1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
|
(* -*- 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 *)
(************************************************************************)
(* Certified Haskell Prelude.
* 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.
Ltac rew H := clrewrite H.
Lemma setoideq_eq [ sa : Setoid a eqa ] : forall x y, eqa x y -> x = y.
Proof.
admit.
Qed.
Implicit Arguments setoideq_eq [[a] [eqa] [sa]].
Ltac setoideq :=
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_tac
end.
|
