source: C-semantics/Csem.ma @ 175

Last change on this file since 175 was 175, checked in by campbell, 9 years ago

Add cost labels, with the semantics that the label is added to the
event trace.

File size: 61.2 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 → trace → Prop ≝
571  | eval_Econst_int:   ∀i,ty.
572      eval_expr ge e m (Expr (Econst_int i) ty) (Vint i) E0
573  | eval_Econst_float:   ∀f,ty.
574      eval_expr ge e m (Expr (Econst_float f) ty) (Vfloat f) E0
575  | eval_Elvalue: ∀a,ty,psp,loc,ofs,v,tr.
576      eval_lvalue ge e m (Expr a ty) psp loc ofs tr →
577      load_value_of_type ty m psp loc ofs = Some ? v →
578      eval_expr ge e m (Expr a ty) v tr
579  | eval_Eaddrof: ∀a,ty,psp,loc,ofs,tr.
580      eval_lvalue ge e m a psp loc ofs tr →
581      eval_expr ge e m (Expr (Eaddrof a) ty) (Vptr psp loc ofs) tr
582  | eval_Esizeof: ∀ty',ty.
583      eval_expr ge e m (Expr (Esizeof ty') ty) (Vint (repr (sizeof ty'))) E0
584  | eval_Eunop:  ∀op,a,ty,v1,v,tr.
585      eval_expr ge e m a v1 tr →
586      sem_unary_operation op v1 (typeof a) = Some ? v →
587      eval_expr ge e m (Expr (Eunop op a) ty) v tr
588  | eval_Ebinop: ∀op,a1,a2,ty,v1,v2,v,tr1,tr2.
589      eval_expr ge e m a1 v1 tr1 →
590      eval_expr ge e m a2 v2 tr2 →
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 (tr1⧺tr2)
593  | eval_Econdition_true: ∀a1,a2,a3,ty,v1,v2,tr1,tr2.
594      eval_expr ge e m a1 v1 tr1 →
595      is_true v1 (typeof a1) →
596      eval_expr ge e m a2 v2 tr2 →
597      eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v2 (tr1⧺tr2)
598  | eval_Econdition_false: ∀a1,a2,a3,ty,v1,v3,tr1,tr2.
599      eval_expr ge e m a1 v1 tr1 →
600      is_false v1 (typeof a1) →
601      eval_expr ge e m a3 v3 tr2 →
602      eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v3 (tr1⧺tr2)
603  | eval_Eorbool_1: ∀a1,a2,ty,v1,tr.
604      eval_expr ge e m a1 v1 tr →
605      is_true v1 (typeof a1) →
606      eval_expr ge e m (Expr (Eorbool a1 a2) ty) Vtrue tr
607  | eval_Eorbool_2: ∀a1,a2,ty,v1,v2,v,tr1,tr2.
608      eval_expr ge e m a1 v1 tr1 →
609      is_false v1 (typeof a1) →
610      eval_expr ge e m a2 v2 tr2 →
611      bool_of_val v2 (typeof a2) v →
612      eval_expr ge e m (Expr (Eorbool a1 a2) ty) v (tr1⧺tr2)
613  | eval_Eandbool_1: ∀a1,a2,ty,v1,tr.
614      eval_expr ge e m a1 v1 tr →
615      is_false v1 (typeof a1) →
616      eval_expr ge e m (Expr (Eandbool a1 a2) ty) Vfalse tr
617  | eval_Eandbool_2: ∀a1,a2,ty,v1,v2,v,tr1,tr2.
618      eval_expr ge e m a1 v1 tr1 →
619      is_true v1 (typeof a1) →
620      eval_expr ge e m a2 v2 tr2 →
621      bool_of_val v2 (typeof a2) v →
622      eval_expr ge e m (Expr (Eandbool a1 a2) ty) v (tr1⧺tr2)
623  | eval_Ecast:   ∀a,ty,ty',v1,v,tr.
624      eval_expr ge e m a v1 tr →
625      cast m v1 (typeof a) ty v →
626      eval_expr ge e m (Expr (Ecast ty a) ty') v tr
627  | eval_Ecost: ∀a,ty,v,l,tr.
628      eval_expr ge e m a v tr →
629      eval_expr ge e m (Expr (Ecost l a) ty) v (tr⧺Echarge l)
630
631(* * [eval_lvalue ge e m a b ofs] defines the evaluation of expression [a]
632  in l-value position.  The result is the memory location [b, ofs]
633  that contains the value of the expression [a]. *)
634
635with eval_lvalue (*(ge:genv) (e:env) (m:mem)*) : expr → memory_space → block → int → trace → Prop ≝
636  | eval_Evar_local:   ∀id,l,ty.
637      (* XXX notation? e!id*) get ??? id e = Some ? l →
638      eval_lvalue ge e m (Expr (Evar id) ty) Any l zero E0
639  | eval_Evar_global: ∀id,sp,l,ty.
640      (* XXX e!id *) get ??? id e = None ? →
641      find_symbol ?? ge id = Some ? 〈sp,l〉 →
642      eval_lvalue ge e m (Expr (Evar id) ty) sp l zero E0
643  | eval_Ederef: ∀a,ty,psp,l,ofs,tr.
644      eval_expr ge e m a (Vptr psp l ofs) tr →
645      eval_lvalue ge e m (Expr (Ederef a) ty) psp l ofs tr
646 | eval_Efield_struct:   ∀a,i,ty,psp,l,ofs,id,fList,delta,tr.
647      eval_lvalue ge e m a psp l ofs tr →
648      typeof a = Tstruct id fList →
649      field_offset i fList = OK ? delta →
650      eval_lvalue ge e m (Expr (Efield a i) ty) psp l (add ofs (repr delta)) tr
651 | eval_Efield_union:   ∀a,i,ty,psp,l,ofs,id,fList,tr.
652      eval_lvalue ge e m a psp l ofs tr →
653      typeof a = Tunion id fList →
654      eval_lvalue ge e m (Expr (Efield a i) ty) psp l ofs tr.
655
656(*
657Scheme eval_expr_ind2 := Minimality for eval_expr Sort Prop
658  with eval_lvalue_ind2 := Minimality for eval_lvalue Sort Prop.
659*)
660
661(* * [eval_exprlist ge e m al vl] evaluates a list of r-value
662  expressions [al] to their values [vl]. *)
663
664ninductive eval_exprlist (ge:genv) (e:env) (m:mem) : list expr → list val → trace → Prop ≝
665  | eval_Enil:
666      eval_exprlist ge e m (nil ?) (nil ?) E0
667  | eval_Econs:   ∀a,bl,v,vl,tr1,tr2.
668      eval_expr ge e m a v tr1 →
669      eval_exprlist ge e m bl vl tr2 →
670      eval_exprlist ge e m (a :: bl) (v :: vl) (tr1⧺tr2).
671
672(*End EXPR.*)
673
674(* * ** Transition semantics for statements and functions *)
675
676(* * Continuations *)
677
678ninductive cont: Type :=
679  | Kstop: cont
680  | Kseq: statement -> cont -> cont
681       (**r [Kseq s2 k] = after [s1] in [s1;s2] *)
682  | Kwhile: expr -> statement -> cont -> cont
683       (**r [Kwhile e s k] = after [s] in [while (e) s] *)
684  | Kdowhile: expr -> statement -> cont -> cont
685       (**r [Kdowhile e s k] = after [s] in [do s while (e)] *)
686  | Kfor2: expr -> statement -> statement -> cont -> cont
687       (**r [Kfor2 e2 e3 s k] = after [s] in [for(e1;e2;e3) s] *)
688  | Kfor3: expr -> statement -> statement -> cont -> cont
689       (**r [Kfor3 e2 e3 s k] = after [e3] in [for(e1;e2;e3) s] *)
690  | Kswitch: cont -> cont
691       (**r catches [break] statements arising out of [switch] *)
692  | Kcall: option (memory_space × block × int × type) ->   (**r where to store result *)
693           function ->                      (**r calling function *)
694           env ->                           (**r local env of calling function *)
695           cont -> cont.
