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/auxiliary_multi_machines.ma".
18 definition rewind ≝ λsrc,dst,sig,n.
19 parmove src dst sig n L · mmove src sig n R · mmove dst sig n R.
21 definition R_rewind_strong ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
23 nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
24 ∀ls0,y,y0,target,rs0.|xs| = |target| →
25 nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
27 (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src)
28 (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst) ∧
30 nth dst ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
31 ∀ls0,y,y0,target,rs0.|xs| = |target| →
32 nth src ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
34 (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) dst)
35 (midtape sig ls0 y0 (reverse ? target@y::rs0)) src) ∧
36 (∀x,rs.nth src ? int (niltape ?) = midtape sig [] x rs →
37 ∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 →
39 (∀x,rs.nth dst ? int (niltape ?) = midtape sig [] x rs →
40 ∀ls0,y,rs0.nth src ? int (niltape ?) = midtape sig ls0 y rs0 →
43 definition R_rewind ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
45 nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
46 ∀ls0,y,y0,target,rs0.|xs| = |target| →
47 nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
49 (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src)
50 (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst) ∧
51 (∀x,rs.nth src ? int (niltape ?) = midtape sig [] x rs →
52 ∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 →
55 lemma sem_rewind_strong : ∀src,dst,sig,n.
56 src ≠ dst → src < S n → dst < S n →
57 rewind src dst sig n ⊨ R_rewind_strong src dst sig n.
58 #src #dst #sig #n #Hneq #Hsrc #Hdst
59 @(sem_seq_app sig n ????? (sem_parmoveL src dst sig n Hneq Hsrc Hdst) ?)
60 [| @(sem_seq_app sig n ????? (sem_move_multi … R ?) (sem_move_multi … R ?)) //
62 #ta #tb * #tc * * * #Htc1 #Htc2 #_ * #td * whd in ⊢ (%→%→?); #Htd #Htb % [ % [ %
63 [ #x #x0 #xs #rs #Hmidta_src #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_dst
64 >(Htc1 ??? Hmidta_src ls0 y (target@[y0]) rs0 ??) in Htd;
66 |>length_append >length_append >Hlen % ]
67 >change_vec_commute [|@sym_not_eq //]
68 >change_vec_change_vec
69 >nth_change_vec_neq [|@sym_not_eq //]
70 >nth_change_vec // >reverse_append >reverse_single
71 >reverse_append >reverse_single normalize in match (tape_move ???);
72 >rev_append_def >append_nil #Htd >Htd in Htb;
73 >change_vec_change_vec >nth_change_vec //
74 cases ls0 [|#l1 #ls1] normalize in match (tape_move ???); //
75 | #x #x0 #xs #rs #Hmidta_dst #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_src
76 >(Htc2 ??? Hmidta_dst ls0 y (target@[y0]) rs0 ??) in Htd;
78 |>length_append >length_append >Hlen % ]
79 >change_vec_change_vec
80 >change_vec_commute [|@sym_not_eq //]
82 >reverse_append >reverse_single
83 >reverse_append >reverse_single
84 cases ls0 [|#l1 #ls1] normalize in match (tape_move ???);
85 #Htd >Htd in Htb; >change_vec_change_vec >nth_change_vec //
86 >rev_append_def >change_vec_commute // normalize in match (tape_move ???); // ]
87 | #x #rs #Hmidta_src #ls0 #y #rs0 #Hmidta_dst
88 lapply (Htc1 … Hmidta_src … (refl ??) Hmidta_dst) -Htc1 #Htc >Htc in Htd;
89 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
90 >nth_change_vec_neq [|@sym_not_eq //]
91 >nth_change_vec // lapply (refl ? ls0) cases ls0 in ⊢ (???%→%);
92 [ #Hls0 #Htd >Htd in Htb;
93 >nth_change_vec // >change_vec_change_vec
94 whd in match (tape_move ???);whd in match (tape_move ???); <Hmidta_src
95 <Hls0 <Hmidta_dst >change_vec_same >change_vec_same //
96 | #l1 #ls1 #Hls0 #Htd >Htd in Htb;
97 >nth_change_vec // >change_vec_change_vec
98 whd in match (tape_move ???);whd in match (tape_move ???); <Hmidta_src
99 <Hls0 <Hmidta_dst >change_vec_same >change_vec_same //
101 | #x #rs #Hmidta_dst #ls0 #y #rs0 #Hmidta_src
102 lapply (Htc2 … Hmidta_dst … (refl ??) Hmidta_src) -Htc2 #Htc >Htc in Htd;
103 >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
104 >nth_change_vec // lapply (refl ? ls0) cases ls0 in ⊢ (???%→%);
105 [ #Hls0 destruct (Hls0) #Htd >Htd in Htb;
106 >nth_change_vec // >change_vec_change_vec
107 whd in match (tape_move ???);whd in match (tape_move ???);
108 <Hmidta_src <Hmidta_dst >change_vec_same >change_vec_same //
109 | #l1 #ls1 #Hls0 destruct (Hls0) #Htd >Htd in Htb;
110 >nth_change_vec // >change_vec_change_vec
111 whd in match (tape_move ???); whd in match (tape_move ???); <Hmidta_src
112 <Hmidta_dst >change_vec_same >change_vec_same //
117 lemma sem_rewind : ∀src,dst,sig,n.
