2 include "turing/multi_to_mono/exec_moves.ma".
4 definition Mono_realize ≝ Realize.
6 definition to_sig ≝ λQ,sig.λc:sig_ext (TA Q sig).
10 [inl x ⇒ None ? (* this is not possible *)
13 definition to_sig_inv ≝ λQ,sig.λc1:(option sig) × move.λc2.
14 match (fst ?? c1) with
16 |Some a ⇒ Some ? (inr Q ? a)].
18 definition transf ≝ λQ,sig:FinSet.λn.
19 λt: Q × (Vector (option sig) n) → Q × (Vector ((option sig) × move) n).
20 λa:MTA (HC Q n) sig (S n).
21 let qM ≝ nth n ? (vec … a) (blank ?) in
22 let a1 ≝ resize_vec ? (S n) a n (blank ?) in
23 let a2 ≝ vec_map ?? (to_sig ? sig) n a1 in
25 [None ⇒ all_blanks (TA (HC Q n) sig) (S n)
29 let 〈q1,actions〉 ≝ t 〈fst ?? p1, a2〉 in
30 let moves ≝ vec_map ?? (snd ??) n actions in
31 let new_chars ≝ pmap_vec ??? (to_sig_inv (HC Q n) ?) n actions a1 in
32 vec_cons_right ? (Some ? (inl ?? 〈q1,moves〉)) n new_chars
33 |inr p2 ⇒ all_blanks (TA (HC Q n) sig) (S n)]
36 lemma transf_eq: ∀Q,sig,n,t.∀a:MTA (HC Q n) sig (S n).
37 ∀a1,a2,q,m,q1,actions,moves,new_chars.
38 nth n ? (vec … a) (blank ?) = Some ? (inl ?? 〈q,m〉) →
39 a1 = resize_vec ? (S n) a n (blank ?) →
40 a2 = vec_map ?? (to_sig ? sig) n a1 →
41 t 〈q, a2〉 = 〈q1,actions〉 →
42 moves = vec_map ?? (snd ??) n actions →
43 new_chars = pmap_vec ??? (to_sig_inv (HC Q n) ?) n actions a1 →
45 vec_cons_right ? (Some ? (inl ?? 〈q1,moves〉)) n new_chars.
46 #Q #sig #n #t #a #a1 #a2 #q #m #q1 #actions #moves #new_chars
47 #Hnth #Ha1 #Ha2 #Ht #Hmoves #Hnew_chars
48 whd in ⊢ (??%?); >Hnth whd in ⊢ (??%?); <Ha1 <Ha2 >Ht
49 @eq_cons_right [<Hmoves // |<Hnew_chars // ]
52 lemma transf_eq_ex: ∀Q,sig,n,t.∀a:MTA (HC Q n) sig (S n).
53 ∀q,m.nth n ? (vec … a) (blank ?) = Some ? (inl ?? 〈q,m〉)→
54 ∃a1.a1 = resize_vec ? (S n) a n (blank ?) ∧
55 ∃a2. a2 = vec_map ?? (to_sig ? sig) n a1 ∧
56 ∃q1,actions. t 〈q, a2〉 = 〈q1,actions〉 ∧
57 ∃moves.moves = vec_map ?? (snd ??) n actions ∧
58 ∃new_chars.new_chars = pmap_vec ??? (to_sig_inv (HC Q n) ?) n actions a1 ∧
60 vec_cons_right ? (Some ? (inl ?? 〈q1,moves〉)) n new_chars.
61 #Q #sig #n #t #a #q #m #H
62 %{(resize_vec ? (S n) a n (blank ?))} % [%]
63 % [2:% [%] | skip] % [2: % [ 2: % [@eq_pair_fst_snd ] |skip] | skip]
64 % [2:% [%] | skip] % [2:% [%] | skip] whd in ⊢ (??%?); >H whd in ⊢ (??%?);
65 >(eq_pair_fst_snd … (t …)) %
69 lemma nth_transf: ∀Q,sig,n,t.∀a:MTA (HC Q n) sig (S n).
70 ∀a1,a2,q,m,q1,actions,moves.
71 nth n ? (vec … a) (blank ?) = Some ? (inl ?? 〈q,m〉) →
72 a1 = resize_vec ? (S n) a n (blank ?) →
73 a2 = vec_map ?? (to_sig ? sig) n a1 →
74 t 〈q, a2〉 = 〈q1,actions〉 →
75 moves = vec_map ?? (snd ??) n actions →
76 nth n ? (vec … (transf Q sig n t a)) (blank ?) = Some ? (inl ?? 〈q1,moves〉).
