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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 *)
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15 include "turing/multi_universal/moves.ma".
16 include "turing/if_multi.ma".
17 include "turing/inject.ma".
18 include "turing/basic_machines.ma".
20 definition compare_states ≝ initN 3.
22 definition comp0 : compare_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
23 definition comp1 : compare_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
24 definition comp2 : compare_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
28 0) (x,x) → (x,x)(R,R) → 1
35 definition trans_compare_step ≝
36 λi,j.λsig:FinSet.λn.λis_endc.
37 λp:compare_states × (Vector (option sig) (S n)).
40 [ O ⇒ match nth i ? a (None ?) with
41 [ None ⇒ 〈comp2,null_action ? n〉
42 | Some ai ⇒ match nth j ? a (None ?) with
43 [ None ⇒ 〈comp2,null_action ? n〉
44 | Some aj ⇒ if notb (is_endc ai) ∧ ai == aj
45 then 〈comp1,change_vec ? (S n)
46 (change_vec ? (S n) (null_action ? n) (Some ? 〈ai,R〉) i)
48 else 〈comp2,null_action ? n〉 ]
51 [ O ⇒ (* 1 *) 〈comp1,null_action ? n〉
52 | S _ ⇒ (* 2 *) 〈comp2,null_action ? n〉 ] ].
54 definition compare_step ≝
56 mk_mTM sig n compare_states (trans_compare_step i j sig n is_endc)
57 comp0 (λq.q == comp1 ∨ q == comp2).
59 definition R_comp_step_true ≝
60 λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
63 current ? (nth i ? int (niltape ?)) = Some ? x ∧
64 current ? (nth j ? int (niltape ?)) = Some ? x ∧
67 (tape_move ? (nth i ? int (niltape ?)) (Some ? 〈x,R〉)) i)
68 (tape_move ? (nth j ? int (niltape ?)) (Some ? 〈x,R〉)) j.
70 definition R_comp_step_false ≝
71 λi,j:nat.λsig,n,is_endc.λint,outt: Vector (tape sig) (S n).
72 ((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
73 current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
74 current ? (nth i ? int (niltape ?)) = None ? ∨
75 current ? (nth j ? int (niltape ?)) = None ?) ∧ outt = int.
77 lemma comp_q0_q2_null :
78 ∀i,j,sig,n,is_endc,v.i < S n → j < S n →
79 (nth i ? (current_chars ?? v) (None ?) = None ? ∨
80 nth j ? (current_chars ?? v) (None ?) = None ?) →
81 step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v)
82 = mk_mconfig ??? comp2 v.
83 #i #j #sig #n #is_endc #v #Hi #Hj
84 whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??%?);
87 [ whd in ⊢ (??(???%)?); >Hcurrent %
88 | whd in ⊢ (??(???????(???%))?); >Hcurrent @tape_move_null_action ]
90 [ whd in ⊢ (??(???%)?); >Hcurrent cases (nth i ?? (None sig)) //
91 | whd in ⊢ (??(???????(???%))?); >Hcurrent
92 cases (nth i ?? (None sig)) [|#x] @tape_move_null_action ] ]
95 lemma comp_q0_q2_neq :
96 ∀i,j,sig,n,is_endc,v.i < S n → j < S n →
97 ((∃x.nth i ? (current_chars ?? v) (None ?) = Some ? x ∧ is_endc x = true) ∨
98 nth i ? (current_chars ?? v) (None ?) ≠ nth j ? (current_chars ?? v) (None ?)) →
99 step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v)
100 = mk_mconfig ??? comp2 v.
101 #i #j #sig #n #is_endc #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?)))
102 cases (nth i ?? (None ?)) in ⊢ (???%→?);
103 [ #Hnth #_ @comp_q0_q2_null // % //
104 | #ai #Hai lapply (refl ? (nth j ?(current_chars ?? v)(None ?)))
