# HG changeset patch
# User Christian Urban <christian dot urban at kcl dot ac dot uk>
# Date 1360124823 0
# Node ID e995ae949731bc05cfeb465ac039d4d36c0a17bb
# Parent  1e89c65f844b88e89bc7ec666088e2c3145a9716
updated

diff -r 1e89c65f844b -r e995ae949731 paper.pdf
Binary file paper.pdf has changed
diff -r 1e89c65f844b -r e995ae949731 thys/UTM.thy
--- a/thys/UTM.thy	Wed Feb 06 04:11:06 2013 +0000
+++ b/thys/UTM.thy	Wed Feb 06 04:27:03 2013 +0000
@@ -1,5 +1,5 @@
 theory UTM
-imports Main uncomputable recursive abacus UF GCD 
+imports Main recursive abacus UF GCD turing_hoare
 begin
 
 section {* Wang coding of input arguments *}
@@ -24,7 +24,7 @@
 \newcommand{\vsep}{5\baseheight}
 
 The TM used to generate the Wang's code of input arguments is divided into three TMs
- executed sequentially, namely $prepare$, $mainwork$ and $adjust$¡£According to the
+ executed sequentially, namely $prepare$, $mainwork$ and $adjust$\<exclamdown>\<pounds>According to the
  convention, start state of ever TM is fixed to state $1$ while the final state is
  fixed to $0$.
 
@@ -509,21 +509,27 @@
   where
   "fourtimes_ly = layout_of abc_fourtimes"
 
-definition t_twice :: "tprog"
+definition t_twice_compile :: "instr list"
+where
+  "t_twice_compile= (tm_of abc_twice @ (shift (mopup 1) (length (tm_of abc_twice) div 2)))"
+
+definition t_twice :: "instr list"
   where
-  "t_twice = change_termi_state (tm_of (abc_twice) @ (tMp 1 (start_of twice_ly (length abc_twice) - Suc 0)))"
-
-definition t_fourtimes :: "tprog"
+  "t_twice = adjust t_twice_compile"
+
+definition t_fourtimes_compile :: "instr list"
+where
+  "t_fourtimes_compile= (tm_of abc_fourtimes @ (shift (mopup 1) (length (tm_of abc_fourtimes) div 2)))"
+
+definition t_fourtimes :: "instr list"
   where
-  "t_fourtimes = change_termi_state (tm_of (abc_fourtimes) @ 
-             (tMp 1 (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)))"
-
+  "t_fourtimes = adjust t_fourtimes_compile"
 
 definition t_twice_len :: "nat"
   where
   "t_twice_len = length t_twice div 2"
 
-definition t_wcode_main_first_part:: "tprog"
+definition t_wcode_main_first_part:: "instr list"
   where
   "t_wcode_main_first_part \<equiv> 
                    [(L, 1), (L, 2), (L, 7), (R, 3),
@@ -533,12 +539,12 @@
                     (R, 10), (W0, 9), (R, 10), (R, 11), 
                     (W1, 12), (R, 11), (R, t_twice_len + 14), (L, 12)]"
 
-definition t_wcode_main :: "tprog"
+definition t_wcode_main :: "instr list"
   where
-  "t_wcode_main = (t_wcode_main_first_part @ tshift t_twice 12 @ [(L, 1), (L, 1)]
-                    @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])"
-
-fun bl_bin :: "block list \<Rightarrow> nat"
+  "t_wcode_main = (t_wcode_main_first_part @ shift t_twice 12 @ [(L, 1), (L, 1)]
+                    @ shift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])"
+
+fun bl_bin :: "cell list \<Rightarrow> nat"
   where
   "bl_bin [] = 0" 
 | "bl_bin (Bk # xs) = 2 * bl_bin xs"
@@ -546,29 +552,29 @@
 
 declare bl_bin.simps[simp del]
 
-type_synonym bin_inv_t = "block list \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+type_synonym bin_inv_t = "cell list \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 
 fun wcode_before_double :: "bin_inv_t"
   where
   "wcode_before_double ires rs (l, r) =
-     (\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> 
-               r = Oc\<^bsup>(Suc (Suc rs))\<^esup> @ Bk\<^bsup>rn \<^esup>)"
+     (\<exists> ln rn. l = Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and> 
+               r = Oc\<up>((Suc (Suc rs))) @ Bk\<up>(rn ))"
 
 declare wcode_before_double.simps[simp del]
 
 fun wcode_after_double :: "bin_inv_t"
   where
   "wcode_after_double ires rs (l, r) = 
-     (\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
-         r = Oc\<^bsup>Suc (Suc (Suc 2*rs))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+     (\<exists> ln rn. l = Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and>
+         r = Oc\<up>(Suc (Suc (Suc 2*rs))) @ Bk\<up>(rn))"
 
 declare wcode_after_double.simps[simp del]
 
 fun wcode_on_left_moving_1_B :: "bin_inv_t"
   where
   "wcode_on_left_moving_1_B ires rs (l, r) = 
-     (\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Oc # ires \<and> 
-               r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+     (\<exists> ml mr rn. l = Bk\<up>(ml) @ Oc # Oc # ires \<and> 
+               r = Bk\<up>(mr) @ Oc\<up>(Suc rs) @ Bk\<up>(rn) \<and>
                ml + mr > Suc 0 \<and> mr > 0)"
 
 declare wcode_on_left_moving_1_B.simps[simp del]
@@ -578,7 +584,7 @@
   "wcode_on_left_moving_1_O ires rs (l, r) = 
      (\<exists> ln rn.
                l = Oc # ires \<and> 
-               r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+               r = Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 
 declare wcode_on_left_moving_1_O.simps[simp del]
 
@@ -593,13 +599,13 @@
   where
    "wcode_on_checking_1 ires rs (l, r) = 
     (\<exists> ln rn. l = ires \<and>
-              r = Oc # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+              r = Oc # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 
 fun wcode_erase1 :: "bin_inv_t"
   where
 "wcode_erase1 ires rs (l, r) = 
        (\<exists> ln rn. l = Oc # ires \<and> 
-                 tl r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+                 tl r = Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 
 declare wcode_erase1.simps [simp del]
 
@@ -607,8 +613,8 @@
   where
   "wcode_on_right_moving_1 ires rs (l, r) = 
        (\<exists> ml mr rn.        
-             l = Bk\<^bsup>ml\<^esup> @ Oc # ires \<and> 
-             r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+             l = Bk\<up>(ml) @ Oc # ires \<and> 
+             r = Bk\<up>(mr) @ Oc\<up>(Suc rs) @ Bk\<up>(rn) \<and>
              ml + mr > Suc 0)"
 
 declare wcode_on_right_moving_1.simps [simp del] 
@@ -619,8 +625,8 @@
   where
   "wcode_goon_right_moving_1 ires rs (l, r) = 
       (\<exists> ml mr ln rn. 
-            l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> 
-            r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+            l = Oc\<up>(ml) @ Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and> 
+            r = Oc\<up>(mr) @ Bk\<up>(rn) \<and>
             ml + mr = Suc rs)"
 
 declare wcode_goon_right_moving_1.simps[simp del]
@@ -628,8 +634,8 @@
 fun wcode_backto_standard_pos_B :: "bin_inv_t"
   where
   "wcode_backto_standard_pos_B ires rs (l, r) = 
-          (\<exists> ln rn. l =  Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> 
-               r =  Bk # Oc\<^bsup>(Suc (Suc rs))\<^esup> @ Bk\<^bsup>rn \<^esup>)"
+          (\<exists> ln rn. l =  Bk # Bk\<up>(ln) @ Oc # ires \<and> 
+               r =  Bk # Oc\<up>((Suc (Suc rs))) @ Bk\<up>(rn ))"
 
 declare wcode_backto_standard_pos_B.simps[simp del]
 
@@ -637,8 +643,8 @@
   where
    "wcode_backto_standard_pos_O ires rs (l, r) = 
         (\<exists> ml mr ln rn. 
-            l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
-            r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+            l = Oc\<up>(ml) @ Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and>
+            r = Oc\<up>(mr) @ Bk\<up>(rn) \<and>
             ml + mr = Suc (Suc rs) \<and> mr > 0)"
 
 declare wcode_backto_standard_pos_O.simps[simp del]
@@ -651,13 +657,11 @@
 declare wcode_backto_standard_pos.simps[simp del]
 
 lemma [simp]: "<0::nat> = [Oc]"
-apply(simp add: tape_of_nat_abv exponent_def tape_of_nat_list.simps)
+apply(simp add: tape_of_nat_abv tape_of_nat_list.simps)
 done
 
 lemma tape_of_Suc_nat: "<Suc (a ::nat)> = replicate a Oc @ [Oc, Oc]"
-apply(simp add: tape_of_nat_abv exp_ind tape_of_nat_list.simps)
-apply(simp only: exp_ind_def[THEN sym])
-apply(simp only: exp_ind, simp, simp add: exponent_def)
+apply(simp only: tape_of_nat_abv exp_ind, simp)
 done
 
 lemma [simp]: "length (<a::nat>) = Suc a"
@@ -665,8 +669,8 @@
 done
 
 lemma [simp]: "<[a::nat]> = <a>"
-apply(simp add: tape_of_nat_abv tape_of_nl_abv exponent_def
-                tape_of_nat_list.simps)
+apply(simp add: tape_of_nat_abv tape_of_nl_abv
+  tape_of_nat_list.simps)
 done
 
 lemma bin_wc_eq: "bl_bin xs = bl2wc xs"
@@ -683,27 +687,30 @@
     done
 qed
 
-declare exp_def[simp del]
-
 lemma bl_bin_nat_Suc:  
   "bl_bin (<Suc a>) = bl_bin (<a>) + 2^(Suc a)"
-apply(simp add: tape_of_nat_abv bin_wc_eq)
-apply(simp add: bl2wc.simps)
-done
-lemma [simp]: " rev (a\<^bsup>aa\<^esup>) = a\<^bsup>aa\<^esup>"
-apply(simp add: exponent_def)
-done
- 
-declare tape_of_nl_abv_cons[simp del]
+apply(simp add: tape_of_nat_abv bl_bin.simps)
+apply(induct a, auto simp: bl_bin.simps)
+done
+
+lemma [simp]: " rev (a\<up>(aa)) = a\<up>(aa)"
+apply(simp)
+done
+
+lemma tape_of_nl_append_one: "lm \<noteq> [] \<Longrightarrow>  <lm @ [a]> = <lm> @ Bk # Oc\<up>Suc a"
+apply(induct lm, auto simp: tape_of_nl_cons split:if_splits)
+done
 
 lemma tape_of_nl_rev: "rev (<lm::nat list>) = (<rev lm>)"
-apply(induct lm rule: list_tl_induct, simp)
-apply(case_tac "list = []", simp add: tape_of_nl_abv tape_of_nat_list.simps)
-apply(simp add: tape_of_nat_list_butlast_last tape_of_nl_abv_cons)
-done
-lemma [simp]: "a\<^bsup>Suc 0\<^esup> = [a]" 
-by(simp add: exp_def)
-lemma tape_of_nl_cons_app1: "(<a # xs @ [b]>) = (Oc\<^bsup>Suc a\<^esup> @ Bk # (<xs@ [b]>))"
+apply(induct lm, simp, auto)
+apply(auto simp: tape_of_nl_cons tape_of_nl_append_one split: if_splits)
+apply(simp add: exp_ind[THEN sym])
+done
+
+lemma [simp]: "a\<up>(Suc 0) = [a]" 
+by(simp)
+
+lemma tape_of_nl_cons_app1: "(<a # xs @ [b]>) = (Oc\<up>(Suc a) @ Bk # (<xs@ [b]>))"
 apply(case_tac xs, simp add: tape_of_nl_abv tape_of_nat_list.simps)
 apply(simp add: tape_of_nl_abv  tape_of_nat_list.simps)
 done
@@ -716,26 +723,27 @@
 apply(simp add: bl2nat_cons_bk bl2wc.simps)
 done
 
-lemma tape_of_nat[simp]: "(<a::nat>) = Oc\<^bsup>Suc a\<^esup>"
+lemma tape_of_nat[simp]: "(<a::nat>) = Oc\<up>(Suc a)"
 apply(simp add: tape_of_nat_abv)
 done
-lemma tape_of_nl_cons_app2: "(<c # xs @ [b]>) = (<c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>)"
+
+lemma tape_of_nl_cons_app2: "(<c # xs @ [b]>) = (<c # xs> @ Bk # Oc\<up>(Suc b))"
 proof(induct "length xs" arbitrary: xs c,
   simp add: tape_of_nl_abv  tape_of_nat_list.simps)
   fix x xs c
   assume ind: "\<And>xs c. x = length xs \<Longrightarrow> <c # xs @ [b]> = 
-    <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>"
+    <c # xs> @ Bk # Oc\<up>(Suc b)"
     and h: "Suc x = length (xs::nat list)" 
-  show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>"
+  show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<up>(Suc b)"
   proof(case_tac xs, simp add: tape_of_nl_abv  tape_of_nat_list.simps)
     fix a list
     assume g: "xs = a # list"
-    hence k: "<a # list @ [b]> =  <a # list> @ Bk # Oc\<^bsup>Suc b\<^esup>"
+    hence k: "<a # list @ [b]> =  <a # list> @ Bk # Oc\<up>(Suc b)"
       apply(rule_tac ind)
       using h
       apply(simp)
       done
-    from g and k show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>"
+    from g and k show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<up>(Suc b)"
       apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
       done
   qed
@@ -745,21 +753,24 @@
 apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
 done
 
-lemma [simp]: "bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista) @ [Bk, Oc]) =
-              bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)) + 
-              2* 2^(length (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)))"
-using bl_bin_bk_oc[of "Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)"]
+lemma [simp]: "bl_bin (Oc\<up>(Suc aa) @ Bk # tape_of_nat_list (a # lista) @ [Bk, Oc]) =
+              bl_bin (Oc\<up>(Suc aa) @ Bk # tape_of_nat_list (a # lista)) + 
+              2* 2^(length (Oc\<up>(Suc aa) @ Bk # tape_of_nat_list (a # lista)))"
+using bl_bin_bk_oc[of "Oc\<up>(Suc aa) @ Bk # tape_of_nat_list (a # lista)"]
 apply(simp)
 done
 
+declare replicate_Suc[simp del]
+
 lemma [simp]: 
   "bl_bin (<aa # list>) + (4 * rs + 4) * 2 ^ (length (<aa # list>) - Suc 0)
-  = bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [0]>) + rs * (2 * 2 ^ (aa + length (<list @ [0]>)))"
-apply(case_tac "list", simp add: add_mult_distrib, simp)
+  = bl_bin (Oc\<up>(Suc aa) @ Bk # <list @ [0]>) + rs * (2 * 2 ^ (aa + length (<list @ [0]>)))"
+
+apply(case_tac "list", simp add: add_mult_distrib)
 apply(simp add: tape_of_nl_cons_app2 add_mult_distrib)
 apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
 done
-  
+
 lemma tape_of_nl_app_Suc: "((<list @ [Suc ab]>)) = (<list @ [ab]>) @ [Oc]"
 apply(induct list)
 apply(simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind)
@@ -767,17 +778,17 @@
 apply(simp_all add:tape_of_nl_abv tape_of_nat_list.simps exp_ind)
 done
 
-lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]> @ [Oc])
-              = bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>) +
-              2^(length (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>))"
+lemma [simp]: "bl_bin (Oc # Oc\<up>(aa) @ Bk # <list @ [ab]> @ [Oc])
+              = bl_bin (Oc # Oc\<up>(aa) @ Bk # <list @ [ab]>) +
+              2^(length (Oc # Oc\<up>(aa) @ Bk # <list @ [ab]>))"
 apply(simp add: bin_wc_eq)
 apply(simp add: bl2nat_cons_oc bl2wc.simps)
-using bl2nat_cons_oc[of "Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>"]
+using bl2nat_cons_oc[of "Oc # Oc\<up>(aa) @ Bk # <list @ [ab]>"]
 apply(simp)
 done
-lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>) + (4 * 2 ^ (aa + length (<list @ [ab]>)) +
+lemma [simp]: "bl_bin (Oc # Oc\<up>(aa) @ Bk # <list @ [ab]>) + (4 * 2 ^ (aa + length (<list @ [ab]>)) +
          4 * (rs * 2 ^ (aa + length (<list @ [ab]>)))) =
-       bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [Suc ab]>) +
+       bl_bin (Oc # Oc\<up>(aa) @ Bk # <list @ [Suc ab]>) +
          rs * (2 * 2 ^ (aa + length (<list @ [Suc ab]>)))"
 apply(simp add: tape_of_nl_app_Suc)
 done
@@ -798,12 +809,12 @@
 
 declare wcode_double_case_inv.simps[simp del]
 
-fun wcode_double_case_state :: "t_conf \<Rightarrow> nat"
+fun wcode_double_case_state :: "config \<Rightarrow> nat"
   where
   "wcode_double_case_state (st, l, r) = 
    13 - st"
 
-fun wcode_double_case_step :: "t_conf \<Rightarrow> nat"
+fun wcode_double_case_step :: "config \<Rightarrow> nat"
   where
   "wcode_double_case_step (st, l, r) = 
       (if st = Suc 0 then (length l)
@@ -815,13 +826,13 @@
       else if st = 6 then (length l)
       else 0)"
 
-fun wcode_double_case_measure :: "t_conf \<Rightarrow> nat \<times> nat"
+fun wcode_double_case_measure :: "config \<Rightarrow> nat \<times> nat"
   where
   "wcode_double_case_measure (st, l, r) = 
      (wcode_double_case_state (st, l, r), 
       wcode_double_case_step (st, l, r))"
 
-definition wcode_double_case_le :: "(t_conf \<times> t_conf) set"
+definition wcode_double_case_le :: "(config \<times> config) set"
   where "wcode_double_case_le \<equiv> (inv_image lex_pair wcode_double_case_measure)"
 
 lemma [intro]: "wf lex_pair"
@@ -857,42 +868,49 @@
 done
 
 lemma [simp]: "fetch t_wcode_main 4 Bk = (R, 4)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
-                fetch.simps nth_of.simps)
+apply(subgoal_tac "4 = Suc 3")
+apply(simp only: t_wcode_main_def t_wcode_main_first_part_def
+                fetch.simps nth_of.simps, auto)
 done
 
 lemma [simp]: "fetch t_wcode_main 4 Oc = (R, 5)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
-                fetch.simps nth_of.simps)
+apply(subgoal_tac "4 = Suc 3")
+apply(simp only: t_wcode_main_def t_wcode_main_first_part_def
+                fetch.simps nth_of.simps, auto)
 done
 
 lemma [simp]: "fetch t_wcode_main 5 Oc = (R, 5)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
-                fetch.simps nth_of.simps)
+apply(subgoal_tac "5 = Suc 4")
+apply(simp only: t_wcode_main_def t_wcode_main_first_part_def
+                fetch.simps nth_of.simps, auto)
 done
 
 lemma [simp]: "fetch t_wcode_main 5 Bk = (W1, 6)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
-                fetch.simps nth_of.simps)
+apply(subgoal_tac "5 = Suc 4")
+apply(simp only: t_wcode_main_def t_wcode_main_first_part_def
+                fetch.simps nth_of.simps, auto)
 done
 
 lemma [simp]: "fetch t_wcode_main 6 Bk = (R, 13)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
-                fetch.simps nth_of.simps)
-done
-
+apply(subgoal_tac "6 = Suc 5")
+apply(simp only: t_wcode_main_def t_wcode_main_first_part_def
+                fetch.simps nth_of.simps, auto)
+done
+  
 lemma [simp]: "fetch t_wcode_main 6 Oc = (L, 6)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
-                fetch.simps nth_of.simps)
-done
-lemma [elim]: "Bk\<^bsup>mr\<^esup> = [] \<Longrightarrow> mr = 0"
-apply(case_tac mr, auto simp: exponent_def)
+apply(subgoal_tac "6 = Suc 5")
+apply(simp only: t_wcode_main_def t_wcode_main_first_part_def
+                fetch.simps nth_of.simps, auto)
+done
+
+lemma [elim]: "Bk\<up>(mr) = [] \<Longrightarrow> mr = 0"
+apply(case_tac mr, auto)
 done
 
 lemma [simp]: "wcode_on_left_moving_1 ires rs (b, []) = False"
 apply(simp add: wcode_on_left_moving_1.simps wcode_on_left_moving_1_B.simps
-                wcode_on_left_moving_1_O.simps, auto)
-done
+                wcode_on_left_moving_1_O.simps) 
+done                                           
 
 
 declare wcode_on_checking_1.simps[simp del]
@@ -921,11 +939,12 @@
 apply(case_tac mr, simp, case_tac nat, simp, simp add: exp_ind)
 apply(rule_tac disjI1)
 apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = rn in exI, 
-      simp add: exp_ind_def)
+      simp, simp add: replicate_Suc)
 apply(erule_tac exE)+
 apply(simp)
 done
 
