blob: 023cb5f59d4515740c95f4cb79ad672e3e122bc7 (
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
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
|
(************************************************************************)
(* 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 *)
(************************************************************************)
(************************************************************************)
Lemma essai : forall x : nat, x = x.
refine
((fun x0 : nat => match x0 with
| O => _
| S p => _
end)).
Restart.
refine
(fun x0 : nat => match x0 as n return (n = n) with
| O => _
| S p => _
end). (* OK *)
Restart.
refine
(fun x0 : nat => match x0 as n return (n = n) with
| O => _
| S p => _
end). (* OK *)
Restart.
(**
Refine [x0:nat]Cases x0 of O => ? | (S p) => ? end. (* cannot be executed *)
**)
Abort.
(************************************************************************)
Lemma T : nat.
refine (S _).
Abort.
(************************************************************************)
Lemma essai2 : forall x : nat, x = x.
refine (fix f (x : nat) : x = x := _).
Restart.
refine
(fix f (x : nat) : x = x :=
match x as n return (n = n :>nat) with
| O => _
| S p => _
end).
Restart.
refine
(fix f (x : nat) : x = x :=
match x as n return (n = n) with
| O => _
| S p => _
end).
Restart.
refine
(fix f (x : nat) : x = x :=
match x as n return (n = n :>nat) with
| O => _
| S p => f_equal S _
end).
Restart.
refine
(fix f (x : nat) : x = x :=
match x as n return (n = n :>nat) with
| O => _
| S p => f_equal S _
end).
Abort.
(************************************************************************)
Parameter f : nat * nat -> nat -> nat.
Lemma essai : nat.
refine (f _ ((fun x : nat => _:nat) 0)).
Restart.
refine (f _ 0).
Abort.
(************************************************************************)
Parameter P : nat -> Prop.
Lemma essai : {x : nat | x = 1}.
refine (exist _ 1 _). (* ECHEC *)
Restart.
(* mais si on contraint par le but alors ca marche : *)
(* Remarque : on peut toujours faire ça *)
refine (exist _ 1 _:{x : nat | x = 1}).
Restart.
refine (exist (fun x : nat => x = 1) 1 _).
Abort.
(************************************************************************)
Lemma essai : forall n : nat, {x : nat | x = S n}.
refine
(fun n : nat =>
match n return {x : nat | x = S n} with
| O => _
| S p => _
end).
Restart.
refine
(fun n : nat => match n with
| O => _
| S p => _
end).
Restart.
refine
(fun n : nat =>
match n return {x : nat | x = S n} with
| O => _
| S p => _
end).
Restart.
refine
(fix f (n : nat) : {x : nat | x = S n} :=
match n return {x : nat | x = S n} with
| O => _
| S p => _
end).
Restart.
refine
(fix f (n : nat) : {x : nat | x = S n} :=
match n return {x : nat | x = S n} with
| O => _
| S p => _
end).
exists 1. trivial.
elim (f p).
refine
(fun (x : nat) (h : x = S p) => exist (fun x : nat => x = S (S p)) (S x) _).
rewrite h. auto.
Qed.
(* Quelques essais de recurrence bien fondée *)
Require Import Wf.
Require Import Wf_nat.
Lemma essai_wf : nat -> nat.
refine
(fun x : nat =>
well_founded_induction _ (fun _ : nat => nat -> nat)
(fun (phi0 : nat) (w : forall phi : nat, phi < phi0 -> nat -> nat) =>
w x _) x x).
exact lt_wf.
Abort.
Require Import Compare_dec.
Require Import Lt.
Lemma fibo : nat -> nat.
refine
(well_founded_induction _ (fun _ : nat => nat)
(fun (x0 : nat) (fib : forall x : nat, x < x0 -> nat) =>
match zerop x0 with
| left _ => 1
| right h1 =>
match zerop (pred x0) with
| left _ => 1
| right h2 => fib (pred x0) _ + fib (pred (pred x0)) _
end
end)).
exact lt_wf.
auto with arith.
apply lt_trans with (m := pred x0); auto with arith.
Qed.
|
