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/simple_machines.ma".
16 include "turing/multi_universal/compare.ma".
17 include "turing/multi_universal/par_test.ma".
18 include "turing/multi_universal/moves_2.ma".
20 lemma eq_vec_change_vec : ∀sig,n.∀v1,v2:Vector sig n.∀i,t,d.
22 (∀j.i ≠ j → nth j ? v1 d = nth j ? v2 d) →
23 v2 = change_vec ?? v1 t i.
24 #sig #n #v1 #v2 #i #t #d #H1 #H2 @(eq_vec … d)
25 #i0 #Hlt cases (decidable_eq_nat i0 i) #Hii0
26 [ >Hii0 >nth_change_vec //
27 | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @H2 @sym_not_eq // ]
31 ∀sig,ls,c,rs.(c = None ? → ls = [ ] ∨ rs = [ ]) → right ? (mk_tape sig ls c rs) = rs.
32 #sig #ls #c #rs cases c // cases ls
34 | #l0 #ls0 #H normalize cases (H (refl ??)) #H1 [ destruct (H1) | >H1 % ] ]
37 lemma left_mk_tape : ∀sig,ls,c,rs.left ? (mk_tape sig ls c rs) = ls.
38 #sig #ls #c #rs cases c // cases ls // cases rs //
41 lemma current_mk_tape : ∀sig,ls,c,rs.current ? (mk_tape sig ls c rs) = c.
42 #sig #ls #c #rs cases c // cases ls // cases rs //
45 lemma length_tail : ∀A,l.0 < |l| → |tail A l| < |l|.
53 b::bl → match rec(al,bl)
58 lemma lists_length_split :
59 ∀A.∀l1,l2:list A.(∃la,lb.(|la| = |l1| ∧ l2 = la@lb) ∨ (|la| = |l2| ∧ l1 = la@lb)).
61 [ #l2 %{[ ]} %{l2} % % %
63 [ %{[ ]} %{(hd1::tl1)} %2 % %
64 | #hd2 #tl2 cases (IH tl2) #x * #y *
65 [ * #IH1 #IH2 %{(hd2::x)} %{y} % normalize % //
66 | * #IH1 #IH2 %{(hd1::x)} %{y} %2 normalize % // ]
71 definition option_cons ≝ λsig.λc:option sig.λl.
72 match c with [ None ⇒ l | Some c0 ⇒ c0::l ].
74 lemma opt_cons_tail_expand : ∀A,l.l = option_cons A (option_hd ? l) (tail ? l).
78 definition match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n.
79 match (nth src (option sig) v (None ?)) with
81 | Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ].
83 definition rewind ≝ λsrc,dst,sig,n.
84 parmove src dst sig n L · mmove src sig n R · mmove dst sig n R.
86 definition R_rewind_strong ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
88 nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
89 ∀ls0,y,y0,target,rs0.|xs| = |target| →
90 nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
92 (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src)
93 (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst) ∧
95 nth dst ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
96 ∀ls0,y,y0,target,rs0.|xs| = |target| →
97 nth src ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
99 (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) dst)
100 (midtape sig ls0 y0 (reverse ? target@y::rs0)) src) ∧
101 (∀x,rs.nth src ? int (niltape ?) = midtape sig [] x rs →
102 ∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 →
104 (∀x,rs.nth dst ? int (niltape ?) = midtape sig [] x rs →
105 ∀ls0,y,rs0.nth src ? int (niltape ?) = midtape sig ls0 y rs0 →
108 definition R_rewind ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
110 nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
111 ∀ls0,y,y0,target,rs0.|xs| = |target| →
112 nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
114 (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src)
115 (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst) ∧
116 (∀x,rs.nth src ? int (niltape ?) = midtape sig [] x rs →
117 ∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 →
121 theorem accRealize_to_Realize :
122 ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
123 M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse.
124 #sig #n #M #Rtrue #Rfalse #acc #HR #t
125 cases (HR t) #k * #outc * * #Hloop
126 #Htrue #Hfalse %{k} %{outc} % //
127 cases (true_or_false (cstate sig (states sig n M) n outc == acc)) #Hcase
128 [ % @Htrue @(\P Hcase) | %2 @Hfalse @(\Pf Hcase) ]
132 lemma sem_rewind_strong : ∀src,dst,sig,n.
133 src ≠ dst → src < S n → dst < S n →
134 rewind src dst sig n ⊨ R_rewind_strong src dst sig n.
135 #src #dst #sig #n #Hneq #Hsrc #Hdst
136 @(sem_seq_app sig n ????? (sem_parmoveL src dst sig n Hneq Hsrc Hdst) ?)
