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 (∃ls0,rs0,xs0. nth dst ? int (niltape ?) = midtape sig ls0 x rs0 ∧
101 current sig (nth dst (tape sig) outt (niltape sig)) = None ?) ∨
103 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
107 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
108 (midtape sig (reverse ? xs@x::ls0) c rsj) dst).
110 definition R_match_step_true ≝
111 λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
112 ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s →
113 current sig (nth dst (tape sig) int (niltape sig)) ≠ None ? ∧
114 (is_startc s = true →
115 (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) →
116 (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 →
117 outt = change_vec ?? int
118 (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧
119 (∀ls,x,xs,ci,cj,rs,ls0,rs0.
120 nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
121 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) →
122 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
124 (outt = change_vec ?? int
125 (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false))).
129 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src)
130 (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)). *)
132 lemma sem_match_step :
133 ∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n →
134 match_step src dst sig n is_startc is_endc ⊨
135 [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
136 R_match_step_true src dst sig n is_startc is_endc,
137 R_match_step_false src dst sig n is_endc ].
138 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst
139 @(acc_sem_seq_app sig n … (sem_compare src dst sig n is_endc Hneq Hsrc Hdst)
140 (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ? is_endc))
142 (sem_parmoveL ???? is_startc Hneq Hsrc Hdst)
143 (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
145 [#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd >Htd -Htd
146 * #te * #Hte #Htb whd
148 [ lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
149 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
150 [| #c #_ % #Hfalse destruct (Hfalse) ]
151 #Hcurta_dst >Hcomp1 in Htest; [| %2 %2 //]
152 whd in ⊢ (??%?→?); change with (current ? (niltape ?)) in match (None ?);
153 <nth_vec_map >Hcurta_src whd in ⊢ (??%?→?); <nth_vec_map
154 >Hcurta_dst cases (is_endc s) normalize in ⊢ (%→?); #H destruct (H)
155 | #Hstart #Hnotstart %
156 [ #s1 #Hcurta_dst #Hneqss1 -Hcomp2
158 [@Hcomp1 %2 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //]
159 #H destruct (H) -Hcomp1 cases Hte #_ -Hte #Hte
160 cut (te = ta) [@Hte %1 %1 %{s} % //] -Hte #H destruct (H) %
161 [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … ))
162 #i #Hi cases (decidable_eq_nat i dst) #Hidst
163 [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
164 #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs //
165 | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ]
166 | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src
167 >Hcurta_src in Htest; whd in ⊢ (??%?→?);
168 cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq //
170 |#ls #x #xs #ci #cj #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc #Hcicj
171 cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) [ * #H destruct (H) ]
172 * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2
173 lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc %
174 [ cases Hte -Hte #Hte #_ whd in Hte;
175 >Htasrc_mid in Hcurta_src; whd in ⊢ (??%?→?); #H destruct (H)
176 lapply (Hte ls ci (reverse ? xs) rs s ??? ls0 cj' (reverse ? xs) s rs0' (refl ??) ?) //
177 [ >Htc >nth_change_vec //
178 | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid cases (orb_true_l … Hc0) -Hc0 #Hc0
179 [@memb_append_l2 >(\P Hc0) @memb_hd
180 |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse //
182 | >Htc >change_vec_commute // >nth_change_vec // ] -Hte
183 >Htc >change_vec_commute // >change_vec_change_vec
184 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
185 >Hte in Htb; * * #_ >reverse_reverse #Htbdst1 #Htbdst2 -Hte @(eq_vec … (niltape ?))
