source: src/Clight/CexecComplete.ma @ 2120

Last change on this file since 2120 was 2120, checked in by campbell, 6 years ago

Fix victim of alloc unfolding.

File size: 17.1 KB
Line 
1include "Clight/Cexec.ma".
2
3definition yields ≝ λA.λa:res A.λv':A.
4match a with [ OK v ⇒ v' = v | _ ⇒ False ].
5
6(* This tells us that some execution of a results in v'.
7   (There may be many possible executions due to I/O, but we're trying to prove
8   that one particular one exists corresponding to a derivation in the inductive
9   semantics.) *)
10let rec yieldsIO (A:Type[0]) (a:IO io_out io_in A) (v':A) on a : Prop ≝
11match a with [ Value v ⇒ v' = v | Interact _ k ⇒ ∃r.yieldsIO A (k r) v' | _ ⇒ False ].
12
13definition yields_sig : ∀A,P. res (Sig A P) → A → Prop ≝
14λA,P,e,v. match e with [ OK v' ⇒ match v' with [ mk_Sig v'' _ ⇒ v = v'' ] | _ ⇒ False].
15
16let rec yieldsIO_sig (A:Type[0]) (P:A → Prop) (e:IO io_out io_in (Sig A P)) (v:A) on e : Prop ≝
17match e with
18[ Value v' ⇒ match v' with [ mk_Sig v'' _ ⇒ v = v'' ]
19| Interact _ k ⇒ ∃r.yieldsIO_sig A P (k r) v
20| _ ⇒ False].
21
22lemma remove_io_sig: ∀A. ∀P:A → Prop. ∀a,v',p.
23yieldsIO A a v' →
24yieldsIO_sig A P (io_inject io_out io_in A P (Some ? a) p) v'.
25#A #P #a elim a;
26[ #a #k #IH #v' #p #H whd in H ⊢ %; elim H; #r #H' %{ r} @IH @H'
27| #v #v' #p #H @H
28| #a #b #c *;
29] qed.
30
31lemma yields_eq: ∀A,a,v'. yields A a v' → a = OK ? v'.
32#A #a #v' cases a; // #m whd in ⊢ (% → ?); *;
33qed.
34
35lemma yields_sig_eq: ∀A,P,e,v. yields_sig A P e v → ∃p. e = OK ? (mk_Sig … v p).
36#A #P #e #v cases e;
37[ #vp cases vp; #v' #p #H whd in H; >H %{ p} @refl
38| #m *;
39] qed.
40
41lemma is_pointer_compat_true: ∀b,sp.
42  pointer_compat b sp →
43  is_pointer_compat b sp = true.
44#b #sp #H whd in ⊢ (??%?);
45elim (pointer_compat_dec b sp);
46[ //
47| #H' @False_ind @(absurd … H H')
48] qed.
49
50theorem is_det: ∀p,s,s'.
51initial_state p s → initial_state p s' → s = s'.
52#p #s #s' #H1 #H2
53inversion H1; #b1 #f1 #ge1 #m1 #e11 #e12 #e13 #e14 #e15 #e16
54inversion H2; #b2 #f2 #ge2 #m2 #e21 #e22 #e23 #e24 #e25 #e26
55>e11 in e21; #e1 >(?:ge1 = ge2) in e13 e14;
56[ 2: destruct (e1) skip (e11); @refl ]
57#e13 #e14
58
59>e12 in e22; #e2 destruct (e2) skip (e12);
60
61>e13 in e23; #e3 >(?:b1 = b2) in e14;
62[ >e24 #e4 >(?:f2 = f1)
63  [ //;
64  | destruct (e4) skip (e24 e25); //;
65  ]
66| destruct (e3) skip (e13); //
67] qed.
68
69
70theorem the_initial_state:
71  ∀p,s. initial_state p s → yields ? (make_initial_state p) s.
