From e142e937772dd7b92152057f7fe9402d240b7b7e Mon Sep 17 00:00:00 2001
From: Andrea Asperti <andrea.asperti@unibo.it>
Date: Fri, 9 Nov 2012 15:09:23 +0000
Subject: [PATCH] done

---
 matita/matita/lib/turing/turing.ma | 401 +++++++++++++----------------
 1 file changed, 183 insertions(+), 218 deletions(-)

diff --git a/matita/matita/lib/turing/turing.ma b/matita/matita/lib/turing/turing.ma
index d76d096c5..c9412adb0 100644
--- a/matita/matita/lib/turing/turing.ma
+++ b/matita/matita/lib/turing/turing.ma
@@ -3,9 +3,10 @@ include "basics/vectors.ma".
 
 (* We do not distinuish an input tape *)
 
-record mTM (sig:FinSet): Type[1] ≝ 
+(* tapes_no = number of ADDITIONAL working tapes *)
+
+record mTM (sig:FinSet) (tapes_no:nat) : Type[1] ≝ 
 { states : FinSet;
-  tapes_no: nat; (* additional working tapes *)
   trans : states × (Vector (option sig) (S tapes_no)) → 
     states  × (Vector (option (sig × move))(S tapes_no));
   start: states;
@@ -17,11 +18,22 @@ record mconfig (sig,states:FinSet) (n:nat): Type[0] ≝
   ctapes : Vector (tape sig) (S n)
 }.
 
+lemma mconfig_expand: ∀sig,n,Q,c. 
+  c = mk_mconfig sig Q n (cstate ??? c) (ctapes ??? c).
+#sig #n #Q * // 
+qed.
+  
+lemma mconfig_eq : ∀sig,n,M,c1,c2.
+  cstate sig n M c1 = cstate sig n M c2 → 
+    ctapes sig n M c1 = ctapes sig n M c2 →  c1 = c2.
+#sig #n #M1 * #s1 #t1 * #s2 #t2 //
+qed.
+
 definition current_chars ≝ λsig.λn.λtapes.
   vec_map ?? (current sig) (S n) tapes.
 
-definition step ≝ λsig.λM:mTM sig.λc:mconfig sig (states ? M) (tapes_no ? M).
-  let 〈news,mvs〉 ≝ trans sig M 〈cstate ??? c,current_chars ?? (ctapes ??? c)〉 in
+definition step ≝ λsig.λn.λM:mTM sig n.λc:mconfig sig (states ?? M) n.
+  let 〈news,mvs〉 ≝ trans sig n M 〈cstate ??? c,current_chars ?? (ctapes ??? c)〉 in
   mk_mconfig ??? 
     news 
     (pmap_vec ??? (tape_move sig) ? (ctapes ??? c) mvs).
@@ -32,121 +44,123 @@ elim n // normalize //
 qed.
 
 (************************** Realizability *************************************)
-definition loopM ≝ λsig.λM:mTM sig.λi,cin.
-  loop ? i (step sig M) (λc.halt sig M (cstate ??? c)) cin.
+definition loopM ≝ λsig,n.λM:mTM sig n.λi,cin.
+  loop ? i (step sig n M) (λc.halt sig n 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.
+lemma loopM_unfold : ∀sig,n,M,i,cin.
+  loopM sig n M i cin = loop ? i (step sig n M) (λc.halt sig n M (cstate ??? c)) cin.
 // qed.
 
-definition initc ≝ λsig.λM:mTM sig.λtapes.
-  mk_mconfig sig (states sig M) (tapes_no sig M) (start sig M) tapes.
+definition initc ≝ λsig,n.λM:mTM sig n.λtapes.
+  mk_mconfig sig (states sig n M) n (start sig n M) tapes.
 
