Set a) (ℓ : Level) where"},{"Add":""},{"Add":" open AlgebraicStructures {A = Pred A ℓ} _≐_"},{"Add":" open AlgebraicLatticeStructures {A = Pred A ℓ} _≐_"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Algebraic structures of _∩_"},{"Add":""},{"Add":" ∩-isMagma : IsMagma _∩_"},{"Add":" ∩-isMagma = record"},{"Add":" { isEquivalence = ≐-isEquivalence"},{"Add":" ; ∙-cong = ∩-cong"},{"Add":" }"},{"Add":""},{"Add":" ∩-isSemigroup : IsSemigroup _∩_"},{"Add":" ∩-isSemigroup = record"},{"Add":" { isMagma = ∩-isMagma"},{"Add":" ; assoc = ∩-assoc"},{"Add":" }"},{"Add":""},{"Add":" ∩-isBand : IsBand _∩_"},{"Add":" ∩-isBand = record"},{"Add":" { isSemigroup = ∩-isSemigroup"},{"Add":" ; idem = ∩-idem"},{"Add":" }"},{"Add":""},{"Add":" ∩-isSemilattice : IsSemilattice _∩_"},{"Add":" ∩-isSemilattice = record"},{"Add":" { isBand = ∩-isBand"},{"Add":" ; comm = ∩-comm"},{"Add":" }"},{"Add":""},{"Add":" ∩-isMonoid : IsMonoid _∩_ U"},{"Add":" ∩-isMonoid = record"},{"Add":" { isSemigroup = ∩-isSemigroup"},{"Add":" ; identity = ∩-identity"},{"Add":" }"},{"Add":""},{"Add":" ∩-isCommutativeMonoid : IsCommutativeMonoid _∩_ U"},{"Add":" ∩-isCommutativeMonoid = record"},{"Add":" { isMonoid = ∩-isMonoid"},{"Add":" ; comm = ∩-comm"},{"Add":" }"},{"Add":""},{"Add":" ∩-isIdempotentCommutativeMonoid : IsIdempotentCommutativeMonoid _∩_ U"},{"Add":" ∩-isIdempotentCommutativeMonoid = record"},{"Add":" { isCommutativeMonoid = ∩-isCommutativeMonoid"},{"Add":" ; idem = ∩-idem"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Algebraic structures of _∪_"},{"Add":""},{"Add":" ∪-isMagma : IsMagma _∪_"},{"Add":" ∪-isMagma = record"},{"Add":" { isEquivalence = ≐-isEquivalence"},{"Add":" ; ∙-cong = ∪-cong"},{"Add":" }"},{"Add":""},{"Add":" ∪-isSemigroup : IsSemigroup _∪_"},{"Add":" ∪-isSemigroup = record"},{"Add":" { isMagma = ∪-isMagma"},{"Add":" ; assoc = ∪-assoc"},{"Add":" }"},{"Add":""},{"Add":" ∪-isBand : IsBand _∪_"},{"Add":" ∪-isBand = record"},{"Add":" { isSemigroup = ∪-isSemigroup"},{"Add":" ; idem = ∪-idem"},{"Add":" }"},{"Add":""},{"Add":" ∪-isSemilattice : IsSemilattice _∪_"},{"Add":" ∪-isSemilattice = record"},{"Add":" { isBand = ∪-isBand"},{"Add":" ; comm = ∪-comm"},{"Add":" }"},{"Add":""},{"Add":" ∪-isMonoid : IsMonoid _∪_ ∅"},{"Add":" ∪-isMonoid = record"},{"Add":" { isSemigroup = ∪-isSemigroup"},{"Add":" ; identity = ∪-identity"},{"Add":" }"},{"Add":""},{"Add":" ∪-isCommutativeMonoid : IsCommutativeMonoid _∪_ ∅"},{"Add":" ∪-isCommutativeMonoid = record"},{"Add":" { isMonoid = ∪-isMonoid"},{"Add":" ; comm = ∪-comm"},{"Add":" }"},{"Add":""},{"Add":" ∪-isIdempotentCommutativeMonoid : IsIdempotentCommutativeMonoid _∪_ ∅"},{"Add":" ∪-isIdempotentCommutativeMonoid = record"},{"Add":" { isCommutativeMonoid = ∪-isCommutativeMonoid"},{"Add":" ; idem = ∪-idem"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Algebraic structures of _∩_ and _∪_"},{"Add":""},{"Add":" ∪-∩-isSemiringWithoutAnnihilatingZero : IsSemiringWithoutAnnihilatingZero _∪_ _∩_ ∅ U"},{"Add":" ∪-∩-isSemiringWithoutAnnihilatingZero = record"},{"Add":" { +-isCommutativeMonoid = ∪-isCommutativeMonoid"},{"Add":" ; *-cong = ∩-cong"},{"Add":" ; *-assoc = ∩-assoc"},{"Add":" ; *-identity = ∩-identity"},{"Add":" ; distrib = ∩-distrib-∪"},{"Add":" }"},{"Add":""},{"Add":" ∪-∩-isSemiring : IsSemiring _∪_ _∩_ ∅ U"},{"Add":" ∪-∩-isSemiring = record"},{"Add":" { isSemiringWithoutAnnihilatingZero = ∪-∩-isSemiringWithoutAnnihilatingZero"},{"Add":" ; zero = ∩-zero"},{"Add":" }"},{"Add":""},{"Add":" ∪-∩-isCommutativeSemiring : IsCommutativeSemiring _∪_ _∩_ ∅ U"},{"Add":" ∪-∩-isCommutativeSemiring = record"},{"Add":" { isSemiring = ∪-∩-isSemiring"},{"Add":" ; *-comm = ∩-comm"},{"Add":" }"},{"Add":""},{"Add":" ∪-∩-isLattice : IsLattice _∪_ _∩_"},{"Add":" ∪-∩-isLattice = record"},{"Add":" { isEquivalence = ≐-isEquivalence"},{"Add":" ; ∨-comm = ∪-comm"},{"Add":" ; ∨-assoc = ∪-assoc"},{"Add":" ; ∨-cong = ∪-cong"},{"Add":" ; ∧-comm = ∩-comm"},{"Add":" ; ∧-assoc = ∩-assoc"},{"Add":" ; ∧-cong = ∩-cong"},{"Add":" ; absorptive = ∪-∩-absorptive"},{"Add":" }"},{"Add":""},{"Add":" ∪-∩-isDistributiveLattice : IsDistributiveLattice _∪_ _∩_"},{"Add":" ∪-∩-isDistributiveLattice = record"},{"Add":" { isLattice = ∪-∩-isLattice"},{"Add":" ; ∨-distrib-∧ = ∪-distrib-∩"},{"Add":" ; ∧-distrib-∨ = ∩-distrib-∪"},{"Add":" }"},{"Add":""},{"Add":" ∩-∪-isSemiringWithoutAnnihilatingZero : IsSemiringWithoutAnnihilatingZero _∩_ _∪_ U ∅"},{"Add":" ∩-∪-isSemiringWithoutAnnihilatingZero = record"},{"Add":" { +-isCommutativeMonoid = ∩-isCommutativeMonoid"},{"Add":" ; *-cong = ∪-cong"},{"Add":" ; *-assoc = ∪-assoc"},{"Add":" ; *-identity = ∪-identity"},{"Add":" ; distrib = ∪-distrib-∩"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Algebraic bundles of _∩_"},{"Add":""},{"Add":" ∩-magma : Magma _ _"},{"Add":" ∩-magma = record"},{"Add":" { isMagma = ∩-isMagma"},{"Add":" }"},{"Add":""},{"Add":" ∩-semigroup : Semigroup _ _"},{"Add":" ∩-semigroup = record"},{"Add":" { isSemigroup = ∩-isSemigroup"},{"Add":" }"},{"Add":""},{"Add":" ∩-band : Band _ _"},{"Add":" ∩-band = record"},{"Add":" { isBand = ∩-isBand"},{"Add":" }"},{"Add":""},{"Add":" ∩-semilattice : Semilattice _ _"},{"Add":" ∩-semilattice = record"},{"Add":" { isSemilattice = ∩-isSemilattice"},{"Add":" }"},{"Add":""},{"Add":" ∩-monoid : Monoid _ _"},{"Add":" ∩-monoid = record"},{"Add":" { isMonoid = ∩-isMonoid"},{"Add":" }"},{"Add":""},{"Add":" ∩-commutativeMonoid : CommutativeMonoid _ _"},{"Add":" ∩-commutativeMonoid = record"},{"Add":" { isCommutativeMonoid = ∩-isCommutativeMonoid"},{"Add":" }"},{"Add":""},{"Add":" ∩-idempotentCommutativeMonoid : IdempotentCommutativeMonoid _ _"},{"Add":" ∩-idempotentCommutativeMonoid = record"},{"Add":" { isIdempotentCommutativeMonoid = ∩-isIdempotentCommutativeMonoid"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Algebraic bundles of _∪_"},{"Add":""},{"Add":" ∪-magma : Magma _ _"},{"Add":" ∪-magma = record"},{"Add":" { isMagma = ∪-isMagma"},{"Add":" }"},{"Add":""},{"Add":" ∪-semigroup : Semigroup _ _"},{"Add":" ∪-semigroup = record"},{"Add":" { isSemigroup = ∪-isSemigroup"},{"Add":" }"},{"Add":""},{"Add":" ∪-band : Band _ _"},{"Add":" ∪-band = record"},{"Add":" { isBand = ∪-isBand"},{"Add":" }"},{"Add":""},{"Add":" ∪-semilattice : Semilattice _ _"},{"Add":" ∪-semilattice = record"},{"Add":" { isSemilattice = ∪-isSemilattice"},{"Add":" }"},{"Add":""},{"Add":" ∪-monoid : Monoid _ _"},{"Add":" ∪-monoid = record"},{"Add":" { isMonoid = ∪-isMonoid"},{"Add":" }"},{"Add":""},{"Add":" ∪-commutativeMonoid : CommutativeMonoid _ _"},{"Add":" ∪-commutativeMonoid = record"},{"Add":" { isCommutativeMonoid = ∪-isCommutativeMonoid"},{"Add":" }"},{"Add":""},{"Add":" ∪-idempotentCommutativeMonoid : IdempotentCommutativeMonoid _ _"},{"Add":" ∪-idempotentCommutativeMonoid = record"},{"Add":" { isIdempotentCommutativeMonoid = ∪-isIdempotentCommutativeMonoid"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Algebraic bundles of _∩_ and _∪_"},{"Add":""},{"Add":" ∪-∩-semiringWithoutAnnihilatingZero : SemiringWithoutAnnihilatingZero _ _"},{"Add":" ∪-∩-semiringWithoutAnnihilatingZero = record"},{"Add":" { isSemiringWithoutAnnihilatingZero = ∪-∩-isSemiringWithoutAnnihilatingZero"},{"Add":" }"},{"Add":""},{"Add":" ∪-∩-semiring : Semiring _ _"},{"Add":" ∪-∩-semiring = record"},{"Add":" { isSemiring = ∪-∩-isSemiring"},{"Add":" }"},{"Add":""},{"Add":" ∪-∩-commutativeSemiring : CommutativeSemiring _ _"},{"Add":" ∪-∩-commutativeSemiring = record"},{"Add":" { isCommutativeSemiring = ∪-∩-isCommutativeSemiring"},{"Add":" }"},{"Add":""},{"Add":" ∪-∩-lattice : Lattice _ _"},{"Add":" ∪-∩-lattice = record"},{"Add":" { isLattice = ∪-∩-isLattice"},{"Add":" }"},{"Add":""},{"Add":" ∪-∩-distributiveLattice : DistributiveLattice _ _"},{"Add":" ∪-∩-distributiveLattice = record"},{"Add":" { isDistributiveLattice = ∪-∩-isDistributiveLattice"},{"Add":" }"},{"Add":""},{"Add":" ∩-∪-semiringWithoutAnnihilatingZero : SemiringWithoutAnnihilatingZero _ _"},{"Add":" ∩-∪-semiringWithoutAnnihilatingZero = record"},{"Add":" { isSemiringWithoutAnnihilatingZero = ∩-∪-isSemiringWithoutAnnihilatingZero"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Nullary.agda","metadata":0},"context":[{"change":0,"pos":320100}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Operations on nullary relations (like negation and decidability)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Some operations on/properties of nullary relations, i.e. sets."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary where"},{"Add":""},{"Add":"open import Agda.Builtin.Equality using (_≡_)"},{"Add":"open import Agda.Builtin.Maybe using (Maybe)"},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" p : Level"},{"Add":" P : Set p"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exports"},{"Add":""},{"Add":"open import Relation.Nullary.Recomputable public using (Recomputable)"},{"Add":"open import Relation.Nullary.Negation.Core public"},{"Add":"open import Relation.Nullary.Reflects public hiding (recompute; recompute-constant)"},{"Add":"open import Relation.Nullary.Decidable.Core public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Irrelevant types"},{"Add":""},{"Add":"Irrelevant : Set p → Set p"},{"Add":"Irrelevant P = ∀ (p₁ p₂ : P) → p₁ ≡ p₂"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Weak decidability"},{"Add":"-- `nothing` is 'don't know'/'give up'; `just` is `yes`/`definitely`"},{"Add":""},{"Add":"WeaklyDecidable : Set p → Set p"},{"Add":"WeaklyDecidable = Maybe"}]}},{"FileAdd":{"meta":{"basename":"Nullary","metadata":576},"context":[{"change":0,"pos":320100}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Universe.agda","metadata":0},"context":[{"change":0,"pos":371196}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- A universe of proposition functors, along with some properties"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Universe where"},{"Add":""},{"Add":"open import Relation.Nullary"},{"Add":"open import Relation.Nullary.Negation"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"import Relation.Binary.Construct.Always as Always"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as PropEq"},{"Add":"import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial"},{"Add":" as Trivial"},{"Add":"open import Data.Sum.Base as Sum hiding (map)"},{"Add":"open import Data.Sum.Relation.Binary.Pointwise using (_⊎ₛ_; inj₁; inj₂)"},{"Add":"open import Data.Product.Base as Product hiding (map)"},{"Add":"open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)"},{"Add":"open import Function.Base using (_∘_; id)"},{"Add":"open import Function.Indexed.Relation.Binary.Equality using (≡-setoid)"},{"Add":"open import Data.Empty"},{"Add":"open import Effect.Applicative"},{"Add":"open import Effect.Monad"},{"Add":"open import Level"},{"Add":""},{"Add":"infix 5 ¬¬_"},{"Add":"infixr 4 _⇒_"},{"Add":"infixr 3 _∧_"},{"Add":"infixr 2 _∨_"},{"Add":"infix 1 ⟨_⟩_≈_"},{"Add":""},{"Add":"-- The universe."},{"Add":""},{"Add":"data PropF p : Set (suc p) where"},{"Add":" Id : PropF p"},{"Add":" K : (P : Set p) → PropF p"},{"Add":" _∨_ : (F₁ F₂ : PropF p) → PropF p"},{"Add":" _∧_ : (F₁ F₂ : PropF p) → PropF p"},{"Add":" _⇒_ : (P₁ : Set p) (F₂ : PropF p) → PropF p"},{"Add":" ¬¬_ : (F : PropF p) → PropF p"},{"Add":""},{"Add":"-- Equalities for universe inhabitants."},{"Add":""},{"Add":"mutual"},{"Add":""},{"Add":" setoid : ∀ {p} → PropF p → Set p → Setoid p p"},{"Add":" setoid Id P = PropEq.setoid P"},{"Add":" setoid (K P) _ = PropEq.setoid P"},{"Add":" setoid (F₁ ∨ F₂) P = (setoid F₁ P) ⊎ₛ (setoid F₂ P)"},{"Add":" setoid (F₁ ∧ F₂) P = (setoid F₁ P) ×ₛ (setoid F₂ P)"},{"Add":" setoid (P₁ ⇒ F₂) P = ≡-setoid P₁"},{"Add":" (Trivial.indexedSetoid (setoid F₂ P))"},{"Add":" setoid (¬¬ F) P = Always.setoid (¬ ¬ ⟦ F ⟧ P) _"},{"Add":""},{"Add":" ⟦_⟧ : ∀ {p} → PropF p → (Set p → Set p)"},{"Add":" ⟦ F ⟧ P = Setoid.Carrier (setoid F P)"},{"Add":""},{"Add":"⟨_⟩_≈_ : ∀ {p} (F : PropF p) {P : Set p} → Rel (⟦ F ⟧ P) p"},{"Add":"⟨_⟩_≈_ F = Setoid._≈_ (setoid F _)"},{"Add":""},{"Add":"-- ⟦ F ⟧ is functorial."},{"Add":""},{"Add":"map : ∀ {p} (F : PropF p) {P Q} → (P → Q) → ⟦ F ⟧ P → ⟦ F ⟧ Q"},{"Add":"map Id f p = f p"},{"Add":"map (K P) f p = p"},{"Add":"map (F₁ ∨ F₂) f FP = Sum.map (map F₁ f) (map F₂ f) FP"},{"Add":"map (F₁ ∧ F₂) f FP = Product.map (map F₁ f) (map F₂ f) FP"},{"Add":"map (P₁ ⇒ F₂) f FP = map F₂ f ∘ FP"},{"Add":"map (¬¬ F) f FP = ¬¬-map (map F f) FP"},{"Add":""},{"Add":"map-id : ∀ {p} (F : PropF p) {P} → ⟨ ⟦ F ⟧ P ⇒ F ⟩ map F id ≈ id"},{"Add":"map-id Id x = refl"},{"Add":"map-id (K P) x = refl"},{"Add":"map-id (F₁ ∨ F₂) (inj₁ x) = inj₁ (map-id F₁ x)"},{"Add":"map-id (F₁ ∨ F₂) (inj₂ y) = inj₂ (map-id F₂ y)"},{"Add":"map-id (F₁ ∧ F₂) (x , y) = (map-id F₁ x , map-id F₂ y)"},{"Add":"map-id (P₁ ⇒ F₂) f = λ x → map-id F₂ (f x)"},{"Add":"map-id (¬¬ F) ¬¬x = _"},{"Add":""},{"Add":"map-∘ : ∀ {p} (F : PropF p) {P Q R} (f : Q → R) (g : P → Q) →"},{"Add":" ⟨ ⟦ F ⟧ P ⇒ F ⟩ map F f ∘ map F g ≈ map F (f ∘ g)"},{"Add":"map-∘ Id f g x = refl"},{"Add":"map-∘ (K P) f g x = refl"},{"Add":"map-∘ (F₁ ∨ F₂) f g (inj₁ x) = inj₁ (map-∘ F₁ f g x)"},{"Add":"map-∘ (F₁ ∨ F₂) f g (inj₂ y) = inj₂ (map-∘ F₂ f g y)"},{"Add":"map-∘ (F₁ ∧ F₂) f g x = (map-∘ F₁ f g (proj₁ x) ,"},{"Add":" map-∘ F₂ f g (proj₂ x))"},{"Add":"map-∘ (P₁ ⇒ F₂) f g h = λ x → map-∘ F₂ f g (h x)"},{"Add":"map-∘ (¬¬ F) f g x = _"},{"Add":""},{"Add":"-- A variant of sequence can be implemented for ⟦ F ⟧."},{"Add":""},{"Add":"sequence : ∀ {p AF} → RawApplicative AF →"},{"Add":" (AF (Lift p ⊥) → ⊥) →"},{"Add":" ({A B : Set p} → (A → AF B) → AF (A → B)) →"},{"Add":" ∀ F {P} → ⟦ F ⟧ (AF P) → AF (⟦ F ⟧ P)"},{"Add":"sequence {AF = AF} A extract-⊥ sequence-⇒ = helper"},{"Add":" where"},{"Add":" open RawApplicative A"},{"Add":""},{"Add":" helper : ∀ F {P} → ⟦ F ⟧ (AF P) → AF (⟦ F ⟧ P)"},{"Add":" helper Id x = x"},{"Add":" helper (K P) x = pure x"},{"Add":" helper (F₁ ∨ F₂) (inj₁ x) = inj₁ <$> helper F₁ x"},{"Add":" helper (F₁ ∨ F₂) (inj₂ y) = inj₂ <$> helper F₂ y"},{"Add":" helper (F₁ ∧ F₂) (x , y) = _,_ <$> helper F₁ x ⊛ helper F₂ y"},{"Add":" helper (P₁ ⇒ F₂) f = sequence-⇒ (helper F₂ ∘ f)"},{"Add":" helper (¬¬ F) x ="},{"Add":" pure (λ ¬FP → x (λ fp → extract-⊥ (lift ∘ ¬FP <$> helper F fp)))"},{"Add":""},{"Add":"-- Some lemmas about double negation."},{"Add":""},{"Add":"private"},{"Add":" open module M {a} = RawMonad (¬¬-Monad {a = a})"},{"Add":""},{"Add":"¬¬-pull : ∀ {p} (F : PropF p) {P} →"},{"Add":" ⟦ F ⟧ (¬ ¬ P) → ¬ ¬ ⟦ F ⟧ P"},{"Add":"¬¬-pull = sequence rawApplicative"},{"Add":" (λ f → f lower)"},{"Add":" (λ f g → g (λ x → ⊥-elim (f x (λ y → g (λ _ → y)))))"},{"Add":""},{"Add":"¬¬-remove : ∀ {p} (F : PropF p) {P} →"},{"Add":" ¬ ¬ ⟦ F ⟧ (¬ ¬ P) → ¬ ¬ ⟦ F ⟧ P"},{"Add":"¬¬-remove F = negated-stable ∘ ¬¬-pull (¬¬ F)"}]}},{"FileAdd":{"meta":{"basename":"Sum.agda","metadata":0},"context":[{"change":0,"pos":371196}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Sum where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Relation.Nullary.Sum was deprecated in v2.0."},{"Add":"Use `Relation.Nullary.Decidable` or `Relation.Nullary` instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Relation.Nullary.Negation.Core public using (_¬-⊎_)"},{"Add":"open import Relation.Nullary.Reflects public using (_⊎-reflects_)"},{"Add":"open import Relation.Nullary.Decidable.Core public using (_⊎-dec_)"}]}},{"FileAdd":{"meta":{"basename":"Reflects.agda","metadata":0},"context":[{"change":0,"pos":371196}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of the `Reflects` construct"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Reflects where"},{"Add":""},{"Add":"open import Agda.Builtin.Equality"},{"Add":""},{"Add":"open import Data.Bool.Base"},{"Add":"open import Data.Sum.Base using (_⊎_; inj₁; inj₂)"},{"Add":"open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)"},{"Add":"open import Level using (Level)"},{"Add":"open import Function.Base using (_$_; _∘_; const; id)"},{"Add":"open import Relation.Nullary.Negation.Core"},{"Add":" using (¬_; contradiction-irr; contradiction; _¬-⊎_)"},{"Add":"open import Relation.Nullary.Recomputable as Recomputable using (Recomputable)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a : Level"},{"Add":" A B : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- `Reflects` idiom."},{"Add":""},{"Add":"-- The truth value of A is reflected by a boolean value."},{"Add":"-- `Reflects A b` is equivalent to `if b then A else ¬ A`."},{"Add":""},{"Add":"data Reflects (A : Set a) : Bool → Set a where"},{"Add":" ofʸ : ( a : A) → Reflects A true"},{"Add":" ofⁿ : (¬a : ¬ A) → Reflects A false"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Constructors and destructors"},{"Add":""},{"Add":"-- These lemmas are intended to be used mostly when `b` is a value, so"},{"Add":"-- that the `if` expressions have already been evaluated away."},{"Add":"-- In this case, `of` works like the relevant constructor (`ofⁿ` or"},{"Add":"-- `ofʸ`), and `invert` strips off the constructor to just give either"},{"Add":"-- the proof of `A` or the proof of `¬ A`."},{"Add":""},{"Add":"of : ∀ {b} → if b then A else ¬ A → Reflects A b"},{"Add":"of {b = false} ¬a = ofⁿ ¬a"},{"Add":"of {b = true } a = ofʸ a"},{"Add":""},{"Add":"invert : ∀ {b} → Reflects A b → if b then A else ¬ A"},{"Add":"invert (ofʸ a) = a"},{"Add":"invert (ofⁿ ¬a) = ¬a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- recompute"},{"Add":""},{"Add":"-- Given an irrelevant proof of a reflected type, a proof can"},{"Add":"-- be recomputed and subsequently used in relevant contexts."},{"Add":""},{"Add":"recompute : ∀ {b} → Reflects A b → Recomputable A"},{"Add":"recompute (ofʸ a) _ = a"},{"Add":"recompute (ofⁿ ¬a) a = contradiction-irr a ¬a"},{"Add":""},{"Add":"recompute-constant : ∀ {b} (r : Reflects A b) (p q : A) →"},{"Add":" recompute r p ≡ recompute r q"},{"Add":"recompute-constant = Recomputable.recompute-constant ∘ recompute"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Interaction with negation, product, sums etc."},{"Add":""},{"Add":"infixr 1 _⊎-reflects_"},{"Add":"infixr 2 _×-reflects_ _→-reflects_"},{"Add":""},{"Add":"T-reflects : ∀ b → Reflects (T b) b"},{"Add":"T-reflects true = of _"},{"Add":"T-reflects false = of id"},{"Add":""},{"Add":"-- If we can decide A, then we can decide its negation."},{"Add":"¬-reflects : ∀ {b} → Reflects A b → Reflects (¬ A) (not b)"},{"Add":"¬-reflects (ofʸ a) = of (_$ a)"},{"Add":"¬-reflects (ofⁿ ¬a) = of ¬a"},{"Add":""},{"Add":"-- If we can decide A and Q then we can decide their product"},{"Add":"_×-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →"},{"Add":" Reflects (A × B) (a ∧ b)"},{"Add":"ofʸ a ×-reflects ofʸ b = of (a , b)"},{"Add":"ofʸ a ×-reflects ofⁿ ¬b = of (¬b ∘ proj₂)"},{"Add":"ofⁿ ¬a ×-reflects _ = of (¬a ∘ proj₁)"},{"Add":""},{"Add":"_⊎-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →"},{"Add":" Reflects (A ⊎ B) (a ∨ b)"},{"Add":"ofʸ a ⊎-reflects _ = of (inj₁ a)"},{"Add":"ofⁿ ¬a ⊎-reflects ofʸ b = of (inj₂ b)"},{"Add":"ofⁿ ¬a ⊎-reflects ofⁿ ¬b = of (¬a ¬-⊎ ¬b)"},{"Add":""},{"Add":"_→-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →"},{"Add":" Reflects (A → B) (not a ∨ b)"},{"Add":"ofʸ a →-reflects ofʸ b = of (const b)"},{"Add":"ofʸ a →-reflects ofⁿ ¬b = of (¬b ∘ (_$ a))"},{"Add":"ofⁿ ¬a →-reflects _ = of (λ a → contradiction a ¬a)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Other lemmas"},{"Add":""},{"Add":"fromEquivalence : ∀ {b} → (T b → A) → (A → T b) → Reflects A b"},{"Add":"fromEquivalence {b = true} sound complete = of (sound _)"},{"Add":"fromEquivalence {b = false} sound complete = of complete"},{"Add":""},{"Add":"-- `Reflects` is deterministic."},{"Add":"det : ∀ {b b′} → Reflects A b → Reflects A b′ → b ≡ b′"},{"Add":"det (ofʸ a) (ofʸ _) = refl"},{"Add":"det (ofʸ a) (ofⁿ ¬a) = contradiction a ¬a"},{"Add":"det (ofⁿ ¬a) (ofʸ a) = contradiction a ¬a"},{"Add":"det (ofⁿ ¬a) (ofⁿ _) = refl"},{"Add":""},{"Add":"T-reflects-elim : ∀ {a b} → Reflects (T a) b → b ≡ a"},{"Add":"T-reflects-elim {a} r = det r (T-reflects a)"}]}},{"FileAdd":{"meta":{"basename":"Recomputable.agda","metadata":0},"context":[{"change":0,"pos":371196}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Recomputable types and their algebra as Harrop formulas"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Recomputable where"},{"Add":""},{"Add":"open import Agda.Builtin.Equality using (_≡_; refl)"},{"Add":"open import Data.Empty using (⊥)"},{"Add":"open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Nullary.Negation.Core using (¬_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":"--"},{"Add":"-- The idea of being 'recomputable' is that, given an *irrelevant* proof"},{"Add":"-- of a proposition `A` (signalled by being a value of type `.A`, all of"},{"Add":"-- whose inhabitants are identified up to definitional equality, and hence"},{"Add":"-- do *not* admit pattern-matching), one may 'promote' such a value to a"},{"Add":"-- 'genuine' value of `A`, available for subsequent eg. pattern-matching."},{"Add":""},{"Add":"Recomputable : (A : Set a) → Set a"},{"Add":"Recomputable A = .A → A"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Fundamental property: 'promotion' is a constant function"},{"Add":""},{"Add":"recompute-constant : (r : Recomputable A) (p q : A) → r p ≡ r q"},{"Add":"recompute-constant r p q = refl"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Constructions"},{"Add":""},{"Add":"⊥-recompute : Recomputable ⊥"},{"Add":"⊥-recompute ()"},{"Add":""},{"Add":"_×-recompute_ : Recomputable A → Recomputable B → Recomputable (A × B)"},{"Add":"(rA ×-recompute rB) p = rA (p .proj₁) , rB (p .proj₂)"},{"Add":""},{"Add":"_→-recompute_ : (A : Set a) → Recomputable B → Recomputable (A → B)"},{"Add":"(A →-recompute rB) f a = rB (f a)"},{"Add":""},{"Add":"Π-recompute : (B : A → Set b) → (∀ x → Recomputable (B x)) → Recomputable (∀ x → B x)"},{"Add":"Π-recompute B rB f a = rB a (f a)"},{"Add":""},{"Add":"∀-recompute : (B : A → Set b) → (∀ {x} → Recomputable (B x)) → Recomputable (∀ {x} → B x)"},{"Add":"∀-recompute B rB f = rB f"},{"Add":""},{"Add":"-- corollary: negated propositions are Recomputable"},{"Add":""},{"Add":"¬-recompute : Recomputable (¬ A)"},{"Add":"¬-recompute {A = A} = A →-recompute ⊥-recompute"},{"Add":""}]}},{"FileAdd":{"meta":{"basename":"Product.agda","metadata":0},"context":[{"change":0,"pos":371196}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Product where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Relation.Nullary.Product was deprecated in v2.0."},{"Add":"Use `Relation.Nullary.Decidable` or `Relation.Nullary` instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Relation.Nullary.Decidable.Core public using (_×-dec_)"},{"Add":"open import Relation.Nullary.Reflects public using (_×-reflects_)"}]}},{"FileAdd":{"meta":{"basename":"Negation.agda","metadata":0},"context":[{"change":0,"pos":371196}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties related to negation"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Negation where"},{"Add":""},{"Add":"open import Data.Bool.Base using (Bool; false; true; if_then_else_)"},{"Add":"open import Data.Product.Base as Product using (_,_; Σ; Σ-syntax; ∃; curry; uncurry)"},{"Add":"open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])"},{"Add":"open import Effect.Monad using (RawMonad; mkRawMonad)"},{"Add":"open import Function.Base using (flip; _∘_; const; _∘′_)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Nullary.Decidable.Core using (Dec; yes; no; ¬¬-excluded-middle)"},{"Add":"open import Relation.Unary using (Universal; Pred)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c d p w : Level"},{"Add":" A B C D : Set a"},{"Add":" P : Pred A p"},{"Add":" Whatever : Set w"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export public definitions"},{"Add":""},{"Add":"open import Relation.Nullary.Negation.Core public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Quantifier juggling"},{"Add":""},{"Add":"∃⟶¬∀¬ : ∃ P → ¬ (∀ x → ¬ P x)"},{"Add":"∃⟶¬∀¬ = flip uncurry"},{"Add":""},{"Add":"∀⟶¬∃¬ : (∀ x → P x) → ¬ ∃ λ x → ¬ P x"},{"Add":"∀⟶¬∃¬ ∀xPx (x , ¬Px) = ¬Px (∀xPx x)"},{"Add":""},{"Add":"¬∃⟶∀¬ : ¬ ∃ (λ x → P x) → ∀ x → ¬ P x"},{"Add":"¬∃⟶∀¬ = curry"},{"Add":""},{"Add":"∀¬⟶¬∃ : (∀ x → ¬ P x) → ¬ ∃ (λ x → P x)"},{"Add":"∀¬⟶¬∃ = uncurry"},{"Add":""},{"Add":"∃¬⟶¬∀ : ∃ (λ x → ¬ P x) → ¬ (∀ x → P x)"},{"Add":"∃¬⟶¬∀ = flip ∀⟶¬∃¬"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Double Negation"},{"Add":""},{"Add":"-- Double-negation is a monad (if we assume that all elements of ¬ ¬ P"},{"Add":"-- are equal)."},{"Add":""},{"Add":"¬¬-Monad : RawMonad {a} DoubleNegation"},{"Add":"¬¬-Monad = mkRawMonad"},{"Add":" DoubleNegation"},{"Add":" contradiction"},{"Add":" (λ x f → negated-stable (¬¬-map f x))"},{"Add":""},{"Add":"¬¬-push : DoubleNegation Π[ P ] → Π[ DoubleNegation ∘ P ]"},{"Add":"¬¬-push ¬¬∀P a ¬Pa = ¬¬∀P (λ ∀P → ¬Pa (∀P a))"},{"Add":""},{"Add":"-- If Whatever is instantiated with ¬ ¬ something, then this function"},{"Add":"-- is call with current continuation in the double-negation monad, or,"},{"Add":"-- if you will, a double-negation translation of Peirce's law."},{"Add":"--"},{"Add":"-- In order to prove ¬ ¬ P one can assume ¬ P and prove ⊥. However,"},{"Add":"-- sometimes it is nice to avoid leaving the double-negation monad; in"},{"Add":"-- that case this function can be used (with Whatever instantiated to"},{"Add":"-- ⊥)."},{"Add":""},{"Add":"call/cc : ((A → Whatever) → DoubleNegation A) → DoubleNegation A"},{"Add":"call/cc hyp ¬a = hyp (flip contradiction ¬a) ¬a"},{"Add":""},{"Add":"-- The \"independence of premise\" rule, in the double-negation monad."},{"Add":"-- It is assumed that the index set (A) is inhabited."},{"Add":""},{"Add":"independence-of-premise : A → (B → Σ A P) → DoubleNegation (Σ[ x ∈ A ] (B → P x))"},{"Add":"independence-of-premise {A = A} {B = B} {P = P} q f = ¬¬-map helper ¬¬-excluded-middle"},{"Add":" where"},{"Add":" helper : Dec B → Σ[ x ∈ A ] (B → P x)"},{"Add":" helper (yes p) = Product.map₂ const (f p)"},{"Add":" helper (no ¬p) = (q , flip contradiction ¬p)"},{"Add":""},{"Add":"-- The independence of premise rule for binary sums."},{"Add":""},{"Add":"independence-of-premise-⊎ : (A → B ⊎ C) → DoubleNegation ((A → B) ⊎ (A → C))"},{"Add":"independence-of-premise-⊎ {A = A} {B = B} {C = C} f = ¬¬-map helper ¬¬-excluded-middle"},{"Add":" where"},{"Add":" helper : Dec A → (A → B) ⊎ (A → C)"},{"Add":" helper (yes p) = Sum.map const const (f p)"},{"Add":" helper (no ¬p) = inj₁ (flip contradiction ¬p)"},{"Add":""},{"Add":"private"},{"Add":""},{"Add":" -- Note that independence-of-premise-⊎ is a consequence of"},{"Add":" -- independence-of-premise (for simplicity it is assumed that Q and"},{"Add":" -- R have the same type here):"},{"Add":""},{"Add":" corollary : {B C : Set b} → (A → B ⊎ C) → DoubleNegation ((A → B) ⊎ (A → C))"},{"Add":" corollary {A = A} {B = B} {C = C} f ="},{"Add":" ¬¬-map helper (independence-of-premise true ([ _,_ true , _,_ false ] ∘′ f))"},{"Add":" where"},{"Add":" helper : ∃ (λ b → A → if b then B else C) → (A → B) ⊎ (A → C)"},{"Add":" helper (true , f) = inj₁ f"},{"Add":" helper (false , f) = inj₂ f"}]}},{"FileAdd":{"meta":{"basename":"Negation","metadata":576},"context":[{"change":0,"pos":371196}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Core.agda","metadata":0},"context":[{"change":0,"pos":388320}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Core properties related to negation"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Negation.Core where"},{"Add":""},{"Add":"open import Data.Empty using (⊥; ⊥-elim-irr)"},{"Add":"open import Data.Sum.Base using (_⊎_; [_,_]; inj₁; inj₂)"},{"Add":"open import Function.Base using (flip; _∘_; const)"},{"Add":"open import Level"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a p q w : Level"},{"Add":" A B C : Set a"},{"Add":" Whatever : Set w"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Negation."},{"Add":""},{"Add":"infix 3 ¬_"},{"Add":"¬_ : Set a → Set a"},{"Add":"¬ A = A → ⊥"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Stability."},{"Add":""},{"Add":"-- Double-negation"},{"Add":"DoubleNegation : Set a → Set a"},{"Add":"DoubleNegation A = ¬ ¬ A"},{"Add":""},{"Add":"-- Stability under double-negation."},{"Add":"Stable : Set a → Set a"},{"Add":"Stable A = ¬ ¬ A → A"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relationship to sum"},{"Add":""},{"Add":"infixr 1 _¬-⊎_"},{"Add":"_¬-⊎_ : ¬ A → ¬ B → ¬ (A ⊎ B)"},{"Add":"_¬-⊎_ = [_,_]"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Uses of negation"},{"Add":""},{"Add":"contradiction-irr : .A → ¬ A → Whatever"},{"Add":"contradiction-irr a ¬a = ⊥-elim-irr (¬a a)"},{"Add":""},{"Add":"contradiction : A → ¬ A → Whatever"},{"Add":"contradiction a = contradiction-irr a"},{"Add":""},{"Add":"contradiction₂ : A ⊎ B → ¬ A → ¬ B → Whatever"},{"Add":"contradiction₂ (inj₁ a) ¬a ¬b = contradiction a ¬a"},{"Add":"contradiction₂ (inj₂ b) ¬a ¬b = contradiction b ¬b"},{"Add":""},{"Add":"contraposition : (A → B) → ¬ B → ¬ A"},{"Add":"contraposition f ¬b a = contradiction (f a) ¬b"},{"Add":""},{"Add":"-- Everything is stable in the double-negation monad."},{"Add":"stable : ¬ ¬ Stable A"},{"Add":"stable ¬[¬¬a→a] = ¬[¬¬a→a] (contradiction (¬[¬¬a→a] ∘ const))"},{"Add":""},{"Add":"-- Negated predicates are stable."},{"Add":"negated-stable : Stable (¬ A)"},{"Add":"negated-stable ¬¬¬a a = ¬¬¬a (contradiction a)"},{"Add":""},{"Add":"¬¬-map : (A → B) → ¬ ¬ A → ¬ ¬ B"},{"Add":"¬¬-map f = contraposition (contraposition f)"},{"Add":""},{"Add":"-- Note also the following use of flip:"},{"Add":"private"},{"Add":" note : (A → ¬ B) → B → ¬ A"},{"Add":" note = flip"},{"Add":""}]}},{"FileAdd":{"meta":{"basename":"Indexed.agda","metadata":0},"context":[{"change":0,"pos":371196}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Negation indexed by a Level"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Indexed where"},{"Add":""},{"Add":"open import Data.Empty hiding (⊥-elim)"},{"Add":"open import Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Negation."},{"Add":""},{"Add":"-- level polymorphic version of ¬"},{"Add":"¬ : ∀ {ℓ} (b : Level) → Set ℓ → Set (ℓ ⊔ b)"},{"Add":"¬ b P = P → Lift b ⊥"}]}},{"FileAdd":{"meta":{"basename":"Indexed","metadata":576},"context":[{"change":0,"pos":371196}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Negation.agda","metadata":0},"context":[{"change":0,"pos":391126}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of indexed negation"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Indexed.Negation where"},{"Add":""},{"Add":"open import Axiom.Extensionality.Propositional using (Extensionality)"},{"Add":"open import Data.Empty.Polymorphic"},{"Add":"open import Function.Bundles using (_↔_)"},{"Add":"open import Function.Properties"},{"Add":"import Function.Construct.Identity as Identity"},{"Add":"open import Relation.Nullary.Indexed"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- ¬_ preserves ↔ (assuming extensionality)"},{"Add":""},{"Add":"¬-cong : ∀ {a b c} {A : Set a} {B : Set b} →"},{"Add":" Extensionality a c → Extensionality b c →"},{"Add":" A ↔ B → (¬ c A) ↔ (¬ c B)"},{"Add":"¬-cong extA extB A≈B = →-cong-↔ extA extB A≈B (Identity.↔-id ⊥)"}]}},{"FileAdd":{"meta":{"basename":"Implication.agda","metadata":0},"context":[{"change":0,"pos":371196}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Implication where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Relation.Nullary.Implication was deprecated in v2.0."},{"Add":"Use `Relation.Nullary.Decidable` or `Relation.Nullary` instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Relation.Nullary.Decidable.Core public using (_→-dec_)"},{"Add":"open import Relation.Nullary.Reflects public using (_→-reflects_)"}]}},{"FileAdd":{"meta":{"basename":"Decidable.agda","metadata":0},"context":[{"change":0,"pos":371196}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Operations on and properties of decidable relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Decidable where"},{"Add":""},{"Add":"open import Level using (Level)"},{"Add":"open import Data.Bool.Base using (true; false)"},{"Add":"open import Data.Product.Base using (∃; _,_)"},{"Add":"open import Function.Bundles using"},{"Add":" (Injection; module Injection; module Equivalence; _⇔_; _↔_; mk↔ₛ′)"},{"Add":"open import Relation.Binary.Bundles using (Setoid; module Setoid)"},{"Add":"open import Relation.Binary.Definitions using (Decidable)"},{"Add":"open import Relation.Nullary using (Irrelevant)"},{"Add":"open import Relation.Nullary.Negation.Core using (¬_; contradiction)"},{"Add":"open import Relation.Nullary.Reflects using (invert)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core"},{"Add":" using (_≡_; refl; sym; trans; cong′)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b ℓ₁ ℓ₂ : Level"},{"Add":" A B : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting the core definitions"},{"Add":""},{"Add":"open import Relation.Nullary.Decidable.Core public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Maps"},{"Add":""},{"Add":"map : A ⇔ B → Dec A → Dec B"},{"Add":"map A⇔B = map′ to from"},{"Add":" where open Equivalence A⇔B"},{"Add":""},{"Add":"-- If there is an injection from one setoid to another, and the"},{"Add":"-- latter's equivalence relation is decidable, then the former's"},{"Add":"-- equivalence relation is also decidable."},{"Add":"via-injection : {S : Setoid a ℓ₁} {T : Setoid b ℓ₂}"},{"Add":" (inj : Injection S T) (open Injection inj) →"},{"Add":" Decidable Eq₂._≈_ → Decidable Eq₁._≈_"},{"Add":"via-injection inj _≟_ x y = map′ injective cong (to x ≟ to y)"},{"Add":" where open Injection inj"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- A lemma relating True and Dec"},{"Add":""},{"Add":"True-↔ : (a? : Dec A) → Irrelevant A → True a? ↔ A"},{"Add":"True-↔ (true because [a]) irr = let a = invert [a] in mk↔ₛ′ (λ _ → a) _ (irr a) cong′"},{"Add":"True-↔ (false because [¬a]) _ = let ¬a = invert [¬a] in mk↔ₛ′ (λ ()) ¬a (λ a → contradiction a ¬a) λ ()"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Result of decidability"},{"Add":""},{"Add":"isYes≗does : (a? : Dec A) → isYes a? ≡ does a?"},{"Add":"isYes≗does (true because _) = refl"},{"Add":"isYes≗does (false because _) = refl"},{"Add":""},{"Add":"dec-true : (a? : Dec A) → A → does a? ≡ true"},{"Add":"dec-true (true because _ ) a = refl"},{"Add":"dec-true (false because [¬a]) a = contradiction a (invert [¬a])"},{"Add":""},{"Add":"dec-false : (a? : Dec A) → ¬ A → does a? ≡ false"},{"Add":"dec-false (false because _ ) ¬a = refl"},{"Add":"dec-false (true because [a]) ¬a = contradiction (invert [a]) ¬a"},{"Add":""},{"Add":"dec-yes : (a? : Dec A) → A → ∃ λ a → a? ≡ yes a"},{"Add":"dec-yes a? a with yes a′ ← a? | refl ← dec-true a? a = a′ , refl"},{"Add":""},{"Add":"dec-no : (a? : Dec A) (¬a : ¬ A) → a? ≡ no ¬a"},{"Add":"dec-no a? ¬a with no _ ← a? | refl ← dec-false a? ¬a = refl"},{"Add":""},{"Add":"dec-yes-irr : (a? : Dec A) → Irrelevant A → (a : A) → a? ≡ yes a"},{"Add":"dec-yes-irr a? irr a with a′ , eq ← dec-yes a? a rewrite irr a a′ = eq"},{"Add":""},{"Add":"⌊⌋-map′ : ∀ t f (a? : Dec A) → ⌊ map′ {B = B} t f a? ⌋ ≡ ⌊ a? ⌋"},{"Add":"⌊⌋-map′ t f a? = trans (isYes≗does (map′ t f a?)) (sym (isYes≗does a?))"}]}},{"FileAdd":{"meta":{"basename":"Decidable","metadata":576},"context":[{"change":0,"pos":371196}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Core.agda","metadata":0},"context":[{"change":0,"pos":396100}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Operations on and properties of decidable relations"},{"Add":"--"},{"Add":"-- This file contains some core definitions which are re-exported by"},{"Add":"-- Relation.Nullary.Decidable"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Decidable.Core where"},{"Add":""},{"Add":"-- decToMaybe was deprecated in v2.1 #2330/#2336"},{"Add":"-- this can go through `Data.Maybe.Base` once that deprecation is fully done."},{"Add":"open import Agda.Builtin.Maybe using (Maybe; just; nothing)"},{"Add":""},{"Add":"open import Agda.Builtin.Equality using (_≡_)"},{"Add":"open import Level using (Level)"},{"Add":"open import Data.Bool.Base using (Bool; T; false; true; not; _∧_; _∨_)"},{"Add":"open import Data.Unit.Polymorphic.Base using (⊤)"},{"Add":"open import Data.Product.Base using (_×_)"},{"Add":"open import Data.Sum.Base using (_⊎_; inj₁; inj₂)"},{"Add":"open import Function.Base using (_∘_; const; _$_; flip)"},{"Add":"open import Relation.Nullary.Recomputable as Recomputable hiding (recompute-constant)"},{"Add":"open import Relation.Nullary.Reflects as Reflects hiding (recompute; recompute-constant)"},{"Add":"open import Relation.Nullary.Negation.Core"},{"Add":" using (¬_; Stable; negated-stable; contradiction; DoubleNegation)"},{"Add":""},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b : Level"},{"Add":" A B : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition."},{"Add":""},{"Add":"-- Decidability proofs have two parts: the `does` term which contains"},{"Add":"-- the boolean result and the `proof` term which contains a proof that"},{"Add":"-- reflects the boolean result. This definition allows the boolean"},{"Add":"-- part of the decision procedure to compute independently from the"},{"Add":"-- proof. This leads to better computational behaviour when we only care"},{"Add":"-- about the result and not the proof. See README.Design.Decidability"},{"Add":"-- for further details."},{"Add":""},{"Add":"infix 2 _because_"},{"Add":""},{"Add":"record Dec (A : Set a) : Set a where"},{"Add":" constructor _because_"},{"Add":" field"},{"Add":" does : Bool"},{"Add":" proof : Reflects A does"},{"Add":""},{"Add":"open Dec public"},{"Add":""},{"Add":"pattern yes a = true because ofʸ a"},{"Add":"pattern no ¬a = false because ofⁿ ¬a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Flattening"},{"Add":""},{"Add":"module _ {A : Set a} where"},{"Add":""},{"Add":" From-yes : Dec A → Set a"},{"Add":" From-yes (true because _) = A"},{"Add":" From-yes (false because _) = ⊤"},{"Add":""},{"Add":" From-no : Dec A → Set a"},{"Add":" From-no (false because _) = ¬ A"},{"Add":" From-no (true because _) = ⊤"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Recompute"},{"Add":""},{"Add":"-- Given an irrelevant proof of a decidable type, a proof can"},{"Add":"-- be recomputed and subsequently used in relevant contexts."},{"Add":""},{"Add":"recompute : Dec A → Recomputable A"},{"Add":"recompute = Reflects.recompute ∘ proof"},{"Add":""},{"Add":"recompute-constant : (a? : Dec A) (p q : A) → recompute a? p ≡ recompute a? q"},{"Add":"recompute-constant = Recomputable.recompute-constant ∘ recompute"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Interaction with negation, sum, product etc."},{"Add":""},{"Add":"infixr 1 _⊎-dec_"},{"Add":"infixr 2 _×-dec_ _→-dec_"},{"Add":""},{"Add":"T? : ∀ x → Dec (T x)"},{"Add":"T? x = x because T-reflects x"},{"Add":""},{"Add":"¬? : Dec A → Dec (¬ A)"},{"Add":"does (¬? a?) = not (does a?)"},{"Add":"proof (¬? a?) = ¬-reflects (proof a?)"},{"Add":""},{"Add":"_×-dec_ : Dec A → Dec B → Dec (A × B)"},{"Add":"does (a? ×-dec b?) = does a? ∧ does b?"},{"Add":"proof (a? ×-dec b?) = proof a? ×-reflects proof b?"},{"Add":""},{"Add":"_⊎-dec_ : Dec A → Dec B → Dec (A ⊎ B)"},{"Add":"does (a? ⊎-dec b?) = does a? ∨ does b?"},{"Add":"proof (a? ⊎-dec b?) = proof a? ⊎-reflects proof b?"},{"Add":""},{"Add":"_→-dec_ : Dec A → Dec B → Dec (A → B)"},{"Add":"does (a? →-dec b?) = not (does a?) ∨ does b?"},{"Add":"proof (a? →-dec b?) = proof a? →-reflects proof b?"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relationship with Maybe"},{"Add":""},{"Add":"dec⇒maybe : Dec A → Maybe A"},{"Add":"dec⇒maybe ( true because [a]) = just (invert [a])"},{"Add":"dec⇒maybe (false because _ ) = nothing"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relationship with Sum"},{"Add":""},{"Add":"toSum : Dec A → A ⊎ ¬ A"},{"Add":"toSum ( true because [p]) = inj₁ (invert [p])"},{"Add":"toSum (false because [¬p]) = inj₂ (invert [¬p])"},{"Add":""},{"Add":"fromSum : A ⊎ ¬ A → Dec A"},{"Add":"fromSum (inj₁ p) = yes p"},{"Add":"fromSum (inj₂ ¬p) = no ¬p"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relationship with booleans"},{"Add":""},{"Add":"-- `isYes` is a stricter version of `does`. The lack of computation"},{"Add":"-- means that we can recover the proposition `P` from `isYes a?` by"},{"Add":"-- unification. This is useful when we are using the decision procedure"},{"Add":"-- for proof automation."},{"Add":""},{"Add":"isYes : Dec A → Bool"},{"Add":"isYes (true because _) = true"},{"Add":"isYes (false because _) = false"},{"Add":""},{"Add":"isNo : Dec A → Bool"},{"Add":"isNo = not ∘ isYes"},{"Add":""},{"Add":"True : Dec A → Set"},{"Add":"True = T ∘ isYes"},{"Add":""},{"Add":"False : Dec A → Set"},{"Add":"False = T ∘ isNo"},{"Add":""},{"Add":"-- The traditional name for isYes is ⌊_⌋, indicating the stripping of evidence."},{"Add":"⌊_⌋ = isYes"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Witnesses"},{"Add":""},{"Add":"-- Gives a witness to the \"truth\"."},{"Add":"toWitness : {a? : Dec A} → True a? → A"},{"Add":"toWitness {a? = true because [a]} _ = invert [a]"},{"Add":"toWitness {a? = false because _ } ()"},{"Add":""},{"Add":"-- Establishes a \"truth\", given a witness."},{"Add":"fromWitness : {a? : Dec A} → A → True a?"},{"Add":"fromWitness {a? = true because _ } = const _"},{"Add":"fromWitness {a? = false because [¬a]} = invert [¬a]"},{"Add":""},{"Add":"-- Variants for False."},{"Add":"toWitnessFalse : {a? : Dec A} → False a? → ¬ A"},{"Add":"toWitnessFalse {a? = true because _ } ()"},{"Add":"toWitnessFalse {a? = false because [¬a]} _ = invert [¬a]"},{"Add":""},{"Add":"fromWitnessFalse : {a? : Dec A} → ¬ A → False a?"},{"Add":"fromWitnessFalse {a? = true because [a]} = flip _$_ (invert [a])"},{"Add":"fromWitnessFalse {a? = false because _ } = const _"},{"Add":""},{"Add":"-- If a decision procedure returns \"yes\", then we can extract the"},{"Add":"-- proof using from-yes."},{"Add":"from-yes : (a? : Dec A) → From-yes a?"},{"Add":"from-yes (true because [a]) = invert [a]"},{"Add":"from-yes (false because _ ) = _"},{"Add":""},{"Add":"-- If a decision procedure returns \"no\", then we can extract the proof"},{"Add":"-- using from-no."},{"Add":"from-no : (a? : Dec A) → From-no a?"},{"Add":"from-no (false because [¬a]) = invert [¬a]"},{"Add":"from-no (true because _ ) = _"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Maps"},{"Add":""},{"Add":"map′ : (A → B) → (B → A) → Dec A → Dec B"},{"Add":"does (map′ A→B B→A a?) = does a?"},{"Add":"proof (map′ A→B B→A (true because [a])) = of (A→B (invert [a]))"},{"Add":"proof (map′ A→B B→A (false because [¬a])) = of (invert [¬a] ∘ B→A)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relationship with double-negation"},{"Add":""},{"Add":"-- Decidable predicates are stable."},{"Add":""},{"Add":"decidable-stable : Dec A → Stable A"},{"Add":"decidable-stable (true because [a]) ¬¬a = invert [a]"},{"Add":"decidable-stable (false because [¬a]) ¬¬a = contradiction (invert [¬a]) ¬¬a"},{"Add":""},{"Add":"¬-drop-Dec : Dec (¬ ¬ A) → Dec (¬ A)"},{"Add":"¬-drop-Dec ¬¬a? = map′ negated-stable contradiction (¬? ¬¬a?)"},{"Add":""},{"Add":"-- A double-negation-translated variant of excluded middle (or: every"},{"Add":"-- nullary relation is decidable in the double-negation monad)."},{"Add":""},{"Add":"¬¬-excluded-middle : DoubleNegation (Dec A)"},{"Add":"¬¬-excluded-middle ¬?a = ¬?a (no (λ a → ¬?a (yes a)))"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- DEPRECATED NAMES"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Please use the new names as continuing support for the old names is"},{"Add":"-- not guaranteed."},{"Add":""},{"Add":"-- Version 2.0"},{"Add":""},{"Add":"excluded-middle = ¬¬-excluded-middle"},{"Add":"{-# WARNING_ON_USAGE excluded-middle"},{"Add":"\"Warning: excluded-middle was deprecated in v2.0."},{"Add":"Please use ¬¬-excluded-middle instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"-- Version 2.1"},{"Add":""},{"Add":"decToMaybe = dec⇒maybe"},{"Add":"{-# WARNING_ON_USAGE decToMaybe"},{"Add":"\"Warning: decToMaybe was deprecated in v2.1."},{"Add":"Please use Relation.Nullary.Decidable.Core.dec⇒maybe instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"fromDec = toSum"},{"Add":"{-# WARNING_ON_USAGE fromDec"},{"Add":"\"Warning: fromDec was deprecated in v2.1."},{"Add":"Please use Relation.Nullary.Decidable.Core.toSum instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"toDec = fromSum"},{"Add":"{-# WARNING_ON_USAGE toDec"},{"Add":"\"Warning: toDec was deprecated in v2.1."},{"Add":"Please use Relation.Nullary.Decidable.Core.fromSum instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Construct","metadata":576},"context":[{"change":0,"pos":371196}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Add","metadata":576},"context":[{"change":0,"pos":404137}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Supremum.agda","metadata":0},"context":[{"change":0,"pos":404160}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Notation for freely adding a supremum to any set"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Construct.Add.Supremum where"},{"Add":""},{"Add":"open import Relation.Nullary.Construct.Add.Point"},{"Add":" renaming (Pointed to _⁺; ∙ to ⊤⁺)"},{"Add":" public"}]}},{"FileAdd":{"meta":{"basename":"Point.agda","metadata":0},"context":[{"change":0,"pos":404160}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Notation for adding an additional point to any set"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Construct.Add.Point where"},{"Add":""},{"Add":"open import Data.Maybe.Base public"},{"Add":" using () renaming (Maybe to Pointed; nothing to ∙; just to [_])"},{"Add":""},{"Add":"open import Data.Maybe.Properties public"},{"Add":" using (≡-dec) renaming (just-injective to []-injective)"}]}},{"FileAdd":{"meta":{"basename":"Infimum.agda","metadata":0},"context":[{"change":0,"pos":404160}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Notation for freely adding an infimum to any set"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Construct.Add.Infimum where"},{"Add":""},{"Add":"open import Relation.Nullary.Construct.Add.Point public"},{"Add":" renaming (Pointed to _₋; ∙ to ⊥₋)"}]}},{"FileAdd":{"meta":{"basename":"Extrema.agda","metadata":0},"context":[{"change":0,"pos":404160}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Notation for freely adding extrema to any set"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nullary.Construct.Add.Extrema where"},{"Add":""},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)"},{"Add":"open import Relation.Nullary.Construct.Add.Infimum as Infimum using (_₋)"},{"Add":"open import Relation.Nullary.Construct.Add.Supremum as Supremum using (_⁺)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"_± : ∀ {a} → Set a → Set a"},{"Add":"A ± = A ₋ ⁺"},{"Add":""},{"Add":"pattern ⊥± = Supremum.[ Infimum.⊥₋ ]"},{"Add":"pattern [_] k = Supremum.[ Infimum.[ k ] ]"},{"Add":"pattern ⊤± = Supremum.⊤⁺"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"[_]-injective : ∀ {a} {A : Set a} {k l : A} → [ k ] ≡ [ l ] → k ≡ l"},{"Add":"[_]-injective refl = refl"}]}},{"FileAdd":{"meta":{"basename":"Nary.agda","metadata":0},"context":[{"change":0,"pos":320100}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Heterogeneous N-ary Relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Nary where"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Concrete examples can be found in README.Nary. This file's comments"},{"Add":"-- are more focused on the implementation details and the motivations"},{"Add":"-- behind the design decisions."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"open import Level using (Level; _⊔_; Lift)"},{"Add":"open import Data.Unit.Base"},{"Add":"open import Data.Bool.Base using (true; false)"},{"Add":"open import Data.Empty"},{"Add":"open import Data.Nat.Base using (zero; suc)"},{"Add":"open import Data.Product.Base as Product using (_×_; _,_)"},{"Add":"open import Data.Product.Nary.NonDependent"},{"Add":"open import Data.Sum.Base using (_⊎_)"},{"Add":"open import Function.Base using (_$_; _∘′_)"},{"Add":"open import Function.Nary.NonDependent"},{"Add":"open import Relation.Nullary.Negation using (¬_)"},{"Add":"open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _because_; _×-dec_)"},{"Add":"import Relation.Unary as Unary"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong; subst)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" r : Level"},{"Add":" R : Set r"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Generic type constructors"},{"Add":""},{"Add":"-- `Relation.Unary` provides users with a wealth of combinators to work"},{"Add":"-- with indexed sets. We can generalise these to n-ary relations."},{"Add":""},{"Add":"-- The crucial thing to notice here is that because we are explicitly"},{"Add":"-- considering that the input function should be a `Set`-ended `Arrows`,"},{"Add":"-- all the other parameters are inferrable. This allows us to make the"},{"Add":"-- number arguments (`n`) implicit."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Quantifiers"},{"Add":""},{"Add":"-- If we already know how to quantify over one variable, we can easily"},{"Add":"-- describe how to quantify over `n` variables by induction over said `n`."},{"Add":""},{"Add":"quantₙ : (∀ {i l} {I : Set i} → (I → Set l) → Set (i ⊔ l)) →"},{"Add":" ∀ n {ls} {as : Sets n ls} →"},{"Add":" Arrows n as (Set r) → Set (r ⊔ (⨆ n ls))"},{"Add":"quantₙ Q zero f = f"},{"Add":"quantₙ Q (suc n) f = Q (λ x → quantₙ Q n (f x))"},{"Add":""},{"Add":"infix 5 ∃⟨_⟩ Π[_] ∀[_]"},{"Add":""},{"Add":"-- existential quantifier"},{"Add":""},{"Add":"∃⟨_⟩ : ∀ {n ls r} {as : Sets n ls} → as ⇉ Set r → Set (r ⊔ (⨆ n ls))"},{"Add":"∃⟨_⟩ = quantₙ Unary.Satisfiable _"},{"Add":""},{"Add":"-- explicit universal quantifiers"},{"Add":""},{"Add":"Π[_] : ∀ {n ls r} {as : Sets n ls} → as ⇉ Set r → Set (r ⊔ (⨆ n ls))"},{"Add":"Π[_] = quantₙ Unary.Universal _"},{"Add":""},{"Add":"-- implicit universal quantifiers"},{"Add":""},{"Add":"∀[_] : ∀ {n ls r} {as : Sets n ls} → as ⇉ Set r → Set (r ⊔ (⨆ n ls))"},{"Add":"∀[_] = quantₙ Unary.IUniversal _"},{"Add":""},{"Add":"-- ≟-mapₙ : ∀ n. (con : A₁ → ⋯ → Aₙ → R) →"},{"Add":"-- Injectiveₙ n con →"},{"Add":"-- ∀ a₁₁ a₁₂ ⋯ aₙ₁ aₙ₂ →"},{"Add":"-- Dec (a₁₁ ≡ a₁₂) → ⋯ → Dec (aₙ₁ ≡ aₙ₂) →"},{"Add":"-- Dec (con a₁₁ ⋯ aₙ₁ ≡ con a₁₂ ⋯ aₙ₂)"},{"Add":""},{"Add":"≟-mapₙ : ∀ n {ls} {as : Sets n ls} (con : Arrows n as R) → Injectiveₙ n con →"},{"Add":" ∀ {l r} → Arrows n (Dec <$> Equalₙ n l r) (Dec (uncurryₙ n con l ≡ uncurryₙ n con r))"},{"Add":"≟-mapₙ n con con-inj ="},{"Add":" curryₙ n λ a?s → let as? = Product-dec n a?s in"},{"Add":" Dec.map′ (cong (uncurryₙ n con) ∘′ fromEqualₙ n) con-inj as?"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Substitution"},{"Add":""},{"Add":"module _ {n r ls} {as : Sets n ls} (P : as ⇉ Set r) where"},{"Add":""},{"Add":"-- Substitutionₙ : ∀ n. ∀ a₁₁ a₁₂ ⋯ aₙ₁ aₙ₂ →"},{"Add":"-- a₁₁ ≡ a₁₂ → ⋯ → aₙ₁ ≡ aₙ₂ →"},{"Add":"-- P a₁₁ ⋯ aₙ₁ → P a₁₂ ⋯ aₙ₂"},{"Add":""},{"Add":" Substitutionₙ : Set (r ⊔ (⨆ n ls))"},{"Add":" Substitutionₙ = ∀ {l r} → Equalₙ n l r ⇉ (uncurryₙ n P l → uncurryₙ n P r)"},{"Add":""},{"Add":" substₙ : Substitutionₙ"},{"Add":" substₙ = curryₙ n (subst (uncurryₙ n P) ∘′ fromEqualₙ n)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Pointwise liftings of k-ary operators"},{"Add":""},{"Add":"-- Rather than having multiple ad-hoc lifting functions for various"},{"Add":"-- arities we have a fully generic liftₙ functional which lifts a k-ary"},{"Add":"-- operator to work with k n-ary functions whose respective codomains"},{"Add":"-- match the domains of the operator."},{"Add":"-- The type of liftₙ is fairly unreadable. Here it is written with ellipsis:"},{"Add":""},{"Add":"-- liftₙ : ∀ k n. (B₁ → ⋯ → Bₖ → R) →"},{"Add":"-- (A₁ → ⋯ → Aₙ → B₁) →"},{"Add":"-- ⋮"},{"Add":"-- (A₁ → ⋯ → Aₙ → B₁) →"},{"Add":"-- (A₁ → ⋯ → Aₙ → R)"},{"Add":""},{"Add":"liftₙ : ∀ k n {ls rs} {as : Sets n ls} {bs : Sets k rs} →"},{"Add":" Arrows k bs R → Arrows k (smap _ (Arrows n as) k bs) (Arrows n as R)"},{"Add":"liftₙ k n op = curry⊤ₙ k λ fs →"},{"Add":" curry⊤ₙ n λ vs →"},{"Add":" uncurry⊤ₙ k op $"},{"Add":" palg _ _ k (λ f → uncurry⊤ₙ n f vs) fs where"},{"Add":""},{"Add":" -- The bulk of the work happens in this auxiliary definition:"},{"Add":" palg : ∀ f (F : ∀ {l} → Set l → Set (f l)) n {ls} {as : Sets n ls} →"},{"Add":" (∀ {l} {r : Set l} → F r → r) → Product⊤ n (smap f F n as) → Product⊤ n as"},{"Add":" palg f F zero alg ps = _"},{"Add":" palg f F (suc n) alg (p , ps) = alg p , palg f F n alg ps"},{"Add":""},{"Add":""},{"Add":"-- implication"},{"Add":""},{"Add":"infixr 6 _⇒_"},{"Add":"_⇒_ : ∀ {n} {ls r s} {as : Sets n ls} →"},{"Add":" as ⇉ Set r → as ⇉ Set s → as ⇉ Set (r ⊔ s)"},{"Add":"_⇒_ = liftₙ 2 _ (λ A B → A → B)"},{"Add":""},{"Add":"-- conjunction"},{"Add":""},{"Add":"infixr 7 _∩_"},{"Add":"_∩_ : ∀ {n} {ls r s} {as : Sets n ls} →"},{"Add":" as ⇉ Set r → as ⇉ Set s → as ⇉ Set (r ⊔ s)"},{"Add":"_∩_ = liftₙ 2 _ _×_"},{"Add":""},{"Add":"-- disjunction"},{"Add":""},{"Add":"infixr 8 _∪_"},{"Add":"_∪_ : ∀ {n} {ls r s} {as : Sets n ls} →"},{"Add":" as ⇉ Set r → as ⇉ Set s → as ⇉ Set (r ⊔ s)"},{"Add":"_∪_ = liftₙ 2 _ _⊎_"},{"Add":""},{"Add":"-- negation"},{"Add":""},{"Add":"∁ : ∀ {n ls r} {as : Sets n ls} → as ⇉ Set r → as ⇉ Set r"},{"Add":"∁ = liftₙ 1 _ ¬_"},{"Add":""},{"Add":""},{"Add":"apply⊤ₙ : ∀ {n ls r} {as : Sets n ls} {R : as ⇉ Set r} →"},{"Add":" Π[ R ] → (vs : Product⊤ n as) → uncurry⊤ₙ n R vs"},{"Add":"apply⊤ₙ {zero} prf vs = prf"},{"Add":"apply⊤ₙ {suc n} prf (v , vs) = apply⊤ₙ (prf v) vs"},{"Add":""},{"Add":"applyₙ : ∀ {n ls r} {as : Sets n ls} {R : as ⇉ Set r} →"},{"Add":" Π[ R ] → (vs : Product n as) → uncurry⊤ₙ n R (toProduct⊤ n vs)"},{"Add":"applyₙ {n} prf vs = apply⊤ₙ prf (toProduct⊤ n vs)"},{"Add":""},{"Add":"iapply⊤ₙ : ∀ {n ls r} {as : Sets n ls} {R : as ⇉ Set r} →"},{"Add":" ∀[ R ] → {vs : Product⊤ n as} → uncurry⊤ₙ n R vs"},{"Add":"iapply⊤ₙ {zero} prf = prf"},{"Add":"iapply⊤ₙ {suc n} prf = iapply⊤ₙ {n} prf"},{"Add":""},{"Add":"iapplyₙ : ∀ {n ls r} {as : Sets n ls} {R : as ⇉ Set r} →"},{"Add":" ∀[ R ] → {vs : Product n as} → uncurry⊤ₙ n R (toProduct⊤ n vs)"},{"Add":"iapplyₙ {n} prf = iapply⊤ₙ {n} prf"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of N-ary relations"},{"Add":""},{"Add":"-- Decidability"},{"Add":""},{"Add":"Decidable : ∀ {n ls r} {as : Sets n ls} → as ⇉ Set r → Set (r ⊔ ⨆ n ls)"},{"Add":"Decidable R = Π[ mapₙ _ Dec R ]"},{"Add":""},{"Add":"-- erasure"},{"Add":""},{"Add":"⌊_⌋ : ∀ {n ls r} {as : Sets n ls} {R : as ⇉ Set r} → Decidable R → as ⇉ Set r"},{"Add":"⌊_⌋ {zero} R? = Lift _ (Dec.True R?)"},{"Add":"⌊_⌋ {suc n} R? a = ⌊ R? a ⌋"},{"Add":""},{"Add":"-- equivalence between R and its erasure"},{"Add":""},{"Add":"fromWitness : ∀ {n ls r} {as : Sets n ls} (R : as ⇉ Set r) (R? : Decidable R) →"},{"Add":" ∀[ ⌊ R? ⌋ ⇒ R ]"},{"Add":"fromWitness {zero} R R? with R?"},{"Add":"... | yes r = λ _ → r"},{"Add":"... | false because _ = λ ()"},{"Add":"fromWitness {suc n} R R? = fromWitness (R _) (R? _)"},{"Add":""},{"Add":"toWitness : ∀ {n ls r} {as : Sets n ls} (R : as ⇉ Set r) (R? : Decidable R) →"},{"Add":" ∀[ R ⇒ ⌊ R? ⌋ ]"},{"Add":"toWitness {zero} R R? with R?"},{"Add":"... | true because _ = _"},{"Add":"... | no ¬r = ⊥-elim ∘′ ¬r"},{"Add":"toWitness {suc n} R R? = toWitness (R _) (R? _)"}]}},{"FileAdd":{"meta":{"basename":"Binary.agda","metadata":0},"context":[{"change":0,"pos":320100}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of homogeneous binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary where"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export various components of the binary relation hierarchy"},{"Add":""},{"Add":"open import Relation.Binary.Core public"},{"Add":"open import Relation.Binary.Definitions public"},{"Add":"open import Relation.Binary.Structures public"},{"Add":"open import Relation.Binary.Structures.Biased public"},{"Add":"open import Relation.Binary.Bundles public"}]}},{"FileAdd":{"meta":{"basename":"Binary","metadata":576},"context":[{"change":0,"pos":320100}],"contents":[]}},{"FileAdd":{"meta":{"basename":"TypeClasses.agda","metadata":0},"context":[{"change":0,"pos":415495}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Typeclasses for use with instance arguments"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.TypeClasses where"},{"Add":""},{"Add":"open import Relation.Binary.Structures using (IsDecEquivalence; IsDecTotalOrder) public"},{"Add":""},{"Add":"open IsDecEquivalence {{...}} using (_≟_) public"},{"Add":"open IsDecTotalOrder {{...}} using (_≤?_) public"}]}},{"FileAdd":{"meta":{"basename":"Structures.agda","metadata":0},"context":[{"change":0,"pos":415495}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Structures for homogeneous binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via `Relation.Binary`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core"},{"Add":""},{"Add":"module Relation.Binary.Structures"},{"Add":" {a ℓ} {A : Set a} -- The underlying set"},{"Add":" (_≈_ : Rel A ℓ) -- The underlying equality relation"},{"Add":" where"},{"Add":""},{"Add":"open import Data.Product.Base using (proj₁; proj₂; _,_)"},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Relation.Nullary.Negation.Core using (¬_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)"},{"Add":"open import Relation.Binary.Consequences"},{"Add":"open import Relation.Binary.Definitions"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ₂ : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Equivalences"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Note all the following equivalences refer to the equality provided"},{"Add":"-- as a module parameter at the top of this file."},{"Add":""},{"Add":"record IsPartialEquivalence : Set (a ⊔ ℓ) where"},{"Add":" field"},{"Add":" sym : Symmetric _≈_"},{"Add":" trans : Transitive _≈_"},{"Add":""},{"Add":"-- The preorders of this library are defined in terms of an underlying"},{"Add":"-- equivalence relation, and hence equivalence relations are not"},{"Add":"-- defined in terms of preorders."},{"Add":""},{"Add":"-- To preserve backwards compatability, equivalence relations are"},{"Add":"-- not defined in terms of their partial counterparts."},{"Add":""},{"Add":"record IsEquivalence : Set (a ⊔ ℓ) where"},{"Add":" field"},{"Add":" refl : Reflexive _≈_"},{"Add":" sym : Symmetric _≈_"},{"Add":" trans : Transitive _≈_"},{"Add":""},{"Add":" reflexive : _≡_ ⇒ _≈_"},{"Add":" reflexive ≡.refl = refl"},{"Add":""},{"Add":" isPartialEquivalence : IsPartialEquivalence"},{"Add":" isPartialEquivalence = record"},{"Add":" { sym = sym"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":""},{"Add":""},{"Add":"record IsDecEquivalence : Set (a ⊔ ℓ) where"},{"Add":" infix 4 _≟_"},{"Add":" field"},{"Add":" isEquivalence : IsEquivalence"},{"Add":" _≟_ : Decidable _≈_"},{"Add":""},{"Add":" open IsEquivalence isEquivalence public"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Preorders"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record IsPreorder (_≲_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isEquivalence : IsEquivalence"},{"Add":" -- Reflexivity is expressed in terms of the underlying equality:"},{"Add":" reflexive : _≈_ ⇒ _≲_"},{"Add":" trans : Transitive _≲_"},{"Add":""},{"Add":" module Eq = IsEquivalence isEquivalence"},{"Add":""},{"Add":" refl : Reflexive _≲_"},{"Add":" refl = reflexive Eq.refl"},{"Add":""},{"Add":" ≲-respˡ-≈ : _≲_ Respectsˡ _≈_"},{"Add":" ≲-respˡ-≈ x≈y x∼z = trans (reflexive (Eq.sym x≈y)) x∼z"},{"Add":""},{"Add":" ≲-respʳ-≈ : _≲_ Respectsʳ _≈_"},{"Add":" ≲-respʳ-≈ x≈y z∼x = trans z∼x (reflexive x≈y)"},{"Add":""},{"Add":" ≲-resp-≈ : _≲_ Respects₂ _≈_"},{"Add":" ≲-resp-≈ = ≲-respʳ-≈ , ≲-respˡ-≈"},{"Add":""},{"Add":" ∼-respˡ-≈ = ≲-respˡ-≈"},{"Add":" {-# WARNING_ON_USAGE ∼-respˡ-≈"},{"Add":" \"Warning: ∼-respˡ-≈ was deprecated in v2.0."},{"Add":" Please use ≲-respˡ-≈ instead. \""},{"Add":" #-}"},{"Add":" ∼-respʳ-≈ = ≲-respʳ-≈"},{"Add":" {-# WARNING_ON_USAGE ∼-respʳ-≈"},{"Add":" \"Warning: ∼-respʳ-≈ was deprecated in v2.0."},{"Add":" Please use ≲-respʳ-≈ instead. \""},{"Add":" #-}"},{"Add":" ∼-resp-≈ = ≲-resp-≈"},{"Add":" {-# WARNING_ON_USAGE ∼-resp-≈"},{"Add":" \"Warning: ∼-resp-≈ was deprecated in v2.0."},{"Add":" Please use ≲-resp-≈ instead. \""},{"Add":" #-}"},{"Add":""},{"Add":""},{"Add":"record IsTotalPreorder (_≲_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isPreorder : IsPreorder _≲_"},{"Add":" total : Total _≲_"},{"Add":""},{"Add":" open IsPreorder isPreorder public"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Partial orders"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record IsPartialOrder (_≤_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isPreorder : IsPreorder _≤_"},{"Add":" antisym : Antisymmetric _≈_ _≤_"},{"Add":""},{"Add":" open IsPreorder isPreorder public"},{"Add":" renaming"},{"Add":" ( ∼-respˡ-≈ to ≤-respˡ-≈"},{"Add":" ; ∼-respʳ-≈ to ≤-respʳ-≈"},{"Add":" ; ∼-resp-≈ to ≤-resp-≈"},{"Add":" )"},{"Add":""},{"Add":""},{"Add":"record IsDecPartialOrder (_≤_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where"},{"Add":" infix 4 _≟_ _≤?_"},{"Add":" field"},{"Add":" isPartialOrder : IsPartialOrder _≤_"},{"Add":" _≟_ : Decidable _≈_"},{"Add":" _≤?_ : Decidable _≤_"},{"Add":""},{"Add":" open IsPartialOrder isPartialOrder public"},{"Add":" hiding (module Eq)"},{"Add":""},{"Add":" module Eq where"},{"Add":""},{"Add":" isDecEquivalence : IsDecEquivalence"},{"Add":" isDecEquivalence = record"},{"Add":" { isEquivalence = isEquivalence"},{"Add":" ; _≟_ = _≟_"},{"Add":" }"},{"Add":""},{"Add":" open IsDecEquivalence isDecEquivalence public"},{"Add":""},{"Add":""},{"Add":"record IsStrictPartialOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isEquivalence : IsEquivalence"},{"Add":" irrefl : Irreflexive _≈_ _<_"},{"Add":" trans : Transitive _<_"},{"Add":" <-resp-≈ : _<_ Respects₂ _≈_"},{"Add":""},{"Add":" module Eq = IsEquivalence isEquivalence"},{"Add":""},{"Add":" asym : Asymmetric _<_"},{"Add":" asym {x} {y} = trans∧irr⇒asym Eq.refl trans irrefl {x = x} {y}"},{"Add":""},{"Add":" <-respʳ-≈ : _<_ Respectsʳ _≈_"},{"Add":" <-respʳ-≈ = proj₁ <-resp-≈"},{"Add":""},{"Add":" <-respˡ-≈ : _<_ Respectsˡ _≈_"},{"Add":" <-respˡ-≈ = proj₂ <-resp-≈"},{"Add":""},{"Add":""},{"Add":"record IsDecStrictPartialOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where"},{"Add":" infix 4 _≟_ _ : ∀ {x y} → x ≰ y → y < x"},{"Add":"≰⇒> = ToStrict.≰⇒> Eq.sym reflexive total"},{"Add":""},{"Add":"≰⇒≥ : ∀ {x y} → x ≰ y → y ≤ x"},{"Add":"≰⇒≥ x≰y = proj₁ (≰⇒> x≰y)"}]}},{"FileAdd":{"meta":{"basename":"StrictTotalOrder.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties satisfied by strict partial orders"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (StrictTotalOrder; DecTotalOrder)"},{"Add":""},{"Add":"module Relation.Binary.Properties.StrictTotalOrder"},{"Add":" {s₁ s₂ s₃} (STO : StrictTotalOrder s₁ s₂ s₃)"},{"Add":" where"},{"Add":""},{"Add":"open StrictTotalOrder STO"},{"Add":"open import Relation.Binary.Construct.StrictToNonStrict _≈_ _<_"},{"Add":"import Relation.Binary.Properties.StrictPartialOrder as SPO"},{"Add":"open import Relation.Binary.Consequences"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- _<_ - the strict version is a decidable total order"},{"Add":""},{"Add":"decTotalOrder : DecTotalOrder _ _ _"},{"Add":"decTotalOrder = record"},{"Add":" { isDecTotalOrder = isDecTotalOrder isStrictTotalOrder"},{"Add":" }"},{"Add":""},{"Add":"open DecTotalOrder decTotalOrder public"}]}},{"FileAdd":{"meta":{"basename":"StrictPartialOrder.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties satisfied by strict partial orders"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (StrictPartialOrder; Poset)"},{"Add":""},{"Add":"module Relation.Binary.Properties.StrictPartialOrder"},{"Add":" {s₁ s₂ s₃} (SPO : StrictPartialOrder s₁ s₂ s₃) where"},{"Add":""},{"Add":"import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd"},{"Add":"import Relation.Binary.Construct.StrictToNonStrict"},{"Add":""},{"Add":"open StrictPartialOrder SPO"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The converse relation is also a strict partial order."},{"Add":""},{"Add":">-strictPartialOrder : StrictPartialOrder s₁ s₂ s₃"},{"Add":">-strictPartialOrder = EqAndOrd.strictPartialOrder SPO"},{"Add":""},{"Add":"open StrictPartialOrder >-strictPartialOrder public"},{"Add":" using ()"},{"Add":" renaming"},{"Add":" ( _<_ to _>_"},{"Add":" ; irrefl to >-irrefl"},{"Add":" ; trans to >-trans"},{"Add":" ; <-resp-≈ to >-resp-≈"},{"Add":" ; isStrictPartialOrder to >-isStrictPartialOrder"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Strict partial orders can be converted to posets"},{"Add":""},{"Add":"open Relation.Binary.Construct.StrictToNonStrict _≈_ _<_"},{"Add":""},{"Add":"poset : Poset _ _ _"},{"Add":"poset = record"},{"Add":" { isPartialOrder = isPartialOrder isStrictPartialOrder"},{"Add":" }"},{"Add":""},{"Add":"open Poset poset public"}]}},{"FileAdd":{"meta":{"basename":"Setoid.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Additional properties for setoids"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base using (_∘_; id; _$_; flip)"},{"Add":"open import Relation.Nullary.Negation.Core using (¬_; contradiction)"},{"Add":"open import Relation.Binary.Core using (_⇒_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid; Preorder; Poset)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (Symmetric; _Respectsˡ_; _Respectsʳ_; _Respects₂_; Irreflexive)"},{"Add":"open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)"},{"Add":"open import Relation.Binary.Construct.Composition"},{"Add":" using (_;_; impliesˡ; transitive⇒≈;≈⊆≈)"},{"Add":""},{"Add":"module Relation.Binary.Properties.Setoid {a ℓ} (S : Setoid a ℓ) where"},{"Add":""},{"Add":"open Setoid S"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Every setoid is a preorder and partial order with respect to"},{"Add":"-- propositional equality"},{"Add":""},{"Add":"isPreorder : IsPreorder _≡_ _≈_"},{"Add":"isPreorder = record"},{"Add":" { isEquivalence = record"},{"Add":" { refl = ≡.refl"},{"Add":" ; sym = ≡.sym"},{"Add":" ; trans = ≡.trans"},{"Add":" }"},{"Add":" ; reflexive = reflexive"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":""},{"Add":"≈-isPreorder : IsPreorder _≈_ _≈_"},{"Add":"≈-isPreorder = record"},{"Add":" { isEquivalence = isEquivalence"},{"Add":" ; reflexive = id"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":""},{"Add":"≈-isPartialOrder : IsPartialOrder _≈_ _≈_"},{"Add":"≈-isPartialOrder = record"},{"Add":" { isPreorder = ≈-isPreorder"},{"Add":" ; antisym = λ i≈j _ → i≈j"},{"Add":" }"},{"Add":""},{"Add":"preorder : Preorder a a ℓ"},{"Add":"preorder = record"},{"Add":" { isPreorder = isPreorder"},{"Add":" }"},{"Add":""},{"Add":"≈-preorder : Preorder a ℓ ℓ"},{"Add":"≈-preorder = record"},{"Add":" { isPreorder = ≈-isPreorder"},{"Add":" }"},{"Add":""},{"Add":"≈-poset : Poset a ℓ ℓ"},{"Add":"≈-poset = record"},{"Add":" { isPartialOrder = ≈-isPartialOrder"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of _≉_"},{"Add":""},{"Add":"≉-sym : Symmetric _≉_"},{"Add":"≉-sym x≉y = x≉y ∘ sym"},{"Add":""},{"Add":"≉-respˡ : _≉_ Respectsˡ _≈_"},{"Add":"≉-respˡ x≈x′ x≉y = x≉y ∘ trans x≈x′"},{"Add":""},{"Add":"≉-respʳ : _≉_ Respectsʳ _≈_"},{"Add":"≉-respʳ y≈y′ x≉y x≈y′ = x≉y $ trans x≈y′ (sym y≈y′)"},{"Add":""},{"Add":"≉-resp₂ : _≉_ Respects₂ _≈_"},{"Add":"≉-resp₂ = ≉-respʳ , ≉-respˡ"},{"Add":""},{"Add":"≉-irrefl : Irreflexive _≈_ _≉_"},{"Add":"≉-irrefl x≈y x≉y = contradiction x≈y x≉y"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Equality is closed under composition"},{"Add":""},{"Add":"≈;≈⇒≈ : _≈_ ; _≈_ ⇒ _≈_"},{"Add":"≈;≈⇒≈ = transitive⇒≈;≈⊆≈ _ trans"},{"Add":""},{"Add":"≈⇒≈;≈ : _≈_ ⇒ _≈_ ; _≈_"},{"Add":"≈⇒≈;≈ = impliesˡ _≈_ _≈_ refl id"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Other properties"},{"Add":""},{"Add":"respʳ-flip : _≈_ Respectsʳ (flip _≈_)"},{"Add":"respʳ-flip y≈z x≈z = trans x≈z (sym y≈z)"},{"Add":""},{"Add":"respˡ-flip : _≈_ Respectsˡ (flip _≈_)"},{"Add":"respˡ-flip = trans"}]}},{"FileAdd":{"meta":{"basename":"Preorder.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties satisfied by preorders"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (Preorder; Setoid)"},{"Add":"open import Relation.Binary.Structures using (IsPreorder)"},{"Add":""},{"Add":"module Relation.Binary.Properties.Preorder"},{"Add":" {p₁ p₂ p₃} (P : Preorder p₁ p₂ p₃) where"},{"Add":""},{"Add":"open import Function.Base using (flip)"},{"Add":"open import Data.Product.Base as Product using (_×_; _,_; swap)"},{"Add":"import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd"},{"Add":""},{"Add":"open Preorder P"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The converse relation is also a preorder."},{"Add":""},{"Add":"converse-isPreorder : IsPreorder _≈_ _≳_"},{"Add":"converse-isPreorder = EqAndOrd.isPreorder isPreorder"},{"Add":""},{"Add":"converse-preorder : Preorder p₁ p₂ p₃"},{"Add":"converse-preorder = EqAndOrd.preorder P"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- For every preorder there is an induced equivalence"},{"Add":""},{"Add":"InducedEquivalence : Setoid _ _"},{"Add":"InducedEquivalence = record"},{"Add":" { _≈_ = λ x y → x ≲ y × x ≳ y"},{"Add":" ; isEquivalence = record"},{"Add":" { refl = (refl , refl)"},{"Add":" ; sym = swap"},{"Add":" ; trans = Product.zip trans (flip trans)"},{"Add":" }"},{"Add":" }"},{"Add":""},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- DEPRECATED NAMES"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Please use the new names as continuing support for the old names is"},{"Add":"-- not guaranteed."},{"Add":""},{"Add":"-- Version 2.0"},{"Add":""},{"Add":"invIsPreorder = converse-isPreorder"},{"Add":"{-# WARNING_ON_USAGE invIsPreorder"},{"Add":"\"Warning: invIsPreorder was deprecated in v2.0."},{"Add":"Please use converse-isPreorder instead.\""},{"Add":"#-}"},{"Add":"invPreorder = converse-preorder"},{"Add":"{-# WARNING_ON_USAGE invPreorder"},{"Add":"\"Warning: invPreorder was deprecated in v2.0."},{"Add":"Please use converse-preorder instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Poset.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties satisfied by posets"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base using (flip; _∘_)"},{"Add":"open import Relation.Binary.Core using (Rel; _Preserves_⟶_)"},{"Add":"open import Relation.Binary.Bundles using (Poset; StrictPartialOrder)"},{"Add":"open import Relation.Binary.Structures"},{"Add":" using (IsPartialOrder; IsStrictPartialOrder; IsDecPartialOrder)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (_Respectsˡ_; _Respectsʳ_; Decidable)"},{"Add":"import Relation.Binary.Consequences as Consequences"},{"Add":"open import Relation.Nullary using (¬_; yes; no)"},{"Add":"open import Relation.Nullary.Negation using (contradiction)"},{"Add":""},{"Add":"module Relation.Binary.Properties.Poset"},{"Add":" {p₁ p₂ p₃} (P : Poset p₁ p₂ p₃) where"},{"Add":""},{"Add":"open Poset P renaming (Carrier to A)"},{"Add":""},{"Add":"import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_ as ToStrict"},{"Add":"import Relation.Binary.Properties.Preorder preorder as PreorderProperties"},{"Add":"open Eq using (_≉_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The _≥_ relation is also a poset."},{"Add":""},{"Add":"open PreorderProperties public"},{"Add":" using () renaming"},{"Add":" ( converse-isPreorder to ≥-isPreorder"},{"Add":" ; converse-preorder to ≥-preorder"},{"Add":" )"},{"Add":""},{"Add":"≥-isPartialOrder : IsPartialOrder _≈_ _≥_"},{"Add":"≥-isPartialOrder = record"},{"Add":" { isPreorder = ≥-isPreorder"},{"Add":" ; antisym = flip antisym"},{"Add":" }"},{"Add":""},{"Add":"≥-poset : Poset p₁ p₂ p₃"},{"Add":"≥-poset = record"},{"Add":" { isPartialOrder = ≥-isPartialOrder"},{"Add":" }"},{"Add":""},{"Add":"open Poset ≥-poset public"},{"Add":" using () renaming"},{"Add":" ( refl to ≥-refl"},{"Add":" ; reflexive to ≥-reflexive"},{"Add":" ; trans to ≥-trans"},{"Add":" ; antisym to ≥-antisym"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Negated order"},{"Add":""},{"Add":"≰-respˡ-≈ : _≰_ Respectsˡ _≈_"},{"Add":"≰-respˡ-≈ x≈y = _∘ ≤-respˡ-≈ (Eq.sym x≈y)"},{"Add":""},{"Add":"≰-respʳ-≈ : _≰_ Respectsʳ _≈_"},{"Add":"≰-respʳ-≈ x≈y = _∘ ≤-respʳ-≈ (Eq.sym x≈y)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Partial orders can be turned into strict partial orders"},{"Add":""},{"Add":"infix 4 _<_"},{"Add":""},{"Add":"_<_ : Rel A _"},{"Add":"_<_ = ToStrict._<_"},{"Add":""},{"Add":"<-isStrictPartialOrder : IsStrictPartialOrder _≈_ _<_"},{"Add":"<-isStrictPartialOrder = ToStrict.<-isStrictPartialOrder isPartialOrder"},{"Add":""},{"Add":"<-strictPartialOrder : StrictPartialOrder _ _ _"},{"Add":"<-strictPartialOrder = record"},{"Add":" { isStrictPartialOrder = <-isStrictPartialOrder"},{"Add":" }"},{"Add":""},{"Add":"open StrictPartialOrder <-strictPartialOrder public"},{"Add":" using (_≮_; <-resp-≈; <-respʳ-≈; <-respˡ-≈)"},{"Add":" renaming"},{"Add":" ( irrefl to <-irrefl"},{"Add":" ; asym to <-asym"},{"Add":" ; trans to <-trans"},{"Add":" )"},{"Add":""},{"Add":"<⇒≉ : ∀ {x y} → x < y → x ≉ y"},{"Add":"<⇒≉ = ToStrict.<⇒≉"},{"Add":""},{"Add":"≤∧≉⇒< : ∀ {x y} → x ≤ y → x ≉ y → x < y"},{"Add":"≤∧≉⇒< = ToStrict.≤∧≉⇒<"},{"Add":""},{"Add":"<⇒≱ : ∀ {x y} → x < y → y ≰ x"},{"Add":"<⇒≱ = ToStrict.<⇒≱ antisym"},{"Add":""},{"Add":"≤⇒≯ : ∀ {x y} → x ≤ y → y ≮ x"},{"Add":"≤⇒≯ = ToStrict.≤⇒≯ antisym"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- If ≤ is decidable then so is ≈"},{"Add":""},{"Add":"≤-dec⇒≈-dec : Decidable _≤_ → Decidable _≈_"},{"Add":"≤-dec⇒≈-dec _≤?_ x y with x ≤? y | y ≤? x"},{"Add":"... | yes x≤y | yes y≤x = yes (antisym x≤y y≤x)"},{"Add":"... | yes x≤y | no y≰x = no λ x≈y → contradiction (reflexive (Eq.sym x≈y)) y≰x"},{"Add":"... | no x≰y | _ = no λ x≈y → contradiction (reflexive x≈y) x≰y"},{"Add":""},{"Add":"≤-dec⇒isDecPartialOrder : Decidable _≤_ → IsDecPartialOrder _≈_ _≤_"},{"Add":"≤-dec⇒isDecPartialOrder _≤?_ = record"},{"Add":" { isPartialOrder = isPartialOrder"},{"Add":" ; _≟_ = ≤-dec⇒≈-dec _≤?_"},{"Add":" ; _≤?_ = _≤?_"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Other properties"},{"Add":""},{"Add":"mono⇒cong : ∀ {f} → f Preserves _≤_ ⟶ _≤_ → f Preserves _≈_ ⟶ _≈_"},{"Add":"mono⇒cong = Consequences.mono⇒cong _≈_ _≈_ Eq.sym reflexive antisym"},{"Add":""},{"Add":"antimono⇒cong : ∀ {f} → f Preserves _≤_ ⟶ _≥_ → f Preserves _≈_ ⟶ _≈_"},{"Add":"antimono⇒cong = Consequences.antimono⇒cong _≈_ _≈_ Eq.sym reflexive antisym"}]}},{"FileAdd":{"meta":{"basename":"MeetSemilattice.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use"},{"Add":"-- `Relation.Binary.Lattice.Properties.MeetSemilattice` instead."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Properties.MeetSemilattice"},{"Add":" {c ℓ₁ ℓ₂} (M : MeetSemilattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open import Relation.Binary.Lattice.Properties.MeetSemilattice M public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Relation.Binary.Properties.MeetSemilattice was deprecated in v2.0."},{"Add":"Use Relation.Binary.Lattice.Properties.MeetSemilattice instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Lattice.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use"},{"Add":"-- `Relation.Binary.Lattice.Properties.Lattice` instead."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Properties.Lattice"},{"Add":" {c ℓ₁ ℓ₂} (L : Lattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open import Relation.Binary.Lattice.Properties.Lattice L public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Relation.Binary.Properties.Lattice was deprecated in v2.0."},{"Add":"Use Relation.Binary.Lattice.Properties.Lattice instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"JoinSemilattice.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use"},{"Add":"-- `Relation.Binary.Lattice.Properties.JoinSemilattice` instead."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Properties.JoinSemilattice"},{"Add":" {c ℓ₁ ℓ₂} (J : JoinSemilattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open import Relation.Binary.Lattice.Properties.JoinSemilattice J public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Relation.Binary.Properties.JoinSemilattice was deprecated in v2.0."},{"Add":"Use Relation.Binary.Lattice.Properties.JoinSemilattice instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"HeytingAlgebra.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use"},{"Add":"-- `Relation.Binary.Lattice.Properties.HeytingAlgebra` instead."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice using (HeytingAlgebra)"},{"Add":""},{"Add":"module Relation.Binary.Properties.HeytingAlgebra"},{"Add":" {c ℓ₁ ℓ₂} (L : HeytingAlgebra c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open import Relation.Binary.Lattice.Properties.HeytingAlgebra L public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Relation.Binary.Properties.HeytingAlgebra was deprecated in v2.0."},{"Add":"Use Relation.Binary.Lattice.Properties.HeytingAlgebra instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"DistributiveLattice.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use"},{"Add":"-- `Relation.Binary.Lattice.Properties.DistributiveLattice` instead."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Properties.DistributiveLattice"},{"Add":" {c ℓ₁ ℓ₂} (L : DistributiveLattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open import Relation.Binary.Lattice.Properties.DistributiveLattice L public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Relation.Binary.Properties.DistributiveLattice was deprecated in v2.0."},{"Add":"Use Relation.Binary.Lattice.Properties.DistributiveLattice instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"DecTotalOrder.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties satisfied by decidable total orders"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Structures"},{"Add":" using (IsDecTotalOrder; IsStrictTotalOrder)"},{"Add":"open import Relation.Binary.Bundles"},{"Add":" using (DecTotalOrder; StrictTotalOrder)"},{"Add":""},{"Add":"module Relation.Binary.Properties.DecTotalOrder"},{"Add":" {d₁ d₂ d₃} (DTO : DecTotalOrder d₁ d₂ d₃) where"},{"Add":""},{"Add":"open DecTotalOrder DTO hiding (trans)"},{"Add":""},{"Add":"import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd"},{"Add":"import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_ as ToStrict"},{"Add":"import Relation.Binary.Properties.TotalOrder totalOrder as TotalOrderProperties"},{"Add":"open import Relation.Nullary.Negation using (¬_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- _≥_ - the flipped relation is also a total order"},{"Add":""},{"Add":"open TotalOrderProperties public"},{"Add":" using"},{"Add":" ( ≥-refl"},{"Add":" ; ≥-reflexive"},{"Add":" ; ≥-trans"},{"Add":" ; ≥-antisym"},{"Add":" ; ≥-total"},{"Add":" ; ≥-isPreorder"},{"Add":" ; ≥-isPartialOrder"},{"Add":" ; ≥-isTotalOrder"},{"Add":" ; ≥-preorder"},{"Add":" ; ≥-poset"},{"Add":" ; ≥-totalOrder"},{"Add":" )"},{"Add":""},{"Add":"≥-isDecTotalOrder : IsDecTotalOrder _≈_ _≥_"},{"Add":"≥-isDecTotalOrder = EqAndOrd.isDecTotalOrder isDecTotalOrder"},{"Add":""},{"Add":"≥-decTotalOrder : DecTotalOrder _ _ _"},{"Add":"≥-decTotalOrder = record"},{"Add":" { isDecTotalOrder = ≥-isDecTotalOrder"},{"Add":" }"},{"Add":""},{"Add":"open DecTotalOrder ≥-decTotalOrder public"},{"Add":" using () renaming (_≤?_ to _≥?_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- _<_ - the strict version is a strict total order"},{"Add":""},{"Add":"open TotalOrderProperties public"},{"Add":" using"},{"Add":" ( _<_"},{"Add":" ; <-resp-≈"},{"Add":" ; <-respʳ-≈"},{"Add":" ; <-respˡ-≈"},{"Add":" ; <-irrefl"},{"Add":" ; <-asym"},{"Add":" ; <-trans"},{"Add":" ; <-isStrictPartialOrder"},{"Add":" ; <-strictPartialOrder"},{"Add":" ; <⇒≉"},{"Add":" ; ≤∧≉⇒<"},{"Add":" ; <⇒≱"},{"Add":" ; ≤⇒≯"},{"Add":" )"},{"Add":""},{"Add":"<-isStrictTotalOrder : IsStrictTotalOrder _≈_ _<_"},{"Add":"<-isStrictTotalOrder = ToStrict.<-isStrictTotalOrder₂ isDecTotalOrder"},{"Add":""},{"Add":"<-strictTotalOrder : StrictTotalOrder _ _ _"},{"Add":"<-strictTotalOrder = record"},{"Add":" { isStrictTotalOrder = <-isStrictTotalOrder"},{"Add":" }"},{"Add":""},{"Add":"open StrictTotalOrder <-strictTotalOrder public"},{"Add":" using (_≮_) renaming (compare to <-compare)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- _≰_ - the negated order"},{"Add":""},{"Add":"open TotalOrderProperties public"},{"Add":" using"},{"Add":" ( ≰-respʳ-≈"},{"Add":" ; ≰-respˡ-≈"},{"Add":" ; ≰⇒>"},{"Add":" ; ≰⇒≥"},{"Add":" )"},{"Add":""},{"Add":"≮⇒≥ : ∀ {x y} → x ≮ y → y ≤ x"},{"Add":"≮⇒≥ = ToStrict.≮⇒≥ Eq.sym _≟_ reflexive total"}]}},{"FileAdd":{"meta":{"basename":"DecSetoid.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Every decidable setoid induces tight apartness relation."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (DecSetoid; ApartnessRelation)"},{"Add":""},{"Add":"module Relation.Binary.Properties.DecSetoid {c ℓ} (S : DecSetoid c ℓ) where"},{"Add":""},{"Add":"open import Data.Product using (_,_)"},{"Add":"open import Data.Sum using (inj₁; inj₂)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (Cotransitive; Tight)"},{"Add":"import Relation.Binary.Properties.Setoid as SetoidProperties"},{"Add":"open import Relation.Binary.Structures"},{"Add":" using (IsApartnessRelation; IsDecEquivalence)"},{"Add":"open import Relation.Nullary.Decidable.Core"},{"Add":" using (yes; no; decidable-stable)"},{"Add":""},{"Add":"open DecSetoid S using (_≈_; _≉_; _≟_; setoid; trans)"},{"Add":"open SetoidProperties setoid"},{"Add":""},{"Add":"≉-cotrans : Cotransitive _≉_"},{"Add":"≉-cotrans {x} {y} x≉y z with x ≟ z | z ≟ y"},{"Add":"... | _ | no z≉y = inj₂ z≉y"},{"Add":"... | no x≉z | _ = inj₁ x≉z"},{"Add":"... | yes x≈z | yes z≈y = inj₁ λ _ → x≉y (trans x≈z z≈y)"},{"Add":""},{"Add":"≉-isApartnessRelation : IsApartnessRelation _≈_ _≉_"},{"Add":"≉-isApartnessRelation = record"},{"Add":" { irrefl = ≉-irrefl"},{"Add":" ; sym = ≉-sym"},{"Add":" ; cotrans = ≉-cotrans"},{"Add":" }"},{"Add":""},{"Add":"≉-apartnessRelation : ApartnessRelation c ℓ ℓ"},{"Add":"≉-apartnessRelation = record { isApartnessRelation = ≉-isApartnessRelation }"},{"Add":""},{"Add":"≉-tight : Tight _≈_ _≉_"},{"Add":"≉-tight x y = decidable-stable (x ≟ y) , ≉-irrefl"}]}},{"FileAdd":{"meta":{"basename":"BoundedMeetSemilattice.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use"},{"Add":"-- `Relation.Binary.Lattice.Properties.BoundedMeetSemilattice` instead."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Properties.BoundedMeetSemilattice"},{"Add":" {c ℓ₁ ℓ₂} (M : BoundedMeetSemilattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open import Relation.Binary.Lattice.Properties.BoundedMeetSemilattice M public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Relation.Binary.Properties.BoundedMeetSemilattice was deprecated in v2.0."},{"Add":"Use Relation.Binary.Lattice.Properties.BoundedMeetSemilattice instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"BoundedLattice.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use"},{"Add":"-- `Relation.Binary.Lattice.Properties.BoundedLattice` instead."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Properties.BoundedLattice"},{"Add":" {c ℓ₁ ℓ₂} (L : BoundedLattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open import Relation.Binary.Lattice.Properties.BoundedLattice L public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Relation.Binary.Properties.BoundedLattice was deprecated in v2.0."},{"Add":"Use Relation.Binary.Lattice.Properties.BoundedLattice instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"BoundedJoinSemilattice.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use"},{"Add":"-- `Relation.Binary.Lattice.Properties.BoundedJoinSemilattice` instead."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Properties.BoundedJoinSemilattice"},{"Add":" {c ℓ₁ ℓ₂} (J : BoundedJoinSemilattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open import Relation.Binary.Lattice.Properties.BoundedJoinSemilattice J public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Relation.Binary.Properties.BoundedJoinSemilattice was deprecated in v2.0."},{"Add":"Use Relation.Binary.Lattice.Properties.BoundedJoinSemilattice instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"ApartnessRelation.agda","metadata":0},"context":[{"change":0,"pos":497280}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Apartness properties"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":""},{"Add":"module Relation.Binary.Properties.ApartnessRelation"},{"Add":" {a ℓ₁ ℓ₂} {A : Set a}"},{"Add":" {_≈_ : Rel A ℓ₁}"},{"Add":" {_#_ : Rel A ℓ₂}"},{"Add":" where"},{"Add":""},{"Add":"open import Function.Base using (_∘₂_)"},{"Add":"open import Relation.Binary.Definitions using (Reflexive)"},{"Add":"open import Relation.Binary.Consequences using (sym⇒¬-sym; cotrans⇒¬-trans)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence; IsApartnessRelation)"},{"Add":"open import Relation.Nullary.Negation using (¬_)"},{"Add":""},{"Add":"¬#-isEquivalence : Reflexive _≈_ → IsApartnessRelation _≈_ _#_ →"},{"Add":" IsEquivalence (¬_ ∘₂ _#_)"},{"Add":"¬#-isEquivalence re apart = record"},{"Add":" { refl = irrefl re"},{"Add":" ; sym = λ {a} {b} → sym⇒¬-sym sym {a} {b}"},{"Add":" ; trans = cotrans⇒¬-trans cotrans"},{"Add":" } where open IsApartnessRelation apart"}]}},{"FileAdd":{"meta":{"basename":"OrderMorphism.agda","metadata":0},"context":[{"change":0,"pos":415495}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use `Relation.Binary.Morphism`"},{"Add":"-- instead."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.OrderMorphism where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Relation.Binary.OrderMorphism was deprecated in v1.5."},{"Add":"Use Relation.Binary.Reasoning.Morphism instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (_=[_]⇒_)"},{"Add":"open import Relation.Binary.Bundles using (Poset; DecTotalOrder)"},{"Add":"open Poset"},{"Add":"import Function.Base as F"},{"Add":"open import Level"},{"Add":""},{"Add":"record _⇒-Poset_ {p₁ p₂ p₃ p₄ p₅ p₆}"},{"Add":" (P₁ : Poset p₁ p₂ p₃)"},{"Add":" (P₂ : Poset p₄ p₅ p₆) : Set (p₁ ⊔ p₃ ⊔ p₄ ⊔ p₆) where"},{"Add":" field"},{"Add":" fun : Carrier P₁ → Carrier P₂"},{"Add":" monotone : _≤_ P₁ =[ fun ]⇒ _≤_ P₂"},{"Add":""},{"Add":""},{"Add":"_⇒-DTO_ : ∀ {p₁ p₂ p₃ p₄ p₅ p₆} →"},{"Add":" DecTotalOrder p₁ p₂ p₃ →"},{"Add":" DecTotalOrder p₄ p₅ p₆ → Set _"},{"Add":"DTO₁ ⇒-DTO DTO₂ = poset DTO₁ ⇒-Poset poset DTO₂"},{"Add":" where open DecTotalOrder"},{"Add":""},{"Add":"open _⇒-Poset_"},{"Add":""},{"Add":"id : ∀ {p₁ p₂ p₃} {P : Poset p₁ p₂ p₃} → P ⇒-Poset P"},{"Add":"id = record"},{"Add":" { fun = F.id"},{"Add":" ; monotone = F.id"},{"Add":" }"},{"Add":""},{"Add":"_∘_ : ∀ {p₁ p₂ p₃ p₄ p₅ p₆ p₇ p₈ p₉}"},{"Add":" {P₁ : Poset p₁ p₂ p₃}"},{"Add":" {P₂ : Poset p₄ p₅ p₆}"},{"Add":" {P₃ : Poset p₇ p₈ p₉} →"},{"Add":" P₂ ⇒-Poset P₃ → P₁ ⇒-Poset P₂ → P₁ ⇒-Poset P₃"},{"Add":"f ∘ g = record"},{"Add":" { fun = F._∘_ (fun f) (fun g)"},{"Add":" ; monotone = F._∘_ (monotone f) (monotone g)"},{"Add":" }"},{"Add":""},{"Add":"const : ∀ {p₁ p₂ p₃ p₄ p₅ p₆}"},{"Add":" {P₁ : Poset p₁ p₂ p₃}"},{"Add":" {P₂ : Poset p₄ p₅ p₆} →"},{"Add":" Carrier P₂ → P₁ ⇒-Poset P₂"},{"Add":"const {P₂ = P₂} x = record"},{"Add":" { fun = F.const x"},{"Add":" ; monotone = F.const (refl P₂)"},{"Add":" }"},{"Add":""},{"Add":"{-# WARNING_ON_USAGE _⇒-Poset_"},{"Add":"\"Warning: _⇒-Poset_ was deprecated in v1.5."},{"Add":"Please use `IsOrderHomomorphism` from `Relation.Binary.Morphism.Structures` instead.\""},{"Add":"#-}"},{"Add":"{-# WARNING_ON_USAGE _⇒-DTO_"},{"Add":"\"Warning: _⇒-DTO_ was deprecated in v1.5."},{"Add":"Please use `IsOrderHomomorphism` from `Relation.Binary.Morphism.Structures` instead.\""},{"Add":"#-}"},{"Add":"{-# WARNING_ON_USAGE id"},{"Add":"\"Warning: id was deprecated in v1.5."},{"Add":"Please use `issOrderHomomorphism` from `Relation.Binary.Morphism.Construct.Constant` instead.\""},{"Add":"#-}"},{"Add":"{-# WARNING_ON_USAGE _∘_"},{"Add":"\"Warning: _∘_ was deprecated in v1.5."},{"Add":"Please use `isOrderHomomorphism` from `Relation.Binary.Morphism.Construct.Composition` instead.\""},{"Add":"#-}"},{"Add":"{-# WARNING_ON_USAGE const"},{"Add":"\"Warning: const was deprecated in v1.5."},{"Add":"Please use `isOrderHomomorphism` from `Relation.Binary.Morphism.Construct.Constant` instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Morphism.agda","metadata":0},"context":[{"change":0,"pos":415495}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Order morphisms"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core"},{"Add":""},{"Add":"module Relation.Binary.Morphism where"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export contents of morphisms"},{"Add":""},{"Add":"open import Relation.Binary.Morphism.Definitions public"},{"Add":"open import Relation.Binary.Morphism.Structures public"},{"Add":"open import Relation.Binary.Morphism.Bundles public"}]}},{"FileAdd":{"meta":{"basename":"Morphism","metadata":576},"context":[{"change":0,"pos":415495}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Structures.agda","metadata":0},"context":[{"change":0,"pos":526801}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Order morphisms"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":""},{"Add":"module Relation.Binary.Morphism.Structures"},{"Add":" {a b} {A : Set a} {B : Set b}"},{"Add":" where"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Definitions"},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Relation.Binary.Morphism.Definitions A B"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record IsRelHomomorphism (_∼₁_ : Rel A ℓ₁) (_∼₂_ : Rel B ℓ₂)"},{"Add":" (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" cong : Homomorphic₂ _∼₁_ _∼₂_ ⟦_⟧"},{"Add":""},{"Add":""},{"Add":"record IsRelMonomorphism (_∼₁_ : Rel A ℓ₁) (_∼₂_ : Rel B ℓ₂)"},{"Add":" (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isHomomorphism : IsRelHomomorphism _∼₁_ _∼₂_ ⟦_⟧"},{"Add":" injective : Injective _∼₁_ _∼₂_ ⟦_⟧"},{"Add":""},{"Add":" open IsRelHomomorphism isHomomorphism public"},{"Add":""},{"Add":""},{"Add":"record IsRelIsomorphism (_∼₁_ : Rel A ℓ₁) (_∼₂_ : Rel B ℓ₂)"},{"Add":" (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isMonomorphism : IsRelMonomorphism _∼₁_ _∼₂_ ⟦_⟧"},{"Add":" surjective : Surjective _∼₁_ _∼₂_ ⟦_⟧"},{"Add":""},{"Add":" open IsRelMonomorphism isMonomorphism public"},{"Add":""},{"Add":" bijective : Bijective _∼₁_ _∼₂_ ⟦_⟧"},{"Add":" bijective = injective , surjective"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Orders"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record IsOrderHomomorphism (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂)"},{"Add":" (_≲₁_ : Rel A ℓ₃) (_≲₂_ : Rel B ℓ₄)"},{"Add":" (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄)"},{"Add":" where"},{"Add":" field"},{"Add":" cong : Homomorphic₂ _≈₁_ _≈₂_ ⟦_⟧"},{"Add":" mono : Homomorphic₂ _≲₁_ _≲₂_ ⟦_⟧"},{"Add":""},{"Add":" module Eq where"},{"Add":" isRelHomomorphism : IsRelHomomorphism _≈₁_ _≈₂_ ⟦_⟧"},{"Add":" isRelHomomorphism = record { cong = cong }"},{"Add":""},{"Add":" isRelHomomorphism : IsRelHomomorphism _≲₁_ _≲₂_ ⟦_⟧"},{"Add":" isRelHomomorphism = record { cong = mono }"},{"Add":""},{"Add":""},{"Add":"record IsOrderMonomorphism (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂)"},{"Add":" (_≲₁_ : Rel A ℓ₃) (_≲₂_ : Rel B ℓ₄)"},{"Add":" (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄)"},{"Add":" where"},{"Add":" field"},{"Add":" isOrderHomomorphism : IsOrderHomomorphism _≈₁_ _≈₂_ _≲₁_ _≲₂_ ⟦_⟧"},{"Add":" injective : Injective _≈₁_ _≈₂_ ⟦_⟧"},{"Add":" cancel : Injective _≲₁_ _≲₂_ ⟦_⟧"},{"Add":""},{"Add":" open IsOrderHomomorphism isOrderHomomorphism public"},{"Add":" hiding (module Eq)"},{"Add":""},{"Add":" module Eq where"},{"Add":" isRelMonomorphism : IsRelMonomorphism _≈₁_ _≈₂_ ⟦_⟧"},{"Add":" isRelMonomorphism = record"},{"Add":" { isHomomorphism = IsOrderHomomorphism.Eq.isRelHomomorphism isOrderHomomorphism"},{"Add":" ; injective = injective"},{"Add":" }"},{"Add":""},{"Add":" isRelMonomorphism : IsRelMonomorphism _≲₁_ _≲₂_ ⟦_⟧"},{"Add":" isRelMonomorphism = record"},{"Add":" { isHomomorphism = isRelHomomorphism"},{"Add":" ; injective = cancel"},{"Add":" }"},{"Add":""},{"Add":""},{"Add":"record IsOrderIsomorphism (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂)"},{"Add":" (_≲₁_ : Rel A ℓ₃) (_≲₂_ : Rel B ℓ₄)"},{"Add":" (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄)"},{"Add":" where"},{"Add":" field"},{"Add":" isOrderMonomorphism : IsOrderMonomorphism _≈₁_ _≈₂_ _≲₁_ _≲₂_ ⟦_⟧"},{"Add":" surjective : Surjective _≈₁_ _≈₂_ ⟦_⟧"},{"Add":""},{"Add":" open IsOrderMonomorphism isOrderMonomorphism public"},{"Add":" hiding (module Eq)"},{"Add":""},{"Add":" module Eq where"},{"Add":" isRelIsomorphism : IsRelIsomorphism _≈₁_ _≈₂_ ⟦_⟧"},{"Add":" isRelIsomorphism = record"},{"Add":" { isMonomorphism = IsOrderMonomorphism.Eq.isRelMonomorphism isOrderMonomorphism"},{"Add":" ; surjective = surjective"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"RelMonomorphism.agda","metadata":0},"context":[{"change":0,"pos":526801}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Consequences of a monomorphism between binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- See Data.Nat.Binary.Properties for examples of how this and similar"},{"Add":"-- modules can be used to easily translate properties between types."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Function.Base"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence; IsDecEquivalence)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (Reflexive; Symmetric; Transitive; Total; Asymmetric; Decidable)"},{"Add":"open import Relation.Binary.Morphism"},{"Add":""},{"Add":"module Relation.Binary.Morphism.RelMonomorphism"},{"Add":" {a b ℓ₁ ℓ₂} {A : Set a} {B : Set b}"},{"Add":" {_∼₁_ : Rel A ℓ₁} {_∼₂_ : Rel B ℓ₂}"},{"Add":" {⟦_⟧ : A → B} (isMonomorphism : IsRelMonomorphism _∼₁_ _∼₂_ ⟦_⟧)"},{"Add":" where"},{"Add":""},{"Add":"open import Data.Sum.Base as Sum"},{"Add":"open import Relation.Nullary.Decidable using (yes; no)"},{"Add":"open import Relation.Nullary.Decidable"},{"Add":""},{"Add":"open IsRelMonomorphism isMonomorphism"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"refl : Reflexive _∼₂_ → Reflexive _∼₁_"},{"Add":"refl refl = injective refl"},{"Add":""},{"Add":"sym : Symmetric _∼₂_ → Symmetric _∼₁_"},{"Add":"sym sym x∼y = injective (sym (cong x∼y))"},{"Add":""},{"Add":"trans : Transitive _∼₂_ → Transitive _∼₁_"},{"Add":"trans trans x∼y y∼z = injective (trans (cong x∼y) (cong y∼z))"},{"Add":""},{"Add":"total : Total _∼₂_ → Total _∼₁_"},{"Add":"total total x y = Sum.map injective injective (total ⟦ x ⟧ ⟦ y ⟧)"},{"Add":""},{"Add":"asym : Asymmetric _∼₂_ → Asymmetric _∼₁_"},{"Add":"asym asym x∼y y∼x = asym (cong x∼y) (cong y∼x)"},{"Add":""},{"Add":"dec : Decidable _∼₂_ → Decidable _∼₁_"},{"Add":"dec _∼?_ x y = map′ injective cong (⟦ x ⟧ ∼? ⟦ y ⟧)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"isEquivalence : IsEquivalence _∼₂_ → IsEquivalence _∼₁_"},{"Add":"isEquivalence isEq = record"},{"Add":" { refl = refl E.refl"},{"Add":" ; sym = sym E.sym"},{"Add":" ; trans = trans E.trans"},{"Add":" } where module E = IsEquivalence isEq"},{"Add":""},{"Add":"isDecEquivalence : IsDecEquivalence _∼₂_ → IsDecEquivalence _∼₁_"},{"Add":"isDecEquivalence isDecEq = record"},{"Add":" { isEquivalence = isEquivalence E.isEquivalence"},{"Add":" ; _≟_ = dec E._≟_"},{"Add":" } where module E = IsDecEquivalence isDecEq"}]}},{"FileAdd":{"meta":{"basename":"OrderMonomorphism.agda","metadata":0},"context":[{"change":0,"pos":526801}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Consequences of a monomorphism between orders"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- See Data.Nat.Binary.Properties for examples of how this and similar"},{"Add":"-- modules can be used to easily translate properties between types."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Algebra.Morphism.Definitions"},{"Add":"open import Function.Base"},{"Add":"open import Data.Product.Base using (_,_; map)"},{"Add":"open import Relation.Binary.Core using (Rel; _⇒_)"},{"Add":"open import Relation.Binary.Structures"},{"Add":" using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (Irreflexive; Antisymmetric; Trichotomous; tri<; tri≈; tri>; _Respectsˡ_; _Respectsʳ_; _Respects₂_)"},{"Add":"open import Relation.Binary.Morphism"},{"Add":"import Relation.Binary.Morphism.RelMonomorphism as RawRelation"},{"Add":""},{"Add":"module Relation.Binary.Morphism.OrderMonomorphism"},{"Add":" {a b ℓ₁ ℓ₂ ℓ₃ ℓ₄} {A : Set a} {B : Set b}"},{"Add":" {_≈₁_ : Rel A ℓ₁} {_≈₂_ : Rel B ℓ₃}"},{"Add":" {_≲₁_ : Rel A ℓ₂} {_≲₂_ : Rel B ℓ₄}"},{"Add":" {⟦_⟧ : A → B}"},{"Add":" (isOrderMonomorphism : IsOrderMonomorphism _≈₁_ _≈₂_ _≲₁_ _≲₂_ ⟦_⟧)"},{"Add":" where"},{"Add":""},{"Add":"open IsOrderMonomorphism isOrderMonomorphism"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export equivalence proofs"},{"Add":""},{"Add":"module EqM = RawRelation Eq.isRelMonomorphism"},{"Add":""},{"Add":"open RawRelation isRelMonomorphism public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"reflexive : _≈₂_ ⇒ _≲₂_ → _≈₁_ ⇒ _≲₁_"},{"Add":"reflexive refl x≈y = cancel (refl (cong x≈y))"},{"Add":""},{"Add":"irrefl : Irreflexive _≈₂_ _≲₂_ → Irreflexive _≈₁_ _≲₁_"},{"Add":"irrefl irrefl x≈y x∼y = irrefl (cong x≈y) (mono x∼y)"},{"Add":""},{"Add":"antisym : Antisymmetric _≈₂_ _≲₂_ → Antisymmetric _≈₁_ _≲₁_"},{"Add":"antisym antisym x∼y y∼x = injective (antisym (mono x∼y) (mono y∼x))"},{"Add":""},{"Add":"compare : Trichotomous _≈₂_ _≲₂_ → Trichotomous _≈₁_ _≲₁_"},{"Add":"compare compare x y with compare ⟦ x ⟧ ⟦ y ⟧"},{"Add":"... | tri< a ¬b ¬c = tri< (cancel a) (¬b ∘ cong) (¬c ∘ mono)"},{"Add":"... | tri≈ ¬a b ¬c = tri≈ (¬a ∘ mono) (injective b) (¬c ∘ mono)"},{"Add":"... | tri> ¬a ¬b c = tri> (¬a ∘ mono) (¬b ∘ cong) (cancel c)"},{"Add":""},{"Add":"respˡ : _≲₂_ Respectsˡ _≈₂_ → _≲₁_ Respectsˡ _≈₁_"},{"Add":"respˡ resp x≈y x∼z = cancel (resp (cong x≈y) (mono x∼z))"},{"Add":""},{"Add":"respʳ : _≲₂_ Respectsʳ _≈₂_ → _≲₁_ Respectsʳ _≈₁_"},{"Add":"respʳ resp x≈y y∼z = cancel (resp (cong x≈y) (mono y∼z))"},{"Add":""},{"Add":"resp : _≲₂_ Respects₂ _≈₂_ → _≲₁_ Respects₂ _≈₁_"},{"Add":"resp = map respʳ respˡ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"isPreorder : IsPreorder _≈₂_ _≲₂_ → IsPreorder _≈₁_ _≲₁_"},{"Add":"isPreorder O = record"},{"Add":" { isEquivalence = EqM.isEquivalence O.isEquivalence"},{"Add":" ; reflexive = reflexive O.reflexive"},{"Add":" ; trans = trans O.trans"},{"Add":" } where module O = IsPreorder O"},{"Add":""},{"Add":"isPartialOrder : IsPartialOrder _≈₂_ _≲₂_ → IsPartialOrder _≈₁_ _≲₁_"},{"Add":"isPartialOrder O = record"},{"Add":" { isPreorder = isPreorder O.isPreorder"},{"Add":" ; antisym = antisym O.antisym"},{"Add":" } where module O = IsPartialOrder O"},{"Add":""},{"Add":"isTotalOrder : IsTotalOrder _≈₂_ _≲₂_ → IsTotalOrder _≈₁_ _≲₁_"},{"Add":"isTotalOrder O = record"},{"Add":" { isPartialOrder = isPartialOrder O.isPartialOrder"},{"Add":" ; total = total O.total"},{"Add":" } where module O = IsTotalOrder O"},{"Add":""},{"Add":"isDecTotalOrder : IsDecTotalOrder _≈₂_ _≲₂_ → IsDecTotalOrder _≈₁_ _≲₁_"},{"Add":"isDecTotalOrder O = record"},{"Add":" { isTotalOrder = isTotalOrder O.isTotalOrder"},{"Add":" ; _≟_ = EqM.dec O._≟_"},{"Add":" ; _≤?_ = dec O._≤?_"},{"Add":" } where module O = IsDecTotalOrder O"},{"Add":""},{"Add":"isStrictPartialOrder : IsStrictPartialOrder _≈₂_ _≲₂_ →"},{"Add":" IsStrictPartialOrder _≈₁_ _≲₁_"},{"Add":"isStrictPartialOrder O = record"},{"Add":" { isEquivalence = EqM.isEquivalence O.isEquivalence"},{"Add":" ; irrefl = irrefl O.irrefl"},{"Add":" ; trans = trans O.trans"},{"Add":" ; <-resp-≈ = resp O.<-resp-≈"},{"Add":" } where module O = IsStrictPartialOrder O"},{"Add":""},{"Add":"isStrictTotalOrder : IsStrictTotalOrder _≈₂_ _≲₂_ →"},{"Add":" IsStrictTotalOrder _≈₁_ _≲₁_"},{"Add":"isStrictTotalOrder O = record"},{"Add":" { isStrictPartialOrder = isStrictPartialOrder O.isStrictPartialOrder"},{"Add":" ; compare = compare O.compare"},{"Add":" } where module O = IsStrictTotalOrder O"}]}},{"FileAdd":{"meta":{"basename":"Definitions.agda","metadata":0},"context":[{"change":0,"pos":526801}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Basic definitions for morphisms between algebraic structures"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core"},{"Add":""},{"Add":"module Relation.Binary.Morphism.Definitions"},{"Add":" {a} (A : Set a) -- The domain of the morphism"},{"Add":" {b} (B : Set b) -- The codomain of the morphism"},{"Add":" where"},{"Add":""},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ₁ ℓ₂ : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Morphism definition in Function.Core"},{"Add":""},{"Add":"open import Function.Core public"},{"Add":" using (Morphism)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Basic definitions"},{"Add":""},{"Add":"Homomorphic₂ : Rel A ℓ₁ → Rel B ℓ₂ → (A → B) → Set _"},{"Add":"Homomorphic₂ _∼₁_ _∼₂_ ⟦_⟧ = ∀ {x y} → x ∼₁ y → ⟦ x ⟧ ∼₂ ⟦ y ⟧"}]}},{"FileAdd":{"meta":{"basename":"Construct","metadata":576},"context":[{"change":0,"pos":526801}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Identity.agda","metadata":0},"context":[{"change":0,"pos":539284}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The identity morphism for binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base using (id)"},{"Add":"import Function.Construct.Identity as Id"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid; Preorder; Poset)"},{"Add":"open import Relation.Binary.Definitions using (Reflexive)"},{"Add":"open import Relation.Binary.Morphism.Structures"},{"Add":"open import Relation.Binary.Morphism.Bundles"},{"Add":""},{"Add":"module Relation.Binary.Morphism.Construct.Identity where"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a ℓ₁ ℓ₂ : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relations"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"module _ (≈ : Rel A ℓ₁) where"},{"Add":""},{"Add":" isRelHomomorphism : IsRelHomomorphism ≈ ≈ id"},{"Add":" isRelHomomorphism = record"},{"Add":" { cong = Id.congruent ≈"},{"Add":" }"},{"Add":""},{"Add":" isRelMonomorphism : IsRelMonomorphism ≈ ≈ id"},{"Add":" isRelMonomorphism = record"},{"Add":" { isHomomorphism = isRelHomomorphism"},{"Add":" ; injective = Id.injective ≈"},{"Add":" }"},{"Add":""},{"Add":" isRelIsomorphism : Reflexive ≈ → IsRelIsomorphism ≈ ≈ id"},{"Add":" isRelIsomorphism refl = record"},{"Add":" { isMonomorphism = isRelMonomorphism"},{"Add":" ; surjective = Id.surjective ≈"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles"},{"Add":""},{"Add":"module _ (S : Setoid a ℓ₁) where"},{"Add":""},{"Add":" open Setoid S"},{"Add":""},{"Add":" setoidHomomorphism : SetoidHomomorphism S S"},{"Add":" setoidHomomorphism = record { isRelHomomorphism = isRelHomomorphism _≈_ }"},{"Add":""},{"Add":" setoidMonomorphism : SetoidMonomorphism S S"},{"Add":" setoidMonomorphism = record { isRelMonomorphism = isRelMonomorphism _≈_ }"},{"Add":""},{"Add":" setoidIsomorphism : SetoidIsomorphism S S"},{"Add":" setoidIsomorphism = record { isRelIsomorphism = isRelIsomorphism _ refl }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Orders"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"module _ (≈ : Rel A ℓ₁) (∼ : Rel A ℓ₂) where"},{"Add":""},{"Add":" isOrderHomomorphism : IsOrderHomomorphism ≈ ≈ ∼ ∼ id"},{"Add":" isOrderHomomorphism = record"},{"Add":" { cong = id"},{"Add":" ; mono = id"},{"Add":" }"},{"Add":""},{"Add":" isOrderMonomorphism : IsOrderMonomorphism ≈ ≈ ∼ ∼ id"},{"Add":" isOrderMonomorphism = record"},{"Add":" { isOrderHomomorphism = isOrderHomomorphism"},{"Add":" ; injective = Id.injective ≈"},{"Add":" ; cancel = id"},{"Add":" }"},{"Add":""},{"Add":" isOrderIsomorphism : Reflexive ≈ → IsOrderIsomorphism ≈ ≈ ∼ ∼ id"},{"Add":" isOrderIsomorphism refl = record"},{"Add":" { isOrderMonomorphism = isOrderMonomorphism"},{"Add":" ; surjective = Id.surjective ≈"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles"},{"Add":""},{"Add":"module _ (P : Preorder a ℓ₁ ℓ₂) where"},{"Add":""},{"Add":" preorderHomomorphism : PreorderHomomorphism P P"},{"Add":" preorderHomomorphism = record { isOrderHomomorphism = isOrderHomomorphism _ _ }"},{"Add":""},{"Add":"module _ (P : Poset a ℓ₁ ℓ₂) where"},{"Add":""},{"Add":" posetHomomorphism : PosetHomomorphism P P"},{"Add":" posetHomomorphism = record { isOrderHomomorphism = isOrderHomomorphism _ _ }"}]}},{"FileAdd":{"meta":{"basename":"Constant.agda","metadata":0},"context":[{"change":0,"pos":539284}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Constant morphisms between binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Function.Base using (const)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid; Preorder)"},{"Add":"open import Relation.Binary.Definitions using (Reflexive)"},{"Add":"open import Relation.Binary.Morphism.Structures"},{"Add":"open import Relation.Binary.Morphism.Bundles"},{"Add":""},{"Add":"module Relation.Binary.Morphism.Construct.Constant where"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level"},{"Add":" A B : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) where"},{"Add":""},{"Add":" isRelHomomorphism : Reflexive ≈₂ →"},{"Add":" ∀ x → IsRelHomomorphism ≈₁ ≈₂ (const x)"},{"Add":" isRelHomomorphism refl x = record"},{"Add":" { cong = const refl"},{"Add":" }"},{"Add":""},{"Add":"module _ (S : Setoid a ℓ₁) (T : Setoid b ℓ₂) where"},{"Add":""},{"Add":" setoidHomomorphism : ∀ x → SetoidHomomorphism S T"},{"Add":" setoidHomomorphism x = record"},{"Add":" { isRelHomomorphism = isRelHomomorphism _ _ T.refl x"},{"Add":" } where module T = Setoid T"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Orders"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) (∼₁ : Rel A ℓ₃) (∼₂ : Rel B ℓ₄) where"},{"Add":""},{"Add":" isOrderHomomorphism : Reflexive ≈₂ → Reflexive ∼₂ →"},{"Add":" ∀ x → IsOrderHomomorphism ≈₁ ≈₂ ∼₁ ∼₂ (const x)"},{"Add":" isOrderHomomorphism ≈-refl ∼-refl x = record"},{"Add":" { cong = const ≈-refl"},{"Add":" ; mono = const ∼-refl"},{"Add":" }"},{"Add":""},{"Add":"module _ (P : Preorder a ℓ₁ ℓ₂) (Q : Preorder b ℓ₃ ℓ₄) where"},{"Add":""},{"Add":" preorderHomomorphism : ∀ x → PreorderHomomorphism P Q"},{"Add":" preorderHomomorphism x = record"},{"Add":" { isOrderHomomorphism = isOrderHomomorphism _ _ _ _ Q.Eq.refl Q.refl x"},{"Add":" } where module Q = Preorder Q"}]}},{"FileAdd":{"meta":{"basename":"Composition.agda","metadata":0},"context":[{"change":0,"pos":539284}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The composition of morphisms between binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Function.Base using (_∘_)"},{"Add":"open import Function.Construct.Composition using (surjective)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid; Preorder; Poset)"},{"Add":"open import Relation.Binary.Definitions using (Transitive)"},{"Add":"open import Relation.Binary.Morphism.Bundles"},{"Add":"open import Relation.Binary.Morphism.Structures"},{"Add":""},{"Add":"module Relation.Binary.Morphism.Construct.Composition where"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c ℓ₁ ℓ₂ ℓ₃ ℓ₄ ℓ₅ ℓ₆ : Level"},{"Add":" A B C : Set a"},{"Add":" ≈₁ ≈₂ ≈₃ ∼₁ ∼₂ ∼₃ : Rel A ℓ₁"},{"Add":" f g : A → B"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relations"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"isRelHomomorphism : IsRelHomomorphism ≈₁ ≈₂ f →"},{"Add":" IsRelHomomorphism ≈₂ ≈₃ g →"},{"Add":" IsRelHomomorphism ≈₁ ≈₃ (g ∘ f)"},{"Add":"isRelHomomorphism m₁ m₂ = record"},{"Add":" { cong = G.cong ∘ F.cong"},{"Add":" } where module F = IsRelHomomorphism m₁; module G = IsRelHomomorphism m₂"},{"Add":""},{"Add":"isRelMonomorphism : IsRelMonomorphism ≈₁ ≈₂ f →"},{"Add":" IsRelMonomorphism ≈₂ ≈₃ g →"},{"Add":" IsRelMonomorphism ≈₁ ≈₃ (g ∘ f)"},{"Add":"isRelMonomorphism m₁ m₂ = record"},{"Add":" { isHomomorphism = isRelHomomorphism F.isHomomorphism G.isHomomorphism"},{"Add":" ; injective = F.injective ∘ G.injective"},{"Add":" } where module F = IsRelMonomorphism m₁; module G = IsRelMonomorphism m₂"},{"Add":""},{"Add":"isRelIsomorphism : Transitive ≈₃ →"},{"Add":" IsRelIsomorphism ≈₁ ≈₂ f →"},{"Add":" IsRelIsomorphism ≈₂ ≈₃ g →"},{"Add":" IsRelIsomorphism ≈₁ ≈₃ (g ∘ f)"},{"Add":"isRelIsomorphism {≈₃ = ≈₃} ≈₃-trans m₁ m₂ = record"},{"Add":" { isMonomorphism = isRelMonomorphism F.isMonomorphism G.isMonomorphism"},{"Add":" ; surjective = surjective _ _ ≈₃ F.surjective G.surjective"},{"Add":" } where module F = IsRelIsomorphism m₁; module G = IsRelIsomorphism m₂"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles"},{"Add":""},{"Add":"module _ {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} {U : Setoid c ℓ₃} where"},{"Add":""},{"Add":" setoidHomomorphism : SetoidHomomorphism S T →"},{"Add":" SetoidHomomorphism T U →"},{"Add":" SetoidHomomorphism S U"},{"Add":" setoidHomomorphism ST TU = record"},{"Add":" { isRelHomomorphism = isRelHomomorphism ST.isRelHomomorphism TU.isRelHomomorphism"},{"Add":" } where module ST = SetoidHomomorphism ST; module TU = SetoidHomomorphism TU"},{"Add":""},{"Add":" setoidMonomorphism : SetoidMonomorphism S T →"},{"Add":" SetoidMonomorphism T U →"},{"Add":" SetoidMonomorphism S U"},{"Add":" setoidMonomorphism ST TU = record"},{"Add":" { isRelMonomorphism = isRelMonomorphism ST.isRelMonomorphism TU.isRelMonomorphism"},{"Add":" } where module ST = SetoidMonomorphism ST; module TU = SetoidMonomorphism TU"},{"Add":""},{"Add":" setoidIsomorphism : SetoidIsomorphism S T →"},{"Add":" SetoidIsomorphism T U →"},{"Add":" SetoidIsomorphism S U"},{"Add":" setoidIsomorphism ST TU = record"},{"Add":" { isRelIsomorphism = isRelIsomorphism (Setoid.trans U) ST.isRelIsomorphism TU.isRelIsomorphism"},{"Add":" } where module ST = SetoidIsomorphism ST; module TU = SetoidIsomorphism TU"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Orders"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"isOrderHomomorphism : IsOrderHomomorphism ≈₁ ≈₂ ∼₁ ∼₂ f →"},{"Add":" IsOrderHomomorphism ≈₂ ≈₃ ∼₂ ∼₃ g →"},{"Add":" IsOrderHomomorphism ≈₁ ≈₃ ∼₁ ∼₃ (g ∘ f)"},{"Add":"isOrderHomomorphism m₁ m₂ = record"},{"Add":" { cong = G.cong ∘ F.cong"},{"Add":" ; mono = G.mono ∘ F.mono"},{"Add":" } where module F = IsOrderHomomorphism m₁; module G = IsOrderHomomorphism m₂"},{"Add":""},{"Add":"isOrderMonomorphism : IsOrderMonomorphism ≈₁ ≈₂ ∼₁ ∼₂ f →"},{"Add":" IsOrderMonomorphism ≈₂ ≈₃ ∼₂ ∼₃ g →"},{"Add":" IsOrderMonomorphism ≈₁ ≈₃ ∼₁ ∼₃ (g ∘ f)"},{"Add":"isOrderMonomorphism m₁ m₂ = record"},{"Add":" { isOrderHomomorphism = isOrderHomomorphism F.isOrderHomomorphism G.isOrderHomomorphism"},{"Add":" ; injective = F.injective ∘ G.injective"},{"Add":" ; cancel = F.cancel ∘ G.cancel"},{"Add":" } where module F = IsOrderMonomorphism m₁; module G = IsOrderMonomorphism m₂"},{"Add":""},{"Add":"isOrderIsomorphism : Transitive ≈₃ →"},{"Add":" IsOrderIsomorphism ≈₁ ≈₂ ∼₁ ∼₂ f →"},{"Add":" IsOrderIsomorphism ≈₂ ≈₃ ∼₂ ∼₃ g →"},{"Add":" IsOrderIsomorphism ≈₁ ≈₃ ∼₁ ∼₃ (g ∘ f)"},{"Add":"isOrderIsomorphism {≈₃ = ≈₃} ≈₃-trans m₁ m₂ = record"},{"Add":" { isOrderMonomorphism = isOrderMonomorphism F.isOrderMonomorphism G.isOrderMonomorphism"},{"Add":" ; surjective = surjective _ _ ≈₃ F.surjective G.surjective"},{"Add":" } where module F = IsOrderIsomorphism m₁; module G = IsOrderIsomorphism m₂"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles"},{"Add":""},{"Add":"module _ {P : Preorder a ℓ₁ ℓ₂} {Q : Preorder b ℓ₃ ℓ₄} {R : Preorder c ℓ₅ ℓ₆} where"},{"Add":""},{"Add":" preorderHomomorphism : PreorderHomomorphism P Q →"},{"Add":" PreorderHomomorphism Q R →"},{"Add":" PreorderHomomorphism P R"},{"Add":" preorderHomomorphism PQ QR = record"},{"Add":" { isOrderHomomorphism = isOrderHomomorphism PQ.isOrderHomomorphism QR.isOrderHomomorphism"},{"Add":" } where module PQ = PreorderHomomorphism PQ; module QR = PreorderHomomorphism QR"},{"Add":""},{"Add":"module _ {P : Poset a ℓ₁ ℓ₂} {Q : Poset b ℓ₃ ℓ₄} {R : Poset c ℓ₅ ℓ₆} where"},{"Add":""},{"Add":" posetHomomorphism : PosetHomomorphism P Q →"},{"Add":" PosetHomomorphism Q R →"},{"Add":" PosetHomomorphism P R"},{"Add":" posetHomomorphism PQ QR = record"},{"Add":" { isOrderHomomorphism = isOrderHomomorphism PQ.isOrderHomomorphism QR.isOrderHomomorphism"},{"Add":" } where module PQ = PosetHomomorphism PQ; module QR = PosetHomomorphism QR"}]}},{"FileAdd":{"meta":{"basename":"Bundles.agda","metadata":0},"context":[{"change":0,"pos":526801}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Bundles for morphisms between binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Relation.Binary.Core using (_Preserves_⟶_)"},{"Add":"open import Relation.Binary.Bundles"},{"Add":"open import Relation.Binary.Morphism.Structures"},{"Add":"open import Relation.Binary.Consequences using (mono⇒cong)"},{"Add":""},{"Add":"module Relation.Binary.Morphism.Bundles where"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ₁ ℓ₂ ℓ₃ ℓ₄ ℓ₅ ℓ₆ : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Setoids"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"module _ (S₁ : Setoid ℓ₁ ℓ₂) (S₂ : Setoid ℓ₃ ℓ₄) where"},{"Add":""},{"Add":" record SetoidHomomorphism : Set (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄) where"},{"Add":" open Setoid"},{"Add":" field"},{"Add":" ⟦_⟧ : Carrier S₁ → Carrier S₂"},{"Add":" isRelHomomorphism : IsRelHomomorphism (_≈_ S₁) (_≈_ S₂) ⟦_⟧"},{"Add":""},{"Add":" open IsRelHomomorphism isRelHomomorphism public"},{"Add":""},{"Add":""},{"Add":" record SetoidMonomorphism : Set (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄) where"},{"Add":" open Setoid"},{"Add":" field"},{"Add":" ⟦_⟧ : Carrier S₁ → Carrier S₂"},{"Add":" isRelMonomorphism : IsRelMonomorphism (_≈_ S₁) (_≈_ S₂) ⟦_⟧"},{"Add":""},{"Add":" open IsRelMonomorphism isRelMonomorphism public"},{"Add":""},{"Add":" homomorphism : SetoidHomomorphism"},{"Add":" homomorphism = record { isRelHomomorphism = isHomomorphism }"},{"Add":""},{"Add":""},{"Add":" record SetoidIsomorphism : Set (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄) where"},{"Add":" open Setoid"},{"Add":" field"},{"Add":" ⟦_⟧ : Carrier S₁ → Carrier S₂"},{"Add":" isRelIsomorphism : IsRelIsomorphism (_≈_ S₁) (_≈_ S₂) ⟦_⟧"},{"Add":""},{"Add":" open IsRelIsomorphism isRelIsomorphism public"},{"Add":""},{"Add":" monomorphism : SetoidMonomorphism"},{"Add":" monomorphism = record { isRelMonomorphism = isMonomorphism }"},{"Add":""},{"Add":" open SetoidMonomorphism monomorphism public"},{"Add":" using (homomorphism)"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Preorders"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record PreorderHomomorphism (S₁ : Preorder ℓ₁ ℓ₂ ℓ₃)"},{"Add":" (S₂ : Preorder ℓ₄ ℓ₅ ℓ₆)"},{"Add":" : Set (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄ ⊔ ℓ₅ ⊔ ℓ₆) where"},{"Add":" open Preorder"},{"Add":" field"},{"Add":" ⟦_⟧ : Carrier S₁ → Carrier S₂"},{"Add":" isOrderHomomorphism : IsOrderHomomorphism (_≈_ S₁) (_≈_ S₂) (_≲_ S₁) (_≲_ S₂) ⟦_⟧"},{"Add":""},{"Add":" open IsOrderHomomorphism isOrderHomomorphism public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Posets"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"module _ (P : Poset ℓ₁ ℓ₂ ℓ₃) (Q : Poset ℓ₄ ℓ₅ ℓ₆) where"},{"Add":""},{"Add":" private"},{"Add":" module P = Poset P"},{"Add":" module Q = Poset Q"},{"Add":""},{"Add":" record PosetHomomorphism : Set (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄ ⊔ ℓ₅ ⊔ ℓ₆) where"},{"Add":" field"},{"Add":" ⟦_⟧ : P.Carrier → Q.Carrier"},{"Add":" isOrderHomomorphism : IsOrderHomomorphism P._≈_ Q._≈_ P._≤_ Q._≤_ ⟦_⟧"},{"Add":""},{"Add":" open IsOrderHomomorphism isOrderHomomorphism public"},{"Add":""},{"Add":""},{"Add":" -- Smart constructor that automatically constructs the congruence"},{"Add":" -- proof from the monotonicity proof"},{"Add":" mkPosetHomo : ∀ f → f Preserves P._≤_ ⟶ Q._≤_ → PosetHomomorphism"},{"Add":" mkPosetHomo f mono = record"},{"Add":" { ⟦_⟧ = f"},{"Add":" ; isOrderHomomorphism = record"},{"Add":" { cong = mono⇒cong P._≈_ Q._≈_ P.Eq.sym P.reflexive Q.antisym mono"},{"Add":" ; mono = mono"},{"Add":" }"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Lattice.agda","metadata":0},"context":[{"change":0,"pos":415495}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Order-theoretic lattices"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Lattice where"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export various components of the lattice hierarchy"},{"Add":""},{"Add":"open import Relation.Binary.Lattice.Definitions public"},{"Add":"open import Relation.Binary.Lattice.Structures public"},{"Add":"open import Relation.Binary.Lattice.Bundles public"}]}},{"FileAdd":{"meta":{"basename":"Lattice","metadata":576},"context":[{"change":0,"pos":415495}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Structures.agda","metadata":0},"context":[{"change":0,"pos":555498}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Structures for order-theoretic lattices"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via"},{"Add":"-- `Relation.Binary.Lattice`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Structures using (IsPartialOrder)"},{"Add":"open import Relation.Binary.Definitions using (Minimum; Maximum)"},{"Add":""},{"Add":"module Relation.Binary.Lattice.Structures"},{"Add":" {a ℓ₁ ℓ₂} {A : Set a}"},{"Add":" (_≈_ : Rel A ℓ₁) -- The underlying equality."