aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories/Relations/Operators_Properties.v
blob: e8d80f4692d78c6710f7d7fe4427ab56d1ea697a (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
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
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(************************************************************************)
(** * Some properties of the operators on relations                     *)
(************************************************************************)
(** * Initial version by Bruno Barras                                   *)
(************************************************************************)

Require Import Relation_Definitions.
Require Import Relation_Operators.

Section Properties.

  Arguments clos_refl [A] R x _.
  Arguments clos_refl_trans [A] R x _.
  Arguments clos_refl_trans_1n [A] R x _.
  Arguments clos_refl_trans_n1 [A] R x _.
  Arguments clos_refl_sym_trans [A] R _ _.
  Arguments clos_refl_sym_trans_1n [A] R x _.
  Arguments clos_refl_sym_trans_n1 [A] R x _.
  Arguments clos_trans [A] R x _.
  Arguments clos_trans_1n [A] R x _.
  Arguments clos_trans_n1 [A] R x _.
  Arguments inclusion [A] R1 R2.
  Arguments preorder [A] R.

  Variable A : Type.
  Variable R : relation A.

  Section Clos_Refl_Trans.

    Local Notation "R *" := (clos_refl_trans R)
      (at level 8, no associativity, format "R *").

    (** Correctness of the reflexive-transitive closure operator *)

    Lemma clos_rt_is_preorder : preorder R*.
    Proof.
      apply Build_preorder.
      exact (rt_refl A R).

      exact (rt_trans A R).
    Qed.

    (** Idempotency of the reflexive-transitive closure operator *)

    Lemma clos_rt_idempotent : inclusion (R*)* R*.
    Proof.
      red.
      induction 1; auto with sets.
      intros.
      apply rt_trans with y; auto with sets.
    Qed.

  End Clos_Refl_Trans.

  Section Clos_Refl_Sym_Trans.

    (** Reflexive-transitive closure is included in the
        reflexive-symmetric-transitive closure *)

    Lemma clos_rt_clos_rst :
      inclusion (clos_refl_trans R) (clos_refl_sym_trans R).
    Proof.
      red.
      induction 1; auto with sets.
      apply rst_trans with y; auto with sets.
    Qed.

    (** Reflexive closure is included in the
        reflexive-transitive closure *)

    Lemma clos_r_clos_rt :
      inclusion (clos_refl R) (clos_refl_trans R).
    Proof.
      induction 1 as [? ?| ].
      constructor; auto.
      constructor 2.
    Qed.

    Lemma clos_rt_t : forall x y z,
      clos_refl_trans R x y -> clos_trans R y z ->
      clos_trans R x z.
    Proof.
      induction 1 as [b d H1|b|a b d H1 H2 IH1 IH2]; auto.
      intro H. apply t_trans with (y:=d); auto.
      constructor. auto.
    Qed.

    (** Correctness of the reflexive-symmetric-transitive closure *)

    Lemma clos_rst_is_equiv : equivalence A (clos_refl_sym_trans R).
    Proof.
      apply Build_equivalence.
      exact (rst_refl A R).
      exact (rst_trans A R).
      exact (rst_sym A R).
    Qed.

    (** Idempotency of the reflexive-symmetric-transitive closure operator *)

    Lemma clos_rst_idempotent :
      inclusion (clos_refl_sym_trans (clos_refl_sym_trans R))
      (clos_refl_sym_trans R).
    Proof.
      red.
      induction 1; auto with sets.
      apply rst_trans with y; auto with sets.
    Qed.

  End Clos_Refl_Sym_Trans.

  Section Equivalences.

  (** *** Equivalences between the different definition of the reflexive,
      symmetric, transitive closures *)

  (** *** Contributed by P. Castéran *)

    (** Direct transitive closure vs left-step extension *)

    Lemma clos_t1n_trans : forall x y, clos_trans_1n R x y -> clos_trans R x y.
    Proof.
     induction 1.
     left; assumption.
     right with y; auto.
     left; auto.
    Qed.

    Lemma clos_trans_t1n : forall x y, clos_trans R x y -> clos_trans_1n R x y.
    Proof.
      induction 1.
      left; assumption.
      generalize IHclos_trans2; clear IHclos_trans2; induction IHclos_trans1.
      right with y; auto.
      right with y; auto.
      eapply IHIHclos_trans1; auto.
      apply clos_t1n_trans; auto.
    Qed.

    Lemma clos_trans_t1n_iff : forall x y,
        clos_trans R x y <-> clos_trans_1n R x y.
    Proof.
      split.
      apply clos_trans_t1n.
      apply clos_t1n_trans.
    Qed.

    (** Direct transitive closure vs right-step extension *)

    Lemma clos_tn1_trans : forall x y, clos_trans_n1 R x y -> clos_trans R x y.
    Proof.
      induction 1.
      left; assumption.
      right with y; auto.
      left; assumption.
    Qed.

