2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
8 \ / This file is distributed under the terms of the
9 \ / GNU General Public License Version 2
10 V_____________________________________________________________*)
12 include "turing/turing.ma".
13 include "turing/inject.ma".
14 include "turing/while_multi.ma".
15 include "turing/while_machine.ma".
16 include "turing/simple_machines.ma".
17 include "turing/if_machine.ma".
19 definition parmove_states ≝ initN 3.
21 definition parmove0 : parmove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
22 definition parmove1 : parmove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
23 definition parmove2 : parmove_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
27 src: a b c ... z ---→ a b c ... z
30 dst: _ _ _ ... _ ---→ a b c ... z
33 0) (x,_) → (x,_)(R,R) → 1
40 definition trans_parmove_step ≝
42 λp:parmove_states × (Vector (option sig) (S n)).
45 [ O ⇒ match nth src ? a (None ?) with
46 [ None ⇒ 〈parmove2,null_action sig n〉
47 | Some a0 ⇒ match nth dst ? a (None ?) with
48 [ None ⇒ 〈parmove2,null_action ? n〉
49 | Some a1 ⇒ 〈parmove1,change_vec ? (S n)
51 (null_action ? n) (〈None ?,D〉) src)
54 [ O ⇒ (* 1 *) 〈parmove1,null_action ? n〉
55 | S _ ⇒ (* 2 *) 〈parmove2,null_action ? n〉 ] ].
57 definition parmove_step ≝
59 mk_mTM sig n parmove_states (trans_parmove_step src dst sig n D)
60 parmove0 (λq.q == parmove1 ∨ q == parmove2).
62 definition R_parmove_step_true ≝
63 λsrc,dst,sig,n,D.λint,outt: Vector (tape sig) (S n).
65 current ? (nth src ? int (niltape ?)) = Some ? x1 ∧
66 current ? (nth dst ? int (niltape ?)) = Some ? x2 ∧
69 (tape_move ? (nth src ? int (niltape ?)) D) src)
70 (tape_move ? (nth dst ? int (niltape ?)) D) dst.
72 definition R_parmove_step_false ≝
73 λsrc,dst:nat.λsig,n.λint,outt: Vector (tape sig) (S n).
74 (current ? (nth src ? int (niltape ?)) = None ? ∨
75 current ? (nth dst ? int (niltape ?)) = None ?) ∧
78 lemma parmove_q0_q2_null_src :
79 ∀src,dst,sig,n,D,v.src < S n → dst < S n →
80 nth src ? (current_chars ?? v) (None ?) = None ? →
81 step sig n (parmove_step src dst sig n D)
82 (mk_mconfig ??? parmove0 v) =
83 mk_mconfig ??? parmove2 v.
84 #src #dst #sig #n #D #v #Hsrc #Hdst #Hcurrent
85 whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?);
87 [ whd in ⊢ (??(???%)?); >Hcurrent %
88 | whd in ⊢ (??(????(???%))?); >Hcurrent @tape_move_null_action ]
91 lemma parmove_q0_q2_null_dst :
92 ∀src,dst,sig,n,D,v,s.src < S n → dst < S n →
93 nth src ? (current_chars ?? v) (None ?) = Some ? s →
94 nth dst ? (current_chars ?? v) (None ?) = None ? →
95 step sig n (parmove_step src dst sig n D)
96 (mk_mconfig ??? parmove0 v) =
97 mk_mconfig ??? parmove2 v.
98 #src #dst #sig #n #D #v #s #Hsrc #Hdst #Hcursrc #Hcurdst
99 whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?);
101 [ whd in ⊢ (??(???%)?); >Hcursrc whd in ⊢ (??(???%)?); >Hcurdst %
102 | whd in ⊢ (??(????(???%))?); >Hcursrc
103 whd in ⊢ (??(????(???%))?); >Hcurdst @tape_move_null_action ]
106 lemma parmove_q0_q1 :
107 ∀src,dst,sig,n,D,v.src ≠ dst → src < S n → dst < S n →
109 nth src ? (current_chars ?? v) (None ?) = Some ? a1 →
110 nth dst ? (current_chars ?? v) (None ?) = Some ? a2 →
111 step sig n (parmove_step src dst sig n D)
112 (mk_mconfig ??? parmove0 v) =
113 mk_mconfig ??? parmove1
116 (tape_move ? (nth src ? v (niltape ?)) D) src)
117 (tape_move ? (nth dst ? v (niltape ?)) D) dst).
