source: Deliverables/D3.1/C-semantics/Values.ma @ 409

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pdata support

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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 pty with
416    [ Any ⇒ match chunk with [ Mint24 ⇒ Vptr pty b ofs | _ ⇒ Vundef ]
417    | Data ⇒ match chunk with [ Mint8unsigned ⇒ Vptr pty b ofs | _ ⇒ Vundef ]
418    | IData ⇒ match chunk with [ Mint8unsigned ⇒ Vptr pty b ofs | _ ⇒ Vundef ]
419    | PData ⇒ match chunk with [ Mint8unsigned ⇒ Vptr pty b ofs | _ ⇒ Vundef ]
420    | XData ⇒ match chunk with [ Mint16unsigned ⇒ Vptr pty b ofs | _ ⇒ Vundef ]
421    | Code ⇒ match chunk with [ Mint16unsigned ⇒ Vptr pty b ofs | _ ⇒ Vundef ]
422    ]
423  | Vfloat f ⇒
424    match chunk with
425    [ Mfloat32 ⇒ Vfloat(singleoffloat f)
426    | Mfloat64 ⇒ Vfloat f
427    | _ ⇒ Vundef
428    ]
429  | _ ⇒ Vundef
430  ].
431
432(*
433(** Theorems on arithmetic operations. *)
434
435Theorem cast8unsigned_and:
436  forall x, zero_ext 8 x = and x (Vint(Int.repr 255)).
437Proof.
438  destruct x; simpl; auto. decEq.
439  change 255 with (two_p 8 - 1). apply Int.zero_ext_and. vm_compute; auto.
440Qed.
441
442Theorem cast16unsigned_and:
443  forall x, zero_ext 16 x = and x (Vint(Int.repr 65535)).
444Proof.
445  destruct x; simpl; auto. decEq.
446  change 65535 with (two_p 16 - 1). apply Int.zero_ext_and. vm_compute; auto.
447Qed.
448
449Theorem istrue_not_isfalse:
450  forall v, is_false v -> is_true (notbool v).
451Proof.
452  destruct v; simpl; try contradiction.
453  intros. subst i. simpl. discriminate.
454Qed.
455
456Theorem isfalse_not_istrue:
457  forall v, is_true v -> is_false (notbool v).
458Proof.
459  destruct v; simpl; try contradiction.
460  intros. generalize (Int.eq_spec i Int.zero).
461  case (Int.eq i Int.zero); intro.
462  contradiction. simpl. auto.
463  auto.
464Qed.
465
466Theorem bool_of_true_val:
467  forall v, is_true v -> bool_of_val v true.
468Proof.
469  intro. destruct v; simpl; intros; try contradiction.
470  constructor; auto. constructor.
471Qed.
472
473Theorem bool_of_true_val2:
474  forall v, bool_of_val v true -> is_true v.
475Proof.
476  intros. inversion H; simpl; auto.
477Qed.
478
479Theorem bool_of_true_val_inv:
480  forall v b, is_true v -> bool_of_val v b -> b = true.
481Proof.
482  intros. inversion H0; subst v b; simpl in H; auto.
483Qed.
484
485Theorem bool_of_false_val:
486  forall v, is_false v -> bool_of_val v false.
487Proof.
488  intro. destruct v; simpl; intros; try contradiction.
489  subst i;  constructor.
490Qed.
491
492Theorem bool_of_false_val2:
493  forall v, bool_of_val v false -> is_false v.
494Proof.
495  intros. inversion H; simpl; auto.
496Qed.
497
498Theorem bool_of_false_val_inv:
499  forall v b, is_false v -> bool_of_val v b -> b = false.
500Proof.
501  intros. inversion H0; subst v b; simpl in H.
502  congruence. auto. contradiction.
503Qed.
504
505Theorem notbool_negb_1:
506  forall b, of_bool (negb b) = notbool (of_bool b).
507Proof.
508  destruct b; reflexivity.
509Qed.
510
511Theorem notbool_negb_2:
512  forall b, of_bool b = notbool (of_bool (negb b)).
513Proof.
514  destruct b; reflexivity.
515Qed.
516
517Theorem notbool_idem2:
518  forall b, notbool(notbool(of_bool b)) = of_bool b.
519Proof.
520  destruct b; reflexivity.
521Qed.
522
523Theorem notbool_idem3:
524  forall x, notbool(notbool(notbool x)) = notbool x.
