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/multi_universal/moves_2.ma".
13 include "turing/multi_universal/match.ma".
14 include "turing/multi_universal/copy.ma".
15 include "turing/multi_universal/alphabet.ma".
16 include "turing/multi_universal/tuples.ma".
29 current (in.obj) = None
40 (if (current(in.obj)) == None
55 definition obj ≝ (0:DeqNat).
56 definition cfg ≝ (1:DeqNat).
57 definition prg ≝ (2:DeqNat).
59 definition obj_to_cfg ≝
60 mmove cfg FSUnialpha 2 L ·
61 (ifTM ?? (inject_TM ? (test_null ?) 2 obj)
62 (copy_step obj cfg FSUnialpha 2 ·
63 mmove cfg FSUnialpha 2 L ·
64 mmove obj FSUnialpha 2 L)
65 (inject_TM ? (write FSUnialpha null) 2 cfg)
67 inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
68 mmove cfg FSUnialpha 2 R.
70 definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3.
72 nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
73 (∀lso,x,rso.nth obj ? t1 (niltape ?) = midtape FSUnialpha lso x rso →
75 (mk_tape ? [ ] (option_hd ? (reverse ? (x::ls))) (tail ? (reverse ? (x::ls)))) cfg) ∧
76 (current ? (nth obj ? t1 (niltape ?)) = None ? →
78 (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (null::ls)))
79 (tail ? (reverse ? (null::ls)))) cfg).
81 axiom sem_move_to_end_l : ∀sig. move_to_end sig L ⊨ R_move_to_end_l sig.
82 axiom accRealize_to_Realize :
83 ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
84 M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse.
86 lemma eq_mk_tape_rightof :
87 ∀alpha,a,al.mk_tape alpha (a::al) (None ?) [ ] = rightof ? a al.
91 axiom daemon : ∀P:Prop.P.
93 definition option_cons ≝ λsig.λc:option sig.λl.
94 match c with [ None ⇒ l | Some c0 ⇒ c0::l ].
96 lemma tape_move_mk_tape_R :
98 (c = None ? → ls = [ ] ∨ rs = [ ]) →
99 tape_move ? (mk_tape sig ls c rs) R =
100 mk_tape ? (option_cons ? c ls) (option_hd ? rs) (tail ? rs).
101 #sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
102 [| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ]
106 lemma sem_obj_to_cfg : obj_to_cfg ⊨ R_obj_to_cfg.
107 @(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
110 (sem_test_null_multi ?? obj ?)
111 (sem_seq ?????? (accRealize_to_Realize … (sem_copy_step …))
112 (sem_seq ?????? (sem_move_multi ? 2 cfg L ?)
113 (sem_move_multi ? 2 obj L ?)))
114 (sem_inject ???? cfg ? (sem_write FSUnialpha null)))
115 (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
116 (sem_move_multi ? 2 cfg R ?)))) //
118 #tc * whd in ⊢ (%→?); #Htc *
120 [ * #te * * #Hcurtc #Hte
121 * destruct (Hte) #te * *
122 [ whd in ⊢ (%→%→?); * #x * #y * * -Hcurtc #Hcurtc1 #Hcurtc2 #Hte
123 * #tf * whd in ⊢ (%→%→?); #Htf #Htd
124 * #tg * * * whd in ⊢ (%→%→%→%→?); #Htg1 #Htg2 #Htg3 #Htb
126 [ #lso #x0 #rso #Hta2 >Hta1 in Htc; >eq_mk_tape_rightof
127 whd in match (tape_move ???); #Htc
128 cut (tg = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[x])) cfg)
129 [@daemon] -Htg1 -Htg2 -Htg3 #Htg destruct (Htg Htf Hte Htd Htc Htb)
130 >change_vec_change_vec >change_vec_change_vec
131 >change_vec_commute // >change_vec_change_vec
132 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
133 >change_vec_commute // >change_vec_change_vec
134 >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
135 >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
136 >change_vec_commute [|@sym_not_eq //] @eq_f3 //
137 [ >Hta2 cases rso in Hta2; whd in match (tape_move_mono ???);
