source: C-semantics/CexecIO.ma @ 156

Last change on this file since 156 was 156, checked in by campbell, 9 years ago

pdata support

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1
2include "Csem.ma".
3
4include "extralib.ma".
5include "IOMonad.ma".
6
7include "Plogic/russell_support.ma".
8
9ndefinition P_to_P_option_res : ∀A:Type[0].∀P:A → CProp[0].option (res A) → CProp[0] ≝
10  λA,P,a.match a with [ None ⇒ False | Some y ⇒ match y return λ_.CProp[0] with [ Error ⇒ True | OK z ⇒ P z ]].
11
12ndefinition err_inject : ∀A.∀P:A → Prop.∀a:option (res A).∀p:P_to_P_option_res A P a.res (sigma A P) ≝
13  λA.λP:A → Prop.λa:option (res A).λp:P_to_P_option_res A P a.
14  (match a return λa'.a=a' → res (sigma A P) with
15   [ None ⇒ λe1.?
16   | Some b ⇒ λe1.(match b return λb'.b=b' → ? with
17     [ Error ⇒ λ_. Error ?
18     | OK c ⇒ λe2. OK ? (sig_intro A P c ?)
19     ]) (refl ? b)
20   ]) (refl ? a).
21##[ nrewrite > e1 in p; nnormalize; *;
22##| nrewrite > e1 in p; nrewrite > e2; nnormalize; //
23##] nqed.
24
25ndefinition err_eject : ∀A.∀P: A → Prop. res (sigma A P) → res A ≝
26  λA,P,a.match a with [ Error ⇒ Error ? | OK b ⇒
27    match b with [ sig_intro w p ⇒ OK ? w] ].
28
29ndefinition sig_eject : ∀A.∀P: A → Prop. sigma A P → A ≝
30  λA,P,a.match a with [ sig_intro w p ⇒ w].
31
32ncoercion err_inject :
33  ∀A.∀P:A → Prop.∀a.∀p:P_to_P_option_res ? P a.res (sigma A P) ≝ err_inject
34  on a:option (res ?) to res (sigma ? ?).
35ncoercion err_eject : ∀A.∀P:A → Prop.∀c:res (sigma A P).res A ≝ err_eject
36  on _c:res (sigma ? ?) to res ?.
37ncoercion sig_eject : ∀A.∀P:A → Prop.∀c:sigma A P.A ≝ sig_eject
38  on _c:sigma ? ? to ?.
39
40ndefinition bool_of_val_3 : ∀v:val. ∀ty:type. res (Σr:bool. bool_of_val v ty (of_bool r)) ≝
41  λv,ty. match v in val with
42  [ Vint i ⇒ match ty with
43    [ Tint _ _ ⇒ Some ? (OK ? (¬eq i zero))
44    | Tpointer _ _ ⇒ Some ? (OK ? (¬eq i zero))
45    | _ ⇒ Some ? (Error ?)
46    ]
47  | Vfloat f ⇒ match ty with
48    [ Tfloat _ ⇒ Some ? (OK ? (¬Fcmp Ceq f Fzero))
49    | _ ⇒ Some ? (Error ?)
50    ]
51  | Vptr _ _ _ ⇒ match ty with
52    [ Tint _ _ ⇒ Some ? (OK ? true)
53    | Tpointer _ _ ⇒ Some ? (OK ? true)
54    | _ ⇒ Some ? (Error ?)
55    ]
56  | _ ⇒ Some ? (Error ?)
57  ]. nwhd; //;
58##[ ##1,2: nlapply (eq_spec c0 zero); nelim (eq c0 zero);
59  ##[ ##1,3: #e; nrewrite > e; napply bool_of_val_false; //;
60  ##| ##2,4: #ne; napply bool_of_val_true; /2/;
61  ##]
62##| nelim (eq_dec c0 Fzero);
63  ##[ #e; nrewrite > e; nrewrite > (Feq_zero_true …); napply bool_of_val_false; //;
64  ##| #ne; nrewrite > (Feq_zero_false …); //; napply bool_of_val_true; /2/;
65  ##]
66##| ##4,5: napply bool_of_val_true; //
67##] nqed.
68
69ndefinition err_eq ≝ λA,P. λx:res (sigma A P). λy:A.
70  match x with [ Error ⇒ False | OK x' ⇒
71    match x' with [ sig_intro x'' _ ⇒ x'' = y ]].
72(* TODO: can I write a coercion for the above? *)
73
74(* Same as before, except we have to use a slightly different "equality". *)
75
76nlemma bool_of_val_3_complete : ∀v,ty,r. bool_of_val v ty r → ∃b. r = of_bool b ∧ err_eq ?? (bool_of_val_3 v ty) b.
77#v ty r H; nelim H; #v t H'; nelim H';
78  ##[ #i is s ne; @ true; @; //; nwhd; nrewrite > (eq_false … ne); //;
79  ##| #p b i i0 s; @ true; @; //
80  ##| #i p t ne; @ true; @; //; nwhd; nrewrite > (eq_false … ne); //;
81  ##| #p b i p0 t0; @ true; @; //
82  ##| #f s ne; @ true; @; //; nwhd; nrewrite > (Feq_zero_false … ne); //;
83  ##| #i s; @ false; @; //; (*nwhd; nrewrite > (eq_true …); //;*)
84  ##| #p t; @ false; @; //; (*nwhd; nrewrite > (eq_true …); //;*)
85  ##| #s; @ false; @; //; nwhd; nrewrite > (Feq_zero_true …); //;
86  ##]
87nqed.
88
89(* Prove a few minor results to make proof obligations easy. *)
90
91nlemma bind_assoc_r: ∀A,B,C,e,f,g.
92  bind B C (bind A B e f) g = bind A C e (λx.bind B C (f x) g).
93#A B C e f g; ncases e; nnormalize; //; nqed.
94
95nlemma bind_OK: ∀A,B,P,e,f.
96  (∀v. e = OK A v → match f v with [ Error ⇒ True | OK v' ⇒ P v' ]) →
97  match bind A B e f with [ Error ⇒ True | OK v ⇒ P v ].
98#A B P e f; nelim e; /2/; nqed.
99
100nlemma sig_bind_OK: ∀A,B. ∀P:A → Prop. ∀P':B → Prop. ∀e:res (sigma A P). ∀f:sigma A P → res B.
101  (∀v:A. ∀p:P v. match f (sig_intro A P v p) with [ Error ⇒ True | OK v' ⇒ P' v'] ) →
102  match bind (sigma A P) B e f with [ Error ⇒ True | OK v' ⇒ P' v' ].
103#A B P P' e f; nelim e;
104##[ #v0; nelim v0; #v Hv IH; napply IH;
105##| #_; napply I;
106##] nqed.
107
108nlemma bind2_OK: ∀A,B,C,P,e,f.
109  (∀v1,v2. e = OK ? 〈v1,v2〉 → match f v1 v2 with [ Error ⇒ True | OK v' ⇒ P v' ]) →
110  match bind2 A B C e f with [ Error ⇒ True | OK v ⇒ P v ].
