definition atm_states ≝ initN 3.
+definition atm0 : initN 3 ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition atm1 : initN 3 ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition atm2 : initN 3 ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
+
definition atmr_step ≝
λalpha:FinSet.λtest:alpha→bool.
mk_TM alpha atm_states
(λp.let 〈q,a〉 ≝ p in
match a with
- [ None ⇒ 〈1, None ?〉
+ [ None ⇒ 〈atm1, None ?〉
| Some a' ⇒
match test a' with
- [ true ⇒ 〈1,None ?〉
- | false ⇒ 〈2,Some ? 〈a',R〉〉 ]])
- O (λx.notb (x == 0)).
+ [ true ⇒ 〈atm1,None ?〉
+ | false ⇒ 〈atm2,Some ? 〈a',R〉〉 ]])
+ atm0 (λx.notb (x == atm0)).
definition Ratmr_step_true ≝
λalpha,test,t1,t2.
lemma atmr_q0_q1 :
∀alpha,test,ls,a0,rs. test a0 = true →
step alpha (atmr_step alpha test)
- (mk_config ?? 0 (midtape … ls a0 rs)) =
- mk_config alpha (states ? (atmr_step alpha test)) 1
+ (mk_config ?? atm0 (midtape … ls a0 rs)) =
+ mk_config alpha (states ? (atmr_step alpha test)) atm1
(midtape … ls a0 rs).
-#alpha #test #ls #a0 #ts #Htest normalize >Htest %
+#alpha #test #ls #a0 #ts #Htest whd in ⊢ (??%?);
+whd in match (trans … 〈?,?〉); >Htest %
qed.
lemma atmr_q0_q2 :
∀alpha,test,ls,a0,rs. test a0 = false →
step alpha (atmr_step alpha test)
- (mk_config ?? 0 (midtape … ls a0 rs)) =
- mk_config alpha (states ? (atmr_step alpha test)) 2
+ (mk_config ?? atm0 (midtape … ls a0 rs)) =
+ mk_config alpha (states ? (atmr_step alpha test)) atm2
(mk_tape … (a0::ls) (option_hd ? rs) (tail ? rs)).
-#alpha #test #ls #a0 #ts #Htest normalize >Htest cases ts //
+#alpha #test #ls #a0 #ts #Htest whd in ⊢ (??%?);
+whd in match (trans … 〈?,?〉); >Htest cases ts //
qed.
lemma sem_atmr_step :
∀alpha,test.
accRealize alpha (atmr_step alpha test)
- 2 (Ratmr_step_true alpha test) (Ratmr_step_false alpha test).
-#alpha #test *
+ atm2 (Ratmr_step_true alpha test) (Ratmr_step_false alpha test).
+#alpha #test cut (∀P:Prop.atm1 = atm2 →P) [#P normalize #Hfalse destruct] #Hfalse
+*
[ @(ex_intro ?? 2)
- @(ex_intro ?? (mk_config ?? 1 (niltape ?))) %
- [ % // #Hfalse destruct | #_ % // % % ]
-| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? 1 (leftof ? a al)))
- % [ % // #Hfalse destruct | #_ % // % % ]
-| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? 1 (rightof ? a al)))
- % [ % // #Hfalse destruct | #_ % // % % ]
+ @(ex_intro ?? (mk_config ?? atm1 (niltape ?))) %
+ [ % // @Hfalse | #_ % // % % ]
+| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (leftof ? a al)))
+ % [ % // @Hfalse | #_ % // % % ]
+| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (rightof ? a al)))
+ % [ % // @Hfalse | #_ % // % % ]
| #ls #c #rs @(ex_intro ?? 2)
cases (true_or_false (test c)) #Htest
- [ @(ex_intro ?? (mk_config ?? 1 ?))
+ [ @(ex_intro ?? (mk_config ?? atm1 ?))
[| %
[ %
[ whd in ⊢ (??%?); >atmr_q0_q1 //
- | #Hfalse destruct ]
+ | @Hfalse]
| #_ % // %2 @(ex_intro ?? c) % // ]
]
- | @(ex_intro ?? (mk_config ?? 2 (mk_tape ? (c::ls) (option_hd ? rs) (tail ? rs))))
+ | @(ex_intro ?? (mk_config ?? atm2 (mk_tape ? (c::ls) (option_hd ? rs) (tail ? rs))))
%
[ %
[ whd in ⊢ (??%?); >atmr_q0_q2 //
t2 = midtape ? (reverse ? rs1@c::ls) b rs2))).
definition adv_to_mark_r ≝
- λalpha,test.whileTM alpha (atmr_step alpha test) 2.
+ λalpha,test.whileTM alpha (atmr_step alpha test) atm2.
lemma wsem_adv_to_mark_r :
∀alpha,test.
/2/
qed.
-(* NO OPERATION
-
- t1 = t2
- *)
-
-definition nop_states ≝ initN 1.
-
-definition nop ≝
- λalpha:FinSet.mk_TM alpha nop_states
- (λp.let 〈q,a〉 ≝ p in 〈q,None ?〉)
- O (λ_.true).
-
-definition R_nop ≝ λalpha.λt1,t2:tape alpha.t2 = t1.
-
-lemma sem_nop :
- ∀alpha.Realize alpha (nop alpha) (R_nop alpha).
-#alpha #intape @(ex_intro ?? 1) @ex_intro [| % normalize % ]
-qed.
-
(*
q0 _ → q1, R
q1 〈a,false〉 → qF, 〈a,true〉, N
definition mark_states ≝ initN 3.
+definition mark0 : initN 3 ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition mark1 : initN 3 ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition mark2 : initN 3 ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
definition mark ≝
λalpha:FinSet.mk_TM (FinProd … alpha FinBool) mark_states
(λp.let 〈q,a〉 ≝ p in
match a with
- [ None ⇒ 〈2,None ?〉
- | Some a' ⇒ match q with
- [ O ⇒ 〈1,Some ? 〈a',R〉〉
+ [ None ⇒ 〈mark2,None ?〉
+ | Some a' ⇒ match pi1 … q with
+ [ O ⇒ 〈mark1,Some ? 〈a',R〉〉
| S q ⇒ match q with
[ O ⇒ let 〈a'',b〉 ≝ a' in
- 〈2,Some ? 〈〈a'',true〉,N〉〉
- | S _ ⇒ 〈2,None ?〉 ] ] ])
- O (λq.q == 2).
+ 〈mark2,Some ? 〈〈a'',true〉,N〉〉
+ | S _ ⇒ 〈mark2,None ?〉 ] ] ])
+ mark0 (λq.q == mark2).
definition R_mark ≝ λalpha,t1,t2.
∀ls,c,d,b,rs.
definition tc_states ≝ initN 3.
+definition tc0 : initN 3 ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition tc1 : initN 3 ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition atm2 : initN 3 ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
definition test_char ≝
λalpha:FinSet.λtest:alpha→bool.
mk_TM alpha tc_states