aboutsummaryrefslogtreecommitdiffhomepage
path: root/plugins/micromega/Tauto.v
blob: 458844e1b9f8d1af228da5a0cd20c6b2180624b1 (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
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
(*                                                                      *)
(* Micromega: A reflexive tactic using the Positivstellensatz           *)
(*                                                                      *)
(*  Frédéric Besson (Irisa/Inria) 2006-20011                            *)
(*                                                                      *)
(************************************************************************)

Require Import List.
Require Import Refl.
Require Import Bool.

Set Implicit Arguments.


  Inductive BFormula (A:Type) : Type :=
  | TT   : BFormula A
  | FF   : BFormula A
  | X : Prop -> BFormula A
  | A : A -> BFormula A
  | Cj  : BFormula A -> BFormula A -> BFormula A
  | D   : BFormula A-> BFormula A -> BFormula A
  | N  : BFormula A -> BFormula A
  | I : BFormula A-> BFormula A-> BFormula A.

  Fixpoint eval_f (A:Type)  (ev:A -> Prop ) (f:BFormula A) {struct f}: Prop :=
    match f with
      | TT _ => True
      | FF _ => False
      | A a =>  ev a
      | X _ p => p
      | Cj e1 e2 => (eval_f  ev e1) /\ (eval_f ev e2)
      | D e1 e2  => (eval_f  ev e1) \/ (eval_f  ev e2)
      | N e     => ~ (eval_f  ev e)
      | I f1 f2 => (eval_f  ev f1) -> (eval_f  ev f2)
    end.

  Lemma eval_f_morph : forall  A (ev ev' : A -> Prop) (f : BFormula A),
    (forall a, ev a <-> ev' a) -> (eval_f ev f <-> eval_f ev' f).
  Proof.
    induction f ; simpl ; try tauto.
    intros.
    assert (H' := H a).
    auto.
  Qed.



  Fixpoint map_bformula (T U : Type) (fct : T -> U) (f : BFormula T) : BFormula U :=
    match f with
      | TT _ => TT _
      | FF _ => FF _
      | X _ p => X _ p
      | A a => A (fct a)
      | Cj f1 f2 => Cj (map_bformula fct f1) (map_bformula fct f2)
      | D f1 f2 => D (map_bformula fct f1) (map_bformula fct f2)
      | N f     => N (map_bformula fct f)
      | I f1 f2 => I (map_bformula fct f1) (map_bformula fct f2)
    end.

  Lemma eval_f_map : forall T U (fct: T-> U) env f ,
    eval_f env  (map_bformula fct f)  = eval_f (fun x => env (fct x)) f.
  Proof.
    induction f ; simpl ; try (rewrite IHf1 ; rewrite IHf2) ; auto.
    rewrite <- IHf.  auto.    
  Qed.



  Lemma map_simpl : forall A B f l, @map A B f l = match l with
                                                 | nil => nil
                                                 | a :: l=> (f a) :: (@map A B f l)
                                               end.
  Proof.
    destruct l ; reflexivity.
  Qed.




  Section S.

    Variable Env   : Type.
    Variable Term  : Type.
    Variable eval  : Env -> Term -> Prop.
    Variable Term' : Type.
    Variable eval'  : Env -> Term' -> Prop.



    Variable no_middle_eval' : forall env d, (eval' env d) \/ ~ (eval' env d).

    Variable unsat : Term'  -> bool.

    Variable unsat_prop : forall t, unsat t  = true ->  
      forall env, eval' env t -> False. 

    Variable deduce : Term' -> Term' -> option Term'.

    Variable deduce_prop : forall env t t' u,
      eval' env t -> eval' env t' -> deduce t t' = Some u -> eval' env u.

    Definition clause := list  Term'.
    Definition cnf := list clause.

    Variable normalise : Term -> cnf.
    Variable negate : Term -> cnf.


    Definition tt : cnf := @nil clause.
    Definition ff : cnf :=  cons (@nil Term') nil.


