V_____________________________________________________________*)
include "turing/turing.ma".
+(* include "turing/basic_machines.ma". *)
(******************* inject a mono machine into a multi tape one **********************)
+
definition inject_trans ≝ λsig,states:FinSet.λn,i:nat.
- λtrans:states × (option sig) → states × (option (sig × move)).
+ λtrans:states × (option sig) → states × (option sig × move).
λp:states × (Vector (option sig) (S n)).
let 〈q,a〉 ≝ p in
let 〈nq,na〉 ≝ trans 〈q,nth i ? a (None ?)〉 in
definition inject_TM ≝ λsig.λM:TM sig.λn,i.
mk_mTM sig n
(states ? M)
- (inject_trans ?? n i (trans ? M))
+ (inject_trans sig (states ? M) n i ?) (* (trans sig M))*)
(start ? M)
(halt ? M).
+(* ????? *)
+lapply (trans sig M) #trans #x lapply (trans x) * *
+#s #a #m % [ @s | % [ @a | @m ] ]
+qed.
-axiom current_chars_change_vec: ∀sig,n,v,a,i. i < S n →
+lemma current_chars_change_vec: ∀sig,n,v,a,i. i < S n →
current_chars sig ? (change_vec ? (S n) v a i) =
change_vec ? (S n)(current_chars ?? v) (current ? a) i.
-
+#sig #n #v #a #i #Hi @(eq_vec … (None ?)) #i0 #Hi0
+change with (vec_map ?????) in match (current_chars ???);
+<(nth_vec_map … (niltape ?))
+cases (decidable_eq_nat i i0) #Hii0
+[ >Hii0 >nth_change_vec // >nth_change_vec //
+| >nth_change_vec_neq // >nth_change_vec_neq // @nth_vec_map ]
+qed.
+
lemma inject_trans_def :∀sig:FinSet.∀n,i:nat.i < S n →
- ∀M,v,s,a,sn,an.
- trans sig M 〈s,a〉 = 〈sn,an〉 →
- cic:/matita/turing/turing/trans.fix(0,2,9) sig n (inject_TM sig M n i) 〈s,change_vec ? (S n) v a i〉 =
- 〈sn,change_vec ? (S n) (null_action ? n) an i〉.
-#sig #n #i #Hlt #trans #v #s #a #sn #an #Htrans
+ ∀M,v,s,a,sn,an,mn.
+ trans sig M 〈s,a〉 = 〈sn,an,mn〉 →
+ cic:/matita/turing/turing/trans#fix:0:2:9 sig n (inject_TM sig M n i) 〈s,change_vec ? (S n) v a i〉 =
+ 〈sn,change_vec ? (S n) (null_action ? n) 〈an,mn〉 i〉.
+#sig #n #i #Hlt #M #v #s #a #sn #an #mn #Htrans
whd in ⊢ (??%?); >nth_change_vec // >Htrans //
qed.
+lemma inj_ter: ∀A,B,C.∀p:A×B×C.
+ p = 〈\fst (\fst p), \snd (\fst p), \snd p〉.
+// qed.
+
lemma inject_step : ∀sig,n,M,i,q,t,nq,nt,v. i < S n →
step sig M (mk_config ?? q t) = mk_config ?? nq nt →
- cic:/matita/turing/turing/step.def(10) sig n (inject_TM sig M n i)
+ cic:/matita/turing/turing/step#def:12 sig n (inject_TM sig M n i)
(mk_mconfig ?? n q (change_vec ? (S n) v t i))
= mk_mconfig ?? n nq (change_vec ? (S n) v nt i).
#sig #n #M #i #q #t #nq #nt #v #lein whd in ⊢ (??%?→?);
whd in match (step ????); >(current_chars_change_vec … lein)
->(eq_pair_fst_snd … (trans sig M ?)) whd in ⊢ (??%?→?); #H
->(inject_trans_def sig n i lein M … (eq_pair_fst_snd … ))
+>(inj_ter … (trans sig M ?)) whd in ⊢ (??%?→?); #H
+>(inject_trans_def sig n i lein M …)
+[|>(eq_pair_fst_snd ?? (trans sig M 〈q,current sig t〉))
+ >(eq_pair_fst_snd ?? (\fst (trans sig M 〈q,current sig t〉))) %
+| *: skip ]
whd in ⊢ (??%?); @eq_f2 [destruct (H) // ]
@(eq_vec … (niltape ?)) #i0 #lei0n
cases (decidable_eq_nat … i i0) #Hii0
-[ >Hii0 >nth_change_vec // >pmap_change >nth_change_vec // destruct (H) //
-| >nth_change_vec_neq // >pmap_change >nth_change_vec_neq
- >tape_move_null_action //
+[ >Hii0 >nth_change_vec // >tape_move_multi_def >pmap_change >nth_change_vec // destruct (H) //
+| >nth_change_vec_neq // >tape_move_multi_def >pmap_change >nth_change_vec_neq //
+ <tape_move_multi_def >tape_move_null_action //
]
-qed.
+qed.
lemma halt_inject: ∀sig,n,M,i,x.
- cic:/matita/turing/turing/halt.fix(0,2,9) sig n (inject_TM sig M n i) x
+ cic:/matita/turing/turing/halt#fix:0:2:9 sig n (inject_TM sig M n i) x
= halt sig M x.
