source: src/Clight/CexecSound.ma @ 961

Last change on this file since 961 was 961, checked in by campbell, 8 years ago

Use precise bitvector sizes throughout the front end, rather than 32bits
everywhere.

File size: 24.4 KB
Line 
1include "Clight/Cexec.ma".
2
3(*include "Plogic/connectives.ma".*)
4
5(* Is rather careful about using destruct because it currently uses excessive
6   normalization. *)
7lemma exec_bool_of_val_sound: ∀v,ty,r.
8 exec_bool_of_val v ty = OK ? r → bool_of_val v ty (of_bool r).
9#v #ty #r
10cases v; [ | #sz #i | #f | #r1 | #r' #b #pc #of ]
11cases ty; [ 2,11,20,29,38: #sz' #sg | 3,12,21,30,39: #sz' | 4,13,22,31,40: #rg #ty | 5,14,23,32,41: #r #ty #n | 6,15,24,33,42: #args #rty | 7,8,16,17,25,26,34,35,43,44: #id #fs | 9,18,27,36,45: #r #id ]
12whd in ⊢ (??%? → ?)
13[ 2: @intsize_eq_elim_elim
14  [ #NE #H destruct
15  | *; whd @eq_bv_elim
16    [ #E1 #E2 destruct @bool_of_val_false @is_false_int
17    | #E1 #E2 >(?:r=¬false) [ @bool_of_val_true @is_true_int_int @E1 | destruct @refl ]
18    ]
19  ]
20| 8: #H cases (eq_dec f Fzero)
21  [ #e >e in H ⊢ % >Feq_zero_true #E destruct @bool_of_val_false @is_false_float
22  | #ne >Feq_zero_false in H // #E >(?:r=¬false) [ @bool_of_val_true @is_true_float @ne | destruct @refl ]
23  ]
24| 14: #H >(?:r=false) [ @bool_of_val_false @is_false_pointer | destruct @refl ]
25| 15: #H >(?:r=true) [ @bool_of_val_true @is_true_pointer_pointer | destruct @refl ]
26| *: #H destruct
27] qed.
28
29lemma bool_val_distinct: Vtrue ≠ Vfalse.
30% #H whd in H:(??%%); destruct; @(absurd ? e0 one_not_zero)
31qed.
32
33lemma bool_of: ∀v,ty,b. bool_of_val v ty (of_bool b) →
34  if b then is_true v ty else is_false v ty.
35#v #ty #b cases b; #H inversion H; #v' #ty' #H' #ev #et #ev //;
36@False_ind @(absurd ? ev ?)
37[ 2: @sym_neq ] @bool_val_distinct
38qed.
39
40lemma try_cast_null_sound: ∀m,sz,i,ty,ty',v'. try_cast_null m sz i ty ty' = OK ? v' → cast m (Vint sz i) ty ty' v'.
41#m #sz #i #ty #ty' #v'
42whd in ⊢ (??%? → ?)
43@eq_bv_elim
44[ #e >e
45    cases ty; [ | #sz' #sg | #fs | #sp #ty | #sp #ty #n | #args #rty | #id #fs | #id #fs | #r #id ]
46    whd in ⊢ (??%? → ?); #H [ 2: | *: destruct ]
47    cases ty' in H ⊢ %; [ | #sz'' #sg | #fs | #sp #ty | #sp #ty #n | #args #rty | #id #fs | #id #fs | #r #id ]
48    try (@eq_intsize_elim #E) whd in ⊢ (??%? → ?); #H destruct @cast_ip_z //
49| #_ whd in ⊢ (??%? → ?); #H destruct
50]
51qed.
52
53definition exec_cast_sound : ∀m:mem. ∀v:val. ∀ty:type. ∀ty':type. ∀v':val. exec_cast m v ty ty' = OK ? v' → cast m v ty ty' v'.
