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 definition match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n.
21 match (nth src (option sig) v (None ?)) with
23 | Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ].
25 definition rewind ≝ λsrc,dst,sig,n.
26 parmove src dst sig n L · mmove src sig n R · mmove dst sig n R.
28 definition R_rewind_strong ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
30 nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
31 ∀ls0,y,y0,target,rs0.|xs| = |target| →
32 nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
34 (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src)
35 (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst) ∧
37 nth dst ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
38 ∀ls0,y,y0,target,rs0.|xs| = |target| →
39 nth src ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
41 (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) dst)
42 (midtape sig ls0 y0 (reverse ? target@y::rs0)) src) ∧
43 (∀x,rs.nth src ? int (niltape ?) = midtape sig [] x rs →
44 ∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 →
46 (∀x,rs.nth dst ? int (niltape ?) = midtape sig [] x rs →
47 ∀ls0,y,rs0.nth src ? int (niltape ?) = midtape sig ls0 y rs0 →
50 definition R_rewind ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
52 nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
53 ∀ls0,y,y0,target,rs0.|xs| = |target| →
54 nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
56 (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src)
57 (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst) ∧
58 (∀x,rs.nth src ? int (niltape ?) = midtape sig [] x rs →
59 ∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 →
63 theorem accRealize_to_Realize :
64 ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
65 M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse.
66 #sig #n #M #Rtrue #Rfalse #acc #HR #t
67 cases (HR t) #k * #outc * * #Hloop
68 #Htrue #Hfalse %{k} %{outc} % //
69 cases (true_or_false (cstate sig (states sig n M) n outc == acc)) #Hcase
70 [ % @Htrue @(\P Hcase) | %2 @Hfalse @(\Pf Hcase) ]
74 lemma sem_rewind_strong : ∀src,dst,sig,n.
75 src ≠ dst → src < S n → dst < S n →
76 rewind src dst sig n ⊨ R_rewind_strong src dst sig n.
77 #src #dst #sig #n #Hneq #Hsrc #Hdst
78 @(sem_seq_app sig n ????? (sem_parmoveL src dst sig n Hneq Hsrc Hdst) ?)
79 [| @(sem_seq_app sig n ????? (sem_move_multi … R ?) (sem_move_multi … R ?)) //
81 #ta #tb * #tc * * * #Htc1 #Htc2 #_ * #td * whd in ⊢ (%→%→?); #Htd #Htb % [ % [ %
82 [ #x #x0 #xs #rs #Hmidta_src #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_dst
83 >(Htc1 ??? Hmidta_src ls0 y (target@[y0]) rs0 ??) in Htd;
85 |>length_append >length_append >Hlen % ]
86 >change_vec_commute [|@sym_not_eq //]
87 >change_vec_change_vec
88 >nth_change_vec_neq [|@sym_not_eq //]
89 >nth_change_vec // >reverse_append >reverse_single
90 >reverse_append >reverse_single normalize in match (tape_move ???);
91 >rev_append_def >append_nil #Htd >Htd in Htb;
92 >change_vec_change_vec >nth_change_vec //
93 cases ls0 [|#l1 #ls1] normalize in match (tape_move ???); //
94 | #x #x0 #xs #rs #Hmidta_dst #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_src
95 >(Htc2 ??? Hmidta_dst ls0 y (target@[y0]) rs0 ??) in Htd;
97 |>length_append >length_append >Hlen % ]
98 >change_vec_change_vec
99 >change_vec_commute [|@sym_not_eq //]
101 >reverse_append >reverse_single
102 >reverse_append >reverse_single
103 cases ls0 [|#l1 #ls1] normalize in match (tape_move ???);
104 #Htd >Htd in Htb; >change_vec_change_vec >nth_change_vec //
105 >rev_append_def >change_vec_commute // normalize in match (tape_move ???); // ]
106 | #x #rs #Hmidta_src #ls0 #y #rs0 #Hmidta_dst
107 lapply (Htc1 … Hmidta_src … (refl ??) Hmidta_dst) -Htc1 #Htc >Htc in Htd;
108 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
109 >nth_change_vec_neq [|@sym_not_eq //]
110 >nth_change_vec // lapply (refl ? ls0) cases ls0 in ⊢ (???%→%);
111 [ #Hls0 #Htd >Htd in Htb;
112 >nth_change_vec // >change_vec_change_vec
113 whd in match (tape_move ???);whd in match (tape_move ???); <Hmidta_src
114 <Hls0 <Hmidta_dst >change_vec_same >change_vec_same //
115 | #l1 #ls1 #Hls0 #Htd >Htd in Htb;
116 >nth_change_vec // >change_vec_change_vec
117 whd in match (tape_move ???);whd in match (tape_move ???); <Hmidta_src
118 <Hls0 <Hmidta_dst >change_vec_same >change_vec_same //
120 | #x #rs #Hmidta_dst #ls0 #y #rs0 #Hmidta_src
121 lapply (Htc2 … Hmidta_dst … (refl ??) Hmidta_src) -Htc2 #Htc >Htc in Htd;
122 >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
123 >nth_change_vec // lapply (refl ? ls0) cases ls0 in ⊢ (???%→%);
124 [ #Hls0 destruct (Hls0) #Htd >Htd in Htb;
125 >nth_change_vec // >change_vec_change_vec
126 whd in match (tape_move ???);whd in match (tape_move ???);
127 <Hmidta_src <Hmidta_dst >change_vec_same >change_vec_same //
128 | #l1 #ls1 #Hls0 destruct (Hls0) #Htd >Htd in Htb;
129 >nth_change_vec // >change_vec_change_vec
130 whd in match (tape_move ???); whd in match (tape_move ???); <Hmidta_src
131 <Hmidta_dst >change_vec_same >change_vec_same //
136 lemma sem_rewind : ∀src,dst,sig,n.