-definition Realize ≝ λsig.λM:mTM sig.λR:relation (Vector (tape sig) ?).
+definition Realize ≝ λsig,n.λM:mTM sig n.λR:relation (Vector (tape sig) ?).
 ∀t.∃i.∃outc.
-  loopM sig M i (initc sig M t) = Some ? outc ∧ R t (ctapes ??? outc).
+  loopM sig n M i (initc sig n M t) = Some ? outc ∧ R t (ctapes ??? outc).
 
-definition WRealize ≝ λsig.λM:mTM sig.λR:relation (Vector (tape sig) ?).
+definition WRealize ≝ λsig,n.λM:mTM sig n.λR:relation (Vector (tape sig) ?).
 ∀t,i,outc.
-  loopM sig M i (initc sig M t) = Some ? outc → R t (ctapes ??? outc).
+  loopM sig n M i (initc sig n M t) = Some ? outc → R t (ctapes ??? outc).
 
-definition Terminate ≝ λsig.λM:mTM sig.λt. ∃i,outc.
-  loopM sig M i (initc sig M t) = Some ? outc.
+definition Terminate ≝ λsig,n.λM:mTM sig n.λt. ∃i,outc.
+  loopM sig n M i (initc sig n M t) = Some ? outc.
   
 (* notation "M \vDash R" non associative with precedence 45 for @{ 'models $M $R}. *)
-interpretation "multi realizability" 'models M R = (Realize ? M R).
+interpretation "multi realizability" 'models M R = (Realize ?? M R).
 
 (* notation "M \VDash R" non associative with precedence 45 for @{ 'wmodels $M $R}. *)
-interpretation "weak multi realizability" 'wmodels M R = (WRealize ? M R).
+interpretation "weak multi realizability" 'wmodels M R = (WRealize ?? M R).
 
-interpretation "multi termination" 'fintersects M t = (Terminate ? M t).
+interpretation "multi termination" 'fintersects M t = (Terminate ?? M t).
 
-lemma WRealize_to_Realize : ∀sig.∀M: mTM sig.∀R.
+lemma WRealize_to_Realize : ∀sig,n .∀M: mTM sig n.∀R.
   (∀t.M ↓ t) → M ⊫ R → M ⊨ R.
-#sig #M #R #HT #HW #t cases (HT … t) #i * #outc #Hloop 
+#sig #n #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:mTM sig.∀R.
+theorem Realize_to_WRealize : ∀sig,n.∀M:mTM sig n.∀R.
   M ⊨ R → M ⊫ R.
-#sig #M #R #H1 #inc #i #outc #Hloop 
+#sig #n #M #R #H1 #inc #i #outc #Hloop 
 cases (H1 inc) #k * #outc1 * #Hloop1 #HR >(loop_eq … Hloop Hloop1) //
 qed.
 
-definition accRealize ≝ λsig.λM:mTM sig.λacc:states sig M.λRtrue,Rfalse.
+definition accRealize ≝ λsig,n.λM:mTM sig n.λacc:states sig n M.λRtrue,Rfalse.
 ∀t.∃i.∃outc.
-  loopM sig M i (initc sig M t) = Some ? outc ∧
+  loopM sig n M i (initc sig n M t) = Some ? outc ∧
     (cstate ??? outc = acc → Rtrue t (ctapes ??? outc)) ∧ 
     (cstate ??? outc ≠ acc → Rfalse t (ctapes ??? outc)).
     
 (* notation "M ⊨ [q: R1,R2]" non associative with precedence 45 for @{ 'cmodels $M $q $R1 $R2}. *)
-interpretation "conditional multi realizability" 'cmodels M q R1 R2 = (accRealize ? M q R1 R2).
+interpretation "conditional multi realizability" 'cmodels M q R1 R2 = (accRealize ?? M q R1 R2).
 
