source: C-semantics/Csem.ma @ 125

Last change on this file since 125 was 125, checked in by campbell, 9 years ago

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

File size: 60.1 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| type_ptr_int : type_pointable (Tint I32 Unsigned)
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_int : type_space (Tint I32 Unsigned) Any
410| type_spc_pointer : ∀s,t. type_space (Tpointer s t) s
411| type_spc_array : ∀s,t,n. type_space (Tarray s t n) s
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_pointable ty →
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_pp_z: ∀m,ty,ty'.
433      type_pointable ty → (* Don't care which space it is for the source type *)
434      type_pointable ty' →
435      cast m (Vint zero) ty ty' (Vint zero).
436(* XXX: other integers?
437  | cast_nn_i: ∀m,n,t1,t2.     (**r no change in data representation *)
438      neutral_for_cast t1 → neutral_for_cast t2 →
439      cast m (Vint n) t1 t2 (Vint n).
440*)
441(* * * Operational semantics *)
442
443(* * The semantics uses two environments.  The global environment
444  maps names of functions and global variables to memory block references,
445  and function pointers to their definitions.  (See module [Globalenvs].) *)
446
447ndefinition genv ≝ (genv_t Genv) fundef.
448
449(* * The local environment maps local variables to block references.
450  The current value of the variable is stored in the associated memory
451  block. *)
452
453ndefinition env ≝ (tree_t ? PTree) block. (* map variable -> location *)
454
455ndefinition empty_env: env ≝ (empty …).
456
457(* * [load_value_of_type ty m b ofs] computes the value of a datum
458  of type [ty] residing in memory [m] at block [b], offset [ofs].
459  If the type [ty] indicates an access by value, the corresponding
460  memory load is performed.  If the type [ty] indicates an access by
461  reference, the pointer [Vptr b ofs] is returned. *)
462
463nlet rec load_value_of_type (ty: type) (m: mem) (psp:memory_space) (b: block) (ofs: int) : option val ≝
464  match access_mode ty with
465  [ By_value chunk ⇒ loadv chunk m (Vptr psp b ofs)
466  | By_reference ⇒ Some ? (Vptr psp b ofs)
467  | By_nothing ⇒ None ?
468  ].
469
470(* * Symmetrically, [store_value_of_type ty m b ofs v] returns the
471  memory state after storing the value [v] in the datum
472  of type [ty] residing in memory [m] at block [b], offset [ofs].
473  This is allowed only if [ty] indicates an access by value. *)
474
475nlet rec store_value_of_type (ty_dest: type) (m: mem) (psp:memory_space) (loc: block) (ofs: int) (v: val) : option mem ≝
476  match access_mode ty_dest with
477  [ By_value chunk ⇒ storev chunk m (Vptr psp loc ofs) v
478  | By_reference ⇒ None ?
479  | By_nothing ⇒ None ?
480  ].
481
482(* * Allocation of function-local variables.
483  [alloc_variables e1 m1 vars e2 m2] allocates one memory block
484  for each variable declared in [vars], and associates the variable
485  name with this block.  [e1] and [m1] are the initial local environment
486  and memory state.  [e2] and [m2] are the final local environment
487  and memory state. *)
488
489ninductive alloc_variables: env → mem →
490                            list (ident × type) →
491                            env → mem → Prop ≝
492  | alloc_variables_nil:
493      ∀e,m.
494      alloc_variables e m (nil ?) e m
495  | alloc_variables_cons:
496      ∀e,m,id,ty,vars,m1,b1,m2,e2.
497      alloc m 0 (sizeof ty) Any = 〈m1, b1〉 →
498      alloc_variables (set … id b1 e) m1 vars e2 m2 →
499      alloc_variables e m (〈id, ty〉 :: vars) e2 m2.
500
501(* * Initialization of local variables that are parameters to a function.
502  [bind_parameters e m1 params args m2] stores the values [args]
503  in the memory blocks corresponding to the variables [params].
504  [m1] is the initial memory state and [m2] the final memory state. *)
505
506ninductive bind_parameters: env →
507                           mem → list (ident × type) → list val →
508                           mem → Prop ≝
509  | bind_parameters_nil:
510      ∀e,m.
511      bind_parameters e m (nil ?) (nil ?) m
512  | bind_parameters_cons:
513      ∀e,m,id,ty,params,v1,vl,b,m1,m2.
514      get ??? id e = Some ? b →
515      store_value_of_type ty m Any b zero v1 = Some ? m1 →
516      bind_parameters e m1 params vl m2 →
517      bind_parameters e m (〈id, ty〉 :: params) (v1 :: vl) m2.
518
519(* XXX: this doesn't look right - we're assigning arbitrary memory spaces to
520   parameters? *)
521
522(* * Return the list of blocks in the codomain of [e]. *)
523
524ndefinition blocks_of_env : env → list block ≝ λe.
525  map ?? (λx. snd ?? x) (elements ??? e).
526
527(* * Selection of the appropriate case of a [switch], given the value [n]
528  of the selector expression. *)
529
530nlet rec select_switch (n: int) (sl: labeled_statements)
531                       on sl : labeled_statements ≝
532  match sl with
533  [ LSdefault _ ⇒ sl
534  | LScase c s sl' ⇒ if eq c n then sl else select_switch n sl'
535  ].
536
537(* * Turn a labeled statement into a sequence *)
538
539nlet rec seq_of_labeled_statement (sl: labeled_statements) : statement ≝
540  match sl with
541  [ LSdefault s ⇒ s
542  | LScase c s sl' ⇒ Ssequence s (seq_of_labeled_statement sl')
543  ].
544
545(*
546Section SEMANTICS.
547
548Variable ge: genv.
549
550(** ** Evaluation of expressions *)
551
552Section EXPR.
553
554Variable e: env.
555Variable m: mem.
556*)
557(* * [eval_expr ge e m a v] defines the evaluation of expression [a]
558  in r-value position.  [v] is the value of the expression.
559  [e] is the current environment and [m] is the current memory state. *)
560
561ninductive eval_expr (ge:genv) (e:env) (m:mem) : expr → val → Prop ≝
562  | eval_Econst_int:   ∀i,ty.
563      eval_expr ge e m (Expr (Econst_int i) ty) (Vint i)
564  | eval_Econst_float:   ∀f,ty.
565      eval_expr ge e m (Expr (Econst_float f) ty) (Vfloat f)
566  | eval_Elvalue: ∀a,ty,psp,loc,ofs,v.
567      eval_lvalue ge e m (Expr a ty) psp loc ofs ->
568      load_value_of_type ty m psp loc ofs = Some ? v ->
569      eval_expr ge e m (Expr a ty) v
570  | eval_Eaddrof: ∀a,ty,psp,loc,ofs.
571      eval_lvalue ge e m a psp loc ofs ->
572      eval_expr ge e m (Expr (Eaddrof a) ty) (Vptr psp loc ofs)
573  | eval_Esizeof: ∀ty',ty.
574      eval_expr ge e m (Expr (Esizeof ty') ty) (Vint (repr (sizeof ty')))
575  | eval_Eunop:  ∀op,a,ty,v1,v.
576      eval_expr ge e m a v1 ->
577      sem_unary_operation op v1 (typeof a) = Some ? v ->
578      eval_expr ge e m (Expr (Eunop op a) ty) v
579  | eval_Ebinop: ∀op,a1,a2,ty,v1,v2,v.
580      eval_expr ge e m a1 v1 ->
581      eval_expr ge e m a2 v2 ->
582      sem_binary_operation op v1 (typeof a1) v2 (typeof a2) m = Some ? v ->
583      eval_expr ge e m (Expr (Ebinop op a1 a2) ty) v
584  | eval_Econdition_true: ∀a1,a2,a3,ty,v1,v2.
585      eval_expr ge e m a1 v1 ->
586      is_true v1 (typeof a1) ->
587      eval_expr ge e m a2 v2 ->
588      eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v2
589  | eval_Econdition_false: ∀a1,a2,a3,ty,v1,v3.
