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
656
657
658
659
660
661
662
|
// RUN: %dafny /compile:0 /dprint:"%t.dprint" "%s" > "%t"
// RUN: %diff "%s.expect" "%t"
/******* State *******/
type State
function GetSt(p: Path, st: State): Artifact
function SetSt(p: Path, a: Artifact, st: State): State
function DomSt(st: State): set<Path>
function Restrict(paths: set<Path>, st: State): State
requires paths <= DomSt(st);
ensures
var st' := Restrict(paths, st);
DomSt(st') == paths && forall p :: p in paths ==> GetSt(p, st) == GetSt(p, st');
function Union(st: State, st': State): State
ensures
var result := Union(st, st');
DomSt(result) == DomSt(st) + DomSt(st') &&
forall p :: p in DomSt(result) ==>
(p in DomSt(st) ==> GetSt(p, result) == GetSt(p, st)) &&
(p in DomSt(st') ==> GetSt(p, result) == GetSt(p, st'));
ghost method StateEqualityProperty(st: State, st': State)
requires DomSt(st) == DomSt(st');
requires forall p :: p in DomSt(st) ==> GetSt(p, st) == GetSt(p, st');
ensures st == st';
/******* Cached state *******/
datatype StateC = S(st: State, c: Cache)
function EmptyCache(): Cache
ensures DomC(EmptyCache()) == {};
function GetC(h: HashValue, c: Cache): Triple<Expression, Expression, string>
function SetC(h: HashValue, cmd: Triple<Expression, Expression, string>, c: Cache): Cache
ensures DomC(SetC(h, cmd, c)) == DomC(c) + {h};
function UpdateC(cmd: Expression, deps: Expression, exts: Expression, stC: StateC): StateC
requires
cmd.exprLiteral? && cmd.lit.litString? &&
deps.exprLiteral? && deps.lit.litArrOfPaths? &&
exts.exprLiteral? && exts.lit.litArrOfStrings?;
ensures
var stC' := UpdateC(cmd, deps, exts, stC);
var hashValues := set e | e in exts.lit.strs :: Hash(Loc(cmd, deps, e));
stC'.st == stC.st &&
DomC(stC.c) + hashValues == DomC(stC'.c);
decreases exts.lit.strs;
{
var strs := exts.lit.strs;
if strs == {} then
stC
else
var e := Choose(strs);
var c' := SetC(Hash(Loc(cmd, deps, e)), Trio(cmd, deps, e), stC.c);
var exts' := exprLiteral(litArrOfStrings(strs - {e}));
UpdateC(cmd, deps, exts', S(stC.st, c'))
}
ghost method UpdateCLemma(cmd: Expression, deps: Expression, exts: Expression, stC: StateC)
requires
cmd.exprLiteral? && cmd.lit.litString? &&
deps.exprLiteral? && deps.lit.litArrOfPaths? &&
exts.exprLiteral? && exts.lit.litArrOfStrings? &&
ConsistentCache(stC) &&
forall e :: e in exts.lit.strs ==> Loc(cmd, deps, e) in DomSt(stC.st);
ensures
var stC' := UpdateC(cmd, deps, exts, stC);
ConsistentCache(stC') &&
forall e :: e in exts.lit.strs ==> Hash(Loc(cmd, deps, e)) in DomC(stC'.c);
decreases exts.lit.strs;
{
var strs := exts.lit.strs;
var stC' := UpdateC(cmd, deps, exts, stC);
if strs == {} {
} else {
var e := Choose(strs);
var c' := SetC(Hash(Loc(cmd, deps, e)), Trio(cmd, deps, e), stC.c);
var exts' := exprLiteral(litArrOfStrings(strs - {e}));
// note: This assertion is necessary.