696
697(* * Pop continuation until a call or stop *)
698
699nlet rec call_cont (k: cont) : cont :=
700  match k with
701  [ Kseq s k => call_cont k
702  | Kwhile e s k => call_cont k
703  | Kdowhile e s k => call_cont k
704  | Kfor2 e2 e3 s k => call_cont k
705  | Kfor3 e2 e3 s k => call_cont k
706  | Kswitch k => call_cont k
707  | _ => k
708  ].
709
710ndefinition is_call_cont : cont → Prop ≝ λk.
711  match k with
712  [ Kstop => True
713  | Kcall _ _ _ _ => True
714  | _ => False
715  ].
716
717(* * States *)
718
719ninductive state: Type :=
720  | State:
721      ∀f: function.
722      ∀s: statement.
723      ∀k: cont.
724      ∀e: env.
725      ∀m: mem.  state
726  | Callstate:
727      ∀fd: fundef.
728      ∀args: list val.
729      ∀k: cont.
730      ∀m: mem. state
731  | Returnstate:
732      ∀res: val.
733      ∀k: cont.
734      ∀m: mem. state.
735                 
736(* * Find the statement and manufacture the continuation
737  corresponding to a label *)
738
739nlet rec find_label (lbl: label) (s: statement) (k: cont)
740                    on s: option (statement × cont) :=
741  match s with
742  [ Ssequence s1 s2 =>
743      match find_label lbl s1 (Kseq s2 k) with
744      [ Some sk => Some ? sk
745      | None => find_label lbl s2 k
746      ]
747  | Sifthenelse a s1 s2 =>
748      match find_label lbl s1 k with
749      [ Some sk => Some ? sk
750      | None => find_label lbl s2 k
751      ]
752  | Swhile a s1 =>
753      find_label lbl s1 (Kwhile a s1 k)
754  | Sdowhile a s1 =>
755      find_label lbl s1 (Kdowhile a s1 k)
756  | Sfor a1 a2 a3 s1 =>
757      match find_label lbl a1 (Kseq (Sfor Sskip a2 a3 s1) k) with
758      [ Some sk => Some ? sk
759      | None =>
760          match find_label lbl s1 (Kfor2 a2 a3 s1 k) with
761          [ Some sk => Some ? sk
762          | None => find_label lbl a3 (Kfor3 a2 a3 s1 k)
763          ]
764      ]
765  | Sswitch e sl =>
766      find_label_ls lbl sl (Kswitch k)
767  | Slabel lbl' s' =>
768      match ident_eq lbl lbl' with
769      [ inl _ ⇒ Some ? 〈s', k〉
770      | inr _ ⇒ find_label lbl s' k
771      ]
772  | _ => None ?
773  ]
774
775and find_label_ls (lbl: label) (sl: labeled_statements) (k: cont)
776                    on sl: option (statement × cont) :=
777  match sl with
778  [ LSdefault s => find_label lbl s k
779  | LScase _ s sl' =>
780      match find_label lbl s (Kseq (seq_of_labeled_statement sl') k) with
781      [ Some sk => Some ? sk
782      | None => find_label_ls lbl sl' k
783      ]
784  ].
785
786(* * Transition relation *)
787
788(* XXX: note that cost labels in exprs expose a particular eval order. *)
789
790ninductive step (ge:genv) : state → trace → state → Prop ≝
791
792  | step_assign:   ∀f,a1,a2,k,e,m,psp,loc,ofs,v2,m',tr1,tr2.
793      eval_lvalue ge e m a1 psp loc ofs tr1 →
794      eval_expr ge e m a2 v2 tr2 →
795      store_value_of_type (typeof a1) m psp loc ofs v2 = Some ? m' →
796      step ge (State f (Sassign a1 a2) k e m)
797           (tr1⧺tr2) (State f Sskip k e m')
798
799  | step_call_none:   ∀f,a,al,k,e,m,vf,vargs,fd,tr1,tr2.
800      eval_expr ge e m a vf tr1 →
801      eval_exprlist ge e m al vargs tr2 →
802      find_funct ?? ge vf = Some ? fd →
803      type_of_fundef fd = typeof a →
804      step ge (State f (Scall (None ?) a al) k e m)
805           (tr1⧺tr2) (Callstate fd vargs (Kcall (None ?) f e k) m)
806
807  | step_call_some:   ∀f,lhs,a,al,k,e,m,psp,loc,ofs,vf,vargs,fd,tr1,tr2,tr3.
808      eval_lvalue ge e m lhs psp loc ofs tr1 →
809      eval_expr ge e m a vf tr2 →
810      eval_exprlist ge e m al vargs tr3 →
811      find_funct ?? ge vf = Some ? fd →
812      type_of_fundef fd = typeof a →
813      step ge (State f (Scall (Some ? lhs) a al) k e m)
814           (tr1⧺tr2⧺tr3) (Callstate fd vargs (Kcall (Some ? 〈〈〈psp, loc〉, ofs〉, typeof lhs〉) f e k) m)
815
816  | step_seq:  ∀f,s1,s2,k,e,m.
817      step ge (State f (Ssequence s1 s2) k e m)
818           E0 (State f s1 (Kseq s2 k) e m)
819  | step_skip_seq: ∀f,s,k,e,m.
820      step ge (State f Sskip (Kseq s k) e m)
821           E0 (State f s k e m)
822  | step_continue_seq: ∀f,s,k,e,m.
823      step ge (State f Scontinue (Kseq s k) e m)
824           E0 (State f Scontinue k e m)
825  | step_break_seq: ∀f,s,k,e,m.
826      step ge (State f Sbreak (Kseq s k) e m)
827           E0 (State f Sbreak k e m)
828
829  | step_ifthenelse_true:  ∀f,a,s1,s2,k,e,m,v1,tr.
830      eval_expr ge e m a v1 tr →
831      is_true v1 (typeof a) →
832      step ge (State f (Sifthenelse a s1 s2) k e m)
833           tr (State f s1 k e m)
834  | step_ifthenelse_false: ∀f,a,s1,s2,k,e,m,v1,tr.
835      eval_expr ge e m a v1 tr →
836      is_false v1 (typeof a) →
837      step ge (State f (Sifthenelse a s1 s2) k e m)
838           tr (State f s2 k e m)
839
840  | step_while_false: ∀f,a,s,k,e,m,v,tr.
841      eval_expr ge e m a v tr →
842      is_false v (typeof a) →
843      step ge (State f (Swhile a s) k e m)
844           tr (State f Sskip k e m)
845  | step_while_true: ∀f,a,s,k,e,m,v,tr.
846      eval_expr ge e m a v tr →
847      is_true v (typeof a) →
848      step ge (State f (Swhile a s) k e m)
849           tr (State f s (Kwhile a s k) e m)
850  | step_skip_or_continue_while: ∀f,x,a,s,k,e,m.
851      x = Sskip ∨ x = Scontinue →
852      step ge (State f x (Kwhile a s k) e m)
853           E0 (State f (Swhile a s) k e m)
854  | step_break_while: ∀f,a,s,k,e,m.
