source: C-semantics/Csem.ma @ 155

Last change on this file since 155 was 155, checked in by campbell, 10 years ago

More sensible handling of integer types and pointer casts.

File size: 60.4 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(* * Dynamic semantics for the Clight language *)
17
18include "Coqlib.ma".
19include "Errors.ma".
20include "Integers.ma".
21include "Floats.ma".
22include "Values.ma".
23include "AST.ma".
24include "Mem.ma".
25include "Globalenvs.ma".
26include "Csyntax.ma".
27include "Maps.ma".
28include "Events.ma".
29include "Smallstep.ma".
30
31(* * * Semantics of type-dependent operations *)
32
33(* * Interpretation of values as truth values.
34  Non-zero integers, non-zero floats and non-null pointers are
35  considered as true.  The integer zero (which also represents
36  the null pointer) and the float 0.0 are false. *)
37
38ninductive is_false: val → type → Prop ≝
39  | is_false_int: ∀sz,sg.
40      is_false (Vint zero) (Tint sz sg)
41  | is_false_pointer: ∀s,t.
42      is_false (Vint zero) (Tpointer s t)
43 | is_false_float: ∀sz.
44      is_false (Vfloat Fzero) (Tfloat sz).
45
46ninductive is_true: val → type → Prop ≝
47  | is_true_int_int: ∀n,sz,sg.
48      n ≠ zero →
49      is_true (Vint n) (Tint sz sg)
50  | is_true_pointer_int: ∀psp,b,ofs,sz,sg.
51      is_true (Vptr psp b ofs) (Tint sz sg)
52  | is_true_int_pointer: ∀n,s,t.
53      n ≠ zero →
54      is_true (Vint n) (Tpointer s t)
55  | is_true_pointer_pointer: ∀psp,b,ofs,s,t.
56      is_true (Vptr psp b ofs) (Tpointer s t)
57  | is_true_float: ∀f,sz.
58      f ≠ Fzero →
59      is_true (Vfloat f) (Tfloat sz).
60
61ninductive bool_of_val : val → type → val → Prop ≝
62  | bool_of_val_true: ∀v,ty.
63         is_true v ty →
64         bool_of_val v ty Vtrue
65  | bool_of_val_false: ∀v,ty.
66        is_false v ty →
67        bool_of_val v ty Vfalse.
68
69(* * The following [sem_] functions compute the result of an operator
70  application.  Since operators are overloaded, the result depends
71  both on the static types of the arguments and on their run-time values.
72  Unlike in C, automatic conversions between integers and floats
73  are not performed.  For instance, [e1 + e2] is undefined if [e1]
74  is a float and [e2] an integer.  The Clight producer must have explicitly
75  promoted [e2] to a float. *)
76
77nlet rec sem_neg (v: val) (ty: type) : option val ≝
78  match ty with
79  [ Tint _ _ ⇒
80      match v with
81      [ Vint n ⇒ Some ? (Vint (neg n))
82      | _ => None ?
83      ]
84  | Tfloat _ ⇒
85      match v with
86      [ Vfloat f ⇒ Some ? (Vfloat (Fneg f))
87      | _ ⇒ None ?
88      ]
89  | _ ⇒ None ?
90  ].
91
92nlet rec sem_notint (v: val) : option val ≝
93  match v with
94  [ Vint n ⇒ Some ? (Vint (xor n mone))
95  | _ ⇒ None ?
96  ].
97
98nlet rec sem_notbool (v: val) (ty: type) : option val ≝
99  match ty with
100  [ Tint _ _ ⇒
101      match v with
102      [ Vint n ⇒ Some ? (of_bool (eq n zero))
103      | Vptr _ _ _ ⇒ Some ? Vfalse
104      | _ ⇒ None ?
105      ]
106  | Tpointer _ _ ⇒
107      match v with
108      [ Vint n ⇒ Some ? (of_bool (eq n zero))
109      | Vptr _ _ _ ⇒ Some ? Vfalse
110      | _ ⇒ None ?
111      ]
112  | Tfloat _ ⇒
113      match v with
114      [ Vfloat f ⇒ Some ? (of_bool (Fcmp Ceq f Fzero))
115      | _ ⇒ None ?
116      ]
117  | _ ⇒ None ?
118  ].
119
120nlet rec sem_add (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝
121  match classify_add t1 t2 with
122  [ add_case_ii ⇒                       (**r integer addition *)
123      match v1 with
124      [ Vint n1 ⇒ match v2 with
125        [ Vint n2 ⇒ Some ? (Vint (add n1 n2))
126        | _ ⇒ None ? ]
127      | _ ⇒ None ? ]
128  | add_case_ff ⇒                       (**r float addition *)
129      match v1 with
130      [ Vfloat n1 ⇒ match v2 with
131        [ Vfloat n2 ⇒ Some ? (Vfloat (Fadd n1 n2))
132        | _ ⇒ None ? ]
133      | _ ⇒ None ? ]
134  | add_case_pi ty ⇒                    (**r pointer plus integer *)
135      match v1 with
136      [ Vptr pcl1 b1 ofs1 ⇒ match v2 with
137        [ Vint n2 ⇒ Some ? (Vptr pcl1 b1 (add ofs1 (mul (repr (sizeof ty)) n2)))
138        | _ ⇒ None ? ]
139      | _ ⇒ None ? ]
140  | add_case_ip ty ⇒                    (**r integer plus pointer *)
141      match v1 with
142      [ Vint n1 ⇒ match v2 with
143        [ Vptr pcl2 b2 ofs2 ⇒ Some ? (Vptr pcl2 b2 (add ofs2 (mul (repr (sizeof ty)) n1)))
144        | _ ⇒ None ? ]
145      | _ ⇒ None ? ]
146  | add_default ⇒ None ?
147].
148
149nlet rec sem_sub (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝
150  match classify_sub t1 t2 with
151  [ sub_case_ii ⇒                (**r integer subtraction *)
152      match v1 with
153      [ Vint n1 ⇒ match v2 with
154        [ Vint n2 ⇒ Some ? (Vint (sub n1 n2))
155        | _ ⇒ None ? ]
156      | _ ⇒ None ? ]
157  | sub_case_ff ⇒                (**r float subtraction *)
158      match v1 with
159      [ Vfloat f1 ⇒ match v2 with
160        [ Vfloat f2 ⇒ Some ? (Vfloat (Fsub f1 f2))
161        | _ ⇒ None ? ]
162      | _ ⇒ None ? ]
163  | sub_case_pi ty ⇒             (**r pointer minus integer *)
164      match v1 with
165      [ Vptr pcl1 b1 ofs1 ⇒ match v2 with
166        [ Vint n2 ⇒ Some ? (Vptr pcl1 b1 (sub ofs1 (mul (repr (sizeof ty)) n2)))
167        | _ ⇒ None ? ]
168      | _ ⇒ None ? ]
169  | sub_case_pp ty ⇒             (**r pointer minus pointer *)
170      match v1 with
171      [ Vptr pcl1 b1 ofs1 ⇒ match v2 with
172        [ Vptr pcl2 b2 ofs2 ⇒
173          if eqZb b1 b2 then
174            if eq (repr (sizeof ty)) zero then None ?
175            else Some ? (Vint (divu (sub ofs1 ofs2) (repr (sizeof ty))))
176          else None ?
177        | _ ⇒ None ? ]
178      | _ ⇒ None ? ]
179  | sub_default ⇒ None ?
180  ].
181
182nlet rec sem_mul (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝
183 match classify_mul t1 t2 with
184  [ mul_case_ii ⇒
185      match v1 with
186      [ Vint n1 ⇒ match v2 with
187        [ Vint n2 ⇒ Some ? (Vint (mul n1 n2))
188        | _ ⇒ None ? ]
189      | _ ⇒ None ? ]
190  | mul_case_ff ⇒
191      match v1 with
192      [ Vfloat f1 ⇒ match v2 with
193        [ Vfloat f2 ⇒ Some ? (Vfloat (Fmul f1 f2))
194        | _ ⇒ None ? ]
195      | _ ⇒ None ? ]
196  | mul_default ⇒
197      None ?
198].
199
200nlet rec sem_div (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝
201  match classify_div t1 t2 with
202  [ div_case_I32unsi ⇒
203      match v1 with
204      [ Vint n1 ⇒ match v2 with
205        [ Vint n2 ⇒ if eq n2 zero then None ? else Some ? (Vint (divu n1 n2))
206        | _ ⇒ None ? ]
207      | _ ⇒ None ? ]
208  | div_case_ii ⇒
209      match v1 with
210       [ Vint n1 ⇒ match v2 with
211         [ Vint n2 ⇒ if eq n2 zero then None ? else Some ? (Vint(divs n1 n2))
212         | _ ⇒ None ? ]
213      | _ ⇒ None ? ]
214  | div_case_ff ⇒
215      match v1 with
216      [ Vfloat f1 ⇒ match v2 with
217        [ Vfloat f2 ⇒ Some ? (Vfloat(Fdiv f1 f2))
218        | _ ⇒ None ? ]
219      | _ ⇒ None ? ]
220  | div_default ⇒
221      None ?
222  ].
223
224nlet rec sem_mod (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝
225  match classify_mod t1 t2 with
226  [ mod_case_I32unsi ⇒
227      match v1 with
228      [ Vint n1 ⇒ match v2 with
229        [ Vint n2 ⇒ if eq n2 zero then None ? else Some ? (Vint (modu n1 n2))
230        | _ ⇒ None ? ]
231      | _ ⇒ None ? ]
232  | mod_case_ii ⇒
233      match v1 with
234      [ Vint n1 ⇒ match v2 with
235        [ Vint n2 ⇒ if eq n2 zero then None ? else Some ? (Vint (mods n1 n2))
236        | _ ⇒ None ? ]
237      | _ ⇒ None ? ]
238  | mod_default ⇒
239      None ?
240  ].
241
242nlet rec sem_and (v1,v2: val) : option val ≝
243  match v1 with
244  [ Vint n1 ⇒ match v2 with
245    [ Vint n2 ⇒ Some ? (Vint(i_and n1 n2))
246    | _ ⇒ None ? ]
247  | _ ⇒ None ?
248  ].
249
250nlet rec sem_or (v1,v2: val) : option val ≝
251  match v1 with
252  [ Vint n1 ⇒ match v2 with
253    [ Vint n2 ⇒ Some ? (Vint(or n1 n2))
254    | _ ⇒ None ? ]
255  | _ ⇒ None ?
256  ].
