definition exec_move ≝
cfg_to_obj · tape_move_obj · restart_tape prg 2 · obj_to_cfg.
-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_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 ?))
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_cfg))) //
+ (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 #Hc #Hm
+#c #m #ls1 #ls2 #rs2 #Hcfg #Hprg #Honlybits #Hc #Hm
(* M1 = cfg_to_obj *)
lapply (semM1 … Hcfg Hc) #Ht1
(* M2 = tape_move *)
>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 %
+ ]
+ |
+
definition unistep ≝
match_m cfg prg FSUnialpha 2 ·
restart_tape cfg 2 · mmove cfg ? 2 R · copy prg cfg FSUnialpha 2 ·
include "turing/multi_universal/alphabet.ma".
include "turing/multi_universal/tuples.ma".
-(*
-
- in.obj : ... x ...
- ^
- in.cfg : ... ? ? ...
- ^
-
- out.cfg : ... 1 x ...
- ^
-
- ---------------------
- current (in.obj) = None
-
- in.cfg : ... ? ? ...
- ^
-
- out.cfg : ... 0 0 ...
- ^
-
- obj_to_cfg ≝
- move_l(cfg);
- move_l(cfg);
- (if (current(in.obj)) == None
- then write(0,cfg);
- move_r(cfg);
- write(0,cfg);
- else write(1,cfg);
- move_r(cfg);
- copy_step(obj,cfg);
- move_l(obj);)
- move_to_end_l(cfg);
- move_r(cfg);
-
-
- cfg_to_obj
-*)
definition obj ≝ (0:DeqNat).
definition cfg ≝ (1:DeqNat).
| >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @H2 @sym_not_eq // ]
qed.
+lemma None_or_Some: ∀A.∀a. a =None A ∨ ∃b. a = Some ? b.
+#A * /2/ #a %2 %{a} %
+qed.
+
lemma not_None_to_Some: ∀A.∀a. a ≠ None A → ∃b. a = Some ? b.
#A * /2/ * #H @False_ind @H %
qed.
]
qed.
+(* another semantics for obj_to_cfg *)
+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_obj_to_cfg1 ≝ λt1,t2:Vector (tape FSUnialpha) 3.
+ ∀c,ls.
+ nth cfg ? t1 (niltape ?) = midtape ? ls c [ ] →
+ let x ≝ current ? (nth obj ? t1 (niltape ?)) in
+ (∀b. x= Some ? b → is_bit b = true) →
+ t2 = change_vec ?? t1
+ (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (low_char' x::ls)))
+ (tail ? (reverse ? (low_char' x::ls)))) cfg.
+
+lemma sem_obj_to_cfg1: obj_to_cfg ⊨ R_obj_to_cfg1.
+@(Realize_to_Realize … sem_obj_to_cfg) #t1 #t2 #Hsem
+#c #ls #Hcfg lapply(Hsem c ls Hcfg) * #HSome #HNone #Hb
+cases (None_or_Some ? (current ? (nth obj ? t1 (niltape ?))))
+ [#Hcur >Hcur @HNone @Hcur
+ |* #b #Hb1 >Hb1
+ cut (low_char' (Some ? b) = b) [whd in ⊢ (??%?); >(Hb b Hb1) %] #Hlow >Hlow
+ lapply(current_to_midtape … Hb1) * #lsobj * #rsobj #Hmid
+ @(HSome … Hmid)
+ ]
+qed.
+
+(* test_null_char *)
definition test_null_char ≝ test_char FSUnialpha (λc.c == null).
definition R_test_null_char_true ≝ λt1,t2.
]
qed.
+(************** list of tape ******************)
definition list_of_tape ≝ λsig.λt:tape sig.
reverse ? (left ? t)@option_cons ? (current ? t) (right ? t).
+lemma list_of_midtape: ∀sig,ls,c,rs.
+ list_of_tape sig (midtape ? ls c rs) = reverse ? ls@c::rs.
+// qed-.
+
+lemma list_of_rightof: ∀sig,ls,c.
+ list_of_tape sig (rightof ? c ls) = reverse ? (c::ls).
+#sig #ls #c <(append_nil ? (reverse ? (c::ls)))
+// qed-.
+
+lemma list_of_tape_move: ∀sig,t,m.
+ list_of_tape sig t = list_of_tape sig (tape_move ? t m).
+#sig #t * // cases t //
+ [(* rightof, move L *) #a #l >list_of_midtape
+ >append_cons <reverse_single <reverse_append %
+ |(* midtape, move L *) * //
+ #a #ls #c #rs >list_of_midtape >list_of_midtape
+ >reverse_cons >associative_append %
+ |(* midtape, move R *) #ls #c *
+ [>list_of_midtape >list_of_rightof >reverse_cons %
+ |#a #rs >list_of_midtape >list_of_midtape >reverse_cons
+ >associative_append %
+ ]
+ ]
+qed.
+
+lemma list_of_tape_write: ∀sig,cond,t,c.
+(∀b. c = Some ? b → cond b =true) →
+(∀x. mem ? x (list_of_tape ? t) → cond x =true ) →
+∀x. mem ? x (list_of_tape sig (tape_write ? t c)) → cond x =true.
+#sig #cond #t #c #Hc #Htape #x lapply Hc cases c
+ [(* c is None *) #_ whd in match (tape_write ???); @Htape
+ |#b #Hb lapply (Hb … (refl ??)) -Hb #Hb
+ whd in match (tape_write ???); >list_of_midtape
+ #Hx cases(mem_append ???? Hx) -Hx
+ [#Hx @Htape @mem_append_l1 @Hx
+ |* [//]
+ #Hx @Htape @mem_append_l2 cases (current sig t)
+ [@Hx | #c1 %2 @Hx]
+ ]
+ ]
+qed.
+
+lemma current_in_list: ∀sig,t,b.
+ current sig t = Some ? b → mem ? b (list_of_tape sig t).
+#sig #t #b cases t
+ [whd in ⊢ (??%?→?); #Htmp destruct
+ |#l #b whd in ⊢ (??%?→?); #Htmp destruct
+ |#l #b whd in ⊢ (??%?→?); #Htmp destruct
+ |#ls #c #rs whd in ⊢ (??%?→?); #Htmp destruct
+ >list_of_midtape @mem_append_l2 % %
+ ]
+qed.
+
definition restart_tape ≝ λi,n.
mmove i FSUnialpha n L ·
inject_TM ? (move_to_end FSUnialpha L) n i ·
]
]
qed.
-
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