 (*************************** guarded realizablity *****************************)
-definition GRealize ≝ λsig.λM:mTM sig.λPre:Vector (tape sig) ? →Prop.λR:relation (Vector (tape sig) ?).
-∀t.Pre t → ∃i.∃outc.
-  loopM sig M i (initc sig M t) = Some ? outc ∧ R t (ctapes ??? outc).
+definition GRealize ≝ λsig,n.λM:mTM sig n.
+ λPre:Vector (tape sig) ? →Prop.λR:relation (Vector (tape sig) ?).
+  ∀t.Pre t → ∃i.∃outc.
+   loopM sig n M i (initc sig n M t) = Some ? outc ∧ R t (ctapes ??? outc).
   
-definition accGRealize ≝ λsig.λM:mTM sig.λacc:states sig M.
+definition accGRealize ≝ λsig,n.λM:mTM sig n.λacc:states sig n M.
 λPre: Vector (tape sig) ? → Prop.λRtrue,Rfalse.
 ∀t.Pre t → ∃i.∃outc.
-  loopM sig M i (initc sig M t) = Some ? outc ∧
+  loopM sig n M i (initc sig n M t) = Some ? outc ∧
     (cstate ??? outc = acc → Rtrue t (ctapes ??? outc)) ∧ 
     (cstate ??? outc ≠ acc → Rfalse t (ctapes ??? outc)).
     
-lemma WRealize_to_GRealize : ∀sig.∀M: mTM 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 
+lemma WRealize_to_GRealize : ∀sig,n.∀M: mTM sig n.∀Pre,R.
+  (∀t.Pre t → M ↓ t) → M ⊫ R → GRealize sig n M Pre R.
+#sig #n #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: mTM sig.∀P,R. 
-  M ⊨ R → GRealize sig M P R.
-#alpha #M #Pre #R #HR #t #HPre
+lemma Realize_to_GRealize : ∀sig,n.∀M: mTM sig n.∀P,R. 
+  M ⊨ R → GRealize sig n M P R.
+#alpha #n #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:mTM sig.∀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
+lemma acc_Realize_to_acc_GRealize: ∀sig,n.∀M:mTM sig n.∀q:states sig n M.∀P,R1,R2. 
+  M ⊨ [q:R1,R2] → accGRealize sig n M q P R1 R2.
+#alpha #n #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 : ∀sig.∀M:mTM sig.∀R1,R2.
-  R1 ⊆ R2 → Realize sig M R1 → Realize sig M R2.
-#alpha #M #R1 #R2 #Himpl #HR1 #intape
+lemma Realize_to_Realize : ∀sig,n.∀M:mTM sig n.∀R1,R2.
+  R1 ⊆ R2 → M ⊨ R1 → M ⊨ R2.
+#alpha #n #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:mTM sig.∀R1,R2.
-  R1 ⊆ R2 → WRealize sig M R1 → WRealize ? M R2.
-#alpha #M #R1 #R2 #Hsub #HR1 #intape #i #outc #Hloop
+lemma WRealize_to_WRealize: ∀sig,n.∀M:mTM sig n.∀R1,R2.
+  R1 ⊆ R2 → WRealize sig n M R1 → WRealize sig n M R2.
+#alpha #n #M #R1 #R2 #Hsub #HR1 #intape #i #outc #Hloop
 @Hsub @(HR1 … i) @Hloop
 qed.
 
-lemma GRealize_to_GRealize : ∀sig.∀M:mTM sig.∀P,R1,R2.
-  R1 ⊆ R2 → GRealize sig M P R1 → GRealize sig M P R2.
-#alpha #M #P #R1 #R2 #Himpl #HR1 #intape #HP
+lemma GRealize_to_GRealize : ∀sig,n.∀M:mTM sig n.∀P,R1,R2.
+  R1 ⊆ R2 → GRealize sig n M P R1 → GRealize sig n M P R2.
+#alpha #n #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 : ∀sig.∀M:mTM sig.∀P1,P2,R1,R2.
-  P2 ⊆ P1 → R1 ⊆ R2 → GRealize sig M P1 R1 → GRealize sig M P2 R2.
-#alpha #M #P1 #P2 #R1 #R2 #Himpl1 #Himpl2 #H1 #intape #HP
+lemma GRealize_to_GRealize_2 : ∀sig,n.∀M:mTM sig n.∀P1,P2,R1,R2.
+  P2 ⊆ P1 → R1 ⊆ R2 → GRealize sig n M P1 R1 → GRealize sig n M P2 R2.
+#alpha #n #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:mTM sig.∀q:states sig M.∀R1,R2,R3,R4. 
+lemma acc_Realize_to_acc_Realize: ∀sig,n.∀M:mTM sig n.∀q:states sig n 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
+#alpha #n #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 //]
@@ -154,14 +168,14 @@ qed.
 