186 #i #Hi cases (decidable_eq_nat i dst) #Hidst
187 [ >Hidst >nth_change_vec // >(Htbdst1 ls0 s (xs@cj'::rs0'))
188 [| >nth_change_vec // ]
189 >Htadst_mid cases xs //
190 | >nth_change_vec_neq [|@sym_not_eq // ]
191 <Htbdst2 [| @sym_not_eq // ] >nth_change_vec_neq [| @sym_not_eq // ]
192 <Htasrc_mid >change_vec_same % ]
193 | >Hcurta_src in Htest; whd in ⊢(??%?→?);
194 >Htc >change_vec_commute //
195 change with (current ? (niltape ?)) in match (None ?);
196 <nth_vec_map >nth_change_vec // whd in ⊢ (??%?→?);
197 cases (is_endc ci) whd in ⊢ (??%?→?); #H destruct (H) %
201 |#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb
202 whd in ⊢ (%→?); #Hout >Hout >Htb whd
203 #ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend
204 lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
205 cases (current … (nth dst ? intape (niltape ?))) in Hcomp1;
206 [#Hcomp1 #_ %1 % % [% | @Hcomp1 %2 %2 % ]
207 |#c_dst cases (true_or_false (c_src == c_dst)) #Hceq
208 [#_ #Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
209 #ls_dst * #rs_dst #Hmid_dst
210 cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * *
211 #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq >Hrs_dst in Hmid_dst; #Hmid_dst
212 cut (∃r1,rs1.rsi = r1::rs1) [@daemon] * #r1 * #rs1 #Hrs1 >Hrs1 in Hrs_src;
213 #Hrs_src >Hrs_src in Hmid_src; #Hmid_src <(\P Hceq) in Hmid_dst; #Hmid_dst
214 lapply (Hcomp2 ??????? Hmid_src Hmid_dst ?)
215 [ #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
216 [ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //] ]
218 [ * #Hrsj >Hrsj #Hta % %2 >Hta >nth_change_vec //
219 %{ls_dst} %{xs1} cut (∃xs0.xs = xs1@xs0)
220 [lapply Hnotendxs1 -Hnotendxs1 lapply Hrs_src lapply xs elim xs1
223 [ whd in ⊢ (??%%→?); #H destruct (H) #Hnotendxs2
224 >Hnotendxs2 in Hend; [ #H destruct (H) |@memb_hd ]
225 | #x2' #xs2' whd in ⊢ (??%%→?); #H destruct (H)
226 #Hnotendxs2 cases (IH xs2' e0 ?)
227 [ #xs0 #Hxs2 %{xs0} @eq_f //
228 |#c #Hc @Hnotendxs2 @memb_cons // ]
231 ] * #xs0 #Hxs0 %{xs0} % [ %
232 [ >Hmid_dst >Hrsj >append_nil %
234 | cases (reverse ? xs1) // ]
235 | * #cj * #rs2 * #Hrsj #Hta lapply (Hta ?)
236 [ cases (Hneq ?? Hrs1) /2/ #Hr1 %2 @(Hr1 ?? Hrsj) ] -Hta #Hta
237 %2 >Hta in Hc; whd in ⊢ (??%?→?);
238 change with (current ? (niltape ?)) in match (None ?);
239 <nth_vec_map >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
240 whd in ⊢ (??%?→?); #Hc cut (is_endc r1 = true)
241 [ cases (is_endc r1) in Hc; whd in ⊢ (??%?→?); //
242 change with (current ? (niltape ?)) in match (None ?);
243 <nth_vec_map >nth_change_vec // normalize #H destruct (H) ]
244 #Hendr1 cut (xs = xs1)
245 [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
246 -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
247 [ * normalize in ⊢ (%→?); //
248 #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
249 lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
251 [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
252 >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
253 normalize in ⊢ (%→?); #H destruct (H)
254 | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
255 #Hnotendc #Hnotendcxs1 @eq_f @IH
256 [ @(cons_injective_r … Heq)
257 | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
259 | @memb_cons @memb_cons // ]
260 | #c #Hc @Hnotendcxs1 @memb_cons // ]
263 | #Hxsxs1 destruct (Hxsxs1) >Hmid_dst %{ls_dst} %{rsj} % //
264 #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0)
265 lapply (append_l2_injective … Hrs_src) // #Hrs' destruct (Hrs') %
268 |#Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hdst
269 @False_ind lapply (Hcomp1 ?) [%2 %1 %1 >Hmid_src normalize
270 @(not_to_not ??? (\Pf Hceq)) #H destruct //] #Hintape >Hintape in Hc;
271 whd in ⊢(??%?→?); >Hmid_src
272 change with (current ? (niltape ?)) in match (None ?);
273 <nth_vec_map >Hmid_src whd in ⊢ (??%?→?);
274 >(Hnotend c_src) [|@memb_hd]
275 change with (current ? (niltape ?)) in match (None ?);
276 <nth_vec_map >Hmid_src whd in ⊢ (??%?→?); >Hdst normalize #H destruct (H)
282 definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
283 whileTM … (match_step src dst sig n is_startc is_endc)
284 (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
286 definition R_match_m ≝
287 λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
288 (* (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧ *)
290 nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
291 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
292 (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
293 (is_startc x = true →
295 nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
296 (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
299 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
300 (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) dst) ∨
301 ∀l,l1.x0::rs0 ≠ l@x::xs@l1)).
