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).
| S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
].
+lemma loop_S_true :
+ ∀A,n,f,p,a. p a = true →
+ loop A (S n) f p a = Some ? a.
+#A #n #f #p #a #pa normalize >pa //
+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 loop_incr : ∀A,f,p,k1,k2,a1,a2.
loop A k1 f p a1 = Some ? a2 →
loop A (k2+k1) f p a1 = Some ? a2.
∀t,i,outc.
loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (initc sig M t) = Some ? outc →
R t (ctape ?? outc).
-
+
+definition Terminate ≝ λsig.λM:TM sig.λt. ∃i,outc.
+ loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (initc sig M t) = Some ? outc.
+
+lemma WRealize_to_Realize : ∀sig.∀M: TM sig.∀R.
+ (∀t.Terminate sig M t) → WRealize sig M R → Realize sig M R.
+#sig #M #R #HT #HW #t cases (HT … t) #i * #outc #Hloop
+@(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
+qed.
+
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)
(cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
(cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
+(* NO OPERATION
+
+ t1 = t2
+ *)
+
+definition nop_states ≝ initN 1.
+definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … (S 0)).
+
+definition nop ≝
+ λalpha:FinSet.mk_TM alpha nop_states
+ (λp.let 〈q,a〉 ≝ p in 〈q,None ?〉)
+ start_nop (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 … (mk_config ?? start_nop intape)) % %
+qed.
+
(* Compositions *)
definition seq_trans ≝ λsig. λM1,M2 : TM sig.
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.