54#m #v #ty #ty' #v'
55cases v;
56[ #H whd in H:(??%?); destruct;
57| #sz #i cases ty;
58  [ #H whd in H:(??%?); destruct;
59  | 3: #a #H whd in H:(??%?); destruct;
60  | 7,8,9: #a #b #H whd in H:(??%?); destruct;
61  | #sz1 #si1 cases ty';
62    [ whd in ⊢ (??%? → ?) @intsize_eq_elim_elim
63      [ #NE #H destruct
64      | *; whd #H whd in H:(??%?); destruct;
65      ]
66    | 3: #a whd in ⊢ (??%? → ?) @intsize_eq_elim_elim
67      [ #E #H whd in H:(??%?); destruct
68      | *; whd #H whd in H:(??%?); destruct; @cast_if
69      ]
70    | 2,7,8,9: #a #b whd in ⊢ (??%? → ?) @intsize_eq_elim_elim
71      [ 1,3,5,7: #NE #H destruct
72      | *: *; whd #H whd in H:(??%?); destruct; //
73      ]
74    | 4,5,6: [ #sp #ty'' letin t ≝ (Tpointer sp ty'')
75                 | #sp #ty'' #n letin t ≝ (Tarray sp ty'' n)
76                 | #args #rty letin t ≝ (Tfunction args rty) ]
77        whd in ⊢ (??%? → ?) @intsize_eq_elim_elim
78        [ 1,3,5: #NE #H destruct
79        | *: *; whd
80          lapply (try_cast_null_sound m sz i (Tint sz si1) t v');
81          cases (try_cast_null m sz i (Tint sz si1) t);
82          [ 1,3,5: #v'' #H' #e @H' @e
83          | *: #m #_ whd in ⊢ (??%? → ?); #H destruct (H);
84          ]
85        ]
86    ]
87  | *: [ #sp #ty'' letin t ≝ (Tpointer sp ty'')
88           | #sp #ty'' #n letin t ≝ (Tarray sp ty'' n)
89           | #args #rty letin t ≝ (Tfunction args rty) ]
90        whd in ⊢ (??%? → ?);
91        lapply (try_cast_null_sound m sz i t ty' v');
92        cases (try_cast_null m sz i t ty');
93        [ 1,3,5: #v'' #H' #e @H' @e
94        | *: #m #_ whd in ⊢ (??%? → ?); #H destruct (H);
95        ]
96  ]
97| #f cases ty;  [ 3: #x | 2,4,6,7,8,9: #x #y | 5: #x #y #z ]
98                    [ cases ty'; [ #e | 3: #a #e | 2,4,6,7,8,9: #a #b #e | #a #b #c #e ]
99                        whd in e:(??%?); destruct; //;
100                    | *: #e whd in e:(??%?); destruct
101                    ]
102| #r cases ty; [ 3: #x | 2,4,6,7,8,9: #x #y | 5: #x #y #z ]
103    whd in ⊢ (??%? → ?); #H destruct; cases (eq_region_dec r ?) in H;
104    whd in ⊢ (? → ??%? → ?); #H1 #H2 destruct;
105    cases ty' in H2; normalize; try #a try #b try #c try #d destruct;
106    @cast_pp_z //;
107| #r #b #pc #of whd in ⊢ (??%? → ?) #e
108    elim (bind_inversion ????? e) #s * #es #e'
109    elim (bind_inversion ????? e') #u * #eu #e'' -e';
110    elim (bind_inversion ????? e'') #s' * #es' #e'''; -e'';
111    cut (type_region ty s);
112    [ cases ty in es:(??%?) ⊢ %; [ #e | 3: #a #e | 2,4,6,7,8,9: #a #b #e | #a #b #c #e ]
113        whd in e:(??%?); destruct; //;
114    | #Hty
115        cut (type_region ty' s');
116        [ cases ty' in es' ⊢ %; [ #e | 3: #a #e | 2,4,6,7,8,9: #a #b #e | #a #b #c #e ]
117            whd in e:(??%?); destruct; //;
118        | #Hty'
119            cut (s = r). elim (eq_region_dec r s) in eu; //; normalize; #_ #e destruct.
120            #e >e in Hty #Hty
121            cases (pointer_compat_dec b s') in e'''
122            #Hcompat #e''' whd in e''':(??%?); destruct (e'''); /2/
123        ]
124    ]
125] qed.
