source: C-semantics/Values.ma @ 125

Last change on this file since 125 was 125, checked in by campbell, 10 years ago

Unify memory space / pointer types.
Implement global variable initialisation and lookup.
Global variables get memory spaces, local variables could be anywhere (for now).

File size: 27.5 KB
Line 
1(* *********************************************************************)
2(*                                                                     *)
3(*              The Compcert verified compiler                         *)
4(*                                                                     *)
5(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
6(*                                                                     *)
7(*  Copyright Institut National de Recherche en Informatique et en     *)
8(*  Automatique.  All rights reserved.  This file is distributed       *)
9(*  under the terms of the GNU General Public License as published by  *)
10(*  the Free Software Foundation, either version 2 of the License, or  *)
11(*  (at your option) any later version.  This file is also distributed *)
12(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
13(*                                                                     *)
14(* *********************************************************************)
15
16(* * This module defines the type of values that is used in the dynamic
17  semantics of all our intermediate languages. *)
18
19include "Coqlib.ma".
20include "AST.ma".
21include "Integers.ma".
22include "Floats.ma".
23
24include "Plogic/connectives.ma".
25
26ndefinition block ≝ Z.
27(*ndefinition eq_block ≝ zeq.*)
28
29(* * A value is either:
30- a machine integer;
31- a floating-point number;
32- a pointer: a pair of a memory address and an integer offset with respect
33  to this address;
34- the [Vundef] value denoting an arbitrary bit pattern, such as the
35  value of an uninitialized variable.
36*)
37
38(* TODO: should comparison and subtraction of pointers of different sorts
39         be supported? *)
40
41ninductive val: Type[0] ≝
42  | Vundef: val
43  | Vint: int -> val
44  | Vfloat: float -> val
45  | Vptr: memory_space → block -> int -> val.
46
47ndefinition Vzero: val ≝ Vint zero.
48ndefinition Vone: val ≝ Vint one.
49ndefinition Vmone: val ≝ Vint mone.
50
51ndefinition Vtrue: val ≝ Vint one.
52ndefinition Vfalse: val ≝ Vint zero.
53
54(*
55(** The module [Val] defines a number of arithmetic and logical operations
56  over type [val].  Most of these operations are straightforward extensions
57  of the corresponding integer or floating-point operations. *)
58
59Module Val.
60*)
61ndefinition of_bool : bool → val ≝ λb. if b then Vtrue else Vfalse.
62
63ndefinition has_type ≝ λv: val. λt: typ.
64  match v with
65  [ Vundef ⇒ True
66  | Vint _ ⇒ match t with [ Tint ⇒ True | _ ⇒ False ]
67  | Vfloat _ ⇒ match t with [ Tfloat ⇒ True | _ ⇒ False ]
68  | Vptr _ _ _ ⇒ match t with [ Tint ⇒ True | _ ⇒ False ]
69  | _ ⇒ False
70  ].
71
72nlet rec has_type_list (vl: list val) (tl: list typ) on vl : Prop ≝
73  match vl with
74  [ nil ⇒ match tl with [ nil ⇒ True | _ ⇒ False ]
75  | cons v1 vs ⇒ match tl with [ nil ⇒ False | cons t1 ts ⇒
76               has_type v1 t1 ∧ has_type_list vs ts ]
77  ].
78
79(* * Truth values.  Pointers and non-zero integers are treated as [True].
80  The integer 0 (also used to represent the null pointer) is [False].
81  [Vundef] and floats are neither true nor false. *)
82
83ndefinition is_true : val → Prop ≝ λv.
84  match v with
85  [ Vint n ⇒ n ≠ zero
86  | Vptr _ b ofs ⇒ True
87  | _ ⇒ False
88  ].
89
90ndefinition is_false : val → Prop ≝ λv.
91  match v with
92  [ Vint n ⇒ n = zero
93  | _ ⇒ False
94  ].
95
96ninductive bool_of_val: val → bool → Prop ≝
97  | bool_of_val_int_true:
98      ∀n. n ≠ zero → bool_of_val (Vint n) true
99  | bool_of_val_int_false:
100      bool_of_val (Vint zero) false
101  | bool_of_val_ptr:
102      ∀pty,b,ofs. bool_of_val (Vptr pty b ofs) true.
103
104ndefinition neg : val → val ≝ λv.
105  match v with
106  [ Vint n ⇒ Vint (neg n)
107  | _ ⇒ Vundef
108  ].
109
110ndefinition negf : val → val ≝ λv.
111  match v with
112  [ Vfloat f ⇒ Vfloat (Fneg f)
113  | _ => Vundef
114  ].
115
116ndefinition absf : val → val ≝ λv.
117  match v with
118  [ Vfloat f ⇒ Vfloat (Fabs f)
119  | _ ⇒ Vundef
120  ].
121
122ndefinition intoffloat : val → val ≝ λv.
123  match v with
124  [ Vfloat f ⇒ Vint (intoffloat f)
125  | _ ⇒ Vundef
126  ].
127
128ndefinition intuoffloat : val → val ≝ λv.
129  match v with
130  [ Vfloat f ⇒ Vint (intuoffloat f)
131  | _ ⇒ Vundef
132  ].
133
134ndefinition floatofint : val → val ≝ λv.