118 src ≠ dst → src < S n → dst < S n →
119 rewind src dst sig n ⊨ R_rewind src dst sig n.
120 #src #dst #sig #n #Hneq #Hsrc #Hdst @(Realize_to_Realize … (sem_rewind_strong …)) //
121 #ta #tb * * * #H1 #H2 #H3 #H4 % /2 by /
126 definition match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n.
127 match (nth src (option sig) v (None ?)) with
129 | Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ].
131 definition match_step ≝ λsrc,dst,sig,n.
132 compare src dst sig n ·
133 (ifTM ?? (partest sig n (match_test src dst sig ?))
135 (rewind src dst sig n · mmove dst ?? R))
139 (* we assume the src is a midtape
141 if the dst is out of bounds (outt = int)
142 or dst.right is shorter than src.right (outt.current → None)
143 or src.right is a prefix of dst.right (out = just right of the common prefix) *)
144 definition R_match_step_false ≝
145 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
147 nth src ? int (niltape ?) = midtape sig ls x xs →
148 ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
149 (∃ls0,rs0,xs0. nth dst ? int (niltape ?) = midtape sig ls0 x rs0 ∧
152 (change_vec ?? int (mk_tape sig (reverse ? rs0@x::ls) (option_hd ? xs0) (tail ? xs0)) src)
153 (mk_tape ? (reverse ? rs0@x::ls0) (None ?) [ ]) dst) ∨
155 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
159 (change_vec ?? int (mk_tape sig (reverse ? xs@x::ls) (None ?) [ ]) src)
160 (mk_tape sig (reverse ? xs@x::ls0) (option_hd ? rs0) (tail ? rs0)) dst).
163 we assume the src is a midtape [ ] s rs
165 then dst.current = Some ? s1
166 and if s ≠ s1 then outt = int.dst.move_right()
168 then int.src.right and int.dst.right have a common prefix
169 and the heads of their suffixes are different
170 and outt = int.dst.move_right().
173 definition R_match_step_true ≝
174 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
175 ∀s,rs.nth src ? int (niltape ?) = midtape ? [ ] s rs →
176 outt = change_vec ?? int
177 (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst ∧
178 (∃s0.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s0 ∧
180 ∃xs,ci,rs',ls0,cj,rs0.
182 nth dst ? int (niltape ?) = midtape sig ls0 s (xs@cj::rs0) ∧
185 lemma sem_match_step :
186 ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
187 match_step src dst sig n ⊨
188 [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
189 R_match_step_true src dst sig n,
190 R_match_step_false src dst sig n ].
191 #src #dst #sig #n #Hneq #Hsrc #Hdst
192 @(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
193 (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
195 (sem_rewind ???? Hneq Hsrc Hdst)
196 (sem_move_multi … R ?))
198 [ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
199 cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
200 [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
201 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
202 >nth_current_chars >Hcurta_src normalize in ⊢ (%→?); #H destruct (H)
203 | #s #Hs lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
204 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
205 [ #Hcurta_dst #Hcomp #_ * #td * >Hcomp [| %2 %]
206 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
207 >nth_current_chars >nth_current_chars >Hs >Hcurta_dst
208 normalize in ⊢ (%→?); #H destruct (H)
210 cases (current_to_midtape … Hs) #ls * #rs #Hmidta_src >Hmidta_src
211 cases (current_to_midtape … Hs0) #ls0 * #rs0 #Hmidta_dst >Hmidta_dst
212 cases (true_or_false (s == s0)) #Hss0
213 [ lapply (\P Hss0) -Hss0 #Hss0 destruct (Hss0)
214 #_ #Hcomp cases (Hcomp ????? (refl ??) (refl ??)) -Hcomp [ *
215 [ * #rs' * #_ #Hcurtc_dst * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
216 >nth_current_chars >nth_current_chars >Hcurtc_dst
217 cases (current ? (nth src …))
218 [normalize in ⊢ (%→?); #H destruct (H)
219 | #x >nth_change_vec // cases (reverse ? rs0)
220 [ normalize in ⊢ (%→?); #H destruct (H)
221 | #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ]
222 | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
223 >(?:nth src ? (current_chars ?? tc) (None ?) = None ?)