137 [| @(sem_seq_app sig n ????? (sem_move_multi … R ?) (sem_move_multi … R ?)) //
139 #ta #tb * #tc * * * #Htc1 #Htc2 #_ * #td * whd in ⊢ (%→%→?); #Htd #Htb % [ % [ %
140 [ #x #x0 #xs #rs #Hmidta_src #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_dst
141 >(Htc1 ??? Hmidta_src ls0 y (target@[y0]) rs0 ??) in Htd;
143 |>length_append >length_append >Hlen % ]
144 >change_vec_commute [|@sym_not_eq //]
145 >change_vec_change_vec
146 >nth_change_vec_neq [|@sym_not_eq //]
147 >nth_change_vec // >reverse_append >reverse_single
148 >reverse_append >reverse_single normalize in match (tape_move ???);
149 >rev_append_def >append_nil #Htd >Htd in Htb;
150 >change_vec_change_vec >nth_change_vec //
151 cases ls0 [|#l1 #ls1] normalize in match (tape_move ???); //
152 | #x #x0 #xs #rs #Hmidta_dst #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_src
153 >(Htc2 ??? Hmidta_dst ls0 y (target@[y0]) rs0 ??) in Htd;
155 |>length_append >length_append >Hlen % ]
156 >change_vec_change_vec
157 >change_vec_commute [|@sym_not_eq //]
159 >reverse_append >reverse_single
160 >reverse_append >reverse_single
161 cases ls0 [|#l1 #ls1] normalize in match (tape_move ???);
162 #Htd >Htd in Htb; >change_vec_change_vec >nth_change_vec //
163 >rev_append_def >change_vec_commute // normalize in match (tape_move ???); // ]
164 | #x #rs #Hmidta_src #ls0 #y #rs0 #Hmidta_dst
165 lapply (Htc1 … Hmidta_src … (refl ??) Hmidta_dst) -Htc1 #Htc >Htc in Htd;
166 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
167 >nth_change_vec_neq [|@sym_not_eq //]
168 >nth_change_vec // lapply (refl ? ls0) cases ls0 in ⊢ (???%→%);
169 [ #Hls0 #Htd >Htd in Htb;
170 >nth_change_vec // >change_vec_change_vec
171 whd in match (tape_move ???);whd in match (tape_move ???); <Hmidta_src
172 <Hls0 <Hmidta_dst >change_vec_same >change_vec_same //
173 | #l1 #ls1 #Hls0 #Htd >Htd in Htb;
174 >nth_change_vec // >change_vec_change_vec
175 whd in match (tape_move ???);whd in match (tape_move ???); <Hmidta_src
176 <Hls0 <Hmidta_dst >change_vec_same >change_vec_same //
178 | #x #rs #Hmidta_dst #ls0 #y #rs0 #Hmidta_src
179 lapply (Htc2 … Hmidta_dst … (refl ??) Hmidta_src) -Htc2 #Htc >Htc in Htd;
180 >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
181 >nth_change_vec // lapply (refl ? ls0) cases ls0 in ⊢ (???%→%);
182 [ #Hls0 destruct (Hls0) #Htd >Htd in Htb;
183 >nth_change_vec // >change_vec_change_vec
184 whd in match (tape_move ???);whd in match (tape_move ???);
185 <Hmidta_src <Hmidta_dst >change_vec_same >change_vec_same //
186 | #l1 #ls1 #Hls0 destruct (Hls0) #Htd >Htd in Htb;
187 >nth_change_vec // >change_vec_change_vec
188 whd in match (tape_move ???); whd in match (tape_move ???); <Hmidta_src
189 <Hmidta_dst >change_vec_same >change_vec_same //
194 lemma sem_rewind : ∀src,dst,sig,n.
195 src ≠ dst → src < S n → dst < S n →
196 rewind src dst sig n ⊨ R_rewind src dst sig n.
197 #src #dst #sig #n #Hneq #Hsrc #Hdst @(Realize_to_Realize … (sem_rewind_strong …)) //
198 #ta #tb * * * #H1 #H2 #H3 #H4 % /2 by /
201 definition match_step ≝ λsrc,dst,sig,n.
202 compare src dst sig n ·
203 (ifTM ?? (partest sig n (match_test src dst sig ?))
205 (rewind src dst sig n · (inject_TM ? (move_r ?) n dst)))
209 (* we assume the src is a midtape
211 if the dst is out of bounds (outt = int)
212 or dst.right is shorter than src.right (outt.current → None)
213 or src.right is a prefix of dst.right (out = just right of the common prefix) *)
214 definition R_match_step_false ≝
215 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
217 nth src ? int (niltape ?) = midtape sig ls x xs →
218 ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
219 (∃ls0,rs0,xs0. nth dst ? int (niltape ?) = midtape sig ls0 x rs0 ∧
222 (change_vec ?? int (mk_tape sig (reverse ? rs0@x::ls) (option_hd ? xs0) (tail ? xs0)) src)
223 (mk_tape ? (reverse ? rs0@x::ls0) (None ?) [ ]) dst) ∨
225 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
229 (change_vec ?? int (mk_tape sig (reverse ? xs@x::ls) (None ?) [ ]) src)
230 (mk_tape sig (reverse ? xs@x::ls0) (option_hd ? rs0) (tail ? rs0)) dst).
233 we assume the src is a midtape [ ] s rs
235 then dst.current = Some ? s1
236 and if s ≠ s1 then outt = int.dst.move_right()
238 then int.src.right and int.dst.right have a common prefix
239 and the heads of their suffixes are different
240 and outt = int.dst.move_right().
