X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fif_machine.ma;h=3d6675d02b5dd4087affdaaf58901935d28c3ceb;hb=75bf98d7d7a16ff8ce2c530e718809e2bc331568;hp=0fe4a4f1173be0d8d142ddc79d99bd264265794c;hpb=100184e7920cc3c70b50b694a17fa40ecde45e77;p=helm.git

diff --git a/matita/matita/lib/turing/if_machine.ma b/matita/matita/lib/turing/if_machine.ma
index 0fe4a4f11..3d6675d02 100644
--- a/matita/matita/lib/turing/if_machine.ma
+++ b/matita/matita/lib/turing/if_machine.ma
@@ -11,6 +11,17 @@
 
 include "turing/mono.ma".
 
+definition single_finalTM ≝ 
+  λsig.λM:TM sig.seq ? M (nop ?).
+
+lemma sem_single_final :
+  ∀sig,M,R.Realize ? M R → Realize ? (single_finalTM sig M) R.
+#sig #M #R #HR #intape 
+cases (sem_seq ????? HR (sem_nop …) intape)
+#k * #outc * #Hloop * #ta * #Hta whd in ⊢ (%→?); #Houtc
+@(ex_intro ?? k) @(ex_intro ?? outc) %  [ @Hloop | >Houtc // ]
+qed.
+
 definition if_trans ≝ λsig. λM1,M2,M3 : TM sig. λq:states sig M1.
 λp. let 〈s,a〉 ≝ p in
   match s with 
@@ -39,365 +50,101 @@ definition ifTM ≝ λsig. λcondM,thenM,elseM : TM sig.
       [ inl _ ⇒ false 
       | inr s' ⇒ match s' with 
         [ inl s2 ⇒ halt sig thenM s2
-        | inr s3 ⇒ halt sig elseM s3 ]]). 
+        | inr s3 ⇒ halt sig elseM s3 ]]).
+        
+axiom daemon : ∀X:Prop.X.
+axiom tdaemon : ∀X:Type[0].X.
 