assert stC' == UpdateC(cmd, deps, exts', S(stC.st, c'));
forall (cmd', deps', e' | Hash(Loc(cmd', deps', e')) == Hash(Loc(cmd, deps, e))) {
HashProperty(cmd', deps', e', cmd, deps, e);
}
}
}
function Choose(ss: set<string>): string
requires ss != {};
{
var s :| s in ss; s
}
function DomC(c: Cache): set<HashValue>
function UnionC(stC: StateC, stC': StateC): StateC
ensures
var result := UnionC(stC, stC');
DomSt(result.st) == DomSt(stC.st) + DomSt(stC'.st) &&
(forall p :: p in DomSt(result.st) ==>
(p in DomSt(stC.st) ==> GetSt(p, result.st) == GetSt(p, stC.st)) &&
(p in DomSt(stC'.st) ==> GetSt(p, result.st) == GetSt(p, stC'.st))) &&
DomC(result.c) == DomC(stC.c) + DomC(stC'.c) &&
(forall h :: h in DomC(result.c) ==>
(h in DomC(stC.c) ==> GetC(h, result.c) == GetC(h, stC.c)) &&
(h in DomC(stC'.c) ==> GetC(h, result.c) == GetC(h, stC'.c)));
predicate CompatibleC(stsC: set<StateC>)
{
forall stC, stC', p, h :: stC in stsC && stC' in stsC &&
p in DomSt(stC.st) && p in DomSt(stC'.st) &&
h in DomC(stC.c) && h in DomC(stC'.c) ==>
GetSt(p, stC.st) == GetSt(p, stC'.st) && GetC(h, stC.c) == GetC(h, stC'.c)
}
function CombineC(stsC: set<StateC>): StateC
requires stsC != {};
ensures
var stCombinedC := CombineC(stsC);
(forall stC :: stC in stsC ==> DomSt(stC.st) <= DomSt(stCombinedC.st)) &&
(forall stC, p :: stC in stsC && p in DomSt(stC.st) ==>
GetSt(p, stC.st) == GetSt(p, stCombinedC.st)) &&
(forall p :: p in DomSt(stCombinedC.st) ==> exists stC :: stC in stsC && p in DomSt(stC.st)) &&
(forall stC :: stC in stsC ==> DomC(stC.c) <= DomC(stCombinedC.c)) &&
(forall stC, h :: stC in stsC && h in DomC(stC.c) ==>
GetC(h, stC.c) == GetC(h, stCombinedC.c)) &&
(forall h :: h in DomC(stCombinedC.c) ==> exists stC :: stC in stsC && h in DomC(stC.c));
{
var stC :| stC in stsC;
if stsC == {stC} then
stC
else
UnionC(stC, CombineC(stsC - {stC}))
}
ghost method CombineCLemma(stsC: set<StateC>)
requires stsC != {};
requires forall stC :: stC in stsC ==> ConsistentCache(stC);
ensures
var stC' := CombineC(stsC);
ConsistentCache(stC');
{
}
predicate ConsistentCache(stC: StateC)
{
forall cmd, deps, e :: Hash(Loc(cmd, deps, e)) in DomC(stC.c) ==>
Loc(cmd, deps, e) in DomSt(stC.st)
}
/******* {true} init {consistent_cache} *******/
function ClearCache(stC: StateC): StateC
ensures
var stC' := ClearCache(stC);
// note: This follows directly from the definition.
stC.st == stC'.st && DomC(stC'.c) == {} &&
ConsistentCache(stC');
{
S(stC.st, EmptyCache())
}
/******* Environment *******/
type Env
function EmptyEnv(): Env
function GetEnv(id: Identifier, env: Env): Expression
ensures Value(GetEnv(id, env));
function SetEnv(id: Identifier, expr: Expression, env: Env): Env
requires Value(expr);
/******* Primitive function 'exec' *******/
function exec(cmd: Expression, deps: Expression, exts: Expression, st: State): Tuple<Expression, State>
ghost method ExecProperty(cmd: Expression, deps: Expression, exts: Expression, st: State)
requires
cmd.exprLiteral? && cmd.lit.litString? &&
deps.exprLiteral? && deps.lit.litArrOfPaths? &&
exts.exprLiteral? && exts.lit.litArrOfStrings? &&
deps.lit.paths <= DomSt(st) &&
Pre(cmd, deps, exts, Restrict(deps.lit.paths, st));
ensures
var result := exec(cmd, deps, exts, st);
var expr', st' := result.fst, result.snd;
expr'.exprLiteral? && expr'.lit.litArrOfPaths? &&
expr'.lit.paths <= DomSt(st') &&
// note: We need this for the precondition of Restrict.