},{"Add":" (_≤_ : Rel A ℓ₂) -- The partial order."},{"Add":" where"},{"Add":""},{"Add":"open import Algebra.Core"},{"Add":"open import Algebra.Definitions"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Level using (suc; _⊔_)"},{"Add":""},{"Add":"open import Relation.Binary.Lattice.Definitions"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Join semilattices"},{"Add":""},{"Add":"record IsJoinSemilattice (_∨_ : Op₂ A) -- The join operation."},{"Add":" : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isPartialOrder : IsPartialOrder _≈_ _≤_"},{"Add":" supremum : Supremum _≤_ _∨_"},{"Add":""},{"Add":" x≤x∨y : ∀ x y → x ≤ (x ∨ y)"},{"Add":" x≤x∨y x y = let pf , _ , _ = supremum x y in pf"},{"Add":""},{"Add":" y≤x∨y : ∀ x y → y ≤ (x ∨ y)"},{"Add":" y≤x∨y x y = let _ , pf , _ = supremum x y in pf"},{"Add":""},{"Add":" ∨-least : ∀ {x y z} → x ≤ z → y ≤ z → (x ∨ y) ≤ z"},{"Add":" ∨-least {x} {y} {z} = let _ , _ , pf = supremum x y in pf z"},{"Add":""},{"Add":" open IsPartialOrder isPartialOrder public"},{"Add":""},{"Add":"record IsBoundedJoinSemilattice (_∨_ : Op₂ A) -- The join operation."},{"Add":" (⊥ : A) -- The minimum."},{"Add":" : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isJoinSemilattice : IsJoinSemilattice _∨_"},{"Add":" minimum : Minimum _≤_ ⊥"},{"Add":""},{"Add":" open IsJoinSemilattice isJoinSemilattice public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Meet semilattices"},{"Add":""},{"Add":"record IsMeetSemilattice (_∧_ : Op₂ A) -- The meet operation."},{"Add":" : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isPartialOrder : IsPartialOrder _≈_ _≤_"},{"Add":" infimum : Infimum _≤_ _∧_"},{"Add":""},{"Add":" x∧y≤x : ∀ x y → (x ∧ y) ≤ x"},{"Add":" x∧y≤x x y = let pf , _ , _ = infimum x y in pf"},{"Add":""},{"Add":" x∧y≤y : ∀ x y → (x ∧ y) ≤ y"},{"Add":" x∧y≤y x y = let _ , pf , _ = infimum x y in pf"},{"Add":""},{"Add":" ∧-greatest : ∀ {x y z} → x ≤ y → x ≤ z → x ≤ (y ∧ z)"},{"Add":" ∧-greatest {x} {y} {z} = let _ , _ , pf = infimum y z in pf x"},{"Add":""},{"Add":" open IsPartialOrder isPartialOrder public"},{"Add":""},{"Add":"record IsBoundedMeetSemilattice (_∧_ : Op₂ A) -- The join operation."},{"Add":" (⊤ : A) -- The maximum."},{"Add":" : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isMeetSemilattice : IsMeetSemilattice _∧_"},{"Add":" maximum : Maximum _≤_ ⊤"},{"Add":""},{"Add":" open IsMeetSemilattice isMeetSemilattice public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Lattices"},{"Add":""},{"Add":"record IsLattice (_∨_ : Op₂ A) -- The join operation."},{"Add":" (_∧_ : Op₂ A) -- The meet operation."},{"Add":" : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isPartialOrder : IsPartialOrder _≈_ _≤_"},{"Add":" supremum : Supremum _≤_ _∨_"},{"Add":" infimum : Infimum _≤_ _∧_"},{"Add":""},{"Add":" isJoinSemilattice : IsJoinSemilattice _∨_"},{"Add":" isJoinSemilattice = record"},{"Add":" { isPartialOrder = isPartialOrder"},{"Add":" ; supremum = supremum"},{"Add":" }"},{"Add":""},{"Add":" isMeetSemilattice : IsMeetSemilattice _∧_"},{"Add":" isMeetSemilattice = record"},{"Add":" { isPartialOrder = isPartialOrder"},{"Add":" ; infimum = infimum"},{"Add":" }"},{"Add":""},{"Add":" open IsJoinSemilattice isJoinSemilattice public"},{"Add":" using (x≤x∨y; y≤x∨y; ∨-least)"},{"Add":" open IsMeetSemilattice isMeetSemilattice public"},{"Add":" using (x∧y≤x; x∧y≤y; ∧-greatest)"},{"Add":" open IsPartialOrder isPartialOrder public"},{"Add":""},{"Add":"record IsDistributiveLattice (_∨_ : Op₂ A) -- The join operation."},{"Add":" (_∧_ : Op₂ A) -- The meet operation."},{"Add":" : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isLattice : IsLattice _∨_ _∧_"},{"Add":" ∧-distribˡ-∨ : _DistributesOverˡ_ _≈_ _∧_ _∨_"},{"Add":""},{"Add":" open IsLattice isLattice public"},{"Add":""},{"Add":"record IsBoundedLattice (_∨_ : Op₂ A) -- The join operation."},{"Add":" (_∧_ : Op₂ A) -- The meet operation."},{"Add":" (⊤ : A) -- The maximum."},{"Add":" (⊥ : A) -- The minimum."},{"Add":" : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isLattice : IsLattice _∨_ _∧_"},{"Add":" maximum : Maximum _≤_ ⊤"},{"Add":" minimum : Minimum _≤_ ⊥"},{"Add":""},{"Add":" open IsLattice isLattice public"},{"Add":""},{"Add":" isBoundedJoinSemilattice : IsBoundedJoinSemilattice _∨_ ⊥"},{"Add":" isBoundedJoinSemilattice = record"},{"Add":" { isJoinSemilattice = isJoinSemilattice"},{"Add":" ; minimum = minimum"},{"Add":" }"},{"Add":""},{"Add":" isBoundedMeetSemilattice : IsBoundedMeetSemilattice _∧_ ⊤"},{"Add":" isBoundedMeetSemilattice = record"},{"Add":" { isMeetSemilattice = isMeetSemilattice"},{"Add":" ; maximum = maximum"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Heyting algebras (a bounded lattice with exponential operator)"},{"Add":""},{"Add":"record IsHeytingAlgebra (_∨_ : Op₂ A) -- The join operation."},{"Add":" (_∧_ : Op₂ A) -- The meet operation."},{"Add":" (_⇨_ : Op₂ A) -- The exponential operation."},{"Add":" (⊤ : A) -- The maximum."},{"Add":" (⊥ : A) -- The minimum."},{"Add":" : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isBoundedLattice : IsBoundedLattice _∨_ _∧_ ⊤ ⊥"},{"Add":" exponential : Exponential _≤_ _∧_ _⇨_"},{"Add":""},{"Add":" transpose-⇨ : ∀ {w x y} → (w ∧ x) ≤ y → w ≤ (x ⇨ y)"},{"Add":" transpose-⇨ {w} {x} {y} = let pf , _ = exponential w x y in pf"},{"Add":""},{"Add":" transpose-∧ : ∀ {w x y} → w ≤ (x ⇨ y) → (w ∧ x) ≤ y"},{"Add":" transpose-∧ {w} {x} {y} = let _ , pf = exponential w x y in pf"},{"Add":""},{"Add":" open IsBoundedLattice isBoundedLattice public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Boolean algebras (a specialized Heyting algebra)"},{"Add":""},{"Add":"record IsBooleanAlgebra (_∨_ : Op₂ A) -- The join operation."},{"Add":" (_∧_ : Op₂ A) -- The meet operation."},{"Add":" (¬_ : Op₁ A) -- The negation operation."},{"Add":" (⊤ : A) -- The maximum."},{"Add":" (⊥ : A) -- The minimum."},{"Add":" : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" infixr 5 _⇨_"},{"Add":" _⇨_ : Op₂ A"},{"Add":" x ⇨ y = (¬ x) ∨ y"},{"Add":""},{"Add":" field"},{"Add":" isHeytingAlgebra : IsHeytingAlgebra _∨_ _∧_ _⇨_ ⊤ ⊥"},{"Add":""},{"Add":" open IsHeytingAlgebra isHeytingAlgebra public"}]}},{"FileAdd":{"meta":{"basename":"Properties","metadata":576},"context":[{"change":0,"pos":555498}],"contents":[]}},{"FileAdd":{"meta":{"basename":"MeetSemilattice.agda","metadata":0},"context":[{"change":0,"pos":562402}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties satisfied by meet semilattices"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Lattice.Properties.MeetSemilattice"},{"Add":" {c ℓ₁ ℓ₂} (M : MeetSemilattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open MeetSemilattice M"},{"Add":""},{"Add":"open import Algebra.Definitions _≈_"},{"Add":"open import Function.Base using (flip)"},{"Add":"open import Relation.Binary.Structures using (IsDecPartialOrder)"},{"Add":"open import Relation.Binary.Definitions using (Decidable)"},{"Add":"open import Relation.Binary.Properties.Poset poset"},{"Add":"import Relation.Binary.Lattice.Properties.JoinSemilattice as J"},{"Add":""},{"Add":"-- The dual construction is a join semilattice."},{"Add":""},{"Add":"dualIsJoinSemilattice : IsJoinSemilattice _≈_ (flip _≤_) _∧_"},{"Add":"dualIsJoinSemilattice = record"},{"Add":" { isPartialOrder = ≥-isPartialOrder"},{"Add":" ; supremum = infimum"},{"Add":" }"},{"Add":""},{"Add":"dualJoinSemilattice : JoinSemilattice c ℓ₁ ℓ₂"},{"Add":"dualJoinSemilattice = record"},{"Add":" { _∨_ = _∧_"},{"Add":" ; isJoinSemilattice = dualIsJoinSemilattice"},{"Add":" }"},{"Add":""},{"Add":"open J dualJoinSemilattice public"},{"Add":" using (isAlgSemilattice; algSemilattice)"},{"Add":" renaming"},{"Add":" ( ∨-monotonic to ∧-monotonic"},{"Add":" ; ∨-cong to ∧-cong"},{"Add":" ; ∨-comm to ∧-comm"},{"Add":" ; ∨-assoc to ∧-assoc"},{"Add":" ; ∨-idempotent to ∧-idempotent"},{"Add":" ; x≤y⇒x∨y≈y to y≤x⇒x∧y≈y"},{"Add":" ; ≈-dec⇒≤-dec to ≈-dec⇒≥-dec"},{"Add":" )"},{"Add":""},{"Add":"-- If ≈ is decidable then so is ≤"},{"Add":""},{"Add":"≈-dec⇒≤-dec : Decidable _≈_ → Decidable _≤_"},{"Add":"≈-dec⇒≤-dec _≟_ = flip (≈-dec⇒≥-dec _≟_)"},{"Add":""},{"Add":"≈-dec⇒isDecPartialOrder : Decidable _≈_ → IsDecPartialOrder _≈_ _≤_"},{"Add":"≈-dec⇒isDecPartialOrder _≟_ = record"},{"Add":" { isPartialOrder = isPartialOrder"},{"Add":" ; _≟_ = _≟_"},{"Add":" ; _≤?_ = ≈-dec⇒≤-dec _≟_"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Lattice.agda","metadata":0},"context":[{"change":0,"pos":562402}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties satisfied by lattices"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Lattice.Properties.Lattice"},{"Add":" {c ℓ₁ ℓ₂} (L : Lattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open Lattice L"},{"Add":""},{"Add":"import Algebra.Lattice as Alg"},{"Add":"import Algebra.Structures as Alg"},{"Add":"open import Algebra.Definitions _≈_"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base using (flip)"},{"Add":"open import Relation.Binary.Properties.Poset poset"},{"Add":"import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice as J"},{"Add":"import Relation.Binary.Lattice.Properties.MeetSemilattice meetSemilattice as M"},{"Add":"import Relation.Binary.Reasoning.Setoid as ≈-Reasoning"},{"Add":"import Relation.Binary.Reasoning.PartialOrder as ≤-Reasoning"},{"Add":""},{"Add":"∨-absorbs-∧ : _∨_ Absorbs _∧_"},{"Add":"∨-absorbs-∧ x y ="},{"Add":" let x≤x∨[x∧y] , _ , least = supremum x (x ∧ y)"},{"Add":" x∧y≤x , _ , _ = infimum x y"},{"Add":" in antisym (least x refl x∧y≤x) x≤x∨[x∧y]"},{"Add":""},{"Add":"∧-absorbs-∨ : _∧_ Absorbs _∨_"},{"Add":"∧-absorbs-∨ x y ="},{"Add":" let x∧[x∨y]≤x , _ , greatest = infimum x (x ∨ y)"},{"Add":" x≤x∨y , _ , _ = supremum x y"},{"Add":" in antisym x∧[x∨y]≤x (greatest x refl x≤x∨y)"},{"Add":""},{"Add":"absorptive : Absorptive _∨_ _∧_"},{"Add":"absorptive = ∨-absorbs-∧ , ∧-absorbs-∨"},{"Add":""},{"Add":"∧≤∨ : ∀ {x y} → x ∧ y ≤ x ∨ y"},{"Add":"∧≤∨ {x} {y} = begin"},{"Add":" x ∧ y ≤⟨ x∧y≤x x y ⟩"},{"Add":" x ≤⟨ x≤x∨y x y ⟩"},{"Add":" x ∨ y ∎"},{"Add":" where open ≤-Reasoning poset"},{"Add":""},{"Add":"-- two quadrilateral arguments"},{"Add":""},{"Add":"quadrilateral₁ : ∀ {x y} → x ∨ y ≈ x → x ∧ y ≈ y"},{"Add":"quadrilateral₁ {x} {y} x∨y≈x = begin"},{"Add":" x ∧ y ≈⟨ M.∧-cong (Eq.sym x∨y≈x) Eq.refl ⟩"},{"Add":" (x ∨ y) ∧ y ≈⟨ M.∧-comm _ _ ⟩"},{"Add":" y ∧ (x ∨ y) ≈⟨ M.∧-cong Eq.refl (J.∨-comm _ _) ⟩"},{"Add":" y ∧ (y ∨ x) ≈⟨ ∧-absorbs-∨ _ _ ⟩"},{"Add":" y ∎"},{"Add":" where open ≈-Reasoning setoid"},{"Add":""},{"Add":"quadrilateral₂ : ∀ {x y} → x ∧ y ≈ y → x ∨ y ≈ x"},{"Add":"quadrilateral₂ {x} {y} x∧y≈y = begin"},{"Add":" x ∨ y ≈⟨ J.∨-cong Eq.refl (Eq.sym x∧y≈y) ⟩"},{"Add":" x ∨ (x ∧ y) ≈⟨ ∨-absorbs-∧ _ _ ⟩"},{"Add":" x ∎"},{"Add":" where open ≈-Reasoning setoid"},{"Add":""},{"Add":"-- collapsing sublattice"},{"Add":""},{"Add":"collapse₁ : ∀ {x y} → x ≈ y → x ∧ y ≈ x ∨ y"},{"Add":"collapse₁ {x} {y} x≈y = begin"},{"Add":" x ∧ y ≈⟨ M.y≤x⇒x∧y≈y y≤x ⟩"},{"Add":" y ≈⟨ Eq.sym x≈y ⟩"},{"Add":" x ≈⟨ Eq.sym (J.x≤y⇒x∨y≈y y≤x) ⟩"},{"Add":" y ∨ x ≈⟨ J.∨-comm _ _ ⟩"},{"Add":" x ∨ y ∎"},{"Add":" where"},{"Add":" y≤x = reflexive (Eq.sym x≈y)"},{"Add":" open ≈-Reasoning setoid"},{"Add":""},{"Add":"-- this can also be proved by quadrilateral argument, but it's much less symmetric."},{"Add":"collapse₂ : ∀ {x y} → x ∨ y ≤ x ∧ y → x ≈ y"},{"Add":"collapse₂ {x} {y} ∨≤∧ = antisym"},{"Add":" (begin x ≤⟨ x≤x∨y _ _ ⟩"},{"Add":" x ∨ y ≤⟨ ∨≤∧ ⟩"},{"Add":" x ∧ y ≤⟨ x∧y≤y _ _ ⟩"},{"Add":" y ∎)"},{"Add":" (begin y ≤⟨ y≤x∨y _ _ ⟩"},{"Add":" x ∨ y ≤⟨ ∨≤∧ ⟩"},{"Add":" x ∧ y ≤⟨ x∧y≤x _ _ ⟩"},{"Add":" x ∎)"},{"Add":" where open ≤-Reasoning poset"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The dual construction is also a lattice."},{"Add":""},{"Add":"∧-∨-isLattice : IsLattice _≈_ (flip _≤_) _∧_ _∨_"},{"Add":"∧-∨-isLattice = record"},{"Add":" { isPartialOrder = ≥-isPartialOrder"},{"Add":" ; supremum = infimum"},{"Add":" ; infimum = supremum"},{"Add":" }"},{"Add":""},{"Add":"∧-∨-lattice : Lattice c ℓ₁ ℓ₂"},{"Add":"∧-∨-lattice = record"},{"Add":" { _∧_ = _∨_"},{"Add":" ; _∨_ = _∧_"},{"Add":" ; isLattice = ∧-∨-isLattice"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Every order-theoretic lattice can be turned into an algebraic one."},{"Add":""},{"Add":"isAlgLattice : Alg.IsLattice _≈_ _∨_ _∧_"},{"Add":"isAlgLattice = record"},{"Add":" { isEquivalence = isEquivalence"},{"Add":" ; ∨-comm = J.∨-comm"},{"Add":" ; ∨-assoc = J.∨-assoc"},{"Add":" ; ∨-cong = J.∨-cong"},{"Add":" ; ∧-comm = M.∧-comm"},{"Add":" ; ∧-assoc = M.∧-assoc"},{"Add":" ; ∧-cong = M.∧-cong"},{"Add":" ; absorptive = absorptive"},{"Add":" }"},{"Add":""},{"Add":"algLattice : Alg.Lattice c ℓ₁"},{"Add":"algLattice = record { isLattice = isAlgLattice }"}]}},{"FileAdd":{"meta":{"basename":"JoinSemilattice.agda","metadata":0},"context":[{"change":0,"pos":562402}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties satisfied by join semilattices"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Lattice.Properties.JoinSemilattice"},{"Add":" {c ℓ₁ ℓ₂} (J : JoinSemilattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open JoinSemilattice J"},{"Add":""},{"Add":"import Algebra.Lattice as Alg"},{"Add":"import Algebra.Structures as Alg"},{"Add":"open import Algebra.Definitions _≈_"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base using (_∘_; flip)"},{"Add":"open import Relation.Binary.Core using (_Preserves₂_⟶_⟶_)"},{"Add":"open import Relation.Binary.Structures using (IsDecPartialOrder)"},{"Add":"open import Relation.Binary.Definitions using (Decidable)"},{"Add":"open import Relation.Binary.Properties.Poset poset"},{"Add":"open import Relation.Nullary using (¬_; yes; no)"},{"Add":"open import Relation.Nullary.Negation using (contraposition)"},{"Add":""},{"Add":"import Relation.Binary.Reasoning.PartialOrder as PoR"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Algebraic properties"},{"Add":""},{"Add":"-- The join operation is monotonic."},{"Add":""},{"Add":"∨-monotonic : _∨_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_"},{"Add":"∨-monotonic {x} {y} {u} {v} x≤y u≤v ="},{"Add":" let _ , _ , least = supremum x u"},{"Add":" y≤y∨v , v≤y∨v , _ = supremum y v"},{"Add":" in least (y ∨ v) (trans x≤y y≤y∨v) (trans u≤v v≤y∨v)"},{"Add":""},{"Add":"∨-cong : _∨_ Preserves₂ _≈_ ⟶ _≈_ ⟶ _≈_"},{"Add":"∨-cong x≈y u≈v = antisym (∨-monotonic (reflexive x≈y) (reflexive u≈v))"},{"Add":" (∨-monotonic (reflexive (Eq.sym x≈y))"},{"Add":" (reflexive (Eq.sym u≈v)))"},{"Add":""},{"Add":"-- The join operation is commutative, associative and idempotent."},{"Add":""},{"Add":"∨-comm : Commutative _∨_"},{"Add":"∨-comm x y ="},{"Add":" let x≤x∨y , y≤x∨y , least = supremum x y"},{"Add":" y≤y∨x , x≤y∨x , least′ = supremum y x"},{"Add":" in antisym (least (y ∨ x) x≤y∨x y≤y∨x) (least′ (x ∨ y) y≤x∨y x≤x∨y)"},{"Add":""},{"Add":"∨-assoc : Associative _∨_"},{"Add":"∨-assoc x y z ="},{"Add":" let x∨y≤[x∨y]∨z , z≤[x∨y]∨z , least = supremum (x ∨ y) z"},{"Add":" x≤x∨[y∨z] , y∨z≤x∨[y∨z] , least′ = supremum x (y ∨ z)"},{"Add":" y≤y∨z , z≤y∨z , _ = supremum y z"},{"Add":" x≤x∨y , y≤x∨y , _ = supremum x y"},{"Add":" in antisym (least (x ∨ (y ∨ z)) (∨-monotonic refl y≤y∨z)"},{"Add":" (trans z≤y∨z y∨z≤x∨[y∨z]))"},{"Add":" (least′ ((x ∨ y) ∨ z) (trans x≤x∨y x∨y≤[x∨y]∨z)"},{"Add":" (∨-monotonic y≤x∨y refl))"},{"Add":""},{"Add":"∨-idempotent : Idempotent _∨_"},{"Add":"∨-idempotent x ="},{"Add":" let x≤x∨x , _ , least = supremum x x"},{"Add":" in antisym (least x refl refl) x≤x∨x"},{"Add":""},{"Add":"x≤y⇒x∨y≈y : ∀ {x y} → x ≤ y → x ∨ y ≈ y"},{"Add":"x≤y⇒x∨y≈y {x} {y} x≤y = antisym"},{"Add":" (begin x ∨ y ≤⟨ ∨-monotonic x≤y refl ⟩"},{"Add":" y ∨ y ≈⟨ ∨-idempotent _ ⟩"},{"Add":" y ∎)"},{"Add":" (y≤x∨y _ _)"},{"Add":" where open PoR poset"},{"Add":""},{"Add":"-- Every order-theoretic semilattice can be turned into an algebraic one."},{"Add":""},{"Add":"isAlgSemilattice : Alg.IsSemilattice _≈_ _∨_"},{"Add":"isAlgSemilattice = record"},{"Add":" { isBand = record"},{"Add":" { isSemigroup = record"},{"Add":" { isMagma = record"},{"Add":" { isEquivalence = isEquivalence"},{"Add":" ; ∙-cong = ∨-cong"},{"Add":" }"},{"Add":" ; assoc = ∨-assoc"},{"Add":" }"},{"Add":" ; idem = ∨-idempotent"},{"Add":" }"},{"Add":" ; comm = ∨-comm"},{"Add":" }"},{"Add":""},{"Add":"algSemilattice : Alg.Semilattice c ℓ₁"},{"Add":"algSemilattice = record { isSemilattice = isAlgSemilattice }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The dual construction is a meet semilattice."},{"Add":""},{"Add":"dualIsMeetSemilattice : IsMeetSemilattice _≈_ (flip _≤_) _∨_"},{"Add":"dualIsMeetSemilattice = record"},{"Add":" { isPartialOrder = ≥-isPartialOrder"},{"Add":" ; infimum = supremum"},{"Add":" }"},{"Add":""},{"Add":"dualMeetSemilattice : MeetSemilattice c ℓ₁ ℓ₂"},{"Add":"dualMeetSemilattice = record"},{"Add":" { _∧_ = _∨_"},{"Add":" ; isMeetSemilattice = dualIsMeetSemilattice"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- If ≈ is decidable then so is ≤"},{"Add":""},{"Add":"≈-dec⇒≤-dec : Decidable _≈_ → Decidable _≤_"},{"Add":"≈-dec⇒≤-dec _≟_ x y with (x ∨ y) ≟ y"},{"Add":"... | yes x∨y≈y = yes (trans (x≤x∨y x y) (reflexive x∨y≈y))"},{"Add":"... | no x∨y≉y = no (contraposition x≤y⇒x∨y≈y x∨y≉y)"},{"Add":""},{"Add":"≈-dec⇒isDecPartialOrder : Decidable _≈_ → IsDecPartialOrder _≈_ _≤_"},{"Add":"≈-dec⇒isDecPartialOrder _≟_ = record"},{"Add":" { isPartialOrder = isPartialOrder"},{"Add":" ; _≟_ = _≟_"},{"Add":" ; _≤?_ = ≈-dec⇒≤-dec _≟_"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"HeytingAlgebra.agda","metadata":0},"context":[{"change":0,"pos":562402}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties satisfied by Heyting Algebra"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Lattice.Properties.HeytingAlgebra"},{"Add":" {c ℓ₁ ℓ₂} (L : HeytingAlgebra c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open HeytingAlgebra L"},{"Add":""},{"Add":"open import Algebra.Core"},{"Add":"open import Algebra.Definitions _≈_"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base using (_$_; flip; _∘_)"},{"Add":"open import Level using (_⊔_)"},{"Add":"open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)"},{"Add":"import Relation.Binary.Reasoning.PartialOrder as ≤-Reasoning"},{"Add":"open import Relation.Binary.Lattice.Properties.MeetSemilattice meetSemilattice"},{"Add":"open import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice"},{"Add":"import Relation.Binary.Lattice.Properties.BoundedMeetSemilattice boundedMeetSemilattice as BM"},{"Add":"open import Relation.Binary.Lattice.Properties.Lattice lattice"},{"Add":"open import Relation.Binary.Lattice.Properties.BoundedLattice boundedLattice"},{"Add":"import Relation.Binary.Reasoning.Setoid as ≈-Reasoning"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Useful lemmas"},{"Add":""},{"Add":"⇨-eval : ∀ {x y} → (x ⇨ y) ∧ x ≤ y"},{"Add":"⇨-eval {x} {y} = transpose-∧ refl"},{"Add":""},{"Add":"swap-transpose-⇨ : ∀ {x y w} → x ∧ w ≤ y → w ≤ x ⇨ y"},{"Add":"swap-transpose-⇨ x∧w≤y = transpose-⇨ $ trans (reflexive $ ∧-comm _ _) x∧w≤y"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of exponential"},{"Add":""},{"Add":"⇨-unit : ∀ {x} → x ⇨ x ≈ ⊤"},{"Add":"⇨-unit = antisym (maximum _) (transpose-⇨ $ reflexive $ BM.identityˡ _)"},{"Add":""},{"Add":"y≤x⇨y : ∀ {x y} → y ≤ x ⇨ y"},{"Add":"y≤x⇨y = transpose-⇨ (x∧y≤x _ _)"},{"Add":""},{"Add":"⇨-drop : ∀ {x y} → (x ⇨ y) ∧ y ≈ y"},{"Add":"⇨-drop = antisym (x∧y≤y _ _) (∧-greatest y≤x⇨y refl)"},{"Add":""},{"Add":"⇨-app : ∀ {x y} → (x ⇨ y) ∧ x ≈ y ∧ x"},{"Add":"⇨-app = antisym (∧-greatest ⇨-eval (x∧y≤y _ _)) (∧-monotonic y≤x⇨y refl)"},{"Add":""},{"Add":"⇨ʳ-covariant : ∀ {x} → (x ⇨_) Preserves _≤_ ⟶ _≤_"},{"Add":"⇨ʳ-covariant y≤z = transpose-⇨ (trans ⇨-eval y≤z)"},{"Add":""},{"Add":"⇨ˡ-contravariant : ∀ {x} → (_⇨ x) Preserves (flip _≤_) ⟶ _≤_"},{"Add":"⇨ˡ-contravariant z≤y = transpose-⇨ (trans (∧-monotonic refl z≤y) ⇨-eval)"},{"Add":""},{"Add":"⇨-relax : _⇨_ Preserves₂ (flip _≤_) ⟶ _≤_ ⟶ _≤_"},{"Add":"⇨-relax {x} {y} {u} {v} y≤x u≤v = begin"},{"Add":" x ⇨ u ≤⟨ ⇨ʳ-covariant u≤v ⟩"},{"Add":" x ⇨ v ≤⟨ ⇨ˡ-contravariant y≤x ⟩"},{"Add":" y ⇨ v ∎"},{"Add":" where open ≤-Reasoning poset"},{"Add":""},{"Add":"⇨-cong : _⇨_ Preserves₂ _≈_ ⟶ _≈_ ⟶ _≈_"},{"Add":"⇨-cong x≈y u≈v = antisym (⇨-relax (reflexive $ Eq.sym x≈y) (reflexive u≈v))"},{"Add":" (⇨-relax (reflexive x≈y) (reflexive $ Eq.sym u≈v))"},{"Add":""},{"Add":"⇨-applyˡ : ∀ {w x y} → w ≤ x → (x ⇨ y) ∧ w ≤ y"},{"Add":"⇨-applyˡ = transpose-∧ ∘ ⇨ˡ-contravariant"},{"Add":""},{"Add":"⇨-applyʳ : ∀ {w x y} → w ≤ x → w ∧ (x ⇨ y) ≤ y"},{"Add":"⇨-applyʳ w≤x = trans (reflexive (∧-comm _ _)) (⇨-applyˡ w≤x)"},{"Add":""},{"Add":"⇨-curry : ∀ {x y z} → x ∧ y ⇨ z ≈ x ⇨ y ⇨ z"},{"Add":"⇨-curry = antisym (transpose-⇨ $ transpose-⇨ $ trans (reflexive $ ∧-assoc _ _ _) ⇨-eval)"},{"Add":" (transpose-⇨ $ trans (reflexive $ Eq.sym $ ∧-assoc _ _ _)"},{"Add":" (transpose-∧ $ ⇨-applyˡ refl))"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Various proofs of distributivity"},{"Add":""},{"Add":"∧-distribˡ-∨-≤ : ∀ x y z → x ∧ (y ∨ z) ≤ x ∧ y ∨ x ∧ z"},{"Add":"∧-distribˡ-∨-≤ x y z = trans (reflexive $ ∧-comm _ _)"},{"Add":" (transpose-∧ $ ∨-least (swap-transpose-⇨ (x≤x∨y _ _)) $ swap-transpose-⇨ (y≤x∨y _ _))"},{"Add":""},{"Add":"∧-distribˡ-∨-≥ : ∀ x y z → x ∧ y ∨ x ∧ z ≤ x ∧ (y ∨ z)"},{"Add":"∧-distribˡ-∨-≥ x y z = let"},{"Add":" x∧y≤x , x∧y≤y , _ = infimum x y"},{"Add":" x∧z≤x , x∧z≤z , _ = infimum x z"},{"Add":" y≤y∨z , z≤y∨z , _ = supremum y z"},{"Add":" in ∧-greatest (∨-least x∧y≤x x∧z≤x)"},{"Add":" (∨-least (trans x∧y≤y y≤y∨z) (trans x∧z≤z z≤y∨z))"},{"Add":""},{"Add":"∧-distribˡ-∨ : _∧_ DistributesOverˡ _∨_"},{"Add":"∧-distribˡ-∨ x y z = antisym (∧-distribˡ-∨-≤ x y z) (∧-distribˡ-∨-≥ x y z)"},{"Add":""},{"Add":"⇨-distribˡ-∧-≤ : ∀ x y z → x ⇨ y ∧ z ≤ (x ⇨ y) ∧ (x ⇨ z)"},{"Add":"⇨-distribˡ-∧-≤ x y z = let"},{"Add":" y∧z≤y , y∧z≤z , _ = infimum y z"},{"Add":" in ∧-greatest (transpose-⇨ $ trans ⇨-eval y∧z≤y)"},{"Add":" (transpose-⇨ $ trans ⇨-eval y∧z≤z)"},{"Add":""},{"Add":"⇨-distribˡ-∧-≥ : ∀ x y z → (x ⇨ y) ∧ (x ⇨ z) ≤ x ⇨ y ∧ z"},{"Add":"⇨-distribˡ-∧-≥ x y z = transpose-⇨ (begin"},{"Add":" (((x ⇨ y) ∧ (x ⇨ z)) ∧ x) ≈⟨ ∧-cong Eq.refl $ Eq.sym $ ∧-idempotent _ ⟩"},{"Add":" (((x ⇨ y) ∧ (x ⇨ z)) ∧ x ∧ x) ≈⟨ Eq.sym $ ∧-assoc _ _ _ ⟩"},{"Add":" (((x ⇨ y) ∧ (x ⇨ z)) ∧ x) ∧ x ≈⟨ ∧-cong (∧-assoc _ _ _) Eq.refl ⟩"},{"Add":" (((x ⇨ y) ∧ (x ⇨ z) ∧ x) ∧ x) ≈⟨ ∧-cong (∧-cong Eq.refl $ ∧-comm _ _) Eq.refl ⟩"},{"Add":" (((x ⇨ y) ∧ x ∧ (x ⇨ z)) ∧ x) ≈⟨ ∧-cong (Eq.sym $ ∧-assoc _ _ _) Eq.refl ⟩"},{"Add":" (((x ⇨ y) ∧ x) ∧ (x ⇨ z)) ∧ x ≈⟨ ∧-assoc _ _ _ ⟩"},{"Add":" (((x ⇨ y) ∧ x) ∧ (x ⇨ z) ∧ x) ≤⟨ ∧-monotonic ⇨-eval ⇨-eval ⟩"},{"Add":" y ∧ z ∎)"},{"Add":" where open ≤-Reasoning poset"},{"Add":""},{"Add":"⇨-distribˡ-∧ : _⇨_ DistributesOverˡ _∧_"},{"Add":"⇨-distribˡ-∧ x y z = antisym (⇨-distribˡ-∧-≤ x y z) (⇨-distribˡ-∧-≥ x y z)"},{"Add":""},{"Add":"⇨-distribˡ-∨-∧-≤ : ∀ x y z → x ∨ y ⇨ z ≤ (x ⇨ z) ∧ (y ⇨ z)"},{"Add":"⇨-distribˡ-∨-∧-≤ x y z = let x≤x∨y , y≤x∨y , _ = supremum x y"},{"Add":" in ∧-greatest (transpose-⇨ $ trans (∧-monotonic refl x≤x∨y) ⇨-eval)"},{"Add":" (transpose-⇨ $ trans (∧-monotonic refl y≤x∨y) ⇨-eval)"},{"Add":""},{"Add":"⇨-distribˡ-∨-∧-≥ : ∀ x y z → (x ⇨ z) ∧ (y ⇨ z) ≤ x ∨ y ⇨ z"},{"Add":"⇨-distribˡ-∨-∧-≥ x y z = transpose-⇨ (trans (reflexive $ ∧-distribˡ-∨ _ _ _)"},{"Add":" (∨-least (trans (transpose-∧ (x∧y≤x _ _)) refl)"},{"Add":" (trans (transpose-∧ (x∧y≤y _ _)) refl)))"},{"Add":""},{"Add":"⇨-distribˡ-∨-∧ : ∀ x y z → x ∨ y ⇨ z ≈ (x ⇨ z) ∧ (y ⇨ z)"},{"Add":"⇨-distribˡ-∨-∧ x y z = antisym (⇨-distribˡ-∨-∧-≤ x y z) (⇨-distribˡ-∨-∧-≥ x y z)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Heyting algebras are distributive lattices"},{"Add":""},{"Add":"isDistributiveLattice : IsDistributiveLattice _≈_ _≤_ _∨_ _∧_"},{"Add":"isDistributiveLattice = record"},{"Add":" { isLattice = isLattice"},{"Add":" ; ∧-distribˡ-∨ = ∧-distribˡ-∨"},{"Add":" }"},{"Add":""},{"Add":"distributiveLattice : DistributiveLattice _ _ _"},{"Add":"distributiveLattice = record"},{"Add":" { isDistributiveLattice = isDistributiveLattice"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Heyting algebras can define pseudo-complement"},{"Add":""},{"Add":"infix 8 ¬_"},{"Add":""},{"Add":"¬_ : Op₁ Carrier"},{"Add":"¬ x = x ⇨ ⊥"},{"Add":""},{"Add":"x≤¬¬x : ∀ x → x ≤ ¬ ¬ x"},{"Add":"x≤¬¬x x = transpose-⇨ (trans (reflexive (∧-comm _ _)) ⇨-eval)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- De-Morgan laws"},{"Add":""},{"Add":"de-morgan₁ : ∀ x y → ¬ (x ∨ y) ≈ ¬ x ∧ ¬ y"},{"Add":"de-morgan₁ x y = ⇨-distribˡ-∨-∧ _ _ _"},{"Add":""},{"Add":"de-morgan₂-≤ : ∀ x y → ¬ (x ∧ y) ≤ ¬ ¬ (¬ x ∨ ¬ y)"},{"Add":"de-morgan₂-≤ x y = transpose-⇨ $ begin"},{"Add":" ¬ (x ∧ y) ∧ ¬ (¬ x ∨ ¬ y) ≈⟨ ∧-cong ⇨-curry (de-morgan₁ _ _) ⟩"},{"Add":" (x ⇨ ¬ y) ∧ ¬ ¬ x ∧ ¬ ¬ y ≈⟨ ∧-cong Eq.refl (∧-comm _ _) ⟩"},{"Add":" (x ⇨ ¬ y) ∧ ¬ ¬ y ∧ ¬ ¬ x ≈⟨ Eq.sym $ ∧-assoc _ _ _ ⟩"},{"Add":" ((x ⇨ ¬ y) ∧ ¬ ¬ y) ∧ ¬ ¬ x ≤⟨ ⇨-applyʳ $ transpose-⇨ $"},{"Add":" begin"},{"Add":" ((x ⇨ ¬ y) ∧ ¬ ¬ y) ∧ x ≈⟨ ∧-cong (∧-comm _ _) Eq.refl ⟩"},{"Add":" ((¬ ¬ y) ∧ (x ⇨ ¬ y)) ∧ x ≈⟨ ∧-assoc _ _ _ ⟩"},{"Add":" (¬ ¬ y) ∧ (x ⇨ ¬ y) ∧ x ≤⟨ ∧-monotonic refl ⇨-eval ⟩"},{"Add":" ¬ ¬ y ∧ ¬ y ≤⟨ ⇨-eval ⟩"},{"Add":" ⊥ ∎ ⟩"},{"Add":" ⊥ ∎"},{"Add":" where open ≤-Reasoning poset"},{"Add":""},{"Add":"de-morgan₂-≥ : ∀ x y → ¬ ¬ (¬ x ∨ ¬ y) ≤ ¬ (x ∧ y)"},{"Add":"de-morgan₂-≥ x y = transpose-⇨ $ ⇨-applyˡ $ transpose-⇨ $ begin"},{"Add":" (x ∧ y) ∧ (¬ x ∨ ¬ y) ≈⟨ ∧-distribˡ-∨ _ _ _ ⟩"},{"Add":" (x ∧ y) ∧ ¬ x ∨ (x ∧ y) ∧ ¬ y ≤⟨ ∨-monotonic (⇨-applyʳ (x∧y≤x _ _))"},{"Add":" (⇨-applyʳ (x∧y≤y _ _)) ⟩"},{"Add":" ⊥ ∨ ⊥ ≈⟨ ∨-idempotent _ ⟩"},{"Add":" ⊥ ∎"},{"Add":" where open ≤-Reasoning poset"},{"Add":""},{"Add":"de-morgan₂ : ∀ x y → ¬ (x ∧ y) ≈ ¬ ¬ (¬ x ∨ ¬ y)"},{"Add":"de-morgan₂ x y = antisym (de-morgan₂-≤ x y) (de-morgan₂-≥ x y)"},{"Add":""},{"Add":"weak-lem : ∀ {x} → ¬ ¬ (¬ x ∨ x) ≈ ⊤"},{"Add":"weak-lem {x} = begin"},{"Add":" ¬ ¬ (¬ x ∨ x) ≈⟨ ⇨-cong (de-morgan₁ _ _) Eq.refl ⟩"},{"Add":" ¬ (¬ ¬ x ∧ ¬ x) ≈⟨ ⇨-cong ⇨-app Eq.refl ⟩"},{"Add":" ⊥ ∧ (x ⇨ ⊥) ⇨ ⊥ ≈⟨ ⇨-cong (∧-zeroˡ _) Eq.refl ⟩"},{"Add":" ⊥ ⇨ ⊥ ≈⟨ ⇨-unit ⟩"},{"Add":" ⊤ ∎"},{"Add":" where open ≈-Reasoning setoid"}]}},{"FileAdd":{"meta":{"basename":"DistributiveLattice.agda","metadata":0},"context":[{"change":0,"pos":562402}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties for distributive lattice"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Lattice.Properties.DistributiveLattice"},{"Add":" {c ℓ₁ ℓ₂} (L : DistributiveLattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open DistributiveLattice L hiding (refl)"},{"Add":""},{"Add":"open import Algebra.Definitions _≈_"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Reasoning.Setoid setoid"},{"Add":"open import Relation.Binary.Lattice.Properties.Lattice lattice"},{"Add":"open import Relation.Binary.Lattice.Properties.MeetSemilattice meetSemilattice"},{"Add":"open import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice"},{"Add":""},{"Add":"open Setoid setoid"},{"Add":""},{"Add":"∧-distribʳ-∨ : _∧_ DistributesOverʳ _∨_"},{"Add":"∧-distribʳ-∨ x y z = begin"},{"Add":" (y ∨ z) ∧ x ≈⟨ ∧-comm _ _ ⟩"},{"Add":" x ∧ (y ∨ z) ≈⟨ ∧-distribˡ-∨ x y z ⟩"},{"Add":" x ∧ y ∨ x ∧ z ≈⟨ ∨-cong (∧-comm _ _) (∧-comm _ _) ⟩"},{"Add":" y ∧ x ∨ z ∧ x ∎"},{"Add":""},{"Add":"∧-distrib-∨ : _∧_ DistributesOver _∨_"},{"Add":"∧-distrib-∨ = ∧-distribˡ-∨ , ∧-distribʳ-∨"},{"Add":""},{"Add":"∨-distribˡ-∧ : _∨_ DistributesOverˡ _∧_"},{"Add":"∨-distribˡ-∧ x y z = begin"},{"Add":" x ∨ y ∧ z ≈⟨ ∨-cong (sym (∨-absorbs-∧ x y)) refl ⟩"},{"Add":" (x ∨ x ∧ y) ∨ y ∧ z ≈⟨ ∨-cong (∨-cong refl (∧-comm _ _)) refl ⟩"},{"Add":" (x ∨ y ∧ x) ∨ y ∧ z ≈⟨ ∨-assoc x (y ∧ x) (y ∧ z) ⟩"},{"Add":" x ∨ y ∧ x ∨ y ∧ z ≈⟨ ∨-cong refl (sym (∧-distribˡ-∨ y x z)) ⟩"},{"Add":" x ∨ y ∧ (x ∨ z) ≈⟨ ∨-cong (sym (∧-absorbs-∨ _ _)) refl ⟩"},{"Add":" x ∧ (x ∨ z) ∨ y ∧ (x ∨ z) ≈⟨ sym (∧-distribʳ-∨ (x ∨ z) x y) ⟩"},{"Add":" (x ∨ y) ∧ (x ∨ z) ∎"},{"Add":""},{"Add":"∨-distribʳ-∧ : _∨_ DistributesOverʳ _∧_"},{"Add":"∨-distribʳ-∧ x y z = begin"},{"Add":" y ∧ z ∨ x ≈⟨ ∨-comm _ _ ⟩"},{"Add":" x ∨ y ∧ z ≈⟨ ∨-distribˡ-∧ _ _ _ ⟩"},{"Add":" (x ∨ y) ∧ (x ∨ z) ≈⟨ ∧-cong (∨-comm _ _) (∨-comm _ _) ⟩"},{"Add":" (y ∨ x) ∧ (z ∨ x) ∎"},{"Add":""},{"Add":"∨-distrib-∧ : _∨_ DistributesOver _∧_"},{"Add":"∨-distrib-∧ = ∨-distribˡ-∧ , ∨-distribʳ-∧"}]}},{"FileAdd":{"meta":{"basename":"BoundedMeetSemilattice.agda","metadata":0},"context":[{"change":0,"pos":562402}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties satisfied by bounded meet semilattices"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Lattice.Properties.BoundedMeetSemilattice"},{"Add":" {c ℓ₁ ℓ₂} (M : BoundedMeetSemilattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open BoundedMeetSemilattice M"},{"Add":""},{"Add":"open import Algebra.Definitions _≈_"},{"Add":"open import Function.Base using (_∘_; flip)"},{"Add":"open import Relation.Binary.Properties.Poset poset"},{"Add":"import Relation.Binary.Lattice.Properties.BoundedJoinSemilattice as J"},{"Add":""},{"Add":"-- The dual construction is a bounded join semilattice."},{"Add":""},{"Add":"dualIsBoundedJoinSemilattice : IsBoundedJoinSemilattice _≈_ (flip _≤_) _∧_ ⊤"},{"Add":"dualIsBoundedJoinSemilattice = record"},{"Add":" { isJoinSemilattice = record"},{"Add":" { isPartialOrder = ≥-isPartialOrder"},{"Add":" ; supremum = infimum"},{"Add":" }"},{"Add":" ; minimum = maximum"},{"Add":" }"},{"Add":""},{"Add":"dualBoundedJoinSemilattice : BoundedJoinSemilattice c ℓ₁ ℓ₂"},{"Add":"dualBoundedJoinSemilattice = record"},{"Add":" { ⊥ = ⊤"},{"Add":" ; isBoundedJoinSemilattice = dualIsBoundedJoinSemilattice"},{"Add":" }"},{"Add":""},{"Add":"open J dualBoundedJoinSemilattice"},{"Add":" hiding (dualIsBoundedMeetSemilattice; dualBoundedMeetSemilattice) public"}]}},{"FileAdd":{"meta":{"basename":"BoundedLattice.agda","metadata":0},"context":[{"change":0,"pos":562402}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties satisfied by bounded lattice"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Lattice.Properties.BoundedLattice"},{"Add":" {c ℓ₁ ℓ₂} (L : BoundedLattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open BoundedLattice L"},{"Add":""},{"Add":"open import Algebra.Definitions _≈_"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Lattice.Properties.MeetSemilattice meetSemilattice"},{"Add":"open import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice"},{"Add":""},{"Add":"open Setoid setoid renaming (trans to ≈-trans)"},{"Add":""},{"Add":"∧-zeroʳ : RightZero ⊥ _∧_"},{"Add":"∧-zeroʳ x = y≤x⇒x∧y≈y (minimum x)"},{"Add":""},{"Add":"∧-zeroˡ : LeftZero ⊥ _∧_"},{"Add":"∧-zeroˡ x = ≈-trans (∧-comm ⊥ x) (∧-zeroʳ x)"},{"Add":""},{"Add":"∧-zero : Zero ⊥ _∧_"},{"Add":"∧-zero = ∧-zeroˡ , ∧-zeroʳ"},{"Add":""},{"Add":"∨-zeroʳ : RightZero ⊤ _∨_"},{"Add":"∨-zeroʳ x = x≤y⇒x∨y≈y (maximum x)"},{"Add":""},{"Add":"∨-zeroˡ : LeftZero ⊤ _∨_"},{"Add":"∨-zeroˡ x = ≈-trans (∨-comm ⊤ x) (∨-zeroʳ x)"},{"Add":""},{"Add":"∨-zero : Zero ⊤ _∨_"},{"Add":"∨-zero = ∨-zeroˡ , ∨-zeroʳ"}]}},{"FileAdd":{"meta":{"basename":"BoundedJoinSemilattice.agda","metadata":0},"context":[{"change":0,"pos":562402}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties satisfied by bounded join semilattices"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Lattice"},{"Add":""},{"Add":"module Relation.Binary.Lattice.Properties.BoundedJoinSemilattice"},{"Add":" {c ℓ₁ ℓ₂} (J : BoundedJoinSemilattice c ℓ₁ ℓ₂) where"},{"Add":""},{"Add":"open BoundedJoinSemilattice J"},{"Add":""},{"Add":"open import Algebra.Definitions _≈_"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base using (_∘_; flip)"},{"Add":"open import Relation.Binary.Properties.Poset poset"},{"Add":"open import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice"},{"Add":" using (∨-comm)"},{"Add":""},{"Add":"-- Bottom is an identity of the meet operation."},{"Add":""},{"Add":"identityˡ : LeftIdentity ⊥ _∨_"},{"Add":"identityˡ x ="},{"Add":" let _ , x≤⊥∨x , least = supremum ⊥ x"},{"Add":" in antisym (least x (minimum x) refl) x≤⊥∨x"},{"Add":""},{"Add":"identityʳ : RightIdentity ⊥ _∨_"},{"Add":"identityʳ x ="},{"Add":" let x≤x∨⊥ , _ , least = supremum x ⊥"},{"Add":" in antisym (least x refl (minimum x)) x≤x∨⊥"},{"Add":""},{"Add":"identity : Identity ⊥ _∨_"},{"Add":"identity = identityˡ , identityʳ"},{"Add":""},{"Add":""},{"Add":"-- The dual construction is a bounded meet semilattice."},{"Add":""},{"Add":"dualIsBoundedMeetSemilattice : IsBoundedMeetSemilattice _≈_ (flip _≤_) _∨_ ⊥"},{"Add":"dualIsBoundedMeetSemilattice = record"},{"Add":" { isMeetSemilattice = record"},{"Add":" { isPartialOrder = ≥-isPartialOrder"},{"Add":" ; infimum = supremum"},{"Add":" }"},{"Add":" ; maximum = minimum"},{"Add":" }"},{"Add":""},{"Add":"dualBoundedMeetSemilattice : BoundedMeetSemilattice c ℓ₁ ℓ₂"},{"Add":"dualBoundedMeetSemilattice = record"},{"Add":" { ⊤ = ⊥"},{"Add":" ; isBoundedMeetSemilattice = dualIsBoundedMeetSemilattice"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Definitions.agda","metadata":0},"context":[{"change":0,"pos":555498}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Definitions for order-theoretic lattices"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via"},{"Add":"-- `Relation.Binary.Lattice`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Lattice.Definitions where"},{"Add":""},{"Add":"open import Algebra.Core"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Function.Base using (flip)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a ℓ : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relationships between orders and operators"},{"Add":""},{"Add":"Supremum : Rel A ℓ → Op₂ A → Set _"},{"Add":"Supremum _≤_ _∨_ ="},{"Add":" ∀ x y → x ≤ (x ∨ y) × y ≤ (x ∨ y) × ∀ z → x ≤ z → y ≤ z → (x ∨ y) ≤ z"},{"Add":""},{"Add":"Infimum : Rel A ℓ → Op₂ A → Set _"},{"Add":"Infimum _≤_ = Supremum (flip _≤_)"},{"Add":""},{"Add":"Exponential : Rel A ℓ → Op₂ A → Op₂ A → Set _"},{"Add":"Exponential _≤_ _∧_ _⇨_ ="},{"Add":" ∀ w x y → ((w ∧ x) ≤ y → w ≤ (x ⇨ y)) × (w ≤ (x ⇨ y) → (w ∧ x) ≤ y)"}]}},{"FileAdd":{"meta":{"basename":"Bundles.agda","metadata":0},"context":[{"change":0,"pos":555498}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Bundles for order-theoretic lattices"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via"},{"Add":"-- `Relation.Binary.Lattice`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Lattice.Bundles where"},{"Add":""},{"Add":"open import Algebra.Core"},{"Add":"open import Level using (suc; _⊔_)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Poset; Setoid)"},{"Add":"open import Relation.Binary.Lattice.Structures"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Join semilattices"},{"Add":""},{"Add":"record JoinSemilattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where"},{"Add":" infix 4 _≈_ _≤_"},{"Add":" infixr 6 _∨_"},{"Add":" field"},{"Add":" Carrier : Set c"},{"Add":" _≈_ : Rel Carrier ℓ₁ -- The underlying equality."},{"Add":" _≤_ : Rel Carrier ℓ₂ -- The partial order."},{"Add":" _∨_ : Op₂ Carrier -- The join operation."},{"Add":" isJoinSemilattice : IsJoinSemilattice _≈_ _≤_ _∨_"},{"Add":""},{"Add":" open IsJoinSemilattice isJoinSemilattice public"},{"Add":""},{"Add":" poset : Poset c ℓ₁ ℓ₂"},{"Add":" poset = record { isPartialOrder = isPartialOrder }"},{"Add":""},{"Add":" open Poset poset public using (preorder)"},{"Add":""},{"Add":""},{"Add":"record BoundedJoinSemilattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where"},{"Add":" infix 4 _≈_ _≤_"},{"Add":" infixr 6 _∨_"},{"Add":" field"},{"Add":" Carrier : Set c"},{"Add":" _≈_ : Rel Carrier ℓ₁ -- The underlying equality."},{"Add":" _≤_ : Rel Carrier ℓ₂ -- The partial order."},{"Add":" _∨_ : Op₂ Carrier -- The join operation."},{"Add":" ⊥ : Carrier -- The minimum."},{"Add":" isBoundedJoinSemilattice : IsBoundedJoinSemilattice _≈_ _≤_ _∨_ ⊥"},{"Add":""},{"Add":" open IsBoundedJoinSemilattice isBoundedJoinSemilattice public"},{"Add":""},{"Add":" joinSemilattice : JoinSemilattice c ℓ₁ ℓ₂"},{"Add":" joinSemilattice = record { isJoinSemilattice = isJoinSemilattice }"},{"Add":""},{"Add":" open JoinSemilattice joinSemilattice public using (preorder; poset)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Meet semilattices"},{"Add":""},{"Add":"record MeetSemilattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where"},{"Add":" infix 4 _≈_ _≤_"},{"Add":" infixr 7 _∧_"},{"Add":" field"},{"Add":" Carrier : Set c"},{"Add":" _≈_ : Rel Carrier ℓ₁ -- The underlying equality."},{"Add":" _≤_ : Rel Carrier ℓ₂ -- The partial order."},{"Add":" _∧_ : Op₂ Carrier -- The meet operation."},{"Add":" isMeetSemilattice : IsMeetSemilattice _≈_ _≤_ _∧_"},{"Add":""},{"Add":" open IsMeetSemilattice isMeetSemilattice public"},{"Add":""},{"Add":" poset : Poset c ℓ₁ ℓ₂"},{"Add":" poset = record { isPartialOrder = isPartialOrder }"},{"Add":""},{"Add":" open Poset poset public using (preorder)"},{"Add":""},{"Add":"record BoundedMeetSemilattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where"},{"Add":" infix 4 _≈_ _≤_"},{"Add":" infixr 7 _∧_"},{"Add":" field"},{"Add":" Carrier : Set c"},{"Add":" _≈_ : Rel Carrier ℓ₁ -- The underlying equality."},{"Add":" _≤_ : Rel Carrier ℓ₂ -- The partial order."},{"Add":" _∧_ : Op₂ Carrier -- The join operation."},{"Add":" ⊤ : Carrier -- The maximum."},{"Add":" isBoundedMeetSemilattice : IsBoundedMeetSemilattice _≈_ _≤_ _∧_ ⊤"},{"Add":""},{"Add":" open IsBoundedMeetSemilattice isBoundedMeetSemilattice public"},{"Add":""},{"Add":" meetSemilattice : MeetSemilattice c ℓ₁ ℓ₂"},{"Add":" meetSemilattice = record { isMeetSemilattice = isMeetSemilattice }"},{"Add":""},{"Add":" open MeetSemilattice meetSemilattice public using (preorder; poset)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Lattices"},{"Add":""},{"Add":"record Lattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where"},{"Add":" infix 4 _≈_ _≤_"},{"Add":" infixr 6 _∨_"},{"Add":" infixr 7 _∧_"},{"Add":" field"},{"Add":" Carrier : Set c"},{"Add":" _≈_ : Rel Carrier ℓ₁ -- The underlying equality."},{"Add":" _≤_ : Rel Carrier ℓ₂ -- The partial order."},{"Add":" _∨_ : Op₂ Carrier -- The join operation."},{"Add":" _∧_ : Op₂ Carrier -- The meet operation."},{"Add":" isLattice : IsLattice _≈_ _≤_ _∨_ _∧_"},{"Add":""},{"Add":" open IsLattice isLattice public"},{"Add":""},{"Add":" setoid : Setoid c ℓ₁"},{"Add":" setoid = record { isEquivalence = isEquivalence }"},{"Add":""},{"Add":" joinSemilattice : JoinSemilattice c ℓ₁ ℓ₂"},{"Add":" joinSemilattice = record { isJoinSemilattice = isJoinSemilattice }"},{"Add":""},{"Add":" meetSemilattice : MeetSemilattice c ℓ₁ ℓ₂"},{"Add":" meetSemilattice = record { isMeetSemilattice = isMeetSemilattice }"},{"Add":""},{"Add":" open JoinSemilattice joinSemilattice public using (poset; preorder)"},{"Add":""},{"Add":"record DistributiveLattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where"},{"Add":" infix 4 _≈_ _≤_"},{"Add":" infixr 6 _∨_"},{"Add":" infixr 7 _∧_"},{"Add":" field"},{"Add":" Carrier : Set c"},{"Add":" _≈_ : Rel Carrier ℓ₁ -- The underlying equality."},{"Add":" _≤_ : Rel Carrier ℓ₂ -- The partial order."},{"Add":" _∨_ : Op₂ Carrier -- The join operation."},{"Add":" _∧_ : Op₂ Carrier -- The meet operation."},{"Add":" isDistributiveLattice : IsDistributiveLattice _≈_ _≤_ _∨_ _∧_"},{"Add":""},{"Add":" open IsDistributiveLattice isDistributiveLattice using (∧-distribˡ-∨) public"},{"Add":" open IsDistributiveLattice isDistributiveLattice using (isLattice)"},{"Add":""},{"Add":" lattice : Lattice c ℓ₁ ℓ₂"},{"Add":" lattice = record { isLattice = isLattice }"},{"Add":""},{"Add":" open Lattice lattice hiding (Carrier; _≈_; _≤_; _∨_; _∧_) public"},{"Add":""},{"Add":"record BoundedLattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where"},{"Add":" infix 4 _≈_ _≤_"},{"Add":" infixr 6 _∨_"},{"Add":" infixr 7 _∧_"},{"Add":" field"},{"Add":" Carrier : Set c"},{"Add":" _≈_ : Rel Carrier ℓ₁ -- The underlying equality."},{"Add":" _≤_ : Rel Carrier ℓ₂ -- The partial order."},{"Add":" _∨_ : Op₂ Carrier -- The join operation."},{"Add":" _∧_ : Op₂ Carrier -- The meet operation."},{"Add":" ⊤ : Carrier -- The maximum."},{"Add":" ⊥ : Carrier -- The minimum."},{"Add":" isBoundedLattice : IsBoundedLattice _≈_ _≤_ _∨_ _∧_ ⊤ ⊥"},{"Add":""},{"Add":" open IsBoundedLattice isBoundedLattice public"},{"Add":""},{"Add":" boundedJoinSemilattice : BoundedJoinSemilattice c ℓ₁ ℓ₂"},{"Add":" boundedJoinSemilattice = record"},{"Add":" { isBoundedJoinSemilattice = isBoundedJoinSemilattice }"},{"Add":""},{"Add":" boundedMeetSemilattice : BoundedMeetSemilattice c ℓ₁ ℓ₂"},{"Add":" boundedMeetSemilattice = record"},{"Add":" { isBoundedMeetSemilattice = isBoundedMeetSemilattice }"},{"Add":""},{"Add":" lattice : Lattice c ℓ₁ ℓ₂"},{"Add":" lattice = record { isLattice = isLattice }"},{"Add":""},{"Add":" open Lattice lattice public"},{"Add":" using (joinSemilattice; meetSemilattice; poset; preorder; setoid)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Heyting algebras (a bounded lattice with exponential operator)"},{"Add":""},{"Add":"record HeytingAlgebra c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where"},{"Add":" infix 4 _≈_ _≤_"},{"Add":" infixr 5 _⇨_"},{"Add":" infixr 6 _∨_"},{"Add":" infixr 7 _∧_"},{"Add":" field"},{"Add":" Carrier : Set c"},{"Add":" _≈_ : Rel Carrier ℓ₁ -- The underlying equality."},{"Add":" _≤_ : Rel Carrier ℓ₂ -- The partial order."},{"Add":" _∨_ : Op₂ Carrier -- The join operation."},{"Add":" _∧_ : Op₂ Carrier -- The meet operation."},{"Add":" _⇨_ : Op₂ Carrier -- The exponential operation."},{"Add":" ⊤ : Carrier -- The maximum."},{"Add":" ⊥ : Carrier -- The minimum."},{"Add":" isHeytingAlgebra : IsHeytingAlgebra _≈_ _≤_ _∨_ _∧_ _⇨_ ⊤ ⊥"},{"Add":""},{"Add":" boundedLattice : BoundedLattice c ℓ₁ ℓ₂"},{"Add":" boundedLattice = record"},{"Add":" { isBoundedLattice = IsHeytingAlgebra.isBoundedLattice isHeytingAlgebra }"},{"Add":""},{"Add":" open IsHeytingAlgebra isHeytingAlgebra"},{"Add":" using (exponential; transpose-⇨; transpose-∧) public"},{"Add":" open BoundedLattice boundedLattice"},{"Add":" hiding (Carrier; _≈_; _≤_; _∨_; _∧_; ⊤; ⊥) public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Boolean algebras (a specialized Heyting algebra)"},{"Add":""},{"Add":"record BooleanAlgebra c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where"},{"Add":" infix 4 _≈_ _≤_"},{"Add":" infixr 6 _∨_"},{"Add":" infixr 7 _∧_"},{"Add":" infix 8 ¬_"},{"Add":" field"},{"Add":" Carrier : Set c"},{"Add":" _≈_ : Rel Carrier ℓ₁ -- The underlying equality."},{"Add":" _≤_ : Rel Carrier ℓ₂ -- The partial order."},{"Add":" _∨_ : Op₂ Carrier -- The join operation."},{"Add":" _∧_ : Op₂ Carrier -- The meet operation."},{"Add":" ¬_ : Op₁ Carrier -- The negation operation."},{"Add":" ⊤ : Carrier -- The maximum."},{"Add":" ⊥ : Carrier -- The minimum."},{"Add":" isBooleanAlgebra : IsBooleanAlgebra _≈_ _≤_ _∨_ _∧_ ¬_ ⊤ ⊥"},{"Add":""},{"Add":" open IsBooleanAlgebra isBooleanAlgebra using (isHeytingAlgebra)"},{"Add":""},{"Add":" heytingAlgebra : HeytingAlgebra c ℓ₁ ℓ₂"},{"Add":" heytingAlgebra = record { isHeytingAlgebra = isHeytingAlgebra }"},{"Add":""},{"Add":" open HeytingAlgebra heytingAlgebra public"},{"Add":" hiding (Carrier; _≈_; _≤_; _∨_; _∧_; ⊤; ⊥)"}]}},{"FileAdd":{"meta":{"basename":"Indexed","metadata":576},"context":[{"change":0,"pos":415495}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Homogeneous.agda","metadata":0},"context":[{"change":0,"pos":599691}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Homogeneously-indexed binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Indexed.Homogeneous where"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Publicly export core definitions"},{"Add":""},{"Add":"open import Relation.Binary.Indexed.Homogeneous.Core public"},{"Add":"open import Relation.Binary.Indexed.Homogeneous.Definitions public"},{"Add":"open import Relation.Binary.Indexed.Homogeneous.Structures public"},{"Add":"open import Relation.Binary.Indexed.Homogeneous.Bundles public"}]}},{"FileAdd":{"meta":{"basename":"Homogeneous","metadata":576},"context":[{"change":0,"pos":599691}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Structures.agda","metadata":0},"context":[{"change":0,"pos":600438}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Homogeneously-indexed binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via"},{"Add":"-- `Relation.Binary.Indexed.Homogeneous`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Indexed.Homogeneous.Core"},{"Add":""},{"Add":"module Relation.Binary.Indexed.Homogeneous.Structures"},{"Add":" {i a ℓ} {I : Set i}"},{"Add":" (A : I → Set a) -- The underlying indexed sets"},{"Add":" (_≈ᵢ_ : IRel A ℓ) -- The underlying indexed equality relation"},{"Add":" where"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base using (_⟨_⟩_)"},{"Add":"open import Level using (Level; _⊔_; suc)"},{"Add":"open import Relation.Binary.Core using (_⇒_)"},{"Add":"import Relation.Binary.Definitions as B"},{"Add":"import Relation.Binary.Structures as B"},{"Add":"open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)"},{"Add":"open import Relation.Binary.Indexed.Homogeneous.Definitions"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Equivalences"},{"Add":""},{"Add":"-- Indexed structures are laid out in a similar manner as to those"},{"Add":"-- in Relation.Binary. The main difference is each structure also"},{"Add":"-- contains proofs for the lifted version of the relation."},{"Add":""},{"Add":"record IsIndexedEquivalence : Set (i ⊔ a ⊔ ℓ) where"},{"Add":" field"},{"Add":" reflᵢ : Reflexive A _≈ᵢ_"},{"Add":" symᵢ : Symmetric A _≈ᵢ_"},{"Add":" transᵢ : Transitive A _≈ᵢ_"},{"Add":""},{"Add":" reflexiveᵢ : ∀ {i} → _≡_ ⟨ _⇒_ ⟩ _≈ᵢ_ {i}"},{"Add":" reflexiveᵢ ≡.refl = reflᵢ"},{"Add":""},{"Add":" -- Lift properties"},{"Add":""},{"Add":" reflexive : _≡_ ⇒ (Lift A _≈ᵢ_)"},{"Add":" reflexive ≡.refl i = reflᵢ"},{"Add":""},{"Add":" refl : B.Reflexive (Lift A _≈ᵢ_)"},{"Add":" refl i = reflᵢ"},{"Add":""},{"Add":" sym : B.Symmetric (Lift A _≈ᵢ_)"},{"Add":" sym x≈y i = symᵢ (x≈y i)"},{"Add":""},{"Add":" trans : B.Transitive (Lift A _≈ᵢ_)"},{"Add":" trans x≈y y≈z i = transᵢ (x≈y i) (y≈z i)"},{"Add":""},{"Add":" isEquivalence : B.IsEquivalence (Lift A _≈ᵢ_)"},{"Add":" isEquivalence = record"},{"Add":" { refl = refl"},{"Add":" ; sym = sym"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":""},{"Add":""},{"Add":"record IsIndexedDecEquivalence : Set (i ⊔ a ⊔ ℓ) where"},{"Add":" infix 4 _≟ᵢ_"},{"Add":" field"},{"Add":" _≟ᵢ_ : Decidable A _≈ᵢ_"},{"Add":" isEquivalenceᵢ : IsIndexedEquivalence"},{"Add":""},{"Add":" open IsIndexedEquivalence isEquivalenceᵢ public"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Preorders"},{"Add":""},{"Add":"record IsIndexedPreorder {ℓ₂} (_∼ᵢ_ : IRel A ℓ₂)"},{"Add":" : Set (i ⊔ a ⊔ ℓ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isEquivalenceᵢ : IsIndexedEquivalence"},{"Add":" reflexiveᵢ : _≈ᵢ_ ⇒[ A ] _∼ᵢ_"},{"Add":" transᵢ : Transitive A _∼ᵢ_"},{"Add":""},{"Add":" module Eq = IsIndexedEquivalence isEquivalenceᵢ"},{"Add":""},{"Add":" reflᵢ : Reflexive A _∼ᵢ_"},{"Add":" reflᵢ = reflexiveᵢ Eq.reflᵢ"},{"Add":""},{"Add":" ∼ᵢ-respˡ-≈ᵢ : Respectsˡ A _∼ᵢ_ _≈ᵢ_"},{"Add":" ∼ᵢ-respˡ-≈ᵢ x≈y x∼z = transᵢ (reflexiveᵢ (Eq.symᵢ x≈y)) x∼z"},{"Add":""},{"Add":" ∼ᵢ-respʳ-≈ᵢ : Respectsʳ A _∼ᵢ_ _≈ᵢ_"},{"Add":" ∼ᵢ-respʳ-≈ᵢ x≈y z∼x = transᵢ z∼x (reflexiveᵢ x≈y)"},{"Add":""},{"Add":" ∼ᵢ-resp-≈ᵢ : Respects₂ A _∼ᵢ_ _≈ᵢ_"},{"Add":" ∼ᵢ-resp-≈ᵢ = ∼ᵢ-respʳ-≈ᵢ , ∼ᵢ-respˡ-≈ᵢ"},{"Add":""},{"Add":" -- Lifted properties"},{"Add":""},{"Add":" reflexive : Lift A _≈ᵢ_ ⇒ Lift A _∼ᵢ_"},{"Add":" reflexive x≈y i = reflexiveᵢ (x≈y i)"},{"Add":""},{"Add":" refl : B.Reflexive (Lift A _∼ᵢ_)"},{"Add":" refl i = reflᵢ"},{"Add":""},{"Add":" trans : B.Transitive (Lift A _∼ᵢ_)"},{"Add":" trans x≈y y≈z i = transᵢ (x≈y i) (y≈z i)"},{"Add":""},{"Add":" ∼-respˡ-≈ : (Lift A _∼ᵢ_) B.Respectsˡ (Lift A _≈ᵢ_)"},{"Add":" ∼-respˡ-≈ x≈y x∼z i = ∼ᵢ-respˡ-≈ᵢ (x≈y i) (x∼z i)"},{"Add":""},{"Add":" ∼-respʳ-≈ : (Lift A _∼ᵢ_) B.Respectsʳ (Lift A _≈ᵢ_)"},{"Add":" ∼-respʳ-≈ x≈y z∼x i = ∼ᵢ-respʳ-≈ᵢ (x≈y i) (z∼x i)"},{"Add":""},{"Add":" ∼-resp-≈ : (Lift A _∼ᵢ_) B.Respects₂ (Lift A _≈ᵢ_)"},{"Add":" ∼-resp-≈ = ∼-respʳ-≈ , ∼-respˡ-≈"},{"Add":""},{"Add":" isPreorder : B.IsPreorder (Lift A _≈ᵢ_) (Lift A _∼ᵢ_)"},{"Add":" isPreorder = record"},{"Add":" { isEquivalence = Eq.isEquivalence"},{"Add":" ; reflexive = reflexive"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Partial orders"},{"Add":""},{"Add":"record IsIndexedPartialOrder {ℓ₂} (_≤ᵢ_ : IRel A ℓ₂)"},{"Add":" : Set (i ⊔ a ⊔ ℓ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isPreorderᵢ : IsIndexedPreorder _≤ᵢ_"},{"Add":" antisymᵢ : Antisymmetric A _≈ᵢ_ _≤ᵢ_"},{"Add":""},{"Add":" open IsIndexedPreorder isPreorderᵢ public"},{"Add":" renaming"},{"Add":" ( ∼ᵢ-respˡ-≈ᵢ to ≤ᵢ-respˡ-≈ᵢ"},{"Add":" ; ∼ᵢ-respʳ-≈ᵢ to ≤ᵢ-respʳ-≈ᵢ"},{"Add":" ; ∼ᵢ-resp-≈ᵢ to ≤ᵢ-resp-≈ᵢ"},{"Add":" ; ∼-respˡ-≈ to ≤-respˡ-≈"},{"Add":" ; ∼-respʳ-≈ to ≤-respʳ-≈"},{"Add":" ; ∼-resp-≈ to ≤-resp-≈"},{"Add":" )"},{"Add":""},{"Add":" antisym : B.Antisymmetric (Lift A _≈ᵢ_) (Lift A _≤ᵢ_)"},{"Add":" antisym x≤y y≤x i = antisymᵢ (x≤y i) (y≤x i)"},{"Add":""},{"Add":" isPartialOrder : B.IsPartialOrder (Lift A _≈ᵢ_) (Lift A _≤ᵢ_)"},{"Add":" isPartialOrder = record"},{"Add":" { isPreorder = isPreorder"},{"Add":" ; antisym = antisym"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Definitions.agda","metadata":0},"context":[{"change":0,"pos":600438}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Homogeneously-indexed binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via"},{"Add":"-- `Relation.Binary.Indexed.Homogeneous`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Indexed.Homogeneous.Definitions where"},{"Add":""},{"Add":"open import Data.Product.Base using (_×_)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (_⇒_)"},{"Add":"import Relation.Binary.Definitions as B"},{"Add":"open import Relation.Unary.Indexed using (IPred)"},{"Add":""},{"Add":"open import Relation.Binary.Indexed.Homogeneous.Core"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" i a ℓ ℓ₁ ℓ₂ : Level"},{"Add":" I : Set i"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definitions"},{"Add":""},{"Add":"module _ (A : I → Set a) where"},{"Add":""},{"Add":" syntax Implies A _∼₁_ _∼₂_ = _∼₁_ ⇒[ A ] _∼₂_"},{"Add":""},{"Add":" Implies : IRel A ℓ₁ → IRel A ℓ₂ → Set _"},{"Add":" Implies _∼₁_ _∼₂_ = ∀ {i} → _∼₁_ ⇒ (_∼₂_ {i})"},{"Add":""},{"Add":" Reflexive : IRel A ℓ → Set _"},{"Add":" Reflexive _∼_ = ∀ {i} → B.Reflexive (_∼_ {i})"},{"Add":""},{"Add":" Symmetric : IRel A ℓ → Set _"},{"Add":" Symmetric _∼_ = ∀ {i} → B.Symmetric (_∼_ {i})"},{"Add":""},{"Add":" Transitive : IRel A ℓ → Set _"},{"Add":" Transitive _∼_ = ∀ {i} → B.Transitive (_∼_ {i})"},{"Add":""},{"Add":" Antisymmetric : IRel A ℓ₁ → IRel A ℓ₂ → Set _"},{"Add":" Antisymmetric _≈_ _∼_ = ∀ {i} → B.Antisymmetric _≈_ (_∼_ {i})"},{"Add":""},{"Add":" Decidable : IRel A ℓ → Set _"},{"Add":" Decidable _∼_ = ∀ {i} → B.Decidable (_∼_ {i})"},{"Add":""},{"Add":" Respects : IPred A ℓ₁ → IRel A ℓ₂ → Set _"},{"Add":" Respects P _∼_ = ∀ {i} {x y : A i} → x ∼ y → P x → P y"},{"Add":""},{"Add":" Respectsˡ : IRel A ℓ₁ → IRel A ℓ₂ → Set _"},{"Add":" Respectsˡ P _∼_ = ∀ {i} {x y z : A i} → x ∼ y → P x z → P y z"},{"Add":""},{"Add":" Respectsʳ : IRel A ℓ₁ → IRel A ℓ₂ → Set _"},{"Add":" Respectsʳ P _∼_ = ∀ {i} {x y z : A i} → x ∼ y → P z x → P z y"},{"Add":""},{"Add":" Respects₂ : IRel A ℓ₁ → IRel A ℓ₂ → Set _"},{"Add":" Respects₂ P _∼_ = (Respectsʳ P _∼_) × (Respectsˡ P _∼_)"}]}},{"FileAdd":{"meta":{"basename":"Core.agda","metadata":0},"context":[{"change":0,"pos":600438}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Homogeneously-indexed binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via"},{"Add":"-- `Relation.Binary.Indexed.Homogeneous`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Indexed.Homogeneous.Core where"},{"Add":""},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Data.Product.Base using (_×_)"},{"Add":"open import Relation.Binary.Core as B using (REL; Rel)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)"},{"Add":"import Relation.Binary.Indexed.Heterogeneous as I"},{"Add":"open import Relation.Unary.Indexed using (IPred)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c ℓ : Level"},{"Add":" I : Set c"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Homegeneously indexed binary relations"},{"Add":""},{"Add":"-- Heterogeneous types"},{"Add":""},{"Add":"IREL : (I → Set a) → (I → Set b) → (ℓ : Level) → Set _"},{"Add":"IREL A B ℓ = ∀ {i} → REL (A i) (B i) ℓ"},{"Add":""},{"Add":"-- Homogeneous types"},{"Add":""},{"Add":"IRel : (I → Set a) → (ℓ : Level) → Set _"},{"Add":"IRel A = IREL A A"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Lifting to non-indexed binary relations"},{"Add":""},{"Add":"-- Ideally this should be in: `Construct.Lift` but we want this relation"},{"Add":"-- to be exported by the various structures & bundles."},{"Add":""},{"Add":"Lift : (A : I → Set a) → IRel A ℓ → Rel (∀ i → A i) _"},{"Add":"Lift _ _∼_ x y = ∀ i → x i ∼ y i"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Conversion between homogeneous and heterogeneously indexed relations"},{"Add":""},{"Add":"module _ {i a b} {I : Set i} {A : I → Set a} {B : I → Set b} where"},{"Add":""},{"Add":" OverPath : ∀ {ℓ} → IREL A B ℓ → ∀ {i j} → i ≡ j → REL (A i) (B j) ℓ"},{"Add":" OverPath _∼_ refl = _∼_"},{"Add":""},{"Add":" toHetIndexed : ∀ {ℓ} → IREL A B ℓ → I.IREL A B (i ⊔ ℓ)"},{"Add":" toHetIndexed _∼_ {i} {j} x y = (p : i ≡ j) → OverPath _∼_ p x y"},{"Add":""},{"Add":" fromHetIndexed : ∀ {ℓ} → I.IREL A B ℓ → IREL A B ℓ"},{"Add":" fromHetIndexed _∼_ = _∼_"}]}},{"FileAdd":{"meta":{"basename":"Construct","metadata":576},"context":[{"change":0,"pos":600438}],"contents":[]}},{"FileAdd":{"meta":{"basename":"At.agda","metadata":0},"context":[{"change":0,"pos":609738}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Instantiating homogeneously indexed bundles at a particular index"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Relation.Binary.Bundles using (Setoid; DecSetoid; Preorder)"},{"Add":"open import Relation.Binary.Structures"},{"Add":" using (IsEquivalence; IsDecEquivalence; IsPreorder)"},{"Add":"open import Relation.Binary.Indexed.Homogeneous"},{"Add":""},{"Add":"module Relation.Binary.Indexed.Homogeneous.Construct.At where"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a i ℓ ℓ₁ ℓ₂ : Level"},{"Add":" I : Set i"},{"Add":" Aᵢ : I → Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"isEquivalence : ∀ {_≈_ : IRel Aᵢ ℓ} → IsIndexedEquivalence Aᵢ _≈_ →"},{"Add":" (index : I) → IsEquivalence (_≈_ {index})"},{"Add":"isEquivalence isEq index = record"},{"Add":" { refl = reflᵢ"},{"Add":" ; sym = symᵢ"},{"Add":" ; trans = transᵢ"},{"Add":" } where open IsIndexedEquivalence isEq"},{"Add":""},{"Add":"isDecEquivalence : ∀ {_≈_ : IRel Aᵢ ℓ} → IsIndexedDecEquivalence Aᵢ _≈_ →"},{"Add":" (index : I) → IsDecEquivalence (_≈_ {index})"},{"Add":"isDecEquivalence isEq index = record"},{"Add":" { isEquivalence = isEquivalence E.isEquivalenceᵢ index"},{"Add":" ; _≟_ = E._≟ᵢ_"},{"Add":" } where module E = IsIndexedDecEquivalence isEq"},{"Add":""},{"Add":"isPreorder : ∀ {_≈_ : IRel Aᵢ ℓ₁} {_∼_ : IRel Aᵢ ℓ₂} →"},{"Add":" IsIndexedPreorder Aᵢ _≈_ _∼_ →"},{"Add":" (index : I) → IsPreorder (_≈_ {index}) _∼_"},{"Add":"isPreorder isPreorder index = record"},{"Add":" { isEquivalence = isEquivalence O.isEquivalenceᵢ index"},{"Add":" ; reflexive = O.reflexiveᵢ"},{"Add":" ; trans = O.transᵢ"},{"Add":" } where module O = IsIndexedPreorder isPreorder"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles"},{"Add":""},{"Add":"setoid : IndexedSetoid I a ℓ → I → Setoid a ℓ"},{"Add":"setoid S index = record"},{"Add":" { isEquivalence = isEquivalence S.isEquivalenceᵢ index"},{"Add":" } where module S = IndexedSetoid S"},{"Add":""},{"Add":"decSetoid : IndexedDecSetoid I a ℓ → I → DecSetoid a ℓ"},{"Add":"decSetoid S index = record"},{"Add":" { isDecEquivalence = isDecEquivalence DS.isDecEquivalenceᵢ index"},{"Add":" } where module DS = IndexedDecSetoid S"},{"Add":""},{"Add":"preorder : IndexedPreorder I a ℓ₁ ℓ₂ → I → Preorder a ℓ₁ ℓ₂"},{"Add":"preorder O index = record"},{"Add":" { isPreorder = isPreorder O.isPreorderᵢ index"},{"Add":" } where module O = IndexedPreorder O"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Some useful shorthand infix notation"},{"Add":""},{"Add":"infixr -1 _atₛ_"},{"Add":""},{"Add":"_atₛ_ : ∀ {ℓ} → IndexedSetoid I a ℓ → I → Setoid a ℓ"},{"Add":"_atₛ_ = setoid"}]}},{"FileAdd":{"meta":{"basename":"Bundles.agda","metadata":0},"context":[{"change":0,"pos":600438}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Homogeneously-indexed binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via"},{"Add":"-- `Relation.Binary.Indexed.Homogeneous`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Indexed.Homogeneous.Bundles where"},{"Add":""},{"Add":"open import Level using (suc; _⊔_)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles as B"},{"Add":"open import Relation.Nullary.Negation using (¬_)"},{"Add":"open import Relation.Binary.Indexed.Homogeneous.Core"},{"Add":"open import Relation.Binary.Indexed.Homogeneous.Structures"},{"Add":""},{"Add":"-- Indexed structures are laid out in a similar manner as to those"},{"Add":"-- in Relation.Binary. The main difference is each structure also"},{"Add":"-- contains proofs for the lifted version of the relation."},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Equivalences"},{"Add":""},{"Add":"record IndexedSetoid {i} (I : Set i) c ℓ : Set (suc (i ⊔ c ⊔ ℓ)) where"},{"Add":" infix 4 _≈ᵢ_ _≈_"},{"Add":" field"},{"Add":" Carrierᵢ : I → Set c"},{"Add":" _≈ᵢ_ : IRel Carrierᵢ ℓ"},{"Add":" isEquivalenceᵢ : IsIndexedEquivalence Carrierᵢ _≈ᵢ_"},{"Add":""},{"Add":" open IsIndexedEquivalence isEquivalenceᵢ public"},{"Add":""},{"Add":" Carrier : Set _"},{"Add":" Carrier = ∀ i → Carrierᵢ i"},{"Add":""},{"Add":" infix 4 _≉_"},{"Add":""},{"Add":" _≈_ : Rel Carrier _"},{"Add":" _≈_ = Lift Carrierᵢ _≈ᵢ_"},{"Add":""},{"Add":" _≉_ : Rel Carrier _"},{"Add":" x ≉ y = ¬ (x ≈ y)"},{"Add":""},{"Add":" setoid : B.Setoid _ _"},{"Add":" setoid = record"},{"Add":" { isEquivalence = isEquivalence"},{"Add":" }"},{"Add":""},{"Add":""},{"Add":"record IndexedDecSetoid {i} (I : Set i) c ℓ : Set (suc (i ⊔ c ⊔ ℓ)) where"},{"Add":" infix 4 _≈ᵢ_"},{"Add":" field"},{"Add":" Carrierᵢ : I → Set c"},{"Add":" _≈ᵢ_ : IRel Carrierᵢ ℓ"},{"Add":" isDecEquivalenceᵢ : IsIndexedDecEquivalence Carrierᵢ _≈ᵢ_"},{"Add":""},{"Add":" open IsIndexedDecEquivalence isDecEquivalenceᵢ public"},{"Add":""},{"Add":" indexedSetoid : IndexedSetoid I c ℓ"},{"Add":" indexedSetoid = record"},{"Add":" { isEquivalenceᵢ = isEquivalenceᵢ"},{"Add":" }"},{"Add":""},{"Add":" open IndexedSetoid indexedSetoid public"},{"Add":" using (Carrier; _≈_; _≉_; setoid)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Preorders"},{"Add":""},{"Add":"record IndexedPreorder {i} (I : Set i) c ℓ₁ ℓ₂ :"},{"Add":" Set (suc (i ⊔ c ⊔ ℓ₁ ⊔ ℓ₂)) where"},{"Add":""},{"Add":" infix 4 _≈ᵢ_ _∼ᵢ_ _≈_ _∼_"},{"Add":""},{"Add":" field"},{"Add":" Carrierᵢ : I → Set c"},{"Add":" _≈ᵢ_ : IRel Carrierᵢ ℓ₁"},{"Add":" _∼ᵢ_ : IRel Carrierᵢ ℓ₂"},{"Add":" isPreorderᵢ : IsIndexedPreorder Carrierᵢ _≈ᵢ_ _∼ᵢ_"},{"Add":""},{"Add":" open IsIndexedPreorder isPreorderᵢ public"},{"Add":""},{"Add":" Carrier : Set _"},{"Add":" Carrier = ∀ i → Carrierᵢ i"},{"Add":""},{"Add":" _≈_ : Rel Carrier _"},{"Add":" x ≈ y = ∀ i → x i ≈ᵢ y i"},{"Add":""},{"Add":" _∼_ : Rel Carrier _"},{"Add":" x ∼ y = ∀ i → x i ∼ᵢ y i"},{"Add":""},{"Add":" preorder : B.Preorder _ _ _"},{"Add":" preorder = record"},{"Add":" { isPreorder = isPreorder"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Partial orders"},{"Add":""},{"Add":"record IndexedPoset {i} (I : Set i) c ℓ₁ ℓ₂ :"},{"Add":" Set (suc (i ⊔ c ⊔ ℓ₁ ⊔ ℓ₂)) where"},{"Add":" infix 4 _≈ᵢ_ _≤ᵢ_"},{"Add":" field"},{"Add":" Carrierᵢ : I → Set c"},{"Add":" _≈ᵢ_ : IRel Carrierᵢ ℓ₁"},{"Add":" _≤ᵢ_ : IRel Carrierᵢ ℓ₂"},{"Add":" isPartialOrderᵢ : IsIndexedPartialOrder Carrierᵢ _≈ᵢ_ _≤ᵢ_"},{"Add":""},{"Add":" open IsIndexedPartialOrder isPartialOrderᵢ public"},{"Add":""},{"Add":" preorderᵢ : IndexedPreorder I c ℓ₁ ℓ₂"},{"Add":" preorderᵢ = record"},{"Add":" { isPreorderᵢ = isPreorderᵢ"},{"Add":" }"},{"Add":""},{"Add":" open IndexedPreorder preorderᵢ public"},{"Add":" using (Carrier; _≈_; preorder) renaming (_∼_ to _≤_)"},{"Add":""},{"Add":" poset : B.Poset _ _ _"},{"Add":" poset = record"},{"Add":" { isPartialOrder = isPartialOrder"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Heterogeneous.agda","metadata":0},"context":[{"change":0,"pos":599691}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Heterogeneously-indexed binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Indexed.Heterogeneous where"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Publicly export core definitions"},{"Add":""},{"Add":"open import Relation.Binary.Indexed.Heterogeneous.Core public"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous.Definitions public"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous.Structures public"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous.Bundles public"}]}},{"FileAdd":{"meta":{"basename":"Heterogeneous","metadata":576},"context":[{"change":0,"pos":599691}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Structures.agda","metadata":0},"context":[{"change":0,"pos":616921}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Indexed binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via"},{"Add":"-- `Relation.Binary.Indexed.Heterogeneous`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Indexed.Heterogeneous.Core"},{"Add":""},{"Add":"module Relation.Binary.Indexed.Heterogeneous.Structures"},{"Add":" {i a ℓ} {I : Set i} (A : I → Set a) (_≈_ : IRel A ℓ)"},{"Add":" where"},{"Add":""},{"Add":"open import Function.Base"},{"Add":"open import Level using (suc; _⊔_)"},{"Add":"open import Relation.Binary.Core using (_⇒_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous.Definitions"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Equivalences"},{"Add":""},{"Add":"record IsIndexedEquivalence : Set (i ⊔ a ⊔ ℓ) where"},{"Add":" field"},{"Add":" refl : Reflexive A _≈_"},{"Add":" sym : Symmetric A _≈_"},{"Add":" trans : Transitive A _≈_"},{"Add":""},{"Add":" reflexive : ∀ {i} → _≡_ ⟨ _⇒_ ⟩ _≈_ {i}"},{"Add":" reflexive ≡.refl = refl"},{"Add":""},{"Add":""},{"Add":"record IsIndexedPreorder {ℓ₂} (_≲_ : IRel A ℓ₂) : Set (i ⊔ a ⊔ ℓ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isEquivalence : IsIndexedEquivalence"},{"Add":" reflexive : ∀ {i j} → (_≈_ {i} {j}) ⟨ _⇒_ ⟩ _≲_"},{"Add":" trans : Transitive A _≲_"},{"Add":""},{"Add":" module Eq = IsIndexedEquivalence isEquivalence"},{"Add":""},{"Add":" refl : Reflexive A _≲_"},{"Add":" refl = reflexive Eq.refl"}]}},{"FileAdd":{"meta":{"basename":"Definitions.agda","metadata":0},"context":[{"change":0,"pos":616921}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Indexed binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via"},{"Add":"-- `Relation.Binary.Indexed.Heterogeneous`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Indexed.Heterogeneous.Definitions where"},{"Add":""},{"Add":"open import Level"},{"Add":"import Relation.Binary.Definitions as B"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous.Core"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" i a ℓ : Level"},{"Add":" I : Set i"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Simple properties of indexed binary relations"},{"Add":""},{"Add":"Reflexive : (A : I → Set a) → IRel A ℓ → Set _"},{"Add":"Reflexive _ _∼_ = ∀ {i} → B.Reflexive (_∼_ {i})"},{"Add":""},{"Add":"Symmetric : (A : I → Set a) → IRel A ℓ → Set _"},{"Add":"Symmetric _ _∼_ = ∀ {i j} → B.Sym (_∼_ {i} {j}) _∼_"},{"Add":""},{"Add":"Transitive : (A : I → Set a) → IRel A ℓ → Set _"},{"Add":"Transitive _ _∼_ = ∀ {i j k} → B.Trans _∼_ (_∼_ {j}) (_∼_ {i} {k})"}]}},{"FileAdd":{"meta":{"basename":"Core.agda","metadata":0},"context":[{"change":0,"pos":616921}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Indexed binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via"},{"Add":"-- `Relation.Binary.Indexed.Heterogeneous`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Indexed.Heterogeneous.Core where"},{"Add":""},{"Add":"open import Level"},{"Add":"import Relation.Binary.Core as B"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Indexed binary relations"},{"Add":""},{"Add":"-- Heterogeneous types"},{"Add":""},{"Add":"IREL : ∀ {i₁ i₂ a₁ a₂} {I₁ : Set i₁} {I₂ : Set i₂} →"},{"Add":" (I₁ → Set a₁) → (I₂ → Set a₂) → (ℓ : Level) → Set _"},{"Add":"IREL A₁ A₂ ℓ = ∀ {i₁ i₂} → A₁ i₁ → A₂ i₂ → Set ℓ"},{"Add":""},{"Add":"-- Homogeneous types"},{"Add":""},{"Add":"IRel : ∀ {i a} {I : Set i} → (I → Set a) → (ℓ : Level) → Set _"},{"Add":"IRel A ℓ = IREL A A ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Generalised implication."},{"Add":""},{"Add":"infixr 4 _=[_]⇒_"},{"Add":""},{"Add":"_=[_]⇒_ : ∀ {a b ℓ₁ ℓ₂} {A : Set a} {B : A → Set b} →"},{"Add":" B.Rel A ℓ₁ → ((x : A) → B x) → IRel B ℓ₂ → Set _"},{"Add":"P =[ f ]⇒ Q = ∀ {i j} → P i j → Q (f i) (f j)"}]}},{"FileAdd":{"meta":{"basename":"Construct","metadata":576},"context":[{"change":0,"pos":616921}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Trivial.agda","metadata":0},"context":[{"change":0,"pos":620885}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Creates trivially indexed records from their non-indexed counterpart."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Indexed.Heterogeneous.Construct.Trivial"},{"Add":" {i} {I : Set i} where"},{"Add":""},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid; Preorder)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence; IsPreorder)"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"module _ {a} {A : Set a} where"},{"Add":""},{"Add":" private"},{"Add":" Aᵢ : I → Set a"},{"Add":" Aᵢ i = A"},{"Add":""},{"Add":" isIndexedEquivalence : ∀ {ℓ} {_≈_ : Rel A ℓ} → IsEquivalence _≈_ →"},{"Add":" IsIndexedEquivalence Aᵢ _≈_"},{"Add":" isIndexedEquivalence isEq = record"},{"Add":" { refl = refl"},{"Add":" ; sym = sym"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":" where open IsEquivalence isEq"},{"Add":""},{"Add":" isIndexedPreorder : ∀ {ℓ₁ ℓ₂} {_≈_ : Rel A ℓ₁} {_∼_ : Rel A ℓ₂} →"},{"Add":" IsPreorder _≈_ _∼_ →"},{"Add":" IsIndexedPreorder Aᵢ _≈_ _∼_"},{"Add":" isIndexedPreorder isPreorder = record"},{"Add":" { isEquivalence = isIndexedEquivalence isEquivalence"},{"Add":" ; reflexive = reflexive"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":" where open IsPreorder isPreorder"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles"},{"Add":""},{"Add":"indexedSetoid : ∀ {a ℓ} → Setoid a ℓ → IndexedSetoid I a ℓ"},{"Add":"indexedSetoid S = record"},{"Add":" { isEquivalence = isIndexedEquivalence isEquivalence"},{"Add":" }"},{"Add":" where open Setoid S"},{"Add":""},{"Add":"indexedPreorder : ∀ {a ℓ₁ ℓ₂} → Preorder a ℓ₁ ℓ₂ →"},{"Add":" IndexedPreorder I a ℓ₁ ℓ₂"},{"Add":"indexedPreorder O = record"},{"Add":" { isPreorder = isIndexedPreorder isPreorder"},{"Add":" }"},{"Add":" where open Preorder O"}]}},{"FileAdd":{"meta":{"basename":"At.agda","metadata":0},"context":[{"change":0,"pos":620885}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Instantiates indexed binary structures at an index to the equivalent"},{"Add":"-- non-indexed structures."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Indexed.Heterogeneous.Construct.At where"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (Setoid; Preorder)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence; IsPreorder)"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"module _ {a i} {I : Set i} {A : I → Set a} where"},{"Add":""},{"Add":" isEquivalence : ∀ {ℓ} {_≈_ : IRel A ℓ} → IsIndexedEquivalence A _≈_ →"},{"Add":" (index : I) → IsEquivalence (_≈_ {index})"},{"Add":" isEquivalence isEq index = record"},{"Add":" { refl = refl"},{"Add":" ; sym = sym"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":" where open IsIndexedEquivalence isEq"},{"Add":""},{"Add":" isPreorder : ∀ {ℓ₁ ℓ₂} {_≈_ : IRel A ℓ₁} {_≲_ : IRel A ℓ₂} →"},{"Add":" IsIndexedPreorder A _≈_ _≲_ →"},{"Add":" (index : I) → IsPreorder (_≈_ {index}) _≲_"},{"Add":" isPreorder isPreorder index = record"},{"Add":" { isEquivalence = isEquivalence O.isEquivalence index"},{"Add":" ; reflexive = O.reflexive"},{"Add":" ; trans = O.trans"},{"Add":" }"},{"Add":" where module O = IsIndexedPreorder isPreorder"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles"},{"Add":""},{"Add":"module _ {a i} {I : Set i} where"},{"Add":""},{"Add":" setoid : ∀ {ℓ} → IndexedSetoid I a ℓ → I → Setoid a ℓ"},{"Add":" setoid S index = record"},{"Add":" { Carrier = S.Carrier index"},{"Add":" ; _≈_ = S._≈_"},{"Add":" ; isEquivalence = isEquivalence S.isEquivalence index"},{"Add":" }"},{"Add":" where module S = IndexedSetoid S"},{"Add":""},{"Add":" preorder : ∀ {ℓ₁ ℓ₂} → IndexedPreorder I a ℓ₁ ℓ₂ → I → Preorder a ℓ₁ ℓ₂"},{"Add":" preorder O index = record"},{"Add":" { Carrier = O.Carrier index"},{"Add":" ; _≈_ = O._≈_"},{"Add":" ; _≲_ = O._≲_"},{"Add":" ; isPreorder = isPreorder O.isPreorder index"},{"Add":" }"},{"Add":" where module O = IndexedPreorder O"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Some useful shorthand infix notation"},{"Add":""},{"Add":"module _ {a i} {I : Set i} where"},{"Add":""},{"Add":" infixr -1 _atₛ_"},{"Add":""},{"Add":" _atₛ_ : ∀ {ℓ} → IndexedSetoid I a ℓ → I → Setoid a ℓ"},{"Add":" _atₛ_ = setoid"}]}},{"FileAdd":{"meta":{"basename":"Bundles.agda","metadata":0},"context":[{"change":0,"pos":616921}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Indexed binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via"},{"Add":"-- `Relation.Binary.Indexed.Heterogeneous`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Indexed.Heterogeneous.Bundles where"},{"Add":""},{"Add":"open import Level using (suc; _⊔_)"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous.Core"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous.Structures"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definitions"},{"Add":""},{"Add":"record IndexedSetoid {i} (I : Set i) c ℓ : Set (suc (i ⊔ c ⊔ ℓ)) where"},{"Add":" infix 4 _≈_"},{"Add":" field"},{"Add":" Carrier : I → Set c"},{"Add":" _≈_ : IRel Carrier ℓ"},{"Add":" isEquivalence : IsIndexedEquivalence Carrier _≈_"},{"Add":""},{"Add":" open IsIndexedEquivalence isEquivalence public"},{"Add":""},{"Add":""},{"Add":"record IndexedPreorder {i} (I : Set i) c ℓ₁ ℓ₂ :"},{"Add":" Set (suc (i ⊔ c ⊔ ℓ₁ ⊔ ℓ₂)) where"},{"Add":" infix 4 _≈_ _≲_"},{"Add":" field"},{"Add":" Carrier : I → Set c"},{"Add":" _≈_ : IRel Carrier ℓ₁ -- The underlying equality."},{"Add":" _≲_ : IRel Carrier ℓ₂ -- The relation."},{"Add":" isPreorder : IsIndexedPreorder Carrier _≈_ _≲_"},{"Add":""},{"Add":" open IsIndexedPreorder isPreorder public"},{"Add":""},{"Add":" infix 4 _∼_"},{"Add":" _∼_ = _≲_"},{"Add":""},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- DEPRECATED"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Please use the new names as continuing support for the old names is"},{"Add":"-- not guaranteed."},{"Add":""},{"Add":"-- Version 2.0"},{"Add":""},{"Add":"{-# WARNING_ON_USAGE IndexedPreorder._∼_"},{"Add":"\"Warning: IndexedPreorder._∼_ was deprecated in v2.0. Please use IndexedPreorder._≲_ instead. \""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"HeterogeneousEquality.agda","metadata":0},"context":[{"change":0,"pos":415495}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Heterogeneous equality"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --with-K --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.HeterogeneousEquality where"},{"Add":""},{"Add":"import Axiom.Extensionality.Heterogeneous as Ext"},{"Add":"open import Data.Unit.NonEta"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base"},{"Add":"open import Function.Bundles using (Inverse)"},{"Add":"open import Level"},{"Add":"open import Relation.Nullary hiding (Irrelevant)"},{"Add":"open import Relation.Unary using (Pred)"},{"Add":"open import Relation.Binary.Core using (Rel; REL; _⇒_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid; DecSetoid; Preorder)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence; IsPreorder)"},{"Add":"open import Relation.Binary.Definitions using (Substitutive; Irrelevant; Decidable; _Respects₂_; Trans; Reflexive)"},{"Add":"open import Relation.Binary.Consequences"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous"},{"Add":" using (IndexedSetoid)"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous.Construct.At"},{"Add":" using (_atₛ_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; refl)"},{"Add":"open import Relation.Binary.Reasoning.Syntax"},{"Add":""},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":"import Relation.Binary.HeterogeneousEquality.Core as Core"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c p r ℓ : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":" C : Set c"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Heterogeneous equality"},{"Add":""},{"Add":"infix 4 _≇_"},{"Add":""},{"Add":"open Core public using (_≅_; refl)"},{"Add":""},{"Add":"-- Nonequality."},{"Add":""},{"Add":"_≇_ : ∀ {A : Set a} → A → {B : Set b} → B → Set a"},{"Add":"x ≇ y = ¬ x ≅ y"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Conversion"},{"Add":""},{"Add":"open Core public using (≅-to-≡; ≡-to-≅)"},{"Add":""},{"Add":"≅-to-type-≡ : ∀ {A B : Set a} {x : A} {y : B} → x ≅ y → A ≡ B"},{"Add":"≅-to-type-≡ refl = refl"},{"Add":""},{"Add":"≅-to-subst-≡ : ∀ {A B : Set a} {x : A} {y : B} → (p : x ≅ y) →"},{"Add":" ≡.subst (λ x → x) (≅-to-type-≡ p) x ≡ y"},{"Add":"≅-to-subst-≡ refl = refl"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Some properties"},{"Add":""},{"Add":"reflexive : _⇒_ {A = A} _≡_ (λ x y → x ≅ y)"},{"Add":"reflexive refl = refl"},{"Add":""},{"Add":"sym : ∀ {x : A} {y : B} → x ≅ y → y ≅ x"},{"Add":"sym refl = refl"},{"Add":""},{"Add":"trans : ∀ {x : A} {y : B} {z : C} → x ≅ y → y ≅ z → x ≅ z"},{"Add":"trans refl eq = eq"},{"Add":""},{"Add":"subst : Substitutive {A = A} (λ x y → x ≅ y) ℓ"},{"Add":"subst P refl p = p"},{"Add":""},{"Add":"subst₂ : ∀ (_∼_ : REL A B r) {x y u v} → x ≅ y → u ≅ v → x ∼ u → y ∼ v"},{"Add":"subst₂ _∼_ refl refl z = z"},{"Add":""},{"Add":"subst-removable : ∀ (P : Pred A p) {x y} (eq : x ≅ y) (z : P x) →"},{"Add":" subst P eq z ≅ z"},{"Add":"subst-removable P refl z = refl"},{"Add":""},{"Add":"subst₂-removable : ∀ (_∼_ : REL A B r) {x y u v} (eq₁ : x ≅ y) (eq₂ : u ≅ v) (z : x ∼ u) →"},{"Add":" subst₂ _∼_ eq₁ eq₂ z ≅ z"},{"Add":"subst₂-removable _∼_ refl refl z = refl"},{"Add":""},{"Add":"≡-subst-removable : ∀ (P : Pred A p) {x y} (eq : x ≡ y) (z : P x) →"},{"Add":" ≡.subst P eq z ≅ z"},{"Add":"≡-subst-removable P refl z = refl"},{"Add":""},{"Add":"cong : ∀ {A : Set a} {B : A → Set b} {x y}"},{"Add":" (f : (x : A) → B x) → x ≅ y → f x ≅ f y"},{"Add":"cong f refl = refl"},{"Add":""},{"Add":"cong-app : ∀ {A : Set a} {B : A → Set b} {f g : (x : A) → B x} →"},{"Add":" f ≅ g → (x : A) → f x ≅ g x"},{"Add":"cong-app refl x = refl"},{"Add":""},{"Add":"cong₂ : ∀ {A : Set a} {B : A → Set b} {C : ∀ x → B x → Set c} {x y u v}"},{"Add":" (f : (x : A) (y : B x) → C x y) → x ≅ y → u ≅ v → f x u ≅ f y v"},{"Add":"cong₂ f refl refl = refl"},{"Add":""},{"Add":"resp₂ : ∀ (∼ : Rel A ℓ) → ∼ Respects₂ (λ x y → x ≅ y)"},{"Add":"resp₂ _∼_ = subst⇒resp₂ _∼_ subst"},{"Add":""},{"Add":"module _ {I : Set ℓ} (A : I → Set a) {B : {k : I} → A k → Set b} where"},{"Add":""},{"Add":" icong : {i j : I} {x : A i} {y : A j} →"},{"Add":" i ≡ j →"},{"Add":" (f : {k : I} → (z : A k) → B z) →"},{"Add":" x ≅ y →"},{"Add":" f x ≅ f y"},{"Add":" icong refl _ refl = refl"},{"Add":""},{"Add":" icong₂ : {C : {k : I} → (a : A k) → B a → Set c}"},{"Add":" {i j : I} {x : A i} {y : A j} {u : B x} {v : B y} →"},{"Add":" i ≡ j →"},{"Add":" (f : {k : I} → (z : A k) → (w : B z) → C z w) →"},{"Add":" x ≅ y → u ≅ v →"},{"Add":" f x u ≅ f y v"},{"Add":" icong₂ refl _ refl refl = refl"},{"Add":""},{"Add":" icong-subst-removable : {i j : I} (eq : i ≅ j)"},{"Add":" (f : {k : I} → (z : A k) → B z)"},{"Add":" (x : A i) →"},{"Add":" f (subst A eq x) ≅ f x"},{"Add":" icong-subst-removable refl _ _ = refl"},{"Add":""},{"Add":" icong-≡-subst-removable : {i j : I} (eq : i ≡ j)"},{"Add":" (f : {k : I} → (z : A k) → B z)"},{"Add":" (x : A i) →"},{"Add":" f (≡.subst A eq x) ≅ f x"},{"Add":" icong-≡-subst-removable refl _ _ = refl"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"--Proof irrelevance"},{"Add":""},{"Add":"≅-irrelevant : {A B : Set ℓ} → Irrelevant ((A → B → Set ℓ) ∋ λ a → a ≅_)"},{"Add":"≅-irrelevant refl refl = refl"},{"Add":""},{"Add":"module _ {A C : Set a} {B D : Set ℓ}"},{"Add":" {w : A} {x : B} {y : C} {z : D} where"},{"Add":""},{"Add":" ≅-heterogeneous-irrelevant : (p : w ≅ x) (q : y ≅ z) → x ≅ y → p ≅ q"},{"Add":" ≅-heterogeneous-irrelevant refl refl refl = refl"},{"Add":""},{"Add":" ≅-heterogeneous-irrelevantˡ : (p : w ≅ x) (q : y ≅ z) → w ≅ y → p ≅ q"},{"Add":" ≅-heterogeneous-irrelevantˡ refl refl refl = refl"},{"Add":""},{"Add":" ≅-heterogeneous-irrelevantʳ : (p : w ≅ x) (q : y ≅ z) → x ≅ z → p ≅ q"},{"Add":" ≅-heterogeneous-irrelevantʳ refl refl refl = refl"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"isEquivalence : IsEquivalence {A = A} (λ x y → x ≅ y)"},{"Add":"isEquivalence = record"},{"Add":" { refl = refl"},{"Add":" ; sym = sym"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":""},{"Add":"setoid : Set ℓ → Setoid ℓ ℓ"},{"Add":"setoid A = record"},{"Add":" { Carrier = A"},{"Add":" ; _≈_ = λ x y → x ≅ y"},{"Add":" ; isEquivalence = isEquivalence"},{"Add":" }"},{"Add":""},{"Add":"indexedSetoid : (A → Set b) → IndexedSetoid A _ _"},{"Add":"indexedSetoid B = record"},{"Add":" { Carrier = B"},{"Add":" ; _≈_ = λ x y → x ≅ y"},{"Add":" ; isEquivalence = record"},{"Add":" { refl = refl"},{"Add":" ; sym = sym"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":" }"},{"Add":""},{"Add":"≡↔≅ : ∀ {A : Set a} (B : A → Set b) {x : A} →"},{"Add":" Inverse (≡.setoid (B x)) ((indexedSetoid B) atₛ x)"},{"Add":"≡↔≅ B = record"},{"Add":" { to = id"},{"Add":" ; to-cong = ≡-to-≅"},{"Add":" ; from = id"},{"Add":" ; from-cong = ≅-to-≡"},{"Add":" ; inverse = (λ { ≡.refl → refl }) , λ { refl → ≡.refl }"},{"Add":" }"},{"Add":""},{"Add":"decSetoid : Decidable {A = A} {B = A} (λ x y → x ≅ y) →"},{"Add":" DecSetoid _ _"},{"Add":"decSetoid dec = record"},{"Add":" { _≈_ = λ x y → x ≅ y"},{"Add":" ; isDecEquivalence = record"},{"Add":" { isEquivalence = isEquivalence"},{"Add":" ; _≟_ = dec"},{"Add":" }"},{"Add":" }"},{"Add":""},{"Add":"isPreorder : IsPreorder {A = A} (λ x y → x ≅ y) (λ x y → x ≅ y)"},{"Add":"isPreorder = record"},{"Add":" { isEquivalence = isEquivalence"},{"Add":" ; reflexive = id"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":""},{"Add":"isPreorder-≡ : IsPreorder {A = A} _≡_ (λ x y → x ≅ y)"},{"Add":"isPreorder-≡ = record"},{"Add":" { isEquivalence = ≡.isEquivalence"},{"Add":" ; reflexive = reflexive"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":""},{"Add":"preorder : Set ℓ → Preorder ℓ ℓ ℓ"},{"Add":"preorder A = record"},{"Add":" { Carrier = A"},{"Add":" ; _≈_ = _≡_"},{"Add":" ; _≲_ = λ x y → x ≅ y"},{"Add":" ; isPreorder = isPreorder-≡"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Convenient syntax for equational reasoning"},{"Add":""},{"Add":"module ≅-Reasoning where"},{"Add":""},{"Add":" -- The code in `Relation.Binary.Reasoning.Setoid` cannot handle"},{"Add":" -- heterogeneous equalities, hence the code duplication here."},{"Add":""},{"Add":" infix 4 _IsRelatedTo_"},{"Add":""},{"Add":" data _IsRelatedTo_ {A : Set ℓ} {B : Set ℓ} (x : A) (y : B) : Set ℓ where"},{"Add":" relTo : (x≅y : x ≅ y) → x IsRelatedTo y"},{"Add":""},{"Add":" start : ∀ {x : A} {y : B} → x IsRelatedTo y → x ≅ y"},{"Add":" start (relTo x≅y) = x≅y"},{"Add":""},{"Add":" ≡-go : ∀ {A : Set a} → Trans {A = A} {C = A} _≡_ _IsRelatedTo_ _IsRelatedTo_"},{"Add":" ≡-go x≡y (relTo y≅z) = relTo (trans (reflexive x≡y) y≅z)"},{"Add":""},{"Add":" -- Combinators with one heterogeneous relation"},{"Add":" module _ {A : Set ℓ} {B : Set ℓ} where"},{"Add":" open begin-syntax (_IsRelatedTo_ {A = A} {B}) start public"},{"Add":""},{"Add":" -- Combinators with homogeneous relations"},{"Add":" module _ {A : Set ℓ} where"},{"Add":" open ≡-syntax (_IsRelatedTo_ {A = A}) ≡-go public"},{"Add":" open end-syntax (_IsRelatedTo_ {A = A}) (relTo refl) public"},{"Add":""},{"Add":" -- Can't create syntax in the standard `Syntax` module for"},{"Add":" -- heterogeneous steps because it would force that module to use"},{"Add":" -- the `--with-k` option."},{"Add":" infixr 2 _≅⟨_⟩_ _≅⟨_⟨_"},{"Add":""},{"Add":" _≅⟨_⟩_ : ∀ (x : A) {y : B} {z : C} →"},{"Add":" x ≅ y → y IsRelatedTo z → x IsRelatedTo z"},{"Add":" _ ≅⟨ x≅y ⟩ relTo y≅z = relTo (trans x≅y y≅z)"},{"Add":""},{"Add":" _≅⟨_⟨_ : ∀ (x : A) {y : B} {z : C} →"},{"Add":" y ≅ x → y IsRelatedTo z → x IsRelatedTo z"},{"Add":" _ ≅⟨ y≅x ⟨ relTo y≅z = relTo (trans (sym y≅x) y≅z)"},{"Add":""},{"Add":" -- Deprecated"},{"Add":" infixr 2 _≅˘⟨_⟩_"},{"Add":" _≅˘⟨_⟩_ = _≅⟨_⟨_"},{"Add":" {-# WARNING_ON_USAGE _≅˘⟨_⟩_"},{"Add":" \"Warning: _≅˘⟨_⟩_ was deprecated in v2.0."},{"Add":" Please use _≅⟨_⟨_ instead.\""},{"Add":" #-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Inspect"},{"Add":""},{"Add":"-- Inspect can be used when you want to pattern match on the result r"},{"Add":"-- of some expression e, and you also need to \"remember\" that r ≡ e."},{"Add":""},{"Add":"record Reveal_·_is_ {A : Set a} {B : A → Set b}"},{"Add":" (f : (x : A) → B x) (x : A) (y : B x) :"},{"Add":" Set (a ⊔ b) where"},{"Add":" constructor [_]"},{"Add":" field eq : f x ≅ y"},{"Add":""},{"Add":"inspect : ∀ {A : Set a} {B : A → Set b}"},{"Add":" (f : (x : A) → B x) (x : A) → Reveal f · x is f x"},{"Add":"inspect f x = [ refl ]"},{"Add":""},{"Add":"-- Example usage:"},{"Add":""},{"Add":"-- f x y with g x | inspect g x"},{"Add":"-- f x y | c z | [ eq ] = ..."}]}},{"FileAdd":{"meta":{"basename":"HeterogeneousEquality","metadata":576},"context":[{"change":0,"pos":415495}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Quotients.agda","metadata":0},"context":[{"change":0,"pos":636924}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Quotients for Heterogeneous equality"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --with-K --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.HeterogeneousEquality.Quotients where"},{"Add":""},{"Add":"open import Function.Base using (_$_; const; _∘_)"},{"Add":"open import Level hiding (lift)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.HeterogeneousEquality"},{"Add":"open ≅-Reasoning"},{"Add":""},{"Add":"record Quotient {c ℓ} (S : Setoid c ℓ) : Set (suc (c ⊔ ℓ)) where"},{"Add":" open Setoid S renaming (Carrier to A)"},{"Add":" field"},{"Add":" Q : Set c"},{"Add":" abs : A → Q"},{"Add":""},{"Add":" compat : (B : Q → Set c) (f : ∀ a → B (abs a)) → Set (c ⊔ ℓ)"},{"Add":" compat _ f = {a a′ : A} → a ≈ a′ → f a ≅ f a′"},{"Add":""},{"Add":" field"},{"Add":" compat-abs : compat _ abs"},{"Add":" lift : (B : Q → Set c) (f : ∀ a → B (abs a)) → compat B f → ∀ q → B q"},{"Add":" lift-conv : {B : Q → Set c} (f : ∀ a → B (abs a)) (p : compat B f) →"},{"Add":" ∀ a → lift B f p (abs a) ≅ f a"},{"Add":""},{"Add":"Quotients : ∀ c ℓ → Set (suc (c ⊔ ℓ))"},{"Add":"Quotients c ℓ = (S : Setoid c ℓ) → Quotient S"},{"Add":""},{"Add":"module Properties {c ℓ} {S : Setoid c ℓ} (Qu : Quotient S) where"},{"Add":""},{"Add":" open Setoid S renaming (Carrier to A) hiding (refl; sym; trans)"},{"Add":" open Quotient Qu"},{"Add":""},{"Add":" module _ {B B′ : Q → Set c} {f : ∀ a → B (abs a)} {p : compat B f} where"},{"Add":""},{"Add":" lift-unique : {g : ∀ q → B′ q} → (∀ a → g (abs a) ≅ f a) →"},{"Add":" ∀ x → lift B f p x ≅ g x"},{"Add":" lift-unique {g} ext = lift _ liftf≅g liftf≅g-ext where"},{"Add":""},{"Add":" liftf≅g : ∀ a → lift B f p (abs a) ≅ g (abs a)"},{"Add":" liftf≅g x = begin"},{"Add":" lift _ f p (abs x) ≅⟨ lift-conv f p x ⟩"},{"Add":" f x ≅⟨ sym (ext x) ⟩"},{"Add":" g (abs x) ∎"},{"Add":""},{"Add":" liftf≅g-ext : ∀ {a a′} → a ≈ a′ → liftf≅g a ≅ liftf≅g a′"},{"Add":" liftf≅g-ext eq = ≅-heterogeneous-irrelevantˡ _ _"},{"Add":" $ cong (lift B f p) (compat-abs eq)"},{"Add":""},{"Add":" lift-ext : {g : ∀ a → B′ (abs a)} {p′ : compat B′ g} → (∀ x → f x ≅ g x) →"},{"Add":" ∀ x → lift B f p x ≅ lift B′ g p′ x"},{"Add":" lift-ext {g} {p′} h = lift-unique $ λ a → begin"},{"Add":" lift B′ g p′ (abs a) ≅⟨ lift-conv g p′ a ⟩"},{"Add":" g a ≅⟨ sym (h a) ⟩"},{"Add":" f a ∎"},{"Add":""},{"Add":" lift-conv-abs : ∀ a → lift (const Q) abs compat-abs a ≅ a"},{"Add":" lift-conv-abs = lift-unique (λ _ → refl)"},{"Add":""},{"Add":" lift-fold : {B : Q → Set c} (f : ∀ q → B q) →"},{"Add":" ∀ a → lift B (f ∘ abs) (cong f ∘ compat-abs) a ≅ f a"},{"Add":" lift-fold f = lift-unique (λ _ → refl)"},{"Add":""},{"Add":" abs-epimorphism : {B : Q → Set c} {f g : ∀ q → B q} →"},{"Add":" (∀ x → f (abs x) ≅ g (abs x)) → ∀ q → f q ≅ g q"},{"Add":" abs-epimorphism {B} {f} {g} p q = begin"},{"Add":" f q ≅⟨ sym (lift-fold f q) ⟩"},{"Add":" lift B (f ∘ abs) (cong f ∘ compat-abs) q ≅⟨ lift-ext p q ⟩"},{"Add":" lift B (g ∘ abs) (cong g ∘ compat-abs) q ≅⟨ lift-fold g q ⟩"},{"Add":" g q ∎"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties provable with extensionality"},{"Add":""},{"Add":"module _ (ext : ∀ {a b} {A : Set a} {B₁ B₂ : A → Set b} {f₁ : ∀ a → B₁ a}"},{"Add":" {f₂ : ∀ a → B₂ a} → (∀ a → f₁ a ≅ f₂ a) → f₁ ≅ f₂) where"},{"Add":""},{"Add":" module Properties₂"},{"Add":" {c ℓ} {S₁ S₂ : Setoid c ℓ} (Qu₁ : Quotient S₁) (Qu₂ : Quotient S₂)"},{"Add":" where"},{"Add":""},{"Add":" module S₁ = Setoid S₁"},{"Add":" module S₂ = Setoid S₂"},{"Add":" module Qu₁ = Quotient Qu₁"},{"Add":" module Qu₂ = Quotient Qu₂"},{"Add":""},{"Add":" module _ {B : _ → _ → Set c} (f : ∀ s₁ s₂ → B (Qu₁.abs s₁) (Qu₂.abs s₂)) where"},{"Add":""},{"Add":" compat₂ : Set _"},{"Add":" compat₂ = ∀ {a b a′ b′} → a S₁.≈ a′ → b S₂.≈ b′ → f a b ≅ f a′ b′"},{"Add":""},{"Add":" lift₂ : compat₂ → ∀ q q′ → B q q′"},{"Add":" lift₂ p = Qu₁.lift _ g (ext ∘ g-ext) module Lift₂ where"},{"Add":""},{"Add":" g : ∀ a q → B (Qu₁.abs a) q"},{"Add":" g a = Qu₂.lift (B (Qu₁.abs a)) (f a) (p S₁.refl)"},{"Add":""},{"Add":" g-ext : ∀ {a a′} → a S₁.≈ a′ → ∀ q → g a q ≅ g a′ q"},{"Add":" g-ext a≈a′ = Properties.lift-ext Qu₂ (λ _ → p a≈a′ S₂.refl)"},{"Add":""},{"Add":" lift₂-conv : (p : compat₂) → ∀ a a′ → lift₂ p (Qu₁.abs a) (Qu₂.abs a′) ≅ f a a′"},{"Add":" lift₂-conv p a a′ = begin"},{"Add":" lift₂ p (Qu₁.abs a) (Qu₂.abs a′)"},{"Add":" ≅⟨ cong (_$ (Qu₂.abs a′)) (Qu₁.lift-conv (Lift₂.g p) (ext ∘ Lift₂.g-ext p) a) ⟩"},{"Add":" Lift₂.g p a (Qu₂.abs a′)"},{"Add":" ≡⟨⟩"},{"Add":" Qu₂.lift (B (Qu₁.abs a)) (f a) (p S₁.refl) (Qu₂.abs a′)"},{"Add":" ≅⟨ Qu₂.lift-conv (f a) (p S₁.refl) a′ ⟩"},{"Add":" f a a′"},{"Add":" ∎"},{"Add":""},{"Add":" module Properties₃ {c ℓ} {S₁ S₂ S₃ : Setoid c ℓ}"},{"Add":" (Qu₁ : Quotient S₁) (Qu₂ : Quotient S₂) (Qu₃ : Quotient S₃)"},{"Add":" where"},{"Add":""},{"Add":" module S₁ = Setoid S₁"},{"Add":" module S₂ = Setoid S₂"},{"Add":" module S₃ = Setoid S₃"},{"Add":" module Qu₁ = Quotient Qu₁"},{"Add":" module Qu₂ = Quotient Qu₂"},{"Add":" module Qu₃ = Quotient Qu₃"},{"Add":""},{"Add":" module _ {B : _ → _ → _ → Set c}"},{"Add":" (f : ∀ s₁ s₂ s₃ → B (Qu₁.abs s₁) (Qu₂.abs s₂) (Qu₃.abs s₃)) where"},{"Add":""},{"Add":" compat₃ : Set _"},{"Add":" compat₃ = ∀ {a b c a′ b′ c′} → a S₁.≈ a′ → b S₂.≈ b′ → c S₃.≈ c′ →"},{"Add":" f a b c ≅ f a′ b′ c′"},{"Add":""},{"Add":" lift₃ : compat₃ → ∀ q₁ q₂ q₃ → B q₁ q₂ q₃"},{"Add":" lift₃ p = Qu₁.lift _ h (ext ∘ h-ext) module Lift₃ where"},{"Add":""},{"Add":" g : ∀ a b q → B (Qu₁.abs a) (Qu₂.abs b) q"},{"Add":" g a b = Qu₃.lift (B (Qu₁.abs a) (Qu₂.abs b)) (f a b) (p S₁.refl S₂.refl)"},{"Add":""},{"Add":" g-ext : ∀ {a a′ b b′} → a S₁.≈ a′ → b S₂.≈ b′ → ∀ c → g a b c ≅ g a′ b′ c"},{"Add":" g-ext a≈a′ b≈b′ = Properties.lift-ext Qu₃ (λ _ → p a≈a′ b≈b′ S₃.refl)"},{"Add":""},{"Add":" h : ∀ a q₂ q₃ → B (Qu₁.abs a) q₂ q₃"},{"Add":" h a = Qu₂.lift (λ b → ∀ q → B (Qu₁.abs a) b q) (g a) (ext ∘ g-ext S₁.refl)"},{"Add":""},{"Add":" h-ext : ∀ {a a′} → a S₁.≈ a′ → ∀ b → h a b ≅ h a′ b"},{"Add":" h-ext a≈a′ = Properties.lift-ext Qu₂ $ λ _ → ext (g-ext a≈a′ S₂.refl)"}]}},{"FileAdd":{"meta":{"basename":"Quotients","metadata":576},"context":[{"change":0,"pos":636924}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Examples.agda","metadata":0},"context":[{"change":0,"pos":643367}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Example of a Quotient: ℤ as (ℕ × ℕ / ∼)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --with-K --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.HeterogeneousEquality.Quotients.Examples where"},{"Add":""},{"Add":"open import Relation.Binary.HeterogeneousEquality.Quotients"},{"Add":"open import Level using (0ℓ)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence)"},{"Add":"open import Relation.Binary.HeterogeneousEquality hiding (isEquivalence)"},{"Add":"import Relation.Binary.PropositionalEquality.Core as ≡"},{"Add":"open import Data.Nat.Base"},{"Add":"open import Data.Nat.Properties"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Function.Base"},{"Add":""},{"Add":"open ≅-Reasoning"},{"Add":""},{"Add":"ℕ² = ℕ × ℕ"},{"Add":""},{"Add":"_∼_ : ℕ² → ℕ² → Set"},{"Add":"(x , y) ∼ (x′ , y′) = x + y′ ≅ x′ + y"},{"Add":""},{"Add":"infix 10 _∼_"},{"Add":""},{"Add":"∼-trans : ∀ {i j k} → i ∼ j → j ∼ k → i ∼ k"},{"Add":"∼-trans {x₁ , y₁} {x₂ , y₂} {x₃ , y₃} p q ="},{"Add":" ≡-to-≅ $ +-cancelˡ-≡ y₂ _ _ $ ≅-to-≡ $ begin"},{"Add":" y₂ + (x₁ + y₃) ≡⟨ ≡.sym (+-assoc y₂ x₁ y₃) ⟩"},{"Add":" y₂ + x₁ + y₃ ≡⟨ ≡.cong (_+ y₃) (+-comm y₂ x₁) ⟩"},{"Add":" x₁ + y₂ + y₃ ≅⟨ cong (_+ y₃) p ⟩"},{"Add":" x₂ + y₁ + y₃ ≡⟨ ≡.cong (_+ y₃) (+-comm x₂ y₁) ⟩"},{"Add":" y₁ + x₂ + y₃ ≡⟨ +-assoc y₁ x₂ y₃ ⟩"},{"Add":" y₁ + (x₂ + y₃) ≅⟨ cong (y₁ +_) q ⟩"},{"Add":" y₁ + (x₃ + y₂) ≡⟨ +-comm y₁ (x₃ + y₂) ⟩"},{"Add":" x₃ + y₂ + y₁ ≡⟨ ≡.cong (_+ y₁) (+-comm x₃ y₂) ⟩"},{"Add":" y₂ + x₃ + y₁ ≡⟨ +-assoc y₂ x₃ y₁ ⟩"},{"Add":" y₂ + (x₃ + y₁) ∎"},{"Add":""},{"Add":"∼-isEquivalence : IsEquivalence _∼_"},{"Add":"∼-isEquivalence = record"},{"Add":" { refl = refl"},{"Add":" ; sym = sym"},{"Add":" ; trans = λ {i} {j} {k} → ∼-trans {i} {j} {k}"},{"Add":" }"},{"Add":""},{"Add":"ℕ²-∼-setoid : Setoid 0ℓ 0ℓ"},{"Add":"ℕ²-∼-setoid = record { isEquivalence = ∼-isEquivalence }"},{"Add":""},{"Add":"module Integers (quot : Quotients 0ℓ 0ℓ) where"},{"Add":""},{"Add":" Int : Quotient ℕ²-∼-setoid"},{"Add":" Int = quot _"},{"Add":""},{"Add":" open Quotient Int renaming (Q to ℤ)"},{"Add":""},{"Add":" infixl 6 _+²_"},{"Add":""},{"Add":" _+²_ : ℕ² → ℕ² → ℕ²"},{"Add":" (x₁ , y₁) +² (x₂ , y₂) = x₁ + x₂ , y₁ + y₂"},{"Add":""},{"Add":" +²-cong : ∀{a b a′ b′} → a ∼ a′ → b ∼ b′ → (a +² b) ∼ (a′ +² b′)"},{"Add":" +²-cong {a₁ , b₁} {c₁ , d₁} {a₂ , b₂} {c₂ , d₂} ab∼cd₁ ab∼cd₂ = begin"},{"Add":" (a₁ + c₁) + (b₂ + d₂) ≡⟨ ≡.cong (_+ (b₂ + d₂)) (+-comm a₁ c₁) ⟩"},{"Add":" (c₁ + a₁) + (b₂ + d₂) ≡⟨ +-assoc c₁ a₁ (b₂ + d₂) ⟩"},{"Add":" c₁ + (a₁ + (b₂ + d₂)) ≡⟨ ≡.cong (c₁ +_) (≡.sym (+-assoc a₁ b₂ d₂)) ⟩"},{"Add":" c₁ + (a₁ + b₂ + d₂) ≅⟨ cong (λ n → c₁ + (n + d₂)) ab∼cd₁ ⟩"},{"Add":" c₁ + (a₂ + b₁ + d₂) ≡⟨ ≡.cong (c₁ +_) (+-assoc a₂ b₁ d₂) ⟩"},{"Add":" c₁ + (a₂ + (b₁ + d₂)) ≡⟨ ≡.cong (λ n → c₁ + (a₂ + n)) (+-comm b₁ d₂) ⟩"},{"Add":" c₁ + (a₂ + (d₂ + b₁)) ≡⟨ ≡.sym (+-assoc c₁ a₂ (d₂ + b₁)) ⟩"},{"Add":" (c₁ + a₂) + (d₂ + b₁) ≡⟨ ≡.cong (_+ (d₂ + b₁)) (+-comm c₁ a₂) ⟩"},{"Add":" (a₂ + c₁) + (d₂ + b₁) ≡⟨ +-assoc a₂ c₁ (d₂ + b₁) ⟩"},{"Add":" a₂ + (c₁ + (d₂ + b₁)) ≡⟨ ≡.cong (a₂ +_) (≡.sym (+-assoc c₁ d₂ b₁)) ⟩"},{"Add":" a₂ + (c₁ + d₂ + b₁) ≅⟨ cong (λ n → a₂ + (n + b₁)) ab∼cd₂ ⟩"},{"Add":" a₂ + (c₂ + d₁ + b₁) ≡⟨ ≡.cong (a₂ +_) (+-assoc c₂ d₁ b₁) ⟩"},{"Add":" a₂ + (c₂ + (d₁ + b₁)) ≡⟨ ≡.cong (λ n → a₂ + (c₂ + n)) (+-comm d₁ b₁) ⟩"},{"Add":" a₂ + (c₂ + (b₁ + d₁)) ≡⟨ ≡.sym (+-assoc a₂ c₂ (b₁ + d₁)) ⟩"},{"Add":" (a₂ + c₂) + (b₁ + d₁) ∎"},{"Add":""},{"Add":" module _ (ext : ∀ {a b} {A : Set a} {B₁ B₂ : A → Set b} {f₁ : ∀ a → B₁ a}"},{"Add":" {f₂ : ∀ a → B₂ a} → (∀ a → f₁ a ≅ f₂ a) → f₁ ≅ f₂) where"},{"Add":""},{"Add":" infixl 6 _+ℤ_"},{"Add":""},{"Add":" _+ℤ_ : ℤ → ℤ → ℤ"},{"Add":" _+ℤ_ = Properties₂.lift₂ ext Int Int (λ i j → abs (i +² j))"},{"Add":" $ λ {a} {b} {c} p p′ → compat-abs (+²-cong {a} {b} {c} p p′)"},{"Add":""},{"Add":" zero² : ℕ²"},{"Add":" zero² = 0 , 0"},{"Add":""},{"Add":" zeroℤ : ℤ"},{"Add":" zeroℤ = abs zero²"},{"Add":""},{"Add":" +²-identityʳ : (i : ℕ²) → (i +² zero²) ∼ i"},{"Add":" +²-identityʳ (x , y) = begin"},{"Add":" (x + 0) + y ≡⟨ ≡.cong (_+ y) (+-identityʳ x) ⟩"},{"Add":" x + y ≡⟨ ≡.cong (x +_) (≡.sym (+-identityʳ y)) ⟩"},{"Add":" x + (y + 0) ∎"},{"Add":""},{"Add":" +ℤ-on-abs≅abs-+₂ : ∀ a b → abs a +ℤ abs b ≅ abs (a +² b)"},{"Add":" +ℤ-on-abs≅abs-+₂ = Properties₂.lift₂-conv ext Int Int _"},{"Add":" $ λ {a} {b} {c} p p′ → compat-abs (+²-cong {a} {b} {c} p p′)"},{"Add":""},{"Add":" +ℤ-identityʳ : ∀ i → i +ℤ zeroℤ ≅ i"},{"Add":" +ℤ-identityʳ = lift _ eq (≅-heterogeneous-irrelevantʳ _ _ ∘ compat-abs) where"},{"Add":""},{"Add":" eq : ∀ a → abs a +ℤ zeroℤ ≅ abs a"},{"Add":" eq a = begin"},{"Add":" abs a +ℤ zeroℤ ≡⟨⟩"},{"Add":" abs a +ℤ abs zero² ≅⟨ +ℤ-on-abs≅abs-+₂ a zero² ⟩"},{"Add":" abs (a +² zero²) ≅⟨ compat-abs (+²-identityʳ a) ⟩"},{"Add":" abs a ∎"},{"Add":""},{"Add":" +²-identityˡ : (i : ℕ²) → (zero² +² i) ∼ i"},{"Add":" +²-identityˡ i = refl"},{"Add":""},{"Add":" +ℤ-identityˡ : (i : ℤ) → zeroℤ +ℤ i ≅ i"},{"Add":" +ℤ-identityˡ = lift _ eq (≅-heterogeneous-irrelevantʳ _ _ ∘ compat-abs) where"},{"Add":""},{"Add":" eq : ∀ a → zeroℤ +ℤ abs a ≅ abs a"},{"Add":" eq a = begin"},{"Add":" zeroℤ +ℤ abs a ≡⟨⟩"},{"Add":" abs zero² +ℤ abs a ≅⟨ +ℤ-on-abs≅abs-+₂ zero² a ⟩"},{"Add":" abs (zero² +² a) ≅⟨ compat-abs (+²-identityˡ a) ⟩"},{"Add":" abs a ∎"},{"Add":""},{"Add":" +²-assoc : (i j k : ℕ²) → ((i +² j) +² k) ∼ (i +² (j +² k))"},{"Add":" +²-assoc (a , b) (c , d) (e , f) = begin"},{"Add":" ((a + c) + e) + (b + (d + f)) ≡⟨ ≡.cong (_+ (b + (d + f))) (+-assoc a c e) ⟩"},{"Add":" (a + (c + e)) + (b + (d + f)) ≡⟨ ≡.cong ((a + (c + e)) +_) (≡.sym (+-assoc b d f)) ⟩"},{"Add":" (a + (c + e)) + ((b + d) + f) ∎"},{"Add":""},{"Add":" +ℤ-assoc : ∀ i j k → (i +ℤ j) +ℤ k ≅ i +ℤ (j +ℤ k)"},{"Add":" +ℤ-assoc = Properties₃.lift₃ ext Int Int Int eq compat₃ where"},{"Add":""},{"Add":" eq : ∀ i j k → (abs i +ℤ abs j) +ℤ abs k ≅ abs i +ℤ (abs j +ℤ abs k)"},{"Add":" eq i j k = begin"},{"Add":" (abs i +ℤ abs j) +ℤ abs k ≅⟨ cong (_+ℤ abs k) (+ℤ-on-abs≅abs-+₂ i j) ⟩"},{"Add":" (abs (i +² j) +ℤ abs k) ≅⟨ +ℤ-on-abs≅abs-+₂ (i +² j) k ⟩"},{"Add":" abs ((i +² j) +² k) ≅⟨ compat-abs (+²-assoc i j k) ⟩"},{"Add":" abs (i +² (j +² k)) ≅⟨ sym (+ℤ-on-abs≅abs-+₂ i (j +² k)) ⟩"},{"Add":" (abs i +ℤ abs (j +² k)) ≅⟨ cong (abs i +ℤ_) (sym (+ℤ-on-abs≅abs-+₂ j k)) ⟩"},{"Add":" abs i +ℤ (abs j +ℤ abs k) ∎"},{"Add":""},{"Add":" compat₃ : ∀ {a a′ b b′ c c′} → a ∼ a′ → b ∼ b′ → c ∼ c′ → eq a b c ≅ eq a′ b′ c′"},{"Add":" compat₃ p q r = ≅-heterogeneous-irrelevantˡ _ _"},{"Add":" $ cong₂ _+ℤ_ (cong₂ _+ℤ_ (compat-abs p) (compat-abs q))"},{"Add":" $ compat-abs r"}]}},{"FileAdd":{"meta":{"basename":"Core.agda","metadata":0},"context":[{"change":0,"pos":636924}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Heterogeneous equality"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- This file contains some core definitions which are reexported by"},{"Add":"-- Relation.Binary.HeterogeneousEquality."},{"Add":""},{"Add":"{-# OPTIONS --with-K --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.HeterogeneousEquality.Core where"},{"Add":""},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Heterogeneous equality"},{"Add":""},{"Add":"infix 4 _≅_"},{"Add":""},{"Add":"data _≅_ {A : Set a} (x : A) : {B : Set b} → B → Set a where"},{"Add":" refl : x ≅ x"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Conversion"},{"Add":""},{"Add":"≅-to-≡ : ∀ {x y : A} → x ≅ y → x ≡ y"},{"Add":"≅-to-≡ refl = refl"},{"Add":""},{"Add":"≡-to-≅ : ∀ {x y : A} → x ≡ y → x ≅ y"},{"Add":"≡-to-≅ refl = refl"}]}},{"FileAdd":{"meta":{"basename":"Definitions.agda","metadata":0},"context":[{"change":0,"pos":415495}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via `Relation.Binary`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Definitions where"},{"Add":""},{"Add":"open import Agda.Builtin.Equality using (_≡_)"},{"Add":""},{"Add":"open import Data.Product.Base using (_×_; ∃-syntax)"},{"Add":"open import Data.Sum.Base using (_⊎_)"},{"Add":"open import Function.Base using (_on_; flip)"},{"Add":"open import Level"},{"Add":"open import Relation.Binary.Core"},{"Add":"open import Relation.Nullary as Nullary using (¬_; Dec)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":" C : Set c"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definitions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Reflexivity - defined without an underlying equality. It could"},{"Add":"-- alternatively be defined as `_≈_ ⇒ _∼_` for some equality `_≈_`."},{"Add":""},{"Add":"-- Confusingly the convention in the library is to use the name \"refl\""},{"Add":"-- for proofs of Reflexive and `reflexive` for proofs of type `_≈_ ⇒ _∼_`,"},{"Add":"-- e.g. in the definition of `IsEquivalence` later in this file. This"},{"Add":"-- convention is a legacy from the early days of the library."},{"Add":""},{"Add":"Reflexive : Rel A ℓ → Set _"},{"Add":"Reflexive _∼_ = ∀ {x} → x ∼ x"},{"Add":""},{"Add":"-- Generalised symmetry."},{"Add":""},{"Add":"Sym : REL A B ℓ₁ → REL B A ℓ₂ → Set _"},{"Add":"Sym P Q = P ⇒ flip Q"},{"Add":""},{"Add":"-- Symmetry."},{"Add":""},{"Add":"Symmetric : Rel A ℓ → Set _"},{"Add":"Symmetric _∼_ = Sym _∼_ _∼_"},{"Add":""},{"Add":"-- Generalised transitivity."},{"Add":""},{"Add":"Trans : REL A B ℓ₁ → REL B C ℓ₂ → REL A C ℓ₃ → Set _"},{"Add":"Trans P Q R = ∀ {i j k} → P i j → Q j k → R i k"},{"Add":""},{"Add":"RightTrans : REL A B ℓ₁ → REL B B ℓ₂ → Set _"},{"Add":"RightTrans R S = Trans R S R"},{"Add":""},{"Add":"LeftTrans : REL A A ℓ₁ → REL A B ℓ₂ → Set _"},{"Add":"LeftTrans S R = Trans S R R"},{"Add":""},{"Add":"-- A flipped variant of generalised transitivity."},{"Add":""},{"Add":"TransFlip : REL A B ℓ₁ → REL B C ℓ₂ → REL A C ℓ₃ → Set _"},{"Add":"TransFlip P Q R = ∀ {i j k} → Q j k → P i j → R i k"},{"Add":""},{"Add":"-- Transitivity."},{"Add":""},{"Add":"Transitive : Rel A ℓ → Set _"},{"Add":"Transitive _∼_ = Trans _∼_ _∼_ _∼_"},{"Add":""},{"Add":"-- Generalised antisymmetry"},{"Add":""},{"Add":"Antisym : REL A B ℓ₁ → REL B A ℓ₂ → REL A B ℓ₃ → Set _"},{"Add":"Antisym R S E = ∀ {i j} → R i j → S j i → E i j"},{"Add":""},{"Add":"-- Antisymmetry."},{"Add":""},{"Add":"Antisymmetric : Rel A ℓ₁ → Rel A ℓ₂ → Set _"},{"Add":"Antisymmetric _≈_ _≤_ = Antisym _≤_ _≤_ _≈_"},{"Add":""},{"Add":"-- Irreflexivity - this is defined terms of the underlying equality."},{"Add":""},{"Add":"Irreflexive : REL A B ℓ₁ → REL A B ℓ₂ → Set _"},{"Add":"Irreflexive _≈_ _<_ = ∀ {x y} → x ≈ y → ¬ (x < y)"},{"Add":""},{"Add":"-- Asymmetry."},{"Add":""},{"Add":"Asymmetric : Rel A ℓ → Set _"},{"Add":"Asymmetric _<_ = ∀ {x y} → x < y → ¬ (y < x)"},{"Add":""},{"Add":"-- Density"},{"Add":""},{"Add":"Dense : Rel A ℓ → Set _"},{"Add":"Dense _<_ = ∀ {x y} → x < y → ∃[ z ] x < z × z < y"},{"Add":""},{"Add":"-- Generalised connex - at least one of the two relations holds."},{"Add":""},{"Add":"Connex : REL A B ℓ₁ → REL B A ℓ₂ → Set _"},{"Add":"Connex P Q = ∀ x y → P x y ⊎ Q y x"},{"Add":""},{"Add":"-- Totality."},{"Add":""},{"Add":"Total : Rel A ℓ → Set _"},{"Add":"Total _∼_ = Connex _∼_ _∼_"},{"Add":""},{"Add":"-- Generalised trichotomy - exactly one of three types has a witness."},{"Add":""},{"Add":"data Tri (A : Set a) (B : Set b) (C : Set c) : Set (a ⊔ b ⊔ c) where"},{"Add":" tri< : ( a : A) (¬b : ¬ B) (¬c : ¬ C) → Tri A B C"},{"Add":" tri≈ : (¬a : ¬ A) ( b : B) (¬c : ¬ C) → Tri A B C"},{"Add":" tri> : (¬a : ¬ A) (¬b : ¬ B) ( c : C) → Tri A B C"},{"Add":""},{"Add":"-- Trichotomy."},{"Add":""},{"Add":"Trichotomous : Rel A ℓ₁ → Rel A ℓ₂ → Set _"},{"Add":"Trichotomous _≈_ _<_ = ∀ x y → Tri (x < y) (x ≈ y) (x > y)"},{"Add":" where _>_ = flip _<_"},{"Add":""},{"Add":"-- Generalised maximum element."},{"Add":""},{"Add":"Max : REL A B ℓ → B → Set _"},{"Add":"Max _≤_ T = ∀ x → x ≤ T"},{"Add":""},{"Add":"-- Maximum element."},{"Add":""},{"Add":"Maximum : Rel A ℓ → A → Set _"},{"Add":"Maximum = Max"},{"Add":""},{"Add":"-- Generalised minimum element."},{"Add":""},{"Add":"Min : REL A B ℓ → A → Set _"},{"Add":"Min R = Max (flip R)"},{"Add":""},{"Add":"-- Minimum element."},{"Add":""},{"Add":"Minimum : Rel A ℓ → A → Set _"},{"Add":"Minimum = Min"},{"Add":""},{"Add":"-- Definitions for apartness relations"},{"Add":""},{"Add":"-- Note that Cotransitive's arguments are permuted with respect to Transitive's."},{"Add":"Cotransitive : Rel A ℓ → Set _"},{"Add":"Cotransitive _#_ = ∀ {x y} → x # y → ∀ z → (x # z) ⊎ (z # y)"},{"Add":""},{"Add":"Tight : Rel A ℓ₁ → Rel A ℓ₂ → Set _"},{"Add":"Tight _≈_ _#_ = ∀ x y → (¬ x # y → x ≈ y) × (x ≈ y → ¬ x # y)"},{"Add":""},{"Add":"-- Properties of order morphisms, aka order-preserving maps"},{"Add":""},{"Add":"Monotonic₁ : Rel A ℓ₁ → Rel B ℓ₂ → (A → B) → Set _"},{"Add":"Monotonic₁ _≤_ _⊑_ f = f Preserves _≤_ ⟶ _⊑_"},{"Add":""},{"Add":"Antitonic₁ : Rel A ℓ₁ → Rel B ℓ₂ → (A → B) → Set _"},{"Add":"Antitonic₁ _≤_ _⊑_ f = f Preserves (flip _≤_) ⟶ _⊑_"},{"Add":""},{"Add":"Monotonic₂ : Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → (A → B → C) → Set _"},{"Add":"Monotonic₂ _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ _≤_ ⟶ _⊑_ ⟶ _≼_"},{"Add":""},{"Add":"MonotonicAntitonic : Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → (A → B → C) → Set _"},{"Add":"MonotonicAntitonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ _≤_ ⟶ (flip _⊑_) ⟶ _≼_"},{"Add":""},{"Add":"AntitonicMonotonic : Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → (A → B → C) → Set _"},{"Add":"AntitonicMonotonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ _⊑_ ⟶ _≼_"},{"Add":""},{"Add":"Antitonic₂ : Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → (A → B → C) → Set _"},{"Add":"Antitonic₂ _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ (flip _⊑_) ⟶ _≼_"},{"Add":""},{"Add":"Adjoint : Rel A ℓ₁ → Rel B ℓ₂ → (A → B) → (B → A) → Set _"},{"Add":"Adjoint _≤_ _⊑_ f g = ∀ {x y} → (f x ⊑ y → x ≤ g y) × (x ≤ g y → f x ⊑ y)"},{"Add":""},{"Add":"-- Unary relations respecting a binary relation."