525Proof.
526  destruct x; simpl; auto.
527  case (Int.eq i Int.zero); reflexivity.
528Qed.
529
530Theorem add_commut: forall x y, add x y = add y x.
531Proof.
532  destruct x; destruct y; simpl; auto.
533  decEq. apply Int.add_commut.
534Qed.
535
536Theorem add_assoc: forall x y z, add (add x y) z = add x (add y z).
537Proof.
538  destruct x; destruct y; destruct z; simpl; auto.
539  rewrite Int.add_assoc; auto.
540  rewrite Int.add_assoc; auto.
541  decEq. decEq. apply Int.add_commut.
542  decEq. rewrite Int.add_commut. rewrite <- Int.add_assoc.
543  decEq. apply Int.add_commut.
544  decEq. rewrite Int.add_assoc. auto.
545Qed.
546
547Theorem add_permut: forall x y z, add x (add y z) = add y (add x z).
548Proof.
549  intros. rewrite (add_commut y z). rewrite <- add_assoc. apply add_commut.
550Qed.
551
552Theorem add_permut_4:
553  forall x y z t, add (add x y) (add z t) = add (add x z) (add y t).
554Proof.
555  intros. rewrite add_permut. rewrite add_assoc.
556  rewrite add_permut. symmetry. apply add_assoc.
557Qed.
558
559Theorem neg_zero: neg Vzero = Vzero.
560Proof.
561  reflexivity.
562Qed.
563
564Theorem neg_add_distr: forall x y, neg(add x y) = add (neg x) (neg y).
565Proof.
566  destruct x; destruct y; simpl; auto. decEq. apply Int.neg_add_distr.
567Qed.
568
569Theorem sub_zero_r: forall x, sub Vzero x = neg x.
570Proof.
571  destruct x; simpl; auto.
572Qed.
573
574Theorem sub_add_opp: forall x y, sub x (Vint y) = add x (Vint (Int.neg y)).
575Proof.
576  destruct x; intro y; simpl; auto; rewrite Int.sub_add_opp; auto.
577Qed.
578
579Theorem sub_opp_add: forall x y, sub x (Vint (Int.neg y)) = add x (Vint y).
580Proof.
581  intros. unfold sub, add.
582  destruct x; auto; rewrite Int.sub_add_opp; rewrite Int.neg_involutive; auto.
583Qed.
584
585Theorem sub_add_l:
586  forall v1 v2 i, sub (add v1 (Vint i)) v2 = add (sub v1 v2) (Vint i).
587Proof.
588  destruct v1; destruct v2; intros; simpl; auto.
589  rewrite Int.sub_add_l. auto.
590  rewrite Int.sub_add_l. auto.
591  case (zeq b b0); intro. rewrite Int.sub_add_l. auto. reflexivity.
592Qed.
593
594Theorem sub_add_r:
595  forall v1 v2 i, sub v1 (add v2 (Vint i)) = add (sub v1 v2) (Vint (Int.neg i)).
596Proof.
597  destruct v1; destruct v2; intros; simpl; auto.
598  rewrite Int.sub_add_r. auto.
599  repeat rewrite Int.sub_add_opp. decEq.
600  repeat rewrite Int.add_assoc. decEq. apply Int.add_commut.
601  decEq. repeat rewrite Int.sub_add_opp.
602  rewrite Int.add_assoc. decEq. apply Int.neg_add_distr.
603  case (zeq b b0); intro. simpl. decEq.
604  repeat rewrite Int.sub_add_opp. rewrite Int.add_assoc. decEq.
605  apply Int.neg_add_distr.
606  reflexivity.
607Qed.
608
609Theorem mul_commut: forall x y, mul x y = mul y x.
610Proof.
611  destruct x; destruct y; simpl; auto. decEq. apply Int.mul_commut.
612Qed.
613
614Theorem mul_assoc: forall x y z, mul (mul x y) z = mul x (mul y z).
615Proof.
616  destruct x; destruct y; destruct z; simpl; auto.
617  decEq. apply Int.mul_assoc.
618Qed.
619
620Theorem mul_add_distr_l:
621  forall x y z, mul (add x y) z = add (mul x z) (mul y z).
622Proof.
623  destruct x; destruct y; destruct z; simpl; auto.
624  decEq. apply Int.mul_add_distr_l.
625Qed.