138 [ #Hta2 whd in match (tape_move ???); <Hta2 @change_vec_same
139 | #r1 #rs1 #Hta2 whd in match (tape_move ???); <Hta2 @change_vec_same ]
140 | >tape_move_mk_tape_R [| #_ % %] >reverse_cons
141 >nth_change_vec_neq in Hcurtc1; [|@sym_not_eq //] >Hta2
142 normalize in ⊢ (%→?); #H destruct (H) %
144 | #Hta2 >Htc in Hcurtc1; >nth_change_vec_neq [| @sym_not_eq //]
145 >Hta2 #H destruct (H)
147 | * #Hcurtc0 #Hte #_ #_ #c #ls #Hta1 >Hta1 in Htc; >eq_mk_tape_rightof
148 whd in match (tape_move ???); #Htc >Htc in Hcurtc0; *
149 [ >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
150 #Hcurtc #Hcurtc0 >Hcurtc0 in Hcurtc; * #H @False_ind @H %
151 | >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H) ]
153 | * #te * * #Hcurtc #Hte
154 * whd in ⊢ (%→%→?); #Htd1 #Htd2
155 * #tf * * * #Htf1 #Htf2 #Htf3 whd in ⊢ (%→?); #Htb
157 [ #lso #x #rso #Hta2 >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
158 >Hta2 normalize in ⊢ (%→?); #H destruct (H)
159 | #_ >Hta1 in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
160 destruct (Hte) cut (td = change_vec ?? tc (midtape ? ls null []) cfg)
161 [@daemon] -Htd1 -Htd2 #Htd
162 -Htf1 cut (tf = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[null])) cfg)
163 [@daemon] -Htf2 -Htf3 #Htf destruct (Htf Htd Htc Htb)
164 >change_vec_change_vec >change_vec_change_vec >change_vec_change_vec
165 >change_vec_change_vec >change_vec_change_vec >nth_change_vec //
166 >reverse_cons >tape_move_mk_tape_R /2/ ]
170 definition test_null_char ≝ test_char FSUnialpha (λc.c == null).
172 definition R_test_null_char_true ≝ λt1,t2.
173 current FSUnialpha t1 = Some ? null ∧ t1 = t2.
175 definition R_test_null_char_false ≝ λt1,t2.
176 current FSUnialpha t1 ≠ Some ? null ∧ t1 = t2.
178 lemma sem_test_null_char :
179 test_null_char ⊨ [ tc_true : R_test_null_char_true, R_test_null_char_false].
180 #t1 cases (sem_test_char FSUnialpha (λc.c == null) t1) #k * #outc * * #Hloop #Htrue
181 #Hfalse %{k} %{outc} % [ %
183 | #Houtc cases (Htrue ?) [| @Houtc] * #c * #Hcurt1 #Hcnull lapply (\P Hcnull)
184 -Hcnull #H destruct (H) #Houtc1 %
185 [ @Hcurt1 | <Houtc1 % ] ]
186 | #Houtc cases (Hfalse ?) [| @Houtc] #Hc #Houtc %
187 [ % #Hcurt1 >Hcurt1 in Hc; #Hc lapply (Hc ? (refl ??))
188 >(?:((null:FSUnialpha) == null) = true) [|@(\b (refl ??)) ]
193 definition cfg_to_obj ≝
194 mmove cfg FSUnialpha 2 L ·
195 (ifTM ?? (inject_TM ? test_null_char 2 cfg)
197 (copy_step cfg obj FSUnialpha 2 ·
198 mmove cfg FSUnialpha 2 L ·
199 mmove obj FSUnialpha 2 L)
201 inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
202 mmove cfg FSUnialpha 2 R.
204 definition R_cfg_to_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
206 nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
208 t2 = change_vec ?? t1
209 (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (c::ls)))
210 (tail ? (reverse ? (c::ls)))) cfg) ∧
214 (midtape ? (left ? (nth obj ? t1 (niltape ?))) c (right ? (nth obj ? t1 (niltape ?)))) obj)
215 (mk_tape ? [ ] (option_hd ? (reverse ? (c::ls))) (tail ? (reverse ? (c::ls)))) cfg).
217 lemma sem_cfg_to_obj : cfg_to_obj ⊨ R_cfg_to_obj.
218 @(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
221 (acc_sem_inject ?????? cfg ? sem_test_null_char)
223 (sem_seq ?????? (accRealize_to_Realize … (sem_copy_step …))
224 (sem_seq ?????? (sem_move_multi ? 2 cfg L ?) (sem_move_multi ? 2 obj L ?))))
