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
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
|
Require Import TestSuite.admit.
Require Coq.Setoids.Setoid.
Axiom BITS : nat -> Set.
Definition n7 := 7.
Definition n15 := 15.
Definition n31 := 31.
Notation n8 := (S n7).
Notation n16 := (S n15).
Notation n32 := (S n31).
Inductive OpSize := OpSize1 | OpSize2 | OpSize4 .
Definition VWORD s := BITS (match s with OpSize1 => n8 | OpSize2 => n16 | OpSize4 => n32 end).
Definition BYTE := VWORD OpSize1.
Definition WORD := VWORD OpSize2.
Definition DWORD := VWORD OpSize4.
Ltac subst_body :=
repeat match goal with
| [ H := _ |- _ ] => subst H
end.
Import Coq.Setoids.Setoid.
Class Equiv (A : Type) := equiv : relation A.
Infix "===" := equiv (at level 70, no associativity).
Class type (A : Type) {e : Equiv A} := eq_equiv : Equivalence equiv.
Definition setoid_resp {T T'} (f : T -> T') `{e : type T} `{e' : type T'} := forall x y, x === y -> f x === f y.
Record morphism T T' `{e : type T} `{e' : type T'} :=
mkMorph {
morph :> T -> T';
morph_resp : setoid_resp morph}.
Arguments mkMorph [T T' e0 e e1 e'].
Infix "-s>" := morphism (at level 45, right associativity).
Section Morphisms.
Context {S T U V} `{eS : type S} `{eT : type T} `{eU : type U} `{eV : type V}.
Global Instance morph_equiv : Equiv (S -s> T).
admit.
Defined.
Global Instance morph_type : type (S -s> T).
admit.
Defined.
Program Definition mcomp (f: T -s> U) (g: S -s> T) : (S -s> U) :=
mkMorph (fun x => f (g x)) _.
Next Obligation.
admit.
Defined.
End Morphisms.
Infix "<<" := mcomp (at level 35).
Section MorphConsts.
Context {S T U V} `{eS : type S} `{eT : type T} `{eU : type U} `{eV : type V}.
Definition lift2s (f : S -> T -> U) p q : (S -s> T -s> U) :=
mkMorph (fun x => mkMorph (f x) (p x)) q.
End MorphConsts.
#[export] Instance Equiv_PropP : Equiv Prop.
admit.
Defined.
Section SetoidProducts.
Context {A B : Type} `{eA : type A} `{eB : type B}.
Global Instance Equiv_prod : Equiv (A * B).
admit.
Defined.
Global Instance type_prod : type (A * B).
admit.
Defined.
Program Definition mfst : (A * B) -s> A :=
mkMorph (fun p => fst p) _.
Next Obligation.
admit.
Defined.
Program Definition msnd : (A * B) -s> B :=
mkMorph (fun p => snd p) _.
Next Obligation.
admit.
Defined.
Context {C} `{eC : type C}.
Program Definition mprod (f: C -s> A) (g: C -s> B) : C -s> (A * B) :=
mkMorph (fun c => (f c, g c)) _.
Next Obligation.
admit.
Defined.
End SetoidProducts.
Section IndexedProducts.
Record ttyp := {carr :> Type; eqc : Equiv carr; eqok : type carr}.
Global Instance ttyp_proj_eq {A : ttyp} : Equiv A.
admit.
Defined.
Global Instance ttyp_proj_prop {A : ttyp} : type A.
admit.
Defined.
Context {I : Type} {P : I -> ttyp}.
Global Program Instance Equiv_prodI : Equiv (forall i, P i) :=
fun p p' : forall i, P i => (forall i : I, @equiv _ (eqc _) (p i) (p' i)).
Global Instance type_prodI : type (forall i, P i).
admit.
Defined.
Program Definition mprojI (i : I) : (forall i, P i) -s> P i :=
mkMorph (fun X => X i) _.
Next Obligation.
admit.
Defined.
Context {C : Type} `{eC : type C}.
