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 · mmove dst ?? R))
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_move_multi … 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; whd in ⊢ (%→?); #Htb >Htb %
308 [ >change_vec_change_vec >nth_change_vec //
309 >reverse_reverse <Hrs <Hmidta_src >change_vec_same <Hrs0 <Hmidta_dst
311 | >Hmidta_dst %{s'} % [%] #_
312 >Hrs0 %{xs} %{ci} %{rs''} %{ls0} %{cj} %{rs0'} % // % //
315 | lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta)
316 [@Htc % % @(not_to_not ??? Hss0) #H destruct (H) %]
317 -Htc #Htc destruct (Htc) #_ * #td * whd in ⊢ (%→?); * #_
318 #Htd destruct (Htd) * #te * * #_ #Hte whd in ⊢ (%→?); #Htb
319 #s1 #rs1 >Hmidta_src #H destruct (H)
320 lapply (Hte … Hmidta_src … Hmidta_dst) -Hte #Hte destruct (Hte) %
322 | >Hs0 %{s0} % // #H destruct (H) @False_ind cases (Hss0) /2/ ]
326 | #ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd destruct (Htd)
327 whd in ⊢ (%→?); #Htb destruct (Htb) #ls #x #xs #Hta_src
328 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
329 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
330 [ #Hcurta_dst % % % // @Hcomp1 %2 //
331 | #x0 #Hcurta_dst cases (current_to_midtape … Hcurta_dst) -Hcurta_dst
332 #ls0 * #rs0 #Hta_dst cases (true_or_false (x == x0)) #Hxx0
333 [ lapply (\P Hxx0) -Hxx0 #Hxx0 destruct (Hxx0)
334 | >(?:tc=ta) in Htest;
335 [|@Hcomp1 % % >Hta_src >Hta_dst @(not_to_not ??? (\Pf Hxx0)) normalize
336 #Hxx0' destruct (Hxx0') % ]
338 >nth_current_chars >Hta_src >nth_current_chars >Hta_dst
339 whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1
340 cases (Hcomp2 … Hta_src Hta_dst) [ *
341 [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} %
343 | * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ]
344 | * #xs0 * #ci * #cj * #rs' * #rs0' * * *
345 #Hci #Hxs #Hrs0 #Htc @False_ind
347 >(?:nth src ? (current_chars ?? tc) (None ?) = Some ? ci) in Htest;
348 [|>nth_current_chars >Htc >nth_change_vec_neq [|@(not_to_not … Hneq) //]
350 >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj)
351 [|>nth_current_chars >Htc >nth_change_vec //]
352 normalize #H destruct (H) ] ] ]
355 definition match_m ≝ λsrc,dst,sig,n.
356 whileTM … (match_step src dst sig n)
357 (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
359 definition R_match_m ≝
360 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
362 nth src ? int (niltape ?) = midtape sig [ ] x rs →
363 (current sig (nth dst (tape sig) int (niltape sig)) = None ? →
364 right ? (nth dst (tape sig) int (niltape sig)) = [ ] → outt = int) ∧
366 nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
367 (∃l,l1.x0::rs0 = l@x::rs@l1 ∧
370 (mk_tape sig (reverse ? rs@[x]) (None ?) [ ]) src)
371 (mk_tape sig ((reverse ? (l@x::rs))@ls0) (option_hd ? l1) (tail ? l1)) dst) ∨
372 ∀l,l1.x0::rs0 ≠ l@x::rs@l1).
374 lemma not_sub_list_merge :
375 ∀T.∀a,b:list T. (∀l1.a ≠ b@l1) → (∀t,l,l1.a ≠ t::l@b@l1) → ∀l,l1.a ≠ l@b@l1.
376 #T #a #b #H1 #H2 #l elim l normalize //
379 lemma not_sub_list_merge_2 :
380 ∀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.
381 #T #a #b #t #H1 #H2 #l elim l //
382 #t0 #l1 #IH #l2 cases (true_or_false (t == t0)) #Htt0
383 [ >(\P Htt0) % normalize #H destruct (H) cases (H2 l1 l2) /2/
384 | normalize % #H destruct (H) cases (\Pf Htt0) /2/ ]
388 lemma wsem_match_m : ∀src,dst,sig,n.
389 src ≠ dst → src < S n → dst < S n →
390 match_m src dst sig n ⊫ R_match_m src dst sig n.
