X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=inline;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2FbinaryTM.ma;h=f2af60587cf9c6b292bec30210c8cefb5e5d9ea2;hb=d6b8021e8c83eb19033cad0aeaeebf95b327e78a;hp=34c9ae4b8f7e2ec38b95ccb55f2bf9043f3a5937;hpb=04b536f1693534e450fde5dc824022321d93d039;p=helm.git

diff --git a/matita/matita/lib/turing/multi_universal/binaryTM.ma b/matita/matita/lib/turing/multi_universal/binaryTM.ma
index 34c9ae4b8..f2af60587 100644
--- a/matita/matita/lib/turing/multi_universal/binaryTM.ma
+++ b/matita/matita/lib/turing/multi_universal/binaryTM.ma
@@ -39,7 +39,7 @@ definition bin5 : binary_base_states ≝ mk_Sig ?? 5 (leb_true_to_le 6 6 (refl 
 
 definition states_binaryTM : FinSet → FinSet → FinSet ≝ λsig,states.
  FinProd (FinProd states binary_base_states) 
-         (FinProd (FinOption sig) (initN (S (2 * (FS_crd sig))))).
+         (FinProd (FinOption sig) (initN (S (S (2 * (FS_crd sig)))))).
 
 axiom daemon : ∀T:Type[0].T.
 
@@ -48,6 +48,16 @@ definition to_initN : ∀n,m.n < m → initN m ≝ λn,m,Hn.mk_Sig … n ….//
 definition initN_pred : ∀n.∀m:initN n.initN n ≝ λn,m.mk_Sig … (pred (pi1 … m)) …. 
 cases m #m0 /2 by le_to_lt_to_lt/ qed.
 
+definition displ_of_move ≝ λsig,mv.
+  match mv with
+  [ L ⇒ 2*FS_crd sig
+  | N ⇒ FS_crd sig
+  | R ⇒ O ].
+  
+lemma le_displ_of_move : ∀sig,mv.displ_of_move sig mv ≤ S (2*FS_crd sig).
+#sig * /2 by le_S/
+qed.
+
 (* controllare i contatori, molti andranno incrementati di uno *)
 definition trans_binaryTM : ∀sig,states:FinSet.
   (states × (option sig) → states × (option sig) × move) → 
@@ -56,19 +66,21 @@ definition trans_binaryTM : ∀sig,states:FinSet.
 ≝ λsig,states,trans,p.
   let 〈s,a〉 ≝ p in
   let 〈s0,phase,ch,count〉 ≝ s in
-  let (H1 : O < S (2*FS_crd sig)) ≝ ? in
-  let (H2 : FS_crd sig < S (2*FS_crd sig)) ≝ ? in
+  let (H1 : O < S (S (2*FS_crd sig))) ≝ ? in
+  let (H2 : FS_crd sig < S (S (2*FS_crd sig))) ≝ ? in
   match pi1 … phase with
   [ O ⇒ (*** PHASE 0: read ***)
-      match a with
-      [ Some a0 ⇒ 
-        match pi1 … count with
-        [ O ⇒ 〈〈s0,bin1,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
-        | S k ⇒ if (a0 == true) 
-                then 〈〈s0,bin0,FS_nth sig k,initN_pred … count〉, None ?,R〉
-                else 〈〈s0,bin0,ch,initN_pred … count〉,None ?,R〉 ]
-      | None ⇒ (* Overflow position! *)
-          〈〈s0,bin4,None ?,to_initN 0 ? H1〉,None ?,R〉 ]
+      match pi1 … count with
+      [ O ⇒ 〈〈s0,bin1,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
+      | S k ⇒ match a with
+        [ Some a0 ⇒ if (a0 == true) 
+                    then 〈〈s0,bin0,FS_nth sig k,initN_pred … count〉, None ?,R〉
+                    else 〈〈s0,bin0,ch,initN_pred … count〉,None ?,R〉 
+        | None ⇒ (* Overflow position! *)
+          let 〈s',a',mv〉 ≝ trans 〈s0,None ?〉 in
+          match a' with
+          [ None ⇒ (* we don't write anything: go to end of 2 *) 〈〈s0,bin2,None ?,to_initN 0 ? H1〉,None ?,N〉
+          | Some _ ⇒ (* maybe extend tape *) 〈〈s0,bin4,None ?,to_initN O ? H1〉,None ?,R〉 ] ] ]
   | S phase ⇒ match phase with
   [ O ⇒ (*** PHASE 1: restart ***)
       match pi1 … count with
@@ -78,9 +90,7 @@ definition trans_binaryTM : ∀sig,states:FinSet.
   [ O ⇒ (*** PHASE 2: write ***)
       let 〈s',a',mv〉 ≝ trans 〈s0,ch〉 in
       match pi1 … count with
-      [ O ⇒ let mv' ≝ match mv with [ R ⇒ N | _ ⇒ L ] in
-            let count' ≝ match mv with [ R ⇒ 0 | N ⇒ FS_crd sig | L ⇒ 2*(FS_crd sig) ] in
-             〈〈s',bin3,ch,to_initN count' ??〉,None ?,mv'〉
+      [ O ⇒ 〈〈s',bin3,ch,to_initN (displ_of_move sig mv) ??〉,None ?,N〉
       | S k ⇒ match a' with
          [ None ⇒ 〈〈s0,bin2,ch,initN_pred … count〉,None ?,R〉
          | Some a0' ⇒ let out ≝ (FS_nth ? k == a') in
@@ -98,15 +108,14 @@ definition trans_binaryTM : ∀sig,states:FinSet.
       | Some _ ⇒ (* leftof *)
         let 〈s',a',mv〉 ≝ trans 〈s0,ch〉 in
         match a' with
-        [ None ⇒ (* we don't write anything: go to end of 2 *) 〈〈s0,bin2,ch,to_initN 0 ? H1〉,None ?,N〉
+        [ None ⇒ (* (vacuous) go to end of 2 *) 〈〈s0,bin2,ch,to_initN 0 ? H1〉,None ?,N〉
         | Some _ ⇒ (* extend tape *) 〈〈s0,bin5,ch,to_initN (FS_crd sig) ? H2〉,None ?,L〉 ]
       ]
   | S _ ⇒ (*** PHASE 5: left extension ***)
       match pi1 … count with
-      [ O ⇒ 〈〈s0,bin2,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
+      [ O ⇒ 〈〈s0,bin2,ch,to_initN (FS_crd sig) ? H2〉,None ?,R〉
       | S k ⇒ 〈〈s0,bin5,ch,initN_pred … count〉,Some ? false,L〉 ]]]]]].
-[2,3: //]
-whd in match count'; cases mv whd in ⊢ (?%?); //
+[2,3: /2 by lt_S_to_lt/] /2 by le_S_S/
 qed.
 
 definition halt_binaryTM : ∀sig,M.states_binaryTM sig (states sig M) → bool ≝ 
@@ -129,15 +138,21 @@ definition mk_binaryTM ≝
   λsig.λM:TM sig.
   mk_TM FinBool (states_binaryTM sig (states sig M)) 
     (trans_binaryTM sig (states sig M) (trans sig M)) 
-    (〈start sig M,bin0,None ?,FS_crd sig〉) (halt_binaryTM sig M).// qed.
+    (〈start sig M,bin0,None ?,FS_crd sig〉) (halt_binaryTM sig M). 
+/2 by lt_S_to_lt/ qed.
+
+definition bin_char ≝ λsig,ch.unary_of_nat (FS_crd sig) (index_of_FS sig ch).
