source: Deliverables/D3.1/C-semantics/CexecComplete.ma @ 535

Last change on this file since 535 was 500, checked in by campbell, 9 years ago

Use dependent pointer type to ensure that the representation is always
compatible with the memory region used.
Add a couple of missing checks as a result...

File size: 16.5 KB
Line 
1include "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 [ dp 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 [ dp 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 *;
29] qed.
30
31lemma yields_eq: ∀A,a,v'. yields A a v' → a = OK ? v'.
32#A #a #v' cases a; //; whd in ⊢ (% → ?); *;
33qed.
34
35lemma yields_sig_eq: ∀A,P,e,v. yields_sig A P e v → ∃p. e = OK ? (dp … v p).
36#A #P #e #v cases e;
37[ #vp cases vp; #v' #p #H whd in H; >H %{ p} @refl
38| *;
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
50lemma dec_true: ∀P:Prop.∀f:P + ¬P.∀p:P.∀Q:(P + ¬P) → Type[0]. (∀p'.Q (inl ?? p')) → Q f.
51#P #f #p #Q #H cases f;
52[ @H
53| #np cut False [ @(absurd ? p np) | * ]
54] qed.
55
56lemma dec_false: ∀P:Prop.∀f:P + ¬P.∀p:¬P.∀Q:(P + ¬P) → Type[0]. (∀p'.Q (inr ?? p')) → Q f.
57#P #f #p #Q #H cases f;
58[ #np cut False [ @(absurd ? np p) | * ]
59| @H
60] qed.
61
62theorem is_det: ∀p,s,s'.
63initial_state p s → initial_state p s' → s = s'.
64#p #s #s' #H1 #H2
65inversion H1; #b1 #f1 #ge1 #m1 #e11 #e12 #e13 #e14 #e15
66inversion H2; #b2 #f2 #ge2 #m2 #e21 #e22 #e23 #e24 #e25
67>e11 in e21 #e1 >(?:ge1 = ge2) in e13 e14
68[ 2: destruct (e1) skip (e11); @refl ]
69#e13 #e14
70
71>e12 in e22 #e2 destruct (e2) skip (e12);
72
73>e13 in e23 #e3 >(?:b1 = b2) in e14
74[ >e24 #e4 >(?:f2 = f1)
75  [ //;
76  | destruct (e4) skip (e24 e25); //;
77  ]
78| destruct (e3) skip (e13); //
79] qed.
80
81lemma remove_res_sig: ∀A. ∀P:A → Prop. ∀a,v',p.
82yields A a v' →
83yields_sig A P (err_inject A P (Some ? a) p) v'.
84#A #P #a cases a;
85[ #v #v' #p #H @H
86| #a #b *;
87] qed.
88
89
90theorem the_initial_state:
91  ∀p,s. initial_state p s → ∃ge. yields ? (make_initial_state p) 〈ge,s〉.
92#p #s cases p; #fns #main #globs #H
93inversion H;
94#b #f #ge #m #e1 #e2 #e3 #e4 #e5 %{ge}
95whd in ⊢ (??%?);
96>e1
97whd in ⊢ (??%?);
98>e2
99whd in ⊢ (??%?);
100>e3
101whd in ⊢ (??%?);
102>e4
103whd; @refl
104qed.
105
106lemma cast_complete: ∀m,v,ty,ty',v'.
107  cast m v ty ty' v' → yields ? (exec_cast m v ty ty') v'.
108#m #v #ty #ty' #v' #H
109elim H;
110[ #m #i #sz1 #sz2 #sg1 #sg2 @refl
111| #m #f #sz #szi #sg @refl
112| #m #i #sz #sz' #sg @refl
113| #m #f #sz #sz' @refl
114| #m #r #r' #ty #ty' #b #pc #ofs #H1 #H2 #pc'
115    elim H1 in pc ⊢ % [ #r1 #ty1 #pc | #r1 #ty1 #n1 #pc | #tys1 #ty1 #pc letin r1 ≝ Code ]
116    whd in ⊢ (??%?)