77 #Q #sig #n #t #a #a1 #a2 #q #m #q1 #actions #moves
78 #Hnth #Ha1 #Ha2 #Ht #Hmoves
81 (*********************************** stepM ************************************)
82 definition optf ≝ λA,B:Type[0].λf:A →B.λd,oa.
87 lemma optf_Some : ∀A,B,f,a,d. optf A B f d (Some A a) = f a.
90 definition stepM ≝ λQ,sig,n,trans.
91 writef ? (optf ?? (transf Q sig (S n) trans) (all_blanks …)) ·
92 exec_moves Q sig (S n) (S n).
94 let rec to_sig_map Q sig l on l ≝
97 | cons a tl ⇒ match to_sig Q sig a with
99 | Some c ⇒ c::(to_sig_map Q sig tl)]].
102 definition to_sig_tape ≝ λQ,sig,t.
104 [ niltape ⇒ niltape ?
105 | leftof a l ⇒ match to_sig Q sig a with
107 | Some x ⇒ leftof ? x (to_sig_map Q sig l) ]
108 | rightof a l ⇒ match to_sig Q sig a with
110 | Some x ⇒ rightof ? x (to_sig_map Q sig l) ]
111 | midtape ls a rs ⇒ match to_sig Q sig a with
113 | Some x ⇒ midtape ? (to_sig_map Q sig ls) x (to_sig_map Q sig rs) ]
116 let rec rb_tapes Q sig n ls (a:MTA Q sig (S n)) rs i on i ≝
118 [ O ⇒ mk_Vector ? 0 (nil ?) (refl ??)
119 | S j ⇒ vec_cons_right ? (to_sig_tape ?? (rb_trace_i ? (S n) ls (vec … a) rs j)) j
120 (rb_tapes Q sig n ls a rs j)].
122 lemma nth_rb_tapes : ∀Q,sig,n,ls.∀a:MTA Q sig (S n).∀rs,j,i. i < j →
123 nth i ? (rb_tapes Q sig n ls (a:MTA Q sig (S n)) rs j) (niltape ?) =
124 to_sig_tape ?? (rb_trace_i ? (S n) ls (vec … a) rs i).
125 #Q #sig #n #ls #a #rs #j elim j
126 [#i #H @False_ind /2/
127 |#k #Hind #i #lti cases (le_to_or_lt_eq … (le_S_S_to_le … lti))
128 [#Hlt >nth_cons_right_lt [@Hind //|//]
129 |#Heq >Heq @nth_cons_right_n
135 (* q0 is a default value *)
136 definition get_state ≝ λQ,sig,n.λa:MTA (HC Q n) sig (S n).λq0.
137 match nth n ? (vec … a) (blank ?) with
138 [ None ⇒ q0 (* impossible *)
139 | Some qM ⇒ match qM with
140 [inl qM1 ⇒ fst ?? qM1
141 |inr _ ⇒ q0 (* impossible *) ]].
143 definition get_chars ≝ λQ,sig,n.λa:MTA (HC Q n) sig (S n).
144 let a1 ≝ resize_vec ? (S n) a n (blank ?) in
145 vec_map ?? (to_sig ? sig) n a1.
147 lemma get_chars_def : ∀Q,sig,n.∀a:MTA (HC Q n) sig (S n).
149 vec_map ?? (to_sig ? sig) n (resize_vec ? (S n) a n (blank ?)).
152 include "turing/turing.ma".
154 definition readback_config ≝
155 λQ,sig,n,q0,ls.λa:MTA (HC Q (S n)) sig (S (S n)).λrs.
157 (get_state Q sig (S n) a q0)
158 (rb_tapes (HC Q (S n)) sig (S n) ls a rs (S n)).
160 lemma state_readback : ∀Q,sig,n,q0,ls,a,rs.
161 cstate … (readback_config Q sig n q0 ls a rs) =
162 get_state Q sig (S n) a q0.
165 lemma tapes_readback : ∀Q,sig,n,q0,ls,a,rs.
166 ctapes … (readback_config Q sig n q0 ls a rs) =
167 rb_tapes (HC Q (S n)) sig (S n) ls a rs (S n).
170 definition R_stepM ≝ λsig.λn.λM:mTM sig n.λt1,t2.