118 #src #dst #sig #n #D #v #Hneq #Hsrc #Hdst
119 #a1 #a2 #Hcursrc #Hcurdst
120 whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
121 [ whd in match (trans ????);
123 | whd in match (trans ????);
124 >Hcursrc >Hcurdst whd in ⊢ (??(????(???%))?);
125 >tape_move_multi_def <(change_vec_same ?? v dst (niltape ?)) in ⊢ (??%?);
126 >pmap_change <(change_vec_same ?? v src (niltape ?)) in ⊢(??%?);
127 >pmap_change <tape_move_multi_def >tape_move_null_action
128 @eq_f2 // >nth_change_vec_neq //
132 lemma sem_parmove_step :
133 ∀src,dst,sig,n,D.src ≠ dst → src < S n → dst < S n →
134 parmove_step src dst sig n D ⊨
135 [ parmove1: R_parmove_step_true src dst sig n D,
136 R_parmove_step_false src dst sig n ].
137 #src #dst #sig #n #D #Hneq #Hsrc #Hdst #int
138 lapply (refl ? (current ? (nth src ? int (niltape ?))))
139 cases (current ? (nth src ? int (niltape ?))) in ⊢ (???%→?);
142 [ whd in ⊢ (??%?); >parmove_q0_q2_null_src /2/
143 | normalize in ⊢ (%→?); #H destruct (H) ]
145 | #a #Ha lapply (refl ? (current ? (nth dst ? int (niltape ?))))
146 cases (current ? (nth dst ? int (niltape ?))) in ⊢ (???%→?);
149 [ whd in ⊢ (??%?); >(parmove_q0_q2_null_dst …) /2/
150 | normalize in ⊢ (%→?); #H destruct (H) ]
154 [whd in ⊢ (??%?); >(parmove_q0_q1 … Hneq Hsrc Hdst ? b ??)
155 [2: <(nth_vec_map ?? (current …) dst ? int (niltape ?)) //
156 |3: <(nth_vec_map ?? (current …) src ? int (niltape ?)) //
158 | #_ %{a} %{b} % // % // ]
159 | * #H @False_ind @H % ]
163 definition parmove ≝ λsrc,dst,sig,n,D.
164 whileTM … (parmove_step src dst sig n D) parmove1.
166 definition R_parmoveL ≝
167 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
169 nth src ? int (niltape ?) = midtape sig xs x rs →
170 ∀ls0,x0,target,rs0.|xs| = |target| →
171 nth dst ? int (niltape ?) = midtape sig (target@ls0) x0 rs0 →
173 (change_vec ?? int (mk_tape sig [] (None ?) (reverse ? xs@x::rs)) src)
174 (mk_tape sig (tail ? ls0) (option_hd ? ls0) (reverse ? target@x0::rs0)) dst) ∧
176 nth dst ? int (niltape ?) = midtape sig xs x rs →
177 ∀ls0,x0,target,rs0.|xs| = |target| →
178 nth src ? int (niltape ?) = midtape sig (target@ls0) x0 rs0 →
180 (change_vec ?? int (mk_tape sig [] (None ?) (reverse ? xs@x::rs)) dst)
181 (mk_tape sig (tail ? ls0) (option_hd ? ls0) (reverse ? target@x0::rs0)) src) ∧
182 ((current ? (nth src ? int (niltape ?)) = None ? ∨
183 current ? (nth dst ? int (niltape ?)) = None ?) →
186 lemma wsem_parmoveL : ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
187 parmove src dst sig n L ⊫ R_parmoveL src dst sig n.
