include "turing/multi_universal/moves_2.ma".
include "turing/multi_universal/match.ma".
include "turing/multi_universal/copy.ma".
+include "turing/multi_universal/alphabet.ma".
+include "turing/multi_universal/tuples.ma".
(*
cfg_to_obj
*)
-inductive unialpha : Type[0] ≝
-| bit : bool → unialpha
-| bar : unialpha.
-
-definition unialpha_eq ≝
- λa1,a2.match a1 with
- [ bit x ⇒ match a2 with [ bit y ⇒ ¬ xorb x y | _ ⇒ false ]
- | bar ⇒ match a2 with [ bar ⇒ true | _ ⇒ false ] ].
-
-definition DeqUnialpha ≝ mk_DeqSet unialpha unialpha_eq ?.
-* [ #x * [ #y cases x cases y normalize % // #Hfalse destruct
- | *: normalize % #Hfalse destruct ]
- | * [ #y ] normalize % #H1 destruct % ]
-qed.
-
-lemma unialpha_unique :
- uniqueb DeqUnialpha [bit true;bit false;bar] = true.
-// qed.
-
-lemma unialpha_complete :∀x:DeqUnialpha.
- memb ? x [bit true;bit false;bar] = true.
-* // * //
-qed.
-
-definition FSUnialpha ≝
- mk_FinSet DeqUnialpha [bit true;bit false;bar]
- unialpha_unique unialpha_complete.
-
-(*************************** testing characters *******************************)
-definition is_bit ≝ λc.match c with [ bit _ ⇒ true | _ ⇒ false ].
-definition is_bar ≝ λc.match c with [ bar ⇒ true | _ ⇒ false ].
-
-definition obj ≝ 0.
-definition cfg ≝ 1.
-definition prg ≝ 2.
+definition obj ≝ (0:DeqNat).
+definition cfg ≝ (1:DeqNat).
+definition prg ≝ (2:DeqNat).
definition obj_to_cfg ≝
- mmove cfg FSUnialpha 2 L ·
mmove cfg FSUnialpha 2 L ·
(ifTM ?? (inject_TM ? (test_null ?) 2 obj)
- (inject_TM ? (write FSUnialpha (bit false)) 2 cfg ·
- inject_TM ? (move_r FSUnialpha) 2 cfg ·
- inject_TM ? (write FSUnialpha (bit false)) 2 cfg)
- (inject_TM ? (write FSUnialpha (bit true)) 2 cfg ·
- inject_TM ? (move_r FSUnialpha) 2 cfg ·
- copy_step obj cfg FSUnialpha 2) tc_true ·
- inject_TM ? (move_l FSUnialpha) 2 cfg) ·
+ (copy_step obj cfg FSUnialpha 2 ·
+ mmove cfg FSUnialpha 2 L ·
+ mmove obj FSUnialpha 2 L)
+ (inject_TM ? (write FSUnialpha null) 2 cfg)
+ tc_true) ·
inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
- inject_TM ? (move_r FSUnialpha) 2 cfg.
+ mmove cfg FSUnialpha 2 R.
definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3.
- ∀c,opt,ls.
- nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::opt::ls) (None ?) [ ] →
+ ∀c,ls.
+ nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
(∀lso,x,rso.nth obj ? t1 (niltape ?) = midtape FSUnialpha lso x rso →
t2 = change_vec ?? t1
- (mk_tape ? [ ] (option_hd ? (reverse ? (c::opt::ls))) (tail ? (reverse ? (c::opt::ls)))) cfg) ∧
+ (mk_tape ? [ ] (option_hd ? (reverse ? (x::ls))) (tail ? (reverse ? (x::ls)))) cfg) ∧
(current ? (nth obj ? t1 (niltape ?)) = None ? →
t2 = change_vec ?? t1
- (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (bit false::bit false::ls)))
- (tail ? (reverse ? (bit false :: bit false::ls)))) cfg).