 
-definition bin_current ≝ λsig,t.match current ? t with
-[ None ⇒ [ ] | Some c ⇒ unary_of_nat (FS_crd sig) (index_of_FS sig c) ].
+definition opt_bin_char ≝ λsig,c.match c with
+[ None ⇒ [ ] | Some c0 ⇒ bin_char sig c0 ].
+
+definition bin_list ≝ λsig,l.flatten ? (map ?? (bin_char sig) l).
+definition rev_bin_list ≝ λsig,l.flatten ? (map ?? (λc.reverse ? (bin_char sig c)) l).
 
 definition tape_bin_lift ≝ λsig,t.
-let ls' ≝ flatten ? (map ?? (unary_of_nat (FS_crd sig) ∘ (index_of_FS sig)) (left ? t)) in
-let c' ≝ option_hd ? (bin_current sig t) in
-let rs' ≝ tail ? (bin_current sig t)@flatten ? (map ?? (unary_of_nat (FS_crd sig) ∘ (index_of_FS sig)) (right ? t)) in
+let ls' ≝ rev_bin_list ? (left ? t) in
+let c' ≝ option_hd ? (opt_bin_char sig (current ? t)) in
+let rs' ≝ (tail ? (opt_bin_char sig (current ? t))@bin_list ? (right ? t)) in
  mk_tape ? ls' c' rs'.
 
 definition R_bin_lift ≝ λsig,R,t1,t2.
@@ -146,17 +161,754 @@ definition R_bin_lift ≝ λsig,R,t1,t2.
   
 definition state_bin_lift :
   ∀sig.∀M:TM sig.states sig M → states ? (mk_binaryTM ? M)
- ≝ λsig,M,q.〈q,bin0,None ?,FS_crd sig〉.// qed.
+ ≝ λsig,M,q.〈q,bin0,None ?,FS_crd sig〉./2 by lt_S_to_lt/ qed.
+
+lemma lift_halt_binaryTM : 
+  ∀sig,M,q.halt sig M q = halt ? (mk_binaryTM sig M) (state_bin_lift ? M q).
+// qed.
+
+lemma binaryTM_bin0_bin1 :
+  ∀sig,M,t,q,ch.
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,O〉) t) 
+  = mk_config ?? (〈q,bin1,ch,to_initN (FS_crd sig) ??〉) t. //
+qed.
+
+lemma binaryTM_bin0_bin2 :
+  ∀sig,M,t,q,ch,k,qn,mv.
+  current ? t = None ? → S k <S (2*FS_crd sig) → 
+  〈qn,None ?,mv〉 = trans sig M 〈q,None ?〉 → 
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,S k〉) t) 
+  = mk_config ?? (〈q,bin2,None ?,to_initN O ??〉) t. [2,3:/2 by transitive_lt/]
+#sig #M #t #q #ch #k #qn #mv #Hcur #Hk #Htrans
+whd in match (step ???); whd in match (trans ???);
+>Hcur <Htrans %
+qed.
+
+lemma binaryTM_bin0_bin4 :
+  ∀sig,M,t,q,ch,k,qn,chn,mv.
+  current ? t = None ? → S k <S (2*FS_crd sig) → 
+  〈qn,Some ? chn,mv〉 = trans sig M 〈q,None ?〉 → 
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,S k〉) t) 
+  = mk_config ?? (〈q,bin4,None ?,to_initN 0 ??〉) (tape_move ? t R). [2,3:/2 by transitive_lt/]
+#sig #M #t #q #ch #k #qn #chn #mv #Hcur #Hk #Htrans
+whd in match (step ???); whd in match (trans ???);
+>Hcur <Htrans %
+qed.
+
+lemma binaryTM_bin0_true :
+  ∀sig,M,t,q,ch,k.
+  current ? t = Some ? true → S k <S (2*FS_crd sig) → 
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,S k〉) t) 
+  = mk_config ?? (〈q,bin0,FS_nth sig k,to_initN k ??〉) (tape_move ? t R).[2,3:@le_S /2 by lt_S_to_lt/]
+#sig #M #t #q #ch #k #Hcur #Hk
+whd in match (step ???); whd in match (trans ???);
+>Hcur %
+qed.
+
+lemma binaryTM_bin0_false :
+  ∀sig,M,t,q,ch,k.
+  current ? t = Some ? false → S k <S (2*FS_crd sig) → 
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,S k〉) t) 
+  = mk_config ?? (〈q,bin0,ch,to_initN k ??〉) (tape_move ? t R).[2,3:@le_S /2 by lt_S_to_lt/]
+#sig #M #t #q #ch #k #Hcur #Hk
+whd in match (step ???); whd in match (trans ???);
+>Hcur %
+qed.
+
+(* to be checked *)
+axiom binary_to_bin_char :∀sig,csl,csr,a.
+  csl@true::csr=bin_char sig a → FS_nth ? (length ? csr) = Some ? a.
+
+lemma binaryTM_phase0_midtape_aux :
+  ∀sig,M,q,ls,a,rs,k.
+  halt sig M q=false → 
+  ∀csr,csl,t,ch.length ? csr < S (2*FS_crd sig) → 
+  t = mk_tape ? (reverse ? csl@ls) (option_hd ? (csr@rs)) (tail ? (csr@rs)) → 
+  csl@csr = bin_char sig a → 
+  |csl@csr| = FS_crd sig → 
+  (index_of_FS ? a < |csl| → ch = Some ? a) → 
+  loopM ? (mk_binaryTM sig M) (S (length ? csr) + k)
+    (mk_config ?? (〈q,bin0,ch,length ? csr〉) t) 
+  = loopM ? (mk_binaryTM sig M) k 
+      (mk_config ?? (〈q,bin1,Some ? a,FS_crd sig〉) 
+        (mk_tape ? (reverse ? (bin_char ? a)@ls) (option_hd ? rs) (tail ? rs))). [2,3:@le_S /2 by O/]
+#sig #M #q #ls #a #rs #k #Hhalt #csr elim csr
+[ #csl #t #ch #Hlen #Ht >append_nil #Hcsl #Hlencsl #Hch >loopM_unfold >loop_S_false [|normalize //]
+  >Hch [| >Hlencsl (* lemmatize *) @daemon]
+  <loopM_unfold @eq_f >binaryTM_bin0_bin1 @eq_f >Ht 
+  whd in match (step ???); whd in match (trans ???); <Hcsl %
+| #c cases c
+  [ #csr0 #IH #csl #t #ch #Hlen #Ht #Heq #Hcrd #Hch >loopM_unfold >loop_S_false [|normalize //]
+    <loopM_unfold lapply (binary_to_bin_char … Heq) #Ha >binaryTM_bin0_true 
+    [| >Ht % ]
+    lapply (IH (csl@[true]) (tape_move FinBool t R) ??????)
+    [ //
+    | >associative_append @Hcrd
+    | >associative_append @Heq
+    | >Ht whd in match (option_hd ??) in ⊢ (??%?); whd in match (tail ??) in ⊢ (??%?);
+      cases csr0
+      [ cases rs
+        [ normalize >rev_append_def >rev_append_def >reverse_append %
+        | #r1 #rs1 normalize >rev_append_def >rev_append_def >reverse_append % ]
+      | #c1 #csr1 normalize >rev_append_def >rev_append_def >reverse_append % ]
+    | /2 by lt_S_to_lt/
+    |]
+    #H whd in match (plus ??); >H @eq_f @eq_f2 %
+  | #csr0 #IH #csl #t #ch #Hlen #Ht #Heq #Hcrd #Hch >loopM_unfold >loop_S_false [|normalize //]
+    <loopM_unfold >binaryTM_bin0_false [| >Ht % ]
+    lapply (IH (csl@[false]) (tape_move FinBool t R) ??????)
