include "turing/multi_universal/match.ma".
include "turing/multi_universal/copy.ma".
include "turing/multi_universal/alphabet.ma".
+include "turing/multi_universal/tuples.ma".
(*
| <Houtc % ] ]
qed.
+definition copy_char_states ≝ initN 3.
+
+definition cc0 : copy_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition cc1 : copy_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+
+definition trans_copy_char ≝
+ λsrc,dst.λsig:FinSet.λn.
+ λp:copy_char_states × (Vector (option sig) (S n)).
+ let 〈q,a〉 ≝ p in
+ match pi1 … q with
+ [ O ⇒ 〈cc1,change_vec ? (S n)
+ (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) src)
+ (〈nth src ? a (None ?),R〉) dst〉
+ | S _ ⇒ 〈cc1,null_action ? n〉 ].
+
+definition copy_char ≝
+ λsrc,dst,sig,n.
+ mk_mTM sig n copy_char_states (trans_copy_char src dst sig n)
+ cc0 (λq.q == cc1).
+
+definition R_copy_char ≝
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ outt = change_vec ??
+ (change_vec ?? int
+ (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src)
+ (tape_move_mono ? (nth dst ? int (niltape ?))
+ 〈current ? (nth src ? int (niltape ?)), R〉) dst.
+
+lemma copy_char_q0_q1 :
+ ∀src,dst,sig,n,v.src ≠ dst → src < S n → dst < S n →
+ step sig n (copy_char src dst sig n) (mk_mconfig ??? cc0 v) =
+ mk_mconfig ??? cc1
+ (change_vec ? (S n)
+ (change_vec ?? v
+ (tape_move_mono ? (nth src ? v (niltape ?)) 〈None ?, R〉) src)
+ (tape_move_mono ? (nth dst ? v (niltape ?)) 〈current ? (nth src ? v (niltape ?)), R〉) dst).
+#src #dst #sig #n #v #Heq #Hsrc #Hdst
+whd in ⊢ (??%?);
+<(change_vec_same … v dst (niltape ?)) in ⊢ (??%?);
+<(change_vec_same … v src (niltape ?)) in ⊢ (??%?);
+>tape_move_multi_def @eq_f2 //
+>pmap_change >pmap_change <tape_move_multi_def
+>tape_move_null_action @eq_f2 // @eq_f2
+[ >change_vec_same %
+| >change_vec_same >change_vec_same // ]
+qed.
+
+lemma sem_copy_char:
+ ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
+ copy_char src dst sig n ⊨ R_copy_char src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst #int
+%{2} % [| % [ % | whd >copy_char_q0_q1 // ]]
+qed.
+
definition cfg_to_obj ≝
mmove cfg FSUnialpha 2 L ·
(ifTM ?? (inject_TM ? test_null_char 2 cfg)
(nop ? 2)
- (copy_step cfg obj FSUnialpha 2 ·
+ (copy_char cfg obj FSUnialpha 2 ·
mmove cfg FSUnialpha 2 L ·
mmove obj FSUnialpha 2 L)
tc_true) ·
(change_vec ?? t1
(midtape ? (left ? (nth obj ? t1 (niltape ?))) c (right ? (nth obj ? t1 (niltape ?)))) obj)
(mk_tape ? [ ] (option_hd ? (reverse ? (c::ls))) (tail ? (reverse ? (c::ls)))) cfg).
+
+lemma tape_move_mk_tape_L :
+ ∀sig,ls,c,rs.
+ (c = None ? → ls = [ ] ∨ rs = [ ]) →
+ tape_move ? (mk_tape sig ls c rs) L =
+ mk_tape ? (tail ? ls) (option_hd ? ls) (option_cons ? c rs).
+#sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
+[| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ]
+normalize //
+qed.
-axiom sem_cfg_to_obj : cfg_to_obj ⊨ R_cfg_to_obj.
-(*@(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
- (sem_seq ??????
- (sem_if ??????????
- (sem_test_null_multi ?? obj ?)
- (sem_seq ?????? (accRealize_to_Realize … (sem_copy_step …))
- (sem_move_multi ? 2 cfg L ?))