257
258nlet rec sem_xor (v1,v2: val) : option val ≝
259  match v1 with
260  [ Vint n1 ⇒ match v2 with
261    [ Vint n2 ⇒ Some ? (Vint(xor n1 n2))
262    | _ ⇒ None ? ]
263  | _ ⇒ None ?
264  ].
265
266nlet rec sem_shl (v1,v2: val): option val ≝
267  match v1 with
268  [ Vint n1 ⇒ match v2 with
269    [ Vint n2 ⇒
270        if ltu n2 iwordsize then Some ? (Vint(shl n1 n2)) else None ?
271    | _ ⇒ None ? ]
272  | _ ⇒ None ? ].
273
274nlet rec sem_shr (v1: val) (t1: type) (v2: val) (t2: type): option val ≝
275  match classify_shr t1 t2 with
276  [ shr_case_I32unsi ⇒
277      match v1 with
278      [ Vint n1 ⇒ match v2 with
279        [ Vint n2 ⇒
280            if ltu n2 iwordsize then Some ? (Vint (shru n1 n2)) else None ?
281        | _ ⇒ None ? ]
282      | _ ⇒ None ? ]
283   | shr_case_ii =>
284      match v1 with
285      [ Vint n1 ⇒ match v2 with
286        [ Vint n2 ⇒
287            if ltu n2 iwordsize then Some ? (Vint (shr n1 n2)) else None ?
288        | _ ⇒ None ? ]
289      | _ ⇒ None ? ]
290   | shr_default ⇒
291      None ?
292   ].
293
294nlet rec sem_cmp_mismatch (c: comparison): option val ≝
295  match c with
296  [ Ceq =>  Some ? Vfalse
297  | Cne =>  Some ? Vtrue
298  | _   => None ?
299  ].
300
301nlet rec sem_cmp (c:comparison)
302                  (v1: val) (t1: type) (v2: val) (t2: type)
303                  (m: mem): option val ≝
304  match classify_cmp t1 t2 with
305  [ cmp_case_I32unsi ⇒
306      match v1 with
307      [ Vint n1 ⇒ match v2 with
308        [ Vint n2 ⇒ Some ? (of_bool (cmpu c n1 n2))
309        | _ ⇒ None ? ]
310      | _ ⇒ None ? ]
311  | cmp_case_ipip ⇒
312      match v1 with
313      [ Vint n1 ⇒ match v2 with
314         [ Vint n2 ⇒ Some ? (of_bool (cmp c n1 n2))
315         | Vptr psp b ofs ⇒ if eq n1 zero then sem_cmp_mismatch c else None ?
316         | _ ⇒ None ?
317         ]
318      | Vptr pcl1 b1 ofs1 ⇒
319        match v2 with
320        [ Vptr pcl2 b2 ofs2 ⇒
321          if valid_pointer m pcl1 b1 (signed ofs1)
322          ∧ valid_pointer m pcl2 b2 (signed ofs2) then
323            if eqZb b1 b2
324            then Some ? (of_bool (cmp c ofs1 ofs2))
325            else sem_cmp_mismatch c
326          else None ?
327        | Vint n ⇒
328          if eq n zero then sem_cmp_mismatch c else None ?
329        | _ ⇒ None ? ]
330      | _ ⇒ None ? ]
331  | cmp_case_ff ⇒
332      match v1 with
333      [ Vfloat f1 ⇒
334        match v2 with
335        [ Vfloat f2 ⇒ Some ? (of_bool (Fcmp c f1 f2))
336        | _ ⇒ None ? ]
337      | _ ⇒ None ? ]
338  | cmp_default ⇒ None ?
339  ].
340
341ndefinition sem_unary_operation
342            : unary_operation → val → type → option val ≝
343  λop,v,ty.
344  match op with
345  [ Onotbool => sem_notbool v ty
346  | Onotint => sem_notint v
347  | Oneg => sem_neg v ty
348  ].
349
350nlet rec sem_binary_operation
351    (op: binary_operation)
352    (v1: val) (t1: type) (v2: val) (t2:type)
353    (m: mem): option val ≝
354  match op with
355  [ Oadd ⇒ sem_add v1 t1 v2 t2
356  | Osub ⇒ sem_sub v1 t1 v2 t2
357  | Omul ⇒ sem_mul v1 t1 v2 t2
358  | Omod ⇒ sem_mod v1 t1 v2 t2
359  | Odiv ⇒ sem_div v1 t1 v2 t2
360  | Oand ⇒ sem_and v1 v2 
361  | Oor  ⇒ sem_or v1 v2
362  | Oxor ⇒ sem_xor v1 v2
363  | Oshl ⇒ sem_shl v1 v2
364  | Oshr ⇒ sem_shr v1 t1 v2 t2
365  | Oeq ⇒ sem_cmp Ceq v1 t1 v2 t2 m
366  | One ⇒ sem_cmp Cne v1 t1 v2 t2 m
367  | Olt ⇒ sem_cmp Clt v1 t1 v2 t2 m
368  | Ogt ⇒ sem_cmp Cgt v1 t1 v2 t2 m
369  | Ole ⇒ sem_cmp Cle v1 t1 v2 t2 m
370  | Oge ⇒ sem_cmp Cge v1 t1 v2 t2 m
371  ].
372
373(* * Semantic of casts.  [cast v1 t1 t2 v2] holds if value [v1],
374  viewed with static type [t1], can be cast to type [t2],
375  resulting in value [v2].  *)
376
377nlet rec cast_int_int (sz: intsize) (sg: signedness) (i: int) : int ≝
378  match sz with
379  [ I8 ⇒ match sg with [ Signed ⇒ sign_ext 8 i | Unsigned ⇒ zero_ext 8 i ]
380  | I16 ⇒ match sg with [ Signed => sign_ext 16 i | Unsigned ⇒ zero_ext 16 i ]
381  | I32 ⇒ i
382  ].
383
384nlet rec cast_int_float (si : signedness) (i: int) : float ≝
385  match si with
386  [ Signed ⇒ floatofint i
387  | Unsigned ⇒ floatofintu i
388  ].
389
390nlet rec cast_float_int (si : signedness) (f: float) : int ≝
391  match si with
392  [ Signed ⇒ intoffloat f
393  | Unsigned ⇒ intuoffloat f
394  ].
395
396nlet rec cast_float_float (sz: floatsize) (f: float) : float ≝
397  match sz with
398  [ F32 ⇒ singleoffloat f
399  | F64 ⇒ f
400  ].
401
402ninductive type_pointable : type → Prop ≝
403          (* All integer sizes can represent at least one kind of pointer *)
404| type_ptr_pointer : ∀s,t. type_pointable (Tpointer s t)
405| type_ptr_array : ∀s,t,n. type_pointable (Tarray s t n)
406| type_ptr_function : ∀tys,ty. type_pointable (Tfunction tys ty).
407
408ninductive type_space : type → memory_space → Prop ≝
409| type_spc_pointer : ∀s,t. type_space (Tpointer s t) s
410| type_spc_array : ∀s,t,n. type_space (Tarray s t n) s
411(* XXX Is the following necessary? *)
412| type_spc_code : ∀tys,ty. type_space (Tfunction tys ty) Code.
413
414ninductive cast : mem → val → type → type → val → Prop ≝
415  | cast_ii:   ∀m,i,sz2,sz1,si1,si2.            (**r int to int  *)
416      cast m (Vint i) (Tint sz1 si1) (Tint sz2 si2)
417           (Vint (cast_int_int sz2 si2 i))
418  | cast_fi:   ∀m,f,sz1,sz2,si2.                (**r float to int *)
419      cast m (Vfloat f) (Tfloat sz1) (Tint sz2 si2)
420           (Vint (cast_int_int sz2 si2 (cast_float_int si2 f)))
421  | cast_if:   ∀m,i,sz1,sz2,si1.                (**r int to float  *)
422      cast m (Vint i) (Tint sz1 si1) (Tfloat sz2)
423          (Vfloat (cast_float_float sz2 (cast_int_float si1 i)))
424  | cast_ff:   ∀m,f,sz1,sz2.                    (**r float to float *)
425      cast m (Vfloat f) (Tfloat sz1) (Tfloat sz2)
426           (Vfloat (cast_float_float sz2 f))
427  | cast_pp: ∀m,psp,psp',ty,ty',b,ofs.
428      type_space ty psp →
429      type_space ty' psp' →
430      pointer_compat (block_space m b) psp' →
431      cast m (Vptr psp b ofs) ty ty' (Vptr psp' b ofs)
432  | cast_ip_z: ∀m,sz,sg,ty'.
433      type_pointable ty' →
434      cast m (Vint zero) (Tint sz sg) ty' (Vint zero)
435  | cast_pp_z: ∀m,ty,ty'.
436      type_pointable ty →
437      type_pointable ty' →
438      cast m (Vint zero) ty ty' (Vint zero).
439(* Should probably also allow pointers to pass through sufficiently large
440   unsigned integers. *)
441(* Perhaps a little too generous?  For example, some integers may not
442   represent a valid generic pointer.
443  | cast_pp_i: ∀m,n,ty,ty',t1,t2.     (**r no change in data representation *)
444      type_pointable ty →
445      type_pointable ty' →
446      sizeof ty ≤ sizeof ty' →
447      cast m (Vint n) t1 t2 (Vint n).
448*)
449
450(* * * Operational semantics *)
451
452(* * The semantics uses two environments.  The global environment
453  maps names of functions and global variables to memory block references,
454  and function pointers to their definitions.  (See module [Globalenvs].) *)
455
456ndefinition genv ≝ (genv_t Genv) fundef.
457
458(* * The local environment maps local variables to block references.
459  The current value of the variable is stored in the associated memory
460  block. *)
461
462ndefinition env ≝ (tree_t ? PTree) block. (* map variable -> location *)
463
464ndefinition empty_env: env ≝ (empty …).
465
466(* * [load_value_of_type ty m b ofs] computes the value of a datum
467  of type [ty] residing in memory [m] at block [b], offset [ofs].
468  If the type [ty] indicates an access by value, the corresponding
469  memory load is performed.  If the type [ty] indicates an access by
470  reference, the pointer [Vptr b ofs] is returned. *)
471
472nlet rec load_value_of_type (ty: type) (m: mem) (psp:memory_space) (b: block) (ofs: int) : option val ≝
473  match access_mode ty with
474  [ By_value chunk ⇒ loadv chunk m (Vptr psp b ofs)
475  | By_reference ⇒ Some ? (Vptr psp b ofs)
476  | By_nothing ⇒ None ?
477  ].