 (**************************** A canonical relation ****************************)
 
-definition R_mTM ≝ λsig.λM:mTM sig.λq.λt1,t2.
+definition R_mTM ≝ λsig,n.λM:mTM sig n.λq.λt1,t2.
 ∃i,outc.
-  loopM ? M i (mk_mconfig ??? q t1) = Some ? outc ∧ 
+  loopM ? n M i (mk_mconfig ??? q t1) = Some ? outc ∧ 
   t2 = (ctapes ??? outc).
   
-lemma R_mTM_to_R: ∀sig.∀M:mTM sig.∀R. ∀t1,t2. 
-  M ⊫ R → R_mTM ? M (start sig M) t1 t2 → R t1 t2.
-#sig #M #R #t1 #t2 whd in ⊢ (%→?); #HMR * #i * #outc *
+lemma R_mTM_to_R: ∀sig,n.∀M:mTM sig n.∀R. ∀t1,t2. 
+  M ⊫ R → R_mTM ?? M (start sig n M) t1 t2 → R t1 t2.
+#sig #n #M #R #t1 #t2 whd in ⊢ (%→?); #HMR * #i * #outc *
 #Hloop #Ht2 >Ht2 @(HMR … Hloop)
 qed.
 
@@ -173,57 +187,69 @@ qed.
 definition nop_states ≝ initN 1.
 definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … 1). *)
 
-definition mnop ≝ 
-  λalpha:FinSet.λn.mk_mTM alpha nop_states n
+definition nop ≝ 
+  λalpha:FinSet.λn.mk_mTM alpha n nop_states
   (λp.let 〈q,a〉 ≝ p in 〈q,mk_Vector ? (S n) (make_list ? (None ?) (S n)) ?〉)
   start_nop (λ_.true).
 elim n normalize //
 qed.
   
-definition R_mnop ≝ λalpha,n.λt1,t2:Vector (tape alpha) (S n).t2 = t1.
+definition R_nop ≝ λalpha,n.λt1,t2:Vector (tape alpha) (S n).t2 = t1.
 
-lemma sem_mnop :
-  ∀alpha,n.mnop alpha n⊨ R_mnop alpha n.
+lemma sem_nop :
+  ∀alpha,n.nop alpha n⊨ R_nop alpha n.
 #alpha #n #intapes @(ex_intro ?? 1) 
 @(ex_intro … (mk_mconfig ??? start_nop intapes)) % % 
 qed.
 
-lemma mnop_single_state: ∀sig,n.∀q1,q2:states ? (mnop sig n). q1 = q2.
+lemma nop_single_state: ∀sig,n.∀q1,q2:states ? n (nop sig n). q1 = q2.
 normalize #sig #n0 * #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 null_action ≝ λsig.λn.
+mk_Vector ? (S n) (make_list (option (sig × move)) (None ?) (S n)) ?.
+elim (S n) // normalize //
+qed.
 
