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
290
291
292
293
294
295
296
|
(************************************************************************)
(* 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 *)
(************************************************************************)
(** Author: Cristina Cornes
From : Constructing Recursion Operators in Type Theory
L. Paulson JSC (1986) 2, 325-355 *)
Require Import List.
Require Import Relation_Operators.
Require Import Operators_Properties.
Require Import Transitive_Closure.
Import ListNotations.
Section Wf_Lexicographic_Exponentiation.
Variable A : Set.
Variable leA : A -> A -> Prop.
Notation Power := (Pow A leA).
Notation Lex_Exp := (lex_exp A leA).
Notation ltl := (Ltl A leA).
Notation Descl := (Desc A leA).
Notation List := (list A).
Notation "<< x , y >>" := (exist Descl x y) (at level 0, x, y at level 100).
(* Hint Resolve d_one d_nil t_step. *)
Lemma left_prefix : forall x y z : List, ltl (x ++ y) z -> ltl x z.
Proof.
simple induction x.
simple induction z.
simpl; intros H.
inversion_clear H.
simpl; intros; apply (Lt_nil A leA).
intros a l HInd.
simpl.
intros.
inversion_clear H.
apply (Lt_hd A leA); auto with sets.
apply (Lt_tl A leA).
apply (HInd y y0); auto with sets.
Qed.
Lemma right_prefix :
forall x y z : List,
ltl x (y ++ z) ->
ltl x y \/ (exists y' : List, x = y ++ y' /\ ltl y' z).
Proof.
intros x y; generalize x.
elim y; simpl.
right.
exists x0; auto with sets.
intros.
inversion H0.
left; apply (Lt_nil A leA).
left; apply (Lt_hd A leA); auto with sets.
generalize (H x1 z H3).
simple induction 1.
left; apply (Lt_tl A leA); auto with sets.
simple induction 1.
simple induction 1; intros.
rewrite H8.
right; exists x2; auto with sets.
Qed.
Lemma desc_prefix : forall (x : List) (a : A), Descl (x ++ [a]) -> Descl x.
Proof.
intros.
inversion H.
- apply app_cons_not_nil in H1 as [].
- assert (x ++ [a] = [x0]) by auto with sets.
apply app_eq_unit in H0 as [(->, _)| (_, [=])].
auto using d_nil.
- apply app_inj_tail in H0 as (<-, _).
assumption.
Qed.
Lemma desc_tail :
forall (x : List) (a b : A),
Descl (b :: x ++ [a]) -> clos_refl_trans A leA a b.
Proof.
intro.
apply rev_ind with
(P :=
fun x : List =>
forall a b : A, Descl (b :: x ++ [a]) -> clos_refl_trans A leA a b);
intros.
- inversion H.
assert ([b; a] = ([] ++ [b]) ++ [a]) by auto with sets.
destruct (app_inj_tail (l ++ [y]) ([] ++ [b]) _ _ H0) as ((?, <-)%app_inj_tail, <-).
inversion H1; subst; [ apply rt_step; assumption | apply rt_refl ].
- inversion H0.
+ apply app_cons_not_nil in H3 as [].
+ rewrite app_comm_cons in H0, H1. apply desc_prefix in H0.
pose proof (H x0 b H0).
apply rt_trans with (y := x0); auto with sets.
enough (H5 : clos_refl A leA a x0)
by (inversion H5; subst; [ apply rt_step | apply rt_refl ];
assumption).
apply app_inj_tail in H1 as (H1, ->).
rewrite app_comm_cons in H1.
apply app_inj_tail in H1 as (_, <-).
assumption.
Qed.
Lemma dist_aux :
forall z : List,
Descl z -> forall x y : List, z = x ++ y -> Descl x /\ Descl y.
Proof.
intros z D.
induction D as [| | * H D Hind]; intros.
- assert (H0 : x ++ y = []) by auto with sets.
apply app_eq_nil in H0 as (->, ->).
split; apply d_nil.
- assert (E : x0 ++ y = [x]) by auto with sets.
apply app_eq_unit in E as [(->, ->)| (->, ->)].
+ split.
apply d_nil.
apply d_one.
+ split.
apply d_one.
apply d_nil.
- induction y0 using rev_ind in x0, H0 |- *.
+ rewrite <- app_nil_end in H0.
rewrite <- H0.
split.
apply d_conc; auto with sets.
apply d_nil.
+ induction y0 using rev_ind in x1, x0, H0 |- *.
* simpl.
split.
apply app_inj_tail in H0 as (<-, _). assumption.
apply d_one.