304 definition R_match_m ≝
305 λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
306 (((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
307 current ? (nth i ? int (niltape ?)) = None ? ∨
308 current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
309 (∀ls,x,xs,ci,rs,ls0,x0,rs0.
310 (∀x. is_startc x ≠ is_endc x) →
311 is_startc x = true → is_endc ci = true →
312 (∀z. memb ? z (x::xs) = true → is_endc x = false) →
313 nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
314 nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
315 (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
318 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
319 (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) j) ∨
320 ∀l,l1.x0::rs0 ≠ l@x::xs@l1).
324 axiom sub_list_dec: ∀A.∀l,ls:list A.
325 ∃l1,l2. l = l1@ls@l2 ∨ ∀l1,l2. l ≠ l1@ls@l2.
328 lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
329 src ≠ dst → src < S n → dst < S n →
330 match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
331 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
332 lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) //
333 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
334 [ #tc #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
335 cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse
336 [(* current dest = None *) *
337 [ * #Hcur_dst #Houtc %
339 |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
340 normalize in ⊢ (%→?); #H destruct (H)
342 | * #ls0 * #rs0 * #xs0 * * #Htc_dst #Hrs0 #HNone %
343 [ >Htc_dst normalize in ⊢ (%→?); #H destruct (H)
344 | #Hstart #ls1 #x1 #rs1 >Htc_dst #H destruct (H)
346 [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %]
347 #cj #ls2 #H destruct (H)
348 | #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs)
349 >length_append whd in ⊢ (??%(??%)→?); >length_append
350 >length_append normalize >commutative_plus whd in ⊢ (???%→?);
351 #H destruct (H) lapply e0 >(plus_n_O (|rs1|)) in ⊢ (??%?→?);
352 >associative_plus >associative_plus
353 #e1 lapply (injective_plus_r ??? e1) whd in ⊢ (???%→?);
358 |* #ls0 * #rs0 * #Hmid_dst #HFalse %
359 [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
360 | #Hstart #ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
361 %1 %{[ ]} %{rs0} % [%] #cj #l2 #Hnotnil
362 >reverse_cons >associative_append @(HFalse ?? Hnotnil)
365 |#ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
366 #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
367 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
368 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
370 [#_ whd in Htrue; >Hmid_src in Htrue; #Htrue
371 cases (Htrue x (refl … )) -Htrue * #Htaneq #_
372 @False_ind >Hmid_dst in Htaneq; /2/
373 |#Hstart #ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?);
376 | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
377 #Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?);
378 #H destruct (H) whd in Htrue; >Hmid_src in Htrue; #Htrue
379 cases (Htrue x (refl …)) -Htrue #_ #Htrue cases (Htrue Hstart ?) -Htrue
380 [2: #z #membz @daemon (*aggiungere l'ipotesi*)]
381 cases (true_or_false (x==c)) #eqx
382 [ #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc)
383 #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1
385 [>append_nil #Hx1 @daemon (* absurd by Hx1 e notendx1 *)]
386 #ci -tl1 #tl1 #Hxs #H cases (H … (refl … ))
387 [(* this is absurd, since Htrue conlcudes is_endc ci =false *)
388 #Hend_ci @daemon (* lapply(Htrue … (refl …)) -Htrue *)
391 | #cj #tl2' #Hrs0 #Hcomp lapply (Htrue ls x x1 ci cj tl1 ls0 tl2' ????)
392 [ @(Hcomp ?? (refl ??))
393 | #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0
394 [ @Hnotend >(\P Hc0) @memb_hd
396 | >Hmid_dst >Hrs0 >(\P eqx) %
398 | * #Htb >Htb #Hendci %2 >Hrs0 >Hxs
399 cases (IH ls x xs end rs ? Hnotend Hend) [|
404 >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
406 >Hrs0 in Hmid_dst; #Hmid_dst
407 cases(Htrue ???????? Hmid_dst) -Htrue #Htb #Hendx
409 cases(IH ls x xs end rs ? Hstart Hnotend Hend)
410 [* #H1 #H2 >Htb in H1; >nth_change_vec //
411 >Hmid_dst cases rs0 [2: #a #tl normalize in ⊢ (%→?); #H destruct (H)]
412 #_ %2 @daemon (* si dimostra *)
414 |>Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src