X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmono.ma;h=fae3c7673c5097bbd9998785f3e5679ac0ef7a4b;hb=08a53e81b883cc19ddec52a662e9c171656ec364;hp=fbe46d90e4f6869c1b84e7c8019e6726da99423e;hpb=e06d2709c5bd9f9af9f42d7026f9da7056b82174;p=helm.git

diff --git a/matita/matita/lib/turing/mono.ma b/matita/matita/lib/turing/mono.ma
index fbe46d90e..fae3c7673 100644
--- a/matita/matita/lib/turing/mono.ma
+++ b/matita/matita/lib/turing/mono.ma
@@ -12,14 +12,53 @@
 include "basics/vectors.ma".
 (* include "basics/relations.ma". *)
 
+(*
 record tape (sig:FinSet): Type[0] ≝ 
-{ left : list sig;
-  right: list sig
+{ left : list (option sig);
+  right: list (option sig)
 }.
+*)
+
+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 ] ].
 
 inductive move : Type[0] ≝
 | L : move 
 | R : move
+| N : move
 .
 
 (* We do not distinuish an input tape *)
@@ -31,29 +70,83 @@ record TM (sig:FinSet): Type[1] ≝
   halt : states → bool
 }.
 
-record config (sig:FinSet) (M:TM sig): Type[0] ≝ 
-{ cstate : states sig M;
+record config (sig,states:FinSet): Type[0] ≝ 
+{ cstate : states;
   ctape: tape sig
 }.
 
-definition option_hd ≝ λA.λl:list A.
+(* definition option_hd ≝ λA.λl:list (option A).
   match l with
   [nil ⇒ None ?
-  |cons a _ ⇒ Some ? a
+  |cons a _ ⇒ a
   ].
+  *)
 
-definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
-  match m with 
+(*definition tape_write ≝ λsig.λt:tape sig.λs:sig.
+  <left ? t) s (right ? t).
   [ 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))
-    ]
-  ].
+  | 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
+  | cons c0 lt0 ⇒ midtape sig lt0 c0 (c::rt) ].
+  
+definition tape_move_right ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
+  match rt with
+  [ nil ⇒ rightof sig c lt
+  | cons c0 rt0 ⇒ midtape sig (c::lt) c0 rt0 ].
 
-definition step ≝ λsig.λM:TM sig.λc:config sig M.
-  let current_char ≝ option_hd ? (right ? (ctape ?? c)) in
+definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
+  match m with
+  [ None ⇒ t
+  | Some m' ⇒ 
+    let 〈s,m1〉 ≝ m' in 
+    match m1 with
+      [ R ⇒ tape_move_right ? (left ? t) s (right ? t)
+      | 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)))
+    ].
+*)
+  
+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).
   
@@ -62,15 +155,114 @@ let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
   [ 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_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 → 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. 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_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.
+  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).
 
+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).
+  
+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) //
+qed.
+
+definition accRealize ≝ λsig.λM:TM sig.λacc:states sig M.λRtrue,Rfalse:relation (tape sig).
+∀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)).
+
 (* Compositions *)
 
 definition seq_trans ≝ λsig. λM1,M2 : TM sig. 
@@ -121,71 +313,204 @@ definition Rlink ≝ λsig.λM1,M2.λc1,c2.
   
 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 //
+definition lift_confL ≝ 
+  λsig,S1,S2,c.match c with 
+  [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inl … s) t ].
+  
+definition lift_confR ≝ 
+  λsig,S1,S2,c.match c with
+  [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inr … s) t ].
+  
+definition halt_liftL ≝ 
+  λS1,S2,halt.λs:FinSum S1 S2.
+  match s with
+  [ inl s1 ⇒ halt s1
+  | inr _ ⇒ true ]. (* should be vacuous in all cases we use halt_liftL *)
+
+definition halt_liftR ≝ 
+  λS1,S2,halt.λs:FinSum S1 S2.
+  match s with
+  [ inl _ ⇒ false 
+  | inr s2 ⇒ halt s2 ].
+      
+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.
 
-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)))
-    [ ].
+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.
 
-definition stop ≝ λsig.λM:TM sig.λc:config sig M.
-  halt sig M (state sig M c).
+lemma trans_liftR : ∀sig,M1,M2,s,a,news,move.
+  halt ? M2 s = false → 
+  trans sig M2 〈s,a〉 = 〈news,move〉 → 
+  trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … news,move〉.
+#sig #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #move
+#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
+qed.
 
-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 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.
 
-(* 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.
+lemma step_lift_confR : ∀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 #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_liftR … Heq) //
+qed.
 
-(* for decision problems, we accept a string if on termination
-output is not empty *)
+lemma step_lift_confL : ∀sig,M1,M2,c0.
+ halt ? M1 (cstate ?? c0) = false → 
+ 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 #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_liftL … Heq) //
+qed.
 
-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).
+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) %
+ | normalize <Hhlift // @IH ]
+qed.
 
-(* 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 
+(* 
+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 ?? (halt sig M1) (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 ⊢ (??%? → ??%?);
+ cases (true_or_false (halt ?? (cstate sig (states ? M1) c0))) #Hc0 >Hc0
+ [ >(?: halt_liftL ?? (halt sig M1) (cstate sig ? (lift_confL … c0)) = true)
+   [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
+   | <Hc0 cases c0 // ]
+ | >(?: halt_liftL ?? (halt sig M1) (cstate ?? (lift_confL … c0)) = false)
+   [whd in ⊢ (??%? → ??%?); #Hc2 <(IH ? Hc2) @eq_f
+    @step_lift_confL //
+   | <Hc0 cases c0 // ]
+qed.
 
-Perche' serve Type[2] se sposto a e b a destra? *)
+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 generalize in match c1;
+elim k
+[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
+|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
+ cases (true_or_false (halt ?? (cstate sig ? c0))) #Hc0 >Hc0
+ [ >(?: halt ? (seq sig M1 M2) (cstate sig ? (lift_confR … c0)) = true)
+   [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
+   | <Hc0 cases c0 // ]
+ | >(?: halt ? (seq sig M1 M2) (cstate sig ? (lift_confR … c0)) = false)
+   [whd in ⊢ (??%? → ??%?); #Hc2 <(IH ? Hc2) @eq_f
+    @step_lift_confR //
+   | <Hc0 cases c0 // ]
+ ]
+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 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 %
+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.
+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.
   
-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)).
+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.
 
-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 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 (states ? M1) outc1)) #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_lift_confL //
+  | #x <p_halt_liftL %
+  |6:cases outc1 #s1 #t1 %
+  |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2) 
+    [ * #s2 #t2 %
+    | #c0 #Hhalt <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) ]
+ ]
+| @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+  % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
+]
+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).