\ / GNU General Public License Version 2
V_____________________________________________________________*)
-include "turing/multi_universal/moves_2.ma".
+include "turing/auxiliary_machines.ma".
+include "turing/auxiliary_multi_machines.ma".
+include "turing/multi_universal/alphabet.ma".
+include "turing/multi_universal/tuples.ma".
-(*
- in.obj : ... x ...
- ^
- in.cfg : ... ? ? ...
- ^
-
- out.cfg : ... 1 x ...
- ^
-
- ---------------------
- current (in.obj) = None
-
- in.cfg : ... ? ? ...
- ^
+definition obj ≝ (0:DeqNat).
+definition cfg ≝ (1:DeqNat).
+definition prg ≝ (2:DeqNat).
- out.cfg : ... 0 0 ...
- ^
-
- obj_to_cfg ≝
- move_l(cfg);
- move_l(cfg);
- (if (current(in.obj)) == None
- then write(0,cfg);
- move_r(cfg);
- write(0,cfg);
- else write(1,cfg);
- move_r(cfg);
- copy_step(obj,cfg);
- move_l(obj);)
- move_to_end_l(cfg);
- move_r(cfg);
-
+definition obj_to_cfg ≝
+ (ifTM ?? (inject_TM ? (test_null ?) 2 obj)
+ (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 ·
+ mmove cfg FSUnialpha 2 R.
- cfg_to_obj
+definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3.
+ ∀c,ls.
+ 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) ∧
+ (current ? (nth obj ? t1 (niltape ?)) = None ? →
+ t2 = change_vec ?? t1
+ (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (null::ls)))
+ (tail ? (reverse ? (null::ls)))) cfg).
+
+(*
+axiom accRealize_to_Realize :
+ ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
+ M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse.
*)
+
+lemma eq_mk_tape_rightof :
+ ∀alpha,a,al.mk_tape alpha (a::al) (None ?) [ ] = rightof ? a al.
+#alpha #a #al %
+qed.
-definition obj_to_cfg ≝
- mmove cfg unialpha 3 L ·
- mmove cfg unialpha 3 L ·
- if_TM ?? (inject_TM ? (test_null ?) 3 obj)
- (
+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 None_or_Some: ∀A.∀a. a =None A ∨ ∃b. a = Some ? b.
+#A * /2/ #a %2 %{a} %
+qed.
+lemma not_None_to_Some: ∀A.∀a. a ≠ None A → ∃b. a = Some ? b.
+#A * /2/ * #H @False_ind @H %
+qed.
+lemma sem_obj_to_cfg : obj_to_cfg ⊨ R_obj_to_cfg.
+@(sem_seq_app FSUnialpha 2 ?????
+ (sem_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 %
+ ]
+ |#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.
-definition o2c_states ≝ initN 3.
+(* another semantics for obj_to_cfg *)
+definition low_char' ≝ λc.
+ match c with
+ [ None ⇒ null
+ | Some b ⇒ if (is_bit b) then b else null
+ ].
+
+lemma low_char_option : ∀s.
+ low_char' (option_map FinBool FSUnialpha bit s) = low_char s.
+* //
+qed.
-definition copy0 : copy_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
-definition copy1 : copy_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
-definition copy2 : copy_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+definition R_obj_to_cfg1 ≝ λt1,t2:Vector (tape FSUnialpha) 3.
+ ∀c,ls.
+ nth cfg ? t1 (niltape ?) = midtape ? ls c [ ] →
+ let x ≝ current ? (nth obj ? t1 (niltape ?)) in
+ (∀b. x= Some ? b → is_bit b = true) →
+ t2 = change_vec ?? t1
+ (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (low_char' x::ls)))
+ (tail ? (reverse ? (low_char' x::ls)))) cfg.
+lemma sem_obj_to_cfg1: obj_to_cfg ⊨ R_obj_to_cfg1.
+@(Realize_to_Realize … sem_obj_to_cfg) #t1 #t2 #Hsem
+#c #ls #Hcfg lapply(Hsem c ls Hcfg) * #HSome #HNone #Hb
+cases (None_or_Some ? (current ? (nth obj ? t1 (niltape ?))))
+ [#Hcur >Hcur @HNone @Hcur
+ |* #b #Hb1 >Hb1
+ cut (low_char' (Some ? b) = b) [whd in ⊢ (??%?); >(Hb b Hb1) %] #Hlow >Hlow
+ lapply(current_to_midtape … Hb1) * #lsobj * #rsobj #Hmid
+ @(HSome … Hmid)
+ ]
+qed.