391 #src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
392 lapply (sem_while … (sem_match_step src dst sig n Hneq Hsrc Hdst) … Hloop) //
393 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
394 [ #Hfalse #x #xs #Hmid_src
395 cases (Hfalse … Hmid_src) -Hfalse
396 [(* current dest = None *) *
397 [ * #Hcur_dst #Houtc %
399 | #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
400 normalize in ⊢ (%→?); #H destruct (H)
402 | * #ls0 * #rs0 * #xs0 * * #Htc_dst #Hrs0 #HNone %
403 [ >Htc_dst normalize in ⊢ (%→?); #H destruct (H)
404 | #ls1 #x1 #rs1 >Htc_dst #H destruct (H)
405 >Hrs0 >HNone cases xs0
406 [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %]
409 | >reverse_append >reverse_cons >reverse_append
410 >associative_append >associative_append % ]
411 | #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs)
412 >length_append whd in ⊢ (??%(??%)→?); >length_append
413 >length_append normalize >commutative_plus whd in ⊢ (???%→?);
414 #H destruct (H) lapply e0 >(plus_n_O (|rs1|)) in ⊢ (??%?→?);
415 >associative_plus >associative_plus
416 #e1 lapply (injective_plus_r ??? e1) whd in ⊢ (???%→?);
421 |* #ls0 * #rs0 * #Hmid_dst #Houtc %
422 [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
423 |#ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
424 %1 %{[ ]} %{rs0} % [%]
425 >reverse_cons >associative_append >Houtc %
428 |-ta #ta #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
430 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
431 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
433 [#Hcurta_dst #Hrightta_dst whd in Htrue; >Hmidta_src in Htrue; #Htrue
434 cases (Htrue ?? (refl ??)) -Htrue #Htc
436 [ >Htc whd in match (tape_move_mono ???); whd in match (tape_write ???);
437 <(change_vec_same … ta dst (niltape ?)) in ⊢ (???%);
438 lapply Hrightta_dst lapply Hcurta_dst -Hrightta_dst -Hcurta_dst
439 cases (nth dst ? ta (niltape ?))
441 | #r0 #rs0 #_ normalize in ⊢ (%→?); #H destruct (H)
443 | #ls #x0 #rs normalize in ⊢ (%→?); #H destruct (H) ] ]
444 -Htc #Htc destruct (Htc) #_
445 cases (IH … Hmidta_src) #Houtc #_ @Houtc //
446 |#ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst;
447 normalize in ⊢ (%→?); #H destruct (H)
449 | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
450 #ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst; normalize in ⊢ (%→?);
451 #H destruct (H) whd in Htrue; >Hmidta_src in Htrue; #Htrue
452 cases (Htrue ?? (refl …)) -Htrue >Hmidta_dst #Htc
453 cases (true_or_false (x==c)) #eqx
454 [ lapply (\P eqx) -eqx #eqx destruct (eqx) * #s0 * whd in ⊢ (??%?→?); #Hs0
455 destruct (Hs0) #Htrue cases (Htrue (refl ??)) -Htrue
456 #xs0 * #ci * #rs' * #ls1 * #cj * #rs1 * * #Hxs #H destruct (H) #Hcicj
457 >Htc in IH; whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
458 #IH cases (IH … Hmidta_src) -IH #_ >nth_change_vec //
459 cut (∃x1,xs1.xs0@cj::rs1 = x1::xs1)
460 [ cases xs0 [ %{cj} %{rs1} % | #x1 #xs1 %{x1} %{(xs1@cj::rs1)} % ] ] * #x1 * #xs1
461 #Hxs1 >Hxs1 #IH cases (IH … (refl ??)) -IH
462 [ * #l * #l1 * #Hxs1'
463 >change_vec_commute // >change_vec_change_vec
464 #Houtc % %{(s0::l)} %{l1} %
466 | >reverse_cons >associative_append >change_vec_commute // @Houtc ]
467 | #H %2 #l #l1 >(?:l@s0::xs@l1 = l@(s0::xs)@l1) [|%]
469 [ #l2 >Hxs <Hxs1 % normalize #H1 lapply (cons_injective_r ????? H1)
