X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmono.ma;h=3477508c15528dad59bdc44fd0cc654db3e4ab20;hb=600fba840c748f67593838673a6eb40eab9b68e5;hp=38361167a695379c48d4301a8b005cebfd6e410a;hpb=f1c79fb5fbba8a90df94f5c64aa46366cb28ac59;p=helm.git

diff --git a/matita/matita/lib/turing/mono.ma b/matita/matita/lib/turing/mono.ma
index 38361167a..3477508c1 100644
--- a/matita/matita/lib/turing/mono.ma
+++ b/matita/matita/lib/turing/mono.ma
@@ -9,109 +9,484 @@
      \ /   GNU General Public License Version 2   
       V_____________________________________________________________*)
 
+include "basics/core_notation/fintersects_2.ma".
+include "basics/finset.ma".
 include "basics/vectors.ma".
+include "basics/finset.ma".
 (* include "basics/relations.ma". *)
 
-record tape (sig:FinSet): Type[0] ≝ 
-{ left : list sig;
-  right: list 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
+| 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 ] ].
+
+lemma right_mk_tape : 
+  ∀sig,ls,c,rs.(c = None ? → ls = [ ] ∨ rs = [ ]) → right ? (mk_tape sig ls c rs) = rs.
+#sig #ls #c #rs cases c // cases ls 
+[ cases rs // 
+| #l0 #ls0 #H normalize cases (H (refl ??)) #H1 [ destruct (H1) | >H1 % ] ]
+qed-.
+
+lemma left_mk_tape : ∀sig,ls,c,rs.left ? (mk_tape sig ls c rs) = ls.
+#sig #ls #c #rs cases c // cases ls // cases rs //
+qed.
+
+lemma current_mk_tape : ∀sig,ls,c,rs.current ? (mk_tape sig ls c rs) = c.
+#sig #ls #c #rs cases c // cases ls // cases rs //
+qed.
+
+lemma current_to_midtape: ∀sig,t,c. current sig t = Some ? c →
+  ∃ls,rs. t = midtape ? ls c rs.
+#sig *
+  [#c whd in ⊢ ((??%?)→?); #Hfalse destruct
+  |#a #l #c whd in ⊢ ((??%?)→?); #Hfalse destruct
+  |#a #l #c whd in ⊢ ((??%?)→?); #Hfalse destruct
+  |#ls #a #rs #c whd in ⊢ ((??%?)→?); #H destruct 
+   @(ex_intro … ls) @(ex_intro … rs) //
+  ]
+qed.
+
+(*********************************** moves ************************************)
 
 inductive move : Type[0] ≝
-| L : move 
-| R : move
-.
+  | L : move | R : move | N : move.
+  
+(*************************** turning moves into a DeqSet **********************)
+
+definition move_eq ≝ λm1,m2:move.
+  match m1 with
+  [R ⇒ match m2 with [R ⇒ true | _ ⇒ false]
+  |L ⇒ match m2 with [L ⇒ true | _ ⇒ false]
+  |N ⇒ match m2 with [N ⇒ true | _ ⇒ false]].
+
+lemma move_eq_true:∀m1,m2.
+  move_eq m1 m2 = true ↔ m1 = m2.
+*
+  [* normalize [% #_ % |2,3: % #H destruct ]
+  |* normalize [1,3: % #H destruct |% #_ % ]
+  |* normalize [1,2: % #H destruct |% #_ % ]
+qed.
+
+definition DeqMove ≝ mk_DeqSet move move_eq move_eq_true.
+
+unification hint 0 ≔ ;
+    X ≟ DeqMove
+(* ---------------------------------------- *) ⊢ 
+    move ≡ carr X.
+
+unification hint  0 ≔ m1,m2; 
+    X ≟ DeqMove
+(* ---------------------------------------- *) ⊢ 
+    move_eq m1 m2 ≡ eqb X m1 m2.
+
+
+(************************ turning DeqMove into a FinSet ***********************)
+
+definition move_enum ≝ [L;R;N].
 
