source: C-semantics/CexecIO.ma @ 189

Last change on this file since 189 was 189, checked in by campbell, 9 years ago

Rework monad notation so that it is displayed well in proof mode.

File size: 38.8 KB
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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) on i : 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
196ndefinition exec_cast : ∀m:mem. ∀v:val. ∀ty:type. ∀ty':type. res (Σv':val. cast m v ty ty' v') ≝
197λm:mem. λv:val. λty:type. λty':type.
198match v with
199[ Vint i ⇒
200  match ty with
201  [ Tint sz1 si1 ⇒
202    match ty' with
203    [ Tint sz2 si2 ⇒ Some ? (OK ? (Vint (cast_int_int sz2 si2 i)))
204    | Tfloat sz2 ⇒ Some ? (OK ? (Vfloat (cast_float_float sz2 (cast_int_float si1 i))))
205    | Tpointer _ _ ⇒ Some (res val) (do r ← try_cast_null m i ty ty'; OK val r)
206    | Tarray _ _ _ ⇒ Some (res val) (do r ← try_cast_null m i ty ty'; OK val r)
207    | Tfunction _ _ ⇒ Some (res val) (do r ← try_cast_null m i ty ty'; OK val r)
208    | _ ⇒ Some ? (Error ?)
209    ]
210  | Tpointer _ _ ⇒ Some (res val) (do r ← try_cast_null m i ty ty'; OK val r)
211  | Tarray _ _ _ ⇒ Some (res val) (do r ← try_cast_null m i ty ty'; OK val r)
212  | Tfunction _ _ ⇒ Some (res val) (do r ← try_cast_null m i ty ty'; OK val r)
213  | _ ⇒ Some ? (Error ?)
214  ]
215| Vfloat f ⇒
216  match ty with
217  [ Tfloat sz ⇒
218    match ty' with
219    [ Tint sz' si' ⇒ Some ? (OK ? (Vint (cast_int_int sz' si' (cast_float_int si' f))))
220    | Tfloat sz' ⇒ Some ? (OK ? (Vfloat (cast_float_float sz' f)))
221    | _ ⇒ Some ? (Error ?)
222    ]
223  | _ ⇒ Some ? (Error ?)
224  ]
225| Vptr p b ofs ⇒
226  Some ? (
227    do s ← match ty with [ Tpointer s _ ⇒ OK ? s | Tarray s _ _ ⇒ OK ? s | Tfunction _ _ ⇒ OK ? Code | _ ⇒ Error ? ];
228    do u ← match ms_eq_dec p s with [ inl _ ⇒ OK ? something | inr _ ⇒ Error ? ];
229    do s' ← match ty' with
230         [ Tpointer s _ ⇒ OK ? s | Tarray s _ _ ⇒ OK ? s | Tfunction _ _ ⇒ OK ? Code
231         | _ ⇒ Error ? ];
232    if is_pointer_compat (block_space m b) s'
233    then OK ? (Vptr s' b ofs)
234    else Error ?)
235| _ ⇒ Some ? (Error ?)
236]. nwhd; //;
237##[ ##1,2,3,4,5,6: napply sig_bind_OK; #v'; #H; ndestruct; napply H;
238##| napply bind_OK; #s es;
239    ncut (type_space ty s);
240    ##[ ncases ty in es ⊢ %;
241      ##[ #e; ##| ##3,9: #a e; ##| ##2,4,6,7,8: #a b e; ##| #a b c e; ##] nwhd in e:(??%?); ndestruct; //;
242    ##| #Hty;
243        napply bind_OK; #u1 eeq;
244        napply bind_OK; #s' es';
245        ncut (type_space ty' s');
246        ##[ ncases ty' in es' ⊢ %; ##[ #e; ##| ##3,9: #a e; ##| ##2,4,6,7,8: #a b e; ##| #a b c e; ##]
247            nwhd in e:(??%?); ndestruct; //;
248        ##| #Hty';
249            ncut (s = c0). nelim (ms_eq_dec c0 s) in eeq; //; nnormalize; #_; #e; ndestruct.
250            #e; nrewrite < e;
251            nwhd in match (is_pointer_compat ??) in ⊢ %;
252            ncases (pointer_compat_dec (block_space m c1) s'); #Hcompat;
253            nwhd; /2/;
254        ##]
255    ##]
256##] nqed.
257
258ndefinition load_value_of_type' ≝
259λty,m,l. match l with [ mk_pair pl ofs ⇒ match pl with [ mk_pair psp loc ⇒
260  load_value_of_type ty m psp loc ofs ] ].