126
127let rec expr_lvalue_ind
128  (P:expr → Prop)
129  (Q:expr_descr → type → Prop)
130  (ci:∀sz,ty,i.P (Expr (Econst_int sz i) ty))
131  (cf:∀ty,f.P (Expr (Econst_float f) ty))
132  (lv:∀e,ty. Q e ty → Plvalue P e ty)
133  (vr:∀v,ty.Q (Evar v) ty)
134  (dr:∀e,ty.P e → Q (Ederef e) ty)
135  (ao:∀ty,e,ty'.Q e ty' → P (Expr (Eaddrof (Expr e ty')) ty))
136  (uo:∀ty,op,e.P e → P (Expr (Eunop op e) ty))
137  (bo:∀ty,op,e1,e2.P e1 → P e2 → P (Expr (Ebinop op e1 e2) ty))
138  (ca:∀ty,ty',e.P e → P (Expr (Ecast ty' e) ty))
139  (cd:∀ty,e1,e2,e3.P e1 → P e2 → P e3 → P (Expr (Econdition e1 e2 e3) ty))
140  (ab:∀ty,e1,e2.P e1 → P e2 → P (Expr (Eandbool e1 e2) ty))
141  (ob:∀ty,e1,e2.P e1 → P e2 → P (Expr (Eorbool e1 e2) ty))
142  (sz:∀ty,ty'. P (Expr (Esizeof ty') ty))
143  (fl:∀ty,e,ty',i. Q e ty' → Q (Efield (Expr e ty') i) ty)
144  (co:∀ty,l,e. P e → P (Expr (Ecost l e) ty))
145  (xx:∀e,ty. is_not_lvalue e → Q e ty)
146  (e:expr) on e : P e ≝
147match e with
148[ Expr e' ty ⇒
149  match e' with
150  [ Econst_int sz i ⇒ ci sz ty i
151  | Econst_float f ⇒ cf ty f
152  | Evar v ⇒ lv (Evar v) ty (vr v ty)
153  | Ederef e'' ⇒ lv (Ederef e'') ty (dr e'' ty (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e''))
154  | Eaddrof e'' ⇒ match e'' with [ Expr e0 ty0 ⇒ ao ty e0 ty0 (lvalue_expr_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e0 ty0) ]
155  | Eunop op e'' ⇒ uo ty op e'' (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'')
156  | Ebinop op e1 e2 ⇒ bo ty op e1 e2 (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e1) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e2)
157  | Ecast ty' e'' ⇒ ca ty ty' e'' (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'')
158  | Econdition e1 e2 e3 ⇒ cd ty e1 e2 e3 (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e1) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e2) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e3)
159  | Eandbool e1 e2 ⇒ ab ty e1 e2 (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e1) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e2)
160  | Eorbool e1 e2 ⇒ ob ty e1 e2 (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e1) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e2)
161  | Esizeof ty' ⇒ sz ty ty'
162  | Efield e'' i ⇒ match e'' with [ Expr ef tyf ⇒ lv (Efield (Expr ef tyf) i) ty (fl ty ef tyf i (lvalue_expr_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx ef tyf)) ]
163  | Ecost l e'' ⇒ co ty l e'' (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'')
164  ]
165]
166and lvalue_expr_ind
167  (P:expr → Prop)
168  (Q:expr_descr → type → Prop)
169  (ci:∀sz,ty,i.P (Expr (Econst_int sz i) ty))
170  (cf:∀ty,f.P (Expr (Econst_float f) ty))
171  (lv:∀e,ty. Q e ty → Plvalue P e ty)
172  (vr:∀v,ty.Q (Evar v) ty)
173  (dr:∀e,ty.P e → Q (Ederef e) ty)
174  (ao:∀ty,e,ty'.Q e ty' → P (Expr (Eaddrof (Expr e ty')) ty))
175  (uo:∀ty,op,e.P e → P (Expr (Eunop op e) ty))
176  (bo:∀ty,op,e1,e2.P e1 → P e2 → P (Expr (Ebinop op e1 e2) ty))
177  (ca:∀ty,ty',e.P e → P (Expr (Ecast ty' e) ty))
178  (cd:∀ty,e1,e2,e3.P e1 → P e2 → P e3 → P (Expr (Econdition e1 e2 e3) ty))
179  (ab:∀ty,e1,e2.P e1 → P e2 → P (Expr (Eandbool e1 e2) ty))
180  (ob:∀ty,e1,e2.P e1 → P e2 → P (Expr (Eorbool e1 e2) ty))
181  (sz:∀ty,ty'. P (Expr (Esizeof ty') ty))
182  (fl:∀ty,e,ty',i. Q e ty' → Q (Efield (Expr e ty') i) ty)
183  (co:∀ty,l,e. P e → P (Expr (Ecost l e) ty))
184  (xx:∀e,ty. is_not_lvalue e → Q e ty)
185  (e:expr_descr) (ty:type) on e : Q e ty ≝
186  match e return λe0. Q e0 ty with
187  [ Evar v ⇒ vr v ty
188  | Ederef e'' ⇒ dr e'' ty (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'')
189  | Efield e' i ⇒ match e' return λe1.Q (Efield e1 i) ty with [ Expr e'' ty'' ⇒ fl ty e'' ty'' i (lvalue_expr_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'' ty'') ]
190  | _ ⇒ xx ? ty ?
191  ]. whd; @I qed.
192
193
194theorem exec_expr_sound: ∀ge:genv. ∀en:env. ∀m:mem. ∀e:expr.
195(P_res ? (λx.eval_expr ge en m e (\fst x) (\snd x)) (exec_expr ge en m e)).
