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
66
67
68
69
70
71
72
73
74
|
(* -*- coding: utf-8 -*- *)
(************************************************************************)
(* 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 *)
(************************************************************************)
(* $Id$ *)
(** * Proofs about standard combinators, exports functional extensionality.
Author: Matthieu Sozeau
Institution: LRI, CNRS UMR 8623 - University Paris Sud
*)
Require Import Coq.Program.Basics.
Require Export FunctionalExtensionality.
Open Scope program_scope.
(** Composition has [id] for neutral element and is associative. *)
Lemma compose_id_left : forall A B (f : A -> B), id ∘ f = f.
Proof.
intros.
unfold id, compose.
symmetry. apply eta_expansion.
Qed.
Lemma compose_id_right : forall A B (f : A -> B), f ∘ id = f.
Proof.
intros.
unfold id, compose.
symmetry ; apply eta_expansion.
Qed.
Lemma compose_assoc : forall A B C D (f : A -> B) (g : B -> C) (h : C -> D),
h ∘ g ∘ f = h ∘ (g ∘ f).
Proof.
intros.
reflexivity.
Qed.
Hint Rewrite @compose_id_left @compose_id_right : core.
Hint Rewrite <- @compose_assoc : core.
(** [flip] is involutive. *)
Lemma flip_flip : forall A B C, @flip A B C ∘ flip = id.
Proof.
unfold flip, compose.
intros.
extensionality x ; extensionality y ; extensionality z.
reflexivity.
Qed.
(** [prod_curry] and [prod_uncurry] are each others inverses. *)
Lemma prod_uncurry_curry : forall A B C, @prod_uncurry A B C ∘ prod_curry = id.
Proof.
simpl ; intros.
unfold prod_uncurry, prod_curry, compose.
extensionality x ; extensionality y ; extensionality z.
reflexivity.
Qed.
Lemma prod_curry_uncurry : forall A B C, @prod_curry A B C ∘ prod_uncurry = id.
Proof.
simpl ; intros.
unfold prod_uncurry, prod_curry, compose.
extensionality x ; extensionality p.
destruct p ; simpl ; reflexivity.
Qed.
|
