1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "turing/multi_universal/compare.ma".
16 include "turing/multi_universal/par_test.ma".
19 definition Rtc_multi_true ≝
20 λalpha,test,n,i.λt1,t2:Vector ? (S n).
21 (∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1.
23 definition Rtc_multi_false ≝
24 λalpha,test,n,i.λt1,t2:Vector ? (S n).
25 (∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
27 lemma sem_test_char_multi :
28 ∀alpha,test,n,i.i ≤ n →
29 inject_TM ? (test_char ? test) n i ⊨
30 [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
31 #alpha #test #n #i #Hin #int
32 cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
33 #k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
35 | #Hqtrue lapply (Htrue Hqtrue) * * * #c *
36 #Hcur #Htestc #Hnth_i #Hnth_j %
38 | @(eq_vec … (niltape ?)) #i0 #Hi0
39 cases (decidable_eq_nat i0 i) #Hi0i
41 | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
42 | #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
44 | @(eq_vec … (niltape ?)) #i0 #Hi0
45 cases (decidable_eq_nat i0 i) #Hi0i
47 | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
50 definition Rm_test_null_true ≝
51 λalpha,n,i.λt1,t2:Vector ? (S n).
52 current alpha (nth i ? t1 (niltape ?)) ≠ None ? ∧ t2 = t1.
54 definition Rm_test_null_false ≝
55 λalpha,n,i.λt1,t2:Vector ? (S n).
56 current alpha (nth i ? t1 (niltape ?)) = None ? ∧ t2 = t1.
58 lemma sem_test_null_multi : ∀alpha,n,i.i ≤ n →
59 inject_TM ? (test_null ?) n i ⊨
60 [ tc_true : Rm_test_null_true alpha n i, Rm_test_null_false alpha n i ].
61 #alpha #n #i #Hin #int
62 cases (acc_sem_inject … Hin (sem_test_null alpha) int)
63 #k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
65 | #Hqtrue lapply (Htrue Hqtrue) * * #Hcur #Hnth_i #Hnth_j % //
66 @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i
67 [ >Hi0i @sym_eq @Hnth_i | @sym_eq @Hnth_j @sym_not_eq // ] ]
68 | #Hqfalse lapply (Hfalse Hqfalse) * * #Hcur #Hnth_i #Hnth_j %
70 | @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) //
71 #Hi0i @sym_eq @Hnth_j @sym_not_eq // ] ]
74 axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S.∀is_endc. ∃l,tl1,tl2.
75 l1 = l@tl1 ∧ l2 = l@tl2 ∧ (∀c.c ∈ l = true → is_endc c = false) ∧
76 ∀a,tla. tl1 = a::tla → is_endc a = true ∨ (∀b,tlb.tl2 = b::tlb → a≠b).
78 axiom daemon : ∀X:Prop.X.
80 definition match_test ≝ λsrc,dst.λsig:DeqSet.λn,is_endc.λv:Vector ? n.
81 match (nth src (option sig) v (None ?)) with
83 | Some x ⇒ notb ((is_endc x) ∨ (nth dst (DeqOption sig) v (None ?) == None ?))].
85 definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
86 compare src dst sig n is_endc ·
87 (ifTM ?? (partest sig n (match_test src dst sig ? is_endc))
89 (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
93 definition R_match_step_false ≝
94 λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
96 nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
97 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
98 ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
99 (current sig (nth dst (tape sig) outt (niltape sig)) = None ?) ∨
101 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
105 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
106 (midtape sig (reverse ? xs@x::ls0) c rsj) dst).
108 definition R_match_step_true ≝
109 λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
110 ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s →
112 (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) →
113 current sig (nth dst (tape sig) int (niltape sig)) ≠ None ? ∧
114 (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 →
115 outt = change_vec ?? int
116 (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧
117 (∀ls,x,xs,ci,cj,rs,ls0,rs0.