196#ge #en #m #e @(expr_lvalue_ind ? (λe',ty.P_res ? (λr.eval_lvalue ge en m (Expr e' ty) (\fst (\fst r)) (\snd (\fst r)) (\snd r)) (exec_lvalue' ge en m e' ty)) … e)
197(* XXX // fails [ 1,2: #ty #c whd //  *)
198[ #sz #ty #c whd in ⊢ (???%) cases ty try #sz' try #sg try #x try @I whd in ⊢ (???%)
199  @eq_intsize_elim #E try @I <E whd %
200| #ty #c whd %2
201(* expressions that are lvalues *)
202| #e' #ty cases e'; //; [ #i #He' | #e #He' | #e #i #He' ] whd in He' ⊢ %;
203    @bind2_OK #x cases x; #y cases y; #sp #loc #ofs #tr #H
204    @opt_bind_OK #vl #evl whd in evl:(??%?); @(eval_Elvalue … evl)
205    >H in He' #He' @He'
206| #v #ty
207    whd in ⊢ (???%);
208    lapply (refl ? (get ident PTree block v en));
209    cases (get ident PTree block v en) in ⊢ (???% → %);
210    [ #eget @opt_bind_OK #sploc cases sploc; #sp #loc #efind
211        whd; @(eval_Evar_global … eget efind)
212    | #loc #eget @(eval_Evar_local … eget)
213    ]
214| #ty #e #He whd in ⊢ (???%)
215    @bind2_OK #v cases v // #r #l #pc #ofs #tr #Hv whd
216    >Hv in He #He
217    @eval_Ederef [ 3: @He | *: skip ]
218| #ty #e'' #ty'' #He'' @bind2_OK * #loc #ofs #tr #H
219  cases ty // * #pty
220  cases loc in H ⊢ % * #loc' #H
221  whd try @I
222  @eval_Eaddrof whd in H:(??%?) >H in He'' #He'' @He''
223| #ty #op #e1 #He1 @bind2_OK #v1 #tr #Hv1
224    @opt_bind_OK #v #ev
225    @(eval_Eunop … ev)
226    >Hv1 in He1 #He1 @He1
227| #ty #op #e1 #e2 #He1 #He2
228    @bind2_OK #v1 #tr1 #ev1 >ev1 in He1 #He1
229    @bind2_OK #v2 #tr2 #ev2 >ev2 in He2 #He2
230    @opt_bind_OK #v #ev whd in He1 He2; whd;
231    @(eval_Ebinop … He1 He2 ev)
232| #ty #ty' #e' #He'
233    @bind2_OK #v #tr #Hv >Hv in He' #He'
234    @bind_OK #v' #ev'
235    @(eval_Ecast … He' ?)
236    (* XXX /2/; *)
237    @(exec_cast_sound … ev')
238| #ty #e1 #e2 #e3 #He1 #He2 #He3
239    @bind2_OK #vb #tr1 #Hvb >Hvb in He1 #He1
240    @bind_OK #b
241    cases b; #eb lapply (exec_bool_of_val_sound … eb); #Hb
242    @bind2_OK #v #tr #Hv whd in Hv:(??%?)
243    [ >Hv in He2 #He2 whd in He2 Hv:(??%?) ⊢%;
244        @(eval_Econdition_true … He1 ? He2) @(bool_of ??? Hb)
245    | >Hv in He3 #He3 whd in He3 Hv:(??%?) ⊢%;
246        @(eval_Econdition_false … He1 ? He3) @(bool_of ??? Hb)
247    ]
248| #ty #e1 #e2 #He1 #He2
249    @bind2_OK #v1 #tr1 #Hv1 >Hv1 in He1 #He1
250    @bind_OK #b1 cases b1; #eb1 lapply (exec_bool_of_val_sound … eb1); #Hb1
251    [ @bind2_OK #v2 #tr2 #Hv2 >Hv2 in He2 #He2
252        @bind_OK #b2 #eb2
253        @(eval_Eandbool_2 … He1 … He2)
254        [ @(bool_of … Hb1) | @(exec_bool_of_val_sound … eb2) ]
255    | @(eval_Eandbool_1 … He1) @(bool_of … Hb1)
256    ]
257| #ty #e1 #e2 #He1 #He2
258    @bind2_OK #v1 #tr1 #Hv1 >Hv1 in He1 #He1
259    @bind_OK #b1 cases b1; #eb1 lapply (exec_bool_of_val_sound … eb1); #Hb1
260    [ @(eval_Eorbool_1 … He1) @(bool_of … Hb1)
261    | @bind2_OK #v2 #tr2 #Hv2 >Hv2 in He2 #He2
262        @bind_OK #b2 #eb2
263        @(eval_Eorbool_2 … He1 … He2)
264        [ @(bool_of … Hb1) | @(exec_bool_of_val_sound … eb2) ]
265    ]
266| #ty #ty' cases ty try #sz try #sg try #x try @I whd; (* XXX //*) @eval_Esizeof
267| #ty #e' #ty' #i cases ty'; //;
268    [ #id #fs #He' @bind2_OK #x cases x; #sp #l #ofs #H
269        @bind_OK #delta #Hdelta whd in H:(??%?) >H in He' #He'
270        @(eval_Efield_struct …  He' (refl ??) Hdelta)
271    | #id #fs #He' @bind2_OK #x cases x; #sp #l #ofs #H whd in H:(??%?)
