source: C-semantics/CexecIOcomplete.ma @ 253

Last change on this file since 253 was 253, checked in by campbell, 9 years ago

Update completeness proof for executable semantics with separate soundness
proofs.
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1include "CexecIO.ma".
2include "Plogic/connectives.ma".
3
4ndefinition yields : ∀A,P. res (Σx:A. P x) → A → Prop ≝
5λA,P,e,v. match e with [ OK v' ⇒ match v' with [ sig_intro v'' _ ⇒ v = v'' ] | _ ⇒ False].
6
7(* This tells us that some execution of e results in v.
8   (There may be many possible executions due to I/O, but we're trying to prove
9   that one particular one exists corresponding to a derivation in the inductive
10   semantics.) *)
11nlet rec yieldsIO (A:Type) (P:A → Prop) (e:IO eventval io_out (Σx:A. P x)) (v:A) on e : Prop ≝
12match e with
13[ Value v' ⇒ match v' with [ sig_intro v'' _ ⇒ v = v'' ]
14| Interact _ k ⇒ ∃r.yieldsIO A P (k r) v
15| _ ⇒ False].
16
17nlemma is_pointer_compat_true: ∀m,b,sp.
18  pointer_compat (block_space m b) sp →
19  is_pointer_compat (block_space m b) sp = true.
20#m b sp H; nwhd in ⊢ (??%?);
21nelim (pointer_compat_dec (block_space m b) sp);
22##[ //
23##| #H'; napply False_ind; napply (absurd … H H');
24##] nqed.
25
26nlemma ms_eq_dec_true: ∀s. ms_eq_dec s s = inl ???.
27##[ #s; ncases s; napply refl;
28##| ##skip
29##] nqed.
30
31notation < "vbox( e break ↓ break e')" with precedence 99 for @{'yields ${e} ${e'}}.
32interpretation "yields" 'yields e e' = (yields ?? e e').
33interpretation "yields IO" 'yields e e' = (yieldsIO ?? e e').
34
35ntheorem is_det: ∀p,s,s'.
36initial_state p s → initial_state p s' → s = s'.
37#p s s' H1 H2;
38ninversion H1; #b1 f1 e11 e12 e13;
39ninversion H2; #b2 f2 e21 e22 e23;
40nrewrite > e11 in e21;
41#e1; nrewrite > (?:b1 = b2) in e12;
42##[ nrewrite > e22; #e2; nrewrite > (?:f2 = f1);
43  ##[ //;
44  ##| ndestruct (e2) skip (e22 e23); //;
45  ##]
46##| ndestruct (e1) skip (e11); //
47##] nqed.
48
49nlet rec yieldsIObare (A:Type) (a:IO eventval io_out A) (v':A) on a : Prop ≝
50match a with [ Value v ⇒ v' = v | Interact _ k ⇒ ∃r.yieldsIObare A (k r) v' | _ ⇒ False ].
51
52nlemma remove_io_sig: ∀A. ∀P:A → Prop. ∀a,v',p.
53yieldsIObare A a v' →
54yieldsIO A P (io_inject eventval io_out A (λx.P x) (Some ? a) p) v'.
55#A P a; nelim a;
56##[ #a k IH v' p H; nwhd in H ⊢ %; nelim H; #r H'; @ r; napply IH; napply H';
57##| #v v' p H; napply H;
58##| #a b; *;
59##] nqed.
60
61
62ntheorem the_initial_state:
63  ∀p,s. initial_state p s → yieldsIObare ? (make_initial_state p) s.
64#p s; ncases p; #fns main globs H;
65ninversion H;
66#b f e1 e2 e3;
67nwhd in ⊢ (??%?);
68nrewrite > e1;
69nwhd in ⊢ (??%?);
70nrewrite > e2;
71nwhd; napply refl;
72nqed.
73
74ndefinition yieldsbare ≝ λA.λa:res A.λv':A.
75match a with [ OK v ⇒ v' = v | _ ⇒ False ].
76
77nlemma yieldsbare_eq: ∀A,a,v'. yieldsbare A a v' → a = OK ? v'.
78#A a v'; ncases a; //; nwhd in ⊢ (% → ?); *;
79nqed.