+declare replicate_Suc[simp]
 
 lemma [elim]: 
   "\<lbrakk>wcode_on_left_moving_1 ires rs (b, Oc # list); tl b = aa \<and> hd b # Oc # list = ba\<rbrakk> 
@@ -933,10 +952,8 @@
 apply(simp only: wcode_double_case_inv_simps)
 apply(erule_tac disjE)
 apply(erule_tac [!] exE)+
-apply(case_tac mr, simp, simp add: exp_ind_def)
-apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
-done
-
+apply(case_tac mr, simp, auto)
+done
 
 lemma [simp]: "wcode_on_checking_1 ires rs (b, []) = False" 
 apply(auto simp: wcode_double_case_inv_simps)
@@ -967,11 +984,11 @@
 done
 
 lemma [simp]: "wcode_on_right_moving_1 ires rs (b, []) = False"
-apply(simp add: wcode_double_case_inv_simps exp_ind_def)
+apply(simp add: wcode_double_case_inv_simps)
 done
 
 lemma [simp]: "wcode_on_right_moving_1 ires rs (b, []) = False"
-apply(simp add: wcode_double_case_inv_simps exp_ind_def)
+apply(simp add: wcode_double_case_inv_simps)
 done
 
 lemma [elim]: "\<lbrakk>wcode_on_right_moving_1 ires rs (b, Bk # ba);  Bk # b = aa \<and> list = b\<rbrakk> \<Longrightarrow> 
@@ -980,8 +997,8 @@
 apply(erule_tac exE)+
 apply(rule_tac x = "Suc ml" in exI, rule_tac x = "mr - Suc 0" in exI,
       rule_tac x = rn in exI)
-apply(simp add: exp_ind_def)
-apply(case_tac mr, simp, simp add: exp_ind_def)
+apply(simp)
+apply(case_tac mr, simp, simp)
 done
 
 lemma [elim]: 
@@ -991,14 +1008,13 @@
 apply(erule_tac exE)+
 apply(rule_tac x = "Suc 0" in exI, rule_tac x = "rs" in exI,
       rule_tac x = "ml - Suc (Suc 0)" in exI, rule_tac x = rn in exI)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all)
 apply(case_tac ml, simp, case_tac nat, simp, simp)
-apply(simp add: exp_ind_def)
 done
 
 lemma [simp]: 
   "wcode_on_right_moving_1 ires rs (b, []) \<Longrightarrow> False"
-apply(simp add: wcode_double_case_inv_simps exponent_def)
+apply(simp add: wcode_double_case_inv_simps)
 done
 
 lemma [elim]: "\<lbrakk>wcode_erase1 ires rs (b, Bk # ba); Bk # b = aa \<and> list = ba; c = Bk # ba\<rbrakk> 
@@ -1006,7 +1022,7 @@
 apply(simp only: wcode_double_case_inv_simps)
 apply(erule_tac exE)+
 apply(rule_tac x = "Suc 0" in exI, rule_tac x = "Suc (Suc ln)" in exI, 
-      rule_tac x = rn in exI, simp add: exp_ind)
+      rule_tac x = rn in exI, simp add: exp_ind del: replicate_Suc)
 done
 
 lemma [elim]: "\<lbrakk>wcode_erase1 ires rs (aa, Oc # list);  b = aa \<and> Bk # list = ba\<rbrakk> \<Longrightarrow> 
@@ -1024,7 +1040,6 @@
 apply(simp only:wcode_backto_standard_pos_O.simps)
 apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI, rule_tac x = ln in exI,
       rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp_all add: exponent_def)
 done
 
 lemma [elim]: 
@@ -1036,7 +1051,7 @@
 apply(simp only:wcode_backto_standard_pos_O.simps)
 apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI, rule_tac x = ln in exI,
       rule_tac x = "rn - Suc 0" in exI, simp)
-apply(case_tac mr, simp, case_tac rn, simp, simp_all add: exp_ind_def)
+apply(case_tac mr, simp, case_tac rn, simp, simp_all)
 done
 
 lemma [elim]: "\<lbrakk>wcode_goon_right_moving_1 ires rs (b, Oc # ba);  Oc # b = aa \<and> list = ba\<rbrakk> 
@@ -1045,14 +1060,13 @@
 apply(erule_tac exE)+
 apply(rule_tac x = "Suc ml" in exI, rule_tac x = "mr - Suc 0" in exI, 
       rule_tac x = ln in exI, rule_tac x = rn in exI)
-apply(simp add: exp_ind_def)
-apply(case_tac mr, simp, case_tac rn, simp_all add: exp_ind_def)
+apply(simp)
+apply(case_tac mr, simp, case_tac rn, simp_all)
 done
 
 lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, []);  Bk # b = aa\<rbrakk> \<Longrightarrow> False"
 apply(auto simp: wcode_double_case_inv_simps wcode_backto_standard_pos_O.simps
                  wcode_backto_standard_pos_B.simps)
-apply(case_tac mr, simp_all add: exp_ind_def)
 done
 
 lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, Bk # ba); Bk # b = aa \<and> list = ba\<rbrakk> 
@@ -1063,7 +1077,7 @@
 apply(erule_tac exE)+
 apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
 apply(auto)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
+apply(case_tac [!] mr, simp_all)
 done
 
 lemma [simp]: "wcode_backto_standard_pos ires rs ([], Oc # list) = False"
@@ -1074,7 +1088,6 @@
 lemma [simp]: "wcode_backto_standard_pos ires rs (b, []) = False"
 apply(auto simp: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps
                  wcode_backto_standard_pos_O.simps)
-apply(case_tac mr, simp, simp add: exp_ind_def)
 done
 
 lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, Oc # list); tl b = aa; hd b # Oc # list =  ba\<rbrakk>
@@ -1090,19 +1103,18 @@
 apply(rule_tac disjI2)
 apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = ln in exI, 
       rule_tac x = rn in exI, simp)
-apply(simp add: exp_ind_def)
-done
-
-declare new_tape.simps[simp del] nth_of.simps[simp del] fetch.simps[simp del]
+done
+
+declare nth_of.simps[simp del] fetch.simps[simp del]
 lemma wcode_double_case_first_correctness:
   "let P = (\<lambda> (st, l, r). st = 13) in 
        let Q = (\<lambda> (st, l, r). wcode_double_case_inv st ires rs (l, r)) in 
-       let f = (\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp) in
+       let f = (\<lambda> stp. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp) in
        \<exists> n .P (f n) \<and> Q (f (n::nat))"
 proof -
   let ?P = "(\<lambda> (st, l, r). st = 13)"
   let ?Q = "(\<lambda> (st, l, r). wcode_double_case_inv st ires rs (l, r))"
-  let ?f = "(\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)"
+  let ?f = "(\<lambda> stp. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp)"
   have "\<exists> n. ?P (?f n) \<and> ?Q (?f (n::nat))"
   proof(rule_tac halt_lemma2)
     show "wf wcode_double_case_le"
@@ -1110,16 +1122,16 @@
   next
     show "\<forall> na. \<not> ?P (?f na) \<and> ?Q (?f na) \<longrightarrow>
                    ?Q (?f (Suc na)) \<and> (?f (Suc na), ?f na) \<in> wcode_double_case_le"
-    proof(rule_tac allI, case_tac "?f na", simp add: tstep_red)
+    proof(rule_tac allI, case_tac "?f na", simp add: step_red)
       fix na a b c
       show "a \<noteq> 13 \<and> wcode_double_case_inv a ires rs (b, c) \<longrightarrow>
-               (case tstep (a, b, c) t_wcode_main of (st, x) \<Rightarrow> 
+               (case step0 (a, b, c) t_wcode_main of (st, x) \<Rightarrow> 
                    wcode_double_case_inv st ires rs x) \<and> 
-                (tstep (a, b, c) t_wcode_main, a, b, c) \<in> wcode_double_case_le"
+                (step0 (a, b, c) t_wcode_main, a, b, c) \<in> wcode_double_case_le"
         apply(rule_tac impI, simp add: wcode_double_case_inv.simps)
-        apply(auto split: if_splits simp: tstep.simps, 
-              case_tac [!] c, simp_all, case_tac [!] "(c::block list)!0")
-        apply(simp_all add: new_tape.simps wcode_double_case_inv.simps wcode_double_case_le_def
+        apply(auto split: if_splits simp: step.simps, 
+              case_tac [!] c, simp_all, case_tac [!] "(c::cell list)!0")
+        apply(simp_all add: wcode_double_case_inv.simps wcode_double_case_le_def
                                         lex_pair_def)
         apply(auto split: if_splits)
         done
@@ -1130,9 +1142,8 @@
                                   wcode_on_left_moving_1.simps
                                   wcode_on_left_moving_1_B.simps)
       apply(rule_tac disjI1)
-      apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def)
-      apply(rule_tac x = "Suc 0" in exI, simp add: exp_ind_def)
-      apply(auto)
+      apply(rule_tac x = "Suc m" in exI, simp)
+      apply(rule_tac x = "Suc 0" in exI, simp)
       done
   next
     show "\<not> ?P (?f 0)"
@@ -1141,101 +1152,39 @@
   qed
   thus "let P = \<lambda>(st, l, r). st = 13;
     Q = \<lambda>(st, l, r). wcode_double_case_inv st ires rs (l, r);
-    f = steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main
+    f = steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main
     in \<exists>n. P (f n) \<and> Q (f n)"
     apply(simp add: Let_def)
     done
 qed
     
-lemma [elim]: "t_ncorrect tp
-    \<Longrightarrow> t_ncorrect (tshift tp a)"
-apply(simp add: t_ncorrect.simps shift_length)
-done
-
-lemma tshift_fetch: "\<lbrakk> fetch tp a b = (aa, st'); 0 < st'\<rbrakk>
-       \<Longrightarrow> fetch (tshift tp (length tp1 div 2)) a b 
-          = (aa, st' + length tp1 div 2)"
-apply(subgoal_tac "a > 0")
-apply(auto simp: fetch.simps nth_of.simps shift_length nth_map
-                 tshift.simps split: block.splits if_splits)
-done
-
-lemma t_steps_steps_eq: "\<lbrakk>steps (st, l, r) tp stp = (st', l', r');
-         0 < st';  
-         0 < st \<and> st \<le> length tp div 2; 
-         t_ncorrect tp1;
-          t_ncorrect tp\<rbrakk>
-    \<Longrightarrow> t_steps (st + length tp1 div 2, l, r) (tshift tp (length tp1 div 2), 
-                                                      length tp1 div 2) stp
-       = (st' + length tp1 div 2, l', r')"
-apply(induct stp arbitrary: st' l' r', simp add: steps.simps t_steps.simps,
-      simp add: tstep_red stepn)
-apply(case_tac "(steps (st, l, r) tp stp)", simp)
+lemma tm_append_shift_append_steps: 
+"\<lbrakk>steps0 (st, l, r) tp stp = (st', l', r'); 
+  0 < st';
+  length tp1 mod 2 = 0
+  \<rbrakk>
+  \<Longrightarrow> steps0 (st + length tp1 div 2, l, r) (tp1 @ shift tp (length tp1 div 2) @ tp2) stp 
+  = (st' + length tp1 div 2, l', r')"
 proof -
-  fix stp st' l' r' a b c
-  assume ind: "\<And>st' l' r'.
-    \<lbrakk>a = st' \<and> b = l' \<and> c = r'; 0 < st'\<rbrakk>
-    \<Longrightarrow> t_steps (st + length tp1 div 2, l, r) 
-    (tshift tp (length tp1 div 2), length tp1 div 2) stp = 
-     (st' + length tp1 div 2, l', r')"
-  and h: "tstep (a, b, c) tp = (st', l', r')" "0 < st'" "t_ncorrect tp1"  "t_ncorrect tp"
-  have k: "t_steps (st + length tp1 div 2, l, r) (tshift tp (length tp1 div 2),
-         length tp1 div 2) stp = (a + length tp1 div 2, b, c)"
-    apply(rule_tac ind, simp)
+  assume h: 
+    "steps0 (st, l, r) tp stp = (st', l', r')"
+    "0 < st'"
+    "length tp1 mod 2 = 0 "
+  from h have 
+    "steps (st + length tp1 div 2, l, r) (tp1 @ shift tp (length tp1 div 2), 0) stp = 
+                            (st' + length tp1 div 2, l', r')"
+    by(rule_tac tm_append_second_steps_eq, simp_all)
+  then have "steps (st + length tp1 div 2, l, r) ((tp1 @ shift tp (length tp1 div 2)) @ tp2, 0) stp = 
+                            (st' + length tp1 div 2, l', r')"
     using h
-    apply(case_tac a, simp_all add: tstep.simps fetch.simps)
+    apply(rule_tac tm_append_first_steps_eq, simp_all)
     done
-  from h and this show "t_step (t_steps (st + length tp1 div 2, l, r) (tshift tp (length tp1 div 2), length tp1 div 2) stp)
-           (tshift tp (length tp1 div 2), length tp1 div 2) =
-          (st' + length tp1 div 2, l', r')"
-    apply(simp add: k)
-    apply(simp add: tstep.simps t_step.simps)
-    apply(case_tac "fetch tp a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
-    apply(subgoal_tac "fetch (tshift tp (length tp1 div 2)) a
-                       (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) = (aa, st' + length tp1 div 2)", simp)
-    apply(simp add: tshift_fetch)
-    done
+  thus "?thesis"
+    by simp
 qed 
 
-lemma t_tshift_lemma: "\<lbrakk> steps (st, l, r) tp stp = (st', l', r'); 
-                         st' \<noteq> 0; 
-                         stp > 0;
-                         0 < st \<and> st \<le> length tp div 2;
-                         t_ncorrect tp1;
-                         t_ncorrect tp;
-                         t_ncorrect tp2
-                         \<rbrakk>
-         \<Longrightarrow> \<exists> stp>0. steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2) stp 
-                  = (st' + length tp1 div 2, l', r')"
-proof -
-  assume h: "steps (st, l, r) tp stp = (st', l', r')"
-    "st' \<noteq> 0" "stp > 0"
-    "0 < st \<and> st \<le> length tp div 2"
-    "t_ncorrect tp1"
-    "t_ncorrect tp"
-    "t_ncorrect tp2"
-  from h have 
-    "\<exists>stp>0. t_steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2, 0) stp = 
-                            (st' + length tp1 div 2, l', r')"
-    apply(rule_tac stp = stp in turing_shift, simp_all add: shift_length)
-    apply(simp add: t_steps_steps_eq)
-    apply(simp add: t_ncorrect.simps shift_length)
-    done
-  thus "\<exists> stp>0. steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2) stp 
-                  = (st' + length tp1 div 2, l', r')"
-    apply(erule_tac exE)
-    apply(rule_tac x = stp in exI, simp)
-    apply(subgoal_tac "length (tp1 @ tshift tp (length tp1 div 2) @ tp2) mod 2 = 0")
-    apply(simp only: steps_eq)
-    using h
-    apply(auto simp: t_ncorrect.simps shift_length)
-    apply arith
-    done
-qed  
-  
-
 lemma t_twice_len_ge: "Suc 0 \<le> length t_twice div 2"
-apply(simp add: t_twice_def tMp.simps shift_length)
+apply(simp add: t_twice_def mopup.simps t_twice_compile_def)
 done
 
 lemma [intro]: "rec_calc_rel (recf.id (Suc 0) 0) [rs] rs"
@@ -1251,15 +1200,19 @@
 using prime_rel_exec_eq[of "rec_mult" "[rs, 2]"  "2*rs"]
 apply(subgoal_tac "primerec rec_mult (Suc (Suc 0))", auto)
 done
-lemma t_twice_correct: "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) 
-            (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) stp =
-       (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+declare start_of.simps[simp del]
+
+lemma t_twice_correct: 
+  "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) 
+  (tm_of abc_twice @ shift (mopup (Suc 0)) ((length (tm_of abc_twice) div 2))) stp =
+  (0, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
 proof(case_tac "rec_ci rec_twice")
   fix a b c
   assume h: "rec_ci rec_twice = (a, b, c)"
-  have "\<exists>stp m l. steps (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<^bsup>n\<^esup>) (tm_of abc_twice @ tMp (Suc 0) 
-    (start_of twice_ly (length abc_twice) - 1)) stp = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2*rs)\<^esup> @ Bk\<^bsup>l\<^esup>)"
-  proof(rule_tac t_compiled_by_rec)
+  have "\<exists>stp m l. steps0 (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<up>(n)) (tm_of abc_twice @ shift (mopup 1) 
+    (length (tm_of abc_twice) div 2)) stp = (0, Bk\<up>(m) @ Bk # Bk # ires, Oc\<up>(Suc (2*rs)) @ Bk\<up>(l))"
+  proof(rule_tac recursive_compile_to_tm_correct)
     show "rec_ci rec_twice = (a, b, c)" by (simp add: h)
   next
     show "rec_calc_rel rec_twice [rs] (2 * rs)"
@@ -1268,187 +1221,221 @@
       apply(rule_tac allI, case_tac k, auto)
       done
   next
-    show "length [rs] = Suc 0" by simp
-  next
-    show "layout_of (a [+] dummy_abc (Suc 0)) = layout_of (a [+] dummy_abc (Suc 0))"
-      by simp
+    show "length [rs] = 1" by simp
+  next	
+    show "layout_of (a [+] dummy_abc 1) = layout_of (a [+] dummy_abc 1)" by simp
   next
-    show "start_of twice_ly (length abc_twice) = 
-      start_of (layout_of (a [+] dummy_abc (Suc 0))) (length (a [+] dummy_abc (Suc 0)))"
-      using h
-      apply(simp add: twice_ly_def abc_twice_def)
-      done
-  next
-    show "tm_of abc_twice = tm_of (a [+] dummy_abc (Suc 0))"
+    show "tm_of abc_twice = tm_of (a [+] dummy_abc 1)"
       using h
       apply(simp add: abc_twice_def)
       done
   qed
-  thus "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) 
-            (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) stp =
-       (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  thus "?thesis"
     apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
     done
 qed
 
-lemma change_termi_state_fetch: "\<lbrakk>fetch ap a b = (aa, st); st > 0\<rbrakk>
-       \<Longrightarrow> fetch (change_termi_state ap) a b = (aa, st)"
-apply(case_tac b, auto simp: fetch.simps nth_of.simps change_termi_state.simps nth_map
-                       split: if_splits block.splits)
-done
-
-lemma change_termi_state_exec_in_range:
-     "\<lbrakk>steps (st, l, r) ap stp = (st', l', r'); st' \<noteq> 0\<rbrakk>
-    \<Longrightarrow> steps (st, l, r) (change_termi_state ap) stp = (st', l', r')"
-proof(induct stp arbitrary: st l r st' l' r', simp add: steps.simps)
-  fix stp st l r st' l' r'
-  assume ind: "\<And>st l r st' l' r'. 
-    \<lbrakk>steps (st, l, r) ap stp = (st', l', r'); st' \<noteq> 0\<rbrakk> \<Longrightarrow>
-    steps (st, l, r) (change_termi_state ap) stp = (st', l', r')"
-  and h: "steps (st, l, r) ap (Suc stp) = (st', l', r')" "st' \<noteq> 0"
-  from h show "steps (st, l, r) (change_termi_state ap) (Suc stp) = (st', l', r')"
-  proof(simp add: tstep_red, case_tac "steps (st, l, r) ap stp", simp)
-    fix a b c
-    assume g: "steps (st, l, r) ap stp = (a, b, c)"
-              "tstep (a, b, c) ap = (st', l', r')" "0 < st'"
-    hence "steps (st, l, r) (change_termi_state ap) stp = (a, b, c)"
-      apply(rule_tac ind, simp)
-      apply(case_tac a, simp_all add: tstep_0)
-      done
-    from g and this show "tstep (steps (st, l, r) (change_termi_state ap) stp)
-      (change_termi_state ap) = (st', l', r')"
-      apply(simp add: tstep.simps)
-      apply(case_tac "fetch ap a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
-      apply(subgoal_tac "fetch (change_termi_state ap) a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)
-                   = (aa, st')", simp)
-      apply(simp add: change_termi_state_fetch)
-      done
-  qed
+
+lemma adjust_fetch0: 
+  "\<lbrakk>0 < a; a \<le> length ap div 2;  fetch ap a b = (aa, 0)\<rbrakk>
+  \<Longrightarrow> fetch (adjust ap) a b = (aa, Suc (length ap div 2))"
+apply(case_tac b, auto simp: fetch.simps nth_of.simps nth_map
+                       split: if_splits)
+apply(case_tac [!] a, auto simp: fetch.simps nth_of.simps)
+done
+
+lemma adjust_fetch_norm: 
+  "\<lbrakk>st > 0;  st \<le> length tp div 2; fetch ap st b = (aa, ns); ns \<noteq> 0\<rbrakk>
+ \<Longrightarrow>  fetch (turing_basic.adjust ap) st b = (aa, ns)"
+ apply(case_tac b, auto simp: fetch.simps nth_of.simps nth_map
+                       split: if_splits)
+apply(case_tac [!] st, auto simp: fetch.simps nth_of.simps)
+done
+
+lemma adjust_step_eq: 
+  assumes exec: "step0 (st,l,r) ap = (st', l', r')"
+  and wf_tm: "tm_wf (ap, 0)"
+  and notfinal: "st' > 0"
+  shows "step0 (st, l, r) (adjust ap) = (st', l', r')"
+  using assms
+proof -
+  have "st > 0"
+    using assms
+    by(case_tac st, simp_all add: step.simps fetch.simps)
+  moreover hence "st \<le> (length ap) div 2"
+    using assms
+    apply(case_tac "st \<le> (length ap) div 2", simp)
+    apply(case_tac st, auto simp: step.simps fetch.simps)
+    apply(case_tac "read r", simp_all add: fetch.simps nth_of.simps)
+    done   
+  ultimately have "fetch (adjust ap) st (read r) = fetch ap st (read r)"
+    using assms
+    apply(case_tac "fetch ap st (read r)")
+    apply(drule_tac adjust_fetch_norm, simp_all)
+    apply(simp add: step.simps)
+    done
+  thus "?thesis"
+    using exec
+    by(simp add: step.simps)
 qed
 