137 src ≠ dst → src < S n → dst < S n →
138 rewind src dst sig n ⊨ R_rewind src dst sig n.
139 #src #dst #sig #n #Hneq #Hsrc #Hdst @(Realize_to_Realize … (sem_rewind_strong …)) //
140 #ta #tb * * * #H1 #H2 #H3 #H4 % /2 by /
143 definition match_step ≝ λsrc,dst,sig,n.
144 compare src dst sig n ·
145 (ifTM ?? (partest sig n (match_test src dst sig ?))
147 (rewind src dst sig n · (inject_TM ? (move_r ?) n dst)))
151 (* we assume the src is a midtape
153 if the dst is out of bounds (outt = int)
154 or dst.right is shorter than src.right (outt.current → None)
155 or src.right is a prefix of dst.right (out = just right of the common prefix) *)
156 definition R_match_step_false ≝
157 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
159 nth src ? int (niltape ?) = midtape sig ls x xs →
160 ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
161 (∃ls0,rs0,xs0. nth dst ? int (niltape ?) = midtape sig ls0 x rs0 ∧
164 (change_vec ?? int (mk_tape sig (reverse ? rs0@x::ls) (option_hd ? xs0) (tail ? xs0)) src)
165 (mk_tape ? (reverse ? rs0@x::ls0) (None ?) [ ]) dst) ∨
167 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
171 (change_vec ?? int (mk_tape sig (reverse ? xs@x::ls) (None ?) [ ]) src)
172 (mk_tape sig (reverse ? xs@x::ls0) (option_hd ? rs0) (tail ? rs0)) dst).
175 we assume the src is a midtape [ ] s rs
177 then dst.current = Some ? s1
178 and if s ≠ s1 then outt = int.dst.move_right()
180 then int.src.right and int.dst.right have a common prefix
181 and the heads of their suffixes are different
182 and outt = int.dst.move_right().
185 definition R_match_step_true ≝
186 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
187 ∀s,rs.nth src ? int (niltape ?) = midtape ? [ ] s rs →
188 outt = change_vec ?? int
189 (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst ∧
190 (∃s0.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s0 ∧
192 ∃xs,ci,rs',ls0,cj,rs0.
194 nth dst ? int (niltape ?) = midtape sig ls0 s (xs@cj::rs0) ∧
197 lemma sem_match_step :
198 ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
199 match_step src dst sig n ⊨
200 [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
201 R_match_step_true src dst sig n,
202 R_match_step_false src dst sig n ].
203 #src #dst #sig #n #Hneq #Hsrc #Hdst
204 @(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
205 (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
207 (sem_rewind ???? Hneq Hsrc Hdst)
208 (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
210 [ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
211 cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
212 [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
213 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
214 >nth_current_chars >Hcurta_src normalize in ⊢ (%→?); #H destruct (H)
215 | #s #Hs lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
216 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
217 [ #Hcurta_dst #Hcomp #_ * #td * >Hcomp [| %2 %]
218 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
219 >nth_current_chars >nth_current_chars >Hs >Hcurta_dst
220 normalize in ⊢ (%→?); #H destruct (H)
222 cases (current_to_midtape … Hs) #ls * #rs #Hmidta_src >Hmidta_src
223 cases (current_to_midtape … Hs0) #ls0 * #rs0 #Hmidta_dst >Hmidta_dst
224 cases (true_or_false (s == s0)) #Hss0
225 [ lapply (\P Hss0) -Hss0 #Hss0 destruct (Hss0)
226 #_ #Hcomp cases (Hcomp ????? (refl ??) (refl ??)) -Hcomp [ *
227 [ * #rs' * #_ #Hcurtc_dst * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
228 >nth_current_chars >nth_current_chars >Hcurtc_dst
229 cases (current ? (nth src …))
230 [normalize in ⊢ (%→?); #H destruct (H)
231 | #x >nth_change_vec // cases (reverse ? rs0)
232 [ normalize in ⊢ (%→?); #H destruct (H)
233 | #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ]
234 | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
235 >(?:nth src ? (current_chars ?? tc) (None ?) = None ?)