80
81nlemma remove_res_sig: ∀A. ∀P:A → Prop. ∀a,v',p.
82yieldsbare A a v' →
83yields A P (err_inject A (λx.P x) (Some ? a) p) v'.
84#A P a; ncases a;
85##[ #v v' p H; napply H;
86##| #a b; *;
87##] nqed.
88
89nlemma cast_complete: ∀m,v,ty,ty',v'.
90  cast m v ty ty' v' → yieldsbare ? (exec_cast m v ty ty') v'.
91#m v ty ty' v' H;
92nelim H;
93##[ #m i sz1 sz2 sg1 sg2; napply refl;
94##| #m f sz szi sg; napply refl;
95##| #m i sz sz' sg; napply refl;
96##| #m f sz sz'; napply refl;
97##| #m sp sp' ty ty' b ofs H1 H2 H3;
98    nelim H1; ##[ #sp1 ty1 ##| #sp1 ty1 n1 ##| #tys1 ty1; nletin sp1 ≝ Code ##]
99    nwhd in ⊢ (??%?);
100    ##[ ##1,2: nrewrite > (ms_eq_dec_true …); nwhd in ⊢ (??%?); ##]
101    nelim H2 in H3 ⊢ %; ##[ ##1,4,7: #sp2 ty2 ##| ##2,5,8: #sp2 ty2 n2 ##| ##3,6,9: #tys2 ty2; nletin sp2 ≝ Code ##]
102    #H3; nwhd in ⊢ (??%?);
103    nrewrite > (is_pointer_compat_true …); //;
104##| #m sz si ty'' H; ncases H; ##[ #sp1 ty1 ##| #sp1 ty1 n1 ##| #args rty ##] napply refl;
105##| #m t t' H H'; ncases H; ncases H'; //;
106##] nqed.
107
108nlemma yields_eq: ∀A,P,e,v. yields A P e v → ∃p. e = OK ? (sig_intro … v p).
109#A P e v; ncases e;
110##[ #vp; ncases vp; #v' p H; nwhd in H; nrewrite > H; @ p; napply refl;
111##| *;
112##] nqed.
113
114(* Use to narrow down the choice of expression to just the lvalues. *)
115nlemma lvalue_expr: ∀ge,env,m,e,ty,sp,l,ofs,tr. ∀P:expr_descr → Prop.
116  eval_lvalue ge env m (Expr e ty) sp l ofs tr →
117  (∀id. P (Evar id)) → (∀e'. P (Ederef e')) → (∀e',id. P (Efield e' id)) →
118  P e.
119#ge env m e ty sp l ofs tr P H; napply (eval_lvalue_inv_ind … H);
120##[ #id l ty e1 e2 e3 e4 e5 e6; ndestruct; //
121##| #id sp l ty e1 e2 e3 e4 e5 e6 e7; ndestruct; //
122##| #e ty sp l ofs tr H e1 e2 e3 e4 e5; ndestruct; //
123##| #e id ty sp l ofs id' fs d tr H e1 e2;(* bogus? *) #_; #e3 e4 e5 e6 e7; ndestruct; //
124##| #e id ty sp l ofs id' fs tr H e1;(* bogus? *) #_; #e2 e3 e4 e5 e6; ndestruct; //
125##] nqed.
126
127nlemma bool_of_val_3_complete : ∀v,ty,r. bool_of_val v ty r → ∃b. r = of_bool b ∧ yieldsbare ? (exec_bool_of_val v ty) b.
128#v ty r H; nelim H; #v t H'; nelim H';
129  ##[ #i is s ne; @ true; @; //; nwhd; nrewrite > (eq_false … ne); //;
130  ##| #p b i i0 s; @ true; @; //
131  ##| #i p t ne; @ true; @; //; nwhd; nrewrite > (eq_false … ne); //;
132  ##| #p b i p0 t0; @ true; @; //
133  ##| #f s ne; @ true; @; //; nwhd; nrewrite > (Feq_zero_false … ne); //;
134  ##| #i s; @ false; @; //;
135  ##| #p t; @ false; @; //;
136  ##| #s; @ false; @; //; nwhd; nrewrite > (Feq_zero_true …); //;
137  ##]
138nqed.