-lemma change_termi_state_fetch0: 
-  "\<lbrakk>0 < a; a \<le> length ap div 2; t_correct ap; fetch ap a b = (aa, 0)\<rbrakk>
-  \<Longrightarrow> fetch (change_termi_state ap) a b = (aa, Suc (length ap div 2))"
-apply(case_tac b, auto simp: fetch.simps nth_of.simps change_termi_state.simps nth_map
-                       split: if_splits block.splits)
-done
-
-lemma turing_change_termi_state: 
-  "\<lbrakk>steps (Suc 0, l, r) ap stp = (0, l', r'); t_correct ap\<rbrakk>
-     \<Longrightarrow> \<exists> stp. steps (Suc 0, l, r) (change_termi_state ap) stp = 
+declare adjust.simps[simp del]
+
+lemma adjust_steps_eq: 
+  assumes exec: "steps0 (st,l,r) ap stp = (st', l', r')"
+  and wf_tm: "tm_wf (ap, 0)"
+  and notfinal: "st' > 0"
+  shows "steps0 (st, l, r) (adjust ap) stp = (st', l', r')"
+  using exec notfinal
+proof(induct stp arbitrary: st' l' r')
+  case 0
+  thus "?case"
+    by(simp add: steps.simps)
+next
+  case (Suc stp st' l' r')
+  have ind: "\<And>st' l' r'. \<lbrakk>steps0 (st, l, r) ap stp = (st', l', r'); 0 < st'\<rbrakk> 
+    \<Longrightarrow> steps0 (st, l, r) (turing_basic.adjust ap) stp = (st', l', r')" by fact
+  have h: "steps0 (st, l, r) ap (Suc stp) = (st', l', r')" by fact
+  have g:   "0 < st'" by fact
+  obtain st'' l'' r'' where a: "steps0 (st, l, r) ap stp = (st'', l'', r'')"
+    by (metis prod_cases3)
+  hence c:"0 < st''"
+    using h g
+    apply(simp add: step_red)
+    apply(case_tac st'', auto)
+    done
+  hence b: "steps0 (st, l, r) (turing_basic.adjust ap) stp = (st'', l'', r'')"
+    using a
+    by(rule_tac ind, simp_all)
+  thus "?case"
+    using assms a b h g
+    apply(simp add: step_red) 
+    apply(rule_tac adjust_step_eq, simp_all)
+    done
+qed 
+
+lemma adjust_halt_eq:
+  assumes exec: "steps0 (1, l, r) ap stp = (0, l', r')"
+  and tm_wf: "tm_wf (ap, 0)" 
+  shows "\<exists> stp. steps0 (Suc 0, l, r) (adjust ap) stp = 
         (Suc (length ap div 2), l', r')"
-apply(drule first_halt_point)
-apply(erule_tac exE)
-apply(rule_tac x = "Suc stp" in exI, simp add: tstep_red)
-apply(case_tac "steps (Suc 0, l, r) ap stp")
-apply(simp add: isS0_def change_termi_state_exec_in_range)
-apply(subgoal_tac "steps (Suc 0, l, r) (change_termi_state ap) stp = (a, b, c)", simp)
-apply(simp add: tstep.simps)
-apply(case_tac "fetch ap a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
-apply(subgoal_tac "fetch (change_termi_state ap) a 
-  (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) = (aa, Suc (length ap div 2))", simp)
-apply(rule_tac ap = ap in change_termi_state_fetch0, simp_all)
-apply(rule_tac tp = "(l, r)" and l = b and r = c  and stp = stp and A = ap in s_keep, simp_all)
-apply(simp add: change_termi_state_exec_in_range)
+proof -
+  have "\<exists> stp. \<not> is_final (steps0 (1, l, r) ap stp) \<and> (steps0 (1, l, r) ap (Suc stp) = (0, l', r'))"
+    thm before_final using exec
+    by(erule_tac before_final)
+  then obtain stpa where a: 
+    "\<not> is_final (steps0 (1, l, r) ap stpa) \<and> (steps0 (1, l, r) ap (Suc stpa) = (0, l', r'))" ..
+  obtain sa la ra where b:"steps0 (1, l, r) ap stpa = (sa, la, ra)"  by (metis prod_cases3)
+  hence c: "steps0 (Suc 0, l, r) (adjust ap) stpa = (sa, la, ra)"
+    using assms a
+    apply(rule_tac adjust_steps_eq, simp_all)
+    done
+  have d: "sa \<le> length ap div 2"
+    using steps_in_range[of  "(l, r)" ap stpa] a tm_wf b
+    by(simp)
+  obtain ac ns where e: "fetch ap sa (read ra) = (ac, ns)"
+    by (metis prod.exhaust)
+  hence f: "ns = 0"
+    using b a
+    apply(simp add: step_red step.simps)
+    done
+  have k: "fetch (adjust ap) sa (read ra) = (ac, Suc (length ap div 2))"
+    using a b c d e f
+    apply(rule_tac adjust_fetch0, simp_all)
+    done
+  from a b e f k and c show "?thesis"
+    apply(rule_tac x = "Suc stpa" in exI)
+    apply(simp add: step_red, auto)
+    apply(simp add: step.simps)
+    done
+qed    
+   
+declare tm_wf.simps[simp del]
+
+lemma [simp]: " tm_wf (t_twice_compile, 0)"
+apply(simp only: t_twice_compile_def)
+apply(rule_tac t_compiled_correct)
+apply(simp_all add: abc_twice_def)
 done
 
 lemma t_twice_change_term_state:
-  "\<exists> stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp
-     = (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-using t_twice_correct[of ires rs n]
-apply(erule_tac exE)
-apply(erule_tac exE)
-apply(erule_tac exE)
-proof(drule_tac turing_change_termi_state)
-  fix stp ln rn
-  show "t_correct (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0))"
-    apply(rule_tac t_compiled_correct, simp_all)
-    apply(simp add: twice_ly_def)
+  "\<exists> stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) t_twice stp
+     = (Suc t_twice_len, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
+proof -
+  have "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) 
+    (tm_of abc_twice @ shift (mopup (Suc 0)) ((length (tm_of abc_twice) div 2))) stp =
+    (0, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
+    by(rule_tac t_twice_correct)
+  then obtain stp ln rn where " steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) 
+    (tm_of abc_twice @ shift (mopup (Suc 0)) ((length (tm_of abc_twice) div 2))) stp =
+    (0, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))" by blast
+  hence "\<exists> stp. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
+    (adjust t_twice_compile) stp
+     = (Suc (length t_twice_compile div 2), Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
+    thm adjust_halt_eq
+    apply(rule_tac stp = stp in adjust_halt_eq)
+    apply(simp add: t_twice_compile_def, auto)
     done
-next
-  fix stp ln rn
-  show "\<exists>stp. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
-    (change_termi_state (tm_of abc_twice @ tMp (Suc 0) 
-    (start_of twice_ly (length abc_twice) - Suc 0))) stp =
-    (Suc (length (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) div 2),
-    Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) \<Longrightarrow>
-    \<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp = 
-    (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-    apply(erule_tac exE)
-    apply(simp add: t_twice_len_def t_twice_def)
-    apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
-    done
+  then obtain stpb where 
+    "steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
+    (adjust t_twice_compile) stpb
+     = (Suc (length t_twice_compile div 2), Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))" ..
+  thus "?thesis"
+    apply(simp add: t_twice_def t_twice_len_def)
+    by metis
 qed
 
+lemma [intro]: "length t_wcode_main_first_part mod 2 = 0"
+apply(auto simp: t_wcode_main_first_part_def)
+done
+
 lemma t_twice_append_pre:
-  "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp
-  = (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)
-   \<Longrightarrow> \<exists> stp>0. steps (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
-     (t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @
-      ([(L, 1), (L, 1)] @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp 
-    = (Suc (t_twice_len) + length t_wcode_main_first_part div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-proof(rule_tac t_tshift_lemma, simp_all add: t_twice_len_ge)
-  assume "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp = 
-    (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-  thus "0 < stp"
-    apply(case_tac stp, simp add: steps.simps t_twice_len_ge t_twice_len_def)
-    using t_twice_len_ge
-    apply(simp, simp)
-    done
-next
-  show "t_ncorrect t_wcode_main_first_part"
-    apply(simp add: t_ncorrect.simps t_wcode_main_first_part_def)
-    done
-next
-  show "t_ncorrect t_twice"
-    using length_tm_even[of abc_twice]
-    apply(auto simp: t_ncorrect.simps t_twice_def)
-    apply(arith)
-    done
-next
-  show "t_ncorrect ((L, Suc 0) # (L, Suc 0) #
-       tshift t_fourtimes (t_twice_len + 13) @ [(L, Suc 0), (L, Suc 0)])"
-    using length_tm_even[of abc_fourtimes]
-    apply(simp add: t_ncorrect.simps shift_length t_fourtimes_def)
-    apply arith
-    done
-qed
-  
+  "steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) t_twice stp
+  = (Suc t_twice_len, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))
+   \<Longrightarrow> steps0 (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
+     (t_wcode_main_first_part @ shift t_twice (length t_wcode_main_first_part div 2) @
+      ([(L, 1), (L, 1)] @ shift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp 
+    = (Suc (t_twice_len) + length t_wcode_main_first_part div 2, 
+             Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
+by(rule_tac tm_append_shift_append_steps, auto)
+
 lemma t_twice_append:
-  "\<exists> stp ln rn. steps (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
-     (t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @
-      ([(L, 1), (L, 1)] @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp 
-    = (Suc (t_twice_len) + length t_wcode_main_first_part div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  "\<exists> stp ln rn. steps0 (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
+     (t_wcode_main_first_part @ shift t_twice (length t_wcode_main_first_part div 2) @
+      ([(L, 1), (L, 1)] @ shift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp 
+    = (Suc (t_twice_len) + length t_wcode_main_first_part div 2, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
   using t_twice_change_term_state[of ires rs n]
   apply(erule_tac exE)
   apply(erule_tac exE)
   apply(erule_tac exE)
   apply(drule_tac t_twice_append_pre)
-  apply(erule_tac exE)
-  apply(rule_tac x = stpa in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
+  apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
   apply(simp)
   done
   
+lemma mopup_mod2: "length (mopup k) mod 2  = 0"
+apply(auto simp: mopup.simps)
+by arith
+
 lemma [simp]: "fetch t_wcode_main (Suc (t_twice_len + length t_wcode_main_first_part div 2)) Oc
      = (L, Suc 0)"
 apply(subgoal_tac "length (t_twice) mod 2 = 0")
 apply(simp add: t_wcode_main_def nth_append fetch.simps t_wcode_main_first_part_def 
-  nth_of.simps shift_length t_twice_len_def, auto)
-apply(simp add: t_twice_def)
-apply(subgoal_tac "length (tm_of abc_twice) mod 2 = 0")
-apply arith
-apply(rule_tac tm_even)
-done
+  nth_of.simps t_twice_len_def, auto)
+apply(simp add: t_twice_def t_twice_compile_def)
+using mopup_mod2[of 1]
+apply(simp)
+by arith
 
 lemma wcode_jump1: 
-  "\<exists> stp ln rn. steps (Suc (t_twice_len) + length t_wcode_main_first_part div 2,
-                       Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>n\<^esup>)
+  "\<exists> stp ln rn. steps0 (Suc (t_twice_len) + length t_wcode_main_first_part div 2,
+                       Bk\<up>(m) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(n))
      t_wcode_main stp 
-    = (Suc 0, Bk\<^bsup>ln\<^esup> @ Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+    = (Suc 0, Bk\<up>(ln) @ Bk # ires, Bk # Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
 apply(rule_tac x = "Suc 0" in exI, rule_tac x = "m" in exI, rule_tac x = n in exI)
-apply(simp add: steps.simps tstep.simps exp_ind_def new_tape.simps)
-apply(case_tac m, simp, simp add: exp_ind_def)
-apply(simp add: exp_ind_def[THEN sym] exp_ind[THEN sym])
+apply(simp add: steps.simps step.simps exp_ind)
+apply(case_tac m, simp_all)
+apply(simp add: exp_ind[THEN sym])
 done
 
 lemma wcode_main_first_part_len:
@@ -1457,27 +1444,27 @@
   done
 
 lemma wcode_double_case: 
-  shows "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-          (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  shows "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+          (Suc 0, Bk # Bk\<up>(ln) @ Oc # ires, Bk # Oc\<up>(Suc (2 * rs + 2)) @ Bk\<up>(rn))"
 proof -
-  have "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-          (13,  Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  have "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+          (13,  Bk # Bk # Bk\<up>(ln) @ Oc # ires, Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
     using wcode_double_case_first_correctness[of ires rs m n]
     apply(simp)
     apply(erule_tac exE)
-    apply(case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, 
-           Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na",
+    apply(case_tac "steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires, 
+           Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main na",
           auto simp: wcode_double_case_inv.simps
                      wcode_before_double.simps)
     apply(rule_tac x = na in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
     apply(simp)
     done    
   from this obtain stpa lna rna where stp1: 
-    "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa = 
-    (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
-  have "\<exists> stp ln rn. steps (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp =
-    (13 + t_twice_len, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-    using t_twice_append[of "Bk\<^bsup>lna\<^esup> @ Oc # ires" "Suc rs" rna]
+    "steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stpa = 
+    (13, Bk # Bk # Bk\<up>(lna) @ Oc # ires, Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rna))" by blast
+  have "\<exists> stp ln rn. steps0 (13, Bk # Bk # Bk\<up>(lna) @ Oc # ires, Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rna)) t_wcode_main stp =
+    (13 + t_twice_len, Bk # Bk # Bk\<up>(ln) @ Oc # ires, Oc\<up>(Suc (Suc (Suc (2 *rs)))) @ Bk\<up>(rn))"
+    using t_twice_append[of "Bk\<up>(lna) @ Oc # ires" "Suc rs" rna]
     apply(erule_tac exE)
     apply(erule_tac exE)
     apply(erule_tac exE)
@@ -1485,14 +1472,14 @@
     apply(rule_tac x = stp in exI, rule_tac x = "ln + lna" in exI, 
           rule_tac x = rn in exI)
     apply(simp add: t_wcode_main_def)
-    apply(simp add: exp_ind_def[THEN sym] exp_add[THEN sym])
+    apply(simp add: replicate_Suc[THEN sym] exp_add[THEN sym] del: replicate_Suc)
     done
   from this obtain stpb lnb rnb where stp2: 
-    "steps (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb =
-    (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>)" by blast
-  have "\<exists>stp ln rn. steps (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires,
-    Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stp = 
-       (Suc 0,  Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+    "steps0 (13, Bk # Bk # Bk\<up>(lna) @ Oc # ires, Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rna)) t_wcode_main stpb =
+    (13 + t_twice_len, Bk # Bk # Bk\<up>(lnb) @ Oc # ires, Oc\<up>(Suc (Suc (Suc (2 *rs)))) @ Bk\<up>(rnb))" by blast
+  have "\<exists>stp ln rn. steps0 (13 + t_twice_len, Bk # Bk # Bk\<up>(lnb) @ Oc # ires,
+    Oc\<up>(Suc (Suc (Suc (2 *rs)))) @ Bk\<up>(rnb)) t_wcode_main stp = 
+       (Suc 0,  Bk # Bk\<up>(ln) @ Oc # ires, Bk # Oc\<up>(Suc (Suc (Suc (2 *rs)))) @ Bk\<up>(rn))"
     using wcode_jump1[of lnb "Oc # ires" "Suc rs" rnb]
     apply(erule_tac exE)
     apply(erule_tac exE)
@@ -1500,15 +1487,15 @@
     apply(rule_tac x = stp in exI, 
           rule_tac x = ln in exI, 
           rule_tac x = rn in exI, simp add:wcode_main_first_part_len t_wcode_main_def)
-    apply(subgoal_tac "Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc # ires = Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires", simp)
-    apply(simp add: exp_ind_def[THEN sym] exp_ind[THEN sym])
+    apply(subgoal_tac "Bk\<up>(lnb) @ Bk # Bk # Oc # ires = Bk # Bk # Bk\<up>(lnb) @ Oc # ires", simp)
+    apply(simp add: replicate_Suc[THEN sym] exp_ind[THEN sym] del: replicate_Suc)
     apply(simp)
-    apply(case_tac lnb, simp, simp add: exp_ind_def[THEN sym] exp_ind)
+    apply(simp add: replicate_Suc[THEN sym] exp_ind del: replicate_Suc)
     done               
   from this obtain stpc lnc rnc where stp3: 
-    "steps (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires,
-    Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stpc = 
-       (Suc 0,  Bk # Bk\<^bsup>lnc\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnc\<^esup>)"
+    "steps0 (13 + t_twice_len, Bk # Bk # Bk\<up>(lnb) @ Oc # ires,
+    Oc\<up>(Suc (Suc (Suc (2 *rs)))) @ Bk\<up>(rnb)) t_wcode_main stpc = 
+       (Suc 0,  Bk # Bk\<up>(lnc) @ Oc # ires, Bk # Oc\<up>(Suc (Suc (Suc (2 *rs)))) @ Bk\<up>(rnc))"
     by blast
   from stp1 stp2 stp3 show "?thesis"
     apply(rule_tac x = "stpa + stpb + stpc" in exI, rule_tac x = lnc in exI,
@@ -1522,15 +1509,15 @@
 fun wcode_on_left_moving_2_B :: "bin_inv_t"
   where
   "wcode_on_left_moving_2_B ires rs (l, r) =
-     (\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Oc # ires \<and>
-                 r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> 
+     (\<exists> ml mr rn. l = Bk\<up>(ml) @ Oc # Bk # Oc # ires \<and>
+                 r = Bk\<up>(mr) @ Oc\<up>(Suc rs) @ Bk\<up>(rn) \<and> 
                  ml + mr > Suc 0 \<and> mr > 0)"
 
 fun wcode_on_left_moving_2_O :: "bin_inv_t"
   where
   "wcode_on_left_moving_2_O ires rs (l, r) =
      (\<exists> ln rn. l = Bk # Oc # ires \<and>
-               r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+               r = Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 
 fun wcode_on_left_moving_2 :: "bin_inv_t"
   where
@@ -1542,49 +1529,49 @@
   where
   "wcode_on_checking_2 ires rs (l, r) =
        (\<exists> ln rn. l = Oc#ires \<and> 
-                 r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+                 r = Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 
 fun wcode_goon_checking :: "bin_inv_t"
   where
   "wcode_goon_checking ires rs (l, r) =
        (\<exists> ln rn. l = ires \<and>
-                 r = Oc # Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+                 r = Oc # Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 
 fun wcode_right_move :: "bin_inv_t"
   where
   "wcode_right_move ires rs (l, r) = 
      (\<exists> ln rn. l = Oc # ires \<and>
-                 r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+                 r = Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 
 fun wcode_erase2 :: "bin_inv_t"
   where
   "wcode_erase2 ires rs (l, r) = 
         (\<exists> ln rn. l = Bk # Oc # ires \<and>
-                 tl r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+                 tl r = Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 
 fun wcode_on_right_moving_2 :: "bin_inv_t"
   where
   "wcode_on_right_moving_2 ires rs (l, r) = 
-        (\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # ires \<and> 
-                     r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr > Suc 0)"
+        (\<exists> ml mr rn. l = Bk\<up>(ml) @ Oc # ires \<and> 
+                     r = Bk\<up>(mr) @ Oc\<up>(Suc rs) @ Bk\<up>(rn) \<and> ml + mr > Suc 0)"
 
 fun wcode_goon_right_moving_2 :: "bin_inv_t"
   where
   "wcode_goon_right_moving_2 ires rs (l, r) = 
-        (\<exists> ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
-                        r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = Suc rs)"
+        (\<exists> ml mr ln rn. l = Oc\<up>(ml) @ Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and>
+                        r = Oc\<up>(mr) @ Bk\<up>(rn) \<and> ml + mr = Suc rs)"
 
 fun wcode_backto_standard_pos_2_B :: "bin_inv_t"
   where
   "wcode_backto_standard_pos_2_B ires rs (l, r) = 
-           (\<exists> ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> 
-                     r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+           (\<exists> ln rn. l = Bk # Bk\<up>(ln) @ Oc # ires \<and> 
+                     r = Bk # Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 
 fun wcode_backto_standard_pos_2_O :: "bin_inv_t"
   where
   "wcode_backto_standard_pos_2_O ires rs (l, r) = 
-          (\<exists> ml mr ln rn. l = Oc\<^bsup>ml \<^esup>@ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> 
-                          r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> 
+          (\<exists> ml mr ln rn. l = Oc\<up>(ml )@ Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and> 
+                          r = Oc\<up>(mr) @ Bk\<up>(rn) \<and> 
                           ml + mr = (Suc (Suc rs)) \<and> mr > 0)"
 
 fun wcode_backto_standard_pos_2 :: "bin_inv_t"
@@ -1596,8 +1583,8 @@
 fun wcode_before_fourtimes :: "bin_inv_t"
   where
   "wcode_before_fourtimes ires rs (l, r) = 
-          (\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> 
-                    r = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+          (\<exists> ln rn. l = Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and> 
+                    r = Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 
 declare wcode_on_left_moving_2_B.simps[simp del] wcode_on_left_moving_2.simps[simp del]
         wcode_on_left_moving_2_O.simps[simp del] wcode_on_checking_2.simps[simp del]
@@ -1632,11 +1619,11 @@
 
 declare wcode_fourtimes_case_inv.simps[simp del]
 