105 cases (nth j ?? (None ?)) in ⊢ (???%→?);
106 [ #Hnth #_ @comp_q0_q2_null // %2 //
108 [ * #c * >Hai #Heq #Hendc whd in ⊢ (??%?);
109 >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
110 [ whd in match (trans ????); >Hai >Haj destruct (Heq)
111 whd in ⊢ (??(???%)?); >Hendc //
112 | whd in match (trans ????); >Hai >Haj destruct (Heq)
113 whd in ⊢ (??(???????(???%))?); >Hendc @tape_move_null_action
116 whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
117 [ whd in match (trans ????); >Hai >Haj
118 whd in ⊢ (??(???%)?); cut ((¬is_endc ai∧ai==aj)=false)
119 [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) // |#Hcut >Hcut //]
120 | whd in match (trans ????); >Hai >Haj
121 whd in ⊢ (??(???????(???%))?); cut ((¬is_endc ai∧ai==aj)=false)
122 [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) //
123 |#Hcut >Hcut @tape_move_null_action
132 ∀i,j,sig,n,is_endc,v,a.i ≠ j → i < S n → j < S n →
133 nth i ? (current_chars ?? v) (None ?) = Some ? a → is_endc a = false →
134 nth j ? (current_chars ?? v) (None ?) = Some ? a →
135 step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) =
139 (tape_move ? (nth i ? v (niltape ?)) (Some ? 〈a,R〉)) i)
140 (tape_move ? (nth j ? v (niltape ?)) (Some ? 〈a,R〉)) j).
141 #i #j #sig #n #is_endc #v #a #Heq #Hi #Hj #Ha1 #Hnotendc #Ha2
142 whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
143 [ whd in match (trans ????);
144 >Ha1 >Ha2 whd in ⊢ (??(???%)?); >Hnotendc >(\b ?) //
145 | whd in match (trans ????);
146 >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >Hnotendc >(\b ?) //
147 change with (change_vec ?????) in ⊢ (??(???????%)?);
148 <(change_vec_same … v j (niltape ?)) in ⊢ (??%?);
149 <(change_vec_same … v i (niltape ?)) in ⊢ (??%?);
150 >pmap_change >pmap_change >tape_move_null_action
151 @eq_f2 // @eq_f2 // >nth_change_vec_neq //
155 lemma sem_comp_step :
156 ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n →
157 compare_step i j sig n is_endc ⊨
158 [ comp1: R_comp_step_true i j sig n is_endc,
159 R_comp_step_false i j sig n is_endc ].
160 #i #j #sig #n #is_endc #Hneq #Hi #Hj #int
161 lapply (refl ? (current ? (nth i ? int (niltape ?))))
162 cases (current ? (nth i ? int (niltape ?))) in ⊢ (???%→?);
165 [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ % <Hcuri in ⊢ (???%);
167 | normalize in ⊢ (%→?); #H destruct (H) ]
168 | #_ % // % %2 // ] ]
169 | #a #Ha lapply (refl ? (current ? (nth j ? int (niltape ?))))
170 cases (current ? (nth j ? int (niltape ?))) in ⊢ (???%→?);
173 [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ %2 <Hcurj in ⊢ (???%);
175 | normalize in ⊢ (%→?); #H destruct (H) ]
176 | #_ % // >Ha >Hcurj % % %2 % #H destruct (H) ] ]
178 cases (true_or_false (is_endc a)) #Haendc
181 [whd in ⊢ (??%?); >comp_q0_q2_neq //
182 % %{a} % // <Ha @sym_eq @nth_vec_map
183 | normalize in ⊢ (%→?); #H destruct (H) ]
184 | #_ % // % % % >Ha %{a} % // ]
186 |cases (true_or_false (a == b)) #Hab
189 [whd in ⊢ (??%?); >(comp_q0_q1 … a Hneq Hi Hj) //
190 [>(\P Hab) <Hb @sym_eq @nth_vec_map
191 |<Ha @sym_eq @nth_vec_map ]
192 | #_ whd >(\P Hab) %{b} % // % // <(\P Hab) % // ]
193 | * #H @False_ind @H %
197 [whd in ⊢ (??%?); >comp_q0_q2_neq //
198 <(nth_vec_map ?? (current …) i ? int (niltape ?))