72#p #s cases p; #globs #fns #main #H
73inversion H;
74#b #f #ge #m #e1 #e2 #e3 #e4 #e5 #e6
75whd in ⊢ (??%?);
76>e2
77whd in ⊢ (??%?);
78change with (make_global (mk_program ?? globs fns main)) in e1:(??%?);
79>e1
80>e3
81whd in ⊢ (??%?);
82>e4
83whd; @refl
84qed.
85
86lemma cast_complete: ∀m,v,ty,ty',v'.
87  cast m v ty ty' v' → yields ? (exec_cast m v ty ty') v'.
88#m #v #ty #ty' #v' #H
89elim H;
90[ #m #sz1 #sz2 #sg1 #sg2 #i whd in ⊢ (??%?); >intsize_eq_elim_true @refl
91| #m #f #sz #szi #sg @refl
92| #m #sz #sz' #sg #i whd in ⊢ (??%?); >intsize_eq_elim_true @refl
93| #m #f #sz #sz' @refl
94| #m #ty #ty' * #r #b #pc #ofs #r' #H1 #H2 #pc'
95    elim H1 in pc ⊢ %; [ #r1 #ty1 #pc | #r1 #ty1 #n1 #pc | #tys1 #ty1 #pc letin r1 ≝ Code ]
96    whd in ⊢ (??%?);
97    [ 1,2: @(dec_true ? (eq_region_dec r1 r1) (refl ??) …) #H0 whd in ⊢ (??%?); ]
98    elim H2 in pc' ⊢ %; [ 1,4,7: #sp2 #ty2 | 2,5,8: #sp2 #ty2 #n2 | 3,6,9: #tys2 #ty2 letin sp2 ≝ Code ]
99    #pc' whd in ⊢ (??%?);
100    @(dec_true ? (pointer_compat_dec b sp2) pc') //
101| #m #sz #si #ty'' #r #H cases H; [ #s #t | #s #t #n | #tys #ty0 ] whd in ⊢ (??%?);
102  >intsize_eq_elim_true whd in ⊢ (??%?); cases sz //;
103| #m #t #t' #r #r' #H #H' cases H; try #a try #b try #c cases H'; try #d try #e try #f
104    whd in ⊢ (??%?); try @refl @(dec_true ? (eq_region_dec a a) (refl ??)) #H0 @refl
105] qed.
106
107(* Use to narrow down the choice of expression to just the lvalues. *)
108lemma lvalue_expr: ∀ge,env,m,e,ty,l,ofs,tr. ∀P:expr_descr → Prop.
109  eval_lvalue ge env m (Expr e ty) l ofs tr →
110  (∀id. P (Evar id)) → (∀e'. P (Ederef e')) → (∀e',id. P (Efield e' id)) →
111  P e.
112#ge #env #m #e #ty #l #ofs #tr #P #H @(eval_lvalue_inv_ind … H)
113[ #id #l' #ty' #e1 #e2 #e3 #e4 #e5 #e6 destruct; //
114| #id #l' #ty' #e1 #e2 #e3 #e4 #e5 #e6 #e7 destruct; //
115| #e' #ty' #sp #l' #pc #ofs #tr #H' #e1 #e2 #e3 #e4 #e5 -H destruct; //
116| #e' #id #ty' #l' #ofs #id' #fs #d #tr #H' #e1 #e2 #e3 #e4 #e5 #e6 #e7 -H destruct; //
117| #e' #id #ty' #l' #ofs #id' #fs #tr #H' #e1 #e2 #e3 #e4 #e5 #e6 -H destruct; //
118] qed.
119
120lemma bool_of_val_3_complete : ∀v,ty,r. bool_of_val v ty r → ∃b. r = of_bool b ∧ yields ? (exec_bool_of_val v ty) b.