224 [|>nth_current_chars >Hcurtc_src >nth_change_vec_neq
225 [>nth_change_vec [cases (append ???) // | @Hsrc]
226 |@(not_to_not … Hneq) //
228 normalize in ⊢ (%→?); #H destruct (H) ]
229 | * #xs * #ci * #cj * #rs'' * #rs0' * * * #Hcicj #Hrs #Hrs0
230 #Htc * #td * * #Hmatch #Htd destruct (Htd) * #te * *
231 >Htc >change_vec_commute // >nth_change_vec //
232 >change_vec_commute [|@sym_not_eq //] >nth_change_vec // #Hte #_ #Htb
233 #s' #rs' >Hmidta_src #H destruct (H)
234 lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
235 >change_vec_commute // >change_vec_change_vec
236 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
237 >Hte in Htb; whd in ⊢ (%→?); #Htb >Htb %
238 [ >change_vec_change_vec >nth_change_vec //
239 >reverse_reverse <Hrs <Hmidta_src >change_vec_same <Hrs0 <Hmidta_dst
241 | >Hmidta_dst %{s'} % [%] #_
242 >Hrs0 %{xs} %{ci} %{rs''} %{ls0} %{cj} %{rs0'} % // % //
245 | lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta)
246 [@Htc % % @(not_to_not ??? Hss0) #H destruct (H) %]
247 -Htc #Htc destruct (Htc) #_ * #td * whd in ⊢ (%→?); * #_
248 #Htd destruct (Htd) * #te * * #_ #Hte whd in ⊢ (%→?); #Htb
249 #s1 #rs1 >Hmidta_src #H destruct (H)
250 lapply (Hte … Hmidta_src … Hmidta_dst) -Hte #Hte destruct (Hte) %
252 | >Hs0 %{s0} % // #H destruct (H) @False_ind cases (Hss0) /2/ ]
256 | #ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd destruct (Htd)
257 whd in ⊢ (%→?); #Htb destruct (Htb) #ls #x #xs #Hta_src
258 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
259 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
260 [ #Hcurta_dst % % % // @Hcomp1 %2 //
261 | #x0 #Hcurta_dst cases (current_to_midtape … Hcurta_dst) -Hcurta_dst
262 #ls0 * #rs0 #Hta_dst cases (true_or_false (x == x0)) #Hxx0
263 [ lapply (\P Hxx0) -Hxx0 #Hxx0 destruct (Hxx0)
264 | >(?:tc=ta) in Htest;
265 [|@Hcomp1 % % >Hta_src >Hta_dst @(not_to_not ??? (\Pf Hxx0)) normalize
266 #Hxx0' destruct (Hxx0') % ]
268 >nth_current_chars >Hta_src >nth_current_chars >Hta_dst
269 whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1
270 cases (Hcomp2 … Hta_src Hta_dst) [ *
271 [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} %
273 | * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ]
274 | * #xs0 * #ci * #cj * #rs' * #rs0' * * *
275 #Hci #Hxs #Hrs0 #Htc @False_ind
277 >(?:nth src ? (current_chars ?? tc) (None ?) = Some ? ci) in Htest;
278 [|>nth_current_chars >Htc >nth_change_vec_neq [|@(not_to_not … Hneq) //]
280 >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj)
281 [|>nth_current_chars >Htc >nth_change_vec //]
282 normalize #H destruct (H) ] ] ]
285 definition match_m ≝ λsrc,dst,sig,n.
286 whileTM … (match_step src dst sig n)
287 (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
289 definition R_match_m ≝
290 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
292 nth src ? int (niltape ?) = midtape sig [ ] x rs →
293 (current sig (nth dst (tape sig) int (niltape sig)) = None ? →
294 right ? (nth dst (tape sig) int (niltape sig)) = [ ] → outt = int) ∧
296 nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
297 (∃l,l1.x0::rs0 = l@x::rs@l1 ∧
300 (mk_tape sig (reverse ? rs@[x]) (None ?) [ ]) src)
301 (mk_tape sig ((reverse ? (l@x::rs))@ls0) (option_hd ? l1) (tail ? l1)) dst) ∨
302 ∀l,l1.x0::rs0 ≠ l@x::rs@l1).