+    [6: @ch
+    | (* by cases: if index < |csl|, then Hch, else False *)
+       @daemon
+    | >associative_append @Hcrd
+    | >associative_append @Heq
+    | >Ht whd in match (option_hd ??) in ⊢ (??%?); whd in match (tail ??) in ⊢ (??%?);
+      cases csr0
+      [ cases rs
+        [ normalize >rev_append_def >rev_append_def >reverse_append %
+        | #r1 #rs1 normalize >rev_append_def >rev_append_def >reverse_append % ]
+      | #c1 #csr1 normalize >rev_append_def >rev_append_def >reverse_append % ]
+    | /2 by lt_S_to_lt/
+    |]
+    #H whd in match (plus ??); >H @eq_f @eq_f2 %
+  ]
+]
+qed.
+
+lemma le_to_eq : ∀m,n.m ≤ n → ∃k. n = m + k. /3 by plus_minus, ex_intro/
+qed.
+
+lemma minus_tech : ∀a,b.a + b - a = b. // qed.
+
+lemma binaryTM_phase0_midtape :
+  ∀sig,M,t,q,ls,a,rs,ch,k.
+  halt sig M q=false → S (FS_crd sig) ≤ k → 
+  t = mk_tape ? ls (option_hd ? (bin_char ? a)) (tail ? (bin_char sig a)@rs) →
+  loopM ? (mk_binaryTM sig M) k
+    (mk_config ?? (〈q,bin0,ch,FS_crd sig〉) t) 
+  = loopM ? (mk_binaryTM sig M) (k - S (FS_crd sig))
+      (mk_config ?? (〈q,bin1,Some ? a,FS_crd sig〉) 
+        (mk_tape ? (reverse ? (bin_char ? a)@ls) (option_hd ? rs) (tail ? rs))). [|*:@le_S //]
+#sig #M #t #q #ls #a #rs #ch #k #Hhalt #Hk #Ht
+cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
+cut (∃c,cl.bin_char sig a = c::cl) [@daemon] * #c * #cl #Ha >Ha
+cut (FS_crd sig = |bin_char sig a|) [@daemon] #Hlen
+@(trans_eq ?? (loopM ? (mk_binaryTM ? M) (S (|c::cl|) + k0)
+   (mk_config ?? 〈q,bin0,〈ch,|c::cl|〉〉 t))) 
+[ /2 by O/ | @eq_f2 // @eq_f2 // @eq_f <Ha >Hlen % ]
+>(binaryTM_phase0_midtape_aux ? M q ls a rs ? ? (c::cl) [ ] t ch) //
+[| normalize #Hfalse @False_ind cases (not_le_Sn_O ?) /2/
+| <Ha (* |bin_char sig ?| = FS_crd sig *) @daemon
+| >Ha %
+| >Ht >Ha % 
+| <Ha <Hlen // ]
+<Ha %
+qed.
+
+lemma binaryTM_phase0_None_None :
+  ∀sig,M,t,q,ch,k,n,qn,mv.
+  O < n →  n < 2*FS_crd sig → O < k → 
+  halt sig M q=false → 
+  current ? t = None ? →
+  〈qn,None ?,mv〉 = trans sig M 〈q,None ?〉 → 
+  loopM ? (mk_binaryTM sig M) k (mk_config ?? (〈q,bin0,ch,n〉) t) 
+  = loopM ? (mk_binaryTM sig M) (k-1)
+      (mk_config ?? (〈q,bin2,None ?,to_initN O ??〉) t). [2,3: /2 by transitive_lt/ ]
+#sig #M #t #q #ch #k #n #qn #mv #HOn #Hn #Hk #Hhalt
+cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
+cases (le_to_eq … HOn) #n0 #Hn0 destruct (Hn0)
+cases t
+[ >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin2 // /2 by refl, transitive_lt/
+| #r0 #rs0 >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin2 // /2 by refl, transitive_lt/
+| #l0 #ls0 >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin2 // /2 by refl, transitive_lt/
+| #ls #cur #rs normalize in ⊢ (%→?); #H destruct (H) ]
+qed.
+
+lemma binaryTM_phase0_None_Some :
+  ∀sig,M,t,q,ch,k,n,qn,chn,mv.
+  O < n →  n < 2*FS_crd sig → O < k → 
+  halt sig M q=false → 
+  current ? t = None ? →
+  〈qn,Some ? chn,mv〉 = trans sig M 〈q,None ?〉 → 
+  loopM ? (mk_binaryTM sig M) k (mk_config ?? (〈q,bin0,ch,n〉) t) 
+  = loopM ? (mk_binaryTM sig M) (k-1) 
+      (mk_config ?? (〈q,bin4,None ?,to_initN O ??〉) (tape_move ? t R)). [2,3: /2 by transitive_lt/ ]  
+#sig #M #t #q #ch #k #n #qn #chn #mv #HOn #Hn #Hk #Hhalt
+cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
+cases (le_to_eq … HOn) #n0 #Hn0 destruct (Hn0)
+cases t
+[ >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin4 // /2 by refl, transitive_lt/
+| #r0 #rs0 >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin4 // /2 by refl, transitive_lt/
+| #l0 #ls0 >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin4 // /2 by refl, transitive_lt/
+| #ls #cur #rs normalize in ⊢ (%→?); #H destruct (H) ]
+qed.
+
+lemma binaryTM_bin1_O :
+  ∀sig,M,t,q,ch.
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin1,ch,O〉) t) 
+  = mk_config ?? (〈q,bin2,ch,to_initN (FS_crd sig) ??〉) t. [2,3:/2 by lt_S_to_lt/]
+#sig #M #t #q #ch %
+qed.
+
+lemma binaryTM_bin1_S :
+  ∀sig,M,t,q,ch,k. S k <S (2*FS_crd sig) → 
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin1,ch,S k〉) t) 
+  = mk_config ?? (〈q,bin1,ch,to_initN k ??〉) (tape_move ? t L). [2,3:@le_S /2 by lt_S_to_lt/]
+#sig #M #t #q #ch #k #HSk %
+qed.
+
+lemma binaryTM_phase1 :
+  ∀sig,M,q,ls1,ls2,cur,rs,ch,k.
+  S (FS_crd sig) ≤ k → |ls1| = FS_crd sig → (cur = None ? → rs = [ ]) → 
+  loopM ? (mk_binaryTM sig M) k
+    (mk_config ?? (〈q,bin1,ch,FS_crd sig〉) (mk_tape ? (ls1@ls2) cur rs)) 
+  = loopM ? (mk_binaryTM sig M) (k - S (FS_crd sig))
+      (mk_config ?? (〈q,bin2,ch,FS_crd sig〉) 
+        (mk_tape ? ls2 (option_hd ? (reverse ? ls1@option_cons ? cur rs)) 
+          (tail ? (reverse ? ls1@option_cons ? cur rs)))). [2,3:/2 by O/]
+cut (∀sig,M,q,ls1,ls2,ch,k,n,cur,rs.