111#A B C P e f; nelim e; //; #v; ncases v; /2/; nqed.
112
113nlemma sig_bind2_OK: ∀A,B,C. ∀P:A×B → Prop. ∀P':C → Prop. ∀e:res (sigma (A×B) P). ∀f:A → B → res C.
114  (∀v1:A.∀v2:B. P 〈v1,v2〉 → match f v1 v2 with [ Error ⇒ True | OK v' ⇒ P' v'] ) →
115  match bind2 A B C e f with [ Error ⇒ True | OK v' ⇒ P' v' ].
116#A B C P P' e f; nelim e; //;
117#v0; nelim v0; #v; nelim v; #v1 v2 Hv IH; napply IH; //; nqed.
118
119nlemma reinject: ∀A. ∀P,P':A → Prop. ∀e:res (sigma A P').
120  (∀v:A. err_eq A P' e v → P' v → P v) →
121  match err_eject A P' e with [ Error ⇒ True | OK v' ⇒ P v' ].
122#A P P' e; ncases e; //;
123#v0; nelim v0; #v Pv' IH; /2/;
124nqed.
125
126nlemma bool_val_distinct: Vtrue ≠ Vfalse.
127@; #H; nwhd in H:(??%%); ndestruct; napply (absurd ? e0 one_not_zero);
128nqed.
129
130nlemma bool_of: ∀v,ty,b. bool_of_val v ty (of_bool b) →
131  if b then is_true v ty else is_false v ty.
132#v ty b; ncases b; #H; ninversion H; #v' ty' H' ev et ev; //;
133napply False_ind; napply (absurd ? ev ?);
134##[ ##2: napply sym_neq ##] napply bool_val_distinct;
135nqed.
136
137ndefinition opt_to_res ≝ λA.λv:option A. match v with [ None ⇒ Error A | Some v ⇒ OK A v ].
138nlemma opt_OK: ∀A,P,e.
139  (∀v. e = Some ? v → P v) →
140  match opt_to_res A e with [ Error ⇒ True | OK v ⇒ P v ].
141#A P e; nelim e; /2/;
142nqed.
143
144nlemma opt_bind_OK: ∀A,B,P,e,f.
145  (∀v. e = Some A v → match f v with [ Error ⇒ True | OK v' ⇒ P v' ]) →
146  match bind A B (opt_to_res A e) f with [ Error ⇒ True | OK v ⇒ P v ].
147#A B P e f; nelim e; nnormalize; /2/; nqed.
148
149nlemma extract_subset_pair: ∀A,B,C,P. ∀e:{e:A×B | P e}. ∀Q:A→B→res C. ∀R:C→Prop.
150  (∀a,b. eject ?? e = 〈a,b〉 → P 〈a,b〉 → match Q a b with [ OK v ⇒ R v | Error ⇒ True]) →
151  match match eject ?? e with [ mk_pair a b ⇒ Q a b ] with [ OK v ⇒ R v | Error ⇒ True ].
152#A B C P e Q R; ncases e; #e'; ncases e'; nnormalize;
153##[ #H; napply (False_ind … H);
154##| #e''; ncases e''; #a b Pab H; nnormalize; /2/;
155##] nqed.
156
157(*
158nremark err_later: ∀A,B. ∀e:res A. match e with [ Error ⇒ Error B | OK v ⇒ Error B ] = Error B.
159#A B e; ncases e; //; nqed.
160*)
161
162nlet rec try_cast_null (m:mem) (i:int) (ty:type) (ty':type) : res (Σv':val. cast m (Vint i) ty ty' v') ≝
163match eq i zero with
164[ true ⇒
165  match ty with
166  [ Tpointer _ _ ⇒
167    match ty' with
168    [ Tpointer _ _ ⇒ Some ? (OK ? (Vint i))
169    | Tarray _ _ _ ⇒ Some ? (OK ? (Vint i))
170    | Tfunction _ _ ⇒ Some ? (OK ? (Vint i))
171    | _ ⇒ Some ? (Error ?)
172    ]
173  | Tarray _ _ _ ⇒
174    match ty' with
175    [ Tpointer _ _ ⇒ Some ? (OK ? (Vint i))
176    | Tarray _ _ _ ⇒ Some ? (OK ? (Vint i))
177    | Tfunction _ _ ⇒ Some ? (OK ? (Vint i))
178    | _ ⇒ Some ? (Error ?)
179    ]
180  | Tfunction _ _ ⇒
181    match ty' with
182    [ Tpointer _ _ ⇒ Some ? (OK ? (Vint i))
183    | Tarray _ _ _ ⇒ Some ? (OK ? (Vint i))
184    | Tfunction _ _ ⇒ Some ? (OK ? (Vint i))
185    | _ ⇒ Some ? (Error ?)
186    ]
187  | _ ⇒ Some ? (Error ?)
188  ]
189| false ⇒ Some ? (Error ?)
190]. nwhd; //; nlapply (eq_spec i zero); nrewrite > c0; #e; nrewrite > e;
191   napply cast_pp_z; //; nqed.
192
193ndefinition ms_eq_dec : ∀s1,s2:memory_space. (s1 = s2) + (s1 ≠ s2).
194#s1; ncases s1; #s2; ncases s2; /2/; @2; @; #H; ndestruct; nqed.
195
196nlet rec exec_cast (m:mem) (v:val) (ty:type) (ty':type) : res (Σv':val. cast m v ty ty' v') ≝
197match v with
198[ Vint i ⇒
199  match ty with
200  [ Tint sz1 si1 ⇒
201    match ty' with
202    [ Tint sz2 si2 ⇒ Some ? (OK ? (Vint (cast_int_int sz2 si2 i)))
203    | Tfloat sz2 ⇒ Some ? (OK ? (Vfloat (cast_float_float sz2 (cast_int_float si1 i))))
204    | Tpointer _ _ ⇒ Some (res val) (r ← try_cast_null m i ty ty';: OK val r)
205    | Tarray _ _ _ ⇒ Some (res val) (r ← try_cast_null m i ty ty';: OK val r)
206    | Tfunction _ _ ⇒ Some (res val) (r ← try_cast_null m i ty ty';: OK val r)
207    | _ ⇒ Some ? (Error ?)
208    ]
209  | Tpointer _ _ ⇒ Some (res val) (r ← try_cast_null m i ty ty';: OK val r)
210  | Tarray _ _ _ ⇒ Some (res val) (r ← try_cast_null m i ty ty';: OK val r)
211  | Tfunction _ _ ⇒ Some (res val) (r ← try_cast_null m i ty ty';: OK val r)
212  | _ ⇒ Some ? (Error ?)
213  ]
214| Vfloat f ⇒
215  match ty with
216  [ Tfloat sz ⇒
217    match ty' with
218    [ Tint sz' si' ⇒ Some ? (OK ? (Vint (cast_int_int sz' si' (cast_float_int si' f))))
219    | Tfloat sz' ⇒ Some ? (OK ? (Vfloat (cast_float_float sz' f)))
220    | _ ⇒ Some ? (Error ?)