478
479(* * Symmetrically, [store_value_of_type ty m b ofs v] returns the
480  memory state after storing the value [v] in the datum
481  of type [ty] residing in memory [m] at block [b], offset [ofs].
482  This is allowed only if [ty] indicates an access by value. *)
483
484nlet rec store_value_of_type (ty_dest: type) (m: mem) (psp:memory_space) (loc: block) (ofs: int) (v: val) : option mem ≝
485  match access_mode ty_dest with
486  [ By_value chunk ⇒ storev chunk m (Vptr psp loc ofs) v
487  | By_reference ⇒ None ?
488  | By_nothing ⇒ None ?
489  ].
490
491(* * Allocation of function-local variables.
492  [alloc_variables e1 m1 vars e2 m2] allocates one memory block
493  for each variable declared in [vars], and associates the variable
494  name with this block.  [e1] and [m1] are the initial local environment
495  and memory state.  [e2] and [m2] are the final local environment
496  and memory state. *)
497
498ninductive alloc_variables: env → mem →
499                            list (ident × type) →
500                            env → mem → Prop ≝
501  | alloc_variables_nil:
502      ∀e,m.
503      alloc_variables e m (nil ?) e m
504  | alloc_variables_cons:
505      ∀e,m,id,ty,vars,m1,b1,m2,e2.
506      alloc m 0 (sizeof ty) Any = 〈m1, b1〉 →
507      alloc_variables (set … id b1 e) m1 vars e2 m2 →
508      alloc_variables e m (〈id, ty〉 :: vars) e2 m2.
509
510(* * Initialization of local variables that are parameters to a function.
511  [bind_parameters e m1 params args m2] stores the values [args]
512  in the memory blocks corresponding to the variables [params].
513  [m1] is the initial memory state and [m2] the final memory state. *)
514
515ninductive bind_parameters: env →
516                           mem → list (ident × type) → list val →
517                           mem → Prop ≝
518  | bind_parameters_nil:
519      ∀e,m.
520      bind_parameters e m (nil ?) (nil ?) m
521  | bind_parameters_cons:
522      ∀e,m,id,ty,params,v1,vl,b,m1,m2.
523      get ??? id e = Some ? b →
524      store_value_of_type ty m Any b zero v1 = Some ? m1 →
525      bind_parameters e m1 params vl m2 →
526      bind_parameters e m (〈id, ty〉 :: params) (v1 :: vl) m2.
527
528(* XXX: this doesn't look right - we're assigning arbitrary memory spaces to
529   parameters? *)
530
531(* * Return the list of blocks in the codomain of [e]. *)
532
533ndefinition blocks_of_env : env → list block ≝ λe.
534  map ?? (λx. snd ?? x) (elements ??? e).
535
536(* * Selection of the appropriate case of a [switch], given the value [n]
537  of the selector expression. *)
538
539nlet rec select_switch (n: int) (sl: labeled_statements)
540                       on sl : labeled_statements ≝
541  match sl with
542  [ LSdefault _ ⇒ sl
543  | LScase c s sl' ⇒ if eq c n then sl else select_switch n sl'
544  ].
545
546(* * Turn a labeled statement into a sequence *)
547
548nlet rec seq_of_labeled_statement (sl: labeled_statements) : statement ≝
549  match sl with
550  [ LSdefault s ⇒ s
551  | LScase c s sl' ⇒ Ssequence s (seq_of_labeled_statement sl')
552  ].
553
554(*
555Section SEMANTICS.
556
557Variable ge: genv.
558
559(** ** Evaluation of expressions *)
560
561Section EXPR.
562
563Variable e: env.
564Variable m: mem.
565*)
566(* * [eval_expr ge e m a v] defines the evaluation of expression [a]
567  in r-value position.  [v] is the value of the expression.
568  [e] is the current environment and [m] is the current memory state. *)
569
570ninductive eval_expr (ge:genv) (e:env) (m:mem) : expr → val → Prop ≝
571  | eval_Econst_int:   ∀i,ty.
572      eval_expr ge e m (Expr (Econst_int i) ty) (Vint i)
573  | eval_Econst_float:   ∀f,ty.
574      eval_expr ge e m (Expr (Econst_float f) ty) (Vfloat f)
575  | eval_Elvalue: ∀a,ty,psp,loc,ofs,v.
576      eval_lvalue ge e m (Expr a ty) psp loc ofs ->
577      load_value_of_type ty m psp loc ofs = Some ? v ->
578      eval_expr ge e m (Expr a ty) v
579  | eval_Eaddrof: ∀a,ty,psp,loc,ofs.
580      eval_lvalue ge e m a psp loc ofs ->
581      eval_expr ge e m (Expr (Eaddrof a) ty) (Vptr psp loc ofs)
582  | eval_Esizeof: ∀ty',ty.
583      eval_expr ge e m (Expr (Esizeof ty') ty) (Vint (repr (sizeof ty')))
584  | eval_Eunop:  ∀op,a,ty,v1,v.
585      eval_expr ge e m a v1 ->
586      sem_unary_operation op v1 (typeof a) = Some ? v ->
587      eval_expr ge e m (Expr (Eunop op a) ty) v
588  | eval_Ebinop: ∀op,a1,a2,ty,v1,v2,v.
589      eval_expr ge e m a1 v1 ->
590      eval_expr ge e m a2 v2 ->
591      sem_binary_operation op v1 (typeof a1) v2 (typeof a2) m = Some ? v ->
592      eval_expr ge e m (Expr (Ebinop op a1 a2) ty) v
593  | eval_Econdition_true: ∀a1,a2,a3,ty,v1,v2.
594      eval_expr ge e m a1 v1 ->
595      is_true v1 (typeof a1) ->
596      eval_expr ge e m a2 v2 ->
597      eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v2
598  | eval_Econdition_false: ∀a1,a2,a3,ty,v1,v3.
599      eval_expr ge e m a1 v1 ->
600      is_false v1 (typeof a1) ->
601      eval_expr ge e m a3 v3 ->
602      eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v3
603  | eval_Eorbool_1: ∀a1,a2,ty,v1.
604      eval_expr ge e m a1 v1 ->
605      is_true v1 (typeof a1) ->
606      eval_expr ge e m (Expr (Eorbool a1 a2) ty) Vtrue
607  | eval_Eorbool_2: ∀a1,a2,ty,v1,v2,v.
608      eval_expr ge e m a1 v1 ->
609      is_false v1 (typeof a1) ->
610      eval_expr ge e m a2 v2 ->
611      bool_of_val v2 (typeof a2) v ->
612      eval_expr ge e m (Expr (Eorbool a1 a2) ty) v
613  | eval_Eandbool_1: ∀a1,a2,ty,v1.
614      eval_expr ge e m a1 v1 ->
615      is_false v1 (typeof a1) ->
616      eval_expr ge e m (Expr (Eandbool a1 a2) ty) Vfalse
617  | eval_Eandbool_2: ∀a1,a2,ty,v1,v2,v.
618      eval_expr ge e m a1 v1 ->
619      is_true v1 (typeof a1) ->
620      eval_expr ge e m a2 v2 ->
621      bool_of_val v2 (typeof a2) v ->
622      eval_expr ge e m (Expr (Eandbool a1 a2) ty) v
623  | eval_Ecast:   ∀a,ty,ty',v1,v.
624      eval_expr ge e m a v1 ->
625      cast m v1 (typeof a) ty v ->
626      eval_expr ge e m (Expr (Ecast ty a) ty') v
627
628(* * [eval_lvalue ge e m a b ofs] defines the evaluation of expression [a]
629  in l-value position.  The result is the memory location [b, ofs]
630  that contains the value of the expression [a]. *)
631
632with eval_lvalue (*(ge:genv) (e:env) (m:mem)*) : expr → memory_space → block -> int -> Prop ≝
633  | eval_Evar_local:   ∀id,l,ty.
634      (* XXX notation? e!id*) get ??? id e = Some ? l →
635      eval_lvalue ge e m (Expr (Evar id) ty) Any l zero
636  | eval_Evar_global: ∀id,sp,l,ty.
637      (* XXX e!id *) get ??? id e = None ? ->
638      find_symbol ?? ge id = Some ? 〈sp,l〉 ->
639      eval_lvalue ge e m (Expr (Evar id) ty) sp l zero
640  | eval_Ederef: ∀a,ty,psp,l,ofs.
641      eval_expr ge e m a (Vptr psp l ofs) ->
642      eval_lvalue ge e m (Expr (Ederef a) ty) psp l ofs
643 | eval_Efield_struct:   ∀a,i,ty,psp,l,ofs,id,fList,delta.
644      eval_lvalue ge e m a psp l ofs ->
645      typeof a = Tstruct id fList ->
646      field_offset i fList = OK ? delta ->
647      eval_lvalue ge e m (Expr (Efield a i) ty) psp l (add ofs (repr delta))
648 | eval_Efield_union:   ∀a,i,ty,psp,l,ofs,id,fList.
649      eval_lvalue ge e m a psp l ofs ->
650      typeof a = Tunion id fList ->
651      eval_lvalue ge e m (Expr (Efield a i) ty) psp l ofs.
652
653(*
654Scheme eval_expr_ind2 := Minimality for eval_expr Sort Prop
655  with eval_lvalue_ind2 := Minimality for eval_lvalue Sort Prop.
656*)
657
658(* * [eval_exprlist ge e m al vl] evaluates a list of r-value
659  expressions [al] to their values [vl]. *)
660
661ninductive eval_exprlist (ge:genv) (e:env) (m:mem) : list expr -> list val -> Prop :=
662  | eval_Enil:
663      eval_exprlist ge e m (nil ?) (nil ?)
664  | eval_Econs:   ∀a,bl,v,vl.
665      eval_expr ge e m a v ->
666      eval_exprlist ge e m bl vl ->
667      eval_exprlist ge e m (a :: bl) (v :: vl).
668
669(*End EXPR.*)
670
671(* * ** Transition semantics for statements and functions *)
672
673(* * Continuations *)
674
675ninductive cont: Type :=
676  | Kstop: cont
677  | Kseq: statement -> cont -> cont
678       (**r [Kseq s2 k] = after [s1] in [s1;s2] *)
679  | Kwhile: expr -> statement -> cont -> cont
680       (**r [Kwhile e s k] = after [s] in [while (e) s] *)
681  | Kdowhile: expr -> statement -> cont -> cont
682       (**r [Kdowhile e s k] = after [s] in [do s while (e)] *)
683  | Kfor2: expr -> statement -> statement -> cont -> cont
684       (**r [Kfor2 e2 e3 s k] = after [s] in [for(e1;e2;e3) s] *)
685  | Kfor3: expr -> statement -> statement -> cont -> cont
686       (**r [Kfor3 e2 e3 s k] = after [e3] in [for(e1;e2;e3) s] *)
687  | Kswitch: cont -> cont
688       (**r catches [break] statements arising out of [switch] *)
689  | Kcall: option (memory_space × block × int × type) ->   (**r where to store result *)
690           function ->                      (**r calling function *)
691           env ->                           (**r local env of calling function *)
692           cont -> cont.