135  match v with
136  [ Vint n ⇒ Vfloat (floatofint n)
137  | _ ⇒ Vundef
138  ].
139
140ndefinition floatofintu : val → val ≝ λv.
141  match v with
142  [ Vint n ⇒ Vfloat (floatofintu n)
143  | _ ⇒ Vundef
144  ].
145
146ndefinition notint : val → val ≝ λv.
147  match v with
148  [ Vint n ⇒ Vint (xor n mone)
149  | _ ⇒ Vundef
150  ].
151 
152(* FIXME: switch to alias, or rename, or … *)
153ndefinition int_eq : int → int → bool ≝ eq.
154ndefinition notbool : val → val ≝ λv.
155  match v with
156  [ Vint n ⇒ of_bool (int_eq n zero)
157  | Vptr _ b ofs ⇒ Vfalse
158  | _ ⇒ Vundef
159  ].
160
161ndefinition zero_ext ≝ λnbits: Z. λv: val.
162  match v with
163  [ Vint n ⇒ Vint (zero_ext nbits n)
164  | _ ⇒ Vundef
165  ].
166
167ndefinition sign_ext ≝ λnbits:Z. λv:val.
168  match v with
169  [ Vint i ⇒ Vint (sign_ext nbits i)
170  | _ ⇒ Vundef
171  ].
172
173ndefinition singleoffloat : val → val ≝ λv.
174  match v with
175  [ Vfloat f ⇒ Vfloat (singleoffloat f)
176  | _ ⇒ Vundef
177  ].
178
179ndefinition add ≝ λv1,v2: val.
180  match v1 with
181  [ Vint n1 ⇒ match v2 with
182    [ Vint n2 ⇒ Vint (add n1 n2)
183    | Vptr pty b2 ofs2 ⇒ Vptr pty b2 (add ofs2 n1)
184    | _ ⇒ Vundef ]
185  | Vptr pty b1 ofs1 ⇒ match v2 with
186    [ Vint n2 ⇒ Vptr pty b1 (add ofs1 n2)
187    | _ ⇒ Vundef ]
188  | _ ⇒ Vundef ].
189
190ndefinition sub ≝ λv1,v2: val.
191  match v1 with
192  [ Vint n1 ⇒ match v2 with
193    [ Vint n2 ⇒ Vint (sub n1 n2)
194    | _ ⇒ Vundef ]
195  | Vptr pty1 b1 ofs1 ⇒ match v2 with
196    [ Vint n2 ⇒ Vptr pty1 b1 (sub ofs1 n2)
197    | Vptr pty2 b2 ofs2 ⇒
198        if eqZb b1 b2 then Vint (sub ofs1 ofs2) else Vundef
199    | _ ⇒ Vundef ]
200  | _ ⇒ Vundef ].
201
202ndefinition mul ≝ λv1, v2: val.
203  match v1 with
204  [ Vint n1 ⇒ match v2 with
205    [ Vint n2 ⇒ Vint (mul n1 n2)
206    | _ ⇒ Vundef ]
207  | _ ⇒ Vundef ].
208(*
209ndefinition divs ≝ λv1, v2: val.
210  match v1 with
211  [ Vint n1 ⇒ match v2 with
212    [ Vint n2 ⇒ Vint (divs n1 n2)
213    | _ ⇒ Vundef ]
214  | _ ⇒ Vundef ].
215
216Definition mods (v1 v2: val): val :=
217  match v1, v2 with
218  | Vint n1, Vint n2 =>
219      if Int.eq n2 Int.zero then Vundef else Vint(Int.mods n1 n2)
220  | _, _ => Vundef
221  end.
222
223Definition divu (v1 v2: val): val :=
224  match v1, v2 with
225  | Vint n1, Vint n2 =>
226      if Int.eq n2 Int.zero then Vundef else Vint(Int.divu n1 n2)
227  | _, _ => Vundef
228  end.
229
230Definition modu (v1 v2: val): val :=
231  match v1, v2 with
232  | Vint n1, Vint n2 =>
233      if Int.eq n2 Int.zero then Vundef else Vint(Int.modu n1 n2)
234  | _, _ => Vundef
235  end.
236*)
237ndefinition v_and ≝ λv1, v2: val.
238  match v1 with
239  [ Vint n1 ⇒ match v2 with
240    [ Vint n2 ⇒ Vint (i_and n1 n2)
241    | _ ⇒ Vundef ]
242  | _ ⇒ Vundef ].
243
244ndefinition or ≝ λv1, v2: val.
245  match v1 with
246  [ Vint n1 ⇒ match v2 with
247    [ Vint n2 ⇒ Vint (or n1 n2)
248    | _ ⇒ Vundef ]
249  | _ ⇒ Vundef ].
250
251ndefinition xor ≝ λv1, v2: val.
252  match v1 with
253  [ Vint n1 ⇒ match v2 with
254    [ Vint n2 ⇒ Vint (xor n1 n2)
255    | _ ⇒ Vundef ]
256  | _ ⇒ Vundef ].
257(*
258Definition shl (v1 v2: val): val :=
259  match v1, v2 with
260  | Vint n1, Vint n2 =>
261     if Int.ltu n2 Int.iwordsize
262     then Vint(Int.shl n1 n2)
263     else Vundef
264  | _, _ => Vundef
265  end.