117    [ 1,2: @(dec_true ? (eq_region_dec r1 r1) (refl ??) …) #H0 whd in ⊢ (??%?) ]
118    elim H2 in pc' ⊢ %; [ 1,4,7: #sp2 #ty2 | 2,5,8: #sp2 #ty2 #n2 | 3,6,9: #tys2 #ty2 letin sp2 ≝ Code ]
119    #pc' whd in ⊢ (??%?)
120    @(dec_true ? (pointer_compat_dec b sp2) pc') //
121| #m #sz #si #ty'' #r #H cases H; //;
122| #m #t #t' #r #r' #H #H' cases H; try #a try #b try #c cases H'; try #d try #e try #f
123    whd in ⊢ (??%?); try @refl @(dec_true ? (eq_region_dec a a) (refl ??)) #H0 @refl
124] qed.
125
126(* Use to narrow down the choice of expression to just the lvalues. *)
127lemma lvalue_expr: ∀ge,env,m,e,ty,l,ofs,tr. ∀P:expr_descr → Prop.
128  eval_lvalue ge env m (Expr e ty) l ofs tr →
129  (∀id. P (Evar id)) → (∀e'. P (Ederef e')) → (∀e',id. P (Efield e' id)) →
130  P e.
131#ge #env #m #e #ty #l #ofs #tr #P #H @(eval_lvalue_inv_ind … H)
132[ #id #l #ty #e1 #e2 #e3 #e4 #e5 #e6 destruct; //
133| #id #sp #l #ty #e1 #e2 #e3 #e4 #e5 #e6 #e7 destruct; //
134| #e #ty #sp #l #ofs #tr #H #e1 #e2 #e3 #e4 #e5 destruct; //
135| #e #id #ty #l #ofs #id' #fs #d #tr #H #e1 #e2 (* bogus? *) #_ #e3 #e4 #e5 #e6 #e7 destruct; //
136| #e #id #ty #l #ofs #id' #fs #tr #H #e1 (* bogus? *) #_ #e2 #e3 #e4 #e5 #e6 destruct; //
137] qed.
138
139lemma 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.
140#v #ty #r #H elim H; #v #t #H' elim H';
141  [ #i #is #s #ne %{ true} % //; whd; >(eq_false … ne) //;
142  | #r #b #pc #i #i0 #s %{ true} % //
143  | #f #s #ne %{ true} % //; whd; >(Feq_zero_false … ne) //;
144  | #i #s %{ false} % //;
145  | #r #r' #t %{ false} % //;
146  | #s %{ false} % //; whd; >(Feq_zero_true …) //;
147  ]
148qed.
149
150lemma bool_of_true: ∀v,ty. is_true v ty → yields ? (exec_bool_of_val v ty) true.
151#v #ty #H elim H;
152  [ #i #is #s #ne whd; >(eq_false … ne) //;
153  | #p #b #i #i0 #s //
154  | #f #s #ne whd; >(Feq_zero_false … ne) //;
155  ]
156qed.
157
158lemma bool_of_false: ∀v,ty. is_false v ty → yields ? (exec_bool_of_val v ty) false.
159#v #ty #H elim H;
160  [ #i #s //;
161  | #r #r' #t //;
162  | #s whd; >(Feq_zero_true …) //;
163  ]
164qed.
165
166lemma expr_lvalue_complete: ∀ge,env,m.
167(∀e,v,tr. eval_expr ge env m e v tr → yields ? (exec_expr ge env m e) (〈v,tr〉)) ∧
168(∀e,l,off,tr. eval_lvalue ge env m e l off tr → yields ? (exec_lvalue ge env m e) (〈〈l,off〉,tr〉)).
169#ge #env #m
170@(combined_expr_lvalue_ind ge env m
171  (λe,v,tr,H. yields ? (exec_expr ge env m e) (〈v,tr〉))
172  (λe,l,off,tr,H. yields ? (exec_lvalue ge env m e) (〈〈l,off〉,tr〉)));
173[ #i #ty @refl
174| #f #ty @refl
175| #e #ty #l #off #v #tr #H1 #H2 @(lvalue_expr … H1)
176    [ #id | #e' | #e' #id ] #H3
177    whd in ⊢ (??%?);
178    >(yields_eq ??? H3)
179    whd in ⊢ (??%?); >H2 @refl
180| #e #ty #r #l #pc #off #tr #H1 #H2 whd in ⊢ (??%?);
181    >(yields_eq ??? H2) whd in ⊢ (??%?)
