#include <tuple>

  friend bool operator==(const valid_t &x, const valid_t &y) {
    return make_tuple(x.valid_int, x.valid_bnd) ==
           make_tuple(y.valid_int, y.valid_bnd);
  }

    if (x.valid_int < y.valid_int)
      return true;
    return x.valid_bnd < y.valid_bnd;

    return make_tuple(x.valid_int, x.valid_bnd) <
           make_tuple(y.valid_int, y.valid_bnd);

#include <tuple>

  }
  friend bool operator==(const vect &x, const vect &y) {
    return x.elts == y.elts;
  }
  friend bool operator<(const vect &x, const vect &y) {
    return x.elts < y.elts;

#include <tuple>

  friend bool operator==(const clause_t &x, const clause_t &y) {
    return make_tuple(x.gi, x.vi, x.tl, x.valid) ==
           make_tuple(y.gi, y.vi, y.tl, y.valid);
  }

    if (x.gi < y.gi)
      return true;
    if (x.gi > y.gi)
      return false;
    if (x.vi < y.vi)
      return true;
    if (x.vi > y.vi)
      return false;
    if (x.tl < y.tl)
      return true;
    if (x.tl > y.tl)
      return false;
    return x.valid < y.valid;

    return make_tuple(x.gi, x.vi, x.tl, x.valid) <
           make_tuple(y.gi, y.vi, y.tl, y.valid);