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
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
|
(*
Example proof script for testing Coq Proof General.
$Id$
Maintainer: Proof General maintainer <proofgen@dcs.ed.ac.uk>
*)
(****************************************************************************)
(* The Calculus of Inductive Constructions *)
(* *)
(* Projet Coq *)
(* *)
(* INRIA ENS-CNRS *)
(* Rocquencourt Lyon *)
(* *)
(* Coq V6.1 *)
(* Oct 1st 1996 *)
(* *)
(****************************************************************************)
(* List.v *)
(****************************************************************************)
Require Le.
(* Some programs and results about lists *)
Parameter A:Set.
Inductive list : Set := nil : list | cons : A -> list -> list.
(*
Fixpoint app [l:list] : list -> list
:= [m:list]Case l of
(* nil *) m
(* cons a l1 *) [a:A][l1:list](cons a (app l1 m)) end.
*)
Fixpoint app [l:list] : list -> list
:= [m:list]<list>Cases l of
nil => m
| (cons a l1) => (cons a (app l1 m))
end.
Lemma app_nil_end : (l:list)(l=(app l nil)).
Proof.
Intro l ; Elim l ; Simpl ; Auto.
Induction 1; Auto.
Qed.
Hint app_nil_end.
Lemma app_ass : (l,m,n : list)(app (app l m) n)=(app l (app m n)).
Proof.
Intros l m n ; Elim l ; Simpl ; Auto.
Induction 1; Auto.
Qed.
Hint app_ass.
Lemma ass_app : (l,m,n : list)(app l (app m n))=(app (app l m) n).
Proof.
Auto.
Qed.
Hint ass_app.
(*
Definition tail := [l:list]Case l of
(* nil *) nil
(* cons a m *) [a:A][m:list]m end : list->list.
*)
Definition tail :=
[l:list] <list>Cases l of (cons _ m) => m | _ => nil end : list->list.
Lemma nil_cons : (a:A)(m:list)~(nil=(cons a m)).
Intros;Discriminate.
Qed.
(****************************************)
(* Length of lists *)
(****************************************)
(*
Fixpoint length [l:list] : nat
:= Case l of (* nil *) O
(* cons a m *) [a:A][m:list](S (length m)) end.
*)
Fixpoint length [l:list] : nat
:= <nat>Cases l of (cons _ m) => (S (length m)) | _ => O end.
(******************************)
(* Length order of lists *)
(******************************)
Section length_order.
Definition lel := [l,m:list](le (length l) (length m)).
Variables a,b:A.
Variables l,m,n:list.
Lemma lel_refl : (lel l l).
Proof.
Unfold lel ; Auto.
Qed.
Lemma lel_trans : (lel l m)->(lel m n)->(lel l n).
Proof.
Unfold lel ; Intros.
Apply le_trans with (length m) ; Auto.
Qed.
Lemma lel_cons_cons : (lel l m)->(lel (cons a l) (cons b m)).
Proof.
Unfold lel ; Simpl ; Auto.
Qed.
Lemma lel_cons : (lel l m)->(lel l (cons b m)).
Proof.
Unfold lel ; Simpl ; Auto.
Qed.
Lemma lel_tail : (lel (cons a l) (cons b m)) -> (lel l m).
Proof.
Unfold lel ; Simpl ; Auto.
Qed.
Lemma lel_nil : (l':list)(lel l' nil)->(nil=l').
Proof.
Intro l' ; Elim l' ; Auto.
Intros a' y H H0.
(* <list>nil=(cons a' y)
============================
H0 : (lel (cons a' y) nil)
H : (lel y nil)->(<list>nil=y)
y : list
a' : A
l' : list *)
Absurd (le (S (length y)) O); Auto.
Qed.
End length_order.
Hint lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons.
Fixpoint In [a:A;l:list] : Prop :=
(Case l of (* nil *) False
(* cons b m *) [b:A][m:list](b=a)\/(In a m) end).
Lemma in_eq : (a:A)(l:list)(In a (cons a l)).
Proof.
Simpl ; Auto.
Qed.
Hint in_eq.
