--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A||
+ \ / This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ V_____________________________________________________________*)
+
+
+include "turing/multi_universal/unistep_aux.ma".
+
+definition unistep ≝
+ match_m cfg prg FSUnialpha 2 ·
+ restart_tape cfg 2 · mmove cfg ? 2 R · copy prg cfg FSUnialpha 2 ·
+ cfg_to_obj · tape_move_obj · restart_tape prg 2 · 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 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.
+
+lemma lt_obj : obj < 3. // qed.
+lemma lt_cfg : cfg < 3. // qed.
+lemma lt_prg : prg < 3. // qed.
+
+definition R_copy_strict ≝
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ ((current ? (nth src ? int (niltape ?)) = None ? ∨
+ current ? (nth dst ? int (niltape ?)) = None ?) → outt = int) ∧
+ (∀ls,x,x0,rs,ls0,rs0.
+ nth src ? int (niltape ?) = midtape sig ls x rs →
+ nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+ |rs0| ≤ |rs| →
+ (∃rs1,rs2.rs = rs1@rs2 ∧ |rs1| = |rs0| ∧
+ outt = change_vec ??
+ (change_vec ?? int
+ (mk_tape sig (reverse sig rs1@x::ls) (option_hd sig rs2)
+ (tail sig rs2)) src)
+ (mk_tape sig (reverse sig rs1@x::ls0) (None sig) []) dst)).
+
+axiom sem_copy_strict : ∀src,dst,sig,n. src ≠ dst → src < S n → dst < S n →
+ copy src dst sig n ⊨ R_copy_strict src dst sig n.
+
+lemma sem_unistep : ∀n,l,h.unistep ⊨ R_unistep n l h.
+#n #l #h
+@(sem_seq_app ??????? (sem_match_m cfg prg FSUnialpha 2 ???)
+ (sem_seq ?????? (sem_restart_tape ???)
+ (sem_seq ?????? (sem_move_multi ? 2 cfg R ?)
+ (sem_seq ?????? (sem_copy_strict prg cfg FSUnialpha 2 ???)
+ (sem_seq ?????? sem_cfg_to_obj
+ (sem_seq ?????? sem_tape_move_obj
+ (sem_seq ?????? (sem_restart_tape ???) sem_obj_to_cfg)))))))
+ /2 by le_n,sym_not_eq/
+#ta #tb #HR #state #char #table #Hta_cfg #Hcfg #Hta_prg #Htable
+#Hbits_obj #Htotaltable
+#nstate #nchar #m #t #Htuple #Hmatch
+cases HR -HR #tc * whd in ⊢ (%→?);
+>Hta_cfg #H cases (H ?? (refl ??)) -H
+(* prg starts with a bar, so it's not empty *) #_
+>Hta_prg #H lapply (H ??? (refl ??)) -H *
+[| cases Htotaltable #ll * #lr #H >H
+ #Hfalse @False_ind cases (Hfalse ll lr) #H1 @H1 //]
+* #ll * #lr * #Hintable -Htotaltable #Htc
+* #td * whd in ⊢ (%→?); >Htc
+>nth_change_vec_neq [|@sym_not_eq //] >(nth_change_vec ?????? lt_cfg)
+#Htd lapply (Htd ? (refl ??)) -Htd
+>change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
+>(?: list_of_tape ? (mk_tape ? (reverse ? (state@[char])@[bar]) (None ?) [ ]) =
+ bar::state@[char])
+[|whd in ⊢ (??%?); >left_mk_tape >reverse_append >reverse_reverse
+ >current_mk_tape >right_mk_tape normalize >append_nil % ]
+whd in ⊢ (???(???(????%?)??)→?); whd in match (tail ??); #Htd
+(* move cfg to R *)
+* #te * whd in ⊢ (%→?); >Htd
+>change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
+>nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
+>Htable in Hintable; #Hintable #Hte
+(* copy *)
+cases (cfg_in_table_to_tuple ???? Hcfg ?? Hintable)
+#newstate * #m0 * #lr0 * * #Hlr destruct (Hlr) #Hnewcfg #Hm0
+cut (∃fo,so,co.state = fo::so@[co] ∧ |so| = n)
+[ @daemon ] * #fo * #so * #co * #Hstate_exp #Hsolen
+cut (∃fn,sn,cn.newstate = fn::sn@[cn] ∧ |sn| = n)
+[ @daemon ] * #fn * #sn * #cn * #Hnewstate_exp #Hsnlen
+* #tf * * #_ >Hte >(nth_change_vec ?????? lt_prg)
+>nth_change_vec_neq [|@sym_not_eq //] >(nth_change_vec ?????? lt_cfg)
+>Hstate_exp >Hnewstate_exp
+whd in match (mk_tape ????); whd in match (tape_move ???);
+#Htf cases (Htf ?????? (refl ??) (refl ??) ?) -Htf
+[| whd in match (tail ??); >length_append >length_append
+ >Hsolen >length_append >length_append >Hsnlen
+ <plus_n_Sm <plus_n_Sm <plus_n_Sm <plus_n_O <plus_n_O normalize // ]
+#rs1 * #rs2 whd in match (tail ??); * *
+>append_cons #Hrs1rs2 #Hrs1len
+>change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
+>change_vec_change_vec #Htf
+(* cfg to obj *)
+* #tg * whd in ⊢ (%→?); >Htf
+>nth_change_vec_neq [|@sym_not_eq //] >(nth_change_vec ?????? lt_cfg)
+>(change_vec_commute ????? cfg prg) [2:@eqb_false_to_not_eq %]
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+lapply (append_l1_injective ?????? Hrs1rs2)
+[ >Hsnlen >Hrs1len >length_append >length_append >length_append >length_append
+ normalize >Hsolen >Hsnlen % ]
+#Hrs1 <Hrs1 >reverse_append #Htg cases (Htg ?? (refl ??)) -Htg
+cases m0
+ [#mv #_ #Htg #_
+
+
+
+
+
+[ *
+
+ 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 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: ∀M:normalTM.∀c:nconfig (no_states M).Vector ? 3 ≝
+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.λt1,t2.
