-(*
+#(*
||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.
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 ·
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 *)
(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_cfg_to_obj1
(sem_seq ?????? sem_tape_move_obj
(sem_seq ?????? (sem_restart_tape ???) sem_obj_to_cfg)))))))
/2 by le_n,sym_not_eq/
(* 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 >reverse_single
+ >associative_append #Htg lapply (Htg … (refl ??) Hm0) -Htg
+ (* simplifying tg *)
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+
+
+
+
+
+ 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 %]
projects / helm.git / blobdiff