199 <(nth_vec_map ?? (current …) j ? int (niltape ?)) %2 >Ha >Hb
200 @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
201 | normalize in ⊢ (%→?); #H destruct (H) ]
202 | #_ % // % % %2 >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ]
209 definition compare ≝ λi,j,sig,n,is_endc.
210 whileTM … (compare_step i j sig n is_endc) comp1.
212 definition R_compare ≝
213 λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
214 ((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
215 (current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
216 current ? (nth i ? int (niltape ?)) = None ? ∨
217 current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
218 (∀ls,x,xs,ci,rs,ls0,rs0.
219 nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
220 nth j ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
221 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
224 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
225 (mk_tape sig (reverse ? xs@x::ls0) (None ?) []) j) ∨
226 ∃cj,rs1.rs0 = cj::rs1 ∧
227 ((is_endc ci = true ∨ ci ≠ cj) →
229 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
230 (midtape sig (reverse ? xs@x::ls0) cj rs1) j)).
232 lemma wsem_compare : ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n →
233 compare i j sig n is_endc ⊫ R_compare i j sig n is_endc.
234 #i #j #sig #n #is_endc #Hneq #Hi #Hj #ta #k #outc #Hloop
235 lapply (sem_while … (sem_comp_step i j sig n is_endc Hneq Hi Hj) … Hloop) //
236 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
237 [ #tc whd in ⊢ (%→?); * * [ * [ *
238 [* #curi * #Hcuri #Hendi #Houtc %
240 | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi #Hnthj #Hnotendc
242 >Hnthi in Hcuri; normalize in ⊢ (%→?); #H destruct (H)
243 >(Hnotendc ? (memb_hd … )) in Hendi; #H destruct (H)
247 | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi #Hnthj
248 >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
252 | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi >Hnthi in Hci;
253 normalize in ⊢ (%→?); #H destruct (H) ] ]
256 | #ls #x #xs #ci #rs #ls0 #rs0 #_ #Hnthj >Hnthj in Hcj;
257 normalize in ⊢ (%→?); #H destruct (H) ] ]
258 | #tc #td #te * #x * * * #Hendcx #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
260 [ >Hci >Hcj * [* #x0 * #H destruct (H) >Hendcx #H destruct (H)
261 |* [* #H @False_ind [cases H -H #H @H % | destruct (H)] | #H destruct (H)]]
262 | #ls #c0 #xs #ci #rs #ls0 #rs0 cases xs
263 [ #Hnthi #Hnthj #Hnotendc cases rs0 in Hnthj;
266 [ @eq_f3 // >(?:c0=x) [ >Hnthi % ]
267 >Hnthi in Hci;normalize #H destruct (H) %
268 | >(?:c0=x) [ >Hnthj % ]
269 >Hnthi in Hci;normalize #H destruct (H) % ]
270 | >Hd %2 %2 >nth_change_vec // >Hnthj % ]
271 | #r1 #rs1 #Hnthj %2 %{r1} %{rs1} % // *
274 [ @eq_f3 // >(?:c0=x) [ >Hnthi % ]
275 >Hnthi in Hci;normalize #H destruct (H) %
276 | >(?:c0=x) [ >Hnthj % ]
277 >Hnthi in Hci;normalize #H destruct (H) % ]
278 | >Hd >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
279 >nth_change_vec // >Hnthi >Hnthj normalize % %{ci} % //
283 [ @eq_f3 // >(?:c0=x) [ >Hnthi % ]
284 >Hnthi in Hci;normalize #H destruct (H) %
285 | >(?:c0=x) [ >Hnthj % ]
286 >Hnthi in Hci;normalize #H destruct (H) % ]
287 | >Hd %2 % % >nth_change_vec //
288 >nth_change_vec_neq [|@sym_not_eq //]
289 >nth_change_vec // >Hnthi >Hnthj normalize @(not_to_not … Hcir1)
293 |#x0 #xs0 #Hnthi #Hnthj #Hnotendc
294 cut (c0 = x) [ >Hnthi in Hci; normalize #H destruct (H) // ]
295 #Hcut destruct (Hcut) cases rs0 in Hnthj;
297 cases (IH2 (x::ls) x0 xs0 ci rs (x::ls0) [ ] ???) -IH2
298 [ * #_ #IH2 >IH2 >Hd >change_vec_commute in ⊢ (??%?); //
299 >change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
301 | * #cj * #rs1 * #H destruct (H)
302 | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
304 | >Hd >nth_change_vec // >Hnthj %
305 | #c0 #Hc0 @Hnotendc @memb_cons @Hc0 ]
306 | #r1 #rs1 #Hnthj %2 %{r1} %{rs1} % // #Hcir1
307 cases(IH2 (x::ls) x0 xs0 ci rs (x::ls0) (r1::rs1) ???)