626
627
628Theorem mul_add_distr_r:
629  forall x y z, mul x (add y z) = add (mul x y) (mul x z).
630Proof.
631  destruct x; destruct y; destruct z; simpl; auto.
632  decEq. apply Int.mul_add_distr_r.
633Qed.
634
635Theorem mul_pow2:
636  forall x n logn,
637  Int.is_power2 n = Some logn ->
638  mul x (Vint n) = shl x (Vint logn).
639Proof.
640  intros; destruct x; simpl; auto.
641  change 32 with (Z_of_nat Int.wordsize).
642  rewrite (Int.is_power2_range _ _ H). decEq. apply Int.mul_pow2. auto.
643Qed. 
644
645Theorem mods_divs:
646  forall x y, mods x y = sub x (mul (divs x y) y).
647Proof.
648  destruct x; destruct y; simpl; auto.
649  case (Int.eq i0 Int.zero); simpl. auto. decEq. apply Int.mods_divs.
650Qed.
651
652Theorem modu_divu:
653  forall x y, modu x y = sub x (mul (divu x y) y).
654Proof.
655  destruct x; destruct y; simpl; auto.
656  generalize (Int.eq_spec i0 Int.zero);
657  case (Int.eq i0 Int.zero); simpl. auto.
658  intro. decEq. apply Int.modu_divu. auto.
659Qed.
660
661Theorem divs_pow2:
662  forall x n logn,
663  Int.is_power2 n = Some logn ->
664  divs x (Vint n) = shrx x (Vint logn).
665Proof.
666  intros; destruct x; simpl; auto.
667  change 32 with (Z_of_nat Int.wordsize).
668  rewrite (Int.is_power2_range _ _ H).
669  generalize (Int.eq_spec n Int.zero);
670  case (Int.eq n Int.zero); intro.
671  subst n. compute in H. discriminate.
672  decEq. apply Int.divs_pow2. auto.
673Qed.
674
675Theorem divu_pow2:
676  forall x n logn,
677  Int.is_power2 n = Some logn ->
678  divu x (Vint n) = shru x (Vint logn).
679Proof.
680  intros; destruct x; simpl; auto.
681  change 32 with (Z_of_nat Int.wordsize).
682  rewrite (Int.is_power2_range _ _ H).
683  generalize (Int.eq_spec n Int.zero);
684  case (Int.eq n Int.zero); intro.
685  subst n. compute in H. discriminate.
686  decEq. apply Int.divu_pow2. auto.
687Qed.
688
689Theorem modu_pow2:
690  forall x n logn,
691  Int.is_power2 n = Some logn ->
692  modu x (Vint n) = and x (Vint (Int.sub n Int.one)).
693Proof.
694  intros; destruct x; simpl; auto.
695  generalize (Int.eq_spec n Int.zero);
696  case (Int.eq n Int.zero); intro.
697  subst n. compute in H. discriminate.
698  decEq. eapply Int.modu_and; eauto.
699Qed.
700
701Theorem and_commut: forall x y, and x y = and y x.
702Proof.
703  destruct x; destruct y; simpl; auto. decEq. apply Int.and_commut.
704Qed.
705
706Theorem and_assoc: forall x y z, and (and x y) z = and x (and y z).
707Proof.
708  destruct x; destruct y; destruct z; simpl; auto.
709  decEq. apply Int.and_assoc.
710Qed.
711
712Theorem or_commut: forall x y, or x y = or y x.
713Proof.
714  destruct x; destruct y; simpl; auto. decEq. apply Int.or_commut.
715Qed.
716
717Theorem or_assoc: forall x y z, or (or x y) z = or x (or y z).
718Proof.
719  destruct x; destruct y; destruct z; simpl; auto.
720  decEq. apply Int.or_assoc.
721Qed.
722
723Theorem xor_commut: forall x y, xor x y = xor y x.
724Proof.
725  destruct x; destruct y; simpl; auto. decEq. apply Int.xor_commut.
726Qed.
727
728Theorem xor_assoc: forall x y z, xor (xor x y) z = xor x (xor y z).
729Proof.
730  destruct x; destruct y; destruct z; simpl; auto.
731  decEq. apply Int.xor_assoc.
732Qed.
733
734Theorem shl_mul: forall x y, Val.mul x (Val.shl Vone y) = Val.shl x y.
735Proof.
736  destruct x; destruct y; simpl; auto.