121#v #ty #r #H elim H; #v #t #H' elim H';
122  [ #sz #sg #i #ne %{ true} % //; whd in ⊢ (??%?); >intsize_eq_elim_true
123    >(eq_bv_false … ne) //
124  | * #r #b #pc #i #i0 #s %{ true} % //
125  | #f #s #ne %{ true} % //; whd; >(Feq_zero_false … ne) //;
126  | #sz #sg %{ false} % // whd in ⊢ (??%?); >intsize_eq_elim_true >eq_bv_true //
127  | #r #r' #t %{ false} % //;
128  | #s %{ false} % //; whd; >(Feq_zero_true …) //;
129  ]
130qed.
131
132lemma bool_of_true: ∀v,ty. is_true v ty → yields ? (exec_bool_of_val v ty) true.
133#v #ty #H elim H;
134  [ #i #is #s #ne whd in ⊢ (??%?); >intsize_eq_elim_true >(eq_bv_false … ne) //;
135  | * #p #b #i #i0 #s //
136  | #f #s #ne whd; >(Feq_zero_false … ne) //;
137  ]
138qed.
139
140lemma bool_of_false: ∀v,ty. is_false v ty → yields ? (exec_bool_of_val v ty) false.
141#v #ty #H elim H;
142  [ #sz #sg whd in ⊢ (??%?); >intsize_eq_elim_true >eq_bv_true //;
143  | #r #r' #t //;
144  | #s whd; >(Feq_zero_true …) //;
145  ]
146qed.
147
148lemma expr_lvalue_complete: ∀ge,env,m.
149(∀e,v,tr. eval_expr ge env m e v tr → yields ? (exec_expr ge env m e) (〈v,tr〉)) ∧
150(∀e,l,off,tr. eval_lvalue ge env m e l off tr → yields ? (exec_lvalue ge env m e) (〈〈l,off〉,tr〉)).
151#ge #env #m
152@(combined_expr_lvalue_ind ge env m
153  (λe,v,tr,H. yields ? (exec_expr ge env m e) (〈v,tr〉))
154  (λe,l,off,tr,H. yields ? (exec_lvalue ge env m e) (〈〈l,off〉,tr〉)));
155[ #sz #sg #i whd in ⊢ (??%?); >eq_intsize_true @refl
156| #f #ty @refl
157| #e #ty #l #off #v #tr #H1 #H2 @(lvalue_expr … H1)
158    [ #id | #e' | #e' #id ] #H3
159    whd in ⊢ (??%?);
160    [ change with (exec_lvalue' ge env m (Evar id) ty) in H3:(??%?);
161    | change with (exec_lvalue' ge env m (Ederef e') ty) in H3:(??%?);
162    | change with (exec_lvalue' ge env m (Efield e' id) ty) in H3:(??%?);
163    ]
164    >(yields_eq ??? H3)
165    whd in ⊢ (??%?); change with (load_value_of_type' ty m 〈l,off〉) in H2:(??%?);
166    >H2 @refl
167| #e #ty #r #l #pc #off #tr #H1 #H2 whd in ⊢ (??%?);
168    >(yields_eq ??? H2) whd in ⊢ (??%?);
169    @(dec_true ? (pointer_compat_dec l r) pc) #pc' whd
170    @refl
171| #ty' #sz #sg @refl
172| #op #e #ty #v1 #v #tr #H1 #H2 #H3 whd in ⊢ (??%?);
173    >(yields_eq ??? H3)
174    whd in ⊢ (??%?); >H2 @refl
175| #op #e1 #e2 #ty #v1 #v2 #v #tr1 #tr2 #H1 #H2 #e3 #H4 #H5 whd in ⊢ (??%?);
176    >(yields_eq ??? H4) whd in ⊢ (??%?);
177    >(yields_eq ??? H5) whd in ⊢ (??%?);
178    >e3 @refl
179| #e1 #e2 #e3 #ty #v1 #v2 #tr1 #tr2 #H1 #H2 #H3 #H4 #H5 whd in ⊢ (??%?);