236 [|>nth_current_chars >Hcurtc_src >nth_change_vec_neq
237 [>nth_change_vec [cases (append ???) // | @Hsrc]
238 |@(not_to_not … Hneq) //
240 normalize in ⊢ (%→?); #H destruct (H) ]
241 | * #xs * #ci * #cj * #rs'' * #rs0' * * * #Hcicj #Hrs #Hrs0
242 #Htc * #td * * #Hmatch #Htd destruct (Htd) * #te * *
243 >Htc >change_vec_commute // >nth_change_vec //
244 >change_vec_commute [|@sym_not_eq //] >nth_change_vec // #Hte #_ #Htb
245 #s' #rs' >Hmidta_src #H destruct (H)
246 lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
247 >change_vec_commute // >change_vec_change_vec
248 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
249 >Hte in Htb; * * #_ >nth_change_vec // #Htb1
250 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2 %
251 [ @(eq_vec … (niltape ?)) #i #Hi
252 cases (true_or_false ((dst : DeqNat) == i)) #Hdsti
253 [ <(\P Hdsti) >Htb1 >nth_change_vec // >Hmidta_dst
254 >Hrs0 >reverse_reverse cases xs [|#r1 #rs1] %
255 | <Htb2 [|@(\Pf Hdsti)] >nth_change_vec_neq [| @(\Pf Hdsti)]
256 >Hrs0 >reverse_reverse >nth_change_vec_neq in ⊢ (???%);
257 <Hrs <Hmidta_src [|@(\Pf Hdsti)] >change_vec_same % ]
258 | >Hmidta_dst %{s'} % [%] #_
259 >Hrs0 %{xs} %{ci} %{rs''} %{ls0} %{cj} %{rs0'} % // % //
262 | lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta)
263 [@Htc % % @(not_to_not ??? Hss0) #H destruct (H) %]
264 -Htc #Htc destruct (Htc) #_ * #td * whd in ⊢ (%→?); * #_
265 #Htd destruct (Htd) * #te * * #_ #Hte * * #_ #Htb1 #Htb2
266 #s1 #rs1 >Hmidta_src #H destruct (H)
267 lapply (Hte … Hmidta_src … Hmidta_dst) -Hte #Hte destruct (Hte) %
268 [ @(eq_vec … (niltape ?)) #i #Hi
269 cases (true_or_false ((dst : DeqNat) == i)) #Hdsti
270 [ <(\P Hdsti) >(Htb1 … Hmidta_dst) >nth_change_vec // >Hmidta_dst
271 cases rs0 [|#r2 #rs2] %
272 | <Htb2 [|@(\Pf Hdsti)] >nth_change_vec_neq [| @(\Pf Hdsti)] % ]
273 | >Hs0 %{s0} % // #H destruct (H) @False_ind cases (Hss0) /2/ ]
277 | #ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd destruct (Htd)
278 whd in ⊢ (%→?); #Htb destruct (Htb) #ls #x #xs #Hta_src
279 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
280 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
281 [ #Hcurta_dst % % % // @Hcomp1 %2 //
282 | #x0 #Hcurta_dst cases (current_to_midtape … Hcurta_dst) -Hcurta_dst
283 #ls0 * #rs0 #Hta_dst cases (true_or_false (x == x0)) #Hxx0
284 [ lapply (\P Hxx0) -Hxx0 #Hxx0 destruct (Hxx0)
285 | >(?:tc=ta) in Htest;
286 [|@Hcomp1 % % >Hta_src >Hta_dst @(not_to_not ??? (\Pf Hxx0)) normalize
287 #Hxx0' destruct (Hxx0') % ]
289 >nth_current_chars >Hta_src >nth_current_chars >Hta_dst
290 whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1
291 cases (Hcomp2 … Hta_src Hta_dst) [ *
292 [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} %
294 | * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ]
295 | * #xs0 * #ci * #cj * #rs' * #rs0' * * *
296 #Hci #Hxs #Hrs0 #Htc @False_ind
298 >(?:nth src ? (current_chars ?? tc) (None ?) = Some ? ci) in Htest;
299 [|>nth_current_chars >Htc >nth_change_vec_neq [|@(not_to_not … Hneq) //]
301 >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj)
302 [|>nth_current_chars >Htc >nth_change_vec //]
303 normalize #H destruct (H) ] ] ]
306 definition match_m ≝ λsrc,dst,sig,n.