243 definition R_match_step_true ≝
244 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
245 ∀s,rs.nth src ? int (niltape ?) = midtape ? [ ] s rs →
246 outt = change_vec ?? int
247 (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst ∧
248 (∃s0.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s0 ∧
250 ∃xs,ci,rs',ls0,cj,rs0.
252 nth dst ? int (niltape ?) = midtape sig ls0 s (xs@cj::rs0) ∧
255 lemma sem_match_step :
256 ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
257 match_step src dst sig n ⊨
258 [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
259 R_match_step_true src dst sig n,
260 R_match_step_false src dst sig n ].
261 #src #dst #sig #n #Hneq #Hsrc #Hdst
262 @(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
263 (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
265 (sem_rewind ???? Hneq Hsrc Hdst)
266 (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
268 [ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
269 cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
270 [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
271 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
272 >nth_current_chars >Hcurta_src normalize in ⊢ (%→?); #H destruct (H)
273 | #s #Hs lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
274 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
275 [ #Hcurta_dst #Hcomp #_ * #td * >Hcomp [| %2 %]
276 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
277 >nth_current_chars >nth_current_chars >Hs >Hcurta_dst
278 normalize in ⊢ (%→?); #H destruct (H)
280 cases (current_to_midtape … Hs) #ls * #rs #Hmidta_src >Hmidta_src
281 cases (current_to_midtape … Hs0) #ls0 * #rs0 #Hmidta_dst >Hmidta_dst
282 cases (true_or_false (s == s0)) #Hss0
283 [ lapply (\P Hss0) -Hss0 #Hss0 destruct (Hss0)
284 #_ #Hcomp cases (Hcomp ????? (refl ??) (refl ??)) -Hcomp [ *
285 [ * #rs' * #_ #Hcurtc_dst * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
286 >nth_current_chars >nth_current_chars >Hcurtc_dst
287 cases (current ? (nth src …))
288 [normalize in ⊢ (%→?); #H destruct (H)
289 | #x >nth_change_vec // cases (reverse ? rs0)
290 [ normalize in ⊢ (%→?); #H destruct (H)
291 | #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ]
292 | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
293 >(?:nth src ? (current_chars ?? tc) (None ?) = None ?)
294 [|>nth_current_chars >Hcurtc_src >nth_change_vec_neq
295 [>nth_change_vec [cases (append ???) // | @Hsrc]
296 |@(not_to_not … Hneq) //
298 normalize in ⊢ (%→?); #H destruct (H) ]
299 | * #xs * #ci * #cj * #rs'' * #rs0' * * * #Hcicj #Hrs #Hrs0
300 #Htc * #td * * #Hmatch #Htd destruct (Htd) * #te * *
301 >Htc >change_vec_commute // >nth_change_vec //
302 >change_vec_commute [|@sym_not_eq //] >nth_change_vec // #Hte #_ #Htb
303 #s' #rs' >Hmidta_src #H destruct (H)
304 lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
305 >change_vec_commute // >change_vec_change_vec
306 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
307 >Hte in Htb; * * #_ >nth_change_vec // #Htb1
308 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2 %
309 [ @(eq_vec … (niltape ?)) #i #Hi
310 cases (true_or_false ((dst : DeqNat) == i)) #Hdsti
311 [ <(\P Hdsti) >Htb1 >nth_change_vec // >Hmidta_dst
312 >Hrs0 >reverse_reverse cases xs [|#r1 #rs1] %
313 | <Htb2 [|@(\Pf Hdsti)] >nth_change_vec_neq [| @(\Pf Hdsti)]
314 >Hrs0 >reverse_reverse >nth_change_vec_neq in ⊢ (???%);
315 <Hrs <Hmidta_src [|@(\Pf Hdsti)] >change_vec_same % ]
316 | >Hmidta_dst %{s'} % [%] #_
317 >Hrs0 %{xs} %{ci} %{rs''} %{ls0} %{cj} %{rs0'} % // % //
320 | lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta)
321 [@Htc % % @(not_to_not ??? Hss0) #H destruct (H) %]
322 -Htc #Htc destruct (Htc) #_ * #td * whd in ⊢ (%→?); * #_
323 #Htd destruct (Htd) * #te * * #_ #Hte * * #_ #Htb1 #Htb2
324 #s1 #rs1 >Hmidta_src #H destruct (H)
325 lapply (Hte … Hmidta_src … Hmidta_dst) -Hte #Hte destruct (Hte) %
326 [ @(eq_vec … (niltape ?)) #i #Hi
327 cases (true_or_false ((dst : DeqNat) == i)) #Hdsti
328 [ <(\P Hdsti) >(Htb1 … Hmidta_dst) >nth_change_vec // >Hmidta_dst
329 cases rs0 [|#r2 #rs2] %
330 | <Htb2 [|@(\Pf Hdsti)] >nth_change_vec_neq [| @(\Pf Hdsti)] % ]
331 | >Hs0 %{s0} % // #H destruct (H) @False_ind cases (Hss0) /2/ ]
335 | #ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd destruct (Htd)
336 whd in ⊢ (%→?); #Htb destruct (Htb) #ls #x #xs #Hta_src
337 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
338 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
339 [ #Hcurta_dst % % % // @Hcomp1 %2 //
340 | #x0 #Hcurta_dst cases (current_to_midtape … Hcurta_dst) -Hcurta_dst
341 #ls0 * #rs0 #Hta_dst cases (true_or_false (x == x0)) #Hxx0
342 [ lapply (\P Hxx0) -Hxx0 #Hxx0 destruct (Hxx0)
343 | >(?:tc=ta) in Htest;
344 [|@Hcomp1 % % >Hta_src >Hta_dst @(not_to_not ??? (\Pf Hxx0)) normalize
345 #Hxx0' destruct (Hxx0') % ]
347 >nth_current_chars >Hta_src >nth_current_chars >Hta_dst
348 whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1
349 cases (Hcomp2 … Hta_src Hta_dst) [ *
350 [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} %
352 | * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ]
353 | * #xs0 * #ci * #cj * #rs' * #rs0' * * *
354 #Hci #Hxs #Hrs0 #Htc @False_ind
356 >(?:nth src ? (current_chars ?? tc) (None ?) = Some ? ci) in Htest;
357 [|>nth_current_chars >Htc >nth_change_vec_neq [|@(not_to_not … Hneq) //]
359 >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj)
360 [|>nth_current_chars >Htc >nth_change_vec //]
361 normalize #H destruct (H) ] ] ]
364 definition match_m ≝ λsrc,dst,sig,n.