139
140nlemma bool_of_true: ∀v,ty. is_true v ty → yieldsbare ? (exec_bool_of_val v ty) true.
141#v ty H; nelim H;
142  ##[ #i is s ne; nwhd; nrewrite > (eq_false … ne); //;
143  ##| #p b i i0 s; //
144  ##| #i p t ne; nwhd; nrewrite > (eq_false … ne); //;
145  ##| #p b i p0 t0; //
146  ##| #f s ne; nwhd; nrewrite > (Feq_zero_false … ne); //;
147  ##]
148nqed.
149
150nlemma bool_of_false: ∀v,ty. is_false v ty → yieldsbare ? (exec_bool_of_val v ty) false.
151#v ty H; nelim H;
152  ##[ #i s; //;
153  ##| #p t; //;
154  ##| #s; nwhd; nrewrite > (Feq_zero_true …); //;
155  ##]
156nqed.
157
158nremark eq_to_jmeq: ∀A. ∀a,b:A. a = b → a ≃ b.
159#A a b H; nrewrite > H; //; nqed.
160
161nlemma dep_option_rewrite: ∀A,B:Type. ∀e:option A. ∀r:B. ∀P:B → Prop. ∀Q:e ≃ None A → res (Σx:B. P x). ∀R:∀v. e ≃ Some A v → res (Σx:B. P x). ∀h: e = None A.
162 yields ?? (Q (eq_to_jmeq ??? h)) r →
163 yields ?? ((match e return λe'.e ≃ e' → ? with [ None ⇒ λp.Q p | Some v ⇒ λp.R v p ]) (refl_jmeq (option A) e)) r.
164#A B e; ncases e;
165##[ #r P Q R h; nwhd in ⊢ (? → ???%?);
166napply (streicherKjmeq ?? (λe. yields ?? (Q e) r → yields ?? (Q (refl_jmeq (option A) (None A))) r));
167//;
168##| #v r P Q R h; ndestruct (h);
169##] nqed.
170
171nlemma expr_lvalue_complete: ∀ge,env,m.
172(∀e,v,tr. eval_expr ge env m e v tr → yieldsbare ? (exec_expr ge env m e) (〈v,tr〉)) ∧
173(∀e,sp,l,off,tr. eval_lvalue ge env m e sp l off tr → yieldsbare ? (exec_lvalue ge env m e) (〈〈〈sp,l〉,off〉,tr〉)).
174#ge env m;
175napply (combined_expr_lvalue_ind ge env m
176  (λe,v,tr,H. yieldsbare ? (exec_expr ge env m e) (〈v,tr〉))
177  (λe,sp,l,off,tr,H. yieldsbare ? (exec_lvalue ge env m e) (〈〈〈sp,l〉,off〉,tr〉)));
178##[ #i ty; napply refl;
179##| #f ty; napply refl;
180##| #e ty sp l off v tr H1 H2; napply (lvalue_expr … H1);
181    ##[ #id ##| #e' ##| #e' id ##] #H3;
182    nwhd in ⊢ (??%?);
183    nrewrite > (yieldsbare_eq ??? H3);
184    nwhd in ⊢ (??%?); nrewrite > H2; napply refl;
185##| #e ty sp l off tr H1 H2; nwhd in ⊢ (??%?);
186    nrewrite > (yieldsbare_eq ??? H2);
187    napply refl;
188##| #ty' ty; napply refl;
189##| #op e ty v1 v tr H1 H2 H3; nwhd in ⊢ (??%?);
190    nrewrite > (yieldsbare_eq ??? H3);
191    nwhd in ⊢ (??%?); nrewrite > H2; napply refl;
192##| #op e1 e2 ty v1 v2 v tr1 tr2 H1 H2 e3 H4 H5; nwhd in ⊢ (??%?);
193    nrewrite > (yieldsbare_eq ??? H4); nwhd in ⊢ (??%?);
194    nrewrite > (yieldsbare_eq ??? H5); nwhd in ⊢ (??%?);
195    nrewrite > e3; napply refl;
196##| #e1 e2 e3 ty v1 v2 tr1 tr2 H1 H2 H3 H4 H5; nwhd in ⊢ (??%?);
197    nrewrite > (yieldsbare_eq ??? H4); nwhd in ⊢ (??%?);
198    nrewrite > (yieldsbare_eq ??? (bool_of_true ?? H2));