693
694(* * Pop continuation until a call or stop *)
695
696nlet rec call_cont (k: cont) : cont :=
697  match k with
698  [ Kseq s k => call_cont k
699  | Kwhile e s k => call_cont k
700  | Kdowhile e s k => call_cont k
701  | Kfor2 e2 e3 s k => call_cont k
702  | Kfor3 e2 e3 s k => call_cont k
703  | Kswitch k => call_cont k
704  | _ => k
705  ].
706
707ndefinition is_call_cont : cont → Prop ≝ λk.
708  match k with
709  [ Kstop => True
710  | Kcall _ _ _ _ => True
711  | _ => False
712  ].
713
714(* * States *)
715
716ninductive state: Type :=
717  | State:
718      ∀f: function.
719      ∀s: statement.
720      ∀k: cont.
721      ∀e: env.
722      ∀m: mem.  state
723  | Callstate:
724      ∀fd: fundef.
725      ∀args: list val.
726      ∀k: cont.
727      ∀m: mem. state
728  | Returnstate:
729      ∀res: val.
730      ∀k: cont.
731      ∀m: mem. state.
732                 
733(* * Find the statement and manufacture the continuation
734  corresponding to a label *)
735
736nlet rec find_label (lbl: label) (s: statement) (k: cont)
737                    on s: option (statement × cont) :=
738  match s with
739  [ Ssequence s1 s2 =>
740      match find_label lbl s1 (Kseq s2 k) with
741      [ Some sk => Some ? sk
742      | None => find_label lbl s2 k
743      ]
744  | Sifthenelse a s1 s2 =>
745      match find_label lbl s1 k with
746      [ Some sk => Some ? sk
747      | None => find_label lbl s2 k
748      ]
749  | Swhile a s1 =>
750      find_label lbl s1 (Kwhile a s1 k)
751  | Sdowhile a s1 =>
752      find_label lbl s1 (Kdowhile a s1 k)
753  | Sfor a1 a2 a3 s1 =>
754      match find_label lbl a1 (Kseq (Sfor Sskip a2 a3 s1) k) with
755      [ Some sk => Some ? sk
756      | None =>
757          match find_label lbl s1 (Kfor2 a2 a3 s1 k) with
758          [ Some sk => Some ? sk
759          | None => find_label lbl a3 (Kfor3 a2 a3 s1 k)
760          ]
761      ]
762  | Sswitch e sl =>
763      find_label_ls lbl sl (Kswitch k)
764  | Slabel lbl' s' =>
765      match ident_eq lbl lbl' with
766      [ inl _ ⇒ Some ? 〈s', k〉
767      | inr _ ⇒ find_label lbl s' k
768      ]
769  | _ => None ?
770  ]
771
772and find_label_ls (lbl: label) (sl: labeled_statements) (k: cont)
773                    on sl: option (statement × cont) :=
774  match sl with
775  [ LSdefault s => find_label lbl s k
776  | LScase _ s sl' =>
777      match find_label lbl s (Kseq (seq_of_labeled_statement sl') k) with
778      [ Some sk => Some ? sk
779      | None => find_label_ls lbl sl' k
780      ]
781  ].
782
783(* * Transition relation *)
784
785ninductive step (ge:genv) : state -> trace -> state -> Prop :=
786
787  | step_assign:   ∀f,a1,a2,k,e,m,psp,loc,ofs,v2,m'.
788      eval_lvalue ge e m a1 psp loc ofs ->
789      eval_expr ge e m a2 v2 ->
790      store_value_of_type (typeof a1) m psp loc ofs v2 = Some ? m' ->
791      step ge (State f (Sassign a1 a2) k e m)
792           E0 (State f Sskip k e m')
793
794  | step_call_none:   ∀f,a,al,k,e,m,vf,vargs,fd.
795      eval_expr ge e m a vf ->
796      eval_exprlist ge e m al vargs ->
797      find_funct ?? ge vf = Some ? fd ->
798      type_of_fundef fd = typeof a ->
799      step ge (State f (Scall (None ?) a al) k e m)
800           E0 (Callstate fd vargs (Kcall (None ?) f e k) m)
801
802  | step_call_some:   ∀f,lhs,a,al,k,e,m,psp,loc,ofs,vf,vargs,fd.
803      eval_lvalue ge e m lhs psp loc ofs ->
804      eval_expr ge e m a vf ->
805      eval_exprlist ge e m al vargs ->
806      find_funct ?? ge vf = Some ? fd ->
807      type_of_fundef fd = typeof a ->
808      step ge (State f (Scall (Some ? lhs) a al) k e m)
809           E0 (Callstate fd vargs (Kcall (Some ? 〈〈〈psp, loc〉, ofs〉, typeof lhs〉) f e k) m)
810
811  | step_seq:  ∀f,s1,s2,k,e,m.
812      step ge (State f (Ssequence s1 s2) k e m)
813           E0 (State f s1 (Kseq s2 k) e m)
814  | step_skip_seq: ∀f,s,k,e,m.
815      step ge (State f Sskip (Kseq s k) e m)
816           E0 (State f s k e m)
817  | step_continue_seq: ∀f,s,k,e,m.
818      step ge (State f Scontinue (Kseq s k) e m)
819           E0 (State f Scontinue k e m)
820  | step_break_seq: ∀f,s,k,e,m.
821      step ge (State f Sbreak (Kseq s k) e m)
822           E0 (State f Sbreak k e m)
823
824  | step_ifthenelse_true:  ∀f,a,s1,s2,k,e,m,v1.
825      eval_expr ge e m a v1 ->
826      is_true v1 (typeof a) ->
827      step ge (State f (Sifthenelse a s1 s2) k e m)
828           E0 (State f s1 k e m)
829  | step_ifthenelse_false: ∀f,a,s1,s2,k,e,m,v1.
830      eval_expr ge e m a v1 ->
831      is_false v1 (typeof a) ->
832      step ge (State f (Sifthenelse a s1 s2) k e m)
833           E0 (State f s2 k e m)
834
835  | step_while_false: ∀f,a,s,k,e,m,v.
836      eval_expr ge e m a v ->
837      is_false v (typeof a) ->
838      step ge (State f (Swhile a s) k e m)
839           E0 (State f Sskip k e m)
840  | step_while_true: ∀f,a,s,k,e,m,v.
841      eval_expr ge e m a v ->
842      is_true v (typeof a) ->
843      step ge (State f (Swhile a s) k e m)
844           E0 (State f s (Kwhile a s k) e m)
845  | step_skip_or_continue_while: ∀f,x,a,s,k,e,m.
846      x = Sskip ∨ x = Scontinue ->
847      step ge (State f x (Kwhile a s k) e m)
848           E0 (State f (Swhile a s) k e m)
849  | step_break_while: ∀f,a,s,k,e,m.
850      step ge (State f Sbreak (Kwhile a s k) e m)
851           E0 (State f Sskip k e m)
852
853  | step_dowhile: ∀f,a,s,k,e,m.
854      step ge (State f (Sdowhile a s) k e m)
855        E0 (State f s (Kdowhile a s k) e m)
856  | step_skip_or_continue_dowhile_false: ∀f,x,a,s,k,e,m,v.
857      x = Sskip ∨ x = Scontinue ->
858      eval_expr ge e m a v ->
859      is_false v (typeof a) ->
860      step ge (State f x (Kdowhile a s k) e m)
861           E0 (State f Sskip k e m)
862  | step_skip_or_continue_dowhile_true: ∀f,x,a,s,k,e,m,v.
863      x = Sskip ∨ x = Scontinue ->
864      eval_expr ge e m a v ->
865      is_true v (typeof a) ->
866      step ge (State f x (Kdowhile a s k) e m)
867           E0 (State f (Sdowhile a s) k e m)
868  | step_break_dowhile: ∀f,a,s,k,e,m.
869      step ge (State f Sbreak (Kdowhile a s k) e m)
870           E0 (State f Sskip k e m)
871
872  | step_for_start: ∀f,a1,a2,a3,s,k,e,m.
873      a1 ≠ Sskip ->
874      step ge (State f (Sfor a1 a2 a3 s) k e m)
875           E0 (State f a1 (Kseq (Sfor Sskip a2 a3 s) k) e m)
876  | step_for_false: ∀f,a2,a3,s,k,e,m,v.
877      eval_expr ge e m a2 v ->
878      is_false v (typeof a2) ->
879      step ge (State f (Sfor Sskip a2 a3 s) k e m)
880           E0 (State f Sskip k e m)
881  | step_for_true: ∀f,a2,a3,s,k,e,m,v.
882      eval_expr ge e m a2 v ->
883      is_true v (typeof a2) ->
884      step ge (State f (Sfor Sskip a2 a3 s) k e m)
885           E0 (State f s (Kfor2 a2 a3 s k) e m)
886  | step_skip_or_continue_for2: ∀f,x,a2,a3,s,k,e,m.
887      x = Sskip ∨ x = Scontinue ->
888      step ge (State f x (Kfor2 a2 a3 s k) e m)
889           E0 (State f a3 (Kfor3 a2 a3 s k) e m)
890  | step_break_for2: ∀f,a2,a3,s,k,e,m.
891      step ge (State f Sbreak (Kfor2 a2 a3 s k) e m)
892           E0 (State f Sskip k e m)
893  | step_skip_for3: ∀f,a2,a3,s,k,e,m.
894      step ge (State f Sskip (Kfor3 a2 a3 s k) e m)
895           E0 (State f (Sfor Sskip a2 a3 s) k e m)
896
897  | step_return_0: ∀f,k,e,m.
898      fn_return f = Tvoid ->
899      step ge (State f (Sreturn (None ?)) k e m)
900           E0 (Returnstate Vundef (call_cont k) (free_list m (blocks_of_env e)))
901  | step_return_1: ∀f,a,k,e,m,v.
902      fn_return f ≠ Tvoid ->
903      eval_expr ge e m a v ->
904      step ge (State f (Sreturn (Some ? a)) k e m)
905           E0 (Returnstate v (call_cont k) (free_list m (blocks_of_env e)))
906  | step_skip_call: ∀f,k,e,m.
