(* *)
(**************************************************************************)
+include "turing/simple_machines.ma".
include "turing/multi_universal/compare.ma".
include "turing/multi_universal/par_test.ma".
include "turing/multi_universal/moves_2.ma".
-definition Rtc_multi_true ≝
- λalpha,test,n,i.λt1,t2:Vector ? (S n).
- (∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1.
-
-definition Rtc_multi_false ≝
- λalpha,test,n,i.λt1,t2:Vector ? (S n).
- (∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
-
-lemma sem_test_char_multi :
- ∀alpha,test,n,i.i ≤ n →
- inject_TM ? (test_char ? test) n i ⊨
- [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
-#alpha #test #n #i #Hin #int
-cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
-#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
-[ @Hloop
-| #Hqtrue lapply (Htrue Hqtrue) * * * #c *
- #Hcur #Htestc #Hnth_i #Hnth_j %
- [ %{c} % //
- | @(eq_vec … (niltape ?)) #i0 #Hi0
- cases (decidable_eq_nat i0 i) #Hi0i
- [ >Hi0i @Hnth_i
- | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
-| #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
- [ @Htestc
- | @(eq_vec … (niltape ?)) #i0 #Hi0
- cases (decidable_eq_nat i0 i) #Hi0i
- [ >Hi0i @Hnth_i
- | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
-qed.
-
-definition Rm_test_null_true ≝
- λalpha,n,i.λt1,t2:Vector ? (S n).
- current alpha (nth i ? t1 (niltape ?)) ≠ None ? ∧ t2 = t1.
-
-definition Rm_test_null_false ≝
- λalpha,n,i.λt1,t2:Vector ? (S n).
- current alpha (nth i ? t1 (niltape ?)) = None ? ∧ t2 = t1.
-
-lemma sem_test_null_multi : ∀alpha,n,i.i ≤ n →
- inject_TM ? (test_null ?) n i ⊨
- [ tc_true : Rm_test_null_true alpha n i, Rm_test_null_false alpha n i ].
-#alpha #n #i #Hin #int
-cases (acc_sem_inject … Hin (sem_test_null alpha) int)
-#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
-[ @Hloop
-| #Hqtrue lapply (Htrue Hqtrue) * * #Hcur #Hnth_i #Hnth_j % //
- @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i
- [ >Hi0i @sym_eq @Hnth_i | @sym_eq @Hnth_j @sym_not_eq // ] ]
-| #Hqfalse lapply (Hfalse Hqfalse) * * #Hcur #Hnth_i #Hnth_j %
- [ @Hcur
- | @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) //
- #Hi0i @sym_eq @Hnth_j @sym_not_eq // ] ]
-qed.
-
definition match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n.
match (nth src (option sig) v (None ?)) with
[ None ⇒ false
| Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ].
-
-definition mmove_states ≝ initN 2.
-
-definition mmove0 : mmove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
-definition mmove1 : mmove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
-
-definition trans_mmove ≝
- λi,sig,n,D.
- λp:mmove_states × (Vector (option sig) (S n)).
- let 〈q,a〉 ≝ p in match (pi1 … q) with
- [ O ⇒ 〈mmove1,change_vec ? (S n) (null_action ? n) (〈None ?,D〉) i〉
- | S _ ⇒ 〈mmove1,null_action sig n〉 ].
-
-definition mmove ≝
- λi,sig,n,D.
- mk_mTM sig n mmove_states (trans_mmove i sig n D)
- mmove0 (λq.q == mmove1).
-
-definition Rm_multi ≝
- λalpha,n,i,D.λt1,t2:Vector ? (S n).
- t2 = change_vec ? (S n) t1 (tape_move alpha (nth i ? t1 (niltape ?)) D) i.
-
-lemma sem_move_multi :
- ∀alpha,n,i,D.i ≤ n →
- mmove i alpha n D ⊨ Rm_multi alpha n i D.