470 >associative_append #H2 lapply (append_l2_injective ????? (refl ??) H2)
471 #H3 lapply (cons_injective_l ????? H3) #H3 >H3 in Hcicj; * /2/
472 |#t #l2 #l3 % normalize #H1 lapply (cons_injective_r ????? H1)
473 -H1 #H1 cases (H l2 l3) #H2 @H2 @H1
476 | #_ cases (IH x xs ?) -IH
477 [| >Htc >nth_change_vec_neq [|@sym_not_eq //] @Hmidta_src ]
478 >Htc >nth_change_vec // cases rs0
479 [ #_ #_ %2 #l #l1 cases l
482 [ normalize % #H destruct (H) cases (\Pf eqx) /2/
483 | #tmp1 #l2 normalize % #H destruct (H) ]
484 | #tmp1 #l2 normalize % #H destruct (H) ]
485 | #tmp1 #l2 normalize % #H destruct (H)cases l2 in e0;
486 [ normalize #H1 destruct (H1)
487 | #tmp2 #l3 normalize #H1 destruct (H1) ] ]
488 | #r1 #rs1 #_ #IH cases (IH … (refl ??)) -IH
489 [ * #l * #l1 * #Hll1 #Houtc % %{(c::l)} %{l1} % [ >Hll1 % ]
490 >Houtc >change_vec_commute // >change_vec_change_vec
491 >change_vec_commute [|@sym_not_eq //]
492 >reverse_cons >associative_append %
493 | #Hll1 %2 @(not_sub_list_merge_2 ?? (x::xs)) normalize [|@Hll1]
494 #l1 % #H destruct (H) cases (\Pf eqx) /2/
502 definition R_match_step_true_naive ≝
503 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
504 |left ? (nth src ? outt (niltape ?))| +
505 |option_cons ? (current ? (nth dst ? outt (niltape ?))) (right ? (nth dst ? outt (niltape ?)))| <
506 |left ? (nth src ? int (niltape ?))| +
507 |option_cons ? (current ? (nth dst ? int (niltape ?))) (right ? (nth dst ? int (niltape ?)))|.
509 lemma sem_match_step_termination :
510 ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
511 match_step src dst sig n ⊨
512 [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
513 R_match_step_true_naive src dst sig n,
514 R_match_step_false src dst sig n ].
515 #src #dst #sig #n #Hneq #Hsrc #Hdst
516 @(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
517 (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
519 (sem_rewind_strong ???? Hneq Hsrc Hdst)
520 (sem_move_multi … R ?))
522 [ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
523 cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
524 [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
525 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
526 >nth_current_chars >Hcurta_src normalize in ⊢ (%→?); #H destruct (H)
527 | #s #Hs lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
528 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
529 [ #Hcurta_dst #Hcomp #_ * #td * >Hcomp [| %2 %]
530 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
531 >nth_current_chars >nth_current_chars >Hs >Hcurta_dst
532 normalize in ⊢ (%→?); #H destruct (H)
534 cases (current_to_midtape … Hs) #ls * #rs #Hmidta_src >Hmidta_src
535 cases (current_to_midtape … Hs0) #ls0 * #rs0 #Hmidta_dst >Hmidta_dst
536 cases (true_or_false (s == s0)) #Hss0
537 [ lapply (\P Hss0) -Hss0 #Hss0 destruct (Hss0)
538 #_ #Hcomp cases (Hcomp ????? (refl ??) (refl ??)) -Hcomp [ *
539 [ * #rs' * #_ #Hcurtc_dst * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
540 >nth_current_chars >nth_current_chars >Hcurtc_dst
541 cases (current ? (nth src …))
542 [normalize in ⊢ (%→?); #H destruct (H)
543 | #x >nth_change_vec [|@Hdst] cases (reverse ? rs0)
544 [ normalize in ⊢ (%→?); #H destruct (H)
545 | #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ]
546 | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
547 >(?:nth src ? (current_chars ?? tc) (None ?) = None ?)