+∀c:nconfig (no_states M).
+ t1 = low_tapes M c →
+ t2 = low_tapes M (step ? M c).
+
+lemma R_unistep_equiv : ∀M,t1,t2.
+ R_unistep (no_states M) (graph_enum ?? (ntrans M)) (nhalt M) t1 t2 →
+ R_unistep_high M t1 t2.
+#M #t1 #t2 #H whd whd in match (nconfig ?); #c #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 @(eq_vec ? 3 ?? (niltape ?) ?) >Ht2 #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.
]
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 R_cfg_to_obj1 ≝ λt1,t2:Vector (tape FSUnialpha) 3.
+ ∀c,ls.
+ nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
+ c ≠ bar →
+ let new_obj ≝
+ tape_write ? (nth obj ? t1 (niltape ?)) (char_to_bit_option c) in
+ t2 = change_vec ??
+ (change_vec ?? t1 new_obj obj)
+ (mk_tape ? [ ] (option_hd ? (reverse ? (c::ls))) (tail ? (reverse ? (c::ls)))) cfg.
+
+lemma sem_cfg_to_obj1: cfg_to_obj ⊨ R_cfg_to_obj1.
+@(Realize_to_Realize … sem_cfg_to_obj) #t1 #t2 #H #c #ls #Hcfg #Hbar
+cases (H c ls Hcfg) cases (true_or_false (c==null)) #Hc
+ [#Ht2 #_ >(Ht2 (\P Hc)) -Ht2 @(eq_vec … (niltape ?))
+ #i #lei2 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
+ [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
+ [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec_neq in ⊢ (???%); [2:@eqb_false_to_not_eq %]
+ >nth_change_vec // >(\P Hc) %
+ |#Hi >Hi >nth_change_vec //
+ ]
+ |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %] %
+ ]
+ |#_ #Ht2 >(Ht2 (\Pf Hc)) -Ht2 @(eq_vec … (niltape ?))
+ #i #lei2 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
+ [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
+ [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec_neq in ⊢ (???%); [2:@eqb_false_to_not_eq %]
+ >nth_change_vec // >nth_change_vec //
+ lapply (\bf Hbar) lapply Hc elim c //
+ [whd in ⊢ (??%?→?); #H destruct (H)
+ |#_ whd in ⊢ (??%?→?); #H destruct (H)
+ ]
+ |#Hi >Hi >nth_change_vec //
+ ]
+ |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %] %
+ ]
+ ]
+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 tape_move_obj : mTM FSUnialpha 2 ≝
ifTM ??
(inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
]
qed.
-definition unistep ≝
- match_m cfg prg FSUnialpha 2 ·
- restart_tape cfg 2 · mmove cfg ? 2 R · copy prg cfg FSUnialpha 2 ·
- cfg_to_obj · tape_move_obj · restart_tape prg 2 · 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 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.
-
-lemma lt_obj : obj < 3. // qed.
-lemma lt_cfg : cfg < 3. // qed.
-lemma lt_prg : prg < 3. // qed.
-
-definition R_copy_strict ≝
- λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
- ((current ? (nth src ? int (niltape ?)) = None ? ∨
- current ? (nth dst ? int (niltape ?)) = None ?) → outt = int) ∧
- (∀ls,x,x0,rs,ls0,rs0.
- nth src ? int (niltape ?) = midtape sig ls x rs →
- nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
- |rs0| ≤ |rs| →
- (∃rs1,rs2.rs = rs1@rs2 ∧ |rs1| = |rs0| ∧
- outt = change_vec ??
- (change_vec ?? int
- (mk_tape sig (reverse sig rs1@x::ls) (option_hd sig rs2)
- (tail sig rs2)) src)
- (mk_tape sig (reverse sig rs1@x::ls0) (None sig) []) dst)).