261
262(* To make the evaluation of bare lvalue expressions invoke exec_lvalue with
263   a structurally smaller value, we break out the surrounding Expr constructor
264   and use exec_lvalue'. *)
265
266nlet rec exec_expr (ge:genv) (en:env) (m:mem) (e:expr) on e : res (Σr:val×trace. eval_expr ge en m e (\fst r) (\snd r)) ≝
267match e with
268[ Expr e' ty ⇒
269  match e' with
270  [ Econst_int i ⇒ Some ? (OK ? 〈Vint i, E0〉)
271  | Econst_float f ⇒ Some ? (OK ? 〈Vfloat f, E0〉)
272  | Evar _ ⇒ Some ? (
273      do 〈l,tr〉 ← exec_lvalue' ge en m e' ty;
274      do v ← opt_to_res ? (load_value_of_type' ty m l);
275      OK ? 〈v,tr〉)
276  | Ederef _ ⇒ Some ? (
277      do 〈l,tr〉 ← exec_lvalue' ge en m e' ty;
278      do v ← opt_to_res ? (load_value_of_type' ty m l);
279      OK ? 〈v,tr〉)
280  | Efield _ _ ⇒ Some ? (
281      do 〈l,tr〉 ← exec_lvalue' ge en m e' ty;
282      do v ← opt_to_res ? (load_value_of_type' ty m l);
283      OK ? 〈v,tr〉)
284  | Eaddrof a ⇒ Some ? (
285      do 〈plo,tr〉 ← exec_lvalue ge en m a;
286      OK ? 〈match plo with [ mk_pair pl ofs ⇒ match pl with [ mk_pair pcl loc ⇒ Vptr pcl loc ofs ] ], tr〉)
287  | Esizeof ty' ⇒ Some ? (OK ? 〈Vint (repr (sizeof ty')), E0〉)
288  | Eunop op a ⇒ Some ? (
289      do 〈v1,tr〉 ← exec_expr ge en m a;
290      do v ← opt_to_res ? (sem_unary_operation op v1 (typeof a));
291      OK ? 〈v,tr〉)
292  | Ebinop op a1 a2 ⇒ Some ? (
293      do 〈v1,tr1〉 ← exec_expr ge en m a1;
294      do 〈v2,tr2〉 ← exec_expr ge en m a2;
295      do v ← opt_to_res ? (sem_binary_operation op v1 (typeof a1) v2 (typeof a2) m);
296      OK ? 〈v,tr1⧺tr2〉)
297  | Econdition a1 a2 a3 ⇒ Some ? (
298      do 〈v,tr1〉 ← exec_expr ge en m a1;
299      do b ← bool_of_val_3 v (typeof a1);
300      do 〈v',tr2〉 ← match b return λ_.res (val×trace) with
301                 [ true ⇒ (exec_expr ge en m a2)
302                 | false ⇒ (exec_expr ge en m a3) ];
303      OK ? 〈v',tr1⧺tr2〉)
304(*      if b then exec_expr ge en m a2 else exec_expr ge en m a3)*)
305  | Eorbool a1 a2 ⇒ Some ? (
306      do 〈v1,tr1〉 ← exec_expr ge en m a1;
307      do b1 ← bool_of_val_3 v1 (typeof a1);
308      match b1 return λ_.res (val×trace) with [ true ⇒ OK ? 〈Vtrue,tr1〉 | false ⇒
309        do 〈v2,tr2〉 ← exec_expr ge en m a2;
310        do b2 ← bool_of_val_3 v2 (typeof a2);
311        OK ? 〈of_bool b2, tr1⧺tr2〉 ])
312  | Eandbool a1 a2 ⇒ Some ? (
313      do 〈v1,tr1〉 ← exec_expr ge en m a1;
314      do b1 ← bool_of_val_3 v1 (typeof a1);
315      match b1 return λ_.res (val×trace) with [ true ⇒
316        do 〈v2,tr2〉 ← exec_expr ge en m a2;
317        do b2 ← bool_of_val_3 v2 (typeof a2);
318        OK ? 〈of_bool b2, tr1⧺tr2〉
319      | false ⇒ OK ? 〈Vfalse,tr1〉 ])
320  | Ecast ty' a ⇒ Some ? (
321      do 〈v,tr〉 ← exec_expr ge en m a;
322      do v' ← exec_cast m v (typeof a) ty';
323      OK ? 〈(* XXX *)sig_eject ?? v',tr〉)
324  | Ecost l a ⇒ Some ? (
325      do 〈v,tr〉 ← exec_expr ge en m a;
326      OK ? 〈v,tr⧺(Echarge l)〉)
327  ]
328]
329and exec_lvalue' (ge:genv) (en:env) (m:mem) (e':expr_descr) (ty:type) on e' : res (Σr:memory_space × block × int × trace. eval_lvalue ge en m (Expr e' ty) (\fst (\fst (\fst r))) (\snd (\fst (\fst r))) (\snd (\fst r)) (\snd r)) ≝
330  match e' with
331  [ Evar id ⇒
332      match (get … id en) with
333      [ None ⇒ Some ? (do 〈sp,l〉 ← opt_to_res ? (find_symbol ? ? ge id); OK ? 〈〈〈sp,l〉,zero〉,E0〉) (* global *)
334      | Some loc ⇒ Some ? (OK ? 〈〈〈Any,loc〉,zero〉,E0〉) (* local *)
335      ]
336  | Ederef a ⇒ Some ? (
337      do 〈v,tr〉 ← exec_expr ge en m a;
338      match v with
339      [ Vptr sp l ofs ⇒ OK ? 〈〈〈sp,l〉,ofs〉,tr〉
340      | _ ⇒ Error ?
341      ])
342  | Efield a i ⇒
343      match (typeof a) with
344      [ Tstruct id fList ⇒ Some ? (
345          do 〈plofs,tr〉 ← exec_lvalue ge en m a;
346          do delta ← field_offset i fList;
347          OK ? 〈〈\fst plofs,add (\snd plofs) (repr delta)〉,tr〉)
348      | Tunion id fList ⇒ Some ? (
349          do 〈plofs,tr〉 ← exec_lvalue ge en m a;
350          OK ? 〈plofs,tr〉)
351      | _ ⇒ Some ? (Error ?)
352      ]
353  | _ ⇒ Some ? (Error ?)
354  ]
355and exec_lvalue (ge:genv) (en:env) (m:mem) (e:expr) on e : res (Σr:memory_space × block × int × trace. eval_lvalue ge en m e (\fst (\fst (\fst r))) (\snd (\fst (\fst r))) (\snd (\fst r)) (\snd r)) ≝
356match e with [ Expr e' ty ⇒ exec_lvalue' ge en m e' ty ].