365 whileTM … (match_step src dst sig n)
366 (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
368 definition R_match_m ≝
369 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
371 nth src ? int (niltape ?) = midtape sig [ ] x rs →
372 (current sig (nth dst (tape sig) int (niltape sig)) = None ? →
373 right ? (nth dst (tape sig) int (niltape sig)) = [ ] → outt = int) ∧
375 nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
376 (∃l,l1.x0::rs0 = l@x::rs@l1 ∧
379 (mk_tape sig (reverse ? rs@[x]) (None ?) [ ]) src)
380 (mk_tape sig ((reverse ? (l@x::rs))@ls0) (option_hd ? l1) (tail ? l1)) dst) ∨
381 ∀l,l1.x0::rs0 ≠ l@x::rs@l1).
383 lemma not_sub_list_merge :
384 ∀T.∀a,b:list T. (∀l1.a ≠ b@l1) → (∀t,l,l1.a ≠ t::l@b@l1) → ∀l,l1.a ≠ l@b@l1.
385 #T #a #b #H1 #H2 #l elim l normalize //
388 lemma not_sub_list_merge_2 :
389 ∀T:DeqSet.∀a,b:list T.∀t. (∀l1.t::a ≠ b@l1) → (∀l,l1.a ≠ l@b@l1) → ∀l,l1.t::a ≠ l@b@l1.
390 #T #a #b #t #H1 #H2 #l elim l //
391 #t0 #l1 #IH #l2 cases (true_or_false (t == t0)) #Htt0
392 [ >(\P Htt0) % normalize #H destruct (H) cases (H2 l1 l2) /2/
393 | normalize % #H destruct (H) cases (\Pf Htt0) /2/ ]
397 lemma wsem_match_m : ∀src,dst,sig,n.
398 src ≠ dst → src < S n → dst < S n →
399 match_m src dst sig n ⊫ R_match_m src dst sig n.
400 #src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
401 lapply (sem_while … (sem_match_step src dst sig n Hneq Hsrc Hdst) … Hloop) //
402 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
403 [ #Hfalse #x #xs #Hmid_src
404 cases (Hfalse … Hmid_src) -Hfalse
405 [(* current dest = None *) *
406 [ * #Hcur_dst #Houtc %
408 | #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
409 normalize in ⊢ (%→?); #H destruct (H)
411 | * #ls0 * #rs0 * #xs0 * * #Htc_dst #Hrs0 #HNone %
412 [ >Htc_dst normalize in ⊢ (%→?); #H destruct (H)
413 | #ls1 #x1 #rs1 >Htc_dst #H destruct (H)
414 >Hrs0 >HNone cases xs0
415 [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %]
418 | >reverse_append >reverse_cons >reverse_append
419 >associative_append >associative_append % ]
420 | #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs)
421 >length_append whd in ⊢ (??%(??%)→?); >length_append
422 >length_append normalize >commutative_plus whd in ⊢ (???%→?);
423 #H destruct (H) lapply e0 >(plus_n_O (|rs1|)) in ⊢ (??%?→?);
424 >associative_plus >associative_plus
425 #e1 lapply (injective_plus_r ??? e1) whd in ⊢ (???%→?);
430 |* #ls0 * #rs0 * #Hmid_dst #Houtc %
431 [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
432 |#ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
433 %1 %{[ ]} %{rs0} % [%]
434 >reverse_cons >associative_append >Houtc %
437 |-ta #ta #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
439 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
440 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
442 [#Hcurta_dst #Hrightta_dst whd in Htrue; >Hmidta_src in Htrue; #Htrue
443 cases (Htrue ?? (refl ??)) -Htrue #Htc
445 [ >Htc whd in match (tape_move_mono ???); whd in match (tape_write ???);
446 <(change_vec_same … ta dst (niltape ?)) in ⊢ (???%);
447 lapply Hrightta_dst lapply Hcurta_dst -Hrightta_dst -Hcurta_dst
448 cases (nth dst ? ta (niltape ?))