+  |ls1| = n →  n<S (2*FS_crd sig) → (cur = None ? → rs = [ ]) → 
+  loopM ? (mk_binaryTM sig M) (S n + k)
+    (mk_config ?? (〈q,bin1,ch,n〉) (mk_tape ? (ls1@ls2) cur rs)) 
+  = loopM ? (mk_binaryTM sig M) k 
+      (mk_config ?? (〈q,bin2,ch,FS_crd sig〉) 
+        (mk_tape ? ls2 (option_hd ? (reverse ? ls1@option_cons ? cur rs)) 
+          (tail ? (reverse ? ls1@option_cons ? cur rs))))) [1,2:@le_S //]
+[ #sig #M #q #ls1 #ls2 #ch #k elim ls1
+  [ #n normalize in ⊢ (%→?); #cur #rs #Hn <Hn #Hcrd #Hcur >loopM_unfold >loop_S_false [| % ]
+    >binaryTM_bin1_O cases cur in Hcur;
+    [ #H >(H (refl ??)) -H %
+    | #cur' #_ % ]
+  | #l0 #ls0 #IH * [ #cur #rs normalize in ⊢ (%→?); #H destruct (H) ]
+    #n #cur #rs normalize in ⊢ (%→?); #H destruct (H) #Hlt #Hcur
+    >loopM_unfold >loop_S_false [|%] >binaryTM_bin1_S
+    <(?:mk_tape ? (ls0@ls2) (Some ? l0) (option_cons ? cur rs) =
+        tape_move FinBool (mk_tape FinBool ((l0::ls0)@ls2) cur rs) L) 
+    [| cases cur in Hcur; [ #H >(H ?) // | #cur' #_ % ] ]
+    >(?:loop (config FinBool (states FinBool (mk_binaryTM sig M))) (S (|ls0|)+k)
+      (step FinBool (mk_binaryTM sig M))
+      (λc:config FinBool (states FinBool (mk_binaryTM sig M))
+       .halt FinBool (mk_binaryTM sig M)
+       (cstate FinBool (states FinBool (mk_binaryTM sig M)) c))
+      (mk_config FinBool (states FinBool (mk_binaryTM sig M))
+       〈q,bin1,ch,to_initN (|ls0|) ?
+        (le_S ?? (lt_S_to_lt (|ls0|) (S (2*FS_crd sig)) Hlt))〉
+       (mk_tape FinBool (ls0@ls2) (Some FinBool l0) (option_cons FinBool cur rs)))
+      = loopM FinBool (mk_binaryTM sig M) k
+         (mk_config FinBool (states FinBool (mk_binaryTM sig M))
+          〈q,bin2,〈ch,FS_crd sig〉〉
+          (mk_tape FinBool ls2
+           (option_hd FinBool (reverse FinBool ls0@l0::option_cons FinBool cur rs))
+           (tail FinBool (reverse FinBool ls0@l0::option_cons FinBool cur rs)))))
+    [| /2/
+    | >(?: l0::option_cons ? cur rs = option_cons ? (Some ? l0) (option_cons ? cur rs)) [| % ]
+      @trans_eq [|| @(IH ??? (refl ??)) [ /2 by lt_S_to_lt/ | #H destruct (H) ] ]
+      %
+    ]
+   >reverse_cons >associative_append %
+ ]
+| #Hcut #sig #M #q #ls1 #ls2 #cur #rs #ch #k #Hk #Hlen 
+  cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech @Hcut // ]
+qed.
+
+lemma binaryTM_bin2_O :
+  ∀sig,M,t,q,qn,ch,chn,mv.
+  〈qn,chn,mv〉 = trans sig M 〈q,ch〉 → 
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin2,ch,O〉) t)
+  = mk_config ?? (〈qn,bin3,ch,to_initN (displ_of_move sig mv) ??〉) t.[2,3:/2 by lt_S_to_lt,le_S_S/]
+#sig #M #t #q #qn #ch #chn #mv #Htrans
+whd in match (step ???); whd in match (trans ???); <Htrans %
+qed.
+
+lemma binaryTM_bin2_S_None :
+  ∀sig,M,t,q,qn,ch,mv,k.
+  k < S (2*FS_crd sig) → 
+  〈qn,None ?,mv〉 = trans sig M 〈q,ch〉 → 
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin2,ch,S k〉) t)
+  = mk_config ?? (〈q,bin2,ch,k〉) (tape_move ? t R).
+[2,3: @le_S_S /2 by lt_to_le/ ]
+#sig #M #t #q #qn #ch #mv #k #Hk #Htrans
+whd in match (step ???); whd in match (trans ???); <Htrans %
+qed.
+
+lemma binaryTM_bin2_S_Some :
+  ∀sig,M,t,q,qn,ch,chn,mv,k.
+  k< S (2*FS_crd sig) → 
+  〈qn,Some ? chn,mv〉 = trans sig M 〈q,ch〉 → 
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin2,ch,S k〉) t)
+  = mk_config ?? (〈q,bin2,ch,k〉) (tape_move ? (tape_write ? t (Some ? (FS_nth ? k == Some ? chn))) R).
+[2,3: @le_S_S /2 by lt_to_le/ ]
+#sig #M #t #q #qn #ch #chn #mv #k #Hk #Htrans
+whd in match (step ???); whd in match (trans ???); <Htrans %
+qed.
+
+let rec iter (T:Type[0]) f n (t:T) on n ≝ 
+  match n with [ O ⇒ t | S n0 ⇒ iter T f n0 (f t) ].
+
+lemma binaryTM_phase2_None :∀sig,M,q,ch,qn,mv,k,n. S n ≤ k → 
+  ∀t.n≤S (2*FS_crd sig) → 
+  〈qn,None ?,mv〉 = trans sig M 〈q,ch〉 → 
+  loopM ? (mk_binaryTM sig M) k
+    (mk_config ?? (〈q,bin2,ch,n〉) t)
+  = loopM ? (mk_binaryTM sig M) (k - S n)
+      (mk_config ?? (〈qn,bin3,ch,to_initN (displ_of_move sig mv) ??〉) 
+        (iter ? (λt0.tape_move ? t0 R) n t)). [2,3: @le_S_S /2 by lt_S_to_lt/]
+#sig #M #q #ch #qn #mv #k #n #Hk
+cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
+elim n
+[ #t #Hle #Htrans >loopM_unfold >loop_S_false //
+  >(binaryTM_bin2_O … Htrans) //
+| #n0 #IH #t #Hn0 #Htrans >loopM_unfold >loop_S_false // 
+  >(binaryTM_bin2_S_None … Htrans) @(trans_eq ???? (IH …)) //
+]
+qed.
+
+lemma binaryTM_phase2_Some_of : ∀sig,M,q,ch,qn,chn,mv,ls,k.
+  S (FS_crd sig) ≤ k → 〈qn,Some ? chn,mv〉 = trans sig M 〈q,ch〉 → 
+  loopM ? (mk_binaryTM sig M) k
+    (mk_config ?? (〈q,bin2,ch,FS_crd sig〉) (mk_tape ? ls (None ?) [ ])) 
+  = loopM ? (mk_binaryTM sig M) (k - S (FS_crd sig))
+      (mk_config ?? (〈qn,bin3,ch,displ_of_move sig mv〉) 
+        (mk_tape ? (reverse ? (bin_char sig chn)@ls) (None ?) [ ])). [2,3:@le_S_S //]
+cut (∀sig,M,q,ch,qn,chn,mv,ls,k,n.