180    >(yields_eq ??? H4) whd in ⊢ (??%?);
181    >(yields_eq ??? (bool_of_true ?? H2))
182    >(yields_eq ??? H5)
183    @refl
184| #e1 #e2 #e3 #ty #v1 #v2 #tr1 #tr2 #H1 #H2 #H3 #H4 #H5 whd in ⊢ (??%?);
185    >(yields_eq ??? H4) whd in ⊢ (??%?);
186    >(yields_eq ??? (bool_of_false ?? H2))
187    >(yields_eq ??? H5)
188    @refl
189| #e1 #e2 #ty #v1 #tr #H1 #H2 #H3 whd in ⊢ (??%?);
190    >(yields_eq ??? H3) whd in ⊢ (??%?);
191    >(yields_eq ??? (bool_of_true ?? H2))
192    @refl
193| #e1 #e2 #ty #v1 #v2 #v #tr1 #tr2 #H1 #H2 #H3 #H4 #H5 #H6 whd in ⊢ (??%?);
194    >(yields_eq ??? H5) whd in ⊢ (??%?);
195    >(yields_eq ??? (bool_of_false ?? H2))
196    >(yields_eq ??? H6) whd in ⊢ (??%?);
197    elim (bool_of_val_3_complete … H4); #b *; #evb #Hb
198    >(yields_eq ??? Hb) whd in ⊢ (??%?); <evb
199    @refl
200| #e1 #e2 #ty #v1 #tr #H1 #H2 #H3 whd in ⊢ (??%?);
201    >(yields_eq ??? H3) whd in ⊢ (??%?);
202    >(yields_eq ??? (bool_of_false ?? H2))
203    @refl
204| #e1 #e2 #ty #v1 #v2 #v #tr1 #tr2 #H1 #H2 #H3 #H4 #H5 #H6 whd in ⊢ (??%?);
205    >(yields_eq ??? H5) whd in ⊢ (??%?);
206    >(yields_eq ??? (bool_of_true ?? H2))
207    >(yields_eq ??? H6) whd in ⊢ (??%?);
208    elim (bool_of_val_3_complete … H4); #b *; #evb #Hb
209    >(yields_eq ??? Hb) whd in ⊢ (??%?); <evb
210    @refl
211| #e #ty #ty' #v1 #v #tr #H1 #H2 #H3 whd in ⊢ (??%?);
212    >(yields_eq ??? H3) whd in ⊢ (??%?);
213    >(yields_eq ??? (cast_complete … H2))
214    @refl
215| #e #ty #v #l #tr #H1 #H2 whd in ⊢ (??%?);
216    >(yields_eq ??? H2) whd in ⊢ (??%?);
217    @refl
218   
219  (* lvalues *)
220| #id #l #ty #e1 whd in ⊢ (??%?); >e1 @refl
221| #id #l #ty #e1 #e2 whd in ⊢ (??%?); >e1
222    >e2 @refl
223| #e #ty #r #l #pc #ofs #tr #H1 #H2 whd in ⊢ (??%?);
224    >(yields_eq ??? H2)
225    @refl
226| #e #i #ty #l #ofs #id #fList #delta #tr #H1 #H2 #H3 #H4 cases e in H2 H4 ⊢ %;
227    #e' #ty' #H2 whd in H2:(??%?); >H2 #H4 whd in ⊢ (??%?);
228    >(yields_eq ??? H4) whd in ⊢ (??%?);
229    >H3 @refl
230| #e #i #ty #l #ofs #id #fList #tr cases e; #e' #ty' #H1 #H2
231    whd in H2:(??%?); >H2 #H3 whd in ⊢ (??%?);
232    >(yields_eq ??? H3) @refl
233] qed.
234
235theorem expr_complete:  ∀ge,env,m.
236 ∀e,v,tr. eval_expr ge env m e v tr → yields ? (exec_expr ge env m e) (〈v,tr〉).
237#ge #env #m elim (expr_lvalue_complete ge env m); /2/; qed.
238
239theorem exprlist_complete: ∀ge,env,m,es,vs,tr.
240  eval_exprlist ge env m es vs tr → yields ? (exec_exprlist ge env m es) (〈vs,tr〉).