199    nrewrite > (yieldsbare_eq ??? H5);
200    napply refl;
201##| #e1 e2 e3 ty v1 v2 tr1 tr2 H1 H2 H3 H4 H5; nwhd in ⊢ (??%?);
202    nrewrite > (yieldsbare_eq ??? H4); nwhd in ⊢ (??%?);
203    nrewrite > (yieldsbare_eq ??? (bool_of_false ?? H2));
204    nrewrite > (yieldsbare_eq ??? H5);
205    napply refl;
206##| #e1 e2 ty v1 tr H1 H2 H3; nwhd in ⊢ (??%?);
207    nrewrite > (yieldsbare_eq ??? H3); nwhd in ⊢ (??%?);
208    nrewrite > (yieldsbare_eq ??? (bool_of_true ?? H2));
209    napply refl;   
210##| #e1 e2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 H5 H6; nwhd in ⊢ (??%?);
211    nrewrite > (yieldsbare_eq ??? H5); nwhd in ⊢ (??%?);
212    nrewrite > (yieldsbare_eq ??? (bool_of_false ?? H2));
213    nrewrite > (yieldsbare_eq ??? H6); nwhd in ⊢ (??%?);
214    nelim (bool_of_val_3_complete … H4); #b; *; #evb Hb;
215    nrewrite > (yieldsbare_eq ??? Hb); nwhd in ⊢ (??%?); nrewrite < evb;
216    napply refl;   
217##| #e1 e2 ty v1 tr H1 H2 H3; nwhd in ⊢ (??%?);
218    nrewrite > (yieldsbare_eq ??? H3); nwhd in ⊢ (??%?);
219    nrewrite > (yieldsbare_eq ??? (bool_of_false ?? H2));
220    napply refl;   
221##| #e1 e2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 H5 H6; nwhd in ⊢ (??%?);
222    nrewrite > (yieldsbare_eq ??? H5); nwhd in ⊢ (??%?);
223    nrewrite > (yieldsbare_eq ??? (bool_of_true ?? H2));
224    nrewrite > (yieldsbare_eq ??? H6); nwhd in ⊢ (??%?);
225    nelim (bool_of_val_3_complete … H4); #b; *; #evb Hb;
226    nrewrite > (yieldsbare_eq ??? Hb); nwhd in ⊢ (??%?); nrewrite < evb;
227    napply refl;
228##| #e ty ty' v1 v tr H1 H2 H3; nwhd in ⊢ (??%?);
229    nrewrite > (yieldsbare_eq ??? H3); nwhd in ⊢ (??%?);
230    nrewrite > (yieldsbare_eq ??? (cast_complete … H2));
231    napply refl;
232##| #e ty v l tr H1 H2; nwhd in ⊢ (??%?);
233    nrewrite > (yieldsbare_eq ??? H2); nwhd in ⊢ (??%?);
234    napply refl;
235   
236  (* lvalues *)
237##| #id l ty e1; nwhd in ⊢ (??%?); nrewrite > e1; napply refl;
238##| #id sp l ty e1 e2; nwhd in ⊢ (??%?); nrewrite > e1;
239    nrewrite > e2; napply refl;
240##| #e ty sp l ofs tr H1 H2; nwhd in ⊢ (??%?);
241    nrewrite > (yieldsbare_eq ??? H2);
242    napply refl;
243##| #e i ty sp l ofs id fList delta tr H1 H2 H3 H4; ncases e in H2 H4 ⊢ %;
244    #e' ty' H2; nwhd in H2:(??%?); nrewrite > H2; #H4; nwhd in ⊢ (??%?);
245    nrewrite > (yieldsbare_eq ??? H4); nwhd in ⊢ (??%?);
246    nrewrite > H3; napply refl;
247##| #e i ty sp l ofs id fList tr; ncases e; #e' ty' H1 H2;
248    nwhd in H2:(??%?); nrewrite > H2; #H3; nwhd in ⊢ (??%?);
249    nrewrite > (yieldsbare_eq ??? H3); napply refl;
250##] nqed.