+  S n ≤ k → 〈qn,Some ? chn,mv〉 = trans sig M 〈q,ch〉 → 
+  ∀csl. n <S (2*FS_crd sig) → 
+  |csl| + n = FS_crd sig →
+  (∃fs.bin_char sig chn = reverse ? csl@fs) → 
+  loopM ? (mk_binaryTM sig M) k
+    (mk_config ?? (〈q,bin2,ch,n〉) (mk_tape ? (csl@ls) (None ?) [ ])) 
+  = loopM ? (mk_binaryTM sig M) (k - S n)
+      (mk_config ?? (〈qn,bin3,ch,displ_of_move sig mv〉) 
+        (mk_tape ? (reverse ? (bin_char sig chn)@ls) (None ?) [ ]))) [1,2:@le_S_S //]
+[ #sig #M #q #ch #qn #chn #mv #ls #k #n #Hk
+  cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
+  #Htrans elim n
+  [ #csl #Hcount #Hcrd * #fs #Hfs >loopM_unfold >loop_S_false // <loopM_unfold 
+    cut (fs = [ ]) 
+    [ cases fs in Hfs; // #f0 #fs0 #H lapply (eq_f ?? (length ?) … H)
+      >length_append >(?:|bin_char sig chn| = FS_crd sig) [|@daemon]
+      <Hcrd >length_reverse #H1 cut (O = |f0::fs0|) [ /2/ ]
+      normalize #H1 destruct (H1) ]
+    #H destruct (H) >append_nil in Hfs; #Hfs
+    >Hfs >reverse_reverse >(binaryTM_bin2_O … Htrans) //
+  | #n0 #IH #csl #Hcount #Hcrd * #fs #Hfs
+    >loopM_unfold >loop_S_false // <loopM_unfold
+    >(?: step FinBool (mk_binaryTM sig M)
+         (mk_config FinBool (states FinBool (mk_binaryTM sig M)) 〈q,bin2,〈ch,S n0〉〉
+         (mk_tape FinBool (csl@ls) (None FinBool) [])) 
+        = mk_config ?? (〈q,bin2,ch,n0〉) 
+          (tape_move ? (tape_write ? 
+            (mk_tape ? (csl@ls) (None ?) [ ]) (Some ? (FS_nth ? n0 == Some ? chn))) R))
+    [| /2 by lt_S_to_lt/ | @(binaryTM_bin2_S_Some … Htrans) ]
+    >(?: tape_move ? (tape_write ???) ? = 
+          mk_tape ? (((FS_nth ? n0 == Some sig chn)::csl)@ls) (None ?) [ ])
+    [| cases csl // cases ls // ]
+    cases fs in Hfs;
+    [ #Hfalse cut (|bin_char ? chn| = |csl|) [ >Hfalse >length_append >length_reverse // ]
+      -Hfalse >(?:|bin_char sig chn| = FS_crd sig) [|@daemon]
+      <Hcrd in ⊢ (%→?); >(?:|csl| = |csl|+ O) in ⊢ (???%→?); //
+      #Hfalse cut (S n0 = O) /2 by injective_plus_r/ #H destruct (H)
+    | #f0 #fs0 #Hbinchar 
+      cut (bin_char ? chn = reverse ? csl@(FS_nth ? n0 == Some ? chn)::fs0) [@daemon]
+      -Hbinchar #Hbinchar >Hbinchar @(trans_eq ???? (IH …)) //
+      [ %{fs0} >reverse_cons >associative_append @Hbinchar
+      | whd in ⊢ (??%?); /2 by / ]
+      @eq_f @eq_f @eq_f3 //
+    ]
+  ]
+| #Hcut #sig #M #q #ch #qn #chn #mv #ls #k #Hk #Htrans 
+  @trans_eq 
+  [3: @(trans_eq ???? (Hcut ??????? ls ? (FS_crd sig) ? Htrans …)) //
+    [3:@([ ]) | %{(bin_char ? chn)} % | % ]
+  || % ]
+]
+qed.
+
+lemma binaryTM_phase2_Some_ow : ∀sig,M,q,ch,qn,chn,mv,ls,k,cs,rs.
+  S (FS_crd sig) ≤ k → 〈qn,Some ? chn,mv〉 = trans sig M 〈q,ch〉 → 
+  |cs| = FS_crd sig → 
+  loopM ? (mk_binaryTM sig M) k
+    (mk_config ?? (〈q,bin2,ch,FS_crd sig〉) 
+      (mk_tape ? ls (option_hd ? (cs@rs)) (tail ? (cs@rs))))
+  = loopM ? (mk_binaryTM sig M) (k - S (FS_crd sig))
+      (mk_config ?? (〈qn,bin3,ch,displ_of_move sig mv〉) 
+        (mk_tape ? (reverse ? (bin_char sig chn)@ls) (option_hd ? rs) (tail ? rs))). [2,3:@le_S_S /2 by O/]
+cut (∀sig,M,q,ch,qn,chn,mv,ls,rs,k,csr.
+     〈qn,Some ? chn,mv〉 = trans sig M 〈q,ch〉 → 
+     ∀csl.|csr|<S (2*FS_crd sig) → 
+     |csl@csr| = FS_crd sig →
+     (∃fs.bin_char sig chn = reverse ? csl@fs) → 
+     loopM ? (mk_binaryTM sig M) (S (|csr|) + k)
+       (mk_config ?? (〈q,bin2,ch,|csr|〉) 
+         (mk_tape ? (csl@ls) (option_hd ? (csr@rs)) (tail ? (csr@rs))))
+     = loopM ? (mk_binaryTM sig M) k 
+         (mk_config ?? (〈qn,bin3,ch,displ_of_move sig mv〉) 
+           (mk_tape ? (reverse ? (bin_char sig chn)@ls) (option_hd ? rs) (tail ? rs)))) [1,2: @le_S_S /2 by le_S/]
+[ #sig #M #q #ch #qn #chn #mv #ls #rs #k #csr #Htrans elim csr
+  [ #csl #Hcount #Hcrd * #fs #Hfs >loopM_unfold >loop_S_false // normalize in match (length ? [ ]);
+    >(binaryTM_bin2_O … Htrans) <loopM_unfold @eq_f @eq_f @eq_f3 //
+    cases fs in Hfs; // #f0 #fs0 #H lapply (eq_f ?? (length ?) … H)
+    >length_append >(?:|bin_char sig chn| = FS_crd sig) [|@daemon]
+    <Hcrd >length_reverse #H1 cut (O = |f0::fs0|) [ /2/ ]
+    normalize #H1 destruct (H1) 
+  | #b0 #bs0 #IH #csl #Hcount #Hcrd * #fs #Hfs
+    >loopM_unfold >loop_S_false // >(binaryTM_bin2_S_Some … Htrans)  
+    >(?: tape_move ? (tape_write ???) ? = 
+          mk_tape ? (((FS_nth ? (|bs0|)==Some sig chn)::csl)@ls) 
+            (option_hd ? (bs0@rs)) (tail ? (bs0@rs)))
+      in match (tape_move ? (tape_write ???) ?);
+    [| cases bs0 // cases rs // ] @IH
+    [ whd in Hcount:(?%?); /2 by lt_S_to_lt/
+    | <Hcrd >length_append >length_append normalize //
+    | cases fs in Hfs;
+      [ #Hfalse cut (|bin_char ? chn| = |csl|) [ >Hfalse >length_append >length_reverse // ]      -Hfalse >(?:|bin_char sig chn| = FS_crd sig) [|@daemon]
+        <Hcrd >length_append normalize >(?:|csl| = |csl|+ O) in ⊢ (???%→?); //
+        #Hfalse cut (S (|bs0|) = O) /2 by injective_plus_r/ #H destruct (H)
+      | #f0 #fs0 #Hbinchar 
+        cut (bin_char ? chn = reverse ? csl@(FS_nth ? (|bs0|) == Some ? chn)::fs0) [@daemon]
+        -Hbinchar #Hbinchar >Hbinchar %{fs0} >reverse_cons >associative_append %
+      ]
+    ]
+  ]
+| #Hcut #sig #M #q #ch #qn #chn #mv #ls #k #cs #rs #Hk #Htrans #Hcrd
+  cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech @trans_eq 
+  [3: @(trans_eq ???? (Hcut ??????? ls ?? cs Htrans [ ] …)) //
+    [ normalize % // | normalize @Hcrd | >Hcrd // ]
+  || @eq_f2 [ >Hcrd % | @eq_f2 // @eq_f cases Hcrd // ] ] ]
+qed.