590      eval_expr ge e m a1 v1 ->
591      is_false v1 (typeof a1) ->
592      eval_expr ge e m a3 v3 ->
593      eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v3
594  | eval_Eorbool_1: ∀a1,a2,ty,v1.
595      eval_expr ge e m a1 v1 ->
596      is_true v1 (typeof a1) ->
597      eval_expr ge e m (Expr (Eorbool a1 a2) ty) Vtrue
598  | eval_Eorbool_2: ∀a1,a2,ty,v1,v2,v.
599      eval_expr ge e m a1 v1 ->
600      is_false v1 (typeof a1) ->
601      eval_expr ge e m a2 v2 ->
602      bool_of_val v2 (typeof a2) v ->
603      eval_expr ge e m (Expr (Eorbool a1 a2) ty) v
604  | eval_Eandbool_1: ∀a1,a2,ty,v1.
605      eval_expr ge e m a1 v1 ->
606      is_false v1 (typeof a1) ->
607      eval_expr ge e m (Expr (Eandbool a1 a2) ty) Vfalse
608  | eval_Eandbool_2: ∀a1,a2,ty,v1,v2,v.
609      eval_expr ge e m a1 v1 ->
610      is_true v1 (typeof a1) ->
611      eval_expr ge e m a2 v2 ->
612      bool_of_val v2 (typeof a2) v ->
613      eval_expr ge e m (Expr (Eandbool a1 a2) ty) v
614  | eval_Ecast:   ∀a,ty,ty',v1,v.
615      eval_expr ge e m a v1 ->
616      cast m v1 (typeof a) ty v ->
617      eval_expr ge e m (Expr (Ecast ty a) ty') v
618
619(* * [eval_lvalue ge e m a b ofs] defines the evaluation of expression [a]
620  in l-value position.  The result is the memory location [b, ofs]
621  that contains the value of the expression [a]. *)
622
623with eval_lvalue (*(ge:genv) (e:env) (m:mem)*) : expr → memory_space → block -> int -> Prop ≝
624  | eval_Evar_local:   ∀id,l,ty.
625      (* XXX notation? e!id*) get ??? id e = Some ? l →
626      eval_lvalue ge e m (Expr (Evar id) ty) Any l zero
627  | eval_Evar_global: ∀id,sp,l,ty.
628      (* XXX e!id *) get ??? id e = None ? ->
629      find_symbol ?? ge id = Some ? 〈sp,l〉 ->
630      eval_lvalue ge e m (Expr (Evar id) ty) sp l zero
631  | eval_Ederef: ∀a,ty,psp,l,ofs.
632      eval_expr ge e m a (Vptr psp l ofs) ->
633      eval_lvalue ge e m (Expr (Ederef a) ty) psp l ofs
634 | eval_Efield_struct:   ∀a,i,ty,psp,l,ofs,id,fList,delta.
635      eval_lvalue ge e m a psp l ofs ->
636      typeof a = Tstruct id fList ->
637      field_offset i fList = OK ? delta ->
638      eval_lvalue ge e m (Expr (Efield a i) ty) psp l (add ofs (repr delta))
639 | eval_Efield_union:   ∀a,i,ty,psp,l,ofs,id,fList.
640      eval_lvalue ge e m a psp l ofs ->
641      typeof a = Tunion id fList ->
642      eval_lvalue ge e m (Expr (Efield a i) ty) psp l ofs.
643
644(*
645Scheme eval_expr_ind2 := Minimality for eval_expr Sort Prop
646  with eval_lvalue_ind2 := Minimality for eval_lvalue Sort Prop.
647*)
648
649(* * [eval_exprlist ge e m al vl] evaluates a list of r-value
650  expressions [al] to their values [vl]. *)
651
652ninductive eval_exprlist (ge:genv) (e:env) (m:mem) : list expr -> list val -> Prop :=
653  | eval_Enil:
654      eval_exprlist ge e m (nil ?) (nil ?)
655  | eval_Econs:   ∀a,bl,v,vl.
656      eval_expr ge e m a v ->
657      eval_exprlist ge e m bl vl ->
658      eval_exprlist ge e m (a :: bl) (v :: vl).
659
660(*End EXPR.*)
661
662(* * ** Transition semantics for statements and functions *)
663
664(* * Continuations *)
665
666ninductive cont: Type :=
667  | Kstop: cont
668  | Kseq: statement -> cont -> cont
669       (**r [Kseq s2 k] = after [s1] in [s1;s2] *)
670  | Kwhile: expr -> statement -> cont -> cont
671       (**r [Kwhile e s k] = after [s] in [while (e) s] *)
672  | Kdowhile: expr -> statement -> cont -> cont
673       (**r [Kdowhile e s k] = after [s] in [do s while (e)] *)
674  | Kfor2: expr -> statement -> statement -> cont -> cont
675       (**r [Kfor2 e2 e3 s k] = after [s] in [for(e1;e2;e3) s] *)
676  | Kfor3: expr -> statement -> statement -> cont -> cont
677       (**r [Kfor3 e2 e3 s k] = after [e3] in [for(e1;e2;e3) s] *)
678  | Kswitch: cont -> cont
679       (**r catches [break] statements arising out of [switch] *)
680  | Kcall: option (memory_space × block × int × type) ->   (**r where to store result *)
681           function ->                      (**r calling function *)
682           env ->                           (**r local env of calling function *)
683           cont -> cont.
684
685(* * Pop continuation until a call or stop *)
686
687nlet rec call_cont (k: cont) : cont :=
688  match k with
689  [ Kseq s k => call_cont k
690  | Kwhile e s k => call_cont k
691  | Kdowhile e s k => call_cont k
692  | Kfor2 e2 e3 s k => call_cont k
693  | Kfor3 e2 e3 s k => call_cont k
694  | Kswitch k => call_cont k
695  | _ => k
696  ].
697
698ndefinition is_call_cont : cont → Prop ≝ λk.
699  match k with
700  [ Kstop => True
701  | Kcall _ _ _ _ => True
702  | _ => False
703  ].
704
705(* * States *)
706
707ninductive state: Type :=
708  | State:
709      ∀f: function.
710      ∀s: statement.
711      ∀k: cont.
712      ∀e: env.
713      ∀m: mem.  state
714  | Callstate:
715      ∀fd: fundef.
716      ∀args: list val.
717      ∀k: cont.
718      ∀m: mem. state
719  | Returnstate:
720      ∀res: val.
721      ∀k: cont.
722      ∀m: mem. state.
723                 
724(* * Find the statement and manufacture the continuation
725  corresponding to a label *)
726
727nlet rec find_label (lbl: label) (s: statement) (k: cont)
728                    on s: option (statement × cont) :=
729  match s with
730  [ Ssequence s1 s2 =>
731      match find_label lbl s1 (Kseq s2 k) with
732      [ Some sk => Some ? sk
733      | None => find_label lbl s2 k
734      ]
735  | Sifthenelse a s1 s2 =>
736      match find_label lbl s1 k with
737      [ Some sk => Some ? sk
738      | None => find_label lbl s2 k
739      ]
740  | Swhile a s1 =>
741      find_label lbl s1 (Kwhile a s1 k)
742  | Sdowhile a s1 =>
743      find_label lbl s1 (Kdowhile a s1 k)
744  | Sfor a1 a2 a3 s1 =>
745      match find_label lbl a1 (Kseq (Sfor Sskip a2 a3 s1) k) with
746      [ Some sk => Some ? sk
747      | None =>
748          match find_label lbl s1 (Kfor2 a2 a3 s1 k) with
749          [ Some sk => Some ? sk
750          | None => find_label lbl a3 (Kfor3 a2 a3 s1 k)
751          ]
752      ]
753  | Sswitch e sl =>
754      find_label_ls lbl sl (Kswitch k)
755  | Slabel lbl' s' =>
756      match ident_eq lbl lbl' with
757      [ inl _ ⇒ Some ? 〈s', k〉
758      | inr _ ⇒ find_label lbl s' k
759      ]
760  | _ => None ?