Program Definition mprodI (f : forall i, C -s> P i) : C -s> (forall i, P i) :=
mkMorph (fun c i => f i c) _.
Next Obligation.
admit.
Defined.
End IndexedProducts.
Section Exponentials.
Context {A B C D} `{eA : type A} `{eB : type B} `{eC : type C} `{eD : type D}.
Program Definition comps : (B -s> C) -s> (A -s> B) -s> A -s> C :=
lift2s (fun f g => f << g) _ _.
Next Obligation.
admit.
Defined.
Next Obligation.
admit.
Defined.
Program Definition muncurry (f : A -s> B -s> C) : A * B -s> C :=
mkMorph (fun p => f (fst p) (snd p)) _.
Next Obligation.
admit.
Defined.
Program Definition mcurry (f : A * B -s> C) : A -s> B -s> C :=
lift2s (fun a b => f (a, b)) _ _.
Next Obligation.
admit.
Defined.
Next Obligation.
admit.
Defined.
Program Definition meval : (B -s> A) * B -s> A :=
mkMorph (fun p => fst p (snd p)) _.
Next Obligation.
admit.
Defined.
Program Definition mid : A -s> A := mkMorph (fun x => x) _.
Next Obligation.
admit.
Defined.
Program Definition mconst (b : B) : A -s> B := mkMorph (fun _ => b) _.
Next Obligation.
admit.
Defined.
End Exponentials.
Inductive empty : Set := .
#[export] Instance empty_Equiv : Equiv empty.
admit.
Defined.
#[export] Instance empty_type : type empty.
admit.
Defined.
Section Initials.
Context {A} `{eA : type A}.
Program Definition mzero_init : empty -s> A := mkMorph (fun x => match x with end) _.
Next Obligation.
admit.
Defined.
End Initials.
Section Subsetoid.
Context {A} `{eA : type A} {P : A -> Prop}.
Global Instance subset_Equiv : Equiv {a : A | P a}.
admit.
Defined.
Global Instance subset_type : type {a : A | P a}.
admit.
Defined.
Program Definition mforget : {a : A | P a} -s> A :=
mkMorph (fun x => x) _.
Next Obligation.
admit.
Defined.
Context {B} `{eB : type B}.
Program Definition minherit (f : B -s> A) (HB : forall b, P (f b)) : B -s> {a : A | P a} :=
mkMorph (fun b => exist P (f b) (HB b)) _.
Next Obligation.
admit.
Defined.
End Subsetoid.
Section Option.
Context {A} `{eA : type A}.
Global Instance option_Equiv : Equiv (option A).
admit.
Defined.
Global Instance option_type : type (option A).
admit.
Defined.
End Option.
Section OptDefs.
Context {A B} `{eA : type A} `{eB : type B}.
Program Definition msome : A -s> option A := mkMorph (fun a => Some a) _.
Next Obligation.
admit.
Defined.
Program Definition moptionbind (f : A -s> option B) : option A -s> option B :=
mkMorph (fun oa => match oa with None => None | Some a => f a end) _.
Next Obligation.
admit.
Defined.
End OptDefs.
Generalizable Variables Frm.
Class ILogicOps Frm := {
lentails: relation Frm;
ltrue: Frm;
lfalse: Frm;
limpl: Frm -> Frm -> Frm;
land: Frm -> Frm -> Frm;
lor: Frm -> Frm -> Frm;
lforall: forall {T}, (T -> Frm) -> Frm;
lexists: forall {T}, (T -> Frm) -> Frm
}.
Infix "|--" := lentails (at level 79, no associativity).
Infix "//\\" := land (at level 75, right associativity).
Infix "\\//" := lor (at level 76, right associativity).
Infix "-->>" := limpl (at level 77, right associativity).
Notation "'Forall' x .. y , p" :=
(lforall (fun x => .. (lforall (fun y => p)) .. )) (at level 78, x binder, y binder, right associativity).
Notation "'Exists' x .. y , p" :=
(lexists (fun x => .. (lexists (fun y => p)) .. )) (at level 78, x binder, y binder, right associativity).
