-
Notifications
You must be signed in to change notification settings - Fork 2
/
Ensf_couple.v
92 lines (80 loc) · 3.8 KB
/
Ensf_couple.v
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
(* This program is free software; you can redistribute it and/or *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1 *)
(* of the License, or (at your option) any later version. *)
(* *)
(* This program is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
(* GNU General Public License for more details. *)
(* *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this program; if not, write to the Free *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA *)
(* Contribution to the Coq Library V6.3 (July 1999) *)
(****************************************************************************)
(* The Calculus of Inductive Constructions *)
(* *)
(* Projet Coq *)
(* *)
(* INRIA ENS-CNRS *)
(* Rocquencourt Lyon *)
(* *)
(* Coq V5.10 *)
(* Nov 25th 1994 *)
(* *)
(****************************************************************************)
(* Ensf_couple.v *)
(****************************************************************************)
(* Formal Language Theory *)
(* *)
(* Judicael Courant - Jean-Christophe Filliatre *)
(* *)
(* Developped in V5.8 June-July 1993 *)
(* Ported to V5.10 October 1994 *)
(****************************************************************************)
Require Import Ensf_types.
(* *)
(* first et second renvoient repsectivement le premier et le deuxieme *)
(* element d'un couple. *)
(* *)
Definition first (x : Elt) : Elt :=
match x return Elt with
| natural n =>
(* natural *) zero
(* couple *)
| couple a b => a
(* up *)
| up e => zero
(* word *)
| word w => zero
end.
Definition second (x : Elt) : Elt :=
match x return Elt with
| natural n =>
(* natural *) zero
(* couple *)
| couple a b => b
(* up *)
| up e => zero
(* word *)
| word w => zero
end.
(* Grace a first et second on recupere facilement le lemme suivant : *)
Lemma equal_couple :
forall x y z t : Elt,
couple x y = couple z t :>Elt -> x = z :>Elt /\ y = t :>Elt.
intros x y z t H.
injection H; auto.
Qed.
Lemma couple_couple_inv1 :
forall a b c d : Elt, couple a c = couple b d :>Elt -> a = b :>Elt.
intros a b c d H.
injection H; auto.
Qed.
Lemma couple_couple_inv2 :
forall a b c d : Elt, couple a c = couple b d :>Elt -> c = d :>Elt.
intros a b c d H.
injection H; auto.
Qed.