+
+(* test_null_char *)
+definition test_null_char ≝ test_char FSUnialpha (λc.c == null).
-definition trans_copy_step ≝
- λsrc,dst.λsig:FinSet.λn.
- λp:copy_states × (Vector (option sig) (S n)).
- let 〈q,a〉 ≝ p in
- match pi1 … q with
- [ O ⇒ match nth src ? a (None ?) with
- [ None ⇒ 〈copy2,null_action sig n〉
- | Some ai ⇒ match nth dst ? a (None ?) with
- [ None ⇒ 〈copy2,null_action ? n〉
- | Some aj ⇒
- 〈copy1,change_vec ? (S n)
- (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) src)
- (〈Some ? ai,R〉) dst〉
- ]
- ]
- | S q ⇒ match q with
- [ O ⇒ (* 1 *) 〈copy1,null_action ? n〉
- | S _ ⇒ (* 2 *) 〈copy2,null_action ? n〉 ] ].
+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 copy_step ≝
- λsrc,dst,sig,n.
- mk_mTM sig n copy_states (trans_copy_step src dst sig n)
- copy0 (λq.q == copy1 ∨ q == copy2).
+definition cfg_to_obj ≝
+ mmove cfg FSUnialpha 2 L ·
+ (ifTM ?? (inject_TM ? test_null_char 2 cfg)
+ (nop ? 2)
+ (copy_char_N cfg obj FSUnialpha 2)
+ 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 (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)
+ (midtape ? ls c [ ]) 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.
-definition R_comp_step_true ≝
- λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
- ∃x,y.
- current ? (nth src ? int (niltape ?)) = Some ? x ∧
- current ? (nth dst ? int (niltape ?)) = Some ? y ∧
- outt = change_vec ??
- (change_vec ?? int
- (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src)
- (tape_move_mono ? (nth dst ? int (niltape ?)) 〈Some ? x, R〉) dst.
+lemma sem_cfg_to_obj : cfg_to_obj ⊨ R_cfg_to_obj.
+@(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
+ (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 *
+[ * #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 //
+ | #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
+ whd in ⊢ (%→?); #Htb
+ #c #ls #Hta % #Hc
+ [ >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 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
+ [ bit b ⇒ if b then R else L
+ | _ ⇒ N].
-definition R_comp_step_false ≝
- λsrc,dst:nat.λsig,n.λint,outt: Vector (tape sig) (S n).
- (current ? (nth src ? int (niltape ?)) = None ? ∨
- current ? (nth dst ? int (niltape ?)) = None ?) ∧ outt = int.
+definition char_to_bit_option ≝ λc.match c with
+ [ bit b ⇒ Some ? (bit b)
+ | _ ⇒ None ?].
-lemma copy_q0_q2_null :
- ∀src,dst,sig,n,v.src < S n → dst < S n →
- (nth src ? (current_chars ?? v) (None ?) = None ? ∨
- nth dst ? (current_chars ?? v) (None ?) = None ?) →
- step sig n (copy_step src dst sig n) (mk_mconfig ??? copy0 v)
- = mk_mconfig ??? copy2 v.
-#src #dst #sig #n #v #Hi #Hj
-whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??%?);
-* #Hcurrent
-[ @eq_f2
- [ whd in ⊢ (??(???%)?); >Hcurrent %
- | whd in ⊢ (??(????(???%))?); >Hcurrent @tape_move_null_action ]
-| @eq_f2
- [ whd in ⊢ (??(???%)?); >Hcurrent cases (nth src ?? (None sig)) //
- | whd in ⊢ (??(????(???%))?); >Hcurrent
- cases (nth src ?? (None sig)) [|#x] @tape_move_null_action ] ]
+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)
+ (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.
-lemma copy_q0_q1 :
- ∀src,dst,sig,n,v,a,b.src ≠ dst → src < S n → dst < S n →
- nth src ? (current_chars ?? v) (None ?) = Some ? a →
- nth dst ? (current_chars ?? v) (None ?) = Some ? b →
- step sig n (copy_step src dst sig n) (mk_mconfig ??? copy0 v) =
- mk_mconfig ??? copy1
- (change_vec ? (S n)
- (change_vec ?? v
- (tape_move_mono ? (nth src ? v (niltape ?)) 〈None ?, R〉) src)
- (tape_move_mono ? (nth dst ? v (niltape ?)) 〈Some ? a, R〉) dst).