251
252ntheorem expr_complete:  ∀ge,env,m.
253 ∀e,v,tr. eval_expr ge env m e v tr → yieldsbare ? (exec_expr ge env m e) (〈v,tr〉).
254#ge env m; nelim (expr_lvalue_complete ge env m); /2/; nqed.
255
256ntheorem exprlist_complete: ∀ge,env,m,es,vs,tr.
257  eval_exprlist ge env m es vs tr → yieldsbare ? (exec_exprlist ge env m es) (〈vs,tr〉).
258#ge env m es vs tr H; nelim H;
259##[ napply refl;
260##| #e et v vt tr trt H1 H2 H3; nwhd in ⊢ (??%?);
261    nrewrite > (yieldsbare_eq ??? (expr_complete … H1)); nwhd in ⊢ (??%?);
262    nrewrite > (yieldsbare_eq ??? H3);
263    napply refl;
264##] nqed.
265
266ntheorem lvalue_complete: ∀ge,env,m.
267 ∀e,sp,l,off,tr. eval_lvalue ge env m e sp l off tr → yieldsbare ? (exec_lvalue ge env m e) (〈〈〈sp,l〉,off〉,tr〉).
268#ge env m; nelim (expr_lvalue_complete ge env m); /2/; nqed.
269
270nlet rec P_typelist (P:type → Prop) (l:typelist) on l : Prop ≝
271match l with
272[ Tnil ⇒ True
273| Tcons h t ⇒ P h ∧ P_typelist P t
274].
275
276nlet rec type_ind2l
277  (P:type → Prop) (Q:typelist → Prop)
278  (vo:P Tvoid)
279  (it:∀i,s. P (Tint i s))
280  (fl:∀f. P (Tfloat f))
281  (pt:∀s,t. P t → P (Tpointer s t))
282  (ar:∀s,t,n. P t → P (Tarray s t n))
283  (fn:∀tl,t. Q tl → P t → P (Tfunction tl t))
284  (st:∀i,fl. P (Tstruct i fl))
285  (un:∀i,fl. P (Tunion i fl))
286  (cp:∀i. P (Tcomp_ptr i))
287  (nl:Q Tnil)
288  (cs:∀t,tl. P t → Q tl → Q (Tcons t tl))
289 (t:type) on t : P t ≝
290  match t return λt'.P t' with
291  [ Tvoid ⇒ vo
292  | Tint i s ⇒ it i s
293  | Tfloat s ⇒ fl s
294  | Tpointer s t' ⇒ pt s t' (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t')
295  | Tarray s t' n ⇒ ar s t' n (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t')
296  | 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')
297  | Tstruct i fs ⇒ st i fs
298  | Tunion i fs ⇒ un i fs
299  | Tcomp_ptr i ⇒ cp i
300  ]
301and typelist_ind2l
302  (P:type → Prop) (Q:typelist → Prop)
303  (vo:P Tvoid)
304  (it:∀i,s. P (Tint i s))
305  (fl:∀f. P (Tfloat f))
306  (pt:∀s,t. P t → P (Tpointer s t))
307  (ar:∀s,t,n. P t → P (Tarray s t n))
308  (fn:∀tl,t. Q tl → P t → P (Tfunction tl t))
309  (st:∀i,fl. P (Tstruct i fl))
310  (un:∀i,fl. P (Tunion i fl))
311  (cp:∀i. P (Tcomp_ptr i))
312  (nl:Q Tnil)
313  (cs:∀t,tl. P t → Q tl → Q (Tcons t tl))
314  (ts:typelist) on ts : Q ts ≝
315  match ts return λts'.Q ts' with
316  [ Tnil ⇒ nl
317  | Tcons t tl ⇒ cs t tl (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t)
318                     (typelist_ind2l P Q vo it fl pt ar fn st un cp nl cs tl)
319  ].
320
321naxiom assert_type_eq_true: ∀t. ∃p.assert_type_eq t t = OK ? p.
322(*nlemma assert_type_eq_true: ∀t. ∃p.assert_type_eq t t = OK ? p.