907      is_call_cont k ->
908      fn_return f = Tvoid ->
909      step ge (State f Sskip k e m)
910           E0 (Returnstate Vundef k (free_list m (blocks_of_env e)))
911
912  | step_switch: ∀f,a,sl,k,e,m,n.
913      eval_expr ge e m a (Vint n) ->
914      step ge (State f (Sswitch a sl) k e m)
915           E0 (State f (seq_of_labeled_statement (select_switch n sl)) (Kswitch k) e m)
916  | step_skip_break_switch: ∀f,x,k,e,m.
917      x = Sskip ∨ x = Sbreak ->
918      step ge (State f x (Kswitch k) e m)
919           E0 (State f Sskip k e m)
920  | step_continue_switch: ∀f,k,e,m.
921      step ge (State f Scontinue (Kswitch k) e m)
922           E0 (State f Scontinue k e m)
923
924  | step_label: ∀f,lbl,s,k,e,m.
925      step ge (State f (Slabel lbl s) k e m)
926           E0 (State f s k e m)
927
928  | step_goto: ∀f,lbl,k,e,m,s',k'.
929      find_label lbl (fn_body f) (call_cont k) = Some ? 〈s', k'〉 ->
930      step ge (State f (Sgoto lbl) k e m)
931           E0 (State f s' k' e m)
932
933  | step_internal_function: ∀f,vargs,k,m,e,m1,m2.
934      alloc_variables empty_env m ((fn_params f) @ (fn_vars f)) e m1 ->
935      bind_parameters e m1 (fn_params f) vargs m2 ->
936      step ge (Callstate (Internal f) vargs k m)
937           E0 (State f (fn_body f) k e m2)
938
939  | step_external_function: ∀id,targs,tres,vargs,k,m,vres,t.
940      event_match (external_function id targs tres) vargs t vres ->
941      step ge (Callstate (External id targs tres) vargs k m)
942            t (Returnstate vres k m)
943
944  | step_returnstate_0: ∀v,f,e,k,m.
945      step ge (Returnstate v (Kcall (None ?) f e k) m)
946           E0 (State f Sskip k e m)
947
948  | step_returnstate_1: ∀v,f,e,k,m,m',psp,loc,ofs,ty.
949      store_value_of_type ty m psp loc ofs v = Some ? m' ->
950      step ge (Returnstate v (Kcall (Some ? 〈〈〈psp,loc〉, ofs〉, ty〉) f e k) m)
951           E0 (State f Sskip k e m').
952(*
953(** * Alternate big-step semantics *)
954
955(** ** Big-step semantics for terminating statements and functions *)
956
957(** The execution of a statement produces an ``outcome'', indicating
958  how the execution terminated: either normally or prematurely
959  through the execution of a [break], [continue] or [return] statement. *)
960
961ninductive outcome: Type :=
962   | Out_break: outcome                 (**r terminated by [break] *)
963   | Out_continue: outcome              (**r terminated by [continue] *)
964   | Out_normal: outcome                (**r terminated normally *)
965   | Out_return: option val -> outcome. (**r terminated by [return] *)
966
967ninductive out_normal_or_continue : outcome -> Prop :=
968  | Out_normal_or_continue_N: out_normal_or_continue Out_normal
969  | Out_normal_or_continue_C: out_normal_or_continue Out_continue.
970
971ninductive out_break_or_return : outcome -> outcome -> Prop :=
972  | Out_break_or_return_B: out_break_or_return Out_break Out_normal
973  | Out_break_or_return_R: ∀ov.
974      out_break_or_return (Out_return ov) (Out_return ov).
975
976Definition outcome_switch (out: outcome) : outcome :=
977  match out with
978  | Out_break => Out_normal
979  | o => o
980  end.
981
982Definition outcome_result_value (out: outcome) (t: type) (v: val) : Prop :=
983  match out, t with
984  | Out_normal, Tvoid => v = Vundef
985  | Out_return None, Tvoid => v = Vundef
986  | Out_return (Some v'), ty => ty <> Tvoid /\ v'=v
987  | _, _ => False
988  end.
989
990(** [exec_stmt ge e m1 s t m2 out] describes the execution of
991  the statement [s].  [out] is the outcome for this execution.
992  [m1] is the initial memory state, [m2] the final memory state.
993  [t] is the trace of input/output events performed during this
994  evaluation. *)
995
996ninductive exec_stmt: env -> mem -> statement -> trace -> mem -> outcome -> Prop :=
997  | exec_Sskip:   ∀e,m.
998      exec_stmt e m Sskip
999               E0 m Out_normal
1000  | exec_Sassign:   ∀e,m,a1,a2,loc,ofs,v2,m'.
1001      eval_lvalue e m a1 loc ofs ->
1002      eval_expr e m a2 v2 ->
1003      store_value_of_type (typeof a1) m loc ofs v2 = Some m' ->
1004      exec_stmt e m (Sassign a1 a2)
1005               E0 m' Out_normal
1006  | exec_Scall_none:   ∀e,m,a,al,vf,vargs,f,t,m',vres.
1007      eval_expr e m a vf ->
1008      eval_exprlist e m al vargs ->
1009      Genv.find_funct ge vf = Some f ->
1010      type_of_fundef f = typeof a ->
1011      eval_funcall m f vargs t m' vres ->
1012      exec_stmt e m (Scall None a al)
1013                t m' Out_normal
1014  | exec_Scall_some:   ∀e,m,lhs,a,al,loc,ofs,vf,vargs,f,t,m',vres,m''.
1015      eval_lvalue e m lhs loc ofs ->
1016      eval_expr e m a vf ->
1017      eval_exprlist e m al vargs ->
1018      Genv.find_funct ge vf = Some f ->
1019      type_of_fundef f = typeof a ->
1020      eval_funcall m f vargs t m' vres ->
1021      store_value_of_type (typeof lhs) m' loc ofs vres = Some m'' ->
1022      exec_stmt e m (Scall (Some lhs) a al)
1023                t m'' Out_normal
1024  | exec_Sseq_1:   ∀e,m,s1,s2,t1,m1,t2,m2,out.
1025      exec_stmt e m s1 t1 m1 Out_normal ->
1026      exec_stmt e m1 s2 t2 m2 out ->
1027      exec_stmt e m (Ssequence s1 s2)
1028                (t1 ** t2) m2 out
1029  | exec_Sseq_2:   ∀e,m,s1,s2,t1,m1,out.
1030      exec_stmt e m s1 t1 m1 out ->
1031      out <> Out_normal ->
1032      exec_stmt e m (Ssequence s1 s2)
1033                t1 m1 out
1034  | exec_Sifthenelse_true: ∀e,m,a,s1,s2,v1,t,m',out.
1035      eval_expr e m a v1 ->
1036      is_true v1 (typeof a) ->
1037      exec_stmt e m s1 t m' out ->
1038      exec_stmt e m (Sifthenelse a s1 s2)
1039                t m' out
1040  | exec_Sifthenelse_false: ∀e,m,a,s1,s2,v1,t,m',out.
1041      eval_expr e m a v1 ->
1042      is_false v1 (typeof a) ->
1043      exec_stmt e m s2 t m' out ->
1044      exec_stmt e m (Sifthenelse a s1 s2)
1045                t m' out
1046  | exec_Sreturn_none:   ∀e,m.
1047      exec_stmt e m (Sreturn None)
1048               E0 m (Out_return None)
1049  | exec_Sreturn_some: ∀e,m,a,v.
1050      eval_expr e m a v ->
1051      exec_stmt e m (Sreturn (Some a))
1052               E0 m (Out_return (Some v))
1053  | exec_Sbreak:   ∀e,m.
1054      exec_stmt e m Sbreak
1055               E0 m Out_break
1056  | exec_Scontinue:   ∀e,m.
1057      exec_stmt e m Scontinue
1058               E0 m Out_continue
1059  | exec_Swhile_false: ∀e,m,a,s,v.
1060      eval_expr e m a v ->
1061      is_false v (typeof a) ->
1062      exec_stmt e m (Swhile a s)
1063               E0 m Out_normal
1064  | exec_Swhile_stop: ∀e,m,a,v,s,t,m',out',out.
1065      eval_expr e m a v ->
1066      is_true v (typeof a) ->
1067      exec_stmt e m s t m' out' ->
1068      out_break_or_return out' out ->
1069      exec_stmt e m (Swhile a s)
1070                t m' out
1071  | exec_Swhile_loop: ∀e,m,a,s,v,t1,m1,out1,t2,m2,out.
1072      eval_expr e m a v ->
1073      is_true v (typeof a) ->
1074      exec_stmt e m s t1 m1 out1 ->
1075      out_normal_or_continue out1 ->
1076      exec_stmt e m1 (Swhile a s) t2 m2 out ->
1077      exec_stmt e m (Swhile a s)
1078                (t1 ** t2) m2 out
1079  | exec_Sdowhile_false: ∀e,m,s,a,t,m1,out1,v.
1080      exec_stmt e m s t m1 out1 ->
1081      out_normal_or_continue out1 ->
1082      eval_expr e m1 a v ->
1083      is_false v (typeof a) ->
1084      exec_stmt e m (Sdowhile a s)
1085                t m1 Out_normal
1086  | exec_Sdowhile_stop: ∀e,m,s,a,t,m1,out1,out.
1087      exec_stmt e m s t m1 out1 ->
1088      out_break_or_return out1 out ->
1089      exec_stmt e m (Sdowhile a s)
1090                t m1 out
1091  | exec_Sdowhile_loop: ∀e,m,s,a,m1,m2,t1,t2,out,out1,v.
1092      exec_stmt e m s t1 m1 out1 ->
1093      out_normal_or_continue out1 ->
1094      eval_expr e m1 a v ->
1095      is_true v (typeof a) ->
1096      exec_stmt e m1 (Sdowhile a s) t2 m2 out ->
1097      exec_stmt e m (Sdowhile a s)
1098                (t1 ** t2) m2 out
1099  | exec_Sfor_start: ∀e,m,s,a1,a2,a3,out,m1,m2,t1,t2.
1100      a1 <> Sskip ->
1101      exec_stmt e m a1 t1 m1 Out_normal ->
1102      exec_stmt e m1 (Sfor Sskip a2 a3 s) t2 m2 out ->
1103      exec_stmt e m (Sfor a1 a2 a3 s)
1104                (t1 ** t2) m2 out
1105  | exec_Sfor_false: ∀e,m,s,a2,a3,v.