221    ]
222  | _ ⇒ Some ? (Error ?)
223  ]
224| Vptr p b ofs ⇒
225  Some ? (
226    s ← match ty with [ Tpointer s _ ⇒ OK ? s | Tarray s _ _ ⇒ OK ? s | Tfunction _ _ ⇒ OK ? Code | _ ⇒ Error ? ];:
227    u ← match ms_eq_dec p s with [ inl _ ⇒ OK ? something | inr _ ⇒ Error ? ];:
228    s' ← match ty' with
229         [ Tpointer s _ ⇒ OK ? s | Tarray s _ _ ⇒ OK ? s | Tfunction _ _ ⇒ OK ? Code
230         | _ ⇒ Error ? ];:
231    if is_pointer_compat (block_space m b) s'
232    then OK ? (Vptr s' b ofs)
233    else Error ?)
234| _ ⇒ Some ? (Error ?)
235]. nwhd; //;
236##[ ##1,2,3,4,5,6: napply sig_bind_OK; #v'; #H; ndestruct; napply H;
237##| napply bind_OK; #s es;
238    ncut (type_space ty s);
239    ##[ ncases ty in es ⊢ %;
240      ##[ #e; ##| ##3,9: #a e; ##| ##2,4,6,7,8: #a b e; ##| #a b c e; ##] nwhd in e:(??%?); ndestruct; //;
241    ##| #Hty;
242        napply bind_OK; #u1 eeq;
243        napply bind_OK; #s' es';
244        ncut (type_space ty' s');
245        ##[ ncases ty' in es' ⊢ %; ##[ #e; ##| ##3,9: #a e; ##| ##2,4,6,7,8: #a b e; ##| #a b c e; ##]
246            nwhd in e:(??%?); ndestruct; //;
247        ##| #Hty';
248            ncut (s = c0). nelim (ms_eq_dec c0 s) in eeq; //; nnormalize; #_; #e; ndestruct.
249            #e; nrewrite < e;
250            nwhd in match (is_pointer_compat ??) in ⊢ %;
251            ncases (pointer_compat_dec (block_space m c1) s'); #Hcompat;
252            nwhd; /2/;
253        ##]
254    ##]
255##] nqed.
256
257ndefinition load_value_of_type' ≝
258λty,m,l. match l with [ mk_pair pl ofs ⇒ match pl with [ mk_pair psp loc ⇒
259  load_value_of_type ty m psp loc ofs ] ].
260
261(* To make the evaluation of bare lvalue expressions invoke exec_lvalue with
262   a structurally smaller value, we break out the surrounding Expr constructor
263   and use exec_lvalue'. *)
264
265nlet rec exec_expr (ge:genv) (en:env) (m:mem) (e:expr) on e : res (Σr:val. eval_expr ge en m e r) ≝
266match e with
267[ Expr e' ty ⇒
268  match e' with
269  [ Econst_int i ⇒ Some ? (OK ? (Vint i))
270  | Econst_float f ⇒ Some ? (OK ? (Vfloat f))
271  | Evar _ ⇒ Some ? (
272      l ← exec_lvalue' ge en m e' ty;:
273      opt_to_res ? (load_value_of_type' ty m l))
274  | Ederef _ ⇒ Some ? (
275      l ← exec_lvalue' ge en m e' ty;:
276      opt_to_res ? (load_value_of_type' ty m l))
277  | Efield _ _ ⇒ Some ? (
278      l ← exec_lvalue' ge en m e' ty;:
279      opt_to_res ? (load_value_of_type' ty m l))
280  | Eaddrof a ⇒ Some ? (
281      〈pl, ofs〉 ← exec_lvalue ge en m a;:
282      OK ? (match pl with [ mk_pair pcl loc ⇒ Vptr pcl loc ofs ]))
283  | Esizeof ty' ⇒ Some ? (OK ? (Vint (repr (sizeof ty'))))
284  | Eunop op a ⇒ Some ? (
285      v1 ← exec_expr ge en m a;:
286      opt_to_res ? (sem_unary_operation op v1 (typeof a)))
287  | Ebinop op a1 a2 ⇒ Some ? (
288      v1 ← exec_expr ge en m a1;:
289      v2 ← exec_expr ge en m a2;:
290      opt_to_res ? (sem_binary_operation op v1 (typeof a1) v2 (typeof a2) m))
291  | Econdition a1 a2 a3 ⇒ Some ? (
292      v ← exec_expr ge en m a1;:
293      b ← bool_of_val_3 v (typeof a1);:
294      match b return λ_.res val with [ true ⇒ (exec_expr ge en m a2) | false ⇒ (exec_expr ge en m a3) ])
295(*      if b then exec_expr ge en m a2 else exec_expr ge en m a3)*)
296  | Eorbool a1 a2 ⇒ Some ? (
297      v1 ← exec_expr ge en m a1;:
298      b1 ← bool_of_val_3 v1 (typeof a1);:
299      match b1 return λ_.res val with [ true ⇒ OK ? Vtrue | false ⇒
300        v2 ← exec_expr ge en m a2;:
301        b2 ← bool_of_val_3 v2 (typeof a2);:
302        OK ? (of_bool b2) ])
303  | Eandbool a1 a2 ⇒ Some ? (
304      v1 ← exec_expr ge en m a1;:
305      b1 ← bool_of_val_3 v1 (typeof a1);:
306      match b1 return λ_.res val with [ true ⇒
307        v2 ← exec_expr ge en m a2;:
308        b2 ← bool_of_val_3 v2 (typeof a2);:
309        OK ? (of_bool b2)
310      | false ⇒ OK ? Vfalse ])
311  | Ecast ty' a ⇒ Some ? (
312      v ← exec_expr ge en m a;:
313      exec_cast m v (typeof a) ty')
314  ]
315]
316and exec_lvalue' (ge:genv) (en:env) (m:mem) (e':expr_descr) (ty:type) on e' : res (Σr:memory_space × block × int. eval_lvalue ge en m (Expr e' ty) (\fst (\fst r)) (\snd (\fst r)) (\snd r)) ≝
317  match e' with
318  [ Evar id ⇒
319      match (get … id en) with
320      [ None ⇒ Some ? (〈sp,l〉 ← opt_to_res ? (find_symbol ? ? ge id);: OK ? 〈〈sp,l〉,zero〉) (* global *)
321      | Some loc ⇒ Some ? (OK ? 〈〈Any,loc〉,zero〉) (* local *)
322      ]
323  | Ederef a ⇒ Some ? (
324      v ← exec_expr ge en m a;:
325      match v with
326      [ Vptr sp l ofs ⇒ OK ? 〈〈sp,l〉,ofs〉
327      | _ ⇒ Error ?
