From 09a348f01d0a42a1936a3e90803fd27dd49984f4 Mon Sep 17 00:00:00 2001
From: Andrea Asperti <andrea.asperti@unibo.it>
Date: Mon, 30 Apr 2012 08:44:34 +0000
Subject: [PATCH] If machine

---
 matita/matita/lib/turing/if_machine.ma | 404 +++++++++++++++++++++++++
 matita/matita/lib/turing/mono.ma       | 122 +++-----
 2 files changed, 449 insertions(+), 77 deletions(-)
 create mode 100644 matita/matita/lib/turing/if_machine.ma

diff --git a/matita/matita/lib/turing/if_machine.ma b/matita/matita/lib/turing/if_machine.ma
new file mode 100644
index 000000000..592c90947
--- /dev/null
+++ b/matita/matita/lib/turing/if_machine.ma
@@ -0,0 +1,404 @@
+(*
+    ||M||  This file is part of HELM, an Hypertextual, Electronic   
+    ||A||  Library of Mathematics, developed at the Computer Science 
+    ||T||  Department of the University of Bologna, Italy.           
+    ||I||                                                            
+    ||T||  
+    ||A||  
+    \   /  This file is distributed under the terms of the       
+     \ /   GNU General Public License Version 2   
+      V_____________________________________________________________*)
+
+include "turing/mono.ma".
+
+definition if_trans ≝ λsig. λM1,M2,M3 : TM sig. λq:states sig M1.
+λp. let 〈s,a〉 ≝ p in
+  match s with 
+  [ inl s1 ⇒ 
+      if halt sig M1 s1 then
+        if s1==q then 〈inr … (inl … (start sig M2)), None ?〉
+        else 〈inr … (inr … (start sig M3)), None ?〉
+      else let 〈news1,m〉 ≝ trans sig M1 〈s1,a〉 in
+       〈inl … news1,m〉
+  | inr s' ⇒ 
+      match s' with
+      [ inl s2 ⇒ let 〈news2,m〉 ≝ trans sig M2 〈s2,a〉 in
+         〈inr … (inl … news2),m〉
+      | inr s3 ⇒ let 〈news3,m〉 ≝ trans sig M3 〈s3,a〉 in
+         〈inr … (inr … news3),m〉
+      ]
+  ]. 
+ 
+definition ifTM ≝ λsig. λcondM,thenM,elseM : TM sig.
+  λqacc: states sig condM.
+  mk_TM sig 
+    (FinSum (states sig condM) (FinSum (states sig thenM) (states sig elseM)))
+    (if_trans sig condM thenM elseM qacc)
+    (inl … (start sig condM))
+    (λs.match s with
+      [ inl _ ⇒ false 
+      | inr s' ⇒ match s' with 
+        [ inl s2 ⇒ halt sig thenM s2
+        | inr s3 ⇒ halt sig elseM s3 ]]). 
+
+theorem sem_seq: ∀sig,M1,M2,M3,P,R2,R3,q1,q2.
+  Frealize sig M1 P → Realize sig M2 R2 → Realize sig M3 R3 → 
+    Realize sig (ifTN sig M1 M2 M2) 
+      λt1.t2. (P t1 q1 t11 → R2 t11 t2) ∨ (P t1 q2 t12 → R3 t12 t2).
+
+(* We do not distinuish an input tape *)
+
+record TM (sig:FinSet): Type[1] ≝ 
+{ states : FinSet;
+  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;
+  ctape: tape sig
+}.
+
+definition option_hd ≝ λA.λl:list A.
+  match l with
+  [nil ⇒ None ?
+  |cons a _ ⇒ Some ? a
+  ].
+
+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 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).
+  
+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_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_split : ∀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. 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.
+
+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).
+
+(* Compositions *)
+
+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〉
+  ].
+ 
+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))
+    (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).
+
+definition lift_confL ≝ 
+  λsig,M1,M2,c.match c with
+  [ mk_config s t ⇒ mk_config ? (seq sig M1 M2) (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 ].
+  
+definition halt_liftL ≝ 
+  λsig.λM1,M2:TM sig.λs:FinSum (states ? M1) (states ? M2).
+  match s with
+  [ inl s1 ⇒ halt sig M1 s1
+  | inr _ ⇒ true ]. (* should be vacuous in all cases we use halt_liftL *)
+
+definition halt_liftR ≝ 
+  λsig.λM1,M2:TM sig.λs:FinSum (states ? M1) (states ? M2).
+  match s with
+  [ inl _ ⇒ false 
+  | inr s2 ⇒ halt sig M2 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 %
+qed.
+
+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.
+
+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_confL : ∀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.
+
+STOP!
+
+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
+elim k
+[normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
+|#k0 #IH whd in ⊢ (??%? → ??%?);
+ lapply (refl ? (halt ?? (cstate sig M2 c1))) 
+ cases (halt ?? (cstate sig M2 c1)) in ⊢ (???% → ?); #Hc0 >Hc0
+ [ >(?: halt ?? (cstate sig (seq ? M1 M2) (lift_confR … c1)) = true)
+   [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2)
+   | (* ... *) ]
+ | >(?: halt ?? (cstate sig (seq ? M1 M2) (lift_confR … c1)) = false)
+   [whd in ⊢ (??%? → ??%?); #Hc2 <IH
+     [@eq_f (* @step_lift_confR // *)
+     | 
+   | // ]
+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 #a #b elim k
+[normalize #Hfalse destruct
+|#k0 #IH whd in ⊢ (??%? → ?); cases (p a)
+ [ normalize #H1 destruct
+
+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.
+
+lemma eq_ctape_lift_conf_L : ∀sig,M1,M2,outc.
+  ctape sig (seq sig M1 M2) (lift_confL … outc) = ctape … outc.
+#sig #M1 #M2 #outc cases outc #s #t %
+qed.
+  
+lemma eq_ctape_lift_conf_R : ∀sig,M1,M2,outc.
+  ctape sig (seq sig M1 M2) (lift_confR … outc) = ctape … outc.
+#sig #M1 #M2 #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).
+#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 … (k1+k2)) @(ex_intro … (lift_confR … outc2))
+%
+[@(loop_split ??????????? (loop_liftL … Hloop1))
+ [* *
+   [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+   | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+ ||4:cases outc1 #s1 #t1 %
+ |5:@(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) ]
+ ]
+| @(ex_intro … (ctape ? (seq sig M1 M2) (lift_confL … outc1)))
+  % //
+]
+qed.
+
+(* boolean machines: machines with two distinguished halting states *)
+
+
+
+(* old stuff *)
+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? *)
+
+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).
diff --git a/matita/matita/lib/turing/mono.ma b/matita/matita/lib/turing/mono.ma
index 3b97bf495..7f261e48d 100644
--- a/matita/matita/lib/turing/mono.ma
+++ b/matita/matita/lib/turing/mono.ma
@@ -201,6 +201,14 @@ 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.
+  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.
+
 lemma config_eq : 
   ∀sig,M,c1,c2.
   cstate sig M c1 = cstate sig M c2 → 
@@ -208,6 +216,21 @@ lemma config_eq :
 #sig #M1 * #s1 #t1 * #s2 #t2 //
 qed.
 