Class ILogic Frm {ILOps: ILogicOps Frm} := {
lentailsPre :: PreOrder lentails;
ltrueR: forall C, C |-- ltrue;
lfalseL: forall C, lfalse |-- C;
lforallL: forall T x (P: T -> Frm) C, P x |-- C -> lforall P |-- C;
lforallR: forall T (P: T -> Frm) C, (forall x, C |-- P x) -> C |-- lforall P;
lexistsL: forall T (P: T -> Frm) C, (forall x, P x |-- C) -> lexists P |-- C;
lexistsR: forall T x (P: T -> Frm) C, C |-- P x -> C |-- lexists P;
landL1: forall P Q C, P |-- C -> P //\\ Q |-- C;
landL2: forall P Q C, Q |-- C -> P //\\ Q |-- C;
lorR1: forall P Q C, C |-- P -> C |-- P \\// Q;
lorR2: forall P Q C, C |-- Q -> C |-- P \\// Q;
landR: forall P Q C, C |-- P -> C |-- Q -> C |-- P //\\ Q;
lorL: forall P Q C, P |-- C -> Q |-- C -> P \\// Q |-- C;
landAdj: forall P Q C, C |-- (P -->> Q) -> C //\\ P |-- Q;
limplAdj: forall P Q C, C //\\ P |-- Q -> C |-- (P -->> Q)
}.
#[export] Hint Extern 0 (?x |-- ?x) => reflexivity.
Section ILogicExtra.
Context `{IL: ILogic Frm}.
Definition lpropand (p: Prop) Q := Exists _: p, Q.
Definition lpropimpl (p: Prop) Q := Forall _: p, Q.
End ILogicExtra.
Infix "/\\" := lpropand (at level 75, right associativity).
Infix "->>" := lpropimpl (at level 77, right associativity).
Section ILogic_Fun.
Context (T: Type) `{TType: type T}.
Context `{IL: ILogic Frm}.
Record ILFunFrm := mkILFunFrm {
ILFunFrm_pred :> T -> Frm;
ILFunFrm_closed: forall t t': T, t === t' ->
ILFunFrm_pred t |-- ILFunFrm_pred t'
}.
Notation "'mk'" := @mkILFunFrm.
Program Definition ILFun_Ops : ILogicOps ILFunFrm := {|
lentails P Q := forall t:T, P t |-- Q t;
ltrue := mk (fun t => ltrue) _;
lfalse := mk (fun t => lfalse) _;
limpl P Q := mk (fun t => P t -->> Q t) _;
land P Q := mk (fun t => P t //\\ Q t) _;
lor P Q := mk (fun t => P t \\// Q t) _;
lforall A P := mk (fun t => Forall a, P a t) _;
lexists A P := mk (fun t => Exists a, P a t) _
|}.
Next Obligation.
admit.
Defined.
Next Obligation.
admit.
Defined.
Next Obligation.
admit.
Defined.
Next Obligation.
admit.
Defined.
Next Obligation.
admit.
Defined.
End ILogic_Fun.
Arguments ILFunFrm _ {e} _ {ILOps}.
Arguments mkILFunFrm [T] _ [Frm ILOps].
Program Definition ILFun_eq {T R} {ILOps: ILogicOps R} {ILogic: ILogic R} (P : T -> R) :
@ILFunFrm T _ R ILOps :=
@mkILFunFrm T eq R ILOps P _.
Next Obligation.
admit.
Defined.
#[export] Instance ILogicOps_Prop : ILogicOps Prop | 2 := {|
lentails P Q := (P : Prop) -> Q;
ltrue := True;
lfalse := False;
limpl P Q := P -> Q;
land P Q := P /\ Q;
lor P Q := P \/ Q;
lforall T F := forall x:T, F x;
lexists T F := exists x:T, F x
|}.
#[export] Instance ILogic_Prop : ILogic Prop.
admit.
Defined.
Section FunEq.
Context A `{eT: type A}.