241#ge #env #m #es #vs #tr #H elim H;
242[ @refl
243| #e #et #v #vt #tr #trt #H1 #H2 #H3 whd in ⊢ (??%?);
244    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
245    >(yields_eq ??? H3)
246    @refl
247] qed.
248
249theorem lvalue_complete: ∀ge,env,m.
250 ∀e,l,off,tr. eval_lvalue ge env m e l off tr → yields ? (exec_lvalue ge env m e) (〈〈l,off〉,tr〉).
251#ge #env #m elim (expr_lvalue_complete ge env m); /2/; qed.
252
253let rec P_typelist (P:type → Prop) (l:typelist) on l : Prop ≝
254match l with
255[ Tnil ⇒ True
256| Tcons h t ⇒ P h ∧ P_typelist P t
257].
258
259let rec type_ind2l
260  (P:type → Prop) (Q:typelist → Prop)
261  (vo:P Tvoid)
262  (it:∀i,s. P (Tint i s))
263  (fl:∀f. P (Tfloat f))
264  (pt:∀s,t. P t → P (Tpointer s t))
265  (ar:∀s,t,n. P t → P (Tarray s t n))
266  (fn:∀tl,t. Q tl → P t → P (Tfunction tl t))
267  (st:∀i,fl. P (Tstruct i fl))
268  (un:∀i,fl. P (Tunion i fl))
269  (cp:∀r,i. P (Tcomp_ptr r i))
270  (nl:Q Tnil)
271  (cs:∀t,tl. P t → Q tl → Q (Tcons t tl))
272 (t:type) on t : P t ≝
273  match t return λt'.P t' with
274  [ Tvoid ⇒ vo
275  | Tint i s ⇒ it i s
276  | Tfloat s ⇒ fl s
277  | Tpointer s t' ⇒ pt s t' (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t')
278  | Tarray s t' n ⇒ ar s t' n (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t')
279  | Tfunction tl t' ⇒ fn tl t' (typelist_ind2l P Q vo it fl pt ar fn st un cp nl cs tl) (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t')
280  | Tstruct i fs ⇒ st i fs
281  | Tunion i fs ⇒ un i fs
282  | Tcomp_ptr r i ⇒ cp r i
283  ]
284and typelist_ind2l
285  (P:type → Prop) (Q:typelist → Prop)
286  (vo:P Tvoid)
287  (it:∀i,s. P (Tint i s))
288  (fl:∀f. P (Tfloat f))
289  (pt:∀s,t. P t → P (Tpointer s t))
290  (ar:∀s,t,n. P t → P (Tarray s t n))
291  (fn:∀tl,t. Q tl → P t → P (Tfunction tl t))
292  (st:∀i,fl. P (Tstruct i fl))
293  (un:∀i,fl. P (Tunion i fl))
294  (cp:∀r,i. P (Tcomp_ptr r i))
295  (nl:Q Tnil)
296  (cs:∀t,tl. P t → Q tl → Q (Tcons t tl))
297  (ts:typelist) on ts : Q ts ≝
298  match ts return λts'.Q ts' with
299  [ Tnil ⇒ nl
300  | Tcons t tl ⇒ cs t tl (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t)
301                     (typelist_ind2l P Q vo it fl pt ar fn st un cp nl cs tl)
302  ].
303
304lemma assert_type_eq_true: ∀t. ∃p.assert_type_eq t t = OK ? p.
305#t whd in ⊢ (??(λ_.??%?)); cases (type_eq_dec t t); #E
306[ %{ E} //
307| @False_ind @(absurd ?? E) //
308] qed.
309
310lemma alloc_vars_complete: ∀env,m,l,env',m'.
311  alloc_variables env m l env' m' →
312  exec_alloc_variables env m l = 〈env', m'〉.
313#env #m #l #env' #m' #H elim H;
314[ #env'' #m'' %
315| #env1 #m1 #id #ty #l1 #m2 #loc #m3 #env2 #H1 #H2 #H3
316  < H3 whd in H1:(??%?) ⊢ (??%?);
317  >H1
318  @refl
319] qed.