172 t1 = midtape ? ls a rs →
173 (∀i.regular_trace (TA (HC (states … M) (S n)) sig) (S(S n)) a ls rs i) →
174 is_head ?? (nth (S n) ? (vec … a) (blank ?)) = true →
178 t2 = midtape (MTA (HC (states … M) (S n)) sig (S(S n))) ls1 a1 rs1 ∧
179 (∀i.regular_trace (TA (HC (states … M) (S n)) sig) (S(S n)) a1 ls1 rs1 i) ∧
180 readback_config ? sig n (start … M) ls1 a1 rs1 =
181 step sig n M (readback_config ? sig n (start … M) ls a rs).
183 lemma nth_vec_map_lt :
184 ∀A,B,f,i,n.∀v:Vector A n.∀d1,d2.i < n →
185 nth i ? (vec_map A B f n v) d1 = f (nth i ? v d2).
186 #A #B #f #i #n #v #d1 #d2 #ltin >(nth_default B i n ? d1 (f d2) ltin) @sym_eq @nth_vec_map
189 lemma ctapes_mk_config : ∀sig,Q,n,s,t.
190 ctapes sig Q n (mk_mconfig sig Q n s t) = t.
193 lemma cstate_rb: ∀sig,n.∀M:mTM sig n.∀ls,a,rs.∀q,m.
194 nth (S n) ? (vec … a) (blank ?) = Some ? (inl ?? 〈q,m〉) →
195 cstate sig (states sig n M) n
196 (readback_config (states sig n M) sig n (start sig n M) ls a rs) = q.
197 #sig #n #M #ls #a #rs #q #m #H
198 >state_readback whd in ⊢ (??%?); >H %
201 axiom eq_cstate_get_state: ∀sig,n.∀M:mTM sig n.∀ls,a,rs.
202 is_head ?? (nth (S n) ? (vec … a) (blank ?)) = true →
203 cstate sig (states sig n M) n
204 (readback_config (states sig n M) sig n (start sig n M) ls a rs) =
205 get_state (states sig n M) sig (S n) a (start sig n M).
207 axiom eq_current_chars_resize: ∀sig,n.∀M:mTM sig n.∀ls,a,rs.
209 (ctapes sig (states sig n M) n
210 (readback_config (states sig n M) sig n (start sig n M) ls a rs)) =
214 lemma rb_trans : ∀sig,Q,n.∀M:mTM sig n.∀ls,a,rs,q,m.
215 nth (S n) ? (vec … a) (blank ?) = Some ? (inl ?? 〈q,m〉) →
216 transf (states sig n M) sig (S n) (trans sig n M) a =
217 trans sig n M 〈q,get_chars … a〉.
221 ∀A,B,C,f,i,n.∀v1:Vector A n.∀v2:Vector B n.∀d1,d2,d3.i < n→
222 f (nth i ? v1 d1) (nth i ? v2 d2) = nth i ? (pmap_vec A B C f n v1 v2) d3.
223 #A #B #C #f #i elim i
225 [ #v1 #v2 #d1 #d2 #d3 #Hlt @False_ind /2/
226 | #n0 #v1 #v2 #d1 #d2 #d3 #_ >(vec_expand … v1)
227 >(vec_expand … v2) >(nth_default … d3 (f d1 d2)) [% | // ]]
229 [ #v1 #v2 #d1 #d2 #d3 #Hlt @False_ind /2/
230 | #n #v1 #v2 #d1 #d2 #d3 #Hlt>(vec_expand … v1)
232 whd in match (nth …d1); whd in match (tail ??);
233 whd in match (nth … d2); whd in match (tail B ?);
234 whd in match (nth … d3); whd in match (tail C ?);
235 <(IH n ?? d1 d2 d3) [2:@le_S_S_to_le @Hlt] %
239 lemma tape_move_mono_def : ∀sig,t,a,m.
240 tape_move_mono sig t 〈a,m〉 = tape_move sig (tape_write sig t a) m.
243 axiom to_sig_move : ∀Q,sig,n,t,m.
244 to_sig_tape (HC Q (S n)) sig
245 (tape_move (sig_ext (TA (HC Q (S n)) sig)) t m)
246 = tape_move sig (to_sig_tape ?? t) m.
248 definition to_sig_conv :∀Q,sig:FinSet.∀n.
249 option sig → option (sig_ext (TA (HC Q (S n)) sig))
253 |Some a ⇒ Some ? (Some ? (inr (HC Q (S n)) ? a))].
255 axiom to_sig_write : ∀Q,sig,n,t,c.