+
+lemma binaryTM_bin3_O :
+  ∀sig,M,t,q,ch.
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin3,ch,O〉) t) 
+  = mk_config ?? (〈q,bin0,None ?,to_initN (FS_crd sig) ??〉) t. [2,3:@le_S //]
+#sig #M #t #q #ch %
+qed.
+
+lemma binaryTM_bin3_S :
+  ∀sig,M,t,q,ch,k. S k ≤ S (2*FS_crd sig) → 
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin3,ch,S k〉) t) 
+  = mk_config ?? (〈q,bin3,ch,to_initN k ??〉) (tape_move ? t L). [2,3: @le_S_S /2 by lt_to_le/]
+#sig #M #t #q #ch #k #HSk %
+qed.
+
+lemma binaryTM_phase3 :∀sig,M,q,ch,k,n.
+  S n ≤ k → n ≤ S (2*FS_crd sig) →
+  ∀t.loopM ? (mk_binaryTM sig M) k
+    (mk_config ?? (〈q,bin3,ch,n〉) t) 
+  = loopM ? (mk_binaryTM sig M) (k - S n)
+      (mk_config ?? (〈q,bin0,None ?,FS_crd sig〉) 
+        (iter ? (λt0.tape_move ? t0 L) n t)). [2,3: /2 by lt_S_to_lt, le_to_lt_to_lt/]
+#sig #M #q #ch #k #n #Hk
+cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech elim n
+[ #Hcrd #t >loopM_unfold >loop_S_false [| % ] >binaryTM_bin3_O // 
+| #n0 #IH #Hlt #t >loopM_unfold >loop_S_false [|%] >binaryTM_bin3_S [|//]  
+  <IH [|/2 by lt_to_le/]
+  <loopM_unfold % ]
+qed.
+
+lemma binaryTM_bin4_None :
+  ∀sig,M,t,q,ch.
+  current ? t = None ? → 
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin4,ch,O〉) t) 
+  = mk_config ?? (〈q,bin2,ch,to_initN (FS_crd sig) ??〉) t. [2,3: @le_S //]
+#sig #M #t #q #ch #Hcur whd in ⊢ (??%?); >Hcur %
+qed.
+
+lemma binaryTM_phase4_write : ∀sig,M,q,ch,k,t.
+  O < k → current ? t = None ? →
+  loopM ? (mk_binaryTM sig M) k
+    (mk_config ?? (〈q,bin4,ch,O〉) t) 
+  = loopM ? (mk_binaryTM sig M) (k-1)
+      (mk_config ?? (〈q,bin2,ch,to_initN (FS_crd sig) ??〉) t). [2,3: @le_S //]
+#sig #M #q #ch #k #t #Hk #Hcur 
+cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
+>loopM_unfold >loop_S_false // <loopM_unfold >binaryTM_bin4_None //
+qed.
+
+(* we don't get here any more! *
+lemma binaryTM_bin4_noextend :
+  ∀sig,M,t,q,ch,cur,qn,mv.
+  current ? t = Some ? cur → 
+  〈qn,None ?,mv〉 = trans sig M 〈q,ch〉 → 
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin4,ch,O〉) t) 
+  = mk_config ?? (〈q,bin2,ch,to_initN O ??〉) t. [2,3://]
+#sig #M #t #q #ch #cur #qn #mv #Hcur #Htrans
+whd in ⊢ (??%?); >Hcur whd in ⊢ (??%?);
+whd in match (trans FinBool ??); <Htrans %
+qed.
+*)
+
+lemma binaryTM_bin4_extend :
+  ∀sig,M,t,q,ch,cur,qn,an,mv.
+  current ? t = Some ? cur → 
+  〈qn,Some ? an,mv〉 = trans sig M 〈q,ch〉 → 
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin4,ch,O〉) t) 
+  = mk_config ?? (〈q,bin5,ch,to_initN (FS_crd sig) ??〉) (tape_move ? t L). [2,3:@le_S //]
+#sig #M #t #q #ch #cur #qn #an #mv #Hcur #Htrans
+whd in ⊢ (??%?); >Hcur whd in ⊢ (??%?);
+whd in match (trans FinBool ??); <Htrans %
+qed.
+
+lemma binaryTM_phase4_extend : ∀sig,M,q,ch,k,t,cur,qn,an,mv.
+  O < k → current ? t = Some ? cur →
+  〈qn,Some ? an,mv〉 = trans sig M 〈q,ch〉 → 
+  loopM ? (mk_binaryTM sig M) k
+    (mk_config ?? (〈q,bin4,ch,O〉) t) 
+  = loopM ? (mk_binaryTM sig M) (k-1)
+      (mk_config ?? (〈q,bin5,ch,to_initN (FS_crd sig) ??〉) (tape_move ? t L)). [2,3: @le_S //]
+#sig #M #q #ch #k #t #cur #qn #an #mv #Hk #Hcur #Htrans 
+cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
+>loopM_unfold >loop_S_false // <loopM_unfold >binaryTM_bin4_extend //
+qed.
+
+lemma binaryTM_bin5_O :
+  ∀sig,M,t,q,ch.
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin5,ch,O〉) t) 
+  = mk_config ?? (〈q,bin2,ch,to_initN (FS_crd sig) ??〉) (tape_move ? t R). [2,3:@le_S //]
+#sig #M #t #q #ch %
+qed.
+
+lemma binaryTM_bin5_S :
+  ∀sig,M,t,q,ch,k. S k <S (2*FS_crd sig) → 
+  step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin5,ch,S k〉) t) 
+  = mk_config ?? (〈q,bin5,ch,to_initN k ??〉) (tape_move ? (tape_write ? t (Some ? false)) L). [2,3:@le_S /2 by lt_S_to_lt/]
+#sig #M #t #q #ch #k #HSk %
+qed.
+
+(* extends the tape towards the left with an unimportant sequence that will be
+   immediately overwritten *)
+lemma binaryTM_phase5 :∀sig,M,q,ch,k,n. S n ≤ k →   
+  ∀rs.n<S (2*FS_crd sig) →
+  ∃bs.|bs| = n ∧
+  loopM ? (mk_binaryTM sig M) k
+    (mk_config ?? (〈q,bin5,ch,n〉) (mk_tape ? [] (None ?) rs)) 
+  = loopM ? (mk_binaryTM sig M) (k - S n)
+      (mk_config ?? (〈q,bin2,ch,FS_crd sig〉) 
+        (mk_tape ? [] (option_hd ? (bs@rs)) (tail ? (bs@rs)))). [2,3:@le_S //]
+#sig #M #q #ch #k #n #Hk
+cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
+elim n
+[ #rs #Hlt %{[]} % // cases rs //
+| #n0 #IH #rs #Hn0 cases (IH (false::rs) ?) [|/2 by lt_S_to_lt/] 
+  #bs * #Hbs -IH #IH
+  %{(bs@[false])} % [ <Hbs >length_append /2 by increasing_to_injective/ ]
+  >loopM_unfold >loop_S_false // >binaryTM_bin5_S
+  >associative_append normalize in match ([false]@?); <IH
+  >loopM_unfold @eq_f @eq_f cases rs //
+]
+qed.