450 | #r0 #rs0 #_ normalize in ⊢ (%→?); #H destruct (H)
452 | #ls #x0 #rs normalize in ⊢ (%→?); #H destruct (H) ] ]
453 -Htc #Htc destruct (Htc) #_
454 cases (IH … Hmidta_src) #Houtc #_ @Houtc //
455 |#ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst;
456 normalize in ⊢ (%→?); #H destruct (H)
458 | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
459 #ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst; normalize in ⊢ (%→?);
460 #H destruct (H) whd in Htrue; >Hmidta_src in Htrue; #Htrue
461 cases (Htrue ?? (refl …)) -Htrue >Hmidta_dst #Htc
462 cases (true_or_false (x==c)) #eqx
463 [ lapply (\P eqx) -eqx #eqx destruct (eqx) * #s0 * whd in ⊢ (??%?→?); #Hs0
464 destruct (Hs0) #Htrue cases (Htrue (refl ??)) -Htrue
465 #xs0 * #ci * #rs' * #ls1 * #cj * #rs1 * * #Hxs #H destruct (H) #Hcicj
466 >Htc in IH; whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
467 #IH cases (IH … Hmidta_src) -IH #_ >nth_change_vec //
468 cut (∃x1,xs1.xs0@cj::rs1 = x1::xs1)
469 [ cases xs0 [ %{cj} %{rs1} % | #x1 #xs1 %{x1} %{(xs1@cj::rs1)} % ] ] * #x1 * #xs1
470 #Hxs1 >Hxs1 #IH cases (IH … (refl ??)) -IH
471 [ * #l * #l1 * #Hxs1'
472 >change_vec_commute // >change_vec_change_vec
473 #Houtc % %{(s0::l)} %{l1} %
475 | >reverse_cons >associative_append >change_vec_commute // @Houtc ]
476 | #H %2 #l #l1 >(?:l@s0::xs@l1 = l@(s0::xs)@l1) [|%]
478 [ #l2 >Hxs <Hxs1 % normalize #H1 lapply (cons_injective_r ????? H1)
479 >associative_append #H2 lapply (append_l2_injective ????? (refl ??) H2)
480 #H3 lapply (cons_injective_l ????? H3) #H3 >H3 in Hcicj; * /2/
481 |#t #l2 #l3 % normalize #H1 lapply (cons_injective_r ????? H1)
482 -H1 #H1 cases (H l2 l3) #H2 @H2 @H1
485 | #_ cases (IH x xs ?) -IH
486 [| >Htc >nth_change_vec_neq [|@sym_not_eq //] @Hmidta_src ]
487 >Htc >nth_change_vec // cases rs0
488 [ #_ #_ %2 #l #l1 cases l
491 [ normalize % #H destruct (H) cases (\Pf eqx) /2/
492 | #tmp1 #l2 normalize % #H destruct (H) ]
493 | #tmp1 #l2 normalize % #H destruct (H) ]
494 | #tmp1 #l2 normalize % #H destruct (H)cases l2 in e0;
495 [ normalize #H1 destruct (H1)
496 | #tmp2 #l3 normalize #H1 destruct (H1) ] ]
497 | #r1 #rs1 #_ #IH cases (IH … (refl ??)) -IH
498 [ * #l * #l1 * #Hll1 #Houtc % %{(c::l)} %{l1} % [ >Hll1 % ]
499 >Houtc >change_vec_commute // >change_vec_change_vec
500 >change_vec_commute [|@sym_not_eq //]
501 >reverse_cons >associative_append %
502 | #Hll1 %2 @(not_sub_list_merge_2 ?? (x::xs)) normalize [|@Hll1]
503 #l1 % #H destruct (H) cases (\Pf eqx) /2/
511 definition R_match_step_true_naive ≝
512 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
513 |left ? (nth src ? outt (niltape ?))| +
514 |option_cons ? (current ? (nth dst ? outt (niltape ?))) (right ? (nth dst ? outt (niltape ?)))| <
515 |left ? (nth src ? int (niltape ?))| +
516 |option_cons ? (current ? (nth dst ? int (niltape ?))) (right ? (nth dst ? int (niltape ?)))|.