-definition seq_trans ≝ λsig. λM1,M2 : TM sig. 
+lemma tape_move_null_action: ∀sig,n,tapes.
+  pmap_vec ??? (tape_move sig) (S n) tapes (null_action sig n) = tapes.
+#sig #n #tapes cases tapes -tapes #tapes whd in match (null_action ??);
+#Heq @Vector_eq <Heq -Heq elim tapes //
+#a #tl #Hind whd in ⊢ (??%?); @eq_f2 // @Hind
+qed.
+
+definition seq_trans ≝ λsig,n. λM1,M2 : mTM sig n. 
 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 n M1 s1 then 〈inr … (start sig n M2), null_action sig n〉
+      else let 〈news1,m〉 ≝ trans sig n M1 〈s1,a〉 in 〈inl … news1,m〉
+  | inr s2 ⇒ let 〈news2,m〉 ≝ trans sig n M2 〈s2,a〉 in 〈inr … news2,m〉
   ].
  
-definition seq ≝ λsig. λM1,M2 : TM sig. 
-  mk_TM sig 
-    (FinSum (states sig M1) (states sig M2))
-    (seq_trans sig M1 M2) 
-    (inl … (start sig M1))
+definition seq ≝ λsig,n. λM1,M2 : mTM sig n. 
+  mk_mTM sig n
+    (FinSum (states sig n M1) (states sig n M2))
+    (seq_trans sig n M1 M2) 
+    (inl … (start sig n M1))
     (λs.match s with 
-      [ inl _ ⇒ false | inr s2 ⇒ halt sig M2 s2]). 
+      [ inl _ ⇒ false | inr s2 ⇒ halt sig n M2 s2]). 
 
-notation "a · b" right associative with precedence 65 for @{ 'middot $a $b}.
-interpretation "sequential composition" 'middot a b = (seq ? a b).
+(* 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 
-  [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inl … s) t ].
+  λsig,n,S1,S2,c.match c with 
+  [ mk_mconfig s t ⇒ mk_mconfig sig (FinSum S1 S2) n (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 ].
-  
+  λsig,n,S1,S2,c.match c with
+  [ mk_mconfig s t ⇒ mk_mconfig sig (FinSum S1 S2) n (inr … s) t ].
+
+(* 
 definition halt_liftL ≝ 
   λS1,S2,halt.λs:FinSum S1 S2.
   match s with
@@ -234,216 +260,155 @@ definition halt_liftR ≝
   λS1,S2,halt.λs:FinSum S1 S2.
   match s with
   [ inl _ ⇒ false 
-  | inr s2 ⇒ halt s2 ].
+  | inr s2 ⇒ halt s2 ]. *)
       
-lemma p_halt_liftL : ∀sig,S1,S2,halt,c.
-  halt (cstate sig S1 c) =
+lemma p_halt_liftL : ∀sig,n,S1,S2,halt,c.
+  halt (cstate sig S1 n c) =
      halt_liftL S1 S2 halt (cstate … (lift_confL … c)).
-#sig #S1 #S2 #halt #c cases c #s #t %
+#sig #n #S1 #S2 #halt #c cases c #s #t %
 qed.
 
-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〉.
-#sig (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #move
+lemma trans_seq_liftL : ∀sig,n,M1,M2,s,a,news,move.
+  halt ?? M1 s = false → 
+  trans sig n M1 〈s,a〉 = 〈news,move〉 → 
+  trans sig n (seq sig n M1 M2) 〈inl … s,a〉 = 〈inl … news,move〉.
+#sig #n (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #move
 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
 qed.
 
-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〉.
-#sig #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #move
+lemma trans_seq_liftR : ∀sig,n,M1,M2,s,a,news,move.
+  halt ?? M2 s = false → 
+  trans sig n M2 〈s,a〉 = 〈news,move〉 → 
+  trans sig n (seq sig n M1 M2) 〈inr … s,a〉 = 〈inr … news,move〉.
+#sig #n #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #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 #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) //
+lemma step_seq_liftR : ∀sig,n,M1,M2,c0.
+ halt ?? M2 (cstate ??? c0) = false → 
+ step sig n (seq sig n M1 M2) (lift_confR sig n (states ?? M1) (states ?? M2) c0) =
+ lift_confR sig n (states ?? M1) (states ?? M2) (step sig n M2 c0).
+#sig #n #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
+lapply (refl ? (trans ??? 〈s,current_chars sig n t〉))
+cases (trans ??? 〈s,current_chars sig n t〉) in ⊢ (???% → %);
+#s0 #m0 #Heq #Hhalt whd in ⊢ (???(?????%)); >Heq  whd in ⊢ (???%);
+whd in ⊢ (??(????%)?); whd in ⊢ (??%?); >(trans_seq_liftR … Heq) //
 qed.
 