118 nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
119 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) →
120 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
122 (outt = change_vec ?? int
123 (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false)).
127 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src)
128 (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)). *)
130 lemma sem_match_step :
131 ∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n →
132 match_step src dst sig n is_startc is_endc ⊨
133 [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
134 R_match_step_true src dst sig n is_startc is_endc,
135 R_match_step_false src dst sig n is_endc ].
136 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst
139 check (acc_sem_seq_app sig n … (sem_compare src dst sig n is_endc Hneq Hsrc Hdst)
140 (acc_sem_if ? n … (sem_test_char_multi sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
141 (sem_if ? n … (sem_test_null_multi sig n dst (le_S_S_to_le … Hdst))
143 (sem_parmoveL ???? is_startc Hneq Hsrc Hdst)
144 (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
149 @(acc_sem_seq_app sig n … (sem_compare src dst sig n is_endc Hneq Hsrc Hdst)
150 (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ? is_endc))
152 (sem_parmoveL ???? is_startc Hneq Hsrc Hdst)
153 (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
155 [#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd >Htd -Htd
156 * #te * #Hte #Htb whd
157 #s #Hcurta_src #Hstart #Hnotstart % [ %
159 | #s1 #Hcurta_dst #Hneqss1 -Hcomp2
161 [@Hcomp1 %2 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //]
162 #H destruct (H) -Hcomp1 cases Hte #_ -Hte #Hte
163 cut (te = ta) [@Hte %1 %1 %{s} % //] -Hte #H destruct (H) %
164 [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … ))
165 #i #Hi cases (decidable_eq_nat i dst) #Hidst
166 [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
167 #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs //
168 | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ]
169 | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src
170 >Hcurta_src in Htest; whd in ⊢ (??%?→?);
171 cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq //
175 #ls #x #xs #ci #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc
176 cases rs00 in Htadst_mid;
177 [(* case rs empty *) #Htadst_mid % [ #cj #rs1 #H destruct (H) ]
178 #_ cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) -Hcomp2
179 [2: * #x0 * #rs1 * #H destruct (H) ]
180 * #_ #Htc cases Htb #td * * #_ #Htd >Htasrc_mid in Hcurta_src;
181 normalize in ⊢ (%→?); #H destruct (H)
182 >Htd [2: %2 >Htc >nth_change_vec // cases (reverse sig ?) //]
183 >Htc * * >nth_change_vec // #Htbdst #_ #Htbelse
184 @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
185 [ >Hidst >nth_change_vec // <Htbdst // cases (reverse sig ?) //
186 |@sym_eq @Htbelse @sym_not_eq //
188 |#cj0 #rs0 #Htadst_mid % [| #H destruct (H) ]
189 #cj #rs1 #H destruct (H) #Hcicj
190 cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) [ * #H destruct (H) ]
191 * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2
192 lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc
193 cases Htb #td * * #Htd #_ >Htasrc_mid in Hcurta_src; normalize in ⊢ (%→?);
195 >(Htd ls ci (reverse ? xs) rs s ??? ls0 cj' (reverse ? xs) s rs0' (refl ??)) //
196 [| >Htc >nth_change_vec //
197 | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid
198 cases (orb_true_l … Hc0) -Hc0 #Hc0
199 [@memb_append_l2 >(\P Hc0) @memb_hd
200 |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse //
202 | >Htc >nth_change_vec_neq [|@sym_not_eq // ] @nth_change_vec // ]
203 * * #_ #Htbdst #Htbelse %
204 [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
205 [ >Hidst >nth_change_vec // >Htadst_mid >(Htbdst ls0 s (xs@cj'::rs0'))
207 | >nth_change_vec // ]
208 | >nth_change_vec_neq [|@sym_not_eq //]
209 <Htbelse [|@sym_not_eq // ]
210 >nth_change_vec_neq [|@sym_not_eq //]
211 cases (decidable_eq_nat i src) #Hisrc
212 [ >Hisrc >nth_change_vec // >Htasrc_mid //
213 | >nth_change_vec_neq [|@sym_not_eq //]
214 <(Htbelse i) [|@sym_not_eq // ]
215 >Htc >nth_change_vec_neq [|@sym_not_eq // ]
216 >nth_change_vec_neq [|@sym_not_eq // ] //
219 | >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