},{"Add":""},{"Add":"_⟶_Respects_ : (A → Set ℓ₁) → (B → Set ℓ₂) → REL A B ℓ₃ → Set _"},{"Add":"P ⟶ Q Respects _∼_ = ∀ {x y} → x ∼ y → P x → Q y"},{"Add":""},{"Add":"-- Unary relation respects a binary relation."},{"Add":""},{"Add":"_Respects_ : (A → Set ℓ₁) → Rel A ℓ₂ → Set _"},{"Add":"P Respects _∼_ = P ⟶ P Respects _∼_"},{"Add":""},{"Add":"-- Right respecting - relatedness is preserved on the right by equality."},{"Add":""},{"Add":"_Respectsʳ_ : REL A B ℓ₁ → Rel B ℓ₂ → Set _"},{"Add":"_∼_ Respectsʳ _≈_ = ∀ {x} → (x ∼_) Respects _≈_"},{"Add":""},{"Add":"-- Left respecting - relatedness is preserved on the left by equality."},{"Add":""},{"Add":"_Respectsˡ_ : REL A B ℓ₁ → Rel A ℓ₂ → Set _"},{"Add":"P Respectsˡ _∼_ = ∀ {y} → (flip P y) Respects _∼_"},{"Add":""},{"Add":"-- Respecting - relatedness is preserved on both sides by equality"},{"Add":""},{"Add":"_Respects₂_ : Rel A ℓ₁ → Rel A ℓ₂ → Set _"},{"Add":"P Respects₂ _∼_ = (P Respectsʳ _∼_) × (P Respectsˡ _∼_)"},{"Add":""},{"Add":"-- Substitutivity - any two related elements satisfy exactly the same"},{"Add":"-- set of unary relations. Note that only the various derivatives"},{"Add":"-- of propositional equality can satisfy this property."},{"Add":""},{"Add":"Substitutive : Rel A ℓ₁ → (ℓ₂ : Level) → Set _"},{"Add":"Substitutive {A = A} _∼_ p = (P : A → Set p) → P Respects _∼_"},{"Add":""},{"Add":"-- Irrelevancy - all proofs that a given pair of elements are related"},{"Add":"-- are indistinguishable."},{"Add":""},{"Add":"Irrelevant : REL A B ℓ → Set _"},{"Add":"Irrelevant _∼_ = ∀ {x y} → Nullary.Irrelevant (x ∼ y)"},{"Add":""},{"Add":"-- Recomputability - we can rebuild a relevant proof given an"},{"Add":"-- irrelevant one."},{"Add":""},{"Add":"Recomputable : REL A B ℓ → Set _"},{"Add":"Recomputable _∼_ = ∀ {x y} → Nullary.Recomputable (x ∼ y)"},{"Add":""},{"Add":"-- Stability"},{"Add":""},{"Add":"Stable : REL A B ℓ → Set _"},{"Add":"Stable _∼_ = ∀ x y → Nullary.Stable (x ∼ y)"},{"Add":""},{"Add":"-- Weak decidability - it is sometimes possible to determine if a given"},{"Add":"-- pair of elements are related."},{"Add":""},{"Add":"WeaklyDecidable : REL A B ℓ → Set _"},{"Add":"WeaklyDecidable _∼_ = ∀ x y → Nullary.WeaklyDecidable (x ∼ y)"},{"Add":""},{"Add":"-- Decidability - it is possible to determine whether a given pair of"},{"Add":"-- elements are related."},{"Add":""},{"Add":"Decidable : REL A B ℓ → Set _"},{"Add":"Decidable _∼_ = ∀ x y → Dec (x ∼ y)"},{"Add":""},{"Add":"-- Propositional equality is decidable for the type."},{"Add":""},{"Add":"DecidableEquality : (A : Set a) → Set _"},{"Add":"DecidableEquality A = Decidable {A = A} _≡_"},{"Add":""},{"Add":"-- Universal - all pairs of elements are related"},{"Add":""},{"Add":"Universal : REL A B ℓ → Set _"},{"Add":"Universal _∼_ = ∀ x y → x ∼ y"},{"Add":""},{"Add":"-- Empty - no elements are related"},{"Add":""},{"Add":"Empty : REL A B ℓ → Set _"},{"Add":"Empty _∼_ = ∀ {x y} → ¬ (x ∼ y)"},{"Add":""},{"Add":"-- Non-emptiness - at least one pair of elements are related."},{"Add":""},{"Add":"record NonEmpty {A : Set a} {B : Set b}"},{"Add":" (T : REL A B ℓ) : Set (a ⊔ b ⊔ ℓ) where"},{"Add":" constructor nonEmpty"},{"Add":" field"},{"Add":" {x} : A"},{"Add":" {y} : B"},{"Add":" proof : T x y"}]}},{"FileAdd":{"meta":{"basename":"Core.agda","metadata":0},"context":[{"change":0,"pos":415495}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via `Relation.Binary`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Core where"},{"Add":""},{"Add":"open import Data.Product.Base using (_×_)"},{"Add":"open import Function.Base using (_on_)"},{"Add":"open import Level using (Level; _⊔_; suc)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":" C : Set c"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definitions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Heterogeneous binary relations"},{"Add":""},{"Add":"REL : Set a → Set b → (ℓ : Level) → Set (a ⊔ b ⊔ suc ℓ)"},{"Add":"REL A B ℓ = A → B → Set ℓ"},{"Add":""},{"Add":"-- Homogeneous binary relations"},{"Add":""},{"Add":"Rel : Set a → (ℓ : Level) → Set (a ⊔ suc ℓ)"},{"Add":"Rel A ℓ = REL A A ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relationships between relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"infix 4 _⇒_ _⇔_ _=[_]⇒_"},{"Add":""},{"Add":"-- Implication/containment - could also be written _⊆_."},{"Add":"-- and corresponding notion of equivalence"},{"Add":""},{"Add":"_⇒_ : REL A B ℓ₁ → REL A B ℓ₂ → Set _"},{"Add":"P ⇒ Q = ∀ {x y} → P x y → Q x y"},{"Add":""},{"Add":"_⇔_ : REL A B ℓ₁ → REL A B ℓ₂ → Set _"},{"Add":"P ⇔ Q = P ⇒ Q × Q ⇒ P"},{"Add":""},{"Add":"-- Generalised implication - if P ≡ Q it can be read as \"f preserves P\"."},{"Add":""},{"Add":"_=[_]⇒_ : Rel A ℓ₁ → (A → B) → Rel B ℓ₂ → Set _"},{"Add":"P =[ f ]⇒ Q = P ⇒ (Q on f)"},{"Add":""},{"Add":"-- A synonym for _=[_]⇒_."},{"Add":""},{"Add":"_Preserves_⟶_ : (A → B) → Rel A ℓ₁ → Rel B ℓ₂ → Set _"},{"Add":"f Preserves P ⟶ Q = P =[ f ]⇒ Q"},{"Add":""},{"Add":"-- A binary variant of _Preserves_⟶_."},{"Add":""},{"Add":"_Preserves₂_⟶_⟶_ : (A → B → C) → Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → Set _"},{"Add":"_∙_ Preserves₂ P ⟶ Q ⟶ R = ∀ {x y u v} → P x y → Q u v → R (x ∙ u) (y ∙ v)"}]}},{"FileAdd":{"meta":{"basename":"Construct","metadata":576},"context":[{"change":0,"pos":415495}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Union.agda","metadata":0},"context":[{"change":0,"pos":661969}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Union of two binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Construct.Union where"},{"Add":""},{"Add":"open import Data.Product.Base"},{"Add":"open import Data.Sum.Base as Sum"},{"Add":"open import Function.Base using (_∘_)"},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Relation.Binary.Core using (REL; Rel; _⇒_)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (Reflexive; Total; Minimum; Maximum; Symmetric; Irreflexive; Decidable; _Respects_; _Respectsˡ_; _Respectsʳ_; _Respects₂_)"},{"Add":"open import Relation.Nullary.Decidable using (yes; no; _⊎-dec_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b ℓ ℓ₁ ℓ₂ ℓ₃ : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"infixr 6 _∪_"},{"Add":""},{"Add":"_∪_ : REL A B ℓ₁ → REL A B ℓ₂ → REL A B (ℓ₁ ⊔ ℓ₂)"},{"Add":"L ∪ R = λ i j → L i j ⊎ R i j"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"module _ (L : Rel A ℓ) (R : Rel A ℓ) where"},{"Add":""},{"Add":" reflexive : Reflexive L ⊎ Reflexive R → Reflexive (L ∪ R)"},{"Add":" reflexive (inj₁ L-refl) = inj₁ L-refl"},{"Add":" reflexive (inj₂ R-refl) = inj₂ R-refl"},{"Add":""},{"Add":" total : Total L ⊎ Total R → Total (L ∪ R)"},{"Add":" total (inj₁ L-total) x y = [ inj₁ ∘ inj₁ , inj₂ ∘ inj₁ ] (L-total x y)"},{"Add":" total (inj₂ R-total) x y = [ inj₁ ∘ inj₂ , inj₂ ∘ inj₂ ] (R-total x y)"},{"Add":""},{"Add":" min : ∀ {⊤} → Minimum L ⊤ ⊎ Minimum R ⊤ → Minimum (L ∪ R) ⊤"},{"Add":" min = [ inj₁ ∘_ , inj₂ ∘_ ]"},{"Add":""},{"Add":" max : ∀ {⊥} → Maximum L ⊥ ⊎ Maximum R ⊥ → Maximum (L ∪ R) ⊥"},{"Add":" max = [ inj₁ ∘_ , inj₂ ∘_ ]"},{"Add":""},{"Add":"module _ {L : Rel A ℓ} {R : Rel A ℓ} where"},{"Add":""},{"Add":" symmetric : Symmetric L → Symmetric R → Symmetric (L ∪ R)"},{"Add":" symmetric L-sym R-sym = [ inj₁ ∘ L-sym , inj₂ ∘ R-sym ]"},{"Add":""},{"Add":" respects : ∀ {p} {P : A → Set p} →"},{"Add":" P Respects L → P Respects R → P Respects (L ∪ R)"},{"Add":" respects resp-L resp-R = [ resp-L , resp-R ]"},{"Add":""},{"Add":"module _ {≈ : Rel A ℓ₁} (L : REL A B ℓ₂) (R : REL A B ℓ₃) where"},{"Add":""},{"Add":" respˡ : L Respectsˡ ≈ → R Respectsˡ ≈ → (L ∪ R) Respectsˡ ≈"},{"Add":" respˡ Lˡ Rˡ x≈y = Sum.map (Lˡ x≈y) (Rˡ x≈y)"},{"Add":""},{"Add":"module _ {≈ : Rel B ℓ₁} (L : REL A B ℓ₂) (R : REL A B ℓ₃) where"},{"Add":""},{"Add":" respʳ : L Respectsʳ ≈ → R Respectsʳ ≈ → (L ∪ R) Respectsʳ ≈"},{"Add":" respʳ Lʳ Rʳ x≈y = Sum.map (Lʳ x≈y) (Rʳ x≈y)"},{"Add":""},{"Add":"module _ {≈ : Rel A ℓ₁} {L : Rel A ℓ₂} {R : Rel A ℓ₃} where"},{"Add":""},{"Add":" resp₂ : L Respects₂ ≈ → R Respects₂ ≈ → (L ∪ R) Respects₂ ≈"},{"Add":" resp₂ (Lʳ , Lˡ) (Rʳ , Rˡ) = respʳ L R Lʳ Rʳ , respˡ L R Lˡ Rˡ"},{"Add":""},{"Add":"module _ (≈ : REL A B ℓ₁) (L : REL A B ℓ₂) (R : REL A B ℓ₃) where"},{"Add":""},{"Add":" implies : (≈ ⇒ L) ⊎ (≈ ⇒ R) → ≈ ⇒ (L ∪ R)"},{"Add":" implies = [ inj₁ ∘_ , inj₂ ∘_ ]"},{"Add":""},{"Add":" irreflexive : Irreflexive ≈ L → Irreflexive ≈ R → Irreflexive ≈ (L ∪ R)"},{"Add":" irreflexive L-irrefl R-irrefl x≈y = [ L-irrefl x≈y , R-irrefl x≈y ]"},{"Add":""},{"Add":"module _ {L : REL A B ℓ₁} {R : REL A B ℓ₂} where"},{"Add":""},{"Add":" decidable : Decidable L → Decidable R → Decidable (L ∪ R)"},{"Add":" decidable L? R? x y = L? x y ⊎-dec R? x y"}]}},{"FileAdd":{"meta":{"basename":"Subst","metadata":576},"context":[{"change":0,"pos":661969}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Equality.agda","metadata":0},"context":[{"change":0,"pos":665356}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Substituting equalities for binary relations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- For more general transformations between binary relations"},{"Add":"-- see `Relation.Binary.Morphisms`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Relation.Binary.Core using (Rel; _⇔_)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence)"},{"Add":"open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)"},{"Add":""},{"Add":"module Relation.Binary.Construct.Subst.Equality"},{"Add":" {a ℓ₁ ℓ₂} {A : Set a} {≈₁ : Rel A ℓ₁} {≈₂ : Rel A ℓ₂}"},{"Add":" (equiv@(to , from) : ≈₁ ⇔ ≈₂)"},{"Add":" where"},{"Add":""},{"Add":"open import Function.Base"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definitions"},{"Add":""},{"Add":"refl : Reflexive ≈₁ → Reflexive ≈₂"},{"Add":"refl refl = to refl"},{"Add":""},{"Add":"sym : Symmetric ≈₁ → Symmetric ≈₂"},{"Add":"sym sym = to ∘′ sym ∘′ from"},{"Add":""},{"Add":"trans : Transitive ≈₁ → Transitive ≈₂"},{"Add":"trans trans x≈y y≈z = to (trans (from x≈y) (from y≈z))"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"isEquivalence : IsEquivalence ≈₁ → IsEquivalence ≈₂"},{"Add":"isEquivalence E = record"},{"Add":" { refl = refl E.refl"},{"Add":" ; sym = sym E.sym"},{"Add":" ; trans = trans E.trans"},{"Add":" } where module E = IsEquivalence E"}]}},{"FileAdd":{"meta":{"basename":"StrictToNonStrict.agda","metadata":0},"context":[{"change":0,"pos":661969}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Conversion of < to ≤, along with a number of properties"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Possible TODO: Prove that a conversion ≤ → < → ≤ returns a"},{"Add":"-- relation equivalent to the original one (and similarly for"},{"Add":"-- < → ≤ → <)."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (Rel; _⇒_)"},{"Add":"open import Relation.Binary.Structures"},{"Add":" using (IsEquivalence; IsPreorder; IsStrictPartialOrder; IsPartialOrder; IsStrictTotalOrder; IsTotalOrder; IsDecTotalOrder)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (Transitive; Symmetric; Irreflexive; Antisymmetric; Trichotomous; Decidable; Trans; Total; _Respects₂_; _Respectsʳ_; _Respectsˡ_; tri<; tri≈; tri>)"},{"Add":""},{"Add":"module Relation.Binary.Construct.StrictToNonStrict"},{"Add":" {a ℓ₁ ℓ₂} {A : Set a}"},{"Add":" (_≈_ : Rel A ℓ₁) (_<_ : Rel A ℓ₂)"},{"Add":" where"},{"Add":""},{"Add":"open import Data.Product.Base"},{"Add":"open import Data.Sum.Base"},{"Add":"open import Data.Empty"},{"Add":"open import Function.Base"},{"Add":"open import Relation.Binary.Consequences"},{"Add":"open import Relation.Nullary.Decidable using (_⊎-dec_; yes; no)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Conversion"},{"Add":""},{"Add":"-- _<_ can be turned into _≤_ as follows:"},{"Add":"infix 4 _≤_"},{"Add":""},{"Add":"_≤_ : Rel A _"},{"Add":"x ≤ y = (x < y) ⊎ (x ≈ y)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The converted relations have certain properties"},{"Add":"-- (if the original relations have certain other properties)"},{"Add":""},{"Add":"<⇒≤ : _<_ ⇒ _≤_"},{"Add":"<⇒≤ = inj₁"},{"Add":""},{"Add":"reflexive : _≈_ ⇒ _≤_"},{"Add":"reflexive = inj₂"},{"Add":""},{"Add":"antisym : IsEquivalence _≈_ → Transitive _<_ → Irreflexive _≈_ _<_ →"},{"Add":" Antisymmetric _≈_ _≤_"},{"Add":"antisym eq trans irrefl = as"},{"Add":" where"},{"Add":" module Eq = IsEquivalence eq"},{"Add":""},{"Add":" as : Antisymmetric _≈_ _≤_"},{"Add":" as (inj₂ x≈y) _ = x≈y"},{"Add":" as (inj₁ _) (inj₂ y≈x) = Eq.sym y≈x"},{"Add":" as (inj₁ x x≮y x≉y x>y = inj₂ (inj₁ x>y)"},{"Add":""},{"Add":"decidable : Decidable _≈_ → Decidable _<_ → Decidable _≤_"},{"Add":"decidable ≈-dec <-dec x y = <-dec x y ⊎-dec ≈-dec x y"},{"Add":""},{"Add":"decidable′ : Trichotomous _≈_ _<_ → Decidable _≤_"},{"Add":"decidable′ compare x y with compare x y"},{"Add":"... | tri< x x≮y x≉y _ = no [ x≮y , x≉y ]′"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Converting structures"},{"Add":""},{"Add":"isPreorder₁ : IsPreorder _≈_ _<_ → IsPreorder _≈_ _≤_"},{"Add":"isPreorder₁ PO = record"},{"Add":" { isEquivalence = S.isEquivalence"},{"Add":" ; reflexive = reflexive"},{"Add":" ; trans = trans S.isEquivalence S.∼-resp-≈ S.trans"},{"Add":" }"},{"Add":" where module S = IsPreorder PO"},{"Add":""},{"Add":"isPreorder₂ : IsStrictPartialOrder _≈_ _<_ → IsPreorder _≈_ _≤_"},{"Add":"isPreorder₂ SPO = record"},{"Add":" { isEquivalence = S.isEquivalence"},{"Add":" ; reflexive = reflexive"},{"Add":" ; trans = trans S.isEquivalence S.<-resp-≈ S.trans"},{"Add":" }"},{"Add":" where module S = IsStrictPartialOrder SPO"},{"Add":""},{"Add":"isPartialOrder : IsStrictPartialOrder _≈_ _<_ → IsPartialOrder _≈_ _≤_"},{"Add":"isPartialOrder SPO = record"},{"Add":" { isPreorder = isPreorder₂ SPO"},{"Add":" ; antisym = antisym S.isEquivalence S.trans S.irrefl"},{"Add":" }"},{"Add":" where module S = IsStrictPartialOrder SPO"},{"Add":""},{"Add":"isTotalOrder : IsStrictTotalOrder _≈_ _<_ → IsTotalOrder _≈_ _≤_"},{"Add":"isTotalOrder STO = record"},{"Add":" { isPartialOrder = isPartialOrder S.isStrictPartialOrder"},{"Add":" ; total = total S.compare"},{"Add":" }"},{"Add":" where module S = IsStrictTotalOrder STO"},{"Add":""},{"Add":"isDecTotalOrder : IsStrictTotalOrder _≈_ _<_ → IsDecTotalOrder _≈_ _≤_"},{"Add":"isDecTotalOrder STO = record"},{"Add":" { isTotalOrder = isTotalOrder STO"},{"Add":" ; _≟_ = S._≟_"},{"Add":" ; _≤?_ = decidable′ S.compare"},{"Add":" }"},{"Add":" where module S = IsStrictTotalOrder STO"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- DEPRECATED"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Please use the new names as continuing support for the old names is"},{"Add":"-- not guaranteed."},{"Add":""},{"Add":"-- Version 1.4"},{"Add":""},{"Add":"decidable' : Trichotomous _≈_ _<_ → Decidable _≤_"},{"Add":"decidable' = decidable′"},{"Add":"{-# WARNING_ON_USAGE decidable'"},{"Add":"\"Warning: decidable' (ending in an apostrophe) was deprecated in v1.4."},{"Add":"Please use decidable′ (ending in a prime) instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"On.agda","metadata":0},"context":[{"change":0,"pos":661969}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Many properties which hold for `_∼_` also hold for `_∼_ on f`"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Construct.On where"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base using (_on_; _∘_)"},{"Add":"open import Induction.WellFounded using (WellFounded; Acc; acc)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (Rel; _⇒_)"},{"Add":"open import Relation.Binary.Bundles"},{"Add":" using (Preorder; Setoid; DecSetoid; Poset; DecPoset; StrictPartialOrder; TotalOrder; DecTotalOrder; StrictTotalOrder)"},{"Add":"open import Relation.Binary.Structures"},{"Add":" using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsDecPartialOrder; IsStrictPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictTotalOrder)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (Reflexive; Irreflexive; Symmetric; Transitive; Antisymmetric; Asymmetric; Decidable; Total; Trichotomous; _Respects_; _Respects₂_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b p ℓ ℓ₁ ℓ₂ : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definitions"},{"Add":""},{"Add":"module _ (f : B → A) where"},{"Add":""},{"Add":" implies : (≈ : Rel A ℓ₁) (∼ : Rel A ℓ₂) →"},{"Add":" ≈ ⇒ ∼ → (≈ on f) ⇒ (∼ on f)"},{"Add":" implies _ _ impl = impl"},{"Add":""},{"Add":" reflexive : (∼ : Rel A ℓ) → Reflexive ∼ → Reflexive (∼ on f)"},{"Add":" reflexive _ refl = refl"},{"Add":""},{"Add":" irreflexive : (≈ : Rel A ℓ₁) (∼ : Rel A ℓ₂) →"},{"Add":" Irreflexive ≈ ∼ → Irreflexive (≈ on f) (∼ on f)"},{"Add":" irreflexive _ _ irrefl = irrefl"},{"Add":""},{"Add":" symmetric : (∼ : Rel A ℓ) → Symmetric ∼ → Symmetric (∼ on f)"},{"Add":" symmetric _ sym = sym"},{"Add":""},{"Add":" transitive : (∼ : Rel A ℓ) → Transitive ∼ → Transitive (∼ on f)"},{"Add":" transitive _ trans = trans"},{"Add":""},{"Add":" antisymmetric : (≈ : Rel A ℓ₁) (≤ : Rel A ℓ₂) →"},{"Add":" Antisymmetric ≈ ≤ → Antisymmetric (≈ on f) (≤ on f)"},{"Add":" antisymmetric _ _ antisym = antisym"},{"Add":""},{"Add":" asymmetric : (< : Rel A ℓ) → Asymmetric < → Asymmetric (< on f)"},{"Add":" asymmetric _ asym = asym"},{"Add":""},{"Add":" respects : (∼ : Rel A ℓ) (P : A → Set p) →"},{"Add":" P Respects ∼ → (P ∘ f) Respects (∼ on f)"},{"Add":" respects _ _ resp = resp"},{"Add":""},{"Add":" respects₂ : (∼₁ : Rel A ℓ₁) (∼₂ : Rel A ℓ₂) →"},{"Add":" ∼₁ Respects₂ ∼₂ → (∼₁ on f) Respects₂ (∼₂ on f)"},{"Add":" respects₂ _ _ (resp₁ , resp₂) = (resp₁ , resp₂)"},{"Add":""},{"Add":" decidable : (∼ : Rel A ℓ) → Decidable ∼ → Decidable (∼ on f)"},{"Add":" decidable _ dec x y = dec (f x) (f y)"},{"Add":""},{"Add":" total : (∼ : Rel A ℓ) → Total ∼ → Total (∼ on f)"},{"Add":" total _ tot x y = tot (f x) (f y)"},{"Add":""},{"Add":" trichotomous : (≈ : Rel A ℓ₁) (< : Rel A ℓ₂) →"},{"Add":" Trichotomous ≈ < → Trichotomous (≈ on f) (< on f)"},{"Add":" trichotomous _ _ compare x y = compare (f x) (f y)"},{"Add":""},{"Add":" accessible : ∀ {∼ : Rel A ℓ} {x} → Acc ∼ (f x) → Acc (∼ on f) x"},{"Add":" accessible (acc rs) = acc (λ fy)"},{"Add":""},{"Add":"module Relation.Binary.Construct.NonStrictToStrict"},{"Add":" {a ℓ₁ ℓ₂} {A : Set a} (_≈_ : Rel A ℓ₁) (_≤_ : Rel A ℓ₂) where"},{"Add":""},{"Add":"open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)"},{"Add":"open import Data.Sum.Base using (inj₁; inj₂)"},{"Add":"open import Function.Base using (_∘_; flip)"},{"Add":"open import Relation.Nullary using (¬_; yes; no)"},{"Add":"open import Relation.Nullary.Negation using (contradiction)"},{"Add":"open import Relation.Nullary.Decidable using (¬?; _×-dec_)"},{"Add":""},{"Add":"private"},{"Add":" _≉_ : Rel A ℓ₁"},{"Add":" x ≉ y = ¬ (x ≈ y)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- _≤_ can be turned into _<_ as follows:"},{"Add":""},{"Add":"infix 4 _<_"},{"Add":""},{"Add":"_<_ : Rel A _"},{"Add":"x < y = x ≤ y × x ≉ y"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relationship between relations"},{"Add":""},{"Add":"<⇒≤ : _<_ ⇒ _≤_"},{"Add":"<⇒≤ = proj₁"},{"Add":""},{"Add":"<⇒≉ : ∀ {x y} → x < y → x ≉ y"},{"Add":"<⇒≉ = proj₂"},{"Add":""},{"Add":"≤∧≉⇒< : ∀ {x y} → x ≤ y → x ≉ y → x < y"},{"Add":"≤∧≉⇒< = _,_"},{"Add":""},{"Add":"<⇒≱ : Antisymmetric _≈_ _≤_ → ∀ {x y} → x < y → ¬ (y ≤ x)"},{"Add":"<⇒≱ antisym (x≤y , x≉y) y≤x = x≉y (antisym x≤y y≤x)"},{"Add":""},{"Add":"≤⇒≯ : Antisymmetric _≈_ _≤_ → ∀ {x y} → x ≤ y → ¬ (y < x)"},{"Add":"≤⇒≯ antisym x≤y y : Symmetric _≈_ → (_≈_ ⇒ _≤_) → Total _≤_ →"},{"Add":" ∀ {x y} → ¬ (x ≤ y) → y < x"},{"Add":"≰⇒> sym refl total {x} {y} x≰y with total x y"},{"Add":"... | inj₁ x≤y = contradiction x≤y x≰y"},{"Add":"... | inj₂ y≤x = y≤x , x≰y ∘ refl ∘ sym"},{"Add":""},{"Add":"≮⇒≥ : Symmetric _≈_ → Decidable _≈_ → _≈_ ⇒ _≤_ → Total _≤_ →"},{"Add":" ∀ {x y} → ¬ (x < y) → y ≤ x"},{"Add":"≮⇒≥ sym _≟_ ≤-refl _≤?_ {x} {y} x≮y with x ≟ y | y ≤? x"},{"Add":"... | yes x≈y | _ = ≤-refl (sym x≈y)"},{"Add":"... | _ | inj₁ y≤x = y≤x"},{"Add":"... | no x≉y | inj₂ x≤y = contradiction (x≤y , x≉y) x≮y"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relational properties"},{"Add":""},{"Add":"<-irrefl : Irreflexive _≈_ _<_"},{"Add":"<-irrefl x≈y (_ , x≉y) = x≉y x≈y"},{"Add":""},{"Add":"<-trans : IsPartialOrder _≈_ _≤_ → Transitive _<_"},{"Add":"<-trans po (x≤y , x≉y) (y≤z , y≉z) ="},{"Add":" (trans x≤y y≤z , x≉y ∘ antisym x≤y ∘ trans y≤z ∘ reflexive ∘ Eq.sym)"},{"Add":" where open IsPartialOrder po"},{"Add":""},{"Add":"<-≤-trans : Symmetric _≈_ → Transitive _≤_ → Antisymmetric _≈_ _≤_ →"},{"Add":" _≤_ Respectsʳ _≈_ → Trans _<_ _≤_ _<_"},{"Add":"<-≤-trans sym trans antisym respʳ (x≤y , x≉y) y≤z ="},{"Add":" trans x≤y y≤z , (λ x≈z → x≉y (antisym x≤y (respʳ (sym x≈z) y≤z)))"},{"Add":""},{"Add":"≤-<-trans : Transitive _≤_ → Antisymmetric _≈_ _≤_ →"},{"Add":" _≤_ Respectsˡ _≈_ → Trans _≤_ _<_ _<_"},{"Add":"≤-<-trans trans antisym respʳ x≤y (y≤z , y≉z) ="},{"Add":" trans x≤y y≤z , (λ x≈z → y≉z (antisym y≤z (respʳ x≈z x≤y)))"},{"Add":""},{"Add":"<-asym : Antisymmetric _≈_ _≤_ → Asymmetric _<_"},{"Add":"<-asym antisym (x≤y , x≉y) (y≤x , _) = x≉y (antisym x≤y y≤x)"},{"Add":""},{"Add":"<-respˡ-≈ : Transitive _≈_ → _≤_ Respectsˡ _≈_ → _<_ Respectsˡ _≈_"},{"Add":"<-respˡ-≈ trans respˡ y≈z (y≤x , y≉x) ="},{"Add":" respˡ y≈z y≤x , y≉x ∘ trans y≈z"},{"Add":""},{"Add":"<-respʳ-≈ : Symmetric _≈_ → Transitive _≈_ →"},{"Add":" _≤_ Respectsʳ _≈_ → _<_ Respectsʳ _≈_"},{"Add":"<-respʳ-≈ sym trans respʳ y≈z (x≤y , x≉y) ="},{"Add":" (respʳ y≈z x≤y) , λ x≈z → x≉y (trans x≈z (sym y≈z))"},{"Add":""},{"Add":"<-resp-≈ : IsEquivalence _≈_ → _≤_ Respects₂ _≈_ → _<_ Respects₂ _≈_"},{"Add":"<-resp-≈ eq (respʳ , respˡ) ="},{"Add":" <-respʳ-≈ sym trans respʳ , <-respˡ-≈ trans respˡ"},{"Add":" where open IsEquivalence eq"},{"Add":""},{"Add":"<-trichotomous : Symmetric _≈_ → Decidable _≈_ →"},{"Add":" Antisymmetric _≈_ _≤_ → Total _≤_ →"},{"Add":" Trichotomous _≈_ _<_"},{"Add":"<-trichotomous ≈-sym _≟_ antisym total x y with x ≟ y"},{"Add":"... | yes x≈y = tri≈ (<-irrefl x≈y) x≈y (<-irrefl (≈-sym x≈y))"},{"Add":"... | no x≉y with total x y"},{"Add":"... | inj₁ x≤y = tri< (x≤y , x≉y) x≉y (x≉y ∘ antisym x≤y ∘ proj₁)"},{"Add":"... | inj₂ y≤x = tri> (x≉y ∘ flip antisym y≤x ∘ proj₁) x≉y (y≤x , x≉y ∘ ≈-sym)"},{"Add":""},{"Add":"<-decidable : Decidable _≈_ → Decidable _≤_ → Decidable _<_"},{"Add":"<-decidable _≟_ _≤?_ x y = x ≤? y ×-dec ¬? (x ≟ y)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"<-isStrictPartialOrder : IsPartialOrder _≈_ _≤_ →"},{"Add":" IsStrictPartialOrder _≈_ _<_"},{"Add":"<-isStrictPartialOrder po = record"},{"Add":" { isEquivalence = isEquivalence"},{"Add":" ; irrefl = <-irrefl"},{"Add":" ; trans = <-trans po"},{"Add":" ; <-resp-≈ = <-resp-≈ isEquivalence ≤-resp-≈"},{"Add":" } where open IsPartialOrder po"},{"Add":""},{"Add":"<-isDecStrictPartialOrder : IsDecPartialOrder _≈_ _≤_ →"},{"Add":" IsDecStrictPartialOrder _≈_ _<_"},{"Add":"<-isDecStrictPartialOrder dpo = record"},{"Add":" { isStrictPartialOrder = <-isStrictPartialOrder isPartialOrder"},{"Add":" ; _≟_ = _≟_"},{"Add":" ; _"},{"Add":""},{"Add":" max : ∀ {⊥} → Maximum L ⊥ → Maximum R ⊥ → Maximum (L ∩ R) ⊥"},{"Add":" max L-max R-max = < L-max , R-max >"},{"Add":""},{"Add":"module _ (≈ : REL A B ℓ₁) {L : REL A B ℓ₂} {R : REL A B ℓ₃} where"},{"Add":""},{"Add":" implies : (≈ ⇒ L) → (≈ ⇒ R) → ≈ ⇒ (L ∩ R)"},{"Add":" implies ≈⇒L ≈⇒R = < ≈⇒L , ≈⇒R >"},{"Add":""},{"Add":"module _ (≈ : REL A B ℓ₁) (L : REL A B ℓ₂) (R : REL A B ℓ₃) where"},{"Add":""},{"Add":" irreflexive : Irreflexive ≈ L ⊎ Irreflexive ≈ R → Irreflexive ≈ (L ∩ R)"},{"Add":" irreflexive irrefl x≈y (Lxy , Rxy) = [ (λ x → x x≈y Lxy) , (λ x → x x≈y Rxy) ] irrefl"},{"Add":""},{"Add":"module _ (≈ : Rel A ℓ₁) (L : Rel A ℓ₂) (R : Rel A ℓ₃) where"},{"Add":""},{"Add":" respectsˡ : L Respectsˡ ≈ → R Respectsˡ ≈ → (L ∩ R) Respectsˡ ≈"},{"Add":" respectsˡ L-resp R-resp x≈y = map (L-resp x≈y) (R-resp x≈y)"},{"Add":""},{"Add":" respectsʳ : L Respectsʳ ≈ → R Respectsʳ ≈ → (L ∩ R) Respectsʳ ≈"},{"Add":" respectsʳ L-resp R-resp x≈y = map (L-resp x≈y) (R-resp x≈y)"},{"Add":""},{"Add":" respects₂ : L Respects₂ ≈ → R Respects₂ ≈ → (L ∩ R) Respects₂ ≈"},{"Add":" respects₂ (Lʳ , Lˡ) (Rʳ , Rˡ) = respectsʳ Lʳ Rʳ , respectsˡ Lˡ Rˡ"},{"Add":""},{"Add":" antisymmetric : Antisymmetric ≈ L ⊎ Antisymmetric ≈ R → Antisymmetric ≈ (L ∩ R)"},{"Add":" antisymmetric (inj₁ L-antisym) (Lxy , _) (Lyx , _) = L-antisym Lxy Lyx"},{"Add":" antisymmetric (inj₂ R-antisym) (_ , Rxy) (_ , Ryx) = R-antisym Rxy Ryx"},{"Add":""},{"Add":"module _ {L : REL A B ℓ₁} {R : REL A B ℓ₂} where"},{"Add":""},{"Add":" decidable : Decidable L → Decidable R → Decidable (L ∩ R)"},{"Add":" decidable L? R? x y = L? x y ×-dec R? x y"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"isEquivalence : IsEquivalence L → IsEquivalence R → IsEquivalence (L ∩ R)"},{"Add":"isEquivalence {L = L} {R = R} eqₗ eqᵣ = record"},{"Add":" { refl = reflexive L R L.refl R.refl"},{"Add":" ; sym = symmetric L R L.sym R.sym"},{"Add":" ; trans = transitive L R L.trans R.trans"},{"Add":" } where module L = IsEquivalence eqₗ; module R = IsEquivalence eqᵣ"},{"Add":""},{"Add":"isDecEquivalence : IsDecEquivalence L → IsDecEquivalence R → IsDecEquivalence (L ∩ R)"},{"Add":"isDecEquivalence eqₗ eqᵣ = record"},{"Add":" { isEquivalence = isEquivalence L.isEquivalence R.isEquivalence"},{"Add":" ; _≟_ = decidable L._≟_ R._≟_"},{"Add":" } where module L = IsDecEquivalence eqₗ; module R = IsDecEquivalence eqᵣ"},{"Add":""},{"Add":"isPreorder : IsPreorder ≈ L → IsPreorder ≈ R → IsPreorder ≈ (L ∩ R)"},{"Add":"isPreorder {≈ = ≈} {L = L} {R = R} Oₗ Oᵣ = record"},{"Add":" { isEquivalence = Oₗ.isEquivalence"},{"Add":" ; reflexive = implies ≈ Oₗ.reflexive Oᵣ.reflexive"},{"Add":" ; trans = transitive L R Oₗ.trans Oᵣ.trans"},{"Add":" }"},{"Add":" where module Oₗ = IsPreorder Oₗ; module Oᵣ = IsPreorder Oᵣ"},{"Add":""},{"Add":"isPartialOrderˡ : IsPartialOrder ≈ L → IsPreorder ≈ R → IsPartialOrder ≈ (L ∩ R)"},{"Add":"isPartialOrderˡ {≈ = ≈} {L = L} {R = R} Oₗ Oᵣ = record"},{"Add":" { isPreorder = isPreorder Oₗ.isPreorder Oᵣ"},{"Add":" ; antisym = antisymmetric ≈ L R (inj₁ Oₗ.antisym)"},{"Add":" } where module Oₗ = IsPartialOrder Oₗ; module Oᵣ = IsPreorder Oᵣ"},{"Add":""},{"Add":"isPartialOrderʳ : IsPreorder ≈ L → IsPartialOrder ≈ R → IsPartialOrder ≈ (L ∩ R)"},{"Add":"isPartialOrderʳ {≈ = ≈} {L = L} {R = R} Oₗ Oᵣ = record"},{"Add":" { isPreorder = isPreorder Oₗ Oᵣ.isPreorder"},{"Add":" ; antisym = antisymmetric ≈ L R (inj₂ Oᵣ.antisym)"},{"Add":" } where module Oₗ = IsPreorder Oₗ; module Oᵣ = IsPartialOrder Oᵣ"},{"Add":""},{"Add":"isStrictPartialOrderˡ : IsStrictPartialOrder ≈ L →"},{"Add":" Transitive R → R Respects₂ ≈ →"},{"Add":" IsStrictPartialOrder ≈ (L ∩ R)"},{"Add":"isStrictPartialOrderˡ {≈ = ≈} {L = L} {R = R} Oₗ transᵣ respᵣ = record"},{"Add":" { isEquivalence = Oₗ.isEquivalence"},{"Add":" ; irrefl = irreflexive ≈ L R (inj₁ Oₗ.irrefl)"},{"Add":" ; trans = transitive L R Oₗ.trans transᵣ"},{"Add":" ; <-resp-≈ = respects₂ ≈ L R Oₗ.<-resp-≈ respᵣ"},{"Add":" } where module Oₗ = IsStrictPartialOrder Oₗ"},{"Add":""},{"Add":"isStrictPartialOrderʳ : Transitive L → L Respects₂ ≈ →"},{"Add":" IsStrictPartialOrder ≈ R →"},{"Add":" IsStrictPartialOrder ≈ (L ∩ R)"},{"Add":"isStrictPartialOrderʳ {L = L} {≈ = ≈} {R = R} transₗ respₗ Oᵣ = record"},{"Add":" { isEquivalence = Oᵣ.isEquivalence"},{"Add":" ; irrefl = irreflexive ≈ L R (inj₂ Oᵣ.irrefl)"},{"Add":" ; trans = transitive L R transₗ Oᵣ.trans"},{"Add":" ; <-resp-≈ = respects₂ ≈ L R respₗ Oᵣ.<-resp-≈"},{"Add":" } where module Oᵣ = IsStrictPartialOrder Oᵣ"}]}},{"FileAdd":{"meta":{"basename":"Interior","metadata":576},"context":[{"change":0,"pos":661969}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Symmetric.agda","metadata":0},"context":[{"change":0,"pos":707388}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Symmetric interior of a binary relation"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Construct.Interior.Symmetric where"},{"Add":""},{"Add":"open import Function.Base using (flip)"},{"Add":"open import Level"},{"Add":"open import Relation.Binary"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c ℓ r s t : Level"},{"Add":" A : Set a"},{"Add":" R S T : Rel A r"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"record SymInterior (R : Rel A ℓ) (x y : A) : Set ℓ where"},{"Add":" constructor _,_"},{"Add":" field"},{"Add":" lhs≤rhs : R x y"},{"Add":" rhs≤lhs : R y x"},{"Add":""},{"Add":"open SymInterior public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"-- The symmetric interior is symmetric."},{"Add":"symmetric : Symmetric (SymInterior R)"},{"Add":"symmetric (r , r′) = r′ , r"},{"Add":""},{"Add":"-- The symmetric interior of R is greater than (or equal to) any other symmetric"},{"Add":"-- relation contained by R."},{"Add":"unfold : Symmetric S → S ⇒ R → S ⇒ SymInterior R"},{"Add":"unfold sym f s = f s , f (sym s)"},{"Add":""},{"Add":"-- SymInterior preserves various properties."},{"Add":"reflexive : Reflexive R → Reflexive (SymInterior R)"},{"Add":"reflexive refl = refl , refl"},{"Add":""},{"Add":"trans : Trans R S T → Trans S R T →"},{"Add":" Trans (SymInterior R) (SymInterior S) (SymInterior T)"},{"Add":"trans trans-rs trans-sr (r , r′) (s , s′) = trans-rs r s , trans-sr s′ r′"},{"Add":""},{"Add":"transitive : Transitive R → Transitive (SymInterior R)"},{"Add":"transitive tr = trans tr tr"},{"Add":""},{"Add":"-- The symmetric interior of a strict relation is empty."},{"Add":"asymmetric⇒empty : Asymmetric R → Empty (SymInterior R)"},{"Add":"asymmetric⇒empty asym (r , r′) = asym r r′"},{"Add":""},{"Add":"-- A reflexive transitive relation _≤_ gives rise to a poset in which the"},{"Add":"-- equivalence relation is SymInterior _≤_."},{"Add":""},{"Add":"isEquivalence : Reflexive R → Transitive R → IsEquivalence (SymInterior R)"},{"Add":"isEquivalence refl trans = record"},{"Add":" { refl = reflexive refl"},{"Add":" ; sym = symmetric"},{"Add":" ; trans = transitive trans"},{"Add":" }"},{"Add":""},{"Add":"isPartialOrder : Reflexive R → Transitive R → IsPartialOrder (SymInterior R) R"},{"Add":"isPartialOrder refl trans = record"},{"Add":" { isPreorder = record"},{"Add":" { isEquivalence = isEquivalence refl trans"},{"Add":" ; reflexive = lhs≤rhs"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":" ; antisym = _,_"},{"Add":" }"},{"Add":""},{"Add":"poset : ∀ {a} {A : Set a} {R : Rel A ℓ} → Reflexive R → Transitive R → Poset a ℓ ℓ"},{"Add":"poset {R = R} refl trans = record"},{"Add":" { _≤_ = R"},{"Add":" ; isPartialOrder = isPartialOrder refl trans"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"FromRel.agda","metadata":0},"context":[{"change":0,"pos":661969}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Every respectful binary relation induces a preorder. No claim is"},{"Add":"-- made that this preorder is unique."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (REL; Rel; _⇒_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid; Preorder)"},{"Add":"open import Relation.Binary.Structures using (IsPreorder)"},{"Add":"open import Relation.Binary.Definitions using (_Respects_; Transitive)"},{"Add":"open Setoid using (Carrier)"},{"Add":""},{"Add":"module Relation.Binary.Construct.FromRel"},{"Add":" {s₁ s₂} (S : Setoid s₁ s₂) -- The underlying equality"},{"Add":" {a r} {A : Set a} (_R_ : REL A (Carrier S) r) -- The relation"},{"Add":" where"},{"Add":""},{"Add":"open import Function.Base"},{"Add":"open import Level using (_⊔_)"},{"Add":""},{"Add":"open module Eq = Setoid S using (_≈_) renaming (Carrier to B)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"Resp : Rel B (a ⊔ r)"},{"Add":"Resp x y = ∀ {a} → a R x → a R y"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"reflexive : (∀ {a} → (a R_) Respects _≈_) → _≈_ ⇒ Resp"},{"Add":"reflexive resp x≈y = resp x≈y"},{"Add":""},{"Add":"trans : Transitive Resp"},{"Add":"trans x∼y y∼z = y∼z ∘ x∼y"},{"Add":""},{"Add":"isPreorder : (∀ {a} → (a R_) Respects _≈_) → IsPreorder _≈_ Resp"},{"Add":"isPreorder resp = record"},{"Add":" { isEquivalence = Eq.isEquivalence"},{"Add":" ; reflexive = reflexive resp"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":""},{"Add":"preorder : (∀ {a} → (a R_) Respects _≈_) → Preorder _ _ _"},{"Add":"preorder resp = record"},{"Add":" { isPreorder = isPreorder resp"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"FromPred.agda","metadata":0},"context":[{"change":0,"pos":661969}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Every respectful unary relation induces a preorder. No claim is"},{"Add":"-- made that this preorder is unique."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (Rel; _⇒_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid; Preorder)"},{"Add":"open import Relation.Binary.Structures using (IsPreorder)"},{"Add":"open import Relation.Binary.Definitions using (_Respects_; Reflexive; Transitive)"},{"Add":"open import Relation.Unary using (Pred)"},{"Add":""},{"Add":"module Relation.Binary.Construct.FromPred"},{"Add":" {s₁ s₂} (S : Setoid s₁ s₂) -- The underlying equality"},{"Add":" {p} (P : Pred (Setoid.Carrier S) p) -- The predicate"},{"Add":" where"},{"Add":""},{"Add":"open import Function.Base"},{"Add":""},{"Add":"open module Eq = Setoid S using (_≈_) renaming (Carrier to A)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"Resp : Rel A p"},{"Add":"Resp x y = P x → P y"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"reflexive : P Respects _≈_ → _≈_ ⇒ Resp"},{"Add":"reflexive resp = resp"},{"Add":""},{"Add":"refl : P Respects _≈_ → Reflexive Resp"},{"Add":"refl resp = resp Eq.refl"},{"Add":""},{"Add":"trans : Transitive Resp"},{"Add":"trans x⇒y y⇒z = y⇒z ∘ x⇒y"},{"Add":""},{"Add":"isPreorder : P Respects _≈_ → IsPreorder _≈_ Resp"},{"Add":"isPreorder resp = record"},{"Add":" { isEquivalence = Eq.isEquivalence"},{"Add":" ; reflexive = reflexive resp"},{"Add":" ; trans = flip _∘′_"},{"Add":" }"},{"Add":""},{"Add":"preorder : P Respects _≈_ → Preorder _ _ _"},{"Add":"preorder resp = record"},{"Add":" { isPreorder = isPreorder resp"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Flip.agda","metadata":0},"context":[{"change":0,"pos":661969}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use"},{"Add":"-- `Relation.Binary.Construct.Flip.Ord` instead."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Construct.Flip where"},{"Add":""},{"Add":"open import Relation.Binary.Construct.Flip.Ord public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Relation.Binary.Construct.Flip was deprecated in v2.0."},{"Add":"Use Relation.Binary.Construct.Flip.Ord instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Flip","metadata":576},"context":[{"change":0,"pos":661969}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Ord.agda","metadata":0},"context":[{"change":0,"pos":713859}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Many properties which hold for `∼` also hold for `flip ∼`. Unlike"},{"Add":"-- the module `Relation.Binary.Construct.Flip.EqAndOrd` this module"},{"Add":"-- flips both the relation and the underlying equality."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (Rel; REL; _⇒_)"},{"Add":"open import Relation.Binary.Bundles"},{"Add":" using (Setoid; DecSetoid; Preorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)"},{"Add":"open import Relation.Binary.Structures"},{"Add":" using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (Reflexive; Symmetric; Transitive; Asymmetric; Total; _Respects_; _Respects₂_; Minimum; Maximum; Irreflexive; Antisymmetric; Trichotomous; Decidable)"},{"Add":""},{"Add":"module Relation.Binary.Construct.Flip.Ord where"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base using (flip; _∘_)"},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b p ℓ ℓ₁ ℓ₂ : Level"},{"Add":" A B : Set a"},{"Add":" ≈ ∼ ≤ < : Rel A ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"module _ (∼ : Rel A ℓ) where"},{"Add":""},{"Add":" reflexive : Reflexive ∼ → Reflexive (flip ∼)"},{"Add":" reflexive refl = refl"},{"Add":""},{"Add":" symmetric : Symmetric ∼ → Symmetric (flip ∼)"},{"Add":" symmetric sym = sym"},{"Add":""},{"Add":" transitive : Transitive ∼ → Transitive (flip ∼)"},{"Add":" transitive trans = flip trans"},{"Add":""},{"Add":" asymmetric : Asymmetric ∼ → Asymmetric (flip ∼)"},{"Add":" asymmetric asym = asym"},{"Add":""},{"Add":" total : Total ∼ → Total (flip ∼)"},{"Add":" total total x y = total y x"},{"Add":""},{"Add":" respects : ∀ (P : A → Set p) → Symmetric ∼ →"},{"Add":" P Respects ∼ → P Respects flip ∼"},{"Add":" respects _ sym resp ∼ = resp (sym ∼)"},{"Add":""},{"Add":" max : ∀ {⊥} → Minimum ∼ ⊥ → Maximum (flip ∼) ⊥"},{"Add":" max min = min"},{"Add":""},{"Add":" min : ∀ {⊤} → Maximum ∼ ⊤ → Minimum (flip ∼) ⊤"},{"Add":" min max = max"},{"Add":""},{"Add":"module _ (≈ : REL A B ℓ₁) (∼ : REL A B ℓ₂) where"},{"Add":""},{"Add":" implies : ≈ ⇒ ∼ → flip ≈ ⇒ flip ∼"},{"Add":" implies impl = impl"},{"Add":""},{"Add":" irreflexive : Irreflexive ≈ ∼ → Irreflexive (flip ≈) (flip ∼)"},{"Add":" irreflexive irrefl = irrefl"},{"Add":""},{"Add":"module _ (≈ : Rel A ℓ₁) (∼ : Rel A ℓ₂) where"},{"Add":""},{"Add":" antisymmetric : Antisymmetric ≈ ∼ → Antisymmetric (flip ≈) (flip ∼)"},{"Add":" antisymmetric antisym = antisym"},{"Add":""},{"Add":" trichotomous : Trichotomous ≈ ∼ → Trichotomous (flip ≈) (flip ∼)"},{"Add":" trichotomous compare x y = compare y x"},{"Add":""},{"Add":"module _ (∼₁ : Rel A ℓ₁) (∼₂ : Rel A ℓ₂) where"},{"Add":""},{"Add":" respects₂ : Symmetric ∼₂ → ∼₁ Respects₂ ∼₂ → flip ∼₁ Respects₂ flip ∼₂"},{"Add":" respects₂ sym (resp₁ , resp₂) = (resp₂ ∘ sym , resp₁ ∘ sym)"},{"Add":""},{"Add":"module _ (∼ : REL A B ℓ) where"},{"Add":""},{"Add":" decidable : Decidable ∼ → Decidable (flip ∼)"},{"Add":" decidable dec x y = dec y x"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"isEquivalence : IsEquivalence ≈ → IsEquivalence (flip ≈)"},{"Add":"isEquivalence {≈ = ≈} eq = record"},{"Add":" { refl = reflexive ≈ Eq.refl"},{"Add":" ; sym = symmetric ≈ Eq.sym"},{"Add":" ; trans = transitive ≈ Eq.trans"},{"Add":" } where module Eq = IsEquivalence eq"},{"Add":""},{"Add":"isDecEquivalence : IsDecEquivalence ≈ → IsDecEquivalence (flip ≈)"},{"Add":"isDecEquivalence {≈ = ≈} dec = record"},{"Add":" { isEquivalence = isEquivalence Dec.isEquivalence"},{"Add":" ; _≟_ = decidable ≈ Dec._≟_"},{"Add":" } where module Dec = IsDecEquivalence dec"},{"Add":""},{"Add":"isPreorder : IsPreorder ≈ ∼ → IsPreorder (flip ≈) (flip ∼)"},{"Add":"isPreorder {≈ = ≈} {∼ = ∼} O = record"},{"Add":" { isEquivalence = isEquivalence O.isEquivalence"},{"Add":" ; reflexive = implies ≈ ∼ O.reflexive"},{"Add":" ; trans = transitive ∼ O.trans"},{"Add":" } where module O = IsPreorder O"},{"Add":""},{"Add":"isPartialOrder : IsPartialOrder ≈ ≤ → IsPartialOrder (flip ≈) (flip ≤)"},{"Add":"isPartialOrder {≈ = ≈} {≤ = ≤} O = record"},{"Add":" { isPreorder = isPreorder O.isPreorder"},{"Add":" ; antisym = antisymmetric ≈ ≤ O.antisym"},{"Add":" } where module O = IsPartialOrder O"},{"Add":""},{"Add":"isTotalOrder : IsTotalOrder ≈ ≤ → IsTotalOrder (flip ≈) (flip ≤)"},{"Add":"isTotalOrder {≈ = ≈} {≤ = ≤} O = record"},{"Add":" { isPartialOrder = isPartialOrder O.isPartialOrder"},{"Add":" ; total = total ≤ O.total"},{"Add":" }"},{"Add":" where module O = IsTotalOrder O"},{"Add":""},{"Add":"isDecTotalOrder : IsDecTotalOrder ≈ ≤ → IsDecTotalOrder (flip ≈) (flip ≤)"},{"Add":"isDecTotalOrder {≈ = ≈} {≤ = ≤} O = record"},{"Add":" { isTotalOrder = isTotalOrder O.isTotalOrder"},{"Add":" ; _≟_ = decidable ≈ O._≟_"},{"Add":" ; _≤?_ = decidable ≤ O._≤?_"},{"Add":" } where module O = IsDecTotalOrder O"},{"Add":""},{"Add":"isStrictPartialOrder : IsStrictPartialOrder ≈ < →"},{"Add":" IsStrictPartialOrder (flip ≈) (flip <)"},{"Add":"isStrictPartialOrder {≈ = ≈} {< = <} O = record"},{"Add":" { isEquivalence = isEquivalence O.isEquivalence"},{"Add":" ; irrefl = irreflexive ≈ < O.irrefl"},{"Add":" ; trans = transitive < O.trans"},{"Add":" ; <-resp-≈ = respects₂ < ≈ O.Eq.sym O.<-resp-≈"},{"Add":" } where module O = IsStrictPartialOrder O"},{"Add":""},{"Add":"isStrictTotalOrder : IsStrictTotalOrder ≈ < →"},{"Add":" IsStrictTotalOrder (flip ≈) (flip <)"},{"Add":"isStrictTotalOrder {≈ = ≈} {< = <} O = record"},{"Add":" { isStrictPartialOrder = isStrictPartialOrder O.isStrictPartialOrder"},{"Add":" ; compare = trichotomous ≈ < O.compare"},{"Add":" } where module O = IsStrictTotalOrder O"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles"},{"Add":""},{"Add":"setoid : Setoid a ℓ → Setoid a ℓ"},{"Add":"setoid S = record"},{"Add":" { _≈_ = flip S._≈_"},{"Add":" ; isEquivalence = isEquivalence S.isEquivalence"},{"Add":" } where module S = Setoid S"},{"Add":""},{"Add":"decSetoid : DecSetoid a ℓ → DecSetoid a ℓ"},{"Add":"decSetoid S = record"},{"Add":" { _≈_ = flip S._≈_"},{"Add":" ; isDecEquivalence = isDecEquivalence S.isDecEquivalence"},{"Add":" } where module S = DecSetoid S"},{"Add":""},{"Add":"preorder : Preorder a ℓ₁ ℓ₂ → Preorder a ℓ₁ ℓ₂"},{"Add":"preorder O = record"},{"Add":" { isPreorder = isPreorder O.isPreorder"},{"Add":" } where module O = Preorder O"},{"Add":""},{"Add":"poset : Poset a ℓ₁ ℓ₂ → Poset a ℓ₁ ℓ₂"},{"Add":"poset O = record"},{"Add":" { isPartialOrder = isPartialOrder O.isPartialOrder"},{"Add":" } where module O = Poset O"},{"Add":""},{"Add":"totalOrder : TotalOrder a ℓ₁ ℓ₂ → TotalOrder a ℓ₁ ℓ₂"},{"Add":"totalOrder O = record"},{"Add":" { isTotalOrder = isTotalOrder O.isTotalOrder"},{"Add":" } where module O = TotalOrder O"},{"Add":""},{"Add":"decTotalOrder : DecTotalOrder a ℓ₁ ℓ₂ → DecTotalOrder a ℓ₁ ℓ₂"},{"Add":"decTotalOrder O = record"},{"Add":" { isDecTotalOrder = isDecTotalOrder O.isDecTotalOrder"},{"Add":" } where module O = DecTotalOrder O"},{"Add":""},{"Add":"strictPartialOrder : StrictPartialOrder a ℓ₁ ℓ₂ →"},{"Add":" StrictPartialOrder a ℓ₁ ℓ₂"},{"Add":"strictPartialOrder O = record"},{"Add":" { isStrictPartialOrder = isStrictPartialOrder O.isStrictPartialOrder"},{"Add":" } where module O = StrictPartialOrder O"},{"Add":""},{"Add":"strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂ →"},{"Add":" StrictTotalOrder a ℓ₁ ℓ₂"},{"Add":"strictTotalOrder O = record"},{"Add":" { isStrictTotalOrder = isStrictTotalOrder O.isStrictTotalOrder"},{"Add":" } where module O = StrictTotalOrder O"}]}},{"FileAdd":{"meta":{"basename":"EqAndOrd.agda","metadata":0},"context":[{"change":0,"pos":713859}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Many properties which hold for `∼` also hold for `flip ∼`. Unlike"},{"Add":"-- the module `Relation.Binary.Construct.Flip.Ord` this module does not"},{"Add":"-- flip the underlying equality."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (Rel; REL; _⇒_)"},{"Add":"open import Relation.Binary.Bundles"},{"Add":" using (Setoid; DecSetoid; Preorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder; TotalPreorder)"},{"Add":"open import Relation.Binary.Structures"},{"Add":" using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder; IsTotalPreorder)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (Reflexive; Symmetric; Transitive; Asymmetric; Total; _Respects_; _Respects₂_; Minimum; Maximum; Irreflexive; Antisymmetric; Trichotomous; Decidable; tri<; tri>; tri≈)"},{"Add":""},{"Add":"module Relation.Binary.Construct.Flip.EqAndOrd where"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base using (flip; _∘_)"},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b p ℓ ℓ₁ ℓ₂ : Level"},{"Add":" A B : Set a"},{"Add":" ≈ ∼ ≤ < : Rel A ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"module _ (∼ : Rel A ℓ) where"},{"Add":""},{"Add":" refl : Reflexive ∼ → Reflexive (flip ∼)"},{"Add":" refl refl = refl"},{"Add":""},{"Add":" sym : Symmetric ∼ → Symmetric (flip ∼)"},{"Add":" sym sym = sym"},{"Add":""},{"Add":" trans : Transitive ∼ → Transitive (flip ∼)"},{"Add":" trans trans = flip trans"},{"Add":""},{"Add":" asym : Asymmetric ∼ → Asymmetric (flip ∼)"},{"Add":" asym asym = asym"},{"Add":""},{"Add":" total : Total ∼ → Total (flip ∼)"},{"Add":" total total x y = total y x"},{"Add":""},{"Add":" resp : ∀ {p} (P : A → Set p) → Symmetric ∼ →"},{"Add":" P Respects ∼ → P Respects (flip ∼)"},{"Add":" resp _ sym resp ∼ = resp (sym ∼)"},{"Add":""},{"Add":" max : ∀ {⊥} → Minimum ∼ ⊥ → Maximum (flip ∼) ⊥"},{"Add":" max min = min"},{"Add":""},{"Add":" min : ∀ {⊤} → Maximum ∼ ⊤ → Minimum (flip ∼) ⊤"},{"Add":" min max = max"},{"Add":""},{"Add":"module _ {≈ : Rel A ℓ₁} (∼ : Rel A ℓ₂) where"},{"Add":""},{"Add":" reflexive : Symmetric ≈ → (≈ ⇒ ∼) → (≈ ⇒ flip ∼)"},{"Add":" reflexive sym impl = impl ∘ sym"},{"Add":""},{"Add":" irrefl : Symmetric ≈ → Irreflexive ≈ ∼ → Irreflexive ≈ (flip ∼)"},{"Add":" irrefl sym irrefl x≈y y∼x = irrefl (sym x≈y) y∼x"},{"Add":""},{"Add":" antisym : Antisymmetric ≈ ∼ → Antisymmetric ≈ (flip ∼)"},{"Add":" antisym antisym = flip antisym"},{"Add":""},{"Add":" compare : Trichotomous ≈ ∼ → Trichotomous ≈ (flip ∼)"},{"Add":" compare cmp x y with cmp x y"},{"Add":" ... | tri< x y≮x x≉y x x≮y x≉y y>=_"},{"Add":""},{"Add":"_>>=_ : ∀ {i t u} {I : Set i} {T : Rel I t} {U : Rel I u} {j k} →"},{"Add":" Star T j k → T ⇒ Star U → Star U j k"},{"Add":"m >>= f = (f ⋆) m"},{"Add":""},{"Add":"-- Note that the monad-like structure above is not an indexed monad"},{"Add":"-- (as defined in Effect.Monad.Indexed). If it were, then _>>=_"},{"Add":"-- would have a type similar to"},{"Add":"--"},{"Add":"-- ∀ {I} {T U : Rel I t} {i j k} →"},{"Add":"-- Star T i j → (T i j → Star U j k) → Star U i k."},{"Add":"-- ^^^^^"},{"Add":"-- Note, however, that there is no scope for applying T to any indices"},{"Add":"-- in the definition used in Effect.Monad.Indexed."}]}},{"FileAdd":{"meta":{"basename":"ReflexiveTransitive","metadata":576},"context":[{"change":0,"pos":734381}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Properties.agda","metadata":0},"context":[{"change":0,"pos":752002}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Some properties of reflexive transitive closures."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Construct.Closure.ReflexiveTransitive.Properties where"},{"Add":""},{"Add":"open import Function.Base using (id; _∘_; _$_)"},{"Add":"open import Relation.Binary.Core using (Rel; _=[_]⇒_; _⇒_)"},{"Add":"open import Relation.Binary.Bundles using (Preorder)"},{"Add":"open import Relation.Binary.Structures using (IsPreorder)"},{"Add":"open import Relation.Binary.Definitions using (Transitive; Reflexive)"},{"Add":"open import Relation.Binary.Construct.Closure.ReflexiveTransitive"},{"Add":"open import Relation.Binary.PropositionalEquality.Core as ≡"},{"Add":" using (_≡_; refl; sym; cong; cong₂)"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":"import Relation.Binary.Reasoning.Preorder as ≲-Reasoning"},{"Add":"open import Relation.Binary.Reasoning.Syntax"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- _◅◅_"},{"Add":""},{"Add":"module _ {i t} {I : Set i} {T : Rel I t} where"},{"Add":""},{"Add":" ◅◅-assoc : ∀ {i j k l}"},{"Add":" (xs : Star T i j) (ys : Star T j k) (zs : Star T k l) →"},{"Add":" (xs ◅◅ ys) ◅◅ zs ≡ xs ◅◅ (ys ◅◅ zs)"},{"Add":" ◅◅-assoc ε ys zs = refl"},{"Add":" ◅◅-assoc (x ◅ xs) ys zs = cong (_◅_ x) (◅◅-assoc xs ys zs)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- gmap"},{"Add":""},{"Add":"gmap-id : ∀ {i t} {I : Set i} {T : Rel I t} {i j} (xs : Star T i j) →"},{"Add":" gmap id id xs ≡ xs"},{"Add":"gmap-id ε = refl"},{"Add":"gmap-id (x ◅ xs) = cong (_◅_ x) (gmap-id xs)"},{"Add":""},{"Add":"gmap-∘ : ∀ {i t} {I : Set i} {T : Rel I t}"},{"Add":" {j u} {J : Set j} {U : Rel J u}"},{"Add":" {k v} {K : Set k} {V : Rel K v}"},{"Add":" (f : J → K) (g : U =[ f ]⇒ V)"},{"Add":" (f′ : I → J) (g′ : T =[ f′ ]⇒ U)"},{"Add":" {i j} (xs : Star T i j) →"},{"Add":" (gmap {U = V} f g ∘ gmap f′ g′) xs ≡ gmap (f ∘ f′) (g ∘ g′) xs"},{"Add":"gmap-∘ f g f′ g′ ε = refl"},{"Add":"gmap-∘ f g f′ g′ (x ◅ xs) = cong (_◅_ (g (g′ x))) (gmap-∘ f g f′ g′ xs)"},{"Add":""},{"Add":"gmap-◅◅ : ∀ {i t j u}"},{"Add":" {I : Set i} {T : Rel I t} {J : Set j} {U : Rel J u}"},{"Add":" (f : I → J) (g : T =[ f ]⇒ U)"},{"Add":" {i j k} (xs : Star T i j) (ys : Star T j k) →"},{"Add":" gmap {U = U} f g (xs ◅◅ ys) ≡ gmap f g xs ◅◅ gmap f g ys"},{"Add":"gmap-◅◅ f g ε ys = refl"},{"Add":"gmap-◅◅ f g (x ◅ xs) ys = cong (_◅_ (g x)) (gmap-◅◅ f g xs ys)"},{"Add":""},{"Add":"gmap-cong : ∀ {i t j u}"},{"Add":" {I : Set i} {T : Rel I t} {J : Set j} {U : Rel J u}"},{"Add":" (f : I → J) (g : T =[ f ]⇒ U) (g′ : T =[ f ]⇒ U) →"},{"Add":" (∀ {i j} (x : T i j) → g x ≡ g′ x) →"},{"Add":" ∀ {i j} (xs : Star T i j) →"},{"Add":" gmap {U = U} f g xs ≡ gmap f g′ xs"},{"Add":"gmap-cong f g g′ eq ε = refl"},{"Add":"gmap-cong f g g′ eq (x ◅ xs) = cong₂ _◅_ (eq x) (gmap-cong f g g′ eq xs)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- fold"},{"Add":""},{"Add":"fold-◅◅ : ∀ {i p} {I : Set i}"},{"Add":" (P : Rel I p) (_⊕_ : Transitive P) (∅ : Reflexive P) →"},{"Add":" (∀ {i j} (x : P i j) → (∅ ⊕ x) ≡ x) →"},{"Add":" (∀ {i j k l} (x : P i j) (y : P j k) (z : P k l) →"},{"Add":" ((x ⊕ y) ⊕ z) ≡ (x ⊕ (y ⊕ z))) →"},{"Add":" ∀ {i j k} (xs : Star P i j) (ys : Star P j k) →"},{"Add":" fold P _⊕_ ∅ (xs ◅◅ ys) ≡ (fold P _⊕_ ∅ xs ⊕ fold P _⊕_ ∅ ys)"},{"Add":"fold-◅◅ P _⊕_ ∅ left-unit assoc ε ys = sym (left-unit _)"},{"Add":"fold-◅◅ P _⊕_ ∅ left-unit assoc (x ◅ xs) ys = begin"},{"Add":" (x ⊕ fold P _⊕_ ∅ (xs ◅◅ ys)) ≡⟨ cong (_⊕_ x) $"},{"Add":" fold-◅◅ P _⊕_ ∅ left-unit assoc xs ys ⟩"},{"Add":" (x ⊕ (fold P _⊕_ ∅ xs ⊕ fold P _⊕_ ∅ ys)) ≡⟨ sym (assoc x _ _) ⟩"},{"Add":" ((x ⊕ fold P _⊕_ ∅ xs) ⊕ fold P _⊕_ ∅ ys) ∎"},{"Add":" where open ≡.≡-Reasoning"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relational properties"},{"Add":""},{"Add":"module _ {i t} {I : Set i} (T : Rel I t) where"},{"Add":""},{"Add":" reflexive : _≡_ ⇒ Star T"},{"Add":" reflexive refl = ε"},{"Add":""},{"Add":" trans : Transitive (Star T)"},{"Add":" trans = _◅◅_"},{"Add":""},{"Add":" isPreorder : IsPreorder _≡_ (Star T)"},{"Add":" isPreorder = record"},{"Add":" { isEquivalence = ≡.isEquivalence"},{"Add":" ; reflexive = reflexive"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":""},{"Add":" preorder : Preorder _ _ _"},{"Add":" preorder = record"},{"Add":" { _≈_ = _≡_"},{"Add":" ; _≲_ = Star T"},{"Add":" ; isPreorder = isPreorder"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Preorder reasoning for Star"},{"Add":""},{"Add":"module StarReasoning {i t} {I : Set i} (T : Rel I t) where"},{"Add":" private module Base = ≲-Reasoning (preorder T)"},{"Add":""},{"Add":" open Base public"},{"Add":" hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨; step-∼; step-≲)"},{"Add":" renaming (≲-go to ⟶-go)"},{"Add":""},{"Add":" open ⟶-syntax _IsRelatedTo_ _IsRelatedTo_ (⟶-go ∘ (_◅ ε)) public"},{"Add":" open ⟶*-syntax _IsRelatedTo_ _IsRelatedTo_ ⟶-go public"}]}},{"FileAdd":{"meta":{"basename":"Properties","metadata":576},"context":[{"change":0,"pos":752002}],"contents":[]}},{"FileAdd":{"meta":{"basename":"WithK.agda","metadata":0},"context":[{"change":0,"pos":757125}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties, related to reflexive transitive closures, that rely on"},{"Add":"-- the K rule"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --with-K --safe #-}"},{"Add":""},{"Add":"module"},{"Add":" Relation.Binary.Construct.Closure.ReflexiveTransitive.Properties.WithK"},{"Add":" where"},{"Add":""},{"Add":"open import Function.Base using (_∋_)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Construct.Closure.ReflexiveTransitive using (Star; _◅_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Equality"},{"Add":""},{"Add":"module _ {i t} {I : Set i} {T : Rel I t} {i j k} {x y : T i j} {xs ys}"},{"Add":" where"},{"Add":""},{"Add":" ◅-injectiveˡ : (Star T i k ∋ x ◅ xs) ≡ y ◅ ys → x ≡ y"},{"Add":" ◅-injectiveˡ refl = refl"},{"Add":""},{"Add":" ◅-injectiveʳ : (Star T i k ∋ x ◅ xs) ≡ y ◅ ys → xs ≡ ys"},{"Add":" ◅-injectiveʳ refl = refl"}]}},{"FileAdd":{"meta":{"basename":"Reflexive.agda","metadata":0},"context":[{"change":0,"pos":734381}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Reflexive closures"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Relation.Binary.Construct.Closure.Reflexive where"},{"Add":""},{"Add":"open import Data.Unit.Base"},{"Add":"open import Level"},{"Add":"open import Function.Base using (_∋_)"},{"Add":"open import Relation.Binary.Core using (Rel; _=[_]⇒_; _⇒_)"},{"Add":"open import Relation.Binary.Definitions using (Reflexive)"},{"Add":"open import Relation.Binary.Construct.Constant.Core using (Const)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_ ; refl)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a ℓ ℓ₁ ℓ₂ : Level"},{"Add":" A B : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"data ReflClosure {A : Set a} (_∼_ : Rel A ℓ) : Rel A (a ⊔ ℓ) where"},{"Add":" refl : Reflexive (ReflClosure _∼_)"},{"Add":" [_] : ∀ {x y} (x∼y : x ∼ y) → ReflClosure _∼_ x y"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Operations"},{"Add":""},{"Add":"map : ∀ {R : Rel A ℓ₁} {S : Rel B ℓ₂} {f : A → B} →"},{"Add":" R =[ f ]⇒ S → ReflClosure R =[ f ]⇒ ReflClosure S"},{"Add":"map R⇒S [ xRy ] = [ R⇒S xRy ]"},{"Add":"map R⇒S refl = refl"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"-- The reflexive closure has no effect on reflexive relations."},{"Add":"drop-refl : {R : Rel A ℓ} → Reflexive R → ReflClosure R ⇒ R"},{"Add":"drop-refl rfl [ xRy ] = xRy"},{"Add":"drop-refl rfl refl = rfl"},{"Add":""},{"Add":"reflexive : {R : Rel A ℓ} → _≡_ ⇒ ReflClosure R"},{"Add":"reflexive refl = refl"},{"Add":""},{"Add":"[]-injective : {R : Rel A ℓ} → ∀ {x y p q} →"},{"Add":" (ReflClosure R x y ∋ [ p ]) ≡ [ q ] → p ≡ q"},{"Add":"[]-injective refl = refl"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Example usage"},{"Add":""},{"Add":"private"},{"Add":" module Maybe where"},{"Add":" Maybe : Set a → Set a"},{"Add":" Maybe A = ReflClosure (Const A) tt tt"},{"Add":""},{"Add":" nothing : Maybe A"},{"Add":" nothing = refl"},{"Add":""},{"Add":" just : A → Maybe A"},{"Add":" just = [_]"},{"Add":""},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Deprecations"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Please use the new names as continuing support for the old names is"},{"Add":"-- not guaranteed."},{"Add":""},{"Add":"-- v1.5"},{"Add":""},{"Add":"Refl = ReflClosure"},{"Add":"{-# WARNING_ON_USAGE Refl"},{"Add":"\"Warning: Refl was deprecated in v1.5."},{"Add":"Please use ReflClosure instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Reflexive","metadata":576},"context":[{"change":0,"pos":734381}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Properties.agda","metadata":0},"context":[{"change":0,"pos":760655}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Some properties of reflexive closures"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --safe --cubical-compatible #-}"},{"Add":""},{"Add":"module Relation.Binary.Construct.Closure.Reflexive.Properties where"},{"Add":""},{"Add":"open import Data.Product.Base as Product"},{"Add":"open import Data.Sum.Base as Sum"},{"Add":"open import Function.Bundles using (_⇔_; mk⇔)"},{"Add":"open import Function.Base using (id)"},{"Add":"open import Level"},{"Add":"open import Relation.Binary.Core using (Rel; REL; _=[_]⇒_)"},{"Add":"open import Relation.Binary.Structures"},{"Add":" using (IsPreorder; IsStrictPartialOrder; IsPartialOrder; IsDecStrictPartialOrder; IsDecPartialOrder; IsStrictTotalOrder; IsTotalOrder; IsDecTotalOrder)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (Symmetric; Transitive; Reflexive; Asymmetric; Antisymmetric; Trichotomous; Total; Decidable; DecidableEquality; tri<; tri≈; tri>; _Respectsˡ_; _Respectsʳ_; _Respects_; _Respects₂_)"},{"Add":"open import Relation.Binary.Construct.Closure.Reflexive"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":"open import Relation.Nullary.Negation.Core using (contradiction)"},{"Add":"open import Relation.Nullary.Decidable as Dec using (yes; no)"},{"Add":"open import Relation.Unary using (Pred)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b ℓ p q : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relational properties"},{"Add":""},{"Add":"module _ {P : Rel A p} {Q : Rel B q} where"},{"Add":""},{"Add":" =[]⇒ : ∀ {f : A → B} → P =[ f ]⇒ Q → ReflClosure P =[ f ]⇒ ReflClosure Q"},{"Add":" =[]⇒ x [ x∼y ] = [ x x∼y ]"},{"Add":" =[]⇒ x refl = refl"},{"Add":""},{"Add":"module _ {_~_ : Rel A ℓ} where"},{"Add":""},{"Add":" private"},{"Add":" _~ᵒ_ = ReflClosure _~_"},{"Add":""},{"Add":" fromSum : ∀ {x y} → x ≡ y ⊎ x ~ y → x ~ᵒ y"},{"Add":" fromSum (inj₁ refl) = refl"},{"Add":" fromSum (inj₂ y) = [ y ]"},{"Add":""},{"Add":" toSum : ∀ {x y} → x ~ᵒ y → x ≡ y ⊎ x ~ y"},{"Add":" toSum [ x∼y ] = inj₂ x∼y"},{"Add":" toSum refl = inj₁ refl"},{"Add":""},{"Add":" ⊎⇔Refl : ∀ {a b} → (a ≡ b ⊎ a ~ b) ⇔ a ~ᵒ b"},{"Add":" ⊎⇔Refl = mk⇔ fromSum toSum"},{"Add":""},{"Add":" sym : Symmetric _~_ → Symmetric _~ᵒ_"},{"Add":" sym ~-sym [ x∼y ] = [ ~-sym x∼y ]"},{"Add":" sym ~-sym refl = refl"},{"Add":""},{"Add":" trans : Transitive _~_ → Transitive _~ᵒ_"},{"Add":" trans ~-trans [ x∼y ] [ x∼y₁ ] = [ ~-trans x∼y x∼y₁ ]"},{"Add":" trans ~-trans [ x∼y ] refl = [ x∼y ]"},{"Add":" trans ~-trans refl [ x∼y ] = [ x∼y ]"},{"Add":" trans ~-trans refl refl = refl"},{"Add":""},{"Add":" antisym : (_≈_ : Rel A p) → Reflexive _≈_ →"},{"Add":" Asymmetric _~_ → Antisymmetric _≈_ _~ᵒ_"},{"Add":" antisym _≈_ ref asym [ x∼y ] [ y∼x ] = contradiction x∼y (asym y∼x)"},{"Add":" antisym _≈_ ref asym [ x∼y ] refl = ref"},{"Add":" antisym _≈_ ref asym refl _ = ref"},{"Add":""},{"Add":" total : Trichotomous _≡_ _~_ → Total _~ᵒ_"},{"Add":" total compare x y with compare x y"},{"Add":" ... | tri< a _ _ = inj₁ [ a ]"},{"Add":" ... | tri≈ _ refl _ = inj₁ refl"},{"Add":" ... | tri> _ _ c = inj₂ [ c ]"},{"Add":""},{"Add":" dec : DecidableEquality A → Decidable _~_ → Decidable _~ᵒ_"},{"Add":" dec ≡-dec ~-dec a b = Dec.map ⊎⇔Refl (≡-dec a b Dec.⊎-dec ~-dec a b)"},{"Add":""},{"Add":" decidable : Trichotomous _≡_ _~_ → Decidable _~ᵒ_"},{"Add":" decidable ~-tri a b with ~-tri a b"},{"Add":" ... | tri< a~b _ _ = yes [ a~b ]"},{"Add":" ... | tri≈ _ refl _ = yes refl"},{"Add":" ... | tri> ¬a ¬b _ = no λ { refl → ¬b refl ; [ p ] → ¬a p }"},{"Add":""},{"Add":" respˡ : ∀ {P : REL A B p} → P Respectsˡ _~_ → P Respectsˡ _~ᵒ_"},{"Add":" respˡ p-respˡ-~ [ x∼y ] = p-respˡ-~ x∼y"},{"Add":" respˡ _ refl = id"},{"Add":""},{"Add":" respʳ : ∀ {P : REL B A p} → P Respectsʳ _~_ → P Respectsʳ _~ᵒ_"},{"Add":" respʳ = respˡ"},{"Add":""},{"Add":"module _ {_~_ : Rel A ℓ} {P : Pred A p} where"},{"Add":""},{"Add":" resp : P Respects _~_ → P Respects (ReflClosure _~_)"},{"Add":" resp p-resp-~ [ x∼y ] = p-resp-~ x∼y"},{"Add":" resp _ refl = id"},{"Add":""},{"Add":"module _ {_~_ : Rel A ℓ} {P : Rel A p} where"},{"Add":""},{"Add":" resp₂ : P Respects₂ _~_ → P Respects₂ (ReflClosure _~_)"},{"Add":" resp₂ = Product.map respˡ respʳ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"module _ {_~_ : Rel A ℓ} where"},{"Add":""},{"Add":" private"},{"Add":" _~ᵒ_ = ReflClosure _~_"},{"Add":""},{"Add":" isPreorder : Transitive _~_ → IsPreorder _≡_ _~ᵒ_"},{"Add":" isPreorder ~-trans = record"},{"Add":" { isEquivalence = ≡.isEquivalence"},{"Add":" ; reflexive = λ { refl → refl }"},{"Add":" ; trans = trans ~-trans"},{"Add":" }"},{"Add":""},{"Add":" isPartialOrder : IsStrictPartialOrder _≡_ _~_ → IsPartialOrder _≡_ _~ᵒ_"},{"Add":" isPartialOrder O = record"},{"Add":" { isPreorder = isPreorder O.trans"},{"Add":" ; antisym = antisym _≡_ refl O.asym"},{"Add":" } where module O = IsStrictPartialOrder O"},{"Add":""},{"Add":" isDecPartialOrder : IsDecStrictPartialOrder _≡_ _~_ → IsDecPartialOrder _≡_ _~ᵒ_"},{"Add":" isDecPartialOrder O = record"},{"Add":" { isPartialOrder = isPartialOrder O.isStrictPartialOrder"},{"Add":" ; _≟_ = O._≟_"},{"Add":" ; _≤?_ = dec O._≟_ O._; tri<; _Respectsˡ_; _Respectsʳ_; _Respects₂_)"},{"Add":""},{"Add":"module Relation.Binary.Construct.Add.Supremum.Strict"},{"Add":" {a r} {A : Set a} (_<_ : Rel A r) where"},{"Add":""},{"Add":"open import Level using (_⊔_)"},{"Add":"open import Data.Product.Base using (_,_; map)"},{"Add":"open import Function.Base"},{"Add":"open import Relation.Nullary hiding (Irrelevant)"},{"Add":"import Relation.Nullary.Decidable as Dec"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; subst)"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":"open import Relation.Nullary.Construct.Add.Supremum"},{"Add":"import Relation.Binary.Construct.Add.Supremum.Equality as Equality"},{"Add":"import Relation.Binary.Construct.Add.Supremum.NonStrict as NonStrict"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"infix 4 _<⁺_"},{"Add":""},{"Add":"data _<⁺_ : Rel (A ⁺) (a ⊔ r) where"},{"Add":" [_] : {k l : A} → k < l → [ k ] <⁺ [ l ]"},{"Add":" [_]<⊤⁺ : (k : A) → [ k ] <⁺ ⊤⁺"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relational properties"},{"Add":""},{"Add":"[<]-injective : ∀ {k l} → [ k ] <⁺ [ l ] → k < l"},{"Add":"[<]-injective [ p ] = p"},{"Add":""},{"Add":"<⁺-asym : Asymmetric _<_ → Asymmetric _<⁺_"},{"Add":"<⁺-asym <-asym [ p ] [ q ] = <-asym p q"},{"Add":""},{"Add":"<⁺-trans : Transitive _<_ → Transitive _<⁺_"},{"Add":"<⁺-trans <-trans [ p ] [ q ] = [ <-trans p q ]"},{"Add":"<⁺-trans <-trans [ p ] [ k ]<⊤⁺ = [ _ ]<⊤⁺"},{"Add":""},{"Add":"<⁺-dec : Decidable _<_ → Decidable _<⁺_"},{"Add":"<⁺-dec _ (λ ()) (λ ()) [ l ]<⊤⁺"},{"Add":"<⁺-cmp-≡ <-cmp [ k ] ⊤⁺ = tri< [ k ]<⊤⁺ (λ ()) (λ ())"},{"Add":"<⁺-cmp-≡ <-cmp [ k ] [ l ] with <-cmp k l"},{"Add":"... | tri< a ¬b ¬c = tri< [ a ] (¬b ∘ []-injective) (¬c ∘ [<]-injective)"},{"Add":"... | tri≈ ¬a refl ¬c = tri≈ (¬a ∘ [<]-injective) refl (¬c ∘ [<]-injective)"},{"Add":"... | tri> ¬a ¬b c = tri> (¬a ∘ [<]-injective) (¬b ∘ []-injective) [ c ]"},{"Add":""},{"Add":"<⁺-irrefl-≡ : Irreflexive _≡_ _<_ → Irreflexive _≡_ _<⁺_"},{"Add":"<⁺-irrefl-≡ <-irrefl refl [ x ] = <-irrefl refl x"},{"Add":""},{"Add":"<⁺-respˡ-≡ : _<⁺_ Respectsˡ _≡_"},{"Add":"<⁺-respˡ-≡ = subst (_<⁺ _)"},{"Add":""},{"Add":"<⁺-respʳ-≡ : _<⁺_ Respectsʳ _≡_"},{"Add":"<⁺-respʳ-≡ = subst (_ <⁺_)"},{"Add":""},{"Add":"<⁺-resp-≡ : _<⁺_ Respects₂ _≡_"},{"Add":"<⁺-resp-≡ = <⁺-respʳ-≡ , <⁺-respˡ-≡"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relational properties + setoid equality"},{"Add":""},{"Add":"module _ {e} {_≈_ : Rel A e} where"},{"Add":""},{"Add":" open Equality _≈_"},{"Add":""},{"Add":" <⁺-cmp : Trichotomous _≈_ _<_ → Trichotomous _≈⁺_ _<⁺_"},{"Add":" <⁺-cmp <-cmp ⊤⁺ ⊤⁺ = tri≈ (λ ()) ⊤⁺≈⊤⁺ (λ ())"},{"Add":" <⁺-cmp <-cmp ⊤⁺ [ l ] = tri> (λ ()) (λ ()) [ l ]<⊤⁺"},{"Add":" <⁺-cmp <-cmp [ k ] ⊤⁺ = tri< [ k ]<⊤⁺ (λ ()) (λ ())"},{"Add":" <⁺-cmp <-cmp [ k ] [ l ] with <-cmp k l"},{"Add":" ... | tri< a ¬b ¬c = tri< [ a ] (¬b ∘ [≈]-injective) (¬c ∘ [<]-injective)"},{"Add":" ... | tri≈ ¬a b ¬c = tri≈ (¬a ∘ [<]-injective) [ b ] (¬c ∘ [<]-injective)"},{"Add":" ... | tri> ¬a ¬b c = tri> (¬a ∘ [<]-injective) (¬b ∘ [≈]-injective) [ c ]"},{"Add":""},{"Add":" <⁺-irrefl : Irreflexive _≈_ _<_ → Irreflexive _≈⁺_ _<⁺_"},{"Add":" <⁺-irrefl <-irrefl [ p ] [ q ] = <-irrefl p q"},{"Add":""},{"Add":" <⁺-respˡ-≈⁺ : _<_ Respectsˡ _≈_ → _<⁺_ Respectsˡ _≈⁺_"},{"Add":" <⁺-respˡ-≈⁺ <-respˡ-≈ [ p ] [ q ] = [ <-respˡ-≈ p q ]"},{"Add":" <⁺-respˡ-≈⁺ <-respˡ-≈ [ p ] ([ l ]<⊤⁺) = [ _ ]<⊤⁺"},{"Add":" <⁺-respˡ-≈⁺ <-respˡ-≈ ⊤⁺≈⊤⁺ q = q"},{"Add":""},{"Add":" <⁺-respʳ-≈⁺ : _<_ Respectsʳ _≈_ → _<⁺_ Respectsʳ _≈⁺_"},{"Add":" <⁺-respʳ-≈⁺ <-respʳ-≈ [ p ] [ q ] = [ <-respʳ-≈ p q ]"},{"Add":" <⁺-respʳ-≈⁺ <-respʳ-≈ ⊤⁺≈⊤⁺ q = q"},{"Add":""},{"Add":" <⁺-resp-≈⁺ : _<_ Respects₂ _≈_ → _<⁺_ Respects₂ _≈⁺_"},{"Add":" <⁺-resp-≈⁺ = map <⁺-respʳ-≈⁺ <⁺-respˡ-≈⁺"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures + propositional equality"},{"Add":""},{"Add":"<⁺-isStrictPartialOrder-≡ : IsStrictPartialOrder _≡_ _<_ →"},{"Add":" IsStrictPartialOrder _≡_ _<⁺_"},{"Add":"<⁺-isStrictPartialOrder-≡ strict = record"},{"Add":" { isEquivalence = ≡.isEquivalence"},{"Add":" ; irrefl = <⁺-irrefl-≡ irrefl"},{"Add":" ; trans = <⁺-trans trans"},{"Add":" ; <-resp-≈ = <⁺-resp-≡"},{"Add":" } where open IsStrictPartialOrder strict"},{"Add":""},{"Add":"<⁺-isDecStrictPartialOrder-≡ : IsDecStrictPartialOrder _≡_ _<_ →"},{"Add":" IsDecStrictPartialOrder _≡_ _<⁺_"},{"Add":"<⁺-isDecStrictPartialOrder-≡ dectot = record"},{"Add":" { isStrictPartialOrder = <⁺-isStrictPartialOrder-≡ isStrictPartialOrder"},{"Add":" ; _≟_ = ≡-dec _≟_"},{"Add":" ; _; _Respectsˡ_; _Respectsʳ_; _Respects₂_)"},{"Add":""},{"Add":"module Relation.Binary.Construct.Add.Infimum.Strict"},{"Add":" {a ℓ} {A : Set a} (_<_ : Rel A ℓ) where"},{"Add":""},{"Add":"open import Level using (_⊔_)"},{"Add":"open import Data.Product.Base using (_,_; map)"},{"Add":"open import Function.Base"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; subst)"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":"import Relation.Binary.Construct.Add.Infimum.Equality as Equality"},{"Add":"import Relation.Binary.Construct.Add.Infimum.NonStrict as NonStrict"},{"Add":"open import Relation.Nullary hiding (Irrelevant)"},{"Add":"open import Relation.Nullary.Construct.Add.Infimum"},{"Add":"import Relation.Nullary.Decidable as Dec"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"infix 4 _<₋_"},{"Add":""},{"Add":"data _<₋_ : Rel (A ₋) (a ⊔ ℓ) where"},{"Add":" ⊥₋<[_] : (l : A) → ⊥₋ <₋ [ l ]"},{"Add":" [_] : {k l : A} → k < l → [ k ] <₋ [ l ]"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relational properties"},{"Add":""},{"Add":"[<]-injective : ∀ {k l} → [ k ] <₋ [ l ] → k < l"},{"Add":"[<]-injective [ p ] = p"},{"Add":""},{"Add":"<₋-asym : Asymmetric _<_ → Asymmetric _<₋_"},{"Add":"<₋-asym <-asym [ p ] [ q ] = <-asym p q"},{"Add":""},{"Add":"<₋-trans : Transitive _<_ → Transitive _<₋_"},{"Add":"<₋-trans <-trans ⊥₋<[ l ] [ q ] = ⊥₋<[ _ ]"},{"Add":"<₋-trans <-trans [ p ] [ q ] = [ <-trans p q ]"},{"Add":""},{"Add":"<₋-dec : Decidable _<_ → Decidable _<₋_"},{"Add":"<₋-dec _ (λ ()) (λ ()) ⊥₋<[ k ]"},{"Add":"<₋-cmp-≡ <-cmp [ k ] [ l ] with <-cmp k l"},{"Add":"... | tri< a ¬b ¬c = tri< [ a ] (¬b ∘ []-injective) (¬c ∘ [<]-injective)"},{"Add":"... | tri≈ ¬a refl ¬c = tri≈ (¬a ∘ [<]-injective) refl (¬c ∘ [<]-injective)"},{"Add":"... | tri> ¬a ¬b c = tri> (¬a ∘ [<]-injective) (¬b ∘ []-injective) [ c ]"},{"Add":""},{"Add":"<₋-irrefl-≡ : Irreflexive _≡_ _<_ → Irreflexive _≡_ _<₋_"},{"Add":"<₋-irrefl-≡ <-irrefl refl [ x ] = <-irrefl refl x"},{"Add":""},{"Add":"<₋-respˡ-≡ : _<₋_ Respectsˡ _≡_"},{"Add":"<₋-respˡ-≡ = subst (_<₋ _)"},{"Add":""},{"Add":"<₋-respʳ-≡ : _<₋_ Respectsʳ _≡_"},{"Add":"<₋-respʳ-≡ = subst (_ <₋_)"},{"Add":""},{"Add":"<₋-resp-≡ : _<₋_ Respects₂ _≡_"},{"Add":"<₋-resp-≡ = <₋-respʳ-≡ , <₋-respˡ-≡"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relational properties + setoid equality"},{"Add":""},{"Add":"module _ {e} {_≈_ : Rel A e} where"},{"Add":""},{"Add":" open Equality _≈_"},{"Add":""},{"Add":" <₋-cmp : Trichotomous _≈_ _<_ → Trichotomous _≈₋_ _<₋_"},{"Add":" <₋-cmp <-cmp ⊥₋ ⊥₋ = tri≈ (λ ()) ⊥₋≈⊥₋ (λ ())"},{"Add":" <₋-cmp <-cmp ⊥₋ [ l ] = tri< ⊥₋<[ l ] (λ ()) (λ ())"},{"Add":" <₋-cmp <-cmp [ k ] ⊥₋ = tri> (λ ()) (λ ()) ⊥₋<[ k ]"},{"Add":" <₋-cmp <-cmp [ k ] [ l ] with <-cmp k l"},{"Add":" ... | tri< a ¬b ¬c = tri< [ a ] (¬b ∘ [≈]-injective) (¬c ∘ [<]-injective)"},{"Add":" ... | tri≈ ¬a b ¬c = tri≈ (¬a ∘ [<]-injective) [ b ] (¬c ∘ [<]-injective)"},{"Add":" ... | tri> ¬a ¬b c = tri> (¬a ∘ [<]-injective) (¬b ∘ [≈]-injective) [ c ]"},{"Add":""},{"Add":" <₋-irrefl : Irreflexive _≈_ _<_ → Irreflexive _≈₋_ _<₋_"},{"Add":" <₋-irrefl <-irrefl [ p ] [ q ] = <-irrefl p q"},{"Add":""},{"Add":" <₋-respˡ-≈₋ : _<_ Respectsˡ _≈_ → _<₋_ Respectsˡ _≈₋_"},{"Add":" <₋-respˡ-≈₋ <-respˡ-≈ ⊥₋≈⊥₋ q = q"},{"Add":" <₋-respˡ-≈₋ <-respˡ-≈ [ p ] [ q ] = [ <-respˡ-≈ p q ]"},{"Add":""},{"Add":" <₋-respʳ-≈₋ : _<_ Respectsʳ _≈_ → _<₋_ Respectsʳ _≈₋_"},{"Add":" <₋-respʳ-≈₋ <-respʳ-≈ ⊥₋≈⊥₋ q = q"},{"Add":" <₋-respʳ-≈₋ <-respʳ-≈ [ p ] ⊥₋<[ l ] = ⊥₋<[ _ ]"},{"Add":" <₋-respʳ-≈₋ <-respʳ-≈ [ p ] [ q ] = [ <-respʳ-≈ p q ]"},{"Add":""},{"Add":" <₋-resp-≈₋ : _<_ Respects₂ _≈_ → _<₋_ Respects₂ _≈₋_"},{"Add":" <₋-resp-≈₋ = map <₋-respʳ-≈₋ <₋-respˡ-≈₋"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures + propositional equality"},{"Add":""},{"Add":"<₋-isStrictPartialOrder-≡ : IsStrictPartialOrder _≡_ _<_ →"},{"Add":" IsStrictPartialOrder _≡_ _<₋_"},{"Add":"<₋-isStrictPartialOrder-≡ strict = record"},{"Add":" { isEquivalence = ≡.isEquivalence"},{"Add":" ; irrefl = <₋-irrefl-≡ irrefl"},{"Add":" ; trans = <₋-trans trans"},{"Add":" ; <-resp-≈ = <₋-resp-≡"},{"Add":" } where open IsStrictPartialOrder strict"},{"Add":""},{"Add":"<₋-isDecStrictPartialOrder-≡ : IsDecStrictPartialOrder _≡_ _<_ →"},{"Add":" IsDecStrictPartialOrder _≡_ _<₋_"},{"Add":"<₋-isDecStrictPartialOrder-≡ dectot = record"},{"Add":" { isStrictPartialOrder = <₋-isStrictPartialOrder-≡ isStrictPartialOrder"},{"Add":" ; _≟_ = ≡-dec _≟_"},{"Add":" ; _y with tri x y"},{"Add":" ... | tri< _ _ x≯y = x≯y x>y"},{"Add":" ... | tri≈ _ _ x≯y = x≯y x>y"},{"Add":" ... | tri> x≮y _ _ = x≮y x x≮y x≉y y _ x≉y _ = no x≉y"},{"Add":""},{"Add":" tri⇒dec< : Trichotomous _≈_ _<_ → Decidable _<_"},{"Add":" tri⇒dec< compare x y with compare x y"},{"Add":" ... | tri< x x≮y _ _ = no x≮y"},{"Add":""},{"Add":" trans∧tri⇒respʳ : Symmetric _≈_ → Transitive _≈_ →"},{"Add":" Transitive _<_ → Trichotomous _≈_ _<_ →"},{"Add":" _<_ Respectsʳ _≈_"},{"Add":" trans∧tri⇒respʳ sym ≈-tr <-tr tri {x} {y} {z} y≈z x _ _ z _ _ z_"},{"Add":" _>_ = flip _<_"},{"Add":""},{"Add":" infix 4 _≯_"},{"Add":" _≯_ = flip _≮_"},{"Add":""},{"Add":""},{"Add":"record DecStrictPartialOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where"},{"Add":" infix 4 _≈_ _<_"},{"Add":" field"},{"Add":" Carrier : Set c"},{"Add":" _≈_ : Rel Carrier ℓ₁"},{"Add":" _<_ : Rel Carrier ℓ₂"},{"Add":" isDecStrictPartialOrder : IsDecStrictPartialOrder _≈_ _<_"},{"Add":""},{"Add":" private module DSPO = IsDecStrictPartialOrder isDecStrictPartialOrder"},{"Add":""},{"Add":" open DSPO public"},{"Add":" using (_>=_; _>>_)"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- DEPRECATED NAMES"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Please use the new names as continuing support for the old names is"},{"Add":"-- not guaranteed."},{"Add":""},{"Add":"import Reflection.AST.Abstraction as Abstraction"},{"Add":"import Reflection.AST.Argument as Argument"},{"Add":"import Reflection.AST.Definition as Definition"},{"Add":"import Reflection.AST.Meta as Meta"},{"Add":"import Reflection.AST.Name as Name"},{"Add":"import Reflection.AST.Literal as Literal"},{"Add":"import Reflection.AST.Pattern as Pattern"},{"Add":"import Reflection.AST.Term as Term"},{"Add":"import Reflection.AST.Argument.Modality as Modality"},{"Add":"import Reflection.AST.Argument.Quantity as Quantity"},{"Add":"import Reflection.AST.Argument.Relevance as Relevance"},{"Add":"import Reflection.AST.Argument.Visibility as Visibility"},{"Add":"import Reflection.AST.Argument.Information as Information"},{"Add":""},{"Add":"-- Version 1.3"},{"Add":""},{"Add":"Arg-info = Information.ArgInfo"},{"Add":"{-# WARNING_ON_USAGE Arg-info"},{"Add":"\"Warning: Arg-info was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Argument.Information's ArgInfo instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"infix 4 _≟-Lit_ _≟-Name_ _≟-Meta_ _≟-Visibility_ _≟-Relevance_ _≟-Arg-info_"},{"Add":" _≟-Pattern_ _≟-ArgPatterns_"},{"Add":""},{"Add":"_≟-Lit_ = Literal._≟_"},{"Add":"{-# WARNING_ON_USAGE _≟-Lit_"},{"Add":"\"Warning: _≟-Lit_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Literal's _≟_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-Name_ = Name._≟_"},{"Add":"{-# WARNING_ON_USAGE _≟-Name_"},{"Add":"\"Warning: _≟-Name_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Name's _≟_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-Meta_ = Meta._≟_"},{"Add":"{-# WARNING_ON_USAGE _≟-Meta_"},{"Add":"\"Warning: _≟-Meta_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Meta's _≟_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-Visibility_ = Visibility._≟_"},{"Add":"{-# WARNING_ON_USAGE _≟-Visibility_"},{"Add":"\"Warning: _≟-Visibility_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Argument.Visibility's _≟_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-Relevance_ = Relevance._≟_"},{"Add":"{-# WARNING_ON_USAGE _≟-Relevance_"},{"Add":"\"Warning: _≟-Relevance_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Argument.Relevance's _≟_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-Arg-info_ = Information._≟_"},{"Add":"{-# WARNING_ON_USAGE _≟-Arg-info_"},{"Add":"\"Warning: _≟-Arg-info_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Argument.Information's _≟_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-Pattern_ = Pattern._≟_"},{"Add":"{-# WARNING_ON_USAGE _≟-Pattern_"},{"Add":"\"Warning: _≟-Pattern_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Pattern's _≟_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-ArgPatterns_ = Pattern._≟s_"},{"Add":"{-# WARNING_ON_USAGE _≟-ArgPatterns_"},{"Add":"\"Warning: _≟-ArgPatterns_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Pattern's _≟s_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"map-Abs = Abstraction.map"},{"Add":"{-# WARNING_ON_USAGE map-Abs"},{"Add":"\"Warning: map-Abs was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Abstraction's map instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"map-Arg = Argument.map"},{"Add":"{-# WARNING_ON_USAGE map-Arg"},{"Add":"\"Warning: map-Arg was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Argument's map instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"map-Args = Argument.map-Args"},{"Add":"{-# WARNING_ON_USAGE map-Args"},{"Add":"\"Warning: map-Args was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Argument's map-Args instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"visibility = Information.visibility"},{"Add":"{-# WARNING_ON_USAGE visibility"},{"Add":"\"Warning: visibility was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Argument.Information's visibility instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"relevance = Modality.relevance"},{"Add":"{-# WARNING_ON_USAGE relevance"},{"Add":"\"Warning: relevance was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Argument.Modality's relevance instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"infix 4 _≟-AbsTerm_ _≟-AbsType_ _≟-ArgTerm_ _≟-ArgType_ _≟-Args_"},{"Add":" _≟-Clause_ _≟-Clauses_ _≟_"},{"Add":" _≟-Sort_"},{"Add":""},{"Add":"_≟-AbsTerm_ = Term._≟-AbsTerm_"},{"Add":"{-# WARNING_ON_USAGE _≟-AbsTerm_"},{"Add":"\"Warning: _≟-AbsTerm_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Term's _≟-AbsTerm_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-AbsType_ = Term._≟-AbsType_"},{"Add":"{-# WARNING_ON_USAGE _≟-AbsType_"},{"Add":"\"Warning: _≟-AbsType_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Term's _≟-AbsType_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-ArgTerm_ = Term._≟-ArgTerm_"},{"Add":"{-# WARNING_ON_USAGE _≟-ArgTerm_"},{"Add":"\"Warning: _≟-ArgTerm_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Term's _≟-ArgTerm_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-ArgType_ = Term._≟-ArgType_"},{"Add":"{-# WARNING_ON_USAGE _≟-ArgType_"},{"Add":"\"Warning: _≟-ArgType_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Term's _≟-ArgType_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-Args_ = Term._≟-Args_"},{"Add":"{-# WARNING_ON_USAGE _≟-Args_"},{"Add":"\"Warning: _≟-Args_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Term's _≟-Args_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-Clause_ = Term._≟-Clause_"},{"Add":"{-# WARNING_ON_USAGE _≟-Clause_"},{"Add":"\"Warning: _≟-Clause_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Term's _≟-Clause_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-Clauses_ = Term._≟-Clauses_"},{"Add":"{-# WARNING_ON_USAGE _≟-Clauses_"},{"Add":"\"Warning: _≟-Clauses_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Term's _≟-Clauses_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟_ = Term._≟_"},{"Add":"{-# WARNING_ON_USAGE _≟_"},{"Add":"\"Warning: _≟_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Term's _≟_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_≟-Sort_ = Term._≟-Sort_"},{"Add":"{-# WARNING_ON_USAGE _≟-Sort_"},{"Add":"\"Warning: _≟-Sort_ was deprecated in v1.3."},{"Add":"Please use Reflection.AST.Term's _≟-Sort_ instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Reflection","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"TCM.agda","metadata":0},"context":[{"change":0,"pos":851965}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The TC (Type Checking) monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.TCM where"},{"Add":""},{"Add":"import Agda.Builtin.Reflection as Builtin"},{"Add":""},{"Add":"open import Reflection.AST.Term"},{"Add":"import Reflection.TCM.Format as Format"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Type errors"},{"Add":""},{"Add":"open Builtin public"},{"Add":" using (ErrorPart; strErr; termErr; nameErr)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The monad"},{"Add":""},{"Add":"open Builtin public"},{"Add":" using"},{"Add":" ( TC; bindTC; unify; typeError; inferType; checkType"},{"Add":" ; normalise; reduce"},{"Add":" ; catchTC; quoteTC; unquoteTC"},{"Add":" ; getContext; extendContext; inContext; freshName"},{"Add":" ; declareDef; declarePostulate; defineFun; getType; getDefinition"},{"Add":" ; blockOnMeta; commitTC; isMacro; withNormalisation"},{"Add":" ; debugPrint; noConstraints; runSpeculative"},{"Add":" ; Blocker; blockerMeta; blockerAny; blockerAll; blockTC"},{"Add":" )"},{"Add":" renaming (returnTC to pure)"},{"Add":""},{"Add":"open Format public"},{"Add":" using (typeErrorFmt; debugPrintFmt; errorPartFmt)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Utility functions"},{"Add":""},{"Add":"newMeta : Type → TC Term"},{"Add":"newMeta = checkType unknown"}]}},{"FileAdd":{"meta":{"basename":"TCM","metadata":576},"context":[{"change":0,"pos":851965}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Utilities.agda","metadata":0},"context":[{"change":0,"pos":853340}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Reflection utilities"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.TCM.Utilities where"},{"Add":""},{"Add":"open import Data.List using (List; []; _∷_; _++_; map)"},{"Add":"open import Data.Unit using (⊤; tt)"},{"Add":"open import Effect.Applicative using (RawApplicative; mkRawApplicative)"},{"Add":"open import Function using (case_of_)"},{"Add":"open import Reflection.AST.Meta using (Meta)"},{"Add":"open import Reflection.AST.Term using (Term)"},{"Add":"open import Reflection.TCM using (TC; pure; blockTC; blockerAll; blockerMeta)"},{"Add":""},{"Add":"import Reflection.AST.Traversal as Traversal"},{"Add":""},{"Add":"blockOnMetas : Term → TC ⊤"},{"Add":"blockOnMetas t ="},{"Add":" case traverseTerm actions (0 , []) t of λ where"},{"Add":" [] → pure tt"},{"Add":" xs@(_ ∷ _) → blockTC (blockerAll (map blockerMeta xs))"},{"Add":" where"},{"Add":" applicative : ∀ {ℓ} → RawApplicative {ℓ} (λ _ → List Meta)"},{"Add":" applicative = mkRawApplicative (λ _ → List Meta) (λ _ → []) _++_"},{"Add":""},{"Add":" open Traversal applicative"},{"Add":""},{"Add":" actions : Actions"},{"Add":" actions = record defaultActions { onMeta = λ _ x → x ∷ [] }"}]}},{"FileAdd":{"meta":{"basename":"Syntax.agda","metadata":0},"context":[{"change":0,"pos":853340}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Monad syntax for the TC monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.TCM.Syntax where"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection"},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Monad syntax"},{"Add":""},{"Add":"infixl 3 _<|>_"},{"Add":"_<|>_ : TC A → TC A → TC A"},{"Add":"_<|>_ = catchTC"},{"Add":"{-# INLINE _<|>_ #-}"},{"Add":""},{"Add":"infixl 1 _>>=_ _>>_ _<&>_"},{"Add":"_>>=_ : TC A → (A → TC B) → TC B"},{"Add":"_>>=_ = bindTC"},{"Add":"{-# INLINE _>>=_ #-}"},{"Add":""},{"Add":"_>>_ : TC A → TC B → TC B"},{"Add":"xs >> ys = bindTC xs (λ _ → ys)"},{"Add":"{-# INLINE _>>_ #-}"},{"Add":""},{"Add":"infixl 4 _<$>_ _<*>_ _<$_"},{"Add":"_<*>_ : TC (A → B) → TC A → TC B"},{"Add":"fs <*> xs = bindTC fs (λ f → bindTC xs (λ x → returnTC (f x)))"},{"Add":"{-# INLINE _<*>_ #-}"},{"Add":""},{"Add":"_<$>_ : (A → B) → TC A → TC B"},{"Add":"f <$> xs = bindTC xs (λ x → returnTC (f x))"},{"Add":"{-# INLINE _<$>_ #-}"},{"Add":""},{"Add":"_<$_ : A → TC B → TC A"},{"Add":"x <$ xs = bindTC xs (λ _ → returnTC x)"},{"Add":"{-# INLINE _<$_ #-}"},{"Add":""},{"Add":"_<&>_ : TC A → (A → B) → TC B"},{"Add":"xs <&> f = bindTC xs (λ x → returnTC (f x))"},{"Add":"{-# INLINE _<&>_ #-}"}]}},{"FileAdd":{"meta":{"basename":"Instances.agda","metadata":0},"context":[{"change":0,"pos":853340}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Typeclass instances for TC"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.TCM.Instances where"},{"Add":""},{"Add":"open import Reflection.TCM.Effectful"},{"Add":""},{"Add":"instance"},{"Add":" tcFunctor = functor"},{"Add":" tcApplicative = applicative"},{"Add":" tcApplicativeZero = applicativeZero"},{"Add":" tcAlternative = alternative"},{"Add":" tcMonad = monad"},{"Add":" tcMonadZero = monadZero"},{"Add":" tcMonadPlus = monadPlus"}]}},{"FileAdd":{"meta":{"basename":"Format.agda","metadata":0},"context":[{"change":0,"pos":853340}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Printf-style versions of typeError and debugPrint"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.TCM.Format where"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection using (Name; Term; ErrorPart; termErr; nameErr; strErr;"},{"Add":" TC; typeError; debugPrint)"},{"Add":"open import Level using (Level)"},{"Add":"open import Function.Base using (_∘_)"},{"Add":"open import Data.List.Base using (List; [_]; concat)"},{"Add":"open import Data.Maybe.Base using (Maybe; nothing; just)"},{"Add":"open import Data.Nat.Base using (ℕ)"},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Data.Sum.Base using (_⊎_; inj₁; inj₂)"},{"Add":"open import Data.Unit.Base using (⊤)"},{"Add":""},{"Add":"import Text.Format as StdFormat using (formatSpec)"},{"Add":"import Text.Printf as StdPrintf using (printfSpec)"},{"Add":""},{"Add":"open import Text.Format.Generic"},{"Add":"open import Text.Printf.Generic"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ : Level"},{"Add":" A : Set ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Format specification."},{"Add":"-- This extends the formats from Text.Printf with"},{"Add":"-- %t - Term"},{"Add":"-- %q - Name"},{"Add":"-- %e - List ErrorPart"},{"Add":"-- Rendering goes to List ErrorPart."},{"Add":""},{"Add":"module Specification where"},{"Add":""},{"Add":" data ErrorChunk : Set where"},{"Add":" `Term `Name `Parts : ErrorChunk"},{"Add":""},{"Add":" errorSpec : FormatSpec"},{"Add":" errorSpec .FormatSpec.ArgChunk = ErrorChunk"},{"Add":" errorSpec .FormatSpec.ArgType `Term = Term"},{"Add":" errorSpec .FormatSpec.ArgType `Name = Name"},{"Add":" errorSpec .FormatSpec.ArgType `Parts = List ErrorPart"},{"Add":" errorSpec .FormatSpec.lexArg 't' = just `Term"},{"Add":" errorSpec .FormatSpec.lexArg 'q' = just `Name"},{"Add":" errorSpec .FormatSpec.lexArg 'e' = just `Parts"},{"Add":" errorSpec .FormatSpec.lexArg _ = nothing"},{"Add":""},{"Add":" formatSpec : FormatSpec"},{"Add":" formatSpec = unionSpec errorSpec StdFormat.formatSpec"},{"Add":""},{"Add":" open PrintfSpec"},{"Add":""},{"Add":" printfSpec : PrintfSpec formatSpec (List ErrorPart)"},{"Add":" printfSpec .renderArg (inj₁ `Term) t = [ termErr t ]"},{"Add":" printfSpec .renderArg (inj₁ `Name) n = [ nameErr n ]"},{"Add":" printfSpec .renderArg (inj₁ `Parts) es = es"},{"Add":" printfSpec .renderArg (inj₂ arg) x = [ strErr (StdPrintf.printfSpec .renderArg arg x) ]"},{"Add":" printfSpec .renderStr s = [ strErr s ]"},{"Add":""},{"Add":"open Specification"},{"Add":""},{"Add":"open Format formatSpec"},{"Add":"open Type formatSpec renaming (map to mapPrintf)"},{"Add":"open Render printfSpec"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Printf versions of typeError and debugPrint"},{"Add":""},{"Add":"typeErrorFmt : (fmt : String) → Printf (lexer fmt) (TC A)"},{"Add":"typeErrorFmt fmt = mapPrintf (lexer fmt) (typeError ∘ concat) (printf fmt)"},{"Add":""},{"Add":"debugPrintFmt : String → ℕ → (fmt : String) → Printf (lexer fmt) (TC ⊤)"},{"Add":"debugPrintFmt tag lvl fmt = mapPrintf (lexer fmt) (debugPrint tag lvl ∘ concat) (printf fmt)"},{"Add":""},{"Add":"-- Combine with \"%e\" format for nested format calls."},{"Add":"errorPartFmt : (fmt : String) → Printf (lexer fmt) (List ErrorPart)"},{"Add":"errorPartFmt fmt = mapPrintf (lexer fmt) concat (printf fmt)"}]}},{"FileAdd":{"meta":{"basename":"Effectful.agda","metadata":0},"context":[{"change":0,"pos":853340}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Typeclass instances for TC"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.TCM.Effectful where"},{"Add":""},{"Add":"open import Effect.Choice"},{"Add":"open import Effect.Empty"},{"Add":"open import Effect.Functor"},{"Add":"open import Effect.Applicative"},{"Add":"open import Effect.Monad"},{"Add":"open import Data.List.Base using ([])"},{"Add":"open import Function.Base using (_∘_)"},{"Add":"open import Level"},{"Add":""},{"Add":"open import Reflection.TCM"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ : Level"},{"Add":""},{"Add":"functor : RawFunctor {ℓ} TC"},{"Add":"functor = record"},{"Add":" { _<$>_ = λ f mx → bindTC mx (pure ∘ f)"},{"Add":" }"},{"Add":""},{"Add":"applicative : RawApplicative {ℓ} TC"},{"Add":"applicative = record"},{"Add":" { rawFunctor = functor"},{"Add":" ; pure = pure"},{"Add":" ; _<*>_ = λ mf mx → bindTC mf λ f → bindTC mx (pure ∘ f)"},{"Add":" }"},{"Add":""},{"Add":"empty : RawEmpty {ℓ} TC"},{"Add":"empty = record { empty = typeError [] }"},{"Add":""},{"Add":"applicativeZero : RawApplicativeZero {ℓ} TC"},{"Add":"applicativeZero = record"},{"Add":" { rawApplicative = applicative"},{"Add":" ; rawEmpty = empty"},{"Add":" }"},{"Add":""},{"Add":"choice : RawChoice {ℓ} TC"},{"Add":"choice = record { _<|>_ = catchTC }"},{"Add":""},{"Add":"alternative : RawAlternative {ℓ} TC"},{"Add":"alternative = record"},{"Add":" { rawApplicativeZero = applicativeZero"},{"Add":" ; rawChoice = choice"},{"Add":" }"},{"Add":""},{"Add":"monad : RawMonad {ℓ} TC"},{"Add":"monad = record"},{"Add":" { rawApplicative = applicative"},{"Add":" ; _>>=_ = bindTC"},{"Add":" }"},{"Add":""},{"Add":"monadZero : RawMonadZero {ℓ} TC"},{"Add":"monadZero = record"},{"Add":" { rawMonad = monad"},{"Add":" ; rawEmpty = empty"},{"Add":" }"},{"Add":""},{"Add":"monadPlus : RawMonadPlus {ℓ} TC"},{"Add":"monadPlus = record"},{"Add":" { rawMonadZero = monadZero"},{"Add":" ; rawChoice = choice"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Categorical.agda","metadata":0},"context":[{"change":0,"pos":853340}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use"},{"Add":"-- `Reflection.TCM.Effectful` instead."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.TCM.Categorical where"},{"Add":""},{"Add":"open import Reflection.TCM.Effectful public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Reflection.TCM.Categorical was deprecated in v2.0."},{"Add":"Use Reflection.TCM.Effectful instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"External.agda","metadata":0},"context":[{"change":0,"pos":851965}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Support for system calls as part of reflection"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.External where"},{"Add":""},{"Add":"import Agda.Builtin.Reflection.External as Builtin"},{"Add":""},{"Add":"open import Data.Nat.Base using (ℕ; suc; zero; NonZero)"},{"Add":"open import Data.List.Base using (List; _∷_; [])"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Data.String.Base as String using (String; _++_)"},{"Add":"open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])"},{"Add":"open import Data.Unit.Base using (⊤; tt)"},{"Add":"open import Function.Base using (case_of_; _$_; _∘_)"},{"Add":"open import Reflection hiding (name)"},{"Add":""},{"Add":"-- Type aliases for the various strings."},{"Add":""},{"Add":"CmdName = String"},{"Add":"StdIn = String"},{"Add":"StdErr = String"},{"Add":"StdOut = String"},{"Add":""},{"Add":"-- Representation for exit codes, assuming 0 is consistently used to"},{"Add":"-- indicate success across platforms."},{"Add":"data ExitCode : Set where"},{"Add":" exitSuccess : ExitCode"},{"Add":" exitFailure : (n : ℕ) {n≢0 : NonZero n} → ExitCode"},{"Add":""},{"Add":"-- Specification for a command."},{"Add":"record CmdSpec : Set where"},{"Add":" constructor cmdSpec"},{"Add":" field"},{"Add":" name : CmdName -- ^ Executable name (see ~/.agda/executables)"},{"Add":" args : List String -- ^ Command-line arguments for executable"},{"Add":" input : StdIn -- ^ Contents of standard input"},{"Add":""},{"Add":"-- Result of running a command."},{"Add":"record Result : Set where"},{"Add":" constructor result"},{"Add":" field"},{"Add":" exitCode : ExitCode -- ^ Exit code returned by the process"},{"Add":" output : StdOut -- ^ Contents of standard output"},{"Add":" error : StdErr -- ^ Contents of standard error"},{"Add":""},{"Add":"-- Convert a natural number to an exit code."},{"Add":"toExitCode : ℕ → ExitCode"},{"Add":"toExitCode zero = exitSuccess"},{"Add":"toExitCode (suc n) = exitFailure (suc n)"},{"Add":""},{"Add":"-- Quote an exit code as an Agda term."},{"Add":"quoteExitCode : ExitCode → Term"},{"Add":"quoteExitCode exitSuccess ="},{"Add":" con (quote exitSuccess) []"},{"Add":"quoteExitCode (exitFailure n) ="},{"Add":" con (quote exitFailure) (vArg (lit (nat n)) ∷ hArg (con (quote tt) []) ∷ [])"},{"Add":""},{"Add":"-- Quote a result as an Agda term."},{"Add":"quoteResult : Result → Term"},{"Add":"quoteResult (result exitCode output error) ="},{"Add":" con (quote result) $ vArg (quoteExitCode exitCode)"},{"Add":" ∷ vArg (lit (string output))"},{"Add":" ∷ vArg (lit (string error))"},{"Add":" ∷ []"},{"Add":""},{"Add":"-- Run command from specification and return the full result."},{"Add":"--"},{"Add":"-- NOTE: If the command fails, this macro still succeeds, and returns the"},{"Add":"-- full result, including exit code and the contents of stderr."},{"Add":"--"},{"Add":"unsafeRunCmdTC : CmdSpec → TC Result"},{"Add":"unsafeRunCmdTC c = do"},{"Add":" (exitCode , (stdOut , stdErr))"},{"Add":" ← Builtin.execTC (CmdSpec.name c) (CmdSpec.args c) (CmdSpec.input c)"},{"Add":" pure $ result (toExitCode exitCode) stdOut stdErr"},{"Add":""},{"Add":"macro"},{"Add":" unsafeRunCmd : CmdSpec → Term → TC ⊤"},{"Add":" unsafeRunCmd c hole = unsafeRunCmdTC c >>= unify hole ∘ quoteResult"},{"Add":""},{"Add":""},{"Add":"-- Show a command for the user."},{"Add":"showCmdSpec : CmdSpec → String"},{"Add":"showCmdSpec c = String.unwords $ CmdSpec.name c ∷ CmdSpec.args c"},{"Add":""},{"Add":""},{"Add":"private"},{"Add":" -- Helper function for throwing an error from reflection."},{"Add":" userError : ∀ {a} {A : Set a} → CmdSpec → StdOut × StdErr → TC A"},{"Add":" userError c (stdout , stderr) = typeError (strErr errMsg ∷ [])"},{"Add":" where"},{"Add":" errMsg : String"},{"Add":" errMsg = String.unlines"},{"Add":" $ (\"Error while running command '\" ++ showCmdSpec c ++ \"'\")"},{"Add":" ∷ (\"Input:\\n\" ++ CmdSpec.input c)"},{"Add":" ∷ (\"Output:\\n\" ++ stdout)"},{"Add":" ∷ (\"Error:\\n\" ++ stderr)"},{"Add":" ∷ []"},{"Add":""},{"Add":""},{"Add":"-- Run command from specification. If the command succeeds, it returns the"},{"Add":"-- contents of stdout. Otherwise, it throws a type error with the contents"},{"Add":"-- of stderr."},{"Add":"runCmdTC : CmdSpec → TC StdOut"},{"Add":"runCmdTC c = do"},{"Add":" r ← unsafeRunCmdTC c"},{"Add":" let debugPrefix = (\"user.\" ++ CmdSpec.name c)"},{"Add":" case Result.exitCode r of λ"},{"Add":" { exitSuccess → do"},{"Add":" debugPrint (debugPrefix ++ \".stderr\") 10 (strErr (Result.error r) ∷ [])"},{"Add":" pure $ Result.output r"},{"Add":" ; (exitFailure n) → do"},{"Add":" userError c (Result.output r , Result.error r)"},{"Add":" }"},{"Add":""},{"Add":"macro"},{"Add":" runCmd : CmdSpec → Term → TC ⊤"},{"Add":" runCmd c hole = runCmdTC c >>= unify hole ∘ lit ∘ string"}]}},{"FileAdd":{"meta":{"basename":"AnnotatedAST.agda","metadata":0},"context":[{"change":0,"pos":851965}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Annotated reflected syntax."},{"Add":"--"},{"Add":"-- NOTE: This file does not check under --cubical-compatible due to"},{"Add":"-- restrictions in the termination checker. In particular"},{"Add":"-- recursive functions over a universe of types is not supported"},{"Add":"-- by --cubical-compatible."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --safe --with-K #-}"},{"Add":""},{"Add":"module Reflection.AnnotatedAST where"},{"Add":""},{"Add":"open import Level using (Level; 0ℓ; suc; _⊔_)"},{"Add":"open import Effect.Applicative using (RawApplicative)"},{"Add":"open import Data.Bool.Base using (Bool; false; true; if_then_else_)"},{"Add":"open import Data.List.Base using (List; []; _∷_)"},{"Add":"open import Data.List.Relation.Unary.All using (All; _∷_; [])"},{"Add":"open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)"},{"Add":"open import Data.String.Base using (String)"},{"Add":""},{"Add":"open import Reflection hiding (pure)"},{"Add":"open import Reflection.AST.Universe"},{"Add":""},{"Add":"open Clause"},{"Add":"open Pattern"},{"Add":"open Sort"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Annotations and annotated types"},{"Add":""},{"Add":"-- An Annotation is a type indexed by a reflected term."},{"Add":"Annotation : ∀ ℓ → Set (suc ℓ)"},{"Add":"Annotation ℓ = ∀ {u} → ⟦ u ⟧ → Set ℓ"},{"Add":""},{"Add":"-- An annotated type is a family over an Annotation and a reflected term."},{"Add":"Typeₐ : ∀ ℓ → Univ → Set (suc (suc ℓ))"},{"Add":"Typeₐ ℓ u = Annotation ℓ → ⟦ u ⟧ → Set (suc ℓ)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ : Level"},{"Add":" u : Univ"},{"Add":" Ann Ann₁ Ann₂ : Annotation ℓ"},{"Add":""},{"Add":"-- ⟪_⟫ packs up an element of an annotated type with a top-level annotation."},{"Add":"infixr 30 ⟨_⟩_"},{"Add":"data ⟪_⟫ {u} (Tyₐ : Typeₐ ℓ u) : Typeₐ ℓ u where"},{"Add":" ⟨_⟩_ : ∀ {t} → Ann t → Tyₐ Ann t → ⟪ Tyₐ ⟫ Ann t"},{"Add":""},{"Add":"ann : {Tyₐ : Typeₐ ℓ u} {t : ⟦ u ⟧} → ⟪ Tyₐ ⟫ Ann t → Ann t"},{"Add":"ann (⟨ α ⟩ _) = α"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Annotated reflected syntax"},{"Add":""},{"Add":"-- Annotated lists are lists (All) of annotated values. No top-level"},{"Add":"-- annotation added, since we don't want an annotation for every tail"},{"Add":"-- of a list. Instead a top-level annotation is added on the outside."},{"Add":"-- See Argsₐ."},{"Add":"Listₐ : Typeₐ ℓ u → Typeₐ ℓ (⟨list⟩ u)"},{"Add":"Listₐ Tyₐ Ann = All (Tyₐ Ann)"},{"Add":""},{"Add":"-- We define the rest of the annotated types in two variants:"},{"Add":"-- The primed variant which has annotations on subterms, and the"},{"Add":"-- non-primed variant which adds a top-level annotation to the primed"},{"Add":"-- one."},{"Add":""},{"Add":"data Absₐ′ (Tyₐ : Typeₐ ℓ u) : Typeₐ ℓ (⟨abs⟩ u) where"},{"Add":" abs : ∀ {t} x → Tyₐ Ann t → Absₐ′ Tyₐ Ann (abs x t)"},{"Add":""},{"Add":"Absₐ : Typeₐ ℓ u → Typeₐ ℓ (⟨abs⟩ u)"},{"Add":"Absₐ Tyₐ = ⟪ Absₐ′ Tyₐ ⟫"},{"Add":""},{"Add":"data Argₐ′ (Tyₐ : Typeₐ ℓ u) : Typeₐ ℓ (⟨arg⟩ u) where"},{"Add":" arg : ∀ {t} i → Tyₐ Ann t → Argₐ′ Tyₐ Ann (arg i t)"},{"Add":""},{"Add":"Argₐ : Typeₐ ℓ u → Typeₐ ℓ (⟨arg⟩ u)"},{"Add":"Argₐ Tyₐ = ⟪ Argₐ′ Tyₐ ⟫"},{"Add":""},{"Add":"data Namedₐ′ (Tyₐ : Typeₐ ℓ u) : Typeₐ ℓ (⟨named⟩ u) where"},{"Add":" _,_ : ∀ {t} x → Tyₐ Ann t → Namedₐ′ Tyₐ Ann (x , t)"},{"Add":"infixr 4 _,_"},{"Add":""},{"Add":"Namedₐ : Typeₐ ℓ u → Typeₐ ℓ (⟨named⟩ u)"},{"Add":"Namedₐ Tyₐ = ⟪ Namedₐ′ Tyₐ ⟫"},{"Add":""},{"Add":"-- Add a top-level annotation for Args."},{"Add":"Argsₐ : Typeₐ ℓ u → Typeₐ ℓ (⟨list⟩ (⟨arg⟩ u))"},{"Add":"Argsₐ Tyₐ = ⟪ Listₐ (Argₐ Tyₐ) ⟫"},{"Add":""},{"Add":"mutual"},{"Add":" Termₐ : Typeₐ ℓ ⟨term⟩"},{"Add":" Termₐ = ⟪ Termₐ′ ⟫"},{"Add":""},{"Add":" Patternₐ : Typeₐ ℓ ⟨pat⟩"},{"Add":" Patternₐ = ⟪ Patternₐ′ ⟫"},{"Add":""},{"Add":" Sortₐ : Typeₐ ℓ ⟨sort⟩"},{"Add":" Sortₐ = ⟪ Sortₐ′ ⟫"},{"Add":""},{"Add":" Clauseₐ : Typeₐ ℓ ⟨clause⟩"},{"Add":" Clauseₐ = ⟪ Clauseₐ′ ⟫"},{"Add":""},{"Add":" Clausesₐ : Typeₐ ℓ (⟨list⟩ ⟨clause⟩)"},{"Add":" Clausesₐ = ⟪ Listₐ Clauseₐ ⟫"},{"Add":""},{"Add":" Telₐ : Typeₐ ℓ ⟨tel⟩"},{"Add":" Telₐ = ⟪ Listₐ (Namedₐ (Argₐ Termₐ)) ⟫"},{"Add":""},{"Add":" data Termₐ′ {ℓ} : Typeₐ ℓ ⟨term⟩ where"},{"Add":" var : ∀ x {args} → Argsₐ Termₐ Ann args → Termₐ′ Ann (var x args)"},{"Add":" con : ∀ c {args} → Argsₐ Termₐ Ann args → Termₐ′ Ann (con c args)"},{"Add":" def : ∀ f {args} → Argsₐ Termₐ Ann args → Termₐ′ Ann (def f args)"},{"Add":" lam : ∀ v {b} → Absₐ Termₐ Ann b → Termₐ′ Ann (lam v b)"},{"Add":" pat-lam : ∀ {cs args} → Clausesₐ Ann cs → Argsₐ Termₐ Ann args →"},{"Add":" Termₐ′ Ann (pat-lam cs args)"},{"Add":" pi : ∀ {a b} → Argₐ Termₐ Ann a → Absₐ Termₐ Ann b → Termₐ′ Ann (pi a b)"},{"Add":" agda-sort : ∀ {s} → Sortₐ Ann s → Termₐ′ Ann (agda-sort s)"},{"Add":" lit : ∀ l → Termₐ′ Ann (lit l)"},{"Add":" meta : ∀ x {args} → Argsₐ Termₐ Ann args → Termₐ′ Ann (meta x args)"},{"Add":" unknown : Termₐ′ Ann unknown"},{"Add":""},{"Add":" data Clauseₐ′ {ℓ} : Typeₐ ℓ ⟨clause⟩ where"},{"Add":" clause : ∀ {tel ps t} → Telₐ Ann tel → Argsₐ Patternₐ Ann ps →"},{"Add":" Termₐ Ann t → Clauseₐ′ Ann (clause tel ps t)"},{"Add":" absurd-clause : ∀ {tel ps} → Telₐ Ann tel → Argsₐ Patternₐ Ann ps →"},{"Add":" Clauseₐ′ Ann (absurd-clause tel ps)"},{"Add":""},{"Add":" data Sortₐ′ {ℓ} : Typeₐ ℓ ⟨sort⟩ where"},{"Add":" set : ∀ {t} → Termₐ Ann t → Sortₐ′ Ann (set t)"},{"Add":" lit : ∀ n → Sortₐ′ Ann (lit n)"},{"Add":" prop : ∀ {t} → Termₐ Ann t → Sortₐ′ Ann (prop t)"},{"Add":" propLit : ∀ n → Sortₐ′ Ann (propLit n)"},{"Add":" inf : ∀ n → Sortₐ′ Ann (inf n)"},{"Add":" unknown : Sortₐ′ Ann unknown"},{"Add":""},{"Add":" data Patternₐ′ {ℓ} : Typeₐ ℓ ⟨pat⟩ where"},{"Add":" con : ∀ c {ps} → Argsₐ Patternₐ Ann ps → Patternₐ′ Ann (con c ps)"},{"Add":" dot : ∀ {t} → Termₐ Ann t → Patternₐ′ Ann (dot t)"},{"Add":" var : ∀ x → Patternₐ′ Ann (var x)"},{"Add":" lit : ∀ l → Patternₐ′ Ann (lit l)"},{"Add":" proj : ∀ f → Patternₐ′ Ann (proj f)"},{"Add":" absurd : ∀ x → Patternₐ′ Ann (absurd x)"},{"Add":""},{"Add":""},{"Add":"-- Mapping a code in the universe to its corresponding primed and"},{"Add":"-- non-primed annotated type."},{"Add":""},{"Add":"mutual"},{"Add":" Annotated′ : Typeₐ ℓ u"},{"Add":" Annotated′ {u = ⟨term⟩} = Termₐ′"},{"Add":" Annotated′ {u = ⟨pat⟩} = Patternₐ′"},{"Add":" Annotated′ {u = ⟨sort⟩} = Sortₐ′"},{"Add":" Annotated′ {u = ⟨clause⟩} = Clauseₐ′"},{"Add":" Annotated′ {u = ⟨list⟩ u} = Listₐ Annotated"},{"Add":" Annotated′ {u = ⟨arg⟩ u} = Argₐ′ Annotated"},{"Add":" Annotated′ {u = ⟨abs⟩ u} = Absₐ′ Annotated"},{"Add":" Annotated′ {u = ⟨named⟩ u} = Namedₐ′ Annotated"},{"Add":""},{"Add":" Annotated : Typeₐ ℓ u"},{"Add":" Annotated = ⟪ Annotated′ ⟫"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Annotating terms"},{"Add":""},{"Add":"-- An annotation function computes the top-level annotation given a"},{"Add":"-- term annotated at all sub-terms."},{"Add":"AnnotationFun : Annotation ℓ → Set (suc ℓ)"},{"Add":"AnnotationFun Ann = ∀ u {t : ⟦ u ⟧} → Annotated′ Ann t → Ann t"},{"Add":""},{"Add":""},{"Add":"-- Given an annotation function we can do the bottom-up traversal of a"},{"Add":"-- reflected term to compute an annotated version."},{"Add":"module _ (annFun : AnnotationFun Ann) where"},{"Add":""},{"Add":" private"},{"Add":" annotated : {t : ⟦ u ⟧} → Annotated′ Ann t → Annotated Ann t"},{"Add":" annotated ps = ⟨ annFun _ ps ⟩ ps"},{"Add":""},{"Add":" mutual"},{"Add":" annotate′ : (t : ⟦ u ⟧) → Annotated′ Ann t"},{"Add":" annotate′ {⟨term⟩} (var x args) = var x (annotate args)"},{"Add":" annotate′ {⟨term⟩} (con c args) = con c (annotate args)"},{"Add":" annotate′ {⟨term⟩} (def f args) = def f (annotate args)"},{"Add":" annotate′ {⟨term⟩} (lam v t) = lam v (annotate t)"},{"Add":" annotate′ {⟨term⟩} (pat-lam cs args) = pat-lam (annotate cs) (annotate args)"},{"Add":" annotate′ {⟨term⟩} (pi a b) = pi (annotate a) (annotate b)"},{"Add":" annotate′ {⟨term⟩} (agda-sort s) = agda-sort (annotate s)"},{"Add":" annotate′ {⟨term⟩} (lit l) = lit l"},{"Add":" annotate′ {⟨term⟩} (meta x args) = meta x (annotate args)"},{"Add":" annotate′ {⟨term⟩} unknown = unknown"},{"Add":" annotate′ {⟨pat⟩} (con c ps) = con c (annotate ps)"},{"Add":" annotate′ {⟨pat⟩} (dot t) = dot (annotate t)"},{"Add":" annotate′ {⟨pat⟩} (var x) = var x"},{"Add":" annotate′ {⟨pat⟩} (lit l) = lit l"},{"Add":" annotate′ {⟨pat⟩} (proj f) = proj f"},{"Add":" annotate′ {⟨pat⟩} (absurd x) = absurd x"},{"Add":" annotate′ {⟨sort⟩} (set t) = set (annotate t)"},{"Add":" annotate′ {⟨sort⟩} (lit n) = lit n"},{"Add":" annotate′ {⟨sort⟩} (prop t) = prop (annotate t)"},{"Add":" annotate′ {⟨sort⟩} (propLit n) = propLit n"},{"Add":" annotate′ {⟨sort⟩} (inf n) = inf n"},{"Add":" annotate′ {⟨sort⟩} unknown = unknown"},{"Add":" annotate′ {⟨clause⟩} (clause tel ps t) = clause (annotate tel) (annotate ps) (annotate t)"},{"Add":" annotate′ {⟨clause⟩} (absurd-clause tel ps) = absurd-clause (annotate tel) (annotate ps)"},{"Add":" annotate′ {⟨abs⟩ u} (abs x t) = abs x (annotate t)"},{"Add":" annotate′ {⟨arg⟩ u} (arg i t) = arg i (annotate t)"},{"Add":" annotate′ {⟨list⟩ u} [] = []"},{"Add":" annotate′ {⟨list⟩ u} (x ∷ xs) = annotate x ∷ annotate′ xs"},{"Add":" annotate′ {⟨named⟩ u} (x , t) = x , annotate t"},{"Add":""},{"Add":" annotate : (t : ⟦ u ⟧) → Annotated Ann t"},{"Add":" annotate t = annotated (annotate′ t)"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Annotation function combinators"},{"Add":""},{"Add":"-- Mapping over annotations"},{"Add":"mutual"},{"Add":" map′ : ∀ u → (∀ {u} {t : ⟦ u ⟧} → Ann₁ t → Ann₂ t) → {t : ⟦ u ⟧} → Annotated′ Ann₁ t → Annotated′ Ann₂ t"},{"Add":" map′ ⟨term⟩ f (var x args) = var x (map f args)"},{"Add":" map′ ⟨term⟩ f (con c args) = con c (map f args)"},{"Add":" map′ ⟨term⟩ f (def h args) = def h (map f args)"},{"Add":" map′ ⟨term⟩ f (lam v b) = lam v (map f b)"},{"Add":" map′ ⟨term⟩ f (pat-lam cs args) = pat-lam (map f cs) (map f args)"},{"Add":" map′ ⟨term⟩ f (pi a b) = pi (map f a) (map f b)"},{"Add":" map′ ⟨term⟩ f (agda-sort s) = agda-sort (map f s)"},{"Add":" map′ ⟨term⟩ f (lit l) = lit l"},{"Add":" map′ ⟨term⟩ f (meta x args) = meta x (map f args)"},{"Add":" map′ ⟨term⟩ f unknown = unknown"},{"Add":" map′ ⟨pat⟩ f (con c ps) = con c (map f ps)"},{"Add":" map′ ⟨pat⟩ f (dot t) = dot (map f t)"},{"Add":" map′ ⟨pat⟩ f (var x) = var x"},{"Add":" map′ ⟨pat⟩ f (lit l) = lit l"},{"Add":" map′ ⟨pat⟩ f (proj g) = proj g"},{"Add":" map′ ⟨pat⟩ f (absurd x) = absurd x"},{"Add":" map′ ⟨sort⟩ f (set t) = set (map f t)"},{"Add":" map′ ⟨sort⟩ f (lit n) = lit n"},{"Add":" map′ ⟨sort⟩ f (prop t) = prop (map f t)"},{"Add":" map′ ⟨sort⟩ f (propLit n) = propLit n"},{"Add":" map′ ⟨sort⟩ f (inf n) = inf n"},{"Add":" map′ ⟨sort⟩ f unknown = unknown"},{"Add":" map′ ⟨clause⟩ f (clause Γ ps args) = clause (map f Γ) (map f ps) (map f args)"},{"Add":" map′ ⟨clause⟩ f (absurd-clause Γ ps) = absurd-clause (map f Γ) (map f ps)"},{"Add":" map′ (⟨list⟩ u) f [] = []"},{"Add":" map′ (⟨list⟩ u) f (x ∷ xs) = map f x ∷ map′ _ f xs"},{"Add":" map′ (⟨arg⟩ u) f (arg i x) = arg i (map f x)"},{"Add":" map′ (⟨abs⟩ u) f (abs x t) = abs x (map f t)"},{"Add":" map′ (⟨named⟩ u) f (x , t) = x , map f t"},{"Add":""},{"Add":" map : (∀ {u} {t : ⟦ u ⟧} → Ann₁ t → Ann₂ t) → ∀ {u} {t : ⟦ u ⟧} → Annotated Ann₁ t → Annotated Ann₂ t"},{"Add":" map f {u = u} (⟨ α ⟩ t) = ⟨ f α ⟩ map′ u f t"},{"Add":""},{"Add":""},{"Add":"module _ {W : Set ℓ} (ε : W) (_∪_ : W → W → W) where"},{"Add":""},{"Add":" -- This annotation function does nothing except combine ε's with _∪_."},{"Add":" -- Lets you skip the boring cases when defining non-dependent"},{"Add":" -- annotation functions by adding a catch-all calling defaultAnn."},{"Add":" defaultAnn : AnnotationFun (λ _ → W)"},{"Add":" defaultAnn ⟨term⟩ (var x args) = ann args"},{"Add":" defaultAnn ⟨term⟩ (con c args) = ann args"},{"Add":" defaultAnn ⟨term⟩ (def f args) = ann args"},{"Add":" defaultAnn ⟨term⟩ (lam v b) = ann b"},{"Add":" defaultAnn ⟨term⟩ (pat-lam cs args) = ann cs ∪ ann args"},{"Add":" defaultAnn ⟨term⟩ (pi a b) = ann a ∪ ann b"},{"Add":" defaultAnn ⟨term⟩ (agda-sort s) = ann s"},{"Add":" defaultAnn ⟨term⟩ (lit l) = ε"},{"Add":" defaultAnn ⟨term⟩ (meta x args) = ann args"},{"Add":" defaultAnn ⟨term⟩ unknown = ε"},{"Add":" defaultAnn ⟨pat⟩ (con c args) = ann args"},{"Add":" defaultAnn ⟨pat⟩ (dot t) = ann t"},{"Add":" defaultAnn ⟨pat⟩ (var x) = ε"},{"Add":" defaultAnn ⟨pat⟩ (lit l) = ε"},{"Add":" defaultAnn ⟨pat⟩ (proj f) = ε"},{"Add":" defaultAnn ⟨pat⟩ (absurd x) = ε"},{"Add":" defaultAnn ⟨sort⟩ (set t) = ann t"},{"Add":" defaultAnn ⟨sort⟩ (lit n) = ε"},{"Add":" defaultAnn ⟨sort⟩ (prop t) = ann t"},{"Add":" defaultAnn ⟨sort⟩ (propLit n) = ε"},{"Add":" defaultAnn ⟨sort⟩ (inf n) = ε"},{"Add":" defaultAnn ⟨sort⟩ unknown = ε"},{"Add":" defaultAnn ⟨clause⟩ (clause Γ ps t) = ann Γ ∪ (ann ps ∪ ann t)"},{"Add":" defaultAnn ⟨clause⟩ (absurd-clause Γ ps) = ann Γ ∪ ann ps"},{"Add":" defaultAnn (⟨list⟩ u) [] = ε"},{"Add":" defaultAnn (⟨list⟩ u) (x ∷ xs) = ann x ∪ defaultAnn _ xs"},{"Add":" defaultAnn (⟨arg⟩ u) (arg i x) = ann x"},{"Add":" defaultAnn (⟨abs⟩ u) (abs x t) = ann t"},{"Add":" defaultAnn (⟨named⟩ u) (x , t) = ann t"},{"Add":""},{"Add":""},{"Add":"-- Cartisian product of two annotation functions."},{"Add":"infixr 4 _⊗_"},{"Add":"_⊗_ : AnnotationFun Ann₁ → AnnotationFun Ann₂ → AnnotationFun (λ t → Ann₁ t × Ann₂ t)"},{"Add":"(f ⊗ g) u t = f u (map′ u proj₁ t) , g u (map′ u proj₂ t)"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Annotation-driven traversal"},{"Add":""},{"Add":"-- Top-down applicative traversal of an annotated term. Applies an"},{"Add":"-- action (without going into subterms) to terms whose annotation"},{"Add":"-- satisfies a given criterion. Returns an unannotated term."},{"Add":""},{"Add":"module Traverse {M : Set → Set} (appl : RawApplicative M) where"},{"Add":""},{"Add":" open RawApplicative appl"},{"Add":""},{"Add":" module _ (apply? : ∀ {u} {t : ⟦ u ⟧} → Ann t → Bool)"},{"Add":" (action : ∀ {u} {t : ⟦ u ⟧} → Annotated Ann t → M ⟦ u ⟧) where"},{"Add":""},{"Add":" mutual"},{"Add":" traverse′ : {t : ⟦ u ⟧} → Annotated′ Ann t → M ⟦ u ⟧"},{"Add":" traverse′ {⟨term⟩} (var x args) = var x <$> traverse args"},{"Add":" traverse′ {⟨term⟩} (con c args) = con c <$> traverse args"},{"Add":" traverse′ {⟨term⟩} (def f args) = def f <$> traverse args"},{"Add":" traverse′ {⟨term⟩} (lam v b) = lam v <$> traverse b"},{"Add":" traverse′ {⟨term⟩} (pat-lam cs args) = pat-lam <$> traverse cs <*> traverse args"},{"Add":" traverse′ {⟨term⟩} (pi a b) = pi <$> traverse a <*> traverse b"},{"Add":" traverse′ {⟨term⟩} (agda-sort s) = agda-sort <$> traverse s"},{"Add":" traverse′ {⟨term⟩} (lit l) = pure (lit l)"},{"Add":" traverse′ {⟨term⟩} (meta x args) = meta x <$> traverse args"},{"Add":" traverse′ {⟨term⟩} unknown = pure unknown"},{"Add":" traverse′ {⟨pat⟩} (con c args) = con c <$> traverse args"},{"Add":" traverse′ {⟨pat⟩} (dot t) = dot <$> traverse t"},{"Add":" traverse′ {⟨pat⟩} (var x) = pure (var x)"},{"Add":" traverse′ {⟨pat⟩} (lit l) = pure (lit l)"},{"Add":" traverse′ {⟨pat⟩} (proj f) = pure (proj f)"},{"Add":" traverse′ {⟨pat⟩} (absurd x) = pure (absurd x)"},{"Add":" traverse′ {⟨sort⟩} (set t) = set <$> traverse t"},{"Add":" traverse′ {⟨sort⟩} (lit n) = pure (lit n)"},{"Add":" traverse′ {⟨sort⟩} (prop t) = prop <$> traverse t"},{"Add":" traverse′ {⟨sort⟩} (propLit n) = pure (propLit n)"},{"Add":" traverse′ {⟨sort⟩} (inf n) = pure (inf n)"},{"Add":" traverse′ {⟨sort⟩} unknown = pure unknown"},{"Add":" traverse′ {⟨clause⟩} (clause Γ ps t) = clause <$> traverse Γ <*> traverse ps <*> traverse t"},{"Add":" traverse′ {⟨clause⟩} (absurd-clause Γ ps) = absurd-clause <$> traverse Γ <*> traverse ps"},{"Add":" traverse′ {⟨list⟩ u} [] = pure []"},{"Add":" traverse′ {⟨list⟩ u} (x ∷ xs) = _∷_ <$> traverse x <*> traverse′ xs"},{"Add":" traverse′ {⟨arg⟩ u} (arg i x) = arg i <$> traverse x"},{"Add":" traverse′ {⟨abs⟩ u} (abs x t) = abs x <$> traverse t"},{"Add":" traverse′ {⟨named⟩ u} (x , t) = x ,_ <$> traverse t"},{"Add":""},{"Add":" traverse : {t : ⟦ u ⟧} → Annotated Ann t → M ⟦ u ⟧"},{"Add":" traverse t@(⟨ α ⟩ tₐ) = if apply? α then action t else traverse′ tₐ"}]}},{"FileAdd":{"meta":{"basename":"AnnotatedAST","metadata":576},"context":[{"change":0,"pos":851965}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Free.agda","metadata":0},"context":[{"change":0,"pos":883446}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Computing free variable annotations on reflected syntax."