182    @(dec_true ? (pointer_compat_dec l r) pc) #pc' whd
183    @refl
184| #ty' #ty @refl
185| #op #e #ty #v1 #v #tr #H1 #H2 #H3 whd in ⊢ (??%?);
186    >(yields_eq ??? H3)
187    whd in ⊢ (??%?); >H2 @refl
188| #op #e1 #e2 #ty #v1 #v2 #v #tr1 #tr2 #H1 #H2 #e3 #H4 #H5 whd in ⊢ (??%?);
189    >(yields_eq ??? H4) whd in ⊢ (??%?);
190    >(yields_eq ??? H5) whd in ⊢ (??%?);
191    >e3 @refl
192| #e1 #e2 #e3 #ty #v1 #v2 #tr1 #tr2 #H1 #H2 #H3 #H4 #H5 whd in ⊢ (??%?);
193    >(yields_eq ??? H4) whd in ⊢ (??%?);
194    >(yields_eq ??? (bool_of_true ?? H2))
195    >(yields_eq ??? H5)
196    @refl
197| #e1 #e2 #e3 #ty #v1 #v2 #tr1 #tr2 #H1 #H2 #H3 #H4 #H5 whd in ⊢ (??%?);
198    >(yields_eq ??? H4) whd in ⊢ (??%?);
199    >(yields_eq ??? (bool_of_false ?? H2))
200    >(yields_eq ??? H5)
201    @refl
202| #e1 #e2 #ty #v1 #tr #H1 #H2 #H3 whd in ⊢ (??%?);
203    >(yields_eq ??? H3) whd in ⊢ (??%?);
204    >(yields_eq ??? (bool_of_true ?? H2))
205    @refl
206| #e1 #e2 #ty #v1 #v2 #v #tr1 #tr2 #H1 #H2 #H3 #H4 #H5 #H6 whd in ⊢ (??%?);
207    >(yields_eq ??? H5) whd in ⊢ (??%?);
208    >(yields_eq ??? (bool_of_false ?? H2))
209    >(yields_eq ??? H6) whd in ⊢ (??%?);
210    elim (bool_of_val_3_complete … H4); #b *; #evb #Hb
211    >(yields_eq ??? Hb) whd in ⊢ (??%?); <evb
212    @refl
213| #e1 #e2 #ty #v1 #tr #H1 #H2 #H3 whd in ⊢ (??%?);
214    >(yields_eq ??? H3) whd in ⊢ (??%?);
215    >(yields_eq ??? (bool_of_false ?? H2))
216    @refl
217| #e1 #e2 #ty #v1 #v2 #v #tr1 #tr2 #H1 #H2 #H3 #H4 #H5 #H6 whd in ⊢ (??%?);
218    >(yields_eq ??? H5) whd in ⊢ (??%?);
219    >(yields_eq ??? (bool_of_true ?? H2))
220    >(yields_eq ??? H6) whd in ⊢ (??%?);
221    elim (bool_of_val_3_complete … H4); #b *; #evb #Hb
222    >(yields_eq ??? Hb) whd in ⊢ (??%?); <evb
223    @refl
224| #e #ty #ty' #v1 #v #tr #H1 #H2 #H3 whd in ⊢ (??%?);
225    >(yields_eq ??? H3) whd in ⊢ (??%?);
226    >(yields_eq ??? (cast_complete … H2))
227    @refl
228| #e #ty #v #l #tr #H1 #H2 whd in ⊢ (??%?);
229    >(yields_eq ??? H2) whd in ⊢ (??%?);
230    @refl
231   
232  (* lvalues *)
233| #id #l #ty #e1 whd in ⊢ (??%?); >e1 @refl
234| #id #l #ty #e1 #e2 whd in ⊢ (??%?); >e1
235    >e2 @refl
236| #e #ty #r #l #pc #ofs #tr #H1 #H2 whd in ⊢ (??%?);
237    >(yields_eq ??? H2)
238    @refl
239| #e #i #ty #l #ofs #id #fList #delta #tr #H1 #H2 #H3 #H4 cases e in H2 H4 ⊢ %;
240    #e' #ty' #H2 whd in H2:(??%?); >H2 #H4 whd in ⊢ (??%?);
241    >(yields_eq ??? H4) whd in ⊢ (??%?);
242    >H3 @refl
243| #e #i #ty #l #ofs #id #fList #tr cases e; #e' #ty' #H1 #H2
244    whd in H2:(??%?); >H2 #H3 whd in ⊢ (??%?);
245    >(yields_eq ??? H3) @refl
246] qed.