-theorem sem_if: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,acc.
+axiom sem_if: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,acc.
   accRealize sig M1 acc Rtrue Rfalse → Realize sig M2 R2 → Realize sig M3 R3 → 
     Realize sig (ifTM sig M1 M2 M3 acc) (λt1,t2. (Rtrue ∘ R2) t1 t2 ∨ (Rfalse ∘ R3) t1 t2).
-
-(* 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)
-  ].
-  
-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.
-#A #f #p #k1 #k2 #a1 #a2 generalize in match a1; elim k1
-[normalize #a0 #Hfalse destruct
-|#k1' #IH #a0 <plus_n_Sm whd in ⊢ (??%? → ??%?);
- cases (true_or_false (p a0)) #Hpa0 >Hpa0 whd in ⊢ (??%? → ??%?); // @IH
-]
-qed.
-
-lemma loop_split : ∀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 → 
-       loop A k2 f q a3 = Some ? a4 →
-         loop A (k1+k2) f q a1 = Some ? a4.
-#Sig #f #p #q #Hpq #k1 elim k1 
-  [normalize #k2 #a1 #a2 #a3 #a4 #H destruct
-  |#k1' #Hind #k2 #a1 #a2 #a3 #a4 normalize in ⊢ (%→?);
-   cases (true_or_false (p a1)) #pa1 >pa1 normalize in ⊢ (%→?);
-   [#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) //
-   ]
- ]
-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) //
-   ]
- ]
-qed.
-*)
-
-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).
-
-definition lift_confL ≝ 
-  λsig,M1,M2,c.match c with
-  [ mk_config s t ⇒ mk_config ? (seq sig M1 M2) (inl … s) t ].
-definition lift_confR ≝ 
-  λsig,M1,M2,c.match c with
-  [ mk_config s t ⇒ mk_config ? (seq sig M1 M2) (inr … s) t ].
-  
-definition halt_liftL ≝ 
-  λsig.λM1,M2:TM sig.λs:FinSum (states ? M1) (states ? M2).
-  match s with
-  [ inl s1 ⇒ halt sig M1 s1
-  | inr _ ⇒ true ]. (* should be vacuous in all cases we use halt_liftL *)
-
-definition halt_liftR ≝ 
-  λsig.λM1,M2:TM sig.λs:FinSum (states ? M1) (states ? M2).
-  match s with
-  [ inl _ ⇒ false 
-  | inr s2 ⇒ halt sig M2 s2 ].
-      
-lemma p_halt_liftL : ∀sig,M1,M2,c.
-  halt sig M1 (cstate … c) =
-     halt_liftL sig M1 M2 (cstate … (lift_confL … c)).
-#sig #M1 #M2 #c cases c #s #t %
-qed.
-
-lemma trans_liftL : ∀sig,M1,M2,s,a,news,move.
-  halt ? M1 s = false → 
-  trans sig M1 〈s,a〉 = 〈news,move〉 → 
-  trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,move〉.
-#sig (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #move
-#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
-qed.
-
-lemma config_eq : 
-  ∀sig,M,c1,c2.
-  cstate sig M c1 = cstate sig M c2 → 
-  ctape sig M c1 = ctape sig M c2 →  c1 = c2.
-#sig #M1 * #s1 #t1 * #s2 #t2 //
-qed.
-
-lemma step_lift_confL : ∀sig,M1,M2,c0.
- halt ? M1 (cstate ?? c0) = false → 
- step sig (seq sig M1 M2) (lift_confL sig M1 M2 c0) =
- lift_confL sig M1 M2 (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)
-[% | //]
-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)) 
-      (λc.halt_liftL sig M1 M2 (cstate ?? c)) (lift_confL … c1) = 
-    Some ? (lift_confL … c2).
-#sig #k #M1 #M2 #c1 #c2 generalize in match c1;
-elim k
-[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
-|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
- lapply (refl ? (halt ?? (cstate sig M1 c0))) 
- cases (halt ?? (cstate sig M1 c0)) in ⊢ (???% → ?); #Hc0 >Hc0
- [ >(?: halt_liftL ??? (cstate sig (seq ? M1 M2) (lift_confL … c0)) = true)
-   [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
-   | // ]
- | >(?: halt_liftL ??? (cstate sig (seq ? M1 M2) (lift_confL … c0)) = false)
-   [whd in ⊢ (??%? → ??%?); #Hc2 <(IH ? Hc2) @eq_f
-    @step_lift_confL //
-   | // ]
-qed.
-
-STOP!
-
-lemma loop_liftR : ∀sig,k,M1,M2,c1,c2.
-  loop ? k (step sig M2) (λc.halt sig M2 (cstate ?? c)) c1 = Some ? c2 →
-    loop ? k (step sig (seq sig M1 M2)) 
-      (λc.halt sig (seq sig M1 M2) (cstate ?? c)) (lift_confR … c1) = 
-    Some ? (lift_confR … c2).
-#sig #k #M1 #M2 #c1 #c2
-elim k
-[normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
-|#k0 #IH whd in ⊢ (??%? → ??%?);
- lapply (refl ? (halt ?? (cstate sig M2 c1))) 
- cases (halt ?? (cstate sig M2 c1)) in ⊢ (???% → ?); #Hc0 >Hc0
- [ >(?: halt ?? (cstate sig (seq ? M1 M2) (lift_confR … c1)) = true)
-   [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2)
-   | (* ... *) ]
- | >(?: halt ?? (cstate sig (seq ? M1 M2) (lift_confR … c1)) = false)
-   [whd in ⊢ (??%? → ??%?); #Hc2 <IH
-     [@eq_f (* @step_lift_confR // *)
-     | 
-   | // ]
-qed. *)
     
-lemma loop_Some : 
-  ∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
-#A #k #f #p #a #b elim k
-[normalize #Hfalse destruct
-|#k0 #IH whd in ⊢ (??%? → ?); cases (p a)
- [ normalize #H1 destruct
-
-lemma trans_liftL_true : ∀sig,M1,M2,s,a.
-  halt ? M1 s = true → 
-  trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?〉.
-#sig #M1 #M2 #s #a
-#Hhalt whd in ⊢ (??%?); >Hhalt %
+lemma sem_if_app: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,acc.
+  accRealize sig M1 acc Rtrue Rfalse → Realize sig M2 R2 → Realize sig M3 R3 → 
+    (∀t1,t2,t3. (Rtrue t1 t3 ∧ R2 t3 t2) ∨ (Rfalse t1 t3 ∧ R3 t3 t2) → R4 t1 t2) →  
+    Realize sig 
+     (ifTM sig M1 M2 M3 acc) R4.
+#sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #R4 #acc
+#HRacc #HRtrue #HRfalse #Hsub
+#t cases (sem_if … HRacc HRtrue HRfalse t)
+#k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
+% [@Hloop] cases Houtc
+  [* #t3 * #Hleft #Hright @(Hsub … t3) %1 /2/
+  |* #t3 * #Hleft #Hright @(Hsub … t3) %2 /2/ ]
 qed.
 