307 whileTM … (match_step src dst sig n)
308 (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
310 definition R_match_m ≝
311 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
313 nth src ? int (niltape ?) = midtape sig [ ] x rs →
314 (current sig (nth dst (tape sig) int (niltape sig)) = None ? →
315 right ? (nth dst (tape sig) int (niltape sig)) = [ ] → outt = int) ∧
317 nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
318 (∃l,l1.x0::rs0 = l@x::rs@l1 ∧
321 (mk_tape sig (reverse ? rs@[x]) (None ?) [ ]) src)
322 (mk_tape sig ((reverse ? (l@x::rs))@ls0) (option_hd ? l1) (tail ? l1)) dst) ∨
323 ∀l,l1.x0::rs0 ≠ l@x::rs@l1).
325 lemma not_sub_list_merge :
326 ∀T.∀a,b:list T. (∀l1.a ≠ b@l1) → (∀t,l,l1.a ≠ t::l@b@l1) → ∀l,l1.a ≠ l@b@l1.
327 #T #a #b #H1 #H2 #l elim l normalize //
330 lemma not_sub_list_merge_2 :
331 ∀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.
332 #T #a #b #t #H1 #H2 #l elim l //
333 #t0 #l1 #IH #l2 cases (true_or_false (t == t0)) #Htt0
334 [ >(\P Htt0) % normalize #H destruct (H) cases (H2 l1 l2) /2/
335 | normalize % #H destruct (H) cases (\Pf Htt0) /2/ ]
339 lemma wsem_match_m : ∀src,dst,sig,n.
340 src ≠ dst → src < S n → dst < S n →
341 match_m src dst sig n ⊫ R_match_m src dst sig n.
342 #src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
343 lapply (sem_while … (sem_match_step src dst sig n Hneq Hsrc Hdst) … Hloop) //
344 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
345 [ #Hfalse #x #xs #Hmid_src
346 cases (Hfalse … Hmid_src) -Hfalse
347 [(* current dest = None *) *
348 [ * #Hcur_dst #Houtc %
350 | #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
351 normalize in ⊢ (%→?); #H destruct (H)
353 | * #ls0 * #rs0 * #xs0 * * #Htc_dst #Hrs0 #HNone %
354 [ >Htc_dst normalize in ⊢ (%→?); #H destruct (H)
355 | #ls1 #x1 #rs1 >Htc_dst #H destruct (H)
356 >Hrs0 >HNone cases xs0
357 [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %]
360 | >reverse_append >reverse_cons >reverse_append
361 >associative_append >associative_append % ]
362 | #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs)
363 >length_append whd in ⊢ (??%(??%)→?); >length_append
364 >length_append normalize >commutative_plus whd in ⊢ (???%→?);
365 #H destruct (H) lapply e0 >(plus_n_O (|rs1|)) in ⊢ (??%?→?);
366 >associative_plus >associative_plus
367 #e1 lapply (injective_plus_r ??? e1) whd in ⊢ (???%→?);
372 |* #ls0 * #rs0 * #Hmid_dst #Houtc %
373 [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
374 |#ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
375 %1 %{[ ]} %{rs0} % [%]
376 >reverse_cons >associative_append >Houtc %
379 |-ta #ta #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
381 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
382 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
384 [#Hcurta_dst #Hrightta_dst whd in Htrue; >Hmidta_src in Htrue; #Htrue
385 cases (Htrue ?? (refl ??)) -Htrue #Htc
387 [ >Htc whd in match (tape_move_mono ???); whd in match (tape_write ???);
388 <(change_vec_same … ta dst (niltape ?)) in ⊢ (???%);
389 lapply Hrightta_dst lapply Hcurta_dst -Hrightta_dst -Hcurta_dst
390 cases (nth dst ? ta (niltape ?))
