include "basics/vectors.ma".
(* include "basics/relations.ma". *)
-(*
-record tape (sig:FinSet): Type[0] ≝
-{ left : list (option sig);
- right: list (option sig)
-}.
-*)
+(******************************** tape ****************************************)
+
+(* A tape is essentially a triple 〈left,current,right〉 where however the current
+symbol could be missing. This may happen for three different reasons: both tapes
+are empty; we are on the left extremity of a non-empty tape (left overflow), or
+we are on the right extremity of a non-empty tape (right overflow). *)
inductive tape (sig:FinSet) : Type[0] ≝
| niltape : tape sig
definition left ≝
λsig.λt:tape sig.match t with
- [ niltape ⇒ []
- | leftof _ _ ⇒ []
- | rightof s l ⇒ s::l
- | midtape l _ _ ⇒ l ].
+ [ 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 ].
+ [ niltape ⇒ [] | leftof s r ⇒ s::r | rightof _ _ ⇒ []| midtape _ _ r ⇒ r ].
definition current ≝
λsig.λt:tape sig.match t with
- [ midtape _ c _ ⇒ Some ? c
- | _ ⇒ None ? ].
+ [ midtape _ c _ ⇒ Some ? c | _ ⇒ None ? ].
definition mk_tape :
∀sig:FinSet.list sig → option sig → list sig → tape sig ≝
| cons l0 ls0 ⇒ rightof ? l0 ls0 ] ].
inductive move : Type[0] ≝
-| L : move
-| R : move
-| N : move
-.
+ | L : move | R : move | N : move.
-(* We do not distinuish an input tape *)
+(********************************** machine ***********************************)
record TM (sig:FinSet): Type[1] ≝
{ states : FinSet;
halt : states → bool
}.
-record config (sig,states:FinSet): Type[0] ≝
-{ cstate : states;
- ctape: tape sig
-}.
-
-(* definition option_hd ≝ λA.λl:list (option A).
- match l with
- [nil ⇒ None ?
- |cons a _ ⇒ a
- ].
- *)
-
-(*definition tape_write ≝ λsig.λt:tape sig.λs:sig.
- <left ? t) s (right ? t).
- [ None ⇒ 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
| 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)))
- ].
-*)
+
+record config (sig,states:FinSet): Type[0] ≝
+{ cstate : states;
+ ctape: tape sig
+}.
definition step ≝ λsig.λM:TM sig.λc:config sig (states sig M).
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).
-
+
+(******************************** loop ****************************************)
let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
match n with
[ O ⇒ None ?
].
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. 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).
+ loop A (S n) f p a = loop A n f p (f a).
normalize #A #n #f #p #a #Hpa >Hpa %
qed.
[#eqa1a2 destruct #eqa2a3 #Hqa2 #H
whd in ⊢ (??(??%???)?); >plus_n_Sm @loop_incr
whd in ⊢ (??%?); >Hqa2 >eqa2a3 @H
- |normalize >(Hpq … pa1) normalize
- #H1 #H2 #H3 @(Hind … H2) //
+ |normalize >(Hpq … pa1) normalize #H1 #H2 #H3 @(Hind … H2) //
]
]
qed.
]
qed.
-(*
-lemma loop_split : ∀A,f,p,q.(∀b. p b = false → q b = false) →
- ∀k1,k2,a1,a2,a3.
- loop A k1 f p a1 = Some ? a2 →
- loop A k2 f q a2 = Some ? a3 →
- loop A (k1+k2) f q a1 = Some ? a3.
-#Sig #f #p #q #Hpq #k1 elim k1
- [normalize #k2 #a1 #a2 #a3 #H destruct
- |#k1' #Hind #k2 #a1 #a2 #a3 normalize in ⊢ (%→?→?);
- cases (true_or_false (p a1)) #pa1 >pa1 normalize in ⊢ (%→?);
- [#eqa1a2 destruct #H @loop_incr //
- |normalize >(Hpq … pa1) normalize
- #H1 #H2 @(Hind … H2) //
- ]
- ]
+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.
-*)
+
+(************************** Realizability *************************************)
+definition loopM ≝ λsig,M,i,cin.
+ loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
definition initc ≝ λsig.λM:TM sig.λt.
mk_config sig (states sig M) (start sig M) t.
definition Realize ≝ 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).
+ loopM sig M i (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).
+ loopM sig M i (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.
+ loopM sig M i (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.
@(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)
-[ #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) //
+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).
+definition accRealize ≝ λsig.λM:TM sig.λacc:states sig M.λRtrue,Rfalse.
∀t.∃i.∃outc.
- loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (initc sig M t) = Some ? outc ∧
- (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
- (cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
+ loopM sig M i (initc sig M t) = Some ? outc ∧
+ (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
+ (cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
-(* NO OPERATION
+(******************************** NOP Machine *********************************)
- t1 = t2
- *)
+(* NO OPERATION
+ t1 = t2 *)
definition nop_states ≝ initN 1.
-definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … (S 0)).
+definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … 1).
definition nop ≝
λalpha:FinSet.mk_TM alpha nop_states
@(ex_intro … (mk_config ?? start_nop intape)) % %
qed.
-(* Compositions *)
+(************************** Sequential Composition ****************************)
definition seq_trans ≝ λsig. λM1,M2 : TM sig.
λp. let 〈s,a〉 ≝ p in
definition Rcomp ≝ λA.λR1,R2:relation A.λa1,a2.
∃am.R1 a1 am ∧ R2 am a2.
-
-(*
-definition injectRl ≝ λsig.λM1.λM2.λR.
- λc1,c2. ∃c11,c12.
- inl … (cstate sig M1 c11) = cstate sig (seq sig M1 M2) c1 ∧
- inl … (cstate sig M1 c12) = cstate sig (seq sig M1 M2) c2 ∧
- ctape sig M1 c11 = ctape sig (seq sig M1 M2) c1 ∧
- ctape sig M1 c12 = ctape sig (seq sig M1 M2) c2 ∧
- R c11 c12.
-
-definition injectRr ≝ λsig.λM1.λM2.λR.
- λc1,c2. ∃c21,c22.
- inr … (cstate sig M2 c21) = cstate sig (seq sig M1 M2) c1 ∧
- inr … (cstate sig M2 c22) = cstate sig (seq sig M1 M2) c2 ∧
- ctape sig M2 c21 = ctape sig (seq sig M1 M2) c1 ∧
- ctape sig M2 c22 = ctape sig (seq sig M1 M2) c2 ∧
- R c21 c22.
-
-definition Rlink ≝ λsig.λM1,M2.λc1,c2.
- ctape sig (seq sig M1 M2) c1 = ctape sig (seq sig M1 M2) c2 ∧
- cstate sig (seq sig M1 M2) c1 = inl … (halt sig M1) ∧
- cstate sig (seq sig M1 M2) c2 = inr … (start sig M2). *)
interpretation "relation composition" 'compose R1 R2 = (Rcomp ? R1 R2).
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