-#alpha #n #i #D #Hin #int %{2}
-%{(mk_mconfig ? mmove_states n mmove1 ?)}
-[| %
- [ whd in ⊢ (??%?); @eq_f whd in ⊢ (??%?); @eq_f %
- | whd >tape_move_multi_def
- <(change_vec_same … (ctapes …) i (niltape ?))
- >pmap_change <tape_move_multi_def >tape_move_null_action % ] ]
- qed.
definition rewind ≝ λsrc,dst,sig,n.
parmove src dst sig n L · mmove src sig n R · mmove dst sig n R.
src ≠ dst → src < S n → dst < S n →
rewind src dst sig n ⊨ R_rewind src dst sig n.
#src #dst #sig #n #Hneq #Hsrc #Hdst @(Realize_to_Realize … (sem_rewind_strong …)) //
-#ta #tb * * * #H1 #H2 #H3 #H4 % /2/
+#ta #tb * * * #H1 #H2 #H3 #H4 % /2 by /
qed.
definition match_step ≝ λsrc,dst,sig,n.
include "turing/multi_universal/unistep_aux.ma".
-definition unistep ≝
- match_m cfg prg FSUnialpha 2 ·
- restart_tape cfg 2 · mmove cfg ? 2 R · copy prg cfg FSUnialpha 2 ·
+definition exec_move ≝
cfg_to_obj · tape_move_obj · restart_tape prg 2 · 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 low_char' ≝ λc.
match c with
[ None ⇒ null
* //
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 →
+ c ≠ bar → m ≠ bar →
+ let new_obj ≝
+ tape_move_mono ? (nth obj ? t1 (niltape ?))
+ 〈char_to_bit_option c, char_to_move m〉 in
+ let next_c ≝ low_char' (current ? new_obj) in
+ let new_cfg ≝ midtape ? [ ] bar ((reverse ? ls1)@[next_c]) in
+ let new_prg ≝ midtape FSUnialpha [ ] bar ((reverse ? ls2)@m::rs2) in
+ t2 = Vector_of_list ? [new_obj;new_cfg;new_prg].
+
+
+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))) //
+#ta #tout * #t1 * #semM1 * #t2 * #semM2 * #t3 * #semM3 #semM4
+#c #m #ls1 #ls2 #rs2 #Hcfg #Hprg #Hc #Hm
+(* M1 = cfg_to_obj *)
+lapply (semM1 … Hcfg Hc) #Ht1
+(* M2 = tape_move *)
+whd in semM2; >Ht1 in semM2; -Ht1
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>Hprg #Ht2 lapply (Ht2 … (refl ??)) -Ht2
+>nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+>nth_change_vec // >change_vec_commute [2:@eqb_false_to_not_eq %]
+>change_vec_change_vec #Ht2
+(* M3 = restart prg *)
+whd in semM3; >Ht2 in semM3; #semM3 lapply (semM3 … (refl ??)); -semM3
+>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 *)
+
+
+definition unistep ≝
+ match_m cfg prg FSUnialpha 2 ·
+ restart_tape cfg 2 · mmove cfg ? 2 R · copy prg cfg FSUnialpha 2 ·
+ cfg_to_obj · tape_move_obj · restart_tape prg 2 · 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 R_unistep ≝ λn,l,h.λt1,t2: Vector ? 3.
∀state,char,table.
(* cfg *)
(sem_seq ?????? (sem_restart_tape ???)
(sem_seq ?????? (sem_move_multi ? 2 cfg R ?)
(sem_seq ?????? (sem_copy_strict prg cfg FSUnialpha 2 ???)
