--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A||
+ \ / This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ V_____________________________________________________________*)
+
+include "basics/vectors.ma".
+(* include "basics/relations.ma". *)
+
+record tape (sig:FinSet): Type[0] ≝
+{ left : list sig;
+ right: list sig
+}.
+
+inductive move : Type[0] ≝
+| L : move
+| R : move
+.
+
+(* We do not distinuish an input tape *)
+
+record TM (sig:FinSet): Type[1] ≝
+{ states : FinSet;
+ trans : states × (option sig) → states × (option (sig × move));
+ start: states;
+ halt : states → bool
+}.
+
+record config (sig:FinSet) (M:TM sig): Type[0] ≝
+{ cstate : states sig M;
+ ctape: tape sig
+}.
+
+definition option_hd ≝ λA.λl:list A.
+ match l with
+ [nil ⇒ None ?
+ |cons a _ ⇒ Some ? a
+ ].
+
+definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
+ match m with
+ [ 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))
+ ]
+ ].
+
+definition step ≝ λsig.λM:TM sig.λc:config sig M.
+ let current_char ≝ option_hd ? (right ? (ctape ?? c)) in
+ let 〈news,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
+ mk_config ?? news (tape_move sig (ctape ?? c) mv).
+
+let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
+ match n with
+ [ O ⇒ None ?
+ | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
+ ].
+
+definition initc ≝ λsig.λM:TM sig.λt.
+ mk_config 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).
+
+(* Compositions *)
+
+definition seq_trans ≝ λsig. λM1,M2 : TM sig.
+λp. let 〈s,a〉 ≝ p in
+ match s with
+ [ inl s1 ⇒
+ if halt sig M1 s1 then 〈inr … (start sig M2), None ?〉
+ else
+ let 〈news1,m〉 ≝ trans sig M1 〈s1,a〉 in
+ 〈inl … news1,m〉
+ | inr s2 ⇒
+ let 〈news2,m〉 ≝ trans sig M2 〈s2,a〉 in
+ 〈inr … news2,m〉
+ ].
+
+definition seq ≝ λsig. λM1,M2 : TM sig.
+ mk_TM sig
+ (FinSum (states sig M1) (states sig M2))
+ (seq_trans sig M1 M2)
+ (inl … (start sig M1))
+ (λs.match s with
+ [ inl _ ⇒ false | inr s2 ⇒ halt sig M2 s2]).
+
+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).
+
+theorem sem_seq: ∀sig,M1,M2,R1,R2.
+ Realize sig M1 R1 → Realize sig M2 R2 →
+ Realize sig (seq sig M1 M2) (R1 ∘ R2).
+#sig #M1 #M2 #R1 #R2 #HR1 #HR2 #t
+cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
+cases (HR2 (ctape sig M1 outc1)) #k2 * #outc2 * #Hloop2 #HM2
+@(ex_intro … (S(k1+k2))) @
+
+
+
+
+definition empty_tapes ≝ λsig.λn.
+mk_Vector ? n (make_list (tape sig) (mk_tape sig [] []) n) ?.
+elim n // normalize //
+qed.
+
+definition init ≝ λsig.λM:TM sig.λi:(list sig).
+ mk_config ??
+ (start sig M)
+ (vec_cons ? (mk_tape sig [] i) ? (empty_tapes sig (tapes_no sig M)))
+ [ ].
+
+definition stop ≝ λsig.λM:TM sig.λc:config sig M.
+ halt sig M (state sig M c).
+
+let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
+ match n with
+ [ O ⇒ None ?
+ | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
+ ].
+
+(* Compute ? M f states that f is computed by M *)
+definition Compute ≝ λsig.λM:TM sig.λf:(list sig) → (list sig).
+∀l.∃i.∃c.
+ loop ? i (step sig M) (stop sig M) (init sig M l) = Some ? c ∧
+ out ?? c = f l.
+
+(* for decision problems, we accept a string if on termination
+output is not empty *)
+
+definition ComputeB ≝ λsig.λM:TM sig.λf:(list sig) → bool.
+∀l.∃i.∃c.
+ loop ? i (step sig M) (stop sig M) (init sig M l) = Some ? c ∧
+ (isnilb ? (out ?? c) = false).
+
+(* alternative approach.
+We define the notion of computation. The notion must be constructive,
+since we want to define functions over it, like lenght and size
+
+Perche' serve Type[2] se sposto a e b a destra? *)
+
+inductive cmove (A:Type[0]) (f:A→A) (p:A →bool) (a,b:A): Type[0] ≝
+ mk_move: p a = false → b = f a → cmove A f p a b.
+
+inductive cstar (A:Type[0]) (M:A→A→Type[0]) : A →A → Type[0] ≝
+| empty : ∀a. cstar A M a a
+| more : ∀a,b,c. M a b → cstar A M b c → cstar A M a c.
+
+definition computation ≝ λsig.λM:TM sig.
+ cstar ? (cmove ? (step sig M) (stop sig M)).
+
+definition Compute_expl ≝ λsig.λM:TM sig.λf:(list sig) → (list sig).
+ ∀l.∃c.computation sig M (init sig M l) c →
+ (stop sig M c = true) ∧ out ?? c = f l.
+
+definition ComputeB_expl ≝ λsig.λM:TM sig.λf:(list sig) → bool.
+ ∀l.∃c.computation sig M (init sig M l) c →
+ (stop sig M c = true) ∧ (isnilb ? (out ?? c) = false).
projects / helm.git / commitdiff