    Fixpoint add_term (t: Term') (cl : clause) : option clause :=
      match cl with
        | nil => 
          match deduce t t with
            | None =>  Some (t ::nil)
            | Some u => if unsat u then None else Some (t::nil)
          end
        | t'::cl => 
          match deduce t t' with
            | None => 
              match add_term t cl with
                | None => None
                | Some cl' => Some (t' :: cl')
              end
            | Some u => 
              if unsat u then None else 
                match add_term t cl with
                  | None => None
                  | Some cl' => Some (t' :: cl')
                end
          end
      end.

    Fixpoint or_clause (cl1 cl2 : clause) : option clause :=
      match cl1 with
        | nil => Some cl2
        | t::cl => match add_term t cl2 with
                     | None => None
                     | Some cl' => or_clause cl cl'
                   end
      end.

(*    Definition or_clause_cnf (t:clause) (f:cnf) : cnf :=
      List.map (fun x => (t++x)) f. *)

    Definition or_clause_cnf (t:clause) (f:cnf) : cnf :=
      List.fold_right (fun e acc => 
        match or_clause t e with
          | None => acc
          | Some cl => cl :: acc
        end) nil f.


    Fixpoint or_cnf (f : cnf) (f' : cnf) {struct f}: cnf :=
      match f with
        | nil => tt
        | e :: rst => (or_cnf rst f') ++ (or_clause_cnf e f')
      end.


    Definition and_cnf (f1 : cnf) (f2 : cnf) : cnf :=
      f1 ++ f2.

    Fixpoint xcnf (pol : bool) (f : BFormula Term)  {struct f}: cnf :=
      match f with
        | TT _ => if pol then tt else ff
        | FF _ => if pol then ff else tt
        | X _ p => if pol then ff else ff (* This is not complete - cannot negate any proposition *)
        | A x => if pol then normalise x else negate x
        | N e  => xcnf (negb pol) e
        | Cj e1 e2 =>
          (if pol then and_cnf else or_cnf) (xcnf pol e1) (xcnf pol e2)
        | D e1 e2  => (if pol then or_cnf else and_cnf) (xcnf pol e1) (xcnf pol e2)
        | I e1 e2 => (if pol then or_cnf else and_cnf) (xcnf (negb pol) e1) (xcnf pol e2)
      end.

    Definition eval_clause (env : Env) (cl : clause) := ~ make_conj  (eval' env) cl.

    Definition eval_cnf (env : Env) (f:cnf) := make_conj  (eval_clause  env) f.

    
    Lemma eval_cnf_app : forall env x y, eval_cnf env (x++y) -> eval_cnf env x /\ eval_cnf env y.
    Proof.
      unfold eval_cnf.
      intros.
      rewrite make_conj_app in H ; auto.
    Qed.


    Definition eval_opt_clause (env : Env) (cl: option clause) :=
      match cl with
        | None => True
        | Some cl => eval_clause env cl
      end.


    Lemma add_term_correct : forall env t cl , eval_opt_clause env (add_term t cl) -> eval_clause env (t::cl).
    Proof.
      induction cl.
      (* BC *)
      simpl.
      case_eq (deduce t t) ; auto.
      intros *.
      case_eq (unsat t0) ; auto.
      unfold eval_clause.
      rewrite make_conj_cons. 
      intros. intro.
      apply unsat_prop with (1:= H) (env := env).
      apply deduce_prop with (3:= H0) ; tauto.
      (* IC *)
      simpl.
      case_eq (deduce t a).
      intro u.
      case_eq (unsat u).
      simpl. intros.
      unfold eval_clause.
      intro.
      apply unsat_prop  with (1:= H) (env:= env).
      repeat rewrite make_conj_cons in H2.
      apply deduce_prop with (3:= H0); tauto.
      intro.
      case_eq (add_term t cl) ; intros.
      simpl in H2.
      rewrite H0 in IHcl.
      simpl in IHcl.
      unfold eval_clause in *.
      intros.
      repeat rewrite make_conj_cons in *.
      tauto.
      rewrite H0 in IHcl ; simpl in *.
      unfold eval_clause in *.
      intros.
      repeat rewrite make_conj_cons in *.
      tauto.
      case_eq (add_term t cl) ; intros.
      simpl in H1.
      unfold eval_clause in *.
      repeat rewrite make_conj_cons in *.
      rewrite H in IHcl.
      simpl in IHcl.
      tauto.
      simpl in *.
      rewrite H in IHcl.
      simpl in IHcl.
      unfold eval_clause in *.
      repeat rewrite make_conj_cons in *.    
      tauto.
    Qed.