    Lemma clos_trans_tn1 :  forall x y, clos_trans R x y -> clos_trans_n1 R x y.
    Proof.
      induction 1.
      left; assumption.
      elim IHclos_trans2.
      intro y0; right with y.
      auto.
      auto.
      intros.
      right with y0; auto.
    Qed.

    Lemma clos_trans_tn1_iff : forall x y,
        clos_trans R x y <-> clos_trans_n1 R x y.
    Proof.
      split.
      apply clos_trans_tn1.
      apply clos_tn1_trans.
    Qed.

    (** Direct reflexive-transitive closure is equivalent to
        transitivity by left-step extension *)

    Lemma clos_rt1n_step : forall x y, R x y -> clos_refl_trans_1n R x y.
    Proof.
      intros x y H.
      right with y;[assumption|left].
    Qed.

    Lemma clos_rtn1_step : forall x y, R x y -> clos_refl_trans_n1 R x y.
    Proof.
      intros x y H.
      right with x;[assumption|left].
    Qed.

    Lemma clos_rt1n_rt : forall x y,
        clos_refl_trans_1n R x y -> clos_refl_trans R x y.
    Proof.
      induction 1.
      constructor 2.
      constructor 3 with y; auto.
      constructor 1; auto.
    Qed.

    Lemma clos_rt_rt1n : forall x y,
        clos_refl_trans R x y -> clos_refl_trans_1n R x y.
    Proof.
      induction 1.
      apply clos_rt1n_step; assumption.
      left.
      generalize IHclos_refl_trans2; clear IHclos_refl_trans2;
        induction IHclos_refl_trans1; auto.

      right with y; auto.
      eapply IHIHclos_refl_trans1; auto.
      apply clos_rt1n_rt; auto.
    Qed.

    Lemma clos_rt_rt1n_iff : forall x y,
      clos_refl_trans R x y <-> clos_refl_trans_1n R x y.
    Proof.
      split.
      apply clos_rt_rt1n.
      apply clos_rt1n_rt.
    Qed.

    (** Direct reflexive-transitive closure is equivalent to
        transitivity by right-step extension *)

    Lemma clos_rtn1_rt : forall x y,
        clos_refl_trans_n1 R x y -> clos_refl_trans R x y.
    Proof.
      induction 1.
      constructor 2.
      constructor 3 with y; auto.
      constructor 1; assumption.
    Qed.

    Lemma clos_rt_rtn1 :  forall x y,
        clos_refl_trans R x y -> clos_refl_trans_n1 R x y.
    Proof.
      induction 1.
      apply clos_rtn1_step; auto.
      left.
      elim IHclos_refl_trans2; auto.
      intros.
      right with y0; auto.
    Qed.

    Lemma clos_rt_rtn1_iff : forall x y,
        clos_refl_trans R x y <-> clos_refl_trans_n1 R x y.
    Proof.
      split.
      apply clos_rt_rtn1.
      apply clos_rtn1_rt.
    Qed.

    (** Induction on the left transitive step *)

    Lemma clos_refl_trans_ind_left :
      forall (x:A) (P:A -> Prop), P x ->
	(forall y z:A, clos_refl_trans R x y -> P y -> R y z -> P z) ->
	forall z:A, clos_refl_trans R x z -> P z.
    Proof.
      intros.
      revert H H0.
      induction H1; intros; auto with sets.
      apply H1 with x; auto with sets.

      apply IHclos_refl_trans2.
      apply IHclos_refl_trans1; auto with sets.

      intros.
      apply H0 with y0; auto with sets.
      apply rt_trans with y; auto with sets.
    Qed.

    (** Induction on the right transitive step *)

    Lemma rt1n_ind_right : forall (P : A -> Prop) (z:A),
      P z ->
      (forall x y, R x y -> clos_refl_trans_1n R y z -> P y -> P x) ->
      forall x, clos_refl_trans_1n R x z -> P x.
      induction 3; auto.
      apply H0 with y; auto.
    Qed.

    Lemma clos_refl_trans_ind_right : forall (P : A -> Prop) (z:A),
      P z ->
      (forall x y, R x y -> P y -> clos_refl_trans R y z -> P x) ->
      forall x, clos_refl_trans R x z -> P x.
      intros P z Hz IH x Hxz.
      apply clos_rt_rt1n_iff in Hxz.
      elim Hxz using rt1n_ind_right; auto.
      clear x Hxz.
      intros x y Hxy Hyz Hy.
      apply clos_rt_rt1n_iff in Hyz.
      eauto.
    Qed.

    (** Direct reflexive-symmetric-transitive closure is equivalent to
        transitivity by symmetric left-step extension *)

    Lemma clos_rst1n_rst  : forall x y,
      clos_refl_sym_trans_1n R x y -> clos_refl_sym_trans R x y.
    Proof.
      induction 1.
      constructor 2.
      constructor 4 with y; auto.
      case H;[constructor 1|constructor 3; constructor 1]; auto.
    Qed.

    Lemma clos_rst1n_trans : forall x y z, clos_refl_sym_trans_1n R x y ->
        clos_refl_sym_trans_1n R y z -> clos_refl_sym_trans_1n R x z.
      induction 1.
      auto.
      intros; right with y; eauto.
    Qed.

