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(* -*- coq-prog-args: ("-emacs-U" "-nois") -*- *)
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Tactics for typeclass-based setoids.
*
* Author: Matthieu Sozeau
* Institution: LRI, CNRS UMR 8623 - Universit�copyright Paris Sud
* 91405 Orsay, France *)
(* $Id: FSetAVL_prog.v 616 2007-08-08 12:28:10Z msozeau $ *)
Require Import Coq.Program.Program.
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Coq.Classes.SetoidClass.
(* Application of the extensionality axiom to turn a goal on leibinz equality to
a setoid equivalence. *)
Lemma setoideq_eq [ sa : Setoid a ] : forall x y : a, x == y -> x = y.
Proof.
admit.
Qed.
Implicit Arguments setoideq_eq [[a] [sa]].
(** Application of the extensionality principle for setoids. *)
Ltac setoideq_ext :=
match goal with
[ |- @eq ?A ?X ?Y ] => apply (setoideq_eq (a:=A) X Y)
end.
Ltac setoid_tactic :=
match goal with
| [ H : ?eq ?x ?y |- ?eq ?y ?x ] => sym ; apply H
| [ |- ?eq ?x ?x ] => refl
| [ H : ?eq ?x ?y, H' : ?eq ?y ?z |- ?eq' ?x ?z ] => trans y ; [ apply H | apply H' ]
| [ H : ?eq ?x ?y, H' : ?eq ?z ?y |- ?eq' ?x ?z ] => trans y ; [ apply H | sym ; apply H' ]
| [ H : ?eq ?y ?x, H' : ?eq ?z ?y |- ?eq' ?x ?z ] => trans y ; [ sym ; apply H | sym ; apply H' ]
| [ H : ?eq ?y ?x, H' : ?eq ?y ?z |- ?eq' ?x ?z ] => trans y ; [ sym ; apply H | apply H' ]
| [ H : ?eq ?x ?y |- @equiv _ _ _ ?y ?x ] => sym ; apply H
| [ |- @equiv _ _ _ ?x ?x ] => refl
| [ H : ?eq ?x ?y, H' : ?eq ?y ?z |- @equiv _ _ _ ?x ?z ] => trans y ; [ apply H | apply H' ]
| [ H : ?eq ?x ?y, H' : ?eq ?z ?y |- @equiv _ _ _ ?x ?z ] => trans y ; [ apply H | sym ; apply H' ]
| [ H : ?eq ?y ?x, H' : ?eq ?z ?y |- @equiv _ _ _ ?x ?z ] => trans y ; [ sym ; apply H | sym ; apply H' ]
| [ H : ?eq ?y ?x, H' : ?eq ?y ?z |- @equiv _ _ _ ?x ?z ] => trans y ; [ sym ; apply H | apply H' ]
| [ H : @equiv ?A ?R ?s ?x ?y |- @equiv _ _ _ ?y ?x ] => sym ; apply H
| [ |- @equiv _ _ _ ?x ?x ] => refl
| [ H : @equiv ?A ?R ?s ?x ?y, H' : @equiv ?A ?R ?s ?y ?z |- @equiv _ _ _ ?x ?z ] => trans y ; [ apply H | apply H' ]
| [ H : @equiv ?A ?R ?s ?x ?y, H' : @equiv ?A ?R ?s ?z ?y |- @equiv _ _ _ ?x ?z ] => trans y ; [ apply H | sym ; apply H' ]
| [ H : @equiv ?A ?R ?s ?y ?x, H' : @equiv ?A ?R ?s ?z ?y |- @equiv _ _ _ ?x ?z ] => trans y ; [ sym ; apply H | sym ; apply H' ]
| [ H : @equiv ?A ?R ?s ?y ?x, H' : @equiv ?A ?R ?s ?y ?z |- @equiv _ _ _ ?x ?z ] => trans y ; [ sym ; apply H | apply H' ]
| [ H : not (@equiv ?A ?R ?s ?X ?X) |- _ ] => elim H ; refl
| [ H : not (@equiv ?A ?R ?s ?X ?Y), H' : @equiv ?A ?R ?s ?Y ?X |- _ ] => elim H ; sym ; apply H
| [ H : not (@equiv ?A ?R ?s ?X ?Y), H' : ?R ?Y ?X |- _ ] => elim H ; sym ; apply H
| [ H : not (@equiv ?A ?R ?s ?X ?Y) |- False ] => elim H ; clear H ; setoid_tactic
end.
(** Need to fix fresh to not fail if some arguments are not identifiers. *)
(* Ltac setoid_sat ::= *)
(* match goal with *)
(* | [ H : ?x == ?y |- _ ] => let name:=fresh "Heq" y x in add_hypothesis name (equiv_sym H) *)
(* | [ H : ?x =/= ?y |- _ ] => let name:=fresh "Hneq" y x in add_hypothesis name (nequiv_sym H) *)
(* | [ H : ?x == ?y, H' : ?y == ?z |- _ ] => let name:=fresh "Heq" x z in add_hypothesis name (equiv_trans H H') *)
(* | [ H : ?x == ?y, H' : ?y =/= ?z |- _ ] => let name:=fresh "Hneq" x z in add_hypothesis name (equiv_nequiv H H') *)
(* | [ H : ?x =/= ?y, H' : ?y == ?z |- _ ] => let name:=fresh "Hneq" x z in add_hypothesis name (nequiv_equiv H H') *)
(* end. *)
Ltac setoid_sat :=
match goal with
| [ H : ?x == ?y |- _ ] => let name:=fresh "Heq" in add_hypothesis name (equiv_sym H)
| [ H : ?x =/= ?y |- _ ] => let name:=fresh "Hneq" in add_hypothesis name (nequiv_sym H)
| [ H : ?x == ?y, H' : ?y == ?z |- _ ] => let name:=fresh "Heq" in add_hypothesis name (equiv_trans H H')
| [ H : ?x == ?y, H' : ?y =/= ?z |- _ ] => let name:=fresh "Hneq" in add_hypothesis name (equiv_nequiv H H')
| [ H : ?x =/= ?y, H' : ?y == ?z |- _ ] => let name:=fresh "Hneq" in add_hypothesis name (nequiv_equiv H H')
end.
Ltac setoid_saturate := repeat setoid_sat.
|
