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
75
76
77
78
79
80
81
82
83
84
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** This module states the axiom of (dependent) functional extensionality and (dependent) eta-expansion.
It introduces a tactic [extensionality] to apply the axiom of extensionality to an equality goal. *)
(** The converse of functional extensionality. *)
Lemma equal_f : forall {A B : Type} {f g : A -> B},
f = g -> forall x, f x = g x.
Proof.
intros.
rewrite H.
auto.
Qed.
Lemma equal_f_dep : forall {A B} {f g : forall (x : A), B x},
f = g -> forall x, f x = g x.
Proof.
intros A B f g <- H; reflexivity.
Qed.
(** Statements of functional extensionality for simple and dependent functions. *)
Axiom functional_extensionality_dep : forall {A} {B : A -> Type},
forall (f g : forall x : A, B x),
(forall x, f x = g x) -> f = g.
Lemma functional_extensionality {A B} (f g : A -> B) :
(forall x, f x = g x) -> f = g.
Proof.
intros ; eauto using @functional_extensionality_dep.
Qed.
(** Extensionality of [forall]s follows from functional extensionality. *)
Lemma forall_extensionality {A} {B C : A -> Type} (H : forall x : A, B x = C x)
: (forall x, B x) = (forall x, C x).
Proof.
apply functional_extensionality in H. destruct H. reflexivity.
Defined.
Lemma forall_extensionalityP {A} {B C : A -> Prop} (H : forall x : A, B x = C x)
: (forall x, B x) = (forall x, C x).
Proof.
apply functional_extensionality in H. destruct H. reflexivity.
Defined.
Lemma forall_extensionalityS {A} {B C : A -> Set} (H : forall x : A, B x = C x)
: (forall x, B x) = (forall x, C x).
Proof.
apply functional_extensionality in H. destruct H. reflexivity.
Defined.
(** Apply [functional_extensionality], introducing variable x. *)
Tactic Notation "extensionality" ident(x) :=
match goal with
[ |- ?X = ?Y ] =>
(apply (@functional_extensionality _ _ X Y) ||
apply (@functional_extensionality_dep _ _ X Y) ||
apply forall_extensionalityP ||
apply forall_extensionalityS ||
apply forall_extensionality) ; intro x
end.
(** Eta expansion follows from extensionality. *)
Lemma eta_expansion_dep {A} {B : A -> Type} (f : forall x : A, B x) :
f = fun x => f x.
Proof.
intros.
extensionality x.
reflexivity.
Qed.
Lemma eta_expansion {A B} (f : A -> B) : f = fun x => f x.
Proof.
apply (eta_expansion_dep f).
Qed.
|