+lemma step_lift_confR : ∀sig,M1,M2,c0.
+ halt ? M2 (cstate ?? c0) = false → 
+ step sig (seq sig M1 M2) (lift_confR sig M1 M2 c0) =
+ lift_confR sig M1 M2 (step sig M2 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_liftR … Heq)
+[% | //]
+qed.
+
 lemma step_lift_confL : ∀sig,M1,M2,c0.
  halt ? M1 (cstate ?? c0) = false → 
  step sig (seq sig M1 M2) (lift_confL sig M1 M2 c0) =
@@ -243,35 +266,37 @@ elim k
    | // ]
 qed.
 
-STOP!
-
 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
+#sig #k #M1 #M2 #c1 #c2 generalize in match c1;
 elim k
-[normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
-|#k0 #IH whd in ⊢ (??%? → ??%?);
- lapply (refl ? (halt ?? (cstate sig M2 c1))) 
- cases (halt ?? (cstate sig M2 c1)) in ⊢ (???% → ?); #Hc0 >Hc0
- [ >(?: halt ?? (cstate sig (seq ? M1 M2) (lift_confR … c1)) = true)
-   [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2)
-   | (* ... *) ]
- | >(?: halt ?? (cstate sig (seq ? M1 M2) (lift_confR … c1)) = false)
-   [whd in ⊢ (??%? → ??%?); #Hc2 <IH
-     [@eq_f (* @step_lift_confR // *)
-     | 
-   | // ]
-qed. *)
+[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
+|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
+ lapply (refl ? (halt ?? (cstate sig M2 c0))) 
+ cases (halt ?? (cstate sig M2 c0)) in ⊢ (???% → ?); #Hc0 >Hc0
+ [ >(?: halt ?? (cstate sig (seq ? M1 M2) (lift_confR … c0)) = true)
+   [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
+   | <Hc0 cases c0 // ]
+ | >(?: halt ?? (cstate sig (seq ? M1 M2) (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 #a #b elim k
-[normalize #Hfalse destruct
-|#k0 #IH whd in ⊢ (??%? → ?); cases (p a)
- [ normalize #H1 destruct
+#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 → 
@@ -316,60 +341,3 @@ cases (HR2 (ctape sig M1 outc1)) #k2 * #outc2 * #Hloop2 #HM2
 ]
 qed.
 
-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? *)
-
-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).
-- 
2.39.2