DomSt(st) <= DomSt(st') && st == Restrict(DomSt(st), st') &&
OneToOne(cmd, deps, exts, expr') &&
Post(cmd, deps, exts, Restrict(deps.lit.paths, st')) &&
forall p :: p !in DomSt(st) && p in DomSt(st') ==> p.OpaquePath?;
predicate Pre(cmd: Expression, deps: Expression, exts: Expression, st: State)
requires
cmd.exprLiteral? && cmd.lit.litString? &&
deps.exprLiteral? && deps.lit.litArrOfPaths? &&
exts.exprLiteral? && exts.lit.litArrOfStrings?;
{
forall e :: e in exts.lit.strs ==>
Loc(cmd, deps, e) in DomSt(st) ==> GetSt(Loc(cmd, deps, e), st) == Res(cmd, deps, e, st)
}
predicate Post(cmd: Expression, deps: Expression, exts: Expression, st: State)
requires
cmd.exprLiteral? && cmd.lit.litString? &&
deps.exprLiteral? && deps.lit.litArrOfPaths? &&
exts.exprLiteral? && exts.lit.litArrOfStrings?;
{
forall e :: e in exts.lit.strs ==>
Loc(cmd, deps, e) in DomSt(st) && GetSt(Loc(cmd, deps, e), st) == Res(cmd, deps, e, st)
}
function Res(cmd: Expression, deps: Expression, ext: string, st: State): Artifact
predicate OneToOne(cmd: Expression, deps: Expression, exts: Expression, paths: Expression)
requires
cmd.exprLiteral? && cmd.lit.litString? &&
deps.exprLiteral? && deps.lit.litArrOfPaths? &&
exts.exprLiteral? && exts.lit.litArrOfStrings? &&
paths.exprLiteral? && paths.lit.litArrOfPaths?;
{
forall e :: e in exts.lit.strs ==> Loc(cmd, deps, e) in paths.lit.paths
}
function Loc(cmd: Expression, deps: Expression, ext: string): Path
/******* Primitive function 'execC' *******/
function execC(cmd: Expression, deps: Expression, exts: Expression, stC: StateC): Tuple<Expression, StateC>
requires
cmd.exprLiteral? && cmd.lit.litString? &&
deps.exprLiteral? && deps.lit.litArrOfPaths? &&
exts.exprLiteral? && exts.lit.litArrOfStrings?;
{
if forall e | e in exts.lit.strs :: Hash(Loc(cmd, deps, e)) in DomC(stC.c) then
var paths := set e | e in exts.lit.strs :: Loc(cmd, deps, e);
var expr' := exprLiteral(litArrOfPaths(paths));
Pair(expr', stC)
else
var result := exec(cmd, deps, exts, stC.st);
var expr', st' := result.fst, result.snd;
var stC' := UpdateC(cmd, deps, exts, S(st', stC.c));
Pair(expr', stC')
}
ghost method ExecCProperty(cmd: Expression, deps: Expression, exts: Expression, stC: StateC)
requires
cmd.exprLiteral? && cmd.lit.litString? &&
deps.exprLiteral? && deps.lit.litArrOfPaths? &&
exts.exprLiteral? && exts.lit.litArrOfStrings? &&
deps.lit.paths <= DomSt(stC.st) &&
PreC(cmd, deps, exts, stC) &&
ConsistentCache(stC);
ensures
var result := execC(cmd, deps, exts, stC);
var expr', stC' := result.fst, result.snd;
expr'.exprLiteral? && expr'.lit.litArrOfPaths? &&
expr'.lit.paths <= DomSt(stC'.st) &&
// note: We need this for the precondition of Restrict.