- (sem_seq ?????? sem_cfg_to_obj
+ (sem_seq ?????? sem_cfg_to_obj1
(sem_seq ?????? sem_tape_move_obj
(sem_seq ?????? (sem_restart_tape ???) sem_obj_to_cfg)))))))
/2 by le_n,sym_not_eq/
(* cfg to obj *)
* #tg * whd in ⊢ (%→?); >Htf
>nth_change_vec_neq [|@sym_not_eq //] >(nth_change_vec ?????? lt_cfg)
+lapply (append_l1_injective ?????? Hrs1rs2)
+[ >Hsnlen >Hrs1len >length_append >length_append >length_append >length_append
+ normalize >Hsolen >Hsnlen % ] #Hrs1 <Hrs1 >reverse_append >reverse_single
+ >associative_append #Htg lapply (Htg … (refl ??) Hm0) -Htg
+ (* simplifying tg *)
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+ >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+
+
+
+
+
+ cut ((mk_tape FSUnialpha []
+ (option_hd FSUnialpha
+ (reverse FSUnialpha (m0::[]@reverse FSUnialpha (sn@[cn])@[fn; bar])))
+ (tail FSUnialpha
+ (reverse FSUnialpha (m0::[]@reverse FSUnialpha (sn@[cn])@[fn; bar])))) =
+ midtape ? [ ] bar (fn::sn@[cn;m0]))
+ [cut (reverse FSUnialpha (m0::[]@reverse FSUnialpha (sn@[cn])@[fn; bar]) =
+ bar::fn::sn@[cn;m0])
+ [>reverse_cons whd in ⊢ (??(??(??%)?)?); >reverse_append >reverse_reverse
+ >append_cons in ⊢ (???%); % ] #Hrev >Hrev % ] #Hmk_tape >Hmk_tape -Hmk_tape
+ >change_vec_commute
+
+ >reverse_append >reverse_append
+ check reverse_cons
+ <reverse_cons >reverse_cons
+ -Htg #Htg
+
>(change_vec_commute ????? cfg prg) [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 %]
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.
| >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @H2 @sym_not_eq // ]
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 %] %
+ ]
+ |#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 %
]
-| * #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 %]
+ <(Htbeq 2) [%|@eqb_false_to_not_eq %]
+ ]
]
qed.
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
\ / GNU General Public License Version 2
V_____________________________________________________________*)
-include "turing/multi_universal/match.ma".
+include "turing/if_multi.ma".
+include "turing/inject.ma".
+include "turing/basic_machines.ma".
+
+definition Rtc_multi_true ≝
+ λalpha,test,n,i.λt1,t2:Vector ? (S n).
+ (∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1.
+
+definition Rtc_multi_false ≝
+ λalpha,test,n,i.λt1,t2:Vector ? (S n).
+ (∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
+
+lemma sem_test_char_multi :
+ ∀alpha,test,n,i.i ≤ n →
+ inject_TM ? (test_char ? test) n i ⊨
+ [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
+#alpha #test #n #i #Hin #int
+cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
+#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Hqtrue lapply (Htrue Hqtrue) * * * #c *
+ #Hcur #Htestc #Hnth_i #Hnth_j %
+ [ %{c} % //
+ | @(eq_vec … (niltape ?)) #i0 #Hi0
+ cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @Hnth_i
+ | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
+| #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
+ [ @Htestc
+ | @(eq_vec … (niltape ?)) #i0 #Hi0
+ cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @Hnth_i
+ | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
+qed.
+
+definition Rm_test_null_true ≝
+ λalpha,n,i.λt1,t2:Vector ? (S n).
+ current alpha (nth i ? t1 (niltape ?)) ≠ None ? ∧ t2 = t1.
+
+definition Rm_test_null_false ≝
+ λalpha,n,i.λt1,t2:Vector ? (S n).
+ current alpha (nth i ? t1 (niltape ?)) = None ? ∧ t2 = t1.
+
+lemma sem_test_null_multi : ∀alpha,n,i.i ≤ n →
+ inject_TM ? (test_null ?) n i ⊨
+ [ tc_true : Rm_test_null_true alpha n i, Rm_test_null_false alpha n i ].
+#alpha #n #i #Hin #int
+cases (acc_sem_inject … Hin (sem_test_null alpha) int)
+#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Hqtrue lapply (Htrue Hqtrue) * * #Hcur #Hnth_i #Hnth_j % //
+ @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @sym_eq @Hnth_i | @sym_eq @Hnth_j @sym_not_eq // ] ]
+| #Hqfalse lapply (Hfalse Hqfalse) * * #Hcur #Hnth_i #Hnth_j %
+ [ @Hcur
+ | @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) //
+ #Hi0i @sym_eq @Hnth_j @sym_not_eq // ] ]
+qed.