518 lemma sem_match_step_termination :
519 ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
520 match_step src dst sig n ⊨
521 [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
522 R_match_step_true_naive src dst sig n,
523 R_match_step_false src dst sig n ].
524 #src #dst #sig #n #Hneq #Hsrc #Hdst
525 @(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
526 (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
528 (sem_rewind_strong ???? Hneq Hsrc Hdst)
529 (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
531 [ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
532 cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
533 [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
534 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
535 >nth_current_chars >Hcurta_src normalize in ⊢ (%→?); #H destruct (H)
536 | #s #Hs lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
537 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
538 [ #Hcurta_dst #Hcomp #_ * #td * >Hcomp [| %2 %]
539 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
540 >nth_current_chars >nth_current_chars >Hs >Hcurta_dst
541 normalize in ⊢ (%→?); #H destruct (H)
543 cases (current_to_midtape … Hs) #ls * #rs #Hmidta_src >Hmidta_src
544 cases (current_to_midtape … Hs0) #ls0 * #rs0 #Hmidta_dst >Hmidta_dst
545 cases (true_or_false (s == s0)) #Hss0
546 [ lapply (\P Hss0) -Hss0 #Hss0 destruct (Hss0)
547 #_ #Hcomp cases (Hcomp ????? (refl ??) (refl ??)) -Hcomp [ *
548 [ * #rs' * #_ #Hcurtc_dst * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
549 >nth_current_chars >nth_current_chars >Hcurtc_dst
550 cases (current ? (nth src …))
551 [normalize in ⊢ (%→?); #H destruct (H)
552 | #x >nth_change_vec // cases (reverse ? rs0)
553 [ normalize in ⊢ (%→?); #H destruct (H)
554 | #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ]
555 | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
556 >(?:nth src ? (current_chars ?? tc) (None ?) = None ?)
557 [|>nth_current_chars >Hcurtc_src >nth_change_vec_neq
558 [>nth_change_vec [cases (append ???) // | @Hsrc]
559 |@(not_to_not … Hneq) //
561 normalize in ⊢ (%→?); #H destruct (H) ]
562 | * #xs * #ci * #cj * #rs'' * #rs0' * * * #Hcicj #Hrs #Hrs0
563 #Htc * #td * * #Hmatch #Htd destruct (Htd) * #te * * *
564 >Htc >change_vec_commute // >nth_change_vec //
565 >change_vec_commute [|@sym_not_eq //] >nth_change_vec //
566 cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb
568 [ #Hte #_ #_ <(reverse_reverse … ls) in Hte; <(reverse_reverse … lsa)
569 cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen'
570 @(list_cases2 … Hlen')
571 [ #H1 #H2 >H1 >H2 -H1 -H2 normalize in match (reverse ? [ ]); #Hte #_
572 lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
573 >change_vec_commute // >change_vec_change_vec
574 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
575 >Hte * * #_ >nth_change_vec // >reverse_reverse
576 #H lapply (H … (refl ??)) -H #Htb1 #Htb2
577 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta (midtape sig [ ] s0 (xs@ci::rs'')) src) (mk_tape sig (s0::lsb) (option_hd sig (xs@cj::rs0')) (tail sig (xs@cj::rs0'))) dst)
578 [ @(eq_vec_change_vec … (niltape ?))
579 [@Htb1| #j #Hj <Htb2 // >(nth_change_vec_neq ??????? Hj) % ] ]
580 -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
581 >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
582 >right_mk_tape [|cases xs [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H)]
583 normalize in match (left ??);
584 >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
585 whd in match (option_cons ???); >Hrs0
586 normalize in ⊢ (?(?%)%); //
587 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
588 >reverse_cons >reverse_cons #Hte
589 lapply (Hte ci hdb (reverse ? xs@s0::reverse ? tlb) rs'' ?
590 lsb cj hda (reverse ? xs@s0::reverse ? tla) rs0' ??)
591 [ /2 by cons_injective_l, nil/
592 | >length_append >length_append @eq_f @(eq_f ?? S)
593 >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
594 >length_reverse >length_reverse destruct (Hlen') //
595 | /2 by refl, trans_eq/ ] -Hte
596 #Hte #_ * * #_ >Hte >nth_change_vec // #Htb1 #Htb2
597 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
598 (mk_tape sig (hda::lsb) (option_hd ? (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0')) (tail ? (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0'))) dst)
599 (midtape ? [ ] hdb (reverse sig (reverse sig xs@s0::reverse sig tlb)@ci::rs'')) src)
600 [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?))
602 | #j #Hj <Htb2 // >change_vec_commute // >change_vec_change_vec
603 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
604 >nth_change_vec_neq in ⊢ (???%); // ] ]
605 -Htb1 -Htb2 #Htb >Htb whd
606 >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
608 [| cases (reverse sig (reverse sig xs@s0::reverse sig tla))
609 [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
610 >Hmidta_src >Hmidta_dst
611 whd in match (left ??); whd in match (left ??); whd in match (right ??);
612 >current_mk_tape <opt_cons_tail_expand whd in match (option_cons ???);
613 >Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
614 >length_append >commutative_plus in match (|reverse ??| + ?);
615 whd in match (|?::?|); >length_reverse >length_reverse
616 <(length_reverse ? ls) <Hlen' >H1 normalize // ]
617 | #_ #Hte #_ <(reverse_reverse … ls0) in Hte; <(reverse_reverse … lsa)
618 cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen'
619 @(list_cases2 … Hlen')
620 [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #Hte
621 lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
622 >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
623 >change_vec_change_vec #Hte #_
624 >Hte * * #_ >nth_change_vec // >reverse_reverse
625 #H lapply (H … (refl ??)) -H #Htb1 #Htb2
626 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta (mk_tape ? [s0] (option_hd ? (xs@cj::rs0')) (tail ? (xs@cj::rs0'))) dst)
627 (midtape ? lsb s0 (xs@ci::rs'')) src)
628 [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?))