+
+lemma current_None_or_midtape : 
+  ∀sig,t.current sig t = None sig ∨ ∃ls,c,rs.t = midtape sig ls c rs.
+#sig * normalize /2/ #ls #c #rs %2 /4 by ex_intro/
+qed.
+
+lemma state_bin_lift_unfold :
+  ∀sig.∀M:TM sig.∀q:states sig M.
+  state_bin_lift sig M q = 〈q,bin0,None ?,FS_crd sig〉.// qed.
+  
+axiom current_tape_bin_list :
+ ∀sig,t.current sig t = None ? → current ? (tape_bin_lift sig t) = None ?.
+
+lemma tape_bin_lift_unfold :
+ ∀sig,t. tape_bin_lift sig t = 
+ mk_tape ? (rev_bin_list ? (left ? t)) (option_hd ? (opt_bin_char sig (current ? t)))
+   (tail ? (opt_bin_char sig (current ? t))@bin_list ? (right ? t)). //
+qed.
+
+lemma reverse_bin_char_list : ∀sig,c,l.
+  reverse ? (bin_char sig c)@rev_bin_list ? l = rev_bin_list ? (c::l). // qed.
+
+lemma left_midtape : ∀sig,ls,c,rs.left ? (midtape sig ls c rs) = ls.// qed.
+lemma current_midtape : ∀sig,ls,c,rs.current ? (midtape sig ls c rs) = Some ? c.// qed.
+lemma right_midtape : ∀sig,ls,c,rs.right ? (midtape sig ls c rs) = rs.// qed.
+lemma opt_bin_char_Some : ∀sig,c.opt_bin_char sig (Some ? c) = bin_char ? c.// qed.
+
+lemma opt_cons_hd_tl : ∀A,l.option_cons A (option_hd ? l) (tail ? l) = l.
+#A * // qed.
+
+lemma le_tech : ∀a,b,c.a ≤ b → a * c ≤ b * c.
+#a #b #c #H /2 by monotonic_le_times_r/
+qed.
+
+lemma iter_split : ∀T,f,m,n,x. 
+  iter T f (m+n) x = iter T f m (iter T f n x).
+#T #f #m #n elim n /2/
+#n0 #IH #x <plus_n_Sm whd in ⊢ (??%(????%)); >IH %
+qed.
+
+lemma iter_tape_move_R : ∀T,n,ls,cs,rs.|cs| = n → 
+  iter ? (λt0.tape_move T t0 R) n (mk_tape ? ls (option_hd ? (cs@rs)) (tail ? (cs@rs)))
+  = mk_tape ? (reverse ? cs@ls) (option_hd ? rs) (tail ? rs).
+#T #n elim n
+[ #ls * [| #c0 #cs0 #rs #H normalize in H; destruct (H) ] #rs #_ %
+| #n0 #IH #ls * [ #rs #H normalize in H; destruct (H) ] #c #cs #rs #Hlen
+  whd in ⊢ (??%?); 
+  >(?: (tape_move T (mk_tape T ls (option_hd T ((c::cs)@rs)) (tail T ((c::cs)@rs))) R)
+    = mk_tape ? (c::ls) (option_hd ? (cs@rs)) (tail ? (cs@rs))) in ⊢ (??(????%)?);
+  [| cases cs // cases rs // ] >IH
+  [ >reverse_cons >associative_append %
+  | normalize in Hlen; destruct (Hlen) % ]
+]
+qed.
+
+lemma tail_tech : ∀T,l1,l2.O < |l1| → tail T (l1@l2) = tail ? l1@l2.
+#T * normalize // #l2 #Hfalse @False_ind cases (not_le_Sn_O O) /2/
+qed.
+
+lemma hd_tech : ∀T,l1,l2.O < |l1| → option_hd T (l1@l2) = option_hd ? l1.
+#T * normalize // #l2 #Hfalse @False_ind cases (not_le_Sn_O O) /2/
+qed.
+
+lemma iter_tape_move_l_nil : ∀T,n,rs.
+  iter ? (λt0.tape_move T t0 L) n (mk_tape ? [ ] (None ?) rs) =
+  mk_tape ? [ ] (None ?) rs.
+#T #n #rs elim n // #n0 #IH <IH in ⊢ (???%); cases rs //
+qed.
+
+lemma iter_tape_move_L_left : ∀T,n,cs,rs. O < n →
+  iter ? (λt0.tape_move T t0 L) n 
+    (mk_tape ? [ ] (option_hd ? cs) (tail ? cs@rs)) =
+  mk_tape ? [ ] (None ?) (cs@rs).
+#T #n #cs #rs * 
+[ cases cs // cases rs //
+| #m #_ whd in ⊢ (??%?); <(iter_tape_move_l_nil ? m) cases cs // cases rs // ]
+qed.
+
+lemma iter_tape_move_L : ∀T,n,ls,cs,rs.|cs| = n → 
+  iter ? (λt0.tape_move T t0 L) n (mk_tape ? (reverse ? cs@ls) (option_hd ? rs) (tail ? rs))
+  = mk_tape ? ls (option_hd ? (cs@rs)) (tail ? (cs@rs)).
+#T #n elim n
+[ #ls * [| #c0 #cs0 #rs #H normalize in H; destruct (H) ] #rs #_ %
+| #n0 #IH #ls #cs #rs @(list_elim_left … cs)
+  [ #H normalize in H; destruct (H) ] -cs 
+  #c #cs #_ #Hlen >reverse_append whd in ⊢ (??%?); 
+  >(?: tape_move T (mk_tape T ((reverse T [c]@reverse T cs)@ls) (option_hd T rs) (tail T rs)) L
+    = mk_tape ? (reverse T cs@ls) (option_hd ? (c::rs)) (tail ? (c::rs))) in ⊢ (??(????%)?);
+  [| cases rs // ] >IH
+  [ >associative_append %
+  | >length_append in Hlen; normalize // ]
+]
+qed.
 
 lemma binaryTM_loop :
  ∀sig,M,i,t,q,tf,qf.
+ O < FS_crd sig → 
  loopM sig M i (mk_config ?? q t) = Some ? (mk_config ?? qf tf) →
  ∃k.loopM ? (mk_binaryTM sig M) k 
   (mk_config ?? (state_bin_lift ? M q) (tape_bin_lift ? t)) = 
   Some ? (mk_config ?? (state_bin_lift ? M qf) (tape_bin_lift ? tf)).