DomSt(stC.st) <= DomSt(stC'.st) && stC.st == Restrict(DomSt(stC.st), stC'.st) &&
OneToOne(cmd, deps, exts, expr') &&
PostC(cmd, deps, exts, stC') &&
(forall p :: p !in DomSt(stC.st) && p in DomSt(stC'.st) ==> p.OpaquePath?) &&
ConsistentCache(stC');
{
var result := execC(cmd, deps, exts, stC);
var expr', stC' := result.fst, result.snd;
if forall e | e in exts.lit.strs :: Hash(Loc(cmd, deps, e)) in DomC(stC.c) {
StateEqualityProperty(stC.st, Restrict(DomSt(stC.st), stC'.st));
} else {
ExecProperty(cmd, deps, exts, stC.st);
var execResult := exec(cmd, deps, exts, stC.st);
var st' := execResult.snd;
assert DomSt(stC.st) <= DomSt(st');
StateEqualityProperty(stC'.st, st');
UpdateCLemma(cmd, deps, exts, S(st', stC.c));
}
}
predicate PreC(cmd: Expression, deps: Expression, exts: Expression, stC: StateC)
requires
cmd.exprLiteral? && cmd.lit.litString? &&
deps.exprLiteral? && deps.lit.litArrOfPaths? &&
exts.exprLiteral? && exts.lit.litArrOfStrings? &&
// note: We need this for the precondition of Restrict.
deps.lit.paths <= DomSt(stC.st);
{
Pre(cmd, deps, exts, Restrict(deps.lit.paths, stC.st)) &&
forall e :: e in exts.lit.strs ==> Hash(Loc(cmd, deps, e)) in DomC(stC.c) ==>
Loc(cmd, deps, e) in deps.lit.paths
}
predicate PostC(cmd: Expression, deps: Expression, exts: Expression, stC: StateC)
requires
cmd.exprLiteral? && cmd.lit.litString? &&
deps.exprLiteral? && deps.lit.litArrOfPaths? &&
exts.exprLiteral? && exts.lit.litArrOfStrings? &&
// note: We need this for the precondition of Restrict.
deps.lit.paths <= DomSt(stC.st);
{
Post(cmd, deps, exts, Restrict(deps.lit.paths, stC.st)) &&
forall e :: e in exts.lit.strs ==> Hash(Loc(cmd, deps, e)) in DomC(stC.c)
}
function Hash(p: Path): HashValue
ghost method HashProperty(cmd: Expression, deps: Expression, ext: string, cmd': Expression, deps': Expression, ext': string)
requires Hash(Loc(cmd, deps, ext)) == Hash(Loc(cmd', deps', ext'));
ensures cmd == cmd' && deps == deps' && ext == ext';
/******* Grammar *******/
datatype Program = Program(stmts: seq<Statement>)
datatype Statement = stmtVariable(id: Identifier, expr: Expression) |
stmtReturn(ret: Expression)
datatype Expression = exprLiteral(lit: Literal) | exprIdentifier(id: Identifier) |
exprIf(cond: Expression, ifTrue: Expression, ifFalse: Expression) |
exprAnd(conj0: Expression, conj1: Expression) |
exprOr(disj0: Expression, disj1: Expression) |
exprInvocation(fun: Expression, args: seq<Expression>) |
exprError(r: Reason)
datatype Literal = litTrue | litFalse | litUndefined | litNull |
litNumber(num: int) | litString(str: string) |
litPrimitive(prim: Primitive) |
litArrOfPaths(paths: set<Path>) |
litArrOfStrings(strs: set<string>) |
litArray(arr: seq<Expression>)
datatype Primitive = primCreatePath | primExec
datatype Reason = rCompatibility | rValidity | rInconsistentCache
datatype Path = OpaquePath(int) | TransparentPath(int)
type Artifact
type Identifier
type Cache
type HashValue
datatype Tuple<A, B> = Pair(fst: A, snd: B)
datatype Triple<A, B, C> = Trio(fst: A, snd: B, trd: C)
/******* Values *******/
predicate Value(expr: Expression)
{
expr.exprLiteral?