320
321lemma bind_params_complete: ∀e,m,params,vs,m2.
322  bind_parameters e m params vs m2 →
323  yields ? (exec_bind_parameters e m params vs) m2.
324#e #m #params #vs #m2 #H elim H;
325[ //;
326| #env1 #m1 #id #ty #l #v #tl #loc #m2 #m3 #H1 #H2 #H3 #H4
327    whd in ⊢ (??%?);
328    >H1 whd in ⊢ (??%?);
329    >H2 whd in ⊢ (??%?);
330    @H4
331] qed.
332
333lemma eventval_match_complete': ∀ev,ty,v.
334  eventval_match ev ty v → yields ? (check_eventval' v ty) ev.
335#ev #ty #v #H elim H; // #sz #sg #i whd in ⊢ (??%?); >eq_intsize_true @refl qed.
336
337lemma eventval_list_match_complete: ∀vs,tys,evs.
338  eventval_list_match evs tys vs → yields ? (check_eventval_list vs tys) evs.
339#vs #tys #evs #H elim H;
340[ //
341| #e #etl #ty #tytl #v #vtl #H1 #H2 #H3 whd in ⊢ (??%?);
342    >(yields_eq ??? (eventval_match_complete' … H1)) whd in ⊢ (??%?);
343    >(yields_eq ??? H3) whd in ⊢ (??%?); //
344] qed.
345
346theorem step_complete: ∀ge,s,tr,s'.
347  step ge s tr s' → yieldsIO ? (exec_step ge s) 〈tr,s'〉.
348#ge #s #tr #s' #H elim H;
349[ #f #e #e1 #k #e2 #m #loc #ofs #v #m' #tr1 #tr2 #H1 #H2 #H3 whd in ⊢ (??%?);
350    >(yields_eq ??? (lvalue_complete … H1)) whd in ⊢ (??%?);
351    >(yields_eq ??? (expr_complete … H2)) whd in ⊢ (??%?);
352    change with (store_value_of_type' (typeof e) m 〈loc,ofs〉 v) in H3:(??%?);
353    >H3 @refl
354| #f #e #eargs #k #ef #m #vf #vargs #f' #tr1 #tr2 #H1 #H2 #H3 #H4 whd in ⊢ (??%?);
355    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
356    >(yields_eq ??? (exprlist_complete … H2)) whd in ⊢ (??%?);
357    >H3 whd in ⊢ (??%?);
358    >H4 elim (assert_type_eq_true (fun_typeof e)); #pty #ety >ety
359    @refl
360| #f #el #ef #eargs #k #env #m #loc #ofs #vf #vargs #f' #tr1 #tr2 #tr3 #H1 #H2 #H3 #H4 #H5 whd in ⊢ (??%?);
361    >(yields_eq ??? (expr_complete … H2)) whd in ⊢ (??%?);
362    >(yields_eq ??? (exprlist_complete … H3)) whd in ⊢ (??%?);
363    >H4 whd in ⊢ (??%?);
364    >H5 elim (assert_type_eq_true (fun_typeof ef)); #pty #ety >ety
365    whd in ⊢ (??%?);
366    >(yields_eq ??? (lvalue_complete … H1)) whd in ⊢ (??%?);
367    @refl
368| #f #s1 #s2 #k #env #m @refl
369| 5,6,7: #f #s #k #env #m @refl
370| #f #e #s1 #s2 #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
371    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
372    >(yields_eq ??? (bool_of_true ?? H2))
373    @refl
374| #f #e #s1 #s2 #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
375    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
376    >(yields_eq ??? (bool_of_false ?? H2))
377    @refl
378| #f #e #s #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
379    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
380    >(yields_eq ??? (bool_of_false ?? H2))
381    @refl
382| #f #e #s #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
383    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
384    >(yields_eq ??? (bool_of_true ?? H2))
385    @refl
