include "basics/vectors.ma".
(* include "basics/relations.ma". *)
+(*
record tape (sig:FinSet): Type[0] ≝
-{ left : list sig;
- right: list sig
+{ left : list (option sig);
+ right: list (option sig)
}.
+*)
+
+inductive tape (sig:FinSet) : Type[0] ≝
+| niltape : tape sig
+| leftof : sig → list sig → tape sig
+| rightof : sig → list sig → tape sig
+| midtape : list sig → sig → list sig → tape sig.
+
+definition left ≝
+ λsig.λt:tape sig.match t with
+ [ niltape ⇒ []
+ | leftof _ _ ⇒ []
+ | rightof s l ⇒ s::l
+ | midtape l _ _ ⇒ l ].
+
+definition right ≝
+ λsig.λt:tape sig.match t with
+ [ niltape ⇒ []
+ | leftof s r ⇒ s::r
+ | rightof _ _ ⇒ []
+ | midtape _ _ r ⇒ r ].
+
+
+definition current ≝
+ λsig.λt:tape sig.match t with
+ [ midtape _ c _ ⇒ Some ? c
+ | _ ⇒ None ? ].
+
+definition mk_tape :
+ ∀sig:FinSet.list sig → option sig → list sig → tape sig ≝
+ λsig,lt,c,rt.match c with
+ [ Some c' ⇒ midtape sig lt c' rt
+ | None ⇒ match lt with
+ [ nil ⇒ match rt with
+ [ nil ⇒ niltape ?
+ | cons r0 rs0 ⇒ leftof ? r0 rs0 ]
+ | cons l0 ls0 ⇒ rightof ? l0 ls0 ] ].
inductive move : Type[0] ≝
| L : move
| R : move
+| N : move
.
(* We do not distinuish an input tape *)
ctape: tape sig
}.
-definition option_hd ≝ λA.λl:list A.
+(* definition option_hd ≝ λA.λl:list (option A).
match l with
[nil ⇒ None ?
- |cons a _ ⇒ Some ? a
+ |cons a _ ⇒ a
].
+ *)
-definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
- match m with
+(*definition tape_write ≝ λsig.λt:tape sig.λs:sig.
+ <left ? t) s (right ? t).
[ None ⇒ t
- | Some m1 ⇒
- match \snd m1 with
- [ R ⇒ mk_tape sig ((\fst m1)::(left ? t)) (tail ? (right ? t))
- | L ⇒ mk_tape sig (tail ? (left ? t)) ((\fst m1)::(right ? t))
- ]
- ].
+ | Some s' ⇒ midtape ? (left ? t) s' (right ? t) ].*)
+
+definition tape_move_left ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
+ match lt with
+ [ nil ⇒ leftof sig c rt
+ | cons c0 lt0 ⇒ midtape sig lt0 c0 (c::rt) ].
+
+definition tape_move_right ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
+ match rt with
+ [ nil ⇒ rightof sig c lt
+ | cons c0 rt0 ⇒ midtape sig (c::lt) c0 rt0 ].
+definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
+ match m with
+ [ None ⇒ t
+ | Some m' ⇒
+ let 〈s,m1〉 ≝ m' in
+ match m1 with
+ [ R ⇒ tape_move_right ? (left ? t) s (right ? t)
+ | L ⇒ tape_move_left ? (left ? t) s (right ? t)
+ | N ⇒ midtape ? (left ? t) s (right ? t)
+ ] ].
+(*
+ (None,[]) → □
+ (None,a::[]) → □
+ (None,a::b::rs) → None::b::rs
+ (Some a,[]) → [Some a]
+ (Some a,b::rs) → Some a::rs
+ *)
+(*
+definition option_cons ≝ λA.λa:option A.λl.
+ match a with
+ [ None ⇒ match l with
+ [ nil ⇒ []
+ | cons _ _ ⇒ a::l ]
+ | Some _ ⇒ a::l ].
+
+(* definition tape_update := λsig.λt: tape sig.λs:option sig.
+ let newright ≝
+ match right ? t with
+ [ nil ⇒ match s with
+ [ None ⇒ []
+ | Some a ⇒ [Some ? a] ]
+ | cons b rs ⇒ match s with
+ [ None ⇒ match rs with
+ [ nil ⇒ []
+ | cons _ _ ⇒ None ?::rs ]
+ | Some a ⇒ Some ? a::rs ] ]
+ in mk_tape ? (left ? t) newright. *)
+
+definition tape_move ≝ λsig.λt:tape sig.λm:option sig × move.