737  case (Int.ltu i0 Int.iwordsize); auto.
738  decEq. symmetry. apply Int.shl_mul.
739Qed.
740
741Theorem shl_rolm:
742  forall x n,
743  Int.ltu n Int.iwordsize = true ->
744  shl x (Vint n) = rolm x n (Int.shl Int.mone n).
745Proof.
746  intros; destruct x; simpl; auto.
747  rewrite H. decEq. apply Int.shl_rolm. exact H.
748Qed.
749
750Theorem shru_rolm:
751  forall x n,
752  Int.ltu n Int.iwordsize = true ->
753  shru x (Vint n) = rolm x (Int.sub Int.iwordsize n) (Int.shru Int.mone n).
754Proof.
755  intros; destruct x; simpl; auto.
756  rewrite H. decEq. apply Int.shru_rolm. exact H.
757Qed.
758
759Theorem shrx_carry:
760  forall x y,
761  add (shr x y) (shr_carry x y) = shrx x y.
762Proof.
763  destruct x; destruct y; simpl; auto.
764  case (Int.ltu i0 Int.iwordsize); auto.
765  simpl. decEq. apply Int.shrx_carry.
766Qed.
767
768Theorem or_rolm:
769  forall x n m1 m2,
770  or (rolm x n m1) (rolm x n m2) = rolm x n (Int.or m1 m2).
771Proof.
772  intros; destruct x; simpl; auto.
773  decEq. apply Int.or_rolm.
774Qed.
775
776Theorem rolm_rolm:
777  forall x n1 m1 n2 m2,
778  rolm (rolm x n1 m1) n2 m2 =
779    rolm x (Int.modu (Int.add n1 n2) Int.iwordsize)
780           (Int.and (Int.rol m1 n2) m2).
781Proof.
782  intros; destruct x; simpl; auto.
783  decEq.
784  apply Int.rolm_rolm. apply int_wordsize_divides_modulus.
785Qed.
786
787Theorem rolm_zero:
788  forall x m,
789  rolm x Int.zero m = and x (Vint m).
790Proof.
791  intros; destruct x; simpl; auto. decEq. apply Int.rolm_zero.
792Qed.
793
794Theorem addf_commut: forall x y, addf x y = addf y x.
795Proof.
796  destruct x; destruct y; simpl; auto. decEq. apply Float.addf_commut.
797Qed.
798
799Lemma negate_cmp_mismatch:
800  forall c,
801  cmp_mismatch (negate_comparison c) = notbool(cmp_mismatch c).
802Proof.
803  destruct c; reflexivity.
804Qed.
805
806Theorem negate_cmp:
807  forall c x y,
808  cmp (negate_comparison c) x y = notbool (cmp c x y).
809Proof.
810  destruct x; destruct y; simpl; auto.
811  rewrite Int.negate_cmp. apply notbool_negb_1.
812  case (Int.eq i Int.zero). apply negate_cmp_mismatch. reflexivity.
813  case (Int.eq i0 Int.zero). apply negate_cmp_mismatch. reflexivity.
814  case (zeq b b0); intro.
815  rewrite Int.negate_cmp. apply notbool_negb_1.
816  apply negate_cmp_mismatch.
817Qed.
818
819Theorem negate_cmpu:
820  forall c x y,
821  cmpu (negate_comparison c) x y = notbool (cmpu c x y).
822Proof.
823  destruct x; destruct y; simpl; auto.
824  rewrite Int.negate_cmpu. apply notbool_negb_1.
825  case (Int.eq i Int.zero). apply negate_cmp_mismatch. reflexivity.
826  case (Int.eq i0 Int.zero). apply negate_cmp_mismatch. reflexivity.
827  case (zeq b b0); intro.
828  rewrite Int.negate_cmpu. apply notbool_negb_1.
829  apply negate_cmp_mismatch.
830Qed.
831
832Lemma swap_cmp_mismatch:
833  forall c, cmp_mismatch (swap_comparison c) = cmp_mismatch c.
834Proof.
835  destruct c; reflexivity.
836Qed.
837 
838Theorem swap_cmp:
839  forall c x y,
840  cmp (swap_comparison c) x y = cmp c y x.
841Proof.
842  destruct x; destruct y; simpl; auto.
843  rewrite Int.swap_cmp. auto.
844  case (Int.eq i Int.zero). apply swap_cmp_mismatch. auto.