-fun wcode_fourtimes_case_state :: "t_conf \<Rightarrow> nat"
+fun wcode_fourtimes_case_state :: "config \<Rightarrow> nat"
   where
   "wcode_fourtimes_case_state (st, l, r) = 13 - st"
 
-fun wcode_fourtimes_case_step :: "t_conf \<Rightarrow> nat"
+fun wcode_fourtimes_case_step :: "config \<Rightarrow> nat"
   where
   "wcode_fourtimes_case_step (st, l, r) = 
          (if st = Suc 0 then length l
@@ -1648,13 +1635,13 @@
           else if st = 12 then length l
           else 0)"
 
-fun wcode_fourtimes_case_measure :: "t_conf \<Rightarrow> nat \<times> nat"
+fun wcode_fourtimes_case_measure :: "config \<Rightarrow> nat \<times> nat"
   where
   "wcode_fourtimes_case_measure (st, l, r) = 
      (wcode_fourtimes_case_state (st, l, r), 
       wcode_fourtimes_case_step (st, l, r))"
 
-definition wcode_fourtimes_case_le :: "(t_conf \<times> t_conf) set"
+definition wcode_fourtimes_case_le :: "(config \<times> config) set"
   where "wcode_fourtimes_case_le \<equiv> (inv_image lex_pair wcode_fourtimes_case_measure)"
 
 lemma wf_wcode_fourtimes_case_le[intro]: "wf wcode_fourtimes_case_le"
@@ -1666,55 +1653,75 @@
 done
 
 lemma [simp]: "fetch t_wcode_main 7 Oc = (R, 8)"
-apply(simp add: t_wcode_main_def fetch.simps 
+apply(subgoal_tac "7 = Suc 6")
+apply(simp only: t_wcode_main_def fetch.simps 
   t_wcode_main_first_part_def nth_of.simps)
+apply(auto)
 done
  
 lemma [simp]: "fetch t_wcode_main 8 Bk = (R, 9)"
-apply(simp add: t_wcode_main_def fetch.simps 
+apply(subgoal_tac "8 = Suc 7")
+apply(simp only: t_wcode_main_def fetch.simps 
   t_wcode_main_first_part_def nth_of.simps)
-done
+apply(auto)
+done
+
 
 lemma [simp]: "fetch t_wcode_main 9 Bk = (R, 10)"
-apply(simp add: t_wcode_main_def fetch.simps 
+apply(subgoal_tac "9 = Suc 8")
+apply(simp only: t_wcode_main_def fetch.simps 
   t_wcode_main_first_part_def nth_of.simps)
+apply(auto)
 done
 
 lemma [simp]: "fetch t_wcode_main 9 Oc = (W0, 9)"
-apply(simp add: t_wcode_main_def fetch.simps 
+apply(subgoal_tac "9 = Suc 8")
+apply(simp only: t_wcode_main_def fetch.simps 
   t_wcode_main_first_part_def nth_of.simps)
+apply(auto)
 done
 
 lemma [simp]: "fetch t_wcode_main 10 Bk = (R, 10)"
-apply(simp add: t_wcode_main_def fetch.simps 
+apply(subgoal_tac "10 = Suc 9")
+apply(simp only: t_wcode_main_def fetch.simps 
   t_wcode_main_first_part_def nth_of.simps)
+apply(auto)
 done
 
 lemma [simp]: "fetch t_wcode_main 10 Oc = (R, 11)"
-apply(simp add: t_wcode_main_def fetch.simps 
+apply(subgoal_tac "10 = Suc 9")
+apply(simp only: t_wcode_main_def fetch.simps 
   t_wcode_main_first_part_def nth_of.simps)
-done 
+apply(auto)
+done
 
 lemma [simp]: "fetch t_wcode_main 11 Bk = (W1, 12)"
-apply(simp add: t_wcode_main_def fetch.simps 
+apply(subgoal_tac "11 = Suc 10")
+apply(simp only: t_wcode_main_def fetch.simps 
   t_wcode_main_first_part_def nth_of.simps)
+apply(auto)
 done
 
 lemma [simp]: "fetch t_wcode_main 11 Oc = (R, 11)"
-apply(simp add: t_wcode_main_def fetch.simps 
+apply(subgoal_tac "11 = Suc 10")
+apply(simp only: t_wcode_main_def fetch.simps 
   t_wcode_main_first_part_def nth_of.simps)
-done 
+apply(auto)
+done
 
 lemma [simp]: "fetch t_wcode_main 12 Oc = (L, 12)"
-apply(simp add: t_wcode_main_def fetch.simps 
+apply(subgoal_tac "12 = Suc 11")
+apply(simp only: t_wcode_main_def fetch.simps 
   t_wcode_main_first_part_def nth_of.simps)
-done 
+apply(auto)
+done
 
 lemma [simp]: "fetch t_wcode_main 12 Bk = (R, t_twice_len + 14)"
-apply(simp add: t_wcode_main_def fetch.simps 
+apply(subgoal_tac "12 = Suc 11")
+apply(simp only: t_wcode_main_def fetch.simps 
   t_wcode_main_first_part_def nth_of.simps)
-done
-
+apply(auto)
+done
 
 lemma [simp]: "wcode_on_left_moving_2 ires rs (b, []) = False"
 apply(auto simp: wcode_fourtimes_invs)
@@ -1737,27 +1744,27 @@
 done
 
 lemma [simp]: "wcode_on_right_moving_2 ires rs (b, []) = False"
-apply(auto simp: wcode_fourtimes_invs exponent_def)
+apply(auto simp: wcode_fourtimes_invs)
 done
 
 lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, []) = False"
-apply(auto simp: wcode_fourtimes_invs exponent_def)
+apply(auto simp: wcode_fourtimes_invs)
 done
     
 lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
 apply(simp add: wcode_fourtimes_invs, auto)
-done
-        
+done     
+
 lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \<Longrightarrow>  wcode_on_left_moving_2 ires rs (tl b, hd b # Bk # list)"
 apply(simp only: wcode_fourtimes_invs)
 apply(erule_tac disjE)
 apply(erule_tac exE)+
 apply(case_tac ml, simp)
 apply(rule_tac x = "mr - (Suc (Suc 0))" in exI, rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp, case_tac nat, simp, simp add: exp_ind)
+apply(case_tac mr, simp, case_tac nat, simp, simp add: exp_ind del: replicate_Suc)
 apply(rule_tac disjI1)
 apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = rn in exI,
-      simp add: exp_ind_def)
+      simp add: replicate_Suc)
 apply(simp)
 done
 
@@ -1791,7 +1798,7 @@
 lemma [simp]: "wcode_erase2 ires rs (b, Bk # list) \<Longrightarrow> wcode_on_right_moving_2 ires rs (Bk # b, list)"
 apply(auto simp:wcode_fourtimes_invs )
 apply(rule_tac x = "Suc (Suc 0)" in exI, simp add: exp_ind)
-apply(rule_tac x =  "Suc (Suc ln)" in exI, simp add: exp_ind, auto)
+apply(rule_tac x =  "Suc (Suc ln)" in exI, simp add: exp_ind del: replicate_Suc)
 done
 
 lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
@@ -1801,8 +1808,8 @@
 lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list)
        \<Longrightarrow> wcode_on_right_moving_2 ires rs (Bk # b, list)"
 apply(auto simp: wcode_fourtimes_invs)
-apply(rule_tac x = "Suc ml" in exI, simp add: exp_ind_def)
-apply(rule_tac x = "mr - 1" in exI, case_tac mr, auto simp: exp_ind_def)
+apply(rule_tac x = "Suc ml" in exI, simp)
+apply(rule_tac x = "mr - 1" in exI, case_tac mr,auto)
 done
 
 lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
@@ -1814,9 +1821,9 @@
 apply(simp add: wcode_fourtimes_invs, auto)
 apply(rule_tac x = ml in exI, auto)
 apply(rule_tac x = "Suc 0" in exI, simp)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all)
 apply(rule_tac x = "rn - 1" in exI, simp)
-apply(case_tac rn, simp, simp add: exp_ind_def)
+apply(case_tac rn, simp, simp)
 done
    
 lemma  [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list) \<Longrightarrow>  b \<noteq> []"
@@ -1830,7 +1837,7 @@
 lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \<Longrightarrow> 
                      wcode_on_checking_2 ires rs (tl b, hd b # Oc # list)"
 apply(auto simp: wcode_fourtimes_invs)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
+apply(case_tac [!] mr, simp_all)
 done
 
 lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \<Longrightarrow> b \<noteq> []"
@@ -1844,13 +1851,12 @@
 apply(rule_tac disjI1)
 apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI, 
       rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp, simp add: exp_ind_def)
 done
 
 lemma "wcode_backto_standard_pos_2 ires rs (b, Bk # list)
-       \<Longrightarrow> (\<exists>ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \<and> (\<exists>rn. list = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+       \<Longrightarrow> (\<exists>ln. b = Bk # Bk\<up>(ln) @ Oc # ires) \<and> (\<exists>rn. list = Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 apply(auto simp: wcode_fourtimes_invs)
-apply(case_tac [!] mr, auto simp: exp_ind_def)
+apply(case_tac [!] mr, auto)
 done
 
 
@@ -1887,10 +1893,10 @@
 lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Oc # list)
        \<Longrightarrow> wcode_goon_right_moving_2 ires rs (Oc # b, list)"
 apply(auto simp: wcode_fourtimes_invs)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all)
 apply(rule_tac x = "Suc 0" in exI, auto)
 apply(rule_tac x = "ml - 2" in exI)
-apply(case_tac ml, simp, case_tac nat, simp_all add: exp_ind_def)
+apply(case_tac ml, simp, case_tac nat, simp_all)
 done
 
 lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
@@ -1898,9 +1904,9 @@
 done
 
 lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list)
-       \<Longrightarrow> (\<exists>ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \<and> (\<exists>rn. list = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+       \<Longrightarrow> (\<exists>ln. b = Bk # Bk\<up>(ln) @ Oc # ires) \<and> (\<exists>rn. list = Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 apply(simp add: wcode_fourtimes_invs, auto)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
+apply(case_tac [!] mr, simp_all)
 done
 
 lemma [simp]: "wcode_on_checking_2 ires rs (b, Oc # list) = False"
@@ -1910,9 +1916,9 @@
 lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow>
        wcode_goon_right_moving_2 ires rs (Oc # b, list)"
 apply(simp only:wcode_fourtimes_invs, auto)
-apply(rule_tac x = "Suc ml" in exI, auto simp: exp_ind_def)
+apply(rule_tac x = "Suc ml" in exI, simp)
 apply(rule_tac x = "mr - 1" in exI)
-apply(case_tac mr, case_tac rn, auto simp: exp_ind_def)
+apply(case_tac mr, case_tac rn, auto)
 done
 
 lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
@@ -1924,25 +1930,20 @@
 apply(simp only: wcode_fourtimes_invs)
 apply(erule_tac disjE)
 apply(erule_tac exE)+
-apply(case_tac ml, simp)
-apply(rule_tac disjI2)
-apply(rule_tac conjI, rule_tac x = ln in exI, simp)
-apply(rule_tac x = rn in exI, simp)
-apply(rule_tac disjI1)
-apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, 
-      rule_tac x = ln in exI, rule_tac x = rn in exI, simp add: exp_ind_def)
-apply(simp)
+apply(case_tac ml, auto)
+apply(rule_tac x = nat in exI, auto)
+apply(rule_tac x = "Suc mr" in exI, simp)
 done
 
 lemma wcode_fourtimes_case_first_correctness:
  shows "let P = (\<lambda> (st, l, r). st = t_twice_len + 14) in 
   let Q = (\<lambda> (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r)) in 
-  let f = (\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp) in
+  let f = (\<lambda> stp. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp) in
   \<exists> n .P (f n) \<and> Q (f (n::nat))"
 proof -
   let ?P = "(\<lambda> (st, l, r). st = t_twice_len + 14)"
   let ?Q = "(\<lambda> (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r))"
-  let ?f = "(\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)"
+  let ?f = "(\<lambda> stp. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp)"
   have "\<exists> n . ?P (?f n) \<and> ?Q (?f (n::nat))"
   proof(rule_tac halt_lemma2)
     show "wf wcode_fourtimes_case_le"
@@ -1951,19 +1952,21 @@
     show "\<forall> na. \<not> ?P (?f na) \<and> ?Q (?f na) \<longrightarrow>
                   ?Q (?f (Suc na)) \<and> (?f (Suc na), ?f na) \<in> wcode_fourtimes_case_le"
     apply(rule_tac allI,
-     case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na", simp,
+     case_tac "steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main na", simp,
      rule_tac impI)
-    apply(simp add: tstep_red tstep.simps, case_tac c, simp, case_tac [2] aa, simp_all)
-    
-    apply(simp_all add: wcode_fourtimes_case_inv.simps new_tape.simps 
+    apply(simp add: step_red step.simps, case_tac c, simp, case_tac [2] aa, simp_all)
+    apply(simp_all add: wcode_fourtimes_case_inv.simps
                         wcode_fourtimes_case_le_def lex_pair_def split: if_splits)
+    apply(auto simp: wcode_backto_standard_pos_2.simps wcode_backto_standard_pos_2_O.simps
+      wcode_backto_standard_pos_2_B.simps)
+    apply(case_tac mr, simp_all)
     done
   next
     show "?Q (?f 0)"
       apply(simp add: steps.simps wcode_fourtimes_case_inv.simps)
       apply(simp add: wcode_on_left_moving_2.simps wcode_on_left_moving_2_B.simps 
                       wcode_on_left_moving_2_O.simps)
-      apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def)
+      apply(rule_tac x = "Suc m" in exI, simp )
       apply(rule_tac x ="Suc 0" in exI, auto)
       done
   next
@@ -1981,196 +1984,199 @@
   "t_fourtimes_len = (length t_fourtimes div 2)"
 
 lemma t_fourtimes_len_gr:  "t_fourtimes_len > 0"
-apply(simp add: t_fourtimes_len_def t_fourtimes_def)
+apply(simp add: t_fourtimes_len_def t_fourtimes_def mopup.simps t_fourtimes_compile_def)
+done
+
+lemma [intro]: "rec_calc_rel (constn 4) [rs] 4"
+using prime_rel_exec_eq[of "constn 4" "[rs]" 4]
+apply(subgoal_tac "primerec (constn 4) 1", auto)
+done
+
+lemma [intro]: "rec_calc_rel rec_mult [rs, 4] (4 * rs)"
+using prime_rel_exec_eq[of "rec_mult" "[rs, 4]"  "4*rs"]
+apply(subgoal_tac "primerec rec_mult 2", auto simp: numeral_2_eq_2)
 done
 
 lemma t_fourtimes_correct: 
-  "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) 
-    (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) stp =
-       (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) 
+    (tm_of abc_fourtimes @ shift (mopup 1) (length (tm_of abc_fourtimes) div 2)) stp =
+       (0, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))"
 proof(case_tac "rec_ci rec_fourtimes")
   fix a b c
   assume h: "rec_ci rec_fourtimes = (a, b, c)"
-  have "\<exists>stp m l. steps (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<^bsup>n\<^esup>) (tm_of abc_fourtimes @ tMp (Suc 0) 
-    (start_of fourtimes_ly (length abc_fourtimes) - 1)) stp = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4*rs)\<^esup> @ Bk\<^bsup>l\<^esup>)"
-  proof(rule_tac t_compiled_by_rec)
+  have "\<exists>stp m l. steps0 (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<up>(n)) (tm_of abc_fourtimes @ shift (mopup 1) 
+    (length (tm_of abc_fourtimes) div 2)) stp = (0, Bk\<up>(m) @ Bk # Bk # ires, Oc\<up>(Suc (4*rs)) @ Bk\<up>(l))"
+  proof(rule_tac recursive_compile_to_tm_correct)
     show "rec_ci rec_fourtimes = (a, b, c)" by (simp add: h)
   next
     show "rec_calc_rel rec_fourtimes [rs] (4 * rs)"
-      using prime_rel_exec_eq [of rec_fourtimes "[rs]" "4 * rs"]
-      apply(subgoal_tac "primerec rec_fourtimes (length [rs])")
-      apply(simp_all add: rec_fourtimes_def rec_exec.simps)
-      apply(auto)
-      apply(simp only: Nat.One_nat_def[THEN sym], auto)
+      apply(simp add: rec_fourtimes_def)
+      apply(rule_tac rs =  "[rs, 4]" in calc_cn, simp_all)
+      apply(rule_tac allI, case_tac k, auto simp: mult_lemma)
       done
   next
-    show "length [rs] = Suc 0" by simp
-  next
-    show "layout_of (a [+] dummy_abc (Suc 0)) = layout_of (a [+] dummy_abc (Suc 0))"
-      by simp
+    show "length [rs] = 1" by simp
+  next	
+    show "layout_of (a [+] dummy_abc 1) = layout_of (a [+] dummy_abc 1)" by simp
   next
-    show "start_of fourtimes_ly (length abc_fourtimes) = 
-      start_of (layout_of (a [+] dummy_abc (Suc 0))) (length (a [+] dummy_abc (Suc 0)))"
-      using h
-      apply(simp add: fourtimes_ly_def abc_fourtimes_def)
-      done
-  next
-    show "tm_of abc_fourtimes = tm_of (a [+] dummy_abc (Suc 0))"
+    show "tm_of abc_fourtimes = tm_of (a [+] dummy_abc 1)"
       using h
       apply(simp add: abc_fourtimes_def)
       done
   qed
-  thus "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) 
-            (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) stp =
-       (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  thus "?thesis"
     apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
     done
 qed
 
+lemma wf_fourtimes[intro]: "tm_wf (t_fourtimes_compile, 0)"
+apply(simp only: t_fourtimes_compile_def)
+apply(rule_tac t_compiled_correct)
+apply(simp_all add: abc_twice_def)
+done
+
 lemma t_fourtimes_change_term_state:
-  "\<exists> stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp
-     = (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-using t_fourtimes_correct[of ires rs n]
-apply(erule_tac exE)
-apply(erule_tac exE)
-apply(erule_tac exE)
-proof(drule_tac turing_change_termi_state)
-  fix stp ln rn
-  show "t_correct (tm_of abc_fourtimes @ tMp (Suc 0) 
-    (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))"
-    apply(rule_tac t_compiled_correct, auto simp: fourtimes_ly_def)
+  "\<exists> stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) t_fourtimes stp
+     = (Suc t_fourtimes_len, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))"
+proof -
+  have "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) 
+    (tm_of abc_fourtimes @ shift (mopup 1) ((length (tm_of abc_fourtimes) div 2))) stp =
+    (0, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))"
+    by(rule_tac t_fourtimes_correct)
+  then obtain stp ln rn where 
+    "steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) 
+    (tm_of abc_fourtimes @ shift (mopup 1) ((length (tm_of abc_fourtimes) div 2))) stp =
+    (0, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))" by blast
+  hence "\<exists> stp. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
+    (adjust t_fourtimes_compile) stp
+     = (Suc (length t_fourtimes_compile div 2), Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))"
+    apply(rule_tac stp = stp in adjust_halt_eq)
+    apply(simp add: t_fourtimes_compile_def, auto)
     done
-next
-  fix stp ln rn
-  show "\<exists>stp. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
-    (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0) 
-        (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) stp =
-    (Suc (length (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly 
-    (length abc_fourtimes) - Suc 0)) div 2), Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) \<Longrightarrow>
-    \<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp =
-    (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-    apply(erule_tac exE)
-    apply(simp add: t_fourtimes_len_def t_fourtimes_def)
-    apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
-    done
+  then obtain stpb where 
+    "steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
+    (adjust t_fourtimes_compile) stpb
+     = (Suc (length t_fourtimes_compile div 2), Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))" ..
+  thus "?thesis"
+    apply(simp add: t_fourtimes_def t_fourtimes_len_def)
+    by metis
 qed
 