}
/******* Semantics *******/
predicate Legal(stmts: seq<Statement>)
{
|stmts| != 0
}
function Arity(prim: Primitive): nat
{
match prim
case primCreatePath => 1
case primExec => 3
}
/******* Function 'buildC' *******/
function buildC(prog: Program, stC: StateC): Tuple<Expression, StateC>
requires Legal(prog.stmts);
{
doC(prog.stmts, stC, EmptyEnv())
}
/******* Function 'doC' *******/
function doC(stmts: seq<Statement>, stC: StateC, env: Env): Tuple<Expression, StateC>
requires Legal(stmts);
{
var stmt := stmts[0];
if stmt.stmtVariable? then
var result := evalC(stmt.expr, stC, env);
var expr', stC' := result.fst, result.snd;
if Value(expr') then
var env' := SetEnv(stmt.id, expr', env);
if Legal(stmts[1..]) then
doC(stmts[1..], stC', env')
else
Pair(expr', stC')
else
Pair(exprError(rValidity), stC)
// todo(maria): Add the recursive case.
else
assert stmt.stmtVariable? || stmt.stmtReturn?;
evalC(stmt.ret, stC, env)
}
/******* Function 'evalC' *******/
function evalC(expr: Expression, stC: StateC, env: Env): Tuple<Expression, StateC>
decreases expr;
{
if Value(expr) then
Pair(expr, stC)
// identifier
else if expr.exprIdentifier? then
Pair(GetEnv(expr.id, env), stC)
// if-expression
else if expr.exprIf? && expr.cond.exprLiteral? && expr.cond.lit == litTrue then
evalC(expr.ifTrue, stC, env)
else if expr.exprIf? && expr.cond.exprLiteral? && expr.cond.lit == litFalse then
evalC(expr.ifFalse, stC, env)
else if expr.exprIf? then
var result := evalC(expr.cond, stC, env);
var cond', stC' := result.fst, result.snd;
if cond'.exprLiteral? && cond'.lit == litTrue then
evalC(expr.ifTrue, stC', env)
else if cond'.exprLiteral? && cond'.lit == litFalse then
evalC(expr.ifFalse, stC', env)
else
Pair(exprError(rValidity), stC)
// and-expression
else if expr.exprAnd? then
var result := evalC(expr.conj0, stC, env);
var conj0', stC' := result.fst, result.snd;
if conj0'.exprLiteral? && conj0'.lit == litTrue then
evalC(expr.conj1, stC', env)
else if conj0'.exprLiteral? && conj0'.lit == litFalse then
Pair(exprLiteral(litFalse), stC')
else
Pair(exprError(rValidity), stC)
// or-expression
else if expr.exprOr? then
var result := evalC(expr.disj0, stC, env);
var disj0', stC' := result.fst, result.snd;
if disj0'.exprLiteral? && disj0'.lit == litTrue then
Pair(exprLiteral(litTrue), stC')
else if disj0'.exprLiteral? && disj0'.lit == litFalse then
evalC(expr.disj1, stC', env)
else
Pair(exprError(rValidity), stC)
// invocation
else if expr.exprInvocation? then
var resultFun := evalC(expr.fun, stC, env);
var fun', stC' := resultFun.fst, resultFun.snd;
var resultArgs := evalArgsC(expr, expr.args, stC, env);
var args', stsC' := resultArgs.fst, resultArgs.snd;
var stsC'' := {stC'} + stsC';
if CompatibleC(stsC'') then
var stCombinedC := CombineC(stsC'');
// primitive functions
if fun'.exprLiteral? && fun'.lit.litPrimitive? then
// primitive function 'execC'
if fun'.lit.prim.primExec? then
if |args'| == Arity(primExec) && ValidArgsC(primExec, args', stCombinedC) then
execC(args'[0], args'[1], args'[2], stCombinedC)
else
if ConsistentCache(stCombinedC) then
Pair(exprError(rValidity), stC)
else
Pair(exprError(rInconsistentCache), stC)
else
// primitive function 'createPath'
// todo(maria): Add primitive function 'createPath'.