386| #f #s1 #e #s2 #k #env #m #H cases H; #es1 >es1 @refl
387| 13,14: #f #e #s #k #env #m @refl
388| #f #s1 #e #s2 #k #env #m #v #tr *; #es1 >es1 #H1 #H2 whd in ⊢ (??%?);
389    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
390    >(yields_eq ??? (bool_of_false ?? H2))
391    @refl
392| #f #s1 #e #s2 #k #env #m #v #tr *; #es1 >es1 #H1 #H2 whd in ⊢ (??%?);
393    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
394    >(yields_eq ??? (bool_of_true ?? H2))
395    @refl
396| #f #e #s #k #env #m @refl
397| #f #s1 #e #s2 #s3 #k #env #m #nskip whd in ⊢ (??%?); cases (is_Sskip s1);
398    [ #H @False_ind @(absurd ? H nskip)
399    | #H whd in ⊢ (??%?); @refl ]
400| #f #e #s1 #s2 #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
401    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
402    >(yields_eq ??? (bool_of_false ?? H2))
403    @refl
404| #f #e #s1 #s2 #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
405    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
406    >(yields_eq ??? (bool_of_true ?? H2))
407    @refl
408| #f #s1 #e #s2 #s3 #k #env #m *; #es1 >es1 @refl
409| 22,23: #f #e #s1 #s2 #k #env #m @refl
410| #f #k #env #m #H whd in ⊢ (??%?); >H @refl
411| #f #e #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
412    @(dec_false ? (type_eq_dec (fn_return f) Tvoid) H1) #pf'
413    whd in ⊢ (??%?);
414    >(yields_eq ??? (expr_complete … H2)) whd in ⊢ (??%?);
415    @refl
416| #f #k #env #m cases k;
417    [ #H1 #H2 whd in ⊢ (??%?); >H2 @refl
418    | #s' #k' whd in ⊢ (% → ?); *;
419    | 3,4: #e' #s' #k' whd in ⊢ (% → ?); *;
420    | 5,6: #e' #s1' #s2' #k' whd in ⊢ (% → ?); *;
421    | #k' whd in ⊢ (% → ?); *;
422    | #r #f' #env' #k' #H1 #H2 whd in ⊢ (??%?); >H2 @refl
423    ]
424| #f #e #s #k #env #m #sz #i #tr #H1 whd in ⊢ (??%?);
425    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
426    @refl
427| #f #s #k #env #m *; #es >es @refl
428| #f #k #env #m @refl
429| #f #l #s #k #env #m @refl
430| #f #l #k #env #m #s #k' #H1 whd in ⊢ (??%?); >H1 @refl
431| #f #args #k #m1 #env #m2 #m3 #H1 #H2 whd in ⊢ (??%?);
432    >(alloc_vars_complete … H1) whd in ⊢ (??%?);
433    >(yields_eq ??? (bind_params_complete … H2))
434    //
435| #id #tys #rty #args #k #m #rv #tr #H whd in ⊢ (??%?);
436    inversion H; #f' #args' #rv' #eargs #erv #H1 #H2 #e1 #e2 #e3 #e4 #e5 <e1 in H1 H2;
437    #H1 #H2
438    >(yields_eq ??? (eventval_list_match_complete … H1)) whd in ⊢ (??%?);
439    whd; inversion H2; [ #sz #sg #x | #x #sz ] #e5 #e6 #e7 #e8 %{ x} whd in ⊢ (??%?);
440    @refl
441| #v #f #env #k #m @refl
442| #v #f #env #k #m1 #m2 #loc #ofs #ty
443    change with (store_value_of_type' ty m1 〈loc,ofs〉 v) in ⊢ (??%? → ?);
444    #H whd in ⊢ (??%?); >H @refl
445| #f #l #s #k #env #m @refl
446| #r #m //
447] qed.
448
449lemma is_final_complete : ∀s,r. final_state s r → is_final s = Some ? r.
450#s0 #r0 * #r @refl qed.
451 
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