- (sem_inject ???? cfg ? (sem_write FSUnialpha null)))
- (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
- (sem_move_multi ? 2 cfg R ?)))) //
+lemma sem_cfg_to_obj : cfg_to_obj ⊨ R_cfg_to_obj.
+@(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
+ (sem_seq ??????
+ (sem_if ??????????
+ (acc_sem_inject ?????? cfg ? sem_test_null_char)
+ (sem_nop …)
+ (sem_seq ?????? (sem_copy_char …)
+ (sem_seq ?????? (sem_move_multi ? 2 cfg L ?) (sem_move_multi ? 2 obj L ?))))
+ (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
+ (sem_move_multi ? 2 cfg R ?)))) // [@sym_not_eq //]
#ta #tb *
#tc * whd in ⊢ (%→?); #Htc *
#td * *
-[ * #te * * #Hcurtc #Hte
- * destruct (Hte) #te * *
- [ whd in ⊢ (%→%→?); * #x * #y * * -Hcurtc #Hcurtc1 #Hcurtc2 #Hte #Htd
- * #tf * * * whd in ⊢ (%→%→%→%→?); #Htf1 #Htf2 #Htf3 #Htb
- #c #ls #Hta1 %
- [ #lso #x0 #rso #Hta2 >Hta1 in Htc; >eq_mk_tape_rightof
- whd in match (tape_move ???); #Htc
- cut (tf = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls@[x])) cfg)
- [@daemon] -Htf1 -Htf2 -Htf3 #Htf destruct (Htf Hte Htd Htc Htb)
- >change_vec_change_vec >change_vec_change_vec >change_vec_change_vec
- >nth_change_vec // >tape_move_mk_tape_R
- @daemon
- | #Hta2 >Htc in Hcurtc1; >nth_change_vec_neq [| @sym_not_eq //]
- >Hta2 #H destruct (H)
- ]
- | * #Hcurtc0 #Hte #_ #_ #c #ls #Hta1 >Hta1 in Htc; >eq_mk_tape_rightof
- whd in match (tape_move ???); #Htc >Htc in Hcurtc0; *
- [ >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
- #Hcurtc #Hcurtc0 >Hcurtc0 in Hcurtc; * #H @False_ind @H %
- | >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H) ]
- ]
-| * #te * * #Hcurtc #Hte
- * whd in ⊢ (%→%→?); #Htd1 #Htd2
- * #tf * * * #Htf1 #Htf2 #Htf3 whd in ⊢ (%→?); #Htb
- #c #ls #Hta1 %
- [ #lso #x #rso #Hta2 >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
- >Hta2 normalize in ⊢ (%→?); #H destruct (H)
- | #_ >Hta1 in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
- destruct (Hte) cut (td = change_vec ?? tc (midtape ? ls null []) cfg)
- [@daemon] -Htd1 -Htd2 #Htd
- -Htf1 cut (tf = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[null])) cfg)
- [@daemon] -Htf2 -Htf3 #Htf destruct (Htf Htd Htc Htb)
+[ * #te * * * #Hcurtc #Hte1 #Hte2 whd in ⊢ (%→?); #Htd destruct (Htd)
+ * #tf * * * #Htf1 #Htf2 #Htf3
+ whd in ⊢ (%→?); #Htb
+ #c #ls #Hta %
+ [ #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
+ cut (te = tc) [@daemon] -Hte1 -Hte2 #Hte
+ cut (tf = change_vec ? 3 te (mk_tape ? [ ] (None ?) (reverse ? ls@[c])) cfg)
+ [@daemon] -Htf1 -Htf2 -Htf3 #Htf
+ destruct (Htf Hte Htc Htb)
>change_vec_change_vec >change_vec_change_vec >change_vec_change_vec
- >change_vec_change_vec >change_vec_change_vec >nth_change_vec //
- >reverse_cons >tape_move_mk_tape_R /2/ ]
+ >nth_change_vec // >tape_move_mk_tape_R [| #_ % % ]
+ >reverse_cons %
+ | #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
+ >Htc in Hcurtc; >nth_change_vec // normalize in ⊢ (%→?);
+ #H destruct (H) @False_ind cases Hc /2/ ]
+ * #tf * *
+| * #te * * * #Hcurtc #Hte1 #Hte2