309 | * #r1' * #rs1' * #H destruct (H) #Hc1r1 >Hc1r1 //
310 >Hd >change_vec_commute in ⊢ (??%?); //
311 >change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
313 | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
315 | >Hd >nth_change_vec // >Hnthi >Hnthj %
316 | #c0 #Hc0 @Hnotendc @memb_cons @Hc0
320 lemma terminate_compare : ∀i,j,sig,n,is_endc,t.
321 i ≠ j → i < S n → j < S n →
322 compare i j sig n is_endc ↓ t.
323 #i #j #sig #n #is_endc #t #Hneq #Hi #Hj
324 @(terminate_while … (sem_comp_step …)) //
325 <(change_vec_same … t i (niltape ?))
326 cases (nth i (tape sig) t (niltape ?))
327 [ % #t1 * #x * * * #_ >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
328 |2,3: #a0 #al0 % #t1 * #x * * * #_ >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
329 | #ls #c #rs lapply c -c lapply ls -ls lapply t -t elim rs
330 [#t #ls #c % #t1 * #x * * * #Hendcx >nth_change_vec // normalize in ⊢ (%→?);
331 #H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 %
332 #t2 * #x0 * * * #Hendcx0 >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
333 >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
334 |#r0 #rs0 #IH #t #ls #c % #t1 * #x * * >nth_change_vec //
335 normalize in ⊢ (%→?); #H destruct (H) #Hcur
336 >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
341 lemma sem_compare : ∀i,j,sig,n,is_endc.
342 i ≠ j → i < S n → j < S n →
343 compare i j sig n is_endc ⊨ R_compare i j sig n is_endc.
344 #i #j #sig #n #is_endc #Hneq #Hi #Hj @WRealize_to_Realize /2/
349 |confin 0/1 confout move
360 definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
361 compare src dst sig n is_endc ·
362 (ifTM ?? (inject_TM ? (test_char ? (λa.is_endc a == false)) n src)
364 (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
368 definition Rtc_multi_true ≝
369 λalpha,test,n,i.λt1,t2:Vector ? (S n).
370 (∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1.
372 definition Rtc_multi_false ≝
373 λalpha,test,n,i.λt1,t2:Vector ? (S n).
374 (∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
376 definition R_match_step_false ≝
377 λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
379 nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
380 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
381 ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
383 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
387 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
388 (midtape sig (reverse ? xs@x::ls0) c rsj) dst).
390 definition R_match_step_false ≝
391 λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
392 (((∃x.current ? (nth src ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
393 current sig (nth src (tape sig) int (niltape sig)) = None ? ∨
394 current sig (nth dst (tape sig) int (niltape sig)) = None ? ) ∧ outt = int) ∨
395 (∃ls,ls0,rs,rs0,x,xs.
396 nth src ? int (niltape ?) = midtape sig ls x (xs@rs) ∧ is_endc x = false ∧
397 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
399 rs = end::rsi → rs0 = c::rsj →
400 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) ∧ is_endc end = true ∧
401 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@c::rsj) ∧
403 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rsi) src)
404 (midtape sig (reverse ? xs@x::ls0) c rsj) dst).
407 definition R_match_step_true ≝
408 λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
409 ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s →
411 (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) →
412 (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
413 (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 →
414 outt = change_vec ?? int
415 (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧
416 (∀ls,x,xs,ci,rs,ls0,rs0.