-lemma eq_ctape_lift_conf_L : ∀sig,M1,M2,outc.
-  ctape sig (seq sig M1 M2) (lift_confL … outc) = ctape … outc.
-#sig #M1 #M2 #outc cases outc #s #t %
-qed.
-  
-lemma eq_ctape_lift_conf_R : ∀sig,M1,M2,outc.
-  ctape sig (seq sig M1 M2) (lift_confR … outc) = ctape … outc.
-#sig #M1 #M2 #outc cases outc #s #t %
+axiom acc_sem_if: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,acc.
+  accRealize sig M1 acc Rtrue Rfalse → Realize sig M2 R2 → Realize sig M3 R3 → 
+    accRealize sig 
+     (ifTM sig M1 (single_finalTM … M2) M3 acc) 
+     (inr … (inl … (inr … start_nop)))
+     (Rtrue ∘ R2) 
+     (Rfalse ∘ R3).
+     
+lemma acc_sem_if_app: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,R5,acc.
+  accRealize sig M1 acc Rtrue Rfalse → Realize sig M2 R2 → Realize sig M3 R3 → 
+    (∀t1,t2,t3. Rtrue t1 t3 → R2 t3 t2 → R4 t1 t2) → 
+    (∀t1,t2,t3. Rfalse t1 t3 → R3 t3 t2 → R5 t1 t2) → 
+    accRealize sig 
+     (ifTM sig M1 (single_finalTM … M2) M3 acc) 
+     (inr … (inl … (inr … start_nop)))
+     R4 R5.    
+#sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #R4 #R5 #acc
+#HRacc #HRtrue #HRfalse #Hsub1 #Hsub2 
+#t cases (acc_sem_if … HRacc HRtrue HRfalse t)
+#k * #outc * * #Hloop #Houtc1 #Houtc2 @(ex_intro … k) @(ex_intro … outc)
+% [% [@Hloop
+     |#H cases (Houtc1 H) #t3 * #Hleft #Hright @Hsub1 // ]
+  |#H cases (Houtc2 H) #t3 * #Hleft #Hright @Hsub2 // ]
 qed.
 
-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 … (k1+k2)) @(ex_intro … (lift_confR … outc2))
-%
-[@(loop_split ??????????? (loop_liftL … 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) ]
- ]
+(*    
+#sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #acc #HaccR #HR2 #HR3 #t 
+cases (HaccR t) #k1 * #outc1 * * #Hloop1 #HMtrue #HMfalse 
+cases (true_or_false (cstate ?? outc1 == acc)) #Hacc
+  [cases (HR2 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM2
+   @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confL … outc2)))
+   %
+   [@(loop_merge ??????????? 
+    (loop_lift ??? 
+      (lift_confL sig (states ? M1) (FinSum (states ? M2) (states ? M3)))
+      (step sig M1) (step sig (ifTM sig M1 M2 M3 acc)) 
+      (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) ]
+     || #c0 #Hhalt @daemon  (* <step_lift_confL // *)
+     | #x <p_halt_liftL %
+     |6:cases outc1 #s1 #t1 %
+     |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2) 
+       [ * #s2 #t2 %
+       | #c0 #Hhalt @daemon (* <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 ⊢ (???%);
+         @daemon
+(*         @config_eq whd in ⊢ (???%); // *)
+       | @(loop_Some ?????? Hloop10)
+       | @tdaemon 
+       | skip ]
+      ]
+     |
+     |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) ]
+           ]
 | @(ex_intro … (ctape ? (seq sig M1 M2) (lift_confL … outc1)))
   % //
 ]
 qed.
-
-(* boolean machines: machines with two distinguished halting states *)
-
-
-
-(* old stuff *)
-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).
+*)
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