761  ]
762
763and find_label_ls (lbl: label) (sl: labeled_statements) (k: cont)
764                    on sl: option (statement × cont) :=
765  match sl with
766  [ LSdefault s => find_label lbl s k
767  | LScase _ s sl' =>
768      match find_label lbl s (Kseq (seq_of_labeled_statement sl') k) with
769      [ Some sk => Some ? sk
770      | None => find_label_ls lbl sl' k
771      ]
772  ].
773
774(* * Transition relation *)
775
776ninductive step (ge:genv) : state -> trace -> state -> Prop :=
777
778  | step_assign:   ∀f,a1,a2,k,e,m,psp,loc,ofs,v2,m'.
779      eval_lvalue ge e m a1 psp loc ofs ->
780      eval_expr ge e m a2 v2 ->
781      store_value_of_type (typeof a1) m psp loc ofs v2 = Some ? m' ->
782      step ge (State f (Sassign a1 a2) k e m)
783           E0 (State f Sskip k e m')
784
785  | step_call_none:   ∀f,a,al,k,e,m,vf,vargs,fd.
786      eval_expr ge e m a vf ->
787      eval_exprlist ge e m al vargs ->
788      find_funct ?? ge vf = Some ? fd ->
789      type_of_fundef fd = typeof a ->
790      step ge (State f (Scall (None ?) a al) k e m)
791           E0 (Callstate fd vargs (Kcall (None ?) f e k) m)
792
793  | step_call_some:   ∀f,lhs,a,al,k,e,m,psp,loc,ofs,vf,vargs,fd.
794      eval_lvalue ge e m lhs psp loc ofs ->
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 (Some ? lhs) a al) k e m)
800           E0 (Callstate fd vargs (Kcall (Some ? 〈〈〈psp, loc〉, ofs〉, typeof lhs〉) f e k) m)
801
802  | step_seq:  ∀f,s1,s2,k,e,m.
803      step ge (State f (Ssequence s1 s2) k e m)
804           E0 (State f s1 (Kseq s2 k) e m)
805  | step_skip_seq: ∀f,s,k,e,m.
806      step ge (State f Sskip (Kseq s k) e m)
807           E0 (State f s k e m)
808  | step_continue_seq: ∀f,s,k,e,m.
809      step ge (State f Scontinue (Kseq s k) e m)
810           E0 (State f Scontinue k e m)
811  | step_break_seq: ∀f,s,k,e,m.
812      step ge (State f Sbreak (Kseq s k) e m)
813           E0 (State f Sbreak k e m)
814
815  | step_ifthenelse_true:  ∀f,a,s1,s2,k,e,m,v1.
816      eval_expr ge e m a v1 ->
817      is_true v1 (typeof a) ->
818      step ge (State f (Sifthenelse a s1 s2) k e m)
819           E0 (State f s1 k e m)
820  | step_ifthenelse_false: ∀f,a,s1,s2,k,e,m,v1.
821      eval_expr ge e m a v1 ->
822      is_false v1 (typeof a) ->
823      step ge (State f (Sifthenelse a s1 s2) k e m)
824           E0 (State f s2 k e m)
825
826  | step_while_false: ∀f,a,s,k,e,m,v.
827      eval_expr ge e m a v ->
828      is_false v (typeof a) ->
829      step ge (State f (Swhile a s) k e m)
830           E0 (State f Sskip k e m)
831  | step_while_true: ∀f,a,s,k,e,m,v.
832      eval_expr ge e m a v ->
833      is_true v (typeof a) ->
834      step ge (State f (Swhile a s) k e m)
835           E0 (State f s (Kwhile a s k) e m)
836  | step_skip_or_continue_while: ∀f,x,a,s,k,e,m.
837      x = Sskip ∨ x = Scontinue ->
838      step ge (State f x (Kwhile a s k) e m)
839           E0 (State f (Swhile a s) k e m)
840  | step_break_while: ∀f,a,s,k,e,m.
841      step ge (State f Sbreak (Kwhile a s k) e m)
842           E0 (State f Sskip k e m)
843
844  | step_dowhile: ∀f,a,s,k,e,m.
845      step ge (State f (Sdowhile a s) k e m)
846        E0 (State f s (Kdowhile a s k) e m)
847  | step_skip_or_continue_dowhile_false: ∀f,x,a,s,k,e,m,v.
848      x = Sskip ∨ x = Scontinue ->
849      eval_expr ge e m a v ->
850      is_false v (typeof a) ->
851      step ge (State f x (Kdowhile a s k) e m)
852           E0 (State f Sskip k e m)
853  | step_skip_or_continue_dowhile_true: ∀f,x,a,s,k,e,m,v.
854      x = Sskip ∨ x = Scontinue ->
855      eval_expr ge e m a v ->
856      is_true v (typeof a) ->
857      step ge (State f x (Kdowhile a s k) e m)
858           E0 (State f (Sdowhile a s) k e m)
859  | step_break_dowhile: ∀f,a,s,k,e,m.
860      step ge (State f Sbreak (Kdowhile a s k) e m)
861           E0 (State f Sskip k e m)
862
863  | step_for_start: ∀f,a1,a2,a3,s,k,e,m.
864      a1 ≠ Sskip ->
865      step ge (State f (Sfor a1 a2 a3 s) k e m)
866           E0 (State f a1 (Kseq (Sfor Sskip a2 a3 s) k) e m)
867  | step_for_false: ∀f,a2,a3,s,k,e,m,v.
868      eval_expr ge e m a2 v ->
869      is_false v (typeof a2) ->
870      step ge (State f (Sfor Sskip a2 a3 s) k e m)
871           E0 (State f Sskip k e m)
872  | step_for_true: ∀f,a2,a3,s,k,e,m,v.
873      eval_expr ge e m a2 v ->
874      is_true v (typeof a2) ->
875      step ge (State f (Sfor Sskip a2 a3 s) k e m)
876           E0 (State f s (Kfor2 a2 a3 s k) e m)
877  | step_skip_or_continue_for2: ∀f,x,a2,a3,s,k,e,m.
878      x = Sskip ∨ x = Scontinue ->
879      step ge (State f x (Kfor2 a2 a3 s k) e m)
880           E0 (State f a3 (Kfor3 a2 a3 s k) e m)
881  | step_break_for2: ∀f,a2,a3,s,k,e,m.
882      step ge (State f Sbreak (Kfor2 a2 a3 s k) e m)
883           E0 (State f Sskip k e m)
884  | step_skip_for3: ∀f,a2,a3,s,k,e,m.
885      step ge (State f Sskip (Kfor3 a2 a3 s k) e m)
886           E0 (State f (Sfor Sskip a2 a3 s) k e m)
887
888  | step_return_0: ∀f,k,e,m.
889      fn_return f = Tvoid ->
890      step ge (State f (Sreturn (None ?)) k e m)
891           E0 (Returnstate Vundef (call_cont k) (free_list m (blocks_of_env e)))
892  | step_return_1: ∀f,a,k,e,m,v.
893      fn_return f ≠ Tvoid ->
894      eval_expr ge e m a v ->
895      step ge (State f (Sreturn (Some ? a)) k e m)
896           E0 (Returnstate v (call_cont k) (free_list m (blocks_of_env e)))
897  | step_skip_call: ∀f,k,e,m.
898      is_call_cont k ->
899      fn_return f = Tvoid ->
900      step ge (State f Sskip k e m)
901           E0 (Returnstate Vundef k (free_list m (blocks_of_env e)))
902
903  | step_switch: ∀f,a,sl,k,e,m,n.
904      eval_expr ge e m a (Vint n) ->
905      step ge (State f (Sswitch a sl) k e m)
906           E0 (State f (seq_of_labeled_statement (select_switch n sl)) (Kswitch k) e m)
907  | step_skip_break_switch: ∀f,x,k,e,m.
908      x = Sskip ∨ x = Sbreak ->
909      step ge (State f x (Kswitch k) e m)
910           E0 (State f Sskip k e m)
911  | step_continue_switch: ∀f,k,e,m.
912      step ge (State f Scontinue (Kswitch k) e m)
913           E0 (State f Scontinue k e m)
914
915  | step_label: ∀f,lbl,s,k,e,m.
916      step ge (State f (Slabel lbl s) k e m)
917           E0 (State f s k e m)
918
919  | step_goto: ∀f,lbl,k,e,m,s',k'.