266
267Definition shr (v1 v2: val): val :=
268  match v1, v2 with
269  | Vint n1, Vint n2 =>
270     if Int.ltu n2 Int.iwordsize
271     then Vint(Int.shr n1 n2)
272     else Vundef
273  | _, _ => Vundef
274  end.
275
276Definition shr_carry (v1 v2: val): val :=
277  match v1, v2 with
278  | Vint n1, Vint n2 =>
279     if Int.ltu n2 Int.iwordsize
280     then Vint(Int.shr_carry n1 n2)
281     else Vundef
282  | _, _ => Vundef
283  end.
284
285Definition shrx (v1 v2: val): val :=
286  match v1, v2 with
287  | Vint n1, Vint n2 =>
288     if Int.ltu n2 Int.iwordsize
289     then Vint(Int.shrx n1 n2)
290     else Vundef
291  | _, _ => Vundef
292  end.
293
294Definition shru (v1 v2: val): val :=
295  match v1, v2 with
296  | Vint n1, Vint n2 =>
297     if Int.ltu n2 Int.iwordsize
298     then Vint(Int.shru n1 n2)
299     else Vundef
300  | _, _ => Vundef
301  end.
302
303Definition rolm (v: val) (amount mask: int): val :=
304  match v with
305  | Vint n => Vint(Int.rolm n amount mask)
306  | _ => Vundef
307  end.
308
309Definition ror (v1 v2: val): val :=
310  match v1, v2 with
311  | Vint n1, Vint n2 =>
312     if Int.ltu n2 Int.iwordsize
313     then Vint(Int.ror n1 n2)
314     else Vundef
315  | _, _ => Vundef
316  end.
317*)
318ndefinition addf ≝ λv1,v2: val.
319  match v1 with
320  [ Vfloat f1 ⇒ match v2 with
321    [ Vfloat f2 ⇒ Vfloat (Fadd f1 f2)
322    | _ ⇒ Vundef ]
323  | _ ⇒ Vundef ].
324
325ndefinition subf ≝ λv1,v2: val.
326  match v1 with
327  [ Vfloat f1 ⇒ match v2 with
328    [ Vfloat f2 ⇒ Vfloat (Fsub f1 f2)
329    | _ ⇒ Vundef ]
330  | _ ⇒ Vundef ].
331
332ndefinition mulf ≝ λv1,v2: val.
333  match v1 with
334  [ Vfloat f1 ⇒ match v2 with
335    [ Vfloat f2 ⇒ Vfloat (Fmul f1 f2)
336    | _ ⇒ Vundef ]
337  | _ ⇒ Vundef ].
338
339ndefinition divf ≝ λv1,v2: val.
340  match v1 with
341  [ Vfloat f1 ⇒ match v2 with
342    [ Vfloat f2 ⇒ Vfloat (Fdiv f1 f2)
343    | _ ⇒ Vundef ]
344  | _ ⇒ Vundef ].
345
346ndefinition cmp_mismatch : comparison → val ≝ λc.
347  match c with
348  [ Ceq ⇒ Vfalse
349  | Cne ⇒ Vtrue
350  | _   ⇒ Vundef
351  ].
352
353ndefinition cmp ≝ λc: comparison. λv1,v2: val.
354  match v1 with
355  [ Vint n1 ⇒ match v2 with
356    [ Vint n2 ⇒ of_bool (cmp c n1 n2)
357    | Vptr pty2 b2 ofs2 ⇒
358        if eq n1 zero then cmp_mismatch c else Vundef
359    | _ ⇒ Vundef ]
360  | Vptr pty1 b1 ofs1 ⇒ match v2 with
361    [ Vptr pty2 b2 ofs2 ⇒
362        if eqZb b1 b2
363        then of_bool (cmp c ofs1 ofs2)
364        else cmp_mismatch c
365    | Vint n2 ⇒
366        if eq n2 zero then cmp_mismatch c else Vundef
367    | _ ⇒ Vundef ]
368  | _ ⇒ Vundef ].
369
370ndefinition cmpu ≝ λc: comparison. λv1,v2: val.
371  match v1 with
372  [ Vint n1 ⇒ match v2 with
373    [ Vint n2 ⇒ of_bool (cmpu c n1 n2)
374    | Vptr pty2 b2 ofs2 ⇒
375        if eq n1 zero then cmp_mismatch c else Vundef
376    | _ ⇒ Vundef ]
377  | Vptr pty1 b1 ofs1 ⇒ match v2 with
378    [ Vptr pty2 b2 ofs2 ⇒
379        if eqZb b1 b2
380        then of_bool (cmpu c ofs1 ofs2)
381        else cmp_mismatch c
382    | Vint n2 ⇒
383        if eq n2 zero then cmp_mismatch c else Vundef
384    | _ ⇒ Vundef ]
385  | _ ⇒ Vundef ].
386
387ndefinition cmpf ≝ λc: comparison. λv1,v2: val.
388  match v1 with
389  [ Vfloat f1 ⇒ match v2 with
390    [ Vfloat f2 ⇒ of_bool (Fcmp c f1 f2)
391    | _ ⇒ Vundef ]
392  | _ ⇒ Vundef ].