+ (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (null::ls)))
+ (tail ? (reverse ? (null::ls)))) cfg).
axiom sem_move_to_end_l : ∀sig. move_to_end sig L ⊨ R_move_to_end_l sig.
+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.
+#alpha #a #al %
+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_app ?? ????? (sem_move_multi ? 2 cfg L ?)
- (sem_seq_app ???????
- (sem_seq_app ???????
- (sem_if ? 2 ????????
- (sem_test_null_multi ?? obj ?)
- (sem_seq_app ??????? (sem_inject ???? cfg ? (sem_write FSUnialpha (bit false)))
- (sem_seq_app ??????? (sem_inject ???? cfg ? (sem_move_r ?))
- (sem_inject ???? cfg ? (sem_write FSUnialpha (bit false))) ?) ?)
- ?)
- ??) ??) ?) ?)
-[|||||||||||||||| @
-
- ??) ??) ??) ?) ?)
- ?) ?) ?) ?)
-
-
-@(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?) ??)
-[||
-@(sem_seq_app ?? ????? (sem_move_multi ? 2 cfg L ?) ??)
-[|| @sem_seq_app
-[|| @sem_seq_app
-[|| @(sem_if ? 2 ???????? (sem_test_null_multi ?? obj ?))
-[|||@(sem_seq_app ??????? (sem_inject ???? cfg ? (sem_write FSUnialpha (bit false))) ?)
-[||@(sem_seq_app ??????? (sem_inject ???? cfg ? (sem_move_r ?))
- (sem_inject ???? cfg ? (sem_write FSUnialpha (bit false))) ?)
-[||
+axiom daemon : ∀P:Prop.P.
+
+definition option_cons ≝ λsig.λc:option sig.λl.
+ match c with [ None ⇒ l | Some c0 ⇒ c0::l ].
+
+lemma tape_move_mk_tape_R :
+ ∀sig,ls,c,rs.
+ (c = None ? → ls = [ ] ∨ rs = [ ]) →
+ tape_move ? (mk_tape sig ls c rs) R =
+ mk_tape ? (option_cons ? c ls) (option_hd ? rs) (tail ? rs).
+#sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
+[| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ]
+normalize //
+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_app ?? ????? (sem_move_multi ? 2 cfg L ?)
- (sem_seq_app ???????
- (sem_if ? 2 ????????
+ (sem_seq ??????
+ (sem_if ??????????
(sem_test_null_multi ?? obj ?)
- (sem_seq_app ??????? (sem_inject ???? cfg ? (sem_write FSUnialpha (bit false)))
- (sem_seq_app ??????? (sem_inject ???? cfg ? (sem_move_r ?))
- (sem_inject ???? cfg ? (sem_write FSUnialpha (bit false))) ?) ?)
- ?)
- (sem_seq_app ??????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
- (sem_inject ???? cfg ? (sem_move_r ?)) ?) ?) ?) ?)
-
-
-lemma wsem_copy : ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
- copy src dst sig n ⊫ R_copy src dst sig n.
-#src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
-lapply (sem_while … (sem_copy_step src dst sig n Hneq Hsrc Hdst) … Hloop) //
--Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ whd in ⊢ (%→?); * #Hnone #Hout %
- [#_ @Hout
- |#ls #x #x0 #rs #ls0 #rs0 #Hsrc1 #Hdst1 @False_ind cases Hnone
- [>Hsrc1 normalize #H destruct (H) | >Hdst1 normalize #H destruct (H)]
- ]
-|#tc #td * #x * #y * * #Hcx #Hcy #Htd #Hstar #IH #He lapply (IH He) -IH *
- #IH1 #IH2 %
- [* [>Hcx #H destruct (H) | >Hcy #H destruct (H)]
- |#ls #x' #y' #rs #ls0 #rs0 #Hnth_src #Hnth_dst
- >Hnth_src in Hcx; whd in ⊢ (??%?→?); #H destruct (H)
- >Hnth_dst in Hcy; whd in ⊢ (??%?→?); #H destruct (H)
- >Hnth_src in Htd; >Hnth_dst -Hnth_src -Hnth_dst
- cases rs
- [(* the source tape is empty after the move *)
- #Htd lapply (IH1 ?)