225 (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
226 (sem_move_multi ? 2 cfg R ?)))) // [@sym_not_eq //]
228 #tc * whd in ⊢ (%→?); #Htc *
230 [ * #te * * * #Hcurtc #Hte1 #Hte2 whd in ⊢ (%→?); #Htd destruct (Htd)
231 * #tf * * * #Htf1 #Htf2 #Htf3
234 [ #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
235 cut (te = tc) [@daemon] -Hte1 -Hte2 #Hte
236 cut (tf = change_vec ? 3 te (mk_tape ? [ ] (None ?) (reverse ? ls@[c])) cfg)
237 [@daemon] -Htf1 -Htf2 -Htf3 #Htf
238 destruct (Htf Hte Htc Htb)
239 >change_vec_change_vec >change_vec_change_vec >change_vec_change_vec
240 >nth_change_vec // >tape_move_mk_tape_R [| #_ % % ]
242 | #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
243 >Htc in Hcurtc; >nth_change_vec // normalize in ⊢ (%→?);
244 #H destruct (H) @False_ind cases Hc /2/ ]
245 | * #te * * * #Hcurtc #Hte1 #Hte2
247 [ (* purtroppo copy_step assume che la destinazione sia Some (almeno come semantica) *)
249 * #x * #y * * #Hcurte_cfg #Hcurte_obj #Htf
250 * #tg * whd in ⊢ (%→%→?); #Htg #Htd
251 * #th * * * #Hth1 #Hth2 #Hth3
254 [ >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
255 >Htc in Hcurtc; >nth_change_vec // normalize in ⊢ (%→?); >Hc
257 | >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
258 cut (te = tc) [@daemon] -Hte1 -Hte2 #Hte
262 (* macchina che muove il nastro obj a destra o sinistra a seconda del valore
263 del current di prg, che codifica la direzione in cui ci muoviamo *)
265 definition char_to_move ≝ λc.match c with
266 [ bit b ⇒ if b then R else L
269 definition tape_move_obj : mTM FSUnialpha 2 ≝
271 (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
272 (mmove obj FSUnialpha 2 L)
274 (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit true)) 2 prg)
275 (mmove obj FSUnialpha 2 R)
280 definition restart_tape ≝ λi.
281 inject_TM ? (move_to_end FSUnialpha L) 2 i ·
282 mmove i FSUnialpha 2 R.
285 match_m cfg prg FSUnialpha 2 ·
286 restart_tape cfg · copy prg cfg FSUnialpha 2 ·
287 cfg_to_obj · tape_move_obj · restart_tape prg · obj_to_cfg.
290 definition legal_tape ≝ λn,l,h,t.
292 nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[char]) →
293 is_config n (bar::state@[char]) →
294 nth prg ? t1 (niltape ?) = midtape ? [ ] bar table →
295 bar::table = table_TM n l h → *)
297 definition list_of_tape ≝ λsig,t.
298 left sig t@option_cons ? (current ? t) (right ? t).
300 definition low_char' ≝ λc.
303 | Some b ⇒ if (is_bit b) then b else null
306 definition R_unistep ≝ λn,l,h.λt1,t2: Vector ? 3.
309 nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[char]) →
310 is_config n (bar::state@[char]) →
312 nth prg ? t1 (niltape ?) = midtape ? [ ] bar table →
313 bar::table = table_TM n l h →
315 only_bits (list_of_tape ? (nth obj ? t1 (niltape ?))) →
316 let conf ≝ (bar::state@[char]) in
317 (∃ll,lr.bar::table = ll@conf@lr) →
318 ∃nstate,nchar,m,t. tuple_encoding n h t = (conf@nstate@[nchar;m]) ∧
321 tape_move_mono ? (nth obj ? t1 (niltape ?))
322 〈Some ? nchar,char_to_move m〉 in
323 let next_char ≝ low_char' (current ? new_obj) in
326 (change_vec ?? t1 (midtape ? [ ] bar (nstate@[next_char])) cfg)
329 definition tape_map ≝ λA,B:FinSet.λf:A→B.λt.
330 mk_tape B (map ?? f (left ? t))
331 (option_map ?? f (current ? t))
332 (map ?? f (right ? t)).
334 definition low_tapes ≝ λM:normalTM.λc:nconfig (no_states M).Vector_of_list ?
335 [tape_map ?? bit (ctape ?? c);
336 midtape ? [ ] bar (bits_of_state ? (nhalt M) (cstate ?? c));
337 midtape ? [ ] bar (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M))
340 definition R_unistep_high ≝ λM:normalTM.λc:nconfig (no_states M).λt1,t2.
342 t2 = low_tapes M (step ? M c).