X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fif_machine.ma;h=3c6d686ea7b4f8359e53d2887df354917f32462c;hb=d9a1ff8259a7882caa0ffd27282838c00a34cab5;hp=7b73d006d7a827d78743b81d633eb9f138e052ff;hpb=1ed95da08b483c7f7e69fc645bee455572cad031;p=helm.git

diff --git a/matita/matita/lib/turing/if_machine.ma b/matita/matita/lib/turing/if_machine.ma
index 7b73d006d..3c6d686ea 100644
--- a/matita/matita/lib/turing/if_machine.ma
+++ b/matita/matita/lib/turing/if_machine.ma
@@ -11,21 +11,48 @@
 
 include "turing/mono.ma".
 
+(**************************** single final machine ****************************)
+
+definition single_finalTM ≝ 
+  λsig.λM:TM sig.seq ? M (nop ?).
+
+lemma sem_single_final: ∀sig.∀M: TM sig.∀R.
+  M ⊨ R → 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.
+
+lemma single_final: ∀sig.∀M: TM sig.∀q1,q2.
+  halt ? (single_finalTM sig M) q1 = true 
+    →  halt ? (single_finalTM sig M) q2 = true → q1=q2.
+#sig #M * 
+  [#q1M #q2 whd in match (halt ???); #H destruct
+  |#q1nop *
+    [#q2M #_ whd in match (halt ???); #H destruct
+    |#q2nop #_ #_ @eq_f normalize @nop_single_state
+    ]
+  ]
+qed.
+  
+(******************************** if machine **********************************)
+
 definition if_trans ≝ λsig. λM1,M2,M3 : TM sig. λq:states sig M1.
 λp. let 〈s,a〉 ≝ p in
   match s with 
   [ inl s1 ⇒ 
       if halt sig M1 s1 then
-        if s1==q then 〈inr … (inl … (start sig M2)), None ?〉
-        else 〈inr … (inr … (start sig M3)), None ?〉
-      else let 〈news1,m〉 ≝ trans sig M1 〈s1,a〉 in
-       〈inl … news1,m〉
+        if s1==q then 〈inr … (inl … (start sig M2)), None ?,N〉
+        else 〈inr … (inr … (start sig M3)), None ?,N〉
+      else let 〈news1,newa,m〉 ≝ trans sig M1 〈s1,a〉 in
+       〈inl … news1,newa,m〉
   | inr s' ⇒ 
       match s' with
-      [ inl s2 ⇒ let 〈news2,m〉 ≝ trans sig M2 〈s2,a〉 in
-         〈inr … (inl … news2),m〉
-      | inr s3 ⇒ let 〈news3,m〉 ≝ trans sig M3 〈s3,a〉 in
-         〈inr … (inr … news3),m〉
+      [ inl s2 ⇒ let 〈news2,newa,m〉 ≝ trans sig M2 〈s2,a〉 in
+         〈inr … (inl … news2),newa,m〉
+      | inr s3 ⇒ let 〈news3,newa,m〉 ≝ trans sig M3 〈s3,a〉 in
+         〈inr … (inr … news3),newa,m〉
       ]
   ]. 
  
@@ -40,408 +67,399 @@ definition ifTM ≝ λsig. λcondM,thenM,elseM : TM sig.
       | inr s' ⇒ match s' with 
         [ inl s2 ⇒ halt sig thenM s2
         | inr s3 ⇒ halt sig elseM s3 ]]).
-        
-axiom daemon : ∀X:Prop.X. 
 
-theorem 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).
-#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_split ??????????? 
-    (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 ⊢ (???%);
-         @config_eq whd in ⊢ (???%); //
-       | @(loop_Some ?????? Hloop10) ]       
-     ||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)))
-  % //
-]
+(****************************** lifting lemmas ********************************)
+lemma trans_if_liftM1 : ∀sig,M1,M2,M3,acc,s,a,news,newa,move.
+  halt ? M1 s = false → 
+  trans sig M1 〈s,a〉 = 〈news,newa,move〉 → 
+  trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inl … news,newa,move〉.
+#sig * #Q1 #T1 #init1 #halt1 #M2 #M3 #acc #s #a #news #newa #move
+#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
 qed.
-(* 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
-]
+lemma trans_if_liftM2 : ∀sig,M1,M2,M3,acc,s,a,news,newa,move.
+  halt ? M2 s = false → 
+  trans sig M2 〈s,a〉 = 〈news,newa,move〉 → 
+  trans sig (ifTM sig M1 M2 M3 acc) 〈inr … (inl … s),a〉 = 〈inr… (inl … news),newa,move〉.