272        >H in He' #He'
273        @(eval_Efield_union … He' (refl ??))
274    ]
275| #ty #l #e' #He' @bind2_OK #v #tr1 #H >H in He' #He'
276    @(eval_Ecost … He')
277(* exec_lvalue fails on non-lvalues. *)
278| #e' #ty cases e';
279    [ 2,5,12: #a #H | 3,4: #a * | 13,14: #a #b * | 1,6,8,10,11: #a #b #H | 7,9: #a #b #c #H ]
280    @I
281] qed.
282
283lemma addrof_eval_lvalue: ∀ge,en,m,e,r,loc,pc,off,tr,ty.
284eval_expr ge en m (Expr (Eaddrof e) ty) (Vptr r loc pc off) tr →
285eval_lvalue ge en m e loc off tr.
286#ge #en #m #e #r #loc #pc #off #tr #ty #H inversion H;
287[ 1,2,5: #a #b #c #H @False_ind destruct (H);
288| #a #b #c #d #e #f #H1 #g #H2 <H2 in H1 #H1 @False_ind
289    @(eval_lvalue_inv_ind … H1)
290    [ #a #b #c #d #bad destruct (bad);
291    | #a #b #c #d #e #bad destruct (bad);
292    | #a #b #c #d #e #f #g #h #bad destruct (bad);
293    | #a #b #c #d #e #f #g #h #i #j #k #l #m #bad @False_ind destruct (bad);
294    | #a #b #c #d #e #f #g #h #i #j #k #bad destruct (bad);
295    ]
296| #e' #ty' #r' #loc' #ofs' #tr' #H' #pc' #e1 #e2 #e3 destruct (e1 e2 e3); @H'
297| #a #b #c #d #e #f #g #h #i #bad destruct (bad);
298| #a #b #c #d #e #f #g #h #i #j #k #l #k #l #bad destruct (bad);
299| #a #b #c #d #e #f #g #h #i #j #k #l #m #bad destruct (bad);
300| #a #b #c #d #e #f #g #h #i #j #k #l #m #bad destruct (bad);
301| #a #b #c #d #e #f #g #h #bad destruct (bad);
302| #a #b #c #d #e #f #g #h #i #j #k #l #m #n #bad destruct (bad);
303| #a #b #c #d #e #f #g #h #bad destruct (bad);
304| #a #b #c #d #e #f #g #h #i #j #k #l #m #n #bad  destruct (bad);
305| #a #b #c #d #e #f #g #h #i #bad destruct (bad);
306| #a #b #c #d #e #f #g #bad destruct (bad);
307] qed.
308
309lemma addrof_exec_lvalue: ∀ge,en,m,e,r,loc,off,tr.
310exec_lvalue ge en m e = OK ? 〈〈mk_block r loc,off〉,tr〉 →
311exec_expr ge en m (Expr (Eaddrof e) (Tpointer r Tvoid)) = OK ? 〈Vptr r (mk_block r loc) (same_compat ??) off, tr〉.
312#ge #en #m #e #r #loc #off #tr #H whd in ⊢ (??%?)
313>H whd in ⊢ (??%?) cases r @refl
314qed.
315
316theorem exec_lvalue_sound: ∀ge,en,m,e.
317P_res ? (λr.eval_lvalue ge en m e (\fst (\fst r)) (\snd (\fst r)) (\snd r)) (exec_lvalue ge en m e).
318#ge #en #m #e lapply (refl ? (exec_lvalue ge en m e));
319cases (exec_lvalue ge en m e) in ⊢ (???% → %);
320[ #x cases x #y cases y #loc #off #tr #H whd cases loc in H ⊢ % #locr #loci #H
321    @(addrof_eval_lvalue … locr … (Tpointer locr Tvoid)) [ @same_compat ]
322    lapply (addrof_exec_lvalue … H) #H'
323    lapply (exec_expr_sound ge en m (Expr (Eaddrof e) (Tpointer locr Tvoid)))
324    >H' #H'' @H''
325| #msg #_ whd @I
326] qed.
327
328(* Plain equality versions of the above *)
329
330definition exec_expr_sound' ≝ λge,en,m,e,v.
331  λH:exec_expr ge en m e = OK ? v.
332  P_res_to_P ???? (exec_expr_sound ge en m e) H.
333
334definition exec_lvalue_sound' ≝ λge,en,m,e,loc,off,tr.