328      ])
329  | Efield a i ⇒
330      match (typeof a) with
331      [ Tstruct id fList ⇒ Some ? (
332          〈pl,ofs〉 ← exec_lvalue ge en m a;:
333          delta ← field_offset i fList;:
334          OK ? 〈pl,add ofs (repr delta)〉)
335      | Tunion id fList ⇒ Some ? (
336          〈pl,ofs〉 ← exec_lvalue ge en m a;:
337          OK ? 〈pl,ofs〉)
338      | _ ⇒ Some ? (Error ?)
339      ]
340  | _ ⇒ Some ? (Error ?)
341  ]
342and exec_lvalue (ge:genv) (en:env) (m:mem) (e:expr) on e : res (Σr:memory_space × block × int. eval_lvalue ge en m e (\fst (\fst r)) (\snd (\fst r)) (\snd r)) ≝
343match e with [ Expr e' ty ⇒ exec_lvalue' ge en m e' ty ].
344nwhd;
345##[ ##1,2: //
346##| ##3,4:
347    napply sig_bind_OK; nrewrite > c4; #x; ncases x; #y; ncases y; #sp loc ofs H;
348    napply opt_OK;  #v ev; nwhd in ev:(??%?); napply (eval_Elvalue … H ev);
349##| napply sig_bind2_OK; #x; ncases x; #sp loc ofs H;
350    nwhd; napply eval_Eaddrof; //;
351##| napply sig_bind_OK; #v1 Hv1;
352    napply opt_OK; #v ev;
353    napply (eval_Eunop … Hv1 ev);
354##| napply sig_bind_OK; #v1 Hv1;
355    napply sig_bind_OK; #v2 Hv2;
356    napply opt_OK; #v ev;
357    napply (eval_Ebinop … Hv1 Hv2 ev);
358##| napply sig_bind_OK; #v Hv;
359    napply sig_bind_OK; #v' Hv';
360    napply (eval_Ecast … Hv Hv');
361##| napply sig_bind_OK; #vb Hvb;
362    napply sig_bind_OK; #b;
363    ncases b; #Hb; napply reinject; #v ev Hv;
364    ##[ napply (eval_Econdition_true … Hvb ? Hv);  napply (bool_of ??? Hb);
365    ##| napply (eval_Econdition_false … Hvb ? Hv);  napply (bool_of ??? Hb);
366    ##]
367##| napply sig_bind_OK; #v1 Hv1;
368    napply sig_bind_OK; #b1; ncases b1; #Hb1;
369    ##[ napply sig_bind_OK; #v2 Hv2;
370        napply sig_bind_OK; #b2 Hb2;
371        napply (eval_Eandbool_2 … Hv1 … Hv2);
372        ##[ napply (bool_of … Hb1); ##| napply Hb2; ##]
373    ##| napply (eval_Eandbool_1 … Hv1); napply (bool_of … Hb1);
374    ##]
375##| napply sig_bind_OK; #v1 Hv1;
376    napply sig_bind_OK; #b1; ncases b1; #Hb1;
377    ##[ napply (eval_Eorbool_1 … Hv1); napply (bool_of … Hb1);
378    ##| napply sig_bind_OK; #v2 Hv2;
379        napply sig_bind_OK; #b2 Hb2;
380        napply (eval_Eorbool_2 … Hv1 … Hv2);
381        ##[ napply (bool_of … Hb1); ##| napply Hb2; ##]
382    ##]
383##| //
384##| napply sig_bind_OK; nrewrite > c5; #x; ncases x; #y; ncases y; #sp l ofs H;
385    napply opt_OK; #v ev; napply (eval_Elvalue … H ev);
386##| //
387##| //
388##| napply opt_bind_OK; #sl; ncases sl; #sp l el; napply eval_Evar_global; /2/;
389##| napply (eval_Evar_local … c3);
390##| napply sig_bind_OK; #v; ncases v; //; #sp l ofs Hv; nwhd;
391    napply eval_Ederef; //
392##| ##19,20,21,22,23,24,25,26,27,28,29,30,31,32: //
393##| napply sig_bind2_OK; #x; ncases x; #sp l ofs H;
394    napply bind_OK; #delta Hdelta;
395    napply (eval_Efield_struct … H c5 Hdelta);
396##| napply sig_bind2_OK; #x; ncases x; #sp l ofs H;
397    napply (eval_Efield_union … H c5);
398##| //
399##] nqed.
400
401(* TODO: Can we do this sensibly with a map combinator? *)
402nlet rec exec_exprlist (ge:genv) (e:env) (m:mem) (l:list expr) on l : res (Σvl:list val. eval_exprlist ge e m l vl) ≝
403match l with
404[ nil ⇒ Some ? (OK ? (nil val))
405| cons e1 es ⇒ Some ? (
406  v ← exec_expr ge e m e1;:
407  vs ← exec_exprlist ge e m es;:
408  OK ? (cons val v vs))
409]. nwhd; //;
410napply sig_bind_OK; #v Hv;
411napply sig_bind_OK; #vs Hvs;
412nnormalize; /2/;
413nqed.
414
415(* Don't really want to use subset rather than sigma here, but can't be bothered
416   with *another* set of coercions. XXX: why do I have to get the recursive
417   call's property manually? *)
418
419nlet rec exec_alloc_variables (en:env) (m:mem) (l:list (ident × type)) on l : { r:env × mem | alloc_variables en m l (\fst r) (\snd r) } ≝
420match l with
421[ nil ⇒ Some ? 〈en, m〉
422| cons h vars ⇒
423  match h with [ mk_pair id ty ⇒
424    match alloc m 0 (sizeof ty) Any with [ mk_pair m1 b1 ⇒
425      match exec_alloc_variables (set … id b1 en) m1 vars with
426      [ sig_intro r p ⇒ r ]
427]]]. nwhd;
428##[ //;
429##| nelim (exec_alloc_variables (set ident ? ? c3 c7 en) c6 c1);
430    #H; nelim H; //; #H0; nelim H0; nnormalize; #en' m' IH;
431napply (alloc_variables_cons … IH); /2/;
432nqed.
433
434(* TODO: can we establish that length params = length vs in advance? *)
435nlet rec exec_bind_parameters (e:env) (m:mem) (params:list (ident × type)) (vs:list val) on params : res (Σm2:mem. bind_parameters e m params vs m2) ≝
436  match params with
437  [ nil ⇒ match vs with [ nil ⇒ Some ? (OK ? m) | cons _ _ ⇒ Some ? (Error ?) ]
438  | cons idty params' ⇒ match idty with [ mk_pair id ty ⇒
439      match vs with
440      [ nil ⇒ Some ? (Error ?)
441      | cons v1 vl ⇒ Some ? (
442          b ← opt_to_res ? (get … id e);:
443          m1 ← opt_to_res ? (store_value_of_type ty m Any b zero v1);:
444          err_eject ?? (exec_bind_parameters e m1 params' vl)) (* FIXME: don't want to have to eject here *)
445      ]
446  ] ].
447nwhd; //;
448napply opt_bind_OK; #b eb;
449napply opt_bind_OK; #m1 em1;
450napply reinject; #m2 em2 Hm2;
451napply (bind_parameters_cons … eb em1 Hm2);
452nqed.