+ let 〈s,m1〉 ≝ m in match m1 with
+ [ R ⇒ mk_tape sig (option_cons ? s (left ? t)) (tail ? (right ? t))
+ | L ⇒ mk_tape sig (tail ? (left ? t))
+ (option_cons ? (option_hd ? (left ? t))
+ (option_cons ? s (tail ? (right ? t))))
+ | N ⇒ mk_tape sig (left ? t) (option_cons ? s (tail ? (right ? t)))
+ ].
+*)
+
definition step ≝ λsig.λM:TM sig.λc:config sig (states sig M).
- let current_char ≝ option_hd ? (right ? (ctape ?? c)) in
+ let current_char ≝ current ? (ctape ?? c) in
let 〈news,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
mk_config ?? news (tape_move sig (ctape ?? c) mv).
]
qed.
-lemma loop_split : ∀A,f,p,q.(∀b. p b = false → q b = false) →
+lemma loop_merge : ∀A,f,p,q.(∀b. p b = false → q b = false) →
∀k1,k2,a1,a2,a3,a4.
loop A k1 f p a1 = Some ? a2 →
f a2 = a3 → q a2 = false →
]
qed.
+lemma loop_split : ∀A,f,p,q.(∀b. q b = true → p b = true) →
+ ∀k,a1,a2.
+ loop A k f q a1 = Some ? a2 →
+ ∃k1,a3.
+ loop A k1 f p a1 = Some ? a3 ∧
+ loop A (S(k-k1)) f q a3 = Some ? a2.
+#A #f #p #q #Hpq #k elim k
+ [#a1 #a2 normalize #Heq destruct
+ |#i #Hind #a1 #a2 normalize
+ cases (true_or_false (q a1)) #Hqa1 >Hqa1 normalize
+ [ #Ha1a2 destruct
+ @(ex_intro … 1) @(ex_intro … a2) %
+ [normalize >(Hpq …Hqa1) // |>Hqa1 //]
+ |#Hloop cases (true_or_false (p a1)) #Hpa1
+ [@(ex_intro … 1) @(ex_intro … a1) %
+ [normalize >Hpa1 // |>Hqa1 <Hloop normalize //]
+ |cases (Hind …Hloop) #k2 * #a3 * #Hloop1 #Hloop2
+ @(ex_intro … (S k2)) @(ex_intro … a3) %
+ [normalize >Hpa1 normalize // | @Hloop2 ]
+ ]
+ ]
+ ]
+qed.
+
(*
lemma loop_split : ∀A,f,p,q.(∀b. p b = false → q b = false) →
∀k1,k2,a1,a2,a3.
loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (initc sig M t) = Some ? outc ∧
R t (ctape ?? outc).
+definition WRealize ≝ λsig.λM:TM sig.λR:relation (tape sig).
+∀t,i,outc.
+ loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (initc sig M t) = Some ? outc →
+ R t (ctape ?? outc).
+
+lemma loop_eq : ∀sig,f,q,i,j,a,x,y.
+ loop sig i f q a = Some ? x → loop sig j f q a = Some ? y → x = y.
+#sig #f #q #i #j @(nat_elim2 … i j)
+[ #n #a #x #y normalize #Hfalse destruct (Hfalse)
+| #n #a #x #y #H1 normalize #Hfalse destruct (Hfalse)
+| #n1 #n2 #IH #a #x #y normalize cases (q a) normalize
+ [ #H1 #H2 destruct %
+ | /2/ ]
+]
+qed.
+
+theorem Realize_to_WRealize : ∀sig,M,R.Realize sig M R → WRealize sig M R.
+#sig #M #R #H1 #inc #i #outc #Hloop
+cases (H1 inc) #k * #outc1 * #Hloop1 #HR
+>(loop_eq … Hloop Hloop1) //
+qed.
definition accRealize ≝ λsig.λM:TM sig.λacc:states sig M.λRtrue,Rfalse:relation (tape sig).
∀t.∃i.∃outc.
halt ? M2 (cstate ?? c0) = false →
step sig (seq sig M1 M2) (lift_confR sig (states ? M1) (states ? M2) c0) =
lift_confR sig (states ? M1) (states ? M2) (step sig M2 c0).