393
394(* * [load_result] is used in the memory model (library [Mem])
395  to post-process the results of a memory read.  For instance,
396  consider storing the integer value [0xFFF] on 1 byte at a
397  given address, and reading it back.  If it is read back with
398  chunk [Mint8unsigned], zero-extension must be performed, resulting
399  in [0xFF].  If it is read back as a [Mint8signed], sign-extension
400  is performed and [0xFFFFFFFF] is returned.   Type mismatches
401  (e.g. reading back a float as a [Mint32]) read back as [Vundef]. *)
402
403nlet rec load_result (chunk: memory_chunk) (v: val) ≝
404  match v with
405  [ Vint n ⇒
406    match chunk with
407    [ Mint8signed ⇒ Vint (sign_ext 8 n)
408    | Mint8unsigned ⇒ Vint (zero_ext 8 n)
409    | Mint16signed ⇒ Vint (sign_ext 16 n)
410    | Mint16unsigned ⇒ Vint (zero_ext 16 n)
411    | Mint32 ⇒ Vint n
412    | _ ⇒ Vundef
413    ]
414  | Vptr pty b ofs ⇒
415    match chunk with
416    [ Mint32 ⇒ Vptr pty b ofs
417    | _ ⇒ Vundef
418    ]
419  | Vfloat f ⇒
420    match chunk with
421    [ Mfloat32 ⇒ Vfloat(singleoffloat f)
422    | Mfloat64 ⇒ Vfloat f
423    | _ ⇒ Vundef
424    ]
425  | _ ⇒ Vundef
426  ].
427
428(*
429(** Theorems on arithmetic operations. *)
430
431Theorem cast8unsigned_and:
432  forall x, zero_ext 8 x = and x (Vint(Int.repr 255)).
433Proof.
434  destruct x; simpl; auto. decEq.
435  change 255 with (two_p 8 - 1). apply Int.zero_ext_and. vm_compute; auto.
436Qed.
437
438Theorem cast16unsigned_and:
439  forall x, zero_ext 16 x = and x (Vint(Int.repr 65535)).
440Proof.
441  destruct x; simpl; auto. decEq.
442  change 65535 with (two_p 16 - 1). apply Int.zero_ext_and. vm_compute; auto.
443Qed.
444
445Theorem istrue_not_isfalse:
446  forall v, is_false v -> is_true (notbool v).
447Proof.
448  destruct v; simpl; try contradiction.
449  intros. subst i. simpl. discriminate.
450Qed.
451
452Theorem isfalse_not_istrue:
453  forall v, is_true v -> is_false (notbool v).
454Proof.
455  destruct v; simpl; try contradiction.
456  intros. generalize (Int.eq_spec i Int.zero).
457  case (Int.eq i Int.zero); intro.
458  contradiction. simpl. auto.
459  auto.
460Qed.
461
462Theorem bool_of_true_val:
463  forall v, is_true v -> bool_of_val v true.
464Proof.
465  intro. destruct v; simpl; intros; try contradiction.
466  constructor; auto. constructor.
467Qed.
468
469Theorem bool_of_true_val2:
470  forall v, bool_of_val v true -> is_true v.
471Proof.
472  intros. inversion H; simpl; auto.
473Qed.
474
475Theorem bool_of_true_val_inv:
476  forall v b, is_true v -> bool_of_val v b -> b = true.
477Proof.
478  intros. inversion H0; subst v b; simpl in H; auto.
479Qed.
480
481Theorem bool_of_false_val:
482  forall v, is_false v -> bool_of_val v false.
483Proof.
484  intro. destruct v; simpl; intros; try contradiction.
485  subst i;  constructor.
486Qed.
487
488Theorem bool_of_false_val2:
489  forall v, bool_of_val v false -> is_false v.
490Proof.
491  intros. inversion H; simpl; auto.
492Qed.
493
494Theorem bool_of_false_val_inv:
495  forall v b, is_false v -> bool_of_val v b -> b = false.
496Proof.
497  intros. inversion H0; subst v b; simpl in H.
498  congruence. auto. contradiction.
499Qed.
500
501Theorem notbool_negb_1:
502  forall b, of_bool (negb b) = notbool (of_bool b).
503Proof.
504  destruct b; reflexivity.
505Qed.
506
507Theorem notbool_negb_2:
508  forall b, of_bool b = notbool (of_bool (negb b)).
509Proof.
510  destruct b; reflexivity.
511Qed.
512
513Theorem notbool_idem2:
514  forall b, notbool(notbool(of_bool b)) = of_bool b.
515Proof.
516  destruct b; reflexivity.
517Qed.
518
519Theorem notbool_idem3:
520  forall x, notbool(notbool(notbool x)) = notbool x.
521Proof.
522  destruct x; simpl; auto.
523  case (Int.eq i Int.zero); reflexivity.
524Qed.
525
526Theorem add_commut: forall x y, add x y = add y x.
527Proof.
528  destruct x; destruct y; simpl; auto.
529  decEq. apply Int.add_commut.
530Qed.
531
532Theorem add_assoc: forall x y z, add (add x y) z = add x (add y z).
533Proof.
534  destruct x; destruct y; destruct z; simpl; auto.
535  rewrite Int.add_assoc; auto.
536  rewrite Int.add_assoc; auto.
537  decEq. decEq. apply Int.add_commut.