335  λH:exec_lvalue ge en m e = OK ? 〈〈loc,off〉,tr〉.
336  P_res_to_P ???? (exec_lvalue_sound ge en m e) H.
337
338lemma exec_exprlist_sound: ∀ge,e,m,l. P_res ? (λvltr:list val×trace. eval_exprlist ge e m l (\fst vltr) (\snd vltr)) (exec_exprlist ge e m l).
339#ge #e #m #l elim l;
340 whd; (* XXX //; *)
341[ %
342| #e1 #es #IH
343  @bind2_OK #v #tr1 #Hv
344  @bind2_OK #vs #tr2 #Hvs
345  whd; @eval_Econs
346  [ @(P_res_to_P … (exec_expr_sound ge e m e1) Hv)
347  | @(P_res_to_P … IH Hvs)
348  ]
349] qed.
350
351lemma exec_alloc_variables_sound : ∀l,en,m,en',m'.
352  exec_alloc_variables en m l = 〈en',m'〉 →
353  alloc_variables en m l en' m'.
354#l elim l
355[ #en #m #en' #m' #EXEC whd in EXEC:(??%?); destruct %
356| * #id #ty #t #IH #en #m #en' #m'
357  lapply (refl ? (alloc m O (sizeof ty) Any)) #ALLOC #EXEC
358  whd in EXEC:(??%?) ALLOC:(???%)
359 @(alloc_variables_cons … ALLOC)
360 @IH @EXEC
361qed.
362
363lemma exec_bind_parameters_sound : ∀ps,vs,en,m.
364  P_res ? (λm'. bind_parameters en m ps vs m') (exec_bind_parameters en m ps vs).
365#ps elim ps
366[ * //
367| * #id #ty #ps' #IH *
368    [ //
369    | #v #vs #en #m
370      @opt_bind_OK #b #GET
371      @opt_bind_OK #m' #STORE
372      lapply (refl ? (exec_bind_parameters en m' ps' vs))
373      cases (exec_bind_parameters en m' ps' vs) in ⊢ (???% → %) [2: #msg #_ %]
374      #m'' #BIND
375      @(bind_parameters_cons … GET STORE)
376      lapply (IH vs en m') whd in ⊢ (% → ?) >BIND //
377    ]
378] qed.
379
380lemma check_eventval_list_sound : ∀vs,tys.
381  P_res ? (λevs. eventval_list_match evs tys vs) (check_eventval_list vs tys).
382#vs0 elim vs0
383[ * //
384| #v #vs #IH *
385  [ //
386  | #ty #tys whd in ⊢ (???%)
387    cases ty [ #sz #sg | #sz | #r ] cases v //
388    [ #sz' #v @bind_OK #ev whd in ⊢ (??%? → ?)
389      @eq_intsize_elim #E #CHECKev whd in CHECKev:(??%?); destruct
390    | #v ] @bind_OK #evs #CHECKevs
391      @(evl_match_cons ??????? (P_res_to_P ???? (IH ?) CHECKevs))
392      //
393  ]
394] qed.
395
396theorem exec_step_sound: ∀ge,st.
397  P_io ??? (λr. step ge st (\fst r) (\snd r)) (exec_step ge st).
398#ge #st cases st;
399[ #f #s #k #e #m cases s;
400  [ cases k;
401    [ whd in ⊢ (?????%);
402        lapply (refl ? (fn_return f)); cases (fn_return f) in ⊢ (???% → %);
403        //; #H whd;
404        @step_skip_call //;
405    | #s' #k' whd; (* XXX //; *) @step_skip_seq
406    | #ex #s' #k' @step_skip_or_continue_while % //;
407    | #ex #s' #k'
408        @res_bindIO2_OK #v #tr #Hv
409        letin bexpr ≝ (exec_bool_of_val v (typeof ex));
410        lapply (refl ? bexpr);
411        cases bexpr in ⊢ (???% → %);
412        [ #b cases b; #eb lapply (exec_bool_of_val_sound … eb); #Hb
413            whd in ⊢ (?????%);
414            [ @(step_skip_or_continue_dowhile_true … (exec_expr_sound' … Hv))
415              [ % // | @(bool_of … Hb) ]
416            | @(step_skip_or_continue_dowhile_false … (exec_expr_sound' … Hv))
417              [ % // | @(bool_of … Hb) ]
418            ]
419        | #msg #_ //;
420        ]
421    | #ex #s1 #s2 #k' @step_skip_or_continue_for2 % //;
422    | #ex #s1 #s2 #k' @step_skip_for3
423    | #k' @step_skip_break_switch % //;
424    | #r #f' #e' #k' whd in ⊢ (?????%);
425        lapply (refl ? (fn_return f)); cases (fn_return f) in ⊢ (???% → %);