548 [|>nth_current_chars >Hcurtc_src >nth_change_vec_neq
549 [>nth_change_vec [cases (append ???) // | @Hsrc]
550 |@(not_to_not … Hneq) //
552 normalize in ⊢ (%→?); #H destruct (H) ]
553 | * #xs * #ci * #cj * #rs'' * #rs0' * * * #Hcicj #Hrs #Hrs0
554 #Htc * #td * * #Hmatch #Htd destruct (Htd) * #te * * *
555 >Htc >change_vec_commute [|//] >nth_change_vec [|//]
556 >change_vec_commute [|@sym_not_eq //] >nth_change_vec [|//]
557 cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb
559 [ #Hte #_ #_ <(reverse_reverse … ls) in Hte; <(reverse_reverse … lsa)
560 cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen'
561 @(list_cases2 … Hlen')
562 [ #H1 #H2 >H1 >H2 -H1 -H2 normalize in match (reverse ? [ ]); #Hte #_
563 lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
564 >change_vec_commute [|//] >change_vec_change_vec
565 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
566 >Hte whd in ⊢ (%→?); >change_vec_change_vec >nth_change_vec [|//]
567 >reverse_reverse #Htb
568 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)
569 [ >Htb @eq_f3 // cases (xs@cj::rs0') // ]
570 -Htb #Htb >Htb whd >nth_change_vec [|//]
571 >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec [|//]
572 >right_mk_tape [|cases xs [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H)]
573 normalize in match (left ??);
574 >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
575 whd in match (option_cons ???); >Hrs0
576 normalize in ⊢ (?(?%)%); //
577 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
578 >reverse_cons >reverse_cons #Hte
579 lapply (Hte ci hdb (reverse ? xs@s0::reverse ? tlb) rs'' ?
580 lsb cj hda (reverse ? xs@s0::reverse ? tla) rs0' ??)
581 [ /2 by cons_injective_l, nil/
582 | >length_append >length_append @eq_f @(eq_f ?? S)
583 >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
584 >length_reverse >length_reverse destruct (Hlen') //
585 | /2 by refl, trans_eq/ ] -Hte
586 #Hte #_ whd in ⊢ (%→?); #Htb
587 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
588 (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)
589 (midtape ? [ ] hdb (reverse sig (reverse sig xs@s0::reverse sig tlb)@ci::rs'')) src)
590 [ >Htb >Hte >nth_change_vec // >change_vec_change_vec >change_vec_commute [|//]
591 >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
592 >change_vec_change_vec >change_vec_commute [|//]
593 @eq_f3 // cases (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0') // ]
595 >nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
597 [| cases (reverse sig (reverse sig xs@s0::reverse sig tla))
598 [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
599 >Hmidta_src >Hmidta_dst
600 whd in match (left ??); whd in match (left ??); whd in match (right ??);
601 >current_mk_tape <opt_cons_tail_expand whd in match (option_cons ???);
602 >Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
603 >length_append >commutative_plus in match (|reverse ??| + ?);
604 whd in match (|?::?|); >length_reverse >length_reverse
605 <(length_reverse ? ls) <Hlen' >H1 normalize // ]
606 | #_ #Hte #_ <(reverse_reverse … ls0) in Hte; <(reverse_reverse … lsa)
607 cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen'
608 @(list_cases2 … Hlen')
609 [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #Hte
610 lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
611 >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
612 >change_vec_change_vec #Hte #_
613 >Hte whd in ⊢ (%→?); >nth_change_vec [|//] >reverse_reverse #Htb
614 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta (mk_tape ? [s0] (option_hd ? (xs@cj::rs0')) (tail ? (xs@cj::rs0'))) dst)
615 (midtape ? lsb s0 (xs@ci::rs'')) src)
616 [ >Htb >change_vec_change_vec >change_vec_commute [|//]
617 @eq_f3 // <Hrs0 cases rs0 // ]
618 -Htb #Htb >Htb whd >nth_change_vec [|//]
619 >nth_change_vec_neq [|//] >nth_change_vec [|//]
621 [| cases xs [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
622 normalize in match (left ??);
623 >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand >Hrs0
624 >length_append normalize >length_append >length_append
625 <(reverse_reverse ? lsa) >H1 normalize //
626 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
627 >reverse_cons >reverse_cons #Hte
628 lapply (Hte cj hdb (reverse ? xs@s0::reverse ? tlb) rs0' ?
629 lsb ci hda (reverse ? xs@s0::reverse ? tla) rs'' ??)