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --safe --with-K #-}"},{"Add":""},{"Add":"module Reflection.AnnotatedAST.Free where"},{"Add":""},{"Add":"open import Data.Bool.Base using (if_then_else_)"},{"Add":"open import Data.Nat.Base using (ℕ; _∸_; compare; _<ᵇ_; less; equal; greater)"},{"Add":"open import Data.List.Base using (List; []; _∷_; [_]; concatMap; length)"},{"Add":"open import Data.List.Relation.Unary.All using (_∷_)"},{"Add":"open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)"},{"Add":"open import Data.String.Base using (String)"},{"Add":""},{"Add":"open import Reflection"},{"Add":"open import Reflection.AST.Universe"},{"Add":"open import Reflection.AnnotatedAST"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Free variable sets as lists of natural numbers"},{"Add":""},{"Add":"FVs : Set"},{"Add":"FVs = List ℕ -- ordered, no duplicates"},{"Add":""},{"Add":"private"},{"Add":""},{"Add":" infixr 3 _∪_"},{"Add":""},{"Add":" _∪_ : FVs → FVs → FVs"},{"Add":" [] ∪ ys = ys"},{"Add":" xs ∪ [] = xs"},{"Add":" x ∷ xs ∪ y ∷ ys with compare x y | x ∷ xs ∪ ys"},{"Add":" ... | less x _ | _ = x ∷ (xs ∪ y ∷ ys)"},{"Add":" ... | equal x | _ = x ∷ (xs ∪ ys)"},{"Add":" ... | greater y _ | rec = y ∷ rec"},{"Add":""},{"Add":" insert : ℕ → FVs → FVs"},{"Add":" insert x [] = x ∷ []"},{"Add":" insert x (y ∷ xs) with compare x y"},{"Add":" ... | less x k = x ∷ y ∷ xs"},{"Add":" ... | equal x = y ∷ xs"},{"Add":" ... | greater y k = y ∷ insert x xs"},{"Add":""},{"Add":" close : ℕ → FVs → FVs"},{"Add":" close k = concatMap λ x → if x <ᵇ k then [] else [ x ∸ k ]"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Annotation function computing free variables"},{"Add":""},{"Add":"freeVars : AnnotationFun (λ _ → FVs)"},{"Add":"freeVars ⟨term⟩ (var x (⟨ fv ⟩ _)) = insert x fv"},{"Add":"freeVars ⟨pat⟩ (var x) = x ∷ []"},{"Add":"freeVars ⟨pat⟩ (absurd x) = x ∷ []"},{"Add":" -- Note: variables are bound in the clause telescope, so we treat pattern variables as free"},{"Add":"freeVars ⟨clause⟩ (clause {tel = Γ} (⟨ fvΓ ⟩ _) (⟨ fvps ⟩ _) (⟨ fvt ⟩ _)) = fvΓ ∪ close (length Γ) (fvps ∪ fvt)"},{"Add":"freeVars ⟨clause⟩ (absurd-clause {tel = Γ} (⟨ fvΓ ⟩ _) (⟨ fvps ⟩ _)) = fvΓ ∪ close (length Γ) fvps"},{"Add":"freeVars (⟨abs⟩ u) (abs _ (⟨ fv ⟩ _)) = close 1 fv"},{"Add":"freeVars ⟨tel⟩ (⟨ fv ⟩ _ ∷ xs) = fv ∪ close 1 (freeVars _ xs)"},{"Add":"freeVars u t = defaultAnn [] _∪_ u t"},{"Add":""},{"Add":"annotateFVs : ∀ {u} → (t : ⟦ u ⟧) → Annotated (λ _ → FVs) t"},{"Add":"annotateFVs = annotate freeVars"}]}},{"FileAdd":{"meta":{"basename":"AST.agda","metadata":0},"context":[{"change":0,"pos":851965}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The reflected abstract syntax tree"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST where"},{"Add":""},{"Add":"import Agda.Builtin.Reflection as Builtin"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Names, Metas, and Literals re-exported publicly"},{"Add":""},{"Add":"open import Reflection.AST.Abstraction as Abstraction public"},{"Add":" using (Abs; abs)"},{"Add":"open import Reflection.AST.Argument as Argument public"},{"Add":" using (Arg; arg; Args; vArg; hArg; iArg; defaultModality)"},{"Add":"open import Reflection.AST.Definition as Definition public"},{"Add":" using (Definition)"},{"Add":"open import Reflection.AST.Meta as Meta public"},{"Add":" using (Meta)"},{"Add":"open import Reflection.AST.Name as Name public"},{"Add":" using (Name; Names)"},{"Add":"open import Reflection.AST.Literal as Literal public"},{"Add":" using (Literal)"},{"Add":"open import Reflection.AST.Pattern as Pattern public"},{"Add":" using (Pattern)"},{"Add":"open import Reflection.AST.Term as Term public"},{"Add":" using (Term; Type; Clause; Clauses; Sort)"},{"Add":""},{"Add":"import Reflection.AST.Argument.Modality as Modality"},{"Add":"import Reflection.AST.Argument.Quantity as Quantity"},{"Add":"import Reflection.AST.Argument.Relevance as Relevance"},{"Add":"import Reflection.AST.Argument.Visibility as Visibility"},{"Add":"import Reflection.AST.Argument.Information as Information"},{"Add":""},{"Add":"open Definition.Definition public"},{"Add":"open Information.ArgInfo public"},{"Add":"open Literal.Literal public"},{"Add":"open Modality.Modality public"},{"Add":"open Quantity.Quantity public"},{"Add":"open Relevance.Relevance public"},{"Add":"open Term.Term public"},{"Add":"open Visibility.Visibility public"},{"Add":""},{"Add":"open import Reflection.AST.Show public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Fixity"},{"Add":""},{"Add":"open Builtin public"},{"Add":" using (non-assoc; related; unrelated; fixity)"},{"Add":" renaming"},{"Add":" ( left-assoc to assocˡ"},{"Add":" ; right-assoc to assocʳ"},{"Add":" ; primQNameFixity to getFixity"},{"Add":" )"}]}},{"FileAdd":{"meta":{"basename":"AST","metadata":576},"context":[{"change":0,"pos":851965}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Universe.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- A universe for the types involved in the reflected syntax."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.Universe where"},{"Add":""},{"Add":"open import Data.List.Base using (List)"},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Data.Product.Base using (_×_)"},{"Add":"open import Reflection.AST.Argument using (Arg)"},{"Add":"open import Reflection.AST.Abstraction using (Abs)"},{"Add":"open import Reflection.AST.Term using (Term; Pattern; Sort; Clause)"},{"Add":""},{"Add":"data Univ : Set where"},{"Add":" ⟨term⟩ : Univ"},{"Add":" ⟨pat⟩ : Univ"},{"Add":" ⟨sort⟩ : Univ"},{"Add":" ⟨clause⟩ : Univ"},{"Add":" ⟨list⟩ : Univ → Univ"},{"Add":" ⟨arg⟩ : Univ → Univ"},{"Add":" ⟨abs⟩ : Univ → Univ"},{"Add":" ⟨named⟩ : Univ → Univ"},{"Add":""},{"Add":"pattern ⟨tel⟩ = ⟨list⟩ (⟨named⟩ (⟨arg⟩ ⟨term⟩))"},{"Add":""},{"Add":"⟦_⟧ : Univ → Set"},{"Add":"⟦ ⟨term⟩ ⟧ = Term"},{"Add":"⟦ ⟨pat⟩ ⟧ = Pattern"},{"Add":"⟦ ⟨sort⟩ ⟧ = Sort"},{"Add":"⟦ ⟨clause⟩ ⟧ = Clause"},{"Add":"⟦ ⟨list⟩ k ⟧ = List ⟦ k ⟧"},{"Add":"⟦ ⟨arg⟩ k ⟧ = Arg ⟦ k ⟧"},{"Add":"⟦ ⟨abs⟩ k ⟧ = Abs ⟦ k ⟧"},{"Add":"⟦ ⟨named⟩ k ⟧ = String × ⟦ k ⟧"}]}},{"FileAdd":{"meta":{"basename":"Traversal.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- de Bruijn-aware generic traversal of reflected terms."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Effect.Applicative using (RawApplicative)"},{"Add":""},{"Add":"module Reflection.AST.Traversal"},{"Add":" {F : Set → Set} (AppF : RawApplicative F) where"},{"Add":""},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc; _+_)"},{"Add":"open import Data.List.Base using (List; []; _∷_; _++_; reverse; length)"},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Function.Base using (_∘_)"},{"Add":"open import Reflection hiding (pure)"},{"Add":""},{"Add":"open RawApplicative AppF"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Context representation"},{"Add":""},{"Add":"-- Track both number of variables and actual context, to avoid having to"},{"Add":"-- compute the length of the context everytime it's needed."},{"Add":"record Cxt : Set where"},{"Add":" constructor _,_"},{"Add":" pattern"},{"Add":" field"},{"Add":" len : ℕ"},{"Add":" context : List (String × Arg Term)"},{"Add":"infixr 4 _,_"},{"Add":""},{"Add":"private"},{"Add":" _∷cxt_ : String × Arg Term → Cxt → Cxt"},{"Add":" e ∷cxt (n , Γ) = (suc n , e ∷ Γ)"},{"Add":""},{"Add":" _++cxt_ : List (String × Arg Term) → Cxt → Cxt"},{"Add":" es ++cxt (n , Γ) = (length es + n , es ++ Γ)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Actions"},{"Add":""},{"Add":"Action : Set → Set"},{"Add":"Action A = Cxt → A → F A"},{"Add":""},{"Add":"-- A traversal gets to operate on variables, metas, and names."},{"Add":"record Actions : Set where"},{"Add":" field"},{"Add":" onVar : Action ℕ"},{"Add":" onMeta : Action Meta"},{"Add":" onCon : Action Name"},{"Add":" onDef : Action Name"},{"Add":""},{"Add":"-- Default action: do nothing."},{"Add":"defaultActions : Actions"},{"Add":"defaultActions .Actions.onVar _ = pure"},{"Add":"defaultActions .Actions.onMeta _ = pure"},{"Add":"defaultActions .Actions.onCon _ = pure"},{"Add":"defaultActions .Actions.onDef _ = pure"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Traversal functions"},{"Add":""},{"Add":"module _ (actions : Actions) where"},{"Add":""},{"Add":" open Actions actions"},{"Add":""},{"Add":" traverseTerm : Action Term"},{"Add":" traverseSort : Action Sort"},{"Add":" traversePattern : Action Pattern"},{"Add":" traverseArgs : Action (List (Arg Term))"},{"Add":" traverseArg : Action (Arg Term)"},{"Add":" traversePats : Action (List (Arg Pattern))"},{"Add":" traverseAbs : Arg Term → Action (Abs Term)"},{"Add":" traverseClauses : Action Clauses"},{"Add":" traverseClause : Action Clause"},{"Add":" traverseTel : Action (List (String × Arg Term))"},{"Add":""},{"Add":" traverseTerm Γ (var x args) = var <$> onVar Γ x ⊛ traverseArgs Γ args"},{"Add":" traverseTerm Γ (con c args) = con <$> onCon Γ c ⊛ traverseArgs Γ args"},{"Add":" traverseTerm Γ (def f args) = def <$> onDef Γ f ⊛ traverseArgs Γ args"},{"Add":" traverseTerm Γ (pat-lam cs args) = pat-lam <$> traverseClauses Γ cs ⊛ traverseArgs Γ args"},{"Add":" traverseTerm Γ (pi a b) = pi <$> traverseArg Γ a ⊛ traverseAbs a Γ b"},{"Add":" traverseTerm Γ (agda-sort s) = agda-sort <$> traverseSort Γ s"},{"Add":" traverseTerm Γ (meta x args) = meta <$> onMeta Γ x ⊛ traverseArgs Γ args"},{"Add":" traverseTerm Γ t@(lit _) = pure t"},{"Add":" traverseTerm Γ t@unknown = pure t"},{"Add":" traverseTerm Γ (lam v t) = lam v <$> traverseAbs (arg (arg-info v m) unknown) Γ t"},{"Add":" where"},{"Add":" m = defaultModality"},{"Add":""},{"Add":" traverseArg Γ (arg i t) = arg i <$> traverseTerm Γ t"},{"Add":" traverseArgs Γ [] = pure []"},{"Add":" traverseArgs Γ (a ∷ as) = _∷_ <$> traverseArg Γ a ⊛ traverseArgs Γ as"},{"Add":""},{"Add":" traverseAbs ty Γ (abs x t) = abs x <$> traverseTerm ((x , ty) ∷cxt Γ) t"},{"Add":""},{"Add":" traverseClauses Γ [] = pure []"},{"Add":" traverseClauses Γ (c ∷ cs) = _∷_ <$> traverseClause Γ c ⊛ traverseClauses Γ cs"},{"Add":""},{"Add":" traverseClause Γ (Clause.clause tel ps t) ="},{"Add":" Clause.clause <$> traverseTel Γ tel"},{"Add":" ⊛ traversePats Γ′ ps"},{"Add":" ⊛ traverseTerm Γ′ t"},{"Add":" where Γ′ = reverse tel ++cxt Γ"},{"Add":" traverseClause Γ (Clause.absurd-clause tel ps) ="},{"Add":" Clause.absurd-clause <$> traverseTel Γ tel"},{"Add":" ⊛ traversePats Γ′ ps"},{"Add":" where Γ′ = reverse tel ++cxt Γ"},{"Add":""},{"Add":" traverseTel Γ [] = pure []"},{"Add":" traverseTel Γ ((x , ty) ∷ tel) ="},{"Add":" _∷_ ∘ (x ,_) <$> traverseArg Γ ty ⊛ traverseTel ((x , ty) ∷cxt Γ) tel"},{"Add":""},{"Add":" traverseSort Γ (Sort.set t) = Sort.set <$> traverseTerm Γ t"},{"Add":" traverseSort Γ t@(Sort.lit _) = pure t"},{"Add":" traverseSort Γ (Sort.prop t) = Sort.prop <$> traverseTerm Γ t"},{"Add":" traverseSort Γ t@(Sort.propLit _) = pure t"},{"Add":" traverseSort Γ t@(Sort.inf _) = pure t"},{"Add":" traverseSort Γ t@Sort.unknown = pure t"},{"Add":""},{"Add":" traversePattern Γ (Pattern.con c ps) = Pattern.con <$> onCon Γ c ⊛ traversePats Γ ps"},{"Add":" traversePattern Γ (Pattern.dot t) = Pattern.dot <$> traverseTerm Γ t"},{"Add":" traversePattern Γ (Pattern.var x) = Pattern.var <$> onVar Γ x"},{"Add":" traversePattern Γ p@(Pattern.lit _) = pure p"},{"Add":" traversePattern Γ p@(Pattern.proj _) = pure p"},{"Add":" traversePattern Γ (Pattern.absurd x) = Pattern.absurd <$> onVar Γ x"},{"Add":""},{"Add":" traversePats Γ [] = pure []"},{"Add":" traversePats Γ (arg i p ∷ ps) = _∷_ ∘ arg i <$> traversePattern Γ p ⊛ traversePats Γ ps"}]}},{"FileAdd":{"meta":{"basename":"Term.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Terms used in the reflection machinery"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.Term where"},{"Add":""},{"Add":"open import Data.List.Base as List hiding (_++_)"},{"Add":"open import Data.List.Properties using (∷-dec)"},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc)"},{"Add":"import Data.Nat.Properties as ℕ"},{"Add":"open import Data.Product.Base using (_×_; _,_; <_,_>; uncurry; map₁)"},{"Add":"open import Data.Product.Properties using (,-injective)"},{"Add":"open import Data.Maybe.Base using (Maybe; just; nothing)"},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Data.String.Properties as String hiding (_≟_)"},{"Add":"open import Relation.Nullary.Decidable using (map′; _×-dec_; yes; no)"},{"Add":"open import Relation.Binary.Definitions using (Decidable; DecidableEquality)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)"},{"Add":""},{"Add":"open import Reflection.AST.Abstraction"},{"Add":"open import Reflection.AST.Argument"},{"Add":"open import Reflection.AST.Argument.Information using (visibility)"},{"Add":"open import Reflection.AST.Argument.Visibility as Visibility hiding (_≟_)"},{"Add":"import Reflection.AST.Literal as Literal"},{"Add":"import Reflection.AST.Meta as Meta"},{"Add":"open import Reflection.AST.Name as Name using (Name)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting the builtin type and constructors"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection as Builtin public"},{"Add":" using (Sort; Type; Term; Clause; Pattern)"},{"Add":""},{"Add":"open Term public"},{"Add":" renaming (agda-sort to sort)"},{"Add":""},{"Add":"open Sort public"},{"Add":"open Clause public"},{"Add":"open Pattern public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Handy synonyms"},{"Add":""},{"Add":"Clauses : Set"},{"Add":"Clauses = List Clause"},{"Add":""},{"Add":"Telescope : Set"},{"Add":"Telescope = List (String × Arg Type)"},{"Add":""},{"Add":"-- Pattern synonyms for more compact presentation"},{"Add":""},{"Add":"pattern vLam s t = lam visible (abs s t)"},{"Add":"pattern hLam s t = lam hidden (abs s t)"},{"Add":"pattern iLam s t = lam instance′ (abs s t)"},{"Add":"pattern Π[_∶_]_ s a ty = pi a (abs s ty)"},{"Add":"pattern vΠ[_∶_]_ s a ty = Π[ s ∶ (vArg a) ] ty"},{"Add":"pattern hΠ[_∶_]_ s a ty = Π[ s ∶ (hArg a) ] ty"},{"Add":"pattern iΠ[_∶_]_ s a ty = Π[ s ∶ (iArg a) ] ty"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Utility functions"},{"Add":""},{"Add":"getName : Term → Maybe Name"},{"Add":"getName (con c args) = just c"},{"Add":"getName (def f args) = just f"},{"Add":"getName _ = nothing"},{"Add":""},{"Add":"-- \"n ⋯⟨∷⟩ xs\" prepends \"n\" visible unknown arguments to the list of"},{"Add":"-- arguments. Useful when constructing the list of arguments for a"},{"Add":"-- function with initial inferrable arguments."},{"Add":"infixr 5 _⋯⟨∷⟩_"},{"Add":"_⋯⟨∷⟩_ : ℕ → Args Term → Args Term"},{"Add":"zero ⋯⟨∷⟩ xs = xs"},{"Add":"suc i ⋯⟨∷⟩ xs = unknown ⟨∷⟩ (i ⋯⟨∷⟩ xs)"},{"Add":"{-# INLINE _⋯⟨∷⟩_ #-}"},{"Add":""},{"Add":"-- \"n ⋯⟅∷⟆ xs\" prepends \"n\" hidden unknown arguments to the list of"},{"Add":"-- arguments. Useful when constructing the list of arguments for a"},{"Add":"-- function with initial implicit arguments."},{"Add":"infixr 5 _⋯⟅∷⟆_"},{"Add":"_⋯⟅∷⟆_ : ℕ → Args Term → Args Term"},{"Add":"zero ⋯⟅∷⟆ xs = xs"},{"Add":"suc i ⋯⟅∷⟆ xs = unknown ⟅∷⟆ (i ⋯⟅∷⟆ xs)"},{"Add":"{-# INLINE _⋯⟅∷⟆_ #-}"},{"Add":""},{"Add":"-- Strips off any pi bindings returning the list of variables removed"},{"Add":"-- and the eventual body of the expression, e.g."},{"Add":"--"},{"Add":"-- stripPis `∀ x {y} → f x y` = ([\"x\", \"y\"], f x y)"},{"Add":"stripPis : Term → List (String × Arg Type) × Term"},{"Add":"stripPis (Π[ s ∶ t ] x) = map₁ ((s , t) ∷_) (stripPis x)"},{"Add":"stripPis x = [] , x"},{"Add":""},{"Add":"prependLams : List (String × Visibility) → Term → Term"},{"Add":"prependLams xs t = foldr (λ {(s , v) t → lam v (abs s t)}) t (reverse xs)"},{"Add":""},{"Add":"prependHLams : List String → Term → Term"},{"Add":"prependHLams vs = prependLams (List.map (_, hidden) vs)"},{"Add":""},{"Add":"prependVLams : List String → Term → Term"},{"Add":"prependVLams vs = prependLams (List.map (_, visible) vs)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Decidable equality"},{"Add":""},{"Add":"clause-injective₁ : ∀ {tel tel′ ps ps′ b b′} → clause tel ps b ≡ clause tel′ ps′ b′ → tel ≡ tel′"},{"Add":"clause-injective₁ refl = refl"},{"Add":""},{"Add":"clause-injective₂ : ∀ {tel tel′ ps ps′ b b′} → clause tel ps b ≡ clause tel′ ps′ b′ → ps ≡ ps′"},{"Add":"clause-injective₂ refl = refl"},{"Add":""},{"Add":"clause-injective₃ : ∀ {tel tel′ ps ps′ b b′} → clause tel ps b ≡ clause tel′ ps′ b′ → b ≡ b′"},{"Add":"clause-injective₃ refl = refl"},{"Add":""},{"Add":"clause-injective : ∀ {tel tel′ ps ps′ b b′} → clause tel ps b ≡ clause tel′ ps′ b′ → tel ≡ tel′ × ps ≡ ps′ × b ≡ b′"},{"Add":"clause-injective = < clause-injective₁ , < clause-injective₂ , clause-injective₃ > >"},{"Add":""},{"Add":"absurd-clause-injective₁ : ∀ {tel tel′ ps ps′} → absurd-clause tel ps ≡ absurd-clause tel′ ps′ → tel ≡ tel′"},{"Add":"absurd-clause-injective₁ refl = refl"},{"Add":""},{"Add":"absurd-clause-injective₂ : ∀ {tel tel′ ps ps′} → absurd-clause tel ps ≡ absurd-clause tel′ ps′ → ps ≡ ps′"},{"Add":"absurd-clause-injective₂ refl = refl"},{"Add":""},{"Add":"absurd-clause-injective : ∀ {tel tel′ ps ps′} → absurd-clause tel ps ≡ absurd-clause tel′ ps′ → tel ≡ tel′ × ps ≡ ps′"},{"Add":"absurd-clause-injective = < absurd-clause-injective₁ , absurd-clause-injective₂ >"},{"Add":""},{"Add":"infix 4 _≟-AbsTerm_ _≟-AbsType_ _≟-ArgTerm_ _≟-ArgType_ _≟-Args_"},{"Add":" _≟-Clause_ _≟-Clauses_ _≟_"},{"Add":" _≟-Sort_ _≟-Pattern_ _≟-Patterns_ _≟-Telescope_"},{"Add":""},{"Add":"_≟-AbsTerm_ : DecidableEquality (Abs Term)"},{"Add":"_≟-AbsType_ : DecidableEquality (Abs Type)"},{"Add":"_≟-ArgTerm_ : DecidableEquality (Arg Term)"},{"Add":"_≟-ArgType_ : DecidableEquality (Arg Type)"},{"Add":"_≟-Args_ : DecidableEquality (Args Term)"},{"Add":"_≟-Clause_ : DecidableEquality Clause"},{"Add":"_≟-Clauses_ : DecidableEquality Clauses"},{"Add":"_≟_ : DecidableEquality Term"},{"Add":"_≟-Sort_ : DecidableEquality Sort"},{"Add":"_≟-Patterns_ : DecidableEquality (Args Pattern)"},{"Add":"_≟-Pattern_ : DecidableEquality Pattern"},{"Add":""},{"Add":"-- Decidable equality 'transformers'"},{"Add":"-- We need to inline these because the terms are not sized so"},{"Add":"-- termination would not obvious if we were to use higher-order"},{"Add":"-- functions such as Data.List.Properties' ≡-dec"},{"Add":""},{"Add":"abs s a ≟-AbsTerm abs s′ a′ = unAbs-dec (a ≟ a′)"},{"Add":"abs s a ≟-AbsType abs s′ a′ = unAbs-dec (a ≟ a′)"},{"Add":"arg i a ≟-ArgTerm arg i′ a′ = unArg-dec (a ≟ a′)"},{"Add":"arg i a ≟-ArgType arg i′ a′ = unArg-dec (a ≟ a′)"},{"Add":""},{"Add":"[] ≟-Args [] = yes refl"},{"Add":"(x ∷ xs) ≟-Args (y ∷ ys) = ∷-dec (x ≟-ArgTerm y) (xs ≟-Args ys)"},{"Add":"[] ≟-Args (_ ∷ _) = no λ()"},{"Add":"(_ ∷ _) ≟-Args [] = no λ()"},{"Add":""},{"Add":"[] ≟-Clauses [] = yes refl"},{"Add":"(x ∷ xs) ≟-Clauses (y ∷ ys) = ∷-dec (x ≟-Clause y) (xs ≟-Clauses ys)"},{"Add":"[] ≟-Clauses (_ ∷ _) = no λ()"},{"Add":"(_ ∷ _) ≟-Clauses [] = no λ()"},{"Add":""},{"Add":"_≟-Telescope_ : DecidableEquality Telescope"},{"Add":"[] ≟-Telescope [] = yes refl"},{"Add":"((x , t) ∷ tel) ≟-Telescope ((x′ , t′) ∷ tel′) = ∷-dec"},{"Add":" (map′ (uncurry (cong₂ _,_)) ,-injective ((x String.≟ x′) ×-dec (t ≟-ArgTerm t′)))"},{"Add":" (tel ≟-Telescope tel′)"},{"Add":"[] ≟-Telescope (_ ∷ _) = no λ ()"},{"Add":"(_ ∷ _) ≟-Telescope [] = no λ ()"},{"Add":""},{"Add":"clause tel ps b ≟-Clause clause tel′ ps′ b′ ="},{"Add":" map′ (λ (tel≡tel′ , ps≡ps′ , b≡b′) → cong₂ (uncurry clause) (cong₂ _,_ tel≡tel′ ps≡ps′) b≡b′)"},{"Add":" clause-injective"},{"Add":" (tel ≟-Telescope tel′ ×-dec ps ≟-Patterns ps′ ×-dec b ≟ b′)"},{"Add":"absurd-clause tel ps ≟-Clause absurd-clause tel′ ps′ ="},{"Add":" map′ (uncurry (cong₂ absurd-clause))"},{"Add":" absurd-clause-injective"},{"Add":" (tel ≟-Telescope tel′ ×-dec ps ≟-Patterns ps′)"},{"Add":"clause _ _ _ ≟-Clause absurd-clause _ _ = no λ()"},{"Add":"absurd-clause _ _ ≟-Clause clause _ _ _ = no λ()"},{"Add":""},{"Add":"var-injective₁ : ∀ {x x′ args args′} → Term.var x args ≡ var x′ args′ → x ≡ x′"},{"Add":"var-injective₁ refl = refl"},{"Add":""},{"Add":"var-injective₂ : ∀ {x x′ args args′} → Term.var x args ≡ var x′ args′ → args ≡ args′"},{"Add":"var-injective₂ refl = refl"},{"Add":""},{"Add":"var-injective : ∀ {x x′ args args′} → var x args ≡ var x′ args′ → x ≡ x′ × args ≡ args′"},{"Add":"var-injective = < var-injective₁ , var-injective₂ >"},{"Add":""},{"Add":"con-injective₁ : ∀ {c c′ args args′} → Term.con c args ≡ con c′ args′ → c ≡ c′"},{"Add":"con-injective₁ refl = refl"},{"Add":""},{"Add":"con-injective₂ : ∀ {c c′ args args′} → Term.con c args ≡ con c′ args′ → args ≡ args′"},{"Add":"con-injective₂ refl = refl"},{"Add":""},{"Add":"con-injective : ∀ {c c′ args args′} → con c args ≡ con c′ args′ → c ≡ c′ × args ≡ args′"},{"Add":"con-injective = < con-injective₁ , con-injective₂ >"},{"Add":""},{"Add":"def-injective₁ : ∀ {f f′ args args′} → def f args ≡ def f′ args′ → f ≡ f′"},{"Add":"def-injective₁ refl = refl"},{"Add":""},{"Add":"def-injective₂ : ∀ {f f′ args args′} → def f args ≡ def f′ args′ → args ≡ args′"},{"Add":"def-injective₂ refl = refl"},{"Add":""},{"Add":"def-injective : ∀ {f f′ args args′} → def f args ≡ def f′ args′ → f ≡ f′ × args ≡ args′"},{"Add":"def-injective = < def-injective₁ , def-injective₂ >"},{"Add":""},{"Add":"meta-injective₁ : ∀ {x x′ args args′} → meta x args ≡ meta x′ args′ → x ≡ x′"},{"Add":"meta-injective₁ refl = refl"},{"Add":""},{"Add":"meta-injective₂ : ∀ {x x′ args args′} → meta x args ≡ meta x′ args′ → args ≡ args′"},{"Add":"meta-injective₂ refl = refl"},{"Add":""},{"Add":"meta-injective : ∀ {x x′ args args′} → meta x args ≡ meta x′ args′ → x ≡ x′ × args ≡ args′"},{"Add":"meta-injective = < meta-injective₁ , meta-injective₂ >"},{"Add":""},{"Add":"lam-injective₁ : ∀ {v v′ t t′} → lam v t ≡ lam v′ t′ → v ≡ v′"},{"Add":"lam-injective₁ refl = refl"},{"Add":""},{"Add":"lam-injective₂ : ∀ {v v′ t t′} → lam v t ≡ lam v′ t′ → t ≡ t′"},{"Add":"lam-injective₂ refl = refl"},{"Add":""},{"Add":"lam-injective : ∀ {v v′ t t′} → lam v t ≡ lam v′ t′ → v ≡ v′ × t ≡ t′"},{"Add":"lam-injective = < lam-injective₁ , lam-injective₂ >"},{"Add":""},{"Add":"pat-lam-injective₁ : ∀ {cs cs′ args args′} → pat-lam cs args ≡ pat-lam cs′ args′ → cs ≡ cs′"},{"Add":"pat-lam-injective₁ refl = refl"},{"Add":""},{"Add":"pat-lam-injective₂ : ∀ {cs cs′ args args′} → pat-lam cs args ≡ pat-lam cs′ args′ → args ≡ args′"},{"Add":"pat-lam-injective₂ refl = refl"},{"Add":""},{"Add":"pat-lam-injective : ∀ {cs cs′ args args′} → pat-lam cs args ≡ pat-lam cs′ args′ → cs ≡ cs′ × args ≡ args′"},{"Add":"pat-lam-injective = < pat-lam-injective₁ , pat-lam-injective₂ >"},{"Add":""},{"Add":"pi-injective₁ : ∀ {t₁ t₁′ t₂ t₂′} → pi t₁ t₂ ≡ pi t₁′ t₂′ → t₁ ≡ t₁′"},{"Add":"pi-injective₁ refl = refl"},{"Add":""},{"Add":"pi-injective₂ : ∀ {t₁ t₁′ t₂ t₂′} → pi t₁ t₂ ≡ pi t₁′ t₂′ → t₂ ≡ t₂′"},{"Add":"pi-injective₂ refl = refl"},{"Add":""},{"Add":"pi-injective : ∀ {t₁ t₁′ t₂ t₂′} → pi t₁ t₂ ≡ pi t₁′ t₂′ → t₁ ≡ t₁′ × t₂ ≡ t₂′"},{"Add":"pi-injective = < pi-injective₁ , pi-injective₂ >"},{"Add":""},{"Add":"sort-injective : ∀ {x y} → sort x ≡ sort y → x ≡ y"},{"Add":"sort-injective refl = refl"},{"Add":""},{"Add":"lit-injective : ∀ {x y} → lit x ≡ lit y → x ≡ y"},{"Add":"lit-injective refl = refl"},{"Add":""},{"Add":"set-injective : ∀ {x y} → set x ≡ set y → x ≡ y"},{"Add":"set-injective refl = refl"},{"Add":""},{"Add":"slit-injective : ∀ {x y} → Sort.lit x ≡ lit y → x ≡ y"},{"Add":"slit-injective refl = refl"},{"Add":""},{"Add":"prop-injective : ∀ {x y} → prop x ≡ prop y → x ≡ y"},{"Add":"prop-injective refl = refl"},{"Add":""},{"Add":"propLit-injective : ∀ {x y} → propLit x ≡ propLit y → x ≡ y"},{"Add":"propLit-injective refl = refl"},{"Add":""},{"Add":"inf-injective : ∀ {x y} → inf x ≡ inf y → x ≡ y"},{"Add":"inf-injective refl = refl"},{"Add":""},{"Add":"var x args ≟ var x′ args′ ="},{"Add":" map′ (uncurry (cong₂ var)) var-injective (x ℕ.≟ x′ ×-dec args ≟-Args args′)"},{"Add":"con c args ≟ con c′ args′ ="},{"Add":" map′ (uncurry (cong₂ con)) con-injective (c Name.≟ c′ ×-dec args ≟-Args args′)"},{"Add":"def f args ≟ def f′ args′ ="},{"Add":" map′ (uncurry (cong₂ def)) def-injective (f Name.≟ f′ ×-dec args ≟-Args args′)"},{"Add":"meta x args ≟ meta x′ args′ ="},{"Add":" map′ (uncurry (cong₂ meta)) meta-injective (x Meta.≟ x′ ×-dec args ≟-Args args′)"},{"Add":"lam v t ≟ lam v′ t′ ="},{"Add":" map′ (uncurry (cong₂ lam)) lam-injective (v Visibility.≟ v′ ×-dec t ≟-AbsTerm t′)"},{"Add":"pat-lam cs args ≟ pat-lam cs′ args′ ="},{"Add":" map′ (uncurry (cong₂ pat-lam)) pat-lam-injective (cs ≟-Clauses cs′ ×-dec args ≟-Args args′)"},{"Add":"pi t₁ t₂ ≟ pi t₁′ t₂′ ="},{"Add":" map′ (uncurry (cong₂ pi)) pi-injective (t₁ ≟-ArgType t₁′ ×-dec t₂ ≟-AbsType t₂′)"},{"Add":"sort s ≟ sort s′ = map′ (cong sort) sort-injective (s ≟-Sort s′)"},{"Add":"lit l ≟ lit l′ = map′ (cong lit) lit-injective (l Literal.≟ l′)"},{"Add":"unknown ≟ unknown = yes refl"},{"Add":""},{"Add":"var x args ≟ con c args′ = no λ()"},{"Add":"var x args ≟ def f args′ = no λ()"},{"Add":"var x args ≟ lam v t = no λ()"},{"Add":"var x args ≟ pi t₁ t₂ = no λ()"},{"Add":"var x args ≟ sort _ = no λ()"},{"Add":"var x args ≟ lit _ = no λ()"},{"Add":"var x args ≟ meta _ _ = no λ()"},{"Add":"var x args ≟ unknown = no λ()"},{"Add":"con c args ≟ var x args′ = no λ()"},{"Add":"con c args ≟ def f args′ = no λ()"},{"Add":"con c args ≟ lam v t = no λ()"},{"Add":"con c args ≟ pi t₁ t₂ = no λ()"},{"Add":"con c args ≟ sort _ = no λ()"},{"Add":"con c args ≟ lit _ = no λ()"},{"Add":"con c args ≟ meta _ _ = no λ()"},{"Add":"con c args ≟ unknown = no λ()"},{"Add":"def f args ≟ var x args′ = no λ()"},{"Add":"def f args ≟ con c args′ = no λ()"},{"Add":"def f args ≟ lam v t = no λ()"},{"Add":"def f args ≟ pi t₁ t₂ = no λ()"},{"Add":"def f args ≟ sort _ = no λ()"},{"Add":"def f args ≟ lit _ = no λ()"},{"Add":"def f args ≟ meta _ _ = no λ()"},{"Add":"def f args ≟ unknown = no λ()"},{"Add":"lam v t ≟ var x args = no λ()"},{"Add":"lam v t ≟ con c args = no λ()"},{"Add":"lam v t ≟ def f args = no λ()"},{"Add":"lam v t ≟ pi t₁ t₂ = no λ()"},{"Add":"lam v t ≟ sort _ = no λ()"},{"Add":"lam v t ≟ lit _ = no λ()"},{"Add":"lam v t ≟ meta _ _ = no λ()"},{"Add":"lam v t ≟ unknown = no λ()"},{"Add":"pi t₁ t₂ ≟ var x args = no λ()"},{"Add":"pi t₁ t₂ ≟ con c args = no λ()"},{"Add":"pi t₁ t₂ ≟ def f args = no λ()"},{"Add":"pi t₁ t₂ ≟ lam v t = no λ()"},{"Add":"pi t₁ t₂ ≟ sort _ = no λ()"},{"Add":"pi t₁ t₂ ≟ lit _ = no λ()"},{"Add":"pi t₁ t₂ ≟ meta _ _ = no λ()"},{"Add":"pi t₁ t₂ ≟ unknown = no λ()"},{"Add":"sort _ ≟ var x args = no λ()"},{"Add":"sort _ ≟ con c args = no λ()"},{"Add":"sort _ ≟ def f args = no λ()"},{"Add":"sort _ ≟ lam v t = no λ()"},{"Add":"sort _ ≟ pi t₁ t₂ = no λ()"},{"Add":"sort _ ≟ lit _ = no λ()"},{"Add":"sort _ ≟ meta _ _ = no λ()"},{"Add":"sort _ ≟ unknown = no λ()"},{"Add":"lit _ ≟ var x args = no λ()"},{"Add":"lit _ ≟ con c args = no λ()"},{"Add":"lit _ ≟ def f args = no λ()"},{"Add":"lit _ ≟ lam v t = no λ()"},{"Add":"lit _ ≟ pi t₁ t₂ = no λ()"},{"Add":"lit _ ≟ sort _ = no λ()"},{"Add":"lit _ ≟ meta _ _ = no λ()"},{"Add":"lit _ ≟ unknown = no λ()"},{"Add":"meta _ _ ≟ var x args = no λ()"},{"Add":"meta _ _ ≟ con c args = no λ()"},{"Add":"meta _ _ ≟ def f args = no λ()"},{"Add":"meta _ _ ≟ lam v t = no λ()"},{"Add":"meta _ _ ≟ pi t₁ t₂ = no λ()"},{"Add":"meta _ _ ≟ sort _ = no λ()"},{"Add":"meta _ _ ≟ lit _ = no λ()"},{"Add":"meta _ _ ≟ unknown = no λ()"},{"Add":"unknown ≟ var x args = no λ()"},{"Add":"unknown ≟ con c args = no λ()"},{"Add":"unknown ≟ def f args = no λ()"},{"Add":"unknown ≟ lam v t = no λ()"},{"Add":"unknown ≟ pi t₁ t₂ = no λ()"},{"Add":"unknown ≟ sort _ = no λ()"},{"Add":"unknown ≟ lit _ = no λ()"},{"Add":"unknown ≟ meta _ _ = no λ()"},{"Add":"pat-lam _ _ ≟ var x args = no λ()"},{"Add":"pat-lam _ _ ≟ con c args = no λ()"},{"Add":"pat-lam _ _ ≟ def f args = no λ()"},{"Add":"pat-lam _ _ ≟ lam v t = no λ()"},{"Add":"pat-lam _ _ ≟ pi t₁ t₂ = no λ()"},{"Add":"pat-lam _ _ ≟ sort _ = no λ()"},{"Add":"pat-lam _ _ ≟ lit _ = no λ()"},{"Add":"pat-lam _ _ ≟ meta _ _ = no λ()"},{"Add":"pat-lam _ _ ≟ unknown = no λ()"},{"Add":"var x args ≟ pat-lam _ _ = no λ()"},{"Add":"con c args ≟ pat-lam _ _ = no λ()"},{"Add":"def f args ≟ pat-lam _ _ = no λ()"},{"Add":"lam v t ≟ pat-lam _ _ = no λ()"},{"Add":"pi t₁ t₂ ≟ pat-lam _ _ = no λ()"},{"Add":"sort _ ≟ pat-lam _ _ = no λ()"},{"Add":"lit _ ≟ pat-lam _ _ = no λ()"},{"Add":"meta _ _ ≟ pat-lam _ _ = no λ()"},{"Add":"unknown ≟ pat-lam _ _ = no λ()"},{"Add":""},{"Add":"set t ≟-Sort set t′ = map′ (cong set) set-injective (t ≟ t′)"},{"Add":"lit n ≟-Sort lit n′ = map′ (cong lit) slit-injective (n ℕ.≟ n′)"},{"Add":"prop t ≟-Sort prop t′ = map′ (cong prop) prop-injective (t ≟ t′)"},{"Add":"propLit n ≟-Sort propLit n′ = map′ (cong propLit) propLit-injective (n ℕ.≟ n′)"},{"Add":"inf n ≟-Sort inf n′ = map′ (cong inf) inf-injective (n ℕ.≟ n′)"},{"Add":"unknown ≟-Sort unknown = yes refl"},{"Add":"set _ ≟-Sort lit _ = no λ()"},{"Add":"set _ ≟-Sort prop _ = no λ()"},{"Add":"set _ ≟-Sort propLit _ = no λ()"},{"Add":"set _ ≟-Sort inf _ = no λ()"},{"Add":"set _ ≟-Sort unknown = no λ()"},{"Add":"lit _ ≟-Sort set _ = no λ()"},{"Add":"lit _ ≟-Sort prop _ = no λ()"},{"Add":"lit _ ≟-Sort propLit _ = no λ()"},{"Add":"lit _ ≟-Sort inf _ = no λ()"},{"Add":"lit _ ≟-Sort unknown = no λ()"},{"Add":"prop _ ≟-Sort set _ = no λ()"},{"Add":"prop _ ≟-Sort lit _ = no λ()"},{"Add":"prop _ ≟-Sort propLit _ = no λ()"},{"Add":"prop _ ≟-Sort inf _ = no λ()"},{"Add":"prop _ ≟-Sort unknown = no λ()"},{"Add":"propLit _ ≟-Sort set _ = no λ()"},{"Add":"propLit _ ≟-Sort lit _ = no λ()"},{"Add":"propLit _ ≟-Sort prop _ = no λ()"},{"Add":"propLit _ ≟-Sort inf _ = no λ()"},{"Add":"propLit _ ≟-Sort unknown = no λ()"},{"Add":"inf _ ≟-Sort set _ = no λ()"},{"Add":"inf _ ≟-Sort lit _ = no λ()"},{"Add":"inf _ ≟-Sort prop _ = no λ()"},{"Add":"inf _ ≟-Sort propLit _ = no λ()"},{"Add":"inf _ ≟-Sort unknown = no λ()"},{"Add":"unknown ≟-Sort set _ = no λ()"},{"Add":"unknown ≟-Sort lit _ = no λ()"},{"Add":"unknown ≟-Sort prop _ = no λ()"},{"Add":"unknown ≟-Sort propLit _ = no λ()"},{"Add":"unknown ≟-Sort inf _ = no λ()"},{"Add":""},{"Add":""},{"Add":"pat-con-injective₁ : ∀ {c c′ args args′} → Pattern.con c args ≡ con c′ args′ → c ≡ c′"},{"Add":"pat-con-injective₁ refl = refl"},{"Add":""},{"Add":"pat-con-injective₂ : ∀ {c c′ args args′} → Pattern.con c args ≡ con c′ args′ → args ≡ args′"},{"Add":"pat-con-injective₂ refl = refl"},{"Add":""},{"Add":"pat-con-injective : ∀ {c c′ args args′} → Pattern.con c args ≡ con c′ args′ → c ≡ c′ × args ≡ args′"},{"Add":"pat-con-injective = < pat-con-injective₁ , pat-con-injective₂ >"},{"Add":""},{"Add":"pat-var-injective : ∀ {x y} → var x ≡ var y → x ≡ y"},{"Add":"pat-var-injective refl = refl"},{"Add":""},{"Add":"pat-lit-injective : ∀ {x y} → Pattern.lit x ≡ lit y → x ≡ y"},{"Add":"pat-lit-injective refl = refl"},{"Add":""},{"Add":"proj-injective : ∀ {x y} → proj x ≡ proj y → x ≡ y"},{"Add":"proj-injective refl = refl"},{"Add":""},{"Add":"dot-injective : ∀ {x y} → dot x ≡ dot y → x ≡ y"},{"Add":"dot-injective refl = refl"},{"Add":""},{"Add":"absurd-injective : ∀ {x y} → absurd x ≡ absurd y → x ≡ y"},{"Add":"absurd-injective refl = refl"},{"Add":""},{"Add":"con c ps ≟-Pattern con c′ ps′ = map′ (uncurry (cong₂ con)) pat-con-injective (c Name.≟ c′ ×-dec ps ≟-Patterns ps′)"},{"Add":"var x ≟-Pattern var x′ = map′ (cong var) pat-var-injective (x ℕ.≟ x′)"},{"Add":"lit l ≟-Pattern lit l′ = map′ (cong lit) pat-lit-injective (l Literal.≟ l′)"},{"Add":"proj a ≟-Pattern proj a′ = map′ (cong proj) proj-injective (a Name.≟ a′)"},{"Add":"dot t ≟-Pattern dot t′ = map′ (cong dot) dot-injective (t ≟ t′)"},{"Add":"absurd x ≟-Pattern absurd x′ = map′ (cong absurd) absurd-injective (x ℕ.≟ x′)"},{"Add":""},{"Add":"con x x₁ ≟-Pattern dot x₂ = no (λ ())"},{"Add":"con x x₁ ≟-Pattern var x₂ = no (λ ())"},{"Add":"con x x₁ ≟-Pattern lit x₂ = no (λ ())"},{"Add":"con x x₁ ≟-Pattern proj x₂ = no (λ ())"},{"Add":"con x x₁ ≟-Pattern absurd x₂ = no (λ ())"},{"Add":"dot x ≟-Pattern con x₁ x₂ = no (λ ())"},{"Add":"dot x ≟-Pattern var x₁ = no (λ ())"},{"Add":"dot x ≟-Pattern lit x₁ = no (λ ())"},{"Add":"dot x ≟-Pattern proj x₁ = no (λ ())"},{"Add":"dot x ≟-Pattern absurd x₁ = no (λ ())"},{"Add":"var s ≟-Pattern con x x₁ = no (λ ())"},{"Add":"var s ≟-Pattern dot x = no (λ ())"},{"Add":"var s ≟-Pattern lit x = no (λ ())"},{"Add":"var s ≟-Pattern proj x = no (λ ())"},{"Add":"var s ≟-Pattern absurd x = no (λ ())"},{"Add":"lit x ≟-Pattern con x₁ x₂ = no (λ ())"},{"Add":"lit x ≟-Pattern dot x₁ = no (λ ())"},{"Add":"lit x ≟-Pattern var _ = no (λ ())"},{"Add":"lit x ≟-Pattern proj x₁ = no (λ ())"},{"Add":"lit x ≟-Pattern absurd x₁ = no (λ ())"},{"Add":"proj x ≟-Pattern con x₁ x₂ = no (λ ())"},{"Add":"proj x ≟-Pattern dot x₁ = no (λ ())"},{"Add":"proj x ≟-Pattern var _ = no (λ ())"},{"Add":"proj x ≟-Pattern lit x₁ = no (λ ())"},{"Add":"proj x ≟-Pattern absurd x₁ = no (λ ())"},{"Add":"absurd x ≟-Pattern con x₁ x₂ = no (λ ())"},{"Add":"absurd x ≟-Pattern dot x₁ = no (λ ())"},{"Add":"absurd x ≟-Pattern var _ = no (λ ())"},{"Add":"absurd x ≟-Pattern lit x₁ = no (λ ())"},{"Add":"absurd x ≟-Pattern proj x₁ = no (λ ())"},{"Add":""},{"Add":"[] ≟-Patterns [] = yes refl"},{"Add":"(arg i p ∷ xs) ≟-Patterns (arg j q ∷ ys) = ∷-dec (unArg-dec (p ≟-Pattern q)) (xs ≟-Patterns ys)"},{"Add":""},{"Add":"[] ≟-Patterns (_ ∷ _) = no λ()"},{"Add":"(_ ∷ _) ≟-Patterns [] = no λ()"}]}},{"FileAdd":{"meta":{"basename":"Show.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Converting reflection machinery to strings"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Note that Reflection.termErr can also be used directly in tactic"},{"Add":"-- error messages."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.Show where"},{"Add":""},{"Add":"import Data.Char.Base as Char"},{"Add":"import Data.Float.Base as Float"},{"Add":"open import Data.List.Base hiding (_++_; intersperse)"},{"Add":"import Data.Nat.Show as ℕ"},{"Add":"open import Data.String.Base as String using (String; _++_; intersperse; braces; parens; _<+>_)"},{"Add":"open import Data.String as String using (parensIfSpace)"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"import Data.Word64.Base as Word64"},{"Add":"open import Function.Base using (id; _∘′_; case_of_)"},{"Add":"open import Relation.Nullary.Decidable.Core using (yes; no)"},{"Add":""},{"Add":"open import Reflection.AST.Abstraction hiding (map)"},{"Add":"open import Reflection.AST.Argument hiding (map)"},{"Add":"open import Reflection.AST.Argument.Relevance"},{"Add":"open import Reflection.AST.Argument.Visibility"},{"Add":"open import Reflection.AST.Argument.Modality"},{"Add":"open import Reflection.AST.Argument.Information"},{"Add":"open import Reflection.AST.Definition"},{"Add":"open import Reflection.AST.Literal"},{"Add":"open import Reflection.AST.Pattern"},{"Add":"open import Reflection.AST.Term"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export primitive show functions"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection public"},{"Add":" using () renaming"},{"Add":" ( primShowMeta to showMeta"},{"Add":" ; primShowQName to showName"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Non-primitive show functions"},{"Add":""},{"Add":"showRelevance : Relevance → String"},{"Add":"showRelevance relevant = \"relevant\""},{"Add":"showRelevance irrelevant = \"irrelevant\""},{"Add":""},{"Add":"showRel : Relevance → String"},{"Add":"showRel relevant = \"\""},{"Add":"showRel irrelevant = \".\""},{"Add":""},{"Add":"showVisibility : Visibility → String"},{"Add":"showVisibility visible = \"visible\""},{"Add":"showVisibility hidden = \"hidden\""},{"Add":"showVisibility instance′ = \"instance\""},{"Add":""},{"Add":"showLiteral : Literal → String"},{"Add":"showLiteral (nat x) = ℕ.show x"},{"Add":"showLiteral (word64 x) = ℕ.show (Word64.toℕ x)"},{"Add":"showLiteral (float x) = Float.show x"},{"Add":"showLiteral (char x) = Char.show x"},{"Add":"showLiteral (string x) = String.show x"},{"Add":"showLiteral (name x) = showName x"},{"Add":"showLiteral (meta x) = showMeta x"},{"Add":""},{"Add":"private"},{"Add":" -- add appropriate parens depending on the given visibility"},{"Add":" visibilityParen : Visibility → String → String"},{"Add":" visibilityParen visible s = parensIfSpace s"},{"Add":" visibilityParen hidden s = braces s"},{"Add":" visibilityParen instance′ s = braces (braces s)"},{"Add":""},{"Add":"mutual"},{"Add":""},{"Add":" showTerms : List (Arg Term) → String"},{"Add":" showTerms [] = \"\""},{"Add":" showTerms (arg i t ∷ ts) = visibilityParen (visibility i) (showTerm t) <+> showTerms ts"},{"Add":""},{"Add":" showTerm : Term → String"},{"Add":" showTerm (var x args) = \"var\" <+> ℕ.show x <+> showTerms args"},{"Add":" showTerm (con c args) = showName c <+> showTerms args"},{"Add":" showTerm (def f args) = showName f <+> showTerms args"},{"Add":" showTerm (lam v (abs s x)) = \"λ\" <+> visibilityParen v s <+> \"→\" <+> showTerm x"},{"Add":" showTerm (pat-lam cs args) ="},{"Add":" \"λ {\" <+> showClauses cs <+> \"}\" <+> showTerms args"},{"Add":" showTerm (Π[ x ∶ arg i a ] b) ="},{"Add":" \"Π (\" ++ visibilityParen (visibility i) x <+> \":\" <+>"},{"Add":" parensIfSpace (showTerm a) ++ \")\" <+> parensIfSpace (showTerm b)"},{"Add":" showTerm (sort s) = showSort s"},{"Add":" showTerm (lit l) = showLiteral l"},{"Add":" showTerm (meta x args) = showMeta x <+> showTerms args"},{"Add":" showTerm unknown = \"unknown\""},{"Add":""},{"Add":" showSort : Sort → String"},{"Add":" showSort (set t) = \"Set\" <+> parensIfSpace (showTerm t)"},{"Add":" showSort (lit n) = \"Set\" ++ ℕ.show n -- no space to disambiguate from set t"},{"Add":" showSort (prop t) = \"Prop\" <+> parensIfSpace (showTerm t)"},{"Add":" showSort (propLit n) = \"Prop\" ++ ℕ.show n -- no space to disambiguate from prop t"},{"Add":" showSort (inf n) = \"Setω\" ++ ℕ.show n"},{"Add":" showSort unknown = \"unknown\""},{"Add":""},{"Add":" showPatterns : List (Arg Pattern) → String"},{"Add":" showPatterns [] = \"\""},{"Add":" showPatterns (a ∷ ps) = showArg a <+> showPatterns ps"},{"Add":" where"},{"Add":" -- Quantities are ignored."},{"Add":" showArg : Arg Pattern → String"},{"Add":" showArg (arg (arg-info h (modality r _)) p) ="},{"Add":" braces? (showRel r ++ showPattern p)"},{"Add":" where"},{"Add":" braces? = case h of λ where"},{"Add":" visible → id"},{"Add":" hidden → braces"},{"Add":" instance′ → braces ∘′ braces"},{"Add":""},{"Add":" showPattern : Pattern → String"},{"Add":" showPattern (con c []) = showName c"},{"Add":" showPattern (con c ps) = parens (showName c <+> showPatterns ps)"},{"Add":" showPattern (dot t) = \".\" ++ parens (showTerm t)"},{"Add":" showPattern (var x) = \"pat-var\" <+> ℕ.show x"},{"Add":" showPattern (lit l) = showLiteral l"},{"Add":" showPattern (proj f) = showName f"},{"Add":" showPattern (absurd _) = \"()\""},{"Add":""},{"Add":" showClause : Clause → String"},{"Add":" showClause (clause tel ps t) = \"[\" <+> showTel tel <+> \"]\" <+> showPatterns ps <+> \"→\" <+> showTerm t"},{"Add":" showClause (absurd-clause tel ps) = \"[\" <+> showTel tel <+> \"]\" <+> showPatterns ps"},{"Add":""},{"Add":" showClauses : List Clause → String"},{"Add":" showClauses [] = \"\""},{"Add":" showClauses (c ∷ cs) = showClause c <+> \";\" <+> showClauses cs"},{"Add":""},{"Add":" showTel : List (String × Arg Type) → String"},{"Add":" showTel [] = \"\""},{"Add":" showTel ((x , arg i t) ∷ tel) = visibilityParen (visibility i) (x <+> \":\" <+> showTerm t) ++ showTel tel"},{"Add":""},{"Add":"showDefinition : Definition → String"},{"Add":"showDefinition (function cs) = \"function\" <+> braces (showClauses cs)"},{"Add":"showDefinition (data-type pars cs) ="},{"Add":" \"datatype\" <+> ℕ.show pars <+> braces (intersperse \", \" (map showName cs))"},{"Add":"showDefinition (record′ c fs) ="},{"Add":" \"record\" <+> showName c <+> braces (intersperse \", \" (map (showName ∘′ unArg) fs))"},{"Add":"showDefinition (constructor′ d) = \"constructor\" <+> showName d"},{"Add":"showDefinition axiom = \"axiom\""},{"Add":"showDefinition primitive′ = \"primitive\""}]}},{"FileAdd":{"meta":{"basename":"Pattern.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Patterns used in the reflection machinery"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.Pattern where"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting the builtin type and constructors"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection public using (Pattern)"},{"Add":"open Pattern public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting definitions that used to be here"},{"Add":""},{"Add":"open import Reflection.AST.Term public"},{"Add":" using ( proj-injective )"},{"Add":" renaming ( pat-con-injective₁ to con-injective₁"},{"Add":" ; pat-con-injective₂ to con-injective₂"},{"Add":" ; pat-con-injective to con-injective"},{"Add":" ; pat-var-injective to var-injective"},{"Add":" ; pat-lit-injective to lit-injective"},{"Add":" ; _≟-Patterns_ to _≟s_"},{"Add":" ; _≟-Pattern_ to _≟_"},{"Add":" )"}]}},{"FileAdd":{"meta":{"basename":"Name.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Names used in the reflection machinery"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.Name where"},{"Add":""},{"Add":"open import Data.List.Base using (List)"},{"Add":"import Data.Product.Properties as Prodₚ using (≡-dec)"},{"Add":"import Data.Word64.Properties as Wₚ using (_≟_)"},{"Add":"open import Function.Base using (_on_)"},{"Add":"open import Relation.Nullary.Decidable.Core using (map′)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Definitions using (Decidable; DecidableEquality)"},{"Add":"open import Relation.Binary.Construct.On using (decidable)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export built-ins"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection public"},{"Add":" using (Name) renaming (primQNameToWord64s to toWords; primQNameEquality to _≡ᵇ_)"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection.Properties public"},{"Add":" renaming (primQNameToWord64sInjective to toWords-injective)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- More definitions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"Names : Set"},{"Add":"Names = List Name"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Decidable equality for names"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"infix 4 _≈?_ _≟_ _≈_"},{"Add":""},{"Add":"_≈_ : Rel Name _"},{"Add":"_≈_ = _≡_ on toWords"},{"Add":""},{"Add":"_≈?_ : Decidable _≈_"},{"Add":"_≈?_ = decidable toWords _≡_ (Prodₚ.≡-dec Wₚ._≟_ Wₚ._≟_)"},{"Add":""},{"Add":"_≟_ : DecidableEquality Name"},{"Add":"m ≟ n = map′ (toWords-injective _ _) (cong toWords) (m ≈? n)"}]}},{"FileAdd":{"meta":{"basename":"Meta.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Metavariables used in the reflection machinery"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.Meta where"},{"Add":""},{"Add":"import Data.Nat.Properties as ℕ"},{"Add":"open import Function.Base using (_on_)"},{"Add":"open import Relation.Nullary.Decidable.Core using (map′)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Definitions using (Decidable; DecidableEquality)"},{"Add":"import Relation.Binary.Construct.On as On"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection public"},{"Add":" using (Meta) renaming (primMetaToNat to toℕ; primMetaEquality to _≡ᵇ_)"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection.Properties public"},{"Add":" renaming (primMetaToNatInjective to toℕ-injective)"},{"Add":""},{"Add":"-- Equality of metas is decidable."},{"Add":""},{"Add":"infix 4 _≈?_ _≟_ _≈_"},{"Add":""},{"Add":"_≈_ : Rel Meta _"},{"Add":"_≈_ = _≡_ on toℕ"},{"Add":""},{"Add":"_≈?_ : Decidable _≈_"},{"Add":"_≈?_ = On.decidable toℕ _≡_ ℕ._≟_"},{"Add":""},{"Add":"_≟_ : DecidableEquality Meta"},{"Add":"m ≟ n = map′ (toℕ-injective _ _) (cong toℕ) (m ≈? n)"}]}},{"FileAdd":{"meta":{"basename":"Literal.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Literals used in the reflection machinery"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.Literal where"},{"Add":""},{"Add":"open import Data.Bool.Base using (Bool; true; false)"},{"Add":"import Data.Char.Properties as Char"},{"Add":"import Data.Float.Properties as Float"},{"Add":"import Data.Nat.Properties as ℕ"},{"Add":"import Data.String.Properties as String"},{"Add":"import Data.Word64.Properties as Word64"},{"Add":"import Reflection.AST.Meta as Meta"},{"Add":"import Reflection.AST.Name as Name"},{"Add":"open import Relation.Nullary.Decidable.Core using (yes; no; map′; isYes)"},{"Add":"open import Relation.Binary.Definitions using (DecidableEquality)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting the builtin type and constructors"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection public"},{"Add":" using ( Literal )"},{"Add":"open Literal public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Decidable equality"},{"Add":""},{"Add":"meta-injective : ∀ {x y} → meta x ≡ meta y → x ≡ y"},{"Add":"meta-injective refl = refl"},{"Add":""},{"Add":"nat-injective : ∀ {x y} → nat x ≡ nat y → x ≡ y"},{"Add":"nat-injective refl = refl"},{"Add":""},{"Add":"word64-injective : ∀ {x y} → word64 x ≡ word64 y → x ≡ y"},{"Add":"word64-injective refl = refl"},{"Add":""},{"Add":"float-injective : ∀ {x y} → float x ≡ float y → x ≡ y"},{"Add":"float-injective refl = refl"},{"Add":""},{"Add":"char-injective : ∀ {x y} → char x ≡ char y → x ≡ y"},{"Add":"char-injective refl = refl"},{"Add":""},{"Add":"string-injective : ∀ {x y} → string x ≡ string y → x ≡ y"},{"Add":"string-injective refl = refl"},{"Add":""},{"Add":"name-injective : ∀ {x y} → name x ≡ name y → x ≡ y"},{"Add":"name-injective refl = refl"},{"Add":""},{"Add":"infix 4 _≟_"},{"Add":"_≟_ : DecidableEquality Literal"},{"Add":"nat x ≟ nat x₁ = map′ (cong nat) nat-injective (x ℕ.≟ x₁)"},{"Add":"nat x ≟ word64 x₁ = no (λ ())"},{"Add":"nat x ≟ float x₁ = no (λ ())"},{"Add":"nat x ≟ char x₁ = no (λ ())"},{"Add":"nat x ≟ string x₁ = no (λ ())"},{"Add":"nat x ≟ name x₁ = no (λ ())"},{"Add":"nat x ≟ meta x₁ = no (λ ())"},{"Add":"word64 x ≟ word64 x₁ = map′ (cong word64) word64-injective (x Word64.≟ x₁)"},{"Add":"word64 x ≟ nat x₁ = no (λ ())"},{"Add":"word64 x ≟ float x₁ = no (λ ())"},{"Add":"word64 x ≟ char x₁ = no (λ ())"},{"Add":"word64 x ≟ string x₁ = no (λ ())"},{"Add":"word64 x ≟ name x₁ = no (λ ())"},{"Add":"word64 x ≟ meta x₁ = no (λ ())"},{"Add":"float x ≟ nat x₁ = no (λ ())"},{"Add":"float x ≟ word64 x₁ = no (λ ())"},{"Add":"float x ≟ float x₁ = map′ (cong float) float-injective (x Float.≟ x₁)"},{"Add":"float x ≟ char x₁ = no (λ ())"},{"Add":"float x ≟ string x₁ = no (λ ())"},{"Add":"float x ≟ name x₁ = no (λ ())"},{"Add":"float x ≟ meta x₁ = no (λ ())"},{"Add":"char x ≟ nat x₁ = no (λ ())"},{"Add":"char x ≟ word64 x₁ = no (λ ())"},{"Add":"char x ≟ float x₁ = no (λ ())"},{"Add":"char x ≟ char x₁ = map′ (cong char) char-injective (x Char.≟ x₁)"},{"Add":"char x ≟ string x₁ = no (λ ())"},{"Add":"char x ≟ name x₁ = no (λ ())"},{"Add":"char x ≟ meta x₁ = no (λ ())"},{"Add":"string x ≟ nat x₁ = no (λ ())"},{"Add":"string x ≟ word64 x₁ = no (λ ())"},{"Add":"string x ≟ float x₁ = no (λ ())"},{"Add":"string x ≟ char x₁ = no (λ ())"},{"Add":"string x ≟ string x₁ = map′ (cong string) string-injective (x String.≟ x₁)"},{"Add":"string x ≟ name x₁ = no (λ ())"},{"Add":"string x ≟ meta x₁ = no (λ ())"},{"Add":"name x ≟ nat x₁ = no (λ ())"},{"Add":"name x ≟ word64 x₁ = no (λ ())"},{"Add":"name x ≟ float x₁ = no (λ ())"},{"Add":"name x ≟ char x₁ = no (λ ())"},{"Add":"name x ≟ string x₁ = no (λ ())"},{"Add":"name x ≟ name x₁ = map′ (cong name) name-injective (x Name.≟ x₁)"},{"Add":"name x ≟ meta x₁ = no (λ ())"},{"Add":"meta x ≟ nat x₁ = no (λ ())"},{"Add":"meta x ≟ word64 x₁ = no (λ ())"},{"Add":"meta x ≟ float x₁ = no (λ ())"},{"Add":"meta x ≟ char x₁ = no (λ ())"},{"Add":"meta x ≟ string x₁ = no (λ ())"},{"Add":"meta x ≟ name x₁ = no (λ ())"},{"Add":"meta x ≟ meta x₁ = map′ (cong meta) meta-injective (x Meta.≟ x₁)"},{"Add":""},{"Add":"infix 4 _≡ᵇ_"},{"Add":""},{"Add":"_≡ᵇ_ : Literal → Literal → Bool"},{"Add":"l ≡ᵇ l′ = isYes (l ≟ l′)"}]}},{"FileAdd":{"meta":{"basename":"Instances.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Instances for reflected syntax"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.Instances where"},{"Add":""},{"Add":"open import Level"},{"Add":""},{"Add":"import Reflection.AST.Literal as Literal"},{"Add":"import Reflection.AST.Name as Name"},{"Add":"import Reflection.AST.Meta as Meta"},{"Add":"import Reflection.AST.Abstraction as Abstraction"},{"Add":"import Reflection.AST.Argument as Argument"},{"Add":"import Reflection.AST.Argument.Visibility as Visibility"},{"Add":"import Reflection.AST.Argument.Relevance as Relevance"},{"Add":"import Reflection.AST.Argument.Information as Information"},{"Add":"import Reflection.AST.Pattern as Pattern"},{"Add":"import Reflection.AST.Term as Term"},{"Add":""},{"Add":"open import Relation.Binary.PropositionalEquality.Core"},{"Add":"open import Relation.Binary.PropositionalEquality.Properties"},{"Add":" using (isDecEquivalence)"},{"Add":"open import Relation.Binary.TypeClasses"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"instance"},{"Add":" Lit-≡-isDecEquivalence = isDecEquivalence Literal._≟_"},{"Add":" Name-≡-isDecEquivalence = isDecEquivalence Name._≟_"},{"Add":" Meta-≡-isDecEquivalence = isDecEquivalence Meta._≟_"},{"Add":" Visibility-≡-isDecEquivalence = isDecEquivalence Visibility._≟_"},{"Add":" Relevance-≡-isDecEquivalence = isDecEquivalence Relevance._≟_"},{"Add":" ArgInfo-≡-isDecEquivalence = isDecEquivalence Information._≟_"},{"Add":" Pattern-≡-isDecEquivalence = isDecEquivalence Pattern._≟_"},{"Add":" Clause-≡-isDecEquivalence = isDecEquivalence Term._≟-Clause_"},{"Add":" Term-≡-isDecEquivalence = isDecEquivalence Term._≟_"},{"Add":" Sort-≡-isDecEquivalence = isDecEquivalence Term._≟-Sort_"},{"Add":""},{"Add":" Abs-≡-isDecEquivalence : {{IsDecEquivalence {A = A} _≡_}} → IsDecEquivalence {A = Abstraction.Abs A} _≡_"},{"Add":" Abs-≡-isDecEquivalence = isDecEquivalence (Abstraction.≡-dec _≟_)"},{"Add":""},{"Add":" Arg-≡-isDecEquivalence : {{IsDecEquivalence {A = A} _≡_}} → IsDecEquivalence {A = Argument.Arg A} _≡_"},{"Add":" Arg-≡-isDecEquivalence = isDecEquivalence (Argument.≡-dec _≟_)"}]}},{"FileAdd":{"meta":{"basename":"Definition.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Definitions used in the reflection machinery"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.Definition where"},{"Add":""},{"Add":"import Data.List.Properties as List using (≡-dec)"},{"Add":"import Data.Nat.Properties as ℕ using (_≟_)"},{"Add":"open import Data.Product.Base using (_×_; <_,_>; uncurry)"},{"Add":"open import Relation.Nullary.Decidable.Core using (map′; _×-dec_; yes; no)"},{"Add":"open import Relation.Binary.Definitions using (DecidableEquality)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)"},{"Add":""},{"Add":"import Reflection.AST.Argument as Arg"},{"Add":"import Reflection.AST.Name as Name"},{"Add":"import Reflection.AST.Term as Term"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting type publically"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection public"},{"Add":" using ( Definition"},{"Add":" ; function"},{"Add":" ; data-type"},{"Add":" ; axiom"},{"Add":" )"},{"Add":" renaming ( record-type to record′"},{"Add":" ; data-cons to constructor′"},{"Add":" ; prim-fun to primitive′ )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Decidable equality"},{"Add":""},{"Add":"function-injective : ∀ {cs cs′} → function cs ≡ function cs′ → cs ≡ cs′"},{"Add":"function-injective refl = refl"},{"Add":""},{"Add":"data-type-injective₁ : ∀ {pars pars′ cs cs′} → data-type pars cs ≡ data-type pars′ cs′ → pars ≡ pars′"},{"Add":"data-type-injective₁ refl = refl"},{"Add":""},{"Add":"data-type-injective₂ : ∀ {pars pars′ cs cs′} → data-type pars cs ≡ data-type pars′ cs′ → cs ≡ cs′"},{"Add":"data-type-injective₂ refl = refl"},{"Add":""},{"Add":"data-type-injective : ∀ {pars pars′ cs cs′} → data-type pars cs ≡ data-type pars′ cs′ → pars ≡ pars′ × cs ≡ cs′"},{"Add":"data-type-injective = < data-type-injective₁ , data-type-injective₂ >"},{"Add":""},{"Add":"record′-injective₁ : ∀ {c c′ fs fs′} → record′ c fs ≡ record′ c′ fs′ → c ≡ c′"},{"Add":"record′-injective₁ refl = refl"},{"Add":""},{"Add":"record′-injective₂ : ∀ {c c′ fs fs′} → record′ c fs ≡ record′ c′ fs′ → fs ≡ fs′"},{"Add":"record′-injective₂ refl = refl"},{"Add":""},{"Add":"record′-injective : ∀ {c c′ fs fs′} → record′ c fs ≡ record′ c′ fs′ → c ≡ c′ × fs ≡ fs′"},{"Add":"record′-injective = < record′-injective₁ , record′-injective₂ >"},{"Add":""},{"Add":"constructor′-injective : ∀ {c c′} → constructor′ c ≡ constructor′ c′ → c ≡ c′"},{"Add":"constructor′-injective refl = refl"},{"Add":""},{"Add":"infix 4 _≟_"},{"Add":""},{"Add":"_≟_ : DecidableEquality Definition"},{"Add":"function cs ≟ function cs′ ="},{"Add":" map′ (cong function) function-injective (cs Term.≟-Clauses cs′)"},{"Add":"data-type pars cs ≟ data-type pars′ cs′ ="},{"Add":" map′ (uncurry (cong₂ data-type)) data-type-injective"},{"Add":" (pars ℕ.≟ pars′ ×-dec List.≡-dec Name._≟_ cs cs′)"},{"Add":"record′ c fs ≟ record′ c′ fs′ ="},{"Add":" map′ (uncurry (cong₂ record′)) record′-injective"},{"Add":" (c Name.≟ c′ ×-dec List.≡-dec (Arg.≡-dec Name._≟_) fs fs′)"},{"Add":"constructor′ d ≟ constructor′ d′ ="},{"Add":" map′ (cong constructor′) constructor′-injective (d Name.≟ d′)"},{"Add":"axiom ≟ axiom = yes refl"},{"Add":"primitive′ ≟ primitive′ = yes refl"},{"Add":""},{"Add":"-- antidiagonal"},{"Add":"function cs ≟ data-type pars cs₁ = no (λ ())"},{"Add":"function cs ≟ record′ c fs = no (λ ())"},{"Add":"function cs ≟ constructor′ d = no (λ ())"},{"Add":"function cs ≟ axiom = no (λ ())"},{"Add":"function cs ≟ primitive′ = no (λ ())"},{"Add":"data-type pars cs ≟ function cs₁ = no (λ ())"},{"Add":"data-type pars cs ≟ record′ c fs = no (λ ())"},{"Add":"data-type pars cs ≟ constructor′ d = no (λ ())"},{"Add":"data-type pars cs ≟ axiom = no (λ ())"},{"Add":"data-type pars cs ≟ primitive′ = no (λ ())"},{"Add":"record′ c fs ≟ function cs = no (λ ())"},{"Add":"record′ c fs ≟ data-type pars cs = no (λ ())"},{"Add":"record′ c fs ≟ constructor′ d = no (λ ())"},{"Add":"record′ c fs ≟ axiom = no (λ ())"},{"Add":"record′ c fs ≟ primitive′ = no (λ ())"},{"Add":"constructor′ d ≟ function cs = no (λ ())"},{"Add":"constructor′ d ≟ data-type pars cs = no (λ ())"},{"Add":"constructor′ d ≟ record′ c fs = no (λ ())"},{"Add":"constructor′ d ≟ axiom = no (λ ())"},{"Add":"constructor′ d ≟ primitive′ = no (λ ())"},{"Add":"axiom ≟ function cs = no (λ ())"},{"Add":"axiom ≟ data-type pars cs = no (λ ())"},{"Add":"axiom ≟ record′ c fs = no (λ ())"},{"Add":"axiom ≟ constructor′ d = no (λ ())"},{"Add":"axiom ≟ primitive′ = no (λ ())"},{"Add":"primitive′ ≟ function cs = no (λ ())"},{"Add":"primitive′ ≟ data-type pars cs = no (λ ())"},{"Add":"primitive′ ≟ record′ c fs = no (λ ())"},{"Add":"primitive′ ≟ constructor′ d = no (λ ())"},{"Add":"primitive′ ≟ axiom = no (λ ())"}]}},{"FileAdd":{"meta":{"basename":"DeBruijn.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Weakening, strengthening and free variable check for reflected terms."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.DeBruijn where"},{"Add":""},{"Add":"open import Data.Bool.Base using (Bool; true; false; _∨_; if_then_else_)"},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc; _+_; _∸_; _<ᵇ_; _≡ᵇ_)"},{"Add":"open import Data.List.Base using (List; []; _∷_; _++_)"},{"Add":"open import Data.Maybe.Base using (Maybe; nothing; just)"},{"Add":"import Data.Maybe.Effectful as Maybe"},{"Add":"import Function.Identity.Effectful as Identity"},{"Add":"open import Effect.Applicative using (RawApplicative; mkRawApplicative)"},{"Add":""},{"Add":"open import Reflection"},{"Add":"open import Reflection.AST.Argument.Visibility using (Visibility)"},{"Add":"import Reflection.AST.Traversal as Traverse"},{"Add":""},{"Add":"private"},{"Add":" variable A : Set"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Weakening"},{"Add":""},{"Add":"module _ where"},{"Add":""},{"Add":" open Traverse Identity.applicative"},{"Add":""},{"Add":" private"},{"Add":""},{"Add":" wkVar : ℕ → Cxt → ℕ → ℕ"},{"Add":" wkVar by (from , _) i = if i <ᵇ from then i else i + by"},{"Add":""},{"Add":" actions : ℕ → Actions"},{"Add":" actions k = record defaultActions { onVar = wkVar k }"},{"Add":""},{"Add":" weakenFrom′ : (Actions → Cxt → A → A) → (from by : ℕ) → A → A"},{"Add":" weakenFrom′ trav from by = trav (actions by) (from , []) -- not using the context part"},{"Add":""},{"Add":" weakenFrom : (from by : ℕ) → Term → Term"},{"Add":" weakenFrom = weakenFrom′ traverseTerm"},{"Add":""},{"Add":" weaken : (by : ℕ) → Term → Term"},{"Add":" weaken = weakenFrom 0"},{"Add":""},{"Add":" weakenArgs : (by : ℕ) → Args Term → Args Term"},{"Add":" weakenArgs = weakenFrom′ traverseArgs 0"},{"Add":""},{"Add":" weakenClauses : (by : ℕ) → Clauses → Clauses"},{"Add":" weakenClauses = weakenFrom′ traverseClauses 0"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- η-expansion"},{"Add":""},{"Add":"private"},{"Add":" η : Visibility → (Args Term → Term) → Args Term → Term"},{"Add":" η h f args ="},{"Add":" lam h (abs \"x\" (f (weakenArgs 1 args ++"},{"Add":" arg (arg-info h defaultModality) (var 0 []) ∷"},{"Add":" [])))"},{"Add":""},{"Add":"η-expand : Visibility → Term → Term"},{"Add":"η-expand h (var x args) = η h (var (suc x)) args"},{"Add":"η-expand h (con c args) = η h (con c) args"},{"Add":"η-expand h (def f args) = η h (def f) args"},{"Add":"η-expand h (pat-lam cs args) = η h (pat-lam (weakenClauses 1 cs)) args"},{"Add":"η-expand h (meta x args) = η h (meta x) args"},{"Add":"η-expand h t@(lam _ _) = t"},{"Add":"η-expand h t@(pi _ _) = t"},{"Add":"η-expand h t@(agda-sort _) = t"},{"Add":"η-expand h t@(lit _) = t"},{"Add":"η-expand h t@unknown = t"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Strengthening"},{"Add":""},{"Add":"module _ where"},{"Add":""},{"Add":" open Traverse Maybe.applicative"},{"Add":""},{"Add":" private"},{"Add":" strVar : ℕ → Cxt → ℕ → Maybe ℕ"},{"Add":" strVar by (from , Γ) i with i <ᵇ from | i <ᵇ from + by"},{"Add":" ... | true | _ = just i"},{"Add":" ... | _ | true = nothing"},{"Add":" ... | _ | _ = just (i ∸ by)"},{"Add":""},{"Add":" actions : ℕ → Actions"},{"Add":" actions by = record defaultActions { onVar = strVar by }"},{"Add":""},{"Add":" strengthenFromBy′ : (Actions → Cxt → A → Maybe A) → (from by : ℕ) → A → Maybe A"},{"Add":" strengthenFromBy′ trav from by = trav (actions by) (from , []) -- not using the context part"},{"Add":""},{"Add":" strengthenFromBy : (from by : ℕ) → Term → Maybe Term"},{"Add":" strengthenFromBy = strengthenFromBy′ traverseTerm"},{"Add":""},{"Add":" strengthenBy : (by : ℕ) → Term → Maybe Term"},{"Add":" strengthenBy = strengthenFromBy 0"},{"Add":""},{"Add":" strengthenFrom : (from : ℕ) → Term → Maybe Term"},{"Add":" strengthenFrom from = strengthenFromBy from 1"},{"Add":""},{"Add":" strengthen : Term → Maybe Term"},{"Add":" strengthen = strengthenFromBy 0 1"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Free variable check"},{"Add":""},{"Add":"module _ where"},{"Add":""},{"Add":" private"},{"Add":" anyApplicative : ∀ {ℓ} → RawApplicative {ℓ} (λ _ → Bool)"},{"Add":" anyApplicative = mkRawApplicative _ (λ _ → false) _∨_"},{"Add":""},{"Add":" open Traverse anyApplicative"},{"Add":""},{"Add":" private"},{"Add":" fvVar : ℕ → Cxt → ℕ → Bool"},{"Add":" fvVar i (n , _) x = i + n ≡ᵇ x"},{"Add":""},{"Add":" actions : ℕ → Actions"},{"Add":" actions i = record defaultActions { onVar = fvVar i }"},{"Add":""},{"Add":" infix 4 _∈FV_"},{"Add":""},{"Add":" _∈FV_ : ℕ → Term → Bool"},{"Add":" i ∈FV t = traverseTerm (actions i) (0 , []) t"}]}},{"FileAdd":{"meta":{"basename":"Argument.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Arguments used in the reflection machinery"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.Argument where"},{"Add":""},{"Add":"open import Data.List.Base as List using (List; []; _∷_)"},{"Add":"open import Data.Product.Base using (_×_; <_,_>; uncurry)"},{"Add":"open import Relation.Nullary.Decidable.Core using (Dec; map′; _×-dec_)"},{"Add":"open import Relation.Binary.Definitions using (DecidableEquality)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)"},{"Add":"open import Level"},{"Add":""},{"Add":"open import Reflection.AST.Argument.Visibility"},{"Add":"open import Reflection.AST.Argument.Relevance"},{"Add":"open import Reflection.AST.Argument.Quantity"},{"Add":"open import Reflection.AST.Argument.Modality"},{"Add":"open import Reflection.AST.Argument.Information as Information"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b : Level"},{"Add":" A B : Set a"},{"Add":" i j : ArgInfo"},{"Add":" x y : A"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting the builtins publicly"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection public using (Arg)"},{"Add":"open Arg public"},{"Add":""},{"Add":"-- Pattern synonyms"},{"Add":""},{"Add":"pattern defaultModality = modality relevant quantity-ω"},{"Add":""},{"Add":"pattern vArg ty = arg (arg-info visible defaultModality) ty"},{"Add":"pattern hArg ty = arg (arg-info hidden defaultModality) ty"},{"Add":"pattern iArg ty = arg (arg-info instance′ defaultModality) ty"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Lists of arguments"},{"Add":""},{"Add":"Args : (A : Set a) → Set a"},{"Add":"Args A = List (Arg A)"},{"Add":""},{"Add":"-- Pattern for appending a visible argument"},{"Add":"infixr 5 _⟨∷⟩_"},{"Add":"pattern _⟨∷⟩_ x args = vArg x ∷ args"},{"Add":""},{"Add":"-- Pattern for appending a hidden argument"},{"Add":"infixr 5 _⟅∷⟆_"},{"Add":"pattern _⟅∷⟆_ x args = hArg x ∷ args"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Operations"},{"Add":""},{"Add":"map : (A → B) → Arg A → Arg B"},{"Add":"map f (arg i x) = arg i (f x)"},{"Add":""},{"Add":"map-Args : (A → B) → Args A → Args B"},{"Add":"map-Args f xs = List.map (map f) xs"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Decidable equality"},{"Add":""},{"Add":"arg-injective₁ : arg i x ≡ arg j y → i ≡ j"},{"Add":"arg-injective₁ refl = refl"},{"Add":""},{"Add":"arg-injective₂ : arg i x ≡ arg j y → x ≡ y"},{"Add":"arg-injective₂ refl = refl"},{"Add":""},{"Add":"arg-injective : arg i x ≡ arg j y → i ≡ j × x ≡ y"},{"Add":"arg-injective = < arg-injective₁ , arg-injective₂ >"},{"Add":""},{"Add":"-- We often need decidability of equality for Arg A when implementing it"},{"Add":"-- for A. Unfortunately ≡-dec makes the termination checker unhappy."},{"Add":"-- Instead, we can match on both Arg A and use unArg-dec for an"},{"Add":"-- obviously decreasing recursive call."},{"Add":""},{"Add":"unArg : Arg A → A"},{"Add":"unArg (arg i a) = a"},{"Add":""},{"Add":"unArg-dec : {arg1 arg2 : Arg A} → Dec (unArg arg1 ≡ unArg arg2) → Dec (arg1 ≡ arg2)"},{"Add":"unArg-dec {arg1 = arg i x} {arg j y} arg1≟arg2 ="},{"Add":" map′ (uncurry (cong₂ arg)) arg-injective (i Information.≟ j ×-dec arg1≟arg2)"},{"Add":""},{"Add":"≡-dec : DecidableEquality A → DecidableEquality (Arg A)"},{"Add":"≡-dec _≟_ x y = unArg-dec (unArg x ≟ unArg y)"}]}},{"FileAdd":{"meta":{"basename":"Argument","metadata":576},"context":[{"change":0,"pos":888325}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Visibility.agda","metadata":0},"context":[{"change":0,"pos":944218}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard 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Quantity"},{"Add":"quantity-ω ≟ quantity-ω = yes refl"},{"Add":"quantity-0 ≟ quantity-0 = yes refl"},{"Add":"quantity-ω ≟ quantity-0 = no λ()"},{"Add":"quantity-0 ≟ quantity-ω = no λ()"}]}},{"FileAdd":{"meta":{"basename":"Modality.agda","metadata":0},"context":[{"change":0,"pos":944218}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Modalities used in the reflection machinery"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.Argument.Modality where"},{"Add":""},{"Add":"open import Data.Product.Base using (_×_; <_,_>; uncurry)"},{"Add":"open import Relation.Nullary.Decidable.Core using (map′; _×-dec_)"},{"Add":"open import Relation.Binary.Definitions using (DecidableEquality)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)"},{"Add":""},{"Add":"open import Reflection.AST.Argument.Relevance as Relevance using (Relevance)"},{"Add":"open import Reflection.AST.Argument.Quantity as Quantity using (Quantity)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" r r′ : Relevance"},{"Add":" q q′ : Quantity"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting the builtins publicly"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection public using (Modality)"},{"Add":"open Modality public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Operations"},{"Add":""},{"Add":"relevance : Modality → Relevance"},{"Add":"relevance (modality r _) = r"},{"Add":""},{"Add":"quantity : Modality → Quantity"},{"Add":"quantity (modality _ q) = q"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Decidable equality"},{"Add":""},{"Add":"modality-injective₁ : modality r q ≡ modality r′ q′ → r ≡ r′"},{"Add":"modality-injective₁ refl = refl"},{"Add":""},{"Add":"modality-injective₂ : modality r q ≡ modality r′ q′ → q ≡ q′"},{"Add":"modality-injective₂ refl = refl"},{"Add":""},{"Add":"modality-injective : modality r q ≡ modality r′ q′ → r ≡ r′ × q ≡ q′"},{"Add":"modality-injective = < modality-injective₁ , modality-injective₂ >"},{"Add":""},{"Add":"infix 4 _≟_"},{"Add":""},{"Add":"_≟_ : DecidableEquality Modality"},{"Add":"modality r q ≟ modality r′ q′ ="},{"Add":" map′"},{"Add":" (uncurry (cong₂ modality))"},{"Add":" modality-injective"},{"Add":" (r Relevance.≟ r′ ×-dec q Quantity.≟ q′)"}]}},{"FileAdd":{"meta":{"basename":"Information.agda","metadata":0},"context":[{"change":0,"pos":944218}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Argument information used in the reflection machinery"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.Argument.Information where"},{"Add":""},{"Add":"open import Data.Product.Base using (_×_; <_,_>; uncurry)"},{"Add":"open import Relation.Nullary.Decidable.Core using (map′; _×-dec_)"},{"Add":"open import Relation.Binary.Definitions using (DecidableEquality)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)"},{"Add":""},{"Add":"open import Reflection.AST.Argument.Modality as Modality using (Modality)"},{"Add":"open import Reflection.AST.Argument.Visibility as Visibility using (Visibility)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" v v′ : Visibility"},{"Add":" m m′ : Modality"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting the builtins publicly"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection public using (ArgInfo)"},{"Add":"open ArgInfo public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Operations"},{"Add":""},{"Add":"visibility : ArgInfo → Visibility"},{"Add":"visibility (arg-info v _) = v"},{"Add":""},{"Add":"modality : ArgInfo → Modality"},{"Add":"modality (arg-info _ m) = m"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Decidable equality"},{"Add":""},{"Add":"arg-info-injective₁ : arg-info v m ≡ arg-info v′ m′ → v ≡ v′"},{"Add":"arg-info-injective₁ refl = refl"},{"Add":""},{"Add":"arg-info-injective₂ : arg-info v m ≡ arg-info v′ m′ → m ≡ m′"},{"Add":"arg-info-injective₂ refl = refl"},{"Add":""},{"Add":"arg-info-injective : arg-info v m ≡ arg-info v′ m′ → v ≡ v′ × m ≡ m′"},{"Add":"arg-info-injective = < arg-info-injective₁ , arg-info-injective₂ >"},{"Add":""},{"Add":"infix 4 _≟_"},{"Add":""},{"Add":"_≟_ : DecidableEquality ArgInfo"},{"Add":"arg-info v m ≟ arg-info v′ m′ ="},{"Add":" map′"},{"Add":" (uncurry (cong₂ arg-info))"},{"Add":" arg-info-injective"},{"Add":" (v Visibility.≟ v′ ×-dec m Modality.≟ m′)"}]}},{"FileAdd":{"meta":{"basename":"AlphaEquality.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Alpha equality over terms"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.AlphaEquality where"},{"Add":""},{"Add":"open import Data.Bool.Base using (Bool; true; false; _∧_)"},{"Add":"open import Data.List.Base using ([]; _∷_)"},{"Add":"open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _≡ᵇ_)"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Relation.Nullary.Decidable.Core using (⌊_⌋)"},{"Add":"open import Relation.Binary.Definitions using (DecidableEquality)"},{"Add":""},{"Add":"open import Reflection.AST.Abstraction"},{"Add":"open import Reflection.AST.Argument"},{"Add":"open import Reflection.AST.Argument.Information as ArgInfo"},{"Add":"open import Reflection.AST.Argument.Modality as Modality"},{"Add":"open import Reflection.AST.Argument.Visibility as Visibility"},{"Add":"open import Reflection.AST.Meta as Meta"},{"Add":"open import Reflection.AST.Name as Name"},{"Add":"open import Reflection.AST.Literal as Literal"},{"Add":"open import Reflection.AST.Term"},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"record AlphaEquality (A : Set) : Set where"},{"Add":" constructor mkAlphaEquality"},{"Add":" infix 4 _=α=_"},{"Add":" field"},{"Add":" _=α=_ : A → A → Bool"},{"Add":""},{"Add":"open AlphaEquality {{...}} public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Utilities"},{"Add":""},{"Add":"≟⇒α : DecidableEquality A → AlphaEquality A"},{"Add":"≟⇒α _≟_ = mkAlphaEquality (λ x y → ⌊ x ≟ y ⌋)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Propositional cases"},{"Add":""},{"Add":"-- In the following cases alpha equality coincides with propositiona"},{"Add":"-- equality"},{"Add":""},{"Add":"instance"},{"Add":" α-Visibility : AlphaEquality Visibility"},{"Add":" α-Visibility = ≟⇒α Visibility._≟_"},{"Add":""},{"Add":" α-Modality : AlphaEquality Modality"},{"Add":" α-Modality = ≟⇒α Modality._≟_"},{"Add":""},{"Add":" α-ArgInfo : AlphaEquality ArgInfo"},{"Add":" α-ArgInfo = ≟⇒α ArgInfo._≟_"},{"Add":""},{"Add":" α-Literal : AlphaEquality Literal"},{"Add":" α-Literal = ≟⇒α Literal._≟_"},{"Add":""},{"Add":" α-Meta : AlphaEquality Meta"},{"Add":" α-Meta = ≟⇒α Meta._≟_"},{"Add":""},{"Add":" α-Name : AlphaEquality Name"},{"Add":" α-Name = ≟⇒α Name._≟_"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Interesting cases"},{"Add":""},{"Add":"-- This is where we deviate from propositional equality and ignore the"},{"Add":"-- names of the binders."},{"Add":""},{"Add":"-- Unfortunately we can't declare these as instances directly as the"},{"Add":"-- termination checker isn't clever enough to peer inside the records."},{"Add":""},{"Add":"mutual"},{"Add":" _=α=-AbsTerm_ : Abs Term → Abs Term → Bool"},{"Add":" (abs s a) =α=-AbsTerm (abs s′ a′) = a =α=-Term a′"},{"Add":""},{"Add":" _=α=-Telescope_ : Telescope → Telescope → Bool"},{"Add":" [] =α=-Telescope [] = true"},{"Add":" ((s , x) ∷ xs) =α=-Telescope ((s' , x′) ∷ xs′) = (x =α=-ArgTerm x′) ∧ (xs =α=-Telescope xs′)"},{"Add":" [] =α=-Telescope (_ ∷ _) = false"},{"Add":" (_ ∷ _) =α=-Telescope [] = false"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Remaining cases"},{"Add":""},{"Add":"-- The following cases simply recurse over the remaining AST in exactly"},{"Add":"-- the same way as propositional equality."},{"Add":""},{"Add":" _=α=-ArgTerm_ : Arg Term → Arg Term → Bool"},{"Add":" (arg i a) =α=-ArgTerm (arg i′ a′) = a =α=-Term a′"},{"Add":""},{"Add":" _=α=-ArgPattern_ : Arg Pattern → Arg Pattern → Bool"},{"Add":" (arg i a) =α=-ArgPattern (arg i′ a′) = a =α=-Pattern a′"},{"Add":""},{"Add":" _=α=-Term_ : Term → Term → Bool"},{"Add":" (var x args) =α=-Term (var x′ args′) = (x ℕ.≡ᵇ x′) ∧ (args =α=-ArgsTerm args′)"},{"Add":" (con c args) =α=-Term (con c′ args′) = (c =α= c′) ∧ (args =α=-ArgsTerm args′)"},{"Add":" (def f args) =α=-Term (def f′ args′) = (f =α= f′) ∧ (args =α=-ArgsTerm args′)"},{"Add":" (meta x args) =α=-Term (meta x′ args′) = (x =α= x′) ∧ (args =α=-ArgsTerm args′)"},{"Add":" (pat-lam cs args) =α=-Term (pat-lam cs′ args′) = (cs =α=-Clauses cs′) ∧ (args =α=-ArgsTerm args′)"},{"Add":" (pi t₁ t₂ ) =α=-Term (pi t₁′ t₂′ ) = (t₁ =α=-ArgTerm t₁′) ∧ (t₂ =α=-AbsTerm t₂′)"},{"Add":" (sort s ) =α=-Term (sort s′ ) = s =α=-Sort s′"},{"Add":" (lam v t ) =α=-Term (lam v′ t′ ) = (v =α= v′) ∧ (t =α=-AbsTerm t′)"},{"Add":" (lit l ) =α=-Term (lit l′ ) = l =α= l′"},{"Add":" (unknown ) =α=-Term (unknown ) = true"},{"Add":""},{"Add":" (var x args ) =α=-Term (con c args′) = false"},{"Add":" (var x args ) =α=-Term (def f args′) = false"},{"Add":" (var x args ) =α=-Term (lam v t ) = false"},{"Add":" (var x args ) =α=-Term (pi t₁ t₂ ) = false"},{"Add":" (var x args ) =α=-Term (sort _ ) = false"},{"Add":" (var x args ) =α=-Term (lit _ ) = false"},{"Add":" (var x args ) =α=-Term (meta _ _ ) = false"},{"Add":" (var x args ) =α=-Term (unknown ) = false"},{"Add":" (con c args ) =α=-Term (var x args′) = false"},{"Add":" (con c args ) =α=-Term (def f args′) = false"},{"Add":" (con c args ) =α=-Term (lam v t ) = false"},{"Add":" (con c args ) =α=-Term (pi t₁ t₂ ) = false"},{"Add":" (con c args ) =α=-Term (sort _ ) = false"},{"Add":" (con c args ) =α=-Term (lit _ ) = false"},{"Add":" (con c args ) =α=-Term (meta _ _ ) = false"},{"Add":" (con c args ) =α=-Term (unknown ) = false"},{"Add":" (def f args ) =α=-Term (var x args′) = false"},{"Add":" (def f args ) =α=-Term (con c args′) = false"},{"Add":" (def f args ) =α=-Term (lam v t ) = false"},{"Add":" (def f args ) =α=-Term (pi t₁ t₂ ) = false"},{"Add":" (def f args ) =α=-Term (sort _ ) = false"},{"Add":" (def f args ) =α=-Term (lit _ ) = false"},{"Add":" (def f args ) =α=-Term (meta _ _ ) = false"},{"Add":" (def f args ) =α=-Term (unknown ) = false"},{"Add":" (lam v t ) =α=-Term (var x args ) = false"},{"Add":" (lam v t ) =α=-Term (con c args ) = false"},{"Add":" (lam v t ) =α=-Term (def f args ) = false"},{"Add":" (lam v t ) =α=-Term (pi t₁ t₂ ) = false"},{"Add":" (lam v t ) =α=-Term (sort _ ) = false"},{"Add":" (lam v t ) =α=-Term (lit _ ) = false"},{"Add":" (lam v t ) =α=-Term (meta _ _ ) = false"},{"Add":" (lam v t ) =α=-Term (unknown ) = false"},{"Add":" (pi t₁ t₂ ) =α=-Term (var x args ) = false"},{"Add":" (pi t₁ t₂ ) =α=-Term (con c args ) = false"},{"Add":" (pi t₁ t₂ ) =α=-Term (def f args ) = false"},{"Add":" (pi t₁ t₂ ) =α=-Term (lam v t ) = false"},{"Add":" (pi t₁ t₂ ) =α=-Term (sort _ ) = false"},{"Add":" (pi t₁ t₂ ) =α=-Term (lit _ ) = false"},{"Add":" (pi t₁ t₂ ) =α=-Term (meta _ _ ) = false"},{"Add":" (pi t₁ t₂ ) =α=-Term (unknown ) = false"},{"Add":" (sort _ ) =α=-Term (var x args ) = false"},{"Add":" (sort _ ) =α=-Term (con c args ) = false"},{"Add":" (sort _ ) =α=-Term (def f args ) = false"},{"Add":" (sort _ ) =α=-Term (lam v t ) = false"},{"Add":" (sort _ ) =α=-Term (pi t₁ t₂ ) = false"},{"Add":" (sort _ ) =α=-Term (lit _ ) = false"},{"Add":" (sort _ ) =α=-Term (meta _ _ ) = false"},{"Add":" (sort _ ) =α=-Term (unknown ) = false"},{"Add":" (lit _ ) =α=-Term (var x args ) = false"},{"Add":" (lit _ ) =α=-Term (con c args ) = false"},{"Add":" (lit _ ) =α=-Term (def f args ) = false"},{"Add":" (lit _ ) =α=-Term (lam v t ) = false"},{"Add":" (lit _ ) =α=-Term (pi t₁ t₂ ) = false"},{"Add":" (lit _ ) =α=-Term (sort _ ) = false"},{"Add":" (lit _ ) =α=-Term (meta _ _ ) = false"},{"Add":" (lit _ ) =α=-Term (unknown ) = false"},{"Add":" (meta _ _ ) =α=-Term (var x args ) = false"},{"Add":" (meta _ _ ) =α=-Term (con c args ) = false"},{"Add":" (meta _ _ ) =α=-Term (def f args ) = false"},{"Add":" (meta _ _ ) =α=-Term (lam v t ) = false"},{"Add":" (meta _ _ ) =α=-Term (pi t₁ t₂ ) = false"},{"Add":" (meta _ _ ) =α=-Term (sort _ ) = false"},{"Add":" (meta _ _ ) =α=-Term (lit _ ) = false"},{"Add":" (meta _ _ ) =α=-Term (unknown ) = false"},{"Add":" (unknown ) =α=-Term (var x args ) = false"},{"Add":" (unknown ) =α=-Term (con c args ) = false"},{"Add":" (unknown ) =α=-Term (def f args ) = false"},{"Add":" (unknown ) =α=-Term (lam v t ) = false"},{"Add":" (unknown ) =α=-Term (pi t₁ t₂ ) = false"},{"Add":" (unknown ) =α=-Term (sort _ ) = false"},{"Add":" (unknown ) =α=-Term (lit _ ) = false"},{"Add":" (unknown ) =α=-Term (meta _ _ ) = false"},{"Add":" (pat-lam _ _) =α=-Term (var x args ) = false"},{"Add":" (pat-lam _ _) =α=-Term (con c args ) = false"},{"Add":" (pat-lam _ _) =α=-Term (def f args ) = false"},{"Add":" (pat-lam _ _) =α=-Term (lam v t ) = false"},{"Add":" (pat-lam _ _) =α=-Term (pi t₁ t₂ ) = false"},{"Add":" (pat-lam _ _) =α=-Term (sort _ ) = false"},{"Add":" (pat-lam _ _) =α=-Term (lit _ ) = false"},{"Add":" (pat-lam _ _) =α=-Term (meta _ _ ) = false"},{"Add":" (pat-lam _ _) =α=-Term (unknown ) = false"},{"Add":" (var x args ) =α=-Term (pat-lam _ _) = false"},{"Add":" (con c args ) =α=-Term (pat-lam _ _) = false"},{"Add":" (def f args ) =α=-Term (pat-lam _ _) = false"},{"Add":" (lam v t ) =α=-Term (pat-lam _ _) = false"},{"Add":" (pi t₁ t₂ ) =α=-Term (pat-lam _ _) = false"},{"Add":" (sort _ ) =α=-Term (pat-lam _ _) = false"},{"Add":" (lit _ ) =α=-Term (pat-lam _ _) = false"},{"Add":" (meta _ _ ) =α=-Term (pat-lam _ _) = false"},{"Add":" (unknown ) =α=-Term (pat-lam _ _) = false"},{"Add":""},{"Add":" _=α=-Sort_ : Sort → Sort → Bool"},{"Add":" (set t ) =α=-Sort (set t′ ) = t =α=-Term t′"},{"Add":" (lit n ) =α=-Sort (lit n′ ) = n ℕ.≡ᵇ n′"},{"Add":" (prop t ) =α=-Sort (prop t′ ) = t =α=-Term t′"},{"Add":" (propLit n) =α=-Sort (propLit n′) = n ℕ.≡ᵇ n′"},{"Add":" (inf n ) =α=-Sort (inf n′ ) = n ℕ.≡ᵇ n′"},{"Add":" (unknown ) =α=-Sort (unknown ) = true"},{"Add":""},{"Add":" (set _ ) =α=-Sort (lit _ ) = false"},{"Add":" (set _ ) =α=-Sort (prop _ ) = false"},{"Add":" (set _ ) =α=-Sort (propLit _) = false"},{"Add":" (set _ ) =α=-Sort (inf _ ) = false"},{"Add":" (set _ ) =α=-Sort (unknown ) = false"},{"Add":" (lit _ ) =α=-Sort (set _ ) = false"},{"Add":" (lit _ ) =α=-Sort (prop _ ) = false"},{"Add":" (lit _ ) =α=-Sort (propLit _) = false"},{"Add":" (lit _ ) =α=-Sort (inf _ ) = false"},{"Add":" (lit _ ) =α=-Sort (unknown ) = false"},{"Add":" (prop _ ) =α=-Sort (set _ ) = false"},{"Add":" (prop _ ) =α=-Sort (lit _ ) = false"},{"Add":" (prop _ ) =α=-Sort (propLit _) = false"},{"Add":" (prop _ ) =α=-Sort (inf _ ) = false"},{"Add":" (prop _ ) =α=-Sort (unknown ) = false"},{"Add":" (propLit _) =α=-Sort (set _ ) = false"},{"Add":" (propLit _) =α=-Sort (lit _ ) = false"},{"Add":" (propLit _) =α=-Sort (prop _ ) = false"},{"Add":" (propLit _) =α=-Sort (inf _ ) = false"},{"Add":" (propLit _) =α=-Sort (unknown ) = false"},{"Add":" (inf _ ) =α=-Sort (set _ ) = false"},{"Add":" (inf _ ) =α=-Sort (lit _ ) = false"},{"Add":" (inf _ ) =α=-Sort (prop _ ) = false"},{"Add":" (inf _ ) =α=-Sort (propLit _) = false"},{"Add":" (inf _ ) =α=-Sort (unknown ) = false"},{"Add":" (unknown ) =α=-Sort (set _ ) = false"},{"Add":" (unknown ) =α=-Sort (lit _ ) = false"},{"Add":" (unknown ) =α=-Sort (prop _ ) = false"},{"Add":" (unknown ) =α=-Sort (propLit _) = false"},{"Add":" (unknown ) =α=-Sort (inf _ ) = false"},{"Add":""},{"Add":" _=α=-Clause_ : Clause → Clause → Bool"},{"Add":" (clause tel ps b) =α=-Clause (clause tel′ ps′ b′) = (tel =α=-Telescope tel′) ∧ (ps =α=-ArgsPattern ps′) ∧ (b =α=-Term b′)"},{"Add":" (absurd-clause tel ps) =α=-Clause (absurd-clause tel′ ps′) = (tel =α=-Telescope tel′) ∧ (ps =α=-ArgsPattern ps′)"},{"Add":" (clause _ _ _) =α=-Clause (absurd-clause _ _) = false"},{"Add":" (absurd-clause _ _) =α=-Clause (clause _ _ _) = false"},{"Add":""},{"Add":" _=α=-Clauses_ : Clauses → Clauses → Bool"},{"Add":" [] =α=-Clauses [] = true"},{"Add":" (x ∷ xs) =α=-Clauses (x′ ∷ xs′) = (x =α=-Clause x′) ∧ (xs =α=-Clauses xs′)"},{"Add":" [] =α=-Clauses (_ ∷ _) = false"},{"Add":" (_ ∷ _) =α=-Clauses [] = false"},{"Add":""},{"Add":" _=α=-ArgsTerm_ : Args Term → Args Term → Bool"},{"Add":" [] =α=-ArgsTerm [] = true"},{"Add":" (x ∷ xs) =α=-ArgsTerm (x′ ∷ xs′) = (x =α=-ArgTerm x′) ∧ (xs =α=-ArgsTerm xs′)"},{"Add":" [] =α=-ArgsTerm (_ ∷ _) = false"},{"Add":" (_ ∷ _) =α=-ArgsTerm [] = false"},{"Add":""},{"Add":" _=α=-Pattern_ : Pattern → Pattern → Bool"},{"Add":" (con c ps) =α=-Pattern (con c′ ps′) = (c =α= c′) ∧ (ps =α=-ArgsPattern ps′)"},{"Add":" (var x ) =α=-Pattern (var x′ ) = x ℕ.≡ᵇ x′"},{"Add":" (lit l ) =α=-Pattern (lit l′ ) = l =α= l′"},{"Add":" (proj a ) =α=-Pattern (proj a′ ) = a =α= a′"},{"Add":" (dot t ) =α=-Pattern (dot t′ ) = t =α=-Term t′"},{"Add":" (absurd x) =α=-Pattern (absurd x′ ) = x ℕ.≡ᵇ x′"},{"Add":""},{"Add":" (con x x₁) =α=-Pattern (dot x₂ ) = false"},{"Add":" (con x x₁) =α=-Pattern (var x₂ ) = false"},{"Add":" (con x x₁) =α=-Pattern (lit x₂ ) = false"},{"Add":" (con x x₁) =α=-Pattern (proj x₂ ) = false"},{"Add":" (con x x₁) =α=-Pattern (absurd x₂ ) = false"},{"Add":" (dot x ) =α=-Pattern (con x₁ x₂ ) = false"},{"Add":" (dot x ) =α=-Pattern (var x₁ ) = false"},{"Add":" (dot x ) =α=-Pattern (lit x₁ ) = false"},{"Add":" (dot x ) =α=-Pattern (proj x₁ ) = false"},{"Add":" (dot x ) =α=-Pattern (absurd x₁ ) = false"},{"Add":" (var s ) =α=-Pattern (con x x₁ ) = false"},{"Add":" (var s ) =α=-Pattern (dot x ) = false"},{"Add":" (var s ) =α=-Pattern (lit x ) = false"},{"Add":" (var s ) =α=-Pattern (proj x ) = false"},{"Add":" (var s ) =α=-Pattern (absurd x ) = false"},{"Add":" (lit x ) =α=-Pattern (con x₁ x₂ ) = false"},{"Add":" (lit x ) =α=-Pattern (dot x₁ ) = false"},{"Add":" (lit x ) =α=-Pattern (var _ ) = false"},{"Add":" (lit x ) =α=-Pattern (proj x₁ ) = false"},{"Add":" (lit x ) =α=-Pattern (absurd x₁ ) = false"},{"Add":" (proj x ) =α=-Pattern (con x₁ x₂ ) = false"},{"Add":" (proj x ) =α=-Pattern (dot x₁ ) = false"},{"Add":" (proj x ) =α=-Pattern (var _ ) = false"},{"Add":" (proj x ) =α=-Pattern (lit x₁ ) = false"},{"Add":" (proj x ) =α=-Pattern (absurd x₁ ) = false"},{"Add":" (absurd x) =α=-Pattern (con x₁ x₂ ) = false"},{"Add":" (absurd x) =α=-Pattern (dot x₁ ) = false"},{"Add":" (absurd x) =α=-Pattern (var _ ) = false"},{"Add":" (absurd x) =α=-Pattern (lit x₁ ) = false"},{"Add":" (absurd x) =α=-Pattern (proj x₁ ) = false"},{"Add":""},{"Add":" _=α=-ArgsPattern_ : Args Pattern → Args Pattern → Bool"},{"Add":" [] =α=-ArgsPattern [] = true"},{"Add":" (x ∷ xs) =α=-ArgsPattern (x′ ∷ xs′) = (x =α=-ArgPattern x′) ∧ (xs =α=-ArgsPattern xs′)"},{"Add":" [] =α=-ArgsPattern (_ ∷ _) = false"},{"Add":" (_ ∷ _) =α=-ArgsPattern [] = false"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Instance declarations for mutually recursive cases"},{"Add":""},{"Add":"instance"},{"Add":" α-AbsTerm : AlphaEquality (Abs Term)"},{"Add":" α-AbsTerm = mkAlphaEquality _=α=-AbsTerm_"},{"Add":""},{"Add":" α-ArgTerm : AlphaEquality (Arg Term)"},{"Add":" α-ArgTerm = mkAlphaEquality _=α=-ArgTerm_"},{"Add":""},{"Add":" α-ArgPattern : AlphaEquality (Arg Pattern)"},{"Add":" α-ArgPattern = mkAlphaEquality _=α=-ArgPattern_"},{"Add":""},{"Add":" α-Telescope : AlphaEquality Telescope"},{"Add":" α-Telescope = mkAlphaEquality _=α=-Telescope_"},{"Add":""},{"Add":" α-Term : AlphaEquality Term"},{"Add":" α-Term = mkAlphaEquality _=α=-Term_"},{"Add":""},{"Add":" α-Sort : AlphaEquality Sort"},{"Add":" α-Sort = mkAlphaEquality _=α=-Sort_"},{"Add":""},{"Add":" α-Clause : AlphaEquality Clause"},{"Add":" α-Clause = mkAlphaEquality _=α=-Clause_"},{"Add":""},{"Add":" α-Clauses : AlphaEquality Clauses"},{"Add":" α-Clauses = mkAlphaEquality _=α=-Clauses_"},{"Add":""},{"Add":" α-ArgsTerm : AlphaEquality (Args Term)"},{"Add":" α-ArgsTerm = mkAlphaEquality _=α=-ArgsTerm_"},{"Add":""},{"Add":" α-Pattern : AlphaEquality Pattern"},{"Add":" α-Pattern = mkAlphaEquality _=α=-Pattern_"},{"Add":""},{"Add":" α-ArgsPattern : AlphaEquality (Args Pattern)"},{"Add":" α-ArgsPattern = mkAlphaEquality _=α=-ArgsPattern_"}]}},{"FileAdd":{"meta":{"basename":"Abstraction.agda","metadata":0},"context":[{"change":0,"pos":888325}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Abstractions used in the reflection machinery"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Reflection.AST.Abstraction where"},{"Add":""},{"Add":"open import Data.String.Base as String using (String)"},{"Add":"open import Data.String.Properties as String using (_≟_)"},{"Add":"open import Data.Product.Base using (_×_; <_,_>; uncurry)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Nullary.Decidable.Core using (Dec; map′; _×-dec_)"},{"Add":"open import Relation.Binary.Definitions using (DecidableEquality)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b : Level"},{"Add":" A B : Set a"},{"Add":" s t : String"},{"Add":" x y : A"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting the builtins publicly"},{"Add":""},{"Add":"open import Agda.Builtin.Reflection public"},{"Add":" using (Abs)"},{"Add":""},{"Add":"open Abs public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Operations"},{"Add":""},{"Add":"map : (A → B) → Abs A → Abs B"},{"Add":"map f (abs s x) = abs s (f x)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Decidable equality"},{"Add":""},{"Add":"abs-injective₁ : abs s x ≡ abs t y → s ≡ t"},{"Add":"abs-injective₁ refl = refl"},{"Add":""},{"Add":"abs-injective₂ : abs s x ≡ abs t y → x ≡ y"},{"Add":"abs-injective₂ refl = refl"},{"Add":""},{"Add":"abs-injective : abs s x ≡ abs t y → s ≡ t × x ≡ y"},{"Add":"abs-injective = < abs-injective₁ , abs-injective₂ >"},{"Add":""},{"Add":"-- We often need decidability of equality for Abs A when implementing it"},{"Add":"-- for A. Unfortunately ≡-dec makes the termination checker unhappy."},{"Add":"-- Instead, we can match on both Abs A and use unAbs-dec for an"},{"Add":"-- obviously decreasing recursive call."},{"Add":""},{"Add":"unAbs : Abs A → A"},{"Add":"unAbs (abs s a) = a"},{"Add":""},{"Add":"unAbs-dec : {abs1 abs2 : Abs A} → Dec (unAbs abs1 ≡ unAbs abs2) → Dec (abs1 ≡ abs2)"},{"Add":"unAbs-dec {abs1 = abs s a} {abs t a′} a≟a′ ="},{"Add":" map′ (uncurry (cong₂ abs)) abs-injective (s String.≟ t ×-dec a≟a′)"},{"Add":""},{"Add":"≡-dec : DecidableEquality A → DecidableEquality (Abs A)"},{"Add":"≡-dec _≟_ x y = unAbs-dec (unAbs x ≟ unAbs y)"}]}},{"FileAdd":{"meta":{"basename":"Level.agda","metadata":0},"context":[{"change":0,"pos":62922}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Universe levels"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Level where"},{"Add":""},{"Add":"-- Levels."},{"Add":""},{"Add":"open import Agda.Primitive as Prim public"},{"Add":" using (Level; _⊔_; Setω)"},{"Add":" renaming (lzero to zero; lsuc to suc)"},{"Add":""},{"Add":"-- Lifting."},{"Add":""},{"Add":"record Lift {a} ℓ (A : Set a) : Set (a ⊔ ℓ) where"},{"Add":" constructor lift"},{"Add":" field lower : A"},{"Add":""},{"Add":"open Lift public"},{"Add":""},{"Add":"-- Synonyms"},{"Add":""},{"Add":"0ℓ : Level"},{"Add":"0ℓ = zero"},{"Add":""},{"Add":"levelOfType : ∀ {a} → Set a → Level"},{"Add":"levelOfType {a} _ = a"},{"Add":""},{"Add":"levelOfTerm : ∀ {a} {A : Set a} → A → Level"},{"Add":"levelOfTerm {a} _ = a"}]}},{"FileAdd":{"meta":{"basename":"Level","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Literals.agda","metadata":0},"context":[{"change":0,"pos":970648}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Conversion from naturals to universe levels"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Level.Literals where"},{"Add":""},{"Add":"open import Agda.Builtin.Nat renaming (Nat to ℕ)"},{"Add":"open import Agda.Builtin.FromNat"},{"Add":"open import Agda.Builtin.Unit"},{"Add":"open import Level using (Level; 0ℓ)"},{"Add":""},{"Add":"-- Increase a Level by a number of sucs."},{"Add":""},{"Add":"infixl 6 _ℕ+_"},{"Add":""},{"Add":"_ℕ+_ : ℕ → Level → Level"},{"Add":"zero ℕ+ ℓ = ℓ"},{"Add":"suc n ℕ+ ℓ = Level.suc (n ℕ+ ℓ)"},{"Add":""},{"Add":"-- Nat-computed Level."},{"Add":""},{"Add":"infix 10 #_"},{"Add":""},{"Add":"#_ : ℕ → Level"},{"Add":"#_ = _ℕ+ 0ℓ"},{"Add":""},{"Add":"-- Literal overloading for levels."},{"Add":""},{"Add":"Levelℕ : Number Level"},{"Add":"Levelℕ .Number.Constraint _ = ⊤"},{"Add":"Levelℕ .Number.fromNat n = # n"}]}},{"FileAdd":{"meta":{"basename":"Induction.agda","metadata":0},"context":[{"change":0,"pos":62922}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- An abstraction of various forms of recursion/induction"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The idea underlying Induction.* comes from Epigram 1, see Section 4"},{"Add":"-- of \"The view from the left\" by McBride and McKinna."},{"Add":""},{"Add":"-- Note: The types in this module can perhaps be easier to understand"},{"Add":"-- if they are normalised. Note also that Agda can do the"},{"Add":"-- normalisation for you."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Induction where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Relation.Unary"},{"Add":"open import Relation.Unary.PredicateTransformer using (PT)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a ℓ ℓ₁ ℓ₂ : Level"},{"Add":" A : Set a"},{"Add":" Q : Pred A ℓ"},{"Add":" Rec : PT A A ℓ₁ ℓ₂"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- A RecStruct describes the allowed structure of recursion. The"},{"Add":"-- examples in Data.Nat.Induction should explain what this is all"},{"Add":"-- about."},{"Add":""},{"Add":"RecStruct : Set a → (ℓ₁ ℓ₂ : Level) → Set _"},{"Add":"RecStruct A = PT A A"},{"Add":""},{"Add":"-- A recursor builder constructs an instance of a recursion structure"},{"Add":"-- for a given input."},{"Add":""},{"Add":"RecursorBuilder : RecStruct A ℓ₁ ℓ₂ → Set _"},{"Add":"RecursorBuilder Rec = ∀ P → Rec P ⊆′ P → Universal (Rec P)"},{"Add":""},{"Add":"-- A recursor can be used to actually compute/prove something useful."},{"Add":""},{"Add":"Recursor : RecStruct A ℓ₁ ℓ₂ → Set _"},{"Add":"Recursor Rec = ∀ P → Rec P ⊆′ P → Universal P"},{"Add":""},{"Add":"-- And recursors can be constructed from recursor builders."},{"Add":""},{"Add":"build : RecursorBuilder Rec → Recursor Rec"},{"Add":"build builder P f x = f x (builder P f x)"},{"Add":""},{"Add":"-- We can repeat the exercise above for subsets of the type we are"},{"Add":"-- recursing over."},{"Add":""},{"Add":"SubsetRecursorBuilder : Pred A ℓ → RecStruct A ℓ₁ ℓ₂ → Set _"},{"Add":"SubsetRecursorBuilder Q Rec = ∀ P → Rec P ⊆′ P → Q ⊆′ Rec P"},{"Add":""},{"Add":"SubsetRecursor : Pred A ℓ → RecStruct A ℓ₁ ℓ₂ → Set _"},{"Add":"SubsetRecursor Q Rec = ∀ P → Rec P ⊆′ P → Q ⊆′ P"},{"Add":""},{"Add":"subsetBuild : SubsetRecursorBuilder Q Rec → SubsetRecursor Q Rec"},{"Add":"subsetBuild builder P f x q = f x (builder P f x q)"}]}},{"FileAdd":{"meta":{"basename":"Induction","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"WellFounded.agda","metadata":0},"context":[{"change":0,"pos":973696}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Well-founded induction"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Induction.WellFounded where"},{"Add":""},{"Add":"open import Data.Product.Base using (Σ; _,_; proj₁; proj₂)"},{"Add":"open import Function.Base using (_∘_; flip; _on_)"},{"Add":"open import Induction"},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (Symmetric; Asymmetric; Irreflexive; _Respects₂_;"},{"Add":" _Respectsʳ_; _Respects_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)"},{"Add":"open import Relation.Binary.Consequences using (asym⇒irr)"},{"Add":"open import Relation.Unary"},{"Add":"open import Relation.Nullary.Negation.Core using (¬_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b ℓ ℓ₁ ℓ₂ r : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definitions"},{"Add":""},{"Add":"-- When using well-founded recursion you can recurse arbitrarily, as"},{"Add":"-- long as the arguments become smaller, and \"smaller\" is"},{"Add":"-- well-founded."},{"Add":""},{"Add":"WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _"},{"Add":"WfRec _<_ P x = ∀ {y} → y < x → P y"},{"Add":""},{"Add":"-- The accessibility predicate: x is accessible if everything which is"},{"Add":"-- smaller than x is also accessible (inductively)."},{"Add":""},{"Add":"data Acc {A : Set a} (_<_ : Rel A ℓ) (x : A) : Set (a ⊔ ℓ) where"},{"Add":" acc : (rs : WfRec _<_ (Acc _<_) x) → Acc _<_ x"},{"Add":""},{"Add":"-- The accessibility predicate encodes what it means to be"},{"Add":"-- well-founded; if all elements are accessible, then _<_ is"},{"Add":"-- well-founded."},{"Add":""},{"Add":"WellFounded : Rel A ℓ → Set _"},{"Add":"WellFounded _<_ = ∀ x → Acc _<_ x"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Basic properties"},{"Add":""},{"Add":"acc-inverse : ∀ {_<_ : Rel A ℓ} {x : A} (q : Acc _<_ x) →"},{"Add":" WfRec _<_ (Acc _<_) x"},{"Add":"acc-inverse (acc rs) y> sequence′ cs"},{"Add":""},{"Add":" mapM : (A → IO B) → List A → IO (List B)"},{"Add":" mapM f = sequence ∘ map f"},{"Add":""},{"Add":" mapM′ : (A → IO B) → List A → IO ⊤"},{"Add":" mapM′ f = sequence′ ∘ map f"},{"Add":""},{"Add":" forM : List A → (A → IO B) → IO (List B)"},{"Add":" forM = flip mapM"},{"Add":""},{"Add":" forM′ : List A → (A → IO B) → IO ⊤"},{"Add":" forM′ = flip mapM′"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Simple lazy IO"},{"Add":""},{"Add":"-- Note that the functions below produce commands which, when"},{"Add":"-- executed, may raise exceptions."},{"Add":""},{"Add":"-- Note also that the semantics of these functions depends on the"},{"Add":"-- version of the Haskell base library. If the version is 4.2.0.0 (or"},{"Add":"-- later?), then the functions use the character encoding specified by"},{"Add":"-- the locale. For older versions of the library (going back to at"},{"Add":"-- least version 3) the functions use ISO-8859-1."},{"Add":""},{"Add":"open import IO.Finite public"},{"Add":" renaming (readFile to readFiniteFile)"},{"Add":""},{"Add":"open import IO.Infinite public"},{"Add":" renaming ( writeFile to writeFile∞"},{"Add":" ; appendFile to appendFile∞"},{"Add":" ; putStr to putStr∞"},{"Add":" ; putStrLn to putStrLn∞"},{"Add":" )"}]}},{"FileAdd":{"meta":{"basename":"IO","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Primitive.agda","metadata":0},"context":[{"change":0,"pos":997513}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use IO.Primitive.Core instead"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module IO.Primitive where"},{"Add":""},{"Add":"open import IO.Primitive.Core public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"IO.Primitive was deprecated in v2.1. 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If the version is 4.2.0.0 (or"},{"Add":"-- later?), then the functions use the character encoding specified by"},{"Add":"-- the locale. For older versions of the library (going back to at"},{"Add":"-- least version 3) the functions use ISO-8859-1."},{"Add":""},{"Add":""},{"Add":"postulate"},{"Add":" getContents : IO Costring"},{"Add":" readFile : String → IO Costring"},{"Add":" writeFile : String → Costring → IO Unit"},{"Add":" appendFile : String → Costring → IO Unit"},{"Add":" putStr : Costring → IO Unit"},{"Add":" putStrLn : Costring → IO Unit"},{"Add":""},{"Add":"{-# FOREIGN GHC import qualified Data.Text #-}"},{"Add":""},{"Add":"{-# FOREIGN GHC"},{"Add":""},{"Add":" fromColist :: MAlonzo.Code.Codata.Musical.Colist.Base.AgdaColist a -> [a]"},{"Add":" fromColist MAlonzo.Code.Codata.Musical.Colist.Base.Nil = []"},{"Add":" fromColist (MAlonzo.Code.Codata.Musical.Colist.Base.Cons x xs) ="},{"Add":" x : fromColist (MAlonzo.RTE.flat xs)"},{"Add":""},{"Add":" toColist :: [a] -> MAlonzo.Code.Codata.Musical.Colist.Base.AgdaColist a"},{"Add":" toColist [] = MAlonzo.Code.Codata.Musical.Colist.Base.Nil"},{"Add":" toColist (x : xs) ="},{"Add":" MAlonzo.Code.Codata.Musical.Colist.Base.Cons x (MAlonzo.RTE.Sharp (toColist xs))"},{"Add":"#-}"},{"Add":""},{"Add":"{-# COMPILE GHC getContents = fmap toColist getContents #-}"},{"Add":"{-# COMPILE GHC readFile = fmap toColist . readFile . Data.Text.unpack #-}"},{"Add":"{-# COMPILE GHC writeFile = \\x -> writeFile (Data.Text.unpack x) . fromColist #-}"},{"Add":"{-# COMPILE GHC appendFile = \\x -> appendFile (Data.Text.unpack x) . fromColist #-}"},{"Add":"{-# COMPILE GHC putStr = putStr . fromColist #-}"},{"Add":"{-# COMPILE GHC putStrLn = putStrLn . fromColist #-}"},{"Add":"{-# COMPILE UHC getContents = UHC.Agda.Builtins.primGetContents #-}"},{"Add":"{-# COMPILE UHC readFile = UHC.Agda.Builtins.primReadFile #-}"},{"Add":"{-# COMPILE UHC writeFile = UHC.Agda.Builtins.primWriteFile #-}"},{"Add":"{-# COMPILE UHC appendFile = UHC.Agda.Builtins.primAppendFile #-}"},{"Add":"{-# COMPILE UHC putStr = UHC.Agda.Builtins.primPutStr #-}"},{"Add":"{-# COMPILE UHC putStrLn = UHC.Agda.Builtins.primPutStrLn #-}"}]}},{"FileAdd":{"meta":{"basename":"Handle.agda","metadata":0},"context":[{"change":0,"pos":998016}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Primitive IO handles: simple bindings to Haskell types and functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- NOTE: the contents of this module should be accessed via `IO`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module IO.Primitive.Handle where"},{"Add":""},{"Add":"open import Data.Maybe.Base using (Maybe)"},{"Add":"open import Data.Nat.Base using (ℕ)"},{"Add":""},{"Add":"data BufferMode : Set where"},{"Add":" noBuffering lineBuffering : BufferMode"},{"Add":" blockBuffering : Maybe ℕ → BufferMode"},{"Add":"{-# FOREIGN GHC import System.IO #-}"},{"Add":"{-# FOREIGN GHC"},{"Add":" data AgdaBufferMode"},{"Add":" = AgdaNoBuffering"},{"Add":" | AgdaLineBuffering"},{"Add":" | AgdaBlockBuffering (Maybe Integer)"},{"Add":" toBufferMode :: AgdaBufferMode -> BufferMode"},{"Add":" toBufferMode x = case x of"},{"Add":" AgdaNoBuffering -> NoBuffering"},{"Add":" AgdaLineBuffering -> LineBuffering"},{"Add":" AgdaBlockBuffering mi -> BlockBuffering (fromIntegral <$> mi)"},{"Add":" fromBufferMode :: BufferMode -> AgdaBufferMode"},{"Add":" fromBufferMode x = case x of"},{"Add":" NoBuffering -> AgdaNoBuffering"},{"Add":" LineBuffering -> AgdaLineBuffering"},{"Add":" BlockBuffering mi -> AgdaBlockBuffering (fromIntegral <$> mi)"},{"Add":"#-}"},{"Add":""},{"Add":"{-# COMPILE GHC BufferMode = data AgdaBufferMode"},{"Add":" ( AgdaNoBuffering"},{"Add":" | AgdaLineBuffering"},{"Add":" | AgdaBlockBuffering"},{"Add":" )"},{"Add":"#-}"},{"Add":""},{"Add":"open import Data.Unit.Base using (⊤)"},{"Add":"open import IO.Primitive.Core using (IO)"},{"Add":""},{"Add":"postulate"},{"Add":" Handle : Set"},{"Add":" stdin stdout stderr : Handle"},{"Add":" hSetBuffering : Handle → BufferMode → IO ⊤"},{"Add":" hGetBuffering : Handle → IO BufferMode"},{"Add":" hFlush : Handle → IO ⊤"},{"Add":""},{"Add":"{-# FOREIGN GHC import System.IO #-}"},{"Add":"{-# COMPILE GHC Handle = type Handle #-}"},{"Add":"{-# COMPILE GHC stdin = stdin #-}"},{"Add":"{-# COMPILE GHC stdout = stdout #-}"},{"Add":"{-# COMPILE GHC stderr = stderr #-}"},{"Add":"{-# COMPILE GHC hSetBuffering = \\ h -> hSetBuffering h . toBufferMode #-}"},{"Add":"{-# COMPILE GHC hGetBuffering = fmap fromBufferMode . hGetBuffering #-}"},{"Add":"{-# COMPILE GHC hFlush = hFlush #-}"}]}},{"FileAdd":{"meta":{"basename":"Finite.agda","metadata":0},"context":[{"change":0,"pos":998016}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Primitive IO: simple bindings to Haskell types and functions"},{"Add":"-- Everything is assumed to be finite"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module IO.Primitive.Finite where"},{"Add":""},{"Add":"-- NOTE: the contents of this module should be accessed via `IO` or"},{"Add":"-- `IO.Finite`."},{"Add":""},{"Add":"open import Agda.Builtin.IO"},{"Add":"open import Agda.Builtin.String"},{"Add":"open import Agda.Builtin.Unit using () renaming (⊤ to Unit)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Simple lazy IO"},{"Add":""},{"Add":"-- Note that the functions below produce commands which, when"},{"Add":"-- executed, may raise exceptions."},{"Add":""},{"Add":"-- Note also that the semantics of these functions depends on the"},{"Add":"-- version of the Haskell base library. If the version is 4.2.0.0 (or"},{"Add":"-- later?), then the functions use the character encoding specified by"},{"Add":"-- the locale. For older versions of the library (going back to at"},{"Add":"-- least version 3) the functions use ISO-8859-1."},{"Add":""},{"Add":"postulate"},{"Add":" getLine : IO String"},{"Add":" readFile : String → IO String"},{"Add":" writeFile : String → String → IO Unit"},{"Add":" appendFile : String → String → IO Unit"},{"Add":" putStr : String → IO Unit"},{"Add":" putStrLn : String → IO Unit"},{"Add":""},{"Add":"{-# FOREIGN GHC import qualified Data.Text as T #-}"},{"Add":"{-# FOREIGN GHC import qualified Data.Text.IO as TIO #-}"},{"Add":"{-# FOREIGN GHC import qualified System.IO #-}"},{"Add":"{-# FOREIGN GHC import qualified Control.Exception #-}"},{"Add":""},{"Add":"{-# FOREIGN GHC"},{"Add":""},{"Add":" -- Reads a finite file. Raises an exception if the file path refers"},{"Add":" -- to a non-physical file (like \"/dev/zero\")."},{"Add":" readFiniteFile :: T.Text -> IO T.Text"},{"Add":" readFiniteFile f = do"},{"Add":" h <- System.IO.openFile (T.unpack f) System.IO.ReadMode"},{"Add":" Control.Exception.bracketOnError (return ()) (\\_ -> System.IO.hClose h)"},{"Add":" (\\_ -> System.IO.hFileSize h)"},{"Add":" TIO.hGetContents h"},{"Add":""},{"Add":"#-}"},{"Add":""},{"Add":"{-# COMPILE GHC getLine = TIO.getLine #-}"},{"Add":"{-# COMPILE GHC readFile = readFiniteFile #-}"},{"Add":"{-# COMPILE GHC writeFile = TIO.writeFile . T.unpack #-}"},{"Add":"{-# COMPILE GHC appendFile = TIO.appendFile . T.unpack #-}"},{"Add":"{-# COMPILE GHC putStr = TIO.putStr #-}"},{"Add":"{-# COMPILE GHC putStrLn = TIO.putStrLn #-}"},{"Add":""},{"Add":"{-# COMPILE UHC readFile = UHC.Agda.Builtins.primReadFiniteFile #-}"}]}},{"FileAdd":{"meta":{"basename":"Core.agda","metadata":0},"context":[{"change":0,"pos":998016}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Primitive IO: simple bindings to Haskell types and functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- NOTE: the contents of this module should be accessed via `IO`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module IO.Primitive.Core where"},{"Add":""},{"Add":"open import Level using (Level)"},{"Add":"private"},{"Add":" variable"},{"Add":" a : Level"},{"Add":" A B : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The IO monad"},{"Add":""},{"Add":"open import Agda.Builtin.IO public"},{"Add":" using (IO)"},{"Add":""},{"Add":"infixl 1 _>>=_"},{"Add":""},{"Add":"postulate"},{"Add":" pure : A → IO A"},{"Add":" _>>=_ : IO A → (A → IO B) → IO B"},{"Add":""},{"Add":"{-# COMPILE GHC pure = \\_ _ -> return #-}"},{"Add":"{-# COMPILE GHC _>>=_ = \\_ _ _ _ -> (>>=) #-}"},{"Add":"{-# COMPILE UHC pure = \\_ _ x -> UHC.Agda.Builtins.primReturn x #-}"},{"Add":"{-# COMPILE UHC _>>=_ = \\_ _ _ _ x y -> UHC.Agda.Builtins.primBind x y #-}"},{"Add":""},{"Add":"-- Haskell-style alternative syntax"},{"Add":"return : A → IO A"},{"Add":"return = pure"},{"Add":""},{"Add":"_>>_ : IO A → IO B → IO B"},{"Add":"a >> b = a >>= λ _ → b"}]}},{"FileAdd":{"meta":{"basename":"Instances.agda","metadata":0},"context":[{"change":0,"pos":997513}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Typeclass instances for IO"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module IO.Instances where"},{"Add":""},{"Add":"open import IO.Base"},{"Add":"open import Effect.Functor"},{"Add":"open import Effect.Applicative"},{"Add":"open import Effect.Monad"},{"Add":"open import IO.Effectful"},{"Add":""},{"Add":"instance"},{"Add":" ioFunctor = functor"},{"Add":" ioApplicative = applicative"},{"Add":" ioMonad = monad"}]}},{"FileAdd":{"meta":{"basename":"Infinite.agda","metadata":0},"context":[{"change":0,"pos":997513}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- IO only involving infinite objects"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module IO.Infinite where"},{"Add":""},{"Add":"open import Codata.Musical.Costring"},{"Add":"open import Agda.Builtin.String"},{"Add":"open import Data.Unit.Polymorphic.Base"},{"Add":"import Data.Unit.Base as Unit0"},{"Add":"open import IO.Base"},{"Add":"import IO.Primitive.Core as Prim"},{"Add":"import IO.Primitive.Infinite as Prim"},{"Add":"open import Level"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Simple lazy IO"},{"Add":""},{"Add":"-- Note that the functions below produce commands which, when"},{"Add":"-- executed, may raise exceptions."},{"Add":""},{"Add":"-- Note also that the semantics of these functions depends on the"},{"Add":"-- version of the Haskell base library. If the version is 4.2.0.0 (or"},{"Add":"-- later?), then the functions use the character encoding specified by"},{"Add":"-- the locale. For older versions of the library (going back to at"},{"Add":"-- least version 3) the functions use ISO-8859-1."},{"Add":""},{"Add":"getContents : IO Costring"},{"Add":"getContents = lift Prim.getContents"},{"Add":""},{"Add":"writeFile : String → Costring → IO {a} ⊤"},{"Add":"writeFile f s = lift′ (Prim.writeFile f s)"},{"Add":""},{"Add":"appendFile : String → Costring → IO {a} ⊤"},{"Add":"appendFile f s = lift′ (Prim.appendFile f s)"},{"Add":""},{"Add":"putStr : Costring → IO {a} ⊤"},{"Add":"putStr s = lift′ (Prim.putStr s)"},{"Add":""},{"Add":"putStrLn : Costring → IO {a} ⊤"},{"Add":"putStrLn s = lift′ (Prim.putStrLn s)"}]}},{"FileAdd":{"meta":{"basename":"Handle.agda","metadata":0},"context":[{"change":0,"pos":997513}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- IO handles: simple bindings to Haskell types and functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module IO.Handle where"},{"Add":""},{"Add":"open import Level using (Level)"},{"Add":"open import Data.Unit.Polymorphic.Base using (⊤)"},{"Add":"open import IO.Base using (IO; lift; lift′)"},{"Add":""},{"Add":"private variable a : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting types and constructors"},{"Add":""},{"Add":"open import IO.Primitive.Handle as Prim public"},{"Add":" using ( BufferMode"},{"Add":" ; noBuffering"},{"Add":" ; lineBuffering"},{"Add":" ; blockBuffering"},{"Add":" ; Handle"},{"Add":" ; stdin"},{"Add":" ; stdout"},{"Add":" ; stderr"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Wrapping functions"},{"Add":""},{"Add":"hSetBuffering : Handle → BufferMode → IO {a} ⊤"},{"Add":"hSetBuffering hdl bm = lift′ (Prim.hSetBuffering hdl bm)"},{"Add":""},{"Add":"hGetBuffering : Handle → IO BufferMode"},{"Add":"hGetBuffering hdl = lift (Prim.hGetBuffering hdl)"},{"Add":""},{"Add":"hFlush : Handle → IO {a} ⊤"},{"Add":"hFlush hdl = lift′ (Prim.hFlush hdl)"}]}},{"FileAdd":{"meta":{"basename":"Finite.agda","metadata":0},"context":[{"change":0,"pos":997513}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- IO only involving finite objects"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module IO.Finite where"},{"Add":""},{"Add":"open import Data.Unit.Polymorphic.Base"},{"Add":"open import Agda.Builtin.String"},{"Add":"import Data.Unit.Base as Unit0"},{"Add":"open import IO.Base"},{"Add":"import IO.Primitive.Core as Prim"},{"Add":"import IO.Primitive.Finite as Prim"},{"Add":"open import Level"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Simple lazy IO"},{"Add":""},{"Add":"-- Note that the functions below produce commands which, when"},{"Add":"-- executed, may raise exceptions."},{"Add":""},{"Add":"-- Note also that the semantics of these functions depends on the"},{"Add":"-- version of the Haskell base library. If the version is 4.2.0.0 (or"},{"Add":"-- later?), then the functions use the character encoding specified by"},{"Add":"-- the locale. For older versions of the library (going back to at"},{"Add":"-- least version 3) the functions use ISO-8859-1."},{"Add":""},{"Add":"getLine : IO String"},{"Add":"getLine = lift Prim.getLine"},{"Add":""},{"Add":"-- Reads a finite file. Raises an exception if the file path refers to"},{"Add":"-- a non-physical file (like \"/dev/zero\")."},{"Add":""},{"Add":"readFile : String → IO String"},{"Add":"readFile f = lift (Prim.readFile f)"},{"Add":""},{"Add":"writeFile : String → String → IO {a} ⊤"},{"Add":"writeFile f s = lift′ (Prim.writeFile f s)"},{"Add":""},{"Add":"appendFile : String → String → IO {a} ⊤"},{"Add":"appendFile f s = lift′ (Prim.appendFile f s)"},{"Add":""},{"Add":"putStr : String → IO {a} ⊤"},{"Add":"putStr s = lift′ (Prim.putStr s)"},{"Add":""},{"Add":"putStrLn : String → IO {a} ⊤"},{"Add":"putStrLn s = lift′ (Prim.putStrLn s)"}]}},{"FileAdd":{"meta":{"basename":"Effectful.agda","metadata":0},"context":[{"change":0,"pos":997513}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- An effectful view of IO"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module IO.Effectful where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Effect.Functor"},{"Add":"open import Effect.Applicative"},{"Add":"open import Effect.Monad"},{"Add":""},{"Add":"open import IO.Base"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" f : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structure"},{"Add":""},{"Add":"functor : RawFunctor {f} IO"},{"Add":"functor = record { _<$>_ = _<$>_ }"},{"Add":""},{"Add":"applicative : RawApplicative {f} IO"},{"Add":"applicative = record"},{"Add":" { rawFunctor = functor"},{"Add":" ; pure = pure"},{"Add":" ; _<*>_ = _<*>_"},{"Add":" }"},{"Add":""},{"Add":"monad : RawMonad {f} IO"},{"Add":"monad = record"},{"Add":" { rawApplicative = applicative"},{"Add":" ; _>>=_ = _>>=_"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Categorical.agda","metadata":0},"context":[{"change":0,"pos":997513}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use"},{"Add":"-- `IO.Effectful` instead."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module IO.Categorical where"},{"Add":""},{"Add":"open import IO.Effectful public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"IO.Categorical was deprecated in v2.0."},{"Add":"Use IO.Effectful instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Base.agda","metadata":0},"context":[{"change":0,"pos":997513}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- IO: basic types and functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module IO.Base where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Codata.Musical.Notation"},{"Add":"open import Data.Bool.Base using (Bool; true; false; not)"},{"Add":"open import Agda.Builtin.Maybe using (Maybe; nothing; just)"},{"Add":"open import Data.Sum.Base using (_⊎_; inj₁; inj₂)"},{"Add":"import Agda.Builtin.Unit as Unit0"},{"Add":"open import Data.Unit.Polymorphic.Base"},{"Add":"open import Function.Base using (_∘′_; const; flip)"},{"Add":"import IO.Primitive.Core as Prim"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b e : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":" E : Set e"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The IO monad"},{"Add":""},{"Add":"-- One cannot write \"infinitely large\" computations with the"},{"Add":"-- postulated IO monad in IO.Primitive without turning off the"},{"Add":"-- termination checker (or going via the FFI, or perhaps abusing"},{"Add":"-- something else). The following coinductive deep embedding is"},{"Add":"-- introduced to avoid this problem. Possible non-termination is"},{"Add":"-- isolated to the run function below."