426        //; #H whd;
427        @step_skip_call //;
428    ]
429  | #ex1 #ex2
430    @res_bindIO2_OK #x cases x; #y cases y; #pcl #loc #ofs #tr1 #Hlval
431    @res_bindIO2_OK #v2 #tr2 #Hv2
432    @opt_bindIO_OK #m' #em'
433    whd; @(step_assign … (exec_lvalue_sound' … Hlval) (exec_expr_sound' … Hv2) em')
434  | #lex #fex #args
435    @res_bindIO2_OK #vf #tr1 #Hvf0 lapply (exec_expr_sound' … Hvf0); #Hvf
436    @res_bindIO2_OK #vargs #tr2 #Hvargs0 lapply (P_res_to_P ???? (exec_exprlist_sound …) Hvargs0); #Hvargs
437    @opt_bindIO_OK #fd #efd
438    @bindIO_OK #ety
439    cases lex; whd;
440    [ @(step_call_none … Hvf Hvargs efd ety)
441    | #lhs'
442        @res_bindIO2_OK #x cases x; #y cases y; #pcl #loc #ofs #tr3 #Hlocofs
443        whd; @(step_call_some … (exec_lvalue_sound' … Hlocofs) Hvf Hvargs efd ety)
444    ]
445  | #s1 #s2 whd; (* XXX //; *) @step_seq
446  | #ex #s1 #s2
447    @res_bindIO2_OK #v #tr #Hv
448    letin bexpr ≝ (exec_bool_of_val v (typeof ex));
449    lapply (refl ? bexpr); cases bexpr in ⊢ (???% → %); //;
450    #b cases b; #eb lapply (exec_bool_of_val_sound … eb); #Hb
451    [ @(step_ifthenelse_true … (exec_expr_sound' … Hv)) @(bool_of … Hb)
452    | @(step_ifthenelse_false … (exec_expr_sound' … Hv)) @(bool_of … Hb)
453    ]
454  | #ex #s'
455    @res_bindIO2_OK #v #tr #Hv
456    letin bexpr ≝ (exec_bool_of_val v (typeof ex));
457    lapply (refl ? bexpr); cases bexpr in ⊢ (???% → %); //;
458    #b cases b; #eb lapply (exec_bool_of_val_sound … eb); #Hb
459    [ @(step_while_true … (exec_expr_sound' … Hv)) @(bool_of … Hb)
460    | @(step_while_false … (exec_expr_sound' … Hv)) @(bool_of … Hb)
461    ]
462  | #ex #s' whd; (* XXX //; *) @step_dowhile
463  | #s1 #ex #s2 #s3 whd in ⊢ (?????%); elim (is_Sskip s1); #Hs1 whd in ⊢ (?????%);
464    [ >Hs1
465      @res_bindIO2_OK #v #tr #Hv
466      letin bexpr ≝ (exec_bool_of_val v (typeof ex));
467      lapply (refl ? bexpr); cases bexpr in ⊢ (???% → %); //;
468      #b cases b; #eb lapply (exec_bool_of_val_sound … eb); #Hb
469      [ @(step_for_true … (exec_expr_sound' … Hv)) @(bool_of … Hb)
470      | @(step_for_false … (exec_expr_sound' … Hv)) @(bool_of … Hb)
471      ]
472    | @step_for_start //;
473    ]
474  | whd in ⊢ (?????%); cases k; //;
475    [ #s' #k' whd (* XXX // *) @step_break_seq
476    | #ex #s' #k' whd (* //; *) @step_break_while
477    | #ex #s' #k' whd (* //; *) @step_break_dowhile
478    | #ex #s1 #s2 #k' whd (* //; *) @step_break_for2
479    | #k' @step_skip_break_switch %2  //
480    ]
481  | whd in ⊢ (?????%); cases k; //;
482    [ #s' #k' whd; (* XXX //;*) @step_continue_seq
483    | #ex #s' #k' whd; @step_skip_or_continue_while %2 ; //;
484    | #ex #s' #k' whd;
485      @res_bindIO2_OK #v #tr #Hv
486      letin bexpr ≝ (exec_bool_of_val v (typeof ex));
487      lapply (refl ? bexpr); cases bexpr in ⊢ (???% → %); //;
488      #b cases b; #eb lapply (exec_bool_of_val_sound … eb); #Hb
489      [ @(step_skip_or_continue_dowhile_true … (exec_expr_sound' … Hv))
490        [ %2 ; // | @(bool_of … Hb) ]
491      | @(step_skip_or_continue_dowhile_false … (exec_expr_sound' … Hv))
492        [ %2 ; // | @(bool_of … Hb) ]
493      ]
494    | #ex #s1 #s2 #k' whd; @step_skip_or_continue_for2 %2 ; //
495    | #k' whd; (* XXX //;*) @step_continue_switch
496    ]
497  | #r whd in ⊢ (?????%); cases r;