188 #src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
189 lapply (sem_while … (sem_parmove_step src dst sig n L Hneq Hsrc Hdst) … Hloop) //
190 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
191 [ whd in ⊢ (%→?); * #H #Houtc % [2: #_ @Houtc ] cases H #Hcurtb
193 [ #x #xs #rs #Hsrctb >Hsrctb in Hcurtb; normalize in ⊢ (%→?);
194 #Hfalse destruct (Hfalse)
195 | #x #xs #rs #Hdsttb #ls0 #x0 #target #rs0 #Hlen #Hsrctb >Hsrctb in Hcurtb;
196 normalize in ⊢ (%→?); #H destruct (H)
199 [ #x #xs #rs #Hsrctb #ls0 #x0 #target
200 #rs0 #Hlen #Hdsttb >Hdsttb in Hcurtb; normalize in ⊢ (%→?); #H destruct (H)
201 | #x #xs #rs #Hdsttb >Hdsttb in Hcurtb; normalize in ⊢ (%→?);
202 #Hfalse destruct (Hfalse)
205 | #td #te * #c0 * #c1 * * #Hc0 #Hc1 #Hd #Hstar #IH #He
206 lapply (IH He) -IH * * #IH1a #IH1b #IH2 % [ %
207 [ #x #xs #rs #Hsrc_td #ls0 #x0 #target
209 >Hsrc_td in Hc0; normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
210 >Hdst_td in Hd; >Hsrc_td @(list_cases2 … Hlen)
211 [ #Hxsnil #Htargetnil >Hxsnil >Htargetnil #Hd >IH2
212 [2: %1 >Hd >nth_change_vec_neq [|@(sym_not_eq … Hneq)]
214 >Hd -Hd @(eq_vec … (niltape ?))
215 #i #Hi cases (decidable_eq_nat i src) #Hisrc
216 [ >Hisrc >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)]
218 >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)]
220 | cases (decidable_eq_nat i dst) #Hidst
221 [ >Hidst >nth_change_vec // >nth_change_vec //
222 >Hdst_td in Hc1; >Htargetnil
223 normalize in ⊢ (%→?); #Hc1 destruct (Hc1) cases ls0 //
224 | >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
225 >nth_change_vec_neq [|@(sym_not_eq … Hisrc)]
226 >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
227 >nth_change_vec_neq [|@(sym_not_eq … Hisrc)] %
230 | #hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd
231 >(IH1a hd1 tl1 (c0::rs) ? ls0 hd2 tl2 (x0::rs0))
232 [ >Hd >(change_vec_commute … ?? td ?? src dst) //
233 >change_vec_change_vec
234 >(change_vec_commute … ?? td ?? dst src) [|@sym_not_eq //]
235 >change_vec_change_vec
236 >reverse_cons >associative_append
237 >reverse_cons >associative_append %
238 | >Hd >nth_change_vec //
239 | >Hxs in Hlen; >Htarget normalize #Hlen destruct (Hlen) //
240 | >Hd >nth_change_vec_neq [|@sym_not_eq //]
243 | #x #xs #rs #Hdst_td #ls0 #x0 #target
245 >Hdst_td in Hc0; normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
246 >Hsrc_td in Hd; >Hdst_td @(list_cases2 … Hlen)
247 [ #Hxsnil #Htargetnil >Hxsnil >Htargetnil #Hd >IH2
248 [2: %2 >Hd >nth_change_vec //]
249 >Hd -Hd @(eq_vec … (niltape ?))
250 #i #Hi cases (decidable_eq_nat i dst) #Hidst
251 [ >Hidst >(nth_change_vec_neq … dst src) //
252 >nth_change_vec // >nth_change_vec //
253 | cases (decidable_eq_nat i src) #Hisrc
254 [ >Hisrc >nth_change_vec // >(nth_change_vec_neq …) [|@sym_not_eq //]
255 >Hsrc_td in Hc1; >Htargetnil
256 normalize in ⊢ (%→?); #Hc1 destruct (Hc1) >nth_change_vec //
258 | >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
259 >nth_change_vec_neq [|@(sym_not_eq … Hisrc)]
260 >nth_change_vec_neq [|@(sym_not_eq … Hisrc)]
261 >nth_change_vec_neq [|@(sym_not_eq … Hidst)] %
264 | #hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd
265 >(IH1b hd1 tl1 (x::rs) ? ls0 hd2 tl2 (x0::rs0))
266 [ >Hd >(change_vec_commute … ?? td ?? dst src) [|@sym_not_eq //]
267 >change_vec_change_vec
268 >(change_vec_commute … ?? td ?? src dst) //
269 >change_vec_change_vec
270 >reverse_cons >associative_append
271 >reverse_cons >associative_append
272 >change_vec_commute [|@sym_not_eq //] %
273 | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
274 | >Hxs in Hlen; >Htarget normalize #Hlen destruct (Hlen) //
275 | >Hd >nth_change_vec // ]
278 | >Hc0 >Hc1 * [ #Hc0 destruct (Hc0) | #Hc1 destruct (Hc1) ]
282 lemma terminate_parmoveL : ∀src,dst,sig,n,t.