-(* We do not distinuish an input tape *)
+lemma move_enum_unique: uniqueb ? [L;R;N] = true.
+// qed.
+
+lemma move_enum_complete: ∀x:move. memb ? x [L;R;N] = true.
+* // qed.
+
+definition FinMove ≝ 
+  mk_FinSet DeqMove [L;R;N] move_enum_unique move_enum_complete.
+
+unification hint  0 ≔ ; 
+    X ≟ FinMove
+(* ---------------------------------------- *) ⊢ 
+    move ≡ FinSetcarr X.
+(********************************** machine ***********************************)
 
 record TM (sig:FinSet): Type[1] ≝ 
 { states : FinSet;
-  trans : states × (option sig) → states × (option (sig × move));
+  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;
+definition tape_move_left ≝ λsig:FinSet.λt:tape sig.
+  match t with 
+  [ niltape ⇒ niltape sig
+  | leftof _ _ ⇒ t
+  | rightof a ls ⇒ midtape sig ls a [ ]
+  | midtape ls a rs ⇒ 
+    match ls with 
+    [ nil ⇒ leftof sig a rs
+    | cons a0 ls0 ⇒ midtape sig ls0 a0 (a::rs)
+    ]
+  ]. 
+  
+definition tape_move_right ≝ λsig:FinSet.λt:tape sig.
+  match t with 
+  [ niltape ⇒ niltape sig
+  | rightof _ _ ⇒ t
+  | leftof a rs ⇒ midtape sig [ ] a rs
+  | midtape ls a rs ⇒ 
+    match rs with 
+    [ nil ⇒ rightof sig a ls
+    | cons a0 rs0 ⇒ midtape sig (a::ls) a0 rs0
+    ]
+  ]. 
+  
+definition tape_write ≝ λsig.λt: tape sig.λs:option sig.
+  match s with 
+  [ None ⇒ t
+  | Some s0 ⇒ midtape ? (left ? t) s0 (right ? t)
+  ].
+
+definition tape_move ≝ λsig.λt: tape sig.λm:move.
+  match m with
+    [ R ⇒ tape_move_right ? t
+    | L ⇒ tape_move_left ? t
+    | N ⇒ t
+    ].
+
+definition tape_move_mono ≝ 
+  λsig,t,mv.
+  tape_move sig (tape_write sig t (\fst mv)) (\snd mv).
+
+record config (sig,states:FinSet): Type[0] ≝ 
+{ cstate : states;
   ctape: tape sig
 }.
 
-definition option_hd ≝ λA.λl:list A.
-  match l with
-  [nil ⇒ None ?
-  |cons a _ ⇒ Some ? a
-  ].
+lemma config_expand: ∀sig,Q,c. 
+  c = mk_config sig Q (cstate ?? c) (ctape ?? c).
+#sig #Q * // 
+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.
 
-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 (states sig M).
+  let current_char ≝ current ? (ctape ?? c) in
+  let 〈news,a,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
+  mk_config ?? news (tape_move sig (tape_write ? (ctape ?? c) a) mv).
 
-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).
+(*
+lemma step_eq : ∀sig,M,c. 
+  let current_char ≝ current ? (ctape ?? c) in
+  let 〈news,a,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
+  step sig M c = 
+    mk_config ?? news (tape_move sig (tape_write ? (ctape ?? c) a) mv).
+#sig #M #c  
+ whd in match (tape_move_mono sig ??);
+*)
   
+(******************************** loop ****************************************)
 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_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.
 
-axiom loop_incr : ∀A,f,p,k1,k2,a1,a2. 
+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.
+#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.
 