304 lemma not_sub_list_merge :
305 ∀T.∀a,b:list T. (∀l1.a ≠ b@l1) → (∀t,l,l1.a ≠ t::l@b@l1) → ∀l,l1.a ≠ l@b@l1.
306 #T #a #b #H1 #H2 #l elim l normalize //
309 lemma not_sub_list_merge_2 :
310 ∀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.
311 #T #a #b #t #H1 #H2 #l elim l //
312 #t0 #l1 #IH #l2 cases (true_or_false (t == t0)) #Htt0
313 [ >(\P Htt0) % normalize #H destruct (H) cases (H2 l1 l2) /2/
314 | normalize % #H destruct (H) cases (\Pf Htt0) /2/ ]
318 lemma wsem_match_m : ∀src,dst,sig,n.
319 src ≠ dst → src < S n → dst < S n →
320 match_m src dst sig n ⊫ R_match_m src dst sig n.
321 #src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
322 lapply (sem_while … (sem_match_step src dst sig n Hneq Hsrc Hdst) … Hloop) //
323 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
324 [ #Hfalse #x #xs #Hmid_src
325 cases (Hfalse … Hmid_src) -Hfalse
326 [(* current dest = None *) *
327 [ * #Hcur_dst #Houtc %
329 | #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
330 normalize in ⊢ (%→?); #H destruct (H)
332 | * #ls0 * #rs0 * #xs0 * * #Htc_dst #Hrs0 #HNone %
333 [ >Htc_dst normalize in ⊢ (%→?); #H destruct (H)
334 | #ls1 #x1 #rs1 >Htc_dst #H destruct (H)
335 >Hrs0 >HNone cases xs0
336 [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %]
339 | >reverse_append >reverse_cons >reverse_append
340 >associative_append >associative_append % ]
341 | #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs)
342 >length_append whd in ⊢ (??%(??%)→?); >length_append
343 >length_append normalize >commutative_plus whd in ⊢ (???%→?);
344 #H destruct (H) lapply e0 >(plus_n_O (|rs1|)) in ⊢ (??%?→?);
345 >associative_plus >associative_plus
346 #e1 lapply (injective_plus_r ??? e1) whd in ⊢ (???%→?);
351 |* #ls0 * #rs0 * #Hmid_dst #Houtc %
352 [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
353 |#ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
354 %1 %{[ ]} %{rs0} % [%]
355 >reverse_cons >associative_append >Houtc %
358 |-ta #ta #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
360 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
361 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
363 [#Hcurta_dst #Hrightta_dst whd in Htrue; >Hmidta_src in Htrue; #Htrue
364 cases (Htrue ?? (refl ??)) -Htrue #Htc
366 [ >Htc whd in match (tape_move_mono ???); whd in match (tape_write ???);
367 <(change_vec_same … ta dst (niltape ?)) in ⊢ (???%);
368 lapply Hrightta_dst lapply Hcurta_dst -Hrightta_dst -Hcurta_dst
369 cases (nth dst ? ta (niltape ?))
371 | #r0 #rs0 #_ normalize in ⊢ (%→?); #H destruct (H)
373 | #ls #x0 #rs normalize in ⊢ (%→?); #H destruct (H) ] ]
374 -Htc #Htc destruct (Htc) #_
375 cases (IH … Hmidta_src) #Houtc #_ @Houtc //
376 |#ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst;
377 normalize in ⊢ (%→?); #H destruct (H)
379 | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
380 #ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst; normalize in ⊢ (%→?);
381 #H destruct (H) whd in Htrue; >Hmidta_src in Htrue; #Htrue
382 cases (Htrue ?? (refl …)) -Htrue >Hmidta_dst #Htc
383 cases (true_or_false (x==c)) #eqx
384 [ lapply (\P eqx) -eqx #eqx destruct (eqx) * #s0 * whd in ⊢ (??%?→?); #Hs0
385 destruct (Hs0) #Htrue cases (Htrue (refl ??)) -Htrue
386 #xs0 * #ci * #rs' * #ls1 * #cj * #rs1 * * #Hxs #H destruct (H) #Hcicj
387 >Htc in IH; whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
388 #IH cases (IH … Hmidta_src) -IH #_ >nth_change_vec //
389 cut (∃x1,xs1.xs0@cj::rs1 = x1::xs1)
390 [ cases xs0 [ %{cj} %{rs1} % | #x1 #xs1 %{x1} %{(xs1@cj::rs1)} % ] ] * #x1 * #xs1
391 #Hxs1 >Hxs1 #IH cases (IH … (refl ??)) -IH
392 [ * #l * #l1 * #Hxs1'