392 | #r0 #rs0 #_ normalize in ⊢ (%→?); #H destruct (H)
394 | #ls #x0 #rs normalize in ⊢ (%→?); #H destruct (H) ] ]
395 -Htc #Htc destruct (Htc) #_
396 cases (IH … Hmidta_src) #Houtc #_ @Houtc //
397 |#ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst;
398 normalize in ⊢ (%→?); #H destruct (H)
400 | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
401 #ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst; normalize in ⊢ (%→?);
402 #H destruct (H) whd in Htrue; >Hmidta_src in Htrue; #Htrue
403 cases (Htrue ?? (refl …)) -Htrue >Hmidta_dst #Htc
404 cases (true_or_false (x==c)) #eqx
405 [ lapply (\P eqx) -eqx #eqx destruct (eqx) * #s0 * whd in ⊢ (??%?→?); #Hs0
406 destruct (Hs0) #Htrue cases (Htrue (refl ??)) -Htrue
407 #xs0 * #ci * #rs' * #ls1 * #cj * #rs1 * * #Hxs #H destruct (H) #Hcicj
408 >Htc in IH; whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
409 #IH cases (IH … Hmidta_src) -IH #_ >nth_change_vec //
410 cut (∃x1,xs1.xs0@cj::rs1 = x1::xs1)
411 [ cases xs0 [ %{cj} %{rs1} % | #x1 #xs1 %{x1} %{(xs1@cj::rs1)} % ] ] * #x1 * #xs1
412 #Hxs1 >Hxs1 #IH cases (IH … (refl ??)) -IH
413 [ * #l * #l1 * #Hxs1'
414 >change_vec_commute // >change_vec_change_vec
415 #Houtc % %{(s0::l)} %{l1} %
417 | >reverse_cons >associative_append >change_vec_commute // @Houtc ]
418 | #H %2 #l #l1 >(?:l@s0::xs@l1 = l@(s0::xs)@l1) [|%]
420 [ #l2 >Hxs <Hxs1 % normalize #H1 lapply (cons_injective_r ????? H1)
421 >associative_append #H2 lapply (append_l2_injective ????? (refl ??) H2)
422 #H3 lapply (cons_injective_l ????? H3) #H3 >H3 in Hcicj; * /2/
423 |#t #l2 #l3 % normalize #H1 lapply (cons_injective_r ????? H1)
424 -H1 #H1 cases (H l2 l3) #H2 @H2 @H1
427 | #_ cases (IH x xs ?) -IH
428 [| >Htc >nth_change_vec_neq [|@sym_not_eq //] @Hmidta_src ]
429 >Htc >nth_change_vec // cases rs0
430 [ #_ #_ %2 #l #l1 cases l
433 [ normalize % #H destruct (H) cases (\Pf eqx) /2/
434 | #tmp1 #l2 normalize % #H destruct (H) ]
435 | #tmp1 #l2 normalize % #H destruct (H) ]
436 | #tmp1 #l2 normalize % #H destruct (H)cases l2 in e0;
437 [ normalize #H1 destruct (H1)
438 | #tmp2 #l3 normalize #H1 destruct (H1) ] ]
439 | #r1 #rs1 #_ #IH cases (IH … (refl ??)) -IH
440 [ * #l * #l1 * #Hll1 #Houtc % %{(c::l)} %{l1} % [ >Hll1 % ]
441 >Houtc >change_vec_commute // >change_vec_change_vec
442 >change_vec_commute [|@sym_not_eq //]
443 >reverse_cons >associative_append %
444 | #Hll1 %2 @(not_sub_list_merge_2 ?? (x::xs)) normalize [|@Hll1]
445 #l1 % #H destruct (H) cases (\Pf eqx) /2/
453 axiom daemon : ∀P:Prop.P.
455 (* XXX: move to turing (or mono) *)
456 definition option_cons ≝ λsig.λc:option sig.λl.
457 match c with [ None ⇒ l | Some c0 ⇒ c0::l ].
459 definition R_match_step_true_naive ≝
460 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
461 |left ? (nth src ? outt (niltape ?))| +
462 |option_cons ? (current ? (nth dst ? outt (niltape ?))) (right ? (nth dst ? outt (niltape ?)))| <
463 |left ? (nth src ? int (niltape ?))| +
464 |option_cons ? (current ? (nth dst ? int (niltape ?))) (right ? (nth dst ? int (niltape ?)))|.
466 axiom right_mk_tape : ∀sig,ls,c,rs.right ? (mk_tape sig ls c rs) = rs.
467 axiom left_mk_tape : ∀sig,ls,c,rs.left ? (mk_tape sig ls c rs) = ls.
468 axiom current_mk_tape : ∀sig,ls,c,rs.current ? (mk_tape sig ls c rs) = c.
469 axiom length_tail : ∀A,l.0 < |l| → |tail A l| < |l|.
470 axiom lists_length_split :
471 ∀A.∀l1,l2:list A.(∃la,lb.(|la| = |l1| ∧ l2 = la@lb) ∨ (|la| = |l2| ∧ l1 = la@lb)).