920      find_label lbl (fn_body f) (call_cont k) = Some ? 〈s', k'〉 ->
921      step ge (State f (Sgoto lbl) k e m)
922           E0 (State f s' k' e m)
923
924  | step_internal_function: ∀f,vargs,k,m,e,m1,m2.
925      alloc_variables empty_env m ((fn_params f) @ (fn_vars f)) e m1 ->
926      bind_parameters e m1 (fn_params f) vargs m2 ->
927      step ge (Callstate (Internal f) vargs k m)
928           E0 (State f (fn_body f) k e m2)
929
930  | step_external_function: ∀id,targs,tres,vargs,k,m,vres,t.
931      event_match (external_function id targs tres) vargs t vres ->
932      step ge (Callstate (External id targs tres) vargs k m)
933            t (Returnstate vres k m)
934
935  | step_returnstate_0: ∀v,f,e,k,m.
936      step ge (Returnstate v (Kcall (None ?) f e k) m)
937           E0 (State f Sskip k e m)
938
939  | step_returnstate_1: ∀v,f,e,k,m,m',psp,loc,ofs,ty.
940      store_value_of_type ty m psp loc ofs v = Some ? m' ->
941      step ge (Returnstate v (Kcall (Some ? 〈〈〈psp,loc〉, ofs〉, ty〉) f e k) m)
942           E0 (State f Sskip k e m').
943(*
944(** * Alternate big-step semantics *)
945
946(** ** Big-step semantics for terminating statements and functions *)
947
948(** The execution of a statement produces an ``outcome'', indicating
949  how the execution terminated: either normally or prematurely
950  through the execution of a [break], [continue] or [return] statement. *)
951
952ninductive outcome: Type :=
953   | Out_break: outcome                 (**r terminated by [break] *)
954   | Out_continue: outcome              (**r terminated by [continue] *)
955   | Out_normal: outcome                (**r terminated normally *)
956   | Out_return: option val -> outcome. (**r terminated by [return] *)
957
958ninductive out_normal_or_continue : outcome -> Prop :=
959  | Out_normal_or_continue_N: out_normal_or_continue Out_normal
960  | Out_normal_or_continue_C: out_normal_or_continue Out_continue.
961
962ninductive out_break_or_return : outcome -> outcome -> Prop :=
963  | Out_break_or_return_B: out_break_or_return Out_break Out_normal
964  | Out_break_or_return_R: ∀ov.
965      out_break_or_return (Out_return ov) (Out_return ov).
966
967Definition outcome_switch (out: outcome) : outcome :=
968  match out with
969  | Out_break => Out_normal
970  | o => o
971  end.
972
973Definition outcome_result_value (out: outcome) (t: type) (v: val) : Prop :=
974  match out, t with
975  | Out_normal, Tvoid => v = Vundef
976  | Out_return None, Tvoid => v = Vundef
977  | Out_return (Some v'), ty => ty <> Tvoid /\ v'=v
978  | _, _ => False
979  end.
980
981(** [exec_stmt ge e m1 s t m2 out] describes the execution of
982  the statement [s].  [out] is the outcome for this execution.
983  [m1] is the initial memory state, [m2] the final memory state.
984  [t] is the trace of input/output events performed during this
985  evaluation. *)
986
987ninductive exec_stmt: env -> mem -> statement -> trace -> mem -> outcome -> Prop :=
988  | exec_Sskip:   ∀e,m.
989      exec_stmt e m Sskip
990               E0 m Out_normal
991  | exec_Sassign:   ∀e,m,a1,a2,loc,ofs,v2,m'.
992      eval_lvalue e m a1 loc ofs ->
993      eval_expr e m a2 v2 ->
994      store_value_of_type (typeof a1) m loc ofs v2 = Some m' ->
995      exec_stmt e m (Sassign a1 a2)
996               E0 m' Out_normal
997  | exec_Scall_none:   ∀e,m,a,al,vf,vargs,f,t,m',vres.
998      eval_expr e m a vf ->
999      eval_exprlist e m al vargs ->
1000      Genv.find_funct ge vf = Some f ->
1001      type_of_fundef f = typeof a ->
1002      eval_funcall m f vargs t m' vres ->
1003      exec_stmt e m (Scall None a al)
1004                t m' Out_normal
1005  | exec_Scall_some:   ∀e,m,lhs,a,al,loc,ofs,vf,vargs,f,t,m',vres,m''.
1006      eval_lvalue e m lhs loc ofs ->
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      store_value_of_type (typeof lhs) m' loc ofs vres = Some m'' ->
1013      exec_stmt e m (Scall (Some lhs) a al)
1014                t m'' Out_normal
1015  | exec_Sseq_1:   ∀e,m,s1,s2,t1,m1,t2,m2,out.
1016      exec_stmt e m s1 t1 m1 Out_normal ->
1017      exec_stmt e m1 s2 t2 m2 out ->
1018      exec_stmt e m (Ssequence s1 s2)
1019                (t1 ** t2) m2 out
1020  | exec_Sseq_2:   ∀e,m,s1,s2,t1,m1,out.
1021      exec_stmt e m s1 t1 m1 out ->
1022      out <> Out_normal ->
1023      exec_stmt e m (Ssequence s1 s2)
1024                t1 m1 out
1025  | exec_Sifthenelse_true: ∀e,m,a,s1,s2,v1,t,m',out.
1026      eval_expr e m a v1 ->
1027      is_true v1 (typeof a) ->
1028      exec_stmt e m s1 t m' out ->
1029      exec_stmt e m (Sifthenelse a s1 s2)
1030                t m' out
1031  | exec_Sifthenelse_false: ∀e,m,a,s1,s2,v1,t,m',out.
1032      eval_expr e m a v1 ->
1033      is_false v1 (typeof a) ->
1034      exec_stmt e m s2 t m' out ->
1035      exec_stmt e m (Sifthenelse a s1 s2)
1036                t m' out
1037  | exec_Sreturn_none:   ∀e,m.
1038      exec_stmt e m (Sreturn None)
1039               E0 m (Out_return None)
1040  | exec_Sreturn_some: ∀e,m,a,v.
1041      eval_expr e m a v ->
1042      exec_stmt e m (Sreturn (Some a))
1043               E0 m (Out_return (Some v))
1044  | exec_Sbreak:   ∀e,m.
1045      exec_stmt e m Sbreak
1046               E0 m Out_break
1047  | exec_Scontinue:   ∀e,m.
1048      exec_stmt e m Scontinue
1049               E0 m Out_continue
1050  | exec_Swhile_false: ∀e,m,a,s,v.
1051      eval_expr e m a v ->
1052      is_false v (typeof a) ->
1053      exec_stmt e m (Swhile a s)
1054               E0 m Out_normal
1055  | exec_Swhile_stop: ∀e,m,a,v,s,t,m',out',out.
1056      eval_expr e m a v ->
1057      is_true v (typeof a) ->
1058      exec_stmt e m s t m' out' ->
1059      out_break_or_return out' out ->
1060      exec_stmt e m (Swhile a s)
1061                t m' out
1062  | exec_Swhile_loop: ∀e,m,a,s,v,t1,m1,out1,t2,m2,out.
1063      eval_expr e m a v ->
1064      is_true v (typeof a) ->
1065      exec_stmt e m s t1 m1 out1 ->
1066      out_normal_or_continue out1 ->
1067      exec_stmt e m1 (Swhile a s) t2 m2 out ->
1068      exec_stmt e m (Swhile a s)
1069                (t1 ** t2) m2 out
1070  | exec_Sdowhile_false: ∀e,m,s,a,t,m1,out1,v.
1071      exec_stmt e m s t m1 out1 ->
1072      out_normal_or_continue out1 ->
1073      eval_expr e m1 a v ->
1074      is_false v (typeof a) ->
1075      exec_stmt e m (Sdowhile a s)
1076                t m1 Out_normal
1077  | exec_Sdowhile_stop: ∀e,m,s,a,t,m1,out1,out.