  Lemma or_clause_correct : forall cl cl' env,  eval_opt_clause env (or_clause cl cl') -> eval_clause env cl \/ eval_clause env cl'.
  Proof.
    induction cl.
    simpl. tauto.
    intros *.
    simpl.
    assert (HH := add_term_correct env a cl').
    case_eq (add_term a cl').
    simpl in *.
    intros.
    apply IHcl in H0.
    rewrite H in HH.
    simpl in HH.
    unfold eval_clause in *.
    destruct H0.
    repeat rewrite make_conj_cons in *.
    tauto.
    apply HH in H0.
    apply not_make_conj_cons in H0 ; auto.
    repeat rewrite make_conj_cons in *.
    tauto.
    simpl.
    intros.
    rewrite H in HH.
    simpl in HH.
    unfold eval_clause in *.
    assert (HH' := HH Coq.Init.Logic.I).
    apply not_make_conj_cons in HH'; auto.
    repeat rewrite make_conj_cons in *.
    tauto.
  Qed.
    

  Lemma or_clause_cnf_correct : forall env t f, eval_cnf env (or_clause_cnf t f) -> (eval_clause env t) \/ (eval_cnf env f).
  Proof.
    unfold eval_cnf.
    unfold or_clause_cnf.
    intros until t.
    set (F := (fun (e : clause) (acc : list clause) =>
         match or_clause t e with
         | Some cl => cl :: acc
         | None => acc
         end)).
    induction f.
    auto.
    (**)
    simpl.
    intros.
    destruct f.
    simpl in H.
    simpl in IHf. 
    unfold F in H.
    revert H.
    intros.
    apply or_clause_correct.
    destruct (or_clause t a) ; simpl in * ; auto.
    unfold F in H at 1.
    revert H.
    assert (HH := or_clause_correct t a env).
    destruct (or_clause t a); simpl in HH ;
    rewrite make_conj_cons in * ; intuition.
    rewrite make_conj_cons in *.
    tauto.
  Qed.


 Lemma eval_cnf_cons : forall env a f,  (~ make_conj  (eval' env) a) -> eval_cnf env f -> eval_cnf env (a::f).
 Proof.
   intros.
   unfold eval_cnf in *.
   rewrite make_conj_cons ; eauto.
 Qed.

  Lemma or_cnf_correct : forall env f f', eval_cnf env (or_cnf f f') -> (eval_cnf env  f) \/ (eval_cnf  env f').
  Proof.
    induction f.
    unfold eval_cnf.
    simpl.
    tauto.
    (**)
    intros.
    simpl in H.
    destruct (eval_cnf_app _ _ _ H).
    clear H.
    destruct (IHf _ H0).
    destruct (or_clause_cnf_correct _ _ _ H1).
    left.
    apply eval_cnf_cons ; auto.
    right ; auto.
    right ; auto.
  Qed.

  Variable normalise_correct : forall env t, eval_cnf  env (normalise t) -> eval env t.

  Variable negate_correct : forall env t, eval_cnf env (negate t) -> ~ eval env t.


