include "turing/multi_universal/match.ma".
include "turing/multi_universal/copy.ma".
include "turing/multi_universal/alphabet.ma".
+include "turing/multi_universal/tuples.ma".
(*
mmove cfg FSUnialpha 2 L ·
(ifTM ?? (inject_TM ? (test_null ?) 2 obj)
(copy_step obj cfg FSUnialpha 2 ·
- mmove cfg FSUnialpha 2 L)
+ 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 ·
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::ls))) (tail ? (reverse ? (c::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 ? (null::ls)))
(sem_if ??????????
(sem_test_null_multi ?? obj ?)
(sem_seq ?????? (accRealize_to_Realize … (sem_copy_step …))
- (sem_move_multi ? 2 cfg L ?))
+ (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 ?)))) //
#td * *
[ * #te * * #Hcurtc #Hte
* destruct (Hte) #te * *
- [ whd in ⊢ (%→%→?); * #x * #y * * -Hcurtc #Hcurtc1 #Hcurtc2 #Hte #Htd
- * #tf * * * whd in ⊢ (%→%→%→%→?); #Htf1 #Htf2 #Htf3 #Htb
+ [ 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 (tf = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls@[x])) cfg)
- [@daemon] -Htf1 -Htf2 -Htf3 #Htf destruct (Htf Hte Htd Htc Htb)
- >change_vec_change_vec >change_vec_change_vec >change_vec_change_vec
- >nth_change_vec // >tape_move_mk_tape_R
- @daemon
+ cut (tg = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[x])) cfg)
+ [@daemon] -Htg1 -Htg2 -Htg3 #Htg destruct (Htg Htf Hte Htd Htc Htb)
+ >change_vec_change_vec >change_vec_change_vec
+ >change_vec_commute // >change_vec_change_vec
+ >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
+ >change_vec_commute // >change_vec_change_vec
+ >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
+ >change_vec_commute [|@sym_not_eq //] @eq_f3 //
+ [ >Hta2 cases rso in Hta2; whd in match (tape_move_mono ???);
+ [ #Hta2 whd in match (tape_move ???); <Hta2 @change_vec_same
+ | #r1 #rs1 #Hta2 whd in match (tape_move ???); <Hta2 @change_vec_same ]
+ | >tape_move_mk_tape_R [| #_ % %] >reverse_cons
+ >nth_change_vec_neq in Hcurtc1; [|@sym_not_eq //] >Hta2
+ normalize in ⊢ (%→?); #H destruct (H) %
+ ]
| #Hta2 >Htc in Hcurtc1; >nth_change_vec_neq [| @sym_not_eq //]
>Hta2 #H destruct (H)
]
(ifTM ?? (inject_TM ? test_null_char 2 cfg)
(nop ? 2)
(copy_step cfg obj FSUnialpha 2 ·
- mmove cfg FSUnialpha 2 L)
+ 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.
(midtape ? (left ? (nth obj ? t1 (niltape ?))) c (right ? (nth obj ? t1 (niltape ?)))) obj)
(mk_tape ? [ ] (option_hd ? (reverse ? (c::ls))) (tail ? (reverse ? (c::ls)))) cfg).
-axiom sem_cfg_to_obj : obj_to_cfg ⊨ R_obj_to_cfg.
+axiom sem_cfg_to_obj : cfg_to_obj ⊨ R_cfg_to_obj.
(*@(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
(sem_seq ??????
(sem_if ??????????
(* 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 *)
-axiom tape_move_obj : mTM FSUnialpha 2.
+definition char_to_move ≝ λc.match c with
+ [ bit b ⇒ if b then R else L
+ | _ ⇒ N].
+
+definition tape_move_obj : mTM FSUnialpha 2 ≝
+ ifTM ??
+ (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
+ (mmove obj FSUnialpha 2 L)
+ (ifTM ??
+ (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit true)) 2 prg)
+ (mmove obj FSUnialpha 2 R)
+ (nop ??)
+ tc_true)
+ tc_true.
+
+definition restart_tape ≝ λi.
+ inject_TM ? (move_to_end FSUnialpha L) 2 i ·
+ mmove i FSUnialpha 2 R.
definition unistep ≝
- obj_to_cfg · match_m cfg prg FSUnialpha 2 · copy prg cfg FSUnialpha 2 ·
- cfg_to_obj · tape_move_obj.
\ No newline at end of file
+ obj_to_cfg · match_m cfg prg FSUnialpha 2 ·
+ restart_tape cfg · copy prg cfg FSUnialpha 2 ·
+ cfg_to_obj · tape_move_obj · restart_tape prg.
+
+(*
+definition legal_tape ≝ λn,l,h,t.
+ ∃state,char,table.
+ nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[char]) →
+ is_config n (bar::state@[char]) →
+ nth prg ? t1 (niltape ?) = midtape ? [ ] bar table →
+ bar::table = table_TM n l h → *)
+
+definition list_of_tape ≝ λsig,t.
+ left sig t@option_cons ? (current ? t) (right ? t).
+
+definition low_char' ≝ λc.
+ match c with
+ [ None ⇒ null
+ | Some b ⇒ if (is_bit b) then b else null
+ ].
+
+definition R_unistep ≝ λn,l,h.λt1,t2: Vector ? 3.
+ ∀state,oldc,table.
+ (* cfg *)
+ nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[oldc]) →
+ is_config n (bar::state@[oldc]) →
+ (* prg *)
+ nth prg ? t1 (niltape ?) = midtape ? [ ] bar table →
+ bar::table = table_TM n l h →
+ (* obj *)
+ only_bits (list_of_tape ? (nth obj ? t1 (niltape ?))) →
+ let char ≝ low_char' (current ? (nth obj ? t1 (niltape ?))) in
+ let conf ≝ (bar::state@[char]) in
+ (∃ll,lr.bar::table = ll@conf@lr) →
+ ∃nstate,nchar,m,t. tuple_encoding n h t = (conf@nstate@[nchar;m]) ∧
+ mem ? t l ∧
+ t2 =
+ change_vec ??
+ (change_vec ?? t1 (midtape ? [ ] bar (nstate@[nchar])) cfg)
+ (tape_move_mono ? (nth obj ? t1 (niltape ?)) 〈Some ? nchar,char_to_move m〉) obj.
+
+definition tape_map ≝ λA,B:FinSet.λf:A→B.λt.
+ mk_tape B (map ?? f (left ? t))
+ (option_map ?? f (current ? t))
+ (map ?? f (right ? t)).
+
+definition low ≝ λM:normalTM.λc:nconfig (no_states M).Vector_of_list ?
+ [tape_map ?? bit (ctape ?? c);
+ midtape ? [ ] bar (bits_of_state ? (nhalt M) (cstate ?? c));
+ midtape ? [ ] bar (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M))
+ ].
+
+
+
+
+
\ No newline at end of file
projects / helm.git / blobdiff