Global Instance FunEquiv {T} : Equiv (T -> A) := {
equiv P Q := forall a, P a === Q a
}.
End FunEq.
Section SepAlgSect.
Class SepAlgOps T `{eT : type T}:= {
sa_unit : T;
sa_mul : T -> T -> T -> Prop
}.
Class SepAlg T `{SAOps: SepAlgOps T} : Type := {
sa_mul_eqL a b c d : sa_mul a b c -> c === d -> sa_mul a b d;
sa_mul_eqR a b c d : sa_mul a b c -> sa_mul a b d -> c === d;
sa_mon a b c : a === b -> sa_mul a c === sa_mul b c;
sa_mulC a b : sa_mul a b === sa_mul b a;
sa_mulA a b c : forall bc abc, sa_mul a bc abc -> sa_mul b c bc ->
exists ac, sa_mul b ac abc /\ sa_mul a c ac;
sa_unitI a : sa_mul a sa_unit a
}.
End SepAlgSect.
Section BILogic.
Class BILOperators (A : Type) := {
empSP : A;
sepSP : A -> A -> A;
wandSP : A -> A -> A
}.
End BILogic.
Notation "a '**' b" := (sepSP a b)
(at level 75, right associativity).
Section BISepAlg.
Context {A} `{sa : SepAlg A}.
Context {B} `{IL: ILogic B}.
Program Instance SABIOps: BILOperators (ILFunFrm A B) := {
empSP := mkILFunFrm e (fun x => sa_unit === x /\\ ltrue) _;
sepSP P Q := mkILFunFrm e (fun x => Exists x1, Exists x2, sa_mul x1 x2 x /\\
P x1 //\\ Q x2) _;
wandSP P Q := mkILFunFrm e (fun x => Forall x1, Forall x2, sa_mul x x1 x2 ->>
P x1 -->> Q x2) _
}.
Next Obligation.
admit.
Defined.
Next Obligation.
admit.
Defined.
Next Obligation.
admit.
Defined.
End BISepAlg.
Set Implicit Arguments.
Definition Chan := WORD.
Definition Data := BYTE.
Inductive Action :=
| Out (c:Chan) (d:Data)
| In (c:Chan) (d:Data).
Definition Actions := list Action.
#[export] Instance ActionsEquiv : Equiv Actions := {
equiv a1 a2 := a1 = a2
}.
Definition OPred := ILFunFrm Actions Prop.
Definition mkOPred (P : Actions -> Prop) : OPred.
admit.
Defined.
Definition eq_opred s := mkOPred (fun s' => s === s').
Definition empOP : OPred.
exact (eq_opred nil).
Defined.
Definition catOP (P Q: OPred) : OPred.
admit.
Defined.
Class IsPointed (T : Type) := point : T.
Generalizable All Variables.
Notation IsPointed_OPred P := (IsPointed (exists x : Actions, (P : OPred) x)).
Record PointedOPred := mkPointedOPred {
OPred_pred :> OPred;
OPred_inhabited: IsPointed_OPred OPred_pred
}.
#[export] Existing Instance OPred_inhabited.
Canonical Structure default_PointedOPred O `{IsPointed_OPred O} : PointedOPred
:= {| OPred_pred := O ; OPred_inhabited := _ |}.
#[export] Instance IsPointed_eq_opred x : IsPointed_OPred (eq_opred x).
admit.
Defined.
#[export] Instance IsPointed_catOP `{IsPointed_OPred P, IsPointed_OPred Q} : IsPointed_OPred (catOP P Q).
admit.
Defined.
Definition Flag := BITS 5.
Definition OF: Flag.
admit.
Defined.
Inductive FlagVal := mkFlag (b: bool) | FlagUnspecified.
Coercion mkFlag : bool >-> FlagVal.
Inductive NonSPReg := EAX | EBX | ECX | EDX | ESI | EDI | EBP.
Inductive Reg := nonSPReg (r: NonSPReg) | ESP.