Pair(exprError(rValidity), stC)
// todo(maria): Add non-primitive invocations.
else
Pair(exprError(rValidity), stC)
else
Pair(exprError(rCompatibility), stC)
// error
else
Pair(exprError(rValidity), stC)
}
function evalArgsC(expr: Expression, args: seq<Expression>, stC: StateC, env: Env):
Tuple<seq<Expression>, set<StateC>>
requires forall arg :: arg in args ==> arg < expr;
decreases expr, |args| + 1;
{
evalArgsC'(expr, args, stC, env, [], {})
}
function evalArgsC'(expr: Expression, args: seq<Expression>, stC: StateC, env: Env,
args': seq<Expression>, stsC': set<StateC>):
Tuple<seq<Expression>, set<StateC>>
requires forall arg :: arg in args ==> arg < expr;
decreases expr, |args|;
{
if args == [] then
Pair(args', stsC')
else
var arg := args[0];
var result := evalC(arg, stC, env);
var arg', stC' := result.fst, result.snd;
evalArgsC'(expr, args[1..], stC, env, args' + [arg'], stsC' + {stC'})
}
predicate ValidArgsC(prim: Primitive, args: seq<Expression>, stC: StateC)
requires prim.primExec? ==> |args| == 3;
requires prim.primCreatePath? ==> |args| == 1;
{
match prim
case primCreatePath => false
case primExec =>
var cmd, deps, exts := args[0], args[1], args[2];
cmd.exprLiteral? && cmd.lit.litString? &&
deps.exprLiteral? && deps.lit.litArrOfPaths? &&
exts.exprLiteral? && exts.lit.litArrOfStrings? &&
deps.lit.paths <= DomSt(stC.st) &&
PreC(cmd, deps, exts, stC)
}
/******* {consistent_cache} buildC {no_bad_cache_error /\ consistent_cache} *******/
ghost method CachedBuildsTheorem(prog: Program, stC: StateC)
requires Legal(prog.stmts);
requires ConsistentCache(stC);
ensures
var result := buildC(prog, stC);
var expr', stC' := result.fst, result.snd;
ConsistentCache(stC') &&
expr'.exprError? ==> expr'.r != rInconsistentCache;
{
BuildCLemma(prog, stC);
}
ghost method BuildCLemma(prog: Program, stC: StateC)
requires Legal(prog.stmts);
requires ConsistentCache(stC);
ensures
var result := buildC(prog, stC);
var expr', stC' := result.fst, result.snd;
ConsistentCache(stC') && (expr'.exprError? ==> expr'.r != rInconsistentCache);
{
DoCLemma(prog.stmts, stC, EmptyEnv());
}
ghost method DoCLemma(stmts: seq<Statement>, stC: StateC, env: Env)
requires Legal(stmts);
requires ConsistentCache(stC);
ensures
var result := doC(stmts, stC, env);
var expr', stC' := result.fst, result.snd;
ConsistentCache(stC') && (expr'.exprError? ==> expr'.r != rInconsistentCache);
{
var stmt := stmts[0];
if stmt.stmtVariable? {
EvalCLemma(stmt.expr, stC, env);
var result := evalC(stmt.expr, stC, env);
var expr', stC' := result.fst, result.snd;
if Value(expr') {
var env' := SetEnv(stmt.id, expr', env);
if Legal(stmts[1..]) {
DoCLemma(stmts[1..], stC', env');
} else { }
} else { }
// todo(maria): Add the recursive case.