+(* move a single tape *)
+definition mmove_states ≝ initN 2.
+
+definition mmove0 : mmove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
+definition mmove1 : mmove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
+
+definition trans_mmove ≝
+ λi,sig,n,D.
+ λp:mmove_states × (Vector (option sig) (S n)).
+ let 〈q,a〉 ≝ p in match (pi1 … q) with
+ [ O ⇒ 〈mmove1,change_vec ? (S n) (null_action ? n) (〈None ?,D〉) i〉
+ | S _ ⇒ 〈mmove1,null_action sig n〉 ].
+
+definition mmove ≝
+ λi,sig,n,D.
+ mk_mTM sig n mmove_states (trans_mmove i sig n D)
+ mmove0 (λq.q == mmove1).
+
+definition Rm_multi ≝
+ λalpha,n,i,D.λt1,t2:Vector ? (S n).
+ t2 = change_vec ? (S n) t1 (tape_move alpha (nth i ? t1 (niltape ?)) D) i.
+
+lemma sem_move_multi :
+ ∀alpha,n,i,D.i ≤ n →
+ mmove i alpha n D ⊨ Rm_multi alpha n i D.
+#alpha #n #i #D #Hin #int %{2}
+%{(mk_mconfig ? mmove_states n mmove1 ?)}
+[| %
+ [ whd in ⊢ (??%?); @eq_f whd in ⊢ (??%?); @eq_f %
+ | whd >tape_move_multi_def
+ <(change_vec_same … (ctapes …) i (niltape ?))
+ >pmap_change <tape_move_multi_def >tape_move_null_action % ] ]
+ qed.
+
+(* simple copy with no move *)
+definition copy_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_N ≝
+ λsrc,dst.λsig:FinSet.λn.
+ λp:copy_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 ?,N〉) src)
+ (〈nth src ? a (None ?),N〉) dst〉
+ | S _ ⇒ 〈cc1,null_action ? n〉 ].
+
+definition copy_char_N ≝
+ λsrc,dst,sig,n.
+ mk_mTM sig n copy_states (trans_copy_char_N src dst sig n)
+ cc0 (λq.q == cc1).
+
+definition R_copy_char_N ≝
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ outt = change_vec ?? int
+ (tape_write ? (nth dst ? int (niltape ?))
+ (current ? (nth src ? int (niltape ?)))) dst.
+
+lemma copy_char_N_q0_q1 :
+ ∀src,dst,sig,n,v.src ≠ dst → src < S n → dst < S n →
+ step sig n (copy_char_N src dst sig n) (mk_mconfig ??? cc0 v) =
+ mk_mconfig ??? cc1
+ (change_vec ?? v
+ (tape_write ? (nth dst ? v (niltape ?))
+ (current ? (nth src ? v (niltape ?)))) dst).
+#src #dst #sig #n #v #Heq #Hsrc #Hdst
+whd in ⊢ (??%?); @eq_f
+<(change_vec_same … v dst (niltape ?)) in ⊢ (??%?);
+<(change_vec_same … v src (niltape ?)) in ⊢ (??%?);
+>tape_move_multi_def
+>pmap_change >pmap_change <tape_move_multi_def
+>tape_move_null_action @eq_f3 //
+[ >change_vec_same %
+| >change_vec_same >change_vec_same >nth_current_chars // ]
+qed.
+
+lemma sem_copy_char_N:
+ ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
+ copy_char_N src dst sig n ⊨ R_copy_char_N src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst #int
+%{2} % [| % [ % | whd >copy_char_N_q0_q1 // ]]
+qed.
+
+(**************** copy and advance ***********************)
+definition copy_char_states ≝ initN 3.
+
+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.
+
+
(********************** look_ahead test *************************)
projects / helm.git / commitdiff