323#t; napply (type_ind2l ? (λtl. ∃p.assert_typelist_eq tl tl = OK ? p) … t);
324##[ @ (refl ??); // ##| #sz si; ncases sz; ncases si; @ (refl ??); //;
325##| #sz; ncases sz; @ ?; //;
326##| #sp ty IH; ncases sp; nwhd in ⊢ (??(λ_.??%?)); nelim IH; #p IH; nrewrite > IH; @ ?; //;
327##| #sp ty n IH; ncases sp; nwhd in ⊢ (??(λ_.??%?)); nelim IH; #p IH; nrewrite > IH;
328    nwhd in ⊢ (??(λ_.??%?)); ncases (decidable_eq_Z_Type n n);
329    ##[ ##1,3,5,7,9,11: #H; nwhd in ⊢ (??(λ_.??%?)); @ ?; //;
330    ##| ##*: #H; napply False_ind; /2/;
331    ##]
332##| #tys ty IH1 IH2; @ ?;
333    ##[ ##2: nwhd in ⊢ (??%?); nelim IH1; #p1 e1;
334    nrewrite > e1; nwhd in ⊢ (??%?);
335    nelim IH2;
336    *)
337
338nlemma is_not_void_true: ∀f. ¬fn_return f = Tvoid → ∃p. is_not_void (fn_return f) = OK ? p.
339#f; ncases f; #ty; #_; #_; #_; ncases ty;
340##[ #H; napply False_ind; /2/;
341##| #sz sg e; @ ?; //; ##| #sz e; @ ?; // ##| #sp ty e; @ ?; // ##| #sp ty n e; @ ?; // ##|
342    #tys ty e; @ ?; // ##| #id fs e; @ ?; // ##| #id fs e; @ ?; // ##| #id e; @ ?; // ##]
343nqed.
344
345nlemma alloc_vars_complete: ∀env,m,l,env',m'.
346  alloc_variables env m l env' m' →
347  ∃p.exec_alloc_variables env m l = sig_intro ?? (Some ? 〈env', m'〉) p.
348#env m l env' m' H; nelim H;
349##[ #env'' m''; @ ?; nwhd; //;
350##| #env1 m1 id ty l1 m2 loc m3 env2 H1 H2 H3;
351    nwhd in H1:(??%?) ⊢ (??(λ_.??%?));
352    ndestruct (H1);
353    nelim H3; #p3 e3; nrewrite > e3; nwhd in ⊢ (??(λ_.??%?)); @ ?; //;
354##] nqed.
355
356nlemma bind_params_complete: ∀e,m,params,vs,m2.
357  bind_parameters e m params vs m2 →
358  yields ?? (exec_bind_parameters e m params vs) m2.
359#e m params vs m2 H; nelim H;
360##[ //;
361##| #env1 m1 id ty l v tl loc m2 m3 H1 H2 H3 H4;
362    napply remove_res_sig;
363    nrewrite > H1; nwhd in ⊢ (??%?);
364    nrewrite > H2; nwhd in ⊢ (??%?);
365    nelim (yields_eq ???? H4); #p4 e4; nrewrite > e4;
366    napply refl;
367##] nqed.
368
369nlemma eventval_match_complete: ∀ev,ty,v.
370  eventval_match ev ty v → yields ?? (check_eventval ev ty) v.
371#ev ty v H; nelim H; //; nqed.
372
373nlemma eventval_match_complete': ∀ev,ty,v.
374  eventval_match ev ty v → yields ?? (check_eventval' v ty) ev.
375#ev ty v H; nelim H; //; nqed.
376
377nlemma eventval_list_match_complete: ∀vs,tys,evs.
378  eventval_list_match evs tys vs → yields ?? (check_eventval_list vs tys) evs.
379#vs tys evs H; nelim H;
380##[ //
381##| #e etl ty tytl v vtl H1 H2 H3; napply remove_res_sig;
382    nelim (yields_eq ???? (eventval_match_complete' … H1)); #p1 e1; nrewrite > e1; nwhd in ⊢ (??%?);
383    nelim (yields_eq ???? H3); #p3 e3; nrewrite > e3; nwhd in ⊢ (??%?);
384    napply refl;
385##] nqed.   