} else {
assert stmt.stmtVariable? || stmt.stmtReturn?;
EvalCLemma(stmt.ret, stC, env);
}
}
ghost method {:induction expr} EvalCLemma(expr: Expression, stC: StateC, env: Env)
requires ConsistentCache(stC);
ensures
var result := evalC(expr, stC, env);
var expr', stC' := result.fst, result.snd;
ConsistentCache(stC') && (expr'.exprError? ==> expr'.r != rInconsistentCache);
decreases expr;
{
if Value(expr) {
} else if expr.exprIdentifier? {
} else if expr.exprIf? && expr.cond.exprLiteral? && expr.cond.lit == litTrue {
} else if expr.exprIf? && expr.cond.exprLiteral? && expr.cond.lit == litFalse {
} else if expr.exprIf? {
var result := evalC(expr.cond, stC, env);
var cond', stC' := result.fst, result.snd;
if cond'.exprLiteral? && cond'.lit == litTrue {
EvalCLemma(expr.ifTrue, stC', env);
} else if cond'.exprLiteral? && cond'.lit == litFalse {
EvalCLemma(expr.ifFalse, stC', env);
} else { }
} else if expr.exprAnd? {
var result := evalC(expr.conj0, stC, env);
var conj0', stC' := result.fst, result.snd;
if conj0'.exprLiteral? && conj0'.lit == litTrue {
EvalCLemma(expr.conj1, stC', env);
} else if conj0'.exprLiteral? && conj0'.lit == litFalse {
} else { }
} else if expr.exprOr? {
var result := evalC(expr.disj0, stC, env);
var disj0', stC' := result.fst, result.snd;
if disj0'.exprLiteral? && disj0'.lit == litTrue {
} else if disj0'.exprLiteral? && disj0'.lit == litFalse {
EvalCLemma(expr.disj1, stC', env);
} else { }
} else if expr.exprInvocation? {
EvalCLemma(expr.fun, stC, env);
var resultFun := evalC(expr.fun, stC, env);
var fun', stC' := resultFun.fst, resultFun.snd;
EvalArgsCLemma(expr, expr.args, stC, env);
var resultArgs := evalArgsC(expr, expr.args, stC, env);
var args', stsC' := resultArgs.fst, resultArgs.snd;
var stsC'' := {stC'} + stsC';
if CompatibleC(stsC'') {
CombineCLemma(stsC'');
var stCombinedC := CombineC(stsC'');
if fun'.exprLiteral? && fun'.lit.litPrimitive? {
if fun'.lit.prim.primExec? {
if |args'| == Arity(primExec) && ValidArgsC(primExec, args', stCombinedC) {
ExecCProperty(args'[0], args'[1], args'[2], stCombinedC);
var resultExec := execC(args'[0], args'[1], args'[2], stCombinedC);
var stExecC := resultExec.snd;
// note: This assertion is necessary.
assert DomSt(stC'.st) <= DomSt(stCombinedC.st);
forall (p | p in DomSt(stCombinedC.st) && p in DomSt(stExecC.st))
ensures GetSt(p, stCombinedC.st) == GetSt(p, stExecC.st);
{
assert DomSt(stCombinedC.st) <= DomSt(stExecC.st);
assert stCombinedC.st == Restrict(DomSt(stCombinedC.st), stExecC.st);
}
} else {
if ConsistentCache(stCombinedC) {
} else { }
}
} else { }
} else { }
} else { }
} else { }
}
ghost method EvalArgsCLemma(expr: Expression, args: seq<Expression>, stC: StateC, env: Env)
requires ConsistentCache(stC);
requires forall arg :: arg in args ==> arg < expr;
ensures
var result := evalArgsC(expr, args, stC, env);
var stsC' := result.snd;
forall stC' :: stC' in stsC' ==> ConsistentCache(stC');
decreases expr, |args| + 1;
{
EvalArgsC'Lemma(expr, args, stC, env, [], {});
}
ghost method EvalArgsC'Lemma(expr: Expression, args: seq<Expression>, stC: StateC, env: Env,
args': seq<Expression>, stsC': set<StateC>)
requires ConsistentCache(stC);
requires forall stC' :: stC' in stsC' ==> ConsistentCache(stC');
requires forall arg :: arg in args ==> arg < expr;
ensures
var result := evalArgsC'(expr, args, stC, env, args', stsC');
var stsC'' := result.snd;
forall stC'' :: stC'' in stsC'' ==> ConsistentCache(stC'');
decreases expr, |args|;
{
if args == [] {
} else {
var arg := args[0];
EvalCLemma(arg, stC, env);
var result := evalC(arg, stC, env);
var arg', stC' := result.fst, result.snd;
EvalArgsC'Lemma(expr, args[1..], stC, env, args' + [arg'], stsC' + {stC'});
}
}
|