-#src #dst #sig #n #v #a #b #Heq #Hsrc #Hdst #Ha1 #Ha2
-whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-[ whd in match (trans ????);
- >Ha1 >Ha2 whd in ⊢ (??(???%)?); >(\b ?) //
-| whd in match (trans ????);
- >Ha1 >Ha2 whd in ⊢ (??(????(???%))?); >(\b ?) //
- change with (change_vec ?????) in ⊢ (??(????%)?);
- <(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_f2 // >nth_change_vec_neq //
+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_step :
- ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
- copy_step src dst sig n ⊨
- [ copy1: R_comp_step_true src dst sig n,
- R_comp_step_false src dst sig n ].
-#src #dst #sig #n #Hneq #Hsrc #Hdst #int
-lapply (refl ? (current ? (nth src ? int (niltape ?))))
-cases (current ? (nth src ? int (niltape ?))) in ⊢ (???%→?);
-[ #Hcur_src %{2} %
- [| % [ %
- [ whd in ⊢ (??%?); >copy_q0_q2_null /2/
- | normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ % // % // ] ]
-| #a #Ha lapply (refl ? (current ? (nth dst ? int (niltape ?))))
- cases (current ? (nth dst ? int (niltape ?))) in ⊢ (???%→?);
- [ #Hcur_dst %{2} %
- [| % [ %
- [ whd in ⊢ (??%?); >copy_q0_q2_null /2/
- | normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ % // %2 >Hcur_dst % ] ]
- | #b #Hb %{2} %
- [| % [ %
- [whd in ⊢ (??%?); >(copy_q0_q1 … a b Hneq Hsrc Hdst) //
- | #_ %{a} %{b} % // % //]
- | * #H @False_ind @H %
- ]
- ]
- ]
+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 copy ≝ λsrc,dst,sig,n.
- whileTM … (copy_step src dst sig n) copy1.
+(************** list of tape ******************)
+definition list_of_tape ≝ λsig.λt:tape sig.
+ reverse ? (left ? t)@option_cons ? (current ? t) (right ? t).
-definition R_copy ≝
- 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 →
- (∃rs01,rs02.rs0 = rs01@rs02 ∧ |rs01| = |rs| ∧
- outt = change_vec ??
- (change_vec ?? int
- (mk_tape sig (reverse sig rs@x::ls) (None sig) []) src)
- (mk_tape sig (reverse sig rs@x::ls0) (option_hd sig rs02)
- (tail sig rs02)) dst) ∨
- (∃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)).
+lemma list_of_midtape: ∀sig,ls,c,rs.
+ list_of_tape sig (midtape ? ls c rs) = reverse ? ls@c::rs.
+// qed-.
-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
- ]
- ]
+lemma list_of_rightof: ∀sig,ls,c.
+ list_of_tape sig (rightof ? c ls) = reverse ? (c::ls).
+#sig #ls #c <(append_nil ? (reverse ? (c::ls)))
+// qed-.
+
+lemma list_of_tape_move: ∀sig,t,m.
+ list_of_tape sig t = list_of_tape sig (tape_move ? t m).
+#sig #t * // cases t //
+ [(* rightof, move L *) #a #l >list_of_midtape
+ >append_cons <reverse_single <reverse_append %
+ |(* midtape, move L *) * //
+ #a #ls #c #rs >list_of_midtape >list_of_midtape
+ >reverse_cons >associative_append %
+ |(* midtape, move R *) #ls #c *
+ [>list_of_midtape >list_of_rightof >reverse_cons %
+ |#a #rs >list_of_midtape >list_of_midtape >reverse_cons
+ >associative_append %
+ ]
+ ]
+qed.
+
+lemma list_of_tape_write: ∀sig,cond,t,c.
+(∀b. c = Some ? b → cond b =true) →
+(∀x. mem ? x (list_of_tape ? t) → cond x =true ) →
+∀x. mem ? x (list_of_tape sig (tape_write ? t c)) → cond x =true.
+#sig #cond #t #c #Hc #Htape #x lapply Hc cases c
+ [(* c is None *) #_ whd in match (tape_write ???); @Htape
+ |#b #Hb lapply (Hb … (refl ??)) -Hb #Hb
+ whd in match (tape_write ???); >list_of_midtape
+ #Hx cases(mem_append ???? Hx) -Hx
+ [#Hx @Htape @mem_append_l1 @Hx
+ |* [//]
+ #Hx @Htape @mem_append_l2 cases (current sig t)
+ [@Hx | #c1 %2 @Hx]
]
]
qed.