    Lemma clos_rst1n_sym : forall x y, clos_refl_sym_trans_1n R x y ->
      clos_refl_sym_trans_1n R y x.
    Proof.
      intros x y H; elim H.
      constructor 1.
      intros x0 y0 z D H0 H1; apply clos_rst1n_trans with y0; auto.
      right with x0.
      tauto.
      left.
    Qed.

    Lemma clos_rst_rst1n  : forall x y,
      clos_refl_sym_trans R x y -> clos_refl_sym_trans_1n R x y.
      induction 1.
      constructor 2 with y; auto.
      constructor 1.
      constructor 1.
      apply clos_rst1n_sym; auto.
      eapply clos_rst1n_trans; eauto.
    Qed.

    Lemma clos_rst_rst1n_iff : forall x y,
      clos_refl_sym_trans R x y <-> clos_refl_sym_trans_1n R x y.
    Proof.
      split.
      apply clos_rst_rst1n.
      apply clos_rst1n_rst.
    Qed.

    (** Direct reflexive-symmetric-transitive closure is equivalent to
        transitivity by symmetric right-step extension *)

    Lemma clos_rstn1_rst : forall x y,
      clos_refl_sym_trans_n1 R x y -> clos_refl_sym_trans R x y.
    Proof.
      induction 1.
      constructor 2.
      constructor 4 with y; auto.
      case H;[constructor 1|constructor 3; constructor 1]; auto.
    Qed.

    Lemma clos_rstn1_trans : forall x y z, clos_refl_sym_trans_n1 R x y ->
      clos_refl_sym_trans_n1 R y z -> clos_refl_sym_trans_n1 R x z.
    Proof.
      intros x y z H1 H2.
      induction H2.
      auto.
      intros.
      right with y0; eauto.
    Qed.

    Lemma clos_rstn1_sym : forall x y, clos_refl_sym_trans_n1 R x y ->
      clos_refl_sym_trans_n1 R y x.
    Proof.
      intros x y H; elim H.
      constructor 1.
      intros y0 z D H0 H1. apply clos_rstn1_trans with y0; auto.
      right with z.
      tauto.
      left.
    Qed.

    Lemma clos_rst_rstn1 : forall x y,
      clos_refl_sym_trans R x y -> clos_refl_sym_trans_n1 R x y.
    Proof.
      induction 1.
      constructor 2 with x; auto.
      constructor 1.
      constructor 1.
      apply clos_rstn1_sym; auto.
      eapply clos_rstn1_trans; eauto.
    Qed.

    Lemma clos_rst_rstn1_iff : forall x y,
      clos_refl_sym_trans R x y <-> clos_refl_sym_trans_n1 R x y.
    Proof.
      split.
      apply clos_rst_rstn1.
      apply clos_rstn1_rst.
    Qed.

  End Equivalences.

  Lemma clos_trans_transp_permute : forall x y,
    transp _ (clos_trans R) x y <-> clos_trans (transp _ R) x y.
  Proof.
    split; induction 1;
    (apply t_step; assumption) || eapply t_trans; eassumption.
  Qed.

End Properties.

(* begin hide *)
(* Compatibility *)
Notation trans_tn1 := clos_trans_tn1 (only parsing).
Notation tn1_trans := clos_tn1_trans (only parsing).
Notation tn1_trans_equiv := clos_trans_tn1_iff (only parsing).

Notation trans_t1n := clos_trans_t1n (only parsing).
Notation t1n_trans := clos_t1n_trans (only parsing).
Notation t1n_trans_equiv := clos_trans_t1n_iff (only parsing).

Notation R_rtn1 := clos_rtn1_step (only parsing).
Notation trans_rt1n := clos_rt_rt1n (only parsing).
Notation rt1n_trans := clos_rt1n_rt (only parsing).
Notation rt1n_trans_equiv := clos_rt_rt1n_iff (only parsing).

Notation R_rt1n := clos_rt1n_step (only parsing).
Notation trans_rtn1 := clos_rt_rtn1 (only parsing).
Notation rtn1_trans := clos_rtn1_rt (only parsing).
Notation rtn1_trans_equiv := clos_rt_rtn1_iff (only parsing).

Notation rts1n_rts := clos_rst1n_rst (only parsing).
Notation rts_1n_trans := clos_rst1n_trans (only parsing).
Notation rts1n_sym := clos_rst1n_sym (only parsing).
Notation rts_rts1n := clos_rst_rst1n (only parsing).
Notation rts_rts1n_equiv := clos_rst_rst1n_iff (only parsing).

Notation rtsn1_rts := clos_rstn1_rst (only parsing).
Notation rtsn1_trans := clos_rstn1_trans (only parsing).
Notation rtsn1_sym := clos_rstn1_sym (only parsing).
Notation rts_rtsn1 := clos_rst_rstn1 (only parsing).
Notation rts_rtsn1_equiv := clos_rst_rstn1_iff (only parsing).
(* end hide *)