357nwhd;
358##[ ##1,2: //
359##| ##3,4:
360    napply sig_bind2_OK; nrewrite > c4; #x; ncases x; #y; ncases y; #sp loc ofs tr H;
361    napply opt_bind_OK;  #v ev; nwhd in ev:(??%?); napply (eval_Elvalue … H ev);
362##| napply sig_bind2_OK; #x; ncases x; #y; ncases y; #sp loc ofs tr H;
363    nwhd; napply eval_Eaddrof; //;
364##| napply sig_bind2_OK; #v1 tr Hv1;
365    napply opt_bind_OK; #v ev;
366    napply (eval_Eunop … Hv1 ev);
367##| napply sig_bind2_OK; #v1 tr1 Hv1;
368    napply sig_bind2_OK; #v2 tr2 Hv2;
369    napply opt_bind_OK; #v ev;
370    napply (eval_Ebinop … Hv1 Hv2 ev);
371##| napply sig_bind2_OK; #v tr Hv;
372    napply sig_bind_OK; #v' Hv';
373    napply (eval_Ecast … Hv Hv');
374##| napply sig_bind2_OK; #vb tr1 Hvb;
375    napply sig_bind_OK; #b;
376    ncases b; #Hb; napply sig_bind2_OK; #v tr Hv;
377    ##[ napply (eval_Econdition_true … Hvb ? Hv);  napply (bool_of ??? Hb);
378    ##| napply (eval_Econdition_false … Hvb ? Hv);  napply (bool_of ??? Hb);
379    ##]
380##| napply sig_bind2_OK; #v1 tr1 Hv1;
381    napply sig_bind_OK; #b1; ncases b1; #Hb1;
382    ##[ napply sig_bind2_OK; #v2 tr2 Hv2;
383        napply sig_bind_OK; #b2 Hb2;
384        napply (eval_Eandbool_2 … Hv1 … Hv2);
385        ##[ napply (bool_of … Hb1); ##| napply Hb2; ##]
386    ##| napply (eval_Eandbool_1 … Hv1); napply (bool_of … Hb1);
387    ##]
388##| napply sig_bind2_OK; #v1 tr1 Hv1;
389    napply sig_bind_OK; #b1; ncases b1; #Hb1;
390    ##[ napply (eval_Eorbool_1 … Hv1); napply (bool_of … Hb1);
391    ##| napply sig_bind2_OK; #v2 tr2 Hv2;
392        napply sig_bind_OK; #b2 Hb2;
393        napply (eval_Eorbool_2 … Hv1 … Hv2);
394        ##[ napply (bool_of … Hb1); ##| napply Hb2; ##]
395    ##]
396##| //
397##| napply sig_bind2_OK; nrewrite > c5; #x; ncases x; #y; ncases y; #sp l ofs tr H;
398    napply opt_bind_OK; #v ev; napply (eval_Elvalue … H ev);
399##| napply sig_bind2_OK; #v tr1 H; napply (eval_Ecost … H);
400##| //
401##| //
402##| napply opt_bind_OK; #sl; ncases sl; #sp l el; napply eval_Evar_global; /2/;
403##| napply (eval_Evar_local … c3);
404##| napply sig_bind2_OK; #v; ncases v; //; #sp l ofs tr Hv; nwhd;
405    napply eval_Ederef; //
406##| ##20,21,22,23,24,25,26,27,28,29,30,31,32,33: //
407##| napply sig_bind2_OK; #x; ncases x; #sp l ofs H;
408    napply bind_OK; #delta Hdelta;
409    napply (eval_Efield_struct … H c5 Hdelta);
410##| napply sig_bind2_OK; #x; ncases x; #sp l ofs H;
411    napply (eval_Efield_union … H c5);
412##| //
413##| //
414##] nqed.
415
416(* TODO: Can we do this sensibly with a map combinator? *)
417nlet rec exec_exprlist (ge:genv) (e:env) (m:mem) (l:list expr) on l : res (Σvltr:list val×trace. eval_exprlist ge e m l (\fst vltr) (\snd vltr)) ≝
418match l with
419[ nil ⇒ Some ? (OK ? 〈nil val, E0〉)
420| cons e1 es ⇒ Some ? (
421  do 〈v,tr1〉 ← exec_expr ge e m e1;
422  do 〈vs,tr2〉 ← exec_exprlist ge e m es;
423  OK ? 〈cons val v vs, tr1⧺tr2〉)
424]. nwhd; //;
425napply sig_bind2_OK; #v tr1 Hv;
426napply sig_bind2_OK; #vs tr2 Hvs;
427nwhd; napply eval_Econs; //;
428nqed.
429
430(* Don't really want to use subset rather than sigma here, but can't be bothered
431   with *another* set of coercions. XXX: why do I have to get the recursive
432   call's property manually? *)
433
434nlet 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) } ≝
435match l with
436[ nil ⇒ Some ? 〈en, m〉
437| cons h vars ⇒
438  match h with [ mk_pair id ty ⇒
439    match alloc m 0 (sizeof ty) Any with [ mk_pair m1 b1 ⇒
440      match exec_alloc_variables (set … id b1 en) m1 vars with
441      [ sig_intro r p ⇒ r ]
442]]]. nwhd;
443##[ //;
444##| nelim (exec_alloc_variables (set ident ? ? c3 c7 en) c6 c1);
445    #H; nelim H; //; #H0; nelim H0; nnormalize; #en' m' IH;
446napply (alloc_variables_cons … IH); /2/;
447nqed.
448
449(* TODO: can we establish that length params = length vs in advance? *)
450nlet 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) ≝
451  match params with
452  [ nil ⇒ match vs with [ nil ⇒ Some ? (OK ? m) | cons _ _ ⇒ Some ? (Error ?) ]
453  | cons idty params' ⇒ match idty with [ mk_pair id ty ⇒
454      match vs with
455      [ nil ⇒ Some ? (Error ?)
456      | cons v1 vl ⇒ Some ? (
457          do b ← opt_to_res ? (get … id e);
458          do m1 ← opt_to_res ? (store_value_of_type ty m Any b zero v1);
459          err_eject ?? (exec_bind_parameters e m1 params' vl)) (* FIXME: don't want to have to eject here *)
460      ]
461  ] ].
