\ / GNU General Public License Version 2
V_____________________________________________________________*)
-include "turing/multi_universal/moves_2.ma".
-include "turing/multi_universal/match.ma".
-include "turing/multi_universal/copy.ma".
+include "turing/auxiliary_machines.ma".
+include "turing/auxiliary_multi_machines.ma".
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).
(mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (null::ls)))
(tail ? (reverse ? (null::ls)))) cfg).
+(*
axiom accRealize_to_Realize :
∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse.
+*)
lemma eq_mk_tape_rightof :
∀alpha,a,al.mk_tape alpha (a::al) (None ?) [ ] = rightof ? a al.
normalize //
qed.
-lemma eq_vec_change_vec : ∀sig,n.∀v1,v2:Vector sig n.∀i,t,d.
- nth i ? v2 d = t →
- (∀j.i ≠ j → nth j ? v1 d = nth j ? v2 d) →
- v2 = change_vec ?? v1 t i.
-#sig #n #v1 #v2 #i #t #d #H1 #H2 @(eq_vec … d)
-#i0 #Hlt cases (decidable_eq_nat i0 i) #Hii0
-[ >Hii0 >nth_change_vec //
-| >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @H2 @sym_not_eq // ]
+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.
]
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 ·
* #td * * * #Htd1 #Htd2 #Htd3
whd in ⊢ (%→?); #Htb *
[ #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
- cut (td = tc) [@daemon]
+ cut (td = tc)
+ [ <(change_vec_same … tc … i … (niltape ?))
+ @(eq_vec_change_vec … (niltape ?))
+ [ @Htd1 >Htc >nth_change_vec //
+ | @Htd3 ] ]
(* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
#Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
#Htb >Htb %
| #r0 #rs0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
- cut (td = tc) [@daemon]
+ cut (td = tc)
+ [ <(change_vec_same … tc … i … (niltape ?))
+ @(eq_vec_change_vec … (niltape ?))
+ [ @Htd1 >Htc >nth_change_vec //
+ | @Htd3 ] ]
(* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
#Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
#Htb >Htb %
| #l0 #ls0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@[l0])) i)
- [@daemon]
+ [ <(change_vec_same … tc … i … (niltape ?))
+ @(eq_vec_change_vec … (niltape ?))
+ [ @Htd2 >Htc >nth_change_vec //
+ | #j #Hij >nth_change_vec_neq // @Htd3 // ]]
#Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
>nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
cases (reverse ? ls0)
whd in ⊢ (??%?); @eq_f >reverse_reverse normalize >append_nil % ] % ]
| *
[ #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
- cut (td = tc) [@daemon]
+ cut (td = tc)
+ [ <(change_vec_same … tc … i … (niltape ?))
+ @(eq_vec_change_vec … (niltape ?))
+ [ @Htd1 >Htc >nth_change_vec //
+ | @Htd3 ] ]
(* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
#Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
#Htb >Htb %
| #l0 #ls0 #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@l0::c::rs)) i)
- [@daemon]
+ [ @(eq_vec_change_vec … (niltape ?))
+ [ @Htd2 >Htc >nth_change_vec //
+ | @Htd3 ] ]
#Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
>nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
cases (reverse ? ls0)
]
]
qed.
-
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