220 >nth_change_vec // whd in ⊢ (??%?→?);
221 #H destruct (H) cases (is_endc c) in Hcend;
222 normalize #H destruct (H) // ]
225 |#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb
226 whd in ⊢ (%→?); #Hout >Hout >Htb whd
227 #ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend
228 lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
229 cases (current … (nth dst ? intape (niltape ?))) in Hcomp1;
230 [#Hcomp1 #_ %1 % [% | @Hcomp1 %2 %2 % ]
231 |#c_dst cases (true_or_false (c_src == c_dst)) #Hceq
232 [#_ #Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
233 #ls_dst * #rs_dst #Hmid_dst %2
234 cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * *
235 #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq %{ls_dst} %{rsj} >Hrs_dst in Hmid_dst; #Hmid_dst
236 cut (∃r1,rs1.rsi = r1::rs1) [@daemon] * #r1 * #rs1 #Hrs1 >Hrs1 in Hrs_src;
237 #Hrs_src >Hrs_src in Hmid_src; #Hmid_src <(\P Hceq) in Hmid_dst; #Hmid_dst
238 lapply (Hcomp2 ??????? Hmid_src Hmid_dst ?)
239 [ #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
240 [ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //]
243 [ >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
244 #Hc lapply (Hc ? (refl ??)) #Hendr1
246 [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
247 -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
248 [ * normalize in ⊢ (%→?); //
249 #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
250 lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
252 [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
253 >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
254 normalize in ⊢ (%→?); #H destruct (H)
255 | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
256 #Hnotendc #Hnotendcxs1 @eq_f @IH
257 [ @(cons_injective_r … Heq)
258 | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
260 | @memb_cons @memb_cons // ]
261 | #c #Hc @Hnotendcxs1 @memb_cons // ]
264 | #Hxsxs1 >Hmid_dst >Hxsxs1 % ]
265 | #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0) ]
266 | * #cj * #rs2 * #Hrs2 #Hta lapply (Hta ?)
267 [ cases (Hneq … Hrs1) /2/ #H %2 @(H ?? Hrs2) ]
268 -Hta #Hta >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //]
269 >nth_change_vec // #Hc lapply (Hc ? (refl ??)) #Hendr1
270 (* lemmatize this proof *) cut (xs = xs1)
271 [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
272 -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
273 [ * normalize in ⊢ (%→?); //
274 #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
275 lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
277 [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
278 >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
279 normalize in ⊢ (%→?); #H destruct (H)
280 | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
281 #Hnotendc #Hnotendcxs1 @eq_f @IH
282 [ @(cons_injective_r … Heq)
283 | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
285 | @memb_cons @memb_cons // ]
286 | #c #Hc @Hnotendcxs1 @memb_cons // ]
289 | #Hxsxs1 >Hmid_dst >Hxsxs1 % //
290 #rsj0 #c #Hcrsj destruct (Hxsxs1 Hrs2 Hcrsj) @eq_f3 //
291 @eq_f3 // lapply (append_l2_injective ?????? Hrs_src) //
292 #Hendr1 destruct (Hendr1) % ]
296 |#Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hdst
297 @False_ind lapply (Hcomp1 ?) [%2 %1 %1 >Hmid_src normalize
298 @(not_to_not ??? (\Pf Hceq)) #H destruct //] #Hintape
299 >Hintape in Hc; >Hmid_src #Hc lapply (Hc ? (refl …)) -Hc
300 >(Hnotend c_src) // normalize #H destruct (H)
306 definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
307 whileTM … (match_step src dst sig n is_startc is_endc)
308 (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
310 definition R_match_m ≝
311 λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
312 (* (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧ *)
314 nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
315 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
316 (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
317 (is_startc x = true →
319 nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
320 (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
323 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
324 (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) dst) ∨
325 ∀l,l1.x0::rs0 ≠ l@x::xs@l1)).