1106      eval_expr e m a2 v ->
1107      is_false v (typeof a2) ->
1108      exec_stmt e m (Sfor Sskip a2 a3 s)
1109               E0 m Out_normal
1110  | exec_Sfor_stop: ∀e,m,s,a2,a3,v,m1,t,out1,out.
1111      eval_expr e m a2 v ->
1112      is_true v (typeof a2) ->
1113      exec_stmt e m s t m1 out1 ->
1114      out_break_or_return out1 out ->
1115      exec_stmt e m (Sfor Sskip a2 a3 s)
1116                t m1 out
1117  | exec_Sfor_loop: ∀e,m,s,a2,a3,v,m1,m2,m3,t1,t2,t3,out1,out.
1118      eval_expr e m a2 v ->
1119      is_true v (typeof a2) ->
1120      exec_stmt e m s t1 m1 out1 ->
1121      out_normal_or_continue out1 ->
1122      exec_stmt e m1 a3 t2 m2 Out_normal ->
1123      exec_stmt e m2 (Sfor Sskip a2 a3 s) t3 m3 out ->
1124      exec_stmt e m (Sfor Sskip a2 a3 s)
1125                (t1 ** t2 ** t3) m3 out
1126  | exec_Sswitch:   ∀e,m,a,t,n,sl,m1,out.
1127      eval_expr e m a (Vint n) ->
1128      exec_stmt e m (seq_of_labeled_statement (select_switch n sl)) t m1 out ->
1129      exec_stmt e m (Sswitch a sl)
1130                t m1 (outcome_switch out)
1131
1132(** [eval_funcall m1 fd args t m2 res] describes the invocation of
1133  function [fd] with arguments [args].  [res] is the value returned
1134  by the call.  *)
1135
1136with eval_funcall: mem -> fundef -> list val -> trace -> mem -> val -> Prop :=
1137  | eval_funcall_internal: ∀m,f,vargs,t,e,m1,m2,m3,out,vres.
1138      alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
1139      bind_parameters e m1 f.(fn_params) vargs m2 ->
1140      exec_stmt e m2 f.(fn_body) t m3 out ->
1141      outcome_result_value out f.(fn_return) vres ->
1142      eval_funcall m (Internal f) vargs t (Mem.free_list m3 (blocks_of_env e)) vres
1143  | eval_funcall_external: ∀m,id,targs,tres,vargs,t,vres.
1144      event_match (external_function id targs tres) vargs t vres ->
1145      eval_funcall m (External id targs tres) vargs t m vres.
1146
1147Scheme exec_stmt_ind2 := Minimality for exec_stmt Sort Prop
1148  with eval_funcall_ind2 := Minimality for eval_funcall Sort Prop.
1149
1150(** ** Big-step semantics for diverging statements and functions *)
1151
1152(** Coinductive semantics for divergence.
1153  [execinf_stmt ge e m s t] holds if the execution of statement [s]
1154  diverges, i.e. loops infinitely.  [t] is the possibly infinite
1155  trace of observable events performed during the execution. *)
1156
1157Coninductive execinf_stmt: env -> mem -> statement -> traceinf -> Prop :=
1158  | execinf_Scall_none:   ∀e,m,a,al,vf,vargs,f,t.
1159      eval_expr e m a vf ->
1160      eval_exprlist e m al vargs ->
1161      Genv.find_funct ge vf = Some f ->
1162      type_of_fundef f = typeof a ->
1163      evalinf_funcall m f vargs t ->
1164      execinf_stmt e m (Scall None a al) t
1165  | execinf_Scall_some:   ∀e,m,lhs,a,al,loc,ofs,vf,vargs,f,t.
1166      eval_lvalue e m lhs loc ofs ->
1167      eval_expr e m a vf ->
1168      eval_exprlist e m al vargs ->
1169      Genv.find_funct ge vf = Some f ->
1170      type_of_fundef f = typeof a ->
1171      evalinf_funcall m f vargs t ->
1172      execinf_stmt e m (Scall (Some lhs) a al) t
1173  | execinf_Sseq_1:   ∀e,m,s1,s2,t.
1174      execinf_stmt e m s1 t ->
1175      execinf_stmt e m (Ssequence s1 s2) t
1176  | execinf_Sseq_2:   ∀e,m,s1,s2,t1,m1,t2.
1177      exec_stmt e m s1 t1 m1 Out_normal ->
1178      execinf_stmt e m1 s2 t2 ->
1179      execinf_stmt e m (Ssequence s1 s2) (t1 *** t2)
1180  | execinf_Sifthenelse_true: ∀e,m,a,s1,s2,v1,t.
1181      eval_expr e m a v1 ->
1182      is_true v1 (typeof a) ->
1183      execinf_stmt e m s1 t ->
1184      execinf_stmt e m (Sifthenelse a s1 s2) t
1185  | execinf_Sifthenelse_false: ∀e,m,a,s1,s2,v1,t.
1186      eval_expr e m a v1 ->
1187      is_false v1 (typeof a) ->
1188      execinf_stmt e m s2 t ->
1189      execinf_stmt e m (Sifthenelse a s1 s2) t
1190  | execinf_Swhile_body: ∀e,m,a,v,s,t.
1191      eval_expr e m a v ->
1192      is_true v (typeof a) ->
1193      execinf_stmt e m s t ->
1194      execinf_stmt e m (Swhile a s) t
1195  | execinf_Swhile_loop: ∀e,m,a,s,v,t1,m1,out1,t2.
1196      eval_expr e m a v ->
1197      is_true v (typeof a) ->
1198      exec_stmt e m s t1 m1 out1 ->
1199      out_normal_or_continue out1 ->
1200      execinf_stmt e m1 (Swhile a s) t2 ->
1201      execinf_stmt e m (Swhile a s) (t1 *** t2)
1202  | execinf_Sdowhile_body: ∀e,m,s,a,t.
1203      execinf_stmt e m s t ->
1204      execinf_stmt e m (Sdowhile a s) t
1205  | execinf_Sdowhile_loop: ∀e,m,s,a,m1,t1,t2,out1,v.
1206      exec_stmt e m s t1 m1 out1 ->
1207      out_normal_or_continue out1 ->
1208      eval_expr e m1 a v ->
1209      is_true v (typeof a) ->
1210      execinf_stmt e m1 (Sdowhile a s) t2 ->
1211      execinf_stmt e m (Sdowhile a s) (t1 *** t2)
1212  | execinf_Sfor_start_1: ∀e,m,s,a1,a2,a3,t.
1213      execinf_stmt e m a1 t ->
1214      execinf_stmt e m (Sfor a1 a2 a3 s) t
1215  | execinf_Sfor_start_2: ∀e,m,s,a1,a2,a3,m1,t1,t2.
1216      a1 <> Sskip ->
1217      exec_stmt e m a1 t1 m1 Out_normal ->
1218      execinf_stmt e m1 (Sfor Sskip a2 a3 s) t2 ->
1219      execinf_stmt e m (Sfor a1 a2 a3 s) (t1 *** t2)
1220  | execinf_Sfor_body: ∀e,m,s,a2,a3,v,t.
1221      eval_expr e m a2 v ->
1222      is_true v (typeof a2) ->
1223      execinf_stmt e m s t ->
1224      execinf_stmt e m (Sfor Sskip a2 a3 s) t
1225  | execinf_Sfor_next: ∀e,m,s,a2,a3,v,m1,t1,t2,out1.
1226      eval_expr e m a2 v ->
1227      is_true v (typeof a2) ->
1228      exec_stmt e m s t1 m1 out1 ->
1229      out_normal_or_continue out1 ->
1230      execinf_stmt e m1 a3 t2 ->
1231      execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2)
1232  | execinf_Sfor_loop: ∀e,m,s,a2,a3,v,m1,m2,t1,t2,t3,out1.
1233      eval_expr e m a2 v ->
1234      is_true v (typeof a2) ->
1235      exec_stmt e m s t1 m1 out1 ->
1236      out_normal_or_continue out1 ->
1237      exec_stmt e m1 a3 t2 m2 Out_normal ->
1238      execinf_stmt e m2 (Sfor Sskip a2 a3 s) t3 ->
1239      execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2 *** t3)
1240  | execinf_Sswitch:   ∀e,m,a,t,n,sl.
1241      eval_expr e m a (Vint n) ->
1242      execinf_stmt e m (seq_of_labeled_statement (select_switch n sl)) t ->
1243      execinf_stmt e m (Sswitch a sl) t
1244
1245(** [evalinf_funcall ge m fd args t] holds if the invocation of function
1246    [fd] on arguments [args] diverges, with observable trace [t]. *)
1247
1248with evalinf_funcall: mem -> fundef -> list val -> traceinf -> Prop :=
1249  | evalinf_funcall_internal: ∀m,f,vargs,t,e,m1,m2.
1250      alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
1251      bind_parameters e m1 f.(fn_params) vargs m2 ->
1252      execinf_stmt e m2 f.(fn_body) t ->
1253      evalinf_funcall m (Internal f) vargs t.
1254
1255End SEMANTICS.
1256*)
1257(* * * Whole-program semantics *)
1258
1259(* * Execution of whole programs are described as sequences of transitions
1260  from an initial state to a final state.  An initial state is a [Callstate]
1261  corresponding to the invocation of the ``main'' function of the program
1262  without arguments and with an empty continuation. *)
1263
1264ninductive initial_state (p: program): state -> Prop :=
1265  | initial_state_intro: ∀b,f.
1266      let ge := globalenv Genv ?? p in
1267      let m0 := init_mem Genv ?? p in
1268      find_symbol ?? ge (prog_main ?? p) = Some ? 〈Code,b〉 ->
1269      find_funct_ptr ?? ge b = Some ? f ->
1270      initial_state p (Callstate f (nil ?) Kstop m0).
1271
1272(* * A final state is a [Returnstate] with an empty continuation. *)
1273
1274ninductive final_state: state -> int -> Prop :=
1275  | final_state_intro: ∀r,m.
1276      final_state (Returnstate (Vint r) Kstop m) r.
1277
1278(* * Execution of a whole program: [exec_program p beh]
1279  holds if the application of [p]'s main function to no arguments
1280  in the initial memory state for [p] has [beh] as observable
1281  behavior. *)
1282
1283ndefinition exec_program : program → program_behavior → Prop ≝ λp,beh.
1284  program_behaves (mk_transrel ?? step) (initial_state p) final_state (globalenv ??? p) beh.
1285(*
1286(** Big-step execution of a whole program.  *)
1287
1288ninductive bigstep_program_terminates (p: program): trace -> int -> Prop :=
1289  | bigstep_program_terminates_intro: ∀b,f,m1,t,r.