247
248theorem expr_complete:  ∀ge,env,m.
249 ∀e,v,tr. eval_expr ge env m e v tr → yields ? (exec_expr ge env m e) (〈v,tr〉).
250#ge #env #m elim (expr_lvalue_complete ge env m); /2/; qed.
251
252theorem exprlist_complete: ∀ge,env,m,es,vs,tr.
253  eval_exprlist ge env m es vs tr → yields ? (exec_exprlist ge env m es) (〈vs,tr〉).
254#ge #env #m #es #vs #tr #H elim H;
255[ @refl
256| #e #et #v #vt #tr #trt #H1 #H2 #H3 whd in ⊢ (??%?);
257    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
258    >(yields_eq ??? H3)
259    @refl
260] qed.
261
262theorem lvalue_complete: ∀ge,env,m.
263 ∀e,l,off,tr. eval_lvalue ge env m e l off tr → yields ? (exec_lvalue ge env m e) (〈〈l,off〉,tr〉).
264#ge #env #m elim (expr_lvalue_complete ge env m); /2/; qed.
265
266let rec P_typelist (P:type → Prop) (l:typelist) on l : Prop ≝
267match l with
268[ Tnil ⇒ True
269| Tcons h t ⇒ P h ∧ P_typelist P t
270].
271
272let rec type_ind2l
273  (P:type → Prop) (Q:typelist → Prop)
274  (vo:P Tvoid)
275  (it:∀i,s. P (Tint i s))
276  (fl:∀f. P (Tfloat f))
277  (pt:∀s,t. P t → P (Tpointer s t))
278  (ar:∀s,t,n. P t → P (Tarray s t n))
279  (fn:∀tl,t. Q tl → P t → P (Tfunction tl t))
280  (st:∀i,fl. P (Tstruct i fl))
281  (un:∀i,fl. P (Tunion i fl))
282  (cp:∀r,i. P (Tcomp_ptr r i))
283  (nl:Q Tnil)
284  (cs:∀t,tl. P t → Q tl → Q (Tcons t tl))
285 (t:type) on t : P t ≝
286  match t return λt'.P t' with
287  [ Tvoid ⇒ vo
288  | Tint i s ⇒ it i s
289  | Tfloat s ⇒ fl s
290  | Tpointer s t' ⇒ pt s t' (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t')
291  | Tarray s t' n ⇒ ar s t' n (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t')
292  | 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')
293  | Tstruct i fs ⇒ st i fs
294  | Tunion i fs ⇒ un i fs
295  | Tcomp_ptr r i ⇒ cp r i
296  ]
297and typelist_ind2l
298  (P:type → Prop) (Q:typelist → Prop)
299  (vo:P Tvoid)
300  (it:∀i,s. P (Tint i s))
301  (fl:∀f. P (Tfloat f))
302  (pt:∀s,t. P t → P (Tpointer s t))
303  (ar:∀s,t,n. P t → P (Tarray s t n))
304  (fn:∀tl,t. Q tl → P t → P (Tfunction tl t))
305  (st:∀i,fl. P (Tstruct i fl))
306  (un:∀i,fl. P (Tunion i fl))
307  (cp:∀r,i. P (Tcomp_ptr r i))
308  (nl:Q Tnil)
309  (cs:∀t,tl. P t → Q tl → Q (Tcons t tl))
310  (ts:typelist) on ts : Q ts ≝
311  match ts return λts'.Q ts' with
312  [ Tnil ⇒ nl
313  | Tcons t tl ⇒ cs t tl (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t)
314                     (typelist_ind2l P Q vo it fl pt ar fn st un cp nl cs tl)
315  ].
