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Require Export Coq.Program.Tactics Coq.Classes.SetoidTactics Coq.Classes.CMorphisms .
Set Universe Polymorphism.
Delimit Scope category_theory_scope with category_theory.
Open Scope category_theory_scope.
Infix "∧" := prod (at level 80, right associativity) : category_theory_scope.
Class Setoid A := {
equiv : crelation A;
setoid_equiv :> Equivalence equiv
}.
Notation "f ≈ g" := (equiv f g) (at level 79) : category_theory_scope.
Open Scope list_scope.
Generalizable All Variables.
Fixpoint list_equiv `{Setoid A} (xs ys : list A) : Type :=
match xs, ys with
| nil, nil => True
| x :: xs, y :: ys => x ≈ y ∧ list_equiv xs ys
| _, _ => False
end.
Axiom proof_admitted : False.
Tactic Notation "admit" := abstract case proof_admitted.
Program Instance list_equivalence `{Setoid A} : Equivalence list_equiv.
Next Obligation.
repeat intro.
induction x; simpl; split; auto.
reflexivity.
Qed.
Next Obligation.
repeat intro.
generalize dependent y.
induction x, y; simpl; intros; auto.
destruct X; split.
now symmetry.
intuition.
Qed.
Next Obligation.
admit.
Defined.
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