498    [ whd; lapply (refl ? (fn_return f)); cases (fn_return f) in ⊢ (???% → %); //; #H
499        @step_return_0 @H
500    | #ex cases (type_eq_dec (fn_return f) Tvoid); //; whd in ⊢ (% → ?????%); #Hnotvoid
501        @res_bindIO2_OK #v #tr #Hv
502        whd; @(step_return_1 … Hnotvoid (exec_expr_sound' … Hv))
503    ]
504  | #ex #ls @res_bindIO2_OK * // #sz #n #tr #Hv
505    @step_switch @(exec_expr_sound' … Hv)
506  | #l #s' whd; @step_label (* XXX //; *)
507  | #l whd in ⊢ (?????%); lapply (refl ? (find_label l (fn_body f) (call_cont k)));
508      cases (find_label l (fn_body f) (call_cont k)) in ⊢ (???% → %); //;
509      #sk cases sk; #s' #k' #H
510      @(step_goto … H)
511  | #l #s' whd; (* XXX //; *) @step_cost
512  ]
513| #f0 #vargs #k #m whd in ⊢ (?????%); cases f0;
514  [ #fn whd in ⊢ (?????%)
515    lapply (refl ? (exec_alloc_variables empty_env m (fn_params fn@fn_vars fn)))
516    cases (exec_alloc_variables empty_env m (fn_params fn@fn_vars fn)) in ⊢ (???% → %)
517    #en' #m' #ALLOC whd in ⊢ (?????%)
518    @res_bindIO_OK #m2 #BIND
519    whd; @(step_internal_function … (exec_alloc_variables_sound … ALLOC))
520    @(P_res_to_P … (exec_bind_parameters_sound …) BIND)
521  | #id #argtys #rty @res_bindIO_OK #evs #Hevs
522    @bindIO_OK #eres whd;
523    @step_external_function % [ @(P_res_to_P … (check_eventval_list_sound …) Hevs)
524                              | @mk_val_correct ]
525  ] 
526| #v #k' #m' whd in ⊢ (?????%); cases k'; //;
527    #r #f #e #k whd in ⊢ (?????%); cases r;
528  [ whd; @step_returnstate_0
529  | #x cases x; #y cases y; #z cases z; #pcl #b #ofs #ty
530    @opt_bindIO_OK #m' #em' @step_returnstate_1 whd in em':(??%?); //;
531  ]
532]
533qed.
534
535lemma make_initial_state_sound : ∀p. P_res ? (λgs.globalenv Genv ?? p = OK ? (\fst gs) ∧ initial_state p (\snd gs)) (make_initial_state p).
536#p cases p; #fns #main #vars
537whd in ⊢ (???%);
538@bind_OK #ge #Ege
539@bind_OK #m #Em
540@opt_bind_OK #x cases x; #sp #b #esb
541@opt_bind_OK #f #ef
542whd; % [ whd in ⊢ (???(??%)) // | @(initial_state_intro … Ege Em esb ef) ]
543qed.
544
545theorem exec_steps_sound: ∀ge,n,st.
546 P_io ??? (λts:trace×state. star (mk_transrel ?? step) ge st (\fst ts) (\snd ts))
547 (exec_steps n ge st).
548#ge #n elim n;
549[ #st whd in ⊢ (?????%); elim (is_final_state st); #H whd; %
550| #n' #IH #st whd in ⊢ (?????%); elim (is_final_state st); #H
551  [ whd; %
552  | @(P_bindIO2_OK ????????? (exec_step_sound …)) #t #s' cases s';
553      [ #f #s #k #e #m | #fd #args #k #m | #r #k #m ]
554      whd in ⊢ (? → ?????(??????%?));
555      cases m; #mc #mn #mp #Hstep
556      whd in ⊢ (?????(??????%?));
557      @(P_bindIO2_OK ????????? (IH …)) #t' #s'' #Hsteps
558      whd; @(star_step ????????? Hsteps) [ 2,5,8: @Hstep | 3,6,9: // ]
559  ]
560qed.
561
562lemma is_final_sound: ∀s,r. is_final s = Some ? r → final_state s r.
563* [ 3: #v * [ #m #r cases v [ 2: * #r' #E normalize in E; destruct %
564                            | *: normalize #x1 try #x2 try #x3 try #x4 try #x5 destruct
565                            ]
566            | *: normalize #x1 try #x2 try #x3 try #x4 try #x5 try #x6 try #x7 destruct
567            ]
568  | *: normalize #x1 #x2 #x3 #x4 #x5 #x6 try #x7 destruct
569  ] qed.
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