453
454ndefinition is_not_void : ∀t:type. res (Σu:unit. t ≠ Tvoid) ≝
455λt. match t with
456[ Tvoid ⇒ Some ? (Error ?)
457| _ ⇒ Some ? (OK ??)
458]. nwhd; //; @; #H; ndestruct; nqed.
459
460ninductive decide : Type ≝
461| dy : decide | dn : decide.
462
463ndefinition dodecide : ∀P:Prop.∀d.∀p:(match d with [ dy ⇒ P | dn ⇒ ¬P ]).P + ¬P.
464#P d;ncases d;/2/; nqed.
465
466ncoercion decide_inject :
467  ∀P:Prop.∀d.∀p:(match d with [ dy ⇒ P | dn ⇒ ¬P ]).P + ¬P ≝ dodecide
468  on d:decide to ? + (¬?).
469
470ndefinition dodecide2 : ∀P:Prop.∀d.∀p:(match d with [ dy ⇒ P | dn ⇒ True ]).res P.
471#P d; ncases d; nnormalize; #p; ##[ napply (OK ? p); ##| napply Error ##] nqed.
472
473ncoercion decide_inject2 :
474  ∀P:Prop.∀d.∀p:(match d with [ dy ⇒ P | dn ⇒ True ]).res P ≝ dodecide2
475  on d:decide to res ?.
476
477alias id "Tint" = "cic:/matita/c-semantics/Csyntax/type.con(0,2,0)".
478alias id "Tfloat" = "cic:/matita/c-semantics/Csyntax/type.con(0,3,0)".
479ndefinition sz_eq_dec : ∀s1,s2:intsize. (s1 = s2) + (s1 ≠ s2).
480#s1; ncases s1; #s2; ncases s2; /2/; @2; @; #H; ndestruct; nqed.
481ndefinition sg_eq_dec : ∀s1,s2:signedness. (s1 = s2) + (s1 ≠ s2).
482#s1; ncases s1; #s2; ncases s2; /2/; @2; @; #H; ndestruct; nqed.
483ndefinition fs_eq_dec : ∀s1,s2:floatsize. (s1 = s2) + (s1 ≠ s2).
484#s1; ncases s1; #s2; ncases s2; /2/; @2; @; #H; ndestruct; nqed.
485
486nlet rec assert_type_eq (t1,t2:type) : res (t1 = t2) ≝
487match t1 with
488[ Tvoid ⇒ match t2 with [ Tvoid ⇒ dy | _ ⇒ dn ]
489| Tint sz sg ⇒ match t2 with [ Tint sz' sg' ⇒ match sz_eq_dec sz sz' with [ inl _ ⇒ match sg_eq_dec sg sg' with [ inl _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ]
490| Tfloat f ⇒ match t2 with [ Tfloat f' ⇒ match fs_eq_dec f f' with [ inl _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ]
491| Tpointer s t ⇒ match t2 with [ Tpointer s' t' ⇒
492    match ms_eq_dec s s' with [ inl _ ⇒
493      match assert_type_eq t t' with [ OK _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ]
494| Tarray s t n ⇒ match t2 with [ Tarray s' t' n' ⇒
495    match ms_eq_dec s s' with [ inl _ ⇒
496      match assert_type_eq t t' with [ OK _ ⇒
497        match decidable_eq_Z_Type n n' with [ inl _ ⇒ dy | inr _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ]
498| Tfunction tl t ⇒ match t2 with [ Tfunction tl' t' ⇒ match assert_typelist_eq tl tl' with [ OK _ ⇒
499    match assert_type_eq t t' with [ OK _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ]
500| Tstruct i fl ⇒
501    match t2 with [ Tstruct i' fl' ⇒ match ident_eq i i' with [ inl _ ⇒
502      match assert_fieldlist_eq fl fl' with [ OK _ ⇒ dy | _ ⇒ dn ] | inr _ ⇒ dn ] |  _ ⇒ dn ]
503| Tunion i fl ⇒
504    match t2 with [ Tunion i' fl' ⇒ match ident_eq i i' with [ inl _ ⇒
505      match assert_fieldlist_eq fl fl' with [ OK _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] |  _ ⇒ dn ]
506| Tcomp_ptr i ⇒ match t2 with [ Tcomp_ptr i' ⇒ match ident_eq i i' with [ inl _ ⇒ dy | inr _ ⇒ dn ] | _ ⇒ dn ]
507]
508and assert_typelist_eq (tl1,tl2:typelist) : res (tl1 = tl2) ≝
509match tl1 with
510[ Tnil ⇒ match tl2 with [ Tnil ⇒ dy | _ ⇒ dn ]
511| Tcons t1 ts1 ⇒ match tl2 with [ Tnil ⇒ dn | Tcons t2 ts2 ⇒
512    match assert_type_eq t1 t2 with [ OK _ ⇒
513      match assert_typelist_eq ts1 ts2 with [ OK _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] ]
514]
515and assert_fieldlist_eq (fl1,fl2:fieldlist) : res (fl1 = fl2) ≝
516match fl1 with
517[ Fnil ⇒ match fl2 with [ Fnil ⇒ dy | _ ⇒ dn ]
518| Fcons i1 t1 fs1 ⇒ match fl2 with [ Fnil ⇒ dn | Fcons i2 t2 fs2 ⇒
519    match ident_eq i1 i2 with [ inl _ ⇒
520      match assert_type_eq t1 t2 with [ OK _ ⇒
521        match assert_fieldlist_eq fs1 fs2 with [ OK _ ⇒ dy | _ ⇒ dn ]
522        | _ ⇒ dn ] | _ ⇒ dn ] ]
523].
524(* A poor man's clear, otherwise automation picks up recursive calls without
525   checking that the argument is smaller. *)
526ngeneralize in assert_type_eq;
527ngeneralize in assert_typelist_eq;
528ngeneralize in assert_fieldlist_eq; #avoid1; #_; #avoid2; #_; #avoid3; #_; nwhd; //;
529(* XXX: I have no idea why the first // didn't catch these. *)
530//; //; //; //; //; //; //; //; //;
531nqed.
532
533nlet rec is_is_call_cont (k:cont) : (is_call_cont k) + (¬is_call_cont k) ≝
534match k with
535[ Kstop ⇒ dy
536| Kcall _ _ _ _ ⇒ dy
537| _ ⇒ dn
538]. nwhd; //; @; #H; nelim H; nqed.
539
540nlet rec is_Sskip (s:statement) : (s = Sskip) + (s ≠ Sskip) ≝
541match s with
542[ Sskip ⇒ dy
543| _ ⇒ dn
544].
545##[ //;
546##| ##*: @; #H; ndestruct;
547##] nqed.
548
549(* IO monad *)
550
551(* Interactions are function calls that return a value and do not change
552   the rest of the Clight program's state. *)
553ndefinition io_out ≝ (ident × (list eventval)).
554
555ndefinition do_io : ident → list eventval → IO eventval io_out eventval ≝
556λfn,args. Interact ?? eventval 〈fn,args〉 (λres. Value ?? eventval res).
