include "turing/multi_universal/unistep_aux.ma".
+definition exec_move ≝
+ cfg_to_obj · tape_move_obj · restart_tape prg 2 · obj_to_cfg.
+
+definition R_exec_move ≝ λt1,t2:Vector (tape FSUnialpha) 3.
+∀c,m,ls1,ls2,rs2.
+ nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls1@[bar]) (None ?) [ ] →
+ nth prg ? t1 (niltape ?) = midtape FSUnialpha (ls2@[bar]) m rs2 →
+ only_bits (list_of_tape ? (nth obj ? t1 (niltape ?))) →
+ c ≠ bar → m ≠ bar →
+ let new_obj ≝
+ tape_move_mono ? (nth obj ? t1 (niltape ?))
+ 〈char_to_bit_option c, char_to_move m〉 in
+ let next_c ≝ low_char' (current ? new_obj) in
+ let new_cfg ≝ midtape ? [ ] bar ((reverse ? ls1)@[next_c]) in
+ let new_prg ≝ midtape FSUnialpha [ ] bar ((reverse ? ls2)@m::rs2) in
+ t2 = Vector_of_list ? [new_obj;new_cfg;new_prg].
+
+
+lemma sem_exec_move: exec_move ⊨ R_exec_move.
+@(sem_seq_app ??????? sem_cfg_to_obj1
+ (sem_seq ?????? sem_tape_move_obj
+ (sem_seq ?????? (sem_restart_tape ???) sem_obj_to_cfg1))) //
+#ta #tout * #t1 * #semM1 * #t2 * #semM2 * #t3 * #semM3 #semM4
+#c #m #ls1 #ls2 #rs2 #Hcfg #Hprg #Honlybits #Hc #Hm
+(* M1 = cfg_to_obj *)
+lapply (semM1 … Hcfg Hc) #Ht1
+(* M2 = tape_move *)
+whd in semM2; >Ht1 in semM2; -Ht1
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>Hprg #Ht2 lapply (Ht2 … (refl ??)) -Ht2
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>nth_change_vec // >change_vec_commute [2:@eqb_false_to_not_eq %]
+>change_vec_change_vec #Ht2
+(* M3 = restart prg *)
+whd in semM3; >Ht2 in semM3; #semM3 lapply (semM3 … (refl ??)); -semM3
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %] #Ht3
+(* M4 = obj_to_cfg *)
+whd in semM4; >Ht3 in semM4;
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>nth_change_vec [2:@leb_true_to_le %] #semM4 lapply (semM4 … (refl ??)) -semM4
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>nth_change_vec [2:@leb_true_to_le %] #semM4 >(semM4 ?)
+ [(* tape by tape *)
+ @(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 [2:@eqb_false_to_not_eq %]
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec [2:@leb_true_to_le %] %
+ |#Hi >Hi (* obj tape *)
+ >nth_change_vec [2:@leb_true_to_le %] whd in ⊢ (???%);
+ >reverse_cons >reverse_append >reverse_single
+ whd in match (option_hd ??); whd in match (tail ??);
+ whd in ⊢ (??%?); %
+ ]
+ |#Hi >Hi (* prg tape *)
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec [2:@leb_true_to_le %] whd in ⊢ (???%);
+ >Hprg whd in match (list_of_tape ??); >reverse_append
+ >reverse_single %
+ ]
+ |#b #Hcurrent @(list_of_tape_write ? is_bit … (char_to_bit_option c) ? Honlybits)
+ [#b0 cases c
+ [#b1 whd in ⊢ (??%?→?); #Hbit destruct (Hbit) %
+ |whd in ⊢ (??%?→?); #Hbit destruct (Hbit)
+ |whd in ⊢ (??%?→?); #Hbit destruct (Hbit)
+ ]
+ |>(list_of_tape_move … (char_to_move m)) @current_in_list @Hcurrent
+ ]
+ ]
+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.
+ exec_move.
(*
definition legal_tape ≝ λn,l,h,t.
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 *)
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.
+axiom daemon : ∀P:Prop.P.