1078      exec_stmt e m s t m1 out1 ->
1079      out_break_or_return out1 out ->
1080      exec_stmt e m (Sdowhile a s)
1081                t m1 out
1082  | exec_Sdowhile_loop: ∀e,m,s,a,m1,m2,t1,t2,out,out1,v.
1083      exec_stmt e m s t1 m1 out1 ->
1084      out_normal_or_continue out1 ->
1085      eval_expr e m1 a v ->
1086      is_true v (typeof a) ->
1087      exec_stmt e m1 (Sdowhile a s) t2 m2 out ->
1088      exec_stmt e m (Sdowhile a s)
1089                (t1 ** t2) m2 out
1090  | exec_Sfor_start: ∀e,m,s,a1,a2,a3,out,m1,m2,t1,t2.
1091      a1 <> Sskip ->
1092      exec_stmt e m a1 t1 m1 Out_normal ->
1093      exec_stmt e m1 (Sfor Sskip a2 a3 s) t2 m2 out ->
1094      exec_stmt e m (Sfor a1 a2 a3 s)
1095                (t1 ** t2) m2 out
1096  | exec_Sfor_false: ∀e,m,s,a2,a3,v.
1097      eval_expr e m a2 v ->
1098      is_false v (typeof a2) ->
1099      exec_stmt e m (Sfor Sskip a2 a3 s)
1100               E0 m Out_normal
1101  | exec_Sfor_stop: ∀e,m,s,a2,a3,v,m1,t,out1,out.
1102      eval_expr e m a2 v ->
1103      is_true v (typeof a2) ->
1104      exec_stmt e m s t m1 out1 ->
1105      out_break_or_return out1 out ->
1106      exec_stmt e m (Sfor Sskip a2 a3 s)
1107                t m1 out
1108  | exec_Sfor_loop: ∀e,m,s,a2,a3,v,m1,m2,m3,t1,t2,t3,out1,out.
1109      eval_expr e m a2 v ->
1110      is_true v (typeof a2) ->
1111      exec_stmt e m s t1 m1 out1 ->
1112      out_normal_or_continue out1 ->
1113      exec_stmt e m1 a3 t2 m2 Out_normal ->
1114      exec_stmt e m2 (Sfor Sskip a2 a3 s) t3 m3 out ->
1115      exec_stmt e m (Sfor Sskip a2 a3 s)
1116                (t1 ** t2 ** t3) m3 out
1117  | exec_Sswitch:   ∀e,m,a,t,n,sl,m1,out.
1118      eval_expr e m a (Vint n) ->
1119      exec_stmt e m (seq_of_labeled_statement (select_switch n sl)) t m1 out ->
1120      exec_stmt e m (Sswitch a sl)
1121                t m1 (outcome_switch out)
1122
1123(** [eval_funcall m1 fd args t m2 res] describes the invocation of
1124  function [fd] with arguments [args].  [res] is the value returned
1125  by the call.  *)
1126
1127with eval_funcall: mem -> fundef -> list val -> trace -> mem -> val -> Prop :=
1128  | eval_funcall_internal: ∀m,f,vargs,t,e,m1,m2,m3,out,vres.
1129      alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
1130      bind_parameters e m1 f.(fn_params) vargs m2 ->
1131      exec_stmt e m2 f.(fn_body) t m3 out ->
1132      outcome_result_value out f.(fn_return) vres ->
1133      eval_funcall m (Internal f) vargs t (Mem.free_list m3 (blocks_of_env e)) vres
1134  | eval_funcall_external: ∀m,id,targs,tres,vargs,t,vres.
1135      event_match (external_function id targs tres) vargs t vres ->
1136      eval_funcall m (External id targs tres) vargs t m vres.
1137
1138Scheme exec_stmt_ind2 := Minimality for exec_stmt Sort Prop
1139  with eval_funcall_ind2 := Minimality for eval_funcall Sort Prop.
1140
1141(** ** Big-step semantics for diverging statements and functions *)
1142
1143(** Coinductive semantics for divergence.
1144  [execinf_stmt ge e m s t] holds if the execution of statement [s]
1145  diverges, i.e. loops infinitely.  [t] is the possibly infinite
1146  trace of observable events performed during the execution. *)
1147
1148Coninductive execinf_stmt: env -> mem -> statement -> traceinf -> Prop :=
1149  | execinf_Scall_none:   ∀e,m,a,al,vf,vargs,f,t.
1150      eval_expr e m a vf ->
1151      eval_exprlist e m al vargs ->
1152      Genv.find_funct ge vf = Some f ->
1153      type_of_fundef f = typeof a ->
1154      evalinf_funcall m f vargs t ->
1155      execinf_stmt e m (Scall None a al) t
1156  | execinf_Scall_some:   ∀e,m,lhs,a,al,loc,ofs,vf,vargs,f,t.
1157      eval_lvalue e m lhs loc ofs ->
1158      eval_expr e m a vf ->
1159      eval_exprlist e m al vargs ->
1160      Genv.find_funct ge vf = Some f ->
1161      type_of_fundef f = typeof a ->
1162      evalinf_funcall m f vargs t ->
1163      execinf_stmt e m (Scall (Some lhs) a al) t
1164  | execinf_Sseq_1:   ∀e,m,s1,s2,t.
1165      execinf_stmt e m s1 t ->
1166      execinf_stmt e m (Ssequence s1 s2) t
1167  | execinf_Sseq_2:   ∀e,m,s1,s2,t1,m1,t2.
1168      exec_stmt e m s1 t1 m1 Out_normal ->
1169      execinf_stmt e m1 s2 t2 ->
1170      execinf_stmt e m (Ssequence s1 s2) (t1 *** t2)
1171  | execinf_Sifthenelse_true: ∀e,m,a,s1,s2,v1,t.
1172      eval_expr e m a v1 ->
1173      is_true v1 (typeof a) ->
1174      execinf_stmt e m s1 t ->
1175      execinf_stmt e m (Sifthenelse a s1 s2) t
1176  | execinf_Sifthenelse_false: ∀e,m,a,s1,s2,v1,t.
1177      eval_expr e m a v1 ->
1178      is_false v1 (typeof a) ->
1179      execinf_stmt e m s2 t ->
1180      execinf_stmt e m (Sifthenelse a s1 s2) t
1181  | execinf_Swhile_body: ∀e,m,a,v,s,t.
1182      eval_expr e m a v ->
1183      is_true v (typeof a) ->
1184      execinf_stmt e m s t ->
1185      execinf_stmt e m (Swhile a s) t
1186  | execinf_Swhile_loop: ∀e,m,a,s,v,t1,m1,out1,t2.
1187      eval_expr e m a v ->
1188      is_true v (typeof a) ->
1189      exec_stmt e m s t1 m1 out1 ->
1190      out_normal_or_continue out1 ->
1191      execinf_stmt e m1 (Swhile a s) t2 ->
1192      execinf_stmt e m (Swhile a s) (t1 *** t2)
1193  | execinf_Sdowhile_body: ∀e,m,s,a,t.
1194      execinf_stmt e m s t ->
1195      execinf_stmt e m (Sdowhile a s) t
1196  | execinf_Sdowhile_loop: ∀e,m,s,a,m1,t1,t2,out1,v.
1197      exec_stmt e m s t1 m1 out1 ->
1198      out_normal_or_continue out1 ->
1199      eval_expr e m1 a v ->
1200      is_true v (typeof a) ->
1201      execinf_stmt e m1 (Sdowhile a s) t2 ->
1202      execinf_stmt e m (Sdowhile a s) (t1 *** t2)
1203  | execinf_Sfor_start_1: ∀e,m,s,a1,a2,a3,t.
1204      execinf_stmt e m a1 t ->
1205      execinf_stmt e m (Sfor a1 a2 a3 s) t
1206  | execinf_Sfor_start_2: ∀e,m,s,a1,a2,a3,m1,t1,t2.
1207      a1 <> Sskip ->
1208      exec_stmt e m a1 t1 m1 Out_normal ->
1209      execinf_stmt e m1 (Sfor Sskip a2 a3 s) t2 ->
1210      execinf_stmt e m (Sfor a1 a2 a3 s) (t1 *** t2)
1211  | execinf_Sfor_body: ∀e,m,s,a2,a3,v,t.