283 src ≠ dst → src < S n → dst < S n →
284 parmove src dst sig n L ↓ t.
285 #src #dst #sig #n #t #Hneq #Hsrc #Hdst
286 @(terminate_while … (sem_parmove_step …)) //
287 <(change_vec_same … t src (niltape ?))
288 cases (nth src (tape sig) t (niltape ?))
289 [ % #t1 * #x1 * #x2 * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
290 |2,3: #a0 #al0 % #t1 * #x1 * #x2 * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
291 | #ls lapply t -t elim ls
292 [#t #c #rs % #t1 * #x1 * #x2 * * >nth_change_vec // normalize in ⊢ (%→?);
293 #H1 destruct (H1) #Hcurdst >change_vec_change_vec #Ht1 %
294 #t2 * #y1 * #y2 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
295 >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
296 |#l0 #ls0 #IH #t #c #rs % #t1 * #x1 * #x2 * * >nth_change_vec //
297 normalize in ⊢ (%→?); #H destruct (H) #Hcurdst
298 >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
303 lemma sem_parmoveL : ∀src,dst,sig,n.
304 src ≠ dst → src < S n → dst < S n →
305 parmove src dst sig n L ⊨ R_parmoveL src dst sig n.
306 #src #dst #sig #n #Hneq #Hsrc #Hdst @WRealize_to_Realize
307 [/2/ | @wsem_parmoveL //]
317 definition mte_step ≝ λalpha,D.
318 ifTM ? (test_null alpha) (single_finalTM ? (move alpha D)) (nop ?) tc_true.
320 definition R_mte_step_true ≝ λalpha,D,t1,t2.
322 t1 = midtape alpha ls c rs ∧ t2 = tape_move ? t1 D.
324 definition R_mte_step_false ≝ λalpha.λt1,t2:tape alpha.
325 current ? t1 = None ? ∧ t1 = t2.
327 definition mte2 : ∀alpha,D.states ? (mte_step alpha D) ≝
328 λalpha,D.(inr … (inl … (inr … start_nop))).
331 ∀alpha,D.mte_step alpha D ⊨
332 [ mte2 … : R_mte_step_true alpha D, R_mte_step_false alpha ] .
334 @(acc_sem_if_app ??????????? (sem_test_null …)
335 (sem_move_single …) (sem_nop alpha) ??)
336 [ #tb #tc #td * #Hcurtb
337 lapply (refl ? (current ? tb)) cases (current ? tb) in ⊢ (???%→?);
338 [ #H @False_ind >H in Hcurtb; * /2/ ]
339 -Hcurtb #c #Hcurtb #Htb whd in ⊢ (%→?); #Htc whd
340 cases (current_to_midtape … Hcurtb) #ls * #rs #Hmidtb
341 %{ls} %{c} %{rs} % //
342 | #tb #tc #td * #Hcurtb #Htb whd in ⊢ (%→?); #Htc whd % // ]
345 definition move_to_end ≝ λsig,D.whileTM sig (mte_step sig D) (mte2 …).
347 definition R_move_to_end_r ≝
349 (current ? int = None ? → outt = int) ∧
350 ∀ls,c,rs.int = midtape sig ls c rs → outt = mk_tape ? (reverse ? rs@c::ls) (None ?) [ ].
352 lemma wsem_move_to_end_r : ∀sig. move_to_end sig R ⊫ R_move_to_end_r sig.