Inductive AnyReg := regToAnyReg (r: Reg) | EIP.
Inductive BYTEReg := AL|BL|CL|DL|AH|BH|CH|DH.
Inductive WORDReg := mkWordReg (r:Reg).
Definition PState : Type.
admit.
Defined.
#[export] Instance PStateEquiv : Equiv PState.
admit.
Defined.
#[export] Instance PStateType : type PState.
admit.
Defined.
#[export] Instance PStateSepAlgOps: SepAlgOps PState.
admit.
Defined.
Definition SPred : Type.
exact (ILFunFrm PState Prop).
Defined.
Local Existing Instance ILFun_Ops.
Local Existing Instance SABIOps.
Axiom BYTEregIs : BYTEReg -> BYTE -> SPred.
Inductive RegOrFlag :=
| RegOrFlagDWORD :> AnyReg -> RegOrFlag
| RegOrFlagWORD :> WORDReg -> RegOrFlag
| RegOrFlagBYTE :> BYTEReg -> RegOrFlag
| RegOrFlagF :> Flag -> RegOrFlag.
Definition RegOrFlag_target rf :=
match rf with
| RegOrFlagDWORD _ => DWORD
| RegOrFlagWORD _ => WORD
| RegOrFlagBYTE _ => BYTE
| RegOrFlagF _ => FlagVal
end.
Inductive Condition :=
| CC_O | CC_B | CC_Z | CC_BE | CC_S | CC_P | CC_L | CC_LE.
Section ILSpecSect.
Axiom spec : Type.
Global Instance ILOps: ILogicOps spec | 2.
admit.
Defined.
End ILSpecSect.
Axiom parameterized_basic : forall {T_OPred} {proj : T_OPred -> OPred} {T} (P : SPred) (c : T) (O : OPred) (Q : SPred), spec.
Global Notation loopy_basic := (@parameterized_basic PointedOPred OPred_pred _).
Axiom program : Type.
Axiom ConditionIs : forall (cc : Condition) (cv : RegOrFlag_target OF), SPred.
Axiom foldl : forall {T R}, (R -> T -> R) -> R -> list T -> R.
Axiom nth : forall {T}, T -> list T -> nat -> T.
Axiom while : forall (ptest: program)
(cond: Condition) (value: bool)
(pbody: program), program.
Lemma while_rule_ind {quantT}
{ptest} {cond : Condition} {value : bool} {pbody}
{S}
{transition_body : quantT -> quantT}
{P : quantT -> SPred} {Otest : quantT -> OPred} {Obody : quantT -> OPred} {O : quantT -> PointedOPred}
{O_after_test : quantT -> PointedOPred}
{I_state : quantT -> bool -> SPred}
{I_logic : quantT -> bool -> bool}
{Q : quantT -> SPred}
(Htest : S |-- (Forall (x : quantT),
(loopy_basic (P x)
ptest
(Otest x)
(Exists b, I_logic x b = true /\\ I_state x b ** ConditionIs cond b))))
(Hbody : S |-- (Forall (x : quantT),
(loopy_basic (I_logic x value = true /\\ I_state x value ** ConditionIs cond value)
pbody
(Obody x)
(P (transition_body x)))))
(H_after_test : forall x, catOP (Otest x) (O_after_test x) |-- O x)
(H_body_after_test : forall x, I_logic x value = true -> catOP (Obody x) (O (transition_body x)) |-- O_after_test x)
(H_empty : forall x, I_logic x (negb value) = true -> empOP |-- O_after_test x)
(Q_correct : forall x, I_logic x (negb value) = true /\\ I_state x (negb value) ** ConditionIs cond (negb value) |-- Q x)
(Q_safe : forall x, I_logic x value = true -> Q (transition_body x) |-- Q x)
: S |-- (Forall (x : quantT),
loopy_basic (P x)
(while ptest cond value pbody)
(O x)
(Q x)).
admit.
Defined.
Axiom behead : forall {T}, list T -> list T.
Axiom all : forall {T}, (T -> bool) -> list T -> bool.