-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
-  [ #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) //
+lemma step_seq_liftL : ∀sig,n,M1,M2,c0.
+ halt ?? M1 (cstate ??? c0) = false → 
+ step sig n (seq sig n M1 M2) (lift_confL sig n (states ?? M1) (states ?? M2) c0) =
+ lift_confL sig n ?? (step sig n M1 c0).
+#sig #n #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
+  lapply (refl ? (trans ??? 〈s,current_chars sig n t〉))
+  cases (trans ??? 〈s,current_chars sig n t〉) in ⊢ (???% → %);
+  #s0 #m0 #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 %
+lemma trans_liftL_true : ∀sig,n,M1,M2,s,a.
+  halt ?? M1 s = true → 
+  trans sig n (seq sig n M1 M2) 〈inl … s,a〉 = 〈inr … (start ?? M2),null_action sig n〉.
+#sig #n #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 %
+lemma eq_ctape_lift_conf_L : ∀sig,n,S1,S2,outc.
+  ctapes sig (FinSum S1 S2) n (lift_confL … outc) = ctapes … outc.
+#sig #n #S1 #S2 #outc cases outc #s #t %
 qed.
   
-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 %
+lemma eq_ctape_lift_conf_R : ∀sig,n,S1,S2,outc.
+  ctapes sig (FinSum S1 S2) n (lift_confR … outc) = ctapes … outc.
+#sig #n #S1 #S2 #outc cases outc #s #t %
 qed.
 