-axiom loop_split : ∀A,f,p,q.(∀b. p b = false → q b = false) →
+lemma loop_merge : ∀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 → 
-     loop A k2 f q a3 = Some ? a4 →
-       loop A (k1+k2) f q a1 = Some ? a4.
-
-(*
-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.
+     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 #H destruct
-  |#k1' #Hind #k2 #a1 #a2 #a3 normalize in ⊢ (%→?→?);
+  [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 #H @loop_incr //
-   |normalize >(Hpq … pa1) normalize 
-    #H1 #H2 @(Hind … H2) //
+   [#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. q b = true → p b = true) →
+ ∀k,a1,a2.
+   loop A k f q a1 = Some ? a2 → 
+   ∃k1,a3.
+    loop A k1 f p a1 = Some ? a3 ∧ 
+      loop A (S(k-k1)) f q a3 = Some ? a2.
+#A #f #p #q #Hpq #k elim k
+  [#a1 #a2 normalize #Heq destruct
+  |#i #Hind #a1 #a2 normalize 
+   cases (true_or_false (q a1)) #Hqa1 >Hqa1 normalize
+    [ #Ha1a2 destruct
+     @(ex_intro … 1) @(ex_intro … a2) % 
+       [normalize >(Hpq …Hqa1) // |>Hqa1 //]
+    |#Hloop cases (true_or_false (p a1)) #Hpa1 
+       [@(ex_intro … 1) @(ex_intro … a1) % 
+         [normalize >Hpa1 // |>Hqa1 <Hloop normalize //]
+       |cases (Hind …Hloop) #k2 * #a3 * #Hloop1 #Hloop2
+        @(ex_intro … (S k2)) @(ex_intro … a3) %
+         [normalize >Hpa1 normalize // | @Hloop2 ]
+       ]
+    ]
+  ]
+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.
+
+lemma loop_p_true : 
+  ∀A,k,f,p,a.p a = true → loop A (S k) f p a = Some ? a.
+#A #k #f #p #a #Ha normalize >Ha %
+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 elim k 
+  [#a #b normalize #Hfalse destruct
+  |#k0 #IH #a #b whd in ⊢ (??%? → ?); cases (true_or_false (p a)) #Hpa
+    [ >Hpa normalize #H1 destruct // | >Hpa normalize @IH ]
+  ]
+qed. 
+
+lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
+  (∀x.hlift (lift x) = h x) → 
+  (∀x.h x = false → lift (f x) = g (lift x)) → 
+  loop A k f h c1 = Some ? c2 → 
+  loop B k g hlift (lift c1) = Some ? (lift … c2).
+#A #B #k #lift #f #g #h #hlift #c1 #c2 #Hfg #Hhlift
+generalize in match c1; elim k
+[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
+|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
+ cases (true_or_false (h c0)) #Hc0 >Hfg >Hc0 normalize
+ [ #Heq destruct (Heq) % | <Hhlift // @IH ]
+qed.
+
+(************************** Realizability *************************************)
+definition loopM ≝ λsig,M,i,cin.
+  loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
+
+lemma loopM_unfold : ∀sig,M,i,cin.
+  loopM sig M i cin = loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
+// qed.
 
 definition initc ≝ λsig.λM:TM sig.λt.
-  mk_config sig M (start sig M) t.
+  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.
+  loopM sig M i (initc sig M t) = Some ? outc → R t (ctape ?? outc).
+
+definition Terminate ≝ λsig.λM:TM sig.λt. ∃i,outc.
+  loopM sig M i (initc sig M t) = Some ? outc.
+  
+notation "M \vDash R" non associative with precedence 45 for @{ 'models $M $R}.
+interpretation "realizability" 'models M R = (Realize ? M R).
 
