Library Coq.Setoids.Setoid

Require Export Coq.Classes.SetoidTactics.

Export Morphisms.ProperNotations.

Require Coq.ssr.ssrsetoid.

Definition Setoid_Theory := @Equivalence.
Definition Build_Setoid_Theory := @Build_Equivalence.

Register Build_Setoid_Theory as plugins.ring.Build_Setoid_Theory.

Definition Seq_refl A Aeq (s : Setoid_Theory A Aeq) : forall x:A, Aeq x x.

Definition Seq_sym A Aeq (s : Setoid_Theory A Aeq) : forall x y:A, Aeq x y -> Aeq y x.

Definition Seq_trans A Aeq (s : Setoid_Theory A Aeq) : forall x y z:A, Aeq x y -> Aeq y z -> Aeq x z.

Ltac trans_st x :=
  idtac "trans_st on Setoid_Theory is OBSOLETE";
  idtac "use transitivity on Equivalence instead";
  match goal with
    | H : Setoid_Theory _ ?eqA |- ?eqA _ _ =>
      apply (Seq_trans _ _ H) with x; auto
  end.

Ltac sym_st :=
  idtac "sym_st on Setoid_Theory is OBSOLETE";
  idtac "use symmetry on Equivalence instead";
  match goal with
    | H : Setoid_Theory _ ?eqA |- ?eqA _ _ =>
      apply (Seq_sym _ _ H); auto
  end.

Ltac refl_st :=
  idtac "refl_st on Setoid_Theory is OBSOLETE";
  idtac "use reflexivity on Equivalence instead";
  match goal with
    | H : Setoid_Theory _ ?eqA |- ?eqA _ _ =>
      apply (Seq_refl _ _ H); auto
  end.

Definition gen_st : forall A : Set, Setoid_Theory _ (@eq A).