+lemma [intro]: "length t_twice mod 2 = 0"
+apply(auto simp: t_twice_def t_twice_compile_def)
+done
+
 lemma t_fourtimes_append_pre:
-  "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp
-  = (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)
-   \<Longrightarrow> \<exists> stp>0. steps (Suc 0 + length (t_wcode_main_first_part @ 
-              tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2,
-       Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+  "steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) t_fourtimes stp
+  = (Suc t_fourtimes_len, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))
+   \<Longrightarrow> steps0 (Suc 0 + length (t_wcode_main_first_part @ 
+              shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2,
+       Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
      ((t_wcode_main_first_part @ 
-  tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) @ 
-  tshift t_fourtimes (length (t_wcode_main_first_part @ 
-  tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2) @ ([(L, 1), (L, 1)])) stp 
-  = (Suc t_fourtimes_len + length (t_wcode_main_first_part @ 
-  tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2,
-  Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-proof(rule_tac t_tshift_lemma, auto)
-  assume "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp =
-    (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-  thus "0 < stp"
-    using t_fourtimes_len_gr
-    apply(case_tac stp, simp_all add: steps.simps)
-    done
-next
-  show "Suc 0 \<le> length t_fourtimes div 2"
-    apply(simp add: t_fourtimes_def shift_length tMp.simps)
-    done
-next
-  show "t_ncorrect (t_wcode_main_first_part @ 
-    tshift t_twice (length t_wcode_main_first_part div 2) @ 
-    [(L, Suc 0), (L, Suc 0)])"
-    apply(simp add: t_ncorrect.simps t_wcode_main_first_part_def shift_length
-                    t_twice_def)
-    using tm_even[of abc_twice]
-    by arith
-next
-  show "t_ncorrect t_fourtimes"
-    apply(simp add: t_fourtimes_def steps.simps t_ncorrect.simps)
-    using tm_even[of abc_fourtimes]
-    by arith
-next
-  show "t_ncorrect [(L, Suc 0), (L, Suc 0)]"
-    apply(simp add: t_ncorrect.simps)
-    done
-qed
+  shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) @ 
+  shift t_fourtimes (length (t_wcode_main_first_part @ 
+  shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2) @ ([(L, 1), (L, 1)])) stp 
+  = ((Suc t_fourtimes_len) + length (t_wcode_main_first_part @ 
+  shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2,
+  Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))"
+apply(rule_tac tm_append_shift_append_steps, simp_all)
+apply(auto simp: t_wcode_main_first_part_def)
+done
+
 
 lemma [simp]: "length t_wcode_main_first_part = 24"
 apply(simp add: t_wcode_main_first_part_def)
 done
 
 lemma [simp]: "(26 + length t_twice) div 2 = (length t_twice) div 2 + 13"
-using tm_even[of abc_twice]
-apply(simp add: t_twice_def)
-done
-
-lemma [simp]: "((26 + length (tshift t_twice 12)) div 2)
-             = (length (tshift t_twice 12) div 2 + 13)"
-using tm_even[of abc_twice]
+apply(simp add: t_twice_def t_twice_def)
+done
+
+lemma [simp]: "((26 + length (shift t_twice 12)) div 2)
+             = (length (shift t_twice 12) div 2 + 13)"
 apply(simp add: t_twice_def)
 done 
 
-lemma [simp]: "t_twice_len + 14 =  14 + length (tshift t_twice 12) div 2"
-using tm_even[of abc_twice]
-apply(simp add: t_twice_def t_twice_len_def shift_length)
+lemma [simp]: "t_twice_len + 14 =  14 + length (shift t_twice 12) div 2"
+apply(simp add: t_twice_def t_twice_len_def)
 done
 
 lemma t_fourtimes_append:
   "\<exists> stp ln rn. 
-  steps (Suc 0 + length (t_wcode_main_first_part @ tshift t_twice
+  steps0 (Suc 0 + length (t_wcode_main_first_part @ shift t_twice
   (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, 
-  Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
-  ((t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @
-  [(L, 1), (L, 1)]) @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)]) stp 
-  = (Suc t_fourtimes_len + length (t_wcode_main_first_part @ tshift t_twice
-  (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires,
-                                                                 Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
+  ((t_wcode_main_first_part @ shift t_twice (length t_wcode_main_first_part div 2) @
+  [(L, 1), (L, 1)]) @ shift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)]) stp 
+  = (Suc t_fourtimes_len + length (t_wcode_main_first_part @ shift t_twice
+  (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, Bk\<up>(ln) @ Bk # Bk # ires,
+                                                                 Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))"
   using t_fourtimes_change_term_state[of ires rs n]
   apply(erule_tac exE)
   apply(erule_tac exE)
   apply(erule_tac exE)
   apply(drule_tac t_fourtimes_append_pre)
-  apply(erule_tac exE)
-  apply(rule_tac x = stpa in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
-  apply(simp add: t_twice_len_def shift_length)
+  apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
+  apply(simp add: t_twice_len_def)
   done
 
 lemma t_wcode_main_len: "length t_wcode_main = length t_twice + length t_fourtimes + 28"
-apply(simp add: t_wcode_main_def shift_length)
-done
- 
+apply(simp add: t_wcode_main_def)
+done
+
+lemma even_twice_len: "length t_twice mod 2 = 0"
+apply(auto simp: t_twice_def t_twice_compile_def)
+done
+
+lemma even_fourtimes_len: "length t_fourtimes mod 2 = 0"
+apply(auto simp: t_fourtimes_def t_fourtimes_compile_def)
+done
+
+lemma [simp]: "2 * (length t_twice div 2) = length t_twice"
+using even_twice_len
+by arith
+
+lemma [simp]: "2 * (length t_fourtimes div 2) = length t_fourtimes"
+using even_fourtimes_len
+by arith
+
+lemma [simp]: "fetch t_wcode_main (14 + length t_twice div 2 + t_fourtimes_len) Oc
+             = (L, Suc 0)" 
+apply(subgoal_tac "14 = Suc 13")
+apply(simp only: fetch.simps add_Suc nth_of.simps t_wcode_main_def)
+apply(simp add:length_append length_shift Parity.two_times_even_div_two even_twice_len t_fourtimes_len_def)
+by arith
+
+lemma [simp]: "fetch t_wcode_main (14 + length t_twice div 2 + t_fourtimes_len) Bk
+             = (L, Suc 0)"
+apply(subgoal_tac "14 = Suc 13")
+apply(simp only: fetch.simps add_Suc nth_of.simps t_wcode_main_def)
+apply(simp add:length_append length_shift Parity.two_times_even_div_two even_twice_len t_fourtimes_len_def nth_append)
+by arith
+
 lemma [simp]: "fetch t_wcode_main (14 + length t_twice div 2 + t_fourtimes_len) b
              = (L, Suc 0)"
-using tm_even[of "abc_twice"] tm_even[of "abc_fourtimes"]
-apply(case_tac b)
-apply(simp_all only: fetch.simps)
-apply(auto simp: nth_of.simps t_wcode_main_len t_twice_len_def
-                 t_fourtimes_def t_twice_def t_fourtimes_def t_fourtimes_len_def)
-apply(auto simp: t_wcode_main_def t_wcode_main_first_part_def shift_length t_twice_def nth_append 
-                    t_fourtimes_def)
+apply(case_tac b, simp_all)
 done
 
 lemma wcode_jump2: 
-  "\<exists> stp ln rn. steps (t_twice_len + 14 + t_fourtimes_len
-  , Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4 * rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stp =
-  (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4 * rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  "\<exists> stp ln rn. steps0 (t_twice_len + 14 + t_fourtimes_len
+  , Bk # Bk # Bk\<up>(lnb) @ Oc # ires, Oc\<up>(Suc (4 * rs + 4)) @ Bk\<up>(rnb)) t_wcode_main stp =
+  (Suc 0, Bk # Bk\<up>(ln) @ Oc # ires, Bk # Oc\<up>(Suc (4 * rs + 4)) @ Bk\<up>(rn))"
 apply(rule_tac x = "Suc 0" in exI)
-apply(simp add: steps.simps shift_length)
+apply(simp add: steps.simps)
 apply(rule_tac x = lnb in exI, rule_tac x = rnb in exI)
-apply(simp add: tstep.simps new_tape.simps)
+apply(simp add: step.simps)
 done
 
 lemma wcode_fourtimes_case:
   shows "\<exists>stp ln rn.
-  steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-  (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+  (Suc 0, Bk # Bk\<up>(ln) @ Oc # ires, Bk # Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rn))"
 proof -
   have "\<exists>stp ln rn.
-  steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-  (t_twice_len + 14, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+  (t_twice_len + 14, Bk # Bk # Bk\<up>(ln) @ Oc # ires, Oc\<up>(Suc (rs + 1)) @ Bk\<up>(rn))"
     using wcode_fourtimes_case_first_correctness[of ires rs m n]
     apply(simp add: wcode_fourtimes_case_inv.simps, auto)
     apply(rule_tac x = na in exI, rule_tac x = ln in exI,
@@ -2178,12 +2184,12 @@
     apply(simp)
     done
   from this obtain stpa lna rna where stp1:
-    "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
-  (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
-  have "\<exists>stp ln rn. steps (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)
+    "steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stpa =
+  (t_twice_len + 14, Bk # Bk # Bk\<up>(lna) @ Oc # ires, Oc\<up>(Suc (rs + 1)) @ Bk\<up>(rna))" by blast
+  have "\<exists>stp ln rn. steps0 (t_twice_len + 14, Bk # Bk # Bk\<up>(lna) @ Oc # ires, Oc\<up>(Suc (rs + 1)) @ Bk\<up>(rna))
                      t_wcode_main stp =
-          (t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires,  Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-    using t_fourtimes_append[of " Bk\<^bsup>lna\<^esup> @ Oc # ires" "rs + 1" rna]
+          (t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<up>(ln) @ Oc # ires,  Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rn))"
+    using t_fourtimes_append[of " Bk\<up>(lna) @ Oc # ires" "rs + 1" rna]
     apply(erule_tac exE)
     apply(erule_tac exE)
     apply(erule_tac exE)
@@ -2191,24 +2197,24 @@
     apply(rule_tac x = stp in exI, 
           rule_tac x = "ln + lna" in exI,
           rule_tac x = rn in exI, simp)
-    apply(simp add: exp_ind_def[THEN sym] exp_add[THEN sym])
+    apply(simp add: replicate_Suc[THEN sym] exp_add[THEN sym] del: replicate_Suc)
     done
   from this obtain stpb lnb rnb where stp2:
-    "steps (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)
+    "steps0 (t_twice_len + 14, Bk # Bk # Bk\<up>(lna) @ Oc # ires, Oc\<up>(Suc (rs + 1)) @ Bk\<up>(rna))
                      t_wcode_main stpb =
-       (t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires,  Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>)"
+       (t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<up>(lnb) @ Oc # ires,  Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rnb))"
     by blast
-  have "\<exists>stp ln rn. steps (t_twice_len + 14 + t_fourtimes_len,
-    Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires,  Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>)
+  have "\<exists>stp ln rn. steps0 (t_twice_len + 14 + t_fourtimes_len,
+    Bk # Bk # Bk\<up>(lnb) @ Oc # ires,  Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rnb))
     t_wcode_main stp =
-    (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+    (Suc 0, Bk # Bk\<up>(ln) @ Oc # ires, Bk # Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rn))"
     apply(rule wcode_jump2)
     done
   from this obtain stpc lnc rnc where stp3: 
-    "steps (t_twice_len + 14 + t_fourtimes_len,
-    Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires,  Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>)
+    "steps0 (t_twice_len + 14 + t_fourtimes_len,
+    Bk # Bk # Bk\<up>(lnb) @ Oc # ires,  Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rnb))
     t_wcode_main stpc =
-    (Suc 0, Bk # Bk\<^bsup>lnc\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnc\<^esup>)"
+    (Suc 0, Bk # Bk\<up>(lnc) @ Oc # ires, Bk # Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rnc))"
     by blast
   from stp1 stp2 stp3 show "?thesis"
     apply(rule_tac x = "stpa + stpb + stpc" in exI,
@@ -2222,15 +2228,15 @@
 fun wcode_on_left_moving_3_B :: "bin_inv_t"
   where
   "wcode_on_left_moving_3_B ires rs (l, r) = 
-       (\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Bk # ires \<and>
-                    r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> 
+       (\<exists> ml mr rn. l = Bk\<up>(ml) @ Oc # Bk # Bk # ires \<and>
+                    r = Bk\<up>(mr) @ Oc\<up>(Suc rs) @ Bk\<up>(rn) \<and> 
                     ml + mr > Suc 0 \<and> mr > 0 )"
 
 fun wcode_on_left_moving_3_O :: "bin_inv_t"
   where
   "wcode_on_left_moving_3_O ires rs (l, r) = 
          (\<exists> ln rn. l = Bk # Bk # ires \<and>
-                   r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+                   r = Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 
 fun wcode_on_left_moving_3 :: "bin_inv_t"
   where
@@ -2242,19 +2248,19 @@
   where
   "wcode_on_checking_3 ires rs (l, r) = 
          (\<exists> ln rn. l = Bk # ires \<and>
-             r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+             r = Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 
 fun wcode_goon_checking_3 :: "bin_inv_t"
   where
   "wcode_goon_checking_3 ires rs (l, r) = 
          (\<exists> ln rn. l = ires \<and>
-             r = Bk # Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+             r = Bk # Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 
 fun wcode_stop :: "bin_inv_t"
   where
   "wcode_stop ires rs (l, r) = 
           (\<exists> ln rn. l = Bk # ires \<and>
-             r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+             r = Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 
 fun wcode_halt_case_inv :: "nat \<Rightarrow> bin_inv_t"
   where
@@ -2265,7 +2271,7 @@
            else if st = 7 then wcode_goon_checking_3 ires rs (l, r)
            else False)"
 
-fun wcode_halt_case_state :: "t_conf \<Rightarrow> nat"
+fun wcode_halt_case_state :: "config \<Rightarrow> nat"
   where
   "wcode_halt_case_state (st, l, r) = 
            (if st = 1 then 5
@@ -2273,19 +2279,19 @@
             else if st = 7 then 3
             else 0)"
 
-fun wcode_halt_case_step :: "t_conf \<Rightarrow> nat"
+fun wcode_halt_case_step :: "config \<Rightarrow> nat"
   where
   "wcode_halt_case_step (st, l, r) = 
          (if st = 1 then length l
          else 0)"
 
-fun wcode_halt_case_measure :: "t_conf \<Rightarrow> nat \<times> nat"
+fun wcode_halt_case_measure :: "config \<Rightarrow> nat \<times> nat"
   where
   "wcode_halt_case_measure (st, l, r) = 
      (wcode_halt_case_state (st, l, r), 
       wcode_halt_case_step (st, l, r))"
 
-definition wcode_halt_case_le :: "(t_conf \<times> t_conf) set"
+definition wcode_halt_case_le :: "(config \<times> config) set"
   where "wcode_halt_case_le \<equiv> (inv_image lex_pair wcode_halt_case_measure)"
 
 lemma wf_wcode_halt_case_le[intro]: "wf wcode_halt_case_le"
@@ -2301,13 +2307,15 @@
   wcode_on_left_moving_3.simps wcode_stop.simps
 
 lemma [simp]: "fetch t_wcode_main 7 Bk = (R, 0)"
-apply(simp add: fetch.simps t_wcode_main_def nth_append nth_of.simps
+apply(subgoal_tac "7 = Suc 6")
+apply(simp only: fetch.simps t_wcode_main_def nth_append nth_of.simps
                 t_wcode_main_first_part_def)
+apply(auto)
 done
 
 lemma [simp]: "wcode_on_left_moving_3 ires rs (b, [])  = False"
 apply(simp only: wcode_halt_invs)
-apply(simp add: exp_ind_def)
+apply(simp)
 done    
 
 lemma [simp]: "wcode_on_checking_3 ires rs (b, []) = False"
@@ -2325,10 +2333,11 @@
 apply(erule_tac exE)+
 apply(case_tac ml, simp)
 apply(rule_tac x = "mr - 2" in exI, rule_tac x = rn in exI)
-apply(case_tac mr, simp, simp add: exp_ind, simp add: exp_ind[THEN sym])
+apply(case_tac mr, simp, simp add: exp_ind del: replicate_Suc)
+apply(case_tac nat, simp, simp add: exp_ind del: replicate_Suc)
 apply(rule_tac disjI1)
 apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, 
-      rule_tac x = rn in exI, simp add: exp_ind_def)
+      rule_tac x = rn in exI, simp)
 apply(simp)
 done
 
@@ -2345,7 +2354,7 @@
 lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Oc # list) \<Longrightarrow> 
                wcode_on_checking_3 ires rs (tl b, hd b # Oc # list)"
 apply(simp add:wcode_halt_invs, auto)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
+apply(case_tac [!] mr, simp_all)
 done     
 
 lemma [simp]: "wcode_on_checking_3 ires rs (b, Oc # list) = False"
@@ -2356,7 +2365,6 @@
 apply(simp add: wcode_halt_invs, auto)
 done
 
-
 lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
 apply(auto simp: wcode_halt_invs)
 done
@@ -2373,12 +2381,12 @@
 lemma t_halt_case_correctness: 
 shows "let P = (\<lambda> (st, l, r). st = 0) in 
        let Q = (\<lambda> (st, l, r). wcode_halt_case_inv st ires rs (l, r)) in 
-       let f = (\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp) in
+       let f = (\<lambda> stp. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp) in
        \<exists> n .P (f n) \<and> Q (f (n::nat))"
 proof -
   let ?P = "(\<lambda> (st, l, r). st = 0)"
   let ?Q = "(\<lambda> (st, l, r). wcode_halt_case_inv st ires rs (l, r))"
-  let ?f = "(\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)"
+  let ?f = "(\<lambda> stp. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp)"
   have "\<exists> n. ?P (?f n) \<and> ?Q (?f (n::nat))"
   proof(rule_tac halt_lemma2)
     show "wf wcode_halt_case_le" by auto
@@ -2386,14 +2394,14 @@
     show "\<forall> na. \<not> ?P (?f na) \<and> ?Q (?f na) \<longrightarrow> 
                     ?Q (?f (Suc na)) \<and> (?f (Suc na), ?f na) \<in> wcode_halt_case_le"
       apply(rule_tac allI, rule_tac impI, case_tac "?f na")
-      apply(simp add: tstep_red tstep.simps)
+      apply(simp add: step_red step.simps)
       apply(case_tac c, simp, case_tac [2] aa)
-      apply(simp_all split: if_splits add: new_tape.simps wcode_halt_case_le_def lex_pair_def)
+      apply(simp_all split: if_splits add: wcode_halt_case_le_def lex_pair_def)
       done      
   next 
     show "?Q (?f 0)"
       apply(simp add: steps.simps wcode_halt_invs)
-      apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def)
+      apply(rule_tac x = "Suc m" in exI, simp)
       apply(rule_tac x = "Suc 0" in exI, auto)
       done
   next
@@ -2407,20 +2415,19 @@
 qed
 
 declare wcode_halt_case_inv.simps[simp del]
-lemma [intro]: "\<exists> xs. (<rev list @ [aa::nat]> :: block list) = Oc # xs"
+lemma [intro]: "\<exists> xs. (<rev list @ [aa::nat]> :: cell list) = Oc # xs"
 apply(case_tac "rev list", simp)
-apply(simp add: tape_of_nat_abv tape_of_nat_list.simps exp_ind_def)
-apply(case_tac list, simp, simp)
+apply(simp add: tape_of_nl_cons)
 done
 
 lemma wcode_halt_case:
-  "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
-  t_wcode_main stp  = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n))
+  t_wcode_main stp  = (0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
   using t_halt_case_correctness[of ires rs m n]
 apply(simp)
 apply(erule_tac exE)
-apply(case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires,
-                Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na")
+apply(case_tac "steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Bk # ires,
+                Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main na")
 apply(auto simp: wcode_halt_case_inv.simps wcode_stop.simps)
 apply(rule_tac x = na in exI, rule_tac x = ln in exI, 
       rule_tac x = rn in exI, simp)
@@ -2430,20 +2437,28 @@
 apply(simp add: bl_bin.simps)
 done
 