462nwhd; //;
463napply opt_bind_OK; #b eb;
464napply opt_bind_OK; #m1 em1;
465napply reinject; #m2 em2 Hm2;
466napply (bind_parameters_cons … eb em1 Hm2);
467nqed.
468
469ndefinition is_not_void : ∀t:type. res (Σu:unit. t ≠ Tvoid) ≝
470λt. match t with
471[ Tvoid ⇒ Some ? (Error ?)
472| _ ⇒ Some ? (OK ??)
473]. nwhd; //; @; #H; ndestruct; nqed.
474
475ninductive decide : Type ≝
476| dy : decide | dn : decide.
477
478ndefinition dodecide : ∀P:Prop.∀d.∀p:(match d with [ dy ⇒ P | dn ⇒ ¬P ]).P + ¬P.
479#P d;ncases d;/2/; nqed.
480
481ncoercion decide_inject :
482  ∀P:Prop.∀d.∀p:(match d with [ dy ⇒ P | dn ⇒ ¬P ]).P + ¬P ≝ dodecide
483  on d:decide to ? + (¬?).
484
485ndefinition dodecide2 : ∀P:Prop.∀d.∀p:(match d with [ dy ⇒ P | dn ⇒ True ]).res P.
486#P d; ncases d; nnormalize; #p; ##[ napply (OK ? p); ##| napply Error ##] nqed.
487
488ncoercion decide_inject2 :
489  ∀P:Prop.∀d.∀p:(match d with [ dy ⇒ P | dn ⇒ True ]).res P ≝ dodecide2
490  on d:decide to res ?.
491
492alias id "Tint" = "cic:/matita/c-semantics/Csyntax/type.con(0,2,0)".
493alias id "Tfloat" = "cic:/matita/c-semantics/Csyntax/type.con(0,3,0)".
494ndefinition sz_eq_dec : ∀s1,s2:intsize. (s1 = s2) + (s1 ≠ s2).
495#s1; ncases s1; #s2; ncases s2; /2/; @2; @; #H; ndestruct; nqed.
496ndefinition sg_eq_dec : ∀s1,s2:signedness. (s1 = s2) + (s1 ≠ s2).
497#s1; ncases s1; #s2; ncases s2; /2/; @2; @; #H; ndestruct; nqed.
498ndefinition fs_eq_dec : ∀s1,s2:floatsize. (s1 = s2) + (s1 ≠ s2).
499#s1; ncases s1; #s2; ncases s2; /2/; @2; @; #H; ndestruct; nqed.
500
501nlet rec assert_type_eq (t1,t2:type) : res (t1 = t2) ≝
502match t1 with
503[ Tvoid ⇒ match t2 with [ Tvoid ⇒ dy | _ ⇒ dn ]
504| 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 ]
505| Tfloat f ⇒ match t2 with [ Tfloat f' ⇒ match fs_eq_dec f f' with [ inl _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ]
506| Tpointer s t ⇒ match t2 with [ Tpointer s' t' ⇒
507    match ms_eq_dec s s' with [ inl _ ⇒
508      match assert_type_eq t t' with [ OK _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ]
509| Tarray s t n ⇒ match t2 with [ Tarray s' t' n' ⇒
510    match ms_eq_dec s s' with [ inl _ ⇒
511      match assert_type_eq t t' with [ OK _ ⇒
512        match decidable_eq_Z_Type n n' with [ inl _ ⇒ dy | inr _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ]
513| Tfunction tl t ⇒ match t2 with [ Tfunction tl' t' ⇒ match assert_typelist_eq tl tl' with [ OK _ ⇒
514    match assert_type_eq t t' with [ OK _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ]
515| Tstruct i fl ⇒
516    match t2 with [ Tstruct i' fl' ⇒ match ident_eq i i' with [ inl _ ⇒
517      match assert_fieldlist_eq fl fl' with [ OK _ ⇒ dy | _ ⇒ dn ] | inr _ ⇒ dn ] |  _ ⇒ dn ]
518| Tunion i fl ⇒
519    match t2 with [ Tunion i' fl' ⇒ match ident_eq i i' with [ inl _ ⇒
520      match assert_fieldlist_eq fl fl' with [ OK _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] |  _ ⇒ dn ]
521| Tcomp_ptr i ⇒ match t2 with [ Tcomp_ptr i' ⇒ match ident_eq i i' with [ inl _ ⇒ dy | inr _ ⇒ dn ] | _ ⇒ dn ]
522]
523and assert_typelist_eq (tl1,tl2:typelist) : res (tl1 = tl2) ≝
524match tl1 with
525[ Tnil ⇒ match tl2 with [ Tnil ⇒ dy | _ ⇒ dn ]
526| Tcons t1 ts1 ⇒ match tl2 with [ Tnil ⇒ dn | Tcons t2 ts2 ⇒
527    match assert_type_eq t1 t2 with [ OK _ ⇒
528      match assert_typelist_eq ts1 ts2 with [ OK _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] ]
529]
530and assert_fieldlist_eq (fl1,fl2:fieldlist) : res (fl1 = fl2) ≝
531match fl1 with
532[ Fnil ⇒ match fl2 with [ Fnil ⇒ dy | _ ⇒ dn ]
533| Fcons i1 t1 fs1 ⇒ match fl2 with [ Fnil ⇒ dn | Fcons i2 t2 fs2 ⇒
534    match ident_eq i1 i2 with [ inl _ ⇒
535      match assert_type_eq t1 t2 with [ OK _ ⇒
536        match assert_fieldlist_eq fs1 fs2 with [ OK _ ⇒ dy | _ ⇒ dn ]
537        | _ ⇒ dn ] | _ ⇒ dn ] ]
538].