630 [ /2 by cons_injective_l, nil/
631 | >length_append >length_append @eq_f @(eq_f ?? S)
632 >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
633 >length_reverse >length_reverse destruct (Hlen') //
634 | /2 by refl, trans_eq/ ] -Hte
635 #Hte #_ whd in ⊢ (%→?); >Hte >nth_change_vec_neq [|//] >nth_change_vec [|//] #Htb
636 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
637 (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)
638 (midtape ? lsb hda (reverse sig (reverse sig xs@s0::reverse sig tla)@ci::rs'')) src)
639 [ >Htb >change_vec_change_vec >change_vec_commute [|//]
640 >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
641 >change_vec_change_vec >change_vec_commute [|//]
642 @eq_f3 // cases (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0') // ]
644 >nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
646 [| cases (reverse sig (reverse sig xs@s0::reverse sig tlb))
647 [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
648 >Hmidta_src >Hmidta_dst
649 whd in match (left ??); whd in match (left ??); whd in match (right ??);
650 >current_mk_tape <opt_cons_tail_expand
651 whd in match (option_cons ???);
652 >Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
653 >length_append >commutative_plus in match (|reverse ??| + ?);
654 whd in match (|?::?|); >length_reverse >length_reverse
655 <(length_reverse ? lsa) >Hlen' >H2 >length_append
660 | lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta)
661 [@Htc % % @(not_to_not ??? Hss0) #H destruct (H) %]
662 -Htc #Htc destruct (Htc) #_ * #td * whd in ⊢ (%→?); * #_
663 #Htd destruct (Htd) * #te * * * * >Hmidta_src >Hmidta_dst
664 cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb
666 [ <(reverse_reverse … ls) <(reverse_reverse … lsa)
667 cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen'
668 @(list_cases2 … Hlen')
669 [ #H1 #H2 >H1 >H2 -H1 -H2 #_ #_ normalize in match (reverse ? [ ]); #Hte #_
670 lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte)
671 whd in ⊢ (%→?); >Hmidta_dst #Htb
672 cut (tb = change_vec ?? ta (mk_tape ? (s0::lsa@lsb) (option_hd ? rs0) (tail ? rs0)) dst)
673 [ >Htb cases rs0 // ]
674 -Htb #Htb >Htb whd >nth_change_vec [|//]
675 >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
677 [| cases rs0 [ #_ %2 % | #x0 #xs0 normalize in ⊢ (??%?→?); #H destruct (H)] ]
678 normalize in match (left ??); normalize in match (right ??);
679 >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
681 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
682 >reverse_cons >reverse_cons >associative_append #Hte
683 lapply (Hte ???? (refl ??) ? s0 ? (reverse ? tla) ?? (refl ??))
684 [ >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
685 normalize #H destruct (H) // ] #Hte #_ #_ #_
686 whd in ⊢ (%→?); >Hte >change_vec_change_vec >nth_change_vec // #Htb
687 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
688 (mk_tape sig (hda::lsb) (option_hd ? (reverse sig (reverse sig tla)@s0::rs0)) (tail ? (reverse sig (reverse sig tla)@s0::rs0))) dst)
689 (midtape ? [ ] hdb (reverse sig (reverse sig tlb)@s::rs)) src)
690 [ >Htb >change_vec_commute [|//] @eq_f3 // cases (reverse sig (reverse sig tla)@s0::rs0) // ]
692 >nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
694 [| cases (reverse sig (reverse sig tla))
695 [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
696 >Hmidta_src >Hmidta_dst
697 whd in match (left ??); whd in match (left ??); whd in match (right ??);
698 >current_mk_tape <opt_cons_tail_expand >length_append
699 >length_reverse >length_reverse <(length_reverse ? ls) <Hlen'
701 | #_ <(reverse_reverse … ls0) <(reverse_reverse … lsa)
702 cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen'
703 @(list_cases2 … Hlen')
704 [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #_ #_ #Hte
705 lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte)
706 whd in ⊢ (%→?); #Htb whd >Hmidta_dst
707 cut (tb = change_vec (tape sig) (S n) ta (mk_tape ? (s0::ls0) (option_hd ? rs0) (tail ? rs0)) dst)
708 [ >Htb >Hmidta_dst cases rs0 // ]
709 -Htb #Htb >Htb whd >nth_change_vec [|//]
710 >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
711 >current_mk_tape >right_mk_tape
712 [| cases rs0 [ #_ %2 % | #x0 #xs0 normalize in ⊢ (??%?→?); #H destruct (H) ]]
713 normalize in ⊢ (??%); <opt_cons_tail_expand
715 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
716 >reverse_cons >reverse_cons #Hte #_ #_
717 lapply (Hte s0 hdb (reverse ? tlb) rs0 ?