+lemma [simp]: "bl_bin [Oc] = 1"
+apply(simp add: bl_bin.simps)
+done
+
+lemma [intro]: "2 * 2 ^ a = Suc (Suc (2 * bl_bin (Oc \<up> a)))"
+apply(induct a, auto simp: bl_bin.simps)
+done
+declare replicate_Suc[simp del]
+
 lemma t_wcode_main_lemma_pre:
   "\<lbrakk>args \<noteq> []; lm = <args::nat list>\<rbrakk> \<Longrightarrow> 
-       \<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main
+       \<exists> stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(m) @ rev lm @ Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main
                     stp
-      = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2^(length lm - 1) \<^esup> @ Bk\<^bsup>rn\<^esup>)"
+      = (0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin lm + rs * 2^(length lm - 1) ) @ Bk\<up>(rn))"
 proof(induct "length args" arbitrary: args lm rs m n, simp)
   fix x args lm rs m n
   assume ind:
     "\<And>args lm rs m n.
     \<lbrakk>x = length args; (args::nat list) \<noteq> []; lm = <args>\<rbrakk>
     \<Longrightarrow> \<exists>stp ln rn.
-    steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-    (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-  
+    steps0 (Suc 0, Bk # Bk\<up>(m) @ rev lm @ Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+    (0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin lm + rs * 2 ^ (length lm - 1)) @ Bk\<up>(rn))"
     and h: "Suc x = length args" "(args::nat list) \<noteq> []" "lm = <args>"
   from h have "\<exists> (a::nat) xs. args = xs @ [a]"
     apply(rule_tac x = "last args" in exI)
@@ -2452,103 +2467,104 @@
   from this obtain a xs where "args = xs @ [a]" by blast
   from h and this show
     "\<exists>stp ln rn.
-    steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-    (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+    steps0 (Suc 0, Bk # Bk\<up>(m) @ rev lm @ Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+    (0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin lm + rs * 2 ^ (length lm - 1)) @ Bk\<up>(rn))"
   proof(case_tac "xs::nat list", simp)
     show "\<exists>stp ln rn.
-      steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-      (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+          steps0 (Suc 0, Bk # Bk \<up> m @ Oc \<up> Suc a @ Bk # Bk # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main stp =
+          (0, Bk # ires, Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> (bl_bin (Oc \<up> Suc a) + rs * 2 ^ a) @ Bk \<up> rn)"
     proof(induct "a" arbitrary: m n rs ires, simp)
       fix m n rs ires
-      show "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
-        t_wcode_main stp  = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin [Oc] + rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-        apply(simp add: bl_bin_one)
-        apply(rule_tac wcode_halt_case)
+      show "\<exists>stp ln rn.
+          steps0 (Suc 0, Bk # Bk \<up> m @ Oc # Bk # Bk # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main stp =
+          (0, Bk # ires, Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> Suc rs @ Bk \<up> rn)"
+          apply(rule_tac wcode_halt_case)
         done
     next
       fix a m n rs ires
-      assume ind2: 
+      assume ind2:
         "\<And>m n rs ires.
-        \<exists>stp ln rn.
-        steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-        (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-      show "\<exists>stp ln rn.
-        steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-        (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<Suc a>) + rs * 2 ^ Suc a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+           \<exists>stp ln rn.
+              steps0 (Suc 0, Bk # Bk \<up> m @ Oc \<up> Suc a @ Bk # Bk # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main stp =
+              (0, Bk # ires, Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> (bl_bin (Oc \<up> Suc a) + rs * 2 ^ a) @ Bk \<up> rn)"
+      show " \<exists>stp ln rn.
+          steps0 (Suc 0, Bk # Bk \<up> m @ Oc \<up> Suc (Suc a) @ Bk # Bk # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main stp =
+          (0, Bk # ires, Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> (bl_bin (Oc \<up> Suc (Suc a)) + rs * 2 ^ Suc a) @ Bk \<up> rn)"
       proof -
         have "\<exists>stp ln rn.
-          steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-          (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+          steps0 (Suc 0, Bk # Bk\<up>(m) @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+          (Suc 0, Bk # Bk\<up>(ln) @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<up>(Suc (2 * rs + 2)) @ Bk\<up>(rn))"
           apply(simp add: tape_of_nat)
-          using wcode_double_case[of m "Oc\<^bsup>a\<^esup> @ Bk # Bk # ires" rs n]
-          apply(simp add: exp_ind_def)
+          using wcode_double_case[of m "Oc\<up>(a) @ Bk # Bk # ires" rs n]
+          apply(simp add: replicate_Suc)
           done
         from this obtain stpa lna rna where stp1:  
-          "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
-          (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
+          "steps0 (Suc 0, Bk # Bk\<up>(m) @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stpa =
+          (Suc 0, Bk # Bk\<up>(lna) @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<up>(Suc (2 * rs + 2)) @ Bk\<up>(rna))" by blast
         moreover have 
           "\<exists>stp ln rn.
-          steps (Suc 0,  Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp =
-          (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + (2*rs + 2)  * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-          using ind2[of lna ires "2*rs + 2" rna] by simp   
+          steps0 (Suc 0,  Bk # Bk\<up>(lna) @ rev (<a::nat>) @ Bk # Bk # ires, Bk # Oc\<up>(Suc (2 * rs + 2)) @ Bk\<up>(rna)) t_wcode_main stp =
+          (0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin (<a>) + (2*rs + 2)  * 2 ^ a) @ Bk\<up>(rn))"
+          using ind2[of lna ires "2*rs + 2" rna] by(simp add: tape_of_nl_abv tape_of_nat_abv)   
         from this obtain stpb lnb rnb where stp2:  
-          "steps (Suc 0,  Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb =
-          (0, Bk # ires, Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + (2*rs + 2)  * 2 ^ a\<^esup> @ Bk\<^bsup>rnb\<^esup>)"
+          "steps0 (Suc 0,  Bk # Bk\<up>(lna) @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<up>(Suc (2 * rs + 2)) @ Bk\<up>(rna)) t_wcode_main stpb =
+          (0, Bk # ires, Bk # Oc # Bk\<up>(lnb) @ Bk # Bk # Oc\<up>(bl_bin (<a>) + (2*rs + 2)  * 2 ^ a) @ Bk\<up>(rnb))"
           by blast
         from stp1 and stp2 show "?thesis"
           apply(rule_tac x = "stpa + stpb" in exI,
-            rule_tac x = lnb in exI, rule_tac x = rnb in exI, simp)
-          apply(simp add: steps_add bl_bin_nat_Suc exponent_def)
+            rule_tac x = lnb in exI, rule_tac x = rnb in exI, simp add: tape_of_nat_abv)
+          apply(simp add:  bl_bin.simps replicate_Suc)
+          apply(auto)
           done
       qed
     qed
   next
     fix aa list
     assume g: "Suc x = length args" "args \<noteq> []" "lm = <args>" "args = xs @ [a::nat]" "xs = (aa::nat) # list"
-    thus "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-      (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+    thus "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(m) @ rev lm @ Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+      (0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin lm + rs * 2 ^ (length lm - 1)) @ Bk\<up>(rn))"
     proof(induct a arbitrary: m n rs args lm, simp_all add: tape_of_nl_rev, 
         simp only: tape_of_nl_cons_app1, simp)
       fix m n rs args lm
       have "\<exists>stp ln rn.
-        steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # rev (<(aa::nat) # list>) @ Bk # Bk # ires,
-        Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-        (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ rev (<aa # list>) @ Bk # Bk # ires, 
-        Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+        steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # rev (<(aa::nat) # list>) @ Bk # Bk # ires,
+        Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+        (Suc 0, Bk # Bk\<up>(ln) @ rev (<aa # list>) @ Bk # Bk # ires, 
+        Bk # Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rn))"
         proof(simp add: tape_of_nl_rev)
           have "\<exists> xs. (<rev list @ [aa]>) = Oc # xs" by auto           
           from this obtain xs where "(<rev list @ [aa]>) = Oc # xs" ..
           thus "\<exists>stp ln rn.
-            steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires,
-            Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-            (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ <rev list @ [aa]> @ Bk # Bk # ires, Bk # Oc\<^bsup>5 + 4 * rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+            steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+            Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+            (Suc 0, Bk # Bk\<up>(ln) @ <rev list @ [aa]> @ Bk # Bk # ires, Bk # Oc\<up>(5 + 4 * rs) @ Bk\<up>(rn))"
             apply(simp)
             using wcode_fourtimes_case[of m "xs @ Bk # Bk # ires" rs n]
             apply(simp)
             done
         qed
       from this obtain stpa lna rna where stp1:
-        "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # rev (<aa # list>) @ Bk # Bk # ires,
-        Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
-        (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<aa # list>) @ Bk # Bk # ires, 
-        Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
+        "steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # rev (<aa # list>) @ Bk # Bk # ires,
+        Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stpa =
+        (Suc 0, Bk # Bk\<up>(lna) @ rev (<aa # list>) @ Bk # Bk # ires, 
+        Bk # Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rna))" by blast
       from g have 
-        "\<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<(aa::nat) # list>) @ Bk # Bk # ires, 
-        Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp = (0, Bk # ires, 
-        Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<aa#list>)+ (4*rs + 4) * 2^(length (<aa#list>) - 1) \<^esup> @ Bk\<^bsup>rn\<^esup>)"
+        "\<exists> stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(lna) @ rev (<(aa::nat) # list>) @ Bk # Bk # ires, 
+        Bk # Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rna)) t_wcode_main stp = (0, Bk # ires, 
+        Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin (<aa#list>)+ (4*rs + 4) * 2^(length (<aa#list>) - 1) ) @ Bk\<up>(rn))"
          apply(rule_tac args = "(aa::nat)#list" in ind, simp_all)
          done
        from this obtain stpb lnb rnb where stp2:
-         "steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<(aa::nat) # list>) @ Bk # Bk # ires, 
-         Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb = (0, Bk # ires, 
-         Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<aa#list>)+ (4*rs + 4) * 2^(length (<aa#list>) - 1) \<^esup> @ Bk\<^bsup>rnb\<^esup>)"
+         "steps0 (Suc 0, Bk # Bk\<up>(lna) @ rev (<(aa::nat) # list>) @ Bk # Bk # ires, 
+         Bk # Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rna)) t_wcode_main stpb = (0, Bk # ires, 
+         Bk # Oc # Bk\<up>(lnb) @ Bk # Bk # Oc\<up>(bl_bin (<aa#list>)+ (4*rs + 4) * 2^(length (<aa#list>) - 1) ) @ Bk\<up>(rnb))"
          by blast
        from stp1 and stp2 and h
        show "\<exists>stp ln rn.
-         steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires,
-         Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-         (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
-         Bk # Oc\<^bsup>bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [0]>) + rs * (2 * 2 ^ (aa + length (<list @ [0]>)))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+         steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+         Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+         (0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk #
+         Bk # Oc\<up>(bl_bin (Oc\<up>(Suc aa) @ Bk # <list @ [0]>) + rs * (2 * 2 ^ (aa + length (<list @ [0]>)))) @ Bk\<up>(rn))"
          apply(rule_tac x = "stpa + stpb" in exI, rule_tac x = lnb in exI,
            rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_rev)
          done
@@ -2558,53 +2574,53 @@
          "\<And> m n rs args lm.
          \<lbrakk>lm = <aa # list @ [ab]>; args = aa # list @ [ab]\<rbrakk>
          \<Longrightarrow> \<exists>stp ln rn.
-         steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ <ab # rev list @ [aa]> @ Bk # Bk # ires,
-         Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-         (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
-         Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]>) + rs * 2 ^ (length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+         steps0 (Suc 0, Bk # Bk\<up>(m) @ <ab # rev list @ [aa]> @ Bk # Bk # ires,
+         Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+         (0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk #
+         Bk # Oc\<up>(bl_bin (<aa # list @ [ab]>) + rs * 2 ^ (length (<aa # list @ [ab]>) - Suc 0)) @ Bk\<up>(rn))"
          and k: "args = aa # list @ [Suc ab]" "lm = <aa # list @ [Suc ab]>"
        show "\<exists>stp ln rn.
-         steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ <Suc ab # rev list @ [aa]> @ Bk # Bk # ires,
-         Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-         (0, Bk # ires,Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # 
-         Bk # Oc\<^bsup>bl_bin (<aa # list @ [Suc ab]>) + rs * 2 ^ (length (<aa # list @ [Suc ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+         steps0 (Suc 0, Bk # Bk\<up>(m) @ <Suc ab # rev list @ [aa]> @ Bk # Bk # ires,
+         Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+         (0, Bk # ires,Bk # Oc # Bk\<up>(ln) @ Bk # 
+         Bk # Oc\<up>(bl_bin (<aa # list @ [Suc ab]>) + rs * 2 ^ (length (<aa # list @ [Suc ab]>) - Suc 0)) @ Bk\<up>(rn))"
        proof(simp add: tape_of_nl_cons_app1)
          have "\<exists>stp ln rn.
-           steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires, 
-           Bk # Oc # Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp
-           = (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc\<^bsup>Suc ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
-           Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-           using wcode_double_case[of m "Oc\<^bsup>ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires"
+           steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc\<up>(Suc (Suc ab)) @ Bk # <rev list @ [aa]> @ Bk # Bk # ires, 
+           Bk # Oc # Oc\<up>(rs) @ Bk\<up>(n)) t_wcode_main stp
+           = (Suc 0, Bk # Bk\<up>(ln) @ Oc\<up>(Suc ab) @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+           Bk # Oc\<up>(Suc (2*rs + 2)) @ Bk\<up>(rn))"
+           using wcode_double_case[of m "Oc\<up>(ab) @ Bk # <rev list @ [aa]> @ Bk # Bk # ires"
                                       rs n]
-           apply(simp add: exp_ind_def)
+           apply(simp add: replicate_Suc)
            done
          from this obtain stpa lna rna where stp1:
-           "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires, 
-           Bk # Oc # Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa
-           = (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ Oc\<^bsup>Suc ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
-           Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
+           "steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc\<up>(Suc (Suc ab)) @ Bk # <rev list @ [aa]> @ Bk # Bk # ires, 
+           Bk # Oc # Oc\<up>(rs) @ Bk\<up>(n)) t_wcode_main stpa
+           = (Suc 0, Bk # Bk\<up>(lna) @ Oc\<up>(Suc ab) @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+           Bk # Oc\<up>(Suc (2*rs + 2)) @ Bk\<up>(rna))" by blast
          from k have 
-           "\<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ <ab # rev list @ [aa]> @ Bk # Bk # ires,
-           Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp
-           = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
-           Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]> ) +  (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+           "\<exists> stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(lna) @ <ab # rev list @ [aa]> @ Bk # Bk # ires,
+           Bk # Oc\<up>(Suc (2*rs + 2)) @ Bk\<up>(rna)) t_wcode_main stp
+           = (0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk #
+           Bk # Oc\<up>(bl_bin (<aa # list @ [ab]> ) +  (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)) @ Bk\<up>(rn))"
            apply(rule_tac ind2, simp_all)
            done
          from this obtain stpb lnb rnb where stp2: 
-           "steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @  <ab # rev list @ [aa]> @ Bk # Bk # ires,
-           Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb
-           = (0, Bk # ires, Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk #
-           Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]> ) +  (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rnb\<^esup>)" 
+           "steps0 (Suc 0, Bk # Bk\<up>(lna) @  <ab # rev list @ [aa]> @ Bk # Bk # ires,
+           Bk # Oc\<up>(Suc (2*rs + 2)) @ Bk\<up>(rna)) t_wcode_main stpb
+           = (0, Bk # ires, Bk # Oc # Bk\<up>(lnb) @ Bk #
+           Bk # Oc\<up>(bl_bin (<aa # list @ [ab]> ) +  (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)) @ Bk\<up>(rnb))" 
            by blast
          from stp1 and stp2 show 
            "\<exists>stp ln rn.
-           steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
-           Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
-           (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # 
-           Oc\<^bsup>bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [Suc ab]>) + rs * (2 * 2 ^ (aa + length (<list @ [Suc ab]>)))\<^esup> 
-           @ Bk\<^bsup>rn\<^esup>)"
+           steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc\<up>(Suc (Suc ab)) @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+           Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
+           (0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk # 
+           Oc\<up>(bl_bin (Oc\<up>(Suc aa) @ Bk # <list @ [Suc ab]>) + rs * (2 * 2 ^ (aa + length (<list @ [Suc ab]>)))) 
+           @ Bk\<up>(rn))"
            apply(rule_tac x = "stpa + stpb" in exI, rule_tac x = lnb in exI,
-             rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_cons_app1 exp_ind_def)
+             rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_cons_app1 replicate_Suc)
            done
        qed
      qed
@@ -2612,10 +2628,7 @@
  qed
 
 
-         
-(* turing_shift can be used*)
-term t_wcode_main
-definition t_wcode_prepare :: "tprog"
+definition t_wcode_prepare :: "instr list"
   where
   "t_wcode_prepare \<equiv> 
          [(W1, 2), (L, 1), (L, 3), (R, 2), (R, 4), (W0, 3),
@@ -2626,33 +2639,33 @@
   where
   "wprepare_add_one m lm (l, r) = 
       (\<exists> rn. l = [] \<and>
-               (r = <m # lm> @ Bk\<^bsup>rn\<^esup> \<or> 
-                r = Bk # <m # lm> @ Bk\<^bsup>rn\<^esup>))"
+               (r = <m # lm> @ Bk\<up>(rn) \<or> 
+                r = Bk # <m # lm> @ Bk\<up>(rn)))"
 
 fun wprepare_goto_first_end :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wprepare_goto_first_end m lm (l, r) = 
-      (\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> \<and>
-                      r = Oc\<^bsup>mr\<^esup> @ Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and>
+      (\<exists> ml mr rn. l = Oc\<up>(ml) \<and>
+                      r = Oc\<up>(mr) @ Bk # <lm> @ Bk\<up>(rn) \<and>
                       ml + mr = Suc (Suc m))"
 
 fun wprepare_erase :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow>  bool"
   where
   "wprepare_erase m lm (l, r) = 
-     (\<exists> rn. l = Oc\<^bsup>Suc m\<^esup> \<and> 
-               tl r = Bk # <lm> @ Bk\<^bsup>rn\<^esup>)"
+     (\<exists> rn. l = Oc\<up>(Suc m) \<and> 
+               tl r = Bk # <lm> @ Bk\<up>(rn))"
 
 fun wprepare_goto_start_pos_B :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wprepare_goto_start_pos_B m lm (l, r) = 
-     (\<exists> rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-               r = Bk # <lm> @ Bk\<^bsup>rn\<^esup>)"
+     (\<exists> rn. l = Bk # Oc\<up>(Suc m) \<and>
+               r = Bk # <lm> @ Bk\<up>(rn))"
 
 fun wprepare_goto_start_pos_O :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wprepare_goto_start_pos_O m lm (l, r) = 
-     (\<exists> rn. l = Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-               r = <lm> @ Bk\<^bsup>rn\<^esup>)"
+     (\<exists> rn. l = Bk # Bk # Oc\<up>(Suc m) \<and>
+               r = <lm> @ Bk\<up>(rn))"
 
 fun wprepare_goto_start_pos :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
   where
@@ -2663,15 +2676,15 @@
 fun wprepare_loop_start_on_rightmost :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wprepare_loop_start_on_rightmost m lm (l, r) = 
-     (\<exists> rn mr. rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and>
-                       r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+     (\<exists> rn mr. rev l @ r = Oc\<up>(Suc m) @ Bk # Bk # <lm> @ Bk\<up>(rn) \<and> l \<noteq> [] \<and>
+                       r = Oc\<up>(mr) @ Bk\<up>(rn))"
 
 fun wprepare_loop_start_in_middle :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wprepare_loop_start_in_middle m lm (l, r) =
      (\<exists> rn (mr:: nat) (lm1::nat list). 
-  rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and>
-  r = Oc\<^bsup>mr\<^esup> @ Bk # <lm1> @ Bk\<^bsup>rn\<^esup> \<and> lm1 \<noteq> [])"
+  rev l @ r = Oc\<up>(Suc m) @ Bk # Bk # <lm> @ Bk\<up>(rn) \<and> l \<noteq> [] \<and>
+  r = Oc\<up>(mr) @ Bk # <lm1> @ Bk\<up>(rn) \<and> lm1 \<noteq> [])"
 
 fun wprepare_loop_start :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
   where
@@ -2681,16 +2694,16 @@
 fun wprepare_loop_goon_on_rightmost :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wprepare_loop_goon_on_rightmost m lm (l, r) = 
-     (\<exists> rn. l = Bk # <rev lm> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-               r = Bk\<^bsup>rn\<^esup>)"
+     (\<exists> rn. l = Bk # <rev lm> @ Bk # Bk # Oc\<up>(Suc m) \<and>
+               r = Bk\<up>(rn))"
 
 fun wprepare_loop_goon_in_middle :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wprepare_loop_goon_in_middle m lm (l, r) = 
      (\<exists> rn (mr:: nat) (lm1::nat list). 
-  rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and>
-                     (if lm1 = [] then r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> 
-                     else r = Oc\<^bsup>mr\<^esup> @ Bk # <lm1> @ Bk\<^bsup>rn\<^esup>) \<and> mr > 0)"
+  rev l @ r = Oc\<up>(Suc m) @ Bk # Bk # <lm> @ Bk\<up>(rn) \<and> l \<noteq> [] \<and>
+                     (if lm1 = [] then r = Oc\<up>(mr) @ Bk\<up>(rn) 
+                     else r = Oc\<up>(mr) @ Bk # <lm1> @ Bk\<up>(rn)) \<and> mr > 0)"
 
 fun wprepare_loop_goon :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
   where
@@ -2701,14 +2714,14 @@
 fun wprepare_add_one2 :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wprepare_add_one2 m lm (l, r) =
-          (\<exists> rn. l = Bk # Bk # <rev lm> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-               (r = [] \<or> tl r = Bk\<^bsup>rn\<^esup>))"
+          (\<exists> rn. l = Bk # Bk # <rev lm> @ Bk # Bk # Oc\<up>(Suc m) \<and>
+               (r = [] \<or> tl r = Bk\<up>(rn)))"
 
 fun wprepare_stop :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wprepare_stop m lm (l, r) = 
-         (\<exists> rn. l = Bk # <rev lm> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-               r = Bk # Oc # Bk\<^bsup>rn\<^esup>)"
+         (\<exists> rn. l = Bk # <rev lm> @ Bk # Bk # Oc\<up>(Suc m) \<and>
+               r = Bk # Oc # Bk\<up>(rn))"
 
 fun wprepare_inv :: "nat \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
   where
@@ -2723,14 +2736,14 @@
          else if st = 7 then wprepare_add_one2 m lm (l, r)
          else False)"
 