393 >change_vec_commute // >change_vec_change_vec
394 #Houtc % %{(s0::l)} %{l1} %
396 | >reverse_cons >associative_append >change_vec_commute // @Houtc ]
397 | #H %2 #l #l1 >(?:l@s0::xs@l1 = l@(s0::xs)@l1) [|%]
399 [ #l2 >Hxs <Hxs1 % normalize #H1 lapply (cons_injective_r ????? H1)
400 >associative_append #H2 lapply (append_l2_injective ????? (refl ??) H2)
401 #H3 lapply (cons_injective_l ????? H3) #H3 >H3 in Hcicj; * /2/
402 |#t #l2 #l3 % normalize #H1 lapply (cons_injective_r ????? H1)
403 -H1 #H1 cases (H l2 l3) #H2 @H2 @H1
406 | #_ cases (IH x xs ?) -IH
407 [| >Htc >nth_change_vec_neq [|@sym_not_eq //] @Hmidta_src ]
408 >Htc >nth_change_vec // cases rs0
409 [ #_ #_ %2 #l #l1 cases l
412 [ normalize % #H destruct (H) cases (\Pf eqx) /2/
413 | #tmp1 #l2 normalize % #H destruct (H) ]
414 | #tmp1 #l2 normalize % #H destruct (H) ]
415 | #tmp1 #l2 normalize % #H destruct (H)cases l2 in e0;
416 [ normalize #H1 destruct (H1)
417 | #tmp2 #l3 normalize #H1 destruct (H1) ] ]
418 | #r1 #rs1 #_ #IH cases (IH … (refl ??)) -IH
419 [ * #l * #l1 * #Hll1 #Houtc % %{(c::l)} %{l1} % [ >Hll1 % ]
420 >Houtc >change_vec_commute // >change_vec_change_vec
421 >change_vec_commute [|@sym_not_eq //]
422 >reverse_cons >associative_append %
423 | #Hll1 %2 @(not_sub_list_merge_2 ?? (x::xs)) normalize [|@Hll1]
424 #l1 % #H destruct (H) cases (\Pf eqx) /2/
432 definition R_match_step_true_naive ≝
433 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
434 |left ? (nth src ? outt (niltape ?))| +
435 |option_cons ? (current ? (nth dst ? outt (niltape ?))) (right ? (nth dst ? outt (niltape ?)))| <
436 |left ? (nth src ? int (niltape ?))| +
437 |option_cons ? (current ? (nth dst ? int (niltape ?))) (right ? (nth dst ? int (niltape ?)))|.
439 lemma sem_match_step_termination :
440 ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
441 match_step src dst sig n ⊨
442 [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
443 R_match_step_true_naive src dst sig n,
444 R_match_step_false src dst sig n ].
445 #src #dst #sig #n #Hneq #Hsrc #Hdst
446 @(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
447 (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
449 (sem_rewind_strong ???? Hneq Hsrc Hdst)
450 (sem_move_multi … R ?))
452 [ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
453 cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
454 [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
455 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
456 >nth_current_chars >Hcurta_src normalize in ⊢ (%→?); #H destruct (H)
457 | #s #Hs lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
458 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
459 [ #Hcurta_dst #Hcomp #_ * #td * >Hcomp [| %2 %]
460 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
461 >nth_current_chars >nth_current_chars >Hs >Hcurta_dst
462 normalize in ⊢ (%→?); #H destruct (H)
464 cases (current_to_midtape … Hs) #ls * #rs #Hmidta_src >Hmidta_src
465 cases (current_to_midtape … Hs0) #ls0 * #rs0 #Hmidta_dst >Hmidta_dst
466 cases (true_or_false (s == s0)) #Hss0
467 [ lapply (\P Hss0) -Hss0 #Hss0 destruct (Hss0)
468 #_ #Hcomp cases (Hcomp ????? (refl ??) (refl ??)) -Hcomp [ *
469 [ * #rs' * #_ #Hcurtc_dst * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
470 >nth_current_chars >nth_current_chars >Hcurtc_dst
471 cases (current ? (nth src …))
472 [normalize in ⊢ (%→?); #H destruct (H)
473 | #x >nth_change_vec [|@Hdst] cases (reverse ? rs0)
474 [ normalize in ⊢ (%→?); #H destruct (H)
475 | #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ]
476 | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
477 >(?:nth src ? (current_chars ?? tc) (None ?) = None ?)