557
558ndefinition ret: ∀T. T → (IO eventval io_out T) ≝
559λT,x.(Value ?? T x).
560
561(* Checking types of values given to / received from an external function call. *)
562
563ndefinition check_eventval : ∀ev:eventval. ∀ty:typ. res (Σv:val. eventval_match ev ty v) ≝
564λev,ty.
565match ty with
566[ Tint ⇒ match ev with [ EVint i ⇒ Some ? (OK ? (Vint i)) | _ ⇒ Some ? (Error ?) ]
567| Tfloat ⇒ match ev with [ EVfloat f ⇒ Some ? (OK ? (Vfloat f)) | _ ⇒ Some ? (Error ?) ]
568| _ ⇒ Some ? (Error ?)
569]. nwhd; //; nqed.
570
571ndefinition check_eventval' : ∀v:val. ∀ty:typ. res (Σev:eventval. eventval_match ev ty v) ≝
572λv,ty.
573match ty with
574[ Tint ⇒ match v with [ Vint i ⇒ Some ? (OK ? (EVint i)) | _ ⇒ Some ? (Error ?) ]
575| Tfloat ⇒ match v with [ Vfloat f ⇒ Some ? (OK ? (EVfloat f)) | _ ⇒ Some ? (Error ?) ]
576| _ ⇒ Some ? (Error ?)
577]. nwhd; //; nqed.
578
579nlet rec check_eventval_list (vs: list val) (tys: list typ) : res (Σevs:list eventval. eventval_list_match evs tys vs) ≝
580match vs with
581[ nil ⇒ match tys with [ nil ⇒ Some ? (OK ? (nil ?)) | _ ⇒ Some ? (Error ?) ]
582| cons v vt ⇒
583  match tys with
584  [ nil ⇒ Some ? (Error ?)
585  | cons ty tyt ⇒ Some ? (
586    ev ← check_eventval' v ty;:
587    evt ← check_eventval_list vt tyt;:
588    OK ? ((sig_eject ?? ev)::evt))
589  ]
590]. nwhd; //;
591napply sig_bind_OK; #ev Hev;
592napply sig_bind_OK; #evt Hevt;
593nnormalize; /2/;
594nqed.
595
596(* execution *)
597
598ndefinition store_value_of_type' ≝
599λty,m,l,v.
600match l with [ mk_pair pl ofs ⇒
601  match pl with [ mk_pair pcl loc ⇒
602    store_value_of_type ty m pcl loc ofs v ] ].
603
604nlet rec exec_step (ge:genv) (st:state) on st : (IO eventval io_out (Σr:trace × state. step ge st (\fst r) (\snd r))) ≝
605match st with
606[ State f s k e m ⇒
607  match s with
608  [ Sassign a1 a2 ⇒ Some ? (
609    ! l ← exec_lvalue ge e m a1;:
610    ! v2 ← exec_expr ge e m a2;:
611    ! m' ← store_value_of_type' (typeof a1) m l v2;:
612    ret ? 〈E0, State f Sskip k e m'〉)
613  | Scall lhs a al ⇒ Some ? (
614    ! vf ← exec_expr ge e m a;:
615    ! vargs ← exec_exprlist ge e m al;:
616    ! fd ← find_funct ? ? ge vf;:
617    ! p ← err_to_io … (assert_type_eq (type_of_fundef fd) (typeof a));:
618(*
619    ! k' ← match lhs with
620         [ None ⇒ ret ? (Kcall (None ?) f e k)
621         | Some lhs' ⇒
622           ! locofs ← exec_lvalue ge e m lhs';:
623           ret ? (Kcall (Some ? 〈sig_eject ?? locofs, typeof lhs'〉) f e k)
624         ];:
625    ret ? 〈E0, Callstate fd vargs k' m〉)
626*)
627    match lhs with
628         [ None ⇒ ret ? 〈E0, Callstate fd vargs (Kcall (None ?) f e k) m〉
629         | Some lhs' ⇒
630           ! locofs ← exec_lvalue ge e m lhs';:
631           ret ? 〈E0, Callstate fd vargs (Kcall (Some ? 〈sig_eject ?? locofs, typeof lhs'〉) f e k) m〉
632         ])
633  | Ssequence s1 s2 ⇒ Some ? (ret ? 〈E0, State f s1 (Kseq s2 k) e m〉)
634  | Sskip ⇒
635      match k with
636      [ Kseq s k' ⇒ Some ? (ret ? 〈E0, State  f s k' e m〉)
637      | Kstop ⇒
638          match fn_return f with
639          [ Tvoid ⇒ Some ? (ret ? 〈E0, Returnstate Vundef k (free_list m (blocks_of_env e))〉)
640          | _ ⇒ Some ? (Wrong ???)
641          ]
642      | Kcall _ _ _ _ ⇒
643          match fn_return f with
644          [ Tvoid ⇒ Some ? (ret ? 〈E0, Returnstate Vundef k (free_list m (blocks_of_env e))〉)
645          | _ ⇒ Some ? (Wrong ???)
646          ]
647      | Kwhile a s' k' ⇒ Some ? (ret ? 〈E0, State f (Swhile a s') k' e m〉)
648      | Kdowhile a s' k' ⇒ Some ? (
649          ! v ← exec_expr ge e m a;:
650          ! b ← bool_of_val_3 v (typeof a);:
651          match b with
652          [ true ⇒ ret ? 〈E0, State f (Sdowhile a s') k' e m〉
653          | false ⇒ ret ? 〈E0, State f Sskip k' e m〉
654          ])
655      | Kfor2 a2 a3 s' k' ⇒ Some ? (ret ? 〈E0, State f a3 (Kfor3 a2 a3 s' k') e m〉)
656      | Kfor3 a2 a3 s' k' ⇒ Some ? (ret ? 〈E0, State f (Sfor Sskip a2 a3 s') k' e m〉)
657      | Kswitch k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉)
658      | _ ⇒ Some ? (Wrong ???)
659      ]
660  | Scontinue ⇒
661      match k with
662      [ Kseq s' k' ⇒ Some ? (ret ? 〈E0, State f Scontinue k' e m〉)
663      | Kwhile a s' k' ⇒ Some ? (ret ? 〈E0, State f (Swhile a s') k' e m〉)
664      | Kdowhile a s' k' ⇒ Some ? (
665          ! v ← exec_expr ge e m a;:
666          ! b ← bool_of_val_3 v (typeof a);:
667          match b with
668          [ true ⇒ ret ? 〈E0, State f (Sdowhile a s') k' e m〉
669          | false ⇒ ret ? 〈E0, State f Sskip k' e m〉
670          ])
671      | Kfor2 a2 a3 s' k' ⇒ Some ? (ret ? 〈E0, State f a3 (Kfor3 a2 a3 s' k') e m〉)
672      | Kswitch k' ⇒ Some ? (ret ? 〈E0, State f Scontinue k' e m〉)
673      | _ ⇒ Some ? (Wrong ???)
