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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
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.
Proof.
unfold Setoid_Theory in s. intros ; reflexivity.
Defined.
Definition Seq_sym A Aeq (s : Setoid_Theory A Aeq) : forall x y:A, Aeq x y -> Aeq y x.
Proof.
unfold Setoid_Theory in s. 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.
Proof.
unfold Setoid_Theory in s. 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idtac "trans_st on Setoid_Theory is OBSOLETE";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idtac "sym_st on Setoid_Theory is OBSOLETE";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idtac "refl_st on Setoid_Theory is OBSOLETE";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).
Proof.
constructor; congruence.
Qed.
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