472 axiom opt_cons_tail_expand : ∀A,l.l = option_cons A (option_hd ? l) (tail ? l).
474 lemma sem_match_step_termination :
475 ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
476 match_step src dst sig n ⊨
477 [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
478 R_match_step_true_naive src dst sig n,
479 R_match_step_false src dst sig n ].
480 #src #dst #sig #n #Hneq #Hsrc #Hdst
481 @(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
482 (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
484 (sem_rewind_strong ???? Hneq Hsrc Hdst)
485 (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
487 [ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
488 cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
489 [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
490 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
491 >nth_current_chars >Hcurta_src normalize in ⊢ (%→?); #H destruct (H)
492 | #s #Hs lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
493 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
494 [ #Hcurta_dst #Hcomp #_ * #td * >Hcomp [| %2 %]
495 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
496 >nth_current_chars >nth_current_chars >Hs >Hcurta_dst
497 normalize in ⊢ (%→?); #H destruct (H)
499 cases (current_to_midtape … Hs) #ls * #rs #Hmidta_src >Hmidta_src
500 cases (current_to_midtape … Hs0) #ls0 * #rs0 #Hmidta_dst >Hmidta_dst
501 cases (true_or_false (s == s0)) #Hss0
502 [ lapply (\P Hss0) -Hss0 #Hss0 destruct (Hss0)
503 #_ #Hcomp cases (Hcomp ????? (refl ??) (refl ??)) -Hcomp [ *
504 [ * #rs' * #_ #Hcurtc_dst * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
505 >nth_current_chars >nth_current_chars >Hcurtc_dst
506 cases (current ? (nth src …))
507 [normalize in ⊢ (%→?); #H destruct (H)
508 | #x >nth_change_vec // cases (reverse ? rs0)
509 [ normalize in ⊢ (%→?); #H destruct (H)
510 | #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ]
511 | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
512 >(?:nth src ? (current_chars ?? tc) (None ?) = None ?)
513 [|>nth_current_chars >Hcurtc_src >nth_change_vec_neq
514 [>nth_change_vec [cases (append ???) // | @Hsrc]
515 |@(not_to_not … Hneq) //
517 normalize in ⊢ (%→?); #H destruct (H) ]
518 | * #xs * #ci * #cj * #rs'' * #rs0' * * * #Hcicj #Hrs #Hrs0
519 #Htc * #td * * #Hmatch #Htd destruct (Htd) * #te * * *
520 >Htc >change_vec_commute // >nth_change_vec //
521 >change_vec_commute [|@sym_not_eq //] >nth_change_vec //
522 cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb
524 [ #Hte #_ #_ <(reverse_reverse … ls) in Hte; <(reverse_reverse … lsa)
525 cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen'
526 @(list_cases2 … Hlen')
527 [ #H1 #H2 >H1 >H2 -H1 -H2 normalize in match (reverse ? [ ]); #Hte #_
528 lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
529 >change_vec_commute // >change_vec_change_vec
530 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
531 >Hte * * #_ >nth_change_vec // >reverse_reverse
532 #H lapply (H … (refl ??)) -H #Htb1 #Htb2
533 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)
534 [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
535 >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
536 >right_mk_tape normalize in match (left ??);
537 >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
538 whd in match (option_cons ???); >Hrs0
539 normalize in ⊢ (?(?%)%); //
540 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
541 >reverse_cons >reverse_cons #Hte
542 lapply (Hte ci hdb (reverse ? xs@s0::reverse ? tlb) rs'' ?
543 lsb cj hda (reverse ? xs@s0::reverse ? tla) rs0' ??)
