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 copy_char_states ≝ initN 3.
-
-definition cc0 : copy_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
-definition cc1 : copy_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
-
-definition trans_copy_char ≝
- λsrc,dst.λsig:FinSet.λn.
- λp:copy_char_states × (Vector (option sig) (S n)).
- let 〈q,a〉 ≝ p in
- match pi1 … q with
- [ O ⇒ 〈cc1,change_vec ? (S n)
- (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) src)
- (〈nth src ? a (None ?),R〉) dst〉
- | S _ ⇒ 〈cc1,null_action ? n〉 ].
-
-definition copy_char ≝
- λsrc,dst,sig,n.
- mk_mTM sig n copy_char_states (trans_copy_char src dst sig n)
- cc0 (λq.q == cc1).
-
-definition R_copy_char ≝
- λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
- outt = change_vec ??
- (change_vec ?? int
- (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src)
- (tape_move_mono ? (nth dst ? int (niltape ?))
- 〈current ? (nth src ? int (niltape ?)), R〉) dst.
-
-lemma copy_char_q0_q1 :
- ∀src,dst,sig,n,v.src ≠ dst → src < S n → dst < S n →
- step sig n (copy_char src dst sig n) (mk_mconfig ??? cc0 v) =
- mk_mconfig ??? cc1
- (change_vec ? (S n)
- (change_vec ?? v
- (tape_move_mono ? (nth src ? v (niltape ?)) 〈None ?, R〉) src)
- (tape_move_mono ? (nth dst ? v (niltape ?)) 〈current ? (nth src ? v (niltape ?)), R〉) dst).
-#src #dst #sig #n #v #Heq #Hsrc #Hdst
-whd in ⊢ (??%?);
-<(change_vec_same … v dst (niltape ?)) in ⊢ (??%?);
-<(change_vec_same … v src (niltape ?)) in ⊢ (??%?);
->tape_move_multi_def @eq_f2 //
->pmap_change >pmap_change <tape_move_multi_def
->tape_move_null_action @eq_f2 // @eq_f2
-[ >change_vec_same %
-| >change_vec_same >change_vec_same // ]
-qed.
-
-lemma sem_copy_char:
- ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
- copy_char src dst sig n ⊨ R_copy_char src dst sig n.
-#src #dst #sig #n #Hneq #Hsrc #Hdst #int
-%{2} % [| % [ % | whd >copy_char_q0_q1 // ]]
-qed.
definition obj ≝ (0:DeqNat).
definition cfg ≝ (1:DeqNat).
definition prg ≝ (2:DeqNat).
definition obj_to_cfg ≝
- mmove cfg FSUnialpha 2 L ·
(ifTM ?? (inject_TM ? (test_null ?) 2 obj)
- (copy_char obj cfg FSUnialpha 2 ·
- mmove cfg FSUnialpha 2 L ·
- mmove obj FSUnialpha 2 L)
+ (copy_char_N obj cfg FSUnialpha 2)
(inject_TM ? (write FSUnialpha null) 2 cfg)
tc_true) ·
inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3.
∀c,ls.
- nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
+ nth cfg ? t1 (niltape ?) = midtape ? ls c [ ] →
(∀lso,x,rso.nth obj ? t1 (niltape ?) = midtape FSUnialpha lso x rso →
t2 = change_vec ?? t1
(mk_tape ? [ ] (option_hd ? (reverse ? (x::ls))) (tail ? (reverse ? (x::ls)))) cfg) ∧
t2 = change_vec ?? t1
(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.
+#A * /2/ * #H @False_ind @H %
+qed.
+
lemma sem_obj_to_cfg : obj_to_cfg ⊨ R_obj_to_cfg.
-@(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
- (sem_seq ??????
- (sem_if ??????????
- (sem_test_null_multi ?? obj ?)
- (sem_seq ?????? (sem_copy_char …)
- (sem_seq ?????? (sem_move_multi ? 2 cfg L ?)
- (sem_move_multi ? 2 obj L ?)))