1212      eval_expr e m a2 v ->
1213      is_true v (typeof a2) ->
1214      execinf_stmt e m s t ->
1215      execinf_stmt e m (Sfor Sskip a2 a3 s) t
1216  | execinf_Sfor_next: ∀e,m,s,a2,a3,v,m1,t1,t2,out1.
1217      eval_expr e m a2 v ->
1218      is_true v (typeof a2) ->
1219      exec_stmt e m s t1 m1 out1 ->
1220      out_normal_or_continue out1 ->
1221      execinf_stmt e m1 a3 t2 ->
1222      execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2)
1223  | execinf_Sfor_loop: ∀e,m,s,a2,a3,v,m1,m2,t1,t2,t3,out1.
1224      eval_expr e m a2 v ->
1225      is_true v (typeof a2) ->
1226      exec_stmt e m s t1 m1 out1 ->
1227      out_normal_or_continue out1 ->
1228      exec_stmt e m1 a3 t2 m2 Out_normal ->
1229      execinf_stmt e m2 (Sfor Sskip a2 a3 s) t3 ->
1230      execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2 *** t3)
1231  | execinf_Sswitch:   ∀e,m,a,t,n,sl.
1232      eval_expr e m a (Vint n) ->
1233      execinf_stmt e m (seq_of_labeled_statement (select_switch n sl)) t ->
1234      execinf_stmt e m (Sswitch a sl) t
1235
1236(** [evalinf_funcall ge m fd args t] holds if the invocation of function
1237    [fd] on arguments [args] diverges, with observable trace [t]. *)
1238
1239with evalinf_funcall: mem -> fundef -> list val -> traceinf -> Prop :=
1240  | evalinf_funcall_internal: ∀m,f,vargs,t,e,m1,m2.
1241      alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
1242      bind_parameters e m1 f.(fn_params) vargs m2 ->
1243      execinf_stmt e m2 f.(fn_body) t ->
1244      evalinf_funcall m (Internal f) vargs t.
1245
1246End SEMANTICS.
1247*)
1248(* * * Whole-program semantics *)
1249
1250(* * Execution of whole programs are described as sequences of transitions
1251  from an initial state to a final state.  An initial state is a [Callstate]
1252  corresponding to the invocation of the ``main'' function of the program
1253  without arguments and with an empty continuation. *)
1254
1255ninductive initial_state (p: program): state -> Prop :=
1256  | initial_state_intro: ∀b,f.
1257      let ge := globalenv Genv ?? p in
1258      let m0 := init_mem Genv ?? p in
1259      find_symbol ?? ge (prog_main ?? p) = Some ? 〈Code,b〉 ->
1260      find_funct_ptr ?? ge b = Some ? f ->
1261      initial_state p (Callstate f (nil ?) Kstop m0).
1262
1263(* * A final state is a [Returnstate] with an empty continuation. *)
1264
1265ninductive final_state: state -> int -> Prop :=
1266  | final_state_intro: ∀r,m.
1267      final_state (Returnstate (Vint r) Kstop m) r.
1268
1269(* * Execution of a whole program: [exec_program p beh]
1270  holds if the application of [p]'s main function to no arguments
1271  in the initial memory state for [p] has [beh] as observable
1272  behavior. *)
1273
1274ndefinition exec_program : program → program_behavior → Prop ≝ λp,beh.
1275  program_behaves (mk_transrel ?? step) (initial_state p) final_state (globalenv ??? p) beh.
1276(*
1277(** Big-step execution of a whole program.  *)
1278
1279ninductive bigstep_program_terminates (p: program): trace -> int -> Prop :=
1280  | bigstep_program_terminates_intro: ∀b,f,m1,t,r.
1281      let ge := Genv.globalenv p in
1282      let m0 := Genv.init_mem p in
1283      Genv.find_symbol ge p.(prog_main) = Some b ->
1284      Genv.find_funct_ptr ge b = Some f ->
1285      eval_funcall ge m0 f nil t m1 (Vint r) ->
1286      bigstep_program_terminates p t r.
1287
1288ninductive bigstep_program_diverges (p: program): traceinf -> Prop :=
1289  | bigstep_program_diverges_intro: ∀b,f,t.
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      evalinf_funcall ge m0 f nil t ->
1295      bigstep_program_diverges p t.
1296
1297(** * Implication from big-step semantics to transition semantics *)
1298
1299Section BIGSTEP_TO_TRANSITIONS.
1300
1301Variable prog: program.
1302Let ge : genv := Genv.globalenv prog.
1303
1304Definition exec_stmt_eval_funcall_ind
1305  (PS: env -> mem -> statement -> trace -> mem -> outcome -> Prop)
1306  (PF: mem -> fundef -> list val -> trace -> mem -> val -> Prop) :=
1307  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 =>
1308  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)
1309       (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).
1310
1311ninductive outcome_state_match
1312       (e: env) (m: mem) (f: function) (k: cont): outcome -> state -> Prop :=
1313  | osm_normal:
1314      outcome_state_match e m f k Out_normal (State f Sskip k e m)
1315  | osm_break:
1316      outcome_state_match e m f k Out_break (State f Sbreak k e m)
1317  | osm_continue:
1318      outcome_state_match e m f k Out_continue (State f Scontinue k e m)
1319  | osm_return_none: ∀k'.
1320      call_cont k' = call_cont k ->
1321      outcome_state_match e m f k
1322        (Out_return None) (State f (Sreturn None) k' e m)
1323  | osm_return_some: ∀a,v,k'.
1324      call_cont k' = call_cont k ->
1325      eval_expr ge e m a v ->
1326      outcome_state_match e m f k
1327        (Out_return (Some v)) (State f (Sreturn (Some a)) k' e m).
1328
1329Lemma is_call_cont_call_cont:
1330  ∀k. is_call_cont k -> call_cont k = k.
1331Proof.
1332  destruct k; simpl; intros; contradiction || auto.
1333Qed.
1334
1335Lemma exec_stmt_eval_funcall_steps:
1336  (∀e,m,s,t,m',out.
1337   exec_stmt ge e m s t m' out ->
1338   ∀f,k. exists S,
1339   star step ge (State f s k e m) t S
1340   /\ outcome_state_match e m' f k out S)
1341/\
1342  (∀m,fd,args,t,m',res.
1343   eval_funcall ge m fd args t m' res ->
1344   ∀k.
1345   is_call_cont k ->
1346   star step ge (Callstate fd args k m) t (Returnstate res k m')).
1347Proof.
1348  apply exec_stmt_eval_funcall_ind; intros.
1349
1350(* skip *)
1351  econstructor; split. apply star_refl. constructor.
1352
1353(* assign *)
1354  econstructor; split. apply star_one. econstructor; eauto. constructor.
1355
1356(* call none *)
1357  econstructor; split.
1358  eapply star_left. econstructor; eauto.
1359  eapply star_right. apply H4. simpl; auto. econstructor. reflexivity. traceEq.
1360  constructor.
1361
1362(* call some *)
1363  econstructor; split.
1364  eapply star_left. econstructor; eauto.
1365  eapply star_right. apply H5. simpl; auto. econstructor; eauto. reflexivity. traceEq.
1366  constructor.
1367
1368(* sequence 2 *)
1369  destruct (H0 f (Kseq s2 k)) as [S1 [A1 B1]]. inv B1.
1370  destruct (H2 f k) as [S2 [A2 B2]].
1371  econstructor; split.
1372  eapply star_left. econstructor.
1373  eapply star_trans. eexact A1.
1374  eapply star_left. constructor. eexact A2.
1375  reflexivity. reflexivity. traceEq.
1376  auto.
1377
1378(* sequence 1 *)
1379  destruct (H0 f (Kseq s2 k)) as [S1 [A1 B1]].
1380  set (S2 :=
1381    match out with
1382    | Out_break => State f Sbreak k e m1
1383    | Out_continue => State f Scontinue k e m1
1384    | _ => S1
1385    end).
1386  exists S2; split.
1387  eapply star_left. econstructor.
1388  eapply star_trans. eexact A1.