538  decEq. rewrite Int.add_commut. rewrite <- Int.add_assoc.
539  decEq. apply Int.add_commut.
540  decEq. rewrite Int.add_assoc. auto.
541Qed.
542
543Theorem add_permut: forall x y z, add x (add y z) = add y (add x z).
544Proof.
545  intros. rewrite (add_commut y z). rewrite <- add_assoc. apply add_commut.
546Qed.
547
548Theorem add_permut_4:
549  forall x y z t, add (add x y) (add z t) = add (add x z) (add y t).
550Proof.
551  intros. rewrite add_permut. rewrite add_assoc.
552  rewrite add_permut. symmetry. apply add_assoc.
553Qed.
554
555Theorem neg_zero: neg Vzero = Vzero.
556Proof.
557  reflexivity.
558Qed.
559
560Theorem neg_add_distr: forall x y, neg(add x y) = add (neg x) (neg y).
561Proof.
562  destruct x; destruct y; simpl; auto. decEq. apply Int.neg_add_distr.
563Qed.
564
565Theorem sub_zero_r: forall x, sub Vzero x = neg x.
566Proof.
567  destruct x; simpl; auto.
568Qed.
569
570Theorem sub_add_opp: forall x y, sub x (Vint y) = add x (Vint (Int.neg y)).
571Proof.
572  destruct x; intro y; simpl; auto; rewrite Int.sub_add_opp; auto.
573Qed.
574
575Theorem sub_opp_add: forall x y, sub x (Vint (Int.neg y)) = add x (Vint y).
576Proof.
577  intros. unfold sub, add.
578  destruct x; auto; rewrite Int.sub_add_opp; rewrite Int.neg_involutive; auto.
579Qed.
580
581Theorem sub_add_l:
582  forall v1 v2 i, sub (add v1 (Vint i)) v2 = add (sub v1 v2) (Vint i).
583Proof.
584  destruct v1; destruct v2; intros; simpl; auto.
585  rewrite Int.sub_add_l. auto.
586  rewrite Int.sub_add_l. auto.
587  case (zeq b b0); intro. rewrite Int.sub_add_l. auto. reflexivity.
588Qed.
589
590Theorem sub_add_r:
591  forall v1 v2 i, sub v1 (add v2 (Vint i)) = add (sub v1 v2) (Vint (Int.neg i)).
592Proof.
593  destruct v1; destruct v2; intros; simpl; auto.
594  rewrite Int.sub_add_r. auto.
595  repeat rewrite Int.sub_add_opp. decEq.
596  repeat rewrite Int.add_assoc. decEq. apply Int.add_commut.
597  decEq. repeat rewrite Int.sub_add_opp.
598  rewrite Int.add_assoc. decEq. apply Int.neg_add_distr.
599  case (zeq b b0); intro. simpl. decEq.
600  repeat rewrite Int.sub_add_opp. rewrite Int.add_assoc. decEq.
601  apply Int.neg_add_distr.
602  reflexivity.
603Qed.
604
605Theorem mul_commut: forall x y, mul x y = mul y x.
606Proof.
607  destruct x; destruct y; simpl; auto. decEq. apply Int.mul_commut.
608Qed.
609
610Theorem mul_assoc: forall x y z, mul (mul x y) z = mul x (mul y z).
611Proof.
612  destruct x; destruct y; destruct z; simpl; auto.
613  decEq. apply Int.mul_assoc.
614Qed.
615
616Theorem mul_add_distr_l:
617  forall x y z, mul (add x y) z = add (mul x z) (mul y z).
618Proof.
619  destruct x; destruct y; destruct z; simpl; auto.
620  decEq. apply Int.mul_add_distr_l.
621Qed.
622
623
624Theorem mul_add_distr_r:
625  forall x y z, mul x (add y z) = add (mul x y) (mul x z).
626Proof.
627  destruct x; destruct y; destruct z; simpl; auto.
628  decEq. apply Int.mul_add_distr_r.
629Qed.
630
631Theorem mul_pow2:
632  forall x n logn,
633  Int.is_power2 n = Some logn ->
634  mul x (Vint n) = shl x (Vint logn).
635Proof.
636  intros; destruct x; simpl; auto.
637  change 32 with (Z_of_nat Int.wordsize).
638  rewrite (Int.is_power2_range _ _ H). decEq. apply Int.mul_pow2. auto.
639Qed. 
640
641Theorem mods_divs:
642  forall x y, mods x y = sub x (mul (divs x y) y).
643Proof.
644  destruct x; destruct y; simpl; auto.
645  case (Int.eq i0 Int.zero); simpl. auto. decEq. apply Int.mods_divs.
646Qed.
647
648Theorem modu_divu:
649  forall x y, modu x y = sub x (mul (divu x y) y).
650Proof.
651  destruct x; destruct y; simpl; auto.
652  generalize (Int.eq_spec i0 Int.zero);
653  case (Int.eq i0 Int.zero); simpl. auto.
654  intro. decEq. apply Int.modu_divu. auto.
655Qed.
656
657Theorem divs_pow2:
658  forall x n logn,
659  Int.is_power2 n = Some logn ->
660  divs x (Vint n) = shrx x (Vint logn).
661Proof.
662  intros; destruct x; simpl; auto.
663  change 32 with (Z_of_nat Int.wordsize).