-(* Compositions *)
+notation "M \VDash R" non associative with precedence 45 for @{ 'wmodels $M $R}.
+interpretation "weak realizability" 'wmodels M R = (WRealize ? M R).
+
+interpretation "termination" 'fintersects M t = (Terminate ? M t).
+
+lemma WRealize_to_Realize : ∀sig.∀M: TM sig.∀R.
+  (∀t.M ↓ t) → M ⊫ R → M ⊨ R.
+#sig #M #R #HT #HW #t cases (HT … t) #i * #outc #Hloop 
+@(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
+qed.
+
+theorem Realize_to_WRealize : ∀sig.∀M:TM sig.∀R.
+  M ⊨ R → 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.
+∀t.∃i.∃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)).
+    
+notation "M ⊨ [q: R1,R2]" non associative with precedence 45 for @{ 'cmodels $M $q $R1 $R2}.
+interpretation "conditional realizability" 'cmodels M q R1 R2 = (accRealize ? M q R1 R2).
+
+(*************************** guarded realizablity *****************************)
+definition GRealize ≝ λsig.λM:TM sig.λPre:tape sig →Prop.λR:relation (tape sig).
+∀t.Pre t → ∃i.∃outc.
+  loopM sig M i (initc sig M t) = Some ? outc ∧ R t (ctape ?? outc).
+  
+definition accGRealize ≝ λsig.λM:TM sig.λacc:states sig M.
+λPre: tape sig → Prop.λRtrue,Rfalse.
+∀t.Pre t → ∃i.∃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)).
+    
+lemma WRealize_to_GRealize : ∀sig.∀M: TM sig.∀Pre,R.
+  (∀t.Pre t → M ↓ t) → M ⊫ R → GRealize sig M Pre R.
+#sig #M #Pre #R #HT #HW #t #HPre cases (HT … t HPre) #i * #outc #Hloop 
+@(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
+qed.
+
+lemma Realize_to_GRealize : ∀sig,M.∀P,R. 
+  M ⊨ R → GRealize sig M P R.
+#alpha #M #Pre #R #HR #t #HPre
+cases (HR t) -HR #k * #outc * #Hloop #HR 
+@(ex_intro ?? k) @(ex_intro ?? outc) % 
+  [ @Hloop | @HR ]
+qed.
+
+lemma acc_Realize_to_acc_GRealize: ∀sig,M.∀q:states sig M.∀P,R1,R2. 
+  M ⊨ [q:R1,R2] → accGRealize sig M q P R1 R2.
+#alpha #M #q #Pre #R1 #R2 #HR #t #HPre
+cases (HR t) -HR #k * #outc * * #Hloop #HRtrue #HRfalse 
+@(ex_intro ?? k) @(ex_intro ?? outc) % 
+  [ % [@Hloop] @HRtrue | @HRfalse]
+qed.
+
+(******************************** monotonicity ********************************)
+lemma Realize_to_Realize : ∀alpha,M,R1,R2.
+  R1 ⊆ R2 → Realize alpha M R1 → Realize alpha M R2.
+#alpha #M #R1 #R2 #Himpl #HR1 #intape
+cases (HR1 intape) -HR1 #k * #outc * #Hloop #HR1
+@(ex_intro ?? k) @(ex_intro ?? outc) % /2/
+qed.
+
+lemma WRealize_to_WRealize: ∀sig,M,R1,R2.
+  R1 ⊆ R2 → WRealize sig M R1 → WRealize ? M R2.
+#alpha #M #R1 #R2 #Hsub #HR1 #intape #i #outc #Hloop
+@Hsub @(HR1 … i) @Hloop
+qed.
+
+lemma GRealize_to_GRealize : ∀alpha,M,P,R1,R2.
+  R1 ⊆ R2 → GRealize alpha M P R1 → GRealize alpha M P R2.
+#alpha #M #P #R1 #R2 #Himpl #HR1 #intape #HP
+cases (HR1 intape HP) -HR1 #k * #outc * #Hloop #HR1
+@(ex_intro ?? k) @(ex_intro ?? outc) % /2/
+qed.
+
+lemma GRealize_to_GRealize_2 : ∀alpha,M,P1,P2,R1,R2.
+  P2 ⊆ P1 → R1 ⊆ R2 → GRealize alpha M P1 R1 → GRealize alpha M P2 R2.
+#alpha #M #P1 #P2 #R1 #R2 #Himpl1 #Himpl2 #H1 #intape #HP
+cases (H1 intape (Himpl1 … HP)) -H1 #k * #outc * #Hloop #H1
+@(ex_intro ?? k) @(ex_intro ?? outc) % /2/
+qed.
+
+lemma acc_Realize_to_acc_Realize: ∀sig,M.∀q:states sig M.∀R1,R2,R3,R4. 
+  R1 ⊆ R3 → R2 ⊆ R4 → M ⊨ [q:R1,R2] → M ⊨ [q:R3,R4].
+#alpha #M #q #R1 #R2 #R3 #R4 #Hsub13 #Hsub24 #HRa #intape
+cases (HRa intape) -HRa #k * #outc * * #Hloop #HRtrue #HRfalse 
+@(ex_intro ?? k) @(ex_intro ?? outc) % 
+  [ % [@Hloop] #Hq @Hsub13 @HRtrue // | #Hq @Hsub24 @HRfalse //]
+qed.
+
+(**************************** A canonical relation ****************************)
+
+definition R_TM ≝ λsig.λM:TM sig.λq.λt1,t2.
+∃i,outc.
+  loopM ? M i (mk_config ?? q t1) = Some ? outc ∧ 
+  t2 = (ctape ?? outc).
+  
+lemma R_TM_to_R: ∀sig,M,R. ∀t1,t2. 
+  M ⊫ R → R_TM ? M (start sig M) t1 t2 → R t1 t2.
+#sig #M #R #t1 #t2 whd in ⊢ (%→?); #HMR * #i * #outc *
+#Hloop #Ht2 >Ht2 @(HMR … Hloop)
+qed.
+
+(******************************** NOP Machine *********************************)
+
+(* NO OPERATION
+   t1 = t2 *)
+  
+definition nop_states ≝ initN 1.
+definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … 1).
+
+definition nop ≝ 
+  λalpha:FinSet.mk_TM alpha nop_states
+  (λp.let 〈q,a〉 ≝ p in 〈q,None ?,N〉)
+  start_nop (λ_.true).
+  
+definition R_nop ≝ λalpha.λt1,t2:tape alpha.t2 = t1.
+
+lemma sem_nop :
+  ∀alpha.nop alpha ⊨ R_nop alpha.
+#alpha #intape @(ex_intro ?? 1) 
+@(ex_intro … (mk_config ?? start_nop intape)) % % 
+qed.
+
+lemma nop_single_state: ∀sig.∀q1,q2:states ? (nop sig). q1 = q2.
+normalize #sig * #n #ltn1 * #m #ltm1 
+generalize in match ltn1; generalize in match ltm1;
+<(le_n_O_to_eq … (le_S_S_to_le … ltn1)) <(le_n_O_to_eq … (le_S_S_to_le … ltm1)) 
+// qed.
+
+(************************** Sequential Composition ****************************)
 
