*)
-include "turing/while_machine.ma".
+include "turing/basic_machines.ma".
+include "turing/if_machine.ma".
-definition mcc_states : FinSet → FinSet ≝ λalpha:FinSet.FinProd (initN 5) alpha.
-
-definition mcc_step ≝
- λalpha:FinSet.λsep:alpha.
- mk_TM alpha (mcc_states alpha)
- (λp.let 〈q,a〉 ≝ p in
- let 〈q',b〉 ≝ q in
- match a with
- [ None ⇒ 〈〈4,sep〉,None ?〉
- | Some a' ⇒
- match q' with
- [ O ⇒ (* qinit *)
- match a' == sep with
- [ true ⇒ 〈〈4,sep〉,None ?〉
- | false ⇒ 〈〈1,a'〉,Some ? 〈a',L〉〉 ]
- | S q' ⇒ match q' with
- [ O ⇒ (* q1 *)
- 〈〈2,a'〉,Some ? 〈b,R〉〉
- | S q' ⇒ match q' with
- [ O ⇒ (* q2 *)
- 〈〈3,sep〉,Some ? 〈b,R〉〉
- | S q' ⇒ match q' with
- [ O ⇒ (* qacc *)
- 〈〈3,sep〉,None ?〉
- | S q' ⇒ (* qfail *)
- 〈〈4,sep〉,None ?〉 ] ] ] ] ])
- 〈0,sep〉
- (λq.let 〈q',a〉 ≝ q in q' == 3 ∨ q' == 4).
-
-lemma mcc_q0_q1 :
- ∀alpha:FinSet.∀sep,a,ls,a0,rs.
- a0 == sep = false →
- step alpha (mcc_step alpha sep)
- (mk_config ?? 〈0,a〉 (mk_tape … ls (Some ? a0) rs)) =
- mk_config alpha (states ? (mcc_step alpha sep)) 〈1,a0〉
- (tape_move_left alpha ls a0 rs).
-#alpha #sep #a *
-[ #a0 #rs #Ha0 whd in ⊢ (??%?);
- normalize in match (trans ???); >Ha0 %
-| #a1 #ls #a0 #rs #Ha0 whd in ⊢ (??%?);
- normalize in match (trans ???); >Ha0 %
-]
-qed.
-
-lemma mcc_q1_q2 :
- ∀alpha:FinSet.∀sep,a,ls,a0,rs.
- step alpha (mcc_step alpha sep)
- (mk_config ?? 〈1,a〉 (mk_tape … ls (Some ? a0) rs)) =
- mk_config alpha (states ? (mcc_step alpha sep)) 〈2,a0〉
- (tape_move_right alpha ls a rs).
-#alpha #sep #a #ls #a0 * //
-qed.
-
-lemma mcc_q2_q3 :
- ∀alpha:FinSet.∀sep,a,ls,a0,rs.
- step alpha (mcc_step alpha sep)
- (mk_config ?? 〈2,a〉 (mk_tape … ls (Some ? a0) rs)) =
- mk_config alpha (states ? (mcc_step alpha sep)) 〈3,sep〉
- (tape_move_right alpha ls a rs).
-#alpha #sep #a #ls #a0 * //
-qed.
+definition mcc_step ≝ λalpha:FinSet.λsep:alpha.
+ ifTM alpha (test_char ? (λc.¬c==sep))
+ (single_finalTM … (seq … (swap_r alpha sep) (move_r ?))) (nop ?) tc_true.
definition Rmcc_step_true ≝
λalpha,sep,t1,t2.
left ? t1 ≠ [] → current alpha t1 ≠ None alpha →
current alpha t1 = Some alpha sep ∧ t2 = t1.
-lemma loop_S_true :
- ∀A,n,f,p,a. p a = true →
- loop A (S n) f p a = Some ? a. /2/
-qed.
-
-lemma loop_S_false :
- ∀A,n,f,p,a. p a = false →
- loop A (S n) f p a = loop A n f p (f a).
-normalize #A #n #f #p #a #Hpa >Hpa %
-qed.
-
-lemma trans_init_sep:
- ∀alpha,sep,x.
- trans ? (mcc_step alpha sep) 〈〈0,x〉,Some ? sep〉 = 〈〈4,sep〉,None ?〉.
-#alpha #sep #x normalize >(\b ?) //
-qed.
-
-lemma trans_init_not_sep:
- ∀alpha,sep,x,y.y == sep = false →
- trans ? (mcc_step alpha sep) 〈〈0,x〉,Some ? y〉 = 〈〈1,y〉,Some ? 〈y,L〉〉.
-#alpha #sep #x #y #H1 normalize >H1 //
-qed.
-
lemma sem_mcc_step :
∀alpha,sep.
- accRealize alpha (mcc_step alpha sep)
- 〈3,sep〉 (Rmcc_step_true alpha sep) (Rmcc_step_false alpha sep).
-#alpha #sep *
-[@(ex_intro ?? 2)
- @(ex_intro … (mk_config ?? 〈4,sep〉 (niltape ?)))
- % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 #H2 @False_ind @(absurd ?? H2) %]
-|#l0 #lt0 @(ex_intro ?? 2)
- @(ex_intro … (mk_config ?? 〈4,sep〉 (leftof ? l0 lt0)))
- % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 #H2 @False_ind @(absurd ?? H2) %]
-|#r0 #rt0 @(ex_intro ?? 2)
- @(ex_intro … (mk_config ?? 〈4,sep〉 (rightof ? r0 rt0)))
- % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 #H2 #H3 @False_ind @(absurd ?? H3) %]
-| #lt #c #rt cases (true_or_false (c == sep)) #Hc
- [ @(ex_intro ?? 2)
- @(ex_intro ?? (mk_config ?? 〈4,sep〉 (midtape ? lt c rt)))
- % [ %
- [ >(\P Hc) >loop_S_false // >loop_S_true
- [ @eq_f whd in ⊢ (??%?); >trans_init_sep %
- |>(\P Hc) whd in ⊢(??(???(???%))?); >trans_init_sep % ]
- | #Hfalse destruct ]
- |#_ #H1 #H2 % // normalize >(\P Hc) % ]
- | @(ex_intro ?? 4) cases lt
- [ @ex_intro
- [|% [ %
- [ >loop_S_false // >mcc_q0_q1 //
- | normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
- | normalize in ⊢ (%→?); #_ #H1 @False_ind @(absurd ?? H1) % ] ]
- | #l0 #lt @ex_intro
- [| % [ %
- [ >loop_S_false // >mcc_q0_q1 //
- | #_ #a #b #ls #rs #Hb destruct %
- [ @(\Pf Hc)
- | >mcc_q1_q2 >mcc_q2_q3 cases rs normalize // ] ]
- | normalize in ⊢ (% → ?); * #Hfalse
- @False_ind /2/ ]
- ]
- ]
- ]
-]
+ mcc_step alpha sep ⊨
+ [inr … (inl … (inr … start_nop)): Rmcc_step_true alpha sep, Rmcc_step_false alpha sep].
+#alpha #sep
+ @(acc_sem_if_app …
+ (sem_test_char …) (sem_seq …(sem_swap_r …) (sem_move_r …)) (sem_nop …))
+ [#intape #outtape #tapea whd in ⊢ (%→%→%);
+ #Htapea * #tapeb * whd in ⊢ (%→%→?);
+ #Htapeb #Houttape #a #b #ls #rs #Hintape
+ >Hintape in Htapea; #Htapea cases (Htapea ? (refl …)) -Htapea
+ #Hbsep #Htapea % [@(\Pf (injective_notb ? false Hbsep))]
+ @Houttape @Htapeb //
+ |#intape #outtape #tapea whd in ⊢ (%→%→%);
+ cases (current alpha intape)
+ [#_ #_ #_ * #Hfalse @False_ind @Hfalse %
+ |#c #H #Htapea #_ #_ cases (H c (refl …)) #csep #Hintape % //
+ lapply (injective_notb ? true csep) -csep #csep >(\P csep) //
+ ]
+ ]
qed.
(* the move_char (variant c) machine *)
definition move_char_c ≝
- λalpha,sep.whileTM alpha (mcc_step alpha sep) 〈3,sep〉.
+ λalpha,sep.whileTM alpha (mcc_step alpha sep) (inr … (inl … (inr … start_nop))).
definition R_move_char_c ≝
λalpha,sep,t1,t2.
b ≠ sep → memb ? sep rs1 = false →
t2 = midtape alpha (a::reverse ? rs1@b::ls) sep rs2).
-lemma sem_while_move_char :
+lemma sem_move_char_c :
∀alpha,sep.
WRealize alpha (move_char_c alpha sep) (R_move_char_c alpha sep).
#alpha #sep #inc #i #outc #Hloop
-Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
[ #tapea whd in ⊢ (% → ?); #H1 #b #a #ls #rs #Htapea
%
- [ #Hb >Htapea in H1; >Hb normalize in ⊢ (%→?); #H1
- cases (H1 ??)
- [#_ #H2 >H2 %
- |*: % #H2 destruct (H2) ]
+ [ #Hb >Htapea in H1; >Hb #H1 cases (H1 ??)
+ [#_ #H2 >H2 % |*: % #H2 normalize in H2; destruct (H2)]
| #rs1 #rs2 #Hrs #Hb #Hrs1
- >Htapea in H1; normalize in ⊢ (% → ?); #H1
- cases (H1 ??)
- [ #Hfalse @False_ind @(absurd ?? Hb) destruct %
- |*:% #H2 destruct (H2) ]
+ >Htapea in H1; #H1 cases (H1 ??)
+ [#Hfalse @False_ind @(absurd ?? Hb) normalize in Hfalse; destruct %
+ |*:% #H2 normalize in H2; destruct (H2) ]
]
| #tapea #tapeb #tapec #Hstar1 #HRtrue #IH #HRfalse
lapply (IH HRfalse) -IH whd in ⊢ (%→%); #IH
#Ha0 #Htapeb %
[ #Hfalse @False_ind @(absurd ?? Ha0) //
| *
- [ #rs2 whd in ⊢ (???%→?); #Hrs #_ #_ normalize
- >Hrs in Htapeb; normalize #Htapeb
- cases (IH … Htapeb)
- #Houtc #_ >Houtc //
+ [ #rs2 whd in ⊢ (???%→?); #Hrs #_ #_ (* normalize *)
+ >Hrs in Htapeb; #Htapeb normalize in Htapeb;
+ cases (IH … Htapeb) #Houtc #_ >Houtc normalize //
| #r0 #rs0 #rs2 #Hrs #_ #Hrs0
cut (r0 ≠ sep ∧ memb … sep rs0 = false)
[ %
[#Hb @False_ind /2/ | #Hmemb cases (orb_true_l … Hmemb)
[#eqsep %1 >(\P eqsep) // | #H %2 //]
]
-qed.
\ No newline at end of file
+qed.
+
+(* NO GOOD: we must stop if current = None too!!! *)
+
+axiom ssem_move_char_c :
+ ∀alpha,sep.
+ Realize alpha (move_char_c alpha sep) (R_move_char_c alpha sep).
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