539(* A poor man's clear, otherwise automation picks up recursive calls without
540   checking that the argument is smaller. *)
541ngeneralize in assert_type_eq;
542ngeneralize in assert_typelist_eq;
543ngeneralize in assert_fieldlist_eq; #avoid1; #_; #avoid2; #_; #avoid3; #_; nwhd; //;
544(* XXX: I have no idea why the first // didn't catch these. *)
545//; //; //; //; //; //; //; //; //;
546nqed.
547
548nlet rec is_is_call_cont (k:cont) : (is_call_cont k) + (¬is_call_cont k) ≝
549match k with
550[ Kstop ⇒ dy
551| Kcall _ _ _ _ ⇒ dy
552| _ ⇒ dn
553]. nwhd; //; @; #H; nelim H; nqed.
554
555nlet rec is_Sskip (s:statement) : (s = Sskip) + (s ≠ Sskip) ≝
556match s with
557[ Sskip ⇒ dy
558| _ ⇒ dn
559].
560##[ //;
561##| ##*: @; #H; ndestruct;
562##] nqed.
563
564(* IO monad *)
565
566(* Interactions are function calls that return a value and do not change
567   the rest of the Clight program's state. *)
568ndefinition io_out ≝ (ident × (list eventval)).
569
570ndefinition do_io : ident → list eventval → IO eventval io_out eventval ≝
571λfn,args. Interact ?? eventval 〈fn,args〉 (λres. Value ?? eventval res).
572
573ndefinition ret: ∀T. T → (IO eventval io_out T) ≝
574λT,x.(Value ?? T x).
575
576(* Checking types of values given to / received from an external function call. *)
577
578ndefinition check_eventval : ∀ev:eventval. ∀ty:typ. res (Σv:val. eventval_match ev ty v) ≝
579λev,ty.
580match ty with
581[ Tint ⇒ match ev with [ EVint i ⇒ Some ? (OK ? (Vint i)) | _ ⇒ Some ? (Error ?) ]
582| Tfloat ⇒ match ev with [ EVfloat f ⇒ Some ? (OK ? (Vfloat f)) | _ ⇒ Some ? (Error ?) ]
583| _ ⇒ Some ? (Error ?)
584]. nwhd; //; nqed.
585
586ndefinition check_eventval' : ∀v:val. ∀ty:typ. res (Σev:eventval. eventval_match ev ty v) ≝
587λv,ty.
588match ty with
589[ Tint ⇒ match v with [ Vint i ⇒ Some ? (OK ? (EVint i)) | _ ⇒ Some ? (Error ?) ]
590| Tfloat ⇒ match v with [ Vfloat f ⇒ Some ? (OK ? (EVfloat f)) | _ ⇒ Some ? (Error ?) ]
591| _ ⇒ Some ? (Error ?)
592]. nwhd; //; nqed.
593
594nlet rec check_eventval_list (vs: list val) (tys: list typ) : res (Σevs:list eventval. eventval_list_match evs tys vs) ≝
595match vs with
596[ nil ⇒ match tys with [ nil ⇒ Some ? (OK ? (nil ?)) | _ ⇒ Some ? (Error ?) ]
597| cons v vt ⇒
598  match tys with
599  [ nil ⇒ Some ? (Error ?)
600  | cons ty tyt ⇒ Some ? (
601    do ev ← check_eventval' v ty;
602    do evt ← check_eventval_list vt tyt;
603    OK ? ((sig_eject ?? ev)::evt))
604  ]
605]. nwhd; //;
606napply sig_bind_OK; #ev Hev;
607napply sig_bind_OK; #evt Hevt;
608nnormalize; /2/;
609nqed.
610
611(* execution *)
612
613ndefinition store_value_of_type' ≝
614λty,m,l,v.
615match l with [ mk_pair pl ofs ⇒
616  match pl with [ mk_pair pcl loc ⇒
617    store_value_of_type ty m pcl loc ofs v ] ].
618
619nlet rec exec_step (ge:genv) (st:state) on st : (IO eventval io_out (Σr:trace × state. step ge st (\fst r) (\snd r))) ≝
620match st with
621[ State f s k e m ⇒
622  match s with
623  [ Sassign a1 a2 ⇒ Some ? (
624    ! 〈l,tr1〉 ← exec_lvalue ge e m a1;:
625    ! 〈v2,tr2〉 ← exec_expr ge e m a2;:
626    ! m' ← store_value_of_type' (typeof a1) m l v2;:
627    ret ? 〈tr1⧺tr2, State f Sskip k e m'〉)
628  | Scall lhs a al ⇒ Some ? (
629    ! 〈vf,tr2〉 ← exec_expr ge e m a;:
630    ! 〈vargs,tr3〉 ← exec_exprlist ge e m al;:
631    ! fd ← find_funct ? ? ge vf;:
632    ! p ← err_to_io … (assert_type_eq (type_of_fundef fd) (typeof a));:
633(*
634    ! k' ← match lhs with
635         [ None ⇒ ret ? (Kcall (None ?) f e k)
636         | Some lhs' ⇒
637           ! locofs ← exec_lvalue ge e m lhs';:
638           ret ? (Kcall (Some ? 〈sig_eject ?? locofs, typeof lhs'〉) f e k)
639         ];:
640    ret ? 〈E0, Callstate fd vargs k' m〉)
641*)
642    match lhs with
643         [ None ⇒ ret ? 〈tr2⧺tr3, Callstate fd vargs (Kcall (None ?) f e k) m〉
644         | Some lhs' ⇒
645           ! 〈locofs,tr1〉 ← exec_lvalue ge e m lhs';:
646           ret ? 〈tr1⧺tr2⧺tr3, Callstate fd vargs (Kcall (Some ? 〈locofs, typeof lhs'〉) f e k) m〉
647         ])
648  | Ssequence s1 s2 ⇒ Some ? (ret ? 〈E0, State f s1 (Kseq s2 k) e m〉)
649  | Sskip ⇒
650      match k with
651      [ Kseq s k' ⇒ Some ? (ret ? 〈E0, State  f s k' e m〉)
652      | Kstop ⇒
653          match fn_return f with
654          [ Tvoid ⇒ Some ? (ret ? 〈E0, Returnstate Vundef k (free_list m (blocks_of_env e))〉)
655          | _ ⇒ Some ? (Wrong ???)