+#sig #M1 * #Q2 #T2 #init2 #halt2 #M3 #acc #s #a #news #newa #move
+#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
 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) //
-   ]
- ]
+lemma trans_if_liftM3 : ∀sig,M1,M2,M3,acc,s,a,news,newa,move.
+  halt ? M3 s = false → 
+  trans sig M3 〈s,a〉 = 〈news,newa,move〉 → 
+  trans sig (ifTM sig M1 M2 M3 acc) 〈inr … (inr … s),a〉 = 〈inr… (inr … news),newa,move〉.
+#sig #M1 * #Q2 #T2 #init2 #halt2 #M3 #acc #s #a #news #newa #move
+#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
 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 step_if_liftM1 : ∀sig,M1,M2,M3,acc,c0.
+ halt ? M1 (cstate ?? c0) = false → 
+ step sig (ifTM sig M1 M2 M3 acc) (lift_confL sig (states ? M1) ? c0) =
+ lift_confL sig (states ? M1) ? (step sig M1 c0).
+#sig #M1 #M2 #M3 #acc * #s #t
+  lapply (refl ? (trans ?? 〈s,current sig t〉))
+  cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
+  * #s0 #a0 #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_if_liftM1 … Hhalt Heq) //
 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 %
+lemma step_if_liftM2 : ∀sig,M1,M2,M3,acc,c0.
+ halt ? M2 (cstate ?? c0) = false → 
+ step sig (ifTM sig M1 M2 M3 acc) (lift_confR sig ?? (lift_confL sig ?? c0)) =
+ lift_confR sig ?? (lift_confL sig ?? (step sig M2 c0)).
+#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 #M3 #acc * #s #t
+  lapply (refl ? (trans ?? 〈s,current sig t〉))
+  cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
+  * #s0 #a0 #m0 cases t
+  [ #Heq #Hhalt
+  | 2,3: #s1 #l1 #Heq #Hhalt 
+  |#ls #s1 #rs #Heq #Hhalt ] 
+  whd in match (step ? M2 ?); >Heq whd in ⊢ (???%);
+  whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_if_liftM2 … Hhalt Heq) //
 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 %
+lemma step_if_liftM3 : ∀sig,M1,M2,M3,acc,c0.
+ halt ? M3 (cstate ?? c0) = false → 
+ step sig (ifTM sig M1 M2 M3 acc) (lift_confR sig ?? (lift_confR sig ?? c0)) =
+ lift_confR sig ?? (lift_confR sig ?? (step sig M3 c0)).
+#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 #M3 #acc * #s #t
+  lapply (refl ? (trans ?? 〈s,current sig t〉))
+  cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
+  * #s0 #a0 #m0 cases t
+  [ #Heq #Hhalt
+  | 2,3: #s1 #l1 #Heq #Hhalt 
+  |#ls #s1 #rs #Heq #Hhalt ] 
+  whd in match (step ? M3 ?); >Heq whd in ⊢ (???%);
+  whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_if_liftM3 … Hhalt Heq) //
 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 //
+lemma trans_if_M1true_acc : ∀sig,M1,M2,M3,acc,s,a.
+  halt ? M1 s = true → s==acc = true →
+  trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inr … (inl … (start ? M2)),None ?,N〉.
+#sig #M1 #M2 #M3 #acc #s #a #Hhalt #Hacc whd in ⊢ (??%?); >Hhalt >Hacc %
 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)
-[% | //]
+lemma trans_if_M1true_notacc : ∀sig,M1,M2,M3,acc,s,a.
+  halt ? M1 s = true → s==acc = false →
+  trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inr … (inr … (start ? M3)),None ?,N〉.
+#sig #M1 #M2 #M3 #acc #s #a #Hhalt #Hacc whd in ⊢ (??%?); >Hhalt >Hacc %
 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 //
-   | // ]
+(******************************** semantics ***********************************)
+lemma sem_if: ∀sig.∀M1,M2,M3:TM sig.∀Rtrue,Rfalse,R2,R3,acc.
+  M1 ⊨ [acc: Rtrue,Rfalse] → M2 ⊨ R2 → M3 ⊨ R3 → 
+    ifTM sig M1 M2 M3 acc ⊨ (Rtrue ∘ R2) ∪ (Rfalse ∘ R3).
+#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 ?????????
+     (mk_config ? (FinSum (states sig M1) (FinSum (states sig M2) (states sig M3)))
+      (inr (states sig M1) ? (inl (states sig M2) (states sig M3) (start sig M2))) (ctape ?? outc1) )
+     ? 