674      ]
675  | Sbreak ⇒
676      match k with
677      [ Kseq s' k' ⇒ Some ? (ret ? 〈E0, State f Sbreak k' e m〉)
678      | Kwhile a s' k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉)
679      | Kdowhile a s' k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉)
680      | Kfor2 a2 a3 s' k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉)
681      | Kswitch k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉)
682      | _ ⇒ Some ? (Wrong ???)
683      ]
684  | Sifthenelse a s1 s2 ⇒ Some ? (
685      ! v ← exec_expr ge e m a;:
686      ! b ← bool_of_val_3 v (typeof a);:
687      ret ? 〈E0, State f (if b then s1 else s2) k e m〉)
688  | Swhile a s' ⇒ Some ? (
689      ! v ← exec_expr ge e m a;:
690      ! b ← bool_of_val_3 v (typeof a);:
691      ret ? 〈E0, if b then State f s' (Kwhile a s' k) e m
692                     else State f Sskip k e m〉)
693  | Sdowhile a s' ⇒ Some ? (ret ? 〈E0, State f s' (Kdowhile a s' k) e m〉)
694  | Sfor a1 a2 a3 s' ⇒
695      match is_Sskip a1 with
696      [ inl _ ⇒ Some ? (
697          ! v ← exec_expr ge e m a2;:
698          ! b ← bool_of_val_3 v (typeof a2);:
699          ret ? 〈E0, State f (if b then s' else Sskip) (if b then (Kfor2 a2 a3 s' k) else k) e m〉)
700      | inr _ ⇒ Some ? (ret ? 〈E0, State f a1 (Kseq (Sfor Sskip a2 a3 s') k) e m〉)
701      ]
702  | Sreturn a_opt ⇒
703    match a_opt with
704    [ None ⇒ match fn_return f with
705      [ Tvoid ⇒ Some ? (ret ? 〈E0, Returnstate Vundef (call_cont k) (free_list m (blocks_of_env e))〉)
706      | _ ⇒ Some ? (Wrong ???)
707      ]
708    | Some a ⇒ Some ? (
709        ! u ← is_not_void (fn_return f);:
710        ! v ← exec_expr ge e m a;:
711        ret ? 〈E0, Returnstate v (call_cont k) (free_list m (blocks_of_env e))〉)
712    ]
713  | Sswitch a sl ⇒ Some ? (
714      ! v ← exec_expr ge e m a;:
715      match v with [ Vint n ⇒ ret ? 〈E0, State f (seq_of_labeled_statement (select_switch n sl)) (Kswitch k) e m〉
716                   | _ ⇒ Wrong ??? ])
717  | Slabel lbl s' ⇒ Some ? (ret ? 〈E0, State f s' k e m〉)
718  | Sgoto lbl ⇒
719      match find_label lbl (fn_body f) (call_cont k) with
720      [ Some sk' ⇒ match sk' with [ mk_pair s' k' ⇒ Some ? (ret ? 〈E0, State f s' k' e m〉) ]
721      | None ⇒ Some ? (Wrong ???)
722      ]
723  ]
724| Callstate f0 vargs k m ⇒
725  match f0 with
726  [ Internal f ⇒ Some ? (
727    match exec_alloc_variables empty_env m ((fn_params f) @ (fn_vars f)) with [ mk_pair e m1 ⇒
728      ! m2 ← exec_bind_parameters e m1 (fn_params f) vargs;:
729      ret ? 〈E0, State f (fn_body f) k e m2〉
730    ])
731  | External f argtys retty ⇒ Some ? (
732      ! evargs ← check_eventval_list vargs (typlist_of_typelist argtys);:
733      ! evres ← do_io f evargs;:
734      ! vres ← check_eventval evres (proj_sig_res (signature_of_type argtys retty));:
735      ret ? 〈(Eextcall f evargs evres), Returnstate vres k m〉)
736  ]
737| Returnstate res k m ⇒
738  match k with
739  [ Kcall r f e k' ⇒
740    match r with
741    [ None ⇒
742      match res with
743      [ Vundef ⇒ Some ? (ret ? 〈E0, (State f Sskip k' e m)〉)
744      | _ ⇒ Some ? (Wrong ???)
745      ]
746    | Some r' ⇒
747      match r' with [ mk_pair l ty ⇒
748        Some ? (
749          ! m' ← store_value_of_type' ty m l res;:
750          ret ? 〈E0, (State f Sskip k' e m')〉)
751      ]
752    ]
753  | _ ⇒ Some ? (Wrong ???)
754  ]
755]. nwhd; //;
756##[ nrewrite > c7; napply step_skip_call; //; napply c8;
757##| napply step_skip_or_continue_while; @; //;
758##| napply sig_bindIO_OK; #v Hv;
759    napply sig_bindIO_OK; #b; ncases b; #Hb;
760    ##[ napply (step_skip_or_continue_dowhile_true … Hv);
761      ##[ @; // ##| napply (bool_of … Hb); ##]
762    ##| napply (step_skip_or_continue_dowhile_false … Hv);
763      ##[ @; // ##| napply (bool_of … Hb); ##]
764    ##]
765##| napply step_skip_or_continue_for2; @; //;
766##| napply step_skip_break_switch; @; //;
767##| nrewrite > c11; napply step_skip_call; //; napply c12;
768##| napply sig_bindIO_OK; #x; ncases x; #y; ncases y; #pcl loc ofs Hlval;
769    napply sig_bindIO_OK; #v2 Hv2;
770    napply opt_bindIO_OK; #m' em';
771    nwhd; napply (step_assign … Hlval Hv2 em');
772##| napply sig_bindIO_OK; #vf Hvf;
773    napply sig_bindIO_OK; #vargs Hvargs;
774    napply opt_bindIO_OK; #fd efd;
775    napply bindIO_OK; #ety;
776    ncases c6; nwhd;
777    ##[ napply (step_call_none … Hvf Hvargs efd ety);
778    ##| #lhs';
779        napply sig_bindIO_OK; #x; ncases x; #y; ncases y; #pcl loc ofs Hlocofs;
780        nwhd; napply (step_call_some … Hlocofs Hvf Hvargs efd ety);
781    ##]
782##| napply sig_bindIO_OK; #v Hv;
783    napply sig_bindIO_OK; #b; ncases b; #Hb;
784    ##[ napply (step_ifthenelse_true … Hv); napply (bool_of … Hb);
785    ##| napply (step_ifthenelse_false … Hv); napply (bool_of … Hb)
786    ##]
787##| napply sig_bindIO_OK; #v Hv;
788    napply sig_bindIO_OK; #b; ncases b; #Hb;
789    ##[ napply (step_while_true … Hv); napply (bool_of … Hb);
790    ##| napply (step_while_false … Hv); napply (bool_of … Hb);
791    ##]
792##| nrewrite > c11;
793    napply sig_bindIO_OK; #v Hv;
794    napply sig_bindIO_OK; #b; ncases b; #Hb;
795    ##[ napply (step_for_true … Hv); napply (bool_of … Hb);
796    ##| napply (step_for_false … Hv); napply (bool_of … Hb);
797    ##]
798##| napply step_for_start; //;
799##| napply step_skip_break_switch; @2; //;
800##| napply step_skip_or_continue_while; @2; //;
801##| napply sig_bindIO_OK; #v Hv;
802    napply sig_bindIO_OK; #b; ncases b; #Hb;
803    ##[ napply (step_skip_or_continue_dowhile_true … Hv);
804      ##[ @2; // ##| napply (bool_of … Hb); ##]
805    ##| napply (step_skip_or_continue_dowhile_false … Hv);
806      ##[ @2; // ##| napply (bool_of … Hb); ##]
807    ##]
808##| napply step_skip_or_continue_for2; @2; //
809##| napply step_return_0; napply c9;
810##| napply sig_bindIO_OK; #u Hnotvoid;
811    napply sig_bindIO_OK; #v Hv;
812    nwhd; napply (step_return_1 … Hnotvoid Hv);
813##| napply sig_bindIO_OK; #v; ncases v; //; #n Hv;
814    napply step_switch; //;
815##| napply step_goto; nrewrite < c12; napply c9;
816##| napply extract_subset_pair_io; #e m1 ealloc Halloc;
817    napply sig_bindIO_OK; #m2 Hbind;
818    nwhd; napply (step_internal_function … Halloc Hbind);
819##| napply sig_bindIO_OK; #evs Hevs;
820    napply bindIO_OK; #eres;
821    napply sig_bindIO_OK; #res Hres;
822    nwhd; napply step_external_function; @; ##[ napply Hevs; ##| napply Hres; ##] 
823##| ncases c11; #x; ncases x; #pcl b ofs;
824    napply opt_bindIO_OK; #m' em'; napply step_returnstate_1; nwhd in em':(??%?); //;
825##]
826nqed.