// qed.
lemma loop_inject: ∀sig,n,M,i,k,ins,int,outs,outt,vt.i < S n →
loopM sig M k (mk_config ?? ins int) = Some ? (mk_config ?? outs outt) →
- cic:/matita/turing/turing/loopM.def(11) sig n (inject_TM sig M n i) k (mk_mconfig ?? n ins (change_vec ?? vt int i))
+ cic:/matita/turing/turing/loopM#def:13 sig n (inject_TM sig M n i) k (mk_mconfig ?? n ins (change_vec ?? vt int i))
=Some ? (mk_mconfig sig ? n outs (change_vec ?? vt outt i)).
#sig #n #M #i #k elim k -k
[#ins #int #outs #outt #vt #Hin normalize in ⊢ (%→?); #H destruct
]
qed.
-(*
-lemma cstate_inject: ∀sig,n,M,i,x. *)
-
definition inject_R ≝ λsig.λR:relation (tape sig).λn,i:nat.
λv1,v2: (Vector (tape sig) (S n)).
R (nth i ? v1 (niltape ?)) (nth i ? v2 (niltape ?)) ∧
∀j. i ≠ j → nth j ? v1 (niltape ?) = nth j ? v2 (niltape ?).
-(*
-lemma nth_make : ∀A,i,n,j,a,d. i < n → nth i ? (make_veci A a n j) d = a (j+i).
-#A #i elim i
- [#n #j #a #d #ltOn @(lt_O_n_elim … ltOn) <plus_n_O //
- |#m #Hind #n #j #a #d #Hlt lapply Hlt @(lt_O_n_elim … (ltn_to_ltO … Hlt))
- #p <plus_n_Sm #ltmp @Hind @le_S_S_to_le //
- ]
-qed. *)
-
-(*
-lemma mk_config_eq_s: ∀S,sig,s1,s2,t1,t2.
- mk_config S sig s1 t1 = mk_config S sig s2 t2 → s1=s2.
-#S #sig #s1 #s2 #t1 #t2 #H destruct //
-qed.
-
-lemma mk_config_eq_t: ∀S,sig,s1,s2,t1,t2.
- mk_config S sig s1 t1 = mk_config S sig s2 t2 → s1=s2.
-#S #sig #s1 #s2 #t1 #t2 #H destruct //
-qed.
-*)
-
theorem sem_inject: ∀sig.∀M:TM sig.∀R.∀n,i.
i≤n → M ⊨ R → inject_TM sig M n i ⊨ inject_R sig R n i.
#sig #M #R #n #i #lein #HR #vt cases (HR (nth i ? vt (niltape ?)))
#k * * #outs #outt * #Hloop #HRout @(ex_intro ?? k)
-@(ex_intro ?? (mk_mconfig ?? n outs (change_vec ? (S n) vt outt i))) %
- [elim k in Hloop;
- [normalize in ⊢ (%\to ?); #H destruct
- |#k0 #Hind whd in ⊢ (??%?→??%?);
- >halt_inject whd in match (cstate ????);
- whd in match (cstate sig (states sig M)
- (initc sig M (nth i (tape sig) vt (niltape sig))));
- cases (true_or_false (halt sig M (start sig M))) #Hhalt >Hhalt
- whd in ⊢ (??%?→??%?);
- [#H @eq_f whd in ⊢ (??%?); lapply (de_option ??? H) -H
- whd in ⊢ (??%?→?); #H @eq_f2
- [whd in ⊢ (??%?); destruct (H) //
- |@(eq_vec … (niltape ?)) #j #lejn
- cases (true_or_false (eqb i j)) #eqij
- [>(eqb_true_to_eq … eqij) >nth_change_vec //
- <(eqb_true_to_eq … eqij) destruct (H) //
- |>nth_change_vec_neq // @(eqb_false_to_not_eq … eqij)
- ]
- ]
- |#H <Hind STOP
- [(* whd in ⊢ (???(?????%)); >loop_inject // *)
- | <H whd in ⊢ (??%?);
+@(ex_intro ?? (mk_mconfig ?? n outs (change_vec ? (S n) vt outt i))) %
+ [whd in ⊢ (??(?????%)?); <(change_vec_same ?? vt i (niltape ?)) in ⊢ (??%?);
+ @loop_inject /2 by refl, trans_eq, le_S_S/
|%[>nth_change_vec // @le_S_S //
|#j #i >nth_change_vec_neq //
]
]
-
-
-
-
-
-
+qed.
+theorem acc_sem_inject: ∀sig.∀M:TM sig.∀Rtrue,Rfalse,acc.∀n,i.
+ i≤n → M ⊨ [ acc : Rtrue, Rfalse ] →
+ inject_TM sig M n i ⊨ [ acc : inject_R sig Rtrue n i, inject_R sig Rfalse n i ].
+#sig #M #Rtrue #Rfalse #acc #n #i #lein #HR #vt cases (HR (nth i ? vt (niltape ?)))
+#k * * #outs #outt * * #Hloop #HRtrue #HRfalse @(ex_intro ?? k)
+@(ex_intro ?? (mk_mconfig ?? n outs (change_vec ? (S n) vt outt i))) % [ %
+ [whd in ⊢ (??(?????%)?); <(change_vec_same ?? vt i (niltape ?)) in ⊢ (??%?);
+ @loop_inject /2 by refl, trans_eq, le_S_S/
+ |#Hqtrue %
+ [>nth_change_vec /2 by le_S_S/
+ |#j #Hneq >nth_change_vec_neq //
+ ] ]
+ |#Hqfalse %
+ [>nth_change_vec /2 by le_S_S/ @HRfalse @Hqfalse
+ |#j #Hneq >nth_change_vec_neq //
+ ] ]
+qed.
\ No newline at end of file
projects / helm.git / blobdiff