855      step ge (State f Sbreak (Kwhile a s k) e m)
856           E0 (State f Sskip k e m)
857
858  | step_dowhile: ∀f,a,s,k,e,m.
859      step ge (State f (Sdowhile a s) k e m)
860        E0 (State f s (Kdowhile a s k) e m)
861  | step_skip_or_continue_dowhile_false: ∀f,x,a,s,k,e,m,v,tr.
862      x = Sskip ∨ x = Scontinue →
863      eval_expr ge e m a v tr →
864      is_false v (typeof a) →
865      step ge (State f x (Kdowhile a s k) e m)
866           tr (State f Sskip k e m)
867  | step_skip_or_continue_dowhile_true: ∀f,x,a,s,k,e,m,v,tr.
868      x = Sskip ∨ x = Scontinue →
869      eval_expr ge e m a v tr →
870      is_true v (typeof a) →
871      step ge (State f x (Kdowhile a s k) e m)
872           tr (State f (Sdowhile a s) k e m)
873  | step_break_dowhile: ∀f,a,s,k,e,m.
874      step ge (State f Sbreak (Kdowhile a s k) e m)
875           E0 (State f Sskip k e m)
876
877  | step_for_start: ∀f,a1,a2,a3,s,k,e,m.
878      a1 ≠ Sskip →
879      step ge (State f (Sfor a1 a2 a3 s) k e m)
880           E0 (State f a1 (Kseq (Sfor Sskip a2 a3 s) k) e m)
881  | step_for_false: ∀f,a2,a3,s,k,e,m,v,tr.
882      eval_expr ge e m a2 v tr →
883      is_false v (typeof a2) →
884      step ge (State f (Sfor Sskip a2 a3 s) k e m)
885           tr (State f Sskip k e m)
886  | step_for_true: ∀f,a2,a3,s,k,e,m,v,tr.
887      eval_expr ge e m a2 v tr →
888      is_true v (typeof a2) →
889      step ge (State f (Sfor Sskip a2 a3 s) k e m)
890           tr (State f s (Kfor2 a2 a3 s k) e m)
891  | step_skip_or_continue_for2: ∀f,x,a2,a3,s,k,e,m.
892      x = Sskip ∨ x = Scontinue →
893      step ge (State f x (Kfor2 a2 a3 s k) e m)
894           E0 (State f a3 (Kfor3 a2 a3 s k) e m)
895  | step_break_for2: ∀f,a2,a3,s,k,e,m.
896      step ge (State f Sbreak (Kfor2 a2 a3 s k) e m)
897           E0 (State f Sskip k e m)
898  | step_skip_for3: ∀f,a2,a3,s,k,e,m.
899      step ge (State f Sskip (Kfor3 a2 a3 s k) e m)
900           E0 (State f (Sfor Sskip a2 a3 s) k e m)
901
902  | step_return_0: ∀f,k,e,m.
903      fn_return f = Tvoid →
904      step ge (State f (Sreturn (None ?)) k e m)
905           E0 (Returnstate Vundef (call_cont k) (free_list m (blocks_of_env e)))
906  | step_return_1: ∀f,a,k,e,m,v,tr.
907      fn_return f ≠ Tvoid →
908      eval_expr ge e m a v tr →
909      step ge (State f (Sreturn (Some ? a)) k e m)
910           tr (Returnstate v (call_cont k) (free_list m (blocks_of_env e)))
911  | step_skip_call: ∀f,k,e,m.
912      is_call_cont k →
913      fn_return f = Tvoid →
914      step ge (State f Sskip k e m)
915           E0 (Returnstate Vundef k (free_list m (blocks_of_env e)))
916
917  | step_switch: ∀f,a,sl,k,e,m,n,tr.
918      eval_expr ge e m a (Vint n) tr →
919      step ge (State f (Sswitch a sl) k e m)
920           tr (State f (seq_of_labeled_statement (select_switch n sl)) (Kswitch k) e m)
921  | step_skip_break_switch: ∀f,x,k,e,m.
922      x = Sskip ∨ x = Sbreak →
923      step ge (State f x (Kswitch k) e m)
924           E0 (State f Sskip k e m)
925  | step_continue_switch: ∀f,k,e,m.
926      step ge (State f Scontinue (Kswitch k) e m)
927           E0 (State f Scontinue k e m)
928
929  | step_label: ∀f,lbl,s,k,e,m.
930      step ge (State f (Slabel lbl s) k e m)
931           E0 (State f s k e m)
932
933  | step_goto: ∀f,lbl,k,e,m,s',k'.
934      find_label lbl (fn_body f) (call_cont k) = Some ? 〈s', k'〉 →
935      step ge (State f (Sgoto lbl) k e m)
936           E0 (State f s' k' e m)
937
938  | step_internal_function: ∀f,vargs,k,m,e,m1,m2.
939      alloc_variables empty_env m ((fn_params f) @ (fn_vars f)) e m1 →
940      bind_parameters e m1 (fn_params f) vargs m2 →
941      step ge (Callstate (Internal f) vargs k m)
942           E0 (State f (fn_body f) k e m2)
943
944  | step_external_function: ∀id,targs,tres,vargs,k,m,vres,t.
945      event_match (external_function id targs tres) vargs t vres →
946      step ge (Callstate (External id targs tres) vargs k m)
947            t (Returnstate vres k m)
948
949  | step_returnstate_0: ∀v,f,e,k,m.
950      step ge (Returnstate v (Kcall (None ?) f e k) m)
951           E0 (State f Sskip k e m)
952
953  | step_returnstate_1: ∀v,f,e,k,m,m',psp,loc,ofs,ty.
954      store_value_of_type ty m psp loc ofs v = Some ? m' →
955      step ge (Returnstate v (Kcall (Some ? 〈〈〈psp,loc〉, ofs〉, ty〉) f e k) m)
956           E0 (State f Sskip k e m')
957           
958  | step_cost: ∀f,lbl,s,k,e,m.
959      step ge (State f (Scost lbl s) k e m)
960           (Echarge lbl) (State f s k e m).
961(*
962(** * Alternate big-step semantics *)
963
964(** ** Big-step semantics for terminating statements and functions *)
965
966(** The execution of a statement produces an ``outcome'', indicating
967  how the execution terminated: either normally or prematurely
968  through the execution of a [break], [continue] or [return] statement. *)
969
970ninductive outcome: Type :=
971   | Out_break: outcome                 (**r terminated by [break] *)
972   | Out_continue: outcome              (**r terminated by [continue] *)
973   | Out_normal: outcome                (**r terminated normally *)
974   | Out_return: option val -> outcome. (**r terminated by [return] *)
975
976ninductive out_normal_or_continue : outcome -> Prop :=
977  | Out_normal_or_continue_N: out_normal_or_continue Out_normal
978  | Out_normal_or_continue_C: out_normal_or_continue Out_continue.
979
980ninductive out_break_or_return : outcome -> outcome -> Prop :=
981  | Out_break_or_return_B: out_break_or_return Out_break Out_normal
982  | Out_break_or_return_R: ∀ov.
983      out_break_or_return (Out_return ov) (Out_return ov).
984
985Definition outcome_switch (out: outcome) : outcome :=
986  match out with
987  | Out_break => Out_normal
988  | o => o
989  end.
990
991Definition outcome_result_value (out: outcome) (t: type) (v: val) : Prop :=
992  match out, t with
993  | Out_normal, Tvoid => v = Vundef
994  | Out_return None, Tvoid => v = Vundef
995  | Out_return (Some v'), ty => ty <> Tvoid /\ v'=v
996  | _, _ => False
997  end.