1389  unfold S2; inv B1.
1390    congruence.
1391    apply star_one. apply step_break_seq.
1392    apply star_one. apply step_continue_seq.
1393    apply star_refl.
1394    apply star_refl.
1395  reflexivity. traceEq.
1396  unfold S2; inv B1; congruence || econstructor; eauto.
1397
1398(* ifthenelse true *)
1399  destruct (H2 f k) as [S1 [A1 B1]].
1400  exists S1; split.
1401  eapply star_left. eapply step_ifthenelse_true; eauto. eexact A1. traceEq.
1402  auto.
1403
1404(* ifthenelse false *)
1405  destruct (H2 f k) as [S1 [A1 B1]].
1406  exists S1; split.
1407  eapply star_left. eapply step_ifthenelse_false; eauto. eexact A1. traceEq.
1408  auto.
1409
1410(* return none *)
1411  econstructor; split. apply star_refl. constructor. auto.
1412
1413(* return some *)
1414  econstructor; split. apply star_refl. econstructor; eauto.
1415
1416(* break *)
1417  econstructor; split. apply star_refl. constructor.
1418
1419(* continue *)
1420  econstructor; split. apply star_refl. constructor.
1421
1422(* while false *)
1423  econstructor; split.
1424  apply star_one. eapply step_while_false; eauto.
1425  constructor.
1426
1427(* while stop *)
1428  destruct (H2 f (Kwhile a s k)) as [S1 [A1 B1]].
1429  set (S2 :=
1430    match out' with
1431    | Out_break => State f Sskip k e m'
1432    | _ => S1
1433    end).
1434  exists S2; split.
1435  eapply star_left. eapply step_while_true; eauto.
1436  eapply star_trans. eexact A1.
1437  unfold S2. inversion H3; subst.
1438  inv B1. apply star_one. constructor.   
1439  apply star_refl.
1440  reflexivity. traceEq.
1441  unfold S2. inversion H3; subst. constructor. inv B1; econstructor; eauto.
1442
1443(* while loop *)
1444  destruct (H2 f (Kwhile a s k)) as [S1 [A1 B1]].
1445  destruct (H5 f k) as [S2 [A2 B2]].
1446  exists S2; split.
1447  eapply star_left. eapply step_while_true; eauto.
1448  eapply star_trans. eexact A1.
1449  eapply star_left.
1450  inv H3; inv B1; apply step_skip_or_continue_while; auto.
1451  eexact A2.
1452  reflexivity. reflexivity. traceEq.
1453  auto.
1454
1455(* dowhile false *)
1456  destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]].
1457  exists (State f Sskip k e m1); split.
1458  eapply star_left. constructor.
1459  eapply star_right. eexact A1.
1460  inv H1; inv B1; eapply step_skip_or_continue_dowhile_false; eauto.
1461  reflexivity. traceEq.
1462  constructor.
1463
1464(* dowhile stop *)
1465  destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]].
1466  set (S2 :=
1467    match out1 with
1468    | Out_break => State f Sskip k e m1
1469    | _ => S1
1470    end).
1471  exists S2; split.
1472  eapply star_left. apply step_dowhile.
1473  eapply star_trans. eexact A1.
1474  unfold S2. inversion H1; subst.
1475  inv B1. apply star_one. constructor.
1476  apply star_refl.
1477  reflexivity. traceEq.
1478  unfold S2. inversion H1; subst. constructor. inv B1; econstructor; eauto.
1479
1480(* dowhile loop *)
1481  destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]].
1482  destruct (H5 f k) as [S2 [A2 B2]].
1483  exists S2; split.
1484  eapply star_left. apply step_dowhile.
1485  eapply star_trans. eexact A1.
1486  eapply star_left.
1487  inv H1; inv B1; eapply step_skip_or_continue_dowhile_true; eauto.
1488  eexact A2.
1489  reflexivity. reflexivity. traceEq.
1490  auto.
1491
1492(* for start *)
1493  destruct (H1 f (Kseq (Sfor Sskip a2 a3 s) k)) as [S1 [A1 B1]]. inv B1.
1494  destruct (H3 f k) as [S2 [A2 B2]].
1495  exists S2; split.
1496  eapply star_left. apply step_for_start; auto.   
1497  eapply star_trans. eexact A1.
1498  eapply star_left. constructor. eexact A2.
1499  reflexivity. reflexivity. traceEq.
1500  auto.
1501
1502(* for false *)
1503  econstructor; split.
1504  eapply star_one. eapply step_for_false; eauto.
1505  constructor.
1506
1507(* for stop *)
1508  destruct (H2 f (Kfor2 a2 a3 s k)) as [S1 [A1 B1]].
1509  set (S2 :=
1510    match out1 with
1511    | Out_break => State f Sskip k e m1
1512    | _ => S1
1513    end).
1514  exists S2; split.
1515  eapply star_left. eapply step_for_true; eauto.
1516  eapply star_trans. eexact A1.
1517  unfold S2. inversion H3; subst.
1518  inv B1. apply star_one. constructor.
1519  apply star_refl.
1520  reflexivity. traceEq.
1521  unfold S2. inversion H3; subst. constructor. inv B1; econstructor; eauto.
1522
1523(* for loop *)
1524  destruct (H2 f (Kfor2 a2 a3 s k)) as [S1 [A1 B1]].
1525  destruct (H5 f (Kfor3 a2 a3 s k)) as [S2 [A2 B2]]. inv B2.
1526  destruct (H7 f k) as [S3 [A3 B3]].
1527  exists S3; split.
1528  eapply star_left. eapply step_for_true; eauto.
1529  eapply star_trans. eexact A1.
1530  eapply star_trans with (s2 := State f a3 (Kfor3 a2 a3 s k) e m1).
1531  inv H3; inv B1.
1532  apply star_one. constructor. auto.
1533  apply star_one. constructor. auto.
1534  eapply star_trans. eexact A2.
1535  eapply star_left. constructor.
1536  eexact A3.
1537  reflexivity. reflexivity. reflexivity. reflexivity. traceEq.
1538  auto.
1539
1540(* switch *)
1541  destruct (H1 f (Kswitch k)) as [S1 [A1 B1]].
1542  set (S2 :=
1543    match out with
1544    | Out_normal => State f Sskip k e m1
1545    | Out_break => State f Sskip k e m1
1546    | Out_continue => State f Scontinue k e m1
1547    | _ => S1
1548    end).
1549  exists S2; split.
1550  eapply star_left. eapply step_switch; eauto.
1551  eapply star_trans. eexact A1.
1552  unfold S2; inv B1.
1553    apply star_one. constructor. auto.
1554    apply star_one. constructor. auto.
1555    apply star_one. constructor.
1556    apply star_refl.
1557    apply star_refl.
1558  reflexivity. traceEq.
1559  unfold S2. inv B1; simpl; econstructor; eauto.
1560
1561(* call internal *)
1562  destruct (H2 f k) as [S1 [A1 B1]].
1563  eapply star_left. eapply step_internal_function; eauto.
1564  eapply star_right. eexact A1.
1565  inv B1; simpl in H3; try contradiction.
1566  (* Out_normal *)
1567  assert (fn_return f = Tvoid /\ vres = Vundef).
1568    destruct (fn_return f); auto || contradiction.
1569  destruct H5. subst vres. apply step_skip_call; auto.
1570  (* Out_return None *)
1571  assert (fn_return f = Tvoid /\ vres = Vundef).
1572    destruct (fn_return f); auto || contradiction.
1573  destruct H6. subst vres.
1574  rewrite <- (is_call_cont_call_cont k H4). rewrite <- H5.
1575  apply step_return_0; auto.
1576  (* Out_return Some *)
1577  destruct H3. subst vres.
1578  rewrite <- (is_call_cont_call_cont k H4). rewrite <- H5.
1579  eapply step_return_1; eauto.
1580  reflexivity. traceEq.
1581
1582(* call external *)
1583  apply star_one. apply step_external_function; auto.
1584Qed.
1585
1586Lemma exec_stmt_steps:
1587   ∀e,m,s,t,m',out.