},{"Add":""},{"Add":"data IO (A : Set a) : Set (suc a) where"},{"Add":" lift : (m : Prim.IO A) → IO A"},{"Add":" pure : (x : A) → IO A"},{"Add":" bind : {B : Set a} (m : ∞ (IO B)) (f : (x : B) → ∞ (IO A)) → IO A"},{"Add":" seq : {B : Set a} (m₁ : ∞ (IO B)) (m₂ : ∞ (IO A)) → IO A"},{"Add":""},{"Add":"lift! : IO A → IO (Lift b A)"},{"Add":"lift! (lift io) = lift (io Prim.>>= λ a → Prim.pure (Level.lift a))"},{"Add":"lift! (pure a) = pure (Level.lift a)"},{"Add":"lift! {b = b} (bind m f) = bind (♯ lift! {b = b} (♭ m))"},{"Add":" (λ x → ♯ lift! (♭ (f (lower x))))"},{"Add":"lift! {b = b} (seq m₁ m₂) = seq (♯ lift! {b = b} (♭ m₁))"},{"Add":" (♯ lift! (♭ m₂))"},{"Add":""},{"Add":"module _ {A B : Set a} where"},{"Add":""},{"Add":" infixl 1 _<$>_ _<*>_ _>>=_ _>>_"},{"Add":" infixr 1 _=<<_"},{"Add":""},{"Add":" _<*>_ : IO (A → B) → IO A → IO B"},{"Add":" mf <*> mx = bind (♯ mf) λ f → ♯ (bind (♯ mx) λ x → ♯ pure (f x))"},{"Add":""},{"Add":" _<$>_ : (A → B) → IO A → IO B"},{"Add":" f <$> m = pure f <*> m"},{"Add":""},{"Add":" _<$_ : B → IO A → IO B"},{"Add":" b <$ m = (const b) <$> m"},{"Add":""},{"Add":" _>>=_ : IO A → (A → IO B) → IO B"},{"Add":" m >>= f = bind (♯ m) λ x → ♯ f x"},{"Add":""},{"Add":" _=<<_ : (A → IO B) → IO A → IO B"},{"Add":" _=<<_ = flip _>>=_"},{"Add":""},{"Add":" _>>_ : IO A → IO B → IO B"},{"Add":" m₁ >> m₂ = seq (♯ m₁) (♯ m₂)"},{"Add":""},{"Add":" _<<_ : IO B → IO A → IO B"},{"Add":" _<<_ = flip _>>_"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Running programs"},{"Add":""},{"Add":"-- A value of type `IO A` is a description of a computation that may"},{"Add":"-- eventually produce an `A`. The `run` function converts this"},{"Add":"-- description of a computation into calls to primitive functions that"},{"Add":"-- will actually perform it."},{"Add":""},{"Add":"{-# NON_TERMINATING #-}"},{"Add":"run : IO A → Prim.IO A"},{"Add":"run (lift m) = m"},{"Add":"run (pure x) = Prim.pure x"},{"Add":"run (bind m f) = run (♭ m ) Prim.>>= λ x → run (♭ (f x))"},{"Add":"run (seq m₁ m₂) = run (♭ m₁) Prim.>>= λ _ → run (♭ m₂)"},{"Add":""},{"Add":"-- The entrypoint of an Agda program will be assigned type `Main` and"},{"Add":"-- implemented using `run` on some `IO ⊤` program."},{"Add":""},{"Add":"Main : Set"},{"Add":"Main = Prim.IO {0ℓ} ⊤"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Utilities"},{"Add":""},{"Add":"-- Make a unit-returning primitive level polymorphic"},{"Add":"lift′ : Prim.IO Unit0.⊤ → IO {a} ⊤"},{"Add":"lift′ io = lift (io Prim.>>= λ _ → Prim.pure _)"},{"Add":""},{"Add":"-- Throw away the result"},{"Add":"ignore : IO A → IO ⊤"},{"Add":"ignore io = io >> pure _"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Conditional executions"},{"Add":""},{"Add":"-- Only run the action if the boolean is true"},{"Add":"when : Bool → IO {a} ⊤ → IO ⊤"},{"Add":"when true m = m"},{"Add":"when false _ = pure _"},{"Add":""},{"Add":"-- Only run the action if the boolean is false"},{"Add":"unless : Bool → IO {a} ⊤ → IO ⊤"},{"Add":"unless = when ∘′ not"},{"Add":""},{"Add":"-- Run the action if the `Maybe` computation was successful"},{"Add":"whenJust : Maybe A → (A → IO {a} ⊤) → IO ⊤"},{"Add":"whenJust (just a) k = k a"},{"Add":"whenJust nothing _ = pure _"},{"Add":""},{"Add":"-- Run the action if the `E ⊎_` computation was successful"},{"Add":"whenInj₂ : E ⊎ A → (A → IO {a} ⊤) → IO ⊤"},{"Add":"whenInj₂ (inj₂ a) k = k a"},{"Add":"whenInj₂ (inj₁ _) _ = pure _"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Loops"},{"Add":""},{"Add":"-- Keep running the action forever"},{"Add":"forever : IO {a} ⊤ → IO {a} ⊤"},{"Add":"forever act = seq (♯ act) (♯ forever act)"},{"Add":""},{"Add":"-- Keep running an IO action until we get a value. Convenient when user"},{"Add":"-- input is involved and it may be malformed."},{"Add":"untilJust : IO (Maybe A) → IO A"},{"Add":"-- Note that here we are forced to use `bind` & the musical notation"},{"Add":"-- explicitly to guarantee that the corecursive call is guarded"},{"Add":"untilJust m = bind (♯ m) λ where"},{"Add":" nothing → ♯ untilJust m"},{"Add":" (just a) → ♯ pure a"},{"Add":""},{"Add":"untilInj₂ : {A B : Set a} → (A → IO (A ⊎ B)) → A → IO B"},{"Add":"untilInj₂ f x = bind (♯ f x) λ where"},{"Add":" (inj₁ x′) → ♯ untilInj₂ f x′"},{"Add":" (inj₂ y) → ♯ pure y"},{"Add":""},{"Add":""},{"Add":""},{"Add":""},{"Add":""},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- DEPRECATIONS"},{"Add":""},{"Add":"untilRight = untilInj₂"},{"Add":"{-# WARNING_ON_USAGE untilRight"},{"Add":"\"Warning: untilRight was deprecated in v2.1."},{"Add":"Please use untilInj₂ instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Function.agda","metadata":0},"context":[{"change":0,"pos":62922}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function where"},{"Add":""},{"Add":"open import Function.Core public"},{"Add":"open import Function.Base public"},{"Add":"open import Function.Strict public"},{"Add":"open import Function.Definitions public"},{"Add":"open import Function.Structures public"},{"Add":"open import Function.Structures.Biased public"},{"Add":"open import Function.Bundles public"}]}},{"FileAdd":{"meta":{"basename":"Function","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Surjection.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":"{-# OPTIONS --warn=noUserWarning #-}"},{"Add":""},{"Add":"module Function.Surjection where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Function.Surjection was deprecated in v2.0."},{"Add":"Use the standard function hierarchy in Function/Function.Bundles instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Function.Equality as F"},{"Add":" using (_⟶_) renaming (_∘_ to _⟪∘⟫_)"},{"Add":"open import Function.Equivalence using (Equivalence)"},{"Add":"open import Function.Injection hiding (id; _∘_; injection)"},{"Add":"open import Function.LeftInverse as Left hiding (id; _∘_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Surjective functions."},{"Add":""},{"Add":"record Surjective {f₁ f₂ t₁ t₂}"},{"Add":" {From : Setoid f₁ f₂} {To : Setoid t₁ t₂}"},{"Add":" (to : From ⟶ To) :"},{"Add":" Set (f₁ ⊔ f₂ ⊔ t₁ ⊔ t₂) where"},{"Add":" field"},{"Add":" from : To ⟶ From"},{"Add":" right-inverse-of : from RightInverseOf to"},{"Add":"{-# WARNING_ON_USAGE Surjective"},{"Add":"\"Warning: Surjective was deprecated in v2.0."},{"Add":"Please use Function.(Definitions.)Surjective instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The set of all surjections from one setoid to another."},{"Add":""},{"Add":"record Surjection {f₁ f₂ t₁ t₂}"},{"Add":" (From : Setoid f₁ f₂) (To : Setoid t₁ t₂) :"},{"Add":" Set (f₁ ⊔ f₂ ⊔ t₁ ⊔ t₂) where"},{"Add":" field"},{"Add":" to : From ⟶ To"},{"Add":" surjective : Surjective to"},{"Add":""},{"Add":" open Surjective surjective public"},{"Add":""},{"Add":" right-inverse : RightInverse From To"},{"Add":" right-inverse = record"},{"Add":" { to = from"},{"Add":" ; from = to"},{"Add":" ; left-inverse-of = right-inverse-of"},{"Add":" }"},{"Add":""},{"Add":" open LeftInverse right-inverse public"},{"Add":" using () renaming (to-from to from-to)"},{"Add":""},{"Add":" injective : Injective from"},{"Add":" injective = LeftInverse.injective right-inverse"},{"Add":""},{"Add":" injection : Injection To From"},{"Add":" injection = LeftInverse.injection right-inverse"},{"Add":""},{"Add":" equivalence : Equivalence From To"},{"Add":" equivalence = record"},{"Add":" { to = to"},{"Add":" ; from = from"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE Surjection"},{"Add":"\"Warning: Surjection was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)Surjection instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"-- Right inverses can be turned into surjections."},{"Add":""},{"Add":"fromRightInverse :"},{"Add":" ∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →"},{"Add":" RightInverse From To → Surjection From To"},{"Add":"fromRightInverse r = record"},{"Add":" { to = from"},{"Add":" ; surjective = record"},{"Add":" { from = to"},{"Add":" ; right-inverse-of = left-inverse-of"},{"Add":" }"},{"Add":" } where open LeftInverse r"},{"Add":"{-# WARNING_ON_USAGE fromRightInverse"},{"Add":"\"Warning: fromRightInverse was deprecated in v2.0."},{"Add":"Please use Function.(Properties.)RightInverse.RightInverse⇒Surjection instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The set of all surjections from one set to another (i.e. sujections"},{"Add":"-- with propositional equality)"},{"Add":""},{"Add":"infix 3 _↠_"},{"Add":""},{"Add":"_↠_ : ∀ {f t} → Set f → Set t → Set _"},{"Add":"From ↠ To = Surjection (≡.setoid From) (≡.setoid To)"},{"Add":"{-# WARNING_ON_USAGE _↠_"},{"Add":"\"Warning: _↠_ was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)_↠_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"surjection : ∀ {f t} {From : Set f} {To : Set t} →"},{"Add":" (to : From → To) (from : To → From) →"},{"Add":" (∀ x → to (from x) ≡ x) →"},{"Add":" From ↠ To"},{"Add":"surjection to from surjective = record"},{"Add":" { to = F.→-to-⟶ to"},{"Add":" ; surjective = record"},{"Add":" { from = F.→-to-⟶ from"},{"Add":" ; right-inverse-of = surjective"},{"Add":" }"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE surjection"},{"Add":"\"Warning: surjection was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)mk↠ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Identity and composition."},{"Add":""},{"Add":"id : ∀ {s₁ s₂} {S : Setoid s₁ s₂} → Surjection S S"},{"Add":"id {S = S} = record"},{"Add":" { to = F.id"},{"Add":" ; surjective = record"},{"Add":" { from = LeftInverse.to id′"},{"Add":" ; right-inverse-of = LeftInverse.left-inverse-of id′"},{"Add":" }"},{"Add":" } where id′ = Left.id {S = S}"},{"Add":"{-# WARNING_ON_USAGE id"},{"Add":"\"Warning: id was deprecated in v2.0."},{"Add":"Please use Function.Properties.Surjection.refl or"},{"Add":"Function.Construct.Identity.surjection instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"infixr 9 _∘_"},{"Add":""},{"Add":"_∘_ : ∀ {f₁ f₂ m₁ m₂ t₁ t₂}"},{"Add":" {F : Setoid f₁ f₂} {M : Setoid m₁ m₂} {T : Setoid t₁ t₂} →"},{"Add":" Surjection M T → Surjection F M → Surjection F T"},{"Add":"f ∘ g = record"},{"Add":" { to = to f ⟪∘⟫ to g"},{"Add":" ; surjective = record"},{"Add":" { from = LeftInverse.to g∘f"},{"Add":" ; right-inverse-of = LeftInverse.left-inverse-of g∘f"},{"Add":" }"},{"Add":" }"},{"Add":" where"},{"Add":" open Surjection"},{"Add":" g∘f = Left._∘_ (right-inverse g) (right-inverse f)"},{"Add":"{-# WARNING_ON_USAGE _∘_"},{"Add":"\"Warning: _∘_ was deprecated in v2.0."},{"Add":"Please use Function.Properties.Surjection.trans or"},{"Add":"Function.Construct.Composition.surjection instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Structures.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Structures for types of functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this file should usually be accessed from `Function`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence)"},{"Add":""},{"Add":"module Function.Structures {a b ℓ₁ ℓ₂}"},{"Add":" {A : Set a} (_≈₁_ : Rel A ℓ₁) -- Equality over the domain"},{"Add":" {B : Set b} (_≈₂_ : Rel B ℓ₂) -- Equality over the codomain"},{"Add":" where"},{"Add":""},{"Add":"open import Data.Product.Base as Product using (∃; _×_; _,_)"},{"Add":"open import Function.Base"},{"Add":"open import Function.Definitions"},{"Add":"open import Level using (_⊔_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- One element structures"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record IsCongruent (to : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" cong : Congruent _≈₁_ _≈₂_ to"},{"Add":" isEquivalence₁ : IsEquivalence _≈₁_"},{"Add":" isEquivalence₂ : IsEquivalence _≈₂_"},{"Add":""},{"Add":" module Eq₁ where"},{"Add":""},{"Add":" setoid : Setoid a ℓ₁"},{"Add":" setoid = record"},{"Add":" { isEquivalence = isEquivalence₁"},{"Add":" }"},{"Add":""},{"Add":" open Setoid setoid public"},{"Add":""},{"Add":" module Eq₂ where"},{"Add":""},{"Add":" setoid : Setoid b ℓ₂"},{"Add":" setoid = record"},{"Add":" { isEquivalence = isEquivalence₂"},{"Add":" }"},{"Add":""},{"Add":" open Setoid setoid public"},{"Add":""},{"Add":""},{"Add":"record IsInjection (to : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isCongruent : IsCongruent to"},{"Add":" injective : Injective _≈₁_ _≈₂_ to"},{"Add":""},{"Add":" open IsCongruent isCongruent public"},{"Add":""},{"Add":""},{"Add":"record IsSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isCongruent : IsCongruent f"},{"Add":" surjective : Surjective _≈₁_ _≈₂_ f"},{"Add":""},{"Add":" open IsCongruent isCongruent public"},{"Add":""},{"Add":" strictlySurjective : StrictlySurjective _≈₂_ f"},{"Add":" strictlySurjective x = Product.map₂ (λ v → v Eq₁.refl) (surjective x)"},{"Add":""},{"Add":""},{"Add":"record IsBijection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isInjection : IsInjection f"},{"Add":" surjective : Surjective _≈₁_ _≈₂_ f"},{"Add":""},{"Add":" open IsInjection isInjection public"},{"Add":""},{"Add":" bijective : Bijective _≈₁_ _≈₂_ f"},{"Add":" bijective = injective , surjective"},{"Add":""},{"Add":" isSurjection : IsSurjection f"},{"Add":" isSurjection = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; surjective = surjective"},{"Add":" }"},{"Add":""},{"Add":" open IsSurjection isSurjection public"},{"Add":" using (strictlySurjective)"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Two element structures"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record IsLeftInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isCongruent : IsCongruent to"},{"Add":" from-cong : Congruent _≈₂_ _≈₁_ from"},{"Add":" inverseˡ : Inverseˡ _≈₁_ _≈₂_ to from"},{"Add":""},{"Add":" open IsCongruent isCongruent public"},{"Add":" renaming (cong to to-cong)"},{"Add":""},{"Add":" strictlyInverseˡ : StrictlyInverseˡ _≈₂_ to from"},{"Add":" strictlyInverseˡ x = inverseˡ Eq₁.refl"},{"Add":""},{"Add":" isSurjection : IsSurjection to"},{"Add":" isSurjection = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; surjective = λ y → from y , inverseˡ"},{"Add":" }"},{"Add":""},{"Add":""},{"Add":"record IsRightInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isCongruent : IsCongruent to"},{"Add":" from-cong : Congruent _≈₂_ _≈₁_ from"},{"Add":" inverseʳ : Inverseʳ _≈₁_ _≈₂_ to from"},{"Add":""},{"Add":" open IsCongruent isCongruent public"},{"Add":" renaming (cong to to-cong)"},{"Add":""},{"Add":" strictlyInverseʳ : StrictlyInverseʳ _≈₁_ to from"},{"Add":" strictlyInverseʳ x = inverseʳ Eq₂.refl"},{"Add":""},{"Add":""},{"Add":"record IsInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isLeftInverse : IsLeftInverse to from"},{"Add":" inverseʳ : Inverseʳ _≈₁_ _≈₂_ to from"},{"Add":""},{"Add":" open IsLeftInverse isLeftInverse public"},{"Add":""},{"Add":" isRightInverse : IsRightInverse to from"},{"Add":" isRightInverse = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; from-cong = from-cong"},{"Add":" ; inverseʳ = inverseʳ"},{"Add":" }"},{"Add":""},{"Add":" open IsRightInverse isRightInverse public"},{"Add":" using (strictlyInverseʳ)"},{"Add":""},{"Add":" inverse : Inverseᵇ _≈₁_ _≈₂_ to from"},{"Add":" inverse = inverseˡ , inverseʳ"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Three element structures"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record IsBiEquivalence"},{"Add":" (to : A → B) (from₁ : B → A) (from₂ : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" to-isCongruent : IsCongruent to"},{"Add":" from₁-cong : Congruent _≈₂_ _≈₁_ from₁"},{"Add":" from₂-cong : Congruent _≈₂_ _≈₁_ from₂"},{"Add":""},{"Add":" open IsCongruent to-isCongruent public"},{"Add":" renaming (cong to to-cong₁)"},{"Add":""},{"Add":""},{"Add":"record IsBiInverse"},{"Add":" (to : A → B) (from₁ : B → A) (from₂ : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" to-isCongruent : IsCongruent to"},{"Add":" from₁-cong : Congruent _≈₂_ _≈₁_ from₁"},{"Add":" from₂-cong : Congruent _≈₂_ _≈₁_ from₂"},{"Add":" inverseˡ : Inverseˡ _≈₁_ _≈₂_ to from₁"},{"Add":" inverseʳ : Inverseʳ _≈₁_ _≈₂_ to from₂"},{"Add":""},{"Add":" open IsCongruent to-isCongruent public"},{"Add":" renaming (cong to to-cong)"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Other"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- See the comment on `SplitSurjection` in `Function.Bundles` for an"},{"Add":"-- explanation of (split) surjections."},{"Add":"record IsSplitSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" from : B → A"},{"Add":" isLeftInverse : IsLeftInverse f from"},{"Add":""},{"Add":" open IsLeftInverse isLeftInverse public"}]}},{"FileAdd":{"meta":{"basename":"Structures","metadata":576},"context":[{"change":0,"pos":1018846}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Biased.agda","metadata":0},"context":[{"change":0,"pos":1029902}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Ways to give instances of certain structures where some fields can"},{"Add":"-- be given in terms of others."},{"Add":"-- The contents of this file should usually be accessed from `Function`."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence)"},{"Add":""},{"Add":"module Function.Structures.Biased {a b ℓ₁ ℓ₂}"},{"Add":" {A : Set a} (_≈₁_ : Rel A ℓ₁) -- Equality over the domain"},{"Add":" {B : Set b} (_≈₂_ : Rel B ℓ₂) -- Equality over the codomain"},{"Add":" where"},{"Add":""},{"Add":"open import Data.Product.Base as Product using (∃; _×_; _,_)"},{"Add":"open import Function.Base"},{"Add":"open import Function.Definitions"},{"Add":"open import Function.Structures _≈₁_ _≈₂_"},{"Add":"open import Function.Consequences.Setoid"},{"Add":"open import Level using (_⊔_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Surjection"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record IsStrictSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isCongruent : IsCongruent f"},{"Add":" strictlySurjective : StrictlySurjective _≈₂_ f"},{"Add":""},{"Add":" open IsCongruent isCongruent public"},{"Add":""},{"Add":" isSurjection : IsSurjection f"},{"Add":" isSurjection = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; surjective = strictlySurjective⇒surjective"},{"Add":" Eq₁.setoid Eq₂.setoid cong strictlySurjective"},{"Add":" }"},{"Add":""},{"Add":"open IsStrictSurjection public"},{"Add":" using () renaming (isSurjection to isStrictSurjection)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bijection"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record IsStrictBijection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isInjection : IsInjection f"},{"Add":" strictlySurjective : StrictlySurjective _≈₂_ f"},{"Add":""},{"Add":" isBijection : IsBijection f"},{"Add":" isBijection = record"},{"Add":" { isInjection = isInjection"},{"Add":" ; surjective = strictlySurjective⇒surjective"},{"Add":" Eq₁.setoid Eq₂.setoid cong strictlySurjective"},{"Add":" } where open IsInjection isInjection"},{"Add":""},{"Add":"open IsStrictBijection public"},{"Add":" using () renaming (isBijection to isStrictBijection)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Left inverse"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record IsStrictLeftInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isCongruent : IsCongruent to"},{"Add":" from-cong : Congruent _≈₂_ _≈₁_ from"},{"Add":" strictlyInverseˡ : StrictlyInverseˡ _≈₂_ to from"},{"Add":""},{"Add":" isLeftInverse : IsLeftInverse to from"},{"Add":" isLeftInverse = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; from-cong = from-cong"},{"Add":" ; inverseˡ = strictlyInverseˡ⇒inverseˡ"},{"Add":" Eq₁.setoid Eq₂.setoid cong strictlyInverseˡ"},{"Add":" } where open IsCongruent isCongruent"},{"Add":""},{"Add":"open IsStrictLeftInverse public"},{"Add":" using () renaming (isLeftInverse to isStrictLeftInverse)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Right inverse"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record IsStrictRightInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isCongruent : IsCongruent to"},{"Add":" from-cong : Congruent _≈₂_ _≈₁_ from"},{"Add":" strictlyInverseʳ : StrictlyInverseʳ _≈₁_ to from"},{"Add":""},{"Add":" isRightInverse : IsRightInverse to from"},{"Add":" isRightInverse = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; from-cong = from-cong"},{"Add":" ; inverseʳ = strictlyInverseʳ⇒inverseʳ"},{"Add":" Eq₁.setoid Eq₂.setoid from-cong strictlyInverseʳ"},{"Add":" } where open IsCongruent isCongruent"},{"Add":""},{"Add":"open IsStrictRightInverse public"},{"Add":" using () renaming (isRightInverse to isStrictRightInverse)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Inverse"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record IsStrictInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" isLeftInverse : IsLeftInverse to from"},{"Add":" strictlyInverseʳ : StrictlyInverseʳ _≈₁_ to from"},{"Add":""},{"Add":" isInverse : IsInverse to from"},{"Add":" isInverse = record"},{"Add":" { isLeftInverse = isLeftInverse"},{"Add":" ; inverseʳ = strictlyInverseʳ⇒inverseʳ"},{"Add":" Eq₁.setoid Eq₂.setoid from-cong strictlyInverseʳ"},{"Add":" } where open IsLeftInverse isLeftInverse"},{"Add":""},{"Add":"open IsStrictInverse public"},{"Add":" using () renaming (isInverse to isStrictInverse)"}]}},{"FileAdd":{"meta":{"basename":"Strict.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Strict combinators (i.e. that use call-by-value)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module is also accessible via the `Function`"},{"Add":"-- module."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Strict where"},{"Add":""},{"Add":"open import Agda.Builtin.Equality using (_≡_)"},{"Add":"open import Function.Base using (flip)"},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b : Level"},{"Add":" A B : Set a"},{"Add":""},{"Add":"infixl 0 _!|>_ _!|>′_"},{"Add":"infixr -1 _$!_ _$!′_"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Dependent combinators"},{"Add":""},{"Add":"-- These are functions whose output has a type that depends on the"},{"Add":"-- value of the input to the function."},{"Add":""},{"Add":"open import Agda.Builtin.Strict public"},{"Add":" renaming"},{"Add":" ( primForce to force"},{"Add":" ; primForceLemma to force-≡"},{"Add":" )"},{"Add":""},{"Add":"-- Application"},{"Add":"_$!_ : ∀ {A : Set a} {B : A → Set b} →"},{"Add":" ((x : A) → B x) → ((x : A) → B x)"},{"Add":"f $! x = force x f"},{"Add":""},{"Add":"-- Flipped application"},{"Add":"_!|>_ : ∀ {A : Set a} {B : A → Set b} →"},{"Add":" (a : A) → (∀ a → B a) → B a"},{"Add":"_!|>_ = flip _$!_"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Non-dependent combinators"},{"Add":""},{"Add":"-- Any of the above operations for dependent functions will also work"},{"Add":"-- for non-dependent functions but sometimes Agda has difficulty"},{"Add":"-- inferring the non-dependency. Primed (′ = \\prime) versions of the"},{"Add":"-- operations are therefore provided below that sometimes have better"},{"Add":"-- inference properties."},{"Add":""},{"Add":"seq : A → B → B"},{"Add":"seq a b = force a (λ _ → b)"},{"Add":""},{"Add":"seq-≡ : (a : A) (b : B) → seq a b ≡ b"},{"Add":"seq-≡ a b = force-≡ a (λ _ → b)"},{"Add":""},{"Add":"force′ : A → (A → B) → B"},{"Add":"force′ = force"},{"Add":""},{"Add":"force′-≡ : (a : A) (f : A → B) → force′ a f ≡ f a"},{"Add":"force′-≡ = force-≡"},{"Add":""},{"Add":"-- Application"},{"Add":"_$!′_ : (A → B) → (A → B)"},{"Add":"_$!′_ = _$!_"},{"Add":""},{"Add":"-- Flipped application"},{"Add":"_!|>′_ : A → (A → B) → B"},{"Add":"_!|>′_ = _!|>_"}]}},{"FileAdd":{"meta":{"basename":"Relation","metadata":576},"context":[{"change":0,"pos":1018846}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Binary","metadata":576},"context":[{"change":0,"pos":1036624}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Setoid","metadata":576},"context":[{"change":0,"pos":1036646}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Equality.agda","metadata":0},"context":[{"change":0,"pos":1036666}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Function Equality setoid"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":""},{"Add":"module Function.Relation.Binary.Setoid.Equality {a₁ a₂ b₁ b₂ : Level}"},{"Add":" (From : Setoid a₁ a₂) (To : Setoid b₁ b₂) where"},{"Add":""},{"Add":"open import Function.Bundles using (Func; _⟨$⟩_)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (Reflexive; Symmetric; Transitive)"},{"Add":"open import Relation.Binary.Structures"},{"Add":" using (IsEquivalence)"},{"Add":""},{"Add":"private"},{"Add":" module To = Setoid To"},{"Add":" module From = Setoid From"},{"Add":""},{"Add":"infix 4 _≈_"},{"Add":"_≈_ : (f g : Func From To) → Set (a₁ ⊔ b₂)"},{"Add":"f ≈ g = ∀ x → f ⟨$⟩ x To.≈ g ⟨$⟩ x"},{"Add":""},{"Add":"refl : Reflexive _≈_"},{"Add":"refl _ = To.refl"},{"Add":""},{"Add":"sym : Symmetric _≈_"},{"Add":"sym f≈g x = To.sym (f≈g x)"},{"Add":""},{"Add":"trans : Transitive _≈_"},{"Add":"trans f≈g g≈h x = To.trans (f≈g x) (g≈h x)"},{"Add":""},{"Add":"isEquivalence : IsEquivalence _≈_"},{"Add":"isEquivalence = record -- need to η-expand else Agda gets confused"},{"Add":" { refl = λ {f} → refl {f}"},{"Add":" ; sym = λ {f} {g} → sym {f} {g}"},{"Add":" ; trans = λ {f} {g} {h} → trans {f} {g} {h}"},{"Add":" }"},{"Add":""},{"Add":"setoid : Setoid _ _"},{"Add":"setoid = record { isEquivalence = isEquivalence }"},{"Add":""},{"Add":"-- most of the time, this infix version is nicer to use"},{"Add":"infixr 9 _⇨_"},{"Add":"_⇨_ : Setoid _ _"},{"Add":"_⇨_ = setoid"}]}},{"FileAdd":{"meta":{"basename":"Related.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":"{-# OPTIONS --warn=noUserWarning #-}"},{"Add":""},{"Add":"module Function.Related where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Function.Related was deprecated in v2.0."},{"Add":"Use Function.Related.Propositional instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Function.Base"},{"Add":"open import Function.Equality using (_⟨$⟩_)"},{"Add":"open import Function.Equivalence as Eq using (Equivalence)"},{"Add":"open import Function.Injection as Inj using (Injection; _↣_)"},{"Add":"open import Function.Inverse as Inv using (Inverse; _↔_)"},{"Add":"open import Function.LeftInverse as LeftInv using (LeftInverse)"},{"Add":"open import Function.Surjection as Surj using (Surjection)"},{"Add":"open import Function.Consequences.Propositional"},{"Add":"open import Relation.Binary.Core using (_⇒_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid; Preorder)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence; IsPreorder)"},{"Add":"open import Relation.Binary.Definitions using (Reflexive; Trans; Sym)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":"open import Data.Product.Base using (_,_; proj₁; proj₂; <_,_>)"},{"Add":""},{"Add":"import Function.Related.Propositional as R"},{"Add":"import Function.Bundles as B"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ₁ ℓ₂ : Level"},{"Add":" A : Set ℓ₁"},{"Add":" B : Set ℓ₂"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export core concepts from non-deprecated Related code"},{"Add":""},{"Add":"open R public using"},{"Add":" ( Kind"},{"Add":" ; implication"},{"Add":" ; equivalence"},{"Add":" ; injection"},{"Add":" ; surjection"},{"Add":" ; bijection"},{"Add":" ) renaming"},{"Add":" ( reverseImplication to reverse-implication"},{"Add":" ; reverseInjection to reverse-injection"},{"Add":" ; leftInverse to left-inverse"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Wrapper types"},{"Add":""},{"Add":"-- Synonyms which are used to make _∼[_]_ below \"constructor-headed\""},{"Add":"-- (which implies that Agda can deduce the universe code from an"},{"Add":"-- expression matching any of the right-hand sides)."},{"Add":""},{"Add":"infix 3 _←_ _↢_"},{"Add":""},{"Add":"record _←_ {a b} (A : Set a) (B : Set b) : Set (a ⊔ b) where"},{"Add":" constructor lam"},{"Add":" field app-← : B → A"},{"Add":""},{"Add":"open _←_ public"},{"Add":""},{"Add":"record _↢_ {a b} (A : Set a) (B : Set b) : Set (a ⊔ b) where"},{"Add":" constructor lam"},{"Add":" field app-↢ : B ↣ A"},{"Add":""},{"Add":"open _↢_ public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relatedness"},{"Add":""},{"Add":"-- There are several kinds of \"relatedness\"."},{"Add":""},{"Add":"-- The idea to include kinds other than equivalence and bijection came"},{"Add":"-- from Simon Thompson and Bengt Nordström. /NAD"},{"Add":""},{"Add":"infix 4 _∼[_]_"},{"Add":""},{"Add":"_∼[_]_ : ∀ {ℓ₁ ℓ₂} → Set ℓ₁ → Kind → Set ℓ₂ → Set _"},{"Add":"A ∼[ implication ] B = A → B"},{"Add":"A ∼[ reverse-implication ] B = A ← B"},{"Add":"A ∼[ equivalence ] B = Equivalence (≡.setoid A) (≡.setoid B)"},{"Add":"A ∼[ injection ] B = Injection (≡.setoid A) (≡.setoid B)"},{"Add":"A ∼[ reverse-injection ] B = A ↢ B"},{"Add":"A ∼[ left-inverse ] B = LeftInverse (≡.setoid A) (≡.setoid B)"},{"Add":"A ∼[ surjection ] B = Surjection (≡.setoid A) (≡.setoid B)"},{"Add":"A ∼[ bijection ] B = Inverse (≡.setoid A) (≡.setoid B)"},{"Add":""},{"Add":"-- A non-infix synonym."},{"Add":""},{"Add":"Related : Kind → ∀ {ℓ₁ ℓ₂} → Set ℓ₁ → Set ℓ₂ → Set _"},{"Add":"Related k A B = A ∼[ k ] B"},{"Add":""},{"Add":"toRelated : {K : Kind} → A R.∼[ K ] B → A ∼[ K ] B"},{"Add":"toRelated {K = implication} rel = B.Func.to rel"},{"Add":"toRelated {K = reverse-implication} rel = lam (B.Func.to rel)"},{"Add":"toRelated {K = equivalence} rel = Eq.equivalence (B.Equivalence.to rel) (B.Equivalence.from rel)"},{"Add":"toRelated {K = injection} rel = Inj.injection (B.Injection.to rel) (B.Injection.injective rel)"},{"Add":"toRelated {K = reverse-injection} rel = lam (Inj.injection (B.Injection.to rel) (B.Injection.injective rel))"},{"Add":"toRelated {K = left-inverse} rel = LeftInv.leftInverse to from strictlyInverseʳ where open B.RightInverse rel"},{"Add":"toRelated {K = surjection} rel = Surj.surjection to (proj₁ ∘ strictlySurjective) (proj₂ ∘ strictlySurjective)"},{"Add":" where open B.Surjection rel"},{"Add":"toRelated {K = bijection} rel ="},{"Add":" Inv.inverse to from strictlyInverseʳ strictlyInverseˡ"},{"Add":" where open B.Inverse rel"},{"Add":""},{"Add":"fromRelated : {K : Kind} → A ∼[ K ] B → A R.∼[ K ] B"},{"Add":"fromRelated {K = implication} rel = B.mk⟶ rel"},{"Add":"fromRelated {K = reverse-implication} rel = B.mk⟶ (app-← rel)"},{"Add":"fromRelated {K = equivalence} record { to = to ; from = from } = B.mk⇔ (to ⟨$⟩_) (from ⟨$⟩_)"},{"Add":"fromRelated {K = injection} rel = B.mk↣ (Inj.Injection.injective rel)"},{"Add":"fromRelated {K = reverse-injection} (lam app-↢) = B.mk↣ (Inj.Injection.injective app-↢)"},{"Add":"fromRelated {K = left-inverse} record { to = to ; from = from ; left-inverse-of = left-inverse-of } ="},{"Add":" B.mk↪ {to = to ⟨$⟩_} {from = from ⟨$⟩_} (strictlyInverseʳ⇒inverseʳ (to ⟨$⟩_) left-inverse-of)"},{"Add":"fromRelated {K = surjection} record { to = to ; surjective = surjective } with surjective"},{"Add":"... | record { from = from ; right-inverse-of = right-inverse-of } ="},{"Add":" B.mk↠ {to = to ⟨$⟩_} < from ⟨$⟩_ , (λ { x ≡.refl → right-inverse-of x }) >"},{"Add":"fromRelated {K = bijection} rel = B.mk↔ₛ′ (to ⟨$⟩_) (from ⟨$⟩_) right-inverse-of left-inverse-of"},{"Add":" where open Inverse rel"},{"Add":""},{"Add":"-- The bijective equality implies any kind of relatedness."},{"Add":""},{"Add":"↔⇒ : ∀ {k x y} {X : Set x} {Y : Set y} →"},{"Add":" X ∼[ bijection ] Y → X ∼[ k ] Y"},{"Add":"↔⇒ {implication} = _⟨$⟩_ ∘ Inverse.to"},{"Add":"↔⇒ {reverse-implication} = lam ∘′ _⟨$⟩_ ∘ Inverse.from"},{"Add":"↔⇒ {equivalence} = Inverse.equivalence"},{"Add":"↔⇒ {injection} = Inverse.injection"},{"Add":"↔⇒ {reverse-injection} = lam ∘′ Inverse.injection ∘ Inv.sym"},{"Add":"↔⇒ {left-inverse} = Inverse.left-inverse"},{"Add":"↔⇒ {surjection} = Inverse.surjection"},{"Add":"↔⇒ {bijection} = id"},{"Add":""},{"Add":"-- Actual equality also implies any kind of relatedness."},{"Add":""},{"Add":"≡⇒ : ∀ {k ℓ} {X Y : Set ℓ} → X ≡ Y → X ∼[ k ] Y"},{"Add":"≡⇒ ≡.refl = ↔⇒ Inv.id"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Special kinds of kinds"},{"Add":""},{"Add":"-- Kinds whose interpretation is symmetric."},{"Add":""},{"Add":"data Symmetric-kind : Set where"},{"Add":" equivalence : Symmetric-kind"},{"Add":" bijection : Symmetric-kind"},{"Add":""},{"Add":"-- Forgetful map."},{"Add":""},{"Add":"⌊_⌋ : Symmetric-kind → Kind"},{"Add":"⌊ equivalence ⌋ = equivalence"},{"Add":"⌊ bijection ⌋ = bijection"},{"Add":""},{"Add":"-- The proof of symmetry can be found below."},{"Add":""},{"Add":"-- Kinds whose interpretation include a function which \"goes in the"},{"Add":"-- forward direction\"."},{"Add":""},{"Add":"data Forward-kind : Set where"},{"Add":" implication : Forward-kind"},{"Add":" equivalence : Forward-kind"},{"Add":" injection : Forward-kind"},{"Add":" left-inverse : Forward-kind"},{"Add":" surjection : Forward-kind"},{"Add":" bijection : Forward-kind"},{"Add":""},{"Add":"-- Forgetful map."},{"Add":""},{"Add":"⌊_⌋→ : Forward-kind → Kind"},{"Add":"⌊ implication ⌋→ = implication"},{"Add":"⌊ equivalence ⌋→ = equivalence"},{"Add":"⌊ injection ⌋→ = injection"},{"Add":"⌊ left-inverse ⌋→ = left-inverse"},{"Add":"⌊ surjection ⌋→ = surjection"},{"Add":"⌊ bijection ⌋→ = bijection"},{"Add":""},{"Add":"-- The function."},{"Add":""},{"Add":"⇒→ : ∀ {k x y} {X : Set x} {Y : Set y} → X ∼[ ⌊ k ⌋→ ] Y → X → Y"},{"Add":"⇒→ {implication} = id"},{"Add":"⇒→ {equivalence} = _⟨$⟩_ ∘ Equivalence.to"},{"Add":"⇒→ {injection} = _⟨$⟩_ ∘ Injection.to"},{"Add":"⇒→ {left-inverse} = _⟨$⟩_ ∘ LeftInverse.to"},{"Add":"⇒→ {surjection} = _⟨$⟩_ ∘ Surjection.to"},{"Add":"⇒→ {bijection} = _⟨$⟩_ ∘ Inverse.to"},{"Add":""},{"Add":"-- Kinds whose interpretation include a function which \"goes backwards\"."},{"Add":""},{"Add":"data Backward-kind : Set where"},{"Add":" reverse-implication : Backward-kind"},{"Add":" equivalence : Backward-kind"},{"Add":" reverse-injection : Backward-kind"},{"Add":" left-inverse : Backward-kind"},{"Add":" surjection : Backward-kind"},{"Add":" bijection : Backward-kind"},{"Add":""},{"Add":"-- Forgetful map."},{"Add":""},{"Add":"⌊_⌋← : Backward-kind → Kind"},{"Add":"⌊ reverse-implication ⌋← = reverse-implication"},{"Add":"⌊ equivalence ⌋← = equivalence"},{"Add":"⌊ reverse-injection ⌋← = reverse-injection"},{"Add":"⌊ left-inverse ⌋← = left-inverse"},{"Add":"⌊ surjection ⌋← = surjection"},{"Add":"⌊ bijection ⌋← = bijection"},{"Add":""},{"Add":"-- The function."},{"Add":""},{"Add":"⇒← : ∀ {k x y} {X : Set x} {Y : Set y} → X ∼[ ⌊ k ⌋← ] Y → Y → X"},{"Add":"⇒← {reverse-implication} = app-←"},{"Add":"⇒← {equivalence} = _⟨$⟩_ ∘ Equivalence.from"},{"Add":"⇒← {reverse-injection} = _⟨$⟩_ ∘ Injection.to ∘ app-↢"},{"Add":"⇒← {left-inverse} = _⟨$⟩_ ∘ LeftInverse.from"},{"Add":"⇒← {surjection} = _⟨$⟩_ ∘ Surjection.from"},{"Add":"⇒← {bijection} = _⟨$⟩_ ∘ Inverse.from"},{"Add":""},{"Add":"-- Kinds whose interpretation include functions going in both"},{"Add":"-- directions."},{"Add":""},{"Add":"data Equivalence-kind : Set where"},{"Add":" equivalence : Equivalence-kind"},{"Add":" left-inverse : Equivalence-kind"},{"Add":" surjection : Equivalence-kind"},{"Add":" bijection : Equivalence-kind"},{"Add":""},{"Add":"-- Forgetful map."},{"Add":""},{"Add":"⌊_⌋⇔ : Equivalence-kind → Kind"},{"Add":"⌊ equivalence ⌋⇔ = equivalence"},{"Add":"⌊ left-inverse ⌋⇔ = left-inverse"},{"Add":"⌊ surjection ⌋⇔ = surjection"},{"Add":"⌊ bijection ⌋⇔ = bijection"},{"Add":""},{"Add":"-- The functions."},{"Add":""},{"Add":"⇒⇔ : ∀ {k x y} {X : Set x} {Y : Set y} →"},{"Add":" X ∼[ ⌊ k ⌋⇔ ] Y → X ∼[ equivalence ] Y"},{"Add":"⇒⇔ {equivalence} = id"},{"Add":"⇒⇔ {left-inverse} = LeftInverse.equivalence"},{"Add":"⇒⇔ {surjection} = Surjection.equivalence"},{"Add":"⇒⇔ {bijection} = Inverse.equivalence"},{"Add":""},{"Add":"-- Conversions between special kinds."},{"Add":""},{"Add":"⇔⌊_⌋ : Symmetric-kind → Equivalence-kind"},{"Add":"⇔⌊ equivalence ⌋ = equivalence"},{"Add":"⇔⌊ bijection ⌋ = bijection"},{"Add":""},{"Add":"→⌊_⌋ : Equivalence-kind → Forward-kind"},{"Add":"→⌊ equivalence ⌋ = equivalence"},{"Add":"→⌊ left-inverse ⌋ = left-inverse"},{"Add":"→⌊ surjection ⌋ = surjection"},{"Add":"→⌊ bijection ⌋ = bijection"},{"Add":""},{"Add":"←⌊_⌋ : Equivalence-kind → Backward-kind"},{"Add":"←⌊ equivalence ⌋ = equivalence"},{"Add":"←⌊ left-inverse ⌋ = left-inverse"},{"Add":"←⌊ surjection ⌋ = surjection"},{"Add":"←⌊ bijection ⌋ = bijection"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Opposites"},{"Add":""},{"Add":"-- For every kind there is an opposite kind."},{"Add":""},{"Add":"_op : Kind → Kind"},{"Add":"implication op = reverse-implication"},{"Add":"reverse-implication op = implication"},{"Add":"equivalence op = equivalence"},{"Add":"injection op = reverse-injection"},{"Add":"reverse-injection op = injection"},{"Add":"left-inverse op = surjection"},{"Add":"surjection op = left-inverse"},{"Add":"bijection op = bijection"},{"Add":""},{"Add":"-- For every morphism there is a corresponding reverse morphism of the"},{"Add":"-- opposite kind."},{"Add":""},{"Add":"reverse : ∀ {k a b} {A : Set a} {B : Set b} →"},{"Add":" A ∼[ k ] B → B ∼[ k op ] A"},{"Add":"reverse {implication} = lam"},{"Add":"reverse {reverse-implication} = app-←"},{"Add":"reverse {equivalence} = Eq.sym"},{"Add":"reverse {injection} = lam"},{"Add":"reverse {reverse-injection} = app-↢"},{"Add":"reverse {left-inverse} = Surj.fromRightInverse"},{"Add":"reverse {surjection} = Surjection.right-inverse"},{"Add":"reverse {bijection} = Inv.sym"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- For a fixed universe level every kind is a preorder and each"},{"Add":"-- symmetric kind is an equivalence"},{"Add":""},{"Add":"K-refl : ∀ {k ℓ} → Reflexive (Related k {ℓ})"},{"Add":"K-refl {implication} = id"},{"Add":"K-refl {reverse-implication} = lam id"},{"Add":"K-refl {equivalence} = Eq.id"},{"Add":"K-refl {injection} = Inj.id"},{"Add":"K-refl {reverse-injection} = lam Inj.id"},{"Add":"K-refl {left-inverse} = LeftInv.id"},{"Add":"K-refl {surjection} = Surj.id"},{"Add":"K-refl {bijection} = Inv.id"},{"Add":""},{"Add":"K-reflexive : ∀ {k ℓ} → _≡_ ⇒ Related k {ℓ}"},{"Add":"K-reflexive ≡.refl = K-refl"},{"Add":""},{"Add":"K-trans : ∀ {k ℓ₁ ℓ₂ ℓ₃} → Trans (Related k {ℓ₁} {ℓ₂})"},{"Add":" (Related k {ℓ₂} {ℓ₃})"},{"Add":" (Related k {ℓ₁} {ℓ₃})"},{"Add":"K-trans {implication} = flip _∘′_"},{"Add":"K-trans {reverse-implication} = λ f g → lam (app-← f ∘ app-← g)"},{"Add":"K-trans {equivalence} = flip Eq._∘_"},{"Add":"K-trans {injection} = flip Inj._∘_"},{"Add":"K-trans {reverse-injection} = λ f g → lam (Inj._∘_ (app-↢ f) (app-↢ g))"},{"Add":"K-trans {left-inverse} = flip LeftInv._∘_"},{"Add":"K-trans {surjection} = flip Surj._∘_"},{"Add":"K-trans {bijection} = flip Inv._∘_"},{"Add":""},{"Add":"SK-sym : ∀ {k ℓ₁ ℓ₂} → Sym (Related ⌊ k ⌋ {ℓ₁} {ℓ₂})"},{"Add":" (Related ⌊ k ⌋ {ℓ₂} {ℓ₁})"},{"Add":"SK-sym {equivalence} = Eq.sym"},{"Add":"SK-sym {bijection} = Inv.sym"},{"Add":""},{"Add":"SK-isEquivalence : ∀ k ℓ → IsEquivalence {ℓ = ℓ} (Related ⌊ k ⌋)"},{"Add":"SK-isEquivalence k ℓ = record"},{"Add":" { refl = K-refl"},{"Add":" ; sym = SK-sym"},{"Add":" ; trans = K-trans"},{"Add":" }"},{"Add":""},{"Add":"SK-setoid : Symmetric-kind → (ℓ : Level) → Setoid _ _"},{"Add":"SK-setoid k ℓ = record { isEquivalence = SK-isEquivalence k ℓ }"},{"Add":""},{"Add":"K-isPreorder : ∀ k ℓ → IsPreorder _↔_ (Related k)"},{"Add":"K-isPreorder k ℓ = record"},{"Add":" { isEquivalence = SK-isEquivalence bijection ℓ"},{"Add":" ; reflexive = ↔⇒"},{"Add":" ; trans = K-trans"},{"Add":" }"},{"Add":""},{"Add":"K-preorder : Kind → (ℓ : Level) → Preorder _ _ _"},{"Add":"K-preorder k ℓ = record { isPreorder = K-isPreorder k ℓ }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Equational reasoning"},{"Add":""},{"Add":"-- Equational reasoning for related things."},{"Add":""},{"Add":"module EquationalReasoning where"},{"Add":""},{"Add":" infix 3 _∎"},{"Add":" infixr 2 _∼⟨_⟩_ _↔⟨_⟩_ _↔⟨⟩_ _≡⟨_⟩_ _≡˘⟨_⟩_"},{"Add":" infix 1 begin_"},{"Add":""},{"Add":" begin_ : ∀ {k x y} {X : Set x} {Y : Set y} →"},{"Add":" X ∼[ k ] Y → X ∼[ k ] Y"},{"Add":" begin_ x∼y = x∼y"},{"Add":""},{"Add":" _∼⟨_⟩_ : ∀ {k x y z} (X : Set x) {Y : Set y} {Z : Set z} →"},{"Add":" X ∼[ k ] Y → Y ∼[ k ] Z → X ∼[ k ] Z"},{"Add":" _ ∼⟨ X↝Y ⟩ Y↝Z = K-trans X↝Y Y↝Z"},{"Add":""},{"Add":" -- Isomorphisms can be combined with any other kind of relatedness."},{"Add":""},{"Add":" _↔⟨_⟩_ : ∀ {k x y z} (X : Set x) {Y : Set y} {Z : Set z} →"},{"Add":" X ↔ Y → Y ∼[ k ] Z → X ∼[ k ] Z"},{"Add":" X ↔⟨ X↔Y ⟩ Y⇔Z = X ∼⟨ ↔⇒ X↔Y ⟩ Y⇔Z"},{"Add":""},{"Add":" _↔⟨⟩_ : ∀ {k x y} (X : Set x) {Y : Set y} →"},{"Add":" X ∼[ k ] Y → X ∼[ k ] Y"},{"Add":" X ↔⟨⟩ X⇔Y = X⇔Y"},{"Add":""},{"Add":" _≡˘⟨_⟩_ : ∀ {k ℓ z} (X : Set ℓ) {Y : Set ℓ} {Z : Set z} →"},{"Add":" Y ≡ X → Y ∼[ k ] Z → X ∼[ k ] Z"},{"Add":" X ≡˘⟨ Y≡X ⟩ Y⇔Z = X ∼⟨ ≡⇒ (≡.sym Y≡X) ⟩ Y⇔Z"},{"Add":""},{"Add":" _≡⟨_⟩_ : ∀ {k ℓ z} (X : Set ℓ) {Y : Set ℓ} {Z : Set z} →"},{"Add":" X ≡ Y → Y ∼[ k ] Z → X ∼[ k ] Z"},{"Add":" X ≡⟨ X≡Y ⟩ Y⇔Z = X ∼⟨ ≡⇒ X≡Y ⟩ Y⇔Z"},{"Add":""},{"Add":" _∎ : ∀ {k x} (X : Set x) → X ∼[ k ] X"},{"Add":" X ∎ = K-refl"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Every unary relation induces a preorder and, for symmetric kinds,"},{"Add":"-- an equivalence. (No claim is made that these relations are unique.)"},{"Add":""},{"Add":"InducedRelation₁ : Kind → ∀ {a s} {A : Set a} →"},{"Add":" (A → Set s) → A → A → Set _"},{"Add":"InducedRelation₁ k S = λ x y → S x ∼[ k ] S y"},{"Add":""},{"Add":"InducedPreorder₁ : Kind → ∀ {a s} {A : Set a} →"},{"Add":" (A → Set s) → Preorder _ _ _"},{"Add":"InducedPreorder₁ k S = record"},{"Add":" { _≈_ = _≡_"},{"Add":" ; _≲_ = InducedRelation₁ k S"},{"Add":" ; isPreorder = record"},{"Add":" { isEquivalence = ≡.isEquivalence"},{"Add":" ; reflexive = reflexive ∘"},{"Add":" K-reflexive ∘"},{"Add":" ≡.cong S"},{"Add":" ; trans = K-trans"},{"Add":" }"},{"Add":" } where open Preorder (K-preorder _ _)"},{"Add":""},{"Add":"InducedEquivalence₁ : Symmetric-kind → ∀ {a s} {A : Set a} →"},{"Add":" (A → Set s) → Setoid _ _"},{"Add":"InducedEquivalence₁ k S = record"},{"Add":" { _≈_ = InducedRelation₁ ⌊ k ⌋ S"},{"Add":" ; isEquivalence = record"},{"Add":" { refl = K-refl"},{"Add":" ; sym = SK-sym"},{"Add":" ; trans = K-trans"},{"Add":" }"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Every binary relation induces a preorder and, for symmetric kinds,"},{"Add":"-- an equivalence. (No claim is made that these relations are unique.)"},{"Add":""},{"Add":"InducedRelation₂ : Kind → ∀ {a b s} {A : Set a} {B : Set b} →"},{"Add":" (A → B → Set s) → B → B → Set _"},{"Add":"InducedRelation₂ k _S_ = λ x y → ∀ {z} → (z S x) ∼[ k ] (z S y)"},{"Add":""},{"Add":"InducedPreorder₂ : Kind → ∀ {a b s} {A : Set a} {B : Set b} →"},{"Add":" (A → B → Set s) → Preorder _ _ _"},{"Add":"InducedPreorder₂ k _S_ = record"},{"Add":" { _≈_ = _≡_"},{"Add":" ; _≲_ = InducedRelation₂ k _S_"},{"Add":" ; isPreorder = record"},{"Add":" { isEquivalence = ≡.isEquivalence"},{"Add":" ; reflexive = λ x≡y {z} →"},{"Add":" reflexive $"},{"Add":" K-reflexive $"},{"Add":" ≡.cong (_S_ z) x≡y"},{"Add":""},{"Add":" ; trans = λ i↝j j↝k → K-trans i↝j j↝k"},{"Add":" }"},{"Add":" } where open Preorder (K-preorder _ _)"},{"Add":""},{"Add":"InducedEquivalence₂ : Symmetric-kind →"},{"Add":" ∀ {a b s} {A : Set a} {B : Set b} →"},{"Add":" (A → B → Set s) → Setoid _ _"},{"Add":"InducedEquivalence₂ k _S_ = record"},{"Add":" { _≈_ = InducedRelation₂ ⌊ k ⌋ _S_"},{"Add":" ; isEquivalence = record"},{"Add":" { refl = refl"},{"Add":" ; sym = λ i↝j → sym i↝j"},{"Add":" ; trans = λ i↝j j↝k → trans i↝j j↝k"},{"Add":" }"},{"Add":" } where open Setoid (SK-setoid _ _)"}]}},{"FileAdd":{"meta":{"basename":"Related","metadata":576},"context":[{"change":0,"pos":1018846}],"contents":[]}},{"FileAdd":{"meta":{"basename":"TypeIsomorphisms.agda","metadata":0},"context":[{"change":0,"pos":1055304}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Basic lemmas showing that various types are related (isomorphic or"},{"Add":"-- equivalent or…)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Related.TypeIsomorphisms where"},{"Add":""},{"Add":"open import Algebra"},{"Add":"open import Algebra.Structures.Biased using (isCommutativeSemiringˡ)"},{"Add":"open import Axiom.Extensionality.Propositional using (Extensionality)"},{"Add":"open import Data.Bool.Base using (true; false)"},{"Add":"open import Data.Empty.Polymorphic using (⊥; ⊥-elim)"},{"Add":"open import Data.Product.Base as Product"},{"Add":" using (_×_; Σ; curry; uncurry; _,_; -,_; <_,_>; proj₁; proj₂; ∃₂; ∃)"},{"Add":"open import Data.Product.Function.NonDependent.Propositional"},{"Add":"open import Data.Sum.Base as Sum"},{"Add":"open import Data.Sum.Properties using (swap-involutive)"},{"Add":"open import Data.Sum.Function.Propositional using (_⊎-cong_)"},{"Add":"open import Data.Unit.Polymorphic.Base using (⊤)"},{"Add":"open import Level using (Level; Lift; 0ℓ; suc)"},{"Add":"open import Function.Base"},{"Add":"open import Function.Bundles"},{"Add":"open import Function.Related.Propositional"},{"Add":"import Function.Construct.Identity as Identity"},{"Add":"open import Relation.Binary hiding (_⇔_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)"},{"Add":"open import Relation.Binary.PropositionalEquality.Properties"},{"Add":" using (module ≡-Reasoning)"},{"Add":"open import Relation.Nullary.Reflects using (invert)"},{"Add":"open import Relation.Nullary using (Dec; ¬_; _because_; ofⁿ; contradiction)"},{"Add":"import Relation.Nullary.Indexed as I"},{"Add":"open import Relation.Nullary.Decidable using (True)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c d : Level"},{"Add":" A B C D : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of Σ and _×_"},{"Add":""},{"Add":"-- Σ is associative"},{"Add":"Σ-assoc : ∀ {A : Set a} {B : A → Set b} {C : (a : A) → B a → Set c} →"},{"Add":" Σ (Σ A B) (uncurry C) ↔ Σ A (λ a → Σ (B a) (C a))"},{"Add":"Σ-assoc = mk↔ₛ′ Product.assocʳ Product.assocˡ (λ _ → refl) (λ _ → refl)"},{"Add":""},{"Add":"-- × is commutative"},{"Add":""},{"Add":"×-comm : ∀ (A : Set a) (B : Set b) → (A × B) ↔ (B × A)"},{"Add":"×-comm _ _ = mk↔ₛ′ Product.swap Product.swap (λ _ → refl) λ _ → refl"},{"Add":""},{"Add":"-- × has ⊤ as its identity"},{"Add":""},{"Add":"×-identityˡ : ∀ ℓ → LeftIdentity {ℓ = ℓ} _↔_ ⊤ _×_"},{"Add":"×-identityˡ _ _ = mk↔ₛ′ proj₂ -,_ (λ _ → refl) (λ _ → refl)"},{"Add":""},{"Add":"×-identityʳ : ∀ ℓ → RightIdentity {ℓ = ℓ} _↔_ ⊤ _×_"},{"Add":"×-identityʳ _ _ = mk↔ₛ′ proj₁ (_, _) (λ _ → refl) (λ _ → refl)"},{"Add":""},{"Add":"×-identity : ∀ ℓ → Identity _↔_ ⊤ _×_"},{"Add":"×-identity ℓ = ×-identityˡ ℓ , ×-identityʳ ℓ"},{"Add":""},{"Add":"-- × has ⊥ has its zero"},{"Add":""},{"Add":"×-zeroˡ : ∀ ℓ → LeftZero {ℓ = ℓ} _↔_ ⊥ _×_"},{"Add":"×-zeroˡ ℓ A = mk↔ₛ′ proj₁ < id , ⊥-elim > (λ _ → refl) (λ { () })"},{"Add":""},{"Add":"×-zeroʳ : ∀ ℓ → RightZero {ℓ = ℓ} _↔_ ⊥ _×_"},{"Add":"×-zeroʳ ℓ A = mk↔ₛ′ proj₂ < ⊥-elim , id > (λ _ → refl) (λ { () })"},{"Add":""},{"Add":"×-zero : ∀ ℓ → Zero _↔_ ⊥ _×_"},{"Add":"×-zero ℓ = ×-zeroˡ ℓ , ×-zeroʳ ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of ⊎"},{"Add":""},{"Add":"-- ⊎ is associative"},{"Add":""},{"Add":"⊎-assoc : ∀ ℓ → Associative {ℓ = ℓ} _↔_ _⊎_"},{"Add":"⊎-assoc ℓ _ _ _ = mk↔ₛ′"},{"Add":" [ [ inj₁ , inj₂ ∘′ inj₁ ]′ , inj₂ ∘′ inj₂ ]′"},{"Add":" [ inj₁ ∘′ inj₁ , [ inj₁ ∘′ inj₂ , inj₂ ]′ ]′"},{"Add":" [ (λ _ → refl) , [ (λ _ → refl) , (λ _ → refl) ] ]"},{"Add":" [ [ (λ _ → refl) , (λ _ → refl) ] , (λ _ → refl) ]"},{"Add":""},{"Add":"-- ⊎ is commutative"},{"Add":""},{"Add":"⊎-comm : ∀ (A : Set a) (B : Set b) → (A ⊎ B) ↔ (B ⊎ A)"},{"Add":"⊎-comm _ _ = mk↔ₛ′ swap swap swap-involutive swap-involutive"},{"Add":""},{"Add":"-- ⊎ has ⊥ as its identity"},{"Add":""},{"Add":"⊎-identityˡ : ∀ ℓ → LeftIdentity _↔_ (⊥ {ℓ}) _⊎_"},{"Add":"⊎-identityˡ _ _ = mk↔ₛ′ [ (λ ()) , id ]′ inj₂ (λ _ → refl)"},{"Add":" [ (λ ()) , (λ _ → refl) ]"},{"Add":""},{"Add":"⊎-identityʳ : ∀ ℓ → RightIdentity _↔_ (⊥ {ℓ}) _⊎_"},{"Add":"⊎-identityʳ _ _ = mk↔ₛ′ [ id , (λ ()) ]′ inj₁ (λ _ → refl)"},{"Add":" [ (λ _ → refl) , (λ ()) ]"},{"Add":""},{"Add":"⊎-identity : ∀ ℓ → Identity _↔_ ⊥ _⊎_"},{"Add":"⊎-identity ℓ = ⊎-identityˡ ℓ , ⊎-identityʳ ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of × and ⊎"},{"Add":""},{"Add":"-- × distributes over ⊎"},{"Add":""},{"Add":"×-distribˡ-⊎ : ∀ ℓ → _DistributesOverˡ_ {ℓ = ℓ} _↔_ _×_ _⊎_"},{"Add":"×-distribˡ-⊎ ℓ _ _ _ = mk↔ₛ′"},{"Add":" (uncurry λ x → [ inj₁ ∘′ (x ,_) , inj₂ ∘′ (x ,_) ]′)"},{"Add":" [ Product.map₂ inj₁ , Product.map₂ inj₂ ]′"},{"Add":" [ (λ _ → refl) , (λ _ → refl) ]"},{"Add":" (uncurry λ _ → [ (λ _ → refl) , (λ _ → refl) ])"},{"Add":""},{"Add":"×-distribʳ-⊎ : ∀ ℓ → _DistributesOverʳ_ {ℓ = ℓ} _↔_ _×_ _⊎_"},{"Add":"×-distribʳ-⊎ ℓ _ _ _ = mk↔ₛ′"},{"Add":" (uncurry [ curry inj₁ , curry inj₂ ]′)"},{"Add":" [ Product.map₁ inj₁ , Product.map₁ inj₂ ]′"},{"Add":" [ (λ _ → refl) , (λ _ → refl) ]"},{"Add":" (uncurry [ (λ _ _ → refl) , (λ _ _ → refl) ])"},{"Add":""},{"Add":"×-distrib-⊎ : ∀ ℓ → _DistributesOver_ {ℓ = ℓ} _↔_ _×_ _⊎_"},{"Add":"×-distrib-⊎ ℓ = ×-distribˡ-⊎ ℓ , ×-distribʳ-⊎ ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- ⊥, ⊤, _×_ and _⊎_ form a commutative semiring"},{"Add":""},{"Add":"-- ⊤, _×_ form a commutative monoid"},{"Add":""},{"Add":"×-isMagma : ∀ k ℓ → IsMagma {Level.suc ℓ} (Related ⌊ k ⌋) _×_"},{"Add":"×-isMagma k ℓ = record"},{"Add":" { isEquivalence = SK-isEquivalence k"},{"Add":" ; ∙-cong = _×-cong_"},{"Add":" }"},{"Add":""},{"Add":"×-magma : SymmetricKind → (ℓ : Level) → Magma _ _"},{"Add":"×-magma k ℓ = record"},{"Add":" { isMagma = ×-isMagma k ℓ"},{"Add":" }"},{"Add":""},{"Add":"×-isSemigroup : ∀ k ℓ → IsSemigroup {Level.suc ℓ} (Related ⌊ k ⌋) _×_"},{"Add":"×-isSemigroup k ℓ = record"},{"Add":" { isMagma = ×-isMagma k ℓ"},{"Add":" ; assoc = λ _ _ _ → ↔⇒ Σ-assoc"},{"Add":" }"},{"Add":""},{"Add":"×-semigroup : SymmetricKind → (ℓ : Level) → Semigroup _ _"},{"Add":"×-semigroup k ℓ = record"},{"Add":" { isSemigroup = ×-isSemigroup k ℓ"},{"Add":" }"},{"Add":""},{"Add":"×-isMonoid : ∀ k ℓ → IsMonoid (Related ⌊ k ⌋) _×_ ⊤"},{"Add":"×-isMonoid k ℓ = record"},{"Add":" { isSemigroup = ×-isSemigroup k ℓ"},{"Add":" ; identity = (↔⇒ ∘ ×-identityˡ ℓ) , (↔⇒ ∘ ×-identityʳ ℓ)"},{"Add":" }"},{"Add":""},{"Add":"×-monoid : SymmetricKind → (ℓ : Level) → Monoid _ _"},{"Add":"×-monoid k ℓ = record"},{"Add":" { isMonoid = ×-isMonoid k ℓ"},{"Add":" }"},{"Add":""},{"Add":"×-isCommutativeMonoid : ∀ k ℓ → IsCommutativeMonoid (Related ⌊ k ⌋) _×_ ⊤"},{"Add":"×-isCommutativeMonoid k ℓ = record"},{"Add":" { isMonoid = ×-isMonoid k ℓ"},{"Add":" ; comm = λ _ _ → ↔⇒ (×-comm _ _)"},{"Add":" }"},{"Add":""},{"Add":"×-commutativeMonoid : SymmetricKind → (ℓ : Level) → CommutativeMonoid _ _"},{"Add":"×-commutativeMonoid k ℓ = record"},{"Add":" { isCommutativeMonoid = ×-isCommutativeMonoid k ℓ"},{"Add":" }"},{"Add":""},{"Add":"-- ⊥, _⊎_ form a commutative monoid"},{"Add":""},{"Add":"⊎-isMagma : ∀ k ℓ → IsMagma {Level.suc ℓ} (Related ⌊ k ⌋) _⊎_"},{"Add":"⊎-isMagma k ℓ = record"},{"Add":" { isEquivalence = SK-isEquivalence k"},{"Add":" ; ∙-cong = _⊎-cong_"},{"Add":" }"},{"Add":""},{"Add":"⊎-magma : SymmetricKind → (ℓ : Level) → Magma _ _"},{"Add":"⊎-magma k ℓ = record"},{"Add":" { isMagma = ⊎-isMagma k ℓ"},{"Add":" }"},{"Add":""},{"Add":"⊎-isSemigroup : ∀ k ℓ → IsSemigroup {Level.suc ℓ} (Related ⌊ k ⌋) _⊎_"},{"Add":"⊎-isSemigroup k ℓ = record"},{"Add":" { isMagma = ⊎-isMagma k ℓ"},{"Add":" ; assoc = λ A B C → ↔⇒ (⊎-assoc ℓ A B C)"},{"Add":" }"},{"Add":""},{"Add":"⊎-semigroup : SymmetricKind → (ℓ : Level) → Semigroup _ _"},{"Add":"⊎-semigroup k ℓ = record"},{"Add":" { isSemigroup = ⊎-isSemigroup k ℓ"},{"Add":" }"},{"Add":""},{"Add":"⊎-isMonoid : ∀ k ℓ → IsMonoid (Related ⌊ k ⌋) _⊎_ ⊥"},{"Add":"⊎-isMonoid k ℓ = record"},{"Add":" { isSemigroup = ⊎-isSemigroup k ℓ"},{"Add":" ; identity = (↔⇒ ∘ ⊎-identityˡ ℓ) , (↔⇒ ∘ ⊎-identityʳ ℓ)"},{"Add":" }"},{"Add":""},{"Add":"⊎-monoid : SymmetricKind → (ℓ : Level) → Monoid _ _"},{"Add":"⊎-monoid k ℓ = record"},{"Add":" { isMonoid = ⊎-isMonoid k ℓ"},{"Add":" }"},{"Add":""},{"Add":"⊎-isCommutativeMonoid : ∀ k ℓ → IsCommutativeMonoid (Related ⌊ k ⌋) _⊎_ ⊥"},{"Add":"⊎-isCommutativeMonoid k ℓ = record"},{"Add":" { isMonoid = ⊎-isMonoid k ℓ"},{"Add":" ; comm = λ _ _ → ↔⇒ (⊎-comm _ _)"},{"Add":" }"},{"Add":""},{"Add":"⊎-commutativeMonoid : SymmetricKind → (ℓ : Level) →"},{"Add":" CommutativeMonoid _ _"},{"Add":"⊎-commutativeMonoid k ℓ = record"},{"Add":" { isCommutativeMonoid = ⊎-isCommutativeMonoid k ℓ"},{"Add":" }"},{"Add":""},{"Add":"×-⊎-isCommutativeSemiring : ∀ k ℓ →"},{"Add":" IsCommutativeSemiring (Related ⌊ k ⌋) _⊎_ _×_ ⊥ ⊤"},{"Add":"×-⊎-isCommutativeSemiring k ℓ = isCommutativeSemiringˡ record"},{"Add":" { +-isCommutativeMonoid = ⊎-isCommutativeMonoid k ℓ"},{"Add":" ; *-isCommutativeMonoid = ×-isCommutativeMonoid k ℓ"},{"Add":" ; distribʳ = λ A B C → ↔⇒ (×-distribʳ-⊎ ℓ A B C)"},{"Add":" ; zeroˡ = ↔⇒ ∘ ×-zeroˡ ℓ"},{"Add":" }"},{"Add":""},{"Add":"×-⊎-commutativeSemiring : SymmetricKind → (ℓ : Level) →"},{"Add":" CommutativeSemiring (Level.suc ℓ) ℓ"},{"Add":"×-⊎-commutativeSemiring k ℓ = record"},{"Add":" { isCommutativeSemiring = ×-⊎-isCommutativeSemiring k ℓ"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Some reordering lemmas"},{"Add":""},{"Add":"ΠΠ↔ΠΠ : ∀ {a b p} {A : Set a} {B : Set b} (P : A → B → Set p) →"},{"Add":" ((x : A) (y : B) → P x y) ↔ ((y : B) (x : A) → P x y)"},{"Add":"ΠΠ↔ΠΠ _ = mk↔ₛ′ flip flip (λ _ → refl) (λ _ → refl)"},{"Add":""},{"Add":"∃∃↔∃∃ : ∀ {a b p} {A : Set a} {B : Set b} (P : A → B → Set p) →"},{"Add":" (∃₂ λ x y → P x y) ↔ (∃₂ λ y x → P x y)"},{"Add":"∃∃↔∃∃ P = mk↔ₛ′ to from (λ _ → refl) (λ _ → refl)"},{"Add":" where"},{"Add":" to : (∃₂ λ x y → P x y) → (∃₂ λ y x → P x y)"},{"Add":" to (x , y , Pxy) = (y , x , Pxy)"},{"Add":""},{"Add":" from : (∃₂ λ y x → P x y) → (∃₂ λ x y → P x y)"},{"Add":" from (y , x , Pxy) = (x , y , Pxy)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Implicit and explicit function spaces are isomorphic"},{"Add":""},{"Add":"Π↔Π : ∀ {A : Set a} {B : A → Set b} →"},{"Add":" ((x : A) → B x) ↔ ({x : A} → B x)"},{"Add":"Π↔Π = mk↔ₛ′ _$- λ- (λ _ → refl) (λ _ → refl)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- _→_ preserves the symmetric relations"},{"Add":""},{"Add":"→-cong-⇔ : A ⇔ B → C ⇔ D → (A → C) ⇔ (B → D)"},{"Add":"→-cong-⇔ A⇔B C⇔D = mk⇔"},{"Add":" (λ f → to C⇔D ∘ f ∘ from A⇔B)"},{"Add":" (λ f → from C⇔D ∘ f ∘ to A⇔B)"},{"Add":" where open Equivalence"},{"Add":""},{"Add":"→-cong-↔ : Extensionality a c → Extensionality b d →"},{"Add":" {A : Set a} {B : Set b} {C : Set c} {D : Set d} →"},{"Add":" A ↔ B → C ↔ D → (A → C) ↔ (B → D)"},{"Add":"→-cong-↔ ext₁ ext₂ A↔B C↔D = mk↔ₛ′"},{"Add":" (λ f → to C↔D ∘ f ∘ from A↔B)"},{"Add":" (λ f → from C↔D ∘ f ∘ to A↔B)"},{"Add":" (λ f → ext₂ λ x → begin"},{"Add":" to C↔D (from C↔D (f (to A↔B (from A↔B x)))) ≡⟨ strictlyInverseˡ C↔D _ ⟩"},{"Add":" f (to A↔B (from A↔B x)) ≡⟨ cong f $ strictlyInverseˡ A↔B x ⟩"},{"Add":" f x ∎)"},{"Add":" (λ f → ext₁ λ x → begin"},{"Add":" from C↔D (to C↔D (f (from A↔B (to A↔B x)))) ≡⟨ strictlyInverseʳ C↔D _ ⟩"},{"Add":" f (from A↔B (to A↔B x)) ≡⟨ cong f $ strictlyInverseʳ A↔B x ⟩"},{"Add":" f x ∎)"},{"Add":" where open Inverse; open ≡-Reasoning"},{"Add":""},{"Add":"→-cong : Extensionality a c → Extensionality b d →"},{"Add":" ∀ {k} {A : Set a} {B : Set b} {C : Set c} {D : Set d} →"},{"Add":" A ∼[ ⌊ k ⌋ ] B → C ∼[ ⌊ k ⌋ ] D → (A → C) ∼[ ⌊ k ⌋ ] (B → D)"},{"Add":"→-cong extAC extBD {equivalence} = →-cong-⇔"},{"Add":"→-cong extAC extBD {bijection} = →-cong-↔ extAC extBD"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- ¬_ (at Level 0) preserves the symmetric relations"},{"Add":""},{"Add":"¬-cong-⇔ : A ⇔ B → (¬ A) ⇔ (¬ B)"},{"Add":"¬-cong-⇔ A⇔B = →-cong-⇔ A⇔B (Identity.⇔-id _)"},{"Add":""},{"Add":"¬-cong : Extensionality a 0ℓ → Extensionality b 0ℓ →"},{"Add":" ∀ {k} {A : Set a} {B : Set b} →"},{"Add":" A ∼[ ⌊ k ⌋ ] B → (¬ A) ∼[ ⌊ k ⌋ ] (¬ B)"},{"Add":"¬-cong extA extB A≈B = →-cong extA extB A≈B (K-reflexive refl)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- _⇔_ preserves _⇔_"},{"Add":""},{"Add":"-- The type of the following proof is a bit more general."},{"Add":""},{"Add":"Related-cong :"},{"Add":" ∀ {k} →"},{"Add":" A ∼[ ⌊ k ⌋ ] B → C ∼[ ⌊ k ⌋ ] D → (A ∼[ ⌊ k ⌋ ] C) ⇔ (B ∼[ ⌊ k ⌋ ] D)"},{"Add":"Related-cong {A = A} {B = B} {C = C} {D = D} A≈B C≈D = mk⇔"},{"Add":" (λ A≈C → B ∼⟨ SK-sym A≈B ⟩"},{"Add":" A ∼⟨ A≈C ⟩"},{"Add":" C ∼⟨ C≈D ⟩"},{"Add":" D ∎)"},{"Add":" (λ B≈D → A ∼⟨ A≈B ⟩"},{"Add":" B ∼⟨ B≈D ⟩"},{"Add":" D ∼⟨ SK-sym C≈D ⟩"},{"Add":" C ∎)"},{"Add":" where open EquationalReasoning"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- A lemma relating True dec and P, where dec : Dec P"},{"Add":""},{"Add":"True↔ : ∀ {p} {P : Set p}"},{"Add":" (dec : Dec P) → ((p₁ p₂ : P) → p₁ ≡ p₂) → True dec ↔ P"},{"Add":"True↔ ( true because [p]) irr ="},{"Add":" mk↔ₛ′ (λ _ → invert [p]) (λ _ → _) (irr _) (λ _ → refl)"},{"Add":"True↔ (false because ofⁿ ¬p) _ ="},{"Add":" mk↔ₛ′ (λ()) (invert (ofⁿ ¬p)) (λ x → flip contradiction ¬p x) (λ ())"}]}},{"FileAdd":{"meta":{"basename":"TypeIsomorphisms","metadata":576},"context":[{"change":0,"pos":1055304}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Solver.agda","metadata":0},"context":[{"change":0,"pos":1068432}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Automatic solver for equations over product and sum types"},{"Add":"--"},{"Add":"-- See examples at the bottom of the file for how to use this solver"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Related.TypeIsomorphisms.Solver where"},{"Add":""},{"Add":"open import Algebra using (CommutativeSemiring)"},{"Add":"import Algebra.Solver.Ring.NaturalCoefficients.Default"},{"Add":"open import Data.Empty.Polymorphic using (⊥)"},{"Add":"open import Data.Product.Base using (_×_)"},{"Add":"open import Data.Sum.Base using (_⊎_)"},{"Add":"open import Data.Unit.Polymorphic using (⊤)"},{"Add":"open import Level using (Level)"},{"Add":"open import Function.Bundles using (_↔_)"},{"Add":"open import Function.Properties.Inverse using (↔-refl)"},{"Add":"open import Function.Related.Propositional as Related"},{"Add":"open import Function.Related.TypeIsomorphisms"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The solver"},{"Add":""},{"Add":"module ×-⊎-Solver (k : SymmetricKind) {ℓ} ="},{"Add":" Algebra.Solver.Ring.NaturalCoefficients.Default"},{"Add":" (×-⊎-commutativeSemiring k ℓ)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Tests"},{"Add":""},{"Add":"private"},{"Add":""},{"Add":" -- A test of the solver above."},{"Add":""},{"Add":" test : {ℓ : Level} (A B C : Set ℓ) →"},{"Add":" (⊤ × A × (B ⊎ C)) ↔ (A × B ⊎ C × (⊥ ⊎ A))"},{"Add":" test = solve 3 (λ A B C → con 1 :* (A :* (B :+ C)) :="},{"Add":" A :* B :+ C :* (con 0 :+ A))"},{"Add":" ↔-refl"},{"Add":" where open ×-⊎-Solver bijection"}]}},{"FileAdd":{"meta":{"basename":"Propositional.agda","metadata":0},"context":[{"change":0,"pos":1055304}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Relatedness for the function hierarchy"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Related.Propositional where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Relation.Binary"},{"Add":" using (Rel; REL; Sym; Reflexive; Trans; IsEquivalence; Setoid; IsPreorder; Preorder)"},{"Add":"open import Function.Bundles"},{"Add":"open import Function.Base"},{"Add":"open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":"open import Relation.Binary.Reasoning.Syntax"},{"Add":""},{"Add":"open import Function.Properties.Surjection using (↠⇒↪; ↠⇒⇔)"},{"Add":"open import Function.Properties.RightInverse using (↪⇒↠)"},{"Add":"open import Function.Properties.Bijection using (⤖⇒↔; ⤖⇒⇔)"},{"Add":"open import Function.Properties.Inverse using (↔⇒⤖; ↔⇒⇔; ↔⇒↣; ↔⇒↠)"},{"Add":""},{"Add":"import Function.Construct.Symmetry as Symmetry"},{"Add":"import Function.Construct.Identity as Identity"},{"Add":"import Function.Construct.Composition as Composition"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Relatedness"},{"Add":""},{"Add":"-- There are several kinds of \"relatedness\"."},{"Add":""},{"Add":"-- The idea to include kinds other than equivalence and bijection came"},{"Add":"-- from Simon Thompson and Bengt Nordström. /NAD"},{"Add":""},{"Add":"data Kind : Set where"},{"Add":" implication : Kind"},{"Add":" reverseImplication : Kind"},{"Add":" equivalence : Kind"},{"Add":" injection : Kind"},{"Add":" reverseInjection : Kind"},{"Add":" leftInverse : Kind"},{"Add":" surjection : Kind"},{"Add":" bijection : Kind"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c p : Level"},{"Add":" A B C : Set a"},{"Add":" k : Kind"},{"Add":""},{"Add":"-- Interpretation of the codes above. The code \"bijection\" is"},{"Add":"-- interpreted as Inverse rather than Bijection; the two types are"},{"Add":"-- equivalent."},{"Add":""},{"Add":"infix 4 _∼[_]_"},{"Add":""},{"Add":"_∼[_]_ : Set a → Kind → Set b → Set _"},{"Add":"A ∼[ implication ] B = A ⟶ B"},{"Add":"A ∼[ reverseImplication ] B = B ⟶ A"},{"Add":"A ∼[ equivalence ] B = A ⇔ B"},{"Add":"A ∼[ injection ] B = A ↣ B"},{"Add":"A ∼[ reverseInjection ] B = B ↣ A"},{"Add":"A ∼[ leftInverse ] B = A ↪ B"},{"Add":"A ∼[ surjection ] B = A ↠ B"},{"Add":"A ∼[ bijection ] B = A ↔ B"},{"Add":""},{"Add":"-- A non-infix synonym."},{"Add":""},{"Add":"Related : Kind → Set a → Set b → Set _"},{"Add":"Related k A B = A ∼[ k ] B"},{"Add":""},{"Add":"-- The bijective equality implies any kind of relatedness."},{"Add":""},{"Add":"↔⇒ : A ∼[ bijection ] B → A ∼[ k ] B"},{"Add":"↔⇒ {k = implication} = mk⟶ ∘ Inverse.to"},{"Add":"↔⇒ {k = reverseImplication} = mk⟶ ∘ Inverse.from"},{"Add":"↔⇒ {k = equivalence} = ↔⇒⇔"},{"Add":"↔⇒ {k = injection} = ↔⇒↣"},{"Add":"↔⇒ {k = reverseInjection} = ↔⇒↣ ∘ Symmetry.inverse"},{"Add":"↔⇒ {k = leftInverse} = Inverse.rightInverse"},{"Add":"↔⇒ {k = surjection} = ↔⇒↠"},{"Add":"↔⇒ {k = bijection} = id"},{"Add":""},{"Add":"-- Propositional equality also implies any kind of relatedness."},{"Add":""},{"Add":"≡⇒ : A ≡ B → A ∼[ k ] B"},{"Add":"≡⇒ ≡.refl = ↔⇒ (Identity.↔-id _)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Special kinds of kinds"},{"Add":""},{"Add":"-- Kinds whose interpretation is symmetric."},{"Add":""},{"Add":"data SymmetricKind : Set where"},{"Add":" equivalence : SymmetricKind"},{"Add":" bijection : SymmetricKind"},{"Add":""},{"Add":"-- Forgetful map."},{"Add":""},{"Add":"⌊_⌋ : SymmetricKind → Kind"},{"Add":"⌊ equivalence ⌋ = equivalence"},{"Add":"⌊ bijection ⌋ = bijection"},{"Add":""},{"Add":"-- The proof of symmetry can be found below."},{"Add":""},{"Add":"-- Kinds whose interpretation include a function which \"goes in the"},{"Add":"-- forward direction\"."},{"Add":""},{"Add":"data ForwardKind : Set where"},{"Add":" implication : ForwardKind"},{"Add":" equivalence : ForwardKind"},{"Add":" injection : ForwardKind"},{"Add":" leftInverse : ForwardKind"},{"Add":" surjection : ForwardKind"},{"Add":" bijection : ForwardKind"},{"Add":""},{"Add":"-- Forgetful map."},{"Add":""},{"Add":"⌊_⌋→ : ForwardKind → Kind"},{"Add":"⌊ implication ⌋→ = implication"},{"Add":"⌊ equivalence ⌋→ = equivalence"},{"Add":"⌊ injection ⌋→ = injection"},{"Add":"⌊ leftInverse ⌋→ = leftInverse"},{"Add":"⌊ surjection ⌋→ = surjection"},{"Add":"⌊ bijection ⌋→ = bijection"},{"Add":""},{"Add":"-- The function."},{"Add":""},{"Add":"⇒→ : ∀ {k} → A ∼[ ⌊ k ⌋→ ] B → A → B"},{"Add":"⇒→ {k = implication} = Func.to"},{"Add":"⇒→ {k = equivalence} = Equivalence.to"},{"Add":"⇒→ {k = injection} = Injection.to"},{"Add":"⇒→ {k = leftInverse} = RightInverse.to"},{"Add":"⇒→ {k = surjection} = Surjection.to"},{"Add":"⇒→ {k = bijection} = Inverse.to"},{"Add":""},{"Add":"-- Kinds whose interpretation include a function which \"goes backwards\"."},{"Add":""},{"Add":"data BackwardKind : Set where"},{"Add":" reverseImplication : BackwardKind"},{"Add":" equivalence : BackwardKind"},{"Add":" reverseInjection : BackwardKind"},{"Add":" leftInverse : BackwardKind"},{"Add":" surjection : BackwardKind"},{"Add":" bijection : BackwardKind"},{"Add":""},{"Add":"-- Forgetful map."},{"Add":""},{"Add":"⌊_⌋← : BackwardKind → Kind"},{"Add":"⌊ reverseImplication ⌋← = reverseImplication"},{"Add":"⌊ equivalence ⌋← = equivalence"},{"Add":"⌊ reverseInjection ⌋← = reverseInjection"},{"Add":"⌊ leftInverse ⌋← = leftInverse"},{"Add":"⌊ surjection ⌋← = surjection"},{"Add":"⌊ bijection ⌋← = bijection"},{"Add":""},{"Add":"-- The function."},{"Add":""},{"Add":"⇒← : ∀ {k} → A ∼[ ⌊ k ⌋← ] B → B → A"},{"Add":"⇒← {k = reverseImplication} = Func.to"},{"Add":"⇒← {k = equivalence} = Equivalence.from"},{"Add":"⇒← {k = reverseInjection} = Injection.to"},{"Add":"⇒← {k = leftInverse} = RightInverse.from"},{"Add":"⇒← {k = surjection} = RightInverse.to ∘ ↠⇒↪"},{"Add":"⇒← {k = bijection} = Inverse.from"},{"Add":""},{"Add":"-- Kinds whose interpretation include functions going in both"},{"Add":"-- directions."},{"Add":""},{"Add":"data EquivalenceKind : Set where"},{"Add":" equivalence : EquivalenceKind"},{"Add":" leftInverse : EquivalenceKind"},{"Add":" surjection : EquivalenceKind"},{"Add":" bijection : EquivalenceKind"},{"Add":""},{"Add":"-- Forgetful map."},{"Add":""},{"Add":"⌊_⌋⇔ : EquivalenceKind → Kind"},{"Add":"⌊ equivalence ⌋⇔ = equivalence"},{"Add":"⌊ leftInverse ⌋⇔ = leftInverse"},{"Add":"⌊ surjection ⌋⇔ = surjection"},{"Add":"⌊ bijection ⌋⇔ = bijection"},{"Add":""},{"Add":"-- The functions."},{"Add":""},{"Add":"⇒⇔ : ∀ {k} → A ∼[ ⌊ k ⌋⇔ ] B → A ∼[ equivalence ] B"},{"Add":"⇒⇔ {k = equivalence} = id"},{"Add":"⇒⇔ {k = leftInverse} = RightInverse.equivalence"},{"Add":"⇒⇔ {k = surjection} = ↠⇒⇔"},{"Add":"⇒⇔ {k = bijection} = ↔⇒⇔"},{"Add":""},{"Add":"-- Conversions between special kinds."},{"Add":""},{"Add":"⇔⌊_⌋ : SymmetricKind → EquivalenceKind"},{"Add":"⇔⌊ equivalence ⌋ = equivalence"},{"Add":"⇔⌊ bijection ⌋ = bijection"},{"Add":""},{"Add":"→⌊_⌋ : EquivalenceKind → ForwardKind"},{"Add":"→⌊ equivalence ⌋ = equivalence"},{"Add":"→⌊ leftInverse ⌋ = leftInverse"},{"Add":"→⌊ surjection ⌋ = surjection"},{"Add":"→⌊ bijection ⌋ = bijection"},{"Add":""},{"Add":"←⌊_⌋ : EquivalenceKind → BackwardKind"},{"Add":"←⌊ equivalence ⌋ = equivalence"},{"Add":"←⌊ leftInverse ⌋ = leftInverse"},{"Add":"←⌊ surjection ⌋ = surjection"},{"Add":"←⌊ bijection ⌋ = bijection"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Opposites"},{"Add":""},{"Add":"-- For every kind there is an opposite kind."},{"Add":""},{"Add":"_op : Kind → Kind"},{"Add":"implication op = reverseImplication"},{"Add":"reverseImplication op = implication"},{"Add":"equivalence op = equivalence"},{"Add":"injection op = reverseInjection"},{"Add":"reverseInjection op = injection"},{"Add":"leftInverse op = surjection"},{"Add":"surjection op = leftInverse"},{"Add":"bijection op = bijection"},{"Add":""},{"Add":"-- For every morphism there is a corresponding reverse morphism of the"},{"Add":"-- opposite kind."},{"Add":""},{"Add":"reverse : A ∼[ k ] B → B ∼[ k op ] A"},{"Add":"reverse {k = implication} = id"},{"Add":"reverse {k = reverseImplication} = id"},{"Add":"reverse {k = equivalence} = Symmetry.⇔-sym"},{"Add":"reverse {k = injection} = id"},{"Add":"reverse {k = reverseInjection} = id"},{"Add":"reverse {k = leftInverse} = ↪⇒↠"},{"Add":"reverse {k = surjection} = ↠⇒↪"},{"Add":"reverse {k = bijection} = Symmetry.↔-sym"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- For a fixed universe level every kind is a preorder and each"},{"Add":"-- symmetric kind is an equivalence"},{"Add":""},{"Add":"K-refl : Reflexive (Related {a} k)"},{"Add":"K-refl {k = implication} = Identity.⟶-id _"},{"Add":"K-refl {k = reverseImplication} = Identity.⟶-id _"},{"Add":"K-refl {k = equivalence} = Identity.⇔-id _"},{"Add":"K-refl {k = injection} = Identity.↣-id _"},{"Add":"K-refl {k = reverseInjection} = Identity.↣-id _"},{"Add":"K-refl {k = leftInverse} = Identity.↪-id _"},{"Add":"K-refl {k = surjection} = Identity.↠-id _"},{"Add":"K-refl {k = bijection} = Identity.↔-id _"},{"Add":""},{"Add":"K-reflexive : _≡_ Relation.Binary.⇒ Related {a} k"},{"Add":"K-reflexive ≡.refl = K-refl"},{"Add":""},{"Add":"K-trans : Trans (Related {a} {b} k)"},{"Add":" (Related {b} {c} k)"},{"Add":" (Related {a} {c} k)"},{"Add":"K-trans {k = implication} = flip Composition._⟶-∘_"},{"Add":"K-trans {k = reverseImplication} = Composition._⟶-∘_"},{"Add":"K-trans {k = equivalence} = flip Composition._⇔-∘_"},{"Add":"K-trans {k = injection} = flip Composition._↣-∘_"},{"Add":"K-trans {k = reverseInjection} = Composition._↣-∘_"},{"Add":"K-trans {k = leftInverse} = flip Composition._↪-∘_"},{"Add":"K-trans {k = surjection} = flip Composition._↠-∘_"},{"Add":"K-trans {k = bijection} = flip Composition._↔-∘_"},{"Add":""},{"Add":"SK-sym : ∀ {k} → Sym (Related {a} {b} ⌊ k ⌋)"},{"Add":" (Related {b} {a} ⌊ k ⌋)"},{"Add":"SK-sym {k = equivalence} = reverse"},{"Add":"SK-sym {k = bijection} = reverse"},{"Add":""},{"Add":"SK-isEquivalence : ∀ k → IsEquivalence {ℓ = a} (Related ⌊ k ⌋)"},{"Add":"SK-isEquivalence k = record"},{"Add":" { refl = K-refl"},{"Add":" ; sym = SK-sym"},{"Add":" ; trans = K-trans"},{"Add":" }"},{"Add":""},{"Add":"SK-setoid : SymmetricKind → (ℓ : Level) → Setoid _ _"},{"Add":"SK-setoid k ℓ = record { isEquivalence = SK-isEquivalence {ℓ} k }"},{"Add":""},{"Add":"K-isPreorder : ∀ k → IsPreorder {ℓ = a} _↔_ (Related k)"},{"Add":"K-isPreorder k = record"},{"Add":" { isEquivalence = SK-isEquivalence bijection"},{"Add":" ; reflexive = ↔⇒"},{"Add":" ; trans = K-trans"},{"Add":" }"},{"Add":""},{"Add":"K-preorder : Kind → (ℓ : Level) → Preorder _ ℓ _"},{"Add":"K-preorder k ℓ = record { isPreorder = K-isPreorder k }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Equational reasoning"},{"Add":""},{"Add":"-- Equational reasoning for related things. Note that we don't use"},{"Add":"-- the `Relation.Binary.Reasoning.Syntax` for this as this relation"},{"Add":"-- is heterogeneous."},{"Add":""},{"Add":"module EquationalReasoning {k : Kind} where"},{"Add":""},{"Add":" -- Combinators with one heterogeneous relation"},{"Add":" module _ {a b : Level} where"},{"Add":" open begin-syntax (Related {a} {b} k) id public"},{"Add":" open ≡-noncomputing-syntax (Related {a} {b} k) public"},{"Add":""},{"Add":" -- Combinators with two heterogeneous relations"},{"Add":" module _ {a b c : Level} where"},{"Add":" private"},{"Add":" rel1 = Related {b} {c} k"},{"Add":" rel2 = Related {a} {c} k"},{"Add":""},{"Add":" open ∼-syntax rel1 rel2 K-trans public"},{"Add":" open ⤖-syntax rel1 rel2 (K-trans ∘′ ↔⇒ ∘′ ⤖⇒↔) Symmetry.⤖-sym public"},{"Add":" open ↔-syntax rel1 rel2 (K-trans ∘′ ↔⇒) Symmetry.↔-sym public"},{"Add":""},{"Add":" -- Combinators with homogeneous relations"},{"Add":" module _ {a : Level} where"},{"Add":" open end-syntax (Related {a} k) K-refl public"},{"Add":""},{"Add":""},{"Add":" infixr 2 _↔⟨⟩_"},{"Add":" _↔⟨⟩_ : (A : Set a) → A ∼[ k ] B → A ∼[ k ] B"},{"Add":" A ↔⟨⟩ A⇔B = A⇔B"},{"Add":" {-# WARNING_ON_USAGE _↔⟨⟩_"},{"Add":" \"Warning: _↔⟨⟩_ was deprecated in v2.0."},{"Add":" Please use _≡⟨⟩_ instead. \""},{"Add":" #-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Every unary relation induces a preorder and, for symmetric kinds,"},{"Add":"-- an equivalence. (No claim is made that these relations are unique.)"},{"Add":""},{"Add":"InducedRelation₁ : Kind → (P : A → Set p) → A → A → Set _"},{"Add":"InducedRelation₁ k P = λ x y → P x ∼[ k ] P y"},{"Add":""},{"Add":"InducedPreorder₁ : Kind → (P : A → Set p) → Preorder _ _ _"},{"Add":"InducedPreorder₁ k P = record"},{"Add":" { _≈_ = _≡_"},{"Add":" ; _≲_ = InducedRelation₁ k P"},{"Add":" ; isPreorder = record"},{"Add":" { isEquivalence = ≡.isEquivalence"},{"Add":" ; reflexive = reflexive ∘"},{"Add":" K-reflexive ∘"},{"Add":" ≡.cong P"},{"Add":" ; trans = K-trans"},{"Add":" }"},{"Add":" } where open Preorder (K-preorder _ _)"},{"Add":""},{"Add":"InducedEquivalence₁ : SymmetricKind → (P : A → Set p) → Setoid _ _"},{"Add":"InducedEquivalence₁ k P = record"},{"Add":" { _≈_ = InducedRelation₁ ⌊ k ⌋ P"},{"Add":" ; isEquivalence = record"},{"Add":" { refl = K-refl"},{"Add":" ; sym = SK-sym"},{"Add":" ; trans = K-trans"},{"Add":" }"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Every binary relation induces a preorder and, for symmetric kinds,"},{"Add":"-- an equivalence. (No claim is made that these relations are unique.)"},{"Add":""},{"Add":"InducedRelation₂ : Kind → ∀ {s} → (A → B → Set s) → B → B → Set _"},{"Add":"InducedRelation₂ k _S_ = λ x y → ∀ {z} → (z S x) ∼[ k ] (z S y)"},{"Add":""},{"Add":"InducedPreorder₂ : Kind → ∀ {s} → (A → B → Set s) → Preorder _ _ _"},{"Add":"InducedPreorder₂ k _S_ = record"},{"Add":" { _≈_ = _≡_"},{"Add":" ; _≲_ = InducedRelation₂ k _S_"},{"Add":" ; isPreorder = record"},{"Add":" { isEquivalence = ≡.isEquivalence"},{"Add":" ; reflexive = λ x≡y {z} →"},{"Add":" reflexive $"},{"Add":" K-reflexive $"},{"Add":" ≡.cong (_S_ z) x≡y"},{"Add":""},{"Add":" ; trans = λ i↝j j↝k → K-trans i↝j j↝k"},{"Add":" }"},{"Add":" } where open Preorder (K-preorder _ _)"},{"Add":""},{"Add":"InducedEquivalence₂ : SymmetricKind → ∀ {s} → (A → B → Set s) → Setoid _ _"},{"Add":"InducedEquivalence₂ k _S_ = record"},{"Add":" { _≈_ = InducedRelation₂ ⌊ k ⌋ _S_"},{"Add":" ; isEquivalence = record"},{"Add":" { refl = refl"},{"Add":" ; sym = λ i↝j → sym i↝j"},{"Add":" ; trans = λ i↝j j↝k → trans i↝j j↝k"},{"Add":" }"},{"Add":" } where open Setoid (SK-setoid _ _)"}]}},{"FileAdd":{"meta":{"basename":"Reasoning.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- A module used for creating function pipelines, see"},{"Add":"-- README.Function.Reasoning for examples"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Reasoning where"},{"Add":""},{"Add":"open import Function.Base using (_∋_)"},{"Add":""},{"Add":"-- Need to give _∋_ a new name as syntax cannot contain underscores"},{"Add":"infixl 0 ∋-syntax"},{"Add":"∋-syntax = _∋_"},{"Add":""},{"Add":"-- Create ∶ syntax"},{"Add":"syntax ∋-syntax A a = a ∶ A"},{"Add":""},{"Add":"-- Export pipeline functions"},{"Add":"open import Function.Base public using (_|>_; _|>′_)"}]}},{"FileAdd":{"meta":{"basename":"Properties.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Basic properties of the function type"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Properties where"},{"Add":""},{"Add":"open import Axiom.Extensionality.Propositional using (Extensionality)"},{"Add":"open import Function.Base using (flip; _∘_)"},{"Add":"open import Function.Bundles using (_↔_; mk↔ₛ′; Inverse)"},{"Add":"open import Level"},{"Add":"open import Relation.Binary.PropositionalEquality.Core"},{"Add":" using (trans; cong; cong′)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c d p : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Implicit and explicit function spaces are isomorphic"},{"Add":""},{"Add":"Π↔Π : {B : A → Set b} → ((x : A) → B x) ↔ ({x : A} → B x)"},{"Add":"Π↔Π = mk↔ₛ′ (λ f {x} → f x) (λ f _ → f) cong′ cong′"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Function spaces can be reordered"},{"Add":""},{"Add":"ΠΠ↔ΠΠ : (R : A → B → Set p) →"},{"Add":" ((x : A) (y : B) → R x y) ↔ ((y : B) (x : A) → R x y)"},{"Add":"ΠΠ↔ΠΠ _ = mk↔ₛ′ flip flip cong′ cong′"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Assuming extensionality then → preserves ↔"},{"Add":""},{"Add":"→-cong-↔ : {A : Set a} {B : Set b} {C : Set c} {D : Set d} →"},{"Add":" Extensionality a c → Extensionality b d →"},{"Add":" A ↔ B → C ↔ D → (A → C) ↔ (B → D)"},{"Add":"→-cong-↔ extAC extBD A↔B C↔D = mk↔ₛ′"},{"Add":" (λ h → C↔D.to ∘ h ∘ A↔B.from)"},{"Add":" (λ g → C↔D.from ∘ g ∘ A↔B.to )"},{"Add":" (λ h → extBD λ x → trans (C↔D.strictlyInverseˡ _) (cong h (A↔B.strictlyInverseˡ x)))"},{"Add":" (λ g → extAC λ x → trans (C↔D.strictlyInverseʳ _) (cong g (A↔B.strictlyInverseʳ x)))"},{"Add":" where module A↔B = Inverse A↔B; module C↔D = Inverse C↔D"}]}},{"FileAdd":{"meta":{"basename":"Properties","metadata":576},"context":[{"change":0,"pos":1018846}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Surjection.agda","metadata":0},"context":[{"change":0,"pos":1085825}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of surjections"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Properties.Surjection where"},{"Add":""},{"Add":"open import Function.Base using (_∘_; _$_)"},{"Add":"open import Function.Definitions using (Surjective; Injective; Congruent)"},{"Add":"open import Function.Bundles using (Func; Surjection; _↠_; _⟶_; _↪_; mk↪;"},{"Add":" _⇔_; mk⇔)"},{"Add":"import Function.Construct.Identity as Identity"},{"Add":"import Function.Construct.Composition as Compose"},{"Add":"open import Level using (Level)"},{"Add":"open import Data.Product.Base using (proj₁; proj₂)"},{"Add":"import Relation.Binary.PropositionalEquality.Core as ≡"},{"Add":"open import Relation.Binary.Definitions using (Reflexive; Trans)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"import Relation.Binary.Reasoning.Setoid as ≈-Reasoning"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level"},{"Add":" A B : Set a"},{"Add":" T S : Setoid a ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Constructors"},{"Add":""},{"Add":"mkSurjection : (f : Func S T) (open Func f) →"},{"Add":" Surjective Eq₁._≈_ Eq₂._≈_ to →"},{"Add":" Surjection S T"},{"Add":"mkSurjection f surjective = record"},{"Add":" { Func f"},{"Add":" ; surjective = surjective"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Conversion functions"},{"Add":""},{"Add":"↠⇒⟶ : A ↠ B → A ⟶ B"},{"Add":"↠⇒⟶ = Surjection.function"},{"Add":""},{"Add":"↠⇒↪ : A ↠ B → B ↪ A"},{"Add":"↠⇒↪ s = mk↪ {from = to} λ { ≡.refl → proj₂ (strictlySurjective _)}"},{"Add":" where open Surjection s"},{"Add":""},{"Add":"↠⇒⇔ : A ↠ B → A ⇔ B"},{"Add":"↠⇒⇔ s = mk⇔ to (proj₁ ∘ surjective)"},{"Add":" where open Surjection s"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Setoid properties"},{"Add":""},{"Add":"refl : Reflexive (Surjection {a} {ℓ})"},{"Add":"refl {x = x} = Identity.surjection x"},{"Add":""},{"Add":"trans : Trans (Surjection {a} {ℓ₁} {b} {ℓ₂})"},{"Add":" (Surjection {b} {ℓ₂} {c} {ℓ₃})"},{"Add":" (Surjection {a} {ℓ₁} {c} {ℓ₃})"},{"Add":"trans = Compose.surjection"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Other"},{"Add":""},{"Add":"injective⇒to⁻-cong : (surj : Surjection S T) →"},{"Add":" (open Surjection surj) →"},{"Add":" Injective Eq₁._≈_ Eq₂._≈_ to →"},{"Add":" Congruent Eq₂._≈_ Eq₁._≈_ to⁻"},{"Add":"injective⇒to⁻-cong {T = T} surj injective {x} {y} x≈y = injective $ begin"},{"Add":" to (to⁻ x) ≈⟨ to∘to⁻ x ⟩"},{"Add":" x ≈⟨ x≈y ⟩"},{"Add":" y ≈⟨ to∘to⁻ y ⟨"},{"Add":" to (to⁻ y) ∎"},{"Add":" where"},{"Add":" open ≈-Reasoning T"},{"Add":" open Surjection surj"},{"Add":""}]}},{"FileAdd":{"meta":{"basename":"RightInverse.agda","metadata":0},"context":[{"change":0,"pos":1085825}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of right inverses"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Properties.RightInverse where"},{"Add":""},{"Add":"open import Function.Base"},{"Add":"open import Function.Definitions"},{"Add":"open import Function.Bundles"},{"Add":"open import Function.Consequences using (inverseˡ⇒surjective)"},{"Add":"open import Level using (Level)"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ₁ ℓ₂ a b : Level"},{"Add":" A B : Set a"},{"Add":" S T : Setoid a ℓ₁"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Constructors"},{"Add":""},{"Add":"mkRightInverse : (e : Equivalence S T) (open Equivalence e) →"},{"Add":" Inverseʳ Eq₁._≈_ Eq₂._≈_ to from →"},{"Add":" RightInverse S T"},{"Add":"mkRightInverse eq invʳ = record"},{"Add":" { Equivalence eq"},{"Add":" ; inverseʳ = invʳ"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Conversion"},{"Add":""},{"Add":"RightInverse⇒LeftInverse : RightInverse S T → LeftInverse T S"},{"Add":"RightInverse⇒LeftInverse I = record"},{"Add":" { to = from"},{"Add":" ; from = to"},{"Add":" ; to-cong = from-cong"},{"Add":" ; from-cong = to-cong"},{"Add":" ; inverseˡ = inverseʳ"},{"Add":" } where open RightInverse I"},{"Add":""},{"Add":"LeftInverse⇒RightInverse : LeftInverse S T → RightInverse T S"},{"Add":"LeftInverse⇒RightInverse I = record"},{"Add":" { to = from"},{"Add":" ; from = to"},{"Add":" ; to-cong = from-cong"},{"Add":" ; from-cong = to-cong"},{"Add":" ; inverseʳ = inverseˡ"},{"Add":" } where open LeftInverse I"},{"Add":""},{"Add":"RightInverse⇒Surjection : RightInverse S T → Surjection T S"},{"Add":"RightInverse⇒Surjection I = record"},{"Add":" { to = from"},{"Add":" ; cong = from-cong"},{"Add":" ; surjective = inverseˡ⇒surjective Eq₁._≈_ inverseʳ"},{"Add":" } where open RightInverse I"},{"Add":""},{"Add":"↪⇒↠ : B ↪ A → A ↠ B"},{"Add":"↪⇒↠ = RightInverse⇒Surjection"},{"Add":""},{"Add":"↪⇒↩ : B ↪ A → A ↩ B"},{"Add":"↪⇒↩ = RightInverse⇒LeftInverse"},{"Add":""},{"Add":"↩⇒↪ : B ↩ A → A ↪ B"},{"Add":"↩⇒↪ = LeftInverse⇒RightInverse"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Other"},{"Add":""},{"Add":"module _ (R : RightInverse S T) where"},{"Add":" open RightInverse R"},{"Add":""},{"Add":" to-from : ∀ {x y} → to x Eq₂.≈ y → from y Eq₁.≈ x"},{"Add":" to-from eq = Eq₁.trans (from-cong (Eq₂.sym eq)) (strictlyInverseʳ _)"}]}},{"FileAdd":{"meta":{"basename":"Inverse.agda","metadata":0},"context":[{"change":0,"pos":1085825}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of inverses."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Properties.Inverse where"},{"Add":""},{"Add":"open import Axiom.Extensionality.Propositional using (Extensionality)"},{"Add":"open import Data.Product.Base using (_,_; proj₁; proj₂)"},{"Add":"open import Function.Bundles"},{"Add":"import Function.Properties.RightInverse as RightInverse"},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Relation.Binary.Core using (REL)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":"import Relation.Binary.Reasoning.Setoid as ≈-Reasoning"},{"Add":"import Function.Consequences.Setoid as Consequences"},{"Add":""},{"Add":"import Function.Construct.Identity as Identity"},{"Add":"import Function.Construct.Symmetry as Symmetry"},{"Add":"import Function.Construct.Composition as Composition"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b ℓ ℓ₁ ℓ₂ : Level"},{"Add":" A B C D : Set a"},{"Add":" S T U V : Setoid a ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Setoid bundles"},{"Add":""},{"Add":"open Identity public using () renaming (inverse to refl)"},{"Add":"open Symmetry public using () renaming (inverse to sym)"},{"Add":"open Composition public using () renaming (inverse to trans)"},{"Add":""},{"Add":"isEquivalence : IsEquivalence (Inverse {a} {b})"},{"Add":"isEquivalence = record"},{"Add":" { refl = λ {x} → Identity.inverse x"},{"Add":" ; sym = sym"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Propositional bundles"},{"Add":""},{"Add":"↔-refl : A ↔ A"},{"Add":"↔-refl = Identity.↔-id _"},{"Add":""},{"Add":"↔-sym : A ↔ B → B ↔ A"},{"Add":"↔-sym = Symmetry.↔-sym"},{"Add":""},{"Add":"↔-trans : A ↔ B → B ↔ C → A ↔ C"},{"Add":"↔-trans = Composition.inverse"},{"Add":""},{"Add":"-- need to η-expand for everything to line up properly"},{"Add":"↔-isEquivalence : IsEquivalence {ℓ = ℓ} _↔_"},{"Add":"↔-isEquivalence = record"},{"Add":" { refl = ↔-refl"},{"Add":" ; sym = ↔-sym"},{"Add":" ; trans = ↔-trans"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Conversion functions"},{"Add":""},{"Add":"toFunction : Inverse S T → Func S T"},{"Add":"toFunction I = record { to = to ; cong = to-cong }"},{"Add":" where open Inverse I"},{"Add":""},{"Add":"fromFunction : Inverse S T → Func T S"},{"Add":"fromFunction I = record { to = from ; cong = from-cong }"},{"Add":" where open Inverse I"},{"Add":""},{"Add":"Inverse⇒Injection : Inverse S T → Injection S T"},{"Add":"Inverse⇒Injection {S = S} {T = T} I = record"},{"Add":" { to = to"},{"Add":" ; cong = to-cong"},{"Add":" ; injective = inverseʳ⇒injective to inverseʳ"},{"Add":" } where open Inverse I; open Consequences S T"},{"Add":""},{"Add":"Inverse⇒Surjection : Inverse S T → Surjection S T"},{"Add":"Inverse⇒Surjection {S = S} {T = T} I = record"},{"Add":" { to = to"},{"Add":" ; cong = to-cong"},{"Add":" ; surjective = inverseˡ⇒surjective inverseˡ"},{"Add":" } where open Inverse I; open Consequences S T"},{"Add":""},{"Add":"Inverse⇒Bijection : Inverse S T → Bijection S T"},{"Add":"Inverse⇒Bijection {S = S} {T = T} I = record"},{"Add":" { to = to"},{"Add":" ; cong = to-cong"},{"Add":" ; bijective = inverseᵇ⇒bijective inverse"},{"Add":" } where open Inverse I; open Consequences S T"},{"Add":""},{"Add":"Inverse⇒Equivalence : Inverse S T → Equivalence S T"},{"Add":"Inverse⇒Equivalence I = record"},{"Add":" { to = to"},{"Add":" ; from = from"},{"Add":" ; to-cong = to-cong"},{"Add":" ; from-cong = from-cong"},{"Add":" } where open Inverse I"},{"Add":""},{"Add":"↔⇒⟶ : A ↔ B → A ⟶ B"},{"Add":"↔⇒⟶ = toFunction"},{"Add":""},{"Add":"↔⇒⟵ : A ↔ B → B ⟶ A"},{"Add":"↔⇒⟵ = fromFunction"},{"Add":""},{"Add":"↔⇒↣ : A ↔ B → A ↣ B"},{"Add":"↔⇒↣ = Inverse⇒Injection"},{"Add":""},{"Add":"↔⇒↠ : A ↔ B → A ↠ B"},{"Add":"↔⇒↠ = Inverse⇒Surjection"},{"Add":""},{"Add":"↔⇒⤖ : A ↔ B → A ⤖ B"},{"Add":"↔⇒⤖ = Inverse⇒Bijection"},{"Add":""},{"Add":"↔⇒⇔ : A ↔ B → A ⇔ B"},{"Add":"↔⇒⇔ = Inverse⇒Equivalence"},{"Add":""},{"Add":"↔⇒↩ : A ↔ B → A ↩ B"},{"Add":"↔⇒↩ = Inverse.leftInverse"},{"Add":""},{"Add":"↔⇒↪ : A ↔ B → A ↪ B"},{"Add":"↔⇒↪ = Inverse.rightInverse"},{"Add":""},{"Add":"-- The functions above can be combined with the following lemma to"},{"Add":"-- transport an arbitrary relation R (e.g. Injection) across"},{"Add":"-- inverses."},{"Add":"transportVia : {R : ∀ {a b ℓ₁ ℓ₂} → REL (Setoid a ℓ₁) (Setoid b ℓ₂) (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)} →"},{"Add":" (∀ {a b c ℓ₁ ℓ₂ ℓ₃} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} {U : Setoid c ℓ₃} → R S T → R T U → R S U) →"},{"Add":" (∀ {a b ℓ₁ ℓ₂} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} → Inverse S T → R S T) →"},{"Add":" Inverse S T → R T U → Inverse U V → R S V"},{"Add":"transportVia R-trans inv⇒R IBA RBC ICD ="},{"Add":" R-trans (inv⇒R IBA) (R-trans RBC (inv⇒R ICD))"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Other"},{"Add":""},{"Add":"module _ (ext : ∀ {a b} → Extensionality a b) where"},{"Add":""},{"Add":" ↔-fun : A ↔ B → C ↔ D → (A → C) ↔ (B → D)"},{"Add":" ↔-fun A↔B C↔D = mk↔ₛ′"},{"Add":" (λ a→c b → to C↔D (a→c (from A↔B b)))"},{"Add":" (λ b→d a → from C↔D (b→d (to A↔B a)))"},{"Add":" (λ b→d → ext λ _ → ≡.trans (strictlyInverseˡ C↔D _ ) (≡.cong b→d (strictlyInverseˡ A↔B _)))"},{"Add":" (λ a→c → ext λ _ → ≡.trans (strictlyInverseʳ C↔D _ ) (≡.cong a→c (strictlyInverseʳ A↔B _)))"},{"Add":" where open Inverse"},{"Add":""},{"Add":"module _ (I : Inverse S T) where"},{"Add":" open Inverse I"},{"Add":""},{"Add":" to-from : ∀ {x y} → to x Eq₂.≈ y → from y Eq₁.≈ x"},{"Add":" to-from = RightInverse.to-from rightInverse"}]}},{"FileAdd":{"meta":{"basename":"Inverse","metadata":576},"context":[{"change":0,"pos":1085825}],"contents":[]}},{"FileAdd":{"meta":{"basename":"HalfAdjointEquivalence.agda","metadata":0},"context":[{"change":0,"pos":1096372}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Half adjoint equivalences"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Properties.Inverse.HalfAdjointEquivalence where"},{"Add":""},{"Add":"open import Function.Base using (id; _∘_)"},{"Add":"open import Function.Bundles using (Inverse; _↔_; mk↔ₛ′)"},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Relation.Binary.PropositionalEquality"},{"Add":" using (_≡_; refl; cong; sym; trans; trans-reflʳ; cong-≡id; cong-∘; naturality;"},{"Add":" cong-id; trans-assoc; trans-symˡ; module ≡-Reasoning)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b : Level"},{"Add":" A B : Set a"},{"Add":""},{"Add":"-- Half adjoint equivalences (see the HoTT book)."},{"Add":"--"},{"Add":"-- They are inverses with an extra coherence condition that the left"},{"Add":"-- and right inversion proofs interact the right way with `cong`."},{"Add":""},{"Add":"infix 4 _≃_"},{"Add":""},{"Add":"record _≃_ (A : Set a) (B : Set b) : Set (a ⊔ b) where"},{"Add":" field"},{"Add":" to : A → B"},{"Add":" from : B → A"},{"Add":" left-inverse-of : ∀ x → from (to x) ≡ x"},{"Add":" right-inverse-of : ∀ x → to (from x) ≡ x"},{"Add":" left-right : ∀ x → cong to (left-inverse-of x) ≡ right-inverse-of (to x)"},{"Add":""},{"Add":" -- The forward direction of a half adjoint equivalence is injective."},{"Add":""},{"Add":" injective : ∀ {x y} → to x ≡ to y → x ≡ y"},{"Add":" injective {x} {y} to-x≡to-y ="},{"Add":" x ≡⟨ sym (left-inverse-of _) ⟩"},{"Add":" from (to x) ≡⟨ cong from to-x≡to-y ⟩"},{"Add":" from (to y) ≡⟨ left-inverse-of _ ⟩"},{"Add":" y ∎"},{"Add":" where open ≡-Reasoning"},{"Add":""},{"Add":"-- Half adjoint equivalences can be turned into inverses."},{"Add":""},{"Add":"≃⇒↔ : A ≃ B → A ↔ B"},{"Add":"≃⇒↔ A≃B = mk↔ₛ′ to from right-inverse-of left-inverse-of"},{"Add":" where open _≃_ A≃B"},{"Add":""},{"Add":"-- Inverses can be turned into half adjoint equivalences."},{"Add":"--"},{"Add":"-- (This proof is based on one in the HoTT book.)"},{"Add":""},{"Add":"↔⇒≃ : A ↔ B → A ≃ B"},{"Add":"↔⇒≃ A↔B = record"},{"Add":" { to = to"},{"Add":" ; from = from"},{"Add":" ; left-inverse-of = strictlyInverseʳ"},{"Add":" ; right-inverse-of = right-inverse-of"},{"Add":" ; left-right = left-right"},{"Add":" }"},{"Add":" where"},{"Add":" open ≡-Reasoning"},{"Add":" open module A↔B = Inverse A↔B"},{"Add":""},{"Add":" right-inverse-of : ∀ x → to (from x) ≡ x"},{"Add":" right-inverse-of x ="},{"Add":" to (from x) ≡⟨ sym (A↔B.strictlyInverseˡ _) ⟩"},{"Add":" to (from (to (from x))) ≡⟨ cong to (strictlyInverseʳ _) ⟩"},{"Add":" to (from x) ≡⟨ A↔B.strictlyInverseˡ _ ⟩"},{"Add":" x ∎"},{"Add":""},{"Add":" left-right :"},{"Add":" ∀ x →"},{"Add":" cong to (strictlyInverseʳ x) ≡ right-inverse-of (to x)"},{"Add":" left-right x ="},{"Add":" cong to (strictlyInverseʳ x) ≡⟨⟩"},{"Add":""},{"Add":" trans refl (cong to (strictlyInverseʳ _)) ≡⟨ cong (λ p → trans p (cong to (strictlyInverseʳ _)))"},{"Add":" (sym (trans-symˡ (A↔B.strictlyInverseˡ _))) ⟩"},{"Add":" trans (trans (sym (A↔B.strictlyInverseˡ _))"},{"Add":" (A↔B.strictlyInverseˡ _))"},{"Add":" (cong to (strictlyInverseʳ _)) ≡⟨ trans-assoc (sym (A↔B.strictlyInverseˡ _)) ⟩"},{"Add":""},{"Add":" trans (sym (A↔B.strictlyInverseˡ _))"},{"Add":" (trans (A↔B.strictlyInverseˡ _)"},{"Add":" (cong to (strictlyInverseʳ _))) ≡⟨ cong (trans (sym (A↔B.strictlyInverseˡ _))) lemma ⟩"},{"Add":""},{"Add":" trans (sym (A↔B.strictlyInverseˡ _))"},{"Add":" (trans (cong to (strictlyInverseʳ _))"},{"Add":" (trans (A↔B.strictlyInverseˡ _) refl)) ≡⟨⟩"},{"Add":""},{"Add":" right-inverse-of (to x) ∎"},{"Add":" where"},{"Add":" lemma ="},{"Add":" trans (A↔B.strictlyInverseˡ _)"},{"Add":" (cong to (strictlyInverseʳ _)) ≡⟨ cong (trans (A↔B.strictlyInverseˡ _)) (sym (cong-id _)) ⟩"},{"Add":""},{"Add":" trans (A↔B.strictlyInverseˡ _)"},{"Add":" (cong id (cong to (strictlyInverseʳ _))) ≡⟨ sym (naturality A↔B.strictlyInverseˡ) ⟩"},{"Add":""},{"Add":" trans (cong (to ∘ from)"},{"Add":" (cong to (strictlyInverseʳ _)))"},{"Add":" (A↔B.strictlyInverseˡ _) ≡⟨ cong (λ p → trans p (A↔B.strictlyInverseˡ _))"},{"Add":" (sym (cong-∘ _)) ⟩"},{"Add":" trans (cong (to ∘ from ∘ to)"},{"Add":" (strictlyInverseʳ _))"},{"Add":" (A↔B.strictlyInverseˡ _) ≡⟨ cong (λ p → trans p (A↔B.strictlyInverseˡ _))"},{"Add":" (cong-∘ _) ⟩"},{"Add":" trans (cong to"},{"Add":" (cong (from ∘ to)"},{"Add":" (strictlyInverseʳ _)))"},{"Add":" (A↔B.strictlyInverseˡ _) ≡⟨ cong (λ p → trans (cong to p) (strictlyInverseˡ (to x)))"},{"Add":" (cong-≡id strictlyInverseʳ) ⟩"},{"Add":" trans (cong to (strictlyInverseʳ _))"},{"Add":" (A↔B.strictlyInverseˡ _) ≡⟨ cong (trans (cong to (strictlyInverseʳ _)))"},{"Add":" (sym (trans-reflʳ _)) ⟩"},{"Add":" trans (cong to (strictlyInverseʳ _))"},{"Add":" (trans (A↔B.strictlyInverseˡ _) refl) ∎"}]}},{"FileAdd":{"meta":{"basename":"Injection.agda","metadata":0},"context":[{"change":0,"pos":1085825}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties for injections"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Properties.Injection where"},{"Add":""},{"Add":"open import Function.Base"},{"Add":"open import Function.Definitions"},{"Add":"open import Function.Bundles"},{"Add":"import Function.Construct.Identity as Identity"},{"Add":"import Function.Construct.Composition as Compose"},{"Add":"open import Level using (Level)"},{"Add":"open import Data.Product.Base using (proj₁; proj₂)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":"open import Relation.Binary using (Setoid)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level"},{"Add":" A B : Set a"},{"Add":" T S U : Setoid a ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Constructors"},{"Add":""},{"Add":"mkInjection : (f : Func S T) (open Func f) →"},{"Add":" Injective Eq₁._≈_ Eq₂._≈_ to →"},{"Add":" Injection S T"},{"Add":"mkInjection f injective = record"},{"Add":" { Func f"},{"Add":" ; injective = injective"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Conversion functions"},{"Add":""},{"Add":"↣⇒⟶ : A ↣ B → A ⟶ B"},{"Add":"↣⇒⟶ = Injection.function"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Setoid properties"},{"Add":""},{"Add":"refl : Reflexive (Injection {a} {ℓ})"},{"Add":"refl {x = x} = Identity.injection x"},{"Add":""},{"Add":"trans : Trans (Injection {a} {ℓ₁} {b} {ℓ₂})"},{"Add":" (Injection {b} {ℓ₂} {c} {ℓ₃})"},{"Add":" (Injection {a} {ℓ₁} {c} {ℓ₃})"},{"Add":"trans = Compose.injection"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Propositonal properties"},{"Add":""},{"Add":"↣-refl : Injection S S"},{"Add":"↣-refl = refl"},{"Add":""},{"Add":"↣-trans : Injection S T → Injection T U → Injection S U"},{"Add":"↣-trans = trans"}]}},{"FileAdd":{"meta":{"basename":"Equivalence.agda","metadata":0},"context":[{"change":0,"pos":1085825}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Some basic properties of equivalences. This file is designed to be"},{"Add":"-- imported qualified."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Properties.Equivalence where"},{"Add":""},{"Add":"open import Function.Bundles"},{"Add":"open import Level"},{"Add":"open import Relation.Binary.Definitions"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence)"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":""},{"Add":"import Function.Construct.Identity as Identity"},{"Add":"import Function.Construct.Symmetry as Symmetry"},{"Add":"import Function.Construct.Composition as Composition"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level"},{"Add":" A B : Set a"},{"Add":" S T : Setoid a ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Constructors"},{"Add":""},{"Add":"mkEquivalence : Func S T → Func T S → Equivalence S T"},{"Add":"mkEquivalence f g = record"},{"Add":" { to = to f"},{"Add":" ; from = to g"},{"Add":" ; to-cong = cong f"},{"Add":" ; from-cong = cong g"},{"Add":" } where open Func"},{"Add":""},{"Add":"⟶×⟵⇒⇔ : A ⟶ B → B ⟶ A → A ⇔ B"},{"Add":"⟶×⟵⇒⇔ = mkEquivalence"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Destructors"},{"Add":""},{"Add":"⇔⇒⟶ : A ⇔ B → A ⟶ B"},{"Add":"⇔⇒⟶ = Equivalence.toFunction"},{"Add":""},{"Add":"⇔⇒⟵ : A ⇔ B → B ⟶ A"},{"Add":"⇔⇒⟵ = Equivalence.fromFunction"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Setoid bundles"},{"Add":""},{"Add":"refl : Reflexive (Equivalence {a} {ℓ})"},{"Add":"refl {x = x} = Identity.equivalence x"},{"Add":""},{"Add":"sym : Sym (Equivalence {a} {ℓ₁} {b} {ℓ₂})"},{"Add":" (Equivalence {b} {ℓ₂} {a} {ℓ₁})"},{"Add":"sym = Symmetry.equivalence"},{"Add":""},{"Add":"trans : Trans (Equivalence {a} {ℓ₁} {b} {ℓ₂})"},{"Add":" (Equivalence {b} {ℓ₂} {c} {ℓ₃})"},{"Add":" (Equivalence {a} {ℓ₁} {c} {ℓ₃})"},{"Add":"trans = Composition.equivalence"},{"Add":""},{"Add":"isEquivalence : IsEquivalence (Equivalence {a} {ℓ})"},{"Add":"isEquivalence = record"},{"Add":" { refl = refl"},{"Add":" ; sym = sym"},{"Add":" ; trans = Composition.equivalence"},{"Add":" }"},{"Add":""},{"Add":"setoid : (s₁ s₂ : Level) → Setoid (suc (s₁ ⊔ s₂)) (s₁ ⊔ s₂)"},{"Add":"setoid s₁ s₂ = record"},{"Add":" { Carrier = Setoid s₁ s₂"},{"Add":" ; _≈_ = Equivalence"},{"Add":" ; isEquivalence = isEquivalence"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Propositional bundles"},{"Add":""},{"Add":"⇔-isEquivalence : IsEquivalence {ℓ = ℓ} _⇔_"},{"Add":"⇔-isEquivalence = record"},{"Add":" { refl = λ {x} → Identity.equivalence (≡.setoid x)"},{"Add":" ; sym = Symmetry.equivalence"},{"Add":" ; trans = Composition.equivalence"},{"Add":" }"},{"Add":""},{"Add":"⇔-setoid : (ℓ : Level) → Setoid (suc ℓ) ℓ"},{"Add":"⇔-setoid ℓ = record"},{"Add":" { Carrier = Set ℓ"},{"Add":" ; _≈_ = _⇔_"},{"Add":" ; isEquivalence = ⇔-isEquivalence"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Bijection.agda","metadata":0},"context":[{"change":0,"pos":1085825}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Some basic properties of bijections."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Properties.Bijection where"},{"Add":""},{"Add":"open import Function.Bundles using (Bijection; Inverse; Equivalence;"},{"Add":" _⤖_; _↔_; _⇔_)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence)"},{"Add":"open import Relation.Binary.Definitions using (Reflexive; Trans)"},{"Add":"open import Relation.Binary.PropositionalEquality.Properties using (setoid)"},{"Add":"open import Data.Product.Base using (_,_; proj₁; proj₂)"},{"Add":"open import Function.Base using (_∘_)"},{"Add":"open import Function.Properties.Surjection using (injective⇒to⁻-cong)"},{"Add":"open import Function.Properties.Inverse using (Inverse⇒Equivalence)"},{"Add":""},{"Add":"import Function.Construct.Identity as Identity"},{"Add":"import Function.Construct.Symmetry as Symmetry"},{"Add":"import Function.Construct.Composition as Composition"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level"},{"Add":" A B : Set a"},{"Add":" T S : Setoid a ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Setoid properties"},{"Add":""},{"Add":"refl : Reflexive (Bijection {a} {ℓ})"},{"Add":"refl = Identity.bijection _"},{"Add":""},{"Add":"-- Can't prove full symmetry as we have no proof that the witness"},{"Add":"-- produced by the surjection proof preserves equality"},{"Add":"sym-≡ : Bijection S (setoid B) → Bijection (setoid B) S"},{"Add":"sym-≡ = Symmetry.bijection-≡"},{"Add":""},{"Add":"trans : Trans (Bijection {a} {ℓ₁} {b} {ℓ₂}) (Bijection {b} {ℓ₂} {c} {ℓ₃}) Bijection"},{"Add":"trans = Composition.bijection"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Propositional properties"},{"Add":""},{"Add":"⤖-isEquivalence : IsEquivalence {ℓ = ℓ} _⤖_"},{"Add":"⤖-isEquivalence = record"},{"Add":" { refl = refl"},{"Add":" ; sym = sym-≡"},{"Add":" ; trans = trans"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Conversion functions"},{"Add":""},{"Add":"Bijection⇒Inverse : Bijection S T → Inverse S T"},{"Add":"Bijection⇒Inverse bij = record"},{"Add":" { to = to"},{"Add":" ; from = to⁻"},{"Add":" ; to-cong = cong"},{"Add":" ; from-cong = injective⇒to⁻-cong surjection injective"},{"Add":" ; inverse = (λ y≈to⁻[x] → Eq₂.trans (cong y≈to⁻[x]) (to∘to⁻ _)) ,"},{"Add":" (λ y≈to[x] → injective (Eq₂.trans (to∘to⁻ _) y≈to[x]))"},{"Add":" }"},{"Add":" where open Bijection bij; to∘to⁻ = proj₂ ∘ strictlySurjective"},{"Add":""},{"Add":"Bijection⇒Equivalence : Bijection T S → Equivalence T S"},{"Add":"Bijection⇒Equivalence = Inverse⇒Equivalence ∘ Bijection⇒Inverse"},{"Add":""},{"Add":"⤖⇒↔ : A ⤖ B → A ↔ B"},{"Add":"⤖⇒↔ = Bijection⇒Inverse"},{"Add":""},{"Add":"⤖⇒⇔ : A ⤖ B → A ⇔ B"},{"Add":"⤖⇒⇔ = Bijection⇒Equivalence"}]}},{"FileAdd":{"meta":{"basename":"Nary","metadata":576},"context":[{"change":0,"pos":1018846}],"contents":[]}},{"FileAdd":{"meta":{"basename":"NonDependent.agda","metadata":0},"context":[{"change":0,"pos":1109005}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Heterogeneous N-ary Functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Nary.NonDependent where"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Concrete examples can be found in README.Nary. This file's comments"},{"Add":"-- are more focused on the implementation details and the motivations"},{"Add":"-- behind the design decisions."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"open import Level using (Level; 0ℓ; _⊔_; Lift)"},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc)"},{"Add":"open import Data.Fin.Base using (Fin; zero; suc)"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Data.Product.Nary.NonDependent"},{"Add":" using (Product; uncurryₙ; Equalₙ; curryₙ; fromEqualₙ; toEqualₙ)"},{"Add":"open import Function.Base using (_∘′_; _$′_; const; flip)"},{"Add":"open import Relation.Unary using (IUniversal)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core"},{"Add":" using (_≡_; cong)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b r : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-exporting the basic operations"},{"Add":""},{"Add":"open import Function.Nary.NonDependent.Base public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Additional operations on Levels"},{"Add":""},{"Add":"ltabulate : ∀ n → (Fin n → Level) → Levels n"},{"Add":"ltabulate zero f = _"},{"Add":"ltabulate (suc n) f = f zero , ltabulate n (f ∘′ suc)"},{"Add":""},{"Add":"lreplicate : ∀ n → Level → Levels n"},{"Add":"lreplicate n ℓ = ltabulate n (const ℓ)"},{"Add":""},{"Add":"0ℓs : ∀[ Levels ]"},{"Add":"0ℓs = lreplicate _ 0ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Congruence"},{"Add":""},{"Add":"module _ n {ls} {as : Sets n ls} {R : Set r} (f : as ⇉ R) where"},{"Add":""},{"Add":" private"},{"Add":" g : Product n as → R"},{"Add":" g = uncurryₙ n f"},{"Add":""},{"Add":"-- Congruentₙ : ∀ n. ∀ a₁₁ a₁₂ ⋯ aₙ₁ aₙ₂ →"},{"Add":"-- a₁₁ ≡ a₁₂ → ⋯ → aₙ₁ ≡ aₙ₂ →"},{"Add":"-- f a₁₁ ⋯ aₙ₁ ≡ f a₁₂ ⋯ aₙ₂"},{"Add":""},{"Add":" Congruentₙ : Set (r Level.⊔ ⨆ n ls)"},{"Add":" Congruentₙ = ∀ {l r} → Equalₙ n l r ⇉ (g l ≡ g r)"},{"Add":""},{"Add":" congₙ : Congruentₙ"},{"Add":" congₙ = curryₙ n (cong g ∘′ fromEqualₙ n)"},{"Add":""},{"Add":"-- Congruence at a specific location"},{"Add":""},{"Add":"module _ m n {ls ls′} {as : Sets m ls} {bs : Sets n ls′}"},{"Add":" (f : as ⇉ (A → bs ⇉ B)) where"},{"Add":""},{"Add":" private"},{"Add":" g : Product m as → A → Product n bs → B"},{"Add":" g vs a ws = uncurryₙ n (uncurryₙ m f vs a) ws"},{"Add":""},{"Add":" congAt : ∀ {vs ws a₁ a₂} → a₁ ≡ a₂ → g vs a₁ ws ≡ g vs a₂ ws"},{"Add":" congAt {vs} {ws} = cong (λ a → g vs a ws)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Injectivity"},{"Add":""},{"Add":"module _ n {ls} {as : Sets n ls} {R : Set r} (con : as ⇉ R) where"},{"Add":""},{"Add":"-- Injectiveₙ : ∀ n. ∀ a₁₁ a₁₂ ⋯ aₙ₁ aₙ₂ →"},{"Add":"-- con a₁₁ ⋯ aₙ₁ ≡ con a₁₂ ⋯ aₙ₂ →"},{"Add":"-- a₁₁ ≡ a₁₂ × ⋯ × aₙ₁ ≡ aₙ₂"},{"Add":""},{"Add":" private"},{"Add":" c : Product n as → R"},{"Add":" c = uncurryₙ n con"},{"Add":""},{"Add":" Injectiveₙ : Set (r Level.⊔ ⨆ n ls)"},{"Add":" Injectiveₙ = ∀ {l r} → c l ≡ c r → Product n (Equalₙ n l r)"},{"Add":""},{"Add":" injectiveₙ : (∀ {l r} → c l ≡ c r → l ≡ r) → Injectiveₙ"},{"Add":" injectiveₙ con-inj eq = toEqualₙ n (con-inj eq)"}]}},{"FileAdd":{"meta":{"basename":"NonDependent","metadata":576},"context":[{"change":0,"pos":1109005}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Base.agda","metadata":0},"context":[{"change":0,"pos":1112499}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Heterogeneous N-ary Functions: basic types and operations"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Nary.NonDependent.Base where"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Concrete examples can be found in README.Nary. This file's comments"},{"Add":"-- are more focused on the implementation details and the motivations"},{"Add":"-- behind the design decisions."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"open import Level using (Level; 0ℓ; _⊔_)"},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc)"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Data.Unit.Polymorphic.Base"},{"Add":"open import Function.Base using (_∘′_; _$′_; const; flip)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":" C : Set c"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Type Definitions"},{"Add":""},{"Add":"-- We want to define n-ary function spaces and generic n-ary operations"},{"Add":"-- on them such as (un)currying, zipWith, alignWith, etc."},{"Add":""},{"Add":"-- We want users to be able to use these seamlessly whenever n is concrete."},{"Add":""},{"Add":"-- In other words, we want Agda to infer the sets `A₁, ⋯, Aₙ` when we"},{"Add":"-- write `uncurryₙ n f` where `f` has type `A₁ → ⋯ → Aₙ → B`. For this"},{"Add":"-- to happen, we need the structure in which these Sets are stored to"},{"Add":"-- effectively η-expand to `A₁, ⋯, Aₙ` when the parameter `n` is known."},{"Add":""},{"Add":"-- Hence the following definitions:"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- First, a \"vector\" of `n` Levels (defined by induction on n so that it"},{"Add":"-- may be built by η-expansion and unification). Each Level will be that"},{"Add":"-- of one of the Sets we want to take the n-ary product of."},{"Add":""},{"Add":"Levels : ℕ → Set"},{"Add":"Levels zero = ⊤"},{"Add":"Levels (suc n) = Level × Levels n"},{"Add":""},{"Add":"-- The overall Level of a `n` Sets of respective levels `l₁, ⋯, lₙ` will"},{"Add":"-- be the least upper bound `l₁ ⊔ ⋯ ⊔ lₙ` of all of the Levels involved."},{"Add":"-- Hence the following definition of n-ary least upper bound:"},{"Add":""},{"Add":"⨆ : ∀ n → Levels n → Level"},{"Add":"⨆ zero _ = Level.zero"},{"Add":"⨆ (suc n) (l , ls) = l ⊔ (⨆ n ls)"},{"Add":""},{"Add":"-- Second, a \"vector\" of `n` Sets whose respective Levels are determined"},{"Add":"-- by the `Levels n` input."},{"Add":""},{"Add":"Sets : ∀ n (ls : Levels n) → Set (Level.suc (⨆ n ls))"},{"Add":"Sets zero _ = ⊤"},{"Add":"Sets (suc n) (l , ls) = Set l × Sets n ls"},{"Add":""},{"Add":"-- Third, a function type whose domains are given by a \"vector\" of `n`"},{"Add":"-- Sets `A₁, ⋯, Aₙ` and whose codomain is `B`. `Arrows` forms such a"},{"Add":"-- type of shape `A₁ → ⋯ → Aₙ → B` by induction on `n`."},{"Add":""},{"Add":"Arrows : ∀ n {r ls} → Sets n ls → Set r → Set (r ⊔ (⨆ n ls))"},{"Add":"Arrows zero _ b = b"},{"Add":"Arrows (suc n) (a , as) b = a → Arrows n as b"},{"Add":""},{"Add":"-- We introduce a notation for this definition"},{"Add":""},{"Add":"infixr 0 _⇉_"},{"Add":"_⇉_ : ∀ {n ls r} → Sets n ls → Set r → Set (r ⊔ (⨆ n ls))"},{"Add":"_⇉_ = Arrows _"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Operations on Sets"},{"Add":""},{"Add":"-- Level-respecting map"},{"Add":""},{"Add":"infixr -1 _<$>_"},{"Add":""},{"Add":"_<$>_ : (∀ {l} → Set l → Set l) →"},{"Add":" ∀ {n ls} → Sets n ls → Sets n ls"},{"Add":"_<$>_ f {zero} as = _"},{"Add":"_<$>_ f {suc n} (a , as) = f a , (f <$> as)"},{"Add":""},{"Add":"-- Level-modifying generalised map"},{"Add":""},{"Add":"lmap : (Level → Level) → ∀ n → Levels n → Levels n"},{"Add":"lmap f zero ls = _"},{"Add":"lmap f (suc n) (l , ls) = f l , lmap f n ls"},{"Add":""},{"Add":"smap : ∀ f → (∀ {l} → Set l → Set (f l)) →"},{"Add":" ∀ n {ls} → Sets n ls → Sets n (lmap f n ls)"},{"Add":"smap f F zero as = _"},{"Add":"smap f F (suc n) (a , as) = F a , smap f F n as"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Operations on Functions"},{"Add":""},{"Add":"-- mapping under n arguments"},{"Add":""},{"Add":"mapₙ : ∀ n {ls} {as : Sets n ls} → (B → C) → as ⇉ B → as ⇉ C"},{"Add":"mapₙ zero f v = f v"},{"Add":"mapₙ (suc n) f g = mapₙ n f ∘′ g"},{"Add":""},{"Add":"-- compose function at the n-th position"},{"Add":""},{"Add":"infix 1 _%=_⊢_"},{"Add":""},{"Add":"_%=_⊢_ : ∀ n {ls} {as : Sets n ls} → (A → B) → as ⇉ (B → C) → as ⇉ (A → C)"},{"Add":"n %= f ⊢ g = mapₙ n (_∘′ f) g"},{"Add":""},{"Add":"-- partially apply function at the n-th position"},{"Add":""},{"Add":"infix 1 _∷=_⊢_"},{"Add":""},{"Add":"_∷=_⊢_ : ∀ n {ls} {as : Sets n ls} → A → as ⇉ (A → B) → as ⇉ B"},{"Add":"n ∷= x ⊢ g = mapₙ n (_$′ x) g"},{"Add":""},{"Add":"-- hole at the n-th position"},{"Add":""},{"Add":"holeₙ : ∀ n {ls} {as : Sets n ls} → (A → as ⇉ B) → as ⇉ (A → B)"},{"Add":"holeₙ zero f = f"},{"Add":"holeₙ (suc n) f = holeₙ n ∘′ flip f"},{"Add":""},{"Add":"-- function constant in its n first arguments"},{"Add":""},{"Add":"-- Note that its type will only be inferred if it is used in a context"},{"Add":"-- specifying what the type of the function ought to be. Just like the"},{"Add":"-- usual const: there is no way to infer its domain from its argument."},{"Add":""},{"Add":"constₙ : ∀ n {ls r} {as : Sets n ls} {b : Set r} → b → as ⇉ b"},{"Add":"constₙ zero v = v"},{"Add":"constₙ (suc n) v = const (constₙ n v)"}]}},{"FileAdd":{"meta":{"basename":"Metric.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Metrics with arbitrary domains and codomains"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Metric where"},{"Add":""},{"Add":"open import Function.Metric.Core public"},{"Add":"open import Function.Metric.Definitions public"},{"Add":"open import Function.Metric.Structures public"},{"Add":"open import Function.Metric.Bundles public"}]}},{"FileAdd":{"meta":{"basename":"Metric","metadata":576},"context":[{"change":0,"pos":1018846}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Structures.agda","metadata":0},"context":[{"change":0,"pos":1118054}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Some metric structures (not packed up with sets, operations, etc.)