845  case (Int.eq i0 Int.zero). apply swap_cmp_mismatch. auto.
846  case (zeq b b0); intro.
847  subst b0. rewrite zeq_true. rewrite Int.swap_cmp. auto.
848  rewrite zeq_false. apply swap_cmp_mismatch. auto.
849Qed.
850
851Theorem swap_cmpu:
852  forall c x y,
853  cmpu (swap_comparison c) x y = cmpu c y x.
854Proof.
855  destruct x; destruct y; simpl; auto.
856  rewrite Int.swap_cmpu. auto.
857  case (Int.eq i Int.zero). apply swap_cmp_mismatch. auto.
858  case (Int.eq i0 Int.zero). apply swap_cmp_mismatch. auto.
859  case (zeq b b0); intro.
860  subst b0. rewrite zeq_true. rewrite Int.swap_cmpu. auto.
861  rewrite zeq_false. apply swap_cmp_mismatch. auto.
862Qed.
863
864Theorem negate_cmpf_eq:
865  forall v1 v2, notbool (cmpf Cne v1 v2) = cmpf Ceq v1 v2.
866Proof.
867  destruct v1; destruct v2; simpl; auto.
868  rewrite Float.cmp_ne_eq. rewrite notbool_negb_1.
869  apply notbool_idem2.
870Qed.
871
872Theorem negate_cmpf_ne:
873  forall v1 v2, notbool (cmpf Ceq v1 v2) = cmpf Cne v1 v2.
874Proof.
875  destruct v1; destruct v2; simpl; auto.
876  rewrite Float.cmp_ne_eq. rewrite notbool_negb_1. auto.
877Qed.
878
879Lemma or_of_bool:
880  forall b1 b2, or (of_bool b1) (of_bool b2) = of_bool (b1 || b2).
881Proof.
882  destruct b1; destruct b2; reflexivity.
883Qed.
884
885Theorem cmpf_le:
886  forall v1 v2, cmpf Cle v1 v2 = or (cmpf Clt v1 v2) (cmpf Ceq v1 v2).
887Proof.
888  destruct v1; destruct v2; simpl; auto.
889  rewrite or_of_bool. decEq. apply Float.cmp_le_lt_eq.
890Qed.
891
892Theorem cmpf_ge:
893  forall v1 v2, cmpf Cge v1 v2 = or (cmpf Cgt v1 v2) (cmpf Ceq v1 v2).
894Proof.
895  destruct v1; destruct v2; simpl; auto.
896  rewrite or_of_bool. decEq. apply Float.cmp_ge_gt_eq.
897Qed.
898
899Definition is_bool (v: val) :=
900  v = Vundef \/ v = Vtrue \/ v = Vfalse.
901
902Lemma of_bool_is_bool:
903  forall b, is_bool (of_bool b).
904Proof.
905  destruct b; unfold is_bool; simpl; tauto.
906Qed.
907
908Lemma undef_is_bool: is_bool Vundef.
909Proof.
910  unfold is_bool; tauto.
911Qed.
912
913Lemma cmp_mismatch_is_bool:
914  forall c, is_bool (cmp_mismatch c).
915Proof.
916  destruct c; simpl; unfold is_bool; tauto.
917Qed.
918
919Lemma cmp_is_bool:
920  forall c v1 v2, is_bool (cmp c v1 v2).
921Proof.
922  destruct v1; destruct v2; simpl; try apply undef_is_bool.
923  apply of_bool_is_bool.
924  case (Int.eq i Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool.
925  case (Int.eq i0 Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool.
926  case (zeq b b0); intro. apply of_bool_is_bool. apply cmp_mismatch_is_bool.
927Qed.
928
929Lemma cmpu_is_bool:
930  forall c v1 v2, is_bool (cmpu c v1 v2).
931Proof.
932  destruct v1; destruct v2; simpl; try apply undef_is_bool.
933  apply of_bool_is_bool.
934  case (Int.eq i Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool.
935  case (Int.eq i0 Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool.
936  case (zeq b b0); intro. apply of_bool_is_bool. apply cmp_mismatch_is_bool.
937Qed.
938
939Lemma cmpf_is_bool:
940  forall c v1 v2, is_bool (cmpf c v1 v2).
941Proof.
942  destruct v1; destruct v2; simpl;
943  apply undef_is_bool || apply of_bool_is_bool.
944Qed.
