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
- let nai ≝ λj.if (eqb j i) then na else (None ?) in
- 〈q, make_veci ? nai (S n) 0〉.
-
+ 〈nq, change_vec ? (S n) (null_action ? n) na i〉.
+
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.
-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. j ≠ i → nth j ? v1 (niltape ?) = nth j ? v2 (niltape ?).
+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,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: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)
+>(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 // >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.
-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 //
+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
+ = halt sig M x.
+// qed.
+
+lemma de_option: ∀A,a,b. Some A a = Some A b → a = b.
+#A #a #b #H destruct //
+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: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
+ |#k #Hind #ins #int #outs #outt #vt #Hin whd in ⊢ (??%?→??%?);
+ >halt_inject whd in match (cstate ????);
+ cases (true_or_false (halt sig M ins)) #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 //
+ destruct (H) >nth_change_vec //
+ |destruct (H) //
+ ]
+ ]
+ |>(config_expand … (step ???)) #H <(Hind … H) //
+ >loopM_unfold @eq_f >inject_step //
+ ]
]
qed.
+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 ?).
+
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 ?) )
- [(* this is the vector of tapes in the last mk_config *)
- @(make_veci ? (λj.if (eqb j i) then outt else (nth j ? vt (niltape ?))) (S n) 0)
- |%
- [
- |%[>nth_make [>(eq_to_eqb_true … (refl ? i)) @HRout|@le_S_S //]
- |#j #neqji >nth_make [>(not_eq_to_eqb_false … neqji) // |@le_S_S //]
- normalize
-
-
-
-
-
-
+@(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
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