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should usually be accessed via"},{"Add":"-- `Function.Metric`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Structures using (IsPartialOrder; IsEquivalence)"},{"Add":""},{"Add":"module Function.Metric.Structures"},{"Add":" {a i ℓ₁ ℓ₂ ℓ₃} {A : Set a} {I : Set i}"},{"Add":" (_≈ₐ_ : Rel A ℓ₁) (_≈ᵢ_ : Rel I ℓ₂) (_≤_ : Rel I ℓ₃) (0# : I) where"},{"Add":""},{"Add":"open import Algebra.Core using (Op₂)"},{"Add":"open import Function.Metric.Core"},{"Add":"open import Function.Metric.Definitions"},{"Add":"open import Level using (_⊔_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Proto-metrics"},{"Add":""},{"Add":"-- We do not insist that the ordering relation is total as otherwise"},{"Add":"-- we would exclude the real numbers."},{"Add":""},{"Add":"record IsProtoMetric (d : DistanceFunction A I)"},{"Add":" : Set (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where"},{"Add":" field"},{"Add":" isPartialOrder : IsPartialOrder _≈ᵢ_ _≤_"},{"Add":" ≈-isEquivalence : IsEquivalence _≈ₐ_"},{"Add":" cong : Congruent _≈ₐ_ _≈ᵢ_ d"},{"Add":" nonNegative : NonNegative _≤_ d 0#"},{"Add":""},{"Add":" open IsPartialOrder isPartialOrder public"},{"Add":" renaming (module Eq to EqI)"},{"Add":""},{"Add":" module EqC = IsEquivalence ≈-isEquivalence"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Pre-metrics"},{"Add":""},{"Add":"record IsPreMetric (d : DistanceFunction A I)"},{"Add":" : Set (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where"},{"Add":" field"},{"Add":" isProtoMetric : IsProtoMetric d"},{"Add":" ≈⇒0 : Definite _≈ₐ_ _≈ᵢ_ d 0#"},{"Add":""},{"Add":" open IsProtoMetric isProtoMetric public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Quasi-semi-metrics"},{"Add":""},{"Add":"record IsQuasiSemiMetric (d : DistanceFunction A I)"},{"Add":" : Set (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where"},{"Add":" field"},{"Add":" isPreMetric : IsPreMetric d"},{"Add":" 0⇒≈ : Indiscernable _≈ₐ_ _≈ᵢ_ d 0#"},{"Add":""},{"Add":" open IsPreMetric isPreMetric public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Semi-metrics"},{"Add":""},{"Add":"record IsSemiMetric (d : DistanceFunction A I)"},{"Add":" : Set (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where"},{"Add":" field"},{"Add":" isQuasiSemiMetric : IsQuasiSemiMetric d"},{"Add":" sym : Symmetric _≈ᵢ_ d"},{"Add":""},{"Add":" open IsQuasiSemiMetric isQuasiSemiMetric public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- General metrics"},{"Add":""},{"Add":"-- A general metric obeys a generalised form of the triangle inequality."},{"Add":"-- It can be specialised to a standard metric/ultrametric/inframetric"},{"Add":"-- etc. by providing the correct operator."},{"Add":"--"},{"Add":"-- Furthermore we do not assume that _∙_ & 0# form a monoid as"},{"Add":"-- associativity does not hold for p-relaxed metrics/p-inframetrics and"},{"Add":"-- the identity laws do not hold for ultrametrics over negative"},{"Add":"-- codomains."},{"Add":"--"},{"Add":"-- See \"Properties of distance spaces with power triangle inequalities\""},{"Add":"-- by Daniel J. Greenhoe, 2016 (arXiv)"},{"Add":""},{"Add":"record IsGeneralMetric (_∙_ : Op₂ I) (d : DistanceFunction A I)"},{"Add":" : Set (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where"},{"Add":" field"},{"Add":" isSemiMetric : IsSemiMetric d"},{"Add":" triangle : TriangleInequality _≤_ _∙_ d"},{"Add":""},{"Add":" open IsSemiMetric isSemiMetric public"}]}},{"FileAdd":{"meta":{"basename":"Rational.agda","metadata":0},"context":[{"change":0,"pos":1118054}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Metrics with ℚ as the codomain of the metric function"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Metric.Rational where"},{"Add":""},{"Add":"open import Function.Metric.Rational.Core public"},{"Add":"open import Function.Metric.Rational.Definitions public"},{"Add":"open import Function.Metric.Rational.Structures public"},{"Add":"open import Function.Metric.Rational.Bundles public"}]}},{"FileAdd":{"meta":{"basename":"Rational","metadata":576},"context":[{"change":0,"pos":1118054}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Structures.agda","metadata":0},"context":[{"change":0,"pos":1122140}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Core definitions for metrics over ℚ"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Data.Rational.Base"},{"Add":"open import Function.Base using (const)"},{"Add":"open import Level using (Level; suc)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_)"},{"Add":""},{"Add":"open import Function.Metric.Rational.Core"},{"Add":"open import Function.Metric.Rational.Definitions"},{"Add":"import Function.Metric.Structures as Base"},{"Add":""},{"Add":"module Function.Metric.Rational.Structures where"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a ℓ : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Proto-metrics"},{"Add":""},{"Add":"IsProtoMetric : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"IsProtoMetric _≈_ = Base.IsProtoMetric _≈_ _≡_ _≤_ 0ℚ"},{"Add":""},{"Add":"open Base using (module IsProtoMetric) public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Pre-metrics"},{"Add":""},{"Add":"IsPreMetric : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"IsPreMetric _≈_ = Base.IsPreMetric _≈_ _≡_ _≤_ 0ℚ"},{"Add":""},{"Add":"open Base using (module IsPreMetric) public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Quasi-semi-metrics"},{"Add":""},{"Add":"IsQuasiSemiMetric : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"IsQuasiSemiMetric _≈_ = Base.IsQuasiSemiMetric _≈_ _≡_ _≤_ 0ℚ"},{"Add":""},{"Add":"open Base using (module IsQuasiSemiMetric) public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Semi-metrics"},{"Add":""},{"Add":"IsSemiMetric : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"IsSemiMetric _≈_ = Base.IsSemiMetric _≈_ _≡_ _≤_ 0ℚ"},{"Add":""},{"Add":"open Base using (module IsSemiMetric) public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Metrics"},{"Add":""},{"Add":"IsMetric : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"IsMetric _≈_ = Base.IsGeneralMetric _≈_ _≡_ _≤_ 0ℚ _+_"},{"Add":""},{"Add":"module IsMetric {_≈_ : Rel A ℓ} {d : DistanceFunction A}"},{"Add":" (M : IsMetric _≈_ d) where"},{"Add":" open Base.IsGeneralMetric M public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Ultra-metrics"},{"Add":""},{"Add":"IsUltraMetric : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"IsUltraMetric _≈_ = Base.IsGeneralMetric _≈_ _≡_ _≤_ 0ℚ _⊔_"},{"Add":""},{"Add":"module IsUltraMetric {_≈_ : Rel A ℓ} {d : DistanceFunction A}"},{"Add":" (UM : IsUltraMetric _≈_ d) where"},{"Add":" open Base.IsGeneralMetric UM public"}]}},{"FileAdd":{"meta":{"basename":"Definitions.agda","metadata":0},"context":[{"change":0,"pos":1122140}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Core definitions for metrics over ℚ"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Algebra.Core using (Op₂)"},{"Add":"open import Data.Rational.Base"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_)"},{"Add":""},{"Add":"open import Function.Metric.Rational.Core"},{"Add":"import Function.Metric.Definitions as Base"},{"Add":""},{"Add":"module Function.Metric.Rational.Definitions where"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a ℓ : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"-- Basic"},{"Add":""},{"Add":"Congruent : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"Congruent _≈ₐ_ d = Base.Congruent _≈ₐ_ _≡_ d"},{"Add":""},{"Add":"Indiscernable : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"Indiscernable _≈ₐ_ d = Base.Indiscernable _≈ₐ_ _≡_ d 0ℚ"},{"Add":""},{"Add":"Definite : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"Definite _≈ₐ_ d = Base.Definite _≈ₐ_ _≡_ d 0ℚ"},{"Add":""},{"Add":"Symmetric : DistanceFunction A → Set _"},{"Add":"Symmetric = Base.Symmetric _≡_"},{"Add":""},{"Add":"Bounded : DistanceFunction A → Set _"},{"Add":"Bounded = Base.Bounded _≤_"},{"Add":""},{"Add":"TranslationInvariant : Op₂ A → DistanceFunction A → Set _"},{"Add":"TranslationInvariant = Base.TranslationInvariant _≡_"},{"Add":""},{"Add":"-- Inequalities"},{"Add":""},{"Add":"TriangleInequality : DistanceFunction A → Set _"},{"Add":"TriangleInequality = Base.TriangleInequality _≤_ _+_"},{"Add":""},{"Add":"MaxTriangleInequality : DistanceFunction A → Set _"},{"Add":"MaxTriangleInequality = Base.TriangleInequality _≤_ _⊔_"},{"Add":""},{"Add":"-- Contractions"},{"Add":""},{"Add":"Contracting : (A → A) → DistanceFunction A → Set _"},{"Add":"Contracting = Base.Contracting _≤_"},{"Add":""},{"Add":"ContractingOnOrbits : (A → A) → DistanceFunction A → Set _"},{"Add":"ContractingOnOrbits = Base.ContractingOnOrbits _≤_"},{"Add":""},{"Add":"StrictlyContracting : Rel A ℓ → (A → A) → DistanceFunction A → Set _"},{"Add":"StrictlyContracting _≈_ = Base.StrictlyContracting _≈_ _<_"},{"Add":""},{"Add":"StrictlyContractingOnOrbits : Rel A ℓ → (A → A) → DistanceFunction A → Set _"},{"Add":"StrictlyContractingOnOrbits _≈_ = Base.StrictlyContractingOnOrbits _≈_ _<_"}]}},{"FileAdd":{"meta":{"basename":"Core.agda","metadata":0},"context":[{"change":0,"pos":1122140}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Core definitions for metrics over ℕ"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Data.Rational.Base using (ℚ)"},{"Add":"import Function.Metric.Core as Base"},{"Add":""},{"Add":"module Function.Metric.Rational.Core where"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"DistanceFunction : ∀ {a} → Set a → Set a"},{"Add":"DistanceFunction A = Base.DistanceFunction A ℚ"}]}},{"FileAdd":{"meta":{"basename":"Bundles.agda","metadata":0},"context":[{"change":0,"pos":1122140}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Bundles for metrics over ℚ"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Unfortunately, unlike definitions and structures, the bundles over"},{"Add":"-- general metric spaces cannot be reused as it is impossible to"},{"Add":"-- constrain the image set to ℚ."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Metric.Rational.Bundles where"},{"Add":""},{"Add":"open import Function.Base using (const)"},{"Add":"open import Level using (Level; suc; _⊔_)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Properties using (isEquivalence)"},{"Add":""},{"Add":"open import Function.Metric.Rational.Core using (DistanceFunction)"},{"Add":"open import Function.Metric.Rational.Structures"},{"Add":"open import Function.Metric.Bundles as Base"},{"Add":" using (GeneralMetric)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Proto-metric"},{"Add":""},{"Add":"record ProtoMetric a ℓ : Set (suc (a ⊔ ℓ)) where"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" _≈_ : Rel Carrier ℓ"},{"Add":" d : DistanceFunction Carrier"},{"Add":" isProtoMetric : IsProtoMetric _≈_ d"},{"Add":""},{"Add":" infix 4 _≈_"},{"Add":""},{"Add":" open IsProtoMetric isProtoMetric public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- PreMetric"},{"Add":""},{"Add":"record PreMetric a ℓ : Set (suc (a ⊔ ℓ)) where"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" _≈_ : Rel Carrier ℓ"},{"Add":" d : DistanceFunction Carrier"},{"Add":" isPreMetric : IsPreMetric _≈_ d"},{"Add":""},{"Add":" infix 4 _≈_"},{"Add":""},{"Add":" open IsPreMetric isPreMetric public"},{"Add":""},{"Add":" protoMetric : ProtoMetric a ℓ"},{"Add":" protoMetric = record"},{"Add":" { isProtoMetric = isProtoMetric"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- QuasiSemiMetric"},{"Add":""},{"Add":"record QuasiSemiMetric a ℓ : Set (suc (a ⊔ ℓ)) where"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" _≈_ : Rel Carrier ℓ"},{"Add":" d : DistanceFunction Carrier"},{"Add":" isQuasiSemiMetric : IsQuasiSemiMetric _≈_ d"},{"Add":""},{"Add":" infix 4 _≈_"},{"Add":""},{"Add":" open IsQuasiSemiMetric isQuasiSemiMetric public"},{"Add":""},{"Add":" preMetric : PreMetric a ℓ"},{"Add":" preMetric = record"},{"Add":" { isPreMetric = isPreMetric"},{"Add":" }"},{"Add":""},{"Add":" open PreMetric preMetric public"},{"Add":" using (protoMetric)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- SemiMetric"},{"Add":""},{"Add":"record SemiMetric a ℓ : Set (suc (a ⊔ ℓ)) where"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" _≈_ : Rel Carrier ℓ"},{"Add":" d : DistanceFunction Carrier"},{"Add":" isSemiMetric : IsSemiMetric _≈_ d"},{"Add":""},{"Add":" infix 4 _≈_"},{"Add":""},{"Add":" open IsSemiMetric isSemiMetric public"},{"Add":""},{"Add":" quasiSemiMetric : QuasiSemiMetric a ℓ"},{"Add":" quasiSemiMetric = record"},{"Add":" { isQuasiSemiMetric = isQuasiSemiMetric"},{"Add":" }"},{"Add":""},{"Add":" open QuasiSemiMetric quasiSemiMetric public"},{"Add":" using (protoMetric; preMetric)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Metrics"},{"Add":""},{"Add":"record Metric a ℓ : Set (suc (a ⊔ ℓ)) where"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" _≈_ : Rel Carrier ℓ"},{"Add":" d : DistanceFunction Carrier"},{"Add":" isMetric : IsMetric _≈_ d"},{"Add":""},{"Add":" infix 4 _≈_"},{"Add":""},{"Add":" open IsMetric isMetric public"},{"Add":""},{"Add":" semiMetric : SemiMetric a ℓ"},{"Add":" semiMetric = record"},{"Add":" { isSemiMetric = isSemiMetric"},{"Add":" }"},{"Add":""},{"Add":" open SemiMetric semiMetric public"},{"Add":" using (protoMetric; preMetric; quasiSemiMetric)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- UltraMetrics"},{"Add":""},{"Add":"record UltraMetric a ℓ : Set (suc (a ⊔ ℓ)) where"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" _≈_ : Rel Carrier ℓ"},{"Add":" d : DistanceFunction Carrier"},{"Add":" isUltraMetric : IsUltraMetric _≈_ d"},{"Add":""},{"Add":" infix 4 _≈_"},{"Add":""},{"Add":" open IsUltraMetric isUltraMetric public"},{"Add":""},{"Add":" semiMetric : SemiMetric a ℓ"},{"Add":" semiMetric = record"},{"Add":" { isSemiMetric = isSemiMetric"},{"Add":" }"},{"Add":""},{"Add":" open SemiMetric semiMetric public"},{"Add":" using (protoMetric; preMetric; quasiSemiMetric)"}]}},{"FileAdd":{"meta":{"basename":"Nat.agda","metadata":0},"context":[{"change":0,"pos":1118054}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Metrics with ℕ as the codomain of the metric function"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Metric.Nat where"},{"Add":""},{"Add":"open import Function.Metric.Nat.Core public"},{"Add":"open import Function.Metric.Nat.Definitions public"},{"Add":"open import Function.Metric.Nat.Structures public"},{"Add":"open import Function.Metric.Nat.Bundles public"}]}},{"FileAdd":{"meta":{"basename":"Nat","metadata":576},"context":[{"change":0,"pos":1118054}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Structures.agda","metadata":0},"context":[{"change":0,"pos":1132017}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Core definitions for metrics over ℕ"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Metric.Nat.Structures where"},{"Add":""},{"Add":"open import Data.Nat.Base hiding (suc)"},{"Add":"open import Function.Base using (const)"},{"Add":"open import Level using (Level; suc)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_)"},{"Add":""},{"Add":"open import Function.Metric.Nat.Core"},{"Add":"open import Function.Metric.Nat.Definitions"},{"Add":"import Function.Metric.Structures as Base"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a ℓ : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Proto-metrics"},{"Add":""},{"Add":"IsProtoMetric : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"IsProtoMetric _≈_ = Base.IsProtoMetric _≈_ _≡_ _≤_ 0"},{"Add":""},{"Add":"open Base using (module IsProtoMetric) public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Pre-metrics"},{"Add":""},{"Add":"IsPreMetric : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"IsPreMetric _≈_ = Base.IsPreMetric _≈_ _≡_ _≤_ 0"},{"Add":""},{"Add":"open Base using (module IsPreMetric) public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Quasi-semi-metrics"},{"Add":""},{"Add":"IsQuasiSemiMetric : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"IsQuasiSemiMetric _≈_ = Base.IsQuasiSemiMetric _≈_ _≡_ _≤_ 0"},{"Add":""},{"Add":"open Base using (module IsQuasiSemiMetric) public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Semi-metrics"},{"Add":""},{"Add":"IsSemiMetric : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"IsSemiMetric _≈_ = Base.IsSemiMetric _≈_ _≡_ _≤_ 0"},{"Add":""},{"Add":"open Base using (module IsSemiMetric) public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Metrics"},{"Add":""},{"Add":"IsMetric : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"IsMetric _≈_ = Base.IsGeneralMetric _≈_ _≡_ _≤_ 0 _+_"},{"Add":""},{"Add":"module IsMetric {_≈_ : Rel A ℓ} {d : DistanceFunction A}"},{"Add":" (M : IsMetric _≈_ d) where"},{"Add":" open Base.IsGeneralMetric M public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Ultra-metrics"},{"Add":""},{"Add":"IsUltraMetric : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"IsUltraMetric _≈_ = Base.IsGeneralMetric _≈_ _≡_ _≤_ 0 _⊔_"},{"Add":""},{"Add":"module IsUltraMetric {_≈_ : Rel A ℓ} {d : DistanceFunction A}"},{"Add":" (UM : IsUltraMetric _≈_ d) where"},{"Add":" open Base.IsGeneralMetric UM public"}]}},{"FileAdd":{"meta":{"basename":"Definitions.agda","metadata":0},"context":[{"change":0,"pos":1132017}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Core definitions for metrics over ℕ"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Metric.Nat.Definitions where"},{"Add":""},{"Add":"open import Algebra.Core using (Op₂)"},{"Add":"open import Data.Nat.Base"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_)"},{"Add":""},{"Add":"open import Function.Metric.Nat.Core"},{"Add":"import Function.Metric.Definitions as Base"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a ℓ : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"-- Basic"},{"Add":""},{"Add":"Congruent : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"Congruent _≈ₐ_ d = Base.Congruent _≈ₐ_ _≡_ d"},{"Add":""},{"Add":"Indiscernable : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"Indiscernable _≈ₐ_ d = Base.Indiscernable _≈ₐ_ _≡_ d 0"},{"Add":""},{"Add":"Definite : Rel A ℓ → DistanceFunction A → Set _"},{"Add":"Definite _≈ₐ_ d = Base.Definite _≈ₐ_ _≡_ d 0"},{"Add":""},{"Add":"Symmetric : DistanceFunction A → Set _"},{"Add":"Symmetric = Base.Symmetric _≡_"},{"Add":""},{"Add":"Bounded : DistanceFunction A → Set _"},{"Add":"Bounded = Base.Bounded _≤_"},{"Add":""},{"Add":"TranslationInvariant : Op₂ A → DistanceFunction A → Set _"},{"Add":"TranslationInvariant = Base.TranslationInvariant _≡_"},{"Add":""},{"Add":"-- Inequalities"},{"Add":""},{"Add":"TriangleInequality : DistanceFunction A → Set _"},{"Add":"TriangleInequality = Base.TriangleInequality _≤_ _+_"},{"Add":""},{"Add":"MaxTriangleInequality : DistanceFunction A → Set _"},{"Add":"MaxTriangleInequality = Base.TriangleInequality _≤_ _⊔_"},{"Add":""},{"Add":"-- Contractions"},{"Add":""},{"Add":"Contracting : (A → A) → DistanceFunction A → Set _"},{"Add":"Contracting = Base.Contracting _≤_"},{"Add":""},{"Add":"ContractingOnOrbits : (A → A) → DistanceFunction A → Set _"},{"Add":"ContractingOnOrbits = Base.ContractingOnOrbits _≤_"},{"Add":""},{"Add":"StrictlyContracting : Rel A ℓ → (A → A) → DistanceFunction A → Set _"},{"Add":"StrictlyContracting _≈_ = Base.StrictlyContracting _≈_ _<_"},{"Add":""},{"Add":"StrictlyContractingOnOrbits : Rel A ℓ → (A → A) → DistanceFunction A → Set _"},{"Add":"StrictlyContractingOnOrbits _≈_ = Base.StrictlyContractingOnOrbits _≈_ _<_"}]}},{"FileAdd":{"meta":{"basename":"Core.agda","metadata":0},"context":[{"change":0,"pos":1132017}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Core definitions for metrics over ℕ"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Metric.Nat.Core where"},{"Add":""},{"Add":"open import Data.Nat.Base using (ℕ)"},{"Add":"import Function.Metric.Core as Base"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"DistanceFunction : ∀ {a} → Set a → Set a"},{"Add":"DistanceFunction A = Base.DistanceFunction A ℕ"}]}},{"FileAdd":{"meta":{"basename":"Bundles.agda","metadata":0},"context":[{"change":0,"pos":1132017}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Bundles for metrics over ℕ"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Unfortunately, unlike definitions and structures, the bundles over"},{"Add":"-- general metric spaces cannot be reused as it is impossible to"},{"Add":"-- constrain the image set to ℕ."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Metric.Nat.Bundles where"},{"Add":""},{"Add":"open import Data.Nat.Base hiding (suc; _⊔_)"},{"Add":"open import Function.Base using (const)"},{"Add":"open import Level using (Level; suc; _⊔_)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Properties using (isEquivalence)"},{"Add":""},{"Add":"open import Function.Metric.Nat.Core using (DistanceFunction)"},{"Add":"open import Function.Metric.Nat.Structures"},{"Add":"open import Function.Metric.Bundles as Base"},{"Add":" using (GeneralMetric)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Proto-metric"},{"Add":""},{"Add":"record ProtoMetric a ℓ : Set (suc (a ⊔ ℓ)) where"},{"Add":" infix 4 _≈_"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" _≈_ : Rel Carrier ℓ"},{"Add":" d : DistanceFunction Carrier"},{"Add":" isProtoMetric : IsProtoMetric _≈_ d"},{"Add":""},{"Add":" open IsProtoMetric isProtoMetric public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- PreMetric"},{"Add":""},{"Add":"record PreMetric a ℓ : Set (suc (a ⊔ ℓ)) where"},{"Add":" infix 4 _≈_"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" _≈_ : Rel Carrier ℓ"},{"Add":" d : DistanceFunction Carrier"},{"Add":" isPreMetric : IsPreMetric _≈_ d"},{"Add":""},{"Add":" open IsPreMetric isPreMetric public"},{"Add":""},{"Add":" protoMetric : ProtoMetric a ℓ"},{"Add":" protoMetric = record"},{"Add":" { isProtoMetric = isProtoMetric"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- QuasiSemiMetric"},{"Add":""},{"Add":"record QuasiSemiMetric a ℓ : Set (suc (a ⊔ ℓ)) where"},{"Add":" infix 4 _≈_"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" _≈_ : Rel Carrier ℓ"},{"Add":" d : DistanceFunction Carrier"},{"Add":" isQuasiSemiMetric : IsQuasiSemiMetric _≈_ d"},{"Add":""},{"Add":" open IsQuasiSemiMetric isQuasiSemiMetric public"},{"Add":""},{"Add":" preMetric : PreMetric a ℓ"},{"Add":" preMetric = record"},{"Add":" { isPreMetric = isPreMetric"},{"Add":" }"},{"Add":""},{"Add":" open PreMetric preMetric public"},{"Add":" using (protoMetric)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- SemiMetric"},{"Add":""},{"Add":"record SemiMetric a ℓ : Set (suc (a ⊔ ℓ)) where"},{"Add":" infix 4 _≈_"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" _≈_ : Rel Carrier ℓ"},{"Add":" d : DistanceFunction Carrier"},{"Add":" isSemiMetric : IsSemiMetric _≈_ d"},{"Add":""},{"Add":" open IsSemiMetric isSemiMetric public"},{"Add":""},{"Add":" quasiSemiMetric : QuasiSemiMetric a ℓ"},{"Add":" quasiSemiMetric = record"},{"Add":" { isQuasiSemiMetric = isQuasiSemiMetric"},{"Add":" }"},{"Add":""},{"Add":" open QuasiSemiMetric quasiSemiMetric public"},{"Add":" using (protoMetric; preMetric)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Metrics"},{"Add":""},{"Add":"record Metric a ℓ : Set (suc (a ⊔ ℓ)) where"},{"Add":" infix 4 _≈_"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" _≈_ : Rel Carrier ℓ"},{"Add":" d : DistanceFunction Carrier"},{"Add":" isMetric : IsMetric _≈_ d"},{"Add":""},{"Add":" open IsMetric isMetric public"},{"Add":""},{"Add":" semiMetric : SemiMetric a ℓ"},{"Add":" semiMetric = record"},{"Add":" { isSemiMetric = isSemiMetric"},{"Add":" }"},{"Add":""},{"Add":" open SemiMetric semiMetric public"},{"Add":" using (protoMetric; preMetric; quasiSemiMetric)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- UltraMetrics"},{"Add":""},{"Add":"record UltraMetric a ℓ : Set (suc (a ⊔ ℓ)) where"},{"Add":" infix 4 _≈_"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" _≈_ : Rel Carrier ℓ"},{"Add":" d : DistanceFunction Carrier"},{"Add":" isUltraMetric : IsUltraMetric _≈_ d"},{"Add":""},{"Add":" open IsUltraMetric isUltraMetric public"},{"Add":""},{"Add":" semiMetric : SemiMetric a ℓ"},{"Add":" semiMetric = record"},{"Add":" { isSemiMetric = isSemiMetric"},{"Add":" }"},{"Add":""},{"Add":" open SemiMetric semiMetric public"},{"Add":" using (protoMetric; preMetric; quasiSemiMetric)"}]}},{"FileAdd":{"meta":{"basename":"Definitions.agda","metadata":0},"context":[{"change":0,"pos":1118054}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Definitions of properties over distance functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via `Function.Metric`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Metric.Definitions where"},{"Add":""},{"Add":"open import Algebra.Core using (Op₂)"},{"Add":"open import Data.Product.Base using (∃)"},{"Add":"open import Function.Metric.Core using (DistanceFunction)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_)"},{"Add":"open import Relation.Nullary.Negation using (¬_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a i ℓ ℓ₁ ℓ₂ : Level"},{"Add":" A : Set a"},{"Add":" I : Set i"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"Congruent : Rel A ℓ₁ → Rel I ℓ₂ → DistanceFunction A I → Set _"},{"Add":"Congruent _≈ₐ_ _≈ᵢ_ d = d Preserves₂ _≈ₐ_ ⟶ _≈ₐ_ ⟶ _≈ᵢ_"},{"Add":""},{"Add":"Indiscernable : Rel A ℓ₁ → Rel I ℓ₂ → DistanceFunction A I → I → Set _"},{"Add":"Indiscernable _≈ₐ_ _≈ᵢ_ d 0# = ∀ {x y} → d x y ≈ᵢ 0# → x ≈ₐ y"},{"Add":""},{"Add":"Definite : Rel A ℓ₁ → Rel I ℓ₂ → DistanceFunction A I → I → Set _"},{"Add":"Definite _≈ₐ_ _≈ᵢ_ d 0# = ∀ {x y} → x ≈ₐ y → d x y ≈ᵢ 0#"},{"Add":""},{"Add":"NonNegative : Rel I ℓ₂ → DistanceFunction A I → I → Set _"},{"Add":"NonNegative _≤_ d 0# = ∀ {x y} → 0# ≤ d x y"},{"Add":""},{"Add":"Symmetric : Rel I ℓ → DistanceFunction A I → Set _"},{"Add":"Symmetric _≈_ d = ∀ x y → d x y ≈ d y x"},{"Add":""},{"Add":"TriangleInequality : Rel I ℓ → Op₂ I → DistanceFunction A I → _"},{"Add":"TriangleInequality _≤_ _∙_ d = ∀ x y z → d x z ≤ (d x y ∙ d y z)"},{"Add":""},{"Add":"Bounded : Rel I ℓ → DistanceFunction A I → Set _"},{"Add":"Bounded _≤_ d = ∃ λ n → ∀ x y → d x y ≤ n"},{"Add":""},{"Add":"TranslationInvariant : Rel I ℓ₂ → Op₂ A → DistanceFunction A I → Set _"},{"Add":"TranslationInvariant _≈_ _∙_ d = ∀ {x y a} → d (x ∙ a) (y ∙ a) ≈ d x y"},{"Add":""},{"Add":"Contracting : Rel I ℓ → (A → A) → DistanceFunction A I → Set _"},{"Add":"Contracting _≤_ f d = ∀ x y → d (f x) (f y) ≤ d x y"},{"Add":""},{"Add":"ContractingOnOrbits : Rel I ℓ → (A → A) → DistanceFunction A I → Set _"},{"Add":"ContractingOnOrbits _≤_ f d = ∀ x → d (f x) (f (f x)) ≤ d x (f x)"},{"Add":""},{"Add":"StrictlyContracting : Rel A ℓ₁ → Rel I ℓ₂ → (A → A) → DistanceFunction A I → Set _"},{"Add":"StrictlyContracting _≈_ _<_ f d = ∀ {x y} → ¬ (y ≈ x) → d (f x) (f y) < d x y"},{"Add":""},{"Add":"StrictlyContractingOnOrbits : Rel A ℓ₁ → Rel I ℓ₂ → (A → A) → DistanceFunction A I → Set _"},{"Add":"StrictlyContractingOnOrbits _≈_ _<_ f d = ∀ {x} → ¬ (f x ≈ x) → d (f x) (f (f x)) < d x (f x)"}]}},{"FileAdd":{"meta":{"basename":"Core.agda","metadata":0},"context":[{"change":0,"pos":1118054}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Core metric definitions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Metric.Core where"},{"Add":""},{"Add":"open import Level using (Level)"},{"Add":"private"},{"Add":" variable"},{"Add":" a i : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Distance functions"},{"Add":""},{"Add":"DistanceFunction : Set a → Set i → Set _"},{"Add":"DistanceFunction A I = A → A → I"}]}},{"FileAdd":{"meta":{"basename":"Bundles.agda","metadata":0},"context":[{"change":0,"pos":1118054}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Bundles for metrics"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module should be accessed via `Function.Metric`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Metric.Bundles where"},{"Add":""},{"Add":"open import Algebra.Core using (Op₂)"},{"Add":"open import Level using (Level; suc; _⊔_)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":""},{"Add":"open import Function.Metric.Structures"},{"Add":"open import Function.Metric.Core"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- ProtoMetric"},{"Add":""},{"Add":"record ProtoMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)"},{"Add":" : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where"},{"Add":" infix 4 _≈_ _≈ᵢ_ _≤_"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" Image : Set i"},{"Add":" _≈_ : Rel Carrier ℓ₁"},{"Add":" _≈ᵢ_ : Rel Image ℓ₂"},{"Add":" _≤_ : Rel Image ℓ₃"},{"Add":" 0# : Image"},{"Add":" d : DistanceFunction Carrier Image"},{"Add":" isProtoMetric : IsProtoMetric _≈_ _≈ᵢ_ _≤_ 0# d"},{"Add":""},{"Add":" open IsProtoMetric isProtoMetric public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- PreMetric"},{"Add":""},{"Add":"record PreMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)"},{"Add":" : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where"},{"Add":" infix 4 _≈_ _≈ᵢ_ _≤_"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" Image : Set i"},{"Add":" _≈_ : Rel Carrier ℓ₁"},{"Add":" _≈ᵢ_ : Rel Image ℓ₂"},{"Add":" _≤_ : Rel Image ℓ₃"},{"Add":" 0# : Image"},{"Add":" d : DistanceFunction Carrier Image"},{"Add":" isPreMetric : IsPreMetric _≈_ _≈ᵢ_ _≤_ 0# d"},{"Add":""},{"Add":" open IsPreMetric isPreMetric public"},{"Add":""},{"Add":" protoMetric : ProtoMetric a i ℓ₁ ℓ₂ ℓ₃"},{"Add":" protoMetric = record"},{"Add":" { isProtoMetric = isProtoMetric"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- QuasiSemiMetric"},{"Add":""},{"Add":"record QuasiSemiMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)"},{"Add":" : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where"},{"Add":""},{"Add":" infix 4 _≈_ _≈ᵢ_ _≤_"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" Image : Set i"},{"Add":" _≈_ : Rel Carrier ℓ₁"},{"Add":" _≈ᵢ_ : Rel Image ℓ₂"},{"Add":" _≤_ : Rel Image ℓ₃"},{"Add":" 0# : Image"},{"Add":" d : DistanceFunction Carrier Image"},{"Add":" isQuasiSemiMetric : IsQuasiSemiMetric _≈_ _≈ᵢ_ _≤_ 0# d"},{"Add":""},{"Add":" open IsQuasiSemiMetric isQuasiSemiMetric public"},{"Add":""},{"Add":" preMetric : PreMetric a i ℓ₁ ℓ₂ ℓ₃"},{"Add":" preMetric = record"},{"Add":" { isPreMetric = isPreMetric"},{"Add":" }"},{"Add":""},{"Add":" open PreMetric preMetric public"},{"Add":" using (protoMetric)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- SemiMetric"},{"Add":""},{"Add":"record SemiMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)"},{"Add":" : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where"},{"Add":" infix 4 _≈_ _≈ᵢ_ _≤_"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" Image : Set i"},{"Add":" _≈_ : Rel Carrier ℓ₁"},{"Add":" _≈ᵢ_ : Rel Image ℓ₂"},{"Add":" _≤_ : Rel Image ℓ₃"},{"Add":" 0# : Image"},{"Add":" d : DistanceFunction Carrier Image"},{"Add":" isSemiMetric : IsSemiMetric _≈_ _≈ᵢ_ _≤_ 0# d"},{"Add":""},{"Add":" open IsSemiMetric isSemiMetric public"},{"Add":""},{"Add":" quasiSemiMetric : QuasiSemiMetric a i ℓ₁ ℓ₂ ℓ₃"},{"Add":" quasiSemiMetric = record"},{"Add":" { isQuasiSemiMetric = isQuasiSemiMetric"},{"Add":" }"},{"Add":""},{"Add":" open QuasiSemiMetric quasiSemiMetric public"},{"Add":" using (protoMetric; preMetric)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- GeneralMetric"},{"Add":""},{"Add":"-- Note that this package is not necessarily a metric in the classical"},{"Add":"-- sense as there is no way to ensure that the _∙_ operator really"},{"Add":"-- represents addition. See `Function.Metric.Nat` and"},{"Add":"-- `Function.Metric.Rational` for more specialised `Metric` and"},{"Add":"-- `UltraMetric` packages."},{"Add":""},{"Add":"-- See the discussion accompanying the `IsGeneralMetric` structure for"},{"Add":"-- more details."},{"Add":""},{"Add":"record GeneralMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)"},{"Add":" : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where"},{"Add":" infix 4 _≈_ _≈ᵢ_ _≤_"},{"Add":" infixl 6 _∙_"},{"Add":" field"},{"Add":" Carrier : Set a"},{"Add":" Image : Set i"},{"Add":" _≈_ : Rel Carrier ℓ₁"},{"Add":" _≈ᵢ_ : Rel Image ℓ₂"},{"Add":" _≤_ : Rel Image ℓ₃"},{"Add":" 0# : Image"},{"Add":" _∙_ : Op₂ Image"},{"Add":" d : DistanceFunction Carrier Image"},{"Add":" isGeneralMetric : IsGeneralMetric _≈_ _≈ᵢ_ _≤_ 0# _∙_ d"},{"Add":""},{"Add":" open IsGeneralMetric isGeneralMetric public"},{"Add":""},{"Add":" semiMetric : SemiMetric a i ℓ₁ ℓ₂ ℓ₃"},{"Add":" semiMetric = record"},{"Add":" { isSemiMetric = isSemiMetric"},{"Add":" }"},{"Add":""},{"Add":" open SemiMetric semiMetric public"},{"Add":" using (protoMetric; preMetric; quasiSemiMetric)"}]}},{"FileAdd":{"meta":{"basename":"LeftInverse.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":"{-# OPTIONS --warn=noUserWarning #-}"},{"Add":""},{"Add":"module Function.LeftInverse where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Function.LeftInverse was deprecated in v2.0."},{"Add":"Use the standard function hierarchy in Function/Function.Bundles instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Level"},{"Add":"import Relation.Binary.Reasoning.Setoid as EqReasoning"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Function.Equality as Eq"},{"Add":" using (_⟶_; _⟨$⟩_) renaming (_∘_ to _⟪∘⟫_)"},{"Add":"open import Function.Equivalence using (Equivalence)"},{"Add":"open import Function.Injection using (Injective; Injection)"},{"Add":"open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Left and right inverses."},{"Add":""},{"Add":"_LeftInverseOf_ :"},{"Add":" ∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →"},{"Add":" To ⟶ From → From ⟶ To → Set _"},{"Add":"_LeftInverseOf_ {From = From} f g = ∀ x → f ⟨$⟩ (g ⟨$⟩ x) ≈ x"},{"Add":" where open Setoid From"},{"Add":"{-# WARNING_ON_USAGE _LeftInverseOf_"},{"Add":"\"Warning: _LeftInverseOf_ was deprecated in v2.0."},{"Add":"Please use Function.(Structures.)IsRightInverse instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_RightInverseOf_ :"},{"Add":" ∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →"},{"Add":" To ⟶ From → From ⟶ To → Set _"},{"Add":"f RightInverseOf g = g LeftInverseOf f"},{"Add":"{-# WARNING_ON_USAGE _RightInverseOf_"},{"Add":"\"Warning: _RightInverseOf_ was deprecated in v2.0."},{"Add":"Please use Function.(Structures.)IsLeftInverse instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The set of all left inverses between two setoids."},{"Add":""},{"Add":"record LeftInverse {f₁ f₂ t₁ t₂}"},{"Add":" (From : Setoid f₁ f₂) (To : Setoid t₁ t₂) :"},{"Add":" Set (f₁ ⊔ f₂ ⊔ t₁ ⊔ t₂) where"},{"Add":" field"},{"Add":" to : From ⟶ To"},{"Add":" from : To ⟶ From"},{"Add":" left-inverse-of : from LeftInverseOf to"},{"Add":""},{"Add":" private"},{"Add":" open module F = Setoid From"},{"Add":" open module T = Setoid To"},{"Add":" open EqReasoning From"},{"Add":""},{"Add":" injective : Injective to"},{"Add":" injective {x} {y} eq = begin"},{"Add":" x ≈⟨ F.sym (left-inverse-of x) ⟩"},{"Add":" from ⟨$⟩ (to ⟨$⟩ x) ≈⟨ Eq.cong from eq ⟩"},{"Add":" from ⟨$⟩ (to ⟨$⟩ y) ≈⟨ left-inverse-of y ⟩"},{"Add":" y ∎"},{"Add":""},{"Add":" injection : Injection From To"},{"Add":" injection = record { to = to; injective = injective }"},{"Add":""},{"Add":" equivalence : Equivalence From To"},{"Add":" equivalence = record"},{"Add":" { to = to"},{"Add":" ; from = from"},{"Add":" }"},{"Add":""},{"Add":" to-from : ∀ {x y} → to ⟨$⟩ x T.≈ y → from ⟨$⟩ y F.≈ x"},{"Add":" to-from {x} {y} to-x≈y = begin"},{"Add":" from ⟨$⟩ y ≈⟨ Eq.cong from (T.sym to-x≈y) ⟩"},{"Add":" from ⟨$⟩ (to ⟨$⟩ x) ≈⟨ left-inverse-of x ⟩"},{"Add":" x ∎"},{"Add":"{-# WARNING_ON_USAGE LeftInverse"},{"Add":"\"Warning: LeftInverse was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)RightInverse instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"-- The set of all right inverses between two setoids."},{"Add":""},{"Add":"RightInverse : ∀ {f₁ f₂ t₁ t₂}"},{"Add":" (From : Setoid f₁ f₂) (To : Setoid t₁ t₂) → Set _"},{"Add":"RightInverse From To = LeftInverse To From"},{"Add":"{-# WARNING_ON_USAGE RightInverse"},{"Add":"\"Warning: RightInverse was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)LeftInverse instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The set of all left inverses from one set to another (i.e. left"},{"Add":"-- inverses with propositional equality)."},{"Add":"--"},{"Add":"-- Read A ↞ B as \"surjection from B to A\"."},{"Add":""},{"Add":"infix 3 _↞_"},{"Add":""},{"Add":"_↞_ : ∀ {f t} → Set f → Set t → Set _"},{"Add":"From ↞ To = LeftInverse (≡.setoid From) (≡.setoid To)"},{"Add":"{-# WARNING_ON_USAGE _↞_"},{"Add":"\"Warning: _↞_ was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)_↪_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"leftInverse : ∀ {f t} {From : Set f} {To : Set t} →"},{"Add":" (to : From → To) (from : To → From) →"},{"Add":" (∀ x → from (to x) ≡ x) →"},{"Add":" From ↞ To"},{"Add":"leftInverse to from invˡ = record"},{"Add":" { to = Eq.→-to-⟶ to"},{"Add":" ; from = Eq.→-to-⟶ from"},{"Add":" ; left-inverse-of = invˡ"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE leftInverse"},{"Add":"\"Warning: leftInverse was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)mk↪ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Identity and composition."},{"Add":""},{"Add":"id : ∀ {s₁ s₂} {S : Setoid s₁ s₂} → LeftInverse S S"},{"Add":"id {S = S} = record"},{"Add":" { to = Eq.id"},{"Add":" ; from = Eq.id"},{"Add":" ; left-inverse-of = λ _ → Setoid.refl S"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE id"},{"Add":"\"Warning: id was deprecated in v2.0."},{"Add":"Please use either Function.Properties.RightInverse.refl or"},{"Add":"Function.Construct.Identity.rightInverse instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"infixr 9 _∘_"},{"Add":""},{"Add":"_∘_ : ∀ {f₁ f₂ m₁ m₂ t₁ t₂}"},{"Add":" {F : Setoid f₁ f₂} {M : Setoid m₁ m₂} {T : Setoid t₁ t₂} →"},{"Add":" LeftInverse M T → LeftInverse F M → LeftInverse F T"},{"Add":"_∘_ {F = F} f g = record"},{"Add":" { to = to f ⟪∘⟫ to g"},{"Add":" ; from = from g ⟪∘⟫ from f"},{"Add":" ; left-inverse-of = λ x → begin"},{"Add":" from g ⟨$⟩ (from f ⟨$⟩ (to f ⟨$⟩ (to g ⟨$⟩ x))) ≈⟨ Eq.cong (from g) (left-inverse-of f (to g ⟨$⟩ x)) ⟩"},{"Add":" from g ⟨$⟩ (to g ⟨$⟩ x) ≈⟨ left-inverse-of g x ⟩"},{"Add":" x ∎"},{"Add":" }"},{"Add":" where"},{"Add":" open LeftInverse"},{"Add":" open EqReasoning F"},{"Add":"{-# WARNING_ON_USAGE _∘_"},{"Add":"\"Warning: _∘_ was deprecated in v2.0."},{"Add":"Please use either Function.Properties.RightInverse.trans or"},{"Add":"Function.Construct.Composition.rightInverse instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Inverse.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":"{-# OPTIONS --warn=noUserWarning #-}"},{"Add":""},{"Add":"module Function.Inverse where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Function.Inverse was deprecated in v2.0."},{"Add":"Use the standard function hierarchy in Function/Function.Bundles instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Function.Base using (flip)"},{"Add":"open import Function.Bijection hiding (id; _∘_; bijection)"},{"Add":"open import Function.Equality as F"},{"Add":" using (_⟶_) renaming (_∘_ to _⟪∘⟫_)"},{"Add":"open import Function.LeftInverse as Left hiding (id; _∘_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Definitions using (Reflexive; TransFlip; Sym)"},{"Add":"open import Relation.Binary.PropositionalEquality as ≡ using (_≗_; _≡_)"},{"Add":"open import Relation.Unary using (Pred)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Inverses"},{"Add":""},{"Add":"record _InverseOf_ {f₁ f₂ t₁ t₂}"},{"Add":" {From : Setoid f₁ f₂} {To : Setoid t₁ t₂}"},{"Add":" (from : To ⟶ From) (to : From ⟶ To) :"},{"Add":" Set (f₁ ⊔ f₂ ⊔ t₁ ⊔ t₂) where"},{"Add":" field"},{"Add":" left-inverse-of : from LeftInverseOf to"},{"Add":" right-inverse-of : from RightInverseOf to"},{"Add":"{-# WARNING_ON_USAGE _InverseOf_"},{"Add":"\"Warning: _InverseOf_ was deprecated in v2.0."},{"Add":"Please use Function.(Structures.)IsInverse instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The set of all inverses between two setoids"},{"Add":""},{"Add":"record Inverse {f₁ f₂ t₁ t₂}"},{"Add":" (From : Setoid f₁ f₂) (To : Setoid t₁ t₂) :"},{"Add":" Set (f₁ ⊔ f₂ ⊔ t₁ ⊔ t₂) where"},{"Add":" field"},{"Add":" to : From ⟶ To"},{"Add":" from : To ⟶ From"},{"Add":" inverse-of : from InverseOf to"},{"Add":""},{"Add":" open _InverseOf_ inverse-of public"},{"Add":""},{"Add":" left-inverse : LeftInverse From To"},{"Add":" left-inverse = record"},{"Add":" { to = to"},{"Add":" ; from = from"},{"Add":" ; left-inverse-of = left-inverse-of"},{"Add":" }"},{"Add":""},{"Add":" open LeftInverse left-inverse public"},{"Add":" using (injective; injection)"},{"Add":""},{"Add":" bijection : Bijection From To"},{"Add":" bijection = record"},{"Add":" { to = to"},{"Add":" ; bijective = record"},{"Add":" { injective = injective"},{"Add":" ; surjective = record"},{"Add":" { from = from"},{"Add":" ; right-inverse-of = right-inverse-of"},{"Add":" }"},{"Add":" }"},{"Add":" }"},{"Add":""},{"Add":" open Bijection bijection public"},{"Add":" using (equivalence; surjective; surjection; right-inverse;"},{"Add":" to-from; from-to)"},{"Add":"{-# WARNING_ON_USAGE Inverse"},{"Add":"\"Warning: Inverse was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)Inverse instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The set of all inverses between two sets (i.e. inverses with"},{"Add":"-- propositional equality)"},{"Add":""},{"Add":"infix 3 _↔_ _↔̇_"},{"Add":""},{"Add":"_↔_ : ∀ {f t} → Set f → Set t → Set _"},{"Add":"From ↔ To = Inverse (≡.setoid From) (≡.setoid To)"},{"Add":"{-# WARNING_ON_USAGE _↔_"},{"Add":"\"Warning: _↔_ was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)_↔_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_↔̇_ : ∀ {i f t} {I : Set i} → Pred I f → Pred I t → Set _"},{"Add":"From ↔̇ To = ∀ {i} → From i ↔ To i"},{"Add":"{-# WARNING_ON_USAGE _↔̇_"},{"Add":"\"Warning: _↔̇_ was deprecated in v2.0."},{"Add":"Please use Function.Indexed.(Bundles.)_↔ᵢ_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"inverse : ∀ {f t} {From : Set f} {To : Set t} →"},{"Add":" (to : From → To) (from : To → From) →"},{"Add":" (∀ x → from (to x) ≡ x) →"},{"Add":" (∀ x → to (from x) ≡ x) →"},{"Add":" From ↔ To"},{"Add":"inverse to from from∘to to∘from = record"},{"Add":" { to = F.→-to-⟶ to"},{"Add":" ; from = F.→-to-⟶ from"},{"Add":" ; inverse-of = record"},{"Add":" { left-inverse-of = from∘to"},{"Add":" ; right-inverse-of = to∘from"},{"Add":" }"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE inverse"},{"Add":"\"Warning: inverse was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)mk↔ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- If two setoids are in bijective correspondence, then there is an"},{"Add":"-- inverse between them"},{"Add":""},{"Add":"fromBijection :"},{"Add":" ∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →"},{"Add":" Bijection From To → Inverse From To"},{"Add":"fromBijection b = record"},{"Add":" { to = Bijection.to b"},{"Add":" ; from = Bijection.from b"},{"Add":" ; inverse-of = record"},{"Add":" { left-inverse-of = Bijection.left-inverse-of b"},{"Add":" ; right-inverse-of = Bijection.right-inverse-of b"},{"Add":" }"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Inverse is an equivalence relation"},{"Add":""},{"Add":"-- Reflexivity"},{"Add":""},{"Add":"id : ∀ {s₁ s₂} → Reflexive (Inverse {s₁} {s₂})"},{"Add":"id {x = S} = record"},{"Add":" { to = F.id"},{"Add":" ; from = F.id"},{"Add":" ; inverse-of = record"},{"Add":" { left-inverse-of = LeftInverse.left-inverse-of id′"},{"Add":" ; right-inverse-of = LeftInverse.left-inverse-of id′"},{"Add":" }"},{"Add":" } where id′ = Left.id {S = S}"},{"Add":"{-# WARNING_ON_USAGE id"},{"Add":"\"Warning: id was deprecated in v2.0."},{"Add":"Please use either Function.Properties.Inverse.refl or"},{"Add":"Function.Construct.Identity.inverse instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"-- Transitivity"},{"Add":""},{"Add":"infixr 9 _∘_"},{"Add":""},{"Add":"_∘_ : ∀ {f₁ f₂ m₁ m₂ t₁ t₂} →"},{"Add":" TransFlip (Inverse {f₁} {f₂} {m₁} {m₂})"},{"Add":" (Inverse {m₁} {m₂} {t₁} {t₂})"},{"Add":" (Inverse {f₁} {f₂} {t₁} {t₂})"},{"Add":"f ∘ g = record"},{"Add":" { to = to f ⟪∘⟫ to g"},{"Add":" ; from = from g ⟪∘⟫ from f"},{"Add":" ; inverse-of = record"},{"Add":" { left-inverse-of = LeftInverse.left-inverse-of (Left._∘_ (left-inverse f) (left-inverse g))"},{"Add":" ; right-inverse-of = LeftInverse.left-inverse-of (Left._∘_ (right-inverse g) (right-inverse f))"},{"Add":" }"},{"Add":" } where open Inverse"},{"Add":"{-# WARNING_ON_USAGE _∘_"},{"Add":"\"Warning: _∘_ was deprecated in v2.0."},{"Add":"Please use either Function.Properties.Inverse.trans or"},{"Add":"Function.Construct.Composition.inverse instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"-- Symmetry."},{"Add":""},{"Add":"sym : ∀ {f₁ f₂ t₁ t₂} →"},{"Add":" Sym (Inverse {f₁} {f₂} {t₁} {t₂}) (Inverse {t₁} {t₂} {f₁} {f₂})"},{"Add":"sym inv = record"},{"Add":" { from = to"},{"Add":" ; to = from"},{"Add":" ; inverse-of = record"},{"Add":" { left-inverse-of = right-inverse-of"},{"Add":" ; right-inverse-of = left-inverse-of"},{"Add":" }"},{"Add":" } where open Inverse inv"},{"Add":"{-# WARNING_ON_USAGE sym"},{"Add":"\"Warning: sym was deprecated in v2.0."},{"Add":"Please use either Function.Properties.Inverse.sym or"},{"Add":"Function.Construct.Symmetry.inverse instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Transformations"},{"Add":""},{"Add":"map : ∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂}"},{"Add":" {f₁′ f₂′ t₁′ t₂′}"},{"Add":" {From′ : Setoid f₁′ f₂′} {To′ : Setoid t₁′ t₂′} →"},{"Add":" (t : (From ⟶ To) → (From′ ⟶ To′)) →"},{"Add":" (f : (To ⟶ From) → (To′ ⟶ From′)) →"},{"Add":" (∀ {to from} → from InverseOf to → f from InverseOf t to) →"},{"Add":" Inverse From To → Inverse From′ To′"},{"Add":"map t f pres eq = record"},{"Add":" { to = t to"},{"Add":" ; from = f from"},{"Add":" ; inverse-of = pres inverse-of"},{"Add":" } where open Inverse eq"},{"Add":""},{"Add":"zip : ∀ {f₁₁ f₂₁ t₁₁ t₂₁}"},{"Add":" {From₁ : Setoid f₁₁ f₂₁} {To₁ : Setoid t₁₁ t₂₁}"},{"Add":" {f₁₂ f₂₂ t₁₂ t₂₂}"},{"Add":" {From₂ : Setoid f₁₂ f₂₂} {To₂ : Setoid t₁₂ t₂₂}"},{"Add":" {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →"},{"Add":" (t : (From₁ ⟶ To₁) → (From₂ ⟶ To₂) → (From ⟶ To)) →"},{"Add":" (f : (To₁ ⟶ From₁) → (To₂ ⟶ From₂) → (To ⟶ From)) →"},{"Add":" (∀ {to₁ from₁ to₂ from₂} →"},{"Add":" from₁ InverseOf to₁ → from₂ InverseOf to₂ →"},{"Add":" f from₁ from₂ InverseOf t to₁ to₂) →"},{"Add":" Inverse From₁ To₁ → Inverse From₂ To₂ → Inverse From To"},{"Add":"zip t f pres eq₁ eq₂ = record"},{"Add":" { to = t (to eq₁) (to eq₂)"},{"Add":" ; from = f (from eq₁) (from eq₂)"},{"Add":" ; inverse-of = pres (inverse-of eq₁) (inverse-of eq₂)"},{"Add":" } where open Inverse"}]}},{"FileAdd":{"meta":{"basename":"Injection.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":"{-# OPTIONS --warn=noUserWarning #-}"},{"Add":""},{"Add":"module Function.Injection where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Function.Injection was deprecated in v2.0."},{"Add":"Use the standard function hierarchy in Function/Function.Bundles instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Function.Base as Fun using () renaming (_∘_ to _⟨∘⟩_)"},{"Add":"open import Level"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Function.Equality as F"},{"Add":" using (_⟶_; _⟨$⟩_ ; Π) renaming (_∘_ to _⟪∘⟫_)"},{"Add":"open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Injective functions"},{"Add":""},{"Add":"Injective : ∀ {a₁ a₂ b₁ b₂} {A : Setoid a₁ a₂} {B : Setoid b₁ b₂} →"},{"Add":" A ⟶ B → Set _"},{"Add":"Injective {A = A} {B} f = ∀ {x y} → f ⟨$⟩ x ≈₂ f ⟨$⟩ y → x ≈₁ y"},{"Add":" where"},{"Add":" open Setoid A renaming (_≈_ to _≈₁_)"},{"Add":" open Setoid B renaming (_≈_ to _≈₂_)"},{"Add":"{-# WARNING_ON_USAGE Injective"},{"Add":"\"Warning: Injective was deprecated in v2.0."},{"Add":"Please use Function.(Definitions.)Injective instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The set of all injections between two setoids"},{"Add":""},{"Add":"record Injection {f₁ f₂ t₁ t₂}"},{"Add":" (From : Setoid f₁ f₂) (To : Setoid t₁ t₂) :"},{"Add":" Set (f₁ ⊔ f₂ ⊔ t₁ ⊔ t₂) where"},{"Add":" field"},{"Add":" to : From ⟶ To"},{"Add":" injective : Injective to"},{"Add":""},{"Add":" open Π to public"},{"Add":"{-# WARNING_ON_USAGE Injection"},{"Add":"\"Warning: Injection was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)Injection instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The set of all injections from one set to another (i.e. injections"},{"Add":"-- with propositional equality)"},{"Add":""},{"Add":"infix 3 _↣_"},{"Add":""},{"Add":"_↣_ : ∀ {f t} → Set f → Set t → Set _"},{"Add":"From ↣ To = Injection (≡.setoid From) (≡.setoid To)"},{"Add":"{-# WARNING_ON_USAGE _↣_"},{"Add":"\"Warning: _↣_ was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)_↣_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"injection : ∀ {f t} {From : Set f} {To : Set t} → (to : From → To) →"},{"Add":" (∀ {x y} → to x ≡ to y → x ≡ y) → From ↣ To"},{"Add":"injection to injective = record"},{"Add":" { to = record"},{"Add":" { _⟨$⟩_ = to"},{"Add":" ; cong = ≡.cong to"},{"Add":" }"},{"Add":" ; injective = injective"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE injection"},{"Add":"\"Warning: injection was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)mk↣ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Identity and composition."},{"Add":""},{"Add":"infixr 9 _∘_"},{"Add":""},{"Add":"id : ∀ {s₁ s₂} {S : Setoid s₁ s₂} → Injection S S"},{"Add":"id = record"},{"Add":" { to = F.id"},{"Add":" ; injective = Fun.id"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE id"},{"Add":"\"Warning: id was deprecated in v2.0."},{"Add":"Please use Function.Properties.Injection.refl or"},{"Add":"Function.Construct.Identity.injection instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_∘_ : ∀ {f₁ f₂ m₁ m₂ t₁ t₂}"},{"Add":" {F : Setoid f₁ f₂} {M : Setoid m₁ m₂} {T : Setoid t₁ t₂} →"},{"Add":" Injection M T → Injection F M → Injection F T"},{"Add":"f ∘ g = record"},{"Add":" { to = to f ⟪∘⟫ to g"},{"Add":" ; injective = (λ {_} → injective g) ⟨∘⟩ injective f"},{"Add":" } where open Injection"},{"Add":"{-# WARNING_ON_USAGE _∘_"},{"Add":"\"Warning: _∘_ was deprecated in v2.0."},{"Add":"Please use Function.Properties.Injection.trans or"},{"Add":"Function.Construct.Composition.injection instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Indexed","metadata":576},"context":[{"change":0,"pos":1018846}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Relation","metadata":576},"context":[{"change":0,"pos":1166817}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Binary","metadata":576},"context":[{"change":0,"pos":1166838}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Equality.agda","metadata":0},"context":[{"change":0,"pos":1166860}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Function setoids and related constructions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Indexed.Relation.Binary.Equality where"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous using (IndexedSetoid)"},{"Add":""},{"Add":"-- A variant of setoid which uses the propositional equality setoid"},{"Add":"-- for the domain, and a more convenient definition of _≈_."},{"Add":""},{"Add":"≡-setoid : ∀ {f t₁ t₂} (From : Set f) → IndexedSetoid From t₁ t₂ → Setoid _ _"},{"Add":"≡-setoid From To = record"},{"Add":" { Carrier = (x : From) → Carrier x"},{"Add":" ; _≈_ = λ f g → ∀ x → f x ≈ g x"},{"Add":" ; isEquivalence = record"},{"Add":" { refl = λ {f} x → refl"},{"Add":" ; sym = λ f∼g x → sym (f∼g x)"},{"Add":" ; trans = λ f∼g g∼h x → trans (f∼g x) (g∼h x)"},{"Add":" }"},{"Add":" } where open IndexedSetoid To"},{"Add":""}]}},{"FileAdd":{"meta":{"basename":"Bundles.agda","metadata":0},"context":[{"change":0,"pos":1166817}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Operations on Relations for Indexed sets"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Indexed.Bundles where"},{"Add":""},{"Add":"open import Relation.Unary using (Pred)"},{"Add":"open import Function.Bundles using (_⟶_; _↣_; _↠_; _⤖_; _⇔_; _↩_; _↪_; _↩↪_; _↔_)"},{"Add":"open import Relation.Binary.Core hiding (_⇔_)"},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b i : Level"},{"Add":" I : Set i"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles specialised for lifting relations to indexed sets"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"infix 3 _⟶ᵢ_ _↣ᵢ_ _↠ᵢ_ _⤖ᵢ_ _⇔ᵢ_ _↩ᵢ_ _↪ᵢ_ _↩↪ᵢ_ _↔ᵢ_"},{"Add":""},{"Add":"_⟶ᵢ_ : Pred I a → Pred I b → Set _"},{"Add":"A ⟶ᵢ B = ∀ {i} → A i ⟶ B i"},{"Add":""},{"Add":"_↣ᵢ_ : Pred I a → Pred I b → Set _"},{"Add":"A ↣ᵢ B = ∀ {i} → A i ↣ B i"},{"Add":""},{"Add":"_↠ᵢ_ : Pred I a → Pred I b → Set _"},{"Add":"A ↠ᵢ B = ∀ {i} → A i ↠ B i"},{"Add":""},{"Add":"_⤖ᵢ_ : Pred I a → Pred I b → Set _"},{"Add":"A ⤖ᵢ B = ∀ {i} → A i ⤖ B i"},{"Add":""},{"Add":"_⇔ᵢ_ : Pred I a → Pred I b → Set _"},{"Add":"A ⇔ᵢ B = ∀ {i} → A i ⇔ B i"},{"Add":""},{"Add":"_↩ᵢ_ : Pred I a → Pred I b → Set _"},{"Add":"A ↩ᵢ B = ∀ {i} → A i ↩ B i"},{"Add":""},{"Add":"_↪ᵢ_ : Pred I a → Pred I b → Set _"},{"Add":"A ↪ᵢ B = ∀ {i} → A i ↪ B i"},{"Add":""},{"Add":"_↩↪ᵢ_ : Pred I a → Pred I b → Set _"},{"Add":"A ↩↪ᵢ B = ∀ {i} → A i ↩↪ B i"},{"Add":""},{"Add":"_↔ᵢ_ : Pred I a → Pred I b → Set _"},{"Add":"A ↔ᵢ B = ∀ {i} → A i ↔ B i"}]}},{"FileAdd":{"meta":{"basename":"Identity","metadata":576},"context":[{"change":0,"pos":1018846}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Effectful.agda","metadata":0},"context":[{"change":0,"pos":1169586}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- An effectful view of the identity function"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Identity.Effectful where"},{"Add":""},{"Add":"open import Effect.Functor"},{"Add":"open import Effect.Applicative"},{"Add":"open import Effect.Monad"},{"Add":"open import Effect.Comonad"},{"Add":"open import Function.Base using (id; _∘′_; _|>′_; _$′_; flip)"},{"Add":"open import Level"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ : Level"},{"Add":""},{"Add":"Identity : (A : Set ℓ) → Set ℓ"},{"Add":"Identity A = A"},{"Add":""},{"Add":"functor : RawFunctor {ℓ} Identity"},{"Add":"functor = record"},{"Add":" { _<$>_ = id"},{"Add":" }"},{"Add":""},{"Add":"applicative : RawApplicative {ℓ} Identity"},{"Add":"applicative = record"},{"Add":" { rawFunctor = functor"},{"Add":" ; pure = id"},{"Add":" ; _<*>_ = _$′_"},{"Add":" }"},{"Add":""},{"Add":"monad : RawMonad {ℓ} Identity"},{"Add":"monad = record"},{"Add":" { rawApplicative = applicative"},{"Add":" ; _>>=_ = _|>′_"},{"Add":" }"},{"Add":""},{"Add":"comonad : RawComonad {ℓ} Identity"},{"Add":"comonad = record"},{"Add":" { extract = id"},{"Add":" ; extend = _$′_"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Categorical.agda","metadata":0},"context":[{"change":0,"pos":1169586}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use"},{"Add":"-- `Function.Identity.Effectful` instead."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Identity.Categorical where"},{"Add":""},{"Add":"open import Function.Identity.Effectful public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Function.Identity.Categorical was deprecated in v2.0."},{"Add":"Use Function.Identity.Effectful instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"HalfAdjointEquivalence.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":"{-# OPTIONS --warn=noUserWarning #-}"},{"Add":""},{"Add":"module Function.HalfAdjointEquivalence where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Function.HalfAdjointEquivalence was deprecated in v2.0."},{"Add":"Use Function.Properties.Inverse.HalfAdjointEquivalence instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Function.Base"},{"Add":"open import Function.Equality using (_⟨$⟩_)"},{"Add":"open import Function.Inverse as Inv using (_↔_; module Inverse)"},{"Add":"open import Level"},{"Add":"open import Relation.Binary.PropositionalEquality"},{"Add":""},{"Add":"-- Half adjoint equivalences (see the HoTT book)."},{"Add":""},{"Add":"infix 4 _≃_"},{"Add":""},{"Add":"record _≃_ {a b} (A : Set a) (B : Set b) : Set (a ⊔ b) where"},{"Add":" field"},{"Add":" to : A → B"},{"Add":" from : B → A"},{"Add":" left-inverse-of : ∀ x → from (to x) ≡ x"},{"Add":" right-inverse-of : ∀ x → to (from x) ≡ x"},{"Add":" left-right :"},{"Add":" ∀ x → cong to (left-inverse-of x) ≡ right-inverse-of (to x)"},{"Add":""},{"Add":" -- Half adjoint equivalences can be turned into inverses."},{"Add":""},{"Add":" inverse : A ↔ B"},{"Add":" inverse = Inv.inverse to from left-inverse-of right-inverse-of"},{"Add":""},{"Add":" -- The forward direction of a half adjoint equivalence is injective."},{"Add":""},{"Add":" injective : ∀ {x y} → to x ≡ to y → x ≡ y"},{"Add":" injective {x} {y} to-x≡to-y ="},{"Add":" x ≡⟨ sym (left-inverse-of _) ⟩"},{"Add":" from (to x) ≡⟨ cong from to-x≡to-y ⟩"},{"Add":" from (to y) ≡⟨ left-inverse-of _ ⟩"},{"Add":" y ∎"},{"Add":" where"},{"Add":" open ≡-Reasoning"},{"Add":""},{"Add":"-- Inverses can be turned into half adjoint equivalences."},{"Add":"--"},{"Add":"-- (This proof is based on one in the HoTT book.)"},{"Add":""},{"Add":"↔→≃ : ∀ {a b} {A : Set a} {B : Set b} → A ↔ B → A ≃ B"},{"Add":"↔→≃ A↔B = record"},{"Add":" { to = to ⟨$⟩_"},{"Add":" ; from = from ⟨$⟩_"},{"Add":" ; left-inverse-of = left-inverse-of"},{"Add":" ; right-inverse-of = right-inverse-of"},{"Add":" ; left-right = left-right"},{"Add":" }"},{"Add":" where"},{"Add":" open ≡-Reasoning"},{"Add":" open module A↔B = Inverse A↔B using (to; from; left-inverse-of)"},{"Add":""},{"Add":" right-inverse-of : ∀ x → to ⟨$⟩ (from ⟨$⟩ x) ≡ x"},{"Add":" right-inverse-of x ="},{"Add":" to ⟨$⟩ (from ⟨$⟩ x) ≡⟨ sym (A↔B.right-inverse-of _) ⟩"},{"Add":" to ⟨$⟩ (from ⟨$⟩ (to ⟨$⟩ (from ⟨$⟩ x))) ≡⟨ cong (to ⟨$⟩_) (left-inverse-of _) ⟩"},{"Add":" to ⟨$⟩ (from ⟨$⟩ x) ≡⟨ A↔B.right-inverse-of _ ⟩"},{"Add":" x ∎"},{"Add":""},{"Add":" left-right :"},{"Add":" ∀ x →"},{"Add":" cong (to ⟨$⟩_) (left-inverse-of x) ≡ right-inverse-of (to ⟨$⟩ x)"},{"Add":" left-right x ="},{"Add":" cong (to ⟨$⟩_) (left-inverse-of x) ≡⟨⟩"},{"Add":""},{"Add":" trans refl (cong (to ⟨$⟩_) (left-inverse-of _)) ≡⟨ cong (λ p → trans p (cong (to ⟨$⟩_) _))"},{"Add":" (sym (trans-symˡ (A↔B.right-inverse-of _))) ⟩"},{"Add":" trans (trans (sym (A↔B.right-inverse-of _))"},{"Add":" (A↔B.right-inverse-of _))"},{"Add":" (cong (to ⟨$⟩_) (left-inverse-of _)) ≡⟨ trans-assoc (sym (A↔B.right-inverse-of _)) ⟩"},{"Add":""},{"Add":" trans (sym (A↔B.right-inverse-of _))"},{"Add":" (trans (A↔B.right-inverse-of _)"},{"Add":" (cong (to ⟨$⟩_) (left-inverse-of _))) ≡⟨ cong (trans (sym (A↔B.right-inverse-of _))) lemma ⟩"},{"Add":""},{"Add":" trans (sym (A↔B.right-inverse-of _))"},{"Add":" (trans (cong (to ⟨$⟩_) (left-inverse-of _))"},{"Add":" (trans (A↔B.right-inverse-of _) refl)) ≡⟨⟩"},{"Add":""},{"Add":" right-inverse-of (to ⟨$⟩ x) ∎"},{"Add":" where"},{"Add":" lemma ="},{"Add":" trans (A↔B.right-inverse-of _)"},{"Add":" (cong (to ⟨$⟩_) (left-inverse-of _)) ≡⟨ cong (trans (A↔B.right-inverse-of _)) (sym (cong-id _)) ⟩"},{"Add":""},{"Add":" trans (A↔B.right-inverse-of _)"},{"Add":" (cong id (cong (to ⟨$⟩_) (left-inverse-of _))) ≡⟨ sym (naturality A↔B.right-inverse-of) ⟩"},{"Add":""},{"Add":" trans (cong ((to ⟨$⟩_) ∘ (from ⟨$⟩_))"},{"Add":" (cong (to ⟨$⟩_) (left-inverse-of _)))"},{"Add":" (A↔B.right-inverse-of _) ≡⟨ cong (λ p → trans p (A↔B.right-inverse-of _))"},{"Add":" (sym (cong-∘ _)) ⟩"},{"Add":" trans (cong ((to ⟨$⟩_) ∘ (from ⟨$⟩_) ∘ (to ⟨$⟩_))"},{"Add":" (left-inverse-of _))"},{"Add":" (A↔B.right-inverse-of _) ≡⟨ cong (λ p → trans p (A↔B.right-inverse-of _))"},{"Add":" (cong-∘ _) ⟩"},{"Add":" trans (cong (to ⟨$⟩_)"},{"Add":" (cong ((from ⟨$⟩_) ∘ (to ⟨$⟩_))"},{"Add":" (left-inverse-of _)))"},{"Add":" (A↔B.right-inverse-of _) ≡⟨ cong (λ p → trans (cong (to ⟨$⟩_) p) _)"},{"Add":" (cong-≡id left-inverse-of) ⟩"},{"Add":" trans (cong (to ⟨$⟩_) (left-inverse-of _))"},{"Add":" (A↔B.right-inverse-of _) ≡⟨ cong (trans (cong (to ⟨$⟩_) (left-inverse-of _)))"},{"Add":" (sym (trans-reflʳ _)) ⟩"},{"Add":" trans (cong (to ⟨$⟩_) (left-inverse-of _))"},{"Add":" (trans (A↔B.right-inverse-of _) refl) ∎"}]}},{"FileAdd":{"meta":{"basename":"Equivalence.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":"{-# OPTIONS --warn=noUserWarning #-}"},{"Add":""},{"Add":"module Function.Equivalence where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Function.Equivalence was deprecated in v2.0."},{"Add":"Use the standard function hierarchy in Function/Function.Bundles instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Function.Base using (flip)"},{"Add":"open import Function.Equality as F"},{"Add":" using (_⟶_; _⟨$⟩_; →-to-⟶) renaming (_∘_ to _⟪∘⟫_)"},{"Add":"open import Level"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Definitions using (Reflexive; TransFlip; Sym)"},{"Add":"import Relation.Binary.PropositionalEquality as ≡"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Setoid equivalence"},{"Add":""},{"Add":"record Equivalence {f₁ f₂ t₁ t₂}"},{"Add":" (From : Setoid f₁ f₂) (To : Setoid t₁ t₂) :"},{"Add":" Set (f₁ ⊔ f₂ ⊔ t₁ ⊔ t₂) where"},{"Add":" field"},{"Add":" to : From ⟶ To"},{"Add":" from : To ⟶ From"},{"Add":"{-# WARNING_ON_USAGE Equivalence"},{"Add":"\"Warning: Equivalence was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)Equivalence instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The set of all equivalences between two sets (i.e. equivalences"},{"Add":"-- with propositional equality)"},{"Add":""},{"Add":"infix 3 _⇔_"},{"Add":""},{"Add":"_⇔_ : ∀ {f t} → Set f → Set t → Set _"},{"Add":"From ⇔ To = Equivalence (≡.setoid From) (≡.setoid To)"},{"Add":"{-# WARNING_ON_USAGE _⇔_"},{"Add":"\"Warning: _⇔_ was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)_⇔_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"equivalence : ∀ {f t} {From : Set f} {To : Set t} →"},{"Add":" (From → To) → (To → From) → From ⇔ To"},{"Add":"equivalence to from = record"},{"Add":" { to = →-to-⟶ to"},{"Add":" ; from = →-to-⟶ from"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE equivalence"},{"Add":"\"Warning: equivalence was deprecated in v2.0."},{"Add":"Please use Function.Properties.Equivalence.mkEquivalence instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Equivalence is an equivalence relation"},{"Add":""},{"Add":"-- Identity and composition (reflexivity and transitivity)."},{"Add":""},{"Add":"id : ∀ {s₁ s₂} → Reflexive (Equivalence {s₁} {s₂})"},{"Add":"id {x = S} = record"},{"Add":" { to = F.id"},{"Add":" ; from = F.id"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE id"},{"Add":"\"Warning: id was deprecated in v2.0."},{"Add":"Please use Function.Properties.Equivalence.refl or"},{"Add":"Function.Construct.Identity.equivalence instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"infixr 9 _∘_"},{"Add":""},{"Add":"_∘_ : ∀ {f₁ f₂ m₁ m₂ t₁ t₂} →"},{"Add":" TransFlip (Equivalence {f₁} {f₂} {m₁} {m₂})"},{"Add":" (Equivalence {m₁} {m₂} {t₁} {t₂})"},{"Add":" (Equivalence {f₁} {f₂} {t₁} {t₂})"},{"Add":"f ∘ g = record"},{"Add":" { to = to f ⟪∘⟫ to g"},{"Add":" ; from = from g ⟪∘⟫ from f"},{"Add":" } where open Equivalence"},{"Add":"{-# WARNING_ON_USAGE _∘_"},{"Add":"\"Warning: _∘_ was deprecated in v2.0."},{"Add":"Please use Function.Properties.Equivalence.trans or"},{"Add":"Function.Construct.Composition.equivalence instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"-- Symmetry."},{"Add":""},{"Add":"sym : ∀ {f₁ f₂ t₁ t₂} →"},{"Add":" Sym (Equivalence {f₁} {f₂} {t₁} {t₂})"},{"Add":" (Equivalence {t₁} {t₂} {f₁} {f₂})"},{"Add":"sym eq = record"},{"Add":" { from = to"},{"Add":" ; to = from"},{"Add":" } where open Equivalence eq"},{"Add":"{-# WARNING_ON_USAGE sym"},{"Add":"\"Warning: sym was deprecated in v2.0."},{"Add":"Please use Function.Properties.Equivalence.sym or"},{"Add":"Function.Construct.Symmetry.equivalence instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"-- For fixed universe levels we can construct setoids."},{"Add":""},{"Add":"setoid : (s₁ s₂ : Level) → Setoid (suc (s₁ ⊔ s₂)) (s₁ ⊔ s₂)"},{"Add":"setoid s₁ s₂ = record"},{"Add":" { Carrier = Setoid s₁ s₂"},{"Add":" ; _≈_ = Equivalence"},{"Add":" ; isEquivalence = record"},{"Add":" { refl = id"},{"Add":" ; sym = sym"},{"Add":" ; trans = flip _∘_"},{"Add":" }"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE setoid"},{"Add":"\"Warning: setoid was deprecated in v2.0."},{"Add":"Please use Function.Properties.Equivalence.setoid instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"⇔-setoid : (ℓ : Level) → Setoid (suc ℓ) ℓ"},{"Add":"⇔-setoid ℓ = record"},{"Add":" { Carrier = Set ℓ"},{"Add":" ; _≈_ = _⇔_"},{"Add":" ; isEquivalence = record"},{"Add":" { refl = id"},{"Add":" ; sym = sym"},{"Add":" ; trans = flip _∘_"},{"Add":" }"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE ⇔-setoid"},{"Add":"\"Warning: ⇔-setoid was deprecated in v2.0."},{"Add":"Please use Function.Properties.Equivalence.⇔-setoid instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Transformations"},{"Add":""},{"Add":"map : ∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂}"},{"Add":" {f₁′ f₂′ t₁′ t₂′}"},{"Add":" {From′ : Setoid f₁′ f₂′} {To′ : Setoid t₁′ t₂′} →"},{"Add":" ((From ⟶ To) → (From′ ⟶ To′)) →"},{"Add":" ((To ⟶ From) → (To′ ⟶ From′)) →"},{"Add":" Equivalence From To → Equivalence From′ To′"},{"Add":"map t f eq = record { to = t to; from = f from }"},{"Add":" where open Equivalence eq"},{"Add":""},{"Add":"zip : ∀ {f₁₁ f₂₁ t₁₁ t₂₁}"},{"Add":" {From₁ : Setoid f₁₁ f₂₁} {To₁ : Setoid t₁₁ t₂₁}"},{"Add":" {f₁₂ f₂₂ t₁₂ t₂₂}"},{"Add":" {From₂ : Setoid f₁₂ f₂₂} {To₂ : Setoid t₁₂ t₂₂}"},{"Add":" {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →"},{"Add":" ((From₁ ⟶ To₁) → (From₂ ⟶ To₂) → (From ⟶ To)) →"},{"Add":" ((To₁ ⟶ From₁) → (To₂ ⟶ From₂) → (To ⟶ From)) →"},{"Add":" Equivalence From₁ To₁ → Equivalence From₂ To₂ →"},{"Add":" Equivalence From To"},{"Add":"zip t f eq₁ eq₂ ="},{"Add":" record { to = t (to eq₁) (to eq₂); from = f (from eq₁) (from eq₂) }"},{"Add":" where open Equivalence"}]}},{"FileAdd":{"meta":{"basename":"Equality.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Equality where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Function.Equality was deprecated in v2.0."},{"Add":"Use the standard function hierarchy in Function/Function.Bundles instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"import Function.Base as Fun"},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous"},{"Add":" using (IndexedSetoid; _=[_]⇒_)"},{"Add":"import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial"},{"Add":" as Trivial"},{"Add":"import Relation.Binary.PropositionalEquality.Core as ≡"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Functions which preserve equality"},{"Add":""},{"Add":"record Π {f₁ f₂ t₁ t₂}"},{"Add":" (From : Setoid f₁ f₂)"},{"Add":" (To : IndexedSetoid (Setoid.Carrier From) t₁ t₂) :"},{"Add":" Set (f₁ ⊔ f₂ ⊔ t₁ ⊔ t₂) where"},{"Add":" infixl 5 _⟨$⟩_"},{"Add":" field"},{"Add":" _⟨$⟩_ : (x : Setoid.Carrier From) → IndexedSetoid.Carrier To x"},{"Add":" cong : Setoid._≈_ From =[ _⟨$⟩_ ]⇒ IndexedSetoid._≈_ To"},{"Add":"{-# WARNING_ON_USAGE Π"},{"Add":"\"Warning: Π was deprecated in v2.0."},{"Add":"Please use Function.Dependent.Bundles.Func instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open Π public"},{"Add":""},{"Add":"infixr 0 _⟶_"},{"Add":""},{"Add":"_⟶_ : ∀ {f₁ f₂ t₁ t₂} → Setoid f₁ f₂ → Setoid t₁ t₂ → Set _"},{"Add":"From ⟶ To = Π From (Trivial.indexedSetoid To)"},{"Add":"{-# WARNING_ON_USAGE _⟶_"},{"Add":"\"Warning: _⟶_ was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)Func instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Identity and composition."},{"Add":""},{"Add":"id : ∀ {a₁ a₂} {A : Setoid a₁ a₂} → A ⟶ A"},{"Add":"id = record { _⟨$⟩_ = Fun.id; cong = Fun.id }"},{"Add":"{-# WARNING_ON_USAGE id"},{"Add":"\"Warning: id was deprecated in v2.0."},{"Add":"Please use Function.Construct.Identity.function instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"infixr 9 _∘_"},{"Add":""},{"Add":"_∘_ : ∀ {a₁ a₂} {A : Setoid a₁ a₂}"},{"Add":" {b₁ b₂} {B : Setoid b₁ b₂}"},{"Add":" {c₁ c₂} {C : Setoid c₁ c₂} →"},{"Add":" B ⟶ C → A ⟶ B → A ⟶ C"},{"Add":"f ∘ g = record"},{"Add":" { _⟨$⟩_ = Fun._∘_ (_⟨$⟩_ f) (_⟨$⟩_ g)"},{"Add":" ; cong = Fun._∘_ (cong f) (cong g)"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE _∘_"},{"Add":"\"Warning: _∘_ was deprecated in v2.0."},{"Add":"Please use Function.Construct.Composition.function instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"-- Constant equality-preserving function."},{"Add":""},{"Add":"const : ∀ {a₁ a₂} {A : Setoid a₁ a₂}"},{"Add":" {b₁ b₂} {B : Setoid b₁ b₂} →"},{"Add":" Setoid.Carrier B → A ⟶ B"},{"Add":"const {B = B} b = record"},{"Add":" { _⟨$⟩_ = Fun.const b"},{"Add":" ; cong = Fun.const (Setoid.refl B)"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE const"},{"Add":"\"Warning: const was deprecated in v2.0."},{"Add":"Please use Function.Construct.Constant.function instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Function setoids"},{"Add":""},{"Add":"-- Dependent."},{"Add":""},{"Add":"setoid : ∀ {f₁ f₂ t₁ t₂}"},{"Add":" (From : Setoid f₁ f₂) →"},{"Add":" IndexedSetoid (Setoid.Carrier From) t₁ t₂ →"},{"Add":" Setoid _ _"},{"Add":"setoid From To = record"},{"Add":" { Carrier = Π From To"},{"Add":" ; _≈_ = λ f g → ∀ {x y} → x ≈₁ y → f ⟨$⟩ x ≈₂ g ⟨$⟩ y"},{"Add":" ; isEquivalence = record"},{"Add":" { refl = λ {f} → cong f"},{"Add":" ; sym = λ f∼g x∼y → To.sym (f∼g (From.sym x∼y))"},{"Add":" ; trans = λ f∼g g∼h x∼y → To.trans (f∼g From.refl) (g∼h x∼y)"},{"Add":" }"},{"Add":" }"},{"Add":" where"},{"Add":" open module From = Setoid From using () renaming (_≈_ to _≈₁_)"},{"Add":" open module To = IndexedSetoid To using () renaming (_≈_ to _≈₂_)"},{"Add":""},{"Add":"-- Non-dependent."},{"Add":""},{"Add":"infixr 0 _⇨_"},{"Add":""},{"Add":"_⇨_ : ∀ {f₁ f₂ t₁ t₂} → Setoid f₁ f₂ → Setoid t₁ t₂ → Setoid _ _"},{"Add":"From ⇨ To = setoid From (Trivial.indexedSetoid To)"},{"Add":""},{"Add":"-- A variant of setoid which uses the propositional equality setoid"},{"Add":"-- for the domain, and a more convenient definition of _≈_."},{"Add":""},{"Add":"≡-setoid : ∀ {f t₁ t₂} (From : Set f) → IndexedSetoid From t₁ t₂ → Setoid _ _"},{"Add":"≡-setoid From To = record"},{"Add":" { Carrier = (x : From) → Carrier x"},{"Add":" ; _≈_ = λ f g → ∀ x → f x ≈ g x"},{"Add":" ; isEquivalence = record"},{"Add":" { refl = λ {f} x → refl"},{"Add":" ; sym = λ f∼g x → sym (f∼g x)"},{"Add":" ; trans = λ f∼g g∼h x → trans (f∼g x) (g∼h x)"},{"Add":" }"},{"Add":" } where open IndexedSetoid To"},{"Add":""},{"Add":"-- Parameter swapping function."},{"Add":""},{"Add":"flip : ∀ {a₁ a₂} {A : Setoid a₁ a₂}"},{"Add":" {b₁ b₂} {B : Setoid b₁ b₂}"},{"Add":" {c₁ c₂} {C : Setoid c₁ c₂} →"},{"Add":" A ⟶ B ⇨ C → B ⟶ A ⇨ C"},{"Add":"flip {B = B} f = record"},{"Add":" { _⟨$⟩_ = λ b → record"},{"Add":" { _⟨$⟩_ = λ a → f ⟨$⟩ a ⟨$⟩ b"},{"Add":" ; cong = λ a₁≈a₂ → cong f a₁≈a₂ (Setoid.refl B) }"},{"Add":" ; cong = λ b₁≈b₂ a₁≈a₂ → cong f a₁≈a₂ b₁≈b₂"},{"Add":" }"},{"Add":""},{"Add":"→-to-⟶ : ∀ {a b ℓ} {A : Set a} {B : Setoid b ℓ} →"},{"Add":" (A → Setoid.Carrier B) → ≡.setoid A ⟶ B"},{"Add":"→-to-⟶ {B = B} to = record"},{"Add":" { _⟨$⟩_ = to"},{"Add":" ; cong = λ { ≡.refl → Setoid.refl B }"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Endomorphism","metadata":576},"context":[{"change":0,"pos":1018846}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Setoid.agda","metadata":0},"context":[{"change":0,"pos":1187206}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (_Preserves_⟶_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":""},{"Add":"module Function.Endomorphism.Setoid {c e} (S : Setoid c e) where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Function.Endomorphism.Setoid was deprecated in v2.1."},{"Add":"Use Function.Endo.Setoid instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Agda.Builtin.Equality"},{"Add":""},{"Add":"open import Algebra"},{"Add":"open import Algebra.Structures"},{"Add":"open import Algebra.Morphism; open Definitions"},{"Add":"open import Function.Equality using (setoid; _⟶_; id; _∘_; cong)"},{"Add":"open import Data.Nat.Base using (ℕ; _+_); open ℕ"},{"Add":"open import Data.Nat.Properties"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":""},{"Add":"import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial as Trivial"},{"Add":""},{"Add":"private"},{"Add":" module E = Setoid (setoid S (Trivial.indexedSetoid S))"},{"Add":" open E hiding (refl)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Basic type and functions"},{"Add":""},{"Add":"Endo : Set _"},{"Add":"Endo = S ⟶ S"},{"Add":""},{"Add":"infixr 8 _^_"},{"Add":""},{"Add":"_^_ : Endo → ℕ → Endo"},{"Add":"f ^ zero = id"},{"Add":"f ^ suc n = f ∘ (f ^ n)"},{"Add":""},{"Add":"^-cong₂ : ∀ f → (f ^_) Preserves _≡_ ⟶ _≈_"},{"Add":"^-cong₂ f {n} refl = cong (f ^ n)"},{"Add":""},{"Add":"^-homo : ∀ f → Homomorphic₂ ℕ Endo _≈_ (f ^_) _+_ _∘_"},{"Add":"^-homo f zero n x≈y = cong (f ^ n) x≈y"},{"Add":"^-homo f (suc m) n x≈y = cong f (^-homo f m n x≈y)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"∘-isMagma : IsMagma _≈_ _∘_"},{"Add":"∘-isMagma = record"},{"Add":" { isEquivalence = isEquivalence"},{"Add":" ; ∙-cong = λ g f x → g (f x)"},{"Add":" }"},{"Add":""},{"Add":"∘-magma : Magma _ _"},{"Add":"∘-magma = record { isMagma = ∘-isMagma }"},{"Add":""},{"Add":"∘-isSemigroup : IsSemigroup _≈_ _∘_"},{"Add":"∘-isSemigroup = record"},{"Add":" { isMagma = ∘-isMagma"},{"Add":" ; assoc = λ h g f x≈y → cong h (cong g (cong f x≈y))"},{"Add":" }"},{"Add":""},{"Add":"∘-semigroup : Semigroup _ _"},{"Add":"∘-semigroup = record { isSemigroup = ∘-isSemigroup }"},{"Add":""},{"Add":"∘-id-isMonoid : IsMonoid _≈_ _∘_ id"},{"Add":"∘-id-isMonoid = record"},{"Add":" { isSemigroup = ∘-isSemigroup"},{"Add":" ; identity = cong , cong"},{"Add":" }"},{"Add":""},{"Add":"∘-id-monoid : Monoid _ _"},{"Add":"∘-id-monoid = record { isMonoid = ∘-id-isMonoid }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Homomorphism"},{"Add":""},{"Add":"^-isSemigroupMorphism : ∀ f → IsSemigroupMorphism +-semigroup ∘-semigroup (f ^_)"},{"Add":"^-isSemigroupMorphism f = record"},{"Add":" { ⟦⟧-cong = ^-cong₂ f"},{"Add":" ; ∙-homo = ^-homo f"},{"Add":" }"},{"Add":""},{"Add":"^-isMonoidMorphism : ∀ f → IsMonoidMorphism +-0-monoid ∘-id-monoid (f ^_)"},{"Add":"^-isMonoidMorphism f = record"},{"Add":" { sm-homo = ^-isSemigroupMorphism f"},{"Add":" ; ε-homo = λ x≈y → x≈y"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Propositional.agda","metadata":0},"context":[{"change":0,"pos":1187206}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Endomorphism.Propositional {a} (A : Set a) where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Function.Endomorphism.Propositional was deprecated in v2.1."},{"Add":"Use Function.Endo.Propositional instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Algebra"},{"Add":"open import Algebra.Morphism; open Definitions"},{"Add":""},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc; _+_)"},{"Add":"open import Data.Nat.Properties using (+-0-monoid; +-semigroup)"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":""},{"Add":"open import Function.Base using (id; _∘′_; _∋_)"},{"Add":"open import Function.Equality using (_⟨$⟩_)"},{"Add":"open import Relation.Binary.Core using (_Preserves_⟶_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":""},{"Add":"import Function.Endomorphism.Setoid (≡.setoid A) as Setoid"},{"Add":""},{"Add":"Endo : Set a"},{"Add":"Endo = A → A"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Conversion back and forth with the Setoid-based notion of Endomorphism"},{"Add":""},{"Add":"fromSetoidEndo : Setoid.Endo → Endo"},{"Add":"fromSetoidEndo = _⟨$⟩_"},{"Add":""},{"Add":"toSetoidEndo : Endo → Setoid.Endo"},{"Add":"toSetoidEndo f = record"},{"Add":" { _⟨$⟩_ = f"},{"Add":" ; cong = cong f"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- N-th composition"},{"Add":""},{"Add":"infixr 8 _^_"},{"Add":""},{"Add":"_^_ : Endo → ℕ → Endo"},{"Add":"f ^ zero = id"},{"Add":"f ^ suc n = f ∘′ (f ^ n)"},{"Add":""},{"Add":"^-homo : ∀ f → Homomorphic₂ ℕ Endo _≡_ (f ^_) _+_ _∘′_"},{"Add":"^-homo f zero n = refl"},{"Add":"^-homo f (suc m) n = cong (f ∘′_) (^-homo f m n)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"∘-isMagma : IsMagma _≡_ (Op₂ Endo ∋ _∘′_)"},{"Add":"∘-isMagma = record"},{"Add":" { isEquivalence = ≡.isEquivalence"},{"Add":" ; ∙-cong = cong₂ _∘′_"},{"Add":" }"},{"Add":""},{"Add":"∘-magma : Magma _ _"},{"Add":"∘-magma = record { isMagma = ∘-isMagma }"},{"Add":""},{"Add":"∘-isSemigroup : IsSemigroup _≡_ (Op₂ Endo ∋ _∘′_)"},{"Add":"∘-isSemigroup = record"},{"Add":" { isMagma = ∘-isMagma"},{"Add":" ; assoc = λ _ _ _ → refl"},{"Add":" }"},{"Add":""},{"Add":"∘-semigroup : Semigroup _ _"},{"Add":"∘-semigroup = record { isSemigroup = ∘-isSemigroup }"},{"Add":""},{"Add":"∘-id-isMonoid : IsMonoid _≡_ _∘′_ id"},{"Add":"∘-id-isMonoid = record"},{"Add":" { isSemigroup = ∘-isSemigroup"},{"Add":" ; identity = (λ _ → refl) , (λ _ → refl)"},{"Add":" }"},{"Add":""},{"Add":"∘-id-monoid : Monoid _ _"},{"Add":"∘-id-monoid = record { isMonoid = ∘-id-isMonoid }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Homomorphism"},{"Add":""},{"Add":"^-isSemigroupMorphism : ∀ f → IsSemigroupMorphism +-semigroup ∘-semigroup (f ^_)"},{"Add":"^-isSemigroupMorphism f = record"},{"Add":" { ⟦⟧-cong = cong (f ^_)"},{"Add":" ; ∙-homo = ^-homo f"},{"Add":" }"},{"Add":""},{"Add":"^-isMonoidMorphism : ∀ f → IsMonoidMorphism +-0-monoid ∘-id-monoid (f ^_)"},{"Add":"^-isMonoidMorphism f = record"},{"Add":" { sm-homo = ^-isSemigroupMorphism f"},{"Add":" ; ε-homo = refl"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Endo","metadata":576},"context":[{"change":0,"pos":1018846}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Setoid.agda","metadata":0},"context":[{"change":0,"pos":1193013}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Endomorphisms on a Setoid"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":""},{"Add":"module Function.Endo.Setoid {c e} (S : Setoid c e) where"},{"Add":""},{"Add":"open import Agda.Builtin.Equality using (_≡_)"},{"Add":""},{"Add":"open import Algebra using (Semigroup; Magma; RawMagma; Monoid; RawMonoid)"},{"Add":"import Algebra.Definitions.RawMonoid as RawMonoidDefinitions"},{"Add":"import Algebra.Properties.Monoid.Mult as MonoidMultProperties"},{"Add":"open import Algebra.Structures using (IsMagma; IsSemigroup; IsMonoid)"},{"Add":"open import Algebra.Morphism"},{"Add":" using (module Definitions; IsMagmaHomomorphism; IsMonoidHomomorphism)"},{"Add":"open Definitions using (Homomorphic₂)"},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc; _+_; +-rawMagma; +-0-rawMonoid)"},{"Add":"open import Data.Nat.Properties using (+-semigroup; +-identityʳ)"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Bundles using (Func; _⟶ₛ_; _⟨$⟩_)"},{"Add":"open import Function.Construct.Identity using () renaming (function to identity)"},{"Add":"open import Function.Construct.Composition using () renaming (function to _∘_)"},{"Add":"open import Function.Relation.Binary.Setoid.Equality as Eq using (_⇨_)"},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Relation.Binary.Core using (_Preserves_⟶_)"},{"Add":""},{"Add":"private"},{"Add":" open module E = Setoid (S ⇨ S) hiding (refl)"},{"Add":" module S = Setoid S"},{"Add":" open Func using (cong)"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Basic type and raw bundles"},{"Add":""},{"Add":"Endo : Set _"},{"Add":"Endo = S ⟶ₛ S"},{"Add":""},{"Add":"private"},{"Add":" id : Endo"},{"Add":" id = identity S"},{"Add":""},{"Add":" ∘-id-rawMonoid : RawMonoid (c ⊔ e) (c ⊔ e)"},{"Add":" ∘-id-rawMonoid = record { Carrier = Endo; _≈_ = _≈_ ; _∙_ = _∘_ ; ε = id }"},{"Add":""},{"Add":" open RawMonoid ∘-id-rawMonoid"},{"Add":" using ()"},{"Add":" renaming (rawMagma to ∘-rawMagma)"},{"Add":""},{"Add":"--------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"∘-isMagma : IsMagma _≈_ _∘_"},{"Add":"∘-isMagma = record"},{"Add":" { isEquivalence = isEquivalence"},{"Add":" ; ∙-cong = λ {_} {_} {_} {k} f≈g h≈k x → S.trans (h≈k _) (cong k (f≈g x))"},{"Add":" }"},{"Add":""},{"Add":"∘-magma : Magma (c ⊔ e) (c ⊔ e)"},{"Add":"∘-magma = record { isMagma = ∘-isMagma }"},{"Add":""},{"Add":"∘-isSemigroup : IsSemigroup _≈_ _∘_"},{"Add":"∘-isSemigroup = record"},{"Add":" { isMagma = ∘-isMagma"},{"Add":" ; assoc = λ _ _ _ _ → S.refl"},{"Add":" }"},{"Add":""},{"Add":"∘-semigroup : Semigroup (c ⊔ e) (c ⊔ e)"},{"Add":"∘-semigroup = record { isSemigroup = ∘-isSemigroup }"},{"Add":""},{"Add":"∘-id-isMonoid : IsMonoid _≈_ _∘_ id"},{"Add":"∘-id-isMonoid = record"},{"Add":" { isSemigroup = ∘-isSemigroup"},{"Add":" ; identity = (λ _ _ → S.refl) , (λ _ _ → S.refl)"},{"Add":" }"},{"Add":""},{"Add":"∘-id-monoid : Monoid (c ⊔ e) (c ⊔ e)"},{"Add":"∘-id-monoid = record { isMonoid = ∘-id-isMonoid }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- -- n-th iterated composition"},{"Add":""},{"Add":"infixr 8 _^_"},{"Add":""},{"Add":"_^_ : Endo → ℕ → Endo"},{"Add":"f ^ n = n × f where open RawMonoidDefinitions ∘-id-rawMonoid using (_×_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Homomorphism"},{"Add":""},{"Add":"module _ (f : Endo) where"},{"Add":""},{"Add":" open MonoidMultProperties ∘-id-monoid using (×-congˡ; ×-homo-+)"},{"Add":""},{"Add":" ^-cong₂ : (f ^_) Preserves _≡_ ⟶ _≈_"},{"Add":" ^-cong₂ = ×-congˡ {f}"},{"Add":""},{"Add":" ^-homo : Homomorphic₂ ℕ Endo _≈_ (f ^_) _+_ _∘_"},{"Add":" ^-homo = ×-homo-+ f"},{"Add":""},{"Add":" ^-isMagmaHomomorphism : IsMagmaHomomorphism +-rawMagma ∘-rawMagma (f ^_)"},{"Add":" ^-isMagmaHomomorphism = record"},{"Add":" { isRelHomomorphism = record { cong = ^-cong₂ }"},{"Add":" ; homo = ^-homo"},{"Add":" }"},{"Add":""},{"Add":" ^-isMonoidHomomorphism : IsMonoidHomomorphism +-0-rawMonoid ∘-id-rawMonoid (f ^_)"},{"Add":" ^-isMonoidHomomorphism = record"},{"Add":" { isMagmaHomomorphism = ^-isMagmaHomomorphism"},{"Add":" ; ε-homo = λ _ → S.refl"},{"Add":" }"},{"Add":""}]}},{"FileAdd":{"meta":{"basename":"Propositional.agda","metadata":0},"context":[{"change":0,"pos":1193013}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Endomorphisms on a Set"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Endo.Propositional {a} (A : Set a) where"},{"Add":""},{"Add":"open import Algebra using (Semigroup; Magma; RawMagma; Monoid; RawMonoid)"},{"Add":"open import Algebra.Core"},{"Add":"import Algebra.Definitions.RawMonoid as RawMonoidDefinitions"},{"Add":"import Algebra.Properties.Monoid.Mult as MonoidMultProperties"},{"Add":"open import Algebra.Structures using (IsMagma; IsSemigroup; IsMonoid)"},{"Add":"open import Algebra.Morphism"},{"Add":" using (module Definitions; IsMagmaHomomorphism; IsMonoidHomomorphism)"},{"Add":"open Definitions using (Homomorphic₂)"},{"Add":""},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc; _+_; +-rawMagma; +-0-rawMonoid)"},{"Add":"open import Data.Nat.Properties using (+-0-monoid; +-semigroup)"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":""},{"Add":"open import Function.Base using (id; _∘′_; _∋_; flip)"},{"Add":"open import Function.Bundles using (Func; _⟶ₛ_; _⟨$⟩_)"},{"Add":"open import Relation.Binary.Core using (_Preserves_⟶_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":""},{"Add":"import Function.Endo.Setoid (≡.setoid A) as Setoid"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Basic type and raw bundles"},{"Add":""},{"Add":"Endo : Set a"},{"Add":"Endo = A → A"},{"Add":""},{"Add":"private"},{"Add":""},{"Add":" _∘_ : Op₂ Endo"},{"Add":" _∘_ = _∘′_"},{"Add":""},{"Add":" ∘-id-rawMonoid : RawMonoid a a"},{"Add":" ∘-id-rawMonoid = record { Carrier = Endo; _≈_ = _≡_ ; _∙_ = _∘_ ; ε = id }"},{"Add":""},{"Add":" open RawMonoid ∘-id-rawMonoid"},{"Add":" using ()"},{"Add":" renaming (rawMagma to ∘-rawMagma)"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Conversion back and forth with the Setoid-based notion of Endomorphism"},{"Add":""},{"Add":"fromSetoidEndo : Setoid.Endo → Endo"},{"Add":"fromSetoidEndo = _⟨$⟩_"},{"Add":""},{"Add":"toSetoidEndo : Endo → Setoid.Endo"},{"Add":"toSetoidEndo f = record"},{"Add":" { to = f"},{"Add":" ; cong = cong f"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"∘-isMagma : IsMagma _≡_ _∘_"},{"Add":"∘-isMagma = record"},{"Add":" { isEquivalence = ≡.isEquivalence"},{"Add":" ; ∙-cong = cong₂ _∘_"},{"Add":" }"},{"Add":""},{"Add":"∘-magma : Magma _ _"},{"Add":"∘-magma = record { isMagma = ∘-isMagma }"},{"Add":""},{"Add":"∘-isSemigroup : IsSemigroup _≡_ _∘_"},{"Add":"∘-isSemigroup = record"},{"Add":" { isMagma = ∘-isMagma"},{"Add":" ; assoc = λ _ _ _ → refl"},{"Add":" }"},{"Add":""},{"Add":"∘-semigroup : Semigroup _ _"},{"Add":"∘-semigroup = record { isSemigroup = ∘-isSemigroup }"},{"Add":""},{"Add":"∘-id-isMonoid : IsMonoid _≡_ _∘_ id"},{"Add":"∘-id-isMonoid = record"},{"Add":" { isSemigroup = ∘-isSemigroup"},{"Add":" ; identity = (λ _ → refl) , (λ _ → refl)"},{"Add":" }"},{"Add":""},{"Add":"∘-id-monoid : Monoid _ _"},{"Add":"∘-id-monoid = record { isMonoid = ∘-id-isMonoid }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- n-th iterated composition"},{"Add":""},{"Add":"infixr 8 _^_"},{"Add":""},{"Add":"_^_ : Endo → ℕ → Endo"},{"Add":"_^_ = flip _×_ where open RawMonoidDefinitions ∘-id-rawMonoid using (_×_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Homomorphism"},{"Add":""},{"Add":"module _ (f : Endo) where"},{"Add":""},{"Add":" open MonoidMultProperties ∘-id-monoid using (×-homo-+)"},{"Add":""},{"Add":" ^-homo : Homomorphic₂ ℕ Endo _≡_ (f ^_) _+_ _∘_"},{"Add":" ^-homo = ×-homo-+ f"},{"Add":""},{"Add":" ^-isMagmaHomomorphism : IsMagmaHomomorphism +-rawMagma ∘-rawMagma (f ^_)"},{"Add":" ^-isMagmaHomomorphism = record"},{"Add":" { isRelHomomorphism = record { cong = cong (f ^_) }"},{"Add":" ; homo = ^-homo"},{"Add":" }"},{"Add":""},{"Add":" ^-isMonoidHomomorphism : IsMonoidHomomorphism +-0-rawMonoid ∘-id-rawMonoid (f ^_)"},{"Add":" ^-isMonoidHomomorphism = record"},{"Add":" { isMagmaHomomorphism = ^-isMagmaHomomorphism"},{"Add":" ; ε-homo = refl"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Dependent","metadata":576},"context":[{"change":0,"pos":1018846}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Bundles.agda","metadata":0},"context":[{"change":0,"pos":1200501}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Bundles for types of functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this file should usually be accessed from `Function`."},{"Add":""},{"Add":"-- Note that these bundles differ from those found elsewhere in other"},{"Add":"-- library hierarchies as they take Setoids as parameters. This is"},{"Add":"-- because a function is of no use without knowing what its domain and"},{"Add":"-- codomain is, as well which equalities are being considered over them."},{"Add":"-- One consequence of this is that they are not built from the"},{"Add":"-- definitions found in `Function.Structures` as is usually the case in"},{"Add":"-- other library hierarchies, as this would duplicate the equality"},{"Add":"-- axioms."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Dependent.Bundles where"},{"Add":""},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Indexed.Heterogeneous using (IndexedSetoid)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b ℓ₁ ℓ₂ : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Setoid bundles"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"module _"},{"Add":" (From : Setoid a ℓ₁)"},{"Add":" (To : IndexedSetoid (Setoid.Carrier From) b ℓ₂)"},{"Add":" where"},{"Add":""},{"Add":" open Setoid From using () renaming (Carrier to A; _≈_ to _≈₁_)"},{"Add":" open IndexedSetoid To using () renaming (Carrier to B; _≈_ to _≈₂_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles with one element"},{"Add":""},{"Add":" -- Called `Func` rather than `Function` in order to avoid clashing"},{"Add":" -- with the top-level module."},{"Add":" record Func : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" to : (x : A) → B x"},{"Add":" cong : ∀ {x y} → x ≈₁ y → to x ≈₂ to y"}]}},{"FileAdd":{"meta":{"basename":"Definitions.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Definitions for types of functions."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this file should usually be accessed from `Function`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Definitions where"},{"Add":""},{"Add":"open import Data.Product.Base using (∃; _×_)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a ℓ₁ ℓ₂ : Level"},{"Add":" A B : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Basic definitions"},{"Add":""},{"Add":"module _"},{"Add":" (_≈₁_ : Rel A ℓ₁) -- Equality over the domain"},{"Add":" (_≈₂_ : Rel B ℓ₂) -- Equality over the codomain"},{"Add":" where"},{"Add":""},{"Add":" Congruent : (A → B) → Set _"},{"Add":" Congruent f = ∀ {x y} → x ≈₁ y → f x ≈₂ f y"},{"Add":""},{"Add":" Injective : (A → B) → Set _"},{"Add":" Injective f = ∀ {x y} → f x ≈₂ f y → x ≈₁ y"},{"Add":""},{"Add":" Surjective : (A → B) → Set _"},{"Add":" Surjective f = ∀ y → ∃ λ x → ∀ {z} → z ≈₁ x → f z ≈₂ y"},{"Add":""},{"Add":" Bijective : (A → B) → Set _"},{"Add":" Bijective f = Injective f × Surjective f"},{"Add":""},{"Add":" Inverseˡ : (A → B) → (B → A) → Set _"},{"Add":" Inverseˡ f g = ∀ {x y} → y ≈₁ g x → f y ≈₂ x"},{"Add":""},{"Add":" Inverseʳ : (A → B) → (B → A) → Set _"},{"Add":" Inverseʳ f g = ∀ {x y} → y ≈₂ f x → g y ≈₁ x"},{"Add":""},{"Add":" Inverseᵇ : (A → B) → (B → A) → Set _"},{"Add":" Inverseᵇ f g = Inverseˡ f g × Inverseʳ f g"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Strict definitions"},{"Add":""},{"Add":"-- These are often easier to use once but much harder to compose and"},{"Add":"-- reason about."},{"Add":""},{"Add":"StrictlySurjective : Rel B ℓ₂ → (A → B) → Set _"},{"Add":"StrictlySurjective _≈₂_ f = ∀ y → ∃ λ x → f x ≈₂ y"},{"Add":""},{"Add":"StrictlyInverseˡ : Rel B ℓ₂ → (A → B) → (B → A) → Set _"},{"Add":"StrictlyInverseˡ _≈₂_ f g = ∀ y → f (g y) ≈₂ y"},{"Add":""},{"Add":"StrictlyInverseʳ : Rel A ℓ₁ → (A → B) → (B → A) → Set _"},{"Add":"StrictlyInverseʳ _≈₁_ f g = ∀ x → g (f x) ≈₁ x"}]}},{"FileAdd":{"meta":{"basename":"Core.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Core definitions for Functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this file should always be accessed from `Function`."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Core where"},{"Add":""},{"Add":"open import Level using (_⊔_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Types"},{"Add":""},{"Add":"Fun₁ : ∀ {a} → Set a → Set a"},{"Add":"Fun₁ A = A → A"},{"Add":""},{"Add":"Fun₂ : ∀ {a} → Set a → Set a"},{"Add":"Fun₂ A = A → A → A"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Morphism"},{"Add":""},{"Add":"Morphism : ∀ {a} → ∀ {b} → Set a → Set b → Set (a ⊔ b)"},{"Add":"Morphism A B = A → B"}]}},{"FileAdd":{"meta":{"basename":"Construct","metadata":576},"context":[{"change":0,"pos":1018846}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Symmetry.agda","metadata":0},"context":[{"change":0,"pos":1205339}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Some functional properties are symmetric"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Construct.Symmetry where"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_; swap; proj₁; proj₂)"},{"Add":"open import Function.Base using (_∘_)"},{"Add":"open import Function.Definitions"},{"Add":" using (Bijective; Injective; Surjective; Inverseˡ; Inverseʳ; Inverseᵇ; Congruent)"},{"Add":"open import Function.Structures"},{"Add":" using (IsBijection; IsCongruent; IsRightInverse; IsLeftInverse; IsInverse)"},{"Add":"open import Function.Bundles"},{"Add":" using (Bijection; Equivalence; LeftInverse; RightInverse; Inverse; _⤖_; _⇔_; _↩_; _↪_; _↔_)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)"},{"Add":"open import Relation.Binary.PropositionalEquality.Properties using (setoid)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c ℓ₁ ℓ₂ ℓ₃ : Level"},{"Add":" A B C : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {f : A → B}"},{"Add":" ((inj , surj) : Bijective ≈₁ ≈₂ f)"},{"Add":" where"},{"Add":""},{"Add":" private"},{"Add":" f⁻¹ = proj₁ ∘ surj"},{"Add":" f∘f⁻¹≡id = proj₂ ∘ surj"},{"Add":""},{"Add":" injective : Reflexive ≈₁ → Symmetric ≈₂ → Transitive ≈₂ →"},{"Add":" Congruent ≈₁ ≈₂ f → Injective ≈₂ ≈₁ f⁻¹"},{"Add":" injective refl sym trans cong gx≈gy ="},{"Add":" trans (trans (sym (f∘f⁻¹≡id _ refl)) (cong gx≈gy)) (f∘f⁻¹≡id _ refl)"},{"Add":""},{"Add":" surjective : Reflexive ≈₁ → Transitive ≈₂ → Surjective ≈₂ ≈₁ f⁻¹"},{"Add":" surjective refl trans x = f x , inj ∘ trans (f∘f⁻¹≡id _ refl)"},{"Add":""},{"Add":" bijective : Reflexive ≈₁ → Symmetric ≈₂ → Transitive ≈₂ →"},{"Add":" Congruent ≈₁ ≈₂ f → Bijective ≈₂ ≈₁ f⁻¹"},{"Add":" bijective refl sym trans cong = injective refl sym trans cong , surjective refl trans"},{"Add":""},{"Add":"module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) {f : A → B} {f⁻¹ : B → A} where"},{"Add":""},{"Add":" inverseʳ : Inverseˡ ≈₁ ≈₂ f f⁻¹ → Inverseʳ ≈₂ ≈₁ f⁻¹ f"},{"Add":" inverseʳ inv = inv"},{"Add":""},{"Add":" inverseˡ : Inverseʳ ≈₁ ≈₂ f f⁻¹ → Inverseˡ ≈₂ ≈₁ f⁻¹ f"},{"Add":" inverseˡ inv = inv"},{"Add":""},{"Add":" inverseᵇ : Inverseᵇ ≈₁ ≈₂ f f⁻¹ → Inverseᵇ ≈₂ ≈₁ f⁻¹ f"},{"Add":" inverseᵇ (invˡ , invʳ) = (invʳ , invˡ)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂}"},{"Add":" {f : A → B} (isBij : IsBijection ≈₁ ≈₂ f)"},{"Add":" where"},{"Add":""},{"Add":" private"},{"Add":" module IB = IsBijection isBij"},{"Add":" f⁻¹ = proj₁ ∘ IB.surjective"},{"Add":""},{"Add":" -- We can only flip a bijection if the witness produced by the"},{"Add":" -- surjection proof respects the equality on the codomain."},{"Add":" isBijection : Congruent ≈₂ ≈₁ f⁻¹ → IsBijection ≈₂ ≈₁ f⁻¹"},{"Add":" isBijection f⁻¹-cong = record"},{"Add":" { isInjection = record"},{"Add":" { isCongruent = record"},{"Add":" { cong = f⁻¹-cong"},{"Add":" ; isEquivalence₁ = IB.Eq₂.isEquivalence"},{"Add":" ; isEquivalence₂ = IB.Eq₁.isEquivalence"},{"Add":" }"},{"Add":" ; injective = injective IB.bijective IB.Eq₁.refl IB.Eq₂.sym IB.Eq₂.trans IB.cong"},{"Add":" }"},{"Add":" ; surjective = surjective IB.bijective IB.Eq₁.refl IB.Eq₂.trans"},{"Add":" }"},{"Add":""},{"Add":"module _ {≈₁ : Rel A ℓ₁} {f : A → B} (isBij : IsBijection ≈₁ _≡_ f) where"},{"Add":""},{"Add":" -- We can always flip a bijection if using the equality over the"},{"Add":" -- codomain is propositional equality."},{"Add":" isBijection-≡ : IsBijection _≡_ ≈₁ _"},{"Add":" isBijection-≡ = isBijection isBij (IB.Eq₁.reflexive ∘ cong _)"},{"Add":" where module IB = IsBijection isBij"},{"Add":""},{"Add":"module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {f : A → B} {f⁻¹ : B → A} where"},{"Add":""},{"Add":" isCongruent : IsCongruent ≈₁ ≈₂ f → Congruent ≈₂ ≈₁ f⁻¹ → IsCongruent ≈₂ ≈₁ f⁻¹"},{"Add":" isCongruent ic cg = record"},{"Add":" { cong = cg"},{"Add":" ; isEquivalence₁ = F.isEquivalence₂"},{"Add":" ; isEquivalence₂ = F.isEquivalence₁"},{"Add":" } where module F = IsCongruent ic"},{"Add":""},{"Add":" isLeftInverse : IsRightInverse ≈₁ ≈₂ f f⁻¹ → IsLeftInverse ≈₂ ≈₁ f⁻¹ f"},{"Add":" isLeftInverse inv = record"},{"Add":" { isCongruent = isCongruent F.isCongruent F.from-cong"},{"Add":" ; from-cong = F.to-cong"},{"Add":" ; inverseˡ = inverseˡ ≈₁ ≈₂ F.inverseʳ"},{"Add":" } where module F = IsRightInverse inv"},{"Add":""},{"Add":" isRightInverse : IsLeftInverse ≈₁ ≈₂ f f⁻¹ → IsRightInverse ≈₂ ≈₁ f⁻¹ f"},{"Add":" isRightInverse inv = record"},{"Add":" { isCongruent = isCongruent F.isCongruent F.from-cong"},{"Add":" ; from-cong = F.to-cong"},{"Add":" ; inverseʳ = inverseʳ ≈₁ ≈₂ F.inverseˡ"},{"Add":" } where module F = IsLeftInverse inv"},{"Add":""},{"Add":" isInverse : IsInverse ≈₁ ≈₂ f f⁻¹ → IsInverse ≈₂ ≈₁ f⁻¹ f"},{"Add":" isInverse f-inv = record"},{"Add":" { isLeftInverse = isLeftInverse F.isRightInverse"},{"Add":" ; inverseʳ = inverseʳ ≈₁ ≈₂ F.inverseˡ"},{"Add":" } where module F = IsInverse f-inv"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Setoid bundles"},{"Add":""},{"Add":"module _ {R : Setoid a ℓ₁} {S : Setoid b ℓ₂} (bij : Bijection R S) where"},{"Add":""},{"Add":" private"},{"Add":" module IB = Bijection bij"},{"Add":" from = proj₁ ∘ IB.surjective"},{"Add":""},{"Add":" -- We can only flip a bijection if the witness produced by the"},{"Add":" -- surjection proof respects the equality on the codomain."},{"Add":" bijection : Congruent IB.Eq₂._≈_ IB.Eq₁._≈_ from → Bijection S R"},{"Add":" bijection cong = record"},{"Add":" { to = from"},{"Add":" ; cong = cong"},{"Add":" ; bijective = bijective IB.bijective IB.Eq₁.refl IB.Eq₂.sym IB.Eq₂.trans IB.cong"},{"Add":" }"},{"Add":""},{"Add":"-- We can always flip a bijection if using the equality over the"},{"Add":"-- codomain is propositional equality."},{"Add":"bijection-≡ : {R : Setoid a ℓ₁} {B : Set b} →"},{"Add":" Bijection R (setoid B) → Bijection (setoid B) R"},{"Add":"bijection-≡ bij = bijection bij (B.Eq₁.reflexive ∘ cong _)"},{"Add":" where module B = Bijection bij"},{"Add":""},{"Add":"module _ {R : Setoid a ℓ₁} {S : Setoid b ℓ₂} where"},{"Add":""},{"Add":" equivalence : Equivalence R S → Equivalence S R"},{"Add":" equivalence equiv = record"},{"Add":" { to = E.from"},{"Add":" ; from = E.to"},{"Add":" ; to-cong = E.from-cong"},{"Add":" ; from-cong = E.to-cong"},{"Add":" } where module E = Equivalence equiv"},{"Add":""},{"Add":" rightInverse : LeftInverse R S → RightInverse S R"},{"Add":" rightInverse left = record"},{"Add":" { to = L.from"},{"Add":" ; from = L.to"},{"Add":" ; to-cong = L.from-cong"},{"Add":" ; from-cong = L.to-cong"},{"Add":" ; inverseʳ = L.inverseˡ"},{"Add":" } where module L = LeftInverse left"},{"Add":""},{"Add":" leftInverse : RightInverse R S → LeftInverse S R"},{"Add":" leftInverse right = record"},{"Add":" { to = R.from"},{"Add":" ; from = R.to"},{"Add":" ; to-cong = R.from-cong"},{"Add":" ; from-cong = R.to-cong"},{"Add":" ; inverseˡ = R.inverseʳ"},{"Add":" } where module R = RightInverse right"},{"Add":""},{"Add":" inverse : Inverse R S → Inverse S R"},{"Add":" inverse inv = record"},{"Add":" { to = I.from"},{"Add":" ; from = I.to"},{"Add":" ; to-cong = I.from-cong"},{"Add":" ; from-cong = I.to-cong"},{"Add":" ; inverse = swap I.inverse"},{"Add":" } where module I = Inverse inv"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Propositional bundles"},{"Add":""},{"Add":"⤖-sym : A ⤖ B → B ⤖ A"},{"Add":"⤖-sym b = bijection b (cong _)"},{"Add":""},{"Add":"⇔-sym : A ⇔ B → B ⇔ A"},{"Add":"⇔-sym = equivalence"},{"Add":""},{"Add":"↩-sym : A ↩ B → B ↪ A"},{"Add":"↩-sym = rightInverse"},{"Add":""},{"Add":"↪-sym : A ↪ B → B ↩ A"},{"Add":"↪-sym = leftInverse"},{"Add":""},{"Add":"↔-sym : A ↔ B → B ↔ A"},{"Add":"↔-sym = inverse"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- DEPRECATED NAMES"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Please use the new names as continuing support for the old names is"},{"Add":"-- not guaranteed."},{"Add":""},{"Add":"-- Version v2.0"},{"Add":""},{"Add":"sym-⤖ = ⤖-sym"},{"Add":"{-# WARNING_ON_USAGE sym-⤖"},{"Add":"\"Warning: sym-⤖ was deprecated in v2.0."},{"Add":"Please use ⤖-sym instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"sym-⇔ = ⇔-sym"},{"Add":"{-# WARNING_ON_USAGE sym-⇔"},{"Add":"\"Warning: sym-⇔ was deprecated in v2.0."},{"Add":"Please use ⇔-sym instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"sym-↩ = ↩-sym"},{"Add":"{-# WARNING_ON_USAGE sym-↩"},{"Add":"\"Warning: sym-↩ was deprecated in v2.0."},{"Add":"Please use ↩-sym instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"sym-↪ = ↪-sym"},{"Add":"{-# WARNING_ON_USAGE sym-↪"},{"Add":"\"Warning: sym-↪ was deprecated in v2.0."},{"Add":"Please use ↪-sym instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"sym-↔ = ↔-sym"},{"Add":"{-# WARNING_ON_USAGE sym-↔"},{"Add":"\"Warning: sym-↔ was deprecated in v2.0."},{"Add":"Please use ↔-sym instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Identity.agda","metadata":0},"context":[{"change":0,"pos":1205339}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The identity function"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Construct.Identity where"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base using (id)"},{"Add":"open import Function.Bundles"},{"Add":"import Function.Definitions as Definitions"},{"Add":"import Function.Structures as Structures"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Structures as B hiding (IsEquivalence)"},{"Add":"open import Relation.Binary.Definitions using (Reflexive)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Properties using (setoid)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a ℓ : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"module _ (_≈_ : Rel A ℓ) where"},{"Add":""},{"Add":" open Definitions"},{"Add":""},{"Add":" congruent : Congruent _≈_ _≈_ id"},{"Add":" congruent = id"},{"Add":""},{"Add":" injective : Injective _≈_ _≈_ id"},{"Add":" injective = id"},{"Add":""},{"Add":" surjective : Surjective _≈_ _≈_ id"},{"Add":" surjective x = x , id"},{"Add":""},{"Add":" bijective : Bijective _≈_ _≈_ id"},{"Add":" bijective = injective , surjective"},{"Add":""},{"Add":" inverseˡ : Inverseˡ _≈_ _≈_ id id"},{"Add":" inverseˡ = id"},{"Add":""},{"Add":" inverseʳ : Inverseʳ _≈_ _≈_ id id"},{"Add":" inverseʳ = id"},{"Add":""},{"Add":" inverseᵇ : Inverseᵇ _≈_ _≈_ id id"},{"Add":" inverseᵇ = inverseˡ , inverseʳ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"module _ {_≈_ : Rel A ℓ} (isEq : B.IsEquivalence _≈_) where"},{"Add":""},{"Add":" open Structures _≈_ _≈_"},{"Add":" open B.IsEquivalence isEq"},{"Add":""},{"Add":" isCongruent : IsCongruent id"},{"Add":" isCongruent = record"},{"Add":" { cong = id"},{"Add":" ; isEquivalence₁ = isEq"},{"Add":" ; isEquivalence₂ = isEq"},{"Add":" }"},{"Add":""},{"Add":" isInjection : IsInjection id"},{"Add":" isInjection = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; injective = injective _≈_"},{"Add":" }"},{"Add":""},{"Add":" isSurjection : IsSurjection id"},{"Add":" isSurjection = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; surjective = surjective _≈_"},{"Add":" }"},{"Add":""},{"Add":" isBijection : IsBijection id"},{"Add":" isBijection = record"},{"Add":" { isInjection = isInjection"},{"Add":" ; surjective = surjective _≈_"},{"Add":" }"},{"Add":""},{"Add":" isLeftInverse : IsLeftInverse id id"},{"Add":" isLeftInverse = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; from-cong = id"},{"Add":" ; inverseˡ = inverseˡ _≈_"},{"Add":" }"},{"Add":""},{"Add":" isRightInverse : IsRightInverse id id"},{"Add":" isRightInverse = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; from-cong = id"},{"Add":" ; inverseʳ = inverseʳ _≈_"},{"Add":" }"},{"Add":""},{"Add":" isInverse : IsInverse id id"},{"Add":" isInverse = record"},{"Add":" { isLeftInverse = isLeftInverse"},{"Add":" ; inverseʳ = inverseʳ _≈_"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Setoid bundles"},{"Add":""},{"Add":"module _ (S : Setoid a ℓ) where"},{"Add":""},{"Add":" open Setoid S"},{"Add":""},{"Add":" function : Func S S"},{"Add":" function = record"},{"Add":" { to = id"},{"Add":" ; cong = id"},{"Add":" }"},{"Add":""},{"Add":" injection : Injection S S"},{"Add":" injection = record"},{"Add":" { to = id"},{"Add":" ; cong = id"},{"Add":" ; injective = injective _≈_"},{"Add":" }"},{"Add":""},{"Add":" surjection : Surjection S S"},{"Add":" surjection = record"},{"Add":" { to = id"},{"Add":" ; cong = id"},{"Add":" ; surjective = surjective _≈_"},{"Add":" }"},{"Add":""},{"Add":" bijection : Bijection S S"},{"Add":" bijection = record"},{"Add":" { to = id"},{"Add":" ; cong = id"},{"Add":" ; bijective = bijective _≈_"},{"Add":" }"},{"Add":""},{"Add":" equivalence : Equivalence S S"},{"Add":" equivalence = record"},{"Add":" { to = id"},{"Add":" ; from = id"},{"Add":" ; to-cong = id"},{"Add":" ; from-cong = id"},{"Add":" }"},{"Add":""},{"Add":" leftInverse : LeftInverse S S"},{"Add":" leftInverse = record"},{"Add":" { to = id"},{"Add":" ; from = id"},{"Add":" ; to-cong = id"},{"Add":" ; from-cong = id"},{"Add":" ; inverseˡ = inverseˡ _≈_"},{"Add":" }"},{"Add":""},{"Add":" rightInverse : RightInverse S S"},{"Add":" rightInverse = record"},{"Add":" { to = id"},{"Add":" ; from = id"},{"Add":" ; to-cong = id"},{"Add":" ; from-cong = id"},{"Add":" ; inverseʳ = inverseʳ _≈_"},{"Add":" }"},{"Add":""},{"Add":" inverse : Inverse S S"},{"Add":" inverse = record"},{"Add":" { to = id"},{"Add":" ; from = id"},{"Add":" ; to-cong = id"},{"Add":" ; from-cong = id"},{"Add":" ; inverse = inverseᵇ _≈_"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Propositional bundles"},{"Add":""},{"Add":"module _ (A : Set a) where"},{"Add":""},{"Add":" ⟶-id : A ⟶ A"},{"Add":" ⟶-id = function (setoid A)"},{"Add":""},{"Add":" ↣-id : A ↣ A"},{"Add":" ↣-id = injection (setoid A)"},{"Add":""},{"Add":" ↠-id : A ↠ A"},{"Add":" ↠-id = surjection (setoid A)"},{"Add":""},{"Add":" ⤖-id : A ⤖ A"},{"Add":" ⤖-id = bijection (setoid A)"},{"Add":""},{"Add":" ⇔-id : A ⇔ A"},{"Add":" ⇔-id = equivalence (setoid A)"},{"Add":""},{"Add":" ↩-id : A ↩ A"},{"Add":" ↩-id = leftInverse (setoid A)"},{"Add":""},{"Add":" ↪-id : A ↪ A"},{"Add":" ↪-id = rightInverse (setoid A)"},{"Add":""},{"Add":" ↔-id : A ↔ A"},{"Add":" ↔-id = inverse (setoid A)"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- DEPRECATED NAMES"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Please use the new names as continuing support for the old names is"},{"Add":"-- not guaranteed."},{"Add":""},{"Add":"-- Version v2.0"},{"Add":""},{"Add":"id-⟶ = ⟶-id"},{"Add":"{-# WARNING_ON_USAGE id-⟶"},{"Add":"\"Warning: id-⟶ was deprecated in v2.0."},{"Add":"Please use ⟶-id instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"id-↣ = ↣-id"},{"Add":"{-# WARNING_ON_USAGE id-↣"},{"Add":"\"Warning: id-↣ was deprecated in v2.0."},{"Add":"Please use ↣-id instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"id-↠ = ↠-id"},{"Add":"{-# WARNING_ON_USAGE id-↠"},{"Add":"\"Warning: id-↠ was deprecated in v2.0."},{"Add":"Please use ↠-id instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"id-⤖ = ⤖-id"},{"Add":"{-# WARNING_ON_USAGE id-⤖"},{"Add":"\"Warning: id-⤖ was deprecated in v2.0."},{"Add":"Please use ⤖-id instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"id-⇔ = ⇔-id"},{"Add":"{-# WARNING_ON_USAGE id-⇔"},{"Add":"\"Warning: id-⇔ was deprecated in v2.0."},{"Add":"Please use ⇔-id instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"id-↩ = ↩-id"},{"Add":"{-# WARNING_ON_USAGE id-↩"},{"Add":"\"Warning: id-↩ was deprecated in v2.0."},{"Add":"Please use ↩-id instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"id-↪ = ↪-id"},{"Add":"{-# WARNING_ON_USAGE id-↪"},{"Add":"\"Warning: id-↪ was deprecated in v2.0."},{"Add":"Please use ↪-id instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"id-↔ = ↔-id"},{"Add":"{-# WARNING_ON_USAGE id-↔"},{"Add":"\"Warning: id-↔ was deprecated in v2.0."