328 definition R_match_m ≝
329 λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
330 (((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
331 current ? (nth i ? int (niltape ?)) = None ? ∨
332 current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
333 (∀ls,x,xs,ci,rs,ls0,x0,rs0.
334 (∀x. is_startc x ≠ is_endc x) →
335 is_startc x = true → is_endc ci = true →
336 (∀z. memb ? z (x::xs) = true → is_endc x = false) →
337 nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
338 nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
339 (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
342 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
343 (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) j) ∨
344 ∀l,l1.x0::rs0 ≠ l@x::xs@l1).
348 axiom sub_list_dec: ∀A.∀l,ls:list A.
349 ∃l1,l2. l = l1@ls@l2 ∨ ∀l1,l2. l ≠ l1@ls@l2.
352 lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
353 src ≠ dst → src < S n → dst < S n →
354 match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
355 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
356 lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) //
357 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
358 [ #tc #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
359 cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse
360 [(* current dest = None *) * #Hcur_dst #Houtc %
362 |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
363 normalize in ⊢ (%→?); #H destruct (H)
365 |* #ls0 * #rs0 * #Hmid_dst #HFalse %
366 [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
367 | #Hstart #ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
368 %1 %{[ ]} %{rs0} % [%] #cj #l2 #Hnotnil
369 >reverse_cons >associative_append @(HFalse ?? Hnotnil)
372 |#ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
373 #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
374 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
375 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
377 [#_ whd in Htrue; >Hmid_src in Htrue; #Htrue
378 cases (Htrue x (refl … ) Hstart ?) -Htrue [2: @daemon]
379 * #Htb #_ #_ >Htb in IH; // #IH
380 cases (IH ls x xs end rs Hmid_src Hstart Hnotend Hend)
381 #Hcur_outc #_ @Hcur_outc //
382 |#ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?);
385 | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
386 #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?);
387 #H destruct (H) whd in Htrue; >Hmid_src in Htrue; #Htrue
388 cases (Htrue x (refl …) Hstart ?) -Htrue
389 [2: #z #membz @daemon (*aggiungere l'ipotesi*)]
390 cases (true_or_false (x==c)) #eqx
391 [ #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc)
392 #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1
394 [>append_nil #Hx1 @daemon (* absurd by Hx1 e notendx1 *)]
395 #ci -tl1 #tl1 #Hxs #H cases (H … (refl … ))
396 [(* this is absurd, since Htrue conlcudes is_endc ci =false *)
397 #Hend_ci @daemon (* lapply(Htrue … (refl …)) -Htrue *)
398 |#Hcomp lapply (Htrue ls x x1 ci tl1 ls0 tl2 ???)
399 [ #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0
400 [ @Hnotend >(\P Hc0) @memb_hd
402 | >Hmid_dst >Hrs0 >(\P eqx) %
404 | * cases tl2 in Hrs0;
405 [ >append_nil #Hrs0 #_ #Htb whd in IH;
406 lapply (IH ls x x1 ci tl1 ? Hstart ??)
409 | >Htb // >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
411 >Hrs0 in Hmid_dst; #Hmid_dst
412 cases(Htrue ???????? Hmid_dst) -Htrue #Htb #Hendx
414 cases(IH ls x xs end rs ? Hstart Hnotend Hend)
415 [* #H1 #H2 >Htb in H1; >nth_change_vec //
416 >Hmid_dst cases rs0 [2: #a #tl normalize in ⊢ (%→?); #H destruct (H)]
417 #_ %2 @daemon (* si dimostra *)
419 |>Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src