* rewrite <- 2!app_assoc_reverse in H0.
apply app_inj_tail in H0 as (H0, <-).
pose proof H0 as H0'.
apply app_inj_tail in H0' as (_, ->).
rewrite app_assoc_reverse in H0.
apply Hind in H0 as [].
split.
assumption.
apply d_conc; auto with sets.
Qed.
Lemma dist_Desc_concat :
forall x y : List, Descl (x ++ y) -> Descl x /\ Descl y.
Proof.
intros.
apply (dist_aux (x ++ y) H x y); auto with sets.
Qed.
Lemma desc_end :
forall (a b : A) (x : List),
Descl (x ++ [a]) /\ ltl (x ++ [a]) [b] -> clos_trans A leA a b.
Proof.
intros a b x.
case x.
simpl.
simple induction 1.
intros.
inversion H1; auto with sets.
inversion H3.
simple induction 1.
generalize (app_comm_cons l [a] a0).
intros E; rewrite <- E; intros.
generalize (desc_tail l a a0 H0); intro.
inversion H1.
eapply clos_rt_t; [ eassumption | apply t_step; assumption ].
inversion H4.
Qed.
Lemma ltl_unit :
forall (x : List) (a b : A),
Descl (x ++ [a]) -> ltl (x ++ [a]) [b] -> ltl x [b].
Proof.
intro.
case x.
intros; apply (Lt_nil A leA).
simpl; intros.
inversion_clear H0.
apply (Lt_hd A leA a b); auto with sets.
inversion_clear H1.
Qed.
Lemma acc_app :
forall (x1 x2 : List) (y1 : Descl (x1 ++ x2)),
Acc Lex_Exp << x1 ++ x2, y1 >> ->
forall (x : List) (y : Descl x),
ltl x (x1 ++ x2) -> Acc Lex_Exp << x, y >>.
Proof.
intros.
apply (Acc_inv (R:=Lex_Exp) (x:=<< x1 ++ x2, y1 >>)).
auto with sets.
unfold lex_exp; simpl; auto with sets.
Qed.
Theorem wf_lex_exp : well_founded leA -> well_founded Lex_Exp.
Proof.
unfold well_founded at 2.
simple induction a; intros x y.
apply Acc_intro.
simple induction y0.
unfold lex_exp at 1; simpl.
apply rev_ind with
(A := A)
(P :=
fun x : List =>
forall (x0 : List) (y : Descl x0),
ltl x0 x -> Acc Lex_Exp << x0, y >>).
intros.
inversion_clear H0.
intro.
generalize (well_founded_ind (wf_clos_trans A leA H)).
intros GR.
apply GR with
(P :=
fun x0 : A =>
forall l : List,
(forall (x1 : List) (y : Descl x1),
ltl x1 l -> Acc Lex_Exp << x1, y >>) ->
forall (x1 : List) (y : Descl x1),
ltl x1 (l ++ [x0]) -> Acc Lex_Exp << x1, y >>).
intro; intros HInd; intros.
generalize (right_prefix x2 l [x1] H1).
simple induction 1.
intro; apply (H0 x2 y1 H3).
simple induction 1.
intro; simple induction 1.
clear H4 H2.
intro; generalize y1; clear y1.
rewrite H2.
apply rev_ind with
(A := A)
(P :=
fun x3 : List =>
forall y1 : Descl (l ++ x3),
ltl x3 [x1] -> Acc Lex_Exp << l ++ x3, y1 >>).
intros.
generalize (app_nil_end l); intros Heq.
generalize y1.
clear y1.
rewrite <- Heq.
intro.
apply Acc_intro.
simple induction y2.
unfold lex_exp at 1.
simpl; intros x4 y3. intros.
apply (H0 x4 y3); auto with sets.
intros.
generalize (dist_Desc_concat l (l0 ++ [x4]) y1).
simple induction 1.
intros.
generalize (desc_end x4 x1 l0 (conj H8 H5)); intros.
generalize y1.
rewrite <- (app_assoc_reverse l l0 [x4]); intro.
generalize (HInd x4 H9 (l ++ l0)); intros HInd2.
generalize (ltl_unit l0 x4 x1 H8 H5); intro.
generalize (dist_Desc_concat (l ++ l0) [x4] y2).
simple induction 1; intros.
generalize (H4 H12 H10); intro.
generalize (Acc_inv H14).
generalize (acc_app l l0 H12 H14).
intros f g.
generalize (HInd2 f); intro.
apply Acc_intro.
simple induction y3.
unfold lex_exp at 1; simpl; intros.
apply H15; auto with sets.
Qed.
End Wf_Lexicographic_Exponentiation.
|