},{"Add":"Please use ↔-id instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Constant.agda","metadata":0},"context":[{"change":0,"pos":1205339}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The constant function"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Construct.Constant where"},{"Add":""},{"Add":"open import Function.Base using (const)"},{"Add":"open import Function.Bundles"},{"Add":"import Function.Definitions as Definitions"},{"Add":"import Function.Structures as Structures"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Structures as B hiding (IsEquivalence)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b ℓ₁ ℓ₂ : Level"},{"Add":" A B : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"module _ (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂) where"},{"Add":""},{"Add":" open Definitions"},{"Add":""},{"Add":" congruent : ∀ {b} → b ≈₂ b → Congruent _≈₁_ _≈₂_ (const b)"},{"Add":" congruent refl _ = refl"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"module _"},{"Add":" {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂}"},{"Add":" (isEq₁ : B.IsEquivalence ≈₁)"},{"Add":" (isEq₂ : B.IsEquivalence ≈₂) where"},{"Add":""},{"Add":" open Structures ≈₁ ≈₂"},{"Add":" open B.IsEquivalence"},{"Add":""},{"Add":" isCongruent : ∀ b → IsCongruent (const b)"},{"Add":" isCongruent b = record"},{"Add":" { cong = congruent ≈₁ ≈₂ (refl isEq₂)"},{"Add":" ; isEquivalence₁ = isEq₁"},{"Add":" ; isEquivalence₂ = isEq₂"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Setoid bundles"},{"Add":""},{"Add":"module _ (S : Setoid a ℓ₂) (T : Setoid b ℓ₂) where"},{"Add":""},{"Add":" open Setoid"},{"Add":""},{"Add":" function : Carrier T → Func S T"},{"Add":" function b = record"},{"Add":" { to = const b"},{"Add":" ; cong = congruent (_≈_ S) (_≈_ T) (refl T)"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Composition.agda","metadata":0},"context":[{"change":0,"pos":1205339}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Composition of functional properties"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Construct.Composition where"},{"Add":""},{"Add":"open import Data.Product.Base as Product using (_,_)"},{"Add":"open import Function.Base using (_∘_; flip)"},{"Add":"open import Function.Bundles"},{"Add":"open import Function.Definitions"},{"Add":"open import Function.Structures"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Definitions using (Transitive)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c ℓ₁ ℓ₂ ℓ₃ : Level"},{"Add":" A B C : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties"},{"Add":""},{"Add":"module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) (≈₃ : Rel C ℓ₃)"},{"Add":" {f : A → B} {g : B → C}"},{"Add":" where"},{"Add":""},{"Add":" congruent : Congruent ≈₁ ≈₂ f → Congruent ≈₂ ≈₃ g →"},{"Add":" Congruent ≈₁ ≈₃ (g ∘ f)"},{"Add":" congruent f-cong g-cong = g-cong ∘ f-cong"},{"Add":""},{"Add":" injective : Injective ≈₁ ≈₂ f → Injective ≈₂ ≈₃ g →"},{"Add":" Injective ≈₁ ≈₃ (g ∘ f)"},{"Add":" injective f-inj g-inj = f-inj ∘ g-inj"},{"Add":""},{"Add":" surjective : Surjective ≈₁ ≈₂ f → Surjective ≈₂ ≈₃ g →"},{"Add":" Surjective ≈₁ ≈₃ (g ∘ f)"},{"Add":" surjective f-sur g-sur x with g-sur x"},{"Add":" ... | y , gproof with f-sur y"},{"Add":" ... | z , fproof = z , gproof ∘ fproof"},{"Add":""},{"Add":" bijective : Bijective ≈₁ ≈₂ f → Bijective ≈₂ ≈₃ g →"},{"Add":" Bijective ≈₁ ≈₃ (g ∘ f)"},{"Add":" bijective = Product.zip′ injective surjective"},{"Add":""},{"Add":"module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) (≈₃ : Rel C ℓ₃)"},{"Add":" {f : A → B} {f⁻¹ : B → A} {g : B → C} {g⁻¹ : C → B}"},{"Add":" where"},{"Add":""},{"Add":" inverseˡ : Inverseˡ ≈₁ ≈₂ f f⁻¹ → Inverseˡ ≈₂ ≈₃ g g⁻¹ →"},{"Add":" Inverseˡ ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)"},{"Add":" inverseˡ f-inv g-inv = g-inv ∘ f-inv"},{"Add":""},{"Add":" inverseʳ : Inverseʳ ≈₁ ≈₂ f f⁻¹ → Inverseʳ ≈₂ ≈₃ g g⁻¹ →"},{"Add":" Inverseʳ ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)"},{"Add":" inverseʳ f-inv g-inv = f-inv ∘ g-inv"},{"Add":""},{"Add":" inverseᵇ : Inverseᵇ ≈₁ ≈₂ f f⁻¹ → Inverseᵇ ≈₂ ≈₃ g g⁻¹ →"},{"Add":" Inverseᵇ ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)"},{"Add":" inverseᵇ = Product.zip′ inverseˡ inverseʳ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structures"},{"Add":""},{"Add":"module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {≈₃ : Rel C ℓ₃}"},{"Add":" {f : A → B} {g : B → C}"},{"Add":" where"},{"Add":""},{"Add":" isCongruent : IsCongruent ≈₁ ≈₂ f → IsCongruent ≈₂ ≈₃ g →"},{"Add":" IsCongruent ≈₁ ≈₃ (g ∘ f)"},{"Add":" isCongruent f-cong g-cong = record"},{"Add":" { cong = G.cong ∘ F.cong"},{"Add":" ; isEquivalence₁ = F.isEquivalence₁"},{"Add":" ; isEquivalence₂ = G.isEquivalence₂"},{"Add":" } where module F = IsCongruent f-cong; module G = IsCongruent g-cong"},{"Add":""},{"Add":" isInjection : IsInjection ≈₁ ≈₂ f → IsInjection ≈₂ ≈₃ g →"},{"Add":" IsInjection ≈₁ ≈₃ (g ∘ f)"},{"Add":" isInjection f-inj g-inj = record"},{"Add":" { isCongruent = isCongruent F.isCongruent G.isCongruent"},{"Add":" ; injective = injective ≈₁ ≈₂ ≈₃ F.injective G.injective"},{"Add":" } where module F = IsInjection f-inj; module G = IsInjection g-inj"},{"Add":""},{"Add":" isSurjection : IsSurjection ≈₁ ≈₂ f → IsSurjection ≈₂ ≈₃ g →"},{"Add":" IsSurjection ≈₁ ≈₃ (g ∘ f)"},{"Add":" isSurjection f-surj g-surj = record"},{"Add":" { isCongruent = isCongruent F.isCongruent G.isCongruent"},{"Add":" ; surjective = surjective ≈₁ ≈₂ ≈₃ F.surjective G.surjective"},{"Add":" } where module F = IsSurjection f-surj; module G = IsSurjection g-surj"},{"Add":""},{"Add":" isBijection : IsBijection ≈₁ ≈₂ f → IsBijection ≈₂ ≈₃ g →"},{"Add":" IsBijection ≈₁ ≈₃ (g ∘ f)"},{"Add":" isBijection f-bij g-bij = record"},{"Add":" { isInjection = isInjection F.isInjection G.isInjection"},{"Add":" ; surjective = surjective ≈₁ ≈₂ ≈₃ F.surjective G.surjective"},{"Add":" } where module F = IsBijection f-bij; module G = IsBijection g-bij"},{"Add":""},{"Add":"module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {≈₃ : Rel C ℓ₃}"},{"Add":" {f : A → B} {g : B → C} {f⁻¹ : B → A} {g⁻¹ : C → B}"},{"Add":" where"},{"Add":""},{"Add":" isLeftInverse : IsLeftInverse ≈₁ ≈₂ f f⁻¹ → IsLeftInverse ≈₂ ≈₃ g g⁻¹ →"},{"Add":" IsLeftInverse ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)"},{"Add":" isLeftInverse f-invˡ g-invˡ = record"},{"Add":" { isCongruent = isCongruent F.isCongruent G.isCongruent"},{"Add":" ; from-cong = congruent ≈₃ ≈₂ ≈₁ G.from-cong F.from-cong"},{"Add":" ; inverseˡ = inverseˡ ≈₁ ≈₂ ≈₃ F.inverseˡ G.inverseˡ"},{"Add":" } where module F = IsLeftInverse f-invˡ; module G = IsLeftInverse g-invˡ"},{"Add":""},{"Add":" isRightInverse : IsRightInverse ≈₁ ≈₂ f f⁻¹ → IsRightInverse ≈₂ ≈₃ g g⁻¹ →"},{"Add":" IsRightInverse ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)"},{"Add":" isRightInverse f-invʳ g-invʳ = record"},{"Add":" { isCongruent = isCongruent F.isCongruent G.isCongruent"},{"Add":" ; from-cong = congruent ≈₃ ≈₂ ≈₁ G.from-cong F.from-cong"},{"Add":" ; inverseʳ = inverseʳ ≈₁ ≈₂ ≈₃ F.inverseʳ G.inverseʳ"},{"Add":" } where module F = IsRightInverse f-invʳ; module G = IsRightInverse g-invʳ"},{"Add":""},{"Add":" isInverse : IsInverse ≈₁ ≈₂ f f⁻¹ → IsInverse ≈₂ ≈₃ g g⁻¹ →"},{"Add":" IsInverse ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)"},{"Add":" isInverse f-inv g-inv = record"},{"Add":" { isLeftInverse = isLeftInverse F.isLeftInverse G.isLeftInverse"},{"Add":" ; inverseʳ = inverseʳ ≈₁ ≈₂ ≈₃ F.inverseʳ G.inverseʳ"},{"Add":" } where module F = IsInverse f-inv; module G = IsInverse g-inv"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Setoid bundles"},{"Add":""},{"Add":"module _ {R : Setoid a ℓ₁} {S : Setoid b ℓ₂} {T : Setoid c ℓ₃} where"},{"Add":""},{"Add":" open Setoid renaming (_≈_ to ≈)"},{"Add":""},{"Add":" function : Func R S → Func S T → Func R T"},{"Add":" function f g = record"},{"Add":" { to = G.to ∘ F.to"},{"Add":" ; cong = congruent (≈ R) (≈ S) (≈ T) F.cong G.cong"},{"Add":" } where module F = Func f; module G = Func g"},{"Add":""},{"Add":" injection : Injection R S → Injection S T → Injection R T"},{"Add":" injection inj₁ inj₂ = record"},{"Add":" { to = G.to ∘ F.to"},{"Add":" ; cong = congruent (≈ R) (≈ S) (≈ T) F.cong G.cong"},{"Add":" ; injective = injective (≈ R) (≈ S) (≈ T) F.injective G.injective"},{"Add":" } where module F = Injection inj₁; module G = Injection inj₂"},{"Add":""},{"Add":" surjection : Surjection R S → Surjection S T → Surjection R T"},{"Add":" surjection surj₁ surj₂ = record"},{"Add":" { to = G.to ∘ F.to"},{"Add":" ; cong = congruent (≈ R) (≈ S) (≈ T) F.cong G.cong"},{"Add":" ; surjective = surjective (≈ R) (≈ S) (≈ T) F.surjective G.surjective"},{"Add":" } where module F = Surjection surj₁; module G = Surjection surj₂"},{"Add":""},{"Add":" bijection : Bijection R S → Bijection S T → Bijection R T"},{"Add":" bijection bij₁ bij₂ = record"},{"Add":" { to = G.to ∘ F.to"},{"Add":" ; cong = congruent (≈ R) (≈ S) (≈ T) F.cong G.cong"},{"Add":" ; bijective = bijective (≈ R) (≈ S) (≈ T) F.bijective G.bijective"},{"Add":" } where module F = Bijection bij₁; module G = Bijection bij₂"},{"Add":""},{"Add":" equivalence : Equivalence R S → Equivalence S T → Equivalence R T"},{"Add":" equivalence equiv₁ equiv₂ = record"},{"Add":" { to = G.to ∘ F.to"},{"Add":" ; from = F.from ∘ G.from"},{"Add":" ; to-cong = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong"},{"Add":" ; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong"},{"Add":" } where module F = Equivalence equiv₁; module G = Equivalence equiv₂"},{"Add":""},{"Add":" leftInverse : LeftInverse R S → LeftInverse S T → LeftInverse R T"},{"Add":" leftInverse invˡ₁ invˡ₂ = record"},{"Add":" { to = G.to ∘ F.to"},{"Add":" ; from = F.from ∘ G.from"},{"Add":" ; to-cong = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong"},{"Add":" ; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong"},{"Add":" ; inverseˡ = inverseˡ (≈ R) (≈ S) (≈ T) F.inverseˡ G.inverseˡ"},{"Add":" } where module F = LeftInverse invˡ₁; module G = LeftInverse invˡ₂"},{"Add":""},{"Add":" rightInverse : RightInverse R S → RightInverse S T → RightInverse R T"},{"Add":" rightInverse invʳ₁ invʳ₂ = record"},{"Add":" { to = G.to ∘ F.to"},{"Add":" ; from = F.from ∘ G.from"},{"Add":" ; to-cong = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong"},{"Add":" ; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong"},{"Add":" ; inverseʳ = inverseʳ (≈ R) (≈ S) (≈ T) F.inverseʳ G.inverseʳ"},{"Add":" } where module F = RightInverse invʳ₁; module G = RightInverse invʳ₂"},{"Add":""},{"Add":" inverse : Inverse R S → Inverse S T → Inverse R T"},{"Add":" inverse inv₁ inv₂ = record"},{"Add":" { to = G.to ∘ F.to"},{"Add":" ; from = F.from ∘ G.from"},{"Add":" ; to-cong = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong"},{"Add":" ; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong"},{"Add":" ; inverse = inverseᵇ (≈ R) (≈ S) (≈ T) F.inverse G.inverse"},{"Add":" } where module F = Inverse inv₁; module G = Inverse inv₂"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Propositional bundles"},{"Add":""},{"Add":"-- Notice the flipped order of the arguments to mirror composition."},{"Add":""},{"Add":"infix 8 _⟶-∘_ _↣-∘_ _↠-∘_ _⤖-∘_ _⇔-∘_ _↩-∘_ _↪-∘_ _↔-∘_"},{"Add":""},{"Add":"_⟶-∘_ : (B ⟶ C) → (A ⟶ B) → (A ⟶ C)"},{"Add":"_⟶-∘_ = flip function"},{"Add":""},{"Add":"_↣-∘_ : B ↣ C → A ↣ B → A ↣ C"},{"Add":"_↣-∘_ = flip injection"},{"Add":""},{"Add":"_↠-∘_ : B ↠ C → A ↠ B → A ↠ C"},{"Add":"_↠-∘_ = flip surjection"},{"Add":""},{"Add":"_⤖-∘_ : B ⤖ C → A ⤖ B → A ⤖ C"},{"Add":"_⤖-∘_ = flip bijection"},{"Add":""},{"Add":"_⇔-∘_ : B ⇔ C → A ⇔ B → A ⇔ C"},{"Add":"_⇔-∘_ = flip equivalence"},{"Add":""},{"Add":"_↩-∘_ : B ↩ C → A ↩ B → A ↩ C"},{"Add":"_↩-∘_ = flip leftInverse"},{"Add":""},{"Add":"_↪-∘_ : B ↪ C → A ↪ B → A ↪ C"},{"Add":"_↪-∘_ = flip rightInverse"},{"Add":""},{"Add":"_↔-∘_ : B ↔ C → A ↔ B → A ↔ C"},{"Add":"_↔-∘_ = flip inverse"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- DEPRECATED NAMES"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Please use the new names as continuing support for the old names is"},{"Add":"-- not guaranteed."},{"Add":""},{"Add":"-- Version v2.0"},{"Add":""},{"Add":"infix 8 _∘-⟶_ _∘-↣_ _∘-↠_ _∘-⤖_ _∘-⇔_ _∘-↩_ _∘-↪_ _∘-↔_"},{"Add":""},{"Add":"_∘-⟶_ = _⟶-∘_"},{"Add":"{-# WARNING_ON_USAGE _∘-⟶_"},{"Add":"\"Warning: _∘-⟶_ was deprecated in v2.0."},{"Add":"Please use _⟶-∘_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_∘-↣_ = _↣-∘_"},{"Add":"{-# WARNING_ON_USAGE _∘-↣_"},{"Add":"\"Warning: _∘-↣_ was deprecated in v2.0."},{"Add":"Please use _↣-∘_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_∘-↠_ = _↠-∘_"},{"Add":"{-# WARNING_ON_USAGE _∘-↠_"},{"Add":"\"Warning: _∘-↠_ was deprecated in v2.0."},{"Add":"Please use _↠-∘_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_∘-⤖_ = _⤖-∘_"},{"Add":"{-# WARNING_ON_USAGE _∘-⤖_"},{"Add":"\"Warning: _∘-⤖_ was deprecated in v2.0."},{"Add":"Please use _⤖-∘_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_∘-⇔_ = _⇔-∘_"},{"Add":"{-# WARNING_ON_USAGE _∘-⇔_"},{"Add":"\"Warning: _∘-⇔_ was deprecated in v2.0."},{"Add":"Please use _⇔-∘_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_∘-↩_ = _↩-∘_"},{"Add":"{-# WARNING_ON_USAGE _∘-↩_"},{"Add":"\"Warning: _∘-↩_ was deprecated in v2.0."},{"Add":"Please use _↩-∘_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_∘-↪_ = _↪-∘_"},{"Add":"{-# WARNING_ON_USAGE _∘-↪_"},{"Add":"\"Warning: _∘-↪_ was deprecated in v2.0."},{"Add":"Please use _↪-∘_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"_∘-↔_ = _↔-∘_"},{"Add":"{-# WARNING_ON_USAGE _∘-↔_"},{"Add":"\"Warning: _∘-↔_ was deprecated in v2.0."},{"Add":"Please use _↔-∘_ instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Consequences.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Relationships between properties of functions. See"},{"Add":"-- `Function.Consequences.Propositional` for specialisations to"},{"Add":"-- propositional equality."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Consequences where"},{"Add":""},{"Add":"open import Data.Product.Base as Product"},{"Add":"open import Function.Definitions"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)"},{"Add":"open import Relation.Nullary.Negation.Core using (¬_; contraposition)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b ℓ₁ ℓ₂ : Level"},{"Add":" A B : Set a"},{"Add":" ≈₁ ≈₂ : Rel A ℓ₁"},{"Add":" f f⁻¹ : A → B"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Injective"},{"Add":""},{"Add":"contraInjective : ∀ (≈₂ : Rel B ℓ₂) → Injective ≈₁ ≈₂ f →"},{"Add":" ∀ {x y} → ¬ (≈₁ x y) → ¬ (≈₂ (f x) (f y))"},{"Add":"contraInjective _ inj p = contraposition inj p"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Inverseˡ"},{"Add":""},{"Add":"inverseˡ⇒surjective : ∀ (≈₂ : Rel B ℓ₂) →"},{"Add":" Inverseˡ ≈₁ ≈₂ f f⁻¹ →"},{"Add":" Surjective ≈₁ ≈₂ f"},{"Add":"inverseˡ⇒surjective ≈₂ invˡ y = (_ , invˡ)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Inverseʳ"},{"Add":""},{"Add":"inverseʳ⇒injective : ∀ (≈₂ : Rel B ℓ₂) f →"},{"Add":" Reflexive ≈₂ →"},{"Add":" Symmetric ≈₁ →"},{"Add":" Transitive ≈₁ →"},{"Add":" Inverseʳ ≈₁ ≈₂ f f⁻¹ →"},{"Add":" Injective ≈₁ ≈₂ f"},{"Add":"inverseʳ⇒injective ≈₂ f refl sym trans invʳ {x} {y} fx≈fy ="},{"Add":" trans (sym (invʳ refl)) (invʳ fx≈fy)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Inverseᵇ"},{"Add":""},{"Add":"inverseᵇ⇒bijective : ∀ (≈₂ : Rel B ℓ₂) →"},{"Add":" Reflexive ≈₂ →"},{"Add":" Symmetric ≈₁ →"},{"Add":" Transitive ≈₁ →"},{"Add":" Inverseᵇ ≈₁ ≈₂ f f⁻¹ →"},{"Add":" Bijective ≈₁ ≈₂ f"},{"Add":"inverseᵇ⇒bijective {f = f} ≈₂ refl sym trans (invˡ , invʳ) ="},{"Add":" (inverseʳ⇒injective ≈₂ f refl sym trans invʳ , inverseˡ⇒surjective ≈₂ invˡ)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- StrictlySurjective"},{"Add":""},{"Add":"surjective⇒strictlySurjective : ∀ (≈₂ : Rel B ℓ₂) →"},{"Add":" Reflexive ≈₁ →"},{"Add":" Surjective ≈₁ ≈₂ f →"},{"Add":" StrictlySurjective ≈₂ f"},{"Add":"surjective⇒strictlySurjective _ refl surj x ="},{"Add":" Product.map₂ (λ v → v refl) (surj x)"},{"Add":""},{"Add":"strictlySurjective⇒surjective : Transitive ≈₂ →"},{"Add":" Congruent ≈₁ ≈₂ f →"},{"Add":" StrictlySurjective ≈₂ f →"},{"Add":" Surjective ≈₁ ≈₂ f"},{"Add":"strictlySurjective⇒surjective trans cong surj x ="},{"Add":" Product.map₂ (λ fy≈x z≈y → trans (cong z≈y) fy≈x) (surj x)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- StrictlyInverseˡ"},{"Add":""},{"Add":"inverseˡ⇒strictlyInverseˡ : ∀ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) →"},{"Add":" Reflexive ≈₁ →"},{"Add":" Inverseˡ ≈₁ ≈₂ f f⁻¹ →"},{"Add":" StrictlyInverseˡ ≈₂ f f⁻¹"},{"Add":"inverseˡ⇒strictlyInverseˡ _ _ refl sinv x = sinv refl"},{"Add":""},{"Add":"strictlyInverseˡ⇒inverseˡ : Transitive ≈₂ →"},{"Add":" Congruent ≈₁ ≈₂ f →"},{"Add":" StrictlyInverseˡ ≈₂ f f⁻¹ →"},{"Add":" Inverseˡ ≈₁ ≈₂ f f⁻¹"},{"Add":"strictlyInverseˡ⇒inverseˡ trans cong sinv {x} y≈f⁻¹x ="},{"Add":" trans (cong y≈f⁻¹x) (sinv x)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- StrictlyInverseʳ"},{"Add":""},{"Add":"inverseʳ⇒strictlyInverseʳ : ∀ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) →"},{"Add":" Reflexive ≈₂ →"},{"Add":" Inverseʳ ≈₁ ≈₂ f f⁻¹ →"},{"Add":" StrictlyInverseʳ ≈₁ f f⁻¹"},{"Add":"inverseʳ⇒strictlyInverseʳ _ _ refl sinv x = sinv refl"},{"Add":""},{"Add":"strictlyInverseʳ⇒inverseʳ : Transitive ≈₁ →"},{"Add":" Congruent ≈₂ ≈₁ f⁻¹ →"},{"Add":" StrictlyInverseʳ ≈₁ f f⁻¹ →"},{"Add":" Inverseʳ ≈₁ ≈₂ f f⁻¹"},{"Add":"strictlyInverseʳ⇒inverseʳ trans cong sinv {x} y≈f⁻¹x ="},{"Add":" trans (cong y≈f⁻¹x) (sinv x)"}]}},{"FileAdd":{"meta":{"basename":"Consequences","metadata":576},"context":[{"change":0,"pos":1018846}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Setoid.agda","metadata":0},"context":[{"change":0,"pos":1237965}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Relationships between properties of functions where the equality"},{"Add":"-- over both the domain and codomain are assumed to be setoids."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":""},{"Add":"module Function.Consequences.Setoid"},{"Add":" {a b ℓ₁ ℓ₂}"},{"Add":" (S : Setoid a ℓ₁)"},{"Add":" (T : Setoid b ℓ₂)"},{"Add":" where"},{"Add":""},{"Add":"open import Function.Definitions"},{"Add":"open import Relation.Nullary.Negation.Core"},{"Add":""},{"Add":"import Function.Consequences as C"},{"Add":""},{"Add":"private"},{"Add":" open module S = Setoid S using () renaming (Carrier to A; _≈_ to ≈₁)"},{"Add":" open module T = Setoid T using () renaming (Carrier to B; _≈_ to ≈₂)"},{"Add":""},{"Add":" variable"},{"Add":" f : A → B"},{"Add":" f⁻¹ : B → A"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Injective"},{"Add":""},{"Add":"contraInjective : Injective ≈₁ ≈₂ f →"},{"Add":" ∀ {x y} → ¬ (≈₁ x y) → ¬ (≈₂ (f x) (f y))"},{"Add":"contraInjective = C.contraInjective ≈₂"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Inverseˡ"},{"Add":""},{"Add":"inverseˡ⇒surjective : Inverseˡ ≈₁ ≈₂ f f⁻¹ → Surjective ≈₁ ≈₂ f"},{"Add":"inverseˡ⇒surjective = C.inverseˡ⇒surjective ≈₂"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Inverseʳ"},{"Add":""},{"Add":"inverseʳ⇒injective : ∀ f → Inverseʳ ≈₁ ≈₂ f f⁻¹ → Injective ≈₁ ≈₂ f"},{"Add":"inverseʳ⇒injective f = C.inverseʳ⇒injective ≈₂ f T.refl S.sym S.trans"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Inverseᵇ"},{"Add":""},{"Add":"inverseᵇ⇒bijective : Inverseᵇ ≈₁ ≈₂ f f⁻¹ → Bijective ≈₁ ≈₂ f"},{"Add":"inverseᵇ⇒bijective = C.inverseᵇ⇒bijective ≈₂ T.refl S.sym S.trans"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- StrictlySurjective"},{"Add":""},{"Add":"surjective⇒strictlySurjective : Surjective ≈₁ ≈₂ f →"},{"Add":" StrictlySurjective ≈₂ f"},{"Add":"surjective⇒strictlySurjective ="},{"Add":" C.surjective⇒strictlySurjective ≈₂ S.refl"},{"Add":""},{"Add":"strictlySurjective⇒surjective : Congruent ≈₁ ≈₂ f →"},{"Add":" StrictlySurjective ≈₂ f →"},{"Add":" Surjective ≈₁ ≈₂ f"},{"Add":"strictlySurjective⇒surjective ="},{"Add":" C.strictlySurjective⇒surjective T.trans"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- StrictlyInverseˡ"},{"Add":""},{"Add":"inverseˡ⇒strictlyInverseˡ : Inverseˡ ≈₁ ≈₂ f f⁻¹ →"},{"Add":" StrictlyInverseˡ ≈₂ f f⁻¹"},{"Add":"inverseˡ⇒strictlyInverseˡ = C.inverseˡ⇒strictlyInverseˡ ≈₁ ≈₂ S.refl"},{"Add":""},{"Add":"strictlyInverseˡ⇒inverseˡ : Congruent ≈₁ ≈₂ f →"},{"Add":" StrictlyInverseˡ ≈₂ f f⁻¹ →"},{"Add":" Inverseˡ ≈₁ ≈₂ f f⁻¹"},{"Add":"strictlyInverseˡ⇒inverseˡ = C.strictlyInverseˡ⇒inverseˡ T.trans"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- StrictlyInverseʳ"},{"Add":""},{"Add":"inverseʳ⇒strictlyInverseʳ : Inverseʳ ≈₁ ≈₂ f f⁻¹ →"},{"Add":" StrictlyInverseʳ ≈₁ f f⁻¹"},{"Add":"inverseʳ⇒strictlyInverseʳ = C.inverseʳ⇒strictlyInverseʳ ≈₁ ≈₂ T.refl"},{"Add":""},{"Add":"strictlyInverseʳ⇒inverseʳ : Congruent ≈₂ ≈₁ f⁻¹ →"},{"Add":" StrictlyInverseʳ ≈₁ f f⁻¹ →"},{"Add":" Inverseʳ ≈₁ ≈₂ f f⁻¹"},{"Add":"strictlyInverseʳ⇒inverseʳ = C.strictlyInverseʳ⇒inverseʳ S.trans"}]}},{"FileAdd":{"meta":{"basename":"Propositional.agda","metadata":0},"context":[{"change":0,"pos":1237965}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Relationships between properties of functions where the equality"},{"Add":"-- over both the domain and codomain is assumed to be _≡_"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Consequences.Propositional"},{"Add":" {a b} {A : Set a} {B : Set b}"},{"Add":" where"},{"Add":""},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≢_; cong)"},{"Add":"open import Relation.Binary.PropositionalEquality.Properties"},{"Add":" using (setoid)"},{"Add":"open import Function.Definitions"},{"Add":"open import Relation.Nullary.Negation.Core using (contraposition)"},{"Add":""},{"Add":"import Function.Consequences.Setoid (setoid A) (setoid B) as Setoid"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export setoid properties"},{"Add":""},{"Add":"open Setoid public"},{"Add":" hiding"},{"Add":" ( strictlySurjective⇒surjective"},{"Add":" ; strictlyInverseˡ⇒inverseˡ"},{"Add":" ; strictlyInverseʳ⇒inverseʳ"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties that rely on congruence"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" f : A → B"},{"Add":" f⁻¹ : B → A"},{"Add":""},{"Add":"strictlySurjective⇒surjective : StrictlySurjective _≡_ f →"},{"Add":" Surjective _≡_ _≡_ f"},{"Add":"strictlySurjective⇒surjective ="},{"Add":" Setoid.strictlySurjective⇒surjective (cong _)"},{"Add":""},{"Add":"strictlyInverseˡ⇒inverseˡ : ∀ f → StrictlyInverseˡ _≡_ f f⁻¹ →"},{"Add":" Inverseˡ _≡_ _≡_ f f⁻¹"},{"Add":"strictlyInverseˡ⇒inverseˡ f ="},{"Add":" Setoid.strictlyInverseˡ⇒inverseˡ (cong _)"},{"Add":""},{"Add":"strictlyInverseʳ⇒inverseʳ : ∀ f → StrictlyInverseʳ _≡_ f f⁻¹ →"},{"Add":" Inverseʳ _≡_ _≡_ f f⁻¹"},{"Add":"strictlyInverseʳ⇒inverseʳ f ="},{"Add":" Setoid.strictlyInverseʳ⇒inverseʳ (cong _)"}]}},{"FileAdd":{"meta":{"basename":"Bundles.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Bundles for types of functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this file should usually be accessed from `Function`."},{"Add":""},{"Add":"-- Note that these bundles differ from those found elsewhere in other"},{"Add":"-- library hierarchies as they take Setoids as parameters. This is"},{"Add":"-- because a function is of no use without knowing what its domain and"},{"Add":"-- codomain is, as well which equalities are being considered over them."},{"Add":"-- One consequence of this is that they are not built from the"},{"Add":"-- definitions found in `Function.Structures` as is usually the case in"},{"Add":"-- other library hierarchies, as this would duplicate the equality"},{"Add":"-- axioms."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Bundles where"},{"Add":""},{"Add":"open import Function.Base using (_∘_)"},{"Add":"open import Function.Definitions"},{"Add":"import Function.Structures as FunctionStructures"},{"Add":"open import Level using (Level; _⊔_; suc)"},{"Add":"open import Data.Product.Base using (_,_; proj₁; proj₂)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.Core using (_Preserves_⟶_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core as ≡"},{"Add":" using (_≡_)"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":"open import Function.Consequences.Propositional"},{"Add":"open Setoid using (isEquivalence)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b ℓ₁ ℓ₂ : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Setoid bundles"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where"},{"Add":""},{"Add":" open Setoid From using () renaming (Carrier to A; _≈_ to _≈₁_)"},{"Add":" open Setoid To using () renaming (Carrier to B; _≈_ to _≈₂_)"},{"Add":" open FunctionStructures _≈₁_ _≈₂_"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles with one element"},{"Add":""},{"Add":" -- Called `Func` rather than `Function` in order to avoid clashing"},{"Add":" -- with the top-level module."},{"Add":" record Func : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" to : A → B"},{"Add":" cong : Congruent _≈₁_ _≈₂_ to"},{"Add":""},{"Add":" isCongruent : IsCongruent to"},{"Add":" isCongruent = record"},{"Add":" { cong = cong"},{"Add":" ; isEquivalence₁ = isEquivalence From"},{"Add":" ; isEquivalence₂ = isEquivalence To"},{"Add":" }"},{"Add":""},{"Add":" open IsCongruent isCongruent public"},{"Add":" using (module Eq₁; module Eq₂)"},{"Add":""},{"Add":""},{"Add":" record Injection : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" to : A → B"},{"Add":" cong : Congruent _≈₁_ _≈₂_ to"},{"Add":" injective : Injective _≈₁_ _≈₂_ to"},{"Add":""},{"Add":" function : Func"},{"Add":" function = record"},{"Add":" { to = to"},{"Add":" ; cong = cong"},{"Add":" }"},{"Add":""},{"Add":" open Func function public"},{"Add":" hiding (to; cong)"},{"Add":""},{"Add":" isInjection : IsInjection to"},{"Add":" isInjection = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; injective = injective"},{"Add":" }"},{"Add":""},{"Add":""},{"Add":" record Surjection : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" to : A → B"},{"Add":" cong : Congruent _≈₁_ _≈₂_ to"},{"Add":" surjective : Surjective _≈₁_ _≈₂_ to"},{"Add":""},{"Add":" function : Func"},{"Add":" function = record"},{"Add":" { to = to"},{"Add":" ; cong = cong"},{"Add":" }"},{"Add":""},{"Add":" open Func function public"},{"Add":" hiding (to; cong)"},{"Add":""},{"Add":" isSurjection : IsSurjection to"},{"Add":" isSurjection = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; surjective = surjective"},{"Add":" }"},{"Add":""},{"Add":" open IsSurjection isSurjection public"},{"Add":" using"},{"Add":" ( strictlySurjective"},{"Add":" )"},{"Add":""},{"Add":" to⁻ : B → A"},{"Add":" to⁻ = proj₁ ∘ surjective"},{"Add":""},{"Add":" to∘to⁻ : ∀ x → to (to⁻ x) ≈₂ x"},{"Add":" to∘to⁻ = proj₂ ∘ strictlySurjective"},{"Add":""},{"Add":""},{"Add":" record Bijection : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" to : A → B"},{"Add":" cong : Congruent _≈₁_ _≈₂_ to"},{"Add":" bijective : Bijective _≈₁_ _≈₂_ to"},{"Add":""},{"Add":" injective : Injective _≈₁_ _≈₂_ to"},{"Add":" injective = proj₁ bijective"},{"Add":""},{"Add":" surjective : Surjective _≈₁_ _≈₂_ to"},{"Add":" surjective = proj₂ bijective"},{"Add":""},{"Add":" injection : Injection"},{"Add":" injection = record"},{"Add":" { cong = cong"},{"Add":" ; injective = injective"},{"Add":" }"},{"Add":""},{"Add":" surjection : Surjection"},{"Add":" surjection = record"},{"Add":" { cong = cong"},{"Add":" ; surjective = surjective"},{"Add":" }"},{"Add":""},{"Add":" open Injection injection public using (isInjection)"},{"Add":" open Surjection surjection public using (isSurjection; to⁻; strictlySurjective)"},{"Add":""},{"Add":" isBijection : IsBijection to"},{"Add":" isBijection = record"},{"Add":" { isInjection = isInjection"},{"Add":" ; surjective = surjective"},{"Add":" }"},{"Add":""},{"Add":" open IsBijection isBijection public using (module Eq₁; module Eq₂)"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles with two elements"},{"Add":""},{"Add":"module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where"},{"Add":""},{"Add":" open Setoid From using () renaming (Carrier to A; _≈_ to _≈₁_)"},{"Add":" open Setoid To using () renaming (Carrier to B; _≈_ to _≈₂_)"},{"Add":" open FunctionStructures _≈₁_ _≈₂_"},{"Add":""},{"Add":" record Equivalence : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" to : A → B"},{"Add":" from : B → A"},{"Add":" to-cong : Congruent _≈₁_ _≈₂_ to"},{"Add":" from-cong : Congruent _≈₂_ _≈₁_ from"},{"Add":""},{"Add":" toFunction : Func From To"},{"Add":" toFunction = record"},{"Add":" { to = to"},{"Add":" ; cong = to-cong"},{"Add":" }"},{"Add":""},{"Add":" open Func toFunction public"},{"Add":" using (module Eq₁; module Eq₂)"},{"Add":" renaming (isCongruent to to-isCongruent)"},{"Add":""},{"Add":" fromFunction : Func To From"},{"Add":" fromFunction = record"},{"Add":" { to = from"},{"Add":" ; cong = from-cong"},{"Add":" }"},{"Add":""},{"Add":" open Func fromFunction public"},{"Add":" using ()"},{"Add":" renaming (isCongruent to from-isCongruent)"},{"Add":""},{"Add":""},{"Add":" record LeftInverse : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" to : A → B"},{"Add":" from : B → A"},{"Add":" to-cong : Congruent _≈₁_ _≈₂_ to"},{"Add":" from-cong : Congruent _≈₂_ _≈₁_ from"},{"Add":" inverseˡ : Inverseˡ _≈₁_ _≈₂_ to from"},{"Add":""},{"Add":" isCongruent : IsCongruent to"},{"Add":" isCongruent = record"},{"Add":" { cong = to-cong"},{"Add":" ; isEquivalence₁ = isEquivalence From"},{"Add":" ; isEquivalence₂ = isEquivalence To"},{"Add":" }"},{"Add":""},{"Add":" isLeftInverse : IsLeftInverse to from"},{"Add":" isLeftInverse = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; from-cong = from-cong"},{"Add":" ; inverseˡ = inverseˡ"},{"Add":" }"},{"Add":""},{"Add":" open IsLeftInverse isLeftInverse public"},{"Add":" using (module Eq₁; module Eq₂; strictlyInverseˡ; isSurjection)"},{"Add":""},{"Add":" equivalence : Equivalence"},{"Add":" equivalence = record"},{"Add":" { to-cong = to-cong"},{"Add":" ; from-cong = from-cong"},{"Add":" }"},{"Add":""},{"Add":" isSplitSurjection : IsSplitSurjection to"},{"Add":" isSplitSurjection = record"},{"Add":" { from = from"},{"Add":" ; isLeftInverse = isLeftInverse"},{"Add":" }"},{"Add":""},{"Add":" surjection : Surjection From To"},{"Add":" surjection = record"},{"Add":" { to = to"},{"Add":" ; cong = to-cong"},{"Add":" ; surjective = λ y → from y , inverseˡ"},{"Add":" }"},{"Add":""},{"Add":""},{"Add":""},{"Add":" record RightInverse : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" to : A → B"},{"Add":" from : B → A"},{"Add":" to-cong : Congruent _≈₁_ _≈₂_ to"},{"Add":" from-cong : from Preserves _≈₂_ ⟶ _≈₁_"},{"Add":" inverseʳ : Inverseʳ _≈₁_ _≈₂_ to from"},{"Add":""},{"Add":" isCongruent : IsCongruent to"},{"Add":" isCongruent = record"},{"Add":" { cong = to-cong"},{"Add":" ; isEquivalence₁ = isEquivalence From"},{"Add":" ; isEquivalence₂ = isEquivalence To"},{"Add":" }"},{"Add":""},{"Add":" isRightInverse : IsRightInverse to from"},{"Add":" isRightInverse = record"},{"Add":" { isCongruent = isCongruent"},{"Add":" ; from-cong = from-cong"},{"Add":" ; inverseʳ = inverseʳ"},{"Add":" }"},{"Add":""},{"Add":" open IsRightInverse isRightInverse public"},{"Add":" using (module Eq₁; module Eq₂; strictlyInverseʳ)"},{"Add":""},{"Add":" equivalence : Equivalence"},{"Add":" equivalence = record"},{"Add":" { to-cong = to-cong"},{"Add":" ; from-cong = from-cong"},{"Add":" }"},{"Add":""},{"Add":""},{"Add":" record Inverse : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" to : A → B"},{"Add":" from : B → A"},{"Add":" to-cong : Congruent _≈₁_ _≈₂_ to"},{"Add":" from-cong : Congruent _≈₂_ _≈₁_ from"},{"Add":" inverse : Inverseᵇ _≈₁_ _≈₂_ to from"},{"Add":""},{"Add":" inverseˡ : Inverseˡ _≈₁_ _≈₂_ to from"},{"Add":" inverseˡ = proj₁ inverse"},{"Add":""},{"Add":" inverseʳ : Inverseʳ _≈₁_ _≈₂_ to from"},{"Add":" inverseʳ = proj₂ inverse"},{"Add":""},{"Add":" leftInverse : LeftInverse"},{"Add":" leftInverse = record"},{"Add":" { to-cong = to-cong"},{"Add":" ; from-cong = from-cong"},{"Add":" ; inverseˡ = inverseˡ"},{"Add":" }"},{"Add":""},{"Add":" rightInverse : RightInverse"},{"Add":" rightInverse = record"},{"Add":" { to-cong = to-cong"},{"Add":" ; from-cong = from-cong"},{"Add":" ; inverseʳ = inverseʳ"},{"Add":" }"},{"Add":""},{"Add":" open LeftInverse leftInverse public using (isLeftInverse; strictlyInverseˡ)"},{"Add":" open RightInverse rightInverse public using (isRightInverse; strictlyInverseʳ)"},{"Add":""},{"Add":" isInverse : IsInverse to from"},{"Add":" isInverse = record"},{"Add":" { isLeftInverse = isLeftInverse"},{"Add":" ; inverseʳ = inverseʳ"},{"Add":" }"},{"Add":""},{"Add":" open IsInverse isInverse public using (module Eq₁; module Eq₂)"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles with three elements"},{"Add":""},{"Add":" record BiEquivalence : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" to : A → B"},{"Add":" from₁ : B → A"},{"Add":" from₂ : B → A"},{"Add":" to-cong : Congruent _≈₁_ _≈₂_ to"},{"Add":" from₁-cong : Congruent _≈₂_ _≈₁_ from₁"},{"Add":" from₂-cong : Congruent _≈₂_ _≈₁_ from₂"},{"Add":""},{"Add":""},{"Add":" record BiInverse : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where"},{"Add":" field"},{"Add":" to : A → B"},{"Add":" from₁ : B → A"},{"Add":" from₂ : B → A"},{"Add":" to-cong : Congruent _≈₁_ _≈₂_ to"},{"Add":" from₁-cong : Congruent _≈₂_ _≈₁_ from₁"},{"Add":" from₂-cong : Congruent _≈₂_ _≈₁_ from₂"},{"Add":" inverseˡ : Inverseˡ _≈₁_ _≈₂_ to from₁"},{"Add":" inverseʳ : Inverseʳ _≈₁_ _≈₂_ to from₂"},{"Add":""},{"Add":" to-isCongruent : IsCongruent to"},{"Add":" to-isCongruent = record"},{"Add":" { cong = to-cong"},{"Add":" ; isEquivalence₁ = isEquivalence From"},{"Add":" ; isEquivalence₂ = isEquivalence To"},{"Add":" }"},{"Add":""},{"Add":" isBiInverse : IsBiInverse to from₁ from₂"},{"Add":" isBiInverse = record"},{"Add":" { to-isCongruent = to-isCongruent"},{"Add":" ; from₁-cong = from₁-cong"},{"Add":" ; from₂-cong = from₂-cong"},{"Add":" ; inverseˡ = inverseˡ"},{"Add":" ; inverseʳ = inverseʳ"},{"Add":" }"},{"Add":""},{"Add":" biEquivalence : BiEquivalence"},{"Add":" biEquivalence = record"},{"Add":" { to-cong = to-cong"},{"Add":" ; from₁-cong = from₁-cong"},{"Add":" ; from₂-cong = from₂-cong"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Other"},{"Add":""},{"Add":" -- A left inverse is also known as a “split surjection”."},{"Add":" --"},{"Add":" -- As the name implies, a split surjection is a special kind of"},{"Add":" -- surjection where the witness generated in the domain in the"},{"Add":" -- function for elements `x₁` and `x₂` are equal if `x₁ ≈ x₂` ."},{"Add":" --"},{"Add":" -- The difference is the `from-cong` law --- generally, the section"},{"Add":" -- (called `Surjection.to⁻` or `SplitSurjection.from`) of a surjection"},{"Add":" -- need not respect equality, whereas it must in a split surjection."},{"Add":" --"},{"Add":" -- The two notions coincide when the equivalence relation on `B` is"},{"Add":" -- propositional equality (because all functions respect propositional"},{"Add":" -- equality)."},{"Add":" --"},{"Add":" -- For further background on (split) surjections, one may consult any"},{"Add":" -- general mathematical references which work without the principle"},{"Add":" -- of choice. For example:"},{"Add":" --"},{"Add":" -- https://ncatlab.org/nlab/show/split+epimorphism."},{"Add":" --"},{"Add":" -- The connection to set-theoretic notions with the same names is"},{"Add":" -- justified by the setoid type theory/homotopy type theory"},{"Add":" -- observation/definition that (∃x : A. P) = ∥ Σx : A. P ∥ --- i.e.,"},{"Add":" -- we can read set-theoretic ∃ as squashed/propositionally truncated Σ."},{"Add":" --"},{"Add":" -- We see working with setoids as working in the MLTT model of a setoid"},{"Add":" -- type theory, in which ∥ X ∥ is interpreted as the setoid with carrier"},{"Add":" -- set X and the equivalence relation that relates all elements."},{"Add":" -- All maps into ∥ X ∥ respect equality, so in the idiomatic definitions"},{"Add":" -- here, we drop the corresponding trivial `cong` field completely."},{"Add":""},{"Add":" SplitSurjection : Set _"},{"Add":" SplitSurjection = LeftInverse"},{"Add":""},{"Add":" module SplitSurjection (splitSurjection : SplitSurjection) ="},{"Add":" LeftInverse splitSurjection"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Infix abbreviations for oft-used items"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Same naming convention as used for propositional equality below, with"},{"Add":"-- appended ₛ (for 'S'etoid)."},{"Add":""},{"Add":"infixr 0 _⟶ₛ_"},{"Add":"_⟶ₛ_ : Setoid a ℓ₁ → Setoid b ℓ₂ → Set _"},{"Add":"_⟶ₛ_ = Func"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bundles specialised for propositional equality"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"infix 3 _⟶_ _↣_ _↠_ _⤖_ _⇔_ _↩_ _↪_ _↩↪_ _↔_"},{"Add":"_⟶_ : Set a → Set b → Set _"},{"Add":"A ⟶ B = Func (≡.setoid A) (≡.setoid B)"},{"Add":""},{"Add":"_↣_ : Set a → Set b → Set _"},{"Add":"A ↣ B = Injection (≡.setoid A) (≡.setoid B)"},{"Add":""},{"Add":"_↠_ : Set a → Set b → Set _"},{"Add":"A ↠ B = Surjection (≡.setoid A) (≡.setoid B)"},{"Add":""},{"Add":"_⤖_ : Set a → Set b → Set _"},{"Add":"A ⤖ B = Bijection (≡.setoid A) (≡.setoid B)"},{"Add":""},{"Add":"_⇔_ : Set a → Set b → Set _"},{"Add":"A ⇔ B = Equivalence (≡.setoid A) (≡.setoid B)"},{"Add":""},{"Add":"_↩_ : Set a → Set b → Set _"},{"Add":"A ↩ B = LeftInverse (≡.setoid A) (≡.setoid B)"},{"Add":""},{"Add":"_↪_ : Set a → Set b → Set _"},{"Add":"A ↪ B = RightInverse (≡.setoid A) (≡.setoid B)"},{"Add":""},{"Add":"_↩↪_ : Set a → Set b → Set _"},{"Add":"A ↩↪ B = BiInverse (≡.setoid A) (≡.setoid B)"},{"Add":""},{"Add":"_↔_ : Set a → Set b → Set _"},{"Add":"A ↔ B = Inverse (≡.setoid A) (≡.setoid B)"},{"Add":""},{"Add":"-- We now define some constructors for the above that"},{"Add":"-- automatically provide the required congruency proofs."},{"Add":""},{"Add":"module _ {A : Set a} {B : Set b} where"},{"Add":""},{"Add":" mk⟶ : (A → B) → A ⟶ B"},{"Add":" mk⟶ to = record"},{"Add":" { to = to"},{"Add":" ; cong = ≡.cong to"},{"Add":" }"},{"Add":""},{"Add":" mk↣ : ∀ {to : A → B} → Injective _≡_ _≡_ to → A ↣ B"},{"Add":" mk↣ {to} inj = record"},{"Add":" { to = to"},{"Add":" ; cong = ≡.cong to"},{"Add":" ; injective = inj"},{"Add":" }"},{"Add":""},{"Add":" mk↠ : ∀ {to : A → B} → Surjective _≡_ _≡_ to → A ↠ B"},{"Add":" mk↠ {to} surj = record"},{"Add":" { to = to"},{"Add":" ; cong = ≡.cong to"},{"Add":" ; surjective = surj"},{"Add":" }"},{"Add":""},{"Add":" mk⤖ : ∀ {to : A → B} → Bijective _≡_ _≡_ to → A ⤖ B"},{"Add":" mk⤖ {to} bij = record"},{"Add":" { to = to"},{"Add":" ; cong = ≡.cong to"},{"Add":" ; bijective = bij"},{"Add":" }"},{"Add":""},{"Add":" mk⇔ : ∀ (to : A → B) (from : B → A) → A ⇔ B"},{"Add":" mk⇔ to from = record"},{"Add":" { to = to"},{"Add":" ; from = from"},{"Add":" ; to-cong = ≡.cong to"},{"Add":" ; from-cong = ≡.cong from"},{"Add":" }"},{"Add":""},{"Add":" mk↩ : ∀ {to : A → B} {from : B → A} → Inverseˡ _≡_ _≡_ to from → A ↩ B"},{"Add":" mk↩ {to} {from} invˡ = record"},{"Add":" { to = to"},{"Add":" ; from = from"},{"Add":" ; to-cong = ≡.cong to"},{"Add":" ; from-cong = ≡.cong from"},{"Add":" ; inverseˡ = invˡ"},{"Add":" }"},{"Add":""},{"Add":" mk↪ : ∀ {to : A → B} {from : B → A} → Inverseʳ _≡_ _≡_ to from → A ↪ B"},{"Add":" mk↪ {to} {from} invʳ = record"},{"Add":" { to = to"},{"Add":" ; from = from"},{"Add":" ; to-cong = ≡.cong to"},{"Add":" ; from-cong = ≡.cong from"},{"Add":" ; inverseʳ = invʳ"},{"Add":" }"},{"Add":""},{"Add":" mk↩↪ : ∀ {to : A → B} {from₁ : B → A} {from₂ : B → A} →"},{"Add":" Inverseˡ _≡_ _≡_ to from₁ → Inverseʳ _≡_ _≡_ to from₂ → A ↩↪ B"},{"Add":" mk↩↪ {to} {from₁} {from₂} invˡ invʳ = record"},{"Add":" { to = to"},{"Add":" ; from₁ = from₁"},{"Add":" ; from₂ = from₂"},{"Add":" ; to-cong = ≡.cong to"},{"Add":" ; from₁-cong = ≡.cong from₁"},{"Add":" ; from₂-cong = ≡.cong from₂"},{"Add":" ; inverseˡ = invˡ"},{"Add":" ; inverseʳ = invʳ"},{"Add":" }"},{"Add":""},{"Add":" mk↔ : ∀ {to : A → B} {from : B → A} → Inverseᵇ _≡_ _≡_ to from → A ↔ B"},{"Add":" mk↔ {to} {from} inv = record"},{"Add":" { to = to"},{"Add":" ; from = from"},{"Add":" ; to-cong = ≡.cong to"},{"Add":" ; from-cong = ≡.cong from"},{"Add":" ; inverse = inv"},{"Add":" }"},{"Add":""},{"Add":""},{"Add":" -- Strict variant of the above."},{"Add":" mk↠ₛ : ∀ {to : A → B} → StrictlySurjective _≡_ to → A ↠ B"},{"Add":" mk↠ₛ = mk↠ ∘ strictlySurjective⇒surjective"},{"Add":""},{"Add":" mk↔ₛ′ : ∀ (to : A → B) (from : B → A) →"},{"Add":" StrictlyInverseˡ _≡_ to from →"},{"Add":" StrictlyInverseʳ _≡_ to from →"},{"Add":" A ↔ B"},{"Add":" mk↔ₛ′ to from invˡ invʳ = mk↔ {to} {from}"},{"Add":" ( strictlyInverseˡ⇒inverseˡ to invˡ"},{"Add":" , strictlyInverseʳ⇒inverseʳ to invʳ"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Other"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Alternative syntax for the application of functions"},{"Add":""},{"Add":"module _ {From : Setoid a ℓ₁} {To : Setoid b ℓ₂} where"},{"Add":" open Setoid"},{"Add":""},{"Add":" infixl 5 _⟨$⟩_"},{"Add":" _⟨$⟩_ : Func From To → Carrier From → Carrier To"},{"Add":" _⟨$⟩_ = Func.to"}]}},{"FileAdd":{"meta":{"basename":"Bijection.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":"{-# OPTIONS --warn=noUserWarning #-}"},{"Add":""},{"Add":"module Function.Bijection where"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Function.Bijection was deprecated in v2.0."},{"Add":"Use the standard function hierarchy in Function/Function.Bundles instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.PropositionalEquality as ≡"},{"Add":"open import Function.Equality as F"},{"Add":" using (_⟶_; _⟨$⟩_) renaming (_∘_ to _⟪∘⟫_)"},{"Add":"open import Function.Injection as Inj hiding (id; _∘_; injection)"},{"Add":"open import Function.Surjection as Surj hiding (id; _∘_; surjection)"},{"Add":"open import Function.LeftInverse as Left hiding (id; _∘_; leftInverse)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Bijective functions."},{"Add":""},{"Add":"record Bijective {f₁ f₂ t₁ t₂}"},{"Add":" {From : Setoid f₁ f₂} {To : Setoid t₁ t₂}"},{"Add":" (to : From ⟶ To) :"},{"Add":" Set (f₁ ⊔ f₂ ⊔ t₁ ⊔ t₂) where"},{"Add":" field"},{"Add":" injective : Injective to"},{"Add":" surjective : Surjective to"},{"Add":""},{"Add":" open Surjective surjective public"},{"Add":""},{"Add":" left-inverse-of : from LeftInverseOf to"},{"Add":" left-inverse-of x = injective (right-inverse-of (to ⟨$⟩ x))"},{"Add":"{-# WARNING_ON_USAGE Bijective"},{"Add":"\"Warning: Bijective was deprecated in v2.0."},{"Add":"Please use Function.(Structures.)IsBijection instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The set of all bijections between two setoids."},{"Add":""},{"Add":"record Bijection {f₁ f₂ t₁ t₂}"},{"Add":" (From : Setoid f₁ f₂) (To : Setoid t₁ t₂) :"},{"Add":" Set (f₁ ⊔ f₂ ⊔ t₁ ⊔ t₂) where"},{"Add":" field"},{"Add":" to : From ⟶ To"},{"Add":" bijective : Bijective to"},{"Add":""},{"Add":" open Bijective bijective public"},{"Add":""},{"Add":" injection : Injection From To"},{"Add":" injection = record"},{"Add":" { to = to"},{"Add":" ; injective = injective"},{"Add":" }"},{"Add":""},{"Add":" surjection : Surjection From To"},{"Add":" surjection = record"},{"Add":" { to = to"},{"Add":" ; surjective = surjective"},{"Add":" }"},{"Add":""},{"Add":" open Surjection surjection public"},{"Add":" using (equivalence; right-inverse; from-to)"},{"Add":""},{"Add":" left-inverse : LeftInverse From To"},{"Add":" left-inverse = record"},{"Add":" { to = to"},{"Add":" ; from = from"},{"Add":" ; left-inverse-of = left-inverse-of"},{"Add":" }"},{"Add":""},{"Add":" open LeftInverse left-inverse public using (to-from)"},{"Add":"{-# WARNING_ON_USAGE Bijection"},{"Add":"\"Warning: Bijection was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)Bijection instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The set of all bijections between two sets (i.e. bijections with"},{"Add":"-- propositional equality)"},{"Add":""},{"Add":"infix 3 _⤖_"},{"Add":""},{"Add":"_⤖_ : ∀ {f t} → Set f → Set t → Set _"},{"Add":"From ⤖ To = Bijection (≡.setoid From) (≡.setoid To)"},{"Add":"{-# WARNING_ON_USAGE _⤖_"},{"Add":"\"Warning: _⤖_ was deprecated in v2.0."},{"Add":"Please use Function.(Bundles.)mk⤖ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"bijection : ∀ {f t} {From : Set f} {To : Set t} →"},{"Add":" (to : From → To) (from : To → From) →"},{"Add":" (∀ {x y} → to x ≡ to y → x ≡ y) →"},{"Add":" (∀ x → to (from x) ≡ x) →"},{"Add":" From ⤖ To"},{"Add":"bijection to from inj invʳ = record"},{"Add":" { to = F.→-to-⟶ to"},{"Add":" ; bijective = record"},{"Add":" { injective = inj"},{"Add":" ; surjective = record"},{"Add":" { from = F.→-to-⟶ from"},{"Add":" ; right-inverse-of = invʳ"},{"Add":" }"},{"Add":" }"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE bijection"},{"Add":"\"Warning: bijection was deprecated in v2.0."},{"Add":"Please use either Function.Properties.Bijection.trans or"},{"Add":"Function.Construct.Composition.bijection instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Identity and composition. (Note that these proofs are superfluous,"},{"Add":"-- given that Bijection is equivalent to Function.Inverse.Inverse.)"},{"Add":""},{"Add":"id : ∀ {s₁ s₂} {S : Setoid s₁ s₂} → Bijection S S"},{"Add":"id {S = S} = record"},{"Add":" { to = F.id"},{"Add":" ; bijective = record"},{"Add":" { injective = Injection.injective (Inj.id {S = S})"},{"Add":" ; surjective = Surjection.surjective (Surj.id {S = S})"},{"Add":" }"},{"Add":" }"},{"Add":"{-# WARNING_ON_USAGE id"},{"Add":"\"Warning: id was deprecated in v2.0."},{"Add":"Please use either Function.Properties.Bijection.refl or"},{"Add":"Function.Construct.Identity.bijection instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"infixr 9 _∘_"},{"Add":""},{"Add":"_∘_ : ∀ {f₁ f₂ m₁ m₂ t₁ t₂}"},{"Add":" {F : Setoid f₁ f₂} {M : Setoid m₁ m₂} {T : Setoid t₁ t₂} →"},{"Add":" Bijection M T → Bijection F M → Bijection F T"},{"Add":"f ∘ g = record"},{"Add":" { to = to f ⟪∘⟫ to g"},{"Add":" ; bijective = record"},{"Add":" { injective = Injection.injective (Inj._∘_ (injection f) (injection g))"},{"Add":" ; surjective = Surjection.surjective (Surj._∘_ (surjection f) (surjection g))"},{"Add":" }"},{"Add":" } where open Bijection"},{"Add":"{-# WARNING_ON_USAGE _∘_"},{"Add":"\"Warning: _∘_ was deprecated in v2.0."},{"Add":"Please use either Function.Properties.Bijection.trans or"},{"Add":"Function.Construct.Composition.bijection instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Base.agda","metadata":0},"context":[{"change":0,"pos":1018846}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Simple combinators working solely on and with functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- The contents of this module is also accessible via the `Function`"},{"Add":"-- module. See `Function.Strict` for strict versions of these"},{"Add":"-- combinators."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Function.Base where"},{"Add":""},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c d e : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":" C : Set c"},{"Add":" D : Set d"},{"Add":" E : Set e"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Some simple functions"},{"Add":""},{"Add":"id : A → A"},{"Add":"id x = x"},{"Add":""},{"Add":"const : A → B → A"},{"Add":"const x = λ _ → x"},{"Add":""},{"Add":"constᵣ : A → B → B"},{"Add":"constᵣ _ = id"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Operations on dependent functions"},{"Add":""},{"Add":"-- These are functions whose output has a type that depends on the"},{"Add":"-- value of the input to the function."},{"Add":""},{"Add":"infixr 9 _∘_ _∘₂_"},{"Add":"infixl 8 _ˢ_"},{"Add":"infixl 0 _|>_"},{"Add":"infix 0 case_return_of_"},{"Add":"infixr -1 _$_"},{"Add":""},{"Add":"-- Composition"},{"Add":""},{"Add":"_∘_ : ∀ {A : Set a} {B : A → Set b} {C : {x : A} → B x → Set c} →"},{"Add":" (∀ {x} (y : B x) → C y) → (g : (x : A) → B x) →"},{"Add":" ((x : A) → C (g x))"},{"Add":"f ∘ g = λ x → f (g x)"},{"Add":"{-# INLINE _∘_ #-}"},{"Add":""},{"Add":"_∘₂_ : ∀ {A₁ : Set a} {A₂ : A₁ → Set d}"},{"Add":" {B : (x : A₁) → A₂ x → Set b}"},{"Add":" {C : {x : A₁} → {y : A₂ x} → B x y → Set c} →"},{"Add":" ({x : A₁} → {y : A₂ x} → (z : B x y) → C z) →"},{"Add":" (g : (x : A₁) → (y : A₂ x) → B x y) →"},{"Add":" ((x : A₁) → (y : A₂ x) → C (g x y))"},{"Add":"f ∘₂ g = λ x y → f (g x y)"},{"Add":""},{"Add":"-- Flipping order of arguments"},{"Add":""},{"Add":"flip : ∀ {A : Set a} {B : Set b} {C : A → B → Set c} →"},{"Add":" ((x : A) (y : B) → C x y) → ((y : B) (x : A) → C x y)"},{"Add":"flip f = λ y x → f x y"},{"Add":"{-# INLINE flip #-}"},{"Add":""},{"Add":"-- Application - note that _$_ is right associative, as in Haskell."},{"Add":"-- If you want a left associative infix application operator, use"},{"Add":"-- RawFunctor._<$>_ from Effect.Functor."},{"Add":""},{"Add":"_$_ : ∀ {A : Set a} {B : A → Set b} →"},{"Add":" ((x : A) → B x) → ((x : A) → B x)"},{"Add":"f $ x = f x"},{"Add":"{-# INLINE _$_ #-}"},{"Add":""},{"Add":"-- Flipped application (aka pipe-forward)"},{"Add":""},{"Add":"_|>_ : ∀ {A : Set a} {B : A → Set b} →"},{"Add":" (a : A) → (∀ a → B a) → B a"},{"Add":"_|>_ = flip _$_"},{"Add":"{-# INLINE _|>_ #-}"},{"Add":""},{"Add":"-- The S combinator - written infix as in Conor McBride's paper"},{"Add":"-- \"Outrageous but Meaningful Coincidences: Dependent type-safe syntax"},{"Add":"-- and evaluation\"."},{"Add":""},{"Add":"_ˢ_ : ∀ {A : Set a} {B : A → Set b} {C : (x : A) → B x → Set c} →"},{"Add":" ((x : A) (y : B x) → C x y) →"},{"Add":" (g : (x : A) → B x) →"},{"Add":" ((x : A) → C x (g x))"},{"Add":"f ˢ g = λ x → f x (g x)"},{"Add":"{-# INLINE _ˢ_ #-}"},{"Add":""},{"Add":"-- Converting between implicit and explicit function spaces."},{"Add":""},{"Add":"_$- : ∀ {A : Set a} {B : A → Set b} → ((x : A) → B x) → ({x : A} → B x)"},{"Add":"f $- = f _"},{"Add":"{-# INLINE _$- #-}"},{"Add":""},{"Add":"λ- : ∀ {A : Set a} {B : A → Set b} → ({x : A} → B x) → ((x : A) → B x)"},{"Add":"λ- f = λ x → f"},{"Add":"{-# INLINE λ- #-}"},{"Add":""},{"Add":"-- Case expressions (to be used with pattern-matching lambdas, see"},{"Add":"-- README.Case)."},{"Add":""},{"Add":"case_returning_of_ : ∀ {A : Set a} (x : A) (B : A → Set b) →"},{"Add":" ((x : A) → B x) → B x"},{"Add":"case x returning B of f = f x"},{"Add":"{-# INLINE case_returning_of_ #-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Non-dependent versions of dependent operations"},{"Add":""},{"Add":"-- Any of the above operations for dependent functions will also work"},{"Add":"-- for non-dependent functions but sometimes Agda has difficulty"},{"Add":"-- inferring the non-dependency. Primed (′ = \\prime) versions of the"},{"Add":"-- operations are therefore provided below that sometimes have better"},{"Add":"-- inference properties."},{"Add":""},{"Add":"infixr 9 _∘′_ _∘₂′_"},{"Add":"infixl 0 _|>′_"},{"Add":"infix 0 case_of_"},{"Add":"infixr -1 _$′_"},{"Add":""},{"Add":"-- Composition"},{"Add":""},{"Add":"_∘′_ : (B → C) → (A → B) → (A → C)"},{"Add":"f ∘′ g = _∘_ f g"},{"Add":""},{"Add":"_∘₂′_ : (C → D) → (A → B → C) → (A → B → D)"},{"Add":"f ∘₂′ g = _∘₂_ f g"},{"Add":""},{"Add":"-- Flipping order of arguments"},{"Add":""},{"Add":"flip′ : (A → B → C) → (B → A → C)"},{"Add":"flip′ = flip"},{"Add":""},{"Add":"-- Application"},{"Add":""},{"Add":"_$′_ : (A → B) → (A → B)"},{"Add":"_$′_ = _$_"},{"Add":""},{"Add":"-- Flipped application (aka pipe-forward)"},{"Add":""},{"Add":"_|>′_ : A → (A → B) → B"},{"Add":"_|>′_ = _|>_"},{"Add":""},{"Add":"-- Case expressions (to be used with pattern-matching lambdas, see"},{"Add":"-- README.Case)."},{"Add":""},{"Add":"case_of_ : A → (A → B) → B"},{"Add":"case x of f = case x returning _ of f"},{"Add":"{-# INLINE case_of_ #-}"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Operations that are only defined for non-dependent functions"},{"Add":""},{"Add":"infixl 1 _⟨_⟩_"},{"Add":"infixl 0 _∋_"},{"Add":""},{"Add":"-- Binary application"},{"Add":""},{"Add":"_⟨_⟩_ : A → (A → B → C) → B → C"},{"Add":"x ⟨ f ⟩ y = f x y"},{"Add":""},{"Add":"-- In Agda you cannot annotate every subexpression with a type"},{"Add":"-- signature. This function can be used instead."},{"Add":""},{"Add":"_∋_ : (A : Set a) → A → A"},{"Add":"A ∋ x = x"},{"Add":""},{"Add":"-- Conversely it is sometimes useful to be able to extract the"},{"Add":"-- type of a given expression."},{"Add":""},{"Add":"typeOf : {A : Set a} → A → Set a"},{"Add":"typeOf {A = A} _ = A"},{"Add":""},{"Add":"-- Construct an element of the given type by instance search."},{"Add":""},{"Add":"it : {A : Set a} → {{A}} → A"},{"Add":"it {{x}} = x"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Composition of a binary function with other functions"},{"Add":""},{"Add":"infixr 0 _-⟪_⟫-_ _-⟨_⟫-_"},{"Add":"infixl 0 _-⟪_⟩-_"},{"Add":"infixr 1 _-⟨_⟩-_ ∣_⟫-_ ∣_⟩-_"},{"Add":"infixl 1 _on_ _on₂_ _-⟪_∣ _-⟨_∣"},{"Add":""},{"Add":"-- Two binary functions"},{"Add":""},{"Add":"_-⟪_⟫-_ : (A → B → C) → (C → D → E) → (A → B → D) → (A → B → E)"},{"Add":"f -⟪ _*_ ⟫- g = λ x y → f x y * g x y"},{"Add":""},{"Add":"-- A single binary function on the left"},{"Add":""},{"Add":"_-⟪_∣ : (A → B → C) → (C → B → D) → (A → B → D)"},{"Add":"f -⟪ _*_ ∣ = f -⟪ _*_ ⟫- constᵣ"},{"Add":""},{"Add":"-- A single binary function on the right"},{"Add":""},{"Add":"∣_⟫-_ : (A → C → D) → (A → B → C) → (A → B → D)"},{"Add":"∣ _*_ ⟫- g = const -⟪ _*_ ⟫- g"},{"Add":""},{"Add":"-- A single unary function on the left"},{"Add":""},{"Add":"_-⟨_∣ : (A → C) → (C → B → D) → (A → B → D)"},{"Add":"f -⟨ _*_ ∣ = f ∘₂ const -⟪ _*_ ∣"},{"Add":""},{"Add":"-- A single unary function on the right"},{"Add":""},{"Add":"∣_⟩-_ : (A → C → D) → (B → C) → (A → B → D)"},{"Add":"∣ _*_ ⟩- g = ∣ _*_ ⟫- g ∘₂ constᵣ"},{"Add":""},{"Add":"-- A binary function and a unary function"},{"Add":""},{"Add":"_-⟪_⟩-_ : (A → B → C) → (C → D → E) → (B → D) → (A → B → E)"},{"Add":"f -⟪ _*_ ⟩- g = f -⟪ _*_ ⟫- ∣ constᵣ ⟩- g"},{"Add":""},{"Add":"-- A unary function and a binary function"},{"Add":""},{"Add":"_-⟨_⟫-_ : (A → C) → (C → D → E) → (A → B → D) → (A → B → E)"},{"Add":"f -⟨ _*_ ⟫- g = f -⟨ const ∣ -⟪ _*_ ⟫- g"},{"Add":""},{"Add":"-- Two unary functions"},{"Add":""},{"Add":"_-⟨_⟩-_ : (A → C) → (C → D → E) → (B → D) → (A → B → E)"},{"Add":"f -⟨ _*_ ⟩- g = f -⟨ const ∣ -⟪ _*_ ⟫- ∣ constᵣ ⟩- g"},{"Add":""},{"Add":"-- A single binary function on both sides"},{"Add":""},{"Add":"_on₂_ : (C → C → D) → (A → B → C) → (A → B → D)"},{"Add":"_*_ on₂ f = f -⟪ _*_ ⟫- f"},{"Add":""},{"Add":"-- A single unary function on both sides"},{"Add":""},{"Add":"_on_ : (B → B → C) → (A → B) → (A → A → C)"},{"Add":"_*_ on f = f -⟨ _*_ ⟩- f"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- DEPRECATED NAMES"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Please use the new names as continuing support for the old names is"},{"Add":"-- not guaranteed."},{"Add":""},{"Add":"-- Version 1.4"},{"Add":""},{"Add":"_-[_]-_ = _-⟪_⟫-_"},{"Add":"{-# WARNING_ON_USAGE _-[_]-_"},{"Add":"\"Warning: Function._-[_]-_ was deprecated in v1.4."},{"Add":"Please use _-⟪_⟫-_ instead.\""},{"Add":"#-}"},{"Add":""},{"Add":"-- Version 2.0"},{"Add":""},{"Add":"case_return_of_ = case_returning_of_"},{"Add":"{-# WARNING_ON_USAGE case_return_of_"},{"Add":"\"case_return_of_ was deprecated in v2.0."},{"Add":"Please use case_returning_of_ instead.\""},{"Add":"#-}"},{"Add":""}]}},{"FileAdd":{"meta":{"basename":"Foreign","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Haskell.agda","metadata":0},"context":[{"change":0,"pos":1272927}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Type(s) used (only) when calling out to Haskell via the FFI"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module Foreign.Haskell where"},{"Add":""},{"Add":"open import Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Pairs"},{"Add":""},{"Add":"open import Foreign.Haskell.Pair public"},{"Add":" renaming"},{"Add":" ( toForeign to toForeignPair"},{"Add":" ; fromForeign to fromForeignPair"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Sums"},{"Add":""},{"Add":"open import Foreign.Haskell.Either public"},{"Add":" renaming"},{"Add":" ( toForeign to toForeignEither"},{"Add":" ; fromForeign to fromForeignEither"},{"Add":" )"}]}},{"FileAdd":{"meta":{"basename":"Effect","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Monad.agda","metadata":0},"context":[{"change":0,"pos":1273735}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Monads"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Note that currently the monad laws are not included here."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad where"},{"Add":""},{"Add":"open import Data.Bool.Base using (Bool; true; false; not)"},{"Add":"open import Data.Unit.Polymorphic.Base using (⊤)"},{"Add":""},{"Add":"open import Effect.Choice"},{"Add":"open import Effect.Empty"},{"Add":"open import Effect.Applicative"},{"Add":"open import Function.Base using (id; flip; _$′_; _∘′_)"},{"Add":"open import Level using (Level; suc; _⊔_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" f g g₁ g₂ : Level"},{"Add":" A B C : Set f"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The type of raw monads"},{"Add":""},{"Add":"record RawMonad (F : Set f → Set g) : Set (suc f ⊔ g) where"},{"Add":" infixl 1 _>>=_ _>>_ _>=>_"},{"Add":" infixr 1 _=<<_ _<=<_"},{"Add":" field"},{"Add":" rawApplicative : RawApplicative F"},{"Add":" _>>=_ : F A → (A → F B) → F B"},{"Add":""},{"Add":" open RawApplicative rawApplicative public"},{"Add":""},{"Add":" _>>_ : F A → F B → F B"},{"Add":" _>>_ = _*>_"},{"Add":""},{"Add":" _=<<_ : (A → F B) → F A → F B"},{"Add":" _=<<_ = flip _>>=_"},{"Add":""},{"Add":" Kleisli : Set f → Set f → Set (f ⊔ g)"},{"Add":" Kleisli A B = A → F B"},{"Add":""},{"Add":" _>=>_ : Kleisli A B → Kleisli B C → Kleisli A C"},{"Add":" (f >=> g) a = f a >>= g"},{"Add":""},{"Add":" _<=<_ : Kleisli B C → Kleisli A B → Kleisli A C"},{"Add":" _<=<_ = flip _>=>_"},{"Add":""},{"Add":" when : Bool → F ⊤ → F ⊤"},{"Add":" when true m = m"},{"Add":" when false m = pure _"},{"Add":""},{"Add":" unless : Bool → F ⊤ → F ⊤"},{"Add":" unless = when ∘′ not"},{"Add":""},{"Add":"-- When level g=f, a join/μ operator is definable"},{"Add":""},{"Add":"module Join {F : Set f → Set f} (M : RawMonad F) where"},{"Add":" open RawMonad M"},{"Add":""},{"Add":" join : F (F A) → F A"},{"Add":" join = _>>= id"},{"Add":""},{"Add":"-- Smart constructor"},{"Add":""},{"Add":"module _ where"},{"Add":""},{"Add":" open RawMonad"},{"Add":" open RawApplicative"},{"Add":""},{"Add":" mkRawMonad :"},{"Add":" (F : Set f → Set g) →"},{"Add":" (pure : ∀ {A} → A → F A) →"},{"Add":" (bind : ∀ {A B} → F A → (A → F B) → F B) →"},{"Add":" RawMonad F"},{"Add":" mkRawMonad F pure _>>=_ .rawApplicative ="},{"Add":" mkRawApplicative _ pure $′ λ mf mx → do"},{"Add":" f ← mf"},{"Add":" x ← mx"},{"Add":" pure (f x)"},{"Add":" mkRawMonad F pure _>>=_ ._>>=_ = _>>=_"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The type of raw monads with a zero"},{"Add":""},{"Add":"record RawMonadZero (F : Set f → Set g) : Set (suc f ⊔ g) where"},{"Add":" field"},{"Add":" rawMonad : RawMonad F"},{"Add":" rawEmpty : RawEmpty F"},{"Add":""},{"Add":" open RawMonad rawMonad public"},{"Add":" open RawEmpty rawEmpty public"},{"Add":""},{"Add":" rawApplicativeZero : RawApplicativeZero F"},{"Add":" rawApplicativeZero = record"},{"Add":" { rawApplicative = rawApplicative"},{"Add":" ; rawEmpty = rawEmpty"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The type of raw monadplus"},{"Add":""},{"Add":"record RawMonadPlus (F : Set f → Set g) : Set (suc f ⊔ g) where"},{"Add":" field"},{"Add":" rawMonadZero : RawMonadZero F"},{"Add":" rawChoice : RawChoice F"},{"Add":""},{"Add":" open RawMonadZero rawMonadZero public"},{"Add":" open RawChoice rawChoice public"},{"Add":""},{"Add":" rawAlternative : RawAlternative F"},{"Add":" rawAlternative = record"},{"Add":" { rawApplicativeZero = rawApplicativeZero"},{"Add":" ; rawChoice = rawChoice"},{"Add":" }"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The type of raw monad transformer"},{"Add":""},{"Add":"-- F has been RawMonadT'd as TF"},{"Add":"record RawMonadTd (F : Set f → Set g₁) (TF : Set f → Set g₂) : Set (suc f ⊔ g₁ ⊔ g₂) where"},{"Add":" field"},{"Add":" lift : F A → TF A"},{"Add":" rawMonad : RawMonad TF"},{"Add":""},{"Add":" open RawMonad rawMonad public"},{"Add":""},{"Add":"RawMonadT : (T : (Set f → Set g₁) → (Set f → Set g₂)) → Set (suc f ⊔ suc g₁ ⊔ g₂)"},{"Add":"RawMonadT T = ∀ {M} → RawMonad M → RawMonadTd M (T M)"}]}},{"FileAdd":{"meta":{"basename":"Monad","metadata":576},"context":[{"change":0,"pos":1273735}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Writer.agda","metadata":0},"context":[{"change":0,"pos":1277307}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The writer monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.Writer where"},{"Add":""},{"Add":"open import Algebra using (RawMonoid)"},{"Add":"open import Data.Product.Base using (_×_)"},{"Add":"open import Effect.Applicative using (RawApplicative)"},{"Add":"open import Effect.Functor using (RawFunctor)"},{"Add":"open import Effect.Monad using (RawMonad; module Join)"},{"Add":"open import Effect.Monad.Identity as Id using (Identity; runIdentity)"},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"import Effect.Monad.Writer.Transformer as Trans"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" w ℓ : Level"},{"Add":" A : Set w"},{"Add":" 𝕎 : RawMonoid w ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export the monad writer operations"},{"Add":""},{"Add":"open Trans public"},{"Add":" using (RawMonadWriter)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Writer monad"},{"Add":""},{"Add":"Writer : (𝕎 : RawMonoid w ℓ) (A : Set w) → Set w"},{"Add":"Writer 𝕎 = Trans.WriterT 𝕎 Identity"},{"Add":""},{"Add":"functor : RawFunctor (Writer 𝕎)"},{"Add":"functor = Trans.functor Id.functor"},{"Add":""},{"Add":"module _ {𝕎 : RawMonoid w ℓ} where"},{"Add":""},{"Add":" open RawMonoid 𝕎 renaming (Carrier to W)"},{"Add":""},{"Add":" runWriter : Writer 𝕎 A → W × A"},{"Add":" runWriter ma = runIdentity (Trans.runWriterT ma ε)"},{"Add":""},{"Add":" applicative : RawApplicative (Writer 𝕎)"},{"Add":" applicative = Trans.applicative Id.applicative"},{"Add":""},{"Add":" monad : RawMonad (Writer 𝕎)"},{"Add":" monad = Trans.monad Id.monad"},{"Add":""},{"Add":" join : Writer 𝕎 (Writer 𝕎 A) → Writer 𝕎 A"},{"Add":" join = Join.join monad"},{"Add":""},{"Add":" ----------------------------------------------------------------------"},{"Add":" -- Writer monad specifics"},{"Add":""},{"Add":" monadWriter : RawMonadWriter 𝕎 (Writer 𝕎)"},{"Add":" monadWriter = Trans.monadWriter Id.monad"}]}},{"FileAdd":{"meta":{"basename":"Writer","metadata":576},"context":[{"change":0,"pos":1277307}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Transformer.agda","metadata":0},"context":[{"change":0,"pos":1279147}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The writer monad transformer"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":""},{"Add":"module Effect.Monad.Writer.Transformer where"},{"Add":""},{"Add":"open import Algebra using (RawMonoid)"},{"Add":"open import Data.Product.Base using (_×_; _,_; proj₂; map₂)"},{"Add":"open import Effect.Applicative using (RawApplicative; RawApplicativeZero; RawAlternative)"},{"Add":"open import Effect.Choice using (RawChoice)"},{"Add":"open import Effect.Empty using (RawEmpty)"},{"Add":"open import Effect.Functor using (RawFunctor)"},{"Add":"open import Effect.Monad using (RawMonad; RawMonadZero; RawMonadPlus; RawMonadT)"},{"Add":"open import Function.Base using (_∘′_; const; _$_)"},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" w g g₁ g₂ : Level"},{"Add":" A B : Set w"},{"Add":" M : Set w → Set g"},{"Add":" 𝕎 : RawMonoid w g"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export the basic type definitions"},{"Add":""},{"Add":"open import Effect.Monad.Writer.Transformer.Base public"},{"Add":" using ( RawMonadWriter"},{"Add":" ; WriterT"},{"Add":" ; mkWriterT"},{"Add":" ; runWriterT"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structure"},{"Add":""},{"Add":"functor : RawFunctor M → RawFunctor (WriterT 𝕎 M)"},{"Add":"functor M = record"},{"Add":" { _<$>_ = λ f ma → mkWriterT λ w → map₂ f <$> runWriterT ma w"},{"Add":" } where open RawFunctor M"},{"Add":""},{"Add":"empty : RawEmpty M → RawEmpty (WriterT 𝕎 M)"},{"Add":"empty M = record"},{"Add":" { empty = mkWriterT (const (RawEmpty.empty M))"},{"Add":" }"},{"Add":""},{"Add":"choice : RawChoice M → RawChoice (WriterT 𝕎 M)"},{"Add":"choice M = record"},{"Add":" { _<|>_ = λ ma₁ ma₂ → mkWriterT λ w →"},{"Add":" WriterT.runWriterT ma₁ w"},{"Add":" <|> WriterT.runWriterT ma₂ w"},{"Add":" } where open RawChoice M"},{"Add":""},{"Add":"module _ {𝕎 : RawMonoid w g} where"},{"Add":""},{"Add":" open RawMonoid 𝕎 renaming (Carrier to W)"},{"Add":""},{"Add":" applicative : RawApplicative M → RawApplicative (WriterT 𝕎 M)"},{"Add":" applicative M = record"},{"Add":" { rawFunctor = functor rawFunctor"},{"Add":" ; pure = λ a → mkWriterT (pure ∘′ (_, a))"},{"Add":" ; _<*>_ = λ mf mx → mkWriterT $ λ w →"},{"Add":" (go <$> runWriterT mf w) <*> runWriterT mx ε"},{"Add":" } where"},{"Add":" open RawApplicative M"},{"Add":" go : W × (A → B) → W × A → W × B"},{"Add":" go (w₁ , f) (w₂ , x) = w₁ ∙ w₂ , f x"},{"Add":""},{"Add":" applicativeZero : RawApplicativeZero M → RawApplicativeZero (WriterT 𝕎 M)"},{"Add":" applicativeZero M = record"},{"Add":" { rawApplicative = applicative rawApplicative"},{"Add":" ; rawEmpty = empty rawEmpty"},{"Add":" } where open RawApplicativeZero M using (rawApplicative; rawEmpty)"},{"Add":""},{"Add":" alternative : RawAlternative M → RawAlternative (WriterT 𝕎 M)"},{"Add":" alternative M = record"},{"Add":" { rawApplicativeZero = applicativeZero rawApplicativeZero"},{"Add":" ; rawChoice = choice rawChoice"},{"Add":" } where open RawAlternative M"},{"Add":""},{"Add":" monad : RawMonad M → RawMonad (WriterT 𝕎 M)"},{"Add":" monad M = record"},{"Add":" { rawApplicative = applicative rawApplicative"},{"Add":" ; _>>=_ = λ ma f → mkWriterT λ w → do"},{"Add":" w₁ , a ← runWriterT ma w"},{"Add":" runWriterT (f a) w₁"},{"Add":" } where open RawMonad M"},{"Add":""},{"Add":" monadZero : RawMonadZero M → RawMonadZero (WriterT 𝕎 M)"},{"Add":" monadZero M = record"},{"Add":" { rawMonad = monad (RawMonadZero.rawMonad M)"},{"Add":" ; rawEmpty = empty (RawMonadZero.rawEmpty M)"},{"Add":" }"},{"Add":""},{"Add":" monadPlus : RawMonadPlus M → RawMonadPlus (WriterT 𝕎 M)"},{"Add":" monadPlus M = record"},{"Add":" { rawMonadZero = monadZero rawMonadZero"},{"Add":" ; rawChoice = choice rawChoice"},{"Add":" } where open RawMonadPlus M"},{"Add":""},{"Add":" ----------------------------------------------------------------------"},{"Add":" -- Monad writer transformer specifics"},{"Add":""},{"Add":" monadT : RawMonadT {g₁ = g₁} {g₂ = _} (WriterT 𝕎)"},{"Add":" monadT M = record"},{"Add":" { lift = mkWriterT ∘′ λ ma w → (w ,_) <$> ma"},{"Add":" ; rawMonad = monad M"},{"Add":" } where open RawMonad M"},{"Add":""},{"Add":" monadWriter : RawMonad M → RawMonadWriter 𝕎 (WriterT 𝕎 M)"},{"Add":" monadWriter M = record"},{"Add":" { writer = mkWriterT ∘′ λ (w' , a) w → pure (w ∙ w' , a)"},{"Add":" ; listen = λ ma → mkWriterT λ w → do"},{"Add":" w , a ← runWriterT ma w"},{"Add":" pure (w , w , a)"},{"Add":" ; pass = λ mx → mkWriterT λ w → do"},{"Add":" w , f , a ← runWriterT mx w"},{"Add":" pure (f w , a)"},{"Add":" } where open RawMonad M"},{"Add":""},{"Add":"module _ {𝕎₁ : RawMonoid w g₁} where"},{"Add":""},{"Add":" open RawMonoid 𝕎₁ renaming (Carrier to W₁)"},{"Add":""},{"Add":" liftWriterT : (𝕎₂ : RawMonoid w g₂) →"},{"Add":" RawFunctor M →"},{"Add":" RawMonadWriter 𝕎₁ M →"},{"Add":" RawMonadWriter 𝕎₁ (WriterT 𝕎₂ M)"},{"Add":" liftWriterT 𝕎₂ M MWrite = record"},{"Add":" { writer = λ (w , a) → mkWriterT λ w' → (writer (w , w' , a ))"},{"Add":" ; listen = λ mx → mkWriterT λ w' → ((λ (w₁ , w₂ , a) → w₂ , w₁ , a) <$> listen (runWriterT mx w'))"},{"Add":" ; pass = λ mx → mkWriterT λ w' → (pass ((λ (w , f , a) → f , w , a) <$> runWriterT mx w'))"},{"Add":" } where open RawMonadWriter MWrite"},{"Add":" open RawFunctor M"},{"Add":""},{"Add":" private"},{"Add":" variable"},{"Add":" W₂ : Set g₂"},{"Add":""},{"Add":" open import Effect.Monad.Reader.Transformer.Base"},{"Add":""},{"Add":" liftReaderT : RawFunctor M →"},{"Add":" RawMonadWriter 𝕎₁ M →"},{"Add":" RawMonadWriter 𝕎₁ (ReaderT W₂ M)"},{"Add":" liftReaderT M MWrite = record"},{"Add":" { writer = mkReaderT ∘′ const ∘′ writer"},{"Add":" ; listen = λ ma → mkReaderT (listen ∘′ runReaderT ma)"},{"Add":" ; pass = λ ma → mkReaderT (pass ∘′ runReaderT ma)"},{"Add":" } where open RawMonadWriter MWrite"},{"Add":""},{"Add":" open import Effect.Monad.State.Transformer.Base"},{"Add":""},{"Add":" liftStateT : RawFunctor M →"},{"Add":" RawMonadWriter 𝕎₁ M →"},{"Add":" RawMonadWriter 𝕎₁ (StateT W₂ M)"},{"Add":" liftStateT M MWrite = record"},{"Add":" { writer = λ x → mkStateT λ w₂ → (w₂ ,_) <$>"},{"Add":" writer x"},{"Add":" ; listen = λ mx → mkStateT λ w₂ → (w₂ ,_) <$>"},{"Add":" listen (proj₂ <$> runStateT mx w₂)"},{"Add":" ; pass = λ mx → mkStateT λ w₂ → (w₂ ,_) <$>"},{"Add":" pass ((λ (_ , f , a) → f , a) <$> runStateT mx w₂)"},{"Add":" } where open RawMonadWriter MWrite"},{"Add":" open RawFunctor M"}]}},{"FileAdd":{"meta":{"basename":"Transformer","metadata":576},"context":[{"change":0,"pos":1279147}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Base.agda","metadata":0},"context":[{"change":0,"pos":1285176}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Basic type and definition of the writer monad transformer"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.Writer.Transformer.Base where"},{"Add":""},{"Add":"open import Algebra using (RawMonoid)"},{"Add":"open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)"},{"Add":"open import Data.Unit.Polymorphic using (⊤; tt)"},{"Add":"open import Function.Base using (id; _∘′_)"},{"Add":"open import Level using (Level; suc; _⊔_)"},{"Add":""},{"Add":"open import Effect.Functor using (RawFunctor)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" w f g : Level"},{"Add":" A : Set g"},{"Add":" M : Set w → Set g"},{"Add":" 𝕎 : RawMonoid w g"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Writer monad operations"},{"Add":""},{"Add":"record RawMonadWriter"},{"Add":" (𝕎 : RawMonoid w f)"},{"Add":" (M : Set w → Set g)"},{"Add":" : Set (suc w ⊔ g) where"},{"Add":" open RawMonoid 𝕎 renaming (Carrier to W)"},{"Add":" field"},{"Add":" writer : W × A → M A"},{"Add":" listen : M A → M (W × A)"},{"Add":" pass : M ((W → W) × A) → M A"},{"Add":""},{"Add":" tell : W → M ⊤"},{"Add":" tell w = writer (w , tt)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Writer monad transformer (CPS-encoded)"},{"Add":""},{"Add":"record WriterT"},{"Add":" (𝕎 : RawMonoid w f)"},{"Add":" (M : Set w → Set g)"},{"Add":" (A : Set w)"},{"Add":" : Set (w ⊔ g) where"},{"Add":" constructor mkWriterT"},{"Add":" open RawMonoid 𝕎 renaming (Carrier to W)"},{"Add":" field runWriterT : W → M (W × A)"},{"Add":"open WriterT public"},{"Add":""},{"Add":"module _ {𝕎 : RawMonoid w f} where"},{"Add":""},{"Add":" open RawMonoid 𝕎 renaming (Carrier to W)"},{"Add":""},{"Add":" evalWriterT : RawFunctor M → WriterT 𝕎 M A → M A"},{"Add":" evalWriterT M ma = proj₂ <$> runWriterT ma ε"},{"Add":" where open RawFunctor M"},{"Add":""},{"Add":" execWriterT : RawFunctor M → WriterT 𝕎 M A → M W"},{"Add":" execWriterT M ma = proj₁ <$> runWriterT ma ε"},{"Add":" where open RawFunctor M"}]}},{"FileAdd":{"meta":{"basename":"Instances.agda","metadata":0},"context":[{"change":0,"pos":1279147}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Instances for the writer monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.Writer.Instances where"},{"Add":""},{"Add":"open import Effect.Monad.Writer.Transformer"},{"Add":""},{"Add":"instance"},{"Add":" writerTFunctor = λ {s} {S} {f} {M} {{fun}} {{mo}} → functor {s} {S} {f} {M} {mo} fun"},{"Add":" writerTApplicative = λ {s} {S} {f} {M} {{mo}} {{mon}} → applicative {s} {S} {f} {M} {mo} mon"},{"Add":" writerTEmpty = λ {s} {S} {f} {M} {{e}} {{mo}} → empty {s} {S} {f} {M} {mo} e"},{"Add":" writerTChoice = λ {s} {S} {f} {M} {{ch}} {{mo}} → choice {s} {S} {f} {M} {mo} ch"},{"Add":" writerTApplicativeZero = λ {s} {S} {f} {M} {{mo}} {{mon}} → applicativeZero {s} {S} {f} {M} {mo} mon"},{"Add":" writerTAlternative = λ {s} {S} {f} {M} {{mo}} {{mpl}} → alternative {s} {S} {f} {M} {mo} mpl"},{"Add":" writerTMonad = λ {s} {S} {f} {M} {{mo}} {{mon}} → monad {s} {S} {f} {M} {mo} mon"},{"Add":" writerTMonadZero = λ {s} {S} {f} {M} {{mo}} {{mz}} → monadZero {s} {S} {f} {M} {mo} mz"},{"Add":" writerTMonadPlus = λ {s} {S} {f} {M} {{mo}} {{mpl}} → monadPlus {s} {S} {f} {M} {mo} mpl"},{"Add":" writerTMonadT = λ {s} {S} {f} {M} {{mo}} {{mon}} → monadT {s} {S} {f} {M} {mo} mon"},{"Add":" writerTMonadWriter = λ {s} {S} {f} {M} {{mo}} {{mon}} → monadWriter {s} {S} {f} {M} {mo} mon"},{"Add":" writerTLiftReaderT = λ {s} {S₁} {S₂} {f} {g} {M} {{fun}} {{mw}} → liftReaderT {s} {S₁} {S₂} {f} {g} {M} fun mw"},{"Add":" writerTLiftStateT = λ {s} {S₁} {S₂} {f} {g} {M} {{fun}} {{mw}} → liftStateT {s} {S₁} {S₂} {f} {g} {M} fun mw"}]}},{"FileAdd":{"meta":{"basename":"Indexed.agda","metadata":0},"context":[{"change":0,"pos":1279147}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The indexed writer monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Level"},{"Add":""},{"Add":"module Effect.Monad.Writer.Indexed (a : Level) where"},{"Add":""},{"Add":"open import Algebra using (RawMonoid)"},{"Add":"open import Data.Product.Base using (_×_; _,_; map₁)"},{"Add":"open import Data.Unit.Polymorphic"},{"Add":"open import Effect.Applicative.Indexed"},{"Add":"open import Effect.Monad"},{"Add":"open import Effect.Monad.Indexed"},{"Add":"open import Function.Base using (_∘′_)"},{"Add":"open import Function.Identity.Effectful as Id using (Identity)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" w ℓ : Level"},{"Add":" A B I : Set ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Indexed writer"},{"Add":""},{"Add":"IWriterT : (𝕎 : RawMonoid w ℓ) → IFun I (w ⊔ a) → IFun I (w ⊔ a)"},{"Add":"IWriterT 𝕎 M i j A = M i j (RawMonoid.Carrier 𝕎 × A)"},{"Add":""},{"Add":"module _ {M : IFun I (w ⊔ a)} {𝕎 : RawMonoid w ℓ} where"},{"Add":""},{"Add":" open RawMonoid 𝕎 renaming (Carrier to W)"},{"Add":""},{"Add":" ----------------------------------------------------------------------"},{"Add":" -- Indexed writer applicative"},{"Add":""},{"Add":" WriterTIApplicative : RawIApplicative M → RawIApplicative (IWriterT 𝕎 M)"},{"Add":" WriterTIApplicative App = record"},{"Add":" { pure = λ x → pure (ε , x)"},{"Add":" ; _⊛_ = λ m n → go <$> m ⊛ n"},{"Add":" } where"},{"Add":" open RawIApplicative App"},{"Add":" go : W × (A → B) → W × A → W × B"},{"Add":" go (w₁ , f) (w₂ , x) = w₁ ∙ w₂ , f x"},{"Add":""},{"Add":" WriterTIApplicativeZero : RawIApplicativeZero M →"},{"Add":" RawIApplicativeZero (IWriterT 𝕎 M)"},{"Add":" WriterTIApplicativeZero App = record"},{"Add":" { applicative = WriterTIApplicative applicative"},{"Add":" ; ∅ = ∅"},{"Add":" } where open RawIApplicativeZero App"},{"Add":""},{"Add":" WriterTIAlternative : RawIAlternative M → RawIAlternative (IWriterT 𝕎 M)"},{"Add":" WriterTIAlternative Alt = record"},{"Add":" { applicativeZero = WriterTIApplicativeZero applicativeZero"},{"Add":" ; _∣_ = _∣_"},{"Add":" } where open RawIAlternative Alt"},{"Add":""},{"Add":" ----------------------------------------------------------------------"},{"Add":" -- Indexed writer monad"},{"Add":""},{"Add":" WriterTIMonad : RawIMonad M → RawIMonad (IWriterT 𝕎 M)"},{"Add":" WriterTIMonad Mon = record"},{"Add":" { return = λ x → return (ε , x)"},{"Add":" ; _>>=_ = λ m f → do"},{"Add":" w₁ , x ← m"},{"Add":" w₂ , fx ← f x"},{"Add":" return (w₁ ∙ w₂ , fx)"},{"Add":" } where open RawIMonad Mon"},{"Add":""},{"Add":" WriterTIMonadZero : RawIMonadZero M → RawIMonadZero (IWriterT 𝕎 M)"},{"Add":" WriterTIMonadZero Mon = record"},{"Add":" { monad = WriterTIMonad monad"},{"Add":" ; applicativeZero = WriterTIApplicativeZero applicativeZero"},{"Add":" } where open RawIMonadZero Mon"},{"Add":""},{"Add":" WriterTIMonadPlus : RawIMonadPlus M → RawIMonadPlus (IWriterT 𝕎 M)"},{"Add":" WriterTIMonadPlus Mon = record"},{"Add":" { monad = WriterTIMonad monad"},{"Add":" ; alternative = WriterTIAlternative alternative"},{"Add":" } where open RawIMonadPlus Mon"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Writer monad operations"},{"Add":""},{"Add":"record RawIMonadWriter {I : Set ℓ} (𝕎 : RawMonoid w ℓ) (M : IFun I (w ⊔ a))"},{"Add":" : Set (ℓ ⊔ suc (w ⊔ a)) where"},{"Add":" open RawMonoid 𝕎 renaming (Carrier to W)"},{"Add":" field"},{"Add":" monad : RawIMonad M"},{"Add":" writer : ∀ {i} → (W × A) → M i i A"},{"Add":" listen : ∀ {i j} → M i j A → M i j (W × A)"},{"Add":" pass : ∀ {i j} → M i j ((W → W) × A) → M i j A"},{"Add":""},{"Add":" open RawIMonad monad public"},{"Add":""},{"Add":" tell : ∀ {i} → W → M i i ⊤"},{"Add":" tell = writer ∘′ (_, tt)"},{"Add":""},{"Add":" listens : ∀ {i j} {Z : Set w} → (W → Z) → M i j A → M i j (Z × A)"},{"Add":" listens f m = listen m >>= return ∘′ map₁ f"},{"Add":""},{"Add":" censor : ∀ {i j} → (W → W) → M i j A → M i j A"},{"Add":" censor f m = pass (m >>= return ∘′ (f ,_))"},{"Add":""},{"Add":""},{"Add":"WriterTIMonadWriter : {I : Set ℓ} {𝕎 : RawMonoid w ℓ} {M : IFun I (w ⊔ a)} →"},{"Add":" RawIMonad M → RawIMonadWriter 𝕎 (IWriterT 𝕎 M)"},{"Add":"WriterTIMonadWriter {𝕎 = 𝕎} Mon = record"},{"Add":" { monad = WriterTIMonad {𝕎 = 𝕎} Mon"},{"Add":" ; writer = return"},{"Add":" ; listen = λ m → do"},{"Add":" w , a ← m"},{"Add":" return (w , w , a)"},{"Add":" ; pass = λ m → do"},{"Add":" w , f , a ← m"},{"Add":" return (f w , a)"},{"Add":" } where open RawIMonad Mon"},{"Add":" open RawMonoid 𝕎 renaming (Carrier to W)"}]}},{"FileAdd":{"meta":{"basename":"State.agda","metadata":0},"context":[{"change":0,"pos":1277307}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The state monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.State where"},{"Add":""},{"Add":"open import Data.Product.Base using (_×_)"},{"Add":"open import Effect.Functor using (RawFunctor)"},{"Add":"open import Effect.Applicative using (RawApplicative)"},{"Add":"open import Effect.Monad using (RawMonad; module Join)"},{"Add":"open import Effect.Monad.Identity as Id using (Identity; runIdentity)"},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"import Effect.Monad.State.Transformer as Trans"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" s : Level"},{"Add":" S A : Set s"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export the state monad operations"},{"Add":""},{"Add":"open Trans public"},{"Add":" using (RawMonadState)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- State monad"},{"Add":""},{"Add":"State : (S : Set s) (A : Set s) → Set s"},{"Add":"State S = Trans.StateT S Identity"},{"Add":""},{"Add":"runState : State S A → S → S × A"},{"Add":"runState ma s = runIdentity (Trans.runStateT ma s)"},{"Add":""},{"Add":"evalState : State S A → S → A"},{"Add":"evalState ma s = runIdentity (Trans.evalStateT Id.functor ma s)"},{"Add":""},{"Add":"execState : State S A → S → S"},{"Add":"execState ma s = runIdentity (Trans.execStateT Id.functor ma s)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structure"},{"Add":""},{"Add":"functor : RawFunctor (State S)"},{"Add":"functor = Trans.functor Id.functor"},{"Add":""},{"Add":"applicative : RawApplicative (State S)"},{"Add":"applicative = Trans.applicative Id.monad"},{"Add":""},{"Add":"monad : RawMonad (State S)"},{"Add":"monad = Trans.monad Id.monad"},{"Add":""},{"Add":"join : State S (State S A) → State S A"},{"Add":"join = Join.join monad"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- State monad specifics"},{"Add":""},{"Add":"monadState : RawMonadState S (State S)"},{"Add":"monadState = Trans.monadState Id.monad"}]}},{"FileAdd":{"meta":{"basename":"State","metadata":576},"context":[{"change":0,"pos":1277307}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Transformer.agda","metadata":0},"context":[{"change":0,"pos":1294855}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The state monad transformer"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":""},{"Add":"module Effect.Monad.State.Transformer where"},{"Add":""},{"Add":"open import Algebra using (RawMonoid)"},{"Add":"open import Data.Product.Base using (_×_; _,_; map₂; proj₁; proj₂)"},{"Add":"open import Data.Unit.Polymorphic.Base using (tt)"},{"Add":"open import Effect.Choice using (RawChoice)"},{"Add":"open import Effect.Empty using (RawEmpty)"},{"Add":"open import Effect.Functor using (RawFunctor)"},{"Add":"open import Effect.Applicative using (RawApplicative; RawApplicativeZero; RawAlternative)"},{"Add":"open import Effect.Monad using (RawMonad; RawMonadZero; RawMonadPlus; RawMonadT; RawMonadTd)"},{"Add":"open import Function.Base using (_∘′_; _$_; const)"},{"Add":"open import Level using (Level; _⊔_)"},{"Add":""},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" f s : Level"},{"Add":" A B : Set s"},{"Add":" S S₁ S₂ : Set s"},{"Add":" M : Set s → Set f"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export the basic type and definitions"},{"Add":""},{"Add":"open import Effect.Monad.State.Transformer.Base public"},{"Add":" using ( RawMonadState"},{"Add":" ; StateT"},{"Add":" ; mkStateT"},{"Add":" ; runStateT"},{"Add":" ; evalStateT"},{"Add":" ; execStateT"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structure"},{"Add":""},{"Add":"functor : RawFunctor M → RawFunctor (StateT S M)"},{"Add":"functor M = record"},{"Add":" { _<$>_ = λ f ma → mkStateT (λ s → map₂ f <$> StateT.runStateT ma s)"},{"Add":" } where open RawFunctor M"},{"Add":""},{"Add":"applicative : RawMonad M → RawApplicative (StateT S M)"},{"Add":"applicative M = record"},{"Add":" { rawFunctor = functor rawFunctor"},{"Add":" ; pure = λ a → mkStateT (pure ∘′ (_, a))"},{"Add":" ; _<*>_ = λ mf mx → mkStateT $ λ s →"},{"Add":" do (s , f) ← StateT.runStateT mf s"},{"Add":" (s , x) ← StateT.runStateT mx s"},{"Add":" pure (s , f x)"},{"Add":" } where open RawMonad M"},{"Add":""},{"Add":"empty : RawEmpty M → RawEmpty (StateT S M)"},{"Add":"empty M = record"},{"Add":" { empty = mkStateT (const (RawEmpty.empty M))"},{"Add":" }"},{"Add":""},{"Add":"choice : RawChoice M → RawChoice (StateT S M)"},{"Add":"choice M = record"},{"Add":" { _<|>_ = λ ma₁ ma₂ → mkStateT $ λ s →"},{"Add":" StateT.runStateT ma₁ s"},{"Add":" <|> StateT.runStateT ma₂ s"},{"Add":" } where open RawChoice M"},{"Add":""},{"Add":"applicativeZero : RawMonadZero M → RawApplicativeZero (StateT S M)"},{"Add":"applicativeZero M = record"},{"Add":" { rawApplicative = applicative (RawMonadZero.rawMonad M)"},{"Add":" ; rawEmpty = empty (RawMonadZero.rawEmpty M)"},{"Add":" }"},{"Add":""},{"Add":"alternative : RawMonadPlus M → RawAlternative (StateT S M)"},{"Add":"alternative M = record"},{"Add":" { rawApplicativeZero = applicativeZero rawMonadZero"},{"Add":" ; rawChoice = choice rawChoice"},{"Add":" } where open RawMonadPlus M"},{"Add":""},{"Add":"monad : RawMonad M → RawMonad (StateT S M)"},{"Add":"monad M = record"},{"Add":" { rawApplicative = applicative M"},{"Add":" ; _>>=_ = λ ma f → mkStateT $ λ s →"},{"Add":" do (s , a) ← StateT.runStateT ma s"},{"Add":" StateT.runStateT (f a) s"},{"Add":" } where open RawMonad M"},{"Add":""},{"Add":"monadZero : RawMonadZero M → RawMonadZero (StateT S M)"},{"Add":"monadZero M = record"},{"Add":" { rawMonad = monad (RawMonadZero.rawMonad M)"},{"Add":" ; rawEmpty = empty (RawMonadZero.rawEmpty M)"},{"Add":" }"},{"Add":""},{"Add":"monadPlus : RawMonadPlus M → RawMonadPlus (StateT S M)"},{"Add":"monadPlus M = record"},{"Add":" { rawMonadZero = monadZero rawMonadZero"},{"Add":" ; rawChoice = choice rawChoice"},{"Add":" } where open RawMonadPlus M"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- State monad transformer specifics"},{"Add":""},{"Add":"monadT : RawMonadT {f = s} {g₁ = f} {g₂ = s ⊔ f} (StateT S)"},{"Add":"monadT M = record"},{"Add":" { lift = λ ma → mkStateT (λ s → (s ,_) <$> ma)"},{"Add":" ; rawMonad = monad M"},{"Add":" } where open RawMonad M"},{"Add":""},{"Add":"monadState : RawMonad M → RawMonadState S (StateT S M)"},{"Add":"monadState M = record"},{"Add":" { gets = λ f → mkStateT (λ s → pure (s , f s))"},{"Add":" ; modify = λ f → mkStateT (λ s → pure (f s , _))"},{"Add":" } where open RawMonad M"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- State monad transformer specifics"},{"Add":""},{"Add":"liftStateT : RawMonad M →"},{"Add":" RawMonadState S₁ M →"},{"Add":" RawMonadState S₁ (StateT S₂ M)"},{"Add":"liftStateT M Mon = record"},{"Add":" { gets = λ f₁ → lift (gets f₁)"},{"Add":" ; modify = λ f₁ → lift (modify f₁)"},{"Add":" } where open RawMonadTd (monadT M) using (lift); open RawMonadState Mon"},{"Add":""},{"Add":"open import Effect.Monad.Reader.Transformer.Base"},{"Add":""},{"Add":"liftReaderT : RawMonadState S₁ M →"},{"Add":" RawMonadState S₁ (ReaderT S₂ M)"},{"Add":"liftReaderT Mon = record"},{"Add":" { gets = λ f → mkReaderT (const (gets f))"},{"Add":" ; modify = λ f → mkReaderT (const (modify f))"},{"Add":" } where open RawMonadState Mon"},{"Add":""},{"Add":"open import Effect.Monad.Writer.Transformer.Base"},{"Add":""},{"Add":"liftWriterT : (MS : RawMonoid s f) →"},{"Add":" RawFunctor M →"},{"Add":" RawMonadState S M →"},{"Add":" RawMonadState S (WriterT MS M)"},{"Add":"liftWriterT MS M Mon = record"},{"Add":" { gets = λ f → mkWriterT λ w → (gets ((w ,_) ∘′ f))"},{"Add":" ; modify = λ f → mkWriterT λ w → (const (w , tt) <$> modify f)"},{"Add":" } where open RawMonadState Mon"},{"Add":" open RawFunctor M"}]}},{"FileAdd":{"meta":{"basename":"Transformer","metadata":576},"context":[{"change":0,"pos":1294855}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Base.agda","metadata":0},"context":[{"change":0,"pos":1299863}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Basic definition and functions on the state monad transformer"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":""},{"Add":"module Effect.Monad.State.Transformer.Base where"},{"Add":""},{"Add":"open import Data.Product.Base using (_×_; proj₁; proj₂)"},{"Add":"open import Data.Unit.Polymorphic.Base using (⊤)"},{"Add":"open import Function.Base using (_∘′_; const; id)"},{"Add":"open import Level using (Level; suc; _⊔_)"},{"Add":""},{"Add":"open import Effect.Functor"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" f s : Level"},{"Add":" A : Set s"},{"Add":" S : Set s"},{"Add":" M : Set s → Set f"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- State monad operations"},{"Add":""},{"Add":"record RawMonadState"},{"Add":" (S : Set s)"},{"Add":" (M : Set s → Set f)"},{"Add":" : Set (suc s ⊔ f) where"},{"Add":" field"},{"Add":" gets : (S → A) → M A"},{"Add":" modify : (S → S) → M ⊤"},{"Add":""},{"Add":" put = modify ∘′ const"},{"Add":" get = gets id"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- State monad transformer"},{"Add":""},{"Add":"record StateT"},{"Add":" (S : Set s)"},{"Add":" (M : Set s → Set f)"},{"Add":" (A : Set s)"},{"Add":" : Set (s ⊔ f) where"},{"Add":" constructor mkStateT"},{"Add":" field runStateT : S → M (S × A)"},{"Add":"open StateT public"},{"Add":""},{"Add":"evalStateT : RawFunctor M → StateT S M A → S → M A"},{"Add":"evalStateT M ma s = let open RawFunctor M in proj₂ <$> runStateT ma s"},{"Add":""},{"Add":"execStateT : RawFunctor M → StateT S M A → S → M S"},{"Add":"execStateT M ma s = let open RawFunctor M in proj₁ <$> runStateT ma s"}]}},{"FileAdd":{"meta":{"basename":"Instances.agda","metadata":0},"context":[{"change":0,"pos":1294855}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Instances for the state monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.State.Instances where"},{"Add":""},{"Add":"open import Effect.Monad.State.Transformer"},{"Add":""},{"Add":"instance"},{"Add":" stateTFunctor = λ {s} {S} {f} {M} {{fun}} → functor {s} {S} {f} {M} fun"},{"Add":" stateTApplicative = λ {s} {S} {f} {M} {{mon}} → applicative {s} {S} {f} {M} mon"},{"Add":" stateTEmpty = λ {s} {S} {f} {M} {{e}} → empty {s} {S} {f} {M} e"},{"Add":" stateTChoice = λ {s} {S} {f} {M} {{ch}} → choice {s} {S} {f} {M} ch"},{"Add":" stateTApplicativeZero = λ {s} {S} {f} {M} {{mon}} → applicativeZero {s} {S} {f} {M} mon"},{"Add":" stateTAlternative = λ {s} {S} {f} {M} {{mpl}} → alternative {s} {S} {f} {M} mpl"},{"Add":" stateTMonad = λ {s} {S} {f} {M} {{mon}} → monad {s} {S} {f} {M} mon"},{"Add":" stateTMonadZero = λ {s} {S} {f} {M} {{mz}} → monadZero {s} {S} {f} {M} mz"},{"Add":" stateTMonadPlus = λ {s} {S} {f} {M} {{mpl}} → monadPlus {s} {S} {f} {M} mpl"},{"Add":" stateTMonadT = λ {s} {S} {f} {M} {{mon}} → monadT {s} {S} {f} {M} mon"},{"Add":" stateTMonadState = λ {s} {S} {f} {M} {{mon}} → monadState {s} {S} {f} {M} mon"},{"Add":" stateTLiftReaderT = λ {R} {s} {S} {f} {M} {{ms}} → liftReaderT {R} {s} {S} {f} {M} ms"},{"Add":" stateTLiftWriterT = λ {R} {s} {S} {f} {M} {{fun}} {{mo}} {{ms}} → liftWriterT {R} {s} {S} {f} {M} mo fun ms"},{"Add":" -- the following instances conflicts with stateTMonadState so we don't include it"},{"Add":" -- stateTLiftStateT = λ {s} {S₁} {S₂} {f} {M} {{mon}} {{ms}} → liftStateT {s} {S₁} {S₂} {f} {M} mon ms"}]}},{"FileAdd":{"meta":{"basename":"Indexed.agda","metadata":0},"context":[{"change":0,"pos":1294855}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The indexed state monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.State.Indexed where"},{"Add":""},{"Add":"open import Effect.Applicative.Indexed"},{"Add":" using (IFun; RawIApplicative; RawIApplicativeZero; RawIAlternative)"},{"Add":"open import Effect.Monad using (RawMonad; RawMonadZero; RawMonadPlus)"},{"Add":"open import Function.Identity.Effectful as Id using (Identity)"},{"Add":"open import Effect.Monad.Indexed using (RawIMonad; RawIMonadZero;"},{"Add":" RawIMonadPlus)"},{"Add":"open import Data.Product.Base using (_×_; _,_; uncurry)"},{"Add":"open import Data.Unit.Polymorphic using (⊤)"},{"Add":"open import Function.Base using (const; _∘_)"},{"Add":"open import Level using (Level; _⊔_; suc)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" i f : Level"},{"Add":" I : Set i"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Indexed state"},{"Add":""},{"Add":"IStateT : (I → Set f) → (Set f → Set f) → IFun I f"},{"Add":"IStateT S M i j A = S i → M (A × S j)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Indexed state applicative"},{"Add":""},{"Add":"StateTIApplicative : ∀ (S : I → Set f) {M} →"},{"Add":" RawMonad M → RawIApplicative (IStateT S M)"},{"Add":"StateTIApplicative S Mon = record"},{"Add":" { pure = λ a s → pure (a , s)"},{"Add":" ; _⊛_ = λ f t s → do"},{"Add":" (f′ , s′) ← f s"},{"Add":" (t′ , s′′) ← t s′"},{"Add":" pure (f′ t′ , s′′)"},{"Add":" } where open RawMonad Mon"},{"Add":""},{"Add":"StateTIApplicativeZero : ∀ (S : I → Set f) {M} →"},{"Add":" RawMonadZero M → RawIApplicativeZero (IStateT S M)"},{"Add":"StateTIApplicativeZero S Mon = record"},{"Add":" { applicative = StateTIApplicative S rawMonad"},{"Add":" ; ∅ = const ∅"},{"Add":" } where open RawMonadZero Mon"},{"Add":""},{"Add":"StateTIAlternative : ∀ (S : I → Set f) {M} →"},{"Add":" RawMonadPlus M → RawIAlternative (IStateT S M)"},{"Add":"StateTIAlternative S Mon = record"},{"Add":" { applicativeZero = StateTIApplicativeZero S rawMonadZero"},{"Add":" ; _∣_ = λ m n s → m s ∣ n s"},{"Add":" } where open RawMonadPlus Mon"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Indexed state monad"},{"Add":""},{"Add":"StateTIMonad : ∀ (S : I → Set f) {M} → RawMonad M → RawIMonad (IStateT S M)"},{"Add":"StateTIMonad S Mon = record"},{"Add":" { return = λ x s → pure (x , s)"},{"Add":" ; _>>=_ = λ m f s → m s >>= uncurry f"},{"Add":" } where open RawMonad Mon"},{"Add":""},{"Add":"StateTIMonadZero : ∀ (S : I → Set f) {M} →"},{"Add":" RawMonadZero M → RawIMonadZero (IStateT S M)"},{"Add":"StateTIMonadZero S Mon = record"},{"Add":" { monad = StateTIMonad S (RawMonadZero.rawMonad Mon)"},{"Add":" ; applicativeZero = StateTIApplicativeZero S Mon"},{"Add":" } where open RawMonadZero Mon"},{"Add":""},{"Add":"StateTIMonadPlus : ∀ (S : I → Set f) {M} →"},{"Add":" RawMonadPlus M → RawIMonadPlus (IStateT S M)"},{"Add":"StateTIMonadPlus S Mon = record"},{"Add":" { monad = StateTIMonad S rawMonad"},{"Add":" ; alternative = StateTIAlternative S Mon"},{"Add":" } where open RawMonadPlus Mon"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- State monad operations"},{"Add":""},{"Add":"record RawIMonadState {I : Set i} (S : I → Set f)"},{"Add":" (M : IFun I f) : Set (i ⊔ suc f) where"},{"Add":" field"},{"Add":" monad : RawIMonad M"},{"Add":" get : ∀ {i} → M i i (S i)"},{"Add":" put : ∀ {i j} → S j → M i j ⊤"},{"Add":""},{"Add":" open RawIMonad monad public"},{"Add":""},{"Add":" modify : ∀ {i j} → (S i → S j) → M i j ⊤"},{"Add":" modify f = get >>= put ∘ f"},{"Add":""},{"Add":"StateTIMonadState : ∀ {i f} {I : Set i} (S : I → Set f) {M} →"},{"Add":" RawMonad M → RawIMonadState S (IStateT S M)"},{"Add":"StateTIMonadState S Mon = record"},{"Add":" { monad = StateTIMonad S Mon"},{"Add":" ; get = λ s → pure (s , s)"},{"Add":" ; put = λ s _ → pure (_ , s)"},{"Add":" }"},{"Add":" where open RawMonad Mon"}]}},{"FileAdd":{"meta":{"basename":"Reader.agda","metadata":0},"context":[{"change":0,"pos":1277307}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The reader monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.Reader where"},{"Add":""},{"Add":"open import Effect.Functor using (RawFunctor)"},{"Add":"open import Effect.Applicative using (RawApplicative)"},{"Add":"open import Effect.Monad using (RawMonad; module Join)"},{"Add":"open import Effect.Monad.Identity as Id using (Identity; runIdentity)"},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"import Effect.Monad.Reader.Transformer as Trans"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" r : Level"},{"Add":" R A : Set r"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export the monad reader operations"},{"Add":""},{"Add":"open Trans public"},{"Add":" using (RawMonadReader)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Reader monad"},{"Add":""},{"Add":"Reader : (R A : Set r) → Set r"},{"Add":"Reader R = Trans.ReaderT R Identity"},{"Add":""},{"Add":"runReader : Reader R A → R → A"},{"Add":"runReader mr r = runIdentity (Trans.runReaderT mr r)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structure"},{"Add":""},{"Add":"functor : RawFunctor (Reader R)"},{"Add":"functor = Trans.functor Id.functor"},{"Add":""},{"Add":"applicative : RawApplicative (Reader R)"},{"Add":"applicative = Trans.applicative Id.applicative"},{"Add":""},{"Add":"monad : RawMonad (Reader R)"},{"Add":"monad = Trans.monad Id.monad"},{"Add":""},{"Add":"join : Reader R (Reader R A) → Reader R A"},{"Add":"join = Join.join monad"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Reader monad specifics"},{"Add":""},{"Add":"monadReader : RawMonadReader R (Reader R)"},{"Add":"monadReader = Trans.monadReader Id.monad"}]}},{"FileAdd":{"meta":{"basename":"Reader","metadata":576},"context":[{"change":0,"pos":1277307}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Transformer.agda","metadata":0},"context":[{"change":0,"pos":1308505}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The reader monad transformer"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":""},{"Add":"module Effect.Monad.Reader.Transformer where"},{"Add":""},{"Add":"open import Algebra using (RawMonoid)"},{"Add":"open import Effect.Choice using (RawChoice)"},{"Add":"open import Effect.Empty using (RawEmpty)"},{"Add":"open import Effect.Functor using (RawFunctor)"},{"Add":"open import Effect.Applicative using (RawApplicative; RawApplicativeZero; RawAlternative)"},{"Add":"open import Effect.Monad using (RawMonad; RawMonadZero; RawMonadPlus; RawMonadT)"},{"Add":"open import Function.Base using (_∘′_; const; _$_)"},{"Add":"open import Level using (Level; _⊔_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" r g g₁ g₂ : Level"},{"Add":" R R₁ R₂ : Set r"},{"Add":" A B : Set r"},{"Add":" M : Set r → Set g"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export the basic type definitions"},{"Add":""},{"Add":"open import Effect.Monad.Reader.Transformer.Base public"},{"Add":" using ( RawMonadReader"},{"Add":" ; ReaderT"},{"Add":" ; mkReaderT"},{"Add":" ; runReaderT"},{"Add":" )"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Structure"},{"Add":""},{"Add":"functor : RawFunctor M → RawFunctor (ReaderT R M)"},{"Add":"functor M = record"},{"Add":" { _<$>_ = λ f ma → mkReaderT (λ r → f <$> ReaderT.runReaderT ma r)"},{"Add":" } where open RawFunctor M"},{"Add":""},{"Add":"applicative : RawApplicative M → RawApplicative (ReaderT R M)"},{"Add":"applicative M = record"},{"Add":" { rawFunctor = functor rawFunctor"},{"Add":" ; pure = mkReaderT ∘′ const ∘′ pure"},{"Add":" ; _<*>_ = λ mf mx → mkReaderT (λ r → ReaderT.runReaderT mf r <*> ReaderT.runReaderT mx r)"},{"Add":" } where open RawApplicative M"},{"Add":""},{"Add":"empty : RawEmpty M → RawEmpty (ReaderT R M)"},{"Add":"empty M = record"},{"Add":" { empty = mkReaderT (const (RawEmpty.empty M))"},{"Add":" }"},{"Add":""},{"Add":"choice : RawChoice M → RawChoice (ReaderT R M)"},{"Add":"choice M = record"},{"Add":" { _<|>_ = λ ma₁ ma₂ → mkReaderT $ λ r →"},{"Add":" ReaderT.runReaderT ma₁ r"},{"Add":" <|> ReaderT.runReaderT ma₂ r"},{"Add":" } where open RawChoice M"},{"Add":""},{"Add":"applicativeZero : RawApplicativeZero M → RawApplicativeZero (ReaderT R M)"},{"Add":"applicativeZero M = record"},{"Add":" { rawApplicative = applicative rawApplicative"},{"Add":" ; rawEmpty = empty rawEmpty"},{"Add":" } where open RawApplicativeZero M using (rawApplicative; rawEmpty)"},{"Add":""},{"Add":"alternative : RawAlternative M → RawAlternative (ReaderT R M)"},{"Add":"alternative M = record"},{"Add":" { rawApplicativeZero = applicativeZero rawApplicativeZero"},{"Add":" ; rawChoice = choice rawChoice"},{"Add":" } where open RawAlternative M"},{"Add":""},{"Add":"monad : RawMonad M → RawMonad (ReaderT R M)"},{"Add":"monad M = record"},{"Add":" { rawApplicative = applicative rawApplicative"},{"Add":" ; _>>=_ = λ ma f → mkReaderT $ λ r →"},{"Add":" do a ← ReaderT.runReaderT ma r"},{"Add":" ReaderT.runReaderT (f a) r"},{"Add":" } where open RawMonad M"},{"Add":""},{"Add":"monadZero : RawMonadZero M → RawMonadZero (ReaderT R M)"},{"Add":"monadZero M = record"},{"Add":" { rawMonad = monad (RawMonadZero.rawMonad M)"},{"Add":" ; rawEmpty = empty (RawMonadZero.rawEmpty M)"},{"Add":" }"},{"Add":""},{"Add":"monadPlus : RawMonadPlus M → RawMonadPlus (ReaderT R M)"},{"Add":"monadPlus M = record"},{"Add":" { rawMonadZero = monadZero rawMonadZero"},{"Add":" ; rawChoice = choice rawChoice"},{"Add":" } where open RawMonadPlus M"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Monad reader transformer specifics"},{"Add":""},{"Add":"monadT : RawMonadT {g₁ = g₁} {g₂ = r ⊔ g₁} (ReaderT {r} R)"},{"Add":"monadT M = record"},{"Add":" { lift = mkReaderT ∘′ const"},{"Add":" ; rawMonad = monad M"},{"Add":" }"},{"Add":""},{"Add":"monadReader : RawMonad M → RawMonadReader R (ReaderT R M)"},{"Add":"monadReader M = record"},{"Add":" { reader = λ f → mkReaderT (pure ∘′ f)"},{"Add":" ; local = λ f ma → mkReaderT (ReaderT.runReaderT ma ∘′ f)"},{"Add":" } where open RawMonad M"},{"Add":""},{"Add":"liftReaderT : RawMonadReader R₁ M →"},{"Add":" RawMonadReader R₁ (ReaderT R₂ M)"},{"Add":"liftReaderT MRead = record"},{"Add":" { reader = λ k → mkReaderT (const (reader k))"},{"Add":" ; local = λ f mx → mkReaderT (λ r₂ → local f (runReaderT mx r₂))"},{"Add":" } where open RawMonadReader MRead"},{"Add":""},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Effect.Monad.Writer.Transformer.Base"},{"Add":""},{"Add":"liftWriterT : (MR : RawMonoid r g) →"},{"Add":" RawFunctor M →"},{"Add":" RawMonadReader R M →"},{"Add":" RawMonadReader R (WriterT MR M)"},{"Add":"liftWriterT MR M MRead = record"},{"Add":" { reader = λ k → mkWriterT λ w → ((w ,_) <$> reader k)"},{"Add":" ; local = λ f mx → mkWriterT λ w → (local f (runWriterT mx w))"},{"Add":" } where open RawMonadReader MRead"},{"Add":" open RawFunctor M"},{"Add":""},{"Add":"open import Effect.Monad.State.Transformer.Base"},{"Add":""},{"Add":"liftStateT : RawFunctor M →"},{"Add":" RawMonadReader R₁ M →"},{"Add":" RawMonadReader R₁ (StateT R₂ M)"},{"Add":"liftStateT M MRead = record"},{"Add":" { reader = λ k → mkStateT (λ s → (s ,_) <$> reader k)"},{"Add":" ; local = λ f mx → mkStateT (λ s → local f (runStateT mx s))"},{"Add":" } where open RawMonadReader MRead"},{"Add":" open RawFunctor M"}]}},{"FileAdd":{"meta":{"basename":"Transformer","metadata":576},"context":[{"change":0,"pos":1308505}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Base.agda","metadata":0},"context":[{"change":0,"pos":1313333}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Basic type and definition of the reader monad transformer"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.Reader.Transformer.Base {r} (R : Set r) where"},{"Add":""},{"Add":"open import Level using (Level; suc; _⊔_)"},{"Add":"open import Function.Base using (id)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" g : Level"},{"Add":" A : Set r"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Reader monad operations"},{"Add":""},{"Add":"record RawMonadReader"},{"Add":" (M : Set r → Set g)"},{"Add":" : Set (suc r ⊔ g) where"},{"Add":" field"},{"Add":" reader : (R → A) → M A"},{"Add":" local : (R → R) → M A → M A"},{"Add":""},{"Add":" ask : M R"},{"Add":" ask = reader id"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Reader monad transformer"},{"Add":""},{"Add":"record ReaderT"},{"Add":" (M : Set r → Set g)"},{"Add":" (A : Set r)"},{"Add":" : Set (r ⊔ g) where"},{"Add":" constructor mkReaderT"},{"Add":" field runReaderT : R → M A"},{"Add":"open ReaderT public"}]}},{"FileAdd":{"meta":{"basename":"Instances.agda","metadata":0},"context":[{"change":0,"pos":1308505}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Instances for the reader monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.Reader.Instances where"},{"Add":""},{"Add":"open import Effect.Monad.Reader.Transformer"},{"Add":""},{"Add":"instance"},{"Add":" readerTFunctor = λ {s} {S} {f} {M} {{fun}} → functor {s} {S} {f} {M} fun"},{"Add":" readerTApplicative = λ {s} {S} {f} {M} {{mon}} → applicative {s} {S} {f} {M} mon"},{"Add":" readerTEmpty = λ {s} {S} {f} {M} {{e}} → empty {s} {S} {f} {M} e"},{"Add":" readerTChoice = λ {s} {S} {f} {M} {{ch}} → choice {s} {S} {f} {M} ch"},{"Add":" readerTApplicativeZero = λ {s} {S} {f} {M} {{mon}} → applicativeZero {s} {S} {f} {M} mon"},{"Add":" readerTAlternative = λ {s} {S} {f} {M} {{mpl}} → alternative {s} {S} {f} {M} mpl"},{"Add":" readerTMonad = λ {s} {S} {f} {M} {{mon}} → monad {s} {S} {f} {M} mon"},{"Add":" readerTMonadZero = λ {s} {S} {f} {M} {{mz}} → monadZero {s} {S} {f} {M} mz"},{"Add":" readerTMonadPlus = λ {s} {S} {f} {M} {{mpl}} → monadPlus {s} {S} {f} {M} mpl"},{"Add":" readerTMonadT = λ {s} {S} {f} {M} {{mon}} → monadT {s} {S} {f} {M} mon"},{"Add":" readerTMonadReader = λ {s} {S} {f} {M} {{mon}} → monadReader {s} {S} {f} {M} mon"},{"Add":" readerTLiftWriterT = λ {s} {S₁} {S₂} {f} {M} {{mo}} {{fun}} {{mr}} → liftWriterT {s} {S₁} {S₂} {f} {M} mo fun mr"},{"Add":" readerTLiftStateT = λ {s} {S₁} {S₂} {f} {M} {{fun}} {{mr}} → liftStateT {s} {S₁} {S₂} {f} {M} fun mr"},{"Add":" -- the following instance conflicts with readerTMonadReader so we don't include it"},{"Add":" -- readerTLiftReaderT = λ {R} {s} {S} {f} {M} {{ms}} → liftReaderT {R} {s} {S} {f} {M} ms"}]}},{"FileAdd":{"meta":{"basename":"Indexed.agda","metadata":0},"context":[{"change":0,"pos":1308505}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The indexed reader monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Level using (Level; _⊔_; suc; Lift; lift)"},{"Add":""},{"Add":"module Effect.Monad.Reader.Indexed {r} (R : Set r) (a : Level) where"},{"Add":""},{"Add":"open import Function.Base using (const; flip; _∘_)"},{"Add":"open import Function.Identity.Effectful as Id using (Identity)"},{"Add":"open import Effect.Applicative.Indexed"},{"Add":" using (IFun; RawIApplicative; RawIApplicativeZero; RawIAlternative)"},{"Add":"open import Effect.Monad.Indexed using (RawIMonad; RawIMonadZero;"},{"Add":" RawIMonadPlus)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ : Level"},{"Add":" A B I : Set ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Indexed reader"},{"Add":""},{"Add":"IReaderT : IFun I (r ⊔ a) → IFun I (r ⊔ a)"},{"Add":"IReaderT M i j A = R → M i j A"},{"Add":""},{"Add":"module _ {M : IFun I (r ⊔ a)} where"},{"Add":""},{"Add":" ----------------------------------------------------------------------"},{"Add":" -- Indexed reader applicative"},{"Add":""},{"Add":" ReaderTIApplicative : RawIApplicative M → RawIApplicative (IReaderT M)"},{"Add":" ReaderTIApplicative App = record"},{"Add":" { pure = λ x r → pure x"},{"Add":" ; _⊛_ = λ m n r → m r ⊛ n r"},{"Add":" } where open RawIApplicative App"},{"Add":""},{"Add":" ReaderTIApplicativeZero : RawIApplicativeZero M →"},{"Add":" RawIApplicativeZero (IReaderT M)"},{"Add":" ReaderTIApplicativeZero App = record"},{"Add":" { applicative = ReaderTIApplicative applicative"},{"Add":" ; ∅ = const ∅"},{"Add":" } where open RawIApplicativeZero App"},{"Add":""},{"Add":" ReaderTIAlternative : RawIAlternative M → RawIAlternative (IReaderT M)"},{"Add":" ReaderTIAlternative Alt = record"},{"Add":" { applicativeZero = ReaderTIApplicativeZero applicativeZero"},{"Add":" ; _∣_ = λ m n r → m r ∣ n r"},{"Add":" } where open RawIAlternative Alt"},{"Add":""},{"Add":" ----------------------------------------------------------------------"},{"Add":" -- Indexed reader monad"},{"Add":""},{"Add":" ReaderTIMonad : RawIMonad M → RawIMonad (IReaderT M)"},{"Add":" ReaderTIMonad Mon = record"},{"Add":" { return = λ x r → return x"},{"Add":" ; _>>=_ = λ m f r → m r >>= flip f r"},{"Add":" } where open RawIMonad Mon"},{"Add":""},{"Add":" ReaderTIMonadZero : RawIMonadZero M → RawIMonadZero (IReaderT M)"},{"Add":" ReaderTIMonadZero Mon = record"},{"Add":" { monad = ReaderTIMonad monad"},{"Add":" ; applicativeZero = ReaderTIApplicativeZero applicativeZero"},{"Add":" } where open RawIMonadZero Mon"},{"Add":""},{"Add":" ReaderTIMonadPlus : RawIMonadPlus M → RawIMonadPlus (IReaderT M)"},{"Add":" ReaderTIMonadPlus Mon = record"},{"Add":" { monad = ReaderTIMonad monad"},{"Add":" ; alternative = ReaderTIAlternative alternative"},{"Add":" } where open RawIMonadPlus Mon"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Reader monad operations"},{"Add":""},{"Add":"record RawIMonadReader {I : Set ℓ} (M : IFun I (r ⊔ a))"},{"Add":" : Set (ℓ ⊔ suc (r ⊔ a)) where"},{"Add":" field"},{"Add":" monad : RawIMonad M"},{"Add":" reader : ∀ {i} → (R → A) → M i i A"},{"Add":" local : ∀ {i j} → (R → R) → M i j A → M i j A"},{"Add":""},{"Add":" open RawIMonad monad public"},{"Add":""},{"Add":" ask : ∀ {i} → M i i (Lift (r ⊔ a) R)"},{"Add":" ask = reader lift"},{"Add":""},{"Add":" asks : ∀ {i} → (R → A) → M i i A"},{"Add":" asks = reader"},{"Add":""},{"Add":"ReaderTIMonadReader : {I : Set ℓ} {M : IFun I (r ⊔ a)} →"},{"Add":" RawIMonad M → RawIMonadReader (IReaderT M)"},{"Add":"ReaderTIMonadReader Mon = record"},{"Add":" { monad = ReaderTIMonad Mon"},{"Add":" ; reader = λ f r → return (f r)"},{"Add":" ; local = λ f m → m ∘ f"},{"Add":" } where open RawIMonad Mon"}]}},{"FileAdd":{"meta":{"basename":"Predicate.agda","metadata":0},"context":[{"change":0,"pos":1277307}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Monads on indexed sets (predicates)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Note that currently the monad laws are not included here."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.Predicate where"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Effect.Monad.Indexed using (RawIMonad)"},{"Add":"open import Function.Base using (const; id; _∘_)"},{"Add":"open import Level using (Level; _⊔_; suc)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (refl)"},{"Add":"open import Relation.Unary using (_⊆_; _⇒_; _∈_; _∩_; ｛_｝)"},{"Add":"open import Relation.Unary.PredicateTransformer using (Pt)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" i ℓ : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record RawPMonad {I : Set i} (M : Pt I (i ⊔ ℓ)) : Set (suc (i ⊔ ℓ)) where"},{"Add":""},{"Add":" infixl 1 _?>=_ _?>_ _>?>_ _?>=′_"},{"Add":" infixr 1 _==_ : ∀ {P Q} → M P ⊆ const (P ⊆ M Q) ⇒ M Q"},{"Add":" m ?>= f = f ==′_ : ∀ {P Q} → M P ⊆ const (∀ j → {_ : P j} → j ∈ M Q) ⇒ M Q"},{"Add":" m ?>=′ f = m ?>= λ {j} p → f j {p}"},{"Add":""},{"Add":" _?>_ : ∀ {P Q} → M P ⊆ const (∀ {j} → j ∈ M Q) ⇒ M Q"},{"Add":" m₁ ?> m₂ = m₁ ?>= λ _ → m₂"},{"Add":""},{"Add":" join? : ∀ {P} → M (M P) ⊆ M P"},{"Add":" join? m = m ?>= id"},{"Add":""},{"Add":" _>?>_ : {P Q R : _} → P ⊆ M Q → Q ⊆ M R → P ⊆ M R"},{"Add":" f >?> g = _=?> g"},{"Add":""},{"Add":" -- ``Angelic'' operations (the player knows the state)."},{"Add":""},{"Add":" rawIMonad : RawIMonad (λ i j A → i ∈ M (const A ∩ ｛ j ｝))"},{"Add":" rawIMonad = record"},{"Add":" { return = λ x → return? (x , refl)"},{"Add":" ; _>>=_ = λ m k → m ?>= λ { {._} (x , refl) → k x }"},{"Add":" }"},{"Add":""},{"Add":" open RawIMonad rawIMonad public"}]}},{"FileAdd":{"meta":{"basename":"Partiality.agda","metadata":0},"context":[{"change":0,"pos":1277307}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The partiality monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe --guardedness #-}"},{"Add":""},{"Add":"module Effect.Monad.Partiality where"},{"Add":""},{"Add":"open import Codata.Musical.Notation using (∞; ♯_; ♭)"},{"Add":"open import Data.Bool.Base using (Bool; false; true)"},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc; _+_)"},{"Add":"open import Data.Product as Prod using (∃; ∄; -,_; ∃₂; _,_; _×_)"},{"Add":"open import Data.Sum.Base using (_⊎_; inj₁; inj₂)"},{"Add":"open import Effect.Functor using (RawFunctor)"},{"Add":"open import Effect.Applicative using (RawApplicative)"},{"Add":"open import Effect.Monad using (RawMonad; module Join)"},{"Add":"open import Function.Base using (_∘′_; flip; id; _∘_; _$_; _⟨_⟩_)"},{"Add":"open import Function.Bundles using (_⇔_; mk⇔)"},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Relation.Binary.Core as B hiding (Rel; _⇔_)"},{"Add":"open import Relation.Binary.Definitions"},{"Add":" using (DecidableEquality; Reflexive; Symmetric; Transitive)"},{"Add":"open import Relation.Binary.Structures"},{"Add":" using (IsPreorder; IsEquivalence)"},{"Add":"open import Relation.Binary.Bundles"},{"Add":" using (Preorder; Setoid; Poset)"},{"Add":"import Relation.Binary.Properties.Setoid as SetoidProperties"},{"Add":"open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)"},{"Add":"import Relation.Binary.PropositionalEquality.Properties as ≡"},{"Add":"open import Relation.Nullary.Decidable using (yes; no; False; Dec; ¬¬-excluded-middle)"},{"Add":"open import Relation.Nullary.Negation using (¬_; ¬¬-Monad)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c f s ℓ : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":" C : Set c"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The partiality monad"},{"Add":""},{"Add":"data _⊥ (A : Set a) : Set a where"},{"Add":" now : (x : A) → A ⊥"},{"Add":" later : (x : ∞ (A ⊥)) → A ⊥"},{"Add":""},{"Add":"bind : A ⊥ → (A → B ⊥) → B ⊥"},{"Add":"bind (now x) f = f x"},{"Add":"bind (later x) f = later (♯ (bind (♭ x) f))"},{"Add":""},{"Add":"functor : RawFunctor {ℓ} _⊥"},{"Add":"functor = record { _<$>_ = map } where"},{"Add":""},{"Add":" map : (A → B) → A ⊥ → B ⊥"},{"Add":" map f (now a) = now (f a)"},{"Add":" map f (later d) = later (♯ map f (♭ d))"},{"Add":""},{"Add":""},{"Add":"applicative : RawApplicative {f = f} _⊥"},{"Add":"applicative = record"},{"Add":" { rawFunctor = functor"},{"Add":" ; pure = now"},{"Add":" ; _<*>_ = λ mf mx → bind mf (λ f → bind mx (now ∘′ f))"},{"Add":" }"},{"Add":""},{"Add":"monad : RawMonad {f = f} _⊥"},{"Add":"monad = record"},{"Add":" { rawApplicative = applicative"},{"Add":" ; _>>=_ = bind"},{"Add":" }"},{"Add":""},{"Add":"join : (A ⊥) ⊥ → A ⊥"},{"Add":"join = Join.join monad"},{"Add":""},{"Add":"private module M {f} = RawMonad (monad {f})"},{"Add":""},{"Add":"-- Non-termination."