998
999(** [exec_stmt ge e m1 s t m2 out] describes the execution of
1000  the statement [s].  [out] is the outcome for this execution.
1001  [m1] is the initial memory state, [m2] the final memory state.
1002  [t] is the trace of input/output events performed during this
1003  evaluation. *)
1004
1005ninductive exec_stmt: env -> mem -> statement -> trace -> mem -> outcome -> Prop :=
1006  | exec_Sskip:   ∀e,m.
1007      exec_stmt e m Sskip
1008               E0 m Out_normal
1009  | exec_Sassign:   ∀e,m,a1,a2,loc,ofs,v2,m'.
1010      eval_lvalue e m a1 loc ofs ->
1011      eval_expr e m a2 v2 ->
1012      store_value_of_type (typeof a1) m loc ofs v2 = Some m' ->
1013      exec_stmt e m (Sassign a1 a2)
1014               E0 m' Out_normal
1015  | exec_Scall_none:   ∀e,m,a,al,vf,vargs,f,t,m',vres.
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      exec_stmt e m (Scall None a al)
1022                t m' Out_normal
1023  | exec_Scall_some:   ∀e,m,lhs,a,al,loc,ofs,vf,vargs,f,t,m',vres,m''.
1024      eval_lvalue e m lhs loc ofs ->
1025      eval_expr e m a vf ->
1026      eval_exprlist e m al vargs ->
1027      Genv.find_funct ge vf = Some f ->
1028      type_of_fundef f = typeof a ->
1029      eval_funcall m f vargs t m' vres ->
1030      store_value_of_type (typeof lhs) m' loc ofs vres = Some m'' ->
1031      exec_stmt e m (Scall (Some lhs) a al)
1032                t m'' Out_normal
1033  | exec_Sseq_1:   ∀e,m,s1,s2,t1,m1,t2,m2,out.
1034      exec_stmt e m s1 t1 m1 Out_normal ->
1035      exec_stmt e m1 s2 t2 m2 out ->
1036      exec_stmt e m (Ssequence s1 s2)
1037                (t1 ** t2) m2 out
1038  | exec_Sseq_2:   ∀e,m,s1,s2,t1,m1,out.
1039      exec_stmt e m s1 t1 m1 out ->
1040      out <> Out_normal ->
1041      exec_stmt e m (Ssequence s1 s2)
1042                t1 m1 out
1043  | exec_Sifthenelse_true: ∀e,m,a,s1,s2,v1,t,m',out.
1044      eval_expr e m a v1 ->
1045      is_true v1 (typeof a) ->
1046      exec_stmt e m s1 t m' out ->
1047      exec_stmt e m (Sifthenelse a s1 s2)
1048                t m' out
1049  | exec_Sifthenelse_false: ∀e,m,a,s1,s2,v1,t,m',out.
1050      eval_expr e m a v1 ->
1051      is_false v1 (typeof a) ->
1052      exec_stmt e m s2 t m' out ->
1053      exec_stmt e m (Sifthenelse a s1 s2)
1054                t m' out
1055  | exec_Sreturn_none:   ∀e,m.
1056      exec_stmt e m (Sreturn None)
1057               E0 m (Out_return None)
1058  | exec_Sreturn_some: ∀e,m,a,v.
1059      eval_expr e m a v ->
1060      exec_stmt e m (Sreturn (Some a))
1061               E0 m (Out_return (Some v))
1062  | exec_Sbreak:   ∀e,m.
1063      exec_stmt e m Sbreak
1064               E0 m Out_break
1065  | exec_Scontinue:   ∀e,m.
1066      exec_stmt e m Scontinue
1067               E0 m Out_continue
1068  | exec_Swhile_false: ∀e,m,a,s,v.
1069      eval_expr e m a v ->
1070      is_false v (typeof a) ->
1071      exec_stmt e m (Swhile a s)
1072               E0 m Out_normal
1073  | exec_Swhile_stop: ∀e,m,a,v,s,t,m',out',out.
1074      eval_expr e m a v ->
1075      is_true v (typeof a) ->
1076      exec_stmt e m s t m' out' ->
1077      out_break_or_return out' out ->
1078      exec_stmt e m (Swhile a s)
1079                t m' out
1080  | exec_Swhile_loop: ∀e,m,a,s,v,t1,m1,out1,t2,m2,out.
1081      eval_expr e m a v ->
1082      is_true v (typeof a) ->
1083      exec_stmt e m s t1 m1 out1 ->
1084      out_normal_or_continue out1 ->
1085      exec_stmt e m1 (Swhile a s) t2 m2 out ->
1086      exec_stmt e m (Swhile a s)
1087                (t1 ** t2) m2 out
1088  | exec_Sdowhile_false: ∀e,m,s,a,t,m1,out1,v.
1089      exec_stmt e m s t m1 out1 ->
1090      out_normal_or_continue out1 ->
1091      eval_expr e m1 a v ->
1092      is_false v (typeof a) ->
1093      exec_stmt e m (Sdowhile a s)
1094                t m1 Out_normal
1095  | exec_Sdowhile_stop: ∀e,m,s,a,t,m1,out1,out.
1096      exec_stmt e m s t m1 out1 ->
1097      out_break_or_return out1 out ->
1098      exec_stmt e m (Sdowhile a s)
1099                t m1 out
1100  | exec_Sdowhile_loop: ∀e,m,s,a,m1,m2,t1,t2,out,out1,v.
1101      exec_stmt e m s t1 m1 out1 ->
1102      out_normal_or_continue out1 ->
1103      eval_expr e m1 a v ->
1104      is_true v (typeof a) ->
1105      exec_stmt e m1 (Sdowhile a s) t2 m2 out ->
1106      exec_stmt e m (Sdowhile a s)
1107                (t1 ** t2) m2 out
1108  | exec_Sfor_start: ∀e,m,s,a1,a2,a3,out,m1,m2,t1,t2.
1109      a1 <> Sskip ->
1110      exec_stmt e m a1 t1 m1 Out_normal ->
1111      exec_stmt e m1 (Sfor Sskip a2 a3 s) t2 m2 out ->
1112      exec_stmt e m (Sfor a1 a2 a3 s)
1113                (t1 ** t2) m2 out
1114  | exec_Sfor_false: ∀e,m,s,a2,a3,v.
1115      eval_expr e m a2 v ->
1116      is_false v (typeof a2) ->
1117      exec_stmt e m (Sfor Sskip a2 a3 s)
1118               E0 m Out_normal
1119  | exec_Sfor_stop: ∀e,m,s,a2,a3,v,m1,t,out1,out.
1120      eval_expr e m a2 v ->
1121      is_true v (typeof a2) ->
1122      exec_stmt e m s t m1 out1 ->
1123      out_break_or_return out1 out ->
1124      exec_stmt e m (Sfor Sskip a2 a3 s)
1125                t m1 out
1126  | exec_Sfor_loop: ∀e,m,s,a2,a3,v,m1,m2,m3,t1,t2,t3,out1,out.
1127      eval_expr e m a2 v ->
1128      is_true v (typeof a2) ->
1129      exec_stmt e m s t1 m1 out1 ->
1130      out_normal_or_continue out1 ->
1131      exec_stmt e m1 a3 t2 m2 Out_normal ->
1132      exec_stmt e m2 (Sfor Sskip a2 a3 s) t3 m3 out ->
1133      exec_stmt e m (Sfor Sskip a2 a3 s)
1134                (t1 ** t2 ** t3) m3 out
1135  | exec_Sswitch:   ∀e,m,a,t,n,sl,m1,out.
