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 copy_char_states ≝ initN 3.
-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.
+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 …)).
-lemma unialpha_unique :
- uniqueb DeqUnialpha [bit true;bit false;bar] = true.
-// qed.
+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〉 ].
-lemma unialpha_complete :∀x:DeqUnialpha.
- memb ? x [bit true;bit false;bar] = true.
-* // * //
-qed.
+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 FSUnialpha ≝
- mk_FinSet DeqUnialpha [bit true;bit false;bar]
- unialpha_unique unialpha_complete.
+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.
-(*************************** testing characters *******************************)
-definition is_bit ≝ λc.match c with [ bit _ ⇒ true | _ ⇒ false ].
-definition is_bar ≝ λc.match c with [ bar ⇒ true | _ ⇒ false ].
+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.
-definition obj ≝ 0.
-definition cfg ≝ 1.
-definition prg ≝ 2.
+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 ·
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_char 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))) ?)
-[||
+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 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 // ]
+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 ?????? (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)
]
+| * #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/ ]
+]
+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.
+
+definition cfg_to_obj ≝
+ 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) ·
+ 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).
+
+lemma tape_move_mk_tape_L :
+ ∀sig,ls,c,rs.
+ (c = None ? → ls = [ ] ∨ rs = [ ]) →
+ tape_move ? (mk_tape sig ls c rs) L =
+ mk_tape ? (tail ? ls) (option_hd ? ls) (option_cons ? c rs).
+#sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
+[| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ]
+normalize //
+qed.
+
+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 //]
+#ta #tb *
+#tc * whd in ⊢ (%→?); #Htc *
+#td * *
+[ * #te * * * #Hcurtc #Hte1 #Hte2 whd in ⊢ (%→?); #Htd destruct (Htd)
+ * #tf * * * #Htf1 #Htf2 #Htf3
+ whd in ⊢ (%→?); #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 * *
+| * #te * * * #Hcurtc #Hte1 #Hte2
+ * #tf * whd in ⊢ (%→?); #Htf
+ * #tg * whd in ⊢ (%→%→?); #Htg #Htd
+ * #th * * * #Hth1 #Hth2 #Hth3
+ 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/
+ | 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 // #_ % % ]
]
+]
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 char_to_bit_option ≝ λc.match c with
+ [ bit b ⇒ Some ? (bit b)
+ | _ ⇒ None ?].
-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 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 R_tape_move_obj' ≝ λt1,t2:Vector (tape FSUnialpha) 3.
+ (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit false) →
+ t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) L) obj) ∧
+ (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit true) →
+ t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) R) obj) ∧
+ (current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit false) →
+ current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit true) →
+ t2 = t1).
+
+lemma sem_tape_move_obj' : tape_move_obj ⊨ R_tape_move_obj'.
+#ta cases (sem_if ??????????
+ (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit false)))
+ (sem_move_multi ? 2 obj L ?)
+ (sem_if ??????????
+ (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit true)))
+ (sem_move_multi ? 2 obj R ?)
+ (sem_nop …)) ta) //
+#i * #outc * #Hloop #HR %{i} %{outc} % [@Hloop] -i
+cases HR -HR
+[ * #tb * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htb1 #Htb2
+ whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
+ [ >Hcurta_prg #H destruct (H) >(?:tb = ta)
+ [| lapply (eq_vec_change_vec ??????? Htb1 Htb2)
+ >change_vec_same // ] %
+ | >Hcurta_prg #H destruct (H) destruct (Hc) ]
+ | >Hcurta_prg >Hc * #H @False_ind /2/ ]
+| * #tb * * * #Hnotfalse #Htb1 #Htb2 cut (tb = ta)
+ [ lapply (eq_vec_change_vec ??????? Htb1 Htb2)
+ >change_vec_same // ] -Htb1 -Htb2 #Htb destruct (Htb) *
+ [ * #tc * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htc1 #Htc2
+ whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
+ [ >Hcurta_prg #H destruct (H) destruct (Hc)
+ | >Hcurta_prg #H destruct (H) >(?:tc = ta)
+ [| lapply (eq_vec_change_vec ??????? Htc1 Htc2)
+ >change_vec_same // ] % ]
+ | >Hcurta_prg >Hc #_ * #H @False_ind /2/ ]
+ | * #tc * * * #Hnottrue #Htc1 #Htc2 cut (tc = ta)
+ [ lapply (eq_vec_change_vec ??????? Htc1 Htc2)
+ >change_vec_same // ] -Htc1 -Htc2
+ #Htc destruct (Htc) whd in ⊢ (%→?); #Houtc % [ %
+ [ #Hcurta_prg lapply (\Pf (Hnotfalse ? Hcurta_prg)) * #H @False_ind /2/
+ | #Hcurta_prg lapply (\Pf (Hnottrue ? Hcurta_prg)) * #H @False_ind /2/ ]
+ | #_ #_ @Houtc ]
]
]
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 // ]
+definition R_tape_move_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
+ ∀c. current ? (nth prg ? t1 (niltape ?)) = Some ? c →
+ t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) (char_to_move c)) obj.
