Setoid.v

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Require Export Coq.Classes.SetoidTactics.

Export Morphisms.ProperNotations.

For backward compatibility

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

Register Build_Setoid_Theory as plugins.setoid_ring.Build_Setoid_Theory.

Definition Seq_refl A Aeq (s : Setoid_Theory A Aeq) : forall x:A, Aeq x x.A:Type
Aeq:relation A
s:Setoid_Theory A Aeq
forall x : A, Aeq x x
Proof.A:Type
Aeq:relation A
s:Setoid_Theory A Aeq
forall x : A, Aeq x x
  unfold Setoid_Theory in s.A:Type
Aeq:relation A
s:Equivalence Aeq
forall x : A, Aeq x x intros ; reflexivity.
Defined.

Definition Seq_sym A Aeq (s : Setoid_Theory A Aeq) : forall x y:A, Aeq x y -> Aeq y x.A:Type
Aeq:relation A
s:Setoid_Theory A Aeq
forall x y : A, Aeq x y -> Aeq y x
Proof.A:Type
Aeq:relation A
s:Setoid_Theory A Aeq
forall x y : A, Aeq x y -> Aeq y x
  unfold Setoid_Theory in s.A:Type
Aeq:relation A
s:Equivalence Aeq
forall x y : A, Aeq x y -> Aeq y x intros ; symmetry ; assumption.
Defined.

Definition Seq_trans A Aeq (s : Setoid_Theory A Aeq) : forall x y z:A, Aeq x y -> Aeq y z -> Aeq x z.A:Type
Aeq:relation A
s:Setoid_Theory A Aeq
forall x y z : A, Aeq x y -> Aeq y z -> Aeq x z
Proof.A:Type
Aeq:relation A
s:Setoid_Theory A Aeq
forall x y z : A, Aeq x y -> Aeq y z -> Aeq x z
  unfold Setoid_Theory in s.A:Type
Aeq:relation A
s:Equivalence Aeq
forall x y z : A, Aeq x y -> Aeq y z -> Aeq x z intros ; transitivity y ; assumption.
Defined.

Some tactics for manipulating Setoid Theory not officially declared as Setoid.

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  idtac "trans_st on Setoid_Theory is OBSOLETE";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  idtac "sym_st on Setoid_Theory is OBSOLETE";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  idtac "refl_st on Setoid_Theory is OBSOLETE";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).forall A : Set, Setoid_Theory A eq
Proof.forall A : Set, Setoid_Theory A eq
  constructor; congruence.
Qed.