1136      eval_expr e m a (Vint n) ->
1137      exec_stmt e m (seq_of_labeled_statement (select_switch n sl)) t m1 out ->
1138      exec_stmt e m (Sswitch a sl)
1139                t m1 (outcome_switch out)
1140
1141(** [eval_funcall m1 fd args t m2 res] describes the invocation of
1142  function [fd] with arguments [args].  [res] is the value returned
1143  by the call.  *)
1144
1145with eval_funcall: mem -> fundef -> list val -> trace -> mem -> val -> Prop :=
1146  | eval_funcall_internal: ∀m,f,vargs,t,e,m1,m2,m3,out,vres.
1147      alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
1148      bind_parameters e m1 f.(fn_params) vargs m2 ->
1149      exec_stmt e m2 f.(fn_body) t m3 out ->
1150      outcome_result_value out f.(fn_return) vres ->
1151      eval_funcall m (Internal f) vargs t (Mem.free_list m3 (blocks_of_env e)) vres
1152  | eval_funcall_external: ∀m,id,targs,tres,vargs,t,vres.
1153      event_match (external_function id targs tres) vargs t vres ->
1154      eval_funcall m (External id targs tres) vargs t m vres.
1155
1156Scheme exec_stmt_ind2 := Minimality for exec_stmt Sort Prop
1157  with eval_funcall_ind2 := Minimality for eval_funcall Sort Prop.
1158
1159(** ** Big-step semantics for diverging statements and functions *)
1160
1161(** Coinductive semantics for divergence.
1162  [execinf_stmt ge e m s t] holds if the execution of statement [s]
1163  diverges, i.e. loops infinitely.  [t] is the possibly infinite
1164  trace of observable events performed during the execution. *)
1165
1166Coninductive execinf_stmt: env -> mem -> statement -> traceinf -> Prop :=
1167  | execinf_Scall_none:   ∀e,m,a,al,vf,vargs,f,t.
1168      eval_expr e m a vf ->
1169      eval_exprlist e m al vargs ->
1170      Genv.find_funct ge vf = Some f ->
1171      type_of_fundef f = typeof a ->
1172      evalinf_funcall m f vargs t ->
1173      execinf_stmt e m (Scall None a al) t
1174  | execinf_Scall_some:   ∀e,m,lhs,a,al,loc,ofs,vf,vargs,f,t.
1175      eval_lvalue e m lhs loc ofs ->
1176      eval_expr e m a vf ->
1177      eval_exprlist e m al vargs ->
1178      Genv.find_funct ge vf = Some f ->
1179      type_of_fundef f = typeof a ->
1180      evalinf_funcall m f vargs t ->
1181      execinf_stmt e m (Scall (Some lhs) a al) t
1182  | execinf_Sseq_1:   ∀e,m,s1,s2,t.
1183      execinf_stmt e m s1 t ->
1184      execinf_stmt e m (Ssequence s1 s2) t
1185  | execinf_Sseq_2:   ∀e,m,s1,s2,t1,m1,t2.
1186      exec_stmt e m s1 t1 m1 Out_normal ->
1187      execinf_stmt e m1 s2 t2 ->
1188      execinf_stmt e m (Ssequence s1 s2) (t1 *** t2)
1189  | execinf_Sifthenelse_true: ∀e,m,a,s1,s2,v1,t.
1190      eval_expr e m a v1 ->
1191      is_true v1 (typeof a) ->
1192      execinf_stmt e m s1 t ->
1193      execinf_stmt e m (Sifthenelse a s1 s2) t
1194  | execinf_Sifthenelse_false: ∀e,m,a,s1,s2,v1,t.
1195      eval_expr e m a v1 ->
1196      is_false v1 (typeof a) ->
1197      execinf_stmt e m s2 t ->
1198      execinf_stmt e m (Sifthenelse a s1 s2) t
1199  | execinf_Swhile_body: ∀e,m,a,v,s,t.
1200      eval_expr e m a v ->
1201      is_true v (typeof a) ->
1202      execinf_stmt e m s t ->
1203      execinf_stmt e m (Swhile a s) t
1204  | execinf_Swhile_loop: ∀e,m,a,s,v,t1,m1,out1,t2.
1205      eval_expr e m a v ->
1206      is_true v (typeof a) ->
1207      exec_stmt e m s t1 m1 out1 ->
1208      out_normal_or_continue out1 ->
1209      execinf_stmt e m1 (Swhile a s) t2 ->
1210      execinf_stmt e m (Swhile a s) (t1 *** t2)
1211  | execinf_Sdowhile_body: ∀e,m,s,a,t.
1212      execinf_stmt e m s t ->
1213      execinf_stmt e m (Sdowhile a s) t
1214  | execinf_Sdowhile_loop: ∀e,m,s,a,m1,t1,t2,out1,v.
1215      exec_stmt e m s t1 m1 out1 ->
1216      out_normal_or_continue out1 ->
1217      eval_expr e m1 a v ->
1218      is_true v (typeof a) ->
1219      execinf_stmt e m1 (Sdowhile a s) t2 ->
1220      execinf_stmt e m (Sdowhile a s) (t1 *** t2)
1221  | execinf_Sfor_start_1: ∀e,m,s,a1,a2,a3,t.
1222      execinf_stmt e m a1 t ->
1223      execinf_stmt e m (Sfor a1 a2 a3 s) t
1224  | execinf_Sfor_start_2: ∀e,m,s,a1,a2,a3,m1,t1,t2.
1225      a1 <> Sskip ->
1226      exec_stmt e m a1 t1 m1 Out_normal ->
1227      execinf_stmt e m1 (Sfor Sskip a2 a3 s) t2 ->
1228      execinf_stmt e m (Sfor a1 a2 a3 s) (t1 *** t2)
1229  | execinf_Sfor_body: ∀e,m,s,a2,a3,v,t.
1230      eval_expr e m a2 v ->
1231      is_true v (typeof a2) ->
1232      execinf_stmt e m s t ->
1233      execinf_stmt e m (Sfor Sskip a2 a3 s) t
1234  | execinf_Sfor_next: ∀e,m,s,a2,a3,v,m1,t1,t2,out1.
1235      eval_expr e m a2 v ->
1236      is_true v (typeof a2) ->
1237      exec_stmt e m s t1 m1 out1 ->
1238      out_normal_or_continue out1 ->
1239      execinf_stmt e m1 a3 t2 ->
1240      execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2)
1241  | execinf_Sfor_loop: ∀e,m,s,a2,a3,v,m1,m2,t1,t2,t3,out1.
1242      eval_expr e m a2 v ->
1243      is_true v (typeof a2) ->
1244      exec_stmt e m s t1 m1 out1 ->
1245      out_normal_or_continue out1 ->
1246      exec_stmt e m1 a3 t2 m2 Out_normal ->
1247      execinf_stmt e m2 (Sfor Sskip a2 a3 s) t3 ->
1248      execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2 *** t3)
1249  | execinf_Sswitch:   ∀e,m,a,t,n,sl.
1250      eval_expr e m a (Vint n) ->
1251      execinf_stmt e m (seq_of_labeled_statement (select_switch n sl)) t ->
1252      execinf_stmt e m (Sswitch a sl) t
1253
1254(** [evalinf_funcall ge m fd args t] holds if the invocation of function
1255    [fd] on arguments [args] diverges, with observable trace [t]. *)
1256
1257with evalinf_funcall: mem -> fundef -> list val -> traceinf -> Prop :=
1258  | evalinf_funcall_internal: ∀m,f,vargs,t,e,m1,m2.
