X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmono.ma;h=37ce2f2dee192414fc174a9783f0fd0fe4a5dd89;hb=5535cd4e08fd8d1e7e6e067eac1bb6c1bf8fcbbf;hp=e89d710c948d2cf157d516e703f9c61ce76033d0;hpb=28c261aad4d710c9f218fbbbaf1ed3b45c1caf72;p=helm.git

diff --git a/matita/matita/lib/turing/mono.ma b/matita/matita/lib/turing/mono.ma
index e89d710c9..37ce2f2de 100644
--- a/matita/matita/lib/turing/mono.ma
+++ b/matita/matita/lib/turing/mono.ma
@@ -12,12 +12,12 @@
 include "basics/vectors.ma".
 (* include "basics/relations.ma". *)
 
-(*
-record tape (sig:FinSet): Type[0] ≝ 
-{ left : list (option sig);
-  right: list (option 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
@@ -27,23 +27,15 @@ inductive tape (sig:FinSet) : Type[0] ≝
 
 definition left ≝ 
  λsig.λt:tape sig.match t with
- [ niltape ⇒ [] 
- | leftof _ _ ⇒ [] 
- | rightof s l ⇒ s::l
- | midtape l _ _ ⇒ l ].
+ [ 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 ].
- 
+ [ niltape ⇒ [] | leftof s r ⇒ s::r | rightof _ _ ⇒ []| midtape _ _ r ⇒ r ].
  
 definition current ≝ 
  λsig.λt:tape sig.match t with
- [ midtape _ c _ ⇒ Some ? c
- | _ ⇒ None ? ].
+ [ midtape _ c _ ⇒ Some ? c | _ ⇒ None ? ].
  
 definition mk_tape : 
   ∀sig:FinSet.list sig → option sig → list sig → tape sig ≝ 
@@ -55,13 +47,23 @@ definition mk_tape :
       | cons r0 rs0 ⇒ leftof ? r0 rs0 ]
     | cons l0 ls0 ⇒ rightof ? l0 ls0 ] ].
 
+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
-| N : move
-.
+  | L : move | R : move | N : move.
 
-(* We do not distinuish an input tape *)
+(********************************** machine ***********************************)
 
 record TM (sig:FinSet): Type[1] ≝ 
 { states : FinSet;
@@ -70,23 +72,6 @@ record TM (sig:FinSet): Type[1] ≝
   halt : states → bool
 }.
 
-record config (sig,states:FinSet): Type[0] ≝ 
-{ cstate : states;
-  ctape: tape sig
-}.
-
-(* definition option_hd ≝ λA.λl:list (option A).
-  match l with
-  [nil ⇒ None ?
-  |cons a _ ⇒ a
-  ].
-  *)
-
-(*definition tape_write ≝ λsig.λt:tape sig.λs:sig.
-  <left ? t) s (right ? t).
-  [ None ⇒ 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
@@ -107,49 +92,29 @@ definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
       | 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)))
-    ].
-*)
+
+record config (sig,states:FinSet): Type[0] ≝ 
+{ cstate : states;
+  ctape: tape sig
+}.
+
+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 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).
-  
+
+(******************************** loop ****************************************)
 let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
   match n with 
   [ O ⇒ None ?
@@ -157,14 +122,14 @@ let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
   ].
   
 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. p a = true → 
+    loop A (S n) f p a = Some ? a.
 #A #n #f #p #a #pa normalize >pa //
 qed.
 
 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).
+    loop A (S n) f p a = loop A n f p (f a).
 normalize #A #n #f #p #a #Hpa >Hpa %
 qed.  
   
@@ -191,8 +156,7 @@ lemma loop_merge : ∀A,f,p,q.(∀b. p b = false → q b = false) →
    [#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) //
+   |normalize >(Hpq … pa1) normalize #H1 #H2 #H3 @(Hind … H2) //
    ]
  ]
 qed.
@@ -221,101 +185,207 @@ lemma loop_split : ∀A,f,p,q.(∀b. q b = true → p b = true) →
   ]
 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 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 (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.
-  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 Terminate ≝ λsig.λM:TM sig.λt. ∃i,outc.
-  loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (initc sig M t) = Some ? 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).
+
+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.Terminate sig M t) → WRealize sig M R → Realize sig M 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.
 
-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/ ]
-]
+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.
 
-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) //
+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.
 
-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)).
+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.
 
-(* NO OPERATION
+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 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 *********************************)
 
-  t1 = t2
-  *)
+(* 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 ?〉)
-  O (λ_.true).
+  start_nop (λ_.true).
   
 definition R_nop ≝ λalpha.λt1,t2:tape alpha.t2 = t1.
 
 lemma sem_nop :
-  ∀alpha.Realize alpha (nop alpha) (R_nop alpha).
-#alpha #intape @(ex_intro ?? 1) @ex_intro [| % normalize % ]
+  ∀alpha.nop alpha ⊨ R_nop alpha.
+#alpha #intape @(ex_intro ?? 1) 
+@(ex_intro … (mk_config ?? start_nop intape)) % % 
 qed.
 
-(* Compositions *)
+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〉
+      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. 
@@ -323,35 +393,11 @@ 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,S1,S2,c.match c with 
@@ -379,7 +425,7 @@ lemma p_halt_liftL : ∀sig,S1,S2,halt,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,move.
   halt ? M1 s = false → 
   trans sig M1 〈s,a〉 = 〈news,move〉 → 
   trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,move〉.
@@ -387,7 +433,7 @@ lemma trans_liftL : ∀sig,M1,M2,s,a,news,move.
 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
 qed.
 
-lemma trans_liftR : ∀sig,M1,M2,s,a,news,move.
+lemma trans_seq_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〉.
@@ -395,14 +441,7 @@ lemma trans_liftR : ∀sig,M1,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_confR : ∀sig,M1,M2,c0.
+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).
@@ -413,13 +452,11 @@ lemma step_lift_confR : ∀sig,M1,M2,c0.
   [ #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) //
+  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 (states ? M1) (states ? M2) c0) =
  lift_confL sig ?? (step sig M1 c0).
@@ -430,84 +467,14 @@ lemma step_lift_confL : ∀sig,M1,M2,c0.
   [ #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) //
+  whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
+  whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftL … Heq) //
 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) %
- | normalize <Hhlift // @IH ]
-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 ?? (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.
-
-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 %
+#sig #M1 #M2 #s #a #Hhalt whd in ⊢ (??%?); >Hhalt %
 qed.
 
 lemma eq_ctape_lift_conf_L : ∀sig,S1,S2,outc.
@@ -520,9 +487,8 @@ lemma eq_ctape_lift_conf_R : ∀sig,S1,S2,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).
+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 (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
@@ -536,12 +502,12 @@ cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
   [ * *
    [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
    | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
-  || #c0 #Hhalt <step_lift_confL //
+  || #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_lift_confR // ]
+    | #c0 #Hhalt <step_seq_liftR // ]
   |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
    generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10 
    >(trans_liftL_true sig M1 M2 ??) 
@@ -554,3 +520,58 @@ cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
 ]
 qed.
 
+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.
+
+(* 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.
+
+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.