664  rewrite (Int.is_power2_range _ _ H).
665  generalize (Int.eq_spec n Int.zero);
666  case (Int.eq n Int.zero); intro.
667  subst n. compute in H. discriminate.
668  decEq. apply Int.divs_pow2. auto.
669Qed.
670
671Theorem divu_pow2:
672  forall x n logn,
673  Int.is_power2 n = Some logn ->
674  divu x (Vint n) = shru x (Vint logn).
675Proof.
676  intros; destruct x; simpl; auto.
677  change 32 with (Z_of_nat Int.wordsize).
678  rewrite (Int.is_power2_range _ _ H).
679  generalize (Int.eq_spec n Int.zero);
680  case (Int.eq n Int.zero); intro.
681  subst n. compute in H. discriminate.
682  decEq. apply Int.divu_pow2. auto.
683Qed.
684
685Theorem modu_pow2:
686  forall x n logn,
687  Int.is_power2 n = Some logn ->
688  modu x (Vint n) = and x (Vint (Int.sub n Int.one)).
689Proof.
690  intros; destruct x; simpl; auto.
691  generalize (Int.eq_spec n Int.zero);
692  case (Int.eq n Int.zero); intro.
693  subst n. compute in H. discriminate.
694  decEq. eapply Int.modu_and; eauto.
695Qed.
696
697Theorem and_commut: forall x y, and x y = and y x.
698Proof.
699  destruct x; destruct y; simpl; auto. decEq. apply Int.and_commut.
700Qed.
701
702Theorem and_assoc: forall x y z, and (and x y) z = and x (and y z).
703Proof.
704  destruct x; destruct y; destruct z; simpl; auto.
705  decEq. apply Int.and_assoc.
706Qed.
707
708Theorem or_commut: forall x y, or x y = or y x.
709Proof.
710  destruct x; destruct y; simpl; auto. decEq. apply Int.or_commut.
711Qed.
712
713Theorem or_assoc: forall x y z, or (or x y) z = or x (or y z).
714Proof.
715  destruct x; destruct y; destruct z; simpl; auto.
716  decEq. apply Int.or_assoc.
717Qed.
718
719Theorem xor_commut: forall x y, xor x y = xor y x.
720Proof.
721  destruct x; destruct y; simpl; auto. decEq. apply Int.xor_commut.
722Qed.
723
724Theorem xor_assoc: forall x y z, xor (xor x y) z = xor x (xor y z).
725Proof.
726  destruct x; destruct y; destruct z; simpl; auto.
727  decEq. apply Int.xor_assoc.
728Qed.
729
730Theorem shl_mul: forall x y, Val.mul x (Val.shl Vone y) = Val.shl x y.
731Proof.
732  destruct x; destruct y; simpl; auto.
733  case (Int.ltu i0 Int.iwordsize); auto.
734  decEq. symmetry. apply Int.shl_mul.
735Qed.
736
737Theorem shl_rolm:
738  forall x n,
739  Int.ltu n Int.iwordsize = true ->
740  shl x (Vint n) = rolm x n (Int.shl Int.mone n).
741Proof.
742  intros; destruct x; simpl; auto.
743  rewrite H. decEq. apply Int.shl_rolm. exact H.
744Qed.
745
746Theorem shru_rolm:
747  forall x n,
748  Int.ltu n Int.iwordsize = true ->
749  shru x (Vint n) = rolm x (Int.sub Int.iwordsize n) (Int.shru Int.mone n).
750Proof.
751  intros; destruct x; simpl; auto.
752  rewrite H. decEq. apply Int.shru_rolm. exact H.
753Qed.
754
755Theorem shrx_carry:
756  forall x y,
757  add (shr x y) (shr_carry x y) = shrx x y.
758Proof.
759  destruct x; destruct y; simpl; auto.
760  case (Int.ltu i0 Int.iwordsize); auto.
761  simpl. decEq. apply Int.shrx_carry.
762Qed.
763
764Theorem or_rolm:
765  forall x n m1 m2,
766  or (rolm x n m1) (rolm x n m2) = rolm x n (Int.or m1 m2).
767Proof.
768  intros; destruct x; simpl; auto.
769  decEq. apply Int.or_rolm.
770Qed.
771
772Theorem rolm_rolm:
773  forall x n1 m1 n2 m2,
774  rolm (rolm x n1 m1) n2 m2 =
775    rolm x (Int.modu (Int.add n1 n2) Int.iwordsize)
776           (Int.and (Int.rol m1 n2) m2).
777Proof.
778  intros; destruct x; simpl; auto.
779  decEq.
780  apply Int.rolm_rolm. apply int_wordsize_divides_modulus.
781Qed.
782
783Theorem rolm_zero:
784  forall x m,
785  rolm x Int.zero m = and x (Vint m).
786Proof.
787  intros; destruct x; simpl; auto. decEq. apply Int.rolm_zero.
788Qed.
789
790Theorem addf_commut: forall x y, addf x y = addf y x.
791Proof.
792  destruct x; destruct y; simpl; auto. decEq. apply Float.addf_commut.
793Qed.
794
795Lemma negate_cmp_mismatch:
796  forall c,
797  cmp_mismatch (negate_comparison c) = notbool(cmp_mismatch c).
798Proof.
799  destruct c; reflexivity.
