blob: e962faf04757871e777b6b3b6ba0239554677925 (
plain)
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
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** * Morphism instances for relations.
Author: Matthieu Sozeau
Institution: LRI, CNRS UMR 8623 - University Paris Sud
*)
Require Import Relation_Definitions.
Require Import Coq.Classes.Morphisms.
Require Import Coq.Program.Program.
Generalizable Variables A l.
(** Morphisms for relations *)
Instance relation_conjunction_morphism : Proper (relation_equivalence (A:=A) ==>
relation_equivalence ==> relation_equivalence) relation_conjunction.
Proof. firstorder. Qed.
Instance relation_disjunction_morphism : Proper (relation_equivalence (A:=A) ==>
relation_equivalence ==> relation_equivalence) relation_disjunction.
Proof. firstorder. Qed.
(* Predicate equivalence is exactly the same as the pointwise lifting of [iff]. *)
Require Import List.
Lemma predicate_equivalence_pointwise (l : list Type) :
Proper (@predicate_equivalence l ==> pointwise_lifting iff l) id.
Proof. do 2 red. unfold predicate_equivalence. auto. Qed.
Lemma predicate_implication_pointwise (l : list Type) :
Proper (@predicate_implication l ==> pointwise_lifting impl l) id.
Proof. do 2 red. unfold predicate_implication. auto. Qed.
(** The instanciation at relation allows to rewrite applications of relations
[R x y] to [R' x y] when [R] and [R'] are in [relation_equivalence]. *)
Instance relation_equivalence_pointwise :
Proper (relation_equivalence ==> pointwise_relation A (pointwise_relation A iff)) id.
Proof. intro. apply (predicate_equivalence_pointwise (cons A (cons A nil))). Qed.
Instance subrelation_pointwise :
Proper (subrelation ==> pointwise_relation A (pointwise_relation A impl)) id.
Proof. intro. apply (predicate_implication_pointwise (cons A (cons A nil))). Qed.
Lemma inverse_pointwise_relation A (R : relation A) :
relation_equivalence (pointwise_relation A (inverse R)) (inverse (pointwise_relation A R)).
Proof. intros. split; firstorder. Qed.
|