+
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)))))))
+ (sem_exec_move …)))))
/2 by le_n,sym_not_eq/
#ta #tb #HR #state #char #table #Hta_cfg #Hcfg #Hta_prg #Htable
#Hbits_obj #Htotaltable
>(?: 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 % ]
+ >current_mk_tape >right_mk_tape [| #_ %2 % ] normalize >append_nil % ]
whd in ⊢ (???(???(????%?)??)→?); whd in match (tail ??); #Htd
(* move cfg to R *)
* #te * whd in ⊢ (%→?); >Htd
>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)
+lapply (cfg_in_table_to_tuple ???? Hcfg ?? Hintable)
+* #newconfig * #m0 * #lr0 * * #Hlr destruct (Hlr) #Hnewcfg #Hm0
+cut (∃fo,so.state = fo::so ∧ |so| = n)
+[ @daemon ] * #fo * #so * #Hstate_exp #Hsolen
+cut (∃fn,sn,cn.newconfig = 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)
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 // ]
+ >Hsolen >length_append >Hsnlen //]
#rs1 * #rs2 whd in match (tail ??); * *
->append_cons #Hrs1rs2 #Hrs1len
+#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_change_vec >append_nil #Htf
+(* exec_move *)
+cut ((sn@[cn])=rs1)
+ [@(append_l1_injective ?????? Hrs1rs2) >Hrs1len
+ >length_append >length_append normalize >Hsolen >Hsnlen % ] #Hrs1
+cut (m0::lr0=rs2) [@(append_l2_injective ?????? Hrs1rs2) >Hrs1 % ] #Hrs2
+cut (∃ll1. ll@[bar] = bar::ll1)
+ [@daemon (* ll is at the begininng of a table *)] * #ll1 #Hll
+whd in ⊢ (%→?); >Htf >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>nth_change_vec [2:@leb_true_to_le %]
+>nth_change_vec [2:@leb_true_to_le %]
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>append_cons <Hrs1 >reverse_append >reverse_single
+<Hrs2 >(append_cons … bar ll) >Hll >reverse_cons
+#sem_exec_move
+lapply
+ (sem_exec_move ? m0 ?
+ (([cn]@reverse FSUnialpha sn)
+ @fn::reverse FSUnialpha (ll1@(fo::so)@[char])) lr0 … (refl ??) ????)
+ [@Hm0
+ |@daemon (* OK *)
+ |@Hbits_obj
+ |whd in ⊢ (??%?); >associative_append >associative_append
+ >associative_append %
+ |#Htb >Htb @(eq_vec … (niltape ?)) (* tape by tape *) #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
+ (* obj tape *)
+ >nth_change_vec [2:@leb_true_to_le %]
+ (* dimostrare che la tabella e' univoca
+ da cui m = m0 e nchar = cn *)
+ |(* cfg tape *) #eqi >eqi
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec [2:@leb_true_to_le %]
+ (* idem *)
+ ]
+ |(* prg tape *) #eqi >eqi
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >Hta_prg whd in ⊢ (??%?); @eq_f @(cons_injective_r ? bar bar)
+ >Htable >Hintable >reverse_append >reverse_cons
+ >reverse_reverse >reverse_cons >reverse_reverse
+ >associative_append >associative_append >associative_append
+ >(append_cons ? bar ll) >Hll @eq_f @eq_f <Hstate_exp @eq_f
+ >Hnewstate_exp %
+ ]
+ ]
+
+
+qed.
+
+
+
+ cut ((mk_tape FSUnialpha []
+ (option_hd FSUnialpha
+ (reverse FSUnialpha (m0::[]@reverse FSUnialpha (sn@[cn])@[fn; bar])))
+ (tail FSUnialpha
+ (reverse FSUnialpha (m0::[]@reverse FSUnialpha (sn@[cn])@[fn; bar])))) =
+ midtape ? [ ] bar (fn::sn@[cn;m0]))
+ [cut (reverse FSUnialpha (m0::[]@reverse FSUnialpha (sn@[cn])@[fn; bar]) =
+ bar::fn::sn@[cn;m0])
+ [>reverse_cons whd in ⊢ (??(??(??%)?)?); >reverse_append >reverse_reverse
+ >append_cons in ⊢ (???%); % ] #Hrev >Hrev % ] #Hmk_tape >Hmk_tape -Hmk_tape
+ >change_vec_commute
+
+ >reverse_append >reverse_append
+ check reverse_cons
+ <reverse_cons >reverse_cons
+ -Htg #Htg
+
>(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 %]
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