  Lemma xcnf_correct : forall f pol env, eval_cnf env (xcnf pol f) -> eval_f (eval env) (if pol then f else N f).
  Proof.
    induction f.
    (* TT *)
    unfold eval_cnf.
    simpl.
    destruct pol ; simpl ; auto.
    (* FF *)
    unfold eval_cnf.
    destruct pol; simpl ; auto.
    unfold eval_clause ; simpl.
    tauto.
    (* P *)
    simpl.
    destruct pol ; intros ;simpl.
    unfold eval_cnf in H.
    (* Here I have to drop the proposition *)
    simpl in H.
    unfold eval_clause in H ; simpl in H.
    tauto.
    (* Here, I could store P in the clause *)
    unfold eval_cnf in H;simpl in H.
    unfold eval_clause in H ; simpl in H.
    tauto.
    (* A *)
    simpl.
    destruct pol ; simpl.
    intros.
    apply normalise_correct  ; auto.
    (* A 2 *)
    intros.
    apply  negate_correct ; auto.
    auto.
    (* Cj *)
    destruct pol ; simpl.
    (* pol = true *)
    intros.
    unfold and_cnf in H.
    destruct (eval_cnf_app  _ _ _ H).
    clear H.
    split.
    apply (IHf1 _ _ H0).
    apply (IHf2 _ _ H1).
    (* pol = false *)
    intros.
    destruct (or_cnf_correct _ _ _ H).
    generalize (IHf1 false  env H0).
    simpl.
    tauto.
    generalize (IHf2 false  env H0).
    simpl.
    tauto.
    (* D *)
    simpl.
    destruct pol.
    (* pol = true *)
    intros.
    destruct (or_cnf_correct _ _ _ H).
    generalize (IHf1 _  env H0).
    simpl.
    tauto.
    generalize (IHf2 _  env H0).
    simpl.
    tauto.
    (* pol = true *)
    unfold and_cnf.
    intros.
    destruct (eval_cnf_app  _ _ _ H).
    clear H.
    simpl.
    generalize (IHf1 _ _ H0).
    generalize (IHf2 _ _ H1).
    simpl.
    tauto.
    (**)
    simpl.
    destruct pol ; simpl.
    intros.
    apply (IHf false) ; auto.
    intros.
    generalize (IHf _ _ H).
    tauto.
    (* I *)
    simpl; intros.
    destruct pol.
    simpl.
    intro.
    destruct (or_cnf_correct _ _ _ H).
    generalize (IHf1 _ _ H1).
    simpl in *.
    tauto.
    generalize (IHf2 _ _ H1).
    auto.
    (* pol = false *)
    unfold and_cnf in H.
    simpl in H.
    destruct (eval_cnf_app _ _ _ H).
    generalize (IHf1 _ _ H0).
    generalize (IHf2 _ _ H1).
    simpl.
    tauto.
  Qed.


  Variable Witness : Type.
  Variable checker : list Term' -> Witness -> bool.

  Variable checker_sound : forall t  w, checker t w = true -> forall env, make_impl (eval' env)  t False.

  Fixpoint cnf_checker (f : cnf) (l : list Witness)  {struct f}: bool :=
    match f with
      | nil => true
      | e::f => match l with
                  | nil => false
                  | c::l => match checker e c with
                              | true => cnf_checker f l
                              |   _  => false
                            end
                end
      end.

  Lemma cnf_checker_sound : forall t  w, cnf_checker t w = true -> forall env, eval_cnf  env  t.
  Proof.
    unfold eval_cnf.
    induction t.
    (* bc *)
    simpl.
    auto.
    (* ic *)
    simpl.
    destruct w.
    intros ; discriminate.
    case_eq (checker a w) ; intros ; try discriminate.
    generalize (@checker_sound _ _ H env).
    generalize (IHt _ H0 env) ; intros.
    destruct t.
    red ; intro.
    rewrite <- make_conj_impl in H2.
    tauto.
    rewrite <- make_conj_impl in H2.
    tauto.
  Qed.


  Definition tauto_checker (f:BFormula Term) (w:list Witness) : bool :=
    cnf_checker (xcnf true f) w.

  Lemma tauto_checker_sound : forall t  w, tauto_checker t w = true -> forall env, eval_f  (eval env)  t.
  Proof.
    unfold tauto_checker.
    intros.
    change (eval_f (eval env) t) with (eval_f (eval env) (if true then t else TT Term)).
    apply (xcnf_correct t true).
    eapply cnf_checker_sound ; eauto.
  Qed.



End S.

(* Local Variables: *)
(* coding: utf-8 *)
(* End: *)