630 | #j #Hj <Htb2 // >nth_change_vec_neq in ⊢ (???%); // ] ]
631 -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
632 >nth_change_vec_neq // >nth_change_vec //
634 [| cases xs [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
635 normalize in match (left ??);
636 >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand >Hrs0
637 >length_append normalize >length_append >length_append
638 <(reverse_reverse ? lsa) >H1 normalize //
639 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
640 >reverse_cons >reverse_cons #Hte
641 lapply (Hte cj hdb (reverse ? xs@s0::reverse ? tlb) rs0' ?
642 lsb ci hda (reverse ? xs@s0::reverse ? tla) rs'' ??)
643 [ /2 by cons_injective_l, nil/
644 | >length_append >length_append @eq_f @(eq_f ?? S)
645 >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
646 >length_reverse >length_reverse destruct (Hlen') //
647 | /2 by refl, trans_eq/ ] -Hte
648 #Hte #_ * * #_ >Hte >nth_change_vec_neq // >nth_change_vec // #Htb1 #Htb2
649 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
650 (mk_tape sig [hdb] (option_hd ? (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0')) (tail ? (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0'))) dst)
651 (midtape ? lsb hda (reverse sig (reverse sig xs@s0::reverse sig tla)@ci::rs'')) src)
652 [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?))
654 | #j #Hj <Htb2 // >change_vec_change_vec
655 >change_vec_commute [|@sym_not_eq //]
656 >change_vec_change_vec >nth_change_vec_neq in ⊢ (???%); // ] ]
657 -Htb1 -Htb2 #Htb >Htb whd
658 >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
660 [| cases (reverse sig (reverse sig xs@s0::reverse sig tlb))
661 [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
662 >Hmidta_src >Hmidta_dst
663 whd in match (left ??); whd in match (left ??); whd in match (right ??);
664 >current_mk_tape <opt_cons_tail_expand
665 whd in match (option_cons ???);
666 >Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
667 >length_append >commutative_plus in match (|reverse ??| + ?);
668 whd in match (|?::?|); >length_reverse >length_reverse
669 <(length_reverse ? lsa) >Hlen' >H2 >length_append
674 | lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta)
675 [@Htc % % @(not_to_not ??? Hss0) #H destruct (H) %]
676 -Htc #Htc destruct (Htc) #_ * #td * whd in ⊢ (%→?); * #_
677 #Htd destruct (Htd) * #te * * * * >Hmidta_src >Hmidta_dst
678 cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb
680 [ <(reverse_reverse … ls) <(reverse_reverse … lsa)
681 cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen'
682 @(list_cases2 … Hlen')
683 [ #H1 #H2 >H1 >H2 -H1 -H2 #_ #_ normalize in match (reverse ? [ ]); #Hte #_
684 lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte) * * #_
685 >Hmidta_dst #Htb1 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2
686 cut (tb = change_vec ?? ta (mk_tape ? (s0::lsa@lsb) (option_hd ? rs0) (tail ? rs0)) dst)
687 [@(eq_vec_change_vec … (niltape ?)) [@Htb1|@Htb2] ]
688 -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
689 >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
691 [| cases rs0 [ #_ %2 % | #x0 #xs0 normalize in ⊢ (??%?→?); #H destruct (H)] ]
692 normalize in match (left ??); normalize in match (right ??);
693 >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
695 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
696 >reverse_cons >reverse_cons >associative_append #Hte
697 lapply (Hte ???? (refl ??) ? s0 ? (reverse ? tla) ?? (refl ??))
698 [ >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
699 normalize #H destruct (H) // ] #Hte #_ #_ #_
700 * * #_ >Hte >nth_change_vec // #Htb1 #Htb2
701 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
702 (mk_tape sig (hda::lsb) (option_hd ? (reverse sig (reverse sig tla)@s0::rs0)) (tail ? (reverse sig (reverse sig tla)@s0::rs0))) dst)
703 (midtape ? [ ] hdb (reverse sig (reverse sig tlb)@s::rs)) src)
704 [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?))
706 | #j #Hj <Htb2 // >nth_change_vec_neq in ⊢ (???%); // ]]
707 -Htb1 -Htb2 #Htb >Htb whd
708 >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
710 [| cases (reverse sig (reverse sig tla))
711 [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
712 >Hmidta_src >Hmidta_dst
713 whd in match (left ??); whd in match (left ??); whd in match (right ??);
714 >current_mk_tape <opt_cons_tail_expand >length_append
715 >length_reverse >length_reverse <(length_reverse ? ls) <Hlen'
717 | #_ <(reverse_reverse … ls0) <(reverse_reverse … lsa)
718 cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen'
719 @(list_cases2 … Hlen')
720 [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #_ #_ #Hte
721 lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte)
722 * * #_ >Hmidta_dst #Htb1 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2
723 cut (tb = change_vec (tape sig) (S n) ta (mk_tape ? (s0::ls0) (option_hd ? rs0) (tail ? rs0)) dst)
724 [ @(eq_vec_change_vec … (niltape ?)) // @Htb2 ]
725 -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
726 >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
727 >current_mk_tape >right_mk_tape
728 [| cases rs0 [ #_ %2 % | #x0 #xs0 normalize in ⊢ (??%?→?); #H destruct (H) ]]
729 normalize in ⊢ (??%); <opt_cons_tail_expand
731 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
732 >reverse_cons >reverse_cons #Hte #_ #_
733 lapply (Hte s0 hdb (reverse ? tlb) rs0 ?