1290      let ge := Genv.globalenv p in
1291      let m0 := Genv.init_mem p in
1292      Genv.find_symbol ge p.(prog_main) = Some b ->
1293      Genv.find_funct_ptr ge b = Some f ->
1294      eval_funcall ge m0 f nil t m1 (Vint r) ->
1295      bigstep_program_terminates p t r.
1296
1297ninductive bigstep_program_diverges (p: program): traceinf -> Prop :=
1298  | bigstep_program_diverges_intro: ∀b,f,t.
1299      let ge := Genv.globalenv p in
1300      let m0 := Genv.init_mem p in
1301      Genv.find_symbol ge p.(prog_main) = Some b ->
1302      Genv.find_funct_ptr ge b = Some f ->
1303      evalinf_funcall ge m0 f nil t ->
1304      bigstep_program_diverges p t.
1305
1306(** * Implication from big-step semantics to transition semantics *)
1307
1308Section BIGSTEP_TO_TRANSITIONS.
1309
1310Variable prog: program.
1311Let ge : genv := Genv.globalenv prog.
1312
1313Definition exec_stmt_eval_funcall_ind
1314  (PS: env -> mem -> statement -> trace -> mem -> outcome -> Prop)
1315  (PF: mem -> fundef -> list val -> trace -> mem -> val -> Prop) :=
1316  fun a b c d e f g h i j k l m n o p q r s t u v w x y =>
1317  conj (exec_stmt_ind2 ge PS PF a b c d e f g h i j k l m n o p q r s t u v w x y)
1318       (eval_funcall_ind2 ge PS PF a b c d e f g h i j k l m n o p q r s t u v w x y).
1319
1320ninductive outcome_state_match
1321       (e: env) (m: mem) (f: function) (k: cont): outcome -> state -> Prop :=
1322  | osm_normal:
1323      outcome_state_match e m f k Out_normal (State f Sskip k e m)
1324  | osm_break:
1325      outcome_state_match e m f k Out_break (State f Sbreak k e m)
1326  | osm_continue:
1327      outcome_state_match e m f k Out_continue (State f Scontinue k e m)
1328  | osm_return_none: ∀k'.
1329      call_cont k' = call_cont k ->
1330      outcome_state_match e m f k
1331        (Out_return None) (State f (Sreturn None) k' e m)
1332  | osm_return_some: ∀a,v,k'.
1333      call_cont k' = call_cont k ->
1334      eval_expr ge e m a v ->
1335      outcome_state_match e m f k
1336        (Out_return (Some v)) (State f (Sreturn (Some a)) k' e m).
1337
1338Lemma is_call_cont_call_cont:
1339  ∀k. is_call_cont k -> call_cont k = k.
1340Proof.
1341  destruct k; simpl; intros; contradiction || auto.
1342Qed.
1343
1344Lemma exec_stmt_eval_funcall_steps:
1345  (∀e,m,s,t,m',out.
1346   exec_stmt ge e m s t m' out ->
1347   ∀f,k. exists S,
1348   star step ge (State f s k e m) t S
1349   /\ outcome_state_match e m' f k out S)
1350/\
1351  (∀m,fd,args,t,m',res.
1352   eval_funcall ge m fd args t m' res ->
1353   ∀k.
1354   is_call_cont k ->
1355   star step ge (Callstate fd args k m) t (Returnstate res k m')).
1356Proof.
1357  apply exec_stmt_eval_funcall_ind; intros.
1358
1359(* skip *)
1360  econstructor; split. apply star_refl. constructor.
1361
1362(* assign *)
1363  econstructor; split. apply star_one. econstructor; eauto. constructor.
1364
1365(* call none *)
1366  econstructor; split.
1367  eapply star_left. econstructor; eauto.
1368  eapply star_right. apply H4. simpl; auto. econstructor. reflexivity. traceEq.
1369  constructor.
1370
1371(* call some *)
1372  econstructor; split.
1373  eapply star_left. econstructor; eauto.
1374  eapply star_right. apply H5. simpl; auto. econstructor; eauto. reflexivity. traceEq.
1375  constructor.
1376
1377(* sequence 2 *)
1378  destruct (H0 f (Kseq s2 k)) as [S1 [A1 B1]]. inv B1.
1379  destruct (H2 f k) as [S2 [A2 B2]].
1380  econstructor; split.
1381  eapply star_left. econstructor.
1382  eapply star_trans. eexact A1.
1383  eapply star_left. constructor. eexact A2.
1384  reflexivity. reflexivity. traceEq.
1385  auto.
1386
1387(* sequence 1 *)
1388  destruct (H0 f (Kseq s2 k)) as [S1 [A1 B1]].
1389  set (S2 :=
1390    match out with
1391    | Out_break => State f Sbreak k e m1
1392    | Out_continue => State f Scontinue k e m1
1393    | _ => S1
1394    end).
1395  exists S2; split.
1396  eapply star_left. econstructor.
1397  eapply star_trans. eexact A1.
1398  unfold S2; inv B1.
1399    congruence.
1400    apply star_one. apply step_break_seq.
1401    apply star_one. apply step_continue_seq.
1402    apply star_refl.
1403    apply star_refl.
1404  reflexivity. traceEq.
1405  unfold S2; inv B1; congruence || econstructor; eauto.
1406
1407(* ifthenelse true *)
1408  destruct (H2 f k) as [S1 [A1 B1]].
1409  exists S1; split.
1410  eapply star_left. eapply step_ifthenelse_true; eauto. eexact A1. traceEq.
1411  auto.
1412
1413(* ifthenelse false *)
1414  destruct (H2 f k) as [S1 [A1 B1]].
1415  exists S1; split.
1416  eapply star_left. eapply step_ifthenelse_false; eauto. eexact A1. traceEq.
1417  auto.
1418
1419(* return none *)
1420  econstructor; split. apply star_refl. constructor. auto.
1421
1422(* return some *)
1423  econstructor; split. apply star_refl. econstructor; eauto.
1424
1425(* break *)
1426  econstructor; split. apply star_refl. constructor.
1427
1428(* continue *)
1429  econstructor; split. apply star_refl. constructor.
1430
1431(* while false *)
1432  econstructor; split.
1433  apply star_one. eapply step_while_false; eauto.
1434  constructor.
1435
1436(* while stop *)
1437  destruct (H2 f (Kwhile a s k)) as [S1 [A1 B1]].
1438  set (S2 :=
1439    match out' with
1440    | Out_break => State f Sskip k e m'
1441    | _ => S1
1442    end).
1443  exists S2; split.
1444  eapply star_left. eapply step_while_true; eauto.
1445  eapply star_trans. eexact A1.
1446  unfold S2. inversion H3; subst.
1447  inv B1. apply star_one. constructor.   
1448  apply star_refl.
1449  reflexivity. traceEq.
1450  unfold S2. inversion H3; subst. constructor. inv B1; econstructor; eauto.
1451
1452(* while loop *)
1453  destruct (H2 f (Kwhile a s k)) as [S1 [A1 B1]].
1454  destruct (H5 f k) as [S2 [A2 B2]].
1455  exists S2; split.
1456  eapply star_left. eapply step_while_true; eauto.
1457  eapply star_trans. eexact A1.
1458  eapply star_left.
1459  inv H3; inv B1; apply step_skip_or_continue_while; auto.
1460  eexact A2.
1461  reflexivity. reflexivity. traceEq.
1462  auto.
1463
1464(* dowhile false *)
1465  destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]].
1466  exists (State f Sskip k e m1); split.
1467  eapply star_left. constructor.
1468  eapply star_right. eexact A1.
1469  inv H1; inv B1; eapply step_skip_or_continue_dowhile_false; eauto.
1470  reflexivity. traceEq.
1471  constructor.
1472
1473(* dowhile stop *)
1474  destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]].
1475  set (S2 :=
1476    match out1 with
1477    | Out_break => State f Sskip k e m1
1478    | _ => S1
1479    end).
1480  exists S2; split.
1481  eapply star_left. apply step_dowhile.
1482  eapply star_trans. eexact A1.
1483  unfold S2. inversion H1; subst.
1484  inv B1. apply star_one. constructor.
1485  apply star_refl.
1486  reflexivity. traceEq.
1487  unfold S2. inversion H1; subst. constructor. inv B1; econstructor; eauto.
1488
1489(* dowhile loop *)
1490  destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]].
1491  destruct (H5 f k) as [S2 [A2 B2]].
1492  exists S2; split.
1493  eapply star_left. apply step_dowhile.
1494  eapply star_trans. eexact A1.
1495  eapply star_left.
1496  inv H1; inv B1; eapply step_skip_or_continue_dowhile_true; eauto.
1497  eexact A2.
1498  reflexivity. reflexivity. traceEq.
1499  auto.
1500
1501(* for start *)
1502  destruct (H1 f (Kseq (Sfor Sskip a2 a3 s) k)) as [S1 [A1 B1]]. inv B1.
1503  destruct (H3 f k) as [S2 [A2 B2]].
1504  exists S2; split.
1505  eapply star_left. apply step_for_start; auto.   
1506  eapply star_trans. eexact A1.
1507  eapply star_left. constructor. eexact A2.
1508  reflexivity. reflexivity. traceEq.
1509  auto.
1510
1511(* for false *)
1512  econstructor; split.
1513  eapply star_one. eapply step_for_false; eauto.
1514  constructor.
1515
1516(* for stop *)
1517  destruct (H2 f (Kfor2 a2 a3 s k)) as [S1 [A1 B1]].
1518  set (S2 :=
1519    match out1 with
1520    | Out_break => State f Sskip k e m1
1521    | _ => S1
1522    end).
1523  exists S2; split.
1524  eapply star_left. eapply step_for_true; eauto.
1525  eapply star_trans. eexact A1.
1526  unfold S2. inversion H3; subst.
1527  inv B1. apply star_one. constructor.
1528  apply star_refl.
1529  reflexivity. traceEq.
1530  unfold S2. inversion H3; subst. constructor. inv B1; econstructor; eauto.
1531
1532(* for loop *)
1533  destruct (H2 f (Kfor2 a2 a3 s k)) as [S1 [A1 B1]].
1534  destruct (H5 f (Kfor3 a2 a3 s k)) as [S2 [A2 B2]]. inv B2.
1535  destruct (H7 f k) as [S3 [A3 B3]].
1536  exists S3; split.
1537  eapply star_left. eapply step_for_true; eauto.
1538  eapply star_trans. eexact A1.