-theorem sem_seq: ∀sig.∀M1,M2:TM sig.∀R1,R2.
+theorem sem_seq: ∀sig,n.∀M1,M2:mTM sig n.∀R1,R2.
   M1 ⊨ R1 → M2 ⊨ R2 → M1 · M2 ⊨ R1 ∘ R2.
-#sig #M1 #M2 #R1 #R2 #HR1 #HR2 #t 
+#sig #n #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
+cases (HR2 (ctapes sig (states ?? M1) n 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))
+   (loop_lift ??? (lift_confL sig n (states sig n M1) (states sig n M2))
+   (step sig n M1) (step sig n (seq sig n M1 M2)) 
+   (λc.halt sig n M1 (cstate … c)) 
+   (λc.halt_liftL ?? (halt sig n 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) 
+  |7:@(loop_lift … (initc ??? (ctapes … outc1)) … Hloop2) 
     [ * #s2 #t2 %
     | #c0 #Hhalt <step_seq_liftR // ]
-  |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
+  |whd in ⊢ (??(????%)?);whd in ⊢ (??%?);
    generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10 
-   >(trans_liftL_true sig M1 M2 ??) 
+   >(trans_liftL_true sig n M1 M2 ??) 
     [ whd in ⊢ (??%?); whd in ⊢ (???%);
-      @config_eq whd in ⊢ (???%); //
+      @mconfig_eq whd in ⊢ (???%); // 
     | @(loop_Some ?????? Hloop10) ]
  ]
-| @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+| @(ex_intro … (ctapes ? (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.
+theorem sem_seq_app: ∀sig,n.∀M1,M2:mTM sig n.∀R1,R2,R3.
   M1 ⊨ R1 → M2 ⊨ R2 → R1 ∘ R2 ⊆ R3 → M1 · M2 ⊨ R3.
-#sig #M1 #M2 #R1 #R2 #R3 #HR1 #HR2 #Hsub
+#sig #n #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 → 
+theorem sem_seq_guarded: ∀sig,n.∀M1,M2:mTM sig n.∀Pre1,Pre2,R1,R2.
+  GRealize sig n M1 Pre1 R1 → GRealize sig n 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
+  GRealize sig n (M1 · M2) Pre1 (R1 ∘ R2).
+#sig #n #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) ?) 
+cases (HGR2 (ctapes sig (states ?? M1) n 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))
+   (loop_lift ??? (lift_confL sig n (states sig n M1) (states sig n M2))
+   (step sig n M1) (step sig n (seq sig n M1 M2)) 
+   (λc.halt sig n M1 (cstate … c)) 
+   (λc.halt_liftL ?? (halt sig n 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) 
+  |7:@(loop_lift … (initc ??? (ctapes … outc1)) … Hloop2) 
     [ * #s2 #t2 %
     | #c0 #Hhalt <step_seq_liftR // ]
-  |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
+  |whd in ⊢ (??(????%)?);whd in ⊢ (??%?);
    generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10 
-   >(trans_liftL_true sig M1 M2 ??) 
+   >(trans_liftL_true sig n M1 M2 ??) 
     [ whd in ⊢ (??%?); whd in ⊢ (???%);
-      @config_eq whd in ⊢ (???%); //
+      @mconfig_eq whd in ⊢ (???%); //
     | @(loop_Some ?????? Hloop10) ]
  ]
-| @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+| @(ex_intro … (ctapes ? (FinSum (states ?? M1) (states ?? M2)) n (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 → 
+theorem sem_seq_app_guarded: ∀sig,n.∀M1,M2:mTM sig n.∀Pre1,Pre2,R1,R2,R3.
+  GRealize sig n M1 Pre1 R1 → GRealize sig n 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
+  GRealize sig n (M1 · M2) Pre1 R3.
+#sig #n #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 stop ≝ λsig.λM:TM sig.λc:config sig M.
-  halt sig M (state sig M c).
-
-let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
-  match n with 
-  [ O ⇒ None ?
-  | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
-  ].
-
-(* Compute ? M f states that f is computed by M *)
-definition Compute ≝ λsig.λM:TM sig.λf:(list sig) → (list sig).
-∀l.∃i.∃c.
-  loop ? i (step sig M) (stop sig M) (init sig M l) = Some ? c ∧
-  out ?? c = f l.
-
-(* for decision problems, we accept a string if on termination
-output is not empty *)
-
-definition ComputeB ≝ λsig.λM:TM sig.λf:(list sig) → bool.
-∀l.∃i.∃c.
-  loop ? i (step sig M) (stop sig M) (init sig M l) = Some ? c ∧
-  (isnilb ? (out ?? c) = false).
-
-(* alternative approach.
-We define the notion of computation. The notion must be constructive,
-since we want to define functions over it, like lenght and size 
-
-Perche' serve Type[2] se sposto a e b a destra? *)
-
-inductive cmove (A:Type[0]) (f:A→A) (p:A →bool) (a,b:A): Type[0] ≝
-  mk_move: p a = false → b = f a → cmove A f p a b.
-  
-inductive cstar (A:Type[0]) (M:A→A→Type[0]) : A →A → Type[0] ≝
-| empty : ∀a. cstar A M a a
-| more : ∀a,b,c. M a b → cstar A M b c → cstar A M a c.
-
-definition computation ≝ λsig.λM:TM sig.
-  cstar ? (cmove ? (step sig M) (stop sig M)).
-
-definition Compute_expl ≝ λsig.λM:TM sig.λf:(list sig) → (list sig).
-  ∀l.∃c.computation sig M (init sig M l) c → 
-   (stop sig M c = true) ∧ out ?? c = f l.
-
-definition ComputeB_expl ≝ λsig.λM:TM sig.λf:(list sig) → bool.
-  ∀l.∃c.computation sig M (init sig M l) c → 
-   (stop sig M c = true) ∧ (isnilb ? (out ?? c) = false).
- 
\ No newline at end of file
-- 
2.39.2