800Qed.
801
802Theorem negate_cmp:
803  forall c x y,
804  cmp (negate_comparison c) x y = notbool (cmp c x y).
805Proof.
806  destruct x; destruct y; simpl; auto.
807  rewrite Int.negate_cmp. apply notbool_negb_1.
808  case (Int.eq i Int.zero). apply negate_cmp_mismatch. reflexivity.
809  case (Int.eq i0 Int.zero). apply negate_cmp_mismatch. reflexivity.
810  case (zeq b b0); intro.
811  rewrite Int.negate_cmp. apply notbool_negb_1.
812  apply negate_cmp_mismatch.
813Qed.
814
815Theorem negate_cmpu:
816  forall c x y,
817  cmpu (negate_comparison c) x y = notbool (cmpu c x y).
818Proof.
819  destruct x; destruct y; simpl; auto.
820  rewrite Int.negate_cmpu. apply notbool_negb_1.
821  case (Int.eq i Int.zero). apply negate_cmp_mismatch. reflexivity.
822  case (Int.eq i0 Int.zero). apply negate_cmp_mismatch. reflexivity.
823  case (zeq b b0); intro.
824  rewrite Int.negate_cmpu. apply notbool_negb_1.
825  apply negate_cmp_mismatch.
826Qed.
827
828Lemma swap_cmp_mismatch:
829  forall c, cmp_mismatch (swap_comparison c) = cmp_mismatch c.
830Proof.
831  destruct c; reflexivity.
832Qed.
833 
834Theorem swap_cmp:
835  forall c x y,
836  cmp (swap_comparison c) x y = cmp c y x.
837Proof.
838  destruct x; destruct y; simpl; auto.
839  rewrite Int.swap_cmp. auto.
840  case (Int.eq i Int.zero). apply swap_cmp_mismatch. auto.
841  case (Int.eq i0 Int.zero). apply swap_cmp_mismatch. auto.
842  case (zeq b b0); intro.
843  subst b0. rewrite zeq_true. rewrite Int.swap_cmp. auto.
844  rewrite zeq_false. apply swap_cmp_mismatch. auto.
845Qed.
846
847Theorem swap_cmpu:
848  forall c x y,
849  cmpu (swap_comparison c) x y = cmpu c y x.
850Proof.
851  destruct x; destruct y; simpl; auto.
852  rewrite Int.swap_cmpu. auto.
853  case (Int.eq i Int.zero). apply swap_cmp_mismatch. auto.
854  case (Int.eq i0 Int.zero). apply swap_cmp_mismatch. auto.
855  case (zeq b b0); intro.
856  subst b0. rewrite zeq_true. rewrite Int.swap_cmpu. auto.
857  rewrite zeq_false. apply swap_cmp_mismatch. auto.
858Qed.
859
860Theorem negate_cmpf_eq:
861  forall v1 v2, notbool (cmpf Cne v1 v2) = cmpf Ceq v1 v2.
862Proof.
863  destruct v1; destruct v2; simpl; auto.
864  rewrite Float.cmp_ne_eq. rewrite notbool_negb_1.
865  apply notbool_idem2.
866Qed.
867
868Theorem negate_cmpf_ne:
869  forall v1 v2, notbool (cmpf Ceq v1 v2) = cmpf Cne v1 v2.
870Proof.
871  destruct v1; destruct v2; simpl; auto.
872  rewrite Float.cmp_ne_eq. rewrite notbool_negb_1. auto.
873Qed.
874
875Lemma or_of_bool:
876  forall b1 b2, or (of_bool b1) (of_bool b2) = of_bool (b1 || b2).
877Proof.
878  destruct b1; destruct b2; reflexivity.
879Qed.
880
881Theorem cmpf_le:
882  forall v1 v2, cmpf Cle v1 v2 = or (cmpf Clt v1 v2) (cmpf Ceq v1 v2).
883Proof.
884  destruct v1; destruct v2; simpl; auto.
885  rewrite or_of_bool. decEq. apply Float.cmp_le_lt_eq.
886Qed.
887
888Theorem cmpf_ge:
889  forall v1 v2, cmpf Cge v1 v2 = or (cmpf Cgt v1 v2) (cmpf Ceq v1 v2).
890Proof.
891  destruct v1; destruct v2; simpl; auto.
892  rewrite or_of_bool. decEq. apply Float.cmp_ge_gt_eq.
893Qed.
894
895Definition is_bool (v: val) :=
896  v = Vundef \/ v = Vtrue \/ v = Vfalse.
897
898Lemma of_bool_is_bool:
899  forall b, is_bool (of_bool b).
900Proof.
901  destruct b; unfold is_bool; simpl; tauto.
902Qed.
903
904Lemma undef_is_bool: is_bool Vundef.
905Proof.
906  unfold is_bool; tauto.
907Qed.
908
909Lemma cmp_mismatch_is_bool:
910  forall c, is_bool (cmp_mismatch c).
911Proof.
912  destruct c; simpl; unfold is_bool; tauto.
913Qed.
914
915Lemma cmp_is_bool:
916  forall c v1 v2, is_bool (cmp c v1 v2).
917Proof.
918  destruct v1; destruct v2; simpl; try apply undef_is_bool.
919  apply of_bool_is_bool.
920  case (Int.eq i Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool.