-#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s * #lt
-#rs #Hhalt
-whd in ⊢ (???(????%));whd in ⊢ (???%);
-lapply (refl ? (trans ?? 〈s,option_hd sig rs〉))
-cases (trans ?? 〈s,option_hd sig rs〉) in ⊢ (???% → %);
-#s0 #m0 #Heq whd in ⊢ (???%);
-whd in ⊢ (??(???%)?); whd in ⊢ (??%?);
->(trans_liftR … Heq)
-[% | //]
+#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
+ lapply (refl ? (trans ?? 〈s,current sig t〉))
+ cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
+ #s0 #m0 cases t
+ [ #Heq #Hhalt
+ | 2,3: #s1 #l1 #Heq #Hhalt
+ |#ls #s1 #rs #Heq #Hhalt ]
+ whd in ⊢ (???(????%)); >Heq
+ whd in ⊢ (???%);
+ whd in ⊢ (??(???%)?); whd in ⊢ (??%?);
+ >(trans_liftR … Heq) //
qed.
lemma step_lift_confL : ∀sig,M1,M2,c0.
halt ? M1 (cstate ?? c0) = false →
step sig (seq sig M1 M2) (lift_confL sig (states ? M1) (states ? M2) c0) =
lift_confL sig ?? (step sig M1 c0).
-#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s * #lt
-#rs #Hhalt
-whd in ⊢ (???(????%));whd in ⊢ (???%);
-lapply (refl ? (trans ?? 〈s,option_hd sig rs〉))
-cases (trans ?? 〈s,option_hd sig rs〉) in ⊢ (???% → %);
-#s0 #m0 #Heq whd in ⊢ (???%);
-whd in ⊢ (??(???%)?); whd in ⊢ (??%?);
->(trans_liftL … Heq)
-[% | //]
+#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
+ lapply (refl ? (trans ?? 〈s,current sig t〉))
+ cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
+ #s0 #m0 cases t
+ [ #Heq #Hhalt
+ | 2,3: #s1 #l1 #Heq #Hhalt
+ |#ls #s1 #rs #Heq #Hhalt ]
+ whd in ⊢ (???(????%)); >Heq
+ whd in ⊢ (???%);
+ whd in ⊢ (??(???%)?); whd in ⊢ (??%?);
+ >(trans_liftL … Heq) //
+qed.
+
+lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
+ (∀x.hlift (lift x) = h x) →
+ (∀x.h x = false → lift (f x) = g (lift x)) →
+ loop A k f h c1 = Some ? c2 →
+ loop B k g hlift (lift c1) = Some ? (lift … c2).
+#A #B #k #lift #f #g #h #hlift #c1 #c2 #Hfg #Hhlift
+generalize in match c1; elim k
+[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
+|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
+ cases (true_or_false (h c0)) #Hc0 >Hfg >Hc0
+ [ normalize #Heq destruct (Heq) %
+ | normalize <Hhlift // @IH ]
qed.
+(*
lemma loop_liftL : ∀sig,k,M1,M2,c1,c2.
loop ? k (step sig M1) (λc.halt sig M1 (cstate ?? c)) c1 = Some ? c2 →
loop ? k (step sig (seq sig M1 M2))
| <Hc0 cases c0 // ]
]
qed.
+
+*)
lemma loop_Some :
∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
@(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
%
-[@(loop_split ??????????? (loop_liftL … Hloop1))
- [* *
+[@(loop_merge ???????????
+ (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
+ (step sig M1) (step sig (seq sig M1 M2))
+ (λc.halt sig M1 (cstate … c))
+ (λc.halt_liftL ?? (halt sig M1) (cstate … c)) … Hloop1))
+ [ * *
[ #sl #tl whd in ⊢ (??%? → ?); #Hl %
| #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
- ||4:cases outc1 #s1 #t1 %
- |5:@(loop_liftR … Hloop2)
- |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
- generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
- >(trans_liftL_true sig M1 M2 ??)
- [ whd in ⊢ (??%?); whd in ⊢ (???%);
- @config_eq //
- | @(loop_Some ?????? Hloop10) ]
+ || #c0 #Hhalt <step_lift_confL //
+ | #x <p_halt_liftL %
+ |6:cases outc1 #s1 #t1 %
+ |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2)
+ [ * #s2 #t2 %
+ | #c0 #Hhalt <step_lift_confR // ]
+ |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
+ generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
+ >(trans_liftL_true sig M1 M2 ??)
+ [ whd in ⊢ (??%?); whd in ⊢ (???%);
+ @config_eq whd in ⊢ (???%); //
+ | @(loop_Some ?????? Hloop10) ]
]
| @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
% // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
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