 #sig #M #i elim i
-[ #t #q #qf #tf change with (None ?) in ⊢ (??%?→?); #H destruct (H)
-| -i #i #IH #t #q #tf #qf
+[ #t #q #qf #tf #Hcrd change with (None ?) in ⊢ (??%?→?); #H destruct (H)
+| -i #i #IH #t #q #tf #qf #Hcrd >loopM_unfold
+  lapply (refl ? (halt sig M (cstate ?? (mk_config ?? q t))))
+  cases (halt ?? q) in ⊢ (???%→?); #Hhalt
+  [ >(loop_S_true ??? (λc.halt ?? (cstate ?? c)) (mk_config ?? q t) Hhalt)
+    #H destruct (H) %{1} >loopM_unfold >loop_S_true // ]
+  (* interesting case: more than one step *)  
+  >(loop_S_false ??? (λc.halt ?? (cstate ?? c)) (mk_config ?? q t) Hhalt)
+  <loopM_unfold >(config_expand ?? (step ???)) #Hloop 
+  lapply (IH … Hloop) [@Hcrd] -IH * #k0 #IH <config_expand in Hloop; #Hloop
+  %{(7*(S (FS_crd sig)) + k0)}
+  (*** PHASE 0 ***)
+  >state_bin_lift_unfold cases (current_None_or_midtape ? t)
+  (* 0.1) current = None *)
+  [ (* #Hcur >state_bin_lift_unfold in ⊢ (??%?); 
+    lapply (current_tape_bin_list … Hcur) #Hcur' 
+    >binaryTM_phase0_None /2 by monotonic_lt_plus_l/
+    >(?: FS_crd sig + k0 = S (FS_crd sig + k0 - 1)) [|@daemon]
+    >loopM_unfold >loop_S_false // lapply (refl ? t) cases t in ⊢ (???%→?);
+    [4: #ls #c #rs normalize in ⊢ (%→?); #H destruct (H) normalize in Hcur; destruct (Hcur)
+    | #Ht >Ht >binaryTM_bin4_None // <loopM_unfold *)
+  (* 0.2) midtape *)
+  | * #ls * #c * #rs #Ht >Ht >tape_bin_lift_unfold
+    >left_midtape >current_midtape >right_midtape >opt_bin_char_Some
+    >(binaryTM_phase0_midtape … Hhalt ? (refl ??)) [| // ]
+    >(?: 7*S (FS_crd sig) + k0 - S (FS_crd sig) = 6*S (FS_crd sig) + k0) [|/2 by plus_minus/]
+    (*** PHASE 1 ***)
+    >binaryTM_phase1
+    [| cases (bin_list ? rs) normalize // #r0 #rs0 #H destruct (H)
+    | >length_reverse @daemon
+    | // ]
+    >(?:6*S (FS_crd sig) + k0 - S (FS_crd sig) = 5*S (FS_crd sig) + k0) [|/2 by plus_minus/]
+    >reverse_reverse >opt_cons_hd_tl
+    cut (∃qn,chn,mv.〈qn,chn,mv〉 = trans ? M 〈q,Some ? c〉)
+    [ cases (trans ? M 〈q,Some ? c〉) * #qn #chn #mv /4 by ex_intro/ ]
+    * #qn * #chn * #mv cases chn -chn
+    [ (* no write! *) #Htrans >(binaryTM_phase2_None … Htrans) [2,3: //]
+      >iter_tape_move_R [|@daemon]
+      >(?:5*S (FS_crd sig) + k0 - S (FS_crd sig) = 4*S (FS_crd sig) + k0) [|/2 by plus_minus/]
+      >binaryTM_phase3 
+      [|//| cut (S (displ_of_move sig mv) ≤ 2*(S (FS_crd sig)))
+        [ /2 by le_S_S/ 
+        | #H @(transitive_le ??? H) -H -Hcrd @(transitive_le ? (4*S (FS_crd sig))) /2 by le_plus_a/ ]
+      ]
+      cut (∀sig,M,m,n,cfg,cfg'.m < n → loopM sig M m cfg = Some ? cfg' → loopM sig M n cfg = Some ? cfg') [@daemon]
+      #Hcut <(Hcut ??? (4*S (FS_crd sig) + k0 - S (displ_of_move sig mv)) ??? IH)
+      [| cases mv 
+        [ >(?:displ_of_move sig L = 2*FS_crd sig) //
+          >eq_minus_S_pred
+          @(transitive_le ? (pred (4*FS_crd sig+k0-2*FS_crd sig)))
+          [ >(?:4*FS_crd sig+k0-2*FS_crd sig = 2*FS_crd sig + k0) 
+            [ cases Hcrd /2 by le_minus_to_plus, le_n/
+            | <plus_minus [2://] 
+              >(commutative_times 4) >(commutative_times 2) 
+              <distributive_times_minus //
+            ]
+          | @monotonic_pred /2 by monotonic_le_minus_l/ ]
+        | whd in match (displ_of_move ??); @(transitive_le ? (4*1+k0-1)) 
+          [ //
+          | change with (pred (4*1+k0)) in ⊢ (?%?);
+             >eq_minus_S_pred <minus_n_O @monotonic_pred // ]
+        | >(?:displ_of_move sig N = FS_crd sig) //
+          >eq_minus_S_pred
+          @(transitive_le ? (pred (4*FS_crd sig+k0-1*FS_crd sig)))
+          [ >(?:4*FS_crd sig+k0-1*FS_crd sig = 3*FS_crd sig + k0) 
+            [ cases Hcrd /2 by le_minus_to_plus, le_n/
+            | <plus_minus [2://]
+              >(commutative_times 4) >(commutative_times 1) 
+              <distributive_times_minus //
+            ]
+          | @monotonic_pred /2 by transitive_le, le_n/ ] ] ] 
+      @eq_f @eq_f2 
+      [ <state_bin_lift_unfold >Ht whd in match (step ???); <Htrans %
+      | (* must distinguish mv *)
+        cases mv in Htrans; #Htrans
+        [ >(?:displ_of_move ? L = FS_crd sig + FS_crd sig) [| normalize // ]
+          >iter_split >iter_tape_move_L [|@daemon] >Ht cases ls
+          [ normalize in match (rev_bin_list ??); 
+            >hd_tech [|@daemon] >tail_tech [|@daemon]
+            >iter_tape_move_L_left // whd in match (step ???);
+            <Htrans whd in match (ctape ???);
+            >tape_bin_lift_unfold %
+          | #l0 #ls0 change with (reverse ? (bin_char ? l0)@rev_bin_list ? ls0) in match (rev_bin_list ??);
+            >iter_tape_move_L [|@daemon]
+            >hd_tech [|@daemon] >tail_tech [|@daemon]
+            whd in match (step ???); <Htrans whd in match (ctape ???);
+            >tape_bin_lift_unfold >left_midtape >current_midtape
+            >opt_bin_char_Some >right_midtape %
+          ]
+        | change with 
+            (mk_tape ? (reverse ? (bin_char ? c)@rev_bin_list ? ls)
+              (option_hd ? (bin_list ? rs)) (tail ? (bin_list ? rs)))
+          in ⊢ (??%?);
+          >reverse_bin_char_list <IH
+          >Ht >tape_bin_lift_unfold @eq_f3
+          whd in match (step ???); <Htrans cases rs //
+          #r0 #rs0 whd in match (ctape ???); >current_midtape >opt_bin_char_Some
+          [ <hd_tech in ⊢(???%); // @daemon 
+          | >right_midtape <tail_tech // @daemon ]
+        | whd in match (displ_of_move ? N); >iter_tape_move_L [|@daemon]
+          >Ht whd in match (step ???); <Htrans whd in match (ctape ???);
+          >tape_bin_lift_unfold >left_midtape >current_midtape >right_midtape
+          >opt_bin_char_Some lapply Hcrd >(?:FS_crd sig = |bin_char ? c|) [| @daemon ]
+          cases (bin_char ? c) // #H normalize in H; @False_ind
+          cases (not_le_Sn_O O) /2/
+        ]
+      ]
+         
+          
+         
 
 
 (*
@@ -165,4 +917,4 @@ theorem sem_binaryTM : ∀sig,M.
 #sig #M #t #i generalize in match t; -t
 @(nat_elim1 … i) #m #IH #intape #outc #Hloop
 
-*)
\ No newline at end of file
+*)  
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