},{"Add":""},{"Add":"never : A ⊥"},{"Add":"never = later (♯ never)"},{"Add":""},{"Add":"-- run x for n steps peels off at most n \"later\" constructors from x."},{"Add":""},{"Add":"run_for_steps : A ⊥ → ℕ → A ⊥"},{"Add":"run now x for n steps = now x"},{"Add":"run later x for zero steps = later x"},{"Add":"run later x for suc n steps = run ♭ x for n steps"},{"Add":""},{"Add":"-- Is the computation done?"},{"Add":""},{"Add":"isNow : A ⊥ → Bool"},{"Add":"isNow (now x) = true"},{"Add":"isNow (later x) = false"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Kinds"},{"Add":""},{"Add":"-- The partiality monad comes with two forms of equality (weak and"},{"Add":"-- strong) and one ordering. Strong equality is stronger than the"},{"Add":"-- ordering, which is stronger than weak equality."},{"Add":""},{"Add":"-- The three relations are defined using a single data type, indexed"},{"Add":"-- by a \"kind\"."},{"Add":""},{"Add":"data OtherKind : Set where"},{"Add":" geq weak : OtherKind"},{"Add":""},{"Add":"data Kind : Set where"},{"Add":" strong : Kind"},{"Add":" other : (k : OtherKind) → Kind"},{"Add":""},{"Add":"-- Kind equality is decidable."},{"Add":""},{"Add":"infix 4 _≟-Kind_"},{"Add":""},{"Add":"_≟-Kind_ : DecidableEquality Kind"},{"Add":"_≟-Kind_ strong strong = yes ≡.refl"},{"Add":"_≟-Kind_ strong (other k) = no λ()"},{"Add":"_≟-Kind_ (other k) strong = no λ()"},{"Add":"_≟-Kind_ (other geq) (other geq) = yes ≡.refl"},{"Add":"_≟-Kind_ (other geq) (other weak) = no λ()"},{"Add":"_≟-Kind_ (other weak) (other geq) = no λ()"},{"Add":"_≟-Kind_ (other weak) (other weak) = yes ≡.refl"},{"Add":""},{"Add":"-- A predicate which is satisfied only for equalities. Note that, for"},{"Add":"-- concrete inputs, this predicate evaluates to ⊤ or ⊥."},{"Add":""},{"Add":"Equality : Kind → Set"},{"Add":"Equality k = False (k ≟-Kind other geq)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Equality/ordering"},{"Add":""},{"Add":"module Equality {A : Set a} -- The \"return type\"."},{"Add":" (_∼_ : A → A → Set ℓ) where"},{"Add":""},{"Add":" -- The three relations."},{"Add":""},{"Add":" data Rel : Kind → A ⊥ → A ⊥ → Set (a ⊔ ℓ) where"},{"Add":" now : ∀ {k x y} (x∼y : x ∼ y) → Rel k (now x) (now y)"},{"Add":" later : ∀ {k x y} (x∼y : ∞ (Rel k (♭ x) (♭ y))) → Rel k (later x) (later y)"},{"Add":" laterʳ : ∀ {x y} (x≈y : Rel (other weak) x (♭ y) ) → Rel (other weak) x (later y)"},{"Add":" laterˡ : ∀ {k x y} (x∼y : Rel (other k) (♭ x) y ) → Rel (other k) (later x) y"},{"Add":""},{"Add":" infix 4 _≅_ _≳_ _≲_ _≈_"},{"Add":""},{"Add":" _≅_ : A ⊥ → A ⊥ → Set _"},{"Add":" _≅_ = Rel strong"},{"Add":""},{"Add":" _≳_ : A ⊥ → A ⊥ → Set _"},{"Add":" _≳_ = Rel (other geq)"},{"Add":""},{"Add":" _≲_ : A ⊥ → A ⊥ → Set _"},{"Add":" _≲_ = flip _≳_"},{"Add":""},{"Add":" _≈_ : A ⊥ → A ⊥ → Set _"},{"Add":" _≈_ = Rel (other weak)"},{"Add":""},{"Add":" -- x ⇓ y means that x terminates with y."},{"Add":""},{"Add":" infix 4 _⇓[_]_ _⇓_"},{"Add":""},{"Add":" _⇓[_]_ : A ⊥ → Kind → A → Set _"},{"Add":" x ⇓[ k ] y = Rel k x (now y)"},{"Add":""},{"Add":" _⇓_ : A ⊥ → A → Set _"},{"Add":" x ⇓ y = x ⇓[ other weak ] y"},{"Add":""},{"Add":" -- x ⇓ means that x terminates."},{"Add":""},{"Add":" infix 4 _⇓"},{"Add":""},{"Add":" _⇓ : A ⊥ → Set _"},{"Add":" x ⇓ = ∃ λ v → x ⇓ v"},{"Add":""},{"Add":" -- x ⇑ means that x does not terminate."},{"Add":""},{"Add":" infix 4 _⇑[_] _⇑"},{"Add":""},{"Add":" _⇑[_] : A ⊥ → Kind → Set _"},{"Add":" x ⇑[ k ] = Rel k x never"},{"Add":""},{"Add":" _⇑ : A ⊥ → Set _"},{"Add":" x ⇑ = x ⇑[ other weak ]"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Lemmas relating the three relations"},{"Add":""},{"Add":"module _ {A : Set a} {_∼_ : A → A → Set ℓ} where"},{"Add":""},{"Add":" open Equality _∼_ using (Rel; _≅_; _≳_; _≲_; _≈_; _⇓[_]_; _⇑[_])"},{"Add":" open Equality.Rel"},{"Add":""},{"Add":" -- All relations include strong equality."},{"Add":""},{"Add":" ≅⇒ : ∀ {k} {x y : A ⊥} → x ≅ y → Rel k x y"},{"Add":" ≅⇒ (now x∼y) = now x∼y"},{"Add":" ≅⇒ (later x≅y) = later (♯ ≅⇒ (♭ x≅y))"},{"Add":""},{"Add":" -- The weak equality includes the ordering."},{"Add":""},{"Add":" ≳⇒ : ∀ {k} {x y : A ⊥} → x ≳ y → Rel (other k) x y"},{"Add":" ≳⇒ (now x∼y) = now x∼y"},{"Add":" ≳⇒ (later x≳y) = later (♯ ≳⇒ (♭ x≳y))"},{"Add":" ≳⇒ (laterˡ x≳y) = laterˡ (≳⇒ x≳y )"},{"Add":""},{"Add":" -- Weak equality includes the other relations."},{"Add":""},{"Add":" ⇒≈ : ∀ {k} {x y : A ⊥} → Rel k x y → x ≈ y"},{"Add":" ⇒≈ {strong} = ≅⇒"},{"Add":" ⇒≈ {other geq} = ≳⇒"},{"Add":" ⇒≈ {other weak} = id"},{"Add":""},{"Add":" -- The relations agree for non-terminating computations."},{"Add":""},{"Add":" never⇒never : ∀ {k₁ k₂} {x : A ⊥} →"},{"Add":" Rel k₁ x never → Rel k₂ x never"},{"Add":" never⇒never (later x∼never) = later (♯ never⇒never (♭ x∼never))"},{"Add":" never⇒never (laterʳ x≈never) = never⇒never x≈never"},{"Add":" never⇒never (laterˡ x∼never) = later (♯ never⇒never x∼never)"},{"Add":""},{"Add":" -- The \"other\" relations agree when the right-hand side is a value."},{"Add":""},{"Add":" now⇒now : ∀ {k₁ k₂} {x} {y : A} →"},{"Add":" Rel (other k₁) x (now y) → Rel (other k₂) x (now y)"},{"Add":" now⇒now (now x∼y) = now x∼y"},{"Add":" now⇒now (laterˡ x∼now) = laterˡ (now⇒now x∼now)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Later can be dropped"},{"Add":""},{"Add":" laterʳ⁻¹ : ∀ {k} {x : A ⊥} {y} →"},{"Add":" Rel (other k) x (later y) → Rel (other k) x (♭ y)"},{"Add":" laterʳ⁻¹ (later x∼y) = laterˡ (♭ x∼y)"},{"Add":" laterʳ⁻¹ (laterʳ x≈y) = x≈y"},{"Add":" laterʳ⁻¹ (laterˡ x∼ly) = laterˡ (laterʳ⁻¹ x∼ly)"},{"Add":""},{"Add":" laterˡ⁻¹ : ∀ {x} {y : A ⊥} → later x ≈ y → ♭ x ≈ y"},{"Add":" laterˡ⁻¹ (later x≈y) = laterʳ (♭ x≈y)"},{"Add":" laterˡ⁻¹ (laterʳ lx≈y) = laterʳ (laterˡ⁻¹ lx≈y)"},{"Add":" laterˡ⁻¹ (laterˡ x≈y) = x≈y"},{"Add":""},{"Add":" later⁻¹ : ∀ {k} {x y : ∞ (A ⊥)} →"},{"Add":" Rel k (later x) (later y) → Rel k (♭ x) (♭ y)"},{"Add":" later⁻¹ (later x∼y) = ♭ x∼y"},{"Add":" later⁻¹ (laterʳ lx≈y) = laterˡ⁻¹ lx≈y"},{"Add":" later⁻¹ (laterˡ x∼ly) = laterʳ⁻¹ x∼ly"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The relations are equivalences or partial orders, given suitable"},{"Add":"-- assumptions about the underlying relation"},{"Add":""},{"Add":" module Equivalence where"},{"Add":""},{"Add":" -- Reflexivity."},{"Add":""},{"Add":" refl : Reflexive _∼_ → ∀ {k} → Reflexive (Rel k)"},{"Add":" refl refl-∼ {x = now v} = now refl-∼"},{"Add":" refl refl-∼ {x = later x} = later (♯ refl refl-∼)"},{"Add":""},{"Add":" -- Symmetry."},{"Add":""},{"Add":" sym : Symmetric _∼_ → ∀ {k} → Equality k → Symmetric (Rel k)"},{"Add":" sym sym-∼ eq (now x∼y) = now (sym-∼ x∼y)"},{"Add":" sym sym-∼ eq (later x∼y) = later (♯ sym sym-∼ eq (♭ x∼y))"},{"Add":" sym sym-∼ eq (laterʳ x≈y) = laterˡ (sym sym-∼ eq x≈y )"},{"Add":" sym sym-∼ eq (laterˡ {weak} x≈y) = laterʳ (sym sym-∼ eq x≈y )"},{"Add":""},{"Add":" -- Transitivity."},{"Add":""},{"Add":" private"},{"Add":" module Trans (trans-∼ : Transitive _∼_) where"},{"Add":""},{"Add":" now-trans : ∀ {k x y} {v : A} →"},{"Add":" Rel k x y → Rel k y (now v) → Rel k x (now v)"},{"Add":" now-trans (now x∼y) (now y∼z) = now (trans-∼ x∼y y∼z)"},{"Add":" now-trans (laterˡ x∼y) y∼z = laterˡ (now-trans x∼y y∼z)"},{"Add":" now-trans x∼ly (laterˡ y∼z) = now-trans (laterʳ⁻¹ x∼ly) y∼z"},{"Add":""},{"Add":" mutual"},{"Add":""},{"Add":" later-trans : ∀ {k} {x y : A ⊥} {z} →"},{"Add":" Rel k x y → Rel k y (later z) → Rel k x (later z)"},{"Add":" later-trans (later x∼y) ly∼lz = later (♯ trans (♭ x∼y) (later⁻¹ ly∼lz))"},{"Add":" later-trans (laterˡ x∼y) y∼lz = later (♯ trans x∼y (laterʳ⁻¹ y∼lz))"},{"Add":" later-trans (laterʳ x≈y) ly≈lz = later-trans x≈y (laterˡ⁻¹ ly≈lz)"},{"Add":" later-trans x≈y (laterʳ y≈z) = laterʳ ( trans x≈y y≈z )"},{"Add":""},{"Add":" trans : ∀ {k} {x y z : A ⊥} → Rel k x y → Rel k y z → Rel k x z"},{"Add":" trans {z = now v} x∼y y∼v = now-trans x∼y y∼v"},{"Add":" trans {z = later z} x∼y y∼lz = later-trans x∼y y∼lz"},{"Add":""},{"Add":" open Trans public using (trans)"},{"Add":""},{"Add":" -- All the relations are preorders."},{"Add":""},{"Add":" preorder : IsPreorder _≡_ _∼_ → Kind → Preorder _ _ _"},{"Add":" preorder pre k = record"},{"Add":" { Carrier = A ⊥"},{"Add":" ; _≈_ = _≡_"},{"Add":" ; _≲_ = Rel k"},{"Add":" ; isPreorder = record"},{"Add":" { isEquivalence = ≡.isEquivalence"},{"Add":" ; reflexive = refl′"},{"Add":" ; trans = Equivalence.trans (IsPreorder.trans pre)"},{"Add":" }"},{"Add":" }"},{"Add":" where"},{"Add":" refl′ : ∀ {k} {x y : A ⊥} → x ≡ y → Rel k x y"},{"Add":" refl′ ≡.refl = Equivalence.refl (IsPreorder.refl pre)"},{"Add":""},{"Add":" private"},{"Add":" preorder′ : IsEquivalence _∼_ → Kind → Preorder _ _ _"},{"Add":" preorder′ equiv ="},{"Add":" preorder (SetoidProperties.isPreorder (record { isEquivalence = equiv }))"},{"Add":""},{"Add":" -- The two equalities are equivalence relations."},{"Add":""},{"Add":" setoid : IsEquivalence _∼_ →"},{"Add":" (k : Kind) {eq : Equality k} → Setoid _ _"},{"Add":" setoid equiv k {eq} = record"},{"Add":" { Carrier = A ⊥"},{"Add":" ; _≈_ = Rel k"},{"Add":" ; isEquivalence = record"},{"Add":" { refl = Pre.refl"},{"Add":" ; sym = Equivalence.sym (IsEquivalence.sym equiv) eq"},{"Add":" ; trans = Pre.trans"},{"Add":" }"},{"Add":" } where module Pre = Preorder (preorder′ equiv k)"},{"Add":""},{"Add":" -- The order is a partial order, with strong equality as the"},{"Add":" -- underlying equality."},{"Add":""},{"Add":" ≳-poset : IsEquivalence _∼_ → Poset _ _ _"},{"Add":" ≳-poset equiv = record"},{"Add":" { Carrier = A ⊥"},{"Add":" ; _≈_ = _≅_"},{"Add":" ; _≤_ = _≳_"},{"Add":" ; isPartialOrder = record"},{"Add":" { antisym = antisym"},{"Add":" ; isPreorder = record"},{"Add":" { isEquivalence = S.isEquivalence"},{"Add":" ; reflexive = ≅⇒"},{"Add":" ; trans = Pre.trans"},{"Add":" }"},{"Add":" }"},{"Add":" }"},{"Add":" where"},{"Add":" module S = Setoid (setoid equiv strong)"},{"Add":" module Pre = Preorder (preorder′ equiv (other geq))"},{"Add":""},{"Add":" antisym : {x y : A ⊥} → x ≳ y → x ≲ y → x ≅ y"},{"Add":" antisym (now x∼y) (now _) = now x∼y"},{"Add":" antisym (later x≳y) (later x≲y) = later (♯ antisym (♭ x≳y) (♭ x≲y))"},{"Add":" antisym (later x≳y) (laterˡ x≲ly) = later (♯ antisym (♭ x≳y) (laterʳ⁻¹ x≲ly))"},{"Add":" antisym (laterˡ x≳ly) (later x≲y) = later (♯ antisym (laterʳ⁻¹ x≳ly) (♭ x≲y))"},{"Add":" antisym (laterˡ x≳ly) (laterˡ x≲ly) = later (♯ antisym (laterʳ⁻¹ x≳ly) (laterʳ⁻¹ x≲ly))"},{"Add":""},{"Add":" -- Equational reasoning."},{"Add":""},{"Add":" module Reasoning (isEquivalence : IsEquivalence _∼_) where"},{"Add":""},{"Add":" private"},{"Add":" module Pre {k} = Preorder (preorder′ isEquivalence k)"},{"Add":" module S {k eq} = Setoid (setoid isEquivalence k {eq})"},{"Add":""},{"Add":" infix 3 _∎"},{"Add":" infixr 2 _≡⟨_⟩_ _≅⟨_⟩_ _≳⟨_⟩_ _≈⟨_⟩_"},{"Add":""},{"Add":" _≡⟨_⟩_ : ∀ {k} x {y z : A ⊥} → x ≡ y → Rel k y z → Rel k x z"},{"Add":" _ ≡⟨ ≡.refl ⟩ y∼z = y∼z"},{"Add":""},{"Add":" _≅⟨_⟩_ : ∀ {k} x {y z : A ⊥} → x ≅ y → Rel k y z → Rel k x z"},{"Add":" _ ≅⟨ x≅y ⟩ y∼z = Pre.trans (≅⇒ x≅y) y∼z"},{"Add":""},{"Add":" _≳⟨_⟩_ : ∀ {k} x {y z : A ⊥} →"},{"Add":" x ≳ y → Rel (other k) y z → Rel (other k) x z"},{"Add":" _ ≳⟨ x≳y ⟩ y∼z = Pre.trans (≳⇒ x≳y) y∼z"},{"Add":""},{"Add":" _≈⟨_⟩_ : ∀ x {y z : A ⊥} → x ≈ y → y ≈ z → x ≈ z"},{"Add":" _ ≈⟨ x≈y ⟩ y≈z = Pre.trans x≈y y≈z"},{"Add":""},{"Add":" sym : ∀ {k} {eq : Equality k} {x y : A ⊥} →"},{"Add":" Rel k x y → Rel k y x"},{"Add":" sym {eq = eq} = S.sym {eq = eq}"},{"Add":""},{"Add":" _∎ : ∀ {k} (x : A ⊥) → Rel k x x"},{"Add":" x ∎ = Pre.refl"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Lemmas related to now and never"},{"Add":""},{"Add":" -- Now is not never."},{"Add":""},{"Add":" now≉never : ∀ {k} {x : A} → ¬ Rel k (now x) never"},{"Add":" now≉never (laterʳ hyp) = now≉never hyp"},{"Add":""},{"Add":" -- A partial value is either now or never (classically, when the"},{"Add":" -- underlying relation is reflexive)."},{"Add":""},{"Add":" now-or-never : Reflexive _∼_ →"},{"Add":" ∀ {k} (x : A ⊥) →"},{"Add":" ¬ ¬ ((∃ λ y → x ⇓[ other k ] y) ⊎ x ⇑[ other k ])"},{"Add":" now-or-never refl x = helper <$> ¬¬-excluded-middle"},{"Add":" where"},{"Add":" open RawMonad ¬¬-Monad"},{"Add":""},{"Add":" not-now-is-never : (x : A ⊥) → (∄ λ y → x ≳ now y) → x ≳ never"},{"Add":" not-now-is-never (now x) hyp with hyp (-, now refl)"},{"Add":" ... | ()"},{"Add":" not-now-is-never (later x) hyp ="},{"Add":" later (♯ not-now-is-never (♭ x) (hyp ∘ Prod.map id laterˡ))"},{"Add":""},{"Add":" helper : Dec (∃ λ y → x ≳ now y) → _"},{"Add":" helper (yes ≳now) = inj₁ $ Prod.map id ≳⇒ ≳now"},{"Add":" helper (no ≵now) = inj₂ $ ≳⇒ $ not-now-is-never x ≵now"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Map-like results"},{"Add":""},{"Add":" -- Map."},{"Add":""},{"Add":" map : ∀ {_∼′_ : A → A → Set a} {k} →"},{"Add":" _∼′_ ⇒ _∼_ → Equality.Rel _∼′_ k ⇒ Equality.Rel _∼_ k"},{"Add":" map ∼′⇒∼ (now x∼y) = now (∼′⇒∼ x∼y)"},{"Add":" map ∼′⇒∼ (later x∼y) = later (♯ map ∼′⇒∼ (♭ x∼y))"},{"Add":" map ∼′⇒∼ (laterʳ x≈y) = laterʳ (map ∼′⇒∼ x≈y)"},{"Add":" map ∼′⇒∼ (laterˡ x∼y) = laterˡ (map ∼′⇒∼ x∼y)"},{"Add":""},{"Add":" -- If a statement can be proved using propositional equality as the"},{"Add":" -- underlying relation, then it can also be proved for any other"},{"Add":" -- reflexive underlying relation."},{"Add":""},{"Add":" ≡⇒ : Reflexive _∼_ →"},{"Add":" ∀ {k x y} → Equality.Rel _≡_ k x y → Rel k x y"},{"Add":" ≡⇒ refl-∼ = map (flip (≡.subst (_∼_ _)) refl-∼)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Steps"},{"Add":""},{"Add":" -- The number of later constructors (steps) in the terminating"},{"Add":" -- computation x."},{"Add":""},{"Add":" steps : ∀ {k} {x : A ⊥} {y} → x ⇓[ k ] y → ℕ"},{"Add":" steps (now _) = zero"},{"Add":" steps .{x = later x} (laterˡ {x = x} x⇓) = suc (steps {x = ♭ x} x⇓)"},{"Add":""},{"Add":" module Steps {trans-∼ : Transitive _∼_} where"},{"Add":""},{"Add":" left-identity :"},{"Add":" ∀ {k x y} {z : A}"},{"Add":" (x≅y : x ≅ y) (y⇓z : y ⇓[ k ] z) →"},{"Add":" steps (Equivalence.trans trans-∼ (≅⇒ x≅y) y⇓z) ≡ steps y⇓z"},{"Add":" left-identity (now _) (now _) = ≡.refl"},{"Add":" left-identity (later x≅y) (laterˡ y⇓z) ="},{"Add":" ≡.cong suc $ left-identity (♭ x≅y) y⇓z"},{"Add":""},{"Add":" right-identity :"},{"Add":" ∀ {k x} {y z : A}"},{"Add":" (x⇓y : x ⇓[ k ] y) (y≈z : now y ⇓[ k ] z) →"},{"Add":" steps (Equivalence.trans trans-∼ x⇓y y≈z) ≡ steps x⇓y"},{"Add":" right-identity (now x∼y) (now y∼z) = ≡.refl"},{"Add":" right-identity (laterˡ x∼y) (now y∼z) ="},{"Add":" ≡.cong suc $ right-identity x∼y (now y∼z)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Laws related to bind"},{"Add":""},{"Add":" -- Never is a left and right \"zero\" of bind."},{"Add":""},{"Add":" left-zero : (f : B → A ⊥) → let open M in"},{"Add":" (never >>= f) ≅ never"},{"Add":" left-zero f = later (♯ left-zero f)"},{"Add":""},{"Add":" right-zero : (x : B ⊥) → let open M in"},{"Add":" (x >>= λ _ → never) ≅ never"},{"Add":" right-zero (later x) = later (♯ right-zero (♭ x))"},{"Add":" right-zero (now x) = never≅never"},{"Add":" where never≅never : never ≅ never"},{"Add":" never≅never = later (♯ never≅never)"},{"Add":""},{"Add":" -- Now is a left and right identity of bind (for a reflexive"},{"Add":" -- underlying relation)."},{"Add":""},{"Add":" left-identity : Reflexive _∼_ →"},{"Add":" (x : B) (f : B → A ⊥) → let open M in"},{"Add":" (now x >>= f) ≅ f x"},{"Add":" left-identity refl-∼ x f = Equivalence.refl refl-∼"},{"Add":""},{"Add":" right-identity : Reflexive _∼_ →"},{"Add":" (x : A ⊥) → let open M in"},{"Add":" (x >>= now) ≅ x"},{"Add":" right-identity refl (now x) = now refl"},{"Add":" right-identity refl (later x) = later (♯ right-identity refl (♭ x))"},{"Add":""},{"Add":" -- Bind is associative (for a reflexive underlying relation)."},{"Add":""},{"Add":" associative : Reflexive _∼_ →"},{"Add":" (x : C ⊥) (f : C → B ⊥) (g : B → A ⊥) →"},{"Add":" let open M in"},{"Add":" (x >>= f >>= g) ≅ (x >>= λ y → f y >>= g)"},{"Add":" associative refl-∼ (now x) f g = Equivalence.refl refl-∼"},{"Add":" associative refl-∼ (later x) f g ="},{"Add":" later (♯ associative refl-∼ (♭ x) f g)"},{"Add":""},{"Add":"module _ {A B : Set s}"},{"Add":" {_∼A_ : A → A → Set ℓ}"},{"Add":" {_∼B_ : B → B → Set ℓ} where"},{"Add":""},{"Add":" open Equality"},{"Add":" private"},{"Add":" open module EqA = Equality _∼A_ using () renaming (_⇓[_]_ to _⇓[_]A_; _⇑[_] to _⇑[_]A)"},{"Add":" open module EqB = Equality _∼B_ using () renaming (_⇓[_]_ to _⇓[_]B_; _⇑[_] to _⇑[_]B)"},{"Add":""},{"Add":" -- Bind preserves all the relations."},{"Add":""},{"Add":" infixl 1 _>>=-cong_"},{"Add":""},{"Add":" _>>=-cong_ :"},{"Add":" ∀ {k} {x₁ x₂ : A ⊥} {f₁ f₂ : A → B ⊥} → let open M in"},{"Add":" Rel _∼A_ k x₁ x₂ →"},{"Add":" (∀ {x₁ x₂} → x₁ ∼A x₂ → Rel _∼B_ k (f₁ x₁) (f₂ x₂)) →"},{"Add":" Rel _∼B_ k (x₁ >>= f₁) (x₂ >>= f₂)"},{"Add":" now x₁∼x₂ >>=-cong f₁∼f₂ = f₁∼f₂ x₁∼x₂"},{"Add":" later x₁∼x₂ >>=-cong f₁∼f₂ = later (♯ (♭ x₁∼x₂ >>=-cong f₁∼f₂))"},{"Add":" laterʳ x₁≈x₂ >>=-cong f₁≈f₂ = laterʳ (x₁≈x₂ >>=-cong f₁≈f₂)"},{"Add":" laterˡ x₁∼x₂ >>=-cong f₁∼f₂ = laterˡ (x₁∼x₂ >>=-cong f₁∼f₂)"},{"Add":""},{"Add":" -- Inversion lemmas for bind."},{"Add":""},{"Add":" >>=-inversion-⇓ :"},{"Add":" Reflexive _∼A_ →"},{"Add":" ∀ {k} x {f : A → B ⊥} {y} → let open M in"},{"Add":" (x>>=f⇓ : (x >>= f) ⇓[ k ]B y) →"},{"Add":" ∃ λ z → ∃₂ λ (x⇓ : x ⇓[ k ]A z) (fz⇓ : f z ⇓[ k ]B y) →"},{"Add":" steps x⇓ + steps fz⇓ ≡ steps x>>=f⇓"},{"Add":" >>=-inversion-⇓ refl (now x) fx⇓ ="},{"Add":" (x , now refl , fx⇓ , ≡.refl)"},{"Add":" >>=-inversion-⇓ refl (later x) (laterˡ x>>=f⇓) ="},{"Add":" Prod.map id (Prod.map laterˡ (Prod.map id (≡.cong suc))) $"},{"Add":" >>=-inversion-⇓ refl (♭ x) x>>=f⇓"},{"Add":""},{"Add":" >>=-inversion-⇑ : IsEquivalence _∼A_ →"},{"Add":" ∀ {k} x {f : A → B ⊥} → let open M in"},{"Add":" Rel _∼B_ (other k) (x >>= f) never →"},{"Add":" ¬ ¬ (x ⇑[ other k ]A ⊎"},{"Add":" ∃ λ y → x ⇓[ other k ]A y × f y ⇑[ other k ]B)"},{"Add":" >>=-inversion-⇑ eqA {k} x {f} ∼never ="},{"Add":" helper <$> now-or-never IsEqA.refl x"},{"Add":" where"},{"Add":" open RawMonad ¬¬-Monad using (_<$>_)"},{"Add":" open M using (_>>=_)"},{"Add":" open Reasoning eqA"},{"Add":" module IsEqA = IsEquivalence eqA"},{"Add":""},{"Add":" k≳ = other geq"},{"Add":""},{"Add":" is-never : ∀ {x y} →"},{"Add":" x ⇓[ k≳ ]A y → (x >>= f) ⇑[ k≳ ]B →"},{"Add":" ∃ λ z → (y ∼A z) × f z ⇑[ k≳ ]B"},{"Add":" is-never (now x∼y) = λ fx⇑ → (_ , IsEqA.sym x∼y , fx⇑)"},{"Add":" is-never (laterˡ ≳now) = is-never ≳now ∘ later⁻¹"},{"Add":""},{"Add":" helper : (∃ λ y → x ⇓[ k≳ ]A y) ⊎ x ⇑[ k≳ ]A →"},{"Add":" x ⇑[ other k ]A ⊎"},{"Add":" ∃ λ y → x ⇓[ other k ]A y × f y ⇑[ other k ]B"},{"Add":" helper (inj₂ ≳never) = inj₁ (≳⇒ ≳never)"},{"Add":" helper (inj₁ (y , ≳now)) with is-never ≳now (never⇒never ∼never)"},{"Add":" ... | (z , y∼z , fz⇑) = inj₂ (z , ≳⇒ (x ≳⟨ ≳now ⟩"},{"Add":" now y ≅⟨ now y∼z ⟩"},{"Add":" now z ∎)"},{"Add":" , ≳⇒ fz⇑)"},{"Add":""},{"Add":"module _ {A B : Set ℓ} {_∼_ : B → B → Set ℓ} where"},{"Add":""},{"Add":" open Equality"},{"Add":""},{"Add":" -- A variant of _>>=-cong_."},{"Add":""},{"Add":" infixl 1 _≡->>=-cong_"},{"Add":""},{"Add":" _≡->>=-cong_ :"},{"Add":" ∀ {k} {x₁ x₂ : A ⊥} {f₁ f₂ : A → B ⊥} → let open M in"},{"Add":" Rel _≡_ k x₁ x₂ →"},{"Add":" (∀ x → Rel _∼_ k (f₁ x) (f₂ x)) →"},{"Add":" Rel _∼_ k (x₁ >>= f₁) (x₂ >>= f₂)"},{"Add":" _≡->>=-cong_ {k} {f₁ = f₁} {f₂} x₁≈x₂ f₁≈f₂ ="},{"Add":" x₁≈x₂ >>=-cong λ {x} x≡x′ →"},{"Add":" ≡.subst (λ y → Rel _∼_ k (f₁ x) (f₂ y)) x≡x′ (f₁≈f₂ x)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Productivity checker workaround"},{"Add":""},{"Add":"-- The monad can be awkward to use, due to the limitations of guarded"},{"Add":"-- coinduction. The following code provides a (limited) workaround."},{"Add":""},{"Add":"module Workaround {a} where"},{"Add":""},{"Add":" infixl 1 _>>=_"},{"Add":""},{"Add":" data _⊥P : Set a → Set (Level.suc a) where"},{"Add":" now : (x : A) → A ⊥P"},{"Add":" later : (x : ∞ (A ⊥P)) → A ⊥P"},{"Add":" _>>=_ : (x : A ⊥P) (f : A → B ⊥P) → B ⊥P"},{"Add":""},{"Add":" private"},{"Add":""},{"Add":" data _⊥W : Set a → Set (Level.suc a) where"},{"Add":" now : (x : A) → A ⊥W"},{"Add":" later : (x : A ⊥P) → A ⊥W"},{"Add":""},{"Add":" mutual"},{"Add":""},{"Add":" _>>=W_ : A ⊥W → (A → B ⊥P) → B ⊥W"},{"Add":" now x >>=W f = whnf (f x)"},{"Add":" later x >>=W f = later (x >>= f)"},{"Add":""},{"Add":" whnf : A ⊥P → A ⊥W"},{"Add":" whnf (now x) = now x"},{"Add":" whnf (later x) = later (♭ x)"},{"Add":" whnf (x >>= f) = whnf x >>=W f"},{"Add":""},{"Add":" mutual"},{"Add":""},{"Add":" private"},{"Add":""},{"Add":" ⟦_⟧W : A ⊥W → A ⊥"},{"Add":" ⟦ now x ⟧W = now x"},{"Add":" ⟦ later x ⟧W = later (♯ ⟦ x ⟧P)"},{"Add":""},{"Add":" ⟦_⟧P : A ⊥P → A ⊥"},{"Add":" ⟦ x ⟧P = ⟦ whnf x ⟧W"},{"Add":""},{"Add":" -- The definitions above make sense. ⟦_⟧P is homomorphic with"},{"Add":" -- respect to now, later and _>>=_."},{"Add":""},{"Add":" module Correct where"},{"Add":""},{"Add":" private"},{"Add":" open module Eq {A : Set a} = Equality {A = A} _≡_"},{"Add":" open module R {A : Set a} = Reasoning (≡.isEquivalence {A = A})"},{"Add":""},{"Add":" now-hom : (x : A) → ⟦ now x ⟧P ≅ now x"},{"Add":" now-hom x = now x ∎"},{"Add":""},{"Add":" later-hom : (x : ∞ (A ⊥P)) → ⟦ later x ⟧P ≅ later (♯ ⟦ ♭ x ⟧P)"},{"Add":" later-hom x = later (♯ (⟦ ♭ x ⟧P ∎))"},{"Add":""},{"Add":" mutual"},{"Add":""},{"Add":" private"},{"Add":""},{"Add":" >>=-homW : (x : B ⊥W) (f : B → A ⊥P) →"},{"Add":" ⟦ x >>=W f ⟧W ≅ M._>>=_ ⟦ x ⟧W (λ y → ⟦ f y ⟧P)"},{"Add":" >>=-homW (now x) f = ⟦ f x ⟧P ∎"},{"Add":" >>=-homW (later x) f = later (♯ >>=-hom x f)"},{"Add":""},{"Add":" >>=-hom : (x : B ⊥P) (f : B → A ⊥P) →"},{"Add":" ⟦ x >>= f ⟧P ≅ M._>>=_ ⟦ x ⟧P (λ y → ⟦ f y ⟧P)"},{"Add":" >>=-hom x f = >>=-homW (whnf x) f"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- An alternative, but equivalent, formulation of equality/ordering"},{"Add":""},{"Add":"module AlternativeEquality {a ℓ} where"},{"Add":""},{"Add":" private"},{"Add":""},{"Add":" El : Setoid a ℓ → Set _"},{"Add":" El = Setoid.Carrier"},{"Add":""},{"Add":" Eq : ∀ S → B.Rel (El S) _"},{"Add":" Eq = Setoid._≈_"},{"Add":""},{"Add":" open Equality using (Rel)"},{"Add":" open Equality.Rel"},{"Add":""},{"Add":" infix 4 _∣_≅P_ _∣_≳P_ _∣_≈P_"},{"Add":" infix 3 _∎"},{"Add":" infixr 2 _≡⟨_⟩_ _≅⟨_⟩_ _≳⟨_⟩_ _≳⟨_⟩≅_ _≳⟨_⟩≈_ _≈⟨_⟩≅_ _≈⟨_⟩≲_"},{"Add":" infixl 1 _>>=_"},{"Add":""},{"Add":" mutual"},{"Add":""},{"Add":" -- Proof \"programs\"."},{"Add":""},{"Add":" _∣_≅P_ : ∀ S → B.Rel (El S ⊥) _"},{"Add":" _∣_≅P_ = flip RelP strong"},{"Add":""},{"Add":" _∣_≳P_ : ∀ S → B.Rel (El S ⊥) _"},{"Add":" _∣_≳P_ = flip RelP (other geq)"},{"Add":""},{"Add":" _∣_≈P_ : ∀ S → B.Rel (El S ⊥) _"},{"Add":" _∣_≈P_ = flip RelP (other weak)"},{"Add":""},{"Add":" data RelP S : Kind → B.Rel (El S ⊥) (Level.suc (a ⊔ ℓ)) where"},{"Add":""},{"Add":" -- Congruences."},{"Add":""},{"Add":" now : ∀ {k x y} (xRy : x ⟨ Eq S ⟩ y) → RelP S k (now x) (now y)"},{"Add":" later : ∀ {k x y} (x∼y : ∞ (RelP S k (♭ x) (♭ y))) →"},{"Add":" RelP S k (later x) (later y)"},{"Add":" _>>=_ : ∀ {S′ : Setoid a ℓ} {k} {x₁ x₂}"},{"Add":" {f₁ f₂ : El S′ → El S ⊥} →"},{"Add":" let open M in"},{"Add":" (x₁∼x₂ : RelP S′ k x₁ x₂)"},{"Add":" (f₁∼f₂ : ∀ {x y} → x ⟨ Eq S′ ⟩ y →"},{"Add":" RelP S k (f₁ x) (f₂ y)) →"},{"Add":" RelP S k (x₁ >>= f₁) (x₂ >>= f₂)"},{"Add":""},{"Add":" -- Ordering/weak equality."},{"Add":""},{"Add":" laterʳ : ∀ {x y} (x≈y : RelP S (other weak) x (♭ y)) → RelP S (other weak) x (later y)"},{"Add":" laterˡ : ∀ {k x y} (x∼y : RelP S (other k) (♭ x) y) → RelP S (other k) (later x) y"},{"Add":""},{"Add":" -- Equational reasoning. Note that including full transitivity"},{"Add":" -- for weak equality would make _∣_≈P_ trivial; a similar"},{"Add":" -- problem applies to _∣_≳P_ (A ∣ never ≳P now x would be"},{"Add":" -- provable). Instead the definition of RelP includes limited"},{"Add":" -- notions of transitivity, similar to weak bisimulation up-to"},{"Add":" -- various things."},{"Add":""},{"Add":" _∎ : ∀ {k} x → RelP S k x x"},{"Add":" sym : ∀ {k x y} {eq : Equality k} (x∼y : RelP S k x y) → RelP S k y x"},{"Add":" _≡⟨_⟩_ : ∀ {k} x {y z} (x≡y : x ≡ y) (y∼z : RelP S k y z) → RelP S k x z"},{"Add":" _≅⟨_⟩_ : ∀ {k} x {y z} (x≅y : S ∣ x ≅P y) (y∼z : RelP S k y z) → RelP S k x z"},{"Add":" _≳⟨_⟩_ : let open Equality (Eq S) in"},{"Add":" ∀ x {y z} (x≳y : x ≳ y) (y≳z : S ∣ y ≳P z) → S ∣ x ≳P z"},{"Add":" _≳⟨_⟩≅_ : ∀ x {y z} (x≳y : S ∣ x ≳P y) (y≅z : S ∣ y ≅P z) → S ∣ x ≳P z"},{"Add":" _≳⟨_⟩≈_ : ∀ x {y z} (x≳y : S ∣ x ≳P y) (y≈z : S ∣ y ≈P z) → S ∣ x ≈P z"},{"Add":" _≈⟨_⟩≅_ : ∀ x {y z} (x≈y : S ∣ x ≈P y) (y≅z : S ∣ y ≅P z) → S ∣ x ≈P z"},{"Add":" _≈⟨_⟩≲_ : ∀ x {y z} (x≈y : S ∣ x ≈P y) (y≲z : S ∣ z ≳P y) → S ∣ x ≈P z"},{"Add":""},{"Add":" -- If any of the following transitivity-like rules were added to"},{"Add":" -- RelP, then RelP and Rel would no longer be equivalent:"},{"Add":" --"},{"Add":" -- x ≳P y → y ≳P z → x ≳P z"},{"Add":" -- x ≳P y → y ≳ z → x ≳P z"},{"Add":" -- x ≲P y → y ≈P z → x ≈P z"},{"Add":" -- x ≈P y → y ≳P z → x ≈P z"},{"Add":" -- x ≲ y → y ≈P z → x ≈P z"},{"Add":" -- x ≈P y → y ≳ z → x ≈P z"},{"Add":" -- x ≈P y → y ≈P z → x ≈P z"},{"Add":" -- x ≈P y → y ≈ z → x ≈P z"},{"Add":" -- x ≈ y → y ≈P z → x ≈P z"},{"Add":" --"},{"Add":" -- The reason is that any of these rules would make it possible"},{"Add":" -- to derive that never and now x are related."},{"Add":""},{"Add":" -- RelP is complete with respect to Rel."},{"Add":""},{"Add":" complete : ∀ {S k} {x y : El S ⊥} →"},{"Add":" Equality.Rel (Eq S) k x y → RelP S k x y"},{"Add":" complete (now xRy) = now xRy"},{"Add":" complete (later x∼y) = later (♯ complete (♭ x∼y))"},{"Add":" complete (laterʳ x≈y) = laterʳ (complete x≈y)"},{"Add":" complete (laterˡ x∼y) = laterˡ (complete x∼y)"},{"Add":""},{"Add":" -- RelP is sound with respect to Rel."},{"Add":""},{"Add":" private"},{"Add":""},{"Add":" -- Proof WHNFs."},{"Add":""},{"Add":" data RelW S : Kind → B.Rel (El S ⊥) (Level.suc (a ⊔ ℓ)) where"},{"Add":" now : ∀ {k x y} (xRy : x ⟨ Eq S ⟩ y) → RelW S k (now x) (now y)"},{"Add":" later : ∀ {k x y} (x∼y : RelP S k (♭ x) (♭ y)) → RelW S k (later x) (later y)"},{"Add":" laterʳ : ∀ {x y} (x≈y : RelW S (other weak) x (♭ y)) → RelW S (other weak) x (later y)"},{"Add":" laterˡ : ∀ {k x y} (x∼y : RelW S (other k) (♭ x) y) → RelW S (other k) (later x) y"},{"Add":""},{"Add":" -- WHNFs can be turned into programs."},{"Add":""},{"Add":" program : ∀ {S k x y} → RelW S k x y → RelP S k x y"},{"Add":" program (now xRy) = now xRy"},{"Add":" program (later x∼y) = later (♯ x∼y)"},{"Add":" program (laterˡ x∼y) = laterˡ (program x∼y)"},{"Add":" program (laterʳ x≈y) = laterʳ (program x≈y)"},{"Add":""},{"Add":" -- Lemmas for WHNFs."},{"Add":""},{"Add":" _>>=W_ : ∀ {A B k x₁ x₂} {f₁ f₂ : El A → El B ⊥} →"},{"Add":" RelW A k x₁ x₂ →"},{"Add":" (∀ {x y} → x ⟨ Eq A ⟩ y → RelW B k (f₁ x) (f₂ y)) →"},{"Add":" RelW B k (M._>>=_ x₁ f₁) (M._>>=_ x₂ f₂)"},{"Add":" now xRy >>=W f₁∼f₂ = f₁∼f₂ xRy"},{"Add":" later x∼y >>=W f₁∼f₂ = later (x∼y >>= program ∘ f₁∼f₂)"},{"Add":" laterʳ x≈y >>=W f₁≈f₂ = laterʳ (x≈y >>=W f₁≈f₂)"},{"Add":" laterˡ x∼y >>=W f₁∼f₂ = laterˡ (x∼y >>=W f₁∼f₂)"},{"Add":""},{"Add":" reflW : ∀ {S k} x → RelW S k x x"},{"Add":" reflW {S} (now x) = now (Setoid.refl S)"},{"Add":" reflW (later x) = later (♭ x ∎)"},{"Add":""},{"Add":" symW : ∀ {S k x y} → Equality k → RelW S k x y → RelW S k y x"},{"Add":" symW {S} eq (now xRy) = now (Setoid.sym S xRy)"},{"Add":" symW eq (later x≈y) = later (sym {eq = eq} x≈y)"},{"Add":" symW eq (laterʳ x≈y) = laterˡ (symW eq x≈y)"},{"Add":" symW eq (laterˡ {weak} x≈y) = laterʳ (symW eq x≈y)"},{"Add":""},{"Add":" trans≅W : ∀ {S x y z} →"},{"Add":" RelW S strong x y → RelW S strong y z → RelW S strong x z"},{"Add":" trans≅W {S} (now xRy) (now yRz) = now (Setoid.trans S xRy yRz)"},{"Add":" trans≅W (later x≅y) (later y≅z) = later (_ ≅⟨ x≅y ⟩ y≅z)"},{"Add":""},{"Add":" trans≳-W : ∀ {S x y z} → let open Equality (Eq S) in"},{"Add":" x ≳ y → RelW S (other geq) y z → RelW S (other geq) x z"},{"Add":" trans≳-W {S} (now xRy) (now yRz) = now (Setoid.trans S xRy yRz)"},{"Add":" trans≳-W (later x≳y) (later y≳z) = later (_ ≳⟨ ♭ x≳y ⟩ y≳z)"},{"Add":" trans≳-W (later x≳y) (laterˡ y≳z) = laterˡ (trans≳-W (♭ x≳y) y≳z)"},{"Add":" trans≳-W (laterˡ x≳y) y≳z = laterˡ (trans≳-W x≳y y≳z)"},{"Add":""},{"Add":" -- Strong equality programs can be turned into WHNFs."},{"Add":""},{"Add":" whnf≅ : ∀ {S x y} → S ∣ x ≅P y → RelW S strong x y"},{"Add":" whnf≅ (now xRy) = now xRy"},{"Add":" whnf≅ (later x≅y) = later (♭ x≅y)"},{"Add":" whnf≅ (x₁≅x₂ >>= f₁≅f₂) = whnf≅ x₁≅x₂ >>=W λ xRy → whnf≅ (f₁≅f₂ xRy)"},{"Add":" whnf≅ (x ∎) = reflW x"},{"Add":" whnf≅ (sym x≅y) = symW _ (whnf≅ x≅y)"},{"Add":" whnf≅ (x ≡⟨ ≡.refl ⟩ y≅z) = whnf≅ y≅z"},{"Add":" whnf≅ (x ≅⟨ x≅y ⟩ y≅z) = trans≅W (whnf≅ x≅y) (whnf≅ y≅z)"},{"Add":""},{"Add":" -- More transitivity lemmas."},{"Add":""},{"Add":" _⟨_⟩≅_ : ∀ {S k} x {y z} →"},{"Add":" RelP S k x y → S ∣ y ≅P z → RelP S k x z"},{"Add":" _⟨_⟩≅_ {k = strong} x x≅y y≅z = x ≅⟨ x≅y ⟩ y≅z"},{"Add":" _⟨_⟩≅_ {k = other geq} x x≳y y≅z = x ≳⟨ x≳y ⟩≅ y≅z"},{"Add":" _⟨_⟩≅_ {k = other weak} x x≈y y≅z = x ≈⟨ x≈y ⟩≅ y≅z"},{"Add":""},{"Add":" trans∼≅W : ∀ {S k x y z} →"},{"Add":" RelW S k x y → RelW S strong y z → RelW S k x z"},{"Add":" trans∼≅W {S} (now xRy) (now yRz) = now (Setoid.trans S xRy yRz)"},{"Add":" trans∼≅W (later x∼y) (later y≅z) = later (_ ⟨ x∼y ⟩≅ y≅z)"},{"Add":" trans∼≅W (laterʳ x≈y) (later y≅z) = laterʳ (trans∼≅W x≈y (whnf≅ y≅z))"},{"Add":" trans∼≅W (laterˡ x∼y) y≅z = laterˡ (trans∼≅W x∼y y≅z)"},{"Add":""},{"Add":" trans≅∼W : ∀ {S k x y z} →"},{"Add":" RelW S strong x y → RelW S k y z → RelW S k x z"},{"Add":" trans≅∼W {S} (now xRy) (now yRz) = now (Setoid.trans S xRy yRz)"},{"Add":" trans≅∼W (later x≅y) (later y∼z) = later (_ ≅⟨ x≅y ⟩ y∼z)"},{"Add":" trans≅∼W (later x≅y) (laterˡ y∼z) = laterˡ (trans≅∼W (whnf≅ x≅y) y∼z)"},{"Add":" trans≅∼W x≅y (laterʳ ly≈z) = laterʳ (trans≅∼W x≅y ly≈z)"},{"Add":""},{"Add":" -- Order programs can be turned into WHNFs."},{"Add":""},{"Add":" whnf≳ : ∀ {S x y} → S ∣ x ≳P y → RelW S (other geq) x y"},{"Add":" whnf≳ (now xRy) = now xRy"},{"Add":" whnf≳ (later x∼y) = later (♭ x∼y)"},{"Add":" whnf≳ (laterˡ x≲y) = laterˡ (whnf≳ x≲y)"},{"Add":" whnf≳ (x₁∼x₂ >>= f₁∼f₂) = whnf≳ x₁∼x₂ >>=W λ xRy → whnf≳ (f₁∼f₂ xRy)"},{"Add":" whnf≳ (x ∎) = reflW x"},{"Add":" whnf≳ (x ≡⟨ ≡.refl ⟩ y≳z) = whnf≳ y≳z"},{"Add":" whnf≳ (x ≅⟨ x≅y ⟩ y≳z) = trans≅∼W (whnf≅ x≅y) (whnf≳ y≳z)"},{"Add":" whnf≳ (x ≳⟨ x≳y ⟩ y≳z) = trans≳-W x≳y (whnf≳ y≳z)"},{"Add":" whnf≳ (x ≳⟨ x≳y ⟩≅ y≅z) = trans∼≅W (whnf≳ x≳y) (whnf≅ y≅z)"},{"Add":""},{"Add":" -- Another transitivity lemma."},{"Add":""},{"Add":" trans≳≈W : ∀ {S x y z} →"},{"Add":" RelW S (other geq) x y → RelW S (other weak) y z →"},{"Add":" RelW S (other weak) x z"},{"Add":" trans≳≈W {S} (now xRy) (now yRz) = now (Setoid.trans S xRy yRz)"},{"Add":" trans≳≈W (later x≳y) (later y≈z) = later (_ ≳⟨ x≳y ⟩≈ y≈z)"},{"Add":" trans≳≈W (laterˡ x≳y) y≈z = laterˡ (trans≳≈W x≳y y≈z)"},{"Add":" trans≳≈W x≳y (laterʳ y≈z) = laterʳ (trans≳≈W x≳y y≈z)"},{"Add":" trans≳≈W (later x≳y) (laterˡ y≈z) = laterˡ (trans≳≈W (whnf≳ x≳y) y≈z)"},{"Add":""},{"Add":" -- All programs can be turned into WHNFs."},{"Add":""},{"Add":" whnf : ∀ {S k x y} → RelP S k x y → RelW S k x y"},{"Add":" whnf (now xRy) = now xRy"},{"Add":" whnf (later x∼y) = later (♭ x∼y)"},{"Add":" whnf (laterʳ x≈y) = laterʳ (whnf x≈y)"},{"Add":" whnf (laterˡ x∼y) = laterˡ (whnf x∼y)"},{"Add":" whnf (x₁∼x₂ >>= f₁∼f₂) = whnf x₁∼x₂ >>=W λ xRy → whnf (f₁∼f₂ xRy)"},{"Add":" whnf (x ∎) = reflW x"},{"Add":" whnf (sym {eq = eq} x≈y) = symW eq (whnf x≈y)"},{"Add":" whnf (x ≡⟨ ≡.refl ⟩ y∼z) = whnf y∼z"},{"Add":" whnf (x ≅⟨ x≅y ⟩ y∼z) = trans≅∼W (whnf x≅y) (whnf y∼z)"},{"Add":" whnf (x ≳⟨ x≳y ⟩ y≳z) = trans≳-W x≳y (whnf y≳z)"},{"Add":" whnf (x ≳⟨ x≳y ⟩≅ y≅z) = trans∼≅W (whnf x≳y) (whnf y≅z)"},{"Add":" whnf (x ≳⟨ x≳y ⟩≈ y≈z) = trans≳≈W (whnf x≳y) (whnf y≈z)"},{"Add":" whnf (x ≈⟨ x≈y ⟩≅ y≅z) = trans∼≅W (whnf x≈y) (whnf y≅z)"},{"Add":" whnf (x ≈⟨ x≈y ⟩≲ y≲z) = symW _ (trans≳≈W (whnf y≲z) (symW _ (whnf x≈y)))"},{"Add":""},{"Add":" mutual"},{"Add":""},{"Add":" -- Soundness."},{"Add":""},{"Add":" private"},{"Add":""},{"Add":" soundW : ∀ {S k x y} → RelW S k x y → Rel (Eq S) k x y"},{"Add":" soundW (now xRy) = now xRy"},{"Add":" soundW (later x∼y) = later (♯ sound x∼y)"},{"Add":" soundW (laterʳ x≈y) = laterʳ (soundW x≈y)"},{"Add":" soundW (laterˡ x∼y) = laterˡ (soundW x∼y)"},{"Add":""},{"Add":" sound : ∀ {S k x y} → RelP S k x y → Rel (Eq S) k x y"},{"Add":" sound x∼y = soundW (whnf x∼y)"},{"Add":""},{"Add":" -- RelP and Rel are equivalent (when the underlying relation is an"},{"Add":" -- equivalence)."},{"Add":""},{"Add":" correct : ∀ {S k x y} → RelP S k x y ⇔ Rel (Eq S) k x y"},{"Add":" correct = mk⇔ sound complete"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Another lemma"},{"Add":""},{"Add":"-- Bind is \"idempotent\"."},{"Add":""},{"Add":"idempotent :"},{"Add":" (B : Setoid ℓ ℓ) →"},{"Add":" let open M; open Setoid B using (_≈_; Carrier); open Equality _≈_ in"},{"Add":" (x : A ⊥) (f : A → A → Carrier ⊥) →"},{"Add":" (x >>= λ y′ → x >>= λ y″ → f y′ y″) ≳ (x >>= λ y′ → f y′ y′)"},{"Add":"idempotent {A = A} B x f = sound (idem x)"},{"Add":" where"},{"Add":" open AlternativeEquality hiding (_>>=_)"},{"Add":" open M"},{"Add":" open Equality.Rel using (laterˡ)"},{"Add":" open Equivalence using (refl)"},{"Add":""},{"Add":" idem : (x : A ⊥) →"},{"Add":" B ∣ (x >>= λ y′ → x >>= λ y″ → f y′ y″) ≳P"},{"Add":" (x >>= λ y′ → f y′ y′)"},{"Add":" idem (now x) = f x x ∎"},{"Add":" idem (later x) = later (♯ ("},{"Add":" (♭ x >>= λ y′ → later x >>= λ y″ → f y′ y″) ≳⟨ (refl ≡.refl {x = ♭ x} ≡->>=-cong λ _ →"},{"Add":" laterˡ (refl (Setoid.refl B))) ⟩"},{"Add":" (♭ x >>= λ y′ → ♭ x >>= λ y″ → f y′ y″) ≳⟨ idem (♭ x) ⟩≅"},{"Add":" (♭ x >>= λ y′ → f y′ y′) ∎))"}]}},{"FileAdd":{"meta":{"basename":"Partiality","metadata":576},"context":[{"change":0,"pos":1277307}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Instances.agda","metadata":0},"context":[{"change":0,"pos":1357991}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Typeclass instances for _⊥"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe --guardedness #-}"},{"Add":""},{"Add":"module Effect.Monad.Partiality.Instances where"},{"Add":""},{"Add":"open import Effect.Monad.Partiality"},{"Add":""},{"Add":"instance"},{"Add":" partialityMonad = monad"}]}},{"FileAdd":{"meta":{"basename":"All.agda","metadata":0},"context":[{"change":0,"pos":1357991}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- An All predicate for the partiality monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe --guardedness #-}"},{"Add":""},{"Add":"module Effect.Monad.Partiality.All where"},{"Add":""},{"Add":"open import Effect.Monad"},{"Add":"open import Effect.Monad.Partiality as Partiality using (_⊥; ⇒≈)"},{"Add":"open import Codata.Musical.Notation"},{"Add":"open import Function.Base using (flip; _∘_)"},{"Add":"open import Level"},{"Add":"open import Relation.Binary.Definitions using (_Respects_)"},{"Add":"open import Relation.Binary.Structures using (IsEquivalence)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)"},{"Add":""},{"Add":"open Partiality._⊥"},{"Add":"open Partiality.Equality using (Rel)"},{"Add":"open Partiality.Equality.Rel"},{"Add":"private"},{"Add":" open module E {a} {A : Set a} = Partiality.Equality (_≡_ {A = A})"},{"Add":" using (_≅_; _≳_)"},{"Add":" open module M {f} = RawMonad (Partiality.monad {f = f})"},{"Add":" using (_>>=_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b p ℓ : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- All, along with some lemmas"},{"Add":""},{"Add":"-- All P x means that if x terminates with the value v, then P v"},{"Add":"-- holds."},{"Add":""},{"Add":"data All {A : Set a} (P : A → Set p) : A ⊥ → Set (a ⊔ p) where"},{"Add":" now : ∀ {v} (p : P v) → All P (now v)"},{"Add":" later : ∀ {x} (p : ∞ (All P (♭ x))) → All P (later x)"},{"Add":""},{"Add":"-- Bind preserves All in the following way:"},{"Add":""},{"Add":"infixl 1 _>>=-cong_"},{"Add":""},{"Add":"_>>=-cong_ : ∀ {p q} {P : A → Set p} {Q : B → Set q}"},{"Add":" {x : A ⊥} {f : A → B ⊥} →"},{"Add":" All P x → (∀ {x} → P x → All Q (f x)) →"},{"Add":" All Q (x >>= f)"},{"Add":"now p >>=-cong f = f p"},{"Add":"later p >>=-cong f = later (♯ (♭ p >>=-cong f))"},{"Add":""},{"Add":"-- All respects all the relations, given that the predicate respects"},{"Add":"-- the underlying relation."},{"Add":""},{"Add":"respects :"},{"Add":" ∀ {k} {P : A → Set p} {_∼_ : A → A → Set ℓ} →"},{"Add":" P Respects _∼_ → All P Respects Rel _∼_ k"},{"Add":"respects resp (now x∼y) (now p) = now (resp x∼y p)"},{"Add":"respects resp (later x∼y) (later p) = later (♯ respects resp (♭ x∼y) (♭ p))"},{"Add":"respects resp (laterˡ x∼y) (later p) = respects resp x∼y (♭ p)"},{"Add":"respects resp (laterʳ x≈y) p = later (♯ respects resp x≈y p)"},{"Add":""},{"Add":"respects-flip :"},{"Add":" ∀ {k} {P : A → Set p} {_∼_ : A → A → Set ℓ} →"},{"Add":" P Respects flip _∼_ → All P Respects flip (Rel _∼_ k)"},{"Add":"respects-flip resp (now x∼y) (now p) = now (resp x∼y p)"},{"Add":"respects-flip resp (later x∼y) (later p) = later (♯ respects-flip resp (♭ x∼y) (♭ p))"},{"Add":"respects-flip resp (laterˡ x∼y) p = later (♯ respects-flip resp x∼y p)"},{"Add":"respects-flip resp (laterʳ x≈y) (later p) = respects-flip resp x≈y (♭ p)"},{"Add":""},{"Add":"-- \"Equational\" reasoning."},{"Add":""},{"Add":"module Reasoning {P : A → Set p}"},{"Add":" {_∼_ : A → A → Set ℓ}"},{"Add":" (resp : P Respects flip _∼_) where"},{"Add":""},{"Add":" infix 3 finally"},{"Add":" infixr 2 _≡⟨_⟩_ _∼⟨_⟩_"},{"Add":""},{"Add":" _≡⟨_⟩_ : ∀ x {y} → x ≡ y → All P y → All P x"},{"Add":" _ ≡⟨ ≡.refl ⟩ p = p"},{"Add":""},{"Add":" _∼⟨_⟩_ : ∀ {k} x {y} → Rel _∼_ k x y → All P y → All P x"},{"Add":" _ ∼⟨ x∼y ⟩ p = respects-flip resp (⇒≈ x∼y) p"},{"Add":""},{"Add":" -- A cosmetic combinator."},{"Add":""},{"Add":" finally : (x : A ⊥) → All P x → All P x"},{"Add":" finally _ p = p"},{"Add":""},{"Add":" syntax finally x p = x ⟨ p ⟩"},{"Add":""},{"Add":"-- \"Equational\" reasoning with _∼_ instantiated to propositional"},{"Add":"-- equality."},{"Add":""},{"Add":"module Reasoning-≡ {a p} {A : Set a} {P : A → Set p}"},{"Add":" = Reasoning {P = P} {_∼_ = _≡_} (≡.subst P ∘ ≡.sym)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- An alternative, but equivalent, formulation of All"},{"Add":""},{"Add":"module Alternative {a p : Level} where"},{"Add":""},{"Add":" infix 3 _⟨_⟩P"},{"Add":" infixr 2 _≅⟨_⟩P_ _≳⟨_⟩P_"},{"Add":""},{"Add":" -- All \"programs\"."},{"Add":""},{"Add":" data AllP {A : Set a} (P : A → Set p) : A ⊥ → Set (suc (a ⊔ p)) where"},{"Add":" now : ∀ {x} (p : P x) → AllP P (now x)"},{"Add":" later : ∀ {x} (p : ∞ (AllP P (♭ x))) → AllP P (later x)"},{"Add":" _>>=-congP_ : ∀ {B : Set a} {Q : B → Set p} {x f}"},{"Add":" (p-x : AllP Q x) (p-f : ∀ {v} → Q v → AllP P (f v)) →"},{"Add":" AllP P (x >>= f)"},{"Add":" _≅⟨_⟩P_ : ∀ x {y} (x≅y : x ≅ y) (p : AllP P y) → AllP P x"},{"Add":" _≳⟨_⟩P_ : ∀ x {y} (x≳y : x ≳ y) (p : AllP P y) → AllP P x"},{"Add":" _⟨_⟩P : ∀ x (p : AllP P x) → AllP P x"},{"Add":""},{"Add":" infixl 1 _>>=-congP_"},{"Add":""},{"Add":" private"},{"Add":""},{"Add":" -- WHNFs."},{"Add":""},{"Add":" data AllW {A} (P : A → Set p) : A ⊥ → Set (suc (a ⊔ p)) where"},{"Add":" now : ∀ {x} (p : P x) → AllW P (now x)"},{"Add":" later : ∀ {x} (p : AllP P (♭ x)) → AllW P (later x)"},{"Add":""},{"Add":" -- A function which turns WHNFs into programs."},{"Add":""},{"Add":" program : ∀ {P : A → Set p} {x} → AllW P x → AllP P x"},{"Add":" program (now p) = now p"},{"Add":" program (later p) = later (♯ p)"},{"Add":""},{"Add":" -- Functions which turn programs into WHNFs."},{"Add":""},{"Add":" trans-≅ : {P : A → Set p} {x y : A ⊥} →"},{"Add":" x ≅ y → AllW P y → AllW P x"},{"Add":" trans-≅ (now ≡.refl) (now p) = now p"},{"Add":" trans-≅ (later x≅y) (later p) = later (_ ≅⟨ ♭ x≅y ⟩P p)"},{"Add":""},{"Add":" trans-≳ : {P : A → Set p} {x y : A ⊥} →"},{"Add":" x ≳ y → AllW P y → AllW P x"},{"Add":" trans-≳ (now ≡.refl) (now p) = now p"},{"Add":" trans-≳ (later x≳y) (later p) = later (_ ≳⟨ ♭ x≳y ⟩P p)"},{"Add":" trans-≳ (laterˡ x≳y) p = later (_ ≳⟨ x≳y ⟩P program p)"},{"Add":""},{"Add":" mutual"},{"Add":""},{"Add":" _>>=-congW_ : ∀ {P : A → Set p} {Q : B → Set p} {x f} →"},{"Add":" AllW P x → (∀ {v} → P v → AllP Q (f v)) →"},{"Add":" AllW Q (x >>= f)"},{"Add":" now p >>=-congW p-f = whnf (p-f p)"},{"Add":" later p >>=-congW p-f = later (p >>=-congP p-f)"},{"Add":""},{"Add":" whnf : ∀ {P : A → Set p} {x} → AllP P x → AllW P x"},{"Add":" whnf (now p) = now p"},{"Add":" whnf (later p) = later (♭ p)"},{"Add":" whnf (p-x >>=-congP p-f) = whnf p-x >>=-congW p-f"},{"Add":" whnf (_ ≅⟨ x≅y ⟩P p) = trans-≅ x≅y (whnf p)"},{"Add":" whnf (_ ≳⟨ x≳y ⟩P p) = trans-≳ x≳y (whnf p)"},{"Add":" whnf (_ ⟨ p ⟩P) = whnf p"},{"Add":""},{"Add":" -- AllP P is sound and complete with respect to All P."},{"Add":""},{"Add":" sound : ∀ {P : A → Set p} {x} → AllP P x → All P x"},{"Add":" sound = λ p → soundW (whnf p)"},{"Add":" where"},{"Add":" soundW : ∀ {A} {P : A → Set p} {x} → AllW P x → All P x"},{"Add":" soundW (now p) = now p"},{"Add":" soundW (later p) = later (♯ sound p)"},{"Add":""},{"Add":" complete : ∀ {P : A → Set p} {x} → All P x → AllP P x"},{"Add":" complete (now p) = now p"},{"Add":" complete (later p) = later (♯ complete (♭ p))"}]}},{"FileAdd":{"meta":{"basename":"Indexed.agda","metadata":0},"context":[{"change":0,"pos":1277307}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Indexed monads"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Note that currently the monad laws are not included here."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.Indexed where"},{"Add":""},{"Add":"open import Effect.Applicative.Indexed"},{"Add":"open import Function.Base"},{"Add":"open import Level"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c i f : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":" C : Set c"},{"Add":" I : Set i"},{"Add":""},{"Add":"record RawIMonad {I : Set i} (M : IFun I f) : Set (i ⊔ suc f) where"},{"Add":" infixl 1 _>>=_ _>>_ _>=>_"},{"Add":" infixr 1 _=<<_ _<=<_"},{"Add":""},{"Add":" field"},{"Add":" return : ∀ {i} → A → M i i A"},{"Add":" _>>=_ : ∀ {i j k} → M i j A → (A → M j k B) → M i k B"},{"Add":""},{"Add":" _>>_ : ∀ {i j k} → M i j A → M j k B → M i k B"},{"Add":" m₁ >> m₂ = m₁ >>= λ _ → m₂"},{"Add":""},{"Add":" _=<<_ : ∀ {i j k} → (A → M j k B) → M i j A → M i k B"},{"Add":" f =<< c = c >>= f"},{"Add":""},{"Add":" _>=>_ : ∀ {i j k} → (A → M i j B) → (B → M j k C) → (A → M i k C)"},{"Add":" f >=> g = _=<<_ g ∘ f"},{"Add":""},{"Add":" _<=<_ : ∀ {i j k} → (B → M j k C) → (A → M i j B) → (A → M i k C)"},{"Add":" g <=< f = f >=> g"},{"Add":""},{"Add":" join : ∀ {i j k} → M i j (M j k A) → M i k A"},{"Add":" join m = m >>= id"},{"Add":""},{"Add":" rawIApplicative : RawIApplicative M"},{"Add":" rawIApplicative = record"},{"Add":" { pure = return"},{"Add":" ; _⊛_ = λ f x → f >>= λ f′ → x >>= λ x′ → return (f′ x′)"},{"Add":" }"},{"Add":""},{"Add":" open RawIApplicative rawIApplicative public"},{"Add":""},{"Add":"RawIMonadT : {I : Set i} (T : IFun I f → IFun I f) → Set (i ⊔ suc f)"},{"Add":"RawIMonadT T = ∀ {M} → RawIMonad M → RawIMonad (T M)"},{"Add":""},{"Add":"record RawIMonadZero {I : Set i} (M : IFun I f) : Set (i ⊔ suc f) where"},{"Add":" field"},{"Add":" monad : RawIMonad M"},{"Add":" applicativeZero : RawIApplicativeZero M"},{"Add":""},{"Add":" open RawIMonad monad public"},{"Add":" open RawIApplicativeZero applicativeZero using (∅) public"},{"Add":""},{"Add":"record RawIMonadPlus {I : Set i} (M : IFun I f) : Set (i ⊔ suc f) where"},{"Add":" field"},{"Add":" monad : RawIMonad M"},{"Add":" alternative : RawIAlternative M"},{"Add":""},{"Add":" open RawIMonad monad public"},{"Add":" open RawIAlternative alternative using (∅; _∣_) public"},{"Add":""},{"Add":" monadZero : RawIMonadZero M"},{"Add":" monadZero = record"},{"Add":" { monad = monad"},{"Add":" ; applicativeZero = RawIAlternative.applicativeZero alternative"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Identity.agda","metadata":0},"context":[{"change":0,"pos":1277307}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- An effectful view of the identity function"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.Identity where"},{"Add":""},{"Add":"open import Effect.Functor using (RawFunctor)"},{"Add":"open import Effect.Applicative using (RawApplicative)"},{"Add":"open import Effect.Monad using (RawMonad; module Join)"},{"Add":"open import Effect.Comonad using (RawComonad)"},{"Add":"open import Function.Base using (id; _∘′_; _|>′_; _$′_; flip)"},{"Add":"open import Level using (Level)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"record Identity (A : Set a) : Set a where"},{"Add":" constructor mkIdentity"},{"Add":" field runIdentity : A"},{"Add":"open Identity public"},{"Add":""},{"Add":"functor : RawFunctor {a} Identity"},{"Add":"functor = record"},{"Add":" { _<$>_ = λ f a → mkIdentity (f (runIdentity a))"},{"Add":" }"},{"Add":""},{"Add":"applicative : RawApplicative {a} Identity"},{"Add":"applicative = record"},{"Add":" { rawFunctor = functor"},{"Add":" ; pure = mkIdentity"},{"Add":" ; _<*>_ = λ f a → mkIdentity (runIdentity f $′ runIdentity a)"},{"Add":" }"},{"Add":""},{"Add":"monad : RawMonad {a} Identity"},{"Add":"monad = record"},{"Add":" { rawApplicative = applicative"},{"Add":" ; _>>=_ = _|>′_ ∘′ runIdentity"},{"Add":" }"},{"Add":""},{"Add":"comonad : RawComonad {a} Identity"},{"Add":"comonad = record"},{"Add":" { extract = runIdentity"},{"Add":" ; extend = λ f a → mkIdentity (f a)"},{"Add":" }"},{"Add":""},{"Add":"join : Identity (Identity A) → Identity A"},{"Add":"join = Join.join monad"}]}},{"FileAdd":{"meta":{"basename":"Identity","metadata":576},"context":[{"change":0,"pos":1277307}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Instances.agda","metadata":0},"context":[{"change":0,"pos":1368754}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Typeclass instances for Identity"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.Identity.Instances where"},{"Add":""},{"Add":"open import Effect.Monad.Identity"},{"Add":""},{"Add":"instance"},{"Add":" identityFunctor = functor"},{"Add":" identityApplicative = applicative"},{"Add":" identityMonad = monad"},{"Add":" identityComonad = comonad"}]}},{"FileAdd":{"meta":{"basename":"IO.agda","metadata":0},"context":[{"change":0,"pos":1277307}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The IO monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module Effect.Monad.IO where"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Effect.Functor using (RawFunctor)"},{"Add":"open import Function.Base using (id)"},{"Add":"open import IO.Base using (IO)"},{"Add":"open import Level using (Level; suc)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" f g : Level"},{"Add":" A : Set f"},{"Add":" M : Set f → Set g"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- IO monad operations"},{"Add":""},{"Add":"record RawMonadIO"},{"Add":" (M : Set f → Set (suc f))"},{"Add":" : Set (suc f) where"},{"Add":" field"},{"Add":" liftIO : IO A → M A"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- IO monad specifics"},{"Add":""},{"Add":"monadIO : RawMonadIO {f} IO"},{"Add":"monadIO = record { liftIO = id }"},{"Add":""},{"Add":"open import Effect.Monad.State.Transformer.Base using (StateT; mkStateT)"},{"Add":""},{"Add":"liftStateT : ∀ {S} → RawFunctor M → RawMonadIO M → RawMonadIO (StateT S M)"},{"Add":"liftStateT M MIO = record"},{"Add":" { liftIO = λ io → mkStateT (λ s → (s ,_) <$> liftIO io)"},{"Add":" } where open RawFunctor M; open RawMonadIO MIO"},{"Add":""},{"Add":"open import Effect.Monad.Reader.Transformer.Base using (ReaderT; mkReaderT)"},{"Add":""},{"Add":"liftReaderT : ∀ {R} → RawMonadIO M → RawMonadIO (ReaderT R M)"},{"Add":"liftReaderT MIO = record"},{"Add":" { liftIO = λ io → mkReaderT (λ r → liftIO io)"},{"Add":" } where open RawMonadIO MIO"},{"Add":""},{"Add":"open import Effect.Monad.Writer.Transformer.Base using (WriterT; mkWriterT)"},{"Add":""},{"Add":"liftWriterT : ∀ {f 𝕎} → RawFunctor M → RawMonadIO M → RawMonadIO (WriterT {f = f} 𝕎 M)"},{"Add":"liftWriterT M MIO = record"},{"Add":" { liftIO = λ io → mkWriterT (λ w → (w ,_) <$> liftIO io)"},{"Add":" } where open RawFunctor M; open RawMonadIO MIO"}]}},{"FileAdd":{"meta":{"basename":"IO","metadata":576},"context":[{"change":0,"pos":1277307}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Instances.agda","metadata":0},"context":[{"change":0,"pos":1371105}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Typeclass instances for the IO monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --guardedness #-}"},{"Add":""},{"Add":"module Effect.Monad.IO.Instances where"},{"Add":""},{"Add":"open import Effect.Monad.IO"},{"Add":""},{"Add":"instance"},{"Add":" ioMonadIO = monadIO"},{"Add":" stateTMonadIO = λ {s} {S} {M} {{m}} {{mio}} → liftStateT {s} {S} {M} m mio"},{"Add":" readerTMonadIO = λ {r} {R} {M} {{mio}} → liftReaderT {r} {R} {M} mio"},{"Add":" writerTMonadIO = λ {f} {w} {W} {M} {{m}} {{mio}} → liftWriterT {f} {w} {W} {M} m mio"}]}},{"FileAdd":{"meta":{"basename":"Error","metadata":576},"context":[{"change":0,"pos":1277307}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Transformer.agda","metadata":0},"context":[{"change":0,"pos":1371779}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- The error monad transformer"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Level using (Level; _⊔_; suc)"},{"Add":""},{"Add":"module Effect.Monad.Error.Transformer {e} (E : Set e) (a : Level) where"},{"Add":""},{"Add":"open import Effect.Monad using (RawMonad)"},{"Add":"open import Function.Base using (_∘′_; _$_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" f ℓ : Level"},{"Add":" A B : Set ℓ"},{"Add":" M : Set f → Set ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Error monad operations"},{"Add":""},{"Add":"record RawMonadError"},{"Add":" (M : Set (e ⊔ a) → Set ℓ)"},{"Add":" : Set (suc (e ⊔ a) ⊔ ℓ) where"},{"Add":" field"},{"Add":" throw : E → M A"},{"Add":" catch : M A → (E → M A) → M A"},{"Add":""},{"Add":" during : (E → E) → M A → M A"},{"Add":" during f ma = catch ma (throw ∘′ f)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Monad error transformer specifics"},{"Add":""},{"Add":"module Sumₗ where"},{"Add":""},{"Add":" open import Data.Sum.Base using (inj₁; inj₂; [_,_]′)"},{"Add":" open import Data.Sum.Effectful.Left.Transformer E a"},{"Add":""},{"Add":" monadError : RawMonad M → RawMonadError (SumₗT M)"},{"Add":" monadError M = record"},{"Add":" { throw = mkSumₗT ∘′ pure ∘′ inj₁"},{"Add":" ; catch = λ ma k → mkSumₗT $ do"},{"Add":" a ← runSumₗT ma"},{"Add":" [ runSumₗT ∘′ k , pure ∘′ inj₂ ]′ a"},{"Add":" } where open RawMonad M"},{"Add":""},{"Add":"module Sumᵣ where"},{"Add":""},{"Add":" open import Data.Sum.Base using (inj₁; inj₂; [_,_]′)"},{"Add":" open import Data.Sum.Effectful.Right.Transformer a E"},{"Add":""},{"Add":" monadError : RawMonad M → RawMonadError (SumᵣT M)"},{"Add":" monadError M = record"},{"Add":" { throw = mkSumᵣT ∘′ pure ∘′ inj₂"},{"Add":" ; catch = λ ma k → mkSumᵣT $ do"},{"Add":" a ← runSumᵣT ma"},{"Add":" [ pure ∘′ inj₁ , runSumᵣT ∘′ k ]′ a"},{"Add":" } where open RawMonad M"}]}},{"FileAdd":{"meta":{"basename":"Continuation.agda","metadata":0},"context":[{"change":0,"pos":1277307}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- A delimited continuation monad"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Monad.Continuation where"},{"Add":""},{"Add":"open import Effect.Applicative.Indexed using (IFun)"},{"Add":"open import Effect.Monad using (RawMonad)"},{"Add":"open import Function.Identity.Effectful as Id using (Identity)"},{"Add":"open import Effect.Monad.Indexed using (RawIMonad)"},{"Add":"open import Function.Base using (flip)"},{"Add":"open import Level using (Level; _⊔_; suc)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" i f : Level"},{"Add":" I : Set i"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Delimited continuation monads"},{"Add":""},{"Add":"DContT : (I → Set f) → (Set f → Set f) → IFun I f"},{"Add":"DContT K M r₂ r₁ a = (a → M (K r₁)) → M (K r₂)"},{"Add":""},{"Add":"DCont : (I → Set f) → IFun I f"},{"Add":"DCont K = DContT K Identity"},{"Add":""},{"Add":"DContTIMonad : ∀ (K : I → Set f) {M} → RawMonad M → RawIMonad (DContT K M)"},{"Add":"DContTIMonad K Mon = record"},{"Add":" { return = λ a k → k a"},{"Add":" ; _>>=_ = λ c f k → c (flip f k)"},{"Add":" }"},{"Add":" where open RawMonad Mon"},{"Add":""},{"Add":"DContIMonad : (K : I → Set f) → RawIMonad (DCont K)"},{"Add":"DContIMonad K = DContTIMonad K Id.monad"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Delimited continuation operations"},{"Add":""},{"Add":"record RawIMonadDCont {I : Set i} (K : I → Set f)"},{"Add":" (M : IFun I f) : Set (i ⊔ suc f) where"},{"Add":" field"},{"Add":" monad : RawIMonad M"},{"Add":" reset : ∀ {r₁ r₂ r₃} → M r₁ r₂ (K r₂) → M r₃ r₃ (K r₁)"},{"Add":" shift : ∀ {a r₁ r₂ r₃ r₄} →"},{"Add":" ((a → M r₁ r₁ (K r₂)) → M r₃ r₄ (K r₄)) → M r₃ r₂ a"},{"Add":""},{"Add":" open RawIMonad monad public"},{"Add":""},{"Add":"DContTIMonadDCont : ∀ (K : I → Set f) {M} →"},{"Add":" RawMonad M → RawIMonadDCont K (DContT K M)"},{"Add":"DContTIMonadDCont K Mon = record"},{"Add":" { monad = DContTIMonad K Mon"},{"Add":" ; reset = λ e k → e pure >>= k"},{"Add":" ; shift = λ e k → e (λ a k′ → (k a) >>= k′) pure"},{"Add":" }"},{"Add":" where"},{"Add":" open RawMonad Mon"},{"Add":""},{"Add":"DContIMonadDCont : (K : I → Set f) → RawIMonadDCont K (DCont K)"},{"Add":"DContIMonadDCont K = DContTIMonadDCont K Id.monad"}]}},{"FileAdd":{"meta":{"basename":"Functor.agda","metadata":0},"context":[{"change":0,"pos":1273735}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Functors"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Note that currently the functor laws are not included here."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Functor where"},{"Add":""},{"Add":"open import Data.Unit.Polymorphic.Base using (⊤)"},{"Add":"open import Function.Base using (const; flip)"},{"Add":"open import Level"},{"Add":""},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ ℓ′ ℓ″ : Level"},{"Add":" A B X Y : Set ℓ"},{"Add":""},{"Add":"record RawFunctor (F : Set ℓ → Set ℓ′) : Set (suc ℓ ⊔ ℓ′) where"},{"Add":" infixl 4 _<$>_ _<$_"},{"Add":" infixl 1 _<&>_"},{"Add":" field"},{"Add":" _<$>_ : (A → B) → F A → F B"},{"Add":""},{"Add":" _<$_ : A → F B → F A"},{"Add":" x <$ y = const x <$> y"},{"Add":""},{"Add":" _<&>_ : F A → (A → B) → F B"},{"Add":" _<&>_ = flip _<$>_"},{"Add":""},{"Add":" ignore : F A → F ⊤"},{"Add":" ignore = _ <$_"},{"Add":""},{"Add":"-- A functor morphism from F₁ to F₂ is an operation op such that"},{"Add":"-- op (F₁ f x) ≡ F₂ f (op x)"},{"Add":""},{"Add":"record Morphism {F₁ : Set ℓ → Set ℓ′} {F₂ : Set ℓ → Set ℓ″}"},{"Add":" (fun₁ : RawFunctor F₁)"},{"Add":" (fun₂ : RawFunctor F₂) : Set (suc ℓ ⊔ ℓ′ ⊔ ℓ″) where"},{"Add":" open RawFunctor"},{"Add":" field"},{"Add":" op : F₁ X → F₂ X"},{"Add":" op-<$> : (f : X → Y) (x : F₁ X) →"},{"Add":" op (fun₁ ._<$>_ f x) ≡ fun₂ ._<$>_ f (op x)"}]}},{"FileAdd":{"meta":{"basename":"Functor","metadata":576},"context":[{"change":0,"pos":1273735}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Predicate.agda","metadata":0},"context":[{"change":0,"pos":1377403}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Functors on indexed sets (predicates)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Note that currently the functor laws are not included here."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Functor.Predicate where"},{"Add":""},{"Add":"open import Function.Base using (const)"},{"Add":"open import Level"},{"Add":"open import Relation.Unary"},{"Add":"open import Relation.Unary.PredicateTransformer using (PT)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" i j ℓ₁ ℓ₂ : Level"},{"Add":""},{"Add":"record RawPFunctor {I : Set i} {J : Set j}"},{"Add":" (F : PT I J ℓ₁ ℓ₂) : Set (i ⊔ j ⊔ suc ℓ₁ ⊔ suc ℓ₂)"},{"Add":" where"},{"Add":" infixl 4 _<$>_ _<$_"},{"Add":""},{"Add":" field"},{"Add":" _<$>_ : ∀ {P Q} → P ⊆ Q → F P ⊆ F Q"},{"Add":""},{"Add":" _<$_ : ∀ {P Q} → (∀ {i} → P i) → F Q ⊆ F P"},{"Add":" x <$ y = const x <$> y"}]}},{"FileAdd":{"meta":{"basename":"Empty.agda","metadata":0},"context":[{"change":0,"pos":1273735}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Empty values (e.g. [] for List, nothing for Maybe)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Empty where"},{"Add":""},{"Add":"open import Level"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ ℓ′ : Level"},{"Add":" A : Set ℓ"},{"Add":""},{"Add":"record RawEmpty (F : Set ℓ → Set ℓ′) : Set (suc ℓ ⊔ ℓ′) where"},{"Add":" field"},{"Add":" empty : F A"},{"Add":""},{"Add":" -- backwards compatibility: unicode variants"},{"Add":" ∅ : F A"},{"Add":" ∅ = empty"}]}},{"FileAdd":{"meta":{"basename":"Comonad.agda","metadata":0},"context":[{"change":0,"pos":1273735}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Comonads"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Note that currently the monad laws are not included here."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Comonad where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Function.Base using (id; _∘′_; flip)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c f : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":" C : Set c"},{"Add":""},{"Add":"record RawComonad (W : Set f → Set f) : Set (suc f) where"},{"Add":""},{"Add":" infixl 1 _=>>_ _=>=_"},{"Add":" infixr 1 _<<=_ _=<=_"},{"Add":""},{"Add":" field"},{"Add":" extract : W A → A"},{"Add":" extend : (W A → B) → (W A → W B)"},{"Add":""},{"Add":" duplicate : W A → W (W A)"},{"Add":" duplicate = extend id"},{"Add":""},{"Add":" liftW : (A → B) → W A → W B"},{"Add":" liftW f = extend (f ∘′ extract)"},{"Add":""},{"Add":" _=>>_ : W A → (W A → B) → W B"},{"Add":" _=>>_ = flip extend"},{"Add":""},{"Add":" _=>=_ : (W A → B) → (W B → C) → W A → C"},{"Add":" f =>= g = g ∘′ extend f"},{"Add":""},{"Add":" _<<=_ : (W A → B) → W A → W B"},{"Add":" _<<=_ = extend"},{"Add":""},{"Add":" _=<=_ : (W B → C) → (W A → B) → W A → C"},{"Add":" _=<=_ = flip _=>=_"}]}},{"FileAdd":{"meta":{"basename":"Choice.agda","metadata":0},"context":[{"change":0,"pos":1273735}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Type constructors giving rise to a semigroup at every type"},{"Add":"-- e.g. (List, _++_)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Choice where"},{"Add":""},{"Add":"open import Level"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ ℓ′ : Level"},{"Add":" A : Set ℓ"},{"Add":""},{"Add":"record RawChoice (F : Set ℓ → Set ℓ′) : Set (suc ℓ ⊔ ℓ′) where"},{"Add":" infixr 3 _<|>_ _∣_"},{"Add":" field"},{"Add":" _<|>_ : F A → F A → F A"},{"Add":""},{"Add":" -- backwards compatibility: unicode variants"},{"Add":" _∣_ : F A → F A → F A"},{"Add":" _∣_ = _<|>_"}]}},{"FileAdd":{"meta":{"basename":"Applicative.agda","metadata":0},"context":[{"change":0,"pos":1273735}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Applicative functors"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Note that currently the applicative functor laws are not included"},{"Add":"-- here."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Applicative where"},{"Add":""},{"Add":"open import Data.Bool.Base using (Bool; true; false)"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Data.Unit.Polymorphic.Base using (⊤)"},{"Add":""},{"Add":"open import Effect.Choice using (RawChoice)"},{"Add":"open import Effect.Empty using (RawEmpty)"},{"Add":"open import Effect.Functor as Fun using (RawFunctor)"},{"Add":""},{"Add":"open import Function.Base using (const; flip; _∘′_)"},{"Add":"open import Level using (Level; suc; _⊔_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" f g : Level"},{"Add":" A B C : Set f"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The type of raw applicatives"},{"Add":""},{"Add":"record RawApplicative (F : Set f → Set g) : Set (suc f ⊔ g) where"},{"Add":" infixl 4 _<*>_ _<*_ _*>_"},{"Add":" infixl 4 _⊛_ _<⊛_ _⊛>_"},{"Add":" infix 4 _⊗_"},{"Add":" field"},{"Add":" rawFunctor : RawFunctor F"},{"Add":" pure : A → F A"},{"Add":" _<*>_ : F (A → B) → F A → F B"},{"Add":""},{"Add":" open RawFunctor rawFunctor public"},{"Add":""},{"Add":" _<*_ : F A → F B → F A"},{"Add":" a <* b = const <$> a <*> b"},{"Add":""},{"Add":" _*>_ : F A → F B → F B"},{"Add":" a *> b = flip const <$> a <*> b"},{"Add":""},{"Add":" zipWith : (A → B → C) → F A → F B → F C"},{"Add":" zipWith f x y = f <$> x <*> y"},{"Add":""},{"Add":" zip : F A → F B → F (A × B)"},{"Add":" zip = zipWith _,_"},{"Add":""},{"Add":" -- Haskell-style alternative name for pure"},{"Add":" return : A → F A"},{"Add":" return = pure"},{"Add":""},{"Add":" -- backwards compatibility: unicode variants"},{"Add":" _⊛_ : F (A → B) → F A → F B"},{"Add":" _⊛_ = _<*>_"},{"Add":""},{"Add":" _<⊛_ : F A → F B → F A"},{"Add":" _<⊛_ = _<*_"},{"Add":""},{"Add":" _⊛>_ : F A → F B → F B"},{"Add":" _⊛>_ = _*>_"},{"Add":""},{"Add":" _⊗_ : F A → F B → F (A × B)"},{"Add":" _⊗_ = zip"},{"Add":""},{"Add":"module _ where"},{"Add":""},{"Add":" open RawApplicative"},{"Add":" open RawFunctor"},{"Add":""},{"Add":" -- Smart constructor"},{"Add":" mkRawApplicative :"},{"Add":" (F : Set f → Set g) →"},{"Add":" (pure : ∀ {A} → A → F A) →"},{"Add":" (app : ∀ {A B} → F (A → B) → F A → F B) →"},{"Add":" RawApplicative F"},{"Add":" mkRawApplicative F pure app .rawFunctor ._<$>_ = app ∘′ pure"},{"Add":" mkRawApplicative F pure app .pure = pure"},{"Add":" mkRawApplicative F pure app ._<*>_ = app"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The type of raw applicatives with a zero"},{"Add":""},{"Add":"record RawApplicativeZero (F : Set f → Set g) : Set (suc f ⊔ g) where"},{"Add":" field"},{"Add":" rawApplicative : RawApplicative F"},{"Add":" rawEmpty : RawEmpty F"},{"Add":""},{"Add":" open RawApplicative rawApplicative public"},{"Add":" open RawEmpty rawEmpty public"},{"Add":""},{"Add":" guard : Bool → F ⊤"},{"Add":" guard true = pure _"},{"Add":" guard false = empty"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The type of raw alternative applicatives"},{"Add":""},{"Add":"record RawAlternative (F : Set f → Set g) : Set (suc f ⊔ g) where"},{"Add":" field"},{"Add":" rawApplicativeZero : RawApplicativeZero F"},{"Add":" rawChoice : RawChoice F"},{"Add":""},{"Add":" open RawApplicativeZero rawApplicativeZero public"},{"Add":" open RawChoice rawChoice public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- The type of applicative morphisms"},{"Add":""},{"Add":"record Morphism {F₁ F₂ : Set f → Set g}"},{"Add":" (A₁ : RawApplicative F₁)"},{"Add":" (A₂ : RawApplicative F₂) : Set (suc f ⊔ g) where"},{"Add":" module A₁ = RawApplicative A₁"},{"Add":" module A₂ = RawApplicative A₂"},{"Add":" field"},{"Add":" functorMorphism : Fun.Morphism A₁.rawFunctor A₂.rawFunctor"},{"Add":""},{"Add":" open Fun.Morphism functorMorphism public"},{"Add":""},{"Add":" field"},{"Add":" op-pure : (x : A) → op (A₁.pure x) ≡ A₂.pure x"},{"Add":" op-<*> : (f : F₁ (A → B)) (x : F₁ A) →"},{"Add":" op (f A₁.⊛ x) ≡ (op f A₂.⊛ op x)"},{"Add":""},{"Add":" -- backwards compatibility: unicode variants"},{"Add":" op-⊛ = op-<*>"}]}},{"FileAdd":{"meta":{"basename":"Applicative","metadata":576},"context":[{"change":0,"pos":1273735}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Predicate.agda","metadata":0},"context":[{"change":0,"pos":1384534}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Applicative functors on indexed sets (predicates)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Note that currently the applicative functor laws are not included"},{"Add":"-- here."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Applicative.Predicate where"},{"Add":""},{"Add":"open import Effect.Functor.Predicate"},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Function.Base using (const; constᵣ)"},{"Add":"open import Level"},{"Add":"open import Relation.Unary"},{"Add":"open import Relation.Unary.PredicateTransformer using (Pt)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" i ℓ : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"record RawPApplicative {I : Set i} (F : Pt I ℓ) :"},{"Add":" Set (i ⊔ suc ℓ) where"},{"Add":" infixl 4 _⊛_ _<⊛_ _⊛>_"},{"Add":" infix 4 _⊗_"},{"Add":""},{"Add":" field"},{"Add":" pure : ∀ {P} → P ⊆ F P"},{"Add":" _⊛_ : ∀ {P Q} → F (P ⇒ Q) ⊆ F P ⇒ F Q"},{"Add":""},{"Add":" rawPFunctor : RawPFunctor F"},{"Add":" rawPFunctor = record"},{"Add":" { _<$>_ = λ g x → pure g ⊛ x"},{"Add":" }"},{"Add":""},{"Add":" private"},{"Add":" open module RF = RawPFunctor rawPFunctor public"},{"Add":""},{"Add":" _<⊛_ : ∀ {P Q} → F P ⊆ const (∀ {j} → F Q j) ⇒ F P"},{"Add":" x <⊛ y = const <$> x ⊛ y"},{"Add":""},{"Add":" _⊛>_ : ∀ {P Q} → const (∀ {i} → F P i) ⊆ F Q ⇒ F Q"},{"Add":" x ⊛> y = constᵣ <$> x ⊛ y"},{"Add":""},{"Add":" _⊗_ : ∀ {P Q} → F P ⊆ F Q ⇒ F (P ∩ Q)"},{"Add":" x ⊗ y = (_,_) <$> x ⊛ y"},{"Add":""},{"Add":" zipWith : ∀ {P Q R} → (P ⊆ Q ⇒ R) → F P ⊆ F Q ⇒ F R"},{"Add":" zipWith f x y = f <$> x ⊛ y"}]}},{"FileAdd":{"meta":{"basename":"Indexed.agda","metadata":0},"context":[{"change":0,"pos":1384534}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Indexed applicative functors"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- Note that currently the applicative functor laws are not included"},{"Add":"-- here."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Effect.Applicative.Indexed where"},{"Add":""},{"Add":"open import Effect.Functor using (RawFunctor)"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Function.Base"},{"Add":"open import Level"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)"},{"Add":"open import Relation.Binary.PropositionalEquality.Properties"},{"Add":" using (module ≡-Reasoning)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c i f : Level"},{"Add":" A : Set a"},{"Add":" B : Set b"},{"Add":" C : Set c"},{"Add":""},{"Add":"IFun : Set i → (ℓ : Level) → Set (i ⊔ suc ℓ)"},{"Add":"IFun I ℓ = I → I → Set ℓ → Set ℓ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Type, and usual combinators"},{"Add":""},{"Add":"record RawIApplicative {I : Set i} (F : IFun I f) :"},{"Add":" Set (i ⊔ suc f) where"},{"Add":" infixl 4 _⊛_ _<⊛_ _⊛>_"},{"Add":" infix 4 _⊗_"},{"Add":""},{"Add":" field"},{"Add":" pure : ∀ {i} → A → F i i A"},{"Add":" _⊛_ : ∀ {i j k} → F i j (A → B) → F j k A → F i k B"},{"Add":""},{"Add":" rawFunctor : ∀ {i j} → RawFunctor (F i j)"},{"Add":" rawFunctor = record"},{"Add":" { _<$>_ = λ g x → pure g ⊛ x"},{"Add":" }"},{"Add":""},{"Add":" private"},{"Add":" open module RF {i j : I} ="},{"Add":" RawFunctor (rawFunctor {i = i} {j = j})"},{"Add":" public"},{"Add":""},{"Add":" _<⊛_ : ∀ {i j k} → F i j A → F j k B → F i k A"},{"Add":" x <⊛ y = const <$> x ⊛ y"},{"Add":""},{"Add":" _⊛>_ : ∀ {i j k} → F i j A → F j k B → F i k B"},{"Add":" x ⊛> y = constᵣ <$> x ⊛ y"},{"Add":""},{"Add":" _⊗_ : ∀ {i j k} → F i j A → F j k B → F i k (A × B)"},{"Add":" x ⊗ y = (_,_) <$> x ⊛ y"},{"Add":""},{"Add":" zipWith : ∀ {i j k} → (A → B → C) → F i j A → F j k B → F i k C"},{"Add":" zipWith f x y = f <$> x ⊛ y"},{"Add":""},{"Add":" zip : ∀ {i j k} → F i j A → F j k B → F i k (A × B)"},{"Add":" zip = zipWith _,_"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Applicative with a zero"},{"Add":""},{"Add":"record RawIApplicativeZero"},{"Add":" {I : Set i} (F : IFun I f) :"},{"Add":" Set (i ⊔ suc f) where"},{"Add":" field"},{"Add":" applicative : RawIApplicative F"},{"Add":" ∅ : ∀ {i j} → F i j A"},{"Add":""},{"Add":" open RawIApplicative applicative public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Alternative functors: `F i j A` is a monoid"},{"Add":""},{"Add":"record RawIAlternative"},{"Add":" {I : Set i} (F : IFun I f) :"},{"Add":" Set (i ⊔ suc f) where"},{"Add":" infixr 3 _∣_"},{"Add":" field"},{"Add":" applicativeZero : RawIApplicativeZero F"},{"Add":" _∣_ : ∀ {i j} → F i j A → F i j A → F i j A"},{"Add":""},{"Add":" open RawIApplicativeZero applicativeZero public"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Applicative functor morphisms, specialised to propositional"},{"Add":"-- equality."},{"Add":""},{"Add":"record Morphism {I : Set i} {F₁ F₂ : IFun I f}"},{"Add":" (A₁ : RawIApplicative F₁)"},{"Add":" (A₂ : RawIApplicative F₂) : Set (i ⊔ suc f) where"},{"Add":" module A₁ = RawIApplicative A₁"},{"Add":" module A₂ = RawIApplicative A₂"},{"Add":" field"},{"Add":" op : ∀ {i j} → F₁ i j A → F₂ i j A"},{"Add":" op-pure : ∀ {i} (x : A) → op (A₁.pure {i = i} x) ≡ A₂.pure x"},{"Add":" op-⊛ : ∀ {i j k} (f : F₁ i j (A → B)) (x : F₁ j k A) →"},{"Add":" op (f A₁.⊛ x) ≡ (op f A₂.⊛ op x)"},{"Add":""},{"Add":" op-<$> : ∀ {i j} (f : A → B) (x : F₁ i j A) →"},{"Add":" op (f A₁.<$> x) ≡ (f A₂.<$> op x)"},{"Add":" op-<$> f x = begin"},{"Add":" op (A₁._⊛_ (A₁.pure f) x) ≡⟨ op-⊛ _ _ ⟩"},{"Add":" A₂._⊛_ (op (A₁.pure f)) (op x) ≡⟨ cong₂ A₂._⊛_ (op-pure _) refl ⟩"},{"Add":" A₂._⊛_ (A₂.pure f) (op x) ∎"},{"Add":" where open ≡-Reasoning"}]}},{"FileAdd":{"meta":{"basename":"Debug","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Trace.agda","metadata":0},"context":[{"change":0,"pos":1389872}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Printing Strings During Evaluation"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --rewriting #-}"},{"Add":""},{"Add":"-- see README.Debug.Trace for a use-case"},{"Add":""},{"Add":"module Debug.Trace where"},{"Add":""},{"Add":"open import Agda.Builtin.String"},{"Add":"open import Agda.Builtin.Equality"},{"Add":""},{"Add":"-- Postulating the `trace` function and explaining how to compile it"},{"Add":""},{"Add":"postulate"},{"Add":" trace : ∀ {a} {A : Set a} → String → A → A"},{"Add":""},{"Add":"{-# FOREIGN GHC import qualified Debug.Trace as Debug #-}"},{"Add":"{-# FOREIGN GHC import qualified Data.Text as Text #-}"},{"Add":"{-# COMPILE GHC trace = const (const (Debug.trace . Text.unpack)) #-}"},{"Add":""},{"Add":"-- Because expressions involving postulates get stuck during evaluation,"},{"Add":"-- we also postulate an equality characterising `trace`'s behaviour. By"},{"Add":"-- declaring it as a rewrite rule we internalise that evaluation rule."},{"Add":""},{"Add":"postulate"},{"Add":" trace-eq : ∀ {a} {A : Set a} (a : A) str → trace str a ≡ a"},{"Add":""},{"Add":"{-# BUILTIN REWRITE _≡_ #-}"},{"Add":"{-# REWRITE trace-eq #-}"},{"Add":""},{"Add":"-- Specialised version of `trace` returning the traced message."},{"Add":""},{"Add":"traceId : String → String"},{"Add":"traceId str = trace str str"}]}},{"FileAdd":{"meta":{"basename":"Data","metadata":576},"context":[{"change":0,"pos":62922}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Wrap.agda","metadata":0},"context":[{"change":0,"pos":1391121}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Turn a relation into a record definition so as to remember the things"},{"Add":"-- being related."},{"Add":"-- This module has a readme file: README.Data.Wrap"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Wrap where"},{"Add":""},{"Add":"open import Data.Product.Nary.NonDependent"},{"Add":"open import Function.Nary.NonDependent"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Nary"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" ℓ : Level"},{"Add":""},{"Add":"record Wrap′ {n} {ls} {A : Sets n ls} (F : A ⇉ Set ℓ) (xs : Product n A)"},{"Add":" : Set ℓ where"},{"Add":" constructor [_]"},{"Add":" field"},{"Add":" get : uncurryₙ n F xs"},{"Add":""},{"Add":"open Wrap′ public"},{"Add":""},{"Add":"Wrap : ∀ {n ls} {A : Sets n ls} → A ⇉ Set ℓ → A ⇉ Set ℓ"},{"Add":"Wrap {n = n} F = curryₙ n (Wrap′ F)"}]}},{"FileAdd":{"meta":{"basename":"Word8","metadata":576},"context":[{"change":0,"pos":1391121}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Show.agda","metadata":0},"context":[{"change":0,"pos":1392042}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Bytes: showing bit patterns"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module Data.Word8.Show where"},{"Add":""},{"Add":"open import Agda.Builtin.String using (String)"},{"Add":""},{"Add":"open import Data.Bool.Show using (showBit)"},{"Add":"open import Data.Fin.Base as Fin using (Fin)"},{"Add":"import Data.Nat.Show as ℕ"},{"Add":"open import Data.String using (_++_; fromVec; padLeft)"},{"Add":"open import Data.Vec.Base as Vec using (Vec; []; _∷_)"},{"Add":"open import Data.Word8.Base"},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"showBits : Word8 → String"},{"Add":"showBits w"},{"Add":" = \"0b\" ++_"},{"Add":" $ fromVec"},{"Add":" $ Vec.reverse"},{"Add":" $ Vec.map showBit"},{"Add":" $ toBits w"},{"Add":""},{"Add":"showHexa : Word8 → String"},{"Add":"showHexa w"},{"Add":" = \"0x\" ++_"},{"Add":" $ padLeft '0' 2"},{"Add":" $ ℕ.showInBase 16 (toℕ w)"}]}},{"FileAdd":{"meta":{"basename":"Primitive.agda","metadata":0},"context":[{"change":0,"pos":1392042}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Bytes: simple bindings to Haskell types and functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module Data.Word8.Primitive where"},{"Add":""},{"Add":"open import Agda.Builtin.Bool using (Bool)"},{"Add":"open import Agda.Builtin.Nat using (Nat)"},{"Add":"open import Agda.Builtin.String using (String)"},{"Add":""},{"Add":"postulate"},{"Add":" Word8 : Set"},{"Add":" testBit : Word8 → Nat → Bool"},{"Add":" setBit : Word8 → Nat → Word8"},{"Add":" clearBit : Word8 → Nat → Word8"},{"Add":" fromNat : Nat → Word8"},{"Add":" toNat : Word8 → Nat"},{"Add":" _+_ : Word8 → Word8 → Word8"},{"Add":" show : Word8 → String"},{"Add":""},{"Add":"{-# FOREIGN GHC import GHC.Word #-}"},{"Add":"{-# FOREIGN GHC import qualified Data.Bits as B #-}"},{"Add":"{-# FOREIGN GHC import qualified Data.Text as T #-}"},{"Add":""},{"Add":"{-# COMPILE GHC Word8 = type Word8 #-}"},{"Add":"{-# COMPILE GHC testBit = \\ w -> B.testBit w . fromIntegral #-}"},{"Add":"{-# COMPILE GHC setBit = \\ w -> B.setBit w . fromIntegral #-}"},{"Add":"{-# COMPILE GHC clearBit = \\ w -> B.clearBit w . fromIntegral #-}"},{"Add":"{-# COMPILE GHC fromNat = fromIntegral #-}"},{"Add":"{-# COMPILE GHC toNat = fromIntegral #-}"},{"Add":"{-# COMPILE GHC _+_ = (+) #-}"},{"Add":"{-# COMPILE GHC show = T.pack . Prelude.show #-}"}]}},{"FileAdd":{"meta":{"basename":"Literals.agda","metadata":0},"context":[{"change":0,"pos":1392042}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Byte Literals"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module Data.Word8.Literals where"},{"Add":""},{"Add":"open import Agda.Builtin.FromNat using (Number)"},{"Add":"open import Data.Unit.Base using (⊤)"},{"Add":"open import Data.Word8.Base using (Word8; fromℕ)"},{"Add":""},{"Add":"number : Number Word8"},{"Add":"number = record"},{"Add":" { Constraint = λ _ → ⊤"},{"Add":" ; fromNat = λ w → fromℕ w"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Base.agda","metadata":0},"context":[{"change":0,"pos":1392042}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Bytes: base type and functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module Data.Word8.Base where"},{"Add":""},{"Add":"open import Agda.Builtin.Bool using (Bool; true; false)"},{"Add":"open import Agda.Builtin.Char using (Char)"},{"Add":"open import Agda.Builtin.Nat using (Nat; _==_)"},{"Add":"open import Agda.Builtin.Unit using (⊤)"},{"Add":""},{"Add":"open import Data.Fin.Base as Fin using (Fin)"},{"Add":"open import Data.Vec.Base as Vec using (Vec; []; _∷_)"},{"Add":""},{"Add":"open import Function.Base using (_$_; _|>_)"},{"Add":""},{"Add":"------------------------------------------------------------------------------"},{"Add":"-- Re-export type and operations"},{"Add":""},{"Add":"open import Data.Word8.Primitive as Prim public"},{"Add":" using ( Word8"},{"Add":" ; _+_"},{"Add":" ; show"},{"Add":" )"},{"Add":" renaming ( fromNat to fromℕ"},{"Add":" ; toNat to toℕ"},{"Add":" )"},{"Add":""},{"Add":"testBit : Word8 → Fin 8 → Bool"},{"Add":"testBit w i = Prim.testBit w (Fin.toℕ i)"},{"Add":""},{"Add":"_[_]≔_ : Word8 → Fin 8 → Bool → Word8"},{"Add":"w [ i ]≔ false = Prim.clearBit w (Fin.toℕ i)"},{"Add":"w [ i ]≔ true = Prim.setBit w (Fin.toℕ i)"},{"Add":""},{"Add":"------------------------------------------------------------------------------"},{"Add":"-- Basic functions"},{"Add":""},{"Add":"toBits : Word8 → Vec Bool 8"},{"Add":"toBits w = Vec.tabulate (testBit w)"},{"Add":""},{"Add":"fromBits : Vec Bool 8 → Word8"},{"Add":"fromBits bs = Vec.foldl′ _|>_ (fromℕ 0)"},{"Add":" $ Vec.zipWith (λ i b → _[ i ]≔ b) (Vec.allFin 8) bs"},{"Add":""},{"Add":"_≡ᵇ_ : Word8 → Word8 → Bool"},{"Add":"w ≡ᵇ w′ = toℕ w == toℕ w′"}]}},{"FileAdd":{"meta":{"basename":"Word64.agda","metadata":0},"context":[{"change":0,"pos":1391121}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Machine words"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Word64 where"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export base definitions and decidability of equality"},{"Add":""},{"Add":"open import Data.Word64.Base public"},{"Add":"open import Data.Word64.Properties using (_≈?_; __)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export primitives publicly"},{"Add":""},{"Add":"open import Data.Word64.Primitive as Prim public"},{"Add":" using ( show )"},{"Add":""},{"Add":"testBit : Word64 → Fin 64 → Bool"},{"Add":"testBit w i = Prim.testBit w (Fin.toℕ i)"},{"Add":""},{"Add":"_[_]≔_ : Word64 → Fin 64 → Bool → Word64"},{"Add":"w [ i ]≔ false = Prim.clearBit w (Fin.toℕ i)"},{"Add":"w [ i ]≔ true = Prim.setBit w (Fin.toℕ i)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Convert to its components"},{"Add":""},{"Add":"toBits : Word64 → Vec Bool 64"},{"Add":"toBits w = Vec.tabulate (testBit w)"},{"Add":""},{"Add":"fromBits : Vec Bool 64 → Word64"},{"Add":"fromBits bs = Vec.foldl′ _|>_ (fromℕ 0)"},{"Add":" $ Vec.zipWith (λ i b → _[ i ]≔ b) (Vec.allFin 64) bs"},{"Add":""},{"Add":"toWord64s : Word64 → Vec Word8 8"},{"Add":"toWord64s w ="},{"Add":" let ws = proj₁ (Vec.group 8 8 (toBits w)) in"},{"Add":" Vec.map Word8.fromBits ws"}]}},{"FileAdd":{"meta":{"basename":"Show.agda","metadata":0},"context":[{"change":0,"pos":1396854}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Bytes: showing bit patterns"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible #-}"},{"Add":""},{"Add":"module Data.Word64.Show where"},{"Add":""},{"Add":"open import Agda.Builtin.String using (String)"},{"Add":""},{"Add":"open import Data.Bool.Show using (showBit)"},{"Add":"open import Data.Fin.Base as Fin using (Fin)"},{"Add":"import Data.Nat.Show as ℕ"},{"Add":"open import Data.String using (_++_; fromVec; padLeft)"},{"Add":"open import Data.Vec.Base as Vec using (Vec; []; _∷_)"},{"Add":""},{"Add":"open import Data.Word64.Base"},{"Add":"open import Data.Word64.Unsafe"},{"Add":""},{"Add":"open import Function.Base using (_$_)"},{"Add":""},{"Add":"showBits : Word64 → String"},{"Add":"showBits w"},{"Add":" = \"0b\" ++_"},{"Add":" $ fromVec"},{"Add":" $ Vec.reverse"},{"Add":" $ Vec.map showBit"},{"Add":" $ toBits w"},{"Add":""},{"Add":"showHexa : Word64 → String"},{"Add":"showHexa w"},{"Add":" = \"0x\" ++_"},{"Add":" $ padLeft '0' 8"},{"Add":" $ ℕ.showInBase 16 (toℕ w)"}]}},{"FileAdd":{"meta":{"basename":"Properties.agda","metadata":0},"context":[{"change":0,"pos":1396854}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of operations on machine words"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Word64.Properties where"},{"Add":""},{"Add":"import Data.Nat.Base as ℕ"},{"Add":"open import Data.Bool.Base using (Bool)"},{"Add":"open import Data.Word64.Base using (_≈_; toℕ; Word64; _<_)"},{"Add":"import Data.Nat.Properties as ℕ"},{"Add":"open import Relation.Nullary.Decidable.Core using (map′; ⌊_⌋)"},{"Add":"open import Relation.Binary"},{"Add":" using ( _⇒_; Reflexive; Symmetric; Transitive; Substitutive"},{"Add":" ; Decidable; DecidableEquality; IsEquivalence; IsDecEquivalence"},{"Add":" ; Setoid; DecSetoid; StrictTotalOrder)"},{"Add":"import Relation.Binary.Construct.On as On"},{"Add":"open import Relation.Binary.PropositionalEquality.Core"},{"Add":" using (_≡_; refl; cong; sym; trans; subst)"},{"Add":"open import Relation.Binary.PropositionalEquality.Properties"},{"Add":" using (setoid; decSetoid)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Primitive properties"},{"Add":""},{"Add":"open import Agda.Builtin.Word.Properties"},{"Add":" renaming (primWord64ToNatInjective to toℕ-injective)"},{"Add":" public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of _≈_"},{"Add":""},{"Add":"≈⇒≡ : _≈_ ⇒ _≡_"},{"Add":"≈⇒≡ = toℕ-injective _ _"},{"Add":""},{"Add":"≈-reflexive : _≡_ ⇒ _≈_"},{"Add":"≈-reflexive = cong toℕ"},{"Add":""},{"Add":"≈-refl : Reflexive _≈_"},{"Add":"≈-refl = refl"},{"Add":""},{"Add":"≈-sym : Symmetric _≈_"},{"Add":"≈-sym = sym"},{"Add":""},{"Add":"≈-trans : Transitive _≈_"},{"Add":"≈-trans = trans"},{"Add":""},{"Add":"≈-subst : ∀ {ℓ} → Substitutive _≈_ ℓ"},{"Add":"≈-subst P x≈y p = subst P (≈⇒≡ x≈y) p"},{"Add":""},{"Add":"infix 4 _≈?_"},{"Add":"_≈?_ : Decidable _≈_"},{"Add":"x ≈? y = toℕ x ℕ.≟ toℕ y"},{"Add":""},{"Add":"≈-isEquivalence : IsEquivalence _≈_"},{"Add":"≈-isEquivalence = record"},{"Add":" { refl = λ {i} → ≈-refl {i}"},{"Add":" ; sym = λ {i j} → ≈-sym {i} {j}"},{"Add":" ; trans = λ {i j k} → ≈-trans {i} {j} {k}"},{"Add":" }"},{"Add":""},{"Add":"≈-setoid : Setoid _ _"},{"Add":"≈-setoid = record"},{"Add":" { isEquivalence = ≈-isEquivalence"},{"Add":" }"},{"Add":""},{"Add":"≈-isDecEquivalence : IsDecEquivalence _≈_"},{"Add":"≈-isDecEquivalence = record"},{"Add":" { isEquivalence = ≈-isEquivalence"},{"Add":" ; _≟_ = _≈?_"},{"Add":" }"},{"Add":""},{"Add":"≈-decSetoid : DecSetoid _ _"},{"Add":"≈-decSetoid = record"},{"Add":" { isDecEquivalence = ≈-isDecEquivalence"},{"Add":" }"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of _≡_"},{"Add":""},{"Add":"infix 4 _≟_"},{"Add":"_≟_ : DecidableEquality Word64"},{"Add":"x ≟ y = map′ ≈⇒≡ ≈-reflexive (x ≈? y)"},{"Add":""},{"Add":"≡-setoid : Setoid _ _"},{"Add":"≡-setoid = setoid Word64"},{"Add":""},{"Add":"≡-decSetoid : DecSetoid _ _"},{"Add":"≡-decSetoid = decSetoid _≟_"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Boolean equality test."},{"Add":""},{"Add":"infix 4 _==_"},{"Add":"_==_ : Word64 → Word64 → Bool"},{"Add":"w₁ == w₂ = ⌊ w₁ ≟ w₂ ⌋"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of _<_"},{"Add":""},{"Add":"infix 4 _ B.testBit w . fromIntegral #-}"},{"Add":"{-# COMPILE GHC setBit = \\ w -> B.setBit w . fromIntegral #-}"},{"Add":"{-# COMPILE GHC clearBit = \\ w -> B.clearBit w . fromIntegral #-}"},{"Add":"{-# COMPILE GHC show = T.pack . Prelude.show #-}"}]}},{"FileAdd":{"meta":{"basename":"Literals.agda","metadata":0},"context":[{"change":0,"pos":1396854}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Word64 Literals"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Word64.Literals where"},{"Add":""},{"Add":"open import Agda.Builtin.FromNat using (Number)"},{"Add":"open import Data.Unit.Base using (⊤)"},{"Add":"open import Data.Word64.Base using (Word64; fromℕ)"},{"Add":""},{"Add":"number : Number Word64"},{"Add":"number = record"},{"Add":" { Constraint = λ _ → ⊤"},{"Add":" ; fromNat = λ w → fromℕ w"},{"Add":" }"}]}},{"FileAdd":{"meta":{"basename":"Instances.agda","metadata":0},"context":[{"change":0,"pos":1396854}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Instances for words"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Word64.Instances where"},{"Add":""},{"Add":"open import Data.Word64.Properties"},{"Add":"open import Relation.Binary.PropositionalEquality.Properties"},{"Add":" using (isDecEquivalence)"},{"Add":""},{"Add":"instance"},{"Add":" Word64-≡-isDecEquivalence = isDecEquivalence _≟_"}]}},{"FileAdd":{"meta":{"basename":"Base.agda","metadata":0},"context":[{"change":0,"pos":1396854}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Machine words: basic type and conversion functions"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Word64.Base where"},{"Add":""},{"Add":"open import Algebra.Core using (Op₂)"},{"Add":"open import Data.Nat.Base as ℕ using (ℕ)"},{"Add":"open import Function.Base using (_on_; _∘₂′_)"},{"Add":"open import Level using (zero)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export built-ins publicly"},{"Add":""},{"Add":"open import Agda.Builtin.Word public"},{"Add":" using (Word64)"},{"Add":" renaming"},{"Add":" ( primWord64ToNat to toℕ"},{"Add":" ; primWord64FromNat to fromℕ"},{"Add":" )"},{"Add":""},{"Add":"liftOp₂ : Op₂ ℕ → Op₂ Word64"},{"Add":"liftOp₂ op = fromℕ ∘₂′ op on toℕ"},{"Add":""},{"Add":"infix 4 _≈_"},{"Add":"_≈_ : Rel Word64 zero"},{"Add":"_≈_ = _≡_ on toℕ"},{"Add":""},{"Add":"infix 4 _<_"},{"Add":"_<_ : Rel Word64 zero"},{"Add":"_<_ = ℕ._<_ on toℕ"},{"Add":""},{"Add":"infix 4 _≤_"},{"Add":"_≤_ : Rel Word64 zero"},{"Add":"_≤_ = ℕ._≤_ on toℕ"}]}},{"FileAdd":{"meta":{"basename":"Word.agda","metadata":0},"context":[{"change":0,"pos":1391121}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use Data.Word64 instead"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Word where"},{"Add":""},{"Add":"open import Data.Word64 public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Data.Word was deprecated in v2.1. Use Data.Word64 instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Word","metadata":576},"context":[{"change":0,"pos":1391121}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Properties.agda","metadata":0},"context":[{"change":0,"pos":1406121}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use Data.Word64.Properties instead"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Word.Properties where"},{"Add":""},{"Add":"open import Data.Word64.Properties public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Data.Word.Properties was deprecated in v2.1. Use Data.Word64.Properties instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Instances.agda","metadata":0},"context":[{"change":0,"pos":1406121}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use Data.Word64.Instances instead"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Word.Instances where"},{"Add":""},{"Add":"open import Data.Word64.Instances public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Data.Word.Instances was deprecated in v2.1. Use Data.Word64.Instances instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"Base.agda","metadata":0},"context":[{"change":0,"pos":1406121}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- This module is DEPRECATED. Please use Data.Word64.Base instead"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Word.Base where"},{"Add":""},{"Add":"open import Data.Word64.Base public"},{"Add":""},{"Add":"{-# WARNING_ON_IMPORT"},{"Add":"\"Data.Word.Base was deprecated in v2.1. Use Data.Word64.Base instead.\""},{"Add":"#-}"}]}},{"FileAdd":{"meta":{"basename":"W.agda","metadata":0},"context":[{"change":0,"pos":1391121}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- W-types"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.W where"},{"Add":""},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Function.Base using (_$_; _∘_; const)"},{"Add":"open import Data.Product.Base using (_,_; -,_; proj₂)"},{"Add":"open import Data.Container.Core using (Container; ⟦_⟧; Shape; Position; _⇒_; ⟪_⟫)"},{"Add":"open import Data.Container.Relation.Unary.All using (□; all)"},{"Add":"open import Relation.Nullary.Negation using (¬_)"},{"Add":"open import Agda.Builtin.Equality using (_≡_; refl)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" s p s₁ s₂ p₁ p₂ ℓ : Level"},{"Add":" C : Container s p"},{"Add":" C₁ : Container s₁ p₁"},{"Add":" C₂ : Container s₂ p₂"},{"Add":""},{"Add":"-- The family of W-types."},{"Add":""},{"Add":"data W (C : Container s p) : Set (s ⊔ p) where"},{"Add":" sup : ⟦ C ⟧ (W C) → W C"},{"Add":""},{"Add":"sup-injective₁ : ∀ {s t : Shape C} {f : Position C s → W C} {g} →"},{"Add":" sup (s , f) ≡ sup (t , g) → s ≡ t"},{"Add":"sup-injective₁ refl = refl"},{"Add":""},{"Add":"-- See also Data.W.WithK.sup-injective₂."},{"Add":""},{"Add":"-- Projections."},{"Add":""},{"Add":"head : W C → Shape C"},{"Add":"head (sup (x , f)) = x"},{"Add":""},{"Add":"tail : (x : W C) → Position C (head x) → W C"},{"Add":"tail (sup (x , f)) = f"},{"Add":""},{"Add":"-- map"},{"Add":""},{"Add":"map : (m : C₁ ⇒ C₂) → W C₁ → W C₂"},{"Add":"map m (sup (x , f)) = sup (⟪ m ⟫ (x , λ p → map m (f p)))"},{"Add":""},{"Add":"-- induction"},{"Add":""},{"Add":"module _ (P : W C → Set ℓ)"},{"Add":" (alg : ∀ {t} → □ C P t → P (sup t)) where"},{"Add":""},{"Add":" induction : (w : W C) → P w"},{"Add":" induction (sup (s , f)) = alg $ all (induction ∘ f)"},{"Add":""},{"Add":"module _ {P : Set ℓ} (alg : ⟦ C ⟧ P → P) where"},{"Add":""},{"Add":" foldr : W C → P"},{"Add":" foldr = induction (const P) (alg ∘ -,_ ∘ □.proof)"},{"Add":""},{"Add":"-- If Position is always inhabited, then W_C is empty."},{"Add":""},{"Add":"inhabited⇒empty : (∀ s → Position C s) → ¬ W C"},{"Add":"inhabited⇒empty b = foldr ((_$ b _) ∘ proj₂)"}]}},{"FileAdd":{"meta":{"basename":"W","metadata":576},"context":[{"change":0,"pos":1391121}],"contents":[]}},{"FileAdd":{"meta":{"basename":"WithK.agda","metadata":0},"context":[{"change":0,"pos":1409577}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Some code related to the W type that relies on the K rule"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --with-K --safe #-}"},{"Add":""},{"Add":"module Data.W.WithK where"},{"Add":""},{"Add":"open import Data.Product.Base using (_,_)"},{"Add":"open import Data.Container.Core"},{"Add":"open import Data.W"},{"Add":"open import Agda.Builtin.Equality"},{"Add":""},{"Add":"module _ {s p} {C : Container s p}"},{"Add":" {s : Shape C} {f : Position C s → W C} where"},{"Add":""},{"Add":" sup-injective₂ : ∀ {g} → sup (s , f) ≡ sup (s , g) → f ≡ g"},{"Add":" sup-injective₂ refl = refl"}]}},{"FileAdd":{"meta":{"basename":"Sized.agda","metadata":0},"context":[{"change":0,"pos":1409577}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Sized W-types"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --sized-types #-}"},{"Add":""},{"Add":"module Data.W.Sized where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Size"},{"Add":"open import Function.Base using (_$_; _∘_; const)"},{"Add":"open import Data.Product.Base using (_,_; -,_; proj₂)"},{"Add":"open import Data.Container.Core as Container using (Container; ⟦_⟧; Shape; Position; _⇒_; ⟪_⟫)"},{"Add":"open import Data.Container.Relation.Unary.All using (□; all)"},{"Add":"open import Relation.Nullary.Negation using (¬_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" i : Size"},{"Add":" s p s₁ s₂ p₁ p₂ ℓ : Level"},{"Add":" C : Container s p"},{"Add":" C₁ : Container s₁ p₁"},{"Add":" C₂ : Container s₂ p₂"},{"Add":""},{"Add":"-- The family of W-types."},{"Add":""},{"Add":"data W (C : Container s p) : Size → Set (s ⊔ p) where"},{"Add":" sup : ⟦ C ⟧ (W C i) → W C (↑ i)"},{"Add":""},{"Add":"sup-injective₁ : ∀ {s t : Shape C} {f : Position C s → W C i} {g} →"},{"Add":" sup (s , f) ≡ sup (t , g) → s ≡ t"},{"Add":"sup-injective₁ refl = refl"},{"Add":""},{"Add":"-- See also Data.W.WithK.sup-injective₂."},{"Add":""},{"Add":"-- Projections."},{"Add":""},{"Add":"head : W C i → Shape C"},{"Add":"head (sup (x , f)) = x"},{"Add":""},{"Add":"tail : (x : W C i) → Position C (head x) → W C i"},{"Add":"tail (sup (x , f)) = f"},{"Add":""},{"Add":"-- map"},{"Add":""},{"Add":"map : (m : C₁ ⇒ C₂) → W C₁ i → W C₂ i"},{"Add":"map m (sup c) = sup (⟪ m ⟫ (Container.map (map m) c))"},{"Add":""},{"Add":"-- induction"},{"Add":""},{"Add":"module _ (P : W C ∞ → Set ℓ)"},{"Add":" (alg : ∀ {t} → □ C P t → P (sup t)) where"},{"Add":""},{"Add":" induction : (w : W C _) → P w"},{"Add":" induction (sup (s , f)) = alg $ all (induction ∘ f)"},{"Add":""},{"Add":"module _ {P : Set ℓ} (alg : ⟦ C ⟧ P → P) where"},{"Add":""},{"Add":" foldr : W C _ → P"},{"Add":" foldr = induction (const P) (alg ∘ -,_ ∘ □.proof)"},{"Add":""},{"Add":"-- If Position is always inhabited, then W_C is empty."},{"Add":""},{"Add":"inhabited⇒empty : (∀ s → Position C s) → ¬ W C i"},{"Add":"inhabited⇒empty b = foldr ((_$ b _) ∘ proj₂)"}]}},{"FileAdd":{"meta":{"basename":"Indexed.agda","metadata":0},"context":[{"change":0,"pos":1409577}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Indexed W-types aka Petersson-Synek trees"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.W.Indexed where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Data.Container.Indexed.Core"},{"Add":"open import Data.Product.Base using (_,_; Σ)"},{"Add":"open import Relation.Unary"},{"Add":""},{"Add":"-- The family of indexed W-types."},{"Add":""},{"Add":"module _ {ℓ c r} {O : Set ℓ} (C : Container O O c r) where"},{"Add":""},{"Add":" open Container C"},{"Add":""},{"Add":" data W (o : O) : Set (ℓ ⊔ c ⊔ r) where"},{"Add":" sup : ⟦ C ⟧ W o → W o"},{"Add":""},{"Add":" -- Projections."},{"Add":""},{"Add":" head : W ⊆ Command"},{"Add":" head (sup (c , _)) = c"},{"Add":""},{"Add":" tail : ∀ {o} (w : W o) (r : Response (head w)) → W (next (head w) r)"},{"Add":" tail (sup (_ , k)) r = k r"},{"Add":""},{"Add":" -- Induction, (primitive) recursion and iteration."},{"Add":""},{"Add":" ind : ∀ {ℓ} (P : Pred (Σ O W) ℓ) →"},{"Add":" (∀ {o} (cs : ⟦ C ⟧ W o) → □ C P (o , cs) → P (o , sup cs)) →"},{"Add":" ∀ {o} (w : W o) → P (o , w)"},{"Add":" ind P φ (sup (c , k)) = φ (c , k) (λ r → ind P φ (k r))"},{"Add":""},{"Add":" rec : ∀ {ℓ} {X : Pred O ℓ} → (⟦ C ⟧ (W ∩ X) ⊆ X) → W ⊆ X"},{"Add":" rec φ (sup (c , k))= φ (c , λ r → (k r , rec φ (k r)))"},{"Add":""},{"Add":" iter : ∀ {ℓ} {X : Pred O ℓ} → (⟦ C ⟧ X ⊆ X) → W ⊆ X"},{"Add":" iter φ (sup (c , k))= φ (c , λ r → iter φ (k r))"}]}},{"FileAdd":{"meta":{"basename":"Vec.agda","metadata":0},"context":[{"change":0,"pos":1391121}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Vectors"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"-- This implementation is designed for reasoning about dynamic"},{"Add":"-- vectors which may increase or decrease in size."},{"Add":""},{"Add":"-- See `Data.Vec.Functional` for an alternative implementation as"},{"Add":"-- functions from finite sets, which is more suitable for reasoning"},{"Add":"-- about fixed sized vectors and for when ease of retrieval is"},{"Add":"-- important."},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Vec where"},{"Add":""},{"Add":"open import Level"},{"Add":"open import Data.Bool.Base"},{"Add":"import Data.Nat.Properties as ℕ"},{"Add":"open import Data.Vec.Bounded.Base as Vec≤"},{"Add":" using (Vec≤; ≤-cast; fromVec)"},{"Add":"open import Function.Base using (_$_)"},{"Add":"open import Relation.Nullary"},{"Add":"open import Relation.Unary"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a p : Level"},{"Add":" A : Set a"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Publicly re-export the contents of the base module"},{"Add":""},{"Add":"open import Data.Vec.Base public"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Additional operations"},{"Add":""},{"Add":"module _ {P : A → Set p} (P? : Decidable P) where"},{"Add":""},{"Add":" filter : ∀ {n} → Vec A n → Vec≤ A n"},{"Add":" filter [] = Vec≤.[]"},{"Add":" filter (a ∷ as) = if does (P? a) then a Vec≤.∷_ else ≤-cast (ℕ.n≤1+n _) $ filter as"},{"Add":""},{"Add":" takeWhile : ∀ {n} → Vec A n → Vec≤ A n"},{"Add":" takeWhile [] = Vec≤.[]"},{"Add":" takeWhile (a ∷ as) = if does (P? a) then a Vec≤.∷ takeWhile as else Vec≤.[]"},{"Add":""},{"Add":" dropWhile : ∀ {n} → Vec A n → Vec≤ A n"},{"Add":" dropWhile Vec.[] = Vec≤.[]"},{"Add":" dropWhile (a Vec.∷ as) ="},{"Add":" if does (P? a) then ≤-cast (ℕ.n≤1+n _) (dropWhile as)"},{"Add":" else fromVec (a Vec.∷ as)"}]}},{"FileAdd":{"meta":{"basename":"Vec","metadata":576},"context":[{"change":0,"pos":1391121}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Show.agda","metadata":0},"context":[{"change":0,"pos":1415450}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Showing vectors"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Vec.Show where"},{"Add":""},{"Add":"import Data.List.Show as List"},{"Add":"open import Data.String.Base using (String)"},{"Add":"open import Data.Vec.Base using (Vec; toList)"},{"Add":"open import Function.Base using (_∘_)"},{"Add":""},{"Add":"show : ∀ {a} {A : Set a} {n} → (A → String) → (Vec A n → String)"},{"Add":"show s = List.show s ∘ toList"}]}},{"FileAdd":{"meta":{"basename":"Relation","metadata":576},"context":[{"change":0,"pos":1415450}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Unary","metadata":576},"context":[{"change":0,"pos":1416043}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Unique","metadata":576},"context":[{"change":0,"pos":1416065}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Setoid.agda","metadata":0},"context":[{"change":0,"pos":1416084}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Vectors made up entirely of unique elements (setoid equality)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":""},{"Add":"module Data.Vec.Relation.Unary.Unique.Setoid {a ℓ} (S : Setoid a ℓ) where"},{"Add":""},{"Add":"open Setoid S renaming (Carrier to A)"},{"Add":""},{"Add":"open import Data.Vec.Base"},{"Add":"import Data.Vec.Relation.Unary.AllPairs as AllPairsM"},{"Add":"open import Level using (_⊔_)"},{"Add":"open import Relation.Unary using (Pred)"},{"Add":"open import Relation.Nullary.Negation using (¬_)"},{"Add":""},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"private"},{"Add":" Distinct : Rel A ℓ"},{"Add":" Distinct x y = ¬ (x ≈ y)"},{"Add":""},{"Add":"open import Data.Vec.Relation.Unary.AllPairs.Core Distinct public"},{"Add":" renaming (AllPairs to Unique)"},{"Add":""},{"Add":"open import Data.Vec.Relation.Unary.AllPairs {R = Distinct} public"},{"Add":" using (head; tail)"}]}},{"FileAdd":{"meta":{"basename":"Setoid","metadata":576},"context":[{"change":0,"pos":1416084}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Properties.agda","metadata":0},"context":[{"change":0,"pos":1417193}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of unique vectors (setoid equality)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Vec.Relation.Unary.Unique.Setoid.Properties where"},{"Add":""},{"Add":"open import Data.Fin.Base using (Fin; zero; suc)"},{"Add":"open import Data.Vec.Base as Vec"},{"Add":"open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)"},{"Add":"import Data.Vec.Relation.Unary.All.Properties as All"},{"Add":"open import Data.Vec.Relation.Unary.AllPairs as AllPairs using (AllPairs)"},{"Add":"open import Data.Vec.Relation.Unary.Unique.Setoid"},{"Add":"import Data.Vec.Relation.Unary.AllPairs.Properties as AllPairs"},{"Add":"open import Data.Nat.Base using (ℕ; _+_)"},{"Add":"open import Function.Base using (_∘_; id)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)"},{"Add":"open import Relation.Nullary.Negation using (contradiction; contraposition)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c p ℓ ℓ₁ ℓ₂ ℓ₃ : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Introduction (⁺) and elimination (⁻) rules for list operations"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- map"},{"Add":""},{"Add":"module _ (S : Setoid a ℓ₁) (R : Setoid b ℓ₂) where"},{"Add":""},{"Add":" open Setoid S renaming (_≈_ to _≈₁_)"},{"Add":" open Setoid R renaming (_≈_ to _≈₂_)"},{"Add":""},{"Add":" map⁺ : ∀ {f} → (∀ {x y} → f x ≈₂ f y → x ≈₁ y) →"},{"Add":" ∀ {n xs} → Unique S {n} xs → Unique R {n} (map f xs)"},{"Add":" map⁺ inj xs! = AllPairs.map⁺ (AllPairs.map (contraposition inj) xs!)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- take & drop"},{"Add":""},{"Add":"module _ (S : Setoid a ℓ) where"},{"Add":""},{"Add":" drop⁺ : ∀ {n} m {xs} → Unique S {m + n} xs → Unique S {n} (drop m xs)"},{"Add":" drop⁺ = AllPairs.drop⁺"},{"Add":""},{"Add":" take⁺ : ∀ {n} m {xs} → Unique S {m + n} xs → Unique S {m} (take m xs)"},{"Add":" take⁺ = AllPairs.take⁺"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- tabulate"},{"Add":""},{"Add":"module _ (S : Setoid a ℓ) where"},{"Add":""},{"Add":" open Setoid S renaming (Carrier to A)"},{"Add":""},{"Add":" tabulate⁺ : ∀ {n} {f : Fin n → A} → (∀ {i j} → f i ≈ f j → i ≡ j) →"},{"Add":" Unique S (tabulate f)"},{"Add":" tabulate⁺ f-inj = AllPairs.tabulate⁺ (_∘ f-inj)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- lookup"},{"Add":""},{"Add":"module _ (S : Setoid a ℓ) where"},{"Add":""},{"Add":" open Setoid S"},{"Add":""},{"Add":" lookup-injective : ∀ {n xs} → Unique S {n} xs → ∀ i j → lookup xs i ≈ lookup xs j → i ≡ j"},{"Add":" lookup-injective (px ∷ pxs) zero zero eq = ≡.refl"},{"Add":" lookup-injective (px ∷ pxs) zero (suc j) eq = contradiction eq (All.lookup⁺ px j)"},{"Add":" lookup-injective (px ∷ pxs) (suc i) zero eq = contradiction (sym eq) (All.lookup⁺ px i)"},{"Add":" lookup-injective (px ∷ pxs) (suc i) (suc j) eq = ≡.cong suc (lookup-injective pxs i j eq)"}]}},{"FileAdd":{"meta":{"basename":"Propositional.agda","metadata":0},"context":[{"change":0,"pos":1416084}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Vectors made up entirely of unique elements (propositional equality)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Vec.Relation.Unary.Unique.Propositional {a} {A : Set a} where"},{"Add":""},{"Add":"open import Relation.Binary.PropositionalEquality.Properties using (setoid)"},{"Add":"open import Data.Vec.Relation.Unary.Unique.Setoid as SetoidUnique"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Re-export the contents of setoid uniqueness"},{"Add":""},{"Add":"open SetoidUnique (setoid A) public"}]}},{"FileAdd":{"meta":{"basename":"Propositional","metadata":576},"context":[{"change":0,"pos":1416084}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Properties.agda","metadata":0},"context":[{"change":0,"pos":1420994}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties of unique vectors (setoid equality)"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Vec.Relation.Unary.Unique.Propositional.Properties where"},{"Add":""},{"Add":"open import Data.Vec.Base using (Vec; map; take; drop; tabulate; lookup)"},{"Add":"open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)"},{"Add":"open import Data.Vec.Relation.Unary.AllPairs as AllPairs using (AllPairs)"},{"Add":"open import Data.Vec.Relation.Unary.Unique.Propositional using (Unique)"},{"Add":"import Data.Vec.Relation.Unary.Unique.Setoid.Properties as Setoid"},{"Add":"open import Data.Fin.Base using (Fin)"},{"Add":"open import Data.Nat.Base using (_+_)"},{"Add":"open import Data.Nat.Properties using (<⇒≢)"},{"Add":"open import Data.Product.Base using (_×_; _,_)"},{"Add":"open import Data.Product.Relation.Binary.Pointwise.NonDependent using (≡⇒≡×≡)"},{"Add":"open import Function.Base using (id; _∘_)"},{"Add":"open import Level using (Level)"},{"Add":"open import Relation.Binary.Core using (Rel)"},{"Add":"open import Relation.Binary.Bundles using (Setoid)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core"},{"Add":" using (refl; _≡_; _≢_; sym)"},{"Add":"open import Relation.Binary.PropositionalEquality.Properties using (setoid)"},{"Add":"open import Relation.Unary using (Pred; Decidable)"},{"Add":"open import Relation.Nullary.Negation.Core using (¬_)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" a b c p : Level"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Introduction (⁺) and elimination (⁻) rules for list operations"},{"Add":"------------------------------------------------------------------------"},{"Add":"-- map"},{"Add":""},{"Add":"module _ {A : Set a} {B : Set b} where"},{"Add":""},{"Add":" map⁺ : ∀ {f} → (∀ {x y} → f x ≡ f y → x ≡ y) →"},{"Add":" ∀ {n} {xs : Vec A n} → Unique xs → Unique (map f xs)"},{"Add":" map⁺ = Setoid.map⁺ (setoid A) (setoid B)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- take & drop"},{"Add":""},{"Add":"module _ {A : Set a} where"},{"Add":""},{"Add":" drop⁺ : ∀ {n} m {xs : Vec A (m + n)} → Unique xs → Unique (drop m xs)"},{"Add":" drop⁺ = Setoid.drop⁺ (setoid A)"},{"Add":""},{"Add":" take⁺ : ∀ {n} m {xs : Vec A (m + n)} → Unique xs → Unique (take m xs)"},{"Add":" take⁺ = Setoid.take⁺ (setoid A)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- tabulate"},{"Add":""},{"Add":"module _ {A : Set a} where"},{"Add":""},{"Add":" tabulate⁺ : ∀ {n} {f : Fin n → A} → (∀ {i j} → f i ≡ f j → i ≡ j) → Unique (tabulate f)"},{"Add":" tabulate⁺ = Setoid.tabulate⁺ (setoid A)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- lookup"},{"Add":""},{"Add":"module _ {A : Set a} where"},{"Add":""},{"Add":" lookup-injective : ∀ {n} {xs : Vec A n} → Unique xs → ∀ i j → lookup xs i ≡ lookup xs j → i ≡ j"},{"Add":" lookup-injective = Setoid.lookup-injective (setoid A)"}]}},{"FileAdd":{"meta":{"basename":"Linked.agda","metadata":0},"context":[{"change":0,"pos":1416065}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Vectors where every consecutative pair of elements is related."},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Vec.Relation.Unary.Linked {a} {A : Set a} where"},{"Add":""},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc; _+_)"},{"Add":"open import Data.Vec.Base as Vec using (Vec; []; _∷_)"},{"Add":"open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)"},{"Add":"open import Data.Product.Base as Product using (_,_; _×_; uncurry; <_,_>)"},{"Add":"open import Function.Base using (id; _∘_)"},{"Add":"open import Level using (Level; _⊔_)"},{"Add":"open import Relation.Binary.Core using (Rel; _⇒_)"},{"Add":"import Relation.Binary.Definitions as B"},{"Add":"open import Relation.Binary.Construct.Intersection renaming (_∩_ to _∩ᵇ_)"},{"Add":"open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)"},{"Add":"open import Relation.Unary as U renaming (_∩_ to _∩ᵘ_) hiding (_⇒_)"},{"Add":"open import Relation.Nullary.Decidable as Dec using (yes; no; _×-dec_; map′)"},{"Add":""},{"Add":"private"},{"Add":" variable"},{"Add":" p q r s ℓ : Level"},{"Add":" P : Rel A p"},{"Add":" Q : Rel A q"},{"Add":" R : Rel A r"},{"Add":" S : Rel A s"},{"Add":" n : ℕ"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Definition"},{"Add":""},{"Add":"-- Linked R xs means that every consecutative pair of elements in xs is"},{"Add":"-- a member of relation R."},{"Add":""},{"Add":"infixr 5 _∷_"},{"Add":""},{"Add":"data Linked (R : Rel A ℓ) : Vec A n → Set (a ⊔ ℓ) where"},{"Add":" [] : Linked R []"},{"Add":" [-] : ∀ {x} → Linked R (x ∷ [])"},{"Add":" _∷_ : ∀ {x} {xs : Vec A (suc n)} → R x (Vec.head xs) → Linked R xs → Linked R (x ∷ xs)"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Operations"},{"Add":""},{"Add":"head : ∀ {x} {xs : Vec A (suc n)} → Linked R (x ∷ xs) → R x (Vec.head xs)"},{"Add":"head (Rxy ∷ Rxs) = Rxy"},{"Add":""},{"Add":"tail : ∀ {xs : Vec A (suc n)} → Linked R xs → Linked R (Vec.tail xs)"},{"Add":"tail [-] = []"},{"Add":"tail (_ ∷ Rxs) = Rxs"},{"Add":""},{"Add":"map : R ⇒ S → Linked R {n} ⊆ Linked S"},{"Add":"map R⇒S [] = []"},{"Add":"map R⇒S [-] = [-]"},{"Add":"map R⇒S (x~xs ∷ pxs) = R⇒S x~xs ∷ map R⇒S pxs"},{"Add":""},{"Add":"zipWith : P ∩ᵇ Q ⇒ R → Linked P {n} ∩ᵘ Linked Q ⊆ Linked R"},{"Add":"zipWith f ([] , []) = []"},{"Add":"zipWith f ([-] , [-]) = [-]"},{"Add":"zipWith f (px ∷ pxs , qx ∷ qxs) = f (px , qx) ∷ zipWith f (pxs , qxs)"},{"Add":""},{"Add":"unzipWith : R ⇒ P ∩ᵇ Q → Linked R {n} ⊆ Linked P ∩ᵘ Linked Q"},{"Add":"unzipWith f [] = [] , []"},{"Add":"unzipWith f [-] = [-] , [-]"},{"Add":"unzipWith f (rx ∷ rxs) = Product.zip _∷_ _∷_ (f rx) (unzipWith f rxs)"},{"Add":""},{"Add":"zip : Linked P {n} ∩ᵘ Linked Q ⊆ Linked (P ∩ᵇ Q)"},{"Add":"zip = zipWith id"},{"Add":""},{"Add":"unzip : Linked (P ∩ᵇ Q) {n} ⊆ Linked P ∩ᵘ Linked Q"},{"Add":"unzip = unzipWith id"},{"Add":""},{"Add":"------------------------------------------------------------------------"},{"Add":"-- Properties of predicates preserved by Linked"},{"Add":""},{"Add":"linked? : B.Decidable R → U.Decidable (Linked R {n})"},{"Add":"linked? R? [] = yes []"},{"Add":"linked? R? (x ∷ []) = yes [-]"},{"Add":"linked? R? (x ∷ y ∷ xs) ="},{"Add":" map′ (uncurry _∷_) < head , tail > (R? x y ×-dec linked? R? (y ∷ xs))"},{"Add":""},{"Add":"irrelevant : B.Irrelevant R → U.Irrelevant (Linked R {n})"},{"Add":"irrelevant irr [] [] = refl"},{"Add":"irrelevant irr [-] [-] = refl"},{"Add":"irrelevant irr (px₁ ∷ pxs₁) (px₂ ∷ pxs₂) ="},{"Add":" cong₂ _∷_ (irr px₁ px₂) (irrelevant irr pxs₁ pxs₂)"},{"Add":""},{"Add":"satisfiable : U.Satisfiable (Linked R)"},{"Add":"satisfiable = [] , []"}]}},{"FileAdd":{"meta":{"basename":"Linked","metadata":576},"context":[{"change":0,"pos":1416065}],"contents":[]}},{"FileAdd":{"meta":{"basename":"Properties.agda","metadata":0},"context":[{"change":0,"pos":1427324}],"contents":[{"Add":"------------------------------------------------------------------------"},{"Add":"-- The Agda standard library"},{"Add":"--"},{"Add":"-- Properties related to Linked"},{"Add":"------------------------------------------------------------------------"},{"Add":""},{"Add":"{-# OPTIONS --cubical-compatible --safe #-}"},{"Add":""},{"Add":"module Data.Vec.Relation.Unary.Linked.Properties where"},{"Add":""},{"Add":"open import Data.Vec.Base as Vec"},{"Add":"open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)"},{"Add":"import Data.Vec.Relation.Unary.All.Properties as All"},{"Add":"open import Data.Vec.Relation.Unary.Linked as Linked"},{"Add":" using (Linked; []; [-]; _∷_)"},{"Add":"open import Data.Fin.Base using (zero; suc; _<_)"},{"Add":"open import Data.Nat.Base using (ℕ; zero; suc; s

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