544 [ /2 by cons_injective_l, nil/
545 | >length_append >length_append @eq_f @(eq_f ?? S)
546 >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
547 >length_reverse >length_reverse destruct (Hlen') //
548 | /2 by refl, trans_eq/ ] -Hte
549 #Hte #_ * * #_ >Hte >nth_change_vec // #Htb1 #Htb2
550 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
551 (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)
552 (midtape ? [ ] hdb (reverse sig (reverse sig xs@s0::reverse sig tlb)@ci::rs'')) src)
553 [@daemon] -Htb1 -Htb2 #Htb >Htb whd
554 >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
555 >right_mk_tape >Hmidta_src >Hmidta_dst
556 whd in match (left ??); whd in match (left ??); whd in match (right ??);
557 >current_mk_tape <opt_cons_tail_expand whd in match (option_cons ???);
558 >Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
559 >length_append >commutative_plus in match (|reverse ??| + ?);
560 whd in match (|?::?|); >length_reverse >length_reverse
561 <(length_reverse ? ls) <Hlen' >H1 normalize // ]
562 | #_ #Hte #_ <(reverse_reverse … ls0) in Hte; <(reverse_reverse … lsa)
563 cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen'
564 @(list_cases2 … Hlen')
565 [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #Hte
566 lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
567 >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
568 >change_vec_change_vec #Hte #_
569 >Hte * * #_ >nth_change_vec // >reverse_reverse
570 #H lapply (H … (refl ??)) -H #Htb1 #Htb2
571 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta (mk_tape ? [s0] (option_hd ? (xs@cj::rs0')) (tail ? (xs@cj::rs0'))) dst)
572 (midtape ? lsb s0 (xs@ci::rs'')) src)
573 [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
574 >nth_change_vec_neq // >nth_change_vec //
575 >right_mk_tape normalize in match (left ??);
576 >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand >Hrs0
577 >length_append normalize >length_append >length_append
578 <(reverse_reverse ? lsa) >H1 normalize //
579 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
580 >reverse_cons >reverse_cons #Hte
581 lapply (Hte cj hdb (reverse ? xs@s0::reverse ? tlb) rs0' ?
582 lsb ci hda (reverse ? xs@s0::reverse ? tla) rs'' ??)
583 [ /2 by cons_injective_l, nil/
584 | >length_append >length_append @eq_f @(eq_f ?? S)
585 >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
586 >length_reverse >length_reverse destruct (Hlen') //
587 | /2 by refl, trans_eq/ ] -Hte
588 #Hte #_ * * #_ >Hte >nth_change_vec_neq // >nth_change_vec // #Htb1 #Htb2
589 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
590 (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)
591 (midtape ? lsb hda (reverse sig (reverse sig xs@s0::reverse sig tla)@ci::rs'')) src)
592 [@daemon] -Htb1 -Htb2 #Htb >Htb whd
593 >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
594 >right_mk_tape >Hmidta_src >Hmidta_dst
595 whd in match (left ??); whd in match (left ??); whd in match (right ??);
596 >current_mk_tape <opt_cons_tail_expand
597 whd in match (option_cons ???);
598 >Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
599 >length_append >commutative_plus in match (|reverse ??| + ?);
600 whd in match (|?::?|); >length_reverse >length_reverse
601 <(length_reverse ? lsa) >Hlen' >H2 >length_append
606 | lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta)
607 [@Htc % % @(not_to_not ??? Hss0) #H destruct (H) %]
608 -Htc #Htc destruct (Htc) #_ * #td * whd in ⊢ (%→?); * #_
609 #Htd destruct (Htd) * #te * * * * >Hmidta_src >Hmidta_dst
610 cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb
612 [ <(reverse_reverse … ls) <(reverse_reverse … lsa)
613 cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen'
614 @(list_cases2 … Hlen')
615 [ #H1 #H2 >H1 >H2 -H1 -H2 #_ #_ normalize in match (reverse ? [ ]); #Hte #_
616 lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte) * * #_
617 >Hmidta_dst #Htb1 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2
618 cut (tb = change_vec ?? ta (mk_tape ? (s0::lsa@lsb) (option_hd ? rs0) (tail ? rs0)) dst)
619 [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
620 >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
621 >right_mk_tape normalize in match (left ??); normalize in match (right ??);
622 >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
624 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
625 >reverse_cons >reverse_cons >associative_append #Hte
626 lapply (Hte ???? (refl ??) ? s0 ? (reverse ? tla) ?? (refl ??))
627 [ >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
628 normalize #H destruct (H) // ] #Hte #_ #_ #_
629 * * #_ >Hte >nth_change_vec // #Htb1 #Htb2
630 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
631 (mk_tape sig (hda::lsb) (option_hd ? (reverse sig (reverse sig tla)@s0::rs0)) (tail ? (reverse sig (reverse sig tla)@s0::rs0))) dst)
632 (midtape ? [ ] hdb (reverse sig (reverse sig tlb)@s::rs)) src)
633 [@daemon] -Htb1 -Htb2 #Htb >Htb whd
634 >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
635 >right_mk_tape >Hmidta_src >Hmidta_dst
636 whd in match (left ??); whd in match (left ??); whd in match (right ??);
637 >current_mk_tape <opt_cons_tail_expand >length_append
638 >length_reverse >length_reverse <(length_reverse ? ls) <Hlen'
640 | #_ <(reverse_reverse … ls0) <(reverse_reverse … lsa)
641 cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen'
642 @(list_cases2 … Hlen')
643 [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #_ #_ #Hte
644 lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte)
645 * * #_ >Hmidta_dst #Htb1 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2
646 cut (tb = change_vec (tape sig) (S n) ta (mk_tape ? (s0::ls0) (option_hd ? rs0) (tail ? rs0)) dst)
647 [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
648 >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
649 >current_mk_tape >right_mk_tape normalize in ⊢ (??%); <opt_cons_tail_expand
651 | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
652 >reverse_cons >reverse_cons #Hte #_ #_
653 lapply (Hte s0 hdb (reverse ? tlb) rs0 ?