256 to_sig_tape (HC Q (S n)) sig
257 (tape_write ? t (to_sig_conv ??? c))
258 = tape_write sig (to_sig_tape ?? t) c.
260 definition opt ≝ λA.λoc1: option A.λc2.
261 match oc1 with [None ⇒ c2 | Some c1 ⇒ c1].
263 axiom rb_write: ∀sig,n,ls,a,rs,i,c1,c2.
264 nth i ? c1 (None ?) = opt ? c2 (nth i ? c1 (None ?)) →
265 rb_trace_i ? n ls c1 rs i =
266 tape_write ? (rb_trace_i sig n ls a rs i) c2.
268 lemma sem_stepM : ∀sig,n.∀M:mTM sig n.
269 stepM (states sig n M) sig n (trans sig n M) ⊨
272 @(sem_seq_app … (sem_writef … ) (sem_exec_moves … (le_n ?)))
273 #tin #tout * #t1 * whd in ⊢ (%→?); #Hwrite #Hmoves
274 #a #ls #rs #Htin #H1 #H2 #H3 #H4 >Htin in Hwrite; #Hwrite
275 lapply (Hwrite … (refl …)) -Hwrite whd in match (right ??); whd in match (left ??);
277 cut (∃q,m. nth (S n) ? (vec … a) (blank ?) = Some ? (inl ?? 〈q,m〉))
278 [lapply H2 cases (nth (S n) ? (vec … a) (blank ?))
279 [whd in ⊢ (??%?→?); #H destruct (H)
280 |* #x whd in ⊢ (??%?→?); #H destruct (H) cases x #q #m %{q} %{m} %
284 lapply (transf_eq … HaSn (refl ??) (refl ??) (eq_pair_fst_snd …) (refl ??) (refl ??))
286 lapply (Hmoves … Ht1 ?? H3 H4)
287 [>(transf_eq … HaSn (refl ??) (refl ??) (eq_pair_fst_snd …) (refl ??) (refl ??))
289 | (* regularity is preserved *) @daemon
290 |* #ls1 * #a1 * #rs1 * * * #Htout #Hreg #Hrb #HtrSn
291 lapply (HtrSn (S n) (le_n ?) (le_n ?)) -HtrSn #HtrSn
292 cut (nth (S n) ? (vec … a1) (blank ?) =
293 nth (S n ) ? (vec … (transf (states sig n M) sig (S n) (trans sig n M) a)) (blank ?))
294 [@daemon (* from HtrSn *)] #Ha1
298 lapply(transf_eq_ex … (trans sig n M) … HaSn)
299 * #c1 * #Hc1 * #c2 * #Hc2 * #q1 * #actions *
300 #Htrans * #moves * #Hmoves * #new_chars * #Hnew_chars #Htransf
302 [(* state *) >state_readback whd in match (step ????);
303 >(cstate_rb … HaSn) >eq_current_chars_resize >get_chars_def
304 <Hc1 <Hc2 >Htrans whd in ⊢ (???%);
305 whd in ⊢ (??%?); >Ha1 >HaSn >Htransf >nth_cons_right_n %
306 |>tapes_readback whd in match (step ????);
307 >(cstate_rb … HaSn) >eq_current_chars_resize >get_chars_def
308 <Hc1 <Hc2 >Htrans >ctapes_mk_config
309 @(eq_vec … (niltape ?)) #i #lti
310 >nth_rb_tapes [2:@lti]
311 >Hrb [2:@lt_to_le @lti|3:@lti]
312 >Htransf whd in match (get_move ?????); (* whd in match (vec_moves ?????); *)
313 >get_moves_cons_right
315 <nth_pmap_lt [2:@lti|3:%[@None|@N]|4:@niltape]
316 >ctapes_mk_config >nth_rb_tapes [2:@lti]
317 (* >nth_vec_map_lt [2:@lti |3:@niltape] *)
318 >(eq_pair_fst_snd … (nth i ? actions ?))
320 cut (snd ?? (nth i ? actions 〈None sig,N〉) = nth i ? moves N)
321 [>Hmoves @nth_vec_map] #Hmoves1 >Hmoves1
322 >to_sig_move @eq_f2 [2://]
323 <to_sig_write @eq_f @rb_write
324 >nth_cons_right_lt [2:@lti]
325 >Hnew_chars <nth_pmap_lt [2:@lti|3:@None |4:%[@None|@N]]
327 inversion (\fst (nth i ? actions 〈None sig,N〉))
328 [#Hcase whd in ⊢ (??%%); >Hcase % |#c #Hcase % ]