+     (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 >(step_if_liftM1 … Hhalt) // 
+      |#x <p_halt_liftL %
+      |whd in ⊢ (??%?); >(config_expand ?? outc1);
+       whd in match (lift_confL ????);
+       >(trans_if_M1true_acc … Hacc) 
+        [% | @(loop_Some ?????? Hloop1)]
+      |cases outc1 #s1 #t1 %
+      |@(loop_lift ??? 
+         (λc.(lift_confR … (lift_confL sig (states ? M2) (states ? M3) c)))
+         … Hloop2) 
+        [ * #s2 #t2 %
+        | #c0 #Hhalt >(step_if_liftM2 … Hhalt) // ]
+      ]
+    |%1 @(ex_intro … (ctape ?? outc1)) % 
+      [@HMtrue @(\P Hacc) | >(config_expand ?? outc2) @HM2 ]
+    ]
+  |cases (HR3 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM3
+   @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confR … outc2))) %
+    [@(loop_merge ?????????
+     (mk_config ? (FinSum (states sig M1) (FinSum (states sig M2) (states sig M3)))
+      (inr (states sig M1) ? (inr (states sig M2) (states sig M3) (start sig M3))) (ctape ?? outc1) )
+     ? 
+     (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 >(step_if_liftM1 … Hhalt) // 
+      |#x <p_halt_liftL %
+      |whd in ⊢ (??%?); >(config_expand ?? outc1);
+       whd in match (lift_confL ????);
+       >(trans_if_M1true_notacc … Hacc) 
+        [% | @(loop_Some ?????? Hloop1)]
+      |cases outc1 #s1 #t1 %
+      |@(loop_lift ??? 
+         (λc.(lift_confR … (lift_confR sig (states ? M2) (states ? M3) c)))
+         … Hloop2) 
+        [ * #s2 #t2 %
+        | #c0 #Hhalt >(step_if_liftM3 … Hhalt) // ]
+      ]
+    |%2 @(ex_intro … (ctape ?? outc1)) % 
+      [@HMfalse @(\Pf Hacc) | >(config_expand ?? outc2) @HM3 ]
+    ]
+  ]
 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 sem_if_app: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,acc.
+  accRealize sig M1 acc Rtrue Rfalse  → M2 ⊨ R2  → M3 ⊨ R3 →  
+    (∀t1,t2,t3. (Rtrue t1 t3 ∧ R2 t3 t2) ∨ (Rfalse t1 t3 ∧ R3 t3 t2) → R4 t1 t2) → 
+    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 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 %
+(* weak 
+lemma sem_if_app: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,acc.
+  accRealize sig M1 acc Rtrue Rfalse  → M2 ⊨ R2  → M3 ⊨ R3 →  
+    (∀t1,t2,t3. (Rtrue t1 t3 → R2 t3 t2) ∨ (Rfalse t1 t3 → R3 t3 t2) → R4 t1 t2) → 
+    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 %
+(* we can probably use acc_sem_if to prove sem_if *)
+(* for sure we can use acc_sem_if_guarded to prove acc_sem_if *)
+lemma acc_sem_if: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,acc.
+  M1 ⊨ [acc: Rtrue, Rfalse]  → M2 ⊨ R2 → M3 ⊨ R3 → 
+  ifTM sig M1 (single_finalTM … M2) M3 acc ⊨
+     [inr … (inl … (inr … start_nop)): Rtrue ∘ R2, Rfalse ∘ R3].
+#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
+  [lapply (sem_single_final … HR2) -HR2 #HR2
+   cases (HR2 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM2
+   @(ex_intro … (k1+k2)) 
+   @(ex_intro … (lift_confR … (lift_confL … outc2))) %
+    [%
+      [@(loop_merge ?????????
+         (mk_config ? (states sig (ifTM sig M1 (single_finalTM … M2) M3 acc))
+          (inr (states sig M1) ? (inl ? (states sig M3) (start sig (single_finalTM sig M2)))) (ctape ?? outc1) )
+         ? 
+         (loop_lift ??? 
+          (lift_confL sig (states ? M1) (FinSum (states ? (single_finalTM … M2)) (states ? M3)))
+          (step sig M1) (step sig (ifTM sig M1 (single_finalTM ? 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 >(step_if_liftM1 … Hhalt) // 
+        |#x <p_halt_liftL %
+        |whd in ⊢ (??%?); >(config_expand ?? outc1);
+         whd in match (lift_confL ????);
+         >(trans_if_M1true_acc … Hacc) 
+          [% | @(loop_Some ?????? Hloop1)]
+        |cases outc1 #s1 #t1 %
+        |@(loop_lift ??? 