+ * #tf * whd in ⊢ (%→?); #Htf
+ * #tg * whd in ⊢ (%→%→?); #Htg #Htd
+ * #th * * * #Hth1 #Hth2 #Hth3
+ whd in ⊢ (%→?); #Htb
+ #c #ls #Hta % #Hc
+ [ >Htc in Hcurtc; >Hta >nth_change_vec // >tape_move_mk_tape_L //
+ >Hc normalize in ⊢ (%→?); * #H @False_ind /2/
+ | cut (te = tc) [ @daemon ] -Hte1 -Hte2 #Hte
+ cut (th = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[c])) cfg)
+ [@daemon] -Hth1 -Hth2 -Hth3 #Hth
+ destruct (Hth Hte Hta Htb Htd Htg Htc Htf)
+ >change_vec_change_vec >change_vec_change_vec
+ >change_vec_commute // >change_vec_change_vec
+ >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
+ >change_vec_commute // >change_vec_change_vec
+ >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
+ >change_vec_commute [|@sym_not_eq //]
+ @eq_f3 //
+ [ >Hta >tape_move_mk_tape_L // >nth_change_vec // whd in match (current ??);
+ @eq_f2 // cases (nth obj ? ta (niltape ?))
+ [| #r0 #rs0 | #l0 #ls0 | #ls0 #c0 #rs0 ] try %
+ cases rs0 //
+ | >reverse_cons >tape_move_mk_tape_R // #_ % % ]
+ ]
]
qed.
-*)
(* macchina che muove il nastro obj a destra o sinistra a seconda del valore
del current di prg, che codifica la direzione in cui ci muoviamo *)
-
+
+definition char_to_move ≝ λc.match c with
+ [ bit b ⇒ if b then R else L
+ | _ ⇒ N].
+
+definition char_to_bit_option ≝ λc.match c with
+ [ bit b ⇒ Some ? (bit b)
+ | _ ⇒ None ?].
+
definition tape_move_obj : mTM FSUnialpha 2 ≝
ifTM ??
(inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
tc_true)
tc_true.
+definition restart_tape ≝ λi.
+ inject_TM ? (move_to_end FSUnialpha L) 2 i ·
+ mmove i FSUnialpha 2 R.
+
definition unistep ≝
- obj_to_cfg · match_m cfg prg FSUnialpha 2 · copy prg cfg FSUnialpha 2 ·
- cfg_to_obj · tape_move_obj.
\ No newline at end of file
+ match_m cfg prg FSUnialpha 2 ·
+ restart_tape cfg · copy prg cfg FSUnialpha 2 ·
+ cfg_to_obj · tape_move_obj · restart_tape prg · obj_to_cfg.
+
+(*
+definition legal_tape ≝ λn,l,h,t.
+ ∃state,char,table.
+ nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[char]) →
+ is_config n (bar::state@[char]) →
+ nth prg ? t1 (niltape ?) = midtape ? [ ] bar table →
+ bar::table = table_TM n l h → *)
+
+definition list_of_tape ≝ λsig,t.
+ left sig t@option_cons ? (current ? t) (right ? t).
+
+definition low_char' ≝ λc.
+ match c with
+ [ None ⇒ null
+ | Some b ⇒ if (is_bit b) then b else null
+ ].
+
+lemma low_char_option : ∀s.
+ low_char' (option_map FinBool FSUnialpha bit s) = low_char s.
+* //
+qed.
+
+definition R_unistep ≝ λn,l,h.λt1,t2: Vector ? 3.
+ ∀state,char,table.
+ (* cfg *)
+ nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[char]) →
+ is_config n (bar::state@[char]) →
+ (* prg *)
+ nth prg ? t1 (niltape ?) = midtape ? [ ] bar table →
+ bar::table = table_TM n l h →
+ (* obj *)
+ only_bits (list_of_tape ? (nth obj ? t1 (niltape ?))) →
+ let conf ≝ (bar::state@[char]) in
+ (∃ll,lr.bar::table = ll@conf@lr) →
+(*
+ ∃nstate,nchar,m,t. tuple_encoding n h t = (conf@nstate@[nchar;m]) ∧
+ mem ? t l ∧ *)
+ ∀nstate,nchar,m,t.