-
-axiom sem_copy_strict : ∀src,dst,sig,n. src ≠ dst → src < S n → dst < S n →
- copy src dst sig n ⊨ R_copy_strict src dst sig n.
-
-lemma sem_unistep : ∀n,l,h.unistep ⊨ R_unistep n l h.
-#n #l #h
-@(sem_seq_app ??????? (sem_match_m cfg prg FSUnialpha 2 ???)
- (sem_seq ?????? (sem_restart_tape ???)
- (sem_seq ?????? (sem_move_multi ? 2 cfg R ?)
- (sem_seq ?????? (sem_copy_strict prg cfg FSUnialpha 2 ???)
- (sem_seq ?????? sem_cfg_to_obj
- (sem_seq ?????? sem_tape_move_obj
- (sem_seq ?????? (sem_restart_tape ???) sem_obj_to_cfg)))))))
- /2 by le_n,sym_not_eq/
-#ta #tb #HR #state #char #table #Hta_cfg #Hcfg #Hta_prg #Htable
-#Hbits_obj #Htotaltable
-#nstate #nchar #m #t #Htuple #Hmatch
-cases HR -HR #tc * whd in ⊢ (%→?);
->Hta_cfg #H cases (H ?? (refl ??)) -H
-(* prg starts with a bar, so it's not empty *) #_
->Hta_prg #H lapply (H ??? (refl ??)) -H *
-[| cases Htotaltable #ll * #lr #H >H
- #Hfalse @False_ind cases (Hfalse ll lr) #H1 @H1 //]
-* #ll * #lr * #Hintable -Htotaltable #Htc
-* #td * whd in ⊢ (%→?); >Htc
->nth_change_vec_neq [|@sym_not_eq //] >(nth_change_vec ?????? lt_cfg)
-#Htd lapply (Htd ? (refl ??)) -Htd
->change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
->(?: list_of_tape ? (mk_tape ? (reverse ? (state@[char])@[bar]) (None ?) [ ]) =
- bar::state@[char])
-[|whd in ⊢ (??%?); >left_mk_tape >reverse_append >reverse_reverse
- >current_mk_tape >right_mk_tape normalize >append_nil % ]
-whd in ⊢ (???(???(????%?)??)→?); whd in match (tail ??); #Htd
-(* move cfg to R *)
-* #te * whd in ⊢ (%→?); >Htd
->change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
->nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
->Htable in Hintable; #Hintable #Hte
-(* copy *)
-cases (cfg_in_table_to_tuple ???? Hcfg ?? Hintable)
-#newstate * #m0 * #lr0 * * #Hlr destruct (Hlr) #Hnewcfg #Hm0
-cut (∃fo,so,co.state = fo::so@[co] ∧ |so| = n)
-[ @daemon ] * #fo * #so * #co * #Hstate_exp #Hsolen
-cut (∃fn,sn,cn.newstate = fn::sn@[cn] ∧ |sn| = n)
-[ @daemon ] * #fn * #sn * #cn * #Hnewstate_exp #Hsnlen
-* #tf * * #_ >Hte >(nth_change_vec ?????? lt_prg)
->nth_change_vec_neq [|@sym_not_eq //] >(nth_change_vec ?????? lt_cfg)
->Hstate_exp >Hnewstate_exp
-whd in match (mk_tape ????); whd in match (tape_move ???);
-#Htf cases (Htf ?????? (refl ??) (refl ??) ?)
-[| whd in match (tail ??); >length_append >length_append
- >Hsolen >length_append >length_append >Hsnlen
- <plus_n_Sm <plus_n_Sm <plus_n_Sm <plus_n_O <plus_n_O normalize // ]
-#rs1 * #rs2 whd in match (tail ??); * *
->append_cons #Hrs1rs2 #Hrs1len
->change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
->change_vec_change_vec #Htf
-(* cfg to obj *)
-* #tg * whd in ⊢ (%→?); >Htf
->nth_change_vec_neq [|@sym_not_eq //]
->(nth_change_vec ?????? lt_cfg)
-lapply (append_l1_injective ?????? Hrs1rs2)
-[ >Hsnlen >Hrs1len >length_append >length_append >length_append >length_append
- normalize >Hsolen >Hsnlen % ]
-#Hrs1 <Hrs1 >reverse_append #Htg cases (Htg ?? (refl ??)) -Htg #Htg1 #Htg2
-
-
-
-
-
-[ *
-
- 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 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: ∀M:normalTM.∀c:nconfig (no_states M).Vector ? 3 ≝
-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.λt1,t2.
-∀c:nconfig (no_states M).
- t1 = low_tapes M c →
- t2 = low_tapes M (step ? M c).
-
-lemma R_unistep_equiv : ∀M,t1,t2.
- R_unistep (no_states M) (graph_enum ?? (ntrans M)) (nhalt M) t1 t2 →
- R_unistep_high M t1 t2.
-#M #t1 #t2 #H whd whd in match (nconfig ?); #c #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 @(eq_vec ? 3 ?? (niltape ?) ?) >Ht2 #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.
projects / helm.git / commitdiff