+           (λc.(lift_confR … (lift_confL sig (states ? (single_finalTM ? M2)) (states ? M3) c)))
+           … Hloop2) 
+          [ * #s2 #t2 %
+          | #c0 #Hhalt >(step_if_liftM2 … Hhalt) // ]
+        ]
+      |#_ @(ex_intro … (ctape ?? outc1)) % 
+        [@HMtrue @(\P Hacc) | >(config_expand ?? outc2) @HM2 ]
+      ]
+    |>(config_expand ?? outc2) whd in match (lift_confR ????);
+     * #H @False_ind @H @eq_f @eq_f >(config_expand ?? outc2)
+     @single_final // @(loop_Some ?????? Hloop2)
+    ]
+  |cases (HR3 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM3
+   @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confR … outc2))) %
+    [%
+      [@(loop_merge ?????????
+         (mk_config ? (states sig (ifTM sig M1 (single_finalTM … M2) M3 acc))
+          (inr (states sig M1) ? (inr (states sig (single_finalTM ? M2)) ? (start sig M3))) (ctape ?? outc1) )
+         ? 
+         (loop_lift ??? 
+          (lift_confL sig (states ? M1) (FinSum (states ? (single_finalTM … M2)) (states ? M3)))
+          (step sig M1) (step sig (ifTM sig M1 (single_finalTM ? 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 >(step_if_liftM1 … Hhalt) // 
+        |#x <p_halt_liftL %
+        |whd in ⊢ (??%?); >(config_expand ?? outc1);
+         whd in match (lift_confL ????);
+         >(trans_if_M1true_notacc … Hacc) 
+          [% | @(loop_Some ?????? Hloop1)]
+        |cases outc1 #s1 #t1 %
+        |@(loop_lift ??? 
+           (λc.(lift_confR … (lift_confR sig (states ? (single_finalTM ? M2)) (states ? M3) c)))
+           … Hloop2) 
+          [ * #s2 #t2 %
+          | #c0 #Hhalt >(step_if_liftM3 … Hhalt) // ]
+        ]
+      |>(config_expand ?? outc2) whd in match (lift_confR ????);
+       #H destruct (H) 
+      ]
+    |#_ @(ex_intro … (ctape ?? outc1)) % 
+      [@HMfalse @(\Pf Hacc) | >(config_expand ?? outc2) @HM3 ]
+    ]
+  ]
 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 %
+    
+lemma acc_sem_if_app: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,R5,acc.
+  M1 ⊨ [acc: Rtrue, Rfalse] → M2 ⊨ R2 → 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) → 
+     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) ]
- ]
-| @(ex_intro … (ctape ? (seq sig M1 M2) (lift_confL … outc1)))
-  % //
-]
+lemma sem_single_final_guarded: ∀sig.∀M: TM sig.∀Pre,R.
+  GRealize sig M Pre R → GRealize sig (single_finalTM sig M) Pre R.
+#sig #M #Pre #R #HR #intape #HPre 
+cases (sem_seq_guarded ??????? HR (Realize_to_GRealize ?? (λt.True) ? (sem_nop …)) ?? HPre) //
+#k * #outc * #Hloop * #ta * #Hta whd in ⊢ (%→?); #Houtc
+@(ex_intro ?? k) @(ex_intro ?? outc) %  [ @Hloop | >Houtc // ]
 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 //
+    
+lemma acc_sem_if_guarded: ∀sig,M1,M2,M3,P,P2,Rtrue,Rfalse,R2,R3,acc.
+  M1 ⊨ [acc: Rtrue, Rfalse]  → 
+  (GRealize ? M2 P2 R2) → (∀t,t0.P t → Rtrue t t0 → P2 t0) → 
+  M3 ⊨ R3 → 
+  accGRealize ? (ifTM sig M1 (single_finalTM … M2) M3 acc) 
+    (inr … (inl … (inr … start_nop))) P (Rtrue ∘ R2) (Rfalse ∘ R3).
+#sig #M1 #M2 #M3 #P #P2 #Rtrue #Rfalse #R2 #R3 #acc #HaccR #HR2 #HP2 #HR3 #t #HPt 
+cases (HaccR t) #k1 * #outc1 * * #Hloop1 #HMtrue #HMfalse 
+cases (true_or_false (cstate ?? outc1 == acc)) #Hacc
+  [lapply (sem_single_final_guarded … HR2) -HR2 #HR2
+   cases (HR2 (ctape sig ? outc1) ?)