478 [|>nth_current_chars >Hcurtc_src >nth_change_vec_neq
479 [>nth_change_vec [cases (append ???) // | @Hsrc]
480 |@(not_to_not … Hneq) //
482 normalize in ⊢ (%→?); #H destruct (H) ]
483 | * #xs * #ci * #cj * #rs'' * #rs0' * * * #Hcicj #Hrs #Hrs0
484 #Htc * #td * * #Hmatch #Htd destruct (Htd) * #te * * *
485 >Htc >change_vec_commute [|//] >nth_change_vec [|//]
486 >change_vec_commute [|@sym_not_eq //] >nth_change_vec [|//]
487 cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb
489 [ #Hte #_ #_ <(reverse_reverse … ls) in Hte; <(reverse_reverse … lsa)
490 cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen'
491 @(list_cases2 … Hlen')
492 [ #H1 #H2 >H1 >H2 -H1 -H2 normalize in match (reverse ? [ ]); #Hte #_
493 lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
494 >change_vec_commute [|//] >change_vec_change_vec
495 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
496 >Hte whd in ⊢ (%→?); >change_vec_change_vec >nth_change_vec [|//]
497 >reverse_reverse #Htb
498 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)
499 [ >Htb @eq_f3 // cases (xs@cj::rs0') // ]
500 -Htb #Htb >Htb whd >nth_change_vec [|//]
501 >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec [|//]
502 >right_mk_tape [|cases xs [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H)]
503 normalize in match (left ??);
504 >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
505 whd in match (option_cons ???); >Hrs0
506 normalize in ⊢ (?(?%)%); //
507 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
508 >reverse_cons >reverse_cons #Hte
509 lapply (Hte ci hdb (reverse ? xs@s0::reverse ? tlb) rs'' ?
510 lsb cj hda (reverse ? xs@s0::reverse ? tla) rs0' ??)
511 [ /2 by cons_injective_l, nil/
512 | >length_append >length_append @eq_f @(eq_f ?? S)
513 >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
514 >length_reverse >length_reverse destruct (Hlen') //
515 | /2 by refl, trans_eq/ ] -Hte
516 #Hte #_ whd in ⊢ (%→?); #Htb
517 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
518 (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)
519 (midtape ? [ ] hdb (reverse sig (reverse sig xs@s0::reverse sig tlb)@ci::rs'')) src)
520 [ >Htb >Hte >nth_change_vec // >change_vec_change_vec >change_vec_commute [|//]
521 >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
522 >change_vec_change_vec >change_vec_commute [|//]
523 @eq_f3 // cases (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0') // ]
525 >nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
527 [| cases (reverse sig (reverse sig xs@s0::reverse sig tla))
528 [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
529 >Hmidta_src >Hmidta_dst
530 whd in match (left ??); whd in match (left ??); whd in match (right ??);
531 >current_mk_tape <opt_cons_tail_expand whd in match (option_cons ???);
532 >Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
533 >length_append >commutative_plus in match (|reverse ??| + ?);
534 whd in match (|?::?|); >length_reverse >length_reverse
535 <(length_reverse ? ls) <Hlen' >H1 normalize // ]
536 | #_ #Hte #_ <(reverse_reverse … ls0) in Hte; <(reverse_reverse … lsa)
537 cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen'
538 @(list_cases2 … Hlen')
539 [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #Hte
540 lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
541 >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
542 >change_vec_change_vec #Hte #_
543 >Hte whd in ⊢ (%→?); >nth_change_vec [|//] >reverse_reverse #Htb
544 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta (mk_tape ? [s0] (option_hd ? (xs@cj::rs0')) (tail ? (xs@cj::rs0'))) dst)
545 (midtape ? lsb s0 (xs@ci::rs'')) src)
546 [ >Htb >change_vec_change_vec >change_vec_commute [|//]
547 @eq_f3 // <Hrs0 cases rs0 // ]
548 -Htb #Htb >Htb whd >nth_change_vec [|//]
549 >nth_change_vec_neq [|//] >nth_change_vec [|//]
551 [| cases xs [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
552 normalize in match (left ??);
553 >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand >Hrs0
554 >length_append normalize >length_append >length_append
555 <(reverse_reverse ? lsa) >H1 normalize //
556 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
557 >reverse_cons >reverse_cons #Hte
558 lapply (Hte cj hdb (reverse ? xs@s0::reverse ? tlb) rs0' ?
559 lsb ci hda (reverse ? xs@s0::reverse ? tla) rs'' ??)