316
317lemma assert_type_eq_true: ∀t. ∃p.assert_type_eq t t = OK ? p.
318#t whd in ⊢ (??(λ_.??%?)); cases (type_eq_dec t t); #E
319[ %{ E} //
320| @False_ind @(absurd ?? E) //
321] qed.
322
323lemma alloc_vars_complete: ∀env,m,l,env',m'.
324  alloc_variables env m l env' m' →
325  exec_alloc_variables env m l = 〈env', m'〉.
326#env #m #l #env' #m' #H elim H;
327[ #env'' #m'' %
328| #env1 #m1 #id #ty #l1 #m2 #loc #m3 #env2 #H1 #H2 #H3
329  < H3 whd in H1:(??%?) ⊢ (??%?)
330    destruct (H1) //
331] qed.
332
333lemma bind_params_complete: ∀e,m,params,vs,m2.
334  bind_parameters e m params vs m2 →
335  yields ? (exec_bind_parameters e m params vs) m2.
336#e #m #params #vs #m2 #H elim H;
337[ //;
338| #env1 #m1 #id #ty #l #v #tl #loc #m2 #m3 #H1 #H2 #H3 #H4
339    whd in ⊢ (??%?)
340    >H1 whd in ⊢ (??%?);
341    >H2 whd in ⊢ (??%?);
342    @H4
343] qed.
344
345lemma eventval_match_complete': ∀ev,ty,v.
346  eventval_match ev ty v → yields ? (check_eventval' v ty) ev.
347#ev #ty #v #H elim H; //; qed.
348
349lemma eventval_list_match_complete: ∀vs,tys,evs.
350  eventval_list_match evs tys vs → yields ? (check_eventval_list vs tys) evs.
351#vs #tys #evs #H elim H;
352[ //
353| #e #etl #ty #tytl #v #vtl #H1 #H2 #H3 whd in ⊢ (??%?)
354    >(yields_eq ??? (eventval_match_complete' … H1)) whd in ⊢ (??%?)
355    >(yields_eq ??? H3) whd in ⊢ (??%?) //
356] qed.
357
358theorem step_complete: ∀ge,s,tr,s'.
359  step ge s tr s' → yieldsIO ? (exec_step ge s) 〈tr,s'〉.
360#ge #s #tr #s' #H elim H;
361[ #f #e #e1 #k #e2 #m #loc #ofs #v #m' #tr1 #tr2 #H1 #H2 #H3 whd in ⊢ (??%?);
362    >(yields_eq ??? (lvalue_complete … H1)) whd in ⊢ (??%?);
363    >(yields_eq ??? (expr_complete … H2)) whd in ⊢ (??%?);
364    >H3 @refl
365| #f #e #eargs #k #ef #m #vf #vargs #f' #tr1 #tr2 #H1 #H2 #H3 #H4 whd in ⊢ (??%?);
366    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
367    >(yields_eq ??? (exprlist_complete … H2)) whd in ⊢ (??%?);
368    >H3 whd in ⊢ (??%?);
369    >H4 elim (assert_type_eq_true (fun_typeof e)); #pty #ety >ety
370    @refl
371| #f #el #ef #eargs #k #env #m #loc #ofs #vf #vargs #f' #tr1 #tr2 #tr3 #H1 #H2 #H3 #H4 #H5 whd in ⊢ (??%?);
372    >(yields_eq ??? (expr_complete … H2)) whd in ⊢ (??%?);
373    >(yields_eq ??? (exprlist_complete … H3)) whd in ⊢ (??%?);
374    >H4 whd in ⊢ (??%?);
375    >H5 elim (assert_type_eq_true (fun_typeof ef)); #pty #ety >ety
376    whd in ⊢ (??%?);
377    >(yields_eq ??? (lvalue_complete … H1)) whd in ⊢ (??%?);
378    @refl
379| #f #s1 #s2 #k #env #m @refl
380| 5,6,7: #f #s #k #env #m @refl
381| #f #e #s1 #s2 #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
382    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
383    >(yields_eq ??? (bool_of_true ?? H2))
384    @refl
385| #f #e #s1 #s2 #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
386    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
387    >(yields_eq ??? (bool_of_false ?? H2))
388    @refl
389| #f #e #s #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