353 #sig #ta #k #outc #Hloop
354 lapply (sem_while … (sem_mte_step sig R) … Hloop) //
355 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
356 [ * #Hcurtb #Houtc % /2/ #ls #c #rs #Htb >Htb in Hcurtb; normalize in ⊢ (%→?); #H destruct (H)
357 | #tc #td * #ls * #c * #rs * #Htc >Htc cases rs
358 [ normalize in ⊢ (%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse
359 lapply (IH Hfalse) -IH * #Htd1 #_ %
360 [ normalize in ⊢ (%→?); #H destruct (H)
361 | #ls0 #c0 #rs0 #H destruct (H) >Htd1 // ]
362 | #r0 #rs0 whd in ⊢ (???%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse
363 lapply (IH Hfalse) -IH * #_ #IH %
364 [ normalize in ⊢ (%→?); #H destruct (H)
365 | #ls1 #c1 #rs1 #H destruct (H) >reverse_cons >associative_append @IH % ] ] ]
368 lemma terminate_move_to_end_r : ∀sig,t.move_to_end sig R ↓ t.
369 #sig #t @(terminate_while … (sem_mte_step sig R …)) //
371 [ % #t1 * #ls * #c * #rs * #H destruct
372 |2,3: #a0 #al0 % #t1 * #ls * #c * #rs * #H destruct
373 | #ls #c #rs lapply c -c lapply ls -ls elim rs
374 [ #ls #c % #t1 * #ls0 * #c0 * #rs0 * #Hmid #Ht1 destruct %
375 #t2 * #ls1 * #c1 * #rs1 * normalize in ⊢ (%→?); #H destruct
376 | #r0 #rs0 #IH #ls #c % #t1 * #ls1 * #c1 * #rs1 * #Hmid #Ht1 destruct @IH
381 lemma sem_move_to_end_r : ∀sig. move_to_end sig R ⊨ R_move_to_end_r sig.
382 #sig @WRealize_to_Realize //
385 definition R_move_to_end_l ≝
387 (current ? int = None ? → outt = int) ∧
388 ∀ls,c,rs.int = midtape sig ls c rs → outt = mk_tape ? [ ] (None ?) (reverse ? ls@c::rs).
390 lemma wsem_move_to_end_l : ∀sig. move_to_end sig L ⊫ R_move_to_end_l sig.
391 #sig #ta #k #outc #Hloop
392 lapply (sem_while … (sem_mte_step sig L) … Hloop) //
393 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
394 [ * #Hcurtb #Houtc % /2/ #ls #c #rs #Htb >Htb in Hcurtb; normalize in ⊢ (%→?); #H destruct (H)
395 | #tc #td * #ls * #c * #rs * #Htc >Htc cases ls
396 [ normalize in ⊢ (%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse
397 lapply (IH Hfalse) -IH * #Htd1 #_ %
398 [ normalize in ⊢ (%→?); #H destruct (H)
399 | #ls0 #c0 #rs0 #H destruct (H) >Htd1 // ]
400 | #l0 #ls0 whd in ⊢ (???%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse
401 lapply (IH Hfalse) -IH * #_ #IH %
402 [ normalize in ⊢ (%→?); #H destruct (H)
403 | #ls1 #c1 #rs1 #H destruct (H) >reverse_cons >associative_append @IH % ] ] ]
406 lemma terminate_move_to_end_l : ∀sig,t.move_to_end sig L ↓ t.
407 #sig #t @(terminate_while … (sem_mte_step sig L …)) //
409 [ % #t1 * #ls * #c * #rs * #H destruct
410 |2,3: #a0 #al0 % #t1 * #ls * #c * #rs * #H destruct
412 [ #c #rs % #t1 * #ls0 * #c0 * #rs0 * #Hmid #Ht1 destruct %
413 #t2 * #ls1 * #c1 * #rs1 * normalize in ⊢ (%→?); #H destruct
414 | #l0 #ls0 #IH #c #rs % #t1 * #ls1 * #c1 * #rs1 * #Hmid #Ht1 destruct @IH
419 lemma sem_move_to_end_l : ∀sig. move_to_end sig L ⊨ R_move_to_end_l sig.
420 #sig @WRealize_to_Realize //