560 [ /2 by cons_injective_l, nil/
561 | >length_append >length_append @eq_f @(eq_f ?? S)
562 >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
563 >length_reverse >length_reverse destruct (Hlen') //
564 | /2 by refl, trans_eq/ ] -Hte
565 #Hte #_ whd in ⊢ (%→?); >Hte >nth_change_vec_neq [|//] >nth_change_vec [|//] #Htb
566 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
567 (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)
568 (midtape ? lsb hda (reverse sig (reverse sig xs@s0::reverse sig tla)@ci::rs'')) src)
569 [ >Htb >change_vec_change_vec >change_vec_commute [|//]
570 >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
571 >change_vec_change_vec >change_vec_commute [|//]
572 @eq_f3 // cases (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0') // ]
574 >nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
576 [| cases (reverse sig (reverse sig xs@s0::reverse sig tlb))
577 [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
578 >Hmidta_src >Hmidta_dst
579 whd in match (left ??); whd in match (left ??); whd in match (right ??);
580 >current_mk_tape <opt_cons_tail_expand
581 whd in match (option_cons ???);
582 >Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
583 >length_append >commutative_plus in match (|reverse ??| + ?);
584 whd in match (|?::?|); >length_reverse >length_reverse
585 <(length_reverse ? lsa) >Hlen' >H2 >length_append
590 | lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta)
591 [@Htc % % @(not_to_not ??? Hss0) #H destruct (H) %]
592 -Htc #Htc destruct (Htc) #_ * #td * whd in ⊢ (%→?); * #_
593 #Htd destruct (Htd) * #te * * * * >Hmidta_src >Hmidta_dst
594 cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb
596 [ <(reverse_reverse … ls) <(reverse_reverse … lsa)
597 cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen'
598 @(list_cases2 … Hlen')
599 [ #H1 #H2 >H1 >H2 -H1 -H2 #_ #_ normalize in match (reverse ? [ ]); #Hte #_
600 lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte)
601 whd in ⊢ (%→?); >Hmidta_dst #Htb
602 cut (tb = change_vec ?? ta (mk_tape ? (s0::lsa@lsb) (option_hd ? rs0) (tail ? rs0)) dst)
603 [ >Htb cases rs0 // ]
604 -Htb #Htb >Htb whd >nth_change_vec [|//]
605 >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
607 [| cases rs0 [ #_ %2 % | #x0 #xs0 normalize in ⊢ (??%?→?); #H destruct (H)] ]
608 normalize in match (left ??); normalize in match (right ??);
609 >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
611 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
612 >reverse_cons >reverse_cons >associative_append #Hte
613 lapply (Hte ???? (refl ??) ? s0 ? (reverse ? tla) ?? (refl ??))
614 [ >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
615 normalize #H destruct (H) // ] #Hte #_ #_ #_
616 whd in ⊢ (%→?); >Hte >change_vec_change_vec >nth_change_vec // #Htb
617 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
618 (mk_tape sig (hda::lsb) (option_hd ? (reverse sig (reverse sig tla)@s0::rs0)) (tail ? (reverse sig (reverse sig tla)@s0::rs0))) dst)
619 (midtape ? [ ] hdb (reverse sig (reverse sig tlb)@s::rs)) src)
620 [ >Htb >change_vec_commute [|//] @eq_f3 // cases (reverse sig (reverse sig tla)@s0::rs0) // ]
622 >nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
624 [| cases (reverse sig (reverse sig tla))
625 [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
626 >Hmidta_src >Hmidta_dst
627 whd in match (left ??); whd in match (left ??); whd in match (right ??);
628 >current_mk_tape <opt_cons_tail_expand >length_append
629 >length_reverse >length_reverse <(length_reverse ? ls) <Hlen'
631 | #_ <(reverse_reverse … ls0) <(reverse_reverse … lsa)
632 cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen'
633 @(list_cases2 … Hlen')
634 [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #_ #_ #Hte
635 lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte)
636 whd in ⊢ (%→?); #Htb whd >Hmidta_dst
637 cut (tb = change_vec (tape sig) (S n) ta (mk_tape ? (s0::ls0) (option_hd ? rs0) (tail ? rs0)) dst)
638 [ >Htb >Hmidta_dst cases rs0 // ]
639 -Htb #Htb >Htb whd >nth_change_vec [|//]
640 >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
641 >current_mk_tape >right_mk_tape
642 [| cases rs0 [ #_ %2 % | #x0 #xs0 normalize in ⊢ (??%?→?); #H destruct (H) ]]
643 normalize in ⊢ (??%); <opt_cons_tail_expand
645 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
646 >reverse_cons >reverse_cons #Hte #_ #_
647 lapply (Hte s0 hdb (reverse ? tlb) rs0 ?