- (sem_inject ???? cfg ? (sem_write FSUnialpha null)))
- (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
- (sem_move_multi ? 2 cfg R ?)))) //
-#ta #tb *
-#tc * whd in ⊢ (%→?); #Htc *
-#td * *
-[ * #te * * #Hcurtc #Hte
- * destruct (Hte) #te * whd in ⊢ (%→?); #Hte
- cut (∃x.current ? (nth obj ? tc (niltape ?)) = Some ? x)
- [ cases (current ? (nth obj ? tc (niltape ?))) in Hcurtc;
- [ * #H @False_ind /2/ | #x #_ %{x} % ] ] * #x #Hcurtc'
-(* [ whd in ⊢ (%→%→?); * #x * #y * * -Hcurtc #Hcurtc1 #Hcurtc2 #Hte *)
- * #tf * whd in ⊢ (%→%→?); #Htf #Htd
- * #tg * * * whd in ⊢ (%→%→%→%→?); #Htg1 #Htg2 #Htg3 #Htb
- #c #ls #Hta1 %
- [ #lso #x0 #rso #Hta2 >Hta1 in Htc; >eq_mk_tape_rightof
- whd in match (tape_move ???); #Htc
- cut (tg = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[x])) cfg)
- [ lapply (eq_vec_change_vec ??????? (Htg2 ls x [ ] ?) Htg3) //
- >Htd >nth_change_vec_neq // >Htf >nth_change_vec //
- >Hte >Hcurtc' >nth_change_vec // >Htc >nth_change_vec // ]
- -Htg1 -Htg2 -Htg3 #Htg destruct
- >change_vec_change_vec >change_vec_change_vec
- >change_vec_commute // >change_vec_change_vec
- >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
- >change_vec_commute // >change_vec_change_vec
- >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
- >change_vec_commute [|@sym_not_eq //] @eq_f3 //
- [ >Hta2 cases rso in Hta2; whd in match (tape_move_mono ???);
- [ #Hta2 whd in match (tape_move ???); <Hta2 @change_vec_same
- | #r1 #rs1 #Hta2 whd in match (tape_move ???); <Hta2 @change_vec_same ]
- | >tape_move_mk_tape_R [| #_ % %] >reverse_cons
- >nth_change_vec_neq in Hcurtc'; [|@sym_not_eq //] >Hta2
- normalize in ⊢ (%→?); #H destruct (H) %
- ]
- | #Hta2 >Htc in Hcurtc'; >nth_change_vec_neq [| @sym_not_eq //]
- >Hta2 #H destruct (H)
+@(sem_seq_app FSUnialpha 2 ?????
+ (sem_if ??????????
+ (sem_test_null_multi ?? obj ?)
+ (sem_copy_char_N …)
+ (sem_inject ???? cfg ? (sem_write FSUnialpha null)))
+ (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
+ (sem_move_multi ? 2 cfg R ?))) //
+#ta #tout *
+#tb * #Hif * #tc * #HM2 #HM3 #c #ls #Hcfg
+(* Hif *)
+cases Hif -Hif
+[ * #t1 * * #Hcurta #Ht1 destruct (Ht1)
+ lapply (not_None_to_Some … Hcurta) * #curta #Hcurtaeq
+ whd in ⊢ (%→?); #Htb % [2: #Hcur @False_ind /2/]
+ #lso #xo #rso #Hobjta cases HM2 whd in ⊢ (%→?); * #_
+ #HM2 #Heq >Htb in HM2; >nth_change_vec [2: @leb_true_to_le %]
+ >Hcfg >Hcurtaeq #HM2 lapply (HM2 … (refl ??)) -HM2
+ whd in match (left ??); whd in match (right ??);
+ >reverse_cons #Htc >HM3 @(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 %]
+ <(Heq 0) [2:@eqb_false_to_not_eq %] >Htb
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec_neq [%|2:@eqb_false_to_not_eq %]
+ |#Hi >Hi >nth_change_vec // >nth_change_vec // >Htc
+ >Hobjta in Hcurtaeq; whd in ⊢ (??%?→?); #Htmp destruct(Htmp)
+ >tape_move_mk_tape_R [2: #_ %1 %] %
]
-| * #te * * #Hcurtc #Hte
- * whd in ⊢ (%→%→?); #Htd1 #Htd2
- * #tf * * * #Htf1 #Htf2 #Htf3 whd in ⊢ (%→?); #Htb
- #c #ls #Hta1 %
- [ #lso #x #rso #Hta2 >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