-fun wprepare_stage :: "t_conf \<Rightarrow> nat"
+fun wprepare_stage :: "config \<Rightarrow> nat"
   where
   "wprepare_stage (st, l, r) = 
       (if st \<ge> 1 \<and> st \<le> 4 then 3
        else if st = 5 \<or> st = 6 then 2
        else 1)"
 
-fun wprepare_state :: "t_conf \<Rightarrow> nat"
+fun wprepare_state :: "config \<Rightarrow> nat"
   where
   "wprepare_state (st, l, r) = 
        (if st = 1 then 4
@@ -2740,7 +2753,7 @@
         else if st = 7 then 2
         else 0)"
 
-fun wprepare_step :: "t_conf \<Rightarrow> nat"
+fun wprepare_step :: "config \<Rightarrow> nat"
   where
   "wprepare_step (st, l, r) = 
       (if st = 1 then (if hd r = Oc then Suc (length l)
@@ -2755,14 +2768,14 @@
                             else 1)
        else 0)"
 
-fun wcode_prepare_measure :: "t_conf \<Rightarrow> nat \<times> nat \<times> nat"
+fun wcode_prepare_measure :: "config \<Rightarrow> nat \<times> nat \<times> nat"
   where
   "wcode_prepare_measure (st, l, r) = 
      (wprepare_stage (st, l, r), 
       wprepare_state (st, l, r), 
       wprepare_step (st, l, r))"
 
-definition wcode_prepare_le :: "(t_conf \<times> t_conf) set"
+definition wcode_prepare_le :: "(config \<times> config) set"
   where "wcode_prepare_le \<equiv> (inv_image lex_triple wcode_prepare_measure)"
 
 lemma [intro]: "wf lex_pair"
@@ -2770,7 +2783,7 @@
 
 lemma wf_wcode_prepare_le[intro]: "wf wcode_prepare_le"
 by(auto intro:wf_inv_image simp: wcode_prepare_le_def 
-           recursive.lex_triple_def)
+           lex_triple_def)
 
 declare wprepare_add_one.simps[simp del] wprepare_goto_first_end.simps[simp del]
         wprepare_erase.simps[simp del] wprepare_goto_start_pos.simps[simp del]
@@ -2808,45 +2821,56 @@
 done
 
 lemma [simp]: "fetch t_wcode_prepare 4 Bk = (R, 4)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(subgoal_tac "4 = Suc 3")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
 done
 
 lemma [simp]: "fetch t_wcode_prepare 4 Oc = (R, 5)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
+apply(subgoal_tac "4 = Suc 3")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
+done
+
 
 lemma [simp]: "fetch t_wcode_prepare 5 Oc = (R, 5)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(subgoal_tac "5 = Suc 4")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
 done
 
 lemma [simp]: "fetch t_wcode_prepare 5 Bk = (R, 6)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(subgoal_tac "5 = Suc 4")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
 done
 
 lemma [simp]: "fetch t_wcode_prepare 6 Oc = (R, 5)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(subgoal_tac "6 = Suc 5")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
 done
 
 lemma [simp]: "fetch t_wcode_prepare 6 Bk = (R, 7)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(subgoal_tac "6 = Suc 5")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
 done
 
 lemma [simp]: "fetch t_wcode_prepare 7 Oc = (L, 0)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(subgoal_tac "7 = Suc 6")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
 done
 
 lemma [simp]: "fetch t_wcode_prepare 7 Bk = (W1, 7)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma tape_of_nl_not_null: "lm \<noteq> [] \<Longrightarrow> <lm::nat list> \<noteq> []"
-apply(case_tac lm, auto)
-apply(case_tac list, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+apply(subgoal_tac "7 = Suc 6")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
 done
 
 lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_add_one m lm (b, []) = False"
 apply(simp add: wprepare_invs)
-apply(simp add: tape_of_nl_not_null)
 done
 
 lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_goto_first_end m lm (b, []) = False"
@@ -2857,19 +2881,20 @@
 apply(simp add: wprepare_invs)
 done
 
-
-
 lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_goto_start_pos m lm (b, []) = False"
-apply(simp add: wprepare_invs tape_of_nl_not_null)
+apply(simp add: wprepare_invs)
 done
 
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [])\<rbrakk> \<Longrightarrow> b \<noteq> []"
-apply(simp add: wprepare_invs tape_of_nl_not_null, auto)
-done
+apply(simp add: wprepare_invs)
+done
+
+lemma rev_eq: "rev xs = rev ys \<Longrightarrow> xs = ys"
+by auto
 
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [])\<rbrakk> \<Longrightarrow> 
                                   wprepare_loop_goon m lm (Bk # b, [])"
-apply(simp only: wprepare_invs tape_of_nl_not_null)
+apply(simp only: wprepare_invs)
 apply(erule_tac disjE)
 apply(rule_tac disjI2)
 apply(simp add: wprepare_loop_start_on_rightmost.simps
@@ -2878,50 +2903,50 @@
 done
 
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, [])\<rbrakk> \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
+apply(simp only: wprepare_invs, auto)
 done
 
 lemma [simp]:"\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, [])\<rbrakk> \<Longrightarrow> 
   wprepare_add_one2 m lm (Bk # b, [])"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto split: if_splits)
-apply(case_tac mr, simp, simp add: exp_ind_def)
+apply(simp only: wprepare_invs, auto split: if_splits)
 done
 
 lemma [simp]: "wprepare_add_one2 m lm (b, []) \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
+apply(simp only: wprepare_invs, auto)
 done
 
 lemma [simp]: "wprepare_add_one2 m lm (b, []) \<Longrightarrow> wprepare_add_one2 m lm (b, [Oc])"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
+apply(simp only: wprepare_invs, auto)
 done
 
 lemma [simp]: "Bk # list = <(m::nat) # lm> @ ys = False"
-apply(case_tac lm, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+apply(case_tac lm, auto simp: tape_of_nl_cons replicate_Suc)
 done
 
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_add_one m lm (b, Bk # list)\<rbrakk>
        \<Longrightarrow> (b = [] \<longrightarrow> wprepare_goto_first_end m lm ([], Oc # list)) \<and> 
            (b \<noteq> [] \<longrightarrow> wprepare_goto_first_end m lm (b, Oc # list))"
-apply(simp only: wprepare_invs, auto)
-apply(rule_tac x = 0 in exI, simp add: exp_ind_def)
-apply(case_tac lm, simp, simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
-apply(rule_tac x = rn in exI, simp)
+apply(simp only: wprepare_invs)
+apply(auto simp: tape_of_nl_cons split: if_splits)
+apply(rule_tac x = 0 in exI, simp add: replicate_Suc)
+apply(case_tac lm, simp, simp add: tape_of_nl_abv tape_of_nat_list.simps replicate_Suc)
 done
 
 lemma [simp]: "wprepare_goto_first_end m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-done
+apply(simp only: wprepare_invs , auto simp: replicate_Suc)
+done
+
+declare replicate_Suc[simp]
 
 lemma [simp]: "wprepare_goto_first_end m lm (b, Bk # list) \<Longrightarrow>
                           wprepare_erase m lm (tl b, hd b # Bk # list)"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac mr, auto simp: exp_ind_def)
+apply(simp only: wprepare_invs, auto)
+apply(case_tac mr, simp_all)
+apply(case_tac mr, auto)
 done
 
 lemma [simp]: "wprepare_erase m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs exp_ind_def, auto)
+apply(simp only: wprepare_invs, auto)
 done
 
 lemma [simp]: "wprepare_erase m lm (b, Bk # list) \<Longrightarrow> 
@@ -2932,18 +2957,16 @@
 lemma [simp]: "\<lbrakk>wprepare_add_one m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
 apply(simp only: wprepare_invs)
 apply(case_tac lm, simp_all add: tape_of_nl_abv 
-                         tape_of_nat_list.simps exp_ind_def, auto)
+                         tape_of_nat_list.simps, auto)
 done
     
 lemma [simp]: "\<lbrakk>lm \<noteq> [];  wprepare_goto_first_end m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
 apply(simp only: wprepare_invs, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(simp add: tape_of_nl_not_null)
+apply(case_tac mr, simp_all)
 done
      
 lemma [simp]: "\<lbrakk>lm \<noteq> [];  wprepare_goto_first_end m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
 apply(simp only: wprepare_invs, auto)
-apply(case_tac mr, simp_all add: exp_ind_def tape_of_nl_not_null)
 done
 
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_erase m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
@@ -2951,14 +2974,13 @@
 done
 
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_erase m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs, auto simp: exp_ind_def)
+apply(simp only: wprepare_invs, auto)
 done
 
 lemma [simp]: "\<lbrakk>lm \<noteq> [];  wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
 apply(simp only: wprepare_invs, auto)
-apply(simp add: tape_of_nl_not_null)
 apply(case_tac lm, simp, case_tac list)
-apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps)
 done
 
 lemma [simp]: "\<lbrakk>lm \<noteq> [];  wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
@@ -2975,10 +2997,10 @@
   (list \<noteq> [] \<longrightarrow> wprepare_add_one2 m lm (Bk # b, list))"
 apply(simp only: wprepare_invs, simp)
 apply(case_tac list, simp_all split: if_splits, auto)
-apply(case_tac [1-3] mr, simp_all add: exp_ind_def)
-apply(case_tac mr, simp_all add: exp_ind_def tape_of_nl_not_null)
-apply(case_tac [1-2] mr, simp_all add: exp_ind_def)
-apply(case_tac rn, simp, case_tac nat, auto simp: exp_ind_def)
+apply(case_tac [1-3] mr, simp_all add: )
+apply(case_tac mr, simp_all)
+apply(case_tac [1-2] mr, simp_all add: )
+apply(case_tac rn, simp, case_tac nat, auto simp: )
 done
 
 lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
@@ -2996,21 +3018,19 @@
            (b \<noteq> [] \<longrightarrow> wprepare_goto_first_end m lm (Oc # b, list))"
 apply(simp only:  wprepare_invs, auto)
 apply(rule_tac x = 1 in exI, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac ml, simp_all add: exp_ind_def)
-apply(rule_tac x = rn in exI, simp)
-apply(rule_tac x = "Suc ml" in exI, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
+apply(case_tac ml, simp_all add: )
+apply(rule_tac x = "Suc ml" in exI, simp_all add: )
 apply(rule_tac x = "mr - 1" in exI, simp)
-apply(case_tac mr, simp_all add: exp_ind_def, auto)
 done
 
 lemma [simp]: "wprepare_erase m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs, auto simp: exp_ind_def)
+apply(simp only: wprepare_invs, auto simp: )
 done
 
 lemma [simp]: "wprepare_erase m lm (b, Oc # list)
   \<Longrightarrow> wprepare_erase m lm (b, Bk # list)"
-apply(simp  only:wprepare_invs, auto simp: exp_ind_def)
+apply(simp  only:wprepare_invs, auto simp: )
 done
 
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk>
@@ -3022,26 +3042,25 @@
 lemma [simp]: "wprepare_loop_start m lm (b, aa) \<Longrightarrow> b \<noteq> []"
 apply(simp only:wprepare_invs, auto)
 done
-lemma [elim]: "Bk # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>  \<Longrightarrow> \<exists>rn. list = Bk\<^bsup>rn\<^esup>"
+lemma [elim]: "Bk # list = Oc\<up>(mr) @ Bk\<up>(rn)  \<Longrightarrow> \<exists>rn. list = Bk\<up>(rn)"
 apply(case_tac mr, simp_all)
-apply(case_tac rn, simp_all add: exp_ind_def, auto)
+apply(case_tac rn, simp_all)
 done
 
 lemma rev_equal_iff: "x = y \<Longrightarrow> rev x = rev y"
 by simp
 
 lemma tape_of_nl_false1:
-  "lm \<noteq> [] \<Longrightarrow> rev b @ [Bk] \<noteq> Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>m\<^esup> @ Bk # Bk # <lm::nat list>"
+  "lm \<noteq> [] \<Longrightarrow> rev b @ [Bk] \<noteq> Bk\<up>(ln) @ Oc # Oc\<up>(m) @ Bk # Bk # <lm::nat list>"
 apply(auto)
 apply(drule_tac rev_equal_iff, simp add: tape_of_nl_rev)
 apply(case_tac "rev lm")
-apply(case_tac [2] list, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+apply(case_tac [2] list, auto simp: tape_of_nl_abv tape_of_nat_list.simps )
 done
 
 lemma [simp]: "wprepare_loop_start_in_middle m lm (b, [Bk]) = False"
 apply(simp add: wprepare_loop_start_in_middle.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac lm1, simp, simp add: tape_of_nl_not_null)
+apply(case_tac mr, simp_all add: )
 done
 
 declare wprepare_loop_start_in_middle.simps[simp del]
@@ -3059,39 +3078,39 @@
 apply(simp only: wprepare_invs, simp)
 apply(simp add: wprepare_loop_goon_on_rightmost.simps 
   wprepare_loop_start_on_rightmost.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 apply(rule_tac rev_eq)
 apply(simp add: tape_of_nl_rev)
-apply(simp add: exp_ind_def[THEN sym] exp_ind)
+apply(simp add: exp_ind replicate_Suc[THEN sym] del: replicate_Suc)
 done
 
 lemma [simp]: "wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista)
  \<Longrightarrow> wprepare_loop_goon_in_middle m lm (Bk # b, a # lista) = False"
 apply(auto simp: wprepare_loop_start_on_rightmost.simps
                  wprepare_loop_goon_in_middle.simps)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
+apply(case_tac [!] mr, simp_all)
 done
 
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista)\<rbrakk>
     \<Longrightarrow> wprepare_loop_goon_on_rightmost m lm (Bk # b, a # lista)"
 apply(simp only: wprepare_loop_start_on_rightmost.simps
                  wprepare_loop_goon_on_rightmost.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 apply(simp add: tape_of_nl_rev)
-apply(simp add: exp_ind_def[THEN sym] exp_ind)
+apply(simp add: replicate_Suc[THEN sym] exp_ind del: replicate_Suc)
 done
 
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\<rbrakk>
   \<Longrightarrow> wprepare_loop_goon_on_rightmost m lm (Bk # b, a # lista) = False"
 apply(simp add: wprepare_loop_start_in_middle.simps
                 wprepare_loop_goon_on_rightmost.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 apply(case_tac  "lm1::nat list", simp_all, case_tac  list, simp)
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps tape_of_nat_abv exp_ind_def)
-apply(case_tac [!] rna, simp_all add: exp_ind_def)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps tape_of_nat_abv )
+apply(case_tac [!] rna, simp_all add: )
+apply(case_tac mr, simp_all add: )
 apply(case_tac lm1, simp, case_tac list, simp)
-apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def tape_of_nat_abv)
+apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps  tape_of_nat_abv)
 done
 
 lemma [simp]: 
@@ -3100,7 +3119,7 @@
 apply(simp add: wprepare_loop_start_in_middle.simps
                wprepare_loop_goon_in_middle.simps, auto)
 apply(rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 apply(case_tac lm1, simp)
 apply(rule_tac x = "Suc aa" in exI, simp)
 apply(rule_tac x = list in exI)
@@ -3137,15 +3156,14 @@
   wprepare_loop_start_on_rightmost m lm (Oc # b, list)"
 apply(simp add: wprepare_loop_start_on_rightmost.simps, auto)
 apply(rule_tac x = rn in exI, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac rn, auto simp: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 done
 
 lemma [simp]: "wprepare_loop_start_in_middle m lm (b, Oc # list) \<Longrightarrow> 
                 wprepare_loop_start_in_middle m lm (Oc # b, list)"
 apply(simp add: wprepare_loop_start_in_middle.simps, auto)
 apply(rule_tac x = rn in exI, auto)
-apply(case_tac mr, simp, simp add: exp_ind_def)
+apply(case_tac mr, simp, simp add: )
 apply(rule_tac x = nat in exI, simp)
 apply(rule_tac x = lm1 in exI, simp)
 done
@@ -3170,20 +3188,20 @@
 lemma [simp]: "wprepare_loop_goon_on_rightmost m lm (b, Oc # list) = False"
 apply(simp add: wprepare_loop_goon_on_rightmost.simps)
 done
-lemma wprepare_loop1: "\<lbrakk>rev b @ Oc\<^bsup>mr\<^esup> =  Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm>; 
-         b \<noteq> []; 0 < mr; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>\<rbrakk>
+lemma wprepare_loop1: "\<lbrakk>rev b @ Oc\<up>(mr) =  Oc\<up>(Suc m) @ Bk # Bk # <lm>; 
+         b \<noteq> []; 0 < mr; Oc # list = Oc\<up>(mr) @ Bk\<up>(rn)\<rbrakk>
        \<Longrightarrow> wprepare_loop_start_on_rightmost m lm (Oc # b, list)"
 apply(simp add: wprepare_loop_start_on_rightmost.simps)
 apply(rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp, simp add: exp_ind_def, auto)
-done
-
-lemma wprepare_loop2: "\<lbrakk>rev b @ Oc\<^bsup>mr\<^esup> @ Bk # <a # lista> = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm>;
-                b \<noteq> []; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk # <(a::nat) # lista> @ Bk\<^bsup>rn\<^esup>\<rbrakk>
+apply(case_tac mr, simp, simp)
+done
+
+lemma wprepare_loop2: "\<lbrakk>rev b @ Oc\<up>(mr) @ Bk # <a # lista> = Oc\<up>(Suc m) @ Bk # Bk # <lm>;
+                b \<noteq> []; Oc # list = Oc\<up>(mr) @ Bk # <(a::nat) # lista> @ Bk\<up>(rn)\<rbrakk>
        \<Longrightarrow>  wprepare_loop_start_in_middle m lm (Oc # b, list)"
 apply(simp add: wprepare_loop_start_in_middle.simps)
 apply(rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 apply(rule_tac x = nat in exI, simp)
 apply(rule_tac x = "a#lista" in exI, simp)
 done
@@ -3212,7 +3230,7 @@
 apply(auto simp: wprepare_goto_start_pos.simps 
                  wprepare_loop_start_on_rightmost.simps)
 apply(rule_tac x = rn in exI, simp)
-apply(simp add: tape_of_nat_abv tape_of_nat_list.simps exp_ind_def, auto)
+apply(simp add: replicate_Suc[THEN sym] exp_ind del: replicate_Suc)
 done
 
 lemma [simp]:  "wprepare_goto_start_pos m (a # aa # listaa) (b, Oc # list)
@@ -3220,8 +3238,9 @@
 apply(auto simp: wprepare_goto_start_pos.simps
                  wprepare_loop_start_in_middle.simps)
 apply(rule_tac x = rn in exI, simp)
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+apply(simp add: exp_ind[THEN sym])
 apply(rule_tac x = a in exI, rule_tac x = "aa#listaa" in exI, simp)
+apply(simp add: tape_of_nl_cons)
 done
 
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Oc # list)\<rbrakk>
@@ -3246,12 +3265,12 @@
   assumes h: "lm \<noteq> []"
   shows "let P = (\<lambda> (st, l, r). st = 0) in 
   let Q = (\<lambda> (st, l, r). wprepare_inv st m lm (l, r)) in 
-  let f = (\<lambda> stp. steps (Suc 0, [], (<m # lm>)) t_wcode_prepare stp) in
+  let f = (\<lambda> stp. steps0 (Suc 0, [], (<m # lm>)) t_wcode_prepare stp) in
     \<exists> n .P (f n) \<and> Q (f n)"
 proof -
   let ?P = "(\<lambda> (st, l, r). st = 0)"
   let ?Q = "(\<lambda> (st, l, r). wprepare_inv st m lm (l, r))"
-  let ?f = "(\<lambda> stp. steps (Suc 0, [], (<m # lm>)) t_wcode_prepare stp)"
+  let ?f = "(\<lambda> stp. steps0 (Suc 0, [], (<m # lm>)) t_wcode_prepare stp)"
   have "\<exists> n. ?P (?f n) \<and> ?Q (?f n)"
   proof(rule_tac halt_lemma2)
     show "wf wcode_prepare_le" by auto
@@ -3260,11 +3279,9 @@
                  ?Q (?f (Suc n)) \<and> (?f (Suc n), ?f n) \<in> wcode_prepare_le"
       using h
       apply(rule_tac allI, rule_tac impI, case_tac "?f n", 
-            simp add: tstep_red tstep.simps)
+            simp add: step_red step.simps)
       apply(case_tac c, simp, case_tac [2] aa)
-      apply(simp_all add: wprepare_inv.simps wcode_prepare_le_def new_tape.simps
-                          lex_triple_def lex_pair_def
-
+      apply(simp_all add: wprepare_inv.simps wcode_prepare_le_def lex_triple_def lex_pair_def
                  split: if_splits)
       apply(simp_all add: start_2_goon  start_2_start
                            add_one_2_add_one add_one_2_stop)
@@ -3284,35 +3301,27 @@
     done
 qed
 