417 nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
418 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
419 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
420 (∃cj,rs1.rs0 = cj::rs1 → ci ≠ cj →
421 (outt = change_vec ?? int
422 (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false)) ∨
425 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src)
426 (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)).
428 lemma sem_test_char_multi :
429 ∀alpha,test,n,i.i ≤ n →
430 inject_TM ? (test_char ? test) n i ⊨
431 [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
432 #alpha #test #n #i #Hin #int
433 cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
434 #k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
436 | #Hqtrue lapply (Htrue Hqtrue) * * * #c *
437 #Hcur #Htestc #Hnth_i #Hnth_j %
439 | @(eq_vec … (niltape ?)) #i0 #Hi0
440 cases (decidable_eq_nat i0 i) #Hi0i
442 | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
443 | #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
445 | @(eq_vec … (niltape ?)) #i0 #Hi0
446 cases (decidable_eq_nat i0 i) #Hi0i
448 | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
451 axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S.∀is_endc. ∃l,tl1,tl2.
452 l1 = l@tl1 ∧ l2 = l@tl2 ∧ (∀c.c ∈ l = true → is_endc c = false) ∧
453 ∀a,tla. tl1 = a::tla → is_endc a = true ∨ (∀b,tlb.tl2 = b::tlb → a≠b).
455 axiom daemon : ∀X:Prop.X.
457 lemma sem_match_step :
458 ∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n →
459 match_step src dst sig n is_startc is_endc ⊨
460 [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
461 R_match_step_true src dst sig n is_startc is_endc,
462 R_match_step_false src dst sig n is_endc ].
463 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst
464 @(acc_sem_seq_app sig n … (sem_compare src dst sig n is_endc Hneq Hsrc Hdst)
465 (acc_sem_if ? n … (sem_test_char_multi sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
467 (sem_parmoveL ???? is_startc Hneq Hsrc Hdst)
468 (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
470 [#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * * #c * #Hcurtc #Hcend #Htd >Htd -Htd
471 #Htb #s #Hcurta_src #Hstart #Hnotstart % [ %
473 | #s1 #Hcurta_dst #Hneqss1
474 lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta)
475 [|@Hcomp1 %2 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ]
476 #Hcurtc * #te * * #_ #Hte >Hte [2: %1 %1 %{s} % //]
477 whd in ⊢ (%→?); * * #_ #Htbdst #Htbelse %
478 [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
479 [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
480 #ls * #rs #Hta_mid >(Htbdst … Hta_mid) >Hta_mid cases rs //
481 | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Htbelse @sym_not_eq // ]
482 | >Hcurtc in Hcurta_src; #H destruct (H) cases (is_endc s) in Hcend;
483 normalize #H destruct (H) // ]
485 |#ls #x #xs #ci #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc
486 cases rs00 in Htadst_mid;
487 [(* case rs empty *) #Htadst_mid %2 #_
488 cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) -Hcomp2
489 [2: * #x0 * #rs1 * #H destruct (H) ]
490 * #_ #Htc cases Htb #td * * #_ #Htd >Htasrc_mid in Hcurta_src;
491 normalize in ⊢ (%→?); #H destruct (H)
492 >Htd [2: %2 >Htc >nth_change_vec // cases (reverse sig ?) //]
493 >Htc * * >nth_change_vec // #Htbdst #_ #Htbelse
494 @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
495 [ >Hidst >nth_change_vec // <Htbdst // cases (reverse sig ?) //
496 |@sym_eq @Htbelse @sym_not_eq //
498 |#cj #rs0 #Htadst_mid % %{cj} %{rs0} #_ #Hcicj
499 cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) [ * #H destruct (H) ]
500 * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2
501 lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc
502 cases Htb #td * * #Htd #_ >Htasrc_mid in Hcurta_src; normalize in ⊢ (%→?);
504 >(Htd ls ci (reverse ? xs) rs s ??? ls0 cj' (reverse ? xs) s rs0' (refl ??)) //
505 [| >Htc >nth_change_vec //
506 | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid
507 cases (orb_true_l … Hc0) -Hc0 #Hc0
508 [@memb_append_l2 >(\P Hc0) @memb_hd
509 |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse //
511 | >Htc >nth_change_vec_neq [|@sym_not_eq // ] @nth_change_vec // ]
512 * * #_ #Htbdst #Htbelse %
513 [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
514 [ >Hidst >nth_change_vec // >Htadst_mid >(Htbdst ls0 s (xs@cj'::rs0'))
516 | >nth_change_vec // ]