654 lsb s hda (reverse ? tla) rs ??)
655 [ /2 by cons_injective_l, nil/
656 | >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
657 normalize #H destruct (H) //
658 | /2 by refl, trans_eq/ ] -Hte
659 #Hte * * #_ >Hte >nth_change_vec_neq // >nth_change_vec // #Htb1 #Htb2
660 cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
661 (mk_tape sig [hdb] (option_hd ? (reverse sig (reverse sig tlb)@s0::rs0)) (tail ? (reverse sig (reverse sig tlb)@s0::rs0))) dst)
662 (midtape ? lsb hda (reverse sig (reverse sig tla)@s::rs)) src)
663 [@daemon] -Htb1 -Htb2 #Htb >Htb whd
664 >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
665 >right_mk_tape >Hmidta_src >Hmidta_dst
666 whd in match (left ??); whd in match (left ??); whd in match (right ??);
667 >current_mk_tape <opt_cons_tail_expand >length_append
668 normalize in ⊢ (??%); >length_append >reverse_reverse
669 <(length_reverse ? lsa) >Hlen' >H2 normalize //
675 | #ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd destruct (Htd)
676 whd in ⊢ (%→?); #Htb destruct (Htb) #ls #x #xs #Hta_src
677 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
678 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
679 [ #Hcurta_dst % % % // @Hcomp1 %2 //
680 | #x0 #Hcurta_dst cases (current_to_midtape … Hcurta_dst) -Hcurta_dst
681 #ls0 * #rs0 #Hta_dst cases (true_or_false (x == x0)) #Hxx0
682 [ lapply (\P Hxx0) -Hxx0 #Hxx0 destruct (Hxx0)
683 | >(?:tc=ta) in Htest;
684 [|@Hcomp1 % % >Hta_src >Hta_dst @(not_to_not ??? (\Pf Hxx0)) normalize
685 #Hxx0' destruct (Hxx0') % ]
687 >nth_current_chars >Hta_src >nth_current_chars >Hta_dst
688 whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1
689 cases (Hcomp2 … Hta_src Hta_dst) [ *
690 [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} %
692 | * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ]
693 | * #xs0 * #ci * #cj * #rs' * #rs0' * * *
694 #Hci #Hxs #Hrs0 #Htc @False_ind
696 >(?:nth src ? (current_chars ?? tc) (None ?) = Some ? ci) in Htest;
697 [|>nth_current_chars >Htc >nth_change_vec_neq [|@(not_to_not … Hneq) //]
699 >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj)
700 [|>nth_current_chars >Htc >nth_change_vec //]
701 normalize #H destruct (H) ] ] ]
704 (* 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. *)
705 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 ≝
706 match Hwf return (λa0,r.f b = a0 → WF B (λx,y:B. R (f x) (f y)) b) with
707 [ wf a Hwfa ⇒ λHeq.? ] (refl ??).
708 % #b1 #HRb @WF_to_WF_f @Hwfa <Heq @HRb
711 lemma lt_WF : ∀n.WF ? lt n.
712 #n @(nat_elim1 n) -n #n #IH % @IH
715 lemma terminate_match_m :
717 src ≠ dst → src < S n → dst < S n →
718 match_m src dst sig n ↓ t.
719 #src #dst #sig #n #ta #Hneq #Hsrc #Hdst
720 @(terminate_while … (sem_match_step_termination src dst sig n Hneq Hsrc Hdst)) //
721 letin f ≝ (λt0:Vector (tape sig) (S n).|left ? (nth src (tape ?) t0 (niltape ?))|
722 +|option_cons ? (current ? (nth dst (tape ?) t0 (niltape ?)))
723 (right ? (nth dst (tape ?) t0 (niltape ?)))|)
724 change with (λx,y.f x < f y) in ⊢ (??%?); @WF_to_WF_f @lt_WF
727 lemma sem_match_m : ∀src,dst,sig,n.
728 src ≠ dst → src < S n → dst < S n →
729 match_m src dst sig n \vDash R_match_m src dst sig n.
730 #src #dst #sig #n #Hneq #Hsrc #Hdst @WRealize_to_Realize [/2/| @wsem_match_m // ]