+
+lemma sem_tape_move_obj : tape_move_obj ⊨ R_tape_move_obj.
+@(Realize_to_Realize … sem_tape_move_obj')
+#ta #tb * * #Htb1 #Htb2 #Htb3 * [ *
+[ @Htb2 | @Htb1 ]
+| #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
+ >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
+| #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
+ >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
+]
+qed.
+
+definition restart_tape ≝ λi.
+ inject_TM ? (move_to_end FSUnialpha L) 2 i ·
+ mmove i FSUnialpha 2 R.
+
+definition unistep ≝
+ match_m cfg prg FSUnialpha 2 ·
+ restart_tape cfg · copy prg cfg FSUnialpha 2 ·
+ cfg_to_obj · tape_move_obj · restart_tape prg · obj_to_cfg.
+
+(*
+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
+ ].
+
+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 *)
+ nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[char]) →
+ is_config n (bar::state@[char]) →
+ (* 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 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 ∧ *)
+ ∀nstate,nchar,m,t.
+ tuple_encoding n h t = (conf@nstate@[nchar;m])→
+ mem ? t l →
+ let new_obj ≝
+ tape_move_mono ? (nth obj ? t1 (niltape ?))
+ 〈char_to_bit_option nchar,char_to_move m〉 in
+ let next_char ≝ low_char' (current ? new_obj) in
+ t2 =
+ change_vec ??
+ (change_vec ?? t1 (midtape ? [ ] bar (nstate@[next_char])) cfg)
+ new_obj 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)).
+
+lemma map_list_of_tape: ∀A,B,f,t.
+ list_of_tape B (tape_map ?? f t) = map ?? f (list_of_tape A t).
+#A #B #f * // normalize // #ls #c #rs <map_append %
+qed.
+
+lemma low_char_current : ∀t.
+ low_char' (current FSUnialpha (tape_map FinBool FSUnialpha bit t))
+ = low_char (current FinBool t).
+* // qed.
+
+definition low_tapes: ∀M:normalTM.∀c:nconfig (no_states M).Vector ? 3 ≝
+λM:normalTM.λc:nconfig (no_states M).Vector_of_list ?
+ [tape_map ?? bit (ctape ?? c);
+ midtape ? [ ] bar
+ ((bits_of_state ? (nhalt M) (cstate ?? c))@[low_char (current ? (ctape ?? c))]);
+ midtape ? [ ] bar (tail ? (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M)))
+ ].
+
+lemma obj_low_tapes: ∀M,c.
+ nth obj ? (low_tapes M c) (niltape ?) = tape_map ?? bit (ctape ?? c).
+// qed.
+
+lemma cfg_low_tapes: ∀M,c.
+ nth cfg ? (low_tapes M c) (niltape ?) =
+ midtape ? [ ] bar
+ ((bits_of_state ? (nhalt M) (cstate ?? c))@[low_char (current ? (ctape ?? c))]).
+// qed.
+
+lemma prg_low_tapes: ∀M,c.
+ nth prg ? (low_tapes M c) (niltape ?) =
+ midtape ? [ ] bar (tail ? (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M))).
+// qed.
+
+(* commutation lemma for write *)
+lemma map_write: ∀t,cout.
+ tape_write ? (tape_map FinBool ? bit t) (char_to_bit_option (low_char cout))
+ = tape_map ?? bit (tape_write ? t cout).
+#t * // #b whd in match (char_to_bit_option ?);
+whd in ⊢ (??%%); @eq_f3 [elim t // | // | elim t //]
+qed.
+
+(* commutation lemma for moves *)
+lemma map_move: ∀t,m.
+ tape_move ? (tape_map FinBool ? bit t) (char_to_move (low_mv m))
+ = tape_map ?? bit (tape_move ? t m).
+#t * // whd in match (char_to_move ?);
+ [cases t // * // | cases t // #ls #a * //]
qed.
+
+(* commutation lemma for actions *)
+lemma map_action: ∀t,cout,m.
+ tape_move ? (tape_write ? (tape_map FinBool ? bit t)
+ (char_to_bit_option (low_char cout))) (char_to_move (low_mv m))
+ = tape_map ?? bit (tape_move ? (tape_write ? t cout) m).
+#t #cout #m >map_write >map_move %
+qed.
+
+lemma map_move_mono: ∀t,cout,m.