 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〉
+      if halt sig M1 s1 then 〈inr … (start sig M2), None ?,N〉
+      else let 〈news1,newa,m〉 ≝ trans sig M1 〈s1,a〉 in 〈inl … news1,newa,m〉
+  | inr s2 ⇒ let 〈news2,newa,m〉 ≝ trans sig M2 〈s2,a〉 in 〈inr … news2,newa,m〉
   ].
  
 definition seq ≝ λsig. λM1,M2 : TM sig. 
@@ -119,204 +494,239 @@ definition seq ≝ λsig. λM1,M2 : TM sig.
     (FinSum (states sig M1) (states sig M2))
     (seq_trans sig M1 M2) 
     (inl … (start sig M1))
-    (λs.match s with
+    (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).
+notation "a · b" right associative with precedence 65 for @{ 'middot $a $b}.
+interpretation "sequential composition" 'middot a b = (seq ? a b).
 
 definition lift_confL ≝ 
-  λsig,M1,M2,c.match c with
-  [ mk_config s t ⇒ mk_config ? (seq sig M1 M2) (inl … s) t ].
+  λsig,S1,S2,c.match c with 
+  [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (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 ].
+  λsig,S1,S2,c.match c with
+  [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inr … s) t ].
   
 definition halt_liftL ≝ 
-  λsig.λM1,M2:TM sig.λs:FinSum (states ? M1) (states ? M2).
+  λS1,S2,halt.λs:FinSum S1 S2.
   match s with
-  [ inl s1 ⇒ halt sig M1 s1
+  [ inl s1 ⇒ halt s1
   | inr _ ⇒ true ]. (* should be vacuous in all cases we use halt_liftL *)
 
-axiom loop_dec : ∀A,k1,k2,f,p,q,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.
+definition halt_liftR ≝ 
+  λS1,S2,halt.λs:FinSum S1 S2.
+  match s with
+  [ inl _ ⇒ false 
+  | inr s2 ⇒ halt 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 p_halt_liftL : ∀sig,S1,S2,halt,c.
+  halt (cstate sig S1 c) =
+     halt_liftL S1 S2 halt (cstate … (lift_confL … c)).
+#sig #S1 #S2 #halt #c cases c #s #t %
 qed.
 