656          ]
657      | Kcall _ _ _ _ ⇒
658          match fn_return f with
659          [ Tvoid ⇒ Some ? (ret ? 〈E0, Returnstate Vundef k (free_list m (blocks_of_env e))〉)
660          | _ ⇒ Some ? (Wrong ???)
661          ]
662      | Kwhile a s' k' ⇒ Some ? (ret ? 〈E0, State f (Swhile a s') k' e m〉)
663      | Kdowhile a s' k' ⇒ Some ? (
664          ! 〈v,tr〉 ← exec_expr ge e m a;:
665          ! b ← bool_of_val_3 v (typeof a);:
666          match b with
667          [ true ⇒ ret ? 〈tr, State f (Sdowhile a s') k' e m〉
668          | false ⇒ ret ? 〈tr, State f Sskip k' e m〉
669          ])
670      | Kfor2 a2 a3 s' k' ⇒ Some ? (ret ? 〈E0, State f a3 (Kfor3 a2 a3 s' k') e m〉)
671      | Kfor3 a2 a3 s' k' ⇒ Some ? (ret ? 〈E0, State f (Sfor Sskip a2 a3 s') k' e m〉)
672      | Kswitch k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉)
673      | _ ⇒ Some ? (Wrong ???)
674      ]
675  | Scontinue ⇒
676      match k with
677      [ Kseq s' k' ⇒ Some ? (ret ? 〈E0, State f Scontinue k' e m〉)
678      | Kwhile a s' k' ⇒ Some ? (ret ? 〈E0, State f (Swhile a s') k' e m〉)
679      | Kdowhile a s' k' ⇒ Some ? (
680          ! 〈v,tr〉 ← exec_expr ge e m a;:
681          ! b ← bool_of_val_3 v (typeof a);:
682          match b with
683          [ true ⇒ ret ? 〈tr, State f (Sdowhile a s') k' e m〉
684          | false ⇒ ret ? 〈tr, State f Sskip k' e m〉
685          ])
686      | Kfor2 a2 a3 s' k' ⇒ Some ? (ret ? 〈E0, State f a3 (Kfor3 a2 a3 s' k') e m〉)
687      | Kswitch k' ⇒ Some ? (ret ? 〈E0, State f Scontinue k' e m〉)
688      | _ ⇒ Some ? (Wrong ???)
689      ]
690  | Sbreak ⇒
691      match k with
692      [ Kseq s' k' ⇒ Some ? (ret ? 〈E0, State f Sbreak k' e m〉)
693      | Kwhile a s' k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉)
694      | Kdowhile a s' k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉)
695      | Kfor2 a2 a3 s' k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉)
696      | Kswitch k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉)
697      | _ ⇒ Some ? (Wrong ???)
698      ]
699  | Sifthenelse a s1 s2 ⇒ Some ? (
700      ! 〈v,tr〉 ← exec_expr ge e m a;:
701      ! b ← bool_of_val_3 v (typeof a);:
702      ret ? 〈tr, State f (if b then s1 else s2) k e m〉)
703  | Swhile a s' ⇒ Some ? (
704      ! 〈v,tr〉 ← exec_expr ge e m a;:
705      ! b ← bool_of_val_3 v (typeof a);:
706      ret ? 〈tr, if b then State f s' (Kwhile a s' k) e m
707                      else State f Sskip k e m〉)
708  | Sdowhile a s' ⇒ Some ? (ret ? 〈E0, State f s' (Kdowhile a s' k) e m〉)
709  | Sfor a1 a2 a3 s' ⇒
710      match is_Sskip a1 with
711      [ inl _ ⇒ Some ? (
712          ! 〈v,tr〉 ← exec_expr ge e m a2;:
713          ! b ← bool_of_val_3 v (typeof a2);:
714          ret ? 〈tr, State f (if b then s' else Sskip) (if b then (Kfor2 a2 a3 s' k) else k) e m〉)
715      | inr _ ⇒ Some ? (ret ? 〈E0, State f a1 (Kseq (Sfor Sskip a2 a3 s') k) e m〉)
716      ]
717  | Sreturn a_opt ⇒
718    match a_opt with
719    [ None ⇒ match fn_return f with
720      [ Tvoid ⇒ Some ? (ret ? 〈E0, Returnstate Vundef (call_cont k) (free_list m (blocks_of_env e))〉)
721      | _ ⇒ Some ? (Wrong ???)
722      ]
723    | Some a ⇒ Some ? (
724        ! u ← is_not_void (fn_return f);:
725        ! 〈v,tr〉 ← exec_expr ge e m a;:
726        ret ? 〈tr, Returnstate v (call_cont k) (free_list m (blocks_of_env e))〉)
727    ]
728  | Sswitch a sl ⇒ Some ? (
729      ! 〈v,tr〉 ← exec_expr ge e m a;:
730      match v with [ Vint n ⇒ ret ? 〈tr, State f (seq_of_labeled_statement (select_switch n sl)) (Kswitch k) e m〉
731                   | _ ⇒ Wrong ??? ])
732  | Slabel lbl s' ⇒ Some ? (ret ? 〈E0, State f s' k e m〉)
733  | Sgoto lbl ⇒
734      match find_label lbl (fn_body f) (call_cont k) with
735      [ Some sk' ⇒ match sk' with [ mk_pair s' k' ⇒ Some ? (ret ? 〈E0, State f s' k' e m〉) ]
736      | None ⇒ Some ? (Wrong ???)