-
-
-lemma 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
+
+lemma current_in_list: ∀sig,t,b.
+ current sig t = Some ? b → mem ? b (list_of_tape sig t).
+#sig #t #b cases t
+ [whd in ⊢ (??%?→?); #Htmp destruct
+ |#l #b whd in ⊢ (??%?→?); #Htmp destruct
+ |#l #b whd in ⊢ (??%?→?); #Htmp destruct
+ |#ls #c #rs whd in ⊢ (??%?→?); #Htmp destruct
+ >list_of_midtape @mem_append_l2 % %
]
-]
qed.
+
+definition restart_tape ≝ λi,n.
+ mmove i FSUnialpha n L ·
+ inject_TM ? (move_to_end FSUnialpha L) n i ·
+ mmove i FSUnialpha n R.
+
+definition R_restart_tape ≝ λi,n.λint,outt:Vector (tape FSUnialpha) (S n).
+ ∀t.t = nth i ? int (niltape ?) →
+ outt = change_vec ?? int
+ (mk_tape ? [ ] (option_hd ? (list_of_tape ? t)) (tail ? (list_of_tape ? t))) i.
-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 // ]
+lemma sem_restart_tape : ∀i,n.i < S n → restart_tape i n ⊨ R_restart_tape i n.
+#i #n #Hleq
+@(sem_seq_app ??????? (sem_move_multi ? n i L ?)
+ (sem_seq ?????? (sem_inject ???? i ? (sem_move_to_end_l ?))
+ (sem_move_multi ? n i R ?))) [1,2,3:@le_S_S_to_le //]
+#ta #tb * #tc * whd in ⊢ (%→?); #Htc
+* #td * * * #Htd1 #Htd2 #Htd3
+whd in ⊢ (%→?); #Htb *
+[ #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
+ cut (td = tc)
+ [ <(change_vec_same … tc … i … (niltape ?))
+ @(eq_vec_change_vec … (niltape ?))
+ [ @Htd1 >Htc >nth_change_vec //
+ | @Htd3 ] ]
+ (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
+ #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
+ #Htb >Htb %
+| #r0 #rs0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
+ cut (td = tc)
+ [ <(change_vec_same … tc … i … (niltape ?))
+ @(eq_vec_change_vec … (niltape ?))
+ [ @Htd1 >Htc >nth_change_vec //
+ | @Htd3 ] ]
+ (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
+ #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
+ #Htb >Htb %
+| #l0 #ls0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
+ cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@[l0])) i)
+ [ <(change_vec_same … tc … i … (niltape ?))
+ @(eq_vec_change_vec … (niltape ?))
+ [ @Htd2 >Htc >nth_change_vec //
+ | #j #Hij >nth_change_vec_neq // @Htd3 // ]]
+ #Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
+ >nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
+ cases (reverse ? ls0)
+ [ %
+ | #l1 #ls1 >reverse_cons
+ >(?: list_of_tape ? (rightof ? l0 (reverse ? ls1@[l1])) =
+ l1::ls1@[l0])
+ [|change with (reverse ??@?) in ⊢ (??%?);
+ whd in match (left ??); >reverse_cons >reverse_append
+ whd in ⊢ (??%?); @eq_f >reverse_reverse normalize >append_nil % ] % ]
+| *
+ [ #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
+ cut (td = tc)
+ [ <(change_vec_same … tc … i … (niltape ?))
+ @(eq_vec_change_vec … (niltape ?))
+ [ @Htd1 >Htc >nth_change_vec //
+ | @Htd3 ] ]
+ (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
+ #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
+ #Htb >Htb %
+ | #l0 #ls0 #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
+ cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@l0::c::rs)) i)
+ [ @(eq_vec_change_vec … (niltape ?))
+ [ @Htd2 >Htc >nth_change_vec //
+ | @Htd3 ] ]
+ #Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
+ >nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
+ cases (reverse ? ls0)
+ [ %
+ | #l1 #ls1 >reverse_cons
+ >(?: list_of_tape ? (midtape ? (l0::reverse ? ls1@[l1]) c rs) =
+ l1::ls1@l0::c::rs)
+ [|change with (reverse ??@?) in ⊢ (??%?);
+ whd in match (left ??); >reverse_cons >reverse_append
+ whd in ⊢ (??%?); @eq_f >reverse_reverse normalize
+ >associative_append % ] % ]
+ ]
+]
qed.
projects / helm.git / blobdiff