-lemma trans_liftL : ∀sig,M1,M2,s,a,news,move.
+lemma trans_seq_liftL : ∀sig,M1,M2,s,a,news,newa,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
+  trans sig M1 〈s,a〉 = 〈news,newa,move〉 → 
+  trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,newa,move〉.
+#sig (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #newa #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 //
+lemma trans_seq_liftR : ∀sig,M1,M2,s,a,news,newa,move.
+  halt ? M2 s = false → 
+  trans sig M2 〈s,a〉 = 〈news,newa,move〉 → 
+  trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … news,newa,move〉.
+#sig #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #newa #move
+#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
+qed.
+
+lemma step_seq_liftR : ∀sig,M1,M2,c0.
+ 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 #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_seq_liftR … Heq) //
 qed.
 
-lemma step_lift_confL : ∀sig,M1,M2,c0.
+lemma step_seq_liftL : ∀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.
-
-axiom 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).
-    
-axiom loop_Some : 
-  ∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
+ 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 #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_seq_liftL … Heq) //
+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 %
+  trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?,N〉.
+#sig #M1 #M2 #s #a #Hhalt whd in ⊢ (??%?); >Hhalt %
+qed.
+
+lemma eq_ctape_lift_conf_L : ∀sig,S1,S2,outc.
+  ctape sig (FinSum S1 S2) (lift_confL … outc) = ctape … outc.
+#sig #S1 #S2 #outc cases outc #s #t %
 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).
+lemma eq_ctape_lift_conf_R : ∀sig,S1,S2,outc.
+  ctape sig (FinSum S1 S2) (lift_confR … outc) = ctape … outc.
+#sig #S1 #S2 #outc cases outc #s #t %
+qed.
+
+theorem sem_seq: ∀sig.∀M1,M2:TM sig.∀R1,R2.
+  M1 ⊨ R1 → M2 ⊨ R2 → 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
+cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
 %
-[@(loop_split ??????????? (loop_liftL … Hloop1))
- [* *
+[@(loop_merge ??????????? 
+   (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
+   (step sig M1) (step sig (seq sig M1 M2)) 
+   (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) ]
- |
- |4:@(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) ]
+  || #c0 #Hhalt <step_seq_liftL //
+  | #x <p_halt_liftL %
+  |6:cases outc1 #s1 #t1 %
+  |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2) 
+    [ * #s2 #t2 %
+    | #c0 #Hhalt <step_seq_liftR // ]
+  |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) ]
  ]
-| STOP! (* ... *)
+| @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+  % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
 ]
-
-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? *)
+theorem sem_seq_app: ∀sig.∀M1,M2:TM sig.∀R1,R2,R3.
+  M1 ⊨ R1 → M2 ⊨ R2 → R1 ∘ R2 ⊆ R3 → M1 · M2 ⊨ R3.
+#sig #M1 #M2 #R1 #R2 #R3 #HR1 #HR2 #Hsub
+#t cases (sem_seq … HR1 HR2 t)
+#k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
+% [@Hloop |@Hsub @Houtc]
+qed.
 