737      ]
738  | Scost lbl s' ⇒ Some ? (ret ? 〈Echarge lbl, State f s' k e m〉)
739  ]
740| Callstate f0 vargs k m ⇒
741  match f0 with
742  [ Internal f ⇒ Some ? (
743    match exec_alloc_variables empty_env m ((fn_params f) @ (fn_vars f)) with [ mk_pair e m1 ⇒
744      ! m2 ← exec_bind_parameters e m1 (fn_params f) vargs;:
745      ret ? 〈E0, State f (fn_body f) k e m2〉
746    ])
747  | External f argtys retty ⇒ Some ? (
748      ! evargs ← check_eventval_list vargs (typlist_of_typelist argtys);:
749      ! evres ← do_io f evargs;:
750      ! vres ← check_eventval evres (proj_sig_res (signature_of_type argtys retty));:
751      ret ? 〈(Eextcall f evargs evres), Returnstate vres k m〉)
752  ]
753| Returnstate res k m ⇒
754  match k with
755  [ Kcall r f e k' ⇒
756    match r with
757    [ None ⇒
758      match res with
759      [ Vundef ⇒ Some ? (ret ? 〈E0, (State f Sskip k' e m)〉)
760      | _ ⇒ Some ? (Wrong ???)
761      ]
762    | Some r' ⇒
763      match r' with [ mk_pair l ty ⇒
764        Some ? (
765          ! m' ← store_value_of_type' ty m l res;:
766          ret ? 〈E0, (State f Sskip k' e m')〉)
767      ]
768    ]
769  | _ ⇒ Some ? (Wrong ???)
770  ]
771]. nwhd; //;
772##[ nrewrite > c7; napply step_skip_call; //; napply c8;
773##| napply step_skip_or_continue_while; @; //;
774##| napply sig_bindIO2_OK; #v tr Hv;
775    napply sig_bindIO_OK; #b; ncases b; #Hb;
776    ##[ napply (step_skip_or_continue_dowhile_true … Hv);
777      ##[ @; // ##| napply (bool_of … Hb); ##]
778    ##| napply (step_skip_or_continue_dowhile_false … Hv);
779      ##[ @; // ##| napply (bool_of … Hb); ##]
780    ##]
781##| napply step_skip_or_continue_for2; @; //;
782##| napply step_skip_break_switch; @; //;
783##| nrewrite > c11; napply step_skip_call; //; napply c12;
784##| napply sig_bindIO2_OK; #x; ncases x; #y; ncases y; #pcl loc ofs tr1 Hlval;
785    napply sig_bindIO2_OK; #v2 tr2 Hv2;
786    napply opt_bindIO_OK; #m' em';
787    nwhd; napply (step_assign … Hlval Hv2 em');
788##| napply sig_bindIO2_OK; #vf tr1 Hvf;
789    napply sig_bindIO2_OK; #vargs tr2 Hvargs;
790    napply opt_bindIO_OK; #fd efd;
791    napply bindIO_OK; #ety;
792    ncases c6; nwhd;
793    ##[ napply (step_call_none … Hvf Hvargs efd ety);
794    ##| #lhs';
795        napply sig_bindIO2_OK; #x; ncases x; #y; ncases y; #pcl loc ofs tr3 Hlocofs;
796        nwhd; napply (step_call_some … Hlocofs Hvf Hvargs efd ety);
797    ##]
798##| napply sig_bindIO2_OK; #v tr Hv;
799    napply sig_bindIO_OK; #b; ncases b; #Hb;
800    ##[ napply (step_ifthenelse_true … Hv); napply (bool_of … Hb);
801    ##| napply (step_ifthenelse_false … Hv); napply (bool_of … Hb)
802    ##]
803##| napply sig_bindIO2_OK; #v tr Hv;
804    napply sig_bindIO_OK; #b; ncases b; #Hb;
805    ##[ napply (step_while_true … Hv); napply (bool_of … Hb);
806    ##| napply (step_while_false … Hv); napply (bool_of … Hb);
807    ##]
808##| nrewrite > c11;
809    napply sig_bindIO2_OK; #v tr Hv;
810    napply sig_bindIO_OK; #b; ncases b; #Hb;
811    ##[ napply (step_for_true … Hv); napply (bool_of … Hb);
812    ##| napply (step_for_false … Hv); napply (bool_of … Hb);
813    ##]
814##| napply step_for_start; //;
815##| napply step_skip_break_switch; @2; //;
816##| napply step_skip_or_continue_while; @2; //;
817##| napply sig_bindIO2_OK; #v tr Hv;
818    napply sig_bindIO_OK; #b; ncases b; #Hb;
819    ##[ napply (step_skip_or_continue_dowhile_true … Hv);
820      ##[ @2; // ##| napply (bool_of … Hb); ##]
821    ##| napply (step_skip_or_continue_dowhile_false … Hv);
822      ##[ @2; // ##| napply (bool_of … Hb); ##]
823    ##]
824##| napply step_skip_or_continue_for2; @2; //
825##| napply step_return_0; napply c9;
826##| napply sig_bindIO_OK; #u Hnotvoid;
827    napply sig_bindIO2_OK; #v tr Hv;
828    nwhd; napply (step_return_1 … Hnotvoid Hv);
829##| napply sig_bindIO2_OK; #v; ncases v; //; #n tr Hv;
830    napply step_switch; //;
831##| napply step_goto; nrewrite < c12; napply c9;
832##| napply extract_subset_pair_io; #e m1 ealloc Halloc;
833    napply sig_bindIO_OK; #m2 Hbind;
834    nwhd; napply (step_internal_function … Halloc Hbind);
835##| napply sig_bindIO_OK; #evs Hevs;
836    napply bindIO_OK; #eres;
837    napply sig_bindIO_OK; #res Hres;
838    nwhd; napply step_external_function; @; ##[ napply Hevs; ##| napply Hres; ##] 
839##| ncases c11; #x; ncases x; #pcl b ofs;
840    napply opt_bindIO_OK; #m' em'; napply step_returnstate_1; nwhd in em':(??%?); //;
841##]
842nqed.