386
387
388ntheorem step_complete: ∀ge,s,tr,s'.
389  step ge s tr s' → yieldsIObare ? (exec_step ge s) 〈tr,s'〉.
390#ge s tr s' H; nelim H;
391##[ #f e e1 k e2 m sp loc ofs v m' tr1 tr2 H1 H2 H3; nwhd in ⊢ (??%?);
392    nrewrite > (yieldsbare_eq ??? (lvalue_complete … H1)); nwhd in ⊢ (??%?);
393    nrewrite > (yieldsbare_eq ??? (expr_complete … H2)); nwhd in ⊢ (??%?);
394    nrewrite > H3; napply refl;
395##| #f e eargs k ef m vf vargs f' tr1 tr2 H1 H2 H3 H4; nwhd in ⊢ (??%?);
396    nrewrite > (yieldsbare_eq ??? (expr_complete … H1)); nwhd in ⊢ (??%?);
397    nrewrite > (yieldsbare_eq ??? (exprlist_complete … H2)); nwhd in ⊢ (??%?);
398    nrewrite > H3; nwhd in ⊢ (??%?);
399    nrewrite > H4; nelim (assert_type_eq_true (typeof e)); #pty ety; nrewrite > ety;
400    napply refl;
401##| #f el ef eargs k env m sp loc ofs vf vargs f' tr1 tr2 tr3 H1 H2 H3 H4 H5; nwhd in ⊢ (??%?);
402    nrewrite > (yieldsbare_eq ??? (expr_complete … H2)); nwhd in ⊢ (??%?);
403    nrewrite > (yieldsbare_eq ??? (exprlist_complete … H3)); nwhd in ⊢ (??%?);
404    nrewrite > H4; nwhd in ⊢ (??%?);
405    nrewrite > H5; nelim (assert_type_eq_true (typeof ef)); #pty ety; nrewrite > ety;
406    nwhd in ⊢ (??%?);
407    nrewrite > (yieldsbare_eq ??? (lvalue_complete … H1)); nwhd in ⊢ (??%?);
408    napply refl;
409##| #f s1 s2 k env m; napply refl
410##| ##5,6,7: #f s k env m; napply refl
411##| #f e s1 s2 k env m v tr H1 H2; nwhd in ⊢ (??%?);
412    nrewrite > (yieldsbare_eq ??? (expr_complete … H1)); nwhd in ⊢ (??%?);
413    nrewrite > (yieldsbare_eq ??? (bool_of_true ?? H2));
414    napply refl
415##| #f e s1 s2 k env m v tr H1 H2; nwhd in ⊢ (??%?);
416    nrewrite > (yieldsbare_eq ??? (expr_complete … H1)); nwhd in ⊢ (??%?);
417    nrewrite > (yieldsbare_eq ??? (bool_of_false ?? H2));
418    napply refl
419##| #f e s k env m v tr H1 H2; nwhd in ⊢ (??%?);
420    nrewrite > (yieldsbare_eq ??? (expr_complete … H1)); nwhd in ⊢ (??%?);
421    nrewrite > (yieldsbare_eq ??? (bool_of_false ?? H2));
422    napply refl
423##| #f e s k env m v tr H1 H2; nwhd in ⊢ (??%?);
424    nrewrite > (yieldsbare_eq ??? (expr_complete … H1)); nwhd in ⊢ (??%?);
425    nrewrite > (yieldsbare_eq ??? (bool_of_true ?? H2));
426    napply refl
427##| #f s1 e s2 k env m H; ncases H; #es1; nrewrite > es1; napply refl;
428##| ##13,14: #f e s k env m; napply refl
429##| #f s1 e s2 k env m v tr; *; #es1; nrewrite > es1; #H1 H2; nwhd in ⊢ (??%?);
430    nrewrite > (yieldsbare_eq ??? (expr_complete … H1)); nwhd in ⊢ (??%?);
431    nrewrite > (yieldsbare_eq ??? (bool_of_false ?? H2));
432    napply refl
433##| #f s1 e s2 k env m v tr; *; #es1; nrewrite > es1; #H1 H2; nwhd in ⊢ (??%?);
434    nrewrite > (yieldsbare_eq ??? (expr_complete … H1)); nwhd in ⊢ (??%?);