1259      alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
1260      bind_parameters e m1 f.(fn_params) vargs m2 ->
1261      execinf_stmt e m2 f.(fn_body) t ->
1262      evalinf_funcall m (Internal f) vargs t.
1263
1264End SEMANTICS.
1265*)
1266(* * * Whole-program semantics *)
1267
1268(* * Execution of whole programs are described as sequences of transitions
1269  from an initial state to a final state.  An initial state is a [Callstate]
1270  corresponding to the invocation of the ``main'' function of the program
1271  without arguments and with an empty continuation. *)
1272
1273ninductive initial_state (p: program): state -> Prop :=
1274  | initial_state_intro: ∀b,f.
1275      let ge := globalenv Genv ?? p in
1276      let m0 := init_mem Genv ?? p in
1277      find_symbol ?? ge (prog_main ?? p) = Some ? 〈Code,b〉 ->
1278      find_funct_ptr ?? ge b = Some ? f ->
1279      initial_state p (Callstate f (nil ?) Kstop m0).
1280
1281(* * A final state is a [Returnstate] with an empty continuation. *)
1282
1283ninductive final_state: state -> int -> Prop :=
1284  | final_state_intro: ∀r,m.
1285      final_state (Returnstate (Vint r) Kstop m) r.
1286
1287(* * Execution of a whole program: [exec_program p beh]
1288  holds if the application of [p]'s main function to no arguments
1289  in the initial memory state for [p] has [beh] as observable
1290  behavior. *)
1291
1292ndefinition exec_program : program → program_behavior → Prop ≝ λp,beh.
1293  program_behaves (mk_transrel ?? step) (initial_state p) final_state (globalenv ??? p) beh.
1294(*
1295(** Big-step execution of a whole program.  *)
1296
1297ninductive bigstep_program_terminates (p: program): trace -> int -> Prop :=
1298  | bigstep_program_terminates_intro: ∀b,f,m1,t,r.
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      eval_funcall ge m0 f nil t m1 (Vint r) ->
1304      bigstep_program_terminates p t r.
1305
1306ninductive bigstep_program_diverges (p: program): traceinf -> Prop :=
1307  | bigstep_program_diverges_intro: ∀b,f,t.
1308      let ge := Genv.globalenv p in
1309      let m0 := Genv.init_mem p in
1310      Genv.find_symbol ge p.(prog_main) = Some b ->
1311      Genv.find_funct_ptr ge b = Some f ->
1312      evalinf_funcall ge m0 f nil t ->
1313      bigstep_program_diverges p t.
1314
1315(** * Implication from big-step semantics to transition semantics *)
1316
1317Section BIGSTEP_TO_TRANSITIONS.
1318
1319Variable prog: program.
1320Let ge : genv := Genv.globalenv prog.
1321
1322Definition exec_stmt_eval_funcall_ind
1323  (PS: env -> mem -> statement -> trace -> mem -> outcome -> Prop)
1324  (PF: mem -> fundef -> list val -> trace -> mem -> val -> Prop) :=
1325  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 =>
1326  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)
1327       (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).
1328
1329ninductive outcome_state_match
1330       (e: env) (m: mem) (f: function) (k: cont): outcome -> state -> Prop :=
1331  | osm_normal:
1332      outcome_state_match e m f k Out_normal (State f Sskip k e m)
1333  | osm_break:
1334      outcome_state_match e m f k Out_break (State f Sbreak k e m)
1335  | osm_continue:
1336      outcome_state_match e m f k Out_continue (State f Scontinue k e m)
1337  | osm_return_none: ∀k'.
1338      call_cont k' = call_cont k ->
1339      outcome_state_match e m f k
1340        (Out_return None) (State f (Sreturn None) k' e m)
1341  | osm_return_some: ∀a,v,k'.
1342      call_cont k' = call_cont k ->
1343      eval_expr ge e m a v ->
1344      outcome_state_match e m f k
1345        (Out_return (Some v)) (State f (Sreturn (Some a)) k' e m).
1346
1347Lemma is_call_cont_call_cont:
1348  ∀k. is_call_cont k -> call_cont k = k.
1349Proof.
1350  destruct k; simpl; intros; contradiction || auto.
1351Qed.
1352
1353Lemma exec_stmt_eval_funcall_steps:
1354  (∀e,m,s,t,m',out.
1355   exec_stmt ge e m s t m' out ->
1356   ∀f,k. exists S,
1357   star step ge (State f s k e m) t S
1358   /\ outcome_state_match e m' f k out S)
1359/\
1360  (∀m,fd,args,t,m',res.
1361   eval_funcall ge m fd args t m' res ->
1362   ∀k.
1363   is_call_cont k ->
1364   star step ge (Callstate fd args k m) t (Returnstate res k m')).
1365Proof.
1366  apply exec_stmt_eval_funcall_ind; intros.
1367
1368(* skip *)
1369  econstructor; split. apply star_refl. constructor.
1370
1371(* assign *)
1372  econstructor; split. apply star_one. econstructor; eauto. constructor.
1373
1374(* call none *)
1375  econstructor; split.
1376  eapply star_left. econstructor; eauto.
1377  eapply star_right. apply H4. simpl; auto. econstructor. reflexivity. traceEq.
1378  constructor.
1379
1380(* call some *)
1381  econstructor; split.
1382  eapply star_left. econstructor; eauto.
1383  eapply star_right. apply H5. simpl; auto. econstructor; eauto. reflexivity. traceEq.
1384  constructor.
1385
1386(* sequence 2 *)
1387  destruct (H0 f (Kseq s2 k)) as [S1 [A1 B1]]. inv B1.
1388  destruct (H2 f k) as [S2 [A2 B2]].
1389  econstructor; split.
1390  eapply star_left. econstructor.
1391  eapply star_trans. eexact A1.
1392  eapply star_left. constructor. eexact A2.
1393  reflexivity. reflexivity. traceEq.
1394  auto.
1395
1396(* sequence 1 *)
1397  destruct (H0 f (Kseq s2 k)) as [S1 [A1 B1]].
1398  set (S2 :=
1399    match out with
1400    | Out_break => State f Sbreak k e m1
1401    | Out_continue => State f Scontinue k e m1
1402    | _ => S1
1403    end).
1404  exists S2; split.
1405  eapply star_left. econstructor.
1406  eapply star_trans. eexact A1.
1407  unfold S2; inv B1.
1408    congruence.
1409    apply star_one. apply step_break_seq.
1410    apply star_one. apply step_continue_seq.
1411    apply star_refl.
1412    apply star_refl.
1413  reflexivity. traceEq.
1414  unfold S2; inv B1; congruence || econstructor; eauto.
1415
1416(* ifthenelse true *)
1417  destruct (H2 f k) as [S1 [A1 B1]].
1418  exists S1; split.
1419  eapply star_left. eapply step_ifthenelse_true; eauto. eexact A1. traceEq.
1420  auto.
1421
1422(* ifthenelse false *)
1423  destruct (H2 f k) as [S1 [A1 B1]].
1424  exists S1; split.
1425  eapply star_left. eapply step_ifthenelse_false; eauto. eexact A1. traceEq.
1426  auto.
1427
1428(* return none *)
1429  econstructor; split. apply star_refl. constructor. auto.
1430
1431(* return some *)
1432  econstructor; split. apply star_refl. econstructor; eauto.
1433
1434(* break *)
1435  econstructor; split. apply star_refl. constructor.