+ tape_move_mono ? (tape_map FinBool ? bit t)
+ 〈char_to_bit_option (low_char cout), char_to_move (low_mv m)〉
+ = tape_map ?? bit (tape_move_mono ? t 〈cout,m〉).
+@map_action
+qed.
+
+definition R_unistep_high ≝ λM:normalTM.λt1,t2.
+∀c:nconfig (no_states M).
+ t1 = low_tapes M c →
+ t2 = low_tapes M (step ? M c).
+
+lemma R_unistep_equiv : ∀M,t1,t2.
+ R_unistep (no_states M) (graph_enum ?? (ntrans M)) (nhalt M) t1 t2 →
+ R_unistep_high M t1 t2.
+#M #t1 #t2 #H whd whd in match (nconfig ?); #c #Ht1
+lapply (initial_bar ? (nhalt M) (graph_enum ?? (ntrans M)) (nTM_nog ?)) #Htable
+(* tup = current tuple *)
+cut (∃t.t = 〈〈cstate … c,current ? (ctape … c)〉,
+ ntrans M 〈cstate … c,current ? (ctape … c)〉〉) [% //] * #tup #Htup
+(* tup is in the graph *)
+cut (mem ? tup (graph_enum ?? (ntrans M)))
+ [@memb_to_mem >Htup @(graph_enum_complete … (ntrans M)) %] #Hingraph
+(* tupe target = 〈qout,cout,m〉 *)
+lapply (decomp_target ? (ntrans M 〈cstate … c,current ? (ctape … c)〉))
+* #qout * #cout * #m #Htg >Htg in Htup; #Htup
+(* new config *)
+cut (step FinBool M c = mk_config ?? qout (tape_move ? (tape_write ? (ctape … c) cout) m))
+ [>(config_expand … c) whd in ⊢ (??%?); (* >Htg ?? why not?? *)
+ cut (trans ? M 〈cstate … c, current ? (ctape … c)〉 = 〈qout,cout,m〉) [<Htg %] #Heq1
+ >Heq1 %] #Hstep
+(* new state *)
+cut (cstate ?? (step FinBool M c) = qout) [>Hstep %] #Hnew_state
+(* new tape *)
+cut (ctape ?? (step FinBool M c) = tape_move ? (tape_write ? (ctape … c) cout) m)
+ [>Hstep %] #Hnew_tape
+lapply(H (bits_of_state ? (nhalt M) (cstate ?? c))
+ (low_char (current ? (ctape ?? c)))
+ (tail ? (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M)))
+ ??????)
+[<Htable
+ lapply(list_to_table … (nhalt M) …Hingraph) * #ll * #lr #Htable1 %{ll}
+ %{(((bits_of_state ? (nhalt M) qout)@[low_char cout;low_mv m])@lr)}
+ >Htable1 @eq_f <associative_append @eq_f2 // >Htup
+ whd in ⊢ (??%?); @eq_f >associative_append %
+|>Ht1 >obj_low_tapes >map_list_of_tape elim (list_of_tape ??)
+ [#b @False_ind | #b #tl #Hind #a * [#Ha >Ha //| @Hind]]
+|@sym_eq @Htable
+|>Ht1 %
+|%{(bits_of_state ? (nhalt M) (cstate ?? c))} %{(low_char (current ? (ctape ?? c)))}
+ % [% [% [// | cases (current ??) normalize [|#b] % #Hd destruct (Hd)]
+ |>length_map whd in match (length ??); @eq_f //]
+ |//]
+|>Ht1 >cfg_low_tapes //] -H #H
+lapply(H (bits_of_state … (nhalt M) qout) (low_char … cout)
+ (low_mv … m) tup ? Hingraph)
+ [>Htup whd in ⊢ (??%?); @eq_f >associative_append %] -H
+#Ht2 @(eq_vec ? 3 ?? (niltape ?) ?) >Ht2 #i #Hi
+cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
+ [cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
+ [cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
+ [@False_ind /2/
+ |>Hi >obj_low_tapes >nth_change_vec //
+ >Ht1 >obj_low_tapes >Hstep @map_action
+ ]
+ |>Hi >cfg_low_tapes >nth_change_vec_neq
+ [|% whd in ⊢ (??%?→?); #H destruct (H)]
+ >nth_change_vec // >Hnew_state @eq_f @eq_f >Hnew_tape
+ @eq_f2 [|2:%] >Ht1 >obj_low_tapes >map_move_mono >low_char_current %
+ ]
+ |(* program tapes do not change *)
+ >Hi >prg_low_tapes
+ >nth_change_vec_neq [|% whd in ⊢ (??%?→?); #H destruct (H)]
+ >nth_change_vec_neq [|% whd in ⊢ (??%?→?); #H destruct (H)]
+ >Ht1 >prg_low_tapes //
+ ]
+qed.
projects / helm.git / blobdiff