The Coq Proof Assistant

# Library Coq.Setoids.Setoid

Require Export Coq.Classes.SetoidTactics.
Export Morphisms.ProperNotations.
Require Coq.ssr.ssrsetoid.
For backward compatibility
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.
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idtac "trans_st on Setoid_Theory is OBSOLETE";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idtac "sym_st on Setoid_Theory is OBSOLETE";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idtac "refl_st on Setoid_Theory is OBSOLETE";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).