- >Hta2 normalize in ⊢ (%→?); #H destruct (H)
- | #_ >Hta1 in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
- destruct (Hte) cut (td = change_vec ?? tc (midtape ? ls null []) cfg)
- [ lapply (eq_vec_change_vec ??????? (Htd1 ls c [ ] ?) Htd2) //
- >Htc >nth_change_vec // ] -Htd1 -Htd2 #Htd
- -Htf1 cut (tf = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[null])) cfg)
- [ lapply (eq_vec_change_vec ??????? (Htf2 ls null [ ] ?) Htf3) //
- >Htd >nth_change_vec // ] -Htf2 -Htf3 #Htf destruct (Htf Htd Htc Htb)
- >change_vec_change_vec >change_vec_change_vec >change_vec_change_vec
- >change_vec_change_vec >change_vec_change_vec >nth_change_vec //
- >reverse_cons >tape_move_mk_tape_R /2/ ]
+ |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ <(Heq 2) [2:@eqb_false_to_not_eq %] >Htb
+ >nth_change_vec_neq [%|2:@eqb_false_to_not_eq %]
+ ]
+| * #t1 * * #Hcurta #Ht1 destruct (Ht1)
+ * whd in ⊢ (%→?); #Htb lapply (Htb … Hcfg) -Htb #Htb
+ #Htbeq %
+ [#lso #xo #rso #Hmid @False_ind >Hmid in Hcurta;
+ whd in ⊢ (??%?→?); #Htmp destruct (Htmp)]
+ #_ cases HM2 whd in ⊢ (%→?); * #_
+ #HM2 #Heq >Htb in HM2; #HM2 lapply (HM2 … (refl ??)) -HM2
+ #Htc >HM3 @(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 %]
+ <(Heq 0) [2:@eqb_false_to_not_eq %] >Htb
+ <(Htbeq 0) [2:@eqb_false_to_not_eq %] %
+ |#Hi >Hi >nth_change_vec // >nth_change_vec // >Htc
+ >tape_move_mk_tape_R [2: #_ %1 %] >reverse_cons %
+ ]
+ |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ <(Heq 2) [2:@eqb_false_to_not_eq %]
+ <(Htbeq 2) [%|@eqb_false_to_not_eq %]
+ ]
]
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.
mmove cfg FSUnialpha 2 L ·
(ifTM ?? (inject_TM ? test_null_char 2 cfg)
(nop ? 2)
- (copy_char cfg obj FSUnialpha 2 ·
- mmove cfg FSUnialpha 2 L ·
- mmove obj FSUnialpha 2 L)
- tc_true) ·
+ (copy_char_N cfg obj FSUnialpha 2)
+ tc_true).
+(* ·
inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
- mmove cfg FSUnialpha 2 R.
+ mmove cfg FSUnialpha 2 R. *)
definition R_cfg_to_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
∀c,ls.
nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
- (c = null →
- t2 = change_vec ?? t1
- (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (c::ls)))
- (tail ? (reverse ? (c::ls)))) cfg) ∧
+ (c = null → t2 = change_vec ?? t1 (midtape ? ls c [ ]) cfg) ∧
(c ≠ null →
t2 = change_vec ??
(change_vec ?? t1
(midtape ? (left ? (nth obj ? t1 (niltape ?))) c (right ? (nth obj ? t1 (niltape ?)))) obj)
- (mk_tape ? [ ] (option_hd ? (reverse ? (c::ls))) (tail ? (reverse ? (c::ls)))) cfg).
+ (midtape ? ls c [ ]) cfg).
lemma tape_move_mk_tape_L :
∀sig,ls,c,rs.
lemma sem_cfg_to_obj : cfg_to_obj ⊨ R_cfg_to_obj.
@(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
- (sem_seq ??????
- (sem_if ??????????
- (acc_sem_inject ?????? cfg ? sem_test_null_char)
- (sem_nop …)
- (sem_seq ?????? (sem_copy_char …)
- (sem_seq ?????? (sem_move_multi ? 2 cfg L ?) (sem_move_multi ? 2 obj L ?))))
- (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
- (sem_move_multi ? 2 cfg R ?)))) // [@sym_not_eq //]
+ (sem_if ??????????