1539  eapply star_trans with (s2 := State f a3 (Kfor3 a2 a3 s k) e m1).
1540  inv H3; inv B1.
1541  apply star_one. constructor. auto.
1542  apply star_one. constructor. auto.
1543  eapply star_trans. eexact A2.
1544  eapply star_left. constructor.
1545  eexact A3.
1546  reflexivity. reflexivity. reflexivity. reflexivity. traceEq.
1547  auto.
1548
1549(* switch *)
1550  destruct (H1 f (Kswitch k)) as [S1 [A1 B1]].
1551  set (S2 :=
1552    match out with
1553    | Out_normal => State f Sskip k e m1
1554    | Out_break => State f Sskip k e m1
1555    | Out_continue => State f Scontinue k e m1
1556    | _ => S1
1557    end).
1558  exists S2; split.
1559  eapply star_left. eapply step_switch; eauto.
1560  eapply star_trans. eexact A1.
1561  unfold S2; inv B1.
1562    apply star_one. constructor. auto.
1563    apply star_one. constructor. auto.
1564    apply star_one. constructor.
1565    apply star_refl.
1566    apply star_refl.
1567  reflexivity. traceEq.
1568  unfold S2. inv B1; simpl; econstructor; eauto.
1569
1570(* call internal *)
1571  destruct (H2 f k) as [S1 [A1 B1]].
1572  eapply star_left. eapply step_internal_function; eauto.
1573  eapply star_right. eexact A1.
1574  inv B1; simpl in H3; try contradiction.
1575  (* Out_normal *)
1576  assert (fn_return f = Tvoid /\ vres = Vundef).
1577    destruct (fn_return f); auto || contradiction.
1578  destruct H5. subst vres. apply step_skip_call; auto.
1579  (* Out_return None *)
1580  assert (fn_return f = Tvoid /\ vres = Vundef).
1581    destruct (fn_return f); auto || contradiction.
1582  destruct H6. subst vres.
1583  rewrite <- (is_call_cont_call_cont k H4). rewrite <- H5.
1584  apply step_return_0; auto.
1585  (* Out_return Some *)
1586  destruct H3. subst vres.
1587  rewrite <- (is_call_cont_call_cont k H4). rewrite <- H5.
1588  eapply step_return_1; eauto.
1589  reflexivity. traceEq.
1590
1591(* call external *)
1592  apply star_one. apply step_external_function; auto.
1593Qed.
1594
1595Lemma exec_stmt_steps:
1596   ∀e,m,s,t,m',out.
1597   exec_stmt ge e m s t m' out ->
1598   ∀f,k. exists S,
1599   star step ge (State f s k e m) t S
1600   /\ outcome_state_match e m' f k out S.
1601Proof (proj1 exec_stmt_eval_funcall_steps).
1602
1603Lemma eval_funcall_steps:
1604   ∀m,fd,args,t,m',res.
1605   eval_funcall ge m fd args t m' res ->
1606   ∀k.
1607   is_call_cont k ->
1608   star step ge (Callstate fd args k m) t (Returnstate res k m').
1609Proof (proj2 exec_stmt_eval_funcall_steps).
1610
1611Definition order (x y: unit) := False.
1612
1613Lemma evalinf_funcall_forever:
1614  ∀m,fd,args,T,k.
1615  evalinf_funcall ge m fd args T ->
1616  forever_N step order ge tt (Callstate fd args k m) T.
1617Proof.
1618  cofix CIH_FUN.
1619  assert (∀e,m,s,T,f,k.
1620          execinf_stmt ge e m s T ->
1621          forever_N step order ge tt (State f s k e m) T).
1622  cofix CIH_STMT.
1623  intros. inv H.
1624
1625(* call none *)
1626  eapply forever_N_plus.
1627  apply plus_one. eapply step_call_none; eauto.
1628  apply CIH_FUN. eauto. traceEq.
1629(* call some *)
1630  eapply forever_N_plus.
1631  apply plus_one. eapply step_call_some; eauto.
1632  apply CIH_FUN. eauto. traceEq.
1633
1634(* seq 1 *)
1635  eapply forever_N_plus.
1636  apply plus_one. econstructor.
1637  apply CIH_STMT; eauto. traceEq.
1638(* seq 2 *)
1639  destruct (exec_stmt_steps _ _ _ _ _ _ H0 f (Kseq s2 k)) as [S1 [A1 B1]].
1640  inv B1.
1641  eapply forever_N_plus.
1642  eapply plus_left. constructor. eapply star_trans. eexact A1.
1643  apply star_one. constructor. reflexivity. reflexivity.
1644  apply CIH_STMT; eauto. traceEq.
1645
1646(* ifthenelse true *)
1647  eapply forever_N_plus.
1648  apply plus_one. eapply step_ifthenelse_true; eauto.
1649  apply CIH_STMT; eauto. traceEq.
1650(* ifthenelse false *)
1651  eapply forever_N_plus.
1652  apply plus_one. eapply step_ifthenelse_false; eauto.
1653  apply CIH_STMT; eauto. traceEq.
1654
1655(* while body *)
1656  eapply forever_N_plus.
1657  eapply plus_one. eapply step_while_true; eauto.
1658  apply CIH_STMT; eauto. traceEq.
1659(* while loop *)
1660  destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kwhile a s0 k)) as [S1 [A1 B1]].
1661  eapply forever_N_plus with (s2 := State f (Swhile a s0) k e m1).
1662  eapply plus_left. eapply step_while_true; eauto.
1663  eapply star_right. eexact A1.
1664  inv H3; inv B1; apply step_skip_or_continue_while; auto.
1665  reflexivity. reflexivity.
1666  apply CIH_STMT; eauto. traceEq.
1667
1668(* dowhile body *)
1669  eapply forever_N_plus.
1670  eapply plus_one. eapply step_dowhile.
1671  apply CIH_STMT; eauto.
1672  traceEq.
1673
1674(* dowhile loop *)
1675  destruct (exec_stmt_steps _ _ _ _ _ _ H0 f (Kdowhile a s0 k)) as [S1 [A1 B1]].
1676  eapply forever_N_plus with (s2 := State f (Sdowhile a s0) k e m1).
1677  eapply plus_left. eapply step_dowhile.
1678  eapply star_right. eexact A1.
1679  inv H1; inv B1; eapply step_skip_or_continue_dowhile_true; eauto.
1680  reflexivity. reflexivity.
1681  apply CIH_STMT. eauto.
1682  traceEq.
1683
1684(* for start 1 *)
1685  assert (a1 <> Sskip). red; intros; subst. inv H0.
1686  eapply forever_N_plus.
1687  eapply plus_one. apply step_for_start; auto.
1688  apply CIH_STMT; eauto.
1689  traceEq.
1690
1691(* for start 2 *)
1692  destruct (exec_stmt_steps _ _ _ _ _ _ H1 f (Kseq (Sfor Sskip a2 a3 s0) k)) as [S1 [A1 B1]].
1693  inv B1.
1694  eapply forever_N_plus.
1695  eapply plus_left. eapply step_for_start; eauto.
1696  eapply star_right. eexact A1.
1697  apply step_skip_seq.
1698  reflexivity. reflexivity.
1699  apply CIH_STMT; eauto.
1700  traceEq.
1701
1702(* for body *)
1703  eapply forever_N_plus.
1704  apply plus_one. eapply step_for_true; eauto.
1705  apply CIH_STMT; eauto.
1706  traceEq.
1707
1708(* for next *)
1709  destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kfor2 a2 a3 s0 k)) as [S1 [A1 B1]].
1710  eapply forever_N_plus.
1711  eapply plus_left. eapply step_for_true; eauto.
1712  eapply star_trans. eexact A1.
1713  apply star_one.
1714  inv H3; inv B1; apply step_skip_or_continue_for2; auto.
1715  reflexivity. reflexivity.
1716  apply CIH_STMT; eauto.
1717  traceEq.
1718
1719(* for body *)
1720  destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kfor2 a2 a3 s0 k)) as [S1 [A1 B1]].
1721  destruct (exec_stmt_steps _ _ _ _ _ _ H4 f (Kfor3 a2 a3 s0 k)) as [S2 [A2 B2]].
1722  inv B2.
1723  eapply forever_N_plus.
1724  eapply plus_left. eapply step_for_true; eauto.
1725  eapply star_trans. eexact A1.
1726  eapply star_left. inv H3; inv B1; apply step_skip_or_continue_for2; auto.
1727  eapply star_right. eexact A2.
1728  constructor.
1729  reflexivity. reflexivity. reflexivity. reflexivity. 
1730  apply CIH_STMT; eauto.
1731  traceEq.
1732
1733(* switch *)
1734  eapply forever_N_plus.
1735  eapply plus_one. eapply step_switch; eauto.
1736  apply CIH_STMT; eauto.
1737  traceEq.
1738
1739(* call internal *)
1740  intros. inv H0.
1741  eapply forever_N_plus.
1742  eapply plus_one. econstructor; eauto.
1743  apply H; eauto.
1744  traceEq.
1745Qed.
1746
1747Theorem bigstep_program_terminates_exec:
1748  ∀t,r. bigstep_program_terminates prog t r -> exec_program prog (Terminates t r).
1749Proof.
1750  intros. inv H. unfold ge0, m0 in *.
1751  econstructor.
1752  econstructor. eauto. eauto.
1753  apply eval_funcall_steps. eauto. red; auto.
1754  econstructor.
1755Qed.
1756
1757Theorem bigstep_program_diverges_exec:
1758  ∀T. bigstep_program_diverges prog T ->
1759  exec_program prog (Reacts T) \/
1760  exists t, exec_program prog (Diverges t) /\ traceinf_prefix t T.
1761Proof.
1762  intros. inv H.
1763  set (st := Callstate f nil Kstop m0).
1764  assert (forever step ge0 st T).
1765    eapply forever_N_forever with (order := order).
1766    red; intros. constructor; intros. red in H. elim H.
1767    eapply evalinf_funcall_forever; eauto.
1768  destruct (forever_silent_or_reactive _ _ _ _ _ _ H)
1769  as [A | [t [s' [T' [B [C D]]]]]].
1770  left. econstructor. econstructor. eauto. eauto. auto.
1771  right. exists t. split.
1772  econstructor. econstructor; eauto. eauto. auto.
1773  subst T. rewrite <- (E0_right t) at 1. apply traceinf_prefix_app. constructor.
1774Qed.
1775
1776End BIGSTEP_TO_TRANSITIONS.
1777
1778
1779
1780*)
1781
1782 
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