-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.
+(* composition with guards *)
+theorem sem_seq_guarded: ∀sig.∀M1,M2:TM sig.∀Pre1,Pre2,R1,R2.
+  GRealize sig M1 Pre1 R1 → GRealize sig M2 Pre2 R2 → 
+  (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) → 
+  GRealize sig (M1 · M2) Pre1 (R1 ∘ R2).
+#sig #M1 #M2 #Pre1 #Pre2 #R1 #R2 #HGR1 #HGR2 #Hinv #t1 #HPre1
+cases (HGR1 t1 HPre1) #k1 * #outc1 * #Hloop1 #HM1
+cases (HGR2 (ctape sig (states ? M1) outc1) ?) 
+  [2: @(Hinv … HPre1 HM1)]  
+#k2 * #outc2 * #Hloop2 #HM2
+@(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
+%
+[@(loop_merge ??????????? 
+   (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
+   (step sig M1) (step sig (seq sig M1 M2)) 
+   (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_seq_liftL //
+  | #x <p_halt_liftL %
+  |6:cases outc1 #s1 #t1 %
+  |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2) 
+    [ * #s2 #t2 %
+    | #c0 #Hhalt <step_seq_liftR // ]
+  |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) ]
+ ]
+| @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+  % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
+]
+qed.
 
-definition computation ≝ λsig.λM:TM sig.
-  cstar ? (cmove ? (step sig M) (stop sig M)).
+theorem sem_seq_app_guarded: ∀sig.∀M1,M2:TM sig.∀Pre1,Pre2,R1,R2,R3.
+  GRealize sig M1 Pre1 R1 → GRealize sig M2 Pre2 R2 → 
+  (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) → R1 ∘ R2 ⊆ R3 →
+  GRealize sig (M1 · M2) Pre1 R3.
+#sig #M1 #M2 #Pre1 #Pre2 #R1 #R2 #R3 #HR1 #HR2 #Hinv #Hsub
+#t #HPre1 cases (sem_seq_guarded … HR1 HR2 Hinv t HPre1)
+#k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
+% [@Hloop |@Hsub @Houtc]
+qed.
 
-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.
+theorem acc_sem_seq : ∀sig.∀M1,M2:TM sig.∀R1,Rtrue,Rfalse,acc.
+  M1 ⊨ R1 → M2 ⊨ [ acc: Rtrue, Rfalse ] → 
+  M1 · M2 ⊨ [ inr … acc: R1 ∘ Rtrue, R1 ∘ Rfalse ].
+#sig #M1 #M2 #R1 #Rtrue #Rfalse #acc #HR1 #HR2 #t 
+cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
+cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * * #Hloop2 
+#HMtrue #HMfalse
+@(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
+% [ %
+[@(loop_merge …
+   (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
+   (step sig M1) (step sig (seq sig M1 M2)) 
+   (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_seq_liftL //
+  | #x <p_halt_liftL %
+  |6:cases outc1 #s1 #t1 %
+  |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2) 
+    [ * #s2 #t2 %
+    | #c0 #Hhalt <step_seq_liftR // ]
+  |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
+   generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10 
+   >(trans_liftL_true sig M1 M2 ??) 
+    [ whd in ⊢ (??%?); whd in ⊢ (???%); //
+    | @(loop_Some ?????? Hloop10) ]
+ ]
+| >(config_expand … outc2) in ⊢ (%→?); whd in ⊢ (??%?→?); 
+  #Hqtrue destruct (Hqtrue)
+  @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+  % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R /2/ ]
+| >(config_expand … outc2) in ⊢ (%→?); whd in ⊢ (?(??%?)→?); #Hqfalse
+  @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+  % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R @HMfalse
+  @(not_to_not … Hqfalse) //
+]
+qed.
 
-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).
+lemma acc_sem_seq_app : ∀sig.∀M1,M2:TM sig.∀R1,Rtrue,Rfalse,R2,R3,acc.
+  M1 ⊨ R1 → M2 ⊨ [acc: Rtrue, Rfalse] → 
+    (∀t1,t2,t3. R1 t1 t3 → Rtrue t3 t2 → R2 t1 t2) → 
+    (∀t1,t2,t3. R1 t1 t3 → Rfalse t3 t2 → R3 t1 t2) → 
+    M1 · M2 ⊨ [inr … acc : R2, R3].    
+#sig #M1 #M2 #R1 #Rtrue #Rfalse #R2 #R3 #acc
+#HR1 #HRacc #Hsub1 #Hsub2 
+#t cases (acc_sem_seq … HR1 HRacc 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.