+ (acc_sem_inject ?????? cfg ? sem_test_null_char)
+ (sem_nop …)
+ (sem_copy_char_N …)))
+// [@sym_not_eq //]
#ta #tb *
#tc * whd in ⊢ (%→?); #Htc *
-#td * *
-[ * #te * * * #Hcurtc #Hte1 #Hte2 whd in ⊢ (%→?); #Htd destruct (Htd)
- * #tf * * * #Htf1 #Htf2 #Htf3
- whd in ⊢ (%→?); #Htb
+[ * #te * * * #Hcurtc #Hte1 #Hte2 whd in ⊢ (%→?); #Htb destruct (Htb)
#c #ls #Hta %
[ #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
cut (te = tc)
[ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2) >change_vec_same // ]
- -Hte1 -Hte2 #Hte
- cut (tf = change_vec ? 3 te (mk_tape ? [ ] (None ?) (reverse ? ls@[c])) cfg)
- [ lapply (eq_vec_change_vec ??????? (Htf2 ls c [ ] ?) Htf3) //
- >Hte >Htc >nth_change_vec // ] -Htf1 -Htf2 -Htf3 #Htf
- destruct (Htf Hte Htc Htb)
- >change_vec_change_vec >change_vec_change_vec >change_vec_change_vec
- >nth_change_vec // >tape_move_mk_tape_R [| #_ % % ]
- >reverse_cons %
- | #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
- >Htc in Hcurtc; >nth_change_vec // normalize in ⊢ (%→?);
- #H destruct (H) @False_ind cases Hc /2/ ]
- * #tf * *
+ -Hte1 -Hte2 #Hte //
+ | #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
+ >Htc in Hcurtc; >nth_change_vec // normalize in ⊢ (%→?);
+ #H destruct (H) @False_ind cases Hc /2/ ]
| * #te * * * #Hcurtc #Hte1 #Hte2
- * #tf * whd in ⊢ (%→?); #Htf
- * #tg * whd in ⊢ (%→%→?); #Htg #Htd
- * #th * * * #Hth1 #Hth2 #Hth3
- whd in ⊢ (%→?); #Htb
+ whd in ⊢ (%→?); #Htb
#c #ls #Hta % #Hc
- [ >Htc in Hcurtc; >Hta >nth_change_vec // >tape_move_mk_tape_L //
- >Hc normalize in ⊢ (%→?); * #H @False_ind /2/
+ [ >Htc in Hcurtc; >Hta >nth_change_vec //
+ normalize in ⊢ (%→?); * #H @False_ind /2/
| cut (te = tc)
[ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2)
- >change_vec_same // ] -Hte1 -Hte2 #Hte
- cut (th = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[c])) cfg)
- [ lapply (eq_vec_change_vec ??????? (Hth2 ls c [ ] ?) Hth3) //
- >Htd >nth_change_vec_neq // >Htg >nth_change_vec //
- >Htf >nth_change_vec_neq // >nth_change_vec //
- >Hte >Htc >nth_change_vec // >Hta // ] -Hth1 -Hth2 -Hth3 #Hth
- destruct (Hth Hte Hta Htb Htd Htg Htc Htf)
- >change_vec_change_vec >change_vec_change_vec
- >change_vec_commute // >change_vec_change_vec
- >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
- >change_vec_commute // >change_vec_change_vec
- >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
- >change_vec_commute [|@sym_not_eq //]
- @eq_f3 //
- [ >Hta >tape_move_mk_tape_L // >nth_change_vec // whd in match (current ??);
- @eq_f2 // cases (nth obj ? ta (niltape ?))
- [| #r0 #rs0 | #l0 #ls0 | #ls0 #c0 #rs0 ] try %
- cases rs0 //
- | >reverse_cons >tape_move_mk_tape_R // #_ % % ]
+ >change_vec_same // ] -Hte1 -Hte2 #Hte destruct (Hte)
+ >Hta in Htc; whd in match (tape_move ???); #Htc
+ >Htc in Htb; >nth_change_vec //
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq //] >Hta
+ #Htb @Htb
]
-]
qed.
definition char_to_move ≝ λc.match c with
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.
+ tape_write ? (nth obj ? t1 (niltape ?)) (char_to_bit_option c) in
+ t2 = change_vec ??
+ (change_vec ?? t1
+ (tape_write ? (nth obj ? t1 (niltape ?)) (char_to_bit_option c)) obj)
+ (midtape ? ls c [ ]) 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
]
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.
-