435    nrewrite > (yieldsbare_eq ??? (bool_of_true ?? H2));
436    napply refl
437##| #f e s k env m; napply refl;
438##| #f s1 e s2 s3 k env m nskip; nwhd in ⊢ (??%?); ncases (is_Sskip s1);
439    ##[ #H; napply False_ind; /2/;
440    ##| #H; nwhd in ⊢ (??%?); napply refl ##]
441##| #f e s1 s2 k env m v tr H1 H2; nwhd in ⊢ (??%?);
442    nrewrite > (yieldsbare_eq ??? (expr_complete … H1)); nwhd in ⊢ (??%?);
443    nrewrite > (yieldsbare_eq ??? (bool_of_false ?? H2));
444    napply refl;
445##| #f e s1 s2 k env m v tr H1 H2; nwhd in ⊢ (??%?);
446    nrewrite > (yieldsbare_eq ??? (expr_complete … H1)); nwhd in ⊢ (??%?);
447    nrewrite > (yieldsbare_eq ??? (bool_of_true ?? H2));
448    napply refl;
449##| #f s1 e s2 s3 k env m; *; #es1; nrewrite > es1; napply refl;
450##| ##22,23: #f e s1 s2 k env m; napply refl;
451##| #f k env m H; nwhd in ⊢ (??%?); nrewrite > H; napply refl;
452##| #f e k env m v tr H1 H2; nwhd in ⊢ (??%?);
453    nelim (is_not_void_true f H1); #pf ef; nrewrite > ef; nwhd in ⊢ (??%?);
454    nrewrite > (yieldsbare_eq ??? (expr_complete … H2)); nwhd in ⊢ (??%?);
455    napply refl;
456##| #f k env m; ncases k;
457    ##[ #H1 H2; nwhd in ⊢ (??%?); nrewrite > H2; napply refl;
458    ##| #s' k'; nwhd in ⊢ (% → ?); *;
459    ##| ##3,4: #e' s' k'; nwhd in ⊢ (% → ?); *;
460    ##| ##5,6: #e' s1' s2' k'; nwhd in ⊢ (% → ?); *;
461    ##| #k'; nwhd in ⊢ (% → ?); *;
462    ##| #r f' env' k' H1 H2; nwhd in ⊢ (??%?); nrewrite > H2; napply refl
463    ##]
464##| #f e s k env m i tr H1; nwhd in ⊢ (??%?);
465    nrewrite > (yieldsbare_eq ??? (expr_complete … H1)); nwhd in ⊢ (??%?);
466    napply refl
467##| #f s k env m; *; #es; nrewrite > es; napply refl;
468##| #f k env m; napply refl
469##| #f l s k env m; napply refl
470##| #f l k env m s k' H1; nwhd in ⊢ (??%?); nrewrite > H1; napply refl;
471##| #f args k m1 env m2 m3 H1 H2; nwhd in ⊢ (??%?);
472    nelim (alloc_vars_complete … H1); #p1 e1; nrewrite > e1; nwhd in ⊢ (??%?);
473    nelim (yields_eq ???? (bind_params_complete … H2)); #p2 e2; nrewrite > e2;
474    napply refl;
475##| #id tys rty args k m rv tr H; nwhd in ⊢ (??%?);
476    ninversion H; #f' args' rv' eargs erv H1 H2 e1 e2 e3 e4; nrewrite < e1 in H1 H2;
477    #H1 H2;
478    nelim (yields_eq ???? (eventval_list_match_complete … H1)); #p1 e1; nrewrite > e1; nwhd in ⊢ (??%?);
479    nwhd; @ erv; nwhd in ⊢ (??%?);
480    nelim (yields_eq ???? (eventval_match_complete … H2)); #p2 e2; nrewrite > e2; napply refl
481##| #v f env k m; nwhd in ⊢ (??%?); napply daemon (* FIXME: inductive semantics allows any value ?! *)
482##| #v f env k m1 m2 sp loc ofs ty H; nwhd in ⊢ (??%?);
483    nrewrite > H; napply refl
484##| #f l s k env m; napply refl
485##] nqed.
486 
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