- [%1 >Htd >nth_change_vec_neq [2:@(not_to_not … Hneq) //] >nth_change_vec //]
- #Hout (* whd in match (tape_move ???); *) %1 %{([])} %{rs0} %
- [% [// | // ]
- |whd in match (reverse ??); whd in match (reverse ??);
- >Hout >Htd @eq_f2 // cases rs0 //
- ]
- |#c1 #tl1 cases rs0
- [(* the dst tape is empty after the move *)
- #Htd lapply (IH1 ?) [%2 >Htd >nth_change_vec //]
- #Hout (* whd in match (tape_move ???); *) %2 %{[ ]} %{(c1::tl1)} %
- [% [// | // ]
- |whd in match (reverse ??); whd in match (reverse ??);
- >Hout >Htd @eq_f2 //
- ]
- |#c2 #tl2 whd in match (tape_move_mono ???); whd in match (tape_move_mono ???);
- #Htd
- cut (nth src (tape sig) td (niltape sig)=midtape sig (x::ls) c1 tl1)
- [>Htd >nth_change_vec_neq [2:@(not_to_not … Hneq) //] @nth_change_vec //]
- #Hsrc_td
- cut (nth dst (tape sig) td (niltape sig)=midtape sig (x::ls0) c2 tl2)
- [>Htd @nth_change_vec //]
- #Hdst_td cases (IH2 … Hsrc_td Hdst_td) -Hsrc_td -Hdst_td
- [* #rs01 * #rs02 * * #H1 #H2 #H3 %1
- %{(c2::rs01)} %{rs02} % [% [@eq_f //|normalize @eq_f @H2]]
- >Htd in H3; >change_vec_commute // >change_vec_change_vec
- >change_vec_commute [2:@(not_to_not … Hneq) //] >change_vec_change_vec
- #H >reverse_cons >associative_append >associative_append @H
- |* #rs11 * #rs12 * * #H1 #H2 #H3 %2
- %{(c1::rs11)} %{rs12} % [% [@eq_f //|normalize @eq_f @H2]]
- >Htd in H3; >change_vec_commute // >change_vec_change_vec
- >change_vec_commute [2:@(not_to_not … Hneq) //] >change_vec_change_vec
- #H >reverse_cons >associative_append >associative_append @H
- ]
+ (sem_seq ?????? (accRealize_to_Realize … (sem_copy_step …))
+ (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 ⊢ (%→%→?); * #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)
+ [@daemon] -Htg1 -Htg2 -Htg3 #Htg destruct (Htg Htf Hte Htd Htc Htb)
+ >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 Hcurtc1; [|@sym_not_eq //] >Hta2
+ normalize in ⊢ (%→?); #H destruct (H) %
]
+ | #Hta2 >Htc in Hcurtc1; >nth_change_vec_neq [| @sym_not_eq //]
+ >Hta2 #H destruct (H)
]
+ | * #Hcurtc0 #Hte #_ #_ #c #ls #Hta1 >Hta1 in Htc; >eq_mk_tape_rightof
+ whd in match (tape_move ???); #Htc >Htc in Hcurtc0; *
+ [ >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
+ #Hcurtc #Hcurtc0 >Hcurtc0 in Hcurtc; * #H @False_ind @H %
+ | >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H) ]
]
+| * #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)
+ [@daemon] -Htd1 -Htd2 #Htd
+ -Htf1 cut (tf = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[null])) cfg)
+ [@daemon] -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/ ]
+]
+qed.
+
+definition test_null_char ≝ test_char FSUnialpha (λc.c == null).
+
+definition R_test_null_char_true ≝ λt1,t2.
+ current FSUnialpha t1 = Some ? null ∧ t1 = t2.
+
+definition R_test_null_char_false ≝ λt1,t2.
+ current FSUnialpha t1 ≠ Some ? null ∧ t1 = t2.