390    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
391    >(yields_eq ??? (bool_of_false ?? H2))
392    @refl
393| #f #e #s #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
394    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
395    >(yields_eq ??? (bool_of_true ?? H2))
396    @refl
397| #f #s1 #e #s2 #k #env #m #H cases H; #es1 >es1 @refl
398| 13,14: #f #e #s #k #env #m @refl
399| #f #s1 #e #s2 #k #env #m #v #tr *; #es1 >es1 #H1 #H2 whd in ⊢ (??%?);
400    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
401    >(yields_eq ??? (bool_of_false ?? H2))
402    @refl
403| #f #s1 #e #s2 #k #env #m #v #tr *; #es1 >es1 #H1 #H2 whd in ⊢ (??%?);
404    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
405    >(yields_eq ??? (bool_of_true ?? H2))
406    @refl
407| #f #e #s #k #env #m @refl
408| #f #s1 #e #s2 #s3 #k #env #m #nskip whd in ⊢ (??%?); cases (is_Sskip s1);
409    [ #H @False_ind @(absurd ? H nskip)
410    | #H whd in ⊢ (??%?); @refl ]
411| #f #e #s1 #s2 #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
412    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
413    >(yields_eq ??? (bool_of_false ?? H2))
414    @refl
415| #f #e #s1 #s2 #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
416    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
417    >(yields_eq ??? (bool_of_true ?? H2))
418    @refl
419| #f #s1 #e #s2 #s3 #k #env #m *; #es1 >es1 @refl
420| 22,23: #f #e #s1 #s2 #k #env #m @refl
421| #f #k #env #m #H whd in ⊢ (??%?); >H @refl
422| #f #e #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?);
423    @(dec_false ? (type_eq_dec (fn_return f) Tvoid) H1) #pf'
424    whd in ⊢ (??%?);
425    >(yields_eq ??? (expr_complete … H2)) whd in ⊢ (??%?);
426    @refl
427| #f #k #env #m cases k;
428    [ #H1 #H2 whd in ⊢ (??%?); >H2 @refl
429    | #s' #k' whd in ⊢ (% → ?); *;
430    | 3,4: #e' #s' #k' whd in ⊢ (% → ?); *;
431    | 5,6: #e' #s1' #s2' #k' whd in ⊢ (% → ?); *;
432    | #k' whd in ⊢ (% → ?); *;
433    | #r #f' #env' #k' #H1 #H2 whd in ⊢ (??%?); >H2 @refl
434    ]
435| #f #e #s #k #env #m #i #tr #H1 whd in ⊢ (??%?);
436    >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?);
437    @refl
438| #f #s #k #env #m *; #es >es @refl
439| #f #k #env #m @refl
440| #f #l #s #k #env #m @refl
441| #f #l #k #env #m #s #k' #H1 whd in ⊢ (??%?); >H1 @refl
442| #f #args #k #m1 #env #m2 #m3 #H1 #H2 whd in ⊢ (??%?);
443    >(alloc_vars_complete … H1) whd in ⊢ (??%?);
444    >(yields_eq ??? (bind_params_complete … H2))
445    //
446| #id #tys #rty #args #k #m #rv #tr #H whd in ⊢ (??%?);
447    inversion H; #f' #args' #rv' #eargs #erv #H1 #H2 #e1 #e2 #e3 #e4 <e1 in H1 H2
448    #H1 #H2
449    >(yields_eq ??? (eventval_list_match_complete … H1)) whd in ⊢ (??%?);
450    whd; inversion H2; #x #e5 #e6 #e7 %{ x} whd in ⊢ (??%?);
451    @refl
452| #v #f #env #k #m @refl
453| #v #f #env #k #m1 #m2 #loc #ofs #ty #H whd in ⊢ (??%?);
454    >H @refl
455| #f #l #s #k #env #m @refl
456] qed.
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