517 | >nth_change_vec_neq [|@sym_not_eq //]
518 <Htbelse [|@sym_not_eq // ]
519 >nth_change_vec_neq [|@sym_not_eq //]
520 cases (decidable_eq_nat i src) #Hisrc
521 [ >Hisrc >nth_change_vec // >Htasrc_mid //
522 | >nth_change_vec_neq [|@sym_not_eq //]
523 <(Htbelse i) [|@sym_not_eq // ]
524 >Htc >nth_change_vec_neq [|@sym_not_eq // ]
525 >nth_change_vec_neq [|@sym_not_eq // ] //
528 | >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
529 >nth_change_vec // whd in ⊢ (??%?→?);
530 #H destruct (H) cases (is_endc c) in Hcend;
531 normalize #H destruct (H) // ]
534 |#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb
535 whd in ⊢ (%→?); #Hout >Hout >Htb whd
536 #ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend
537 lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
538 cases (current … (nth dst ? intape (niltape ?))) in Hcomp1;
539 [#Hcomp1 #_ %1 % [% | @Hcomp1 %2 %2 % ]
540 |#c_dst cases (true_or_false (c_src == c_dst)) #Hceq
541 [#_ #Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
542 #ls_dst * #rs_dst #Hmid_dst %2
543 cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * *
544 #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq %{ls_dst} %{rsj} >Hrs_dst in Hmid_dst; #Hmid_dst
545 cut (∃r1,rs1.rsi = r1::rs1) [@daemon] * #r1 * #rs1 #Hrs1 >Hrs1 in Hrs_src;
546 #Hrs_src >Hrs_src in Hmid_src; #Hmid_src <(\P Hceq) in Hmid_dst; #Hmid_dst
547 lapply (Hcomp2 ??????? Hmid_src Hmid_dst ?)
548 [ #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
549 [ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //]
552 [ >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
553 #Hc lapply (Hc ? (refl ??)) #Hendr1
555 [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
556 -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
557 [ * normalize in ⊢ (%→?); //
558 #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
559 lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
561 [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
562 >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
563 normalize in ⊢ (%→?); #H destruct (H)
564 | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
565 #Hnotendc #Hnotendcxs1 @eq_f @IH
566 [ @(cons_injective_r … Heq)
567 | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
569 | @memb_cons @memb_cons // ]
570 | #c #Hc @Hnotendcxs1 @memb_cons // ]
573 | #Hxsxs1 >Hmid_dst >Hxsxs1 % ]
574 | #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0) ]
575 | * #cj * #rs2 * #Hrs2 #Hta lapply (Hta ?)
576 [ cases (Hneq … Hrs1) /2/ #H %2 @(H ?? Hrs2) ]
577 -Hta #Hta >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //]
578 >nth_change_vec // #Hc lapply (Hc ? (refl ??)) #Hendr1
579 (* lemmatize this proof *) cut (xs = xs1)
580 [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
581 -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
582 [ * normalize in ⊢ (%→?); //
583 #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
584 lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
586 [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
587 >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
588 normalize in ⊢ (%→?); #H destruct (H)
589 | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
590 #Hnotendc #Hnotendcxs1 @eq_f @IH
591 [ @(cons_injective_r … Heq)
592 | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
594 | @memb_cons @memb_cons // ]
595 | #c #Hc @Hnotendcxs1 @memb_cons // ]
598 | #Hxsxs1 >Hmid_dst >Hxsxs1 % //
599 #rsj0 #c #Hcrsj destruct (Hxsxs1 Hrs2 Hcrsj) @eq_f3 //
600 @eq_f3 // lapply (append_l2_injective ?????? Hrs_src) //
601 #Hendr1 destruct (Hendr1) % ]
605 |#Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hdst
606 @False_ind lapply (Hcomp1 ?) [%2 %1 %1 >Hmid_src normalize
607 @(not_to_not ??? (\Pf Hceq)) #H destruct //] #Hintape
608 >Hintape in Hc; >Hmid_src #Hc lapply (Hc ? (refl …)) -Hc
609 >(Hnotend c_src) // normalize #H destruct (H)
615 definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
616 whileTM … (match_step src dst sig n is_startc is_endc)
617 (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
619 definition R_match_m ≝
620 λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
622 nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
624 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
625 ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) →
626 current sig (nth dst (tape sig) outt (niltape sig)) = None ?)