+ tuple_encoding n h t = (conf@nstate@[nchar;m])→
+ mem ? t l →
+ let new_obj ≝
+ tape_move_mono ? (nth obj ? t1 (niltape ?))
+ 〈char_to_bit_option nchar,char_to_move m〉 in
+ let next_char ≝ low_char' (current ? new_obj) in
+ t2 =
+ change_vec ??
+ (change_vec ?? t1 (midtape ? [ ] bar (nstate@[next_char])) cfg)
+ new_obj obj.
+
+definition tape_map ≝ λA,B:FinSet.λf:A→B.λt.
+ mk_tape B (map ?? f (left ? t))
+ (option_map ?? f (current ? t))
+ (map ?? f (right ? t)).
+
+lemma map_list_of_tape: ∀A,B,f,t.
+ list_of_tape B (tape_map ?? f t) = map ?? f (list_of_tape A t).
+#A #B #f * // normalize // #ls #c #rs <map_append %
+qed.
+
+lemma low_char_current : ∀t.
+ low_char' (current FSUnialpha (tape_map FinBool FSUnialpha bit t))
+ = low_char (current FinBool t).
+* // qed.
+
+definition low_tapes ≝ λM:normalTM.λc:nconfig (no_states M).Vector_of_list ?
+ [tape_map ?? bit (ctape ?? c);
+ midtape ? [ ] bar
+ ((bits_of_state ? (nhalt M) (cstate ?? c))@[low_char (current ? (ctape ?? c))]);
+ midtape ? [ ] bar (tail ? (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M)))
+ ].
+
+lemma obj_low_tapes: ∀M,c.
+ nth obj ? (low_tapes M c) (niltape ?) = tape_map ?? bit (ctape ?? c).
+// qed.
+
+lemma cfg_low_tapes: ∀M,c.
+ nth cfg ? (low_tapes M c) (niltape ?) =
+ midtape ? [ ] bar
+ ((bits_of_state ? (nhalt M) (cstate ?? c))@[low_char (current ? (ctape ?? c))]).
+// qed.
+
+lemma prg_low_tapes: ∀M,c.
+ nth prg ? (low_tapes M c) (niltape ?) =
+ midtape ? [ ] bar (tail ? (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M))).
+// qed.
+
+(* commutation lemma for write *)
+lemma map_write: ∀t,cout.
+ tape_write ? (tape_map FinBool ? bit t) (char_to_bit_option (low_char cout))
+ = tape_map ?? bit (tape_write ? t cout).
+#t * // #b whd in match (char_to_bit_option ?);
+whd in ⊢ (??%%); @eq_f3 [elim t // | // | elim t //]
+qed.
+
+(* commutation lemma for moves *)
+lemma map_move: ∀t,m.
+ tape_move ? (tape_map FinBool ? bit t) (char_to_move (low_mv m))
+ = tape_map ?? bit (tape_move ? t m).
+#t * // whd in match (char_to_move ?);
+ [cases t // * // | cases t // #ls #a * //]
+qed.
+
+(* commutation lemma for actions *)
+lemma map_action: ∀t,cout,m.
+ tape_move ? (tape_write ? (tape_map FinBool ? bit t)
+ (char_to_bit_option (low_char cout))) (char_to_move (low_mv m))
+ = tape_map ?? bit (tape_move ? (tape_write ? t cout) m).
+#t #cout #m >map_write >map_move %
+qed.
+
+lemma map_move_mono: ∀t,cout,m.
+ tape_move_mono ? (tape_map FinBool ? bit t)
+ 〈char_to_bit_option (low_char cout), char_to_move (low_mv m)〉
+ = tape_map ?? bit (tape_move_mono ? t 〈cout,m〉).
+@map_action
+qed.
+
+definition R_unistep_high ≝ λM:normalTM.λc:nconfig (no_states M).λt1,t2.
+ t1 = low_tapes M c →
+ t2 = low_tapes M (step ? M c).
+
+lemma R_unistep_equiv : ∀M,c,t1,t2.
+ R_unistep (no_states M) (graph_enum ?? (ntrans M)) (nhalt M) t1 t2 →
+ R_unistep_high M c t1 t2.