718 lsb s hda (reverse ? tla) rs ??)
719 [ /2 by cons_injective_l, nil/
720 | >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
721 normalize #H destruct (H) //
722 | /2 by refl, trans_eq/ ] -Hte
723 #Hte whd in ⊢ (%→?); >Hte >nth_change_vec_neq [|//] >nth_change_vec [|//] #Htb
724 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
725 (mk_tape sig [hdb] (option_hd ? (reverse sig (reverse sig tlb)@s0::rs0)) (tail ? (reverse sig (reverse sig tlb)@s0::rs0))) dst)
726 (midtape ? lsb hda (reverse sig (reverse sig tla)@s::rs)) src)
727 [ >Htb >change_vec_commute [|//] >change_vec_change_vec
728 @eq_f3 // cases (reverse sig (reverse sig tlb)@s0::rs0) // ]
730 >nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
732 [| cases (reverse ? (reverse ? tlb)) [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
733 >Hmidta_src >Hmidta_dst
734 whd in match (left ??); whd in match (left ??); whd in match (right ??);
735 >current_mk_tape <opt_cons_tail_expand >length_append
736 normalize in ⊢ (??%); >length_append >reverse_reverse
737 <(length_reverse ? lsa) >Hlen' >H2 normalize //
743 | #ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd destruct (Htd)
744 whd in ⊢ (%→?); #Htb destruct (Htb) #ls #x #xs #Hta_src
745 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
746 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
747 [ #Hcurta_dst % % % // @Hcomp1 %2 //
748 | #x0 #Hcurta_dst cases (current_to_midtape … Hcurta_dst) -Hcurta_dst
749 #ls0 * #rs0 #Hta_dst cases (true_or_false (x == x0)) #Hxx0
750 [ lapply (\P Hxx0) -Hxx0 #Hxx0 destruct (Hxx0)
751 | >(?:tc=ta) in Htest;
752 [|@Hcomp1 % % >Hta_src >Hta_dst @(not_to_not ??? (\Pf Hxx0)) normalize
753 #Hxx0' destruct (Hxx0') % ]
755 >nth_current_chars >Hta_src >nth_current_chars >Hta_dst
756 whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1
757 cases (Hcomp2 … Hta_src Hta_dst) [ *
758 [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} %
760 | * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ]
761 | * #xs0 * #ci * #cj * #rs' * #rs0' * * *
762 #Hci #Hxs #Hrs0 #Htc @False_ind
764 >(?:nth src ? (current_chars ?? tc) (None ?) = Some ? ci) in Htest;
765 [|>nth_current_chars >Htc >nth_change_vec_neq [|@(not_to_not … Hneq) //]
767 >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj)
768 [|>nth_current_chars >Htc >nth_change_vec //]
769 normalize #H destruct (H) ] ] ]
772 (* 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. *)
773 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 ≝
774 match Hwf return (λa0,r.f b = a0 → WF B (λx,y:B. R (f x) (f y)) b) with
775 [ wf a Hwfa ⇒ λHeq.? ] (refl ??).
776 % #b1 #HRb @WF_to_WF_f @Hwfa <Heq @HRb
779 lemma lt_WF : ∀n.WF ? lt n.
780 #n @(nat_elim1 n) -n #n #IH % @IH
783 lemma terminate_match_m :
785 src ≠ dst → src < S n → dst < S n →
786 match_m src dst sig n ↓ t.
787 #src #dst #sig #n #ta #Hneq #Hsrc #Hdst
788 @(terminate_while … (sem_match_step_termination src dst sig n Hneq Hsrc Hdst)) //
789 letin f ≝ (λt0:Vector (tape sig) (S n).|left ? (nth src (tape ?) t0 (niltape ?))|
790 +|option_cons ? (current ? (nth dst (tape ?) t0 (niltape ?)))
791 (right ? (nth dst (tape ?) t0 (niltape ?)))|)
792 change with (λx,y.f x < f y) in ⊢ (??%?); @WF_to_WF_f @lt_WF
795 lemma sem_match_m : ∀src,dst,sig,n.
796 src ≠ dst → src < S n → dst < S n →
797 match_m src dst sig n \vDash R_match_m src dst sig n.
798 #src #dst #sig #n #Hneq #Hsrc #Hdst @WRealize_to_Realize [/2/| @wsem_match_m // ]