Lemma in_cons : (a,b:A)(l:list)(In b l)->(In b (cons a l)).
Proof.
Simpl ; Auto.
Qed.
Hint in_cons.
Lemma in_app_or : (l,m:list)(a:A)(In a (app l m))->((In a l)\/(In a m)).
Proof.
Intros l m a.
Elim l ; Simpl ; Auto.
Intros a0 y H H0.
(* ((<A>a0=a)\/(In a y))\/(In a m)
============================
H0 : (<A>a0=a)\/(In a (app y m))
H : (In a (app y m))->((In a y)\/(In a m))
y : list
a0 : A
a : A
m : list
l : list *)
Elim H0 ; Auto.
Intro H1.
(* ((<A>a0=a)\/(In a y))\/(In a m)
============================
H1 : (In a (app y m)) *)
Elim (H H1) ; Auto.
Qed.
Immediate in_app_or.
Lemma in_or_app : (l,m:list)(a:A)((In a l)\/(In a m))->(In a (app l m)).
Proof.
Intros l m a.
Elim l ; Simpl ; Intro H.
(* 1 (In a m)
============================
H : False\/(In a m)
a : A
m : list
l : list *)
Elim H ; Auto ; Intro H0.
(* (In a m)
============================
H0 : False *)
Elim H0. (* subProof completed *)
Intros y H0 H1.
(* 2 (<A>H=a)\/(In a (app y m))
============================
H1 : ((<A>H=a)\/(In a y))\/(In a m)
H0 : ((In a y)\/(In a m))->(In a (app y m))
y : list *)
Elim H1 ; Auto 4.
Intro H2.
(* (<A>H=a)\/(In a (app y m))
============================
H2 : (<A>H=a)\/(In a y) *)
Elim H2 ; Auto.
Qed.
Hint in_or_app.
Definition incl := [l,m:list](a:A)(In a l)->(In a m).
Hint Unfold incl.
Lemma incl_refl : (l:list)(incl l l).
Proof.
Auto.
Qed.
Hint incl_refl.
Lemma incl_tl : (a:A)(l,m:list)(incl l m)->(incl l (cons a m)).
Proof.
Auto.
Qed.
Immediate incl_tl.
Lemma incl_tran : (l,m,n:list)(incl l m)->(incl m n)->(incl l n).
Proof.
Auto.
Qed.
Lemma incl_appl : (l,m,n:list)(incl l n)->(incl l (app n m)).
Proof.
Auto.
Qed.
Immediate incl_appl.
Lemma incl_appr : (l,m,n:list)(incl l n)->(incl l (app m n)).
Proof.
Auto.
Qed.
Immediate incl_appr.
Lemma incl_cons : (a:A)(l,m:list)(In a m)->(incl l m)->(incl (cons a l) m).
Proof.
Unfold incl ; Simpl ; Intros a l m H H0 a0 H1.
(* (In a0 m)
============================
H1 : (<A>a=a0)\/(In a0 l)
a0 : A
H0 : (a:A)(In a l)->(In a m)
H : (In a m)
m : list
l : list
a : A *)
Elim H1.
(* 1 (<A>a=a0)->(In a0 m) *)
Elim H1 ; Auto ; Intro H2.
(* (<A>a=a0)->(In a0 m)
============================
H2 : <A>a=a0 *)
Elim H2 ; Auto. (* solves subgoal *)
(* 2 (In a0 l)->(In a0 m) *)
Auto.
Qed.
Hint incl_cons.
Lemma incl_app : (l,m,n:list)(incl l n)->(incl m n)->(incl (app l m) n).
Proof.
Unfold incl ; Simpl ; Intros l m n H H0 a H1.
(* (In a n)
============================
H1 : (In a (app l m))
a : A
H0 : (a:A)(In a m)->(In a n)
H : (a:A)(In a l)->(In a n)
n : list
m : list
l : list *)
Elim (in_app_or l m a) ; Auto.
Qed.
Hint incl_app.
(* Id: List.v,v 1.4 1996/10/07 07:50:36 ccornes Exp *)
|