+#M #c #t1 #t2 #H #Ht1
+lapply (initial_bar ? (nhalt M) (graph_enum ?? (ntrans M)) (nTM_nog ?)) #Htable
+(* tup = current tuple *)
+cut (∃t.t = 〈〈cstate … c,current ? (ctape … c)〉,
+ ntrans M 〈cstate … c,current ? (ctape … c)〉〉) [% //] * #tup #Htup
+(* tup is in the graph *)
+cut (mem ? tup (graph_enum ?? (ntrans M)))
+ [@memb_to_mem >Htup @(graph_enum_complete … (ntrans M)) %] #Hingraph
+(* tupe target = 〈qout,cout,m〉 *)
+lapply (decomp_target ? (ntrans M 〈cstate … c,current ? (ctape … c)〉))
+* #qout * #cout * #m #Htg >Htg in Htup; #Htup
+(* new config *)
+cut (step FinBool M c = mk_config ?? qout (tape_move ? (tape_write ? (ctape … c) cout) m))
+ [>(config_expand … c) whd in ⊢ (??%?); (* >Htg ?? why not?? *)
+ cut (trans ? M 〈cstate … c, current ? (ctape … c)〉 = 〈qout,cout,m〉) [<Htg %] #Heq1
+ >Heq1 %] #Hstep
+(* new state *)
+cut (cstate ?? (step FinBool M c) = qout) [>Hstep %] #Hnew_state
+(* new tape *)
+cut (ctape ?? (step FinBool M c) = tape_move ? (tape_write ? (ctape … c) cout) m)
+ [>Hstep %] #Hnew_tape
+lapply(H (bits_of_state ? (nhalt M) (cstate ?? c))
+ (low_char (current ? (ctape ?? c)))
+ (tail ? (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M)))
+ ??????)
+[<Htable
+ lapply(list_to_table … (nhalt M) …Hingraph) * #ll * #lr #Htable1 %{ll}
+ %{(((bits_of_state ? (nhalt M) qout)@[low_char cout;low_mv m])@lr)}
+ >Htable1 @eq_f <associative_append @eq_f2 // >Htup
+ whd in ⊢ (??%?); @eq_f >associative_append %
+|>Ht1 >obj_low_tapes >map_list_of_tape elim (list_of_tape ??)
+ [#b @False_ind | #b #tl #Hind #a * [#Ha >Ha //| @Hind]]
+|@sym_eq @Htable
+|>Ht1 %
+|%{(bits_of_state ? (nhalt M) (cstate ?? c))} %{(low_char (current ? (ctape ?? c)))}
+ % [% [% [// | cases (current ??) normalize [|#b] % #Hd destruct (Hd)]
+ |>length_map whd in match (length ??); @eq_f //]
+ |//]
+|>Ht1 >cfg_low_tapes //] -H #H
+lapply(H (bits_of_state … (nhalt M) qout) (low_char … cout)
+ (low_mv … m) tup ? Hingraph)
+ [>Htup whd in ⊢ (??%?); @eq_f >associative_append %] -H
+#Ht2 >Ht2 @(eq_vec ? 3 … (niltape ?)) #i #Hi
+cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
+ [cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
+ [cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
+ [@False_ind /2/
+ |>Hi >obj_low_tapes >nth_change_vec //
+ >Ht1 >obj_low_tapes >Hstep @map_action
+ ]
+ |>Hi >cfg_low_tapes >nth_change_vec_neq
+ [|% whd in ⊢ (??%?→?); #H destruct (H)]
+ >nth_change_vec // >Hnew_state @eq_f @eq_f >Hnew_tape
+ @eq_f2 [|2:%] >Ht1 >obj_low_tapes >map_move_mono >low_char_current %
+ ]
+ |(* program tapes do not change *)
+ >Hi >prg_low_tapes
+ >nth_change_vec_neq [|% whd in ⊢ (??%?→?); #H destruct (H)]
+ >nth_change_vec_neq [|% whd in ⊢ (??%?→?); #H destruct (H)]
+ >Ht1 >prg_low_tapes //
+ ]
+qed.
+
+
+
+
\ No newline at end of file
projects / helm.git / blobdiff