+
+lemma sem_test_null_char :
+ test_null_char ⊨ [ tc_true : R_test_null_char_true, R_test_null_char_false].
+#t1 cases (sem_test_char FSUnialpha (λc.c == null) t1) #k * #outc * * #Hloop #Htrue
+#Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Houtc cases (Htrue ?) [| @Houtc] * #c * #Hcurt1 #Hcnull lapply (\P Hcnull)
+ -Hcnull #H destruct (H) #Houtc1 %
+ [ @Hcurt1 | <Houtc1 % ] ]
+| #Houtc cases (Hfalse ?) [| @Houtc] #Hc #Houtc %
+ [ % #Hcurt1 >Hcurt1 in Hc; #Hc lapply (Hc ? (refl ??))
+ >(?:((null:FSUnialpha) == null) = true) [|@(\b (refl ??)) ]
+ #H destruct (H)
+ | <Houtc % ] ]
qed.
-
-
-lemma terminate_copy : ∀src,dst,sig,n,t.
- src ≠ dst → src < S n → dst < S n → copy src dst sig n ↓ t.
-#src #dst #sig #n #t #Hneq #Hsrc #Hdts
-@(terminate_while … (sem_copy_step …)) //
-<(change_vec_same … t src (niltape ?))
-cases (nth src (tape sig) t (niltape ?))
-[ % #t1 * #x * #y * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
-|2,3: #a0 #al0 % #t1 * #x * #y * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
-| #ls #c #rs lapply c -c lapply ls -ls lapply t -t elim rs
- [#t #ls #c % #t1 * #x * #y * * >nth_change_vec // normalize in ⊢ (%→?);
- #H1 destruct (H1) #_ >change_vec_change_vec #Ht1 %
- #t2 * #x0 * #y0 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
- |#r0 #rs0 #IH #t #ls #c % #t1 * #x * #y * * >nth_change_vec //
- normalize in ⊢ (%→?); #H destruct (H) #Hcur
- >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
+
+definition cfg_to_obj ≝
+ mmove cfg FSUnialpha 2 L ·
+ (ifTM ?? (inject_TM ? test_null_char 2 cfg)
+ (nop ? 2)
+ (copy_step cfg obj FSUnialpha 2 ·
+ mmove cfg FSUnialpha 2 L ·
+ mmove obj FSUnialpha 2 L)
+ tc_true) ·
+ inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
+ 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 ??
+ (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).
+
+axiom 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 ??????????
+ (sem_test_null_multi ?? obj ?)
+ (sem_seq ?????? (accRealize_to_Realize … (sem_copy_step …))
+ (sem_move_multi ? 2 cfg 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 ⊢ (%→%→?); * #x * #y * * -Hcurtc #Hcurtc1 #Hcurtc2 #Hte #Htd
+ * #tf * * * whd in ⊢ (%→%→%→%→?); #Htf1 #Htf2 #Htf3 #Htb
+ #c #ls #Hta1 %
+ [ #lso #x0 #rso #Hta2 >Hta1 in Htc; >eq_mk_tape_rightof
+ whd in match (tape_move ???); #Htc
+ cut (tf = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls@[x])) cfg)
+ [@daemon] -Htf1 -Htf2 -Htf3 #Htf destruct (Htf Hte Htd Htc Htb)
+ >change_vec_change_vec >change_vec_change_vec >change_vec_change_vec
+ >nth_change_vec // >tape_move_mk_tape_R
+ @daemon
+ | #Hta2 >Htc in Hcurtc1; >nth_change_vec_neq [| @sym_not_eq //]
+ >Hta2 #H destruct (H)
+ ]
+ | * #Hcurtc0 #Hte #_ #_ #c #ls #Hta1 >Hta1 in Htc; >eq_mk_tape_rightof
+ whd in match (tape_move ???); #Htc >Htc in Hcurtc0; *
+ [ >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
+ #Hcurtc #Hcurtc0 >Hcurtc0 in Hcurtc; * #H @False_ind @H %
+ | >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H) ]
]
+| * #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)
+ [@daemon] -Htd1 -Htd2 #Htd
+ -Htf1 cut (tf = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[null])) cfg)
+ [@daemon] -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/ ]
]
qed.
