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.
-(* 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 ?].
-
+
+definition R_cfg_to_obj1 ≝ λt1,t2:Vector (tape FSUnialpha) 3.
+ ∀c,ls.
+ 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
+ (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
+cases (H c ls Hcfg) cases (true_or_false (c==null)) #Hc
+ [#Ht2 #_ >(Ht2 (\P Hc)) -Ht2 @(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 in ⊢ (???%); [2:@eqb_false_to_not_eq %]
+ >nth_change_vec // >(\P Hc) %
+ |#Hi >Hi >nth_change_vec //
+ ]
+ |#Hi >Hi >nth_change_vec_neq [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 %] %
+ ]
+ |#_ #Ht2 >(Ht2 (\Pf Hc)) -Ht2 @(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 in ⊢ (???%); [2:@eqb_false_to_not_eq %]
+ >nth_change_vec // >nth_change_vec //
+ lapply (\bf Hbar) lapply Hc elim c //
+ [whd in ⊢ (??%?→?); #H destruct (H)
+ |#_ whd in ⊢ (??%?→?); #H destruct (H)
+ ]
+ |#Hi >Hi >nth_change_vec //
+ ]
+ |#Hi >Hi >nth_change_vec_neq [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 %] %
+ ]
+ ]
+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 tape_move_obj : mTM FSUnialpha 2 ≝
ifTM ??
(inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
]
qed.
-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 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.
-
-lemma lt_obj : obj < 3. // qed.
-lemma lt_cfg : cfg < 3. // qed.
-lemma lt_prg : prg < 3. // qed.
-
-definition R_copy_strict ≝
- λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
- ((current ? (nth src ? int (niltape ?)) = None ? ∨
- current ? (nth dst ? int (niltape ?)) = None ?) → outt = int) ∧
- (∀ls,x,x0,rs,ls0,rs0.
- nth src ? int (niltape ?) = midtape sig ls x rs →
- nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
- |rs0| ≤ |rs| →
- (∃rs1,rs2.rs = rs1@rs2 ∧ |rs1| = |rs0| ∧
- outt = change_vec ??
- (change_vec ?? int
- (mk_tape sig (reverse sig rs1@x::ls) (option_hd sig rs2)
- (tail sig rs2)) src)
- (mk_tape sig (reverse sig rs1@x::ls0) (None sig) []) dst)).
-
-axiom sem_copy_strict : ∀src,dst,sig,n. src ≠ dst → src < S n → dst < S n →
- copy src dst sig n ⊨ R_copy_strict src dst sig n.
-
-lemma sem_unistep : ∀n,l,h.unistep ⊨ R_unistep n l h.
-#n #l #h
-@(sem_seq_app ??????? (sem_match_m cfg prg FSUnialpha 2 ???)
- (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_tape_move_obj
- (sem_seq ?????? (sem_restart_tape ???) sem_obj_to_cfg)))))))
- /2 by le_n,sym_not_eq/
-#ta #tb #HR #state #char #table #Hta_cfg #Hcfg #Hta_prg #Htable
-#Hbits_obj #Htotaltable
-#nstate #nchar #m #t #Htuple #Hmatch
-cases HR -HR #tc * whd in ⊢ (%→?);
->Hta_cfg #H cases (H ?? (refl ??)) -H
-(* prg starts with a bar, so it's not empty *) #_
->Hta_prg #H lapply (H ??? (refl ??)) -H *
-[| cases Htotaltable #ll * #lr #H >H
- #Hfalse @False_ind cases (Hfalse ll lr) #H1 @H1 //]
-* #ll * #lr * #Hintable -Htotaltable #Htc
-* #td * whd in ⊢ (%→?); >Htc
->nth_change_vec_neq [|@sym_not_eq //] >(nth_change_vec ?????? lt_cfg)
-#Htd lapply (Htd ? (refl ??)) -Htd
->change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
->(?: list_of_tape ? (mk_tape ? (reverse ? (state@[char])@[bar]) (None ?) [ ]) =
- bar::state@[char])
-[|whd in ⊢ (??%?); >left_mk_tape >reverse_append >reverse_reverse
- >current_mk_tape >right_mk_tape normalize >append_nil % ]
-whd in ⊢ (???(???(????%?)??)→?); whd in match (tail ??); #Htd
-(* move cfg to R *)
-* #te * whd in ⊢ (%→?); >Htd
->change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
->nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
->Htable in Hintable; #Hintable #Hte
-(* copy *)
-cases (cfg_in_table_to_tuple ???? Hcfg ?? Hintable)
-#newstate * #m0 * #lr0 * * #Hlr destruct (Hlr) #Hnewcfg #Hm0
-cut (∃fo,so,co.state = fo::so@[co] ∧ |so| = n)
-[ @daemon ] * #fo * #so * #co * #Hstate_exp #Hsolen
-cut (∃fn,sn,cn.newstate = fn::sn@[cn] ∧ |sn| = n)
-[ @daemon ] * #fn * #sn * #cn * #Hnewstate_exp #Hsnlen
-* #tf * * #_ >Hte >(nth_change_vec ?????? lt_prg)
->nth_change_vec_neq [|@sym_not_eq //] >(nth_change_vec ?????? lt_cfg)
->Hstate_exp >Hnewstate_exp
-whd in match (mk_tape ????); whd in match (tape_move ???);
-#Htf cases (Htf ?????? (refl ??) (refl ??) ?)
-[| whd in match (tail ??); >length_append >length_append
- >Hsolen >length_append >length_append >Hsnlen
- <plus_n_Sm <plus_n_Sm <plus_n_Sm <plus_n_O <plus_n_O normalize // ]
-#rs1 * #rs2 whd in match (tail ??); * *
->append_cons #Hrs1rs2 #Hrs1len
->change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
->change_vec_change_vec #Htf
-(* 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 #Htg cases (Htg ?? (refl ??)) -Htg #Htg1 #Htg2
-
-
-
-
-
-[ *
-
- 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 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