+   [|@HP2 [||@HMtrue @(\P Hacc)] // ]
+   #k2 * #outc2 * #Hloop2 #HM2
+   @(ex_intro … (k1+k2)) 
+   @(ex_intro … (lift_confR … (lift_confL … outc2))) %
+    [%
+      [@(loop_merge ?????????
+         (mk_config ? (states sig (ifTM sig M1 (single_finalTM … M2) M3 acc))
+          (inr (states sig M1) ? (inl ? (states sig M3) (start sig (single_finalTM sig M2)))) (ctape ?? outc1) )
+         ? 
+         (loop_lift ??? 
+          (lift_confL sig (states ? M1) (FinSum (states ? (single_finalTM … M2)) (states ? M3)))
+          (step sig M1) (step sig (ifTM sig M1 (single_finalTM ? 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 >(step_if_liftM1 … Hhalt) // 
+        |#x <p_halt_liftL %
+        |whd in ⊢ (??%?); >(config_expand ?? outc1);
+         whd in match (lift_confL ????);
+         >(trans_if_M1true_acc … Hacc) 
+          [% | @(loop_Some ?????? Hloop1)]
+        |cases outc1 #s1 #t1 %
+        |@(loop_lift ??? 
+           (λc.(lift_confR … (lift_confL sig (states ? (single_finalTM ? M2)) (states ? M3) c)))
+           … Hloop2) 
+          [ * #s2 #t2 %
+          | #c0 #Hhalt >(step_if_liftM2 … Hhalt) // ]
+        ]
+      |#_ @(ex_intro … (ctape ?? outc1)) % 
+        [@HMtrue @(\P Hacc) | >(config_expand ?? outc2) @HM2 ]
+      ]
+    |>(config_expand ?? outc2) whd in match (lift_confR ????);
+     * #H @False_ind @H @eq_f @eq_f >(config_expand ?? outc2)
+     @single_final // @(loop_Some ?????? Hloop2)
+    ]
+  |cases (HR3 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM3
+   @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confR … outc2))) %
+    [%
+      [@(loop_merge ?????????
+         (mk_config ? (states sig (ifTM sig M1 (single_finalTM … M2) M3 acc))
+          (inr (states sig M1) ? (inr (states sig (single_finalTM ? M2)) ? (start sig M3))) (ctape ?? outc1) )
+         ? 
+         (loop_lift ??? 
+          (lift_confL sig (states ? M1) (FinSum (states ? (single_finalTM … M2)) (states ? M3)))
+          (step sig M1) (step sig (ifTM sig M1 (single_finalTM ? 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 >(step_if_liftM1 … Hhalt) // 
+        |#x <p_halt_liftL %
+        |whd in ⊢ (??%?); >(config_expand ?? outc1);
+         whd in match (lift_confL ????);
+         >(trans_if_M1true_notacc … Hacc) 
+          [% | @(loop_Some ?????? Hloop1)]
+        |cases outc1 #s1 #t1 %
+        |@(loop_lift ??? 
+           (λc.(lift_confR … (lift_confR sig (states ? (single_finalTM ? M2)) (states ? M3) c)))
+           … Hloop2) 
+          [ * #s2 #t2 %
+          | #c0 #Hhalt >(step_if_liftM3 … Hhalt) // ]
+        ]
+      |>(config_expand ?? outc2) whd in match (lift_confR ????);
+       #H destruct (H) 
+      ]
+    |#_ @(ex_intro … (ctape ?? outc1)) % 
+      [@HMfalse @(\Pf Hacc) | >(config_expand ?? outc2) @HM3 ]
+    ]
+  ]
+qed.
+    
+lemma acc_sem_if_app_guarded: ∀sig,M1,M2,M3,P,P2,Rtrue,Rfalse,R2,R3,R4,R5,acc.
+  M1 ⊨ [acc: Rtrue, Rfalse] → 
+  (GRealize ? M2 P2 R2) → (∀t,t0.P t → Rtrue t t0 → P2 t0) → 
+  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) → 
+  accGRealize ? (ifTM sig M1 (single_finalTM … M2) M3 acc) 
+    (inr … (inl … (inr … start_nop))) P R4 R5 .
+#sig #M1 #M2 #M3 #P #P2 #Rtrue #Rfalse #R2 #R3 #R4 #R5 #acc
+#HRacc #HRtrue #Hinv #HRfalse #Hsub1 #Hsub2 
+#t #HPt cases (acc_sem_if_guarded … HRacc HRtrue Hinv HRfalse t HPt)
+#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.
 
-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).