629 nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
630 (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
633 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
634 (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) dst) ∨
635 ∀l,l1.x0::rs0 ≠ l@x::xs@l1).
638 definition R_match_m ≝
639 λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
640 (((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
641 current ? (nth i ? int (niltape ?)) = None ? ∨
642 current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
643 (∀ls,x,xs,ci,rs,ls0,x0,rs0.
644 (∀x. is_startc x ≠ is_endc x) →
645 is_startc x = true → is_endc ci = true →
646 (∀z. memb ? z (x::xs) = true → is_endc x = false) →
647 nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
648 nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
649 (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
652 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
653 (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) j) ∨
654 ∀l,l1.x0::rs0 ≠ l@x::xs@l1).
658 axiom sub_list_dec: ∀A.∀l,ls:list A.
659 ∃l1,l2. l = l1@ls@l2 ∨ ∀l1,l2. l ≠ l1@ls@l2.
662 lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
663 src ≠ dst → src < S n → dst < S n →
664 match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
665 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
666 lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) //
667 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
668 [ #tc #Hfalse #ls #x #xs #end #rs #Hmid_src #Hstart #Hnotend #Hend
669 cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse
670 [(* current dest = None *) * #Hcur_dst #Houtc %
672 |#ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
673 normalize in ⊢ (%→?); #H destruct (H)
675 |* #ls0 * #rs0 * #Hmid_dst #HFalse %
676 [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
677 |#ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
678 %1 %{[ ]} %{rs0} % [%] #cj #l2 #Hnotnil
679 >reverse_cons >associative_append @(HFalse ?? Hnotnil)
682 |#ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
683 #ls #x #xs #end #rs #Hmid_src #Hstart #Hnotend #Hend
684 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
685 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
686 [#Hmid_dst % [#_ whd in Htrue; >Hmid_src in Htrue; #Htrue
687 cases (Htrue x (refl … ) Hstart ?) -Htrue [2: @daemon]
688 * #Htb #_ #_ >Htb in IH; // #IH
689 cases (IH ls x xs end rs Hmid_src Hstart Hnotend Hend)
693 |#cur_dst #Hcur_dst %2 #ls0 #x0 #rs0 #Hmid_dst
694 whd in Htrue; >Hmid_src in Htrue; #Htrue
695 cases (Htrue x (refl …) Hstart ?) -Htrue
696 [2: #z #membz @daemon (*aggiungere l'ipotesi*)]
697 cases (true_or_false (x==cur_dst)) #eqx
698 [#_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc)
699 #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1
701 [>append_nil #Hx1 @daemon (* absurd by Hxs e notendx1 *)]
702 #ci -tl1 #tl1 #Hxs #H cases (H … (refl … ))
703 [(* this is absurd, since Htrue conlcudes is_endc ci =false *)
706 @daemon (* lapply(Htrue … (refl …)) -Htrue *)
707 |#Htrue #_ cases(Htrue cur_dst Hcur_dst (\Pf eqx)) -Htrue #Htb #Hendx
709 cases(IH ls x xs end rs ? Hstart Hnotend Hend)
710 [* #H1 #H2 >Htb in H1; >nth_change_vec //
711 >Hmid_dst cases rs0 [2: #a #tl normalize in ⊢ (%→?); #H destruct (H)]
712 #_ %2 @daemon (* si dimostra *)
714 |>Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src