1436
1437(* continue *)
1438  econstructor; split. apply star_refl. constructor.
1439
1440(* while false *)
1441  econstructor; split.
1442  apply star_one. eapply step_while_false; eauto.
1443  constructor.
1444
1445(* while stop *)
1446  destruct (H2 f (Kwhile a s k)) as [S1 [A1 B1]].
1447  set (S2 :=
1448    match out' with
1449    | Out_break => State f Sskip k e m'
1450    | _ => S1
1451    end).
1452  exists S2; split.
1453  eapply star_left. eapply step_while_true; eauto.
1454  eapply star_trans. eexact A1.
1455  unfold S2. inversion H3; subst.
1456  inv B1. apply star_one. constructor.   
1457  apply star_refl.
1458  reflexivity. traceEq.
1459  unfold S2. inversion H3; subst. constructor. inv B1; econstructor; eauto.
1460
1461(* while loop *)
1462  destruct (H2 f (Kwhile a s k)) as [S1 [A1 B1]].
1463  destruct (H5 f k) as [S2 [A2 B2]].
1464  exists S2; split.
1465  eapply star_left. eapply step_while_true; eauto.
1466  eapply star_trans. eexact A1.
1467  eapply star_left.
1468  inv H3; inv B1; apply step_skip_or_continue_while; auto.
1469  eexact A2.
1470  reflexivity. reflexivity. traceEq.
1471  auto.
1472
1473(* dowhile false *)
1474  destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]].
1475  exists (State f Sskip k e m1); split.
1476  eapply star_left. constructor.
1477  eapply star_right. eexact A1.
1478  inv H1; inv B1; eapply step_skip_or_continue_dowhile_false; eauto.
1479  reflexivity. traceEq.
1480  constructor.
1481
1482(* dowhile stop *)
1483  destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]].
1484  set (S2 :=
1485    match out1 with
1486    | Out_break => State f Sskip k e m1
1487    | _ => S1
1488    end).
1489  exists S2; split.
1490  eapply star_left. apply step_dowhile.
1491  eapply star_trans. eexact A1.
1492  unfold S2. inversion H1; subst.
1493  inv B1. apply star_one. constructor.
1494  apply star_refl.
1495  reflexivity. traceEq.
1496  unfold S2. inversion H1; subst. constructor. inv B1; econstructor; eauto.
1497
1498(* dowhile loop *)
1499  destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]].
1500  destruct (H5 f k) as [S2 [A2 B2]].
1501  exists S2; split.
1502  eapply star_left. apply step_dowhile.
1503  eapply star_trans. eexact A1.
1504  eapply star_left.
1505  inv H1; inv B1; eapply step_skip_or_continue_dowhile_true; eauto.
1506  eexact A2.
1507  reflexivity. reflexivity. traceEq.
1508  auto.
1509
1510(* for start *)
1511  destruct (H1 f (Kseq (Sfor Sskip a2 a3 s) k)) as [S1 [A1 B1]]. inv B1.
1512  destruct (H3 f k) as [S2 [A2 B2]].
1513  exists S2; split.
1514  eapply star_left. apply step_for_start; auto.   
1515  eapply star_trans. eexact A1.
1516  eapply star_left. constructor. eexact A2.
1517  reflexivity. reflexivity. traceEq.
1518  auto.
1519
1520(* for false *)
1521  econstructor; split.
1522  eapply star_one. eapply step_for_false; eauto.
1523  constructor.
1524
1525(* for stop *)
1526  destruct (H2 f (Kfor2 a2 a3 s k)) as [S1 [A1 B1]].
1527  set (S2 :=
1528    match out1 with
1529    | Out_break => State f Sskip k e m1
1530    | _ => S1
1531    end).
1532  exists S2; split.
1533  eapply star_left. eapply step_for_true; eauto.
1534  eapply star_trans. eexact A1.
1535  unfold S2. inversion H3; subst.
1536  inv B1. apply star_one. constructor.
1537  apply star_refl.
1538  reflexivity. traceEq.
1539  unfold S2. inversion H3; subst. constructor. inv B1; econstructor; eauto.
1540
1541(* for loop *)
1542  destruct (H2 f (Kfor2 a2 a3 s k)) as [S1 [A1 B1]].
1543  destruct (H5 f (Kfor3 a2 a3 s k)) as [S2 [A2 B2]]. inv B2.
1544  destruct (H7 f k) as [S3 [A3 B3]].
1545  exists S3; split.
1546  eapply star_left. eapply step_for_true; eauto.
1547  eapply star_trans. eexact A1.
1548  eapply star_trans with (s2 := State f a3 (Kfor3 a2 a3 s k) e m1).
1549  inv H3; inv B1.
1550  apply star_one. constructor. auto.
1551  apply star_one. constructor. auto.
1552  eapply star_trans. eexact A2.
1553  eapply star_left. constructor.
1554  eexact A3.
1555  reflexivity. reflexivity. reflexivity. reflexivity. traceEq.
1556  auto.
1557
1558(* switch *)
1559  destruct (H1 f (Kswitch k)) as [S1 [A1 B1]].
1560  set (S2 :=
1561    match out with
1562    | Out_normal => State f Sskip k e m1
1563    | Out_break => State f Sskip k e m1
1564    | Out_continue => State f Scontinue k e m1
1565    | _ => S1
1566    end).
1567  exists S2; split.
1568  eapply star_left. eapply step_switch; eauto.
1569  eapply star_trans. eexact A1.
1570  unfold S2; inv B1.
1571    apply star_one. constructor. auto.
1572    apply star_one. constructor. auto.
1573    apply star_one. constructor.
1574    apply star_refl.
1575    apply star_refl.
1576  reflexivity. traceEq.
1577  unfold S2. inv B1; simpl; econstructor; eauto.
1578
1579(* call internal *)
1580  destruct (H2 f k) as [S1 [A1 B1]].
1581  eapply star_left. eapply step_internal_function; eauto.
1582  eapply star_right. eexact A1.
1583  inv B1; simpl in H3; try contradiction.
1584  (* Out_normal *)
1585  assert (fn_return f = Tvoid /\ vres = Vundef).
1586    destruct (fn_return f); auto || contradiction.
1587  destruct H5. subst vres. apply step_skip_call; auto.
1588  (* Out_return None *)
1589  assert (fn_return f = Tvoid /\ vres = Vundef).
1590    destruct (fn_return f); auto || contradiction.
1591  destruct H6. subst vres.
1592  rewrite <- (is_call_cont_call_cont k H4). rewrite <- H5.
1593  apply step_return_0; auto.
1594  (* Out_return Some *)
1595  destruct H3. subst vres.
1596  rewrite <- (is_call_cont_call_cont k H4). rewrite <- H5.
1597  eapply step_return_1; eauto.
1598  reflexivity. traceEq.
1599
1600(* call external *)
1601  apply star_one. apply step_external_function; auto.
1602Qed.
1603
1604Lemma exec_stmt_steps:
1605   ∀e,m,s,t,m',out.
1606   exec_stmt ge e m s t m' out ->
1607   ∀f,k. exists S,
1608   star step ge (State f s k e m) t S
1609   /\ outcome_state_match e m' f k out S.
1610Proof (proj1 exec_stmt_eval_funcall_steps).
1611
1612Lemma eval_funcall_steps:
1613   ∀m,fd,args,t,m',res.
1614   eval_funcall ge m fd args t m' res ->
1615   ∀k.
1616   is_call_cont k ->
1617   star step ge (Callstate fd args k m) t (Returnstate res k m').
1618Proof (proj2 exec_stmt_eval_funcall_steps).