648 lsb s hda (reverse ? tla) rs ??)
649 [ /2 by cons_injective_l, nil/
650 | >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
651 normalize #H destruct (H) //
652 | /2 by refl, trans_eq/ ] -Hte
653 #Hte whd in ⊢ (%→?); >Hte >nth_change_vec_neq [|//] >nth_change_vec [|//] #Htb
654 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
655 (mk_tape sig [hdb] (option_hd ? (reverse sig (reverse sig tlb)@s0::rs0)) (tail ? (reverse sig (reverse sig tlb)@s0::rs0))) dst)
656 (midtape ? lsb hda (reverse sig (reverse sig tla)@s::rs)) src)
657 [ >Htb >change_vec_commute [|//] >change_vec_change_vec
658 @eq_f3 // cases (reverse sig (reverse sig tlb)@s0::rs0) // ]
660 >nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
662 [| cases (reverse ? (reverse ? tlb)) [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
663 >Hmidta_src >Hmidta_dst
664 whd in match (left ??); whd in match (left ??); whd in match (right ??);
665 >current_mk_tape <opt_cons_tail_expand >length_append
666 normalize in ⊢ (??%); >length_append >reverse_reverse
667 <(length_reverse ? lsa) >Hlen' >H2 normalize //
673 | #ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd destruct (Htd)
674 whd in ⊢ (%→?); #Htb destruct (Htb) #ls #x #xs #Hta_src
675 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
676 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
677 [ #Hcurta_dst % % % // @Hcomp1 %2 //
678 | #x0 #Hcurta_dst cases (current_to_midtape … Hcurta_dst) -Hcurta_dst
679 #ls0 * #rs0 #Hta_dst cases (true_or_false (x == x0)) #Hxx0
680 [ lapply (\P Hxx0) -Hxx0 #Hxx0 destruct (Hxx0)
681 | >(?:tc=ta) in Htest;
682 [|@Hcomp1 % % >Hta_src >Hta_dst @(not_to_not ??? (\Pf Hxx0)) normalize
683 #Hxx0' destruct (Hxx0') % ]
685 >nth_current_chars >Hta_src >nth_current_chars >Hta_dst
686 whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1
687 cases (Hcomp2 … Hta_src Hta_dst) [ *
688 [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} %
690 | * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ]
691 | * #xs0 * #ci * #cj * #rs' * #rs0' * * *
692 #Hci #Hxs #Hrs0 #Htc @False_ind
694 >(?:nth src ? (current_chars ?? tc) (None ?) = Some ? ci) in Htest;
695 [|>nth_current_chars >Htc >nth_change_vec_neq [|@(not_to_not … Hneq) //]
697 >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj)
698 [|>nth_current_chars >Htc >nth_change_vec //]
699 normalize #H destruct (H) ] ] ]
702 (* 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. *)
703 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 ≝
704 match Hwf return (λa0,r.f b = a0 → WF B (λx,y:B. R (f x) (f y)) b) with
705 [ wf a Hwfa ⇒ λHeq.? ] (refl ??).
706 % #b1 #HRb @WF_to_WF_f @Hwfa <Heq @HRb
709 lemma lt_WF : ∀n.WF ? lt n.
710 #n @(nat_elim1 n) -n #n #IH % @IH
713 lemma terminate_match_m :
715 src ≠ dst → src < S n → dst < S n →
716 match_m src dst sig n ↓ t.
717 #src #dst #sig #n #ta #Hneq #Hsrc #Hdst
718 @(terminate_while … (sem_match_step_termination src dst sig n Hneq Hsrc Hdst)) //
719 letin f ≝ (λt0:Vector (tape sig) (S n).|left ? (nth src (tape ?) t0 (niltape ?))|
720 +|option_cons ? (current ? (nth dst (tape ?) t0 (niltape ?)))
721 (right ? (nth dst (tape ?) t0 (niltape ?)))|)
722 change with (λx,y.f x < f y) in ⊢ (??%?); @WF_to_WF_f @lt_WF
725 lemma sem_match_m : ∀src,dst,sig,n.
726 src ≠ dst → src < S n → dst < S n →
727 match_m src dst sig n \vDash R_match_m src dst sig n.
728 #src #dst #sig #n #Hneq #Hsrc #Hdst @WRealize_to_Realize [/2/| @wsem_match_m // ]