+*)
-lemma sem_copy : ∀src,dst,sig,n.
- src ≠ dst → src < S n → dst < S n →
- copy src dst sig n ⊨ R_copy src dst sig n.
-#i #j #sig #n #Hneq #Hi #Hj @WRealize_to_Realize [/2/| @wsem_copy // ]
-qed.
+(* macchina che muove il nastro obj a destra o sinistra a seconda del valore
+ del current di prg, che codifica la direzione in cui ci muoviamo *)
+
+definition char_to_move ≝ λc.match c with
+ [ bit b ⇒ if b then R else L
+ | _ ⇒ N].
+
+definition tape_move_obj : mTM FSUnialpha 2 ≝
+ ifTM ??
+ (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
+ (mmove obj FSUnialpha 2 L)
+ (ifTM ??
+ (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit true)) 2 prg)
+ (mmove obj FSUnialpha 2 R)
+ (nop ??)
+ tc_true)
+ tc_true.
+
+definition restart_tape ≝ λi.
+ inject_TM ? (move_to_end FSUnialpha L) 2 i ·
+ mmove i FSUnialpha 2 R.
+
+definition unistep ≝
+ obj_to_cfg · match_m cfg prg FSUnialpha 2 ·
+ restart_tape cfg · copy prg cfg FSUnialpha 2 ·
+ cfg_to_obj · tape_move_obj · restart_tape prg.
+
+(*
+definition legal_tape ≝ λn,l,h,t.
+ ∃state,char,table.
+ nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[char]) →
+ is_config n (bar::state@[char]) →
+ nth prg ? t1 (niltape ?) = midtape ? [ ] bar table →
+ bar::table = table_TM n l h → *)
+
+definition list_of_tape ≝ λsig,t.
+ left sig t@option_cons ? (current ? t) (right ? t).
+
+definition low_char' ≝ λc.
+ match c with
+ [ None ⇒ null
+ | Some b ⇒ if (is_bit b) then b else null
+ ].
+
+definition R_unistep ≝ λn,l,h.λt1,t2: Vector ? 3.
+ ∀state,oldc,table.
+ (* cfg *)
+ nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[oldc]) →
+ is_config n (bar::state@[oldc]) →
+ (* prg *)
+ nth prg ? t1 (niltape ?) = midtape ? [ ] bar table →
+ bar::table = table_TM n l h →
+ (* obj *)
+ only_bits (list_of_tape ? (nth obj ? t1 (niltape ?))) →
+ let char ≝ low_char' (current ? (nth obj ? t1 (niltape ?))) in
+ let conf ≝ (bar::state@[char]) in
+ (∃ll,lr.bar::table = ll@conf@lr) →
+ ∃nstate,nchar,m,t. tuple_encoding n h t = (conf@nstate@[nchar;m]) ∧
+ mem ? t l ∧
+ t2 =
+ change_vec ??
+ (change_vec ?? t1 (midtape ? [ ] bar (nstate@[nchar])) cfg)
+ (tape_move_mono ? (nth obj ? t1 (niltape ?)) 〈Some ? nchar,char_to_move m〉) obj.
+
+definition tape_map ≝ λA,B:FinSet.λf:A→B.λt.
+ mk_tape B (map ?? f (left ? t))
+ (option_map ?? f (current ? t))
+ (map ?? f (right ? t)).
+
+definition low ≝ λM:normalTM.λc:nconfig (no_states M).Vector_of_list ?
+ [tape_map ?? bit (ctape ?? c);
+ midtape ? [ ] bar (bits_of_state ? (nhalt M) (cstate ?? c));
+ midtape ? [ ] bar (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M))
+ ].
+
+
+
+
+
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projects / helm.git / blobdiff