734 lsb s hda (reverse ? tla) rs ??)
735 [ /2 by cons_injective_l, nil/
736 | >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
737 normalize #H destruct (H) //
738 | /2 by refl, trans_eq/ ] -Hte
739 #Hte * * #_ >Hte >nth_change_vec_neq // >nth_change_vec // #Htb1 #Htb2
740 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
741 (mk_tape sig [hdb] (option_hd ? (reverse sig (reverse sig tlb)@s0::rs0)) (tail ? (reverse sig (reverse sig tlb)@s0::rs0))) dst)
742 (midtape ? lsb hda (reverse sig (reverse sig tla)@s::rs)) src)
743 [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?))
745 | #j #Hj <Htb2 // >change_vec_commute [|@sym_not_eq //]
746 >nth_change_vec_neq in ⊢ (???%); // ]]
747 -Htb1 -Htb2 #Htb >Htb whd
748 >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
750 [| cases (reverse ? (reverse ? tlb)) [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
751 >Hmidta_src >Hmidta_dst
752 whd in match (left ??); whd in match (left ??); whd in match (right ??);
753 >current_mk_tape <opt_cons_tail_expand >length_append
754 normalize in ⊢ (??%); >length_append >reverse_reverse
755 <(length_reverse ? lsa) >Hlen' >H2 normalize //
761 | #ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd destruct (Htd)
762 whd in ⊢ (%→?); #Htb destruct (Htb) #ls #x #xs #Hta_src
763 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
764 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
765 [ #Hcurta_dst % % % // @Hcomp1 %2 //
766 | #x0 #Hcurta_dst cases (current_to_midtape … Hcurta_dst) -Hcurta_dst
767 #ls0 * #rs0 #Hta_dst cases (true_or_false (x == x0)) #Hxx0
768 [ lapply (\P Hxx0) -Hxx0 #Hxx0 destruct (Hxx0)
769 | >(?:tc=ta) in Htest;
770 [|@Hcomp1 % % >Hta_src >Hta_dst @(not_to_not ??? (\Pf Hxx0)) normalize
771 #Hxx0' destruct (Hxx0') % ]
773 >nth_current_chars >Hta_src >nth_current_chars >Hta_dst
774 whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1
775 cases (Hcomp2 … Hta_src Hta_dst) [ *
776 [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} %
778 | * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ]
779 | * #xs0 * #ci * #cj * #rs' * #rs0' * * *
780 #Hci #Hxs #Hrs0 #Htc @False_ind
782 >(?:nth src ? (current_chars ?? tc) (None ?) = Some ? ci) in Htest;
783 [|>nth_current_chars >Htc >nth_change_vec_neq [|@(not_to_not … Hneq) //]
785 >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj)
786 [|>nth_current_chars >Htc >nth_change_vec //]
787 normalize #H destruct (H) ] ] ]
790 (* lemma WF_to_WF_f : ∀A,B,R,f,b. WF A R (f b) → WF B (λx,y.R (f x) (f y)) b. *)
791 let rec WF_to_WF_f A B R f b (Hwf: WF A R (f b)) on Hwf: WF B (λx,y.R (f x) (f y)) b ≝
792 match Hwf return (λa0,r.f b = a0 → WF B (λx,y:B. R (f x) (f y)) b) with
793 [ wf a Hwfa ⇒ λHeq.? ] (refl ??).
794 % #b1 #HRb @WF_to_WF_f @Hwfa <Heq @HRb
797 lemma lt_WF : ∀n.WF ? lt n.
798 #n @(nat_elim1 n) -n #n #IH % @IH
801 lemma terminate_match_m :
803 src ≠ dst → src < S n → dst < S n →
804 match_m src dst sig n ↓ t.
805 #src #dst #sig #n #ta #Hneq #Hsrc #Hdst
806 @(terminate_while … (sem_match_step_termination src dst sig n Hneq Hsrc Hdst)) //
807 letin f ≝ (λt0:Vector (tape sig) (S n).|left ? (nth src (tape ?) t0 (niltape ?))|
808 +|option_cons ? (current ? (nth dst (tape ?) t0 (niltape ?)))
809 (right ? (nth dst (tape ?) t0 (niltape ?)))|)
810 change with (λx,y.f x < f y) in ⊢ (??%?); @WF_to_WF_f @lt_WF
813 lemma sem_match_m : ∀src,dst,sig,n.
814 src ≠ dst → src < S n → dst < S n →
815 match_m src dst sig n \vDash R_match_m src dst sig n.
816 #src #dst #sig #n #Hneq #Hsrc #Hdst @WRealize_to_Realize [/2/| @wsem_match_m // ]