blob: 857ffe947a827037530808f435621520172bbb5b (
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
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
85
86
87
88
89
90
91
92
93
94
95
|
(************************************************************************)
(* 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 *)
(************************************************************************)
(*i $Id: Logic_Type.v 8642 2006-03-17 10:09:02Z notin $ i*)
(** This module defines type constructors for types in [Type]
([Datatypes.v] and [Logic.v] defined them for types in [Set]) *)
Set Implicit Arguments.
Require Import Datatypes.
Require Export Logic.
(** Negation of a type in [Type] *)
Definition notT (A:Type) := A -> False.
(** Conjunction of types in [Type] *)
Inductive prodT (A B:Type) : Type :=
pairT : A -> B -> prodT A B.
Section prodT_proj.
Variables A B : Type.
Definition fstT (H:prodT A B) := match H with
| pairT x _ => x
end.
Definition sndT (H:prodT A B) := match H with
| pairT _ y => y
end.
End prodT_proj.
Definition prodT_uncurry (A B C:Type) (f:prodT A B -> C)
(x:A) (y:B) : C := f (pairT x y).
Definition prodT_curry (A B C:Type) (f:A -> B -> C)
(p:prodT A B) : C := match p with
| pairT x y => f x y
end.
(** Properties of [identity] *)
Section identity_is_a_congruence.
Variables A B : Type.
Variable f : A -> B.
Variables x y z : A.
Lemma sym_id : identity x y -> identity y x.
Proof.
destruct 1; trivial.
Qed.
Lemma trans_id : identity x y -> identity y z -> identity x z.
Proof.
destruct 2; trivial.
Qed.
Lemma congr_id : identity x y -> identity (f x) (f y).
Proof.
destruct 1; trivial.
Qed.
Lemma sym_not_id : notT (identity x y) -> notT (identity y x).
Proof.
red in |- *; intros H H'; apply H; destruct H'; trivial.
Qed.
End identity_is_a_congruence.
Definition identity_ind_r :
forall (A:Type) (a:A) (P:A -> Prop), P a -> forall y:A, identity y a -> P y.
intros A x P H y H0; case sym_id with (1 := H0); trivial.
Defined.
Definition identity_rec_r :
forall (A:Type) (a:A) (P:A -> Set), P a -> forall y:A, identity y a -> P y.
intros A x P H y H0; case sym_id with (1 := H0); trivial.
Defined.
Definition identity_rect_r :
forall (A:Type) (a:A) (P:A -> Type), P a -> forall y:A, identity y a -> P y.
intros A x P H y H0; case sym_id with (1 := H0); trivial.
Defined.
Hint Immediate sym_id sym_not_id: core v62.
|