945
946Lemma notbool_is_bool:
947  forall v, is_bool (notbool v).
948Proof.
949  destruct v; simpl.
950  apply undef_is_bool. apply of_bool_is_bool.
951  apply undef_is_bool. unfold is_bool; tauto.
952Qed.
953
954Lemma notbool_xor:
955  forall v, is_bool v -> v = xor (notbool v) Vone.
956Proof.
957  intros. elim H; intro. 
958  subst v. reflexivity.
959  elim H0; intro; subst v; reflexivity.
960Qed.
961
962Lemma rolm_lt_zero:
963  forall v, rolm v Int.one Int.one = cmp Clt v (Vint Int.zero).
964Proof.
965  intros. destruct v; simpl; auto.
966  transitivity (Vint (Int.shru i (Int.repr (Z_of_nat Int.wordsize - 1)))).
967  decEq. symmetry. rewrite Int.shru_rolm. auto. auto.
968  rewrite Int.shru_lt_zero. destruct (Int.lt i Int.zero); auto.
969Qed.
970
971Lemma rolm_ge_zero:
972  forall v,
973  xor (rolm v Int.one Int.one) (Vint Int.one) = cmp Cge v (Vint Int.zero).
974Proof.
975  intros. rewrite rolm_lt_zero. destruct v; simpl; auto.
976  destruct (Int.lt i Int.zero); auto.
977Qed.
978*)
979(* * The ``is less defined'' relation between values.
980    A value is less defined than itself, and [Vundef] is
981    less defined than any value. *)
982
983ninductive Val_lessdef: val → val → Prop ≝
984  | lessdef_refl: ∀v. Val_lessdef v v
985  | lessdef_undef: ∀v. Val_lessdef Vundef v.
986
987ninductive lessdef_list: list val → list val → Prop ≝
988  | lessdef_list_nil:
989      lessdef_list (nil ?) (nil ?)
990  | lessdef_list_cons:
991      ∀v1,v2,vl1,vl2.
992      Val_lessdef v1 v2 → lessdef_list vl1 vl2 →
993      lessdef_list (v1 :: vl1) (v2 :: vl2).
994
995(*Hint Resolve lessdef_refl lessdef_undef lessdef_list_nil lessdef_list_cons.*)
996
997nlemma lessdef_list_inv:
998  ∀vl1,vl2. lessdef_list vl1 vl2 → vl1 = vl2 ∨ in_list ? Vundef vl1.
999#vl1; nelim vl1;
1000##[ #vl2; #H; ninversion H; /2/; #h1;#h2;#t1;#t2;#H1;#H2;#H3;#Hbad; ndestruct
1001##| #h;#t;#IH;#vl2;#H;
1002    ninversion H;
1003    ##[ #H'; ndestruct
1004    ##| #h1;#h2;#t1;#t2;#H1;#H2;#H3;#e1;#e2; ndestruct;
1005        nelim H1;
1006        ##[ nelim (IH t2 H2);
1007            ##[ #e; ndestruct; /2/;
1008            ##| /3/ ##]
1009        ##| /3/ ##]
1010    ##]
1011##] nqed.
1012
1013nlemma load_result_lessdef:
1014  ∀chunk,v1,v2.
1015  Val_lessdef v1 v2 → Val_lessdef (load_result chunk v1) (load_result chunk v2).
1016#chunk;#v1;#v2;#H; ninversion H; //; #v e1 e2; ncases chunk; nwhd in ⊢ (?%?); //;
1017nqed.
1018
1019(*
1020Lemma zero_ext_lessdef:
1021  forall n v1 v2, lessdef v1 v2 -> lessdef (zero_ext n v1) (zero_ext n v2).
1022Proof.
1023  intros; inv H; simpl; auto.
1024Qed.
1025*)
1026nlemma sign_ext_lessdef:
1027  ∀n,v1,v2. Val_lessdef v1 v2 → Val_lessdef (sign_ext n v1) (sign_ext n v2).
1028#n;#v1;#v2;#H;ninversion H;//;#v;#e1;#e2;nrewrite < e1 in H; nrewrite > e2; //;
1029nqed.
1030(*
1031Lemma singleoffloat_lessdef:
1032  forall v1 v2, lessdef v1 v2 -> lessdef (singleoffloat v1) (singleoffloat v2).
1033Proof.
1034  intros; inv H; simpl; auto.
1035Qed.
1036
1037End Val.
1038*)
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