921  case (Int.eq i0 Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool.
922  case (zeq b b0); intro. apply of_bool_is_bool. apply cmp_mismatch_is_bool.
923Qed.
924
925Lemma cmpu_is_bool:
926  forall c v1 v2, is_bool (cmpu c v1 v2).
927Proof.
928  destruct v1; destruct v2; simpl; try apply undef_is_bool.
929  apply of_bool_is_bool.
930  case (Int.eq i Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool.
931  case (Int.eq i0 Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool.
932  case (zeq b b0); intro. apply of_bool_is_bool. apply cmp_mismatch_is_bool.
933Qed.
934
935Lemma cmpf_is_bool:
936  forall c v1 v2, is_bool (cmpf c v1 v2).
937Proof.
938  destruct v1; destruct v2; simpl;
939  apply undef_is_bool || apply of_bool_is_bool.
940Qed.
941
942Lemma notbool_is_bool:
943  forall v, is_bool (notbool v).
944Proof.
945  destruct v; simpl.
946  apply undef_is_bool. apply of_bool_is_bool.
947  apply undef_is_bool. unfold is_bool; tauto.
948Qed.
949
950Lemma notbool_xor:
951  forall v, is_bool v -> v = xor (notbool v) Vone.
952Proof.
953  intros. elim H; intro. 
954  subst v. reflexivity.
955  elim H0; intro; subst v; reflexivity.
956Qed.
957
958Lemma rolm_lt_zero:
959  forall v, rolm v Int.one Int.one = cmp Clt v (Vint Int.zero).
960Proof.
961  intros. destruct v; simpl; auto.
962  transitivity (Vint (Int.shru i (Int.repr (Z_of_nat Int.wordsize - 1)))).
963  decEq. symmetry. rewrite Int.shru_rolm. auto. auto.
964  rewrite Int.shru_lt_zero. destruct (Int.lt i Int.zero); auto.
965Qed.
966
967Lemma rolm_ge_zero:
968  forall v,
969  xor (rolm v Int.one Int.one) (Vint Int.one) = cmp Cge v (Vint Int.zero).
970Proof.
971  intros. rewrite rolm_lt_zero. destruct v; simpl; auto.
972  destruct (Int.lt i Int.zero); auto.
973Qed.
974*)
975(* * The ``is less defined'' relation between values.
976    A value is less defined than itself, and [Vundef] is
977    less defined than any value. *)
978
979ninductive Val_lessdef: val → val → Prop ≝
980  | lessdef_refl: ∀v. Val_lessdef v v
981  | lessdef_undef: ∀v. Val_lessdef Vundef v.
982
983ninductive lessdef_list: list val → list val → Prop ≝
984  | lessdef_list_nil:
985      lessdef_list (nil ?) (nil ?)
986  | lessdef_list_cons:
987      ∀v1,v2,vl1,vl2.
988      Val_lessdef v1 v2 → lessdef_list vl1 vl2 →
989      lessdef_list (v1 :: vl1) (v2 :: vl2).
990
991(*Hint Resolve lessdef_refl lessdef_undef lessdef_list_nil lessdef_list_cons.*)
992
993nlemma lessdef_list_inv:
994  ∀vl1,vl2. lessdef_list vl1 vl2 → vl1 = vl2 ∨ in_list ? Vundef vl1.
995#vl1; nelim vl1;
996##[ #vl2; #H; ninversion H; /2/; #h1;#h2;#t1;#t2;#H1;#H2;#H3;#Hbad; ndestruct
997##| #h;#t;#IH;#vl2;#H;
998    ninversion H;
999    ##[ #H'; ndestruct
1000    ##| #h1;#h2;#t1;#t2;#H1;#H2;#H3;#e1;#e2; ndestruct;
1001        nelim H1;
1002        ##[ nelim (IH t2 H2);
1003            ##[ #e; ndestruct; /2/;
1004            ##| /3/ ##]
1005        ##| /3/ ##]
1006    ##]
1007##] nqed.
1008
1009nlemma load_result_lessdef:
1010  ∀chunk,v1,v2.
1011  Val_lessdef v1 v2 → Val_lessdef (load_result chunk v1) (load_result chunk v2).
1012#chunk;#v1;#v2;#H; ninversion H; //; #v e1 e2; ncases chunk; nwhd in ⊢ (?%?); //;
1013nqed.
1014
1015(*
1016Lemma zero_ext_lessdef:
1017  forall n v1 v2, lessdef v1 v2 -> lessdef (zero_ext n v1) (zero_ext n v2).
1018Proof.
1019  intros; inv H; simpl; auto.
1020Qed.
1021*)
1022nlemma sign_ext_lessdef:
1023  ∀n,v1,v2. Val_lessdef v1 v2 → Val_lessdef (sign_ext n v1) (sign_ext n v2).
1024#n;#v1;#v2;#H;ninversion H;//;#v;#e1;#e2;nrewrite < e1 in H; nrewrite > e2; //;
1025nqed.
1026(*
1027Lemma singleoffloat_lessdef:
1028  forall v1 v2, lessdef v1 v2 -> lessdef (singleoffloat v1) (singleoffloat v2).
1029Proof.
1030  intros; inv H; simpl; auto.
1031Qed.
1032
1033End Val.
1034*)
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