X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmono.ma;h=3477508c15528dad59bdc44fd0cc654db3e4ab20;hb=21de0d35017656c5a55528390b54b0b2ae395b44;hp=88ec04e1c77f1e99d56a291b05693eb53b71dbe5;hpb=dd882e640319d8117644986cc0e824d1d3156c5e;p=helm.git

diff --git a/matita/matita/lib/turing/mono.ma b/matita/matita/lib/turing/mono.ma
index 88ec04e1c..3477508c1 100644
--- a/matita/matita/lib/turing/mono.ma
+++ b/matita/matita/lib/turing/mono.ma
@@ -9,7 +9,10 @@
      \ /   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". *)
 
 (******************************** tape ****************************************)
@@ -47,49 +50,164 @@ definition mk_tape :
       | 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 | 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].
+
+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
 }.
 
-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_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.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 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:option (sig × move).
+definition tape_move ≝ λsig.λt: tape sig.λm: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)
-      ] ].
+    [ 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
 }.
+
+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).
+  let 〈news,a,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
+  mk_config ?? news (tape_move sig (tape_write ? (ctape ?? c) a) 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 
@@ -172,10 +290,41 @@ lemma loop_eq : ∀sig,f,q,i,j,a,x,y.
 ]
 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.
 
@@ -215,6 +364,92 @@ definition accRealize ≝ λsig.λM:TM sig.λacc:states sig M.λRtrue,Rfalse.
   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 *********************************)
 
@@ -226,7 +461,7 @@ 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 ?〉)
+  (λp.let 〈q,a〉 ≝ p in 〈q,None ?,N〉)
   start_nop (λ_.true).
   
 definition R_nop ≝ λalpha.λt1,t2:tape alpha.t2 = t1.
@@ -237,15 +472,21 @@ lemma sem_nop :
 @(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. 
@@ -256,14 +497,9 @@ definition seq ≝ λsig. λM1,M2 : TM sig.
     (λs.match s with 
       [ inl _ ⇒ false | inr s2 ⇒ halt sig M2 s2]). 
 
-notation "a · b" non associative with precedence 65 for @{ 'middot $a $b}.
+notation "a · b" right associative with precedence 65 for @{ 'middot $a $b}.
 interpretation "sequential composition" 'middot a b = (seq ? a b).
 
-definition Rcomp ≝ λA.λR1,R2:relation A.λa1,a2.
-  ∃am.R1 a1 am ∧ R2 am a2.
-  
-interpretation "relation composition" 'compose R1 R2 = (Rcomp ? R1 R2).
-
 definition lift_confL ≝ 
   λsig,S1,S2,c.match c with 
   [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inl … s) t ].
@@ -290,84 +526,55 @@ 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,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 trans_liftR : ∀sig,M1,M2,s,a,news,move.
+lemma trans_seq_liftR : ∀sig,M1,M2,s,a,news,newa,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
+  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 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).
 #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
+  * #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_liftR … Heq) //
+  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).
 #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
+  * #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_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) % | <Hhlift // @IH ]
+  whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftL … Heq) //
 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 ?〉.
+  trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?,N〉.
 #sig #M1 #M2 #s #a #Hhalt whd in ⊢ (??%?); >Hhalt %
 qed.
 
@@ -396,12 +603,58 @@ 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_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: ∀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_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 ??) 
@@ -414,3 +667,66 @@ cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
 ]
 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.
+
+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.
+
+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.