827
828nlet rec make_initial_state (p:program) : IO eventval io_out (Σs:state. initial_state p s) ≝
829  let ge ≝ globalenv Genv ?? p in
830  let m0 ≝ init_mem Genv ?? p in
831  Some ? (
832    ! 〈sp,b〉 ← find_symbol ? ? ge (prog_main ?? p);:
833    ! u ← opt_to_io … (match ms_eq_dec sp Code with [ inl _ ⇒ Some ? something | inr _ ⇒ None ? ]);:
834    ! f ← find_funct_ptr ? ? ge b;:
835    ret ? (Callstate f (nil ?) Kstop m0)).
836nwhd;
837napply opt_bindIO2_OK; #sp b esb;
838napply opt_bindIO_OK; #u ecode;
839napply opt_bindIO_OK; #f ef;
840ncases sp in esb ecode; #esb ecode; nwhd in ecode:(??%%); ##[ ##1,2,3,4,5: ndestruct (ecode); ##]
841nwhd; napply (initial_state_intro … esb ef);
842nqed.
843
844ndefinition is_final_state : ∀st:state. (∃r. final_state st r) + (¬∃r. final_state st r).
845#st; nelim st;
846##[ #f s k e m; @2; @;*; #r H; ninversion H; #i m e; ndestruct;
847##| #f l k m; @2; @;*; #r H; ninversion H; #i m e; ndestruct;
848##| #v k m; ncases k;
849  ##[ ncases v;
850    ##[ ##2: #i; @1; @ i; //;
851    ##| ##1: @2; @; *;   #r H; ninversion H; #i m e; ndestruct;
852    ##| #f; @2; @; *;   #r H; ninversion H; #i m e; ndestruct;
853    ##| #pcl b of; @2; @; *;   #r H; ninversion H; #i m e; ndestruct;
854    ##]
855  ##| #a b; @2; @; *; #r H; ninversion H; #i m e; ndestruct;
856  ##| ##3,4: #a b c; @2; @; *; #r H; ninversion H; #i m e; ndestruct;
857  ##| ##5,6,8: #a b c d; @2; @; *; #r H; ninversion H; #i m e; ndestruct;
858  ##| #a; @2; @; *; #r H; ninversion H; #i m e; ndestruct;
859  ##]
860##] nqed.
861
862nlet rec exec_steps (n:nat) (ge:genv) (s:state) :
863 IO eventval io_out (Σts:trace×state. star (mk_transrel ?? step) ge s (\fst ts) (\snd ts)) ≝
864match is_final_state s with
865[ inl _ ⇒ Some ? (ret ? 〈E0, s〉)
866| inr _ ⇒
867  match n with
868  [ O ⇒ Some ? (ret ? 〈E0, s〉)
869  | S n' ⇒ Some ? (
870      ! 〈t,s'〉 ← exec_step ge s;:
871(*      ! 〈t,s'〉 ← match s with
872                 [ State f s k e m ⇒ match m with [ mk_mem c n p ⇒ exec_step ge (State f s k e (mk_mem c n p)) ]
873                 | Callstate fd args k m ⇒ match m with [ mk_mem c n p ⇒ exec_step ge (Callstate fd args k (mk_mem c n p)) ]
874                 | Returnstate r k m ⇒ match m with [ mk_mem c n p ⇒ exec_step ge (Returnstate r k (mk_mem c n p)) ]
875                 ] ;:*)
876      ! 〈t',s''〉 ← match s' with
877                 [ State f s k e m ⇒ match m with [ mk_mem c n p ⇒ exec_steps n' ge (State f s k e (mk_mem c n p)) ]
878                 | Callstate fd args k m ⇒ match m with [ mk_mem c n p ⇒ exec_steps n' ge (Callstate fd args k (mk_mem c n p)) ]
879                 | Returnstate r k m ⇒ match m with [ mk_mem c n p ⇒ exec_steps n' ge (Returnstate r k (mk_mem c n p)) ]
880                 ] ;:
881(*      ! 〈t',s''〉 ← exec_steps n' ge s';:*)
882      ret ? 〈t ⧺ t',s''〉)
883  ]
884]. nwhd; /2/;
885napply sig_bindIO2_OK; #t s'; ncases s'; ##[ #f st k e m; ##| #fd args k m; ##| #r k m; ##]
886nwhd in ⊢ (? → ?????(??????%?));
887ncases m; #mc mn mp; #H1;
888nwhd in ⊢ (?????(??????%?));
889napply sig_bindIO2_OK; #t' s'' IH;
890nwhd; napply (star_step … IH); //;
891nqed.
892(*
893nlet rec exec_steps_without_proof (n:nat) (ge:genv) (s:state) :
894 res (trace×state) ≝
895match is_final_state s with
896[ inl _ ⇒ OK ? 〈E0, s〉
897| inr _ ⇒
898  match n with
899  [ O ⇒ OK ? 〈E0, s〉
900  | S n' ⇒
901      〈t,s'〉 ← exec_step ge s;:
902      〈t',s''〉 ← exec_steps_without_proof n' ge s';:
903      OK ? 〈t ⧺ t',s''〉
904  ]
905].
906*)
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