Axiom all_behead : forall {T} (xs : list T) P, all P xs = true -> all P (behead xs) = true.
#[export] Instance IsPointed_foldlOP A B C f g (init : A * B) `{IsPointed_OPred (g init)}
`{forall a acc, IsPointed_OPred (g acc) -> IsPointed_OPred (g (f acc a))}
(ls : list C)
: IsPointed_OPred (g (foldl f init ls)).
admit.
Defined.
Goal forall (ptest : program) (cond : Condition) (value : bool)
(pbody : program) (T ioT : Type) (P : T -> SPred)
(I : T -> bool -> SPred) (accumulate : T -> ioT -> T)
(Otest Obody : T -> ioT -> PointedOPred)
(coq_test__is_finished : ioT -> bool) (S : spec)
(al : BYTE),
(forall (initial : T) (xs : list ioT) (x : ioT),
all (fun t : ioT => negb (coq_test__is_finished t)) xs = true ->
coq_test__is_finished x = true ->
S
|-- loopy_basic (P initial ** BYTEregIs AL al) ptest
(Otest initial (nth x xs 0))
(I initial
(match coq_test__is_finished (nth x xs 0) with true => negb value | false => value end) **
ConditionIs cond
(match coq_test__is_finished (nth x xs 0) with true => negb value | false => value end))) ->
(forall (initial : T) (xs : list ioT) (x : ioT),
all (fun t : ioT => negb (coq_test__is_finished t)) xs = true ->
xs <> nil ->
coq_test__is_finished x = true ->
S
|-- loopy_basic (I initial value ** ConditionIs cond value) pbody
(Obody initial (nth x xs 0))
(P (accumulate initial (nth x xs 0)) ** BYTEregIs AL al)) ->
forall x : ioT,
coq_test__is_finished x = true ->
S
|-- Forall ixsp : {init_xs : T * list ioT &
all (fun t : ioT => negb (coq_test__is_finished t))
(snd init_xs) = true},
loopy_basic (P (fst (projT1 ixsp)) ** BYTEregIs AL al)
(while ptest cond value pbody)
(catOP
(snd
(foldl
(fun (xy : T * OPred) (v : ioT) =>
(accumulate (fst xy) v,
catOP (catOP (Otest (fst xy) v) (Obody (fst xy) v))
(snd xy))) (fst (projT1 ixsp), empOP)
(snd (projT1 ixsp))))
(Otest (foldl accumulate (fst (projT1 ixsp)) (snd (projT1 ixsp)))
x))
(I (foldl accumulate (fst (projT1 ixsp)) (snd (projT1 ixsp)))
(negb value) ** ConditionIs cond (negb value)).
intros.
eapply @while_rule_ind
with (I_logic := fun ixsp b => match (match (coq_test__is_finished (nth x (snd (projT1 ixsp)) 0)) with true => negb value | false => value end), b with true, true => true | false, false => true | _, _ => false end)
(Otest := fun ixsp => Otest (fst (projT1 ixsp)) (nth x (snd (projT1 ixsp)) 0))
(Obody := fun ixsp => Obody (fst (projT1 ixsp)) (nth x (snd (projT1 ixsp)) 0))
(I_state := fun ixsp => I (fst (projT1 ixsp)))
(transition_body := fun ixsp => let initial := fst (projT1 ixsp) in
let xs := snd (projT1 ixsp) in
existT _ (accumulate initial (nth x xs 0), behead xs) _)
(O_after_test := fun ixsp => let initial := fst (projT1 ixsp) in
let xs := snd (projT1 ixsp) in
match xs with | nil => default_PointedOPred empOP | _ => Obody initial (nth x xs 0) end);
simpl projT1; simpl projT2; simpl fst; simpl snd; clear; let H := fresh in assert (H : False) by (clear; admit); destruct H.
Unshelve.
subst_body; simpl.
Fail refine (all_behead (projT2 _)).
Unset Solve Unification Constraints. refine (all_behead (projT2 _)).
Abort.
|