1588   exec_stmt ge e m s t m' out ->
1589   ∀f,k. exists S,
1590   star step ge (State f s k e m) t S
1591   /\ outcome_state_match e m' f k out S.
1592Proof (proj1 exec_stmt_eval_funcall_steps).
1593
1594Lemma eval_funcall_steps:
1595   ∀m,fd,args,t,m',res.
1596   eval_funcall ge m fd args t m' res ->
1597   ∀k.
1598   is_call_cont k ->
1599   star step ge (Callstate fd args k m) t (Returnstate res k m').
1600Proof (proj2 exec_stmt_eval_funcall_steps).
1601
1602Definition order (x y: unit) := False.
1603
1604Lemma evalinf_funcall_forever:
1605  ∀m,fd,args,T,k.
1606  evalinf_funcall ge m fd args T ->
1607  forever_N step order ge tt (Callstate fd args k m) T.
1608Proof.
1609  cofix CIH_FUN.
1610  assert (∀e,m,s,T,f,k.
1611          execinf_stmt ge e m s T ->
1612          forever_N step order ge tt (State f s k e m) T).
1613  cofix CIH_STMT.
1614  intros. inv H.
1615
1616(* call none *)
1617  eapply forever_N_plus.
1618  apply plus_one. eapply step_call_none; eauto.
1619  apply CIH_FUN. eauto. traceEq.
1620(* call some *)
1621  eapply forever_N_plus.
1622  apply plus_one. eapply step_call_some; eauto.
1623  apply CIH_FUN. eauto. traceEq.
1624
1625(* seq 1 *)
1626  eapply forever_N_plus.
1627  apply plus_one. econstructor.
1628  apply CIH_STMT; eauto. traceEq.
1629(* seq 2 *)
1630  destruct (exec_stmt_steps _ _ _ _ _ _ H0 f (Kseq s2 k)) as [S1 [A1 B1]].
1631  inv B1.
1632  eapply forever_N_plus.
1633  eapply plus_left. constructor. eapply star_trans. eexact A1.
1634  apply star_one. constructor. reflexivity. reflexivity.
1635  apply CIH_STMT; eauto. traceEq.
1636
1637(* ifthenelse true *)
1638  eapply forever_N_plus.
1639  apply plus_one. eapply step_ifthenelse_true; eauto.
1640  apply CIH_STMT; eauto. traceEq.
1641(* ifthenelse false *)
1642  eapply forever_N_plus.
1643  apply plus_one. eapply step_ifthenelse_false; eauto.
1644  apply CIH_STMT; eauto. traceEq.
1645
1646(* while body *)
1647  eapply forever_N_plus.
1648  eapply plus_one. eapply step_while_true; eauto.
1649  apply CIH_STMT; eauto. traceEq.
1650(* while loop *)
1651  destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kwhile a s0 k)) as [S1 [A1 B1]].
1652  eapply forever_N_plus with (s2 := State f (Swhile a s0) k e m1).
1653  eapply plus_left. eapply step_while_true; eauto.
1654  eapply star_right. eexact A1.
1655  inv H3; inv B1; apply step_skip_or_continue_while; auto.
1656  reflexivity. reflexivity.
1657  apply CIH_STMT; eauto. traceEq.
1658
1659(* dowhile body *)
1660  eapply forever_N_plus.
1661  eapply plus_one. eapply step_dowhile.
1662  apply CIH_STMT; eauto.
1663  traceEq.
1664
1665(* dowhile loop *)
1666  destruct (exec_stmt_steps _ _ _ _ _ _ H0 f (Kdowhile a s0 k)) as [S1 [A1 B1]].
1667  eapply forever_N_plus with (s2 := State f (Sdowhile a s0) k e m1).
1668  eapply plus_left. eapply step_dowhile.
1669  eapply star_right. eexact A1.
1670  inv H1; inv B1; eapply step_skip_or_continue_dowhile_true; eauto.
1671  reflexivity. reflexivity.
1672  apply CIH_STMT. eauto.
1673  traceEq.
1674
1675(* for start 1 *)
1676  assert (a1 <> Sskip). red; intros; subst. inv H0.
1677  eapply forever_N_plus.
1678  eapply plus_one. apply step_for_start; auto.
1679  apply CIH_STMT; eauto.
1680  traceEq.
1681
1682(* for start 2 *)
1683  destruct (exec_stmt_steps _ _ _ _ _ _ H1 f (Kseq (Sfor Sskip a2 a3 s0) k)) as [S1 [A1 B1]].
1684  inv B1.
1685  eapply forever_N_plus.
1686  eapply plus_left. eapply step_for_start; eauto.
1687  eapply star_right. eexact A1.
1688  apply step_skip_seq.
1689  reflexivity. reflexivity.
1690  apply CIH_STMT; eauto.
1691  traceEq.
1692
1693(* for body *)
1694  eapply forever_N_plus.
1695  apply plus_one. eapply step_for_true; eauto.
1696  apply CIH_STMT; eauto.
1697  traceEq.
1698
1699(* for next *)
1700  destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kfor2 a2 a3 s0 k)) as [S1 [A1 B1]].
1701  eapply forever_N_plus.
1702  eapply plus_left. eapply step_for_true; eauto.
1703  eapply star_trans. eexact A1.
1704  apply star_one.
1705  inv H3; inv B1; apply step_skip_or_continue_for2; auto.
1706  reflexivity. reflexivity.
1707  apply CIH_STMT; eauto.
1708  traceEq.
1709
1710(* for body *)
1711  destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kfor2 a2 a3 s0 k)) as [S1 [A1 B1]].
1712  destruct (exec_stmt_steps _ _ _ _ _ _ H4 f (Kfor3 a2 a3 s0 k)) as [S2 [A2 B2]].
1713  inv B2.
1714  eapply forever_N_plus.
1715  eapply plus_left. eapply step_for_true; eauto.
1716  eapply star_trans. eexact A1.
1717  eapply star_left. inv H3; inv B1; apply step_skip_or_continue_for2; auto.
1718  eapply star_right. eexact A2.
1719  constructor.
1720  reflexivity. reflexivity. reflexivity. reflexivity. 
1721  apply CIH_STMT; eauto.
1722  traceEq.
1723
1724(* switch *)
1725  eapply forever_N_plus.
1726  eapply plus_one. eapply step_switch; eauto.
1727  apply CIH_STMT; eauto.
1728  traceEq.
1729
1730(* call internal *)
1731  intros. inv H0.
1732  eapply forever_N_plus.
1733  eapply plus_one. econstructor; eauto.
1734  apply H; eauto.
1735  traceEq.
1736Qed.
1737
1738Theorem bigstep_program_terminates_exec:
1739  ∀t,r. bigstep_program_terminates prog t r -> exec_program prog (Terminates t r).
1740Proof.
1741  intros. inv H. unfold ge0, m0 in *.
1742  econstructor.
1743  econstructor. eauto. eauto.
1744  apply eval_funcall_steps. eauto. red; auto.
1745  econstructor.
1746Qed.
1747
1748Theorem bigstep_program_diverges_exec:
1749  ∀T. bigstep_program_diverges prog T ->
1750  exec_program prog (Reacts T) \/
1751  exists t, exec_program prog (Diverges t) /\ traceinf_prefix t T.
1752Proof.
1753  intros. inv H.
1754  set (st := Callstate f nil Kstop m0).
1755  assert (forever step ge0 st T).
1756    eapply forever_N_forever with (order := order).
1757    red; intros. constructor; intros. red in H. elim H.
1758    eapply evalinf_funcall_forever; eauto.
1759  destruct (forever_silent_or_reactive _ _ _ _ _ _ H)
1760  as [A | [t [s' [T' [B [C D]]]]]].
1761  left. econstructor. econstructor. eauto. eauto. auto.
1762  right. exists t. split.
1763  econstructor. econstructor; eauto. eauto. auto.
1764  subst T. rewrite <- (E0_right t) at 1. apply traceinf_prefix_app. constructor.
1765Qed.
1766
1767End BIGSTEP_TO_TRANSITIONS.
1768
1769
1770
1771*)
1772
1773 
Note: See TracBrowser for help on using the repository browser.