843
844nlet rec make_initial_state (p:program) : IO eventval io_out (Σs:state. initial_state p s) ≝
845  let ge ≝ globalenv Genv ?? p in
846  let m0 ≝ init_mem Genv ?? p in
847  Some ? (
848    ! 〈sp,b〉 ← find_symbol ? ? ge (prog_main ?? p);:
849    ! u ← opt_to_io … (match ms_eq_dec sp Code with [ inl _ ⇒ Some ? something | inr _ ⇒ None ? ]);:
850    ! f ← find_funct_ptr ? ? ge b;:
851    ret ? (Callstate f (nil ?) Kstop m0)).
852nwhd;
853napply opt_bindIO2_OK; #sp b esb;
854napply opt_bindIO_OK; #u ecode;
855napply opt_bindIO_OK; #f ef;
856ncases sp in esb ecode; #esb ecode; nwhd in ecode:(??%%); ##[ ##1,2,3,4,5: ndestruct (ecode); ##]
857nwhd; napply (initial_state_intro … esb ef);
858nqed.
859
860ndefinition is_final_state : ∀st:state. (∃r. final_state st r) + (¬∃r. final_state st r).
861#st; nelim st;
862##[ #f s k e m; @2; @;*; #r H; ninversion H; #i m e; ndestruct;
863##| #f l k m; @2; @;*; #r H; ninversion H; #i m e; ndestruct;
864##| #v k m; ncases k;
865  ##[ ncases v;
866    ##[ ##2: #i; @1; @ i; //;
867    ##| ##1: @2; @; *;   #r H; ninversion H; #i m e; ndestruct;
868    ##| #f; @2; @; *;   #r H; ninversion H; #i m e; ndestruct;
869    ##| #pcl b of; @2; @; *;   #r H; ninversion H; #i m e; ndestruct;
870    ##]
871  ##| #a b; @2; @; *; #r H; ninversion H; #i m e; ndestruct;
872  ##| ##3,4: #a b c; @2; @; *; #r H; ninversion H; #i m e; ndestruct;
873  ##| ##5,6,8: #a b c d; @2; @; *; #r H; ninversion H; #i m e; ndestruct;
874  ##| #a; @2; @; *; #r H; ninversion H; #i m e; ndestruct;
875  ##]
876##] nqed.
877
878nlet rec exec_steps (n:nat) (ge:genv) (s:state) :
879 IO eventval io_out (Σts:trace×state. star (mk_transrel ?? step) ge s (\fst ts) (\snd ts)) ≝
880match is_final_state s with
881[ inl _ ⇒ Some ? (ret ? 〈E0, s〉)
882| inr _ ⇒
883  match n with
884  [ O ⇒ Some ? (ret ? 〈E0, s〉)
885  | S n' ⇒ Some ? (
886      ! 〈t,s'〉 ← exec_step ge s;:
887(*      ! 〈t,s'〉 ← match s with
888                 [ 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)) ]
889                 | 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)) ]
890                 | Returnstate r k m ⇒ match m with [ mk_mem c n p ⇒ exec_step ge (Returnstate r k (mk_mem c n p)) ]
891                 ] ;:*)
892      ! 〈t',s''〉 ← match s' with
893                 [ 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)) ]
894                 | 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)) ]
895                 | Returnstate r k m ⇒ match m with [ mk_mem c n p ⇒ exec_steps n' ge (Returnstate r k (mk_mem c n p)) ]
896                 ] ;:
897(*      ! 〈t',s''〉 ← exec_steps n' ge s';:*)
898      ret ? 〈t ⧺ t',s''〉)
899  ]
900]. nwhd; /2/;
901napply sig_bindIO2_OK; #t s'; ncases s'; ##[ #f st k e m; ##| #fd args k m; ##| #r k m; ##]
902nwhd in ⊢ (? → ?????(??????%?));
903ncases m; #mc mn mp; #H1;
904nwhd in ⊢ (?????(??????%?));
905napply sig_bindIO2_OK; #t' s'' IH;
906nwhd; napply (star_step … IH); //;
907nqed.
908(*
909nlet rec exec_steps_without_proof (n:nat) (ge:genv) (s:state) :
910 res (trace×state) ≝
911match is_final_state s with
912[ inl _ ⇒ OK ? 〈E0, s〉
913| inr _ ⇒
914  match n with
915  [ O ⇒ OK ? 〈E0, s〉
916  | S n' ⇒
917      〈t,s'〉 ← exec_step ge s;:
918      〈t',s''〉 ← exec_steps_without_proof n' ge s';:
919      OK ? 〈t ⧺ t',s''〉
920  ]
921].
922*)
923
924(* A (possibly non-terminating) execution. *)
925ncoinductive execution : Type ≝
926| e_stop : trace → state → execution
927| e_step : trace → state → execution → execution
928| e_wrong : execution
929| e_interact : io_out → (eventval → execution) → execution.
930
931nlet corec exec_inf_aux (ge:genv) (s:IO eventval io_out (trace×state)) : execution ≝
932match s with
933[ Wrong ⇒ e_wrong
934| Value v ⇒ match v with [ mk_pair t s' ⇒
935    match is_final_state s' with
936    [ inl _ ⇒ e_stop t s'
937    | inr _ ⇒ e_step t s' (exec_inf_aux ge (exec_step ge s')) ] ]
938| Interact out k' ⇒ e_interact out (λv. exec_inf_aux ge (k' v))
939].
940
941ndefinition exec_inf : program → execution ≝
942λp. exec_inf_aux (globalenv Genv ?? p) (! s ← make_initial_state p;: ret ? 〈E0,sig_eject ?? s〉).
943
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