1619
1620Definition order (x y: unit) := False.
1621
1622Lemma evalinf_funcall_forever:
1623  ∀m,fd,args,T,k.
1624  evalinf_funcall ge m fd args T ->
1625  forever_N step order ge tt (Callstate fd args k m) T.
1626Proof.
1627  cofix CIH_FUN.
1628  assert (∀e,m,s,T,f,k.
1629          execinf_stmt ge e m s T ->
1630          forever_N step order ge tt (State f s k e m) T).
1631  cofix CIH_STMT.
1632  intros. inv H.
1633
1634(* call none *)
1635  eapply forever_N_plus.
1636  apply plus_one. eapply step_call_none; eauto.
1637  apply CIH_FUN. eauto. traceEq.
1638(* call some *)
1639  eapply forever_N_plus.
1640  apply plus_one. eapply step_call_some; eauto.
1641  apply CIH_FUN. eauto. traceEq.
1642
1643(* seq 1 *)
1644  eapply forever_N_plus.
1645  apply plus_one. econstructor.
1646  apply CIH_STMT; eauto. traceEq.
1647(* seq 2 *)
1648  destruct (exec_stmt_steps _ _ _ _ _ _ H0 f (Kseq s2 k)) as [S1 [A1 B1]].
1649  inv B1.
1650  eapply forever_N_plus.
1651  eapply plus_left. constructor. eapply star_trans. eexact A1.
1652  apply star_one. constructor. reflexivity. reflexivity.
1653  apply CIH_STMT; eauto. traceEq.
1654
1655(* ifthenelse true *)
1656  eapply forever_N_plus.
1657  apply plus_one. eapply step_ifthenelse_true; eauto.
1658  apply CIH_STMT; eauto. traceEq.
1659(* ifthenelse false *)
1660  eapply forever_N_plus.
1661  apply plus_one. eapply step_ifthenelse_false; eauto.
1662  apply CIH_STMT; eauto. traceEq.
1663
1664(* while body *)
1665  eapply forever_N_plus.
1666  eapply plus_one. eapply step_while_true; eauto.
1667  apply CIH_STMT; eauto. traceEq.
1668(* while loop *)
1669  destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kwhile a s0 k)) as [S1 [A1 B1]].
1670  eapply forever_N_plus with (s2 := State f (Swhile a s0) k e m1).
1671  eapply plus_left. eapply step_while_true; eauto.
1672  eapply star_right. eexact A1.
1673  inv H3; inv B1; apply step_skip_or_continue_while; auto.
1674  reflexivity. reflexivity.
1675  apply CIH_STMT; eauto. traceEq.
1676
1677(* dowhile body *)
1678  eapply forever_N_plus.
1679  eapply plus_one. eapply step_dowhile.
1680  apply CIH_STMT; eauto.
1681  traceEq.
1682
1683(* dowhile loop *)
1684  destruct (exec_stmt_steps _ _ _ _ _ _ H0 f (Kdowhile a s0 k)) as [S1 [A1 B1]].
1685  eapply forever_N_plus with (s2 := State f (Sdowhile a s0) k e m1).
1686  eapply plus_left. eapply step_dowhile.
1687  eapply star_right. eexact A1.
1688  inv H1; inv B1; eapply step_skip_or_continue_dowhile_true; eauto.
1689  reflexivity. reflexivity.
1690  apply CIH_STMT. eauto.
1691  traceEq.
1692
1693(* for start 1 *)
1694  assert (a1 <> Sskip). red; intros; subst. inv H0.
1695  eapply forever_N_plus.
1696  eapply plus_one. apply step_for_start; auto.
1697  apply CIH_STMT; eauto.
1698  traceEq.
1699
1700(* for start 2 *)
1701  destruct (exec_stmt_steps _ _ _ _ _ _ H1 f (Kseq (Sfor Sskip a2 a3 s0) k)) as [S1 [A1 B1]].
1702  inv B1.
1703  eapply forever_N_plus.
1704  eapply plus_left. eapply step_for_start; eauto.
1705  eapply star_right. eexact A1.
1706  apply step_skip_seq.
1707  reflexivity. reflexivity.
1708  apply CIH_STMT; eauto.
1709  traceEq.
1710
1711(* for body *)
1712  eapply forever_N_plus.
1713  apply plus_one. eapply step_for_true; eauto.
1714  apply CIH_STMT; eauto.
1715  traceEq.
1716
1717(* for next *)
1718  destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kfor2 a2 a3 s0 k)) as [S1 [A1 B1]].
1719  eapply forever_N_plus.
1720  eapply plus_left. eapply step_for_true; eauto.
1721  eapply star_trans. eexact A1.
1722  apply star_one.
1723  inv H3; inv B1; apply step_skip_or_continue_for2; auto.
1724  reflexivity. reflexivity.
1725  apply CIH_STMT; eauto.
1726  traceEq.
1727
1728(* for body *)
1729  destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kfor2 a2 a3 s0 k)) as [S1 [A1 B1]].
1730  destruct (exec_stmt_steps _ _ _ _ _ _ H4 f (Kfor3 a2 a3 s0 k)) as [S2 [A2 B2]].
1731  inv B2.
1732  eapply forever_N_plus.
1733  eapply plus_left. eapply step_for_true; eauto.
1734  eapply star_trans. eexact A1.
1735  eapply star_left. inv H3; inv B1; apply step_skip_or_continue_for2; auto.
1736  eapply star_right. eexact A2.
1737  constructor.
1738  reflexivity. reflexivity. reflexivity. reflexivity. 
1739  apply CIH_STMT; eauto.
1740  traceEq.
1741
1742(* switch *)
1743  eapply forever_N_plus.
1744  eapply plus_one. eapply step_switch; eauto.
1745  apply CIH_STMT; eauto.
1746  traceEq.
1747
1748(* call internal *)
1749  intros. inv H0.
1750  eapply forever_N_plus.
1751  eapply plus_one. econstructor; eauto.
1752  apply H; eauto.
1753  traceEq.
1754Qed.
1755
1756Theorem bigstep_program_terminates_exec:
1757  ∀t,r. bigstep_program_terminates prog t r -> exec_program prog (Terminates t r).
1758Proof.
1759  intros. inv H. unfold ge0, m0 in *.
1760  econstructor.
1761  econstructor. eauto. eauto.
1762  apply eval_funcall_steps. eauto. red; auto.
1763  econstructor.
1764Qed.
1765
1766Theorem bigstep_program_diverges_exec:
1767  ∀T. bigstep_program_diverges prog T ->
1768  exec_program prog (Reacts T) \/
1769  exists t, exec_program prog (Diverges t) /\ traceinf_prefix t T.
1770Proof.
1771  intros. inv H.
1772  set (st := Callstate f nil Kstop m0).
1773  assert (forever step ge0 st T).
1774    eapply forever_N_forever with (order := order).
1775    red; intros. constructor; intros. red in H. elim H.
1776    eapply evalinf_funcall_forever; eauto.
1777  destruct (forever_silent_or_reactive _ _ _ _ _ _ H)
1778  as [A | [t [s' [T' [B [C D]]]]]].
1779  left. econstructor. econstructor. eauto. eauto. auto.
1780  right. exists t. split.
1781  econstructor. econstructor; eauto. eauto. auto.
1782  subst T. rewrite <- (E0_right t) at 1. apply traceinf_prefix_app. constructor.
1783Qed.
1784
1785End BIGSTEP_TO_TRANSITIONS.
1786
1787
1788
1789*)
1790
1791 
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