-lemma [intro]: "t_correct t_wcode_prepare"
-apply(simp add: t_correct.simps t_wcode_prepare_def iseven_def)
-apply(rule_tac x = 7 in exI, simp)
-done
-    
-lemma twice_len_even: "length (tm_of abc_twice) mod 2 = 0"
-apply(simp add: tm_even)
-done
-
-lemma fourtimes_len_even: "length (tm_of abc_fourtimes) mod 2 = 0"
-apply(simp add: tm_even)
-done
-
+lemma [intro]: "tm_wf (t_wcode_prepare, 0)"
+apply(simp add:tm_wf.simps t_wcode_prepare_def)
+done
+   
+(* 
 lemma t_correct_termi: "t_correct tp \<Longrightarrow> 
       list_all (\<lambda>(acn, st). (st \<le> Suc (length tp div 2))) (change_termi_state tp)"
 apply(auto simp: t_correct.simps List.list_all_length)
 apply(erule_tac x = n in allE, simp)
 apply(case_tac "tp!n", auto simp: change_termi_state.simps split: if_splits)
 done
-
+*)
 
 lemma t_correct_shift:
          "list_all (\<lambda>(acn, st). (st \<le> y)) tp \<Longrightarrow>
-          list_all (\<lambda>(acn, st). (st \<le> y + off)) (tshift tp off) "
-apply(auto simp: t_correct.simps List.list_all_length)
-apply(erule_tac x = n in allE, simp add: shift_length)
-apply(case_tac "tp!n", auto simp: tshift.simps)
-done
-
+          list_all (\<lambda>(acn, st). (st \<le> y + off)) (shift tp off) "
+apply(auto simp: List.list_all_length)
+apply(erule_tac x = n in allE, simp add: length_shift)
+apply(case_tac "tp!n", auto simp: shift.simps)
+done
+(*
 lemma [intro]: 
   "t_correct (tm_of abc_twice @ tMp (Suc 0) 
         (start_of twice_ly (length abc_twice) - Suc 0))"
@@ -3325,177 +3334,237 @@
 apply(rule_tac t_compiled_correct, simp_all)
 apply(simp add: fourtimes_ly_def)
 done
-
-
-lemma [intro]: "t_correct t_wcode_main"
-apply(auto simp: t_wcode_main_def t_correct.simps shift_length 
-                 t_twice_def t_fourtimes_def)
+*)
+
+lemma [intro]: "(28 + (length t_twice_compile + length t_fourtimes_compile)) mod 2 = 0"
+apply(auto simp: t_twice_compile_def t_fourtimes_compile_def)
+by arith
+
+lemma [elim]: "(a, b) \<in> set t_wcode_main_first_part \<Longrightarrow>
+  b \<le> (28 + (length t_twice_compile + length t_fourtimes_compile)) div 2"
+apply(auto simp: t_wcode_main_first_part_def t_twice_def)
+done
+
+
+
+lemma tm_wf_change_termi: "tm_wf (tp, 0) \<Longrightarrow> 
+      list_all (\<lambda>(acn, st). (st \<le> Suc (length tp div 2))) (adjust tp)"
+apply(auto simp: tm_wf.simps List.list_all_length)
+apply(case_tac "tp!n", auto simp: adjust.simps split: if_splits)
+apply(erule_tac x = "(a, b)" in ballE, auto)
+by (metis in_set_conv_nth)
+
+lemma tm_wf_shift:
+         "list_all (\<lambda>(acn, st). (st \<le> y)) tp \<Longrightarrow>
+          list_all (\<lambda>(acn, st). (st \<le> y + off)) (shift tp off) "
+apply(auto simp: tm_wf.simps List.list_all_length)
+apply(erule_tac x = n in allE, simp add: length_shift)
+apply(case_tac "tp!n", auto simp: shift.simps)
+done
+
+declare length_tp'[simp del]
+
+lemma [simp]: "length (mopup (Suc 0)) = 16"
+apply(auto simp: mopup.simps)
+done
+
+lemma [elim]: "(a, b) \<in> set (shift (turing_basic.adjust t_twice_compile) 12) \<Longrightarrow> 
+  b \<le> (28 + (length t_twice_compile + length t_fourtimes_compile)) div 2"
+apply(simp add: t_twice_compile_def t_fourtimes_compile_def)
 proof -
-  show "iseven (60 + (length (tm_of abc_twice) +
-                 length (tm_of abc_fourtimes)))"
-    using twice_len_even fourtimes_len_even
-    apply(auto simp: iseven_def)
-    apply(rule_tac x = "30 + q + qa" in exI, simp)
-    done
-next
-  show " list_all (\<lambda>(acn, s). s \<le> (60 + (length (tm_of abc_twice) + 
-           length (tm_of abc_fourtimes))) div 2) t_wcode_main_first_part"
-    apply(auto simp: t_wcode_main_first_part_def shift_length t_twice_def)
-    done
-next
-  have "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_twice @ tMp (Suc 0)
-    (start_of twice_ly (length abc_twice) - Suc 0)) div 2))
-    (change_termi_state (tm_of abc_twice @ tMp (Suc 0) 
-    (start_of twice_ly (length abc_twice) - Suc 0)))"
-    apply(rule_tac t_correct_termi, auto)
+  assume g: "(a, b) \<in> set (shift (turing_basic.adjust (tm_of abc_twice @ shift (mopup (Suc 0)) (length (tm_of abc_twice) div 2))) 12)"
+  moreover have "length (tm_of abc_twice) mod 2 = 0" by auto
+  moreover have "length (tm_of abc_fourtimes) mod 2 = 0" by auto
+  ultimately have "list_all (\<lambda>(acn, st). (st \<le> (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2)) 
+    (shift (turing_basic.adjust t_twice_compile) 12)"
+  proof(auto simp: mod_ex1)
+    fix q qa
+    assume h: "length (tm_of abc_twice) = 2 * q" "length (tm_of abc_fourtimes) = 2 * qa"
+    hence "list_all (\<lambda>(acn, st). st \<le> (18 + (q + qa)) + 12) (shift (turing_basic.adjust t_twice_compile) 12)"
+    proof(rule_tac tm_wf_shift t_twice_compile_def)
+      have "list_all (\<lambda>(acn, st). st \<le> Suc (length t_twice_compile div 2)) (adjust t_twice_compile)"
+        by(rule_tac tm_wf_change_termi, auto)
+      thus "list_all (\<lambda>(acn, st). st \<le> 18 + (q + qa)) (turing_basic.adjust t_twice_compile)"
+        using h
+        apply(simp add: t_twice_compile_def, auto simp: List.list_all_length)
+        done
+    qed
+    thus "list_all (\<lambda>(acn, st). st \<le> 30 + (q + qa)) (shift (turing_basic.adjust t_twice_compile) 12)"
+      by simp
+  qed
+  thus "b \<le> (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2"
+    using g
+    apply(auto simp:t_twice_compile_def)
+    apply(simp add: Ball_set[THEN sym])
+    apply(erule_tac x = "(a, b)" in ballE, simp, simp)
     done
-  hence "list_all (\<lambda>(acn, s). s \<le>  Suc (length (tm_of abc_twice @ tMp (Suc 0)
-    (start_of twice_ly (length abc_twice) - Suc 0)) div 2) + 12)
-     (tshift (change_termi_state (tm_of abc_twice @ tMp (Suc 0) 
-           (start_of twice_ly (length abc_twice) - Suc 0))) 12)"
-    apply(rule_tac t_correct_shift, simp)
-    done
-  thus  "list_all (\<lambda>(acn, s). s \<le> 
-           (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2)
-     (tshift (change_termi_state (tm_of abc_twice @ tMp (Suc 0)
-                 (start_of twice_ly (length abc_twice) - Suc 0))) 12)"
-    apply(simp)
-    apply(simp add: list_all_length, auto)
-    done
-next
-  have "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_fourtimes @ tMp (Suc 0) 
-    (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) div 2))
-      (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0) 
-    (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) "
-    apply(rule_tac t_correct_termi, auto)
-    done
-  hence "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_fourtimes @ tMp (Suc 0) 
-    (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) div 2) + (t_twice_len + 13))
-    (tshift (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0) 
-    (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) (t_twice_len + 13))"
-    apply(rule_tac t_correct_shift, simp)
-    done
-  thus "list_all (\<lambda>(acn, s). s \<le> (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2)
-    (tshift (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0)
-    (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) (t_twice_len + 13))"
-    apply(simp add: t_twice_len_def t_twice_def)
-    using twice_len_even fourtimes_len_even
-    apply(auto simp: list_all_length)
+qed 
+
+lemma [elim]: "(a, b) \<in> set (shift (turing_basic.adjust t_fourtimes_compile) (t_twice_len + 13)) 
+  \<Longrightarrow> b \<le> (28 + (length t_twice_compile + length t_fourtimes_compile)) div 2"
+apply(simp add: t_twice_compile_def t_fourtimes_compile_def t_twice_len_def)
+proof -
+  assume g: "(a, b) \<in> set (shift (turing_basic.adjust (tm_of abc_fourtimes @ shift (mopup (Suc 0))
+    (length (tm_of abc_fourtimes) div 2))) (length t_twice div 2 + 13))"
+  moreover have "length (tm_of abc_twice) mod 2 = 0" by auto
+  moreover have "length (tm_of abc_fourtimes) mod 2 = 0" by auto
+  ultimately have "list_all (\<lambda>(acn, st). (st \<le> (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2)) 
+    (shift (turing_basic.adjust (tm_of abc_fourtimes @ shift (mopup (Suc 0))
+    (length (tm_of abc_fourtimes) div 2))) (length t_twice div 2 + 13))"
+  proof(auto simp: mod_ex1 t_twice_def t_twice_compile_def)
+    fix q qa
+    assume h: "length (tm_of abc_twice) = 2 * q" "length (tm_of abc_fourtimes) = 2 * qa"
+    hence "list_all (\<lambda>(acn, st). st \<le> (9 + qa + (21 + q)))
+      (shift (turing_basic.adjust (tm_of abc_fourtimes @ shift (mopup (Suc 0)) qa)) (21 + q))"
+    proof(rule_tac tm_wf_shift t_twice_compile_def)
+      have "list_all (\<lambda>(acn, st). st \<le> Suc (length (tm_of abc_fourtimes @ shift 
+        (mopup (Suc 0)) qa) div 2)) (adjust (tm_of abc_fourtimes @ shift (mopup (Suc 0)) qa))"
+        apply(rule_tac tm_wf_change_termi)
+        using wf_fourtimes h
+        apply(simp add: t_fourtimes_compile_def)
+        done        
+      thus "list_all (\<lambda>(acn, st). st \<le> 9 + qa) ((turing_basic.adjust (tm_of abc_fourtimes @ shift (mopup (Suc 0)) qa)))"
+        using h
+        apply(simp)
+        done
+    qed
+    thus "list_all (\<lambda>(acn, st). st \<le> 30 + (q + qa)) (shift (turing_basic.adjust (tm_of abc_fourtimes @ shift (mopup (Suc 0)) qa)) (21 + q))"
+      apply(subgoal_tac "qa + q = q + qa")
+      apply(simp, simp)
+      done
+  qed
+  thus "b \<le> (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2"
+    using g
+    apply(simp add: Ball_set[THEN sym])
+    apply(erule_tac x = "(a, b)" in ballE, simp, simp)
     done
 qed
 
-lemma [intro]: "t_correct (t_wcode_prepare |+| t_wcode_main)"
-apply(auto intro: t_correct_add)
+lemma [intro]: "tm_wf (t_wcode_main, 0)"
+apply(auto simp: t_wcode_main_def tm_wf.simps
+                 t_twice_def t_fourtimes_def del: List.list_all_iff)
+done
+
+declare tm_comp.simps[simp del]
+lemma tm_wf_comp: "\<lbrakk>tm_wf (A, 0); tm_wf (B, 0)\<rbrakk> \<Longrightarrow> tm_wf (A |+| B, 0)"
+apply(auto simp: tm_wf.simps shift.simps adjust.simps tm_comp_length
+                 tm_comp.simps)
+done
+
+lemma [intro]: "tm_wf (t_wcode_prepare |+| t_wcode_main, 0)"
+apply(rule_tac tm_wf_comp, auto)
 done
 
 lemma prepare_mainpart_lemma:
   "args \<noteq> [] \<Longrightarrow> 
-  \<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp
-              = (0,  Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  \<exists> stp ln rn. steps0 (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp
+              = (0,  Bk # Oc\<up>(Suc m), Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin (<args>)) @ Bk\<up>(rn))"
 proof -
-  let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <m # args>"
-  let ?Q1 = "\<lambda> (l, r). wprepare_stop m args (l, r)"
+  let ?P1 = "(\<lambda> (l, r). (l::cell list) = [] \<and> r = <m # args>)"
+  let ?Q1 = "(\<lambda> (l, r). wprepare_stop m args (l, r))"
   let ?P2 = ?Q1
-  let ?Q2 = "\<lambda> (l, r). (\<exists> ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-                           r =  Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  let ?Q2 = "(\<lambda> (l, r). (\<exists> ln rn. l = Bk # Oc\<up>(Suc m) \<and>
+                           r =  Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin (<args>)) @ Bk\<up>(rn)))"
   let ?P3 = "\<lambda> tp. False"
   assume h: "args \<noteq> []"
-  have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp)
-                      (t_wcode_prepare |+| t_wcode_main) stp = (0, tp') \<and> ?Q2 tp')"
-  proof(rule_tac turing_merge.t_merge_halt[of t_wcode_prepare t_wcode_main ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2], 
-        auto simp: turing_merge_def)
-    show "\<exists>stp. case steps (Suc 0, [], <m # args>) t_wcode_prepare stp of (st, tp')
-                  \<Rightarrow> st = 0 \<and> wprepare_stop m args tp'"
-      using wprepare_correctness[of args m] h
-      apply(simp, auto)
-      apply(rule_tac x = n in exI, simp add: wprepare_inv.simps)
-      done
+  have "{?P1} t_wcode_prepare |+| t_wcode_main {?Q2}"
+  proof(rule_tac Hoare_plus_halt)
+    show "?Q1 \<mapsto> ?P2"
+      by(simp add: assert_imp_def)
+  next
+    show "tm_wf (t_wcode_prepare, 0)"
+      by auto
+  next
+    show "{?P1} t_wcode_prepare {?Q1}"
+    proof(rule_tac HoareI, auto)
+      show "\<exists>n. is_final (steps0 (Suc 0, [], <m # args>) t_wcode_prepare n) \<and>
+        wprepare_stop m args holds_for steps0 (Suc 0, [], <m # args>) t_wcode_prepare n"
+        using wprepare_correctness[of args m] h
+        apply(auto)
+        apply(rule_tac x = n in exI, simp add: wprepare_inv.simps)
+        done
+    qed
   next
-    fix a b
-    assume "wprepare_stop m args (a, b)"
-    thus "\<exists>stp. case steps (Suc 0, a, b) t_wcode_main stp of
-      (st, tp') \<Rightarrow> (st = 0) \<and> (case tp' of (l, r) \<Rightarrow> l = Bk # Oc\<^bsup>Suc m\<^esup> \<and> 
-      (\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>))"
-      proof(simp only: wprepare_stop.simps, erule_tac exE)
+    show "{?P2} t_wcode_main {?Q2}"
+    proof(rule_tac HoareI, auto)
+      fix l r
+      assume "wprepare_stop m args (l, r)"
+      thus "\<exists>n. is_final (steps0 (Suc 0, l, r) t_wcode_main n) \<and>
+              (\<lambda>(l, r). l = Bk # Oc # Oc \<up> m \<and> (\<exists>ln rn. r = Bk # Oc # Bk \<up> ln @ 
+        Bk # Bk # Oc \<up> bl_bin (<args>) @ Bk \<up> rn)) holds_for steps0 (Suc 0, l, r) t_wcode_main n"
+      proof(auto simp: wprepare_stop.simps)
         fix rn
-        assume "a = Bk # <rev args> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and> 
-                   b = Bk # Oc # Bk\<^bsup>rn\<^esup>"
-        thus "?thesis"
-          using t_wcode_main_lemma_pre[of "args" "<args>" 0 "Oc\<^bsup>Suc m\<^esup>" 0 rn] h
-          apply(simp)
-          apply(erule_tac exE)+
-          apply(rule_tac x = stp in exI, simp add: tape_of_nl_rev, auto)
+        show " \<exists>n. is_final (steps0 (Suc 0, Bk # <rev args> @ Bk # Bk # Oc # Oc \<up> m, Bk # Oc # Bk \<up> rn) t_wcode_main n) \<and>
+          (\<lambda>(l, r). l = Bk # Oc # Oc \<up> m \<and>
+          (\<exists>ln rn. r = Bk # Oc # Bk \<up> ln @
+          Bk # Bk # Oc \<up> bl_bin (<args>) @
+          Bk \<up> rn)) holds_for steps0 (Suc 0, Bk # <rev args> @ Bk # Bk # Oc # Oc \<up> m, Bk # Oc # Bk \<up> rn) t_wcode_main n"
+          using t_wcode_main_lemma_pre[of "args" "<args>" 0 "Oc\<up>(Suc m)" 0 rn] h
+          apply(auto simp: tape_of_nl_rev)
+          apply(rule_tac x = stp in exI, auto)
           done
       qed
-  next
-    show "wprepare_stop m args \<turnstile>-> wprepare_stop m args"
-      by(simp add: t_imply_def)
+    qed
   qed
-  thus "\<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp
-              = (0,  Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-    apply(simp add: t_imply_def)
-    apply(erule_tac exE)+
-    apply(auto)
+  thus "?thesis"
+    apply(auto simp: Hoare_def)
+    apply(rule_tac x = n in exI)
+    apply(case_tac "(steps0 (Suc 0, [], <m # args>)
+      (turing_basic.adjust t_wcode_prepare @ shift t_wcode_main (length t_wcode_prepare div 2)) n)")
+    apply(auto simp: tm_comp.simps)
     done
 qed
-      
-
+   
 lemma [simp]:  "tinres r r' \<Longrightarrow> 
-  fetch t ss (case r of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) = 
-  fetch t ss (case r' of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)"
+  fetch t ss (read r) = 
+  fetch t ss (read r')"
 apply(simp add: fetch.simps, auto split: if_splits simp: tinres_def)
-apply(case_tac [!] r', simp_all)
-apply(case_tac [!] n, simp_all add: exp_ind_def)
-apply(case_tac [!] r, simp_all)
-done
-
-lemma [intro]: "\<exists> n. (a::block)\<^bsup>n\<^esup> = []"
+apply(case_tac [!] n, simp_all)
+done
+
+lemma [intro]: "\<exists> n. (a::cell)\<up>(n) = []"
 by auto
 
 lemma [simp]: "\<lbrakk>tinres r r'; r \<noteq> []; r' \<noteq> []\<rbrakk> \<Longrightarrow> hd r = hd r'"
 apply(auto simp: tinres_def)
 done
 
-lemma [intro]: "hd (Bk\<^bsup>Suc n\<^esup>) = Bk"
-apply(simp add: exp_ind_def)
+lemma [intro]: "hd (Bk\<up>(Suc n)) = Bk"
+apply(simp add: )
 done
 
 lemma [simp]: "\<lbrakk>tinres r []; r \<noteq> []\<rbrakk> \<Longrightarrow> hd r = Bk"
 apply(auto simp: tinres_def)
-apply(case_tac n, auto)
 done
 
 lemma [simp]: "\<lbrakk>tinres [] r'; r' \<noteq> []\<rbrakk> \<Longrightarrow> hd r' = Bk"
 apply(auto simp: tinres_def)
 done
 
-lemma [intro]: "\<exists>na. tl r = tl (r @ Bk\<^bsup>n\<^esup>) @ Bk\<^bsup>na\<^esup> \<or> tl (r @ Bk\<^bsup>n\<^esup>) = tl r @ Bk\<^bsup>na\<^esup>"
+lemma [intro]: "\<exists>na. tl r = tl (r @ Bk\<up>(n)) @ Bk\<up>(na) \<or> tl (r @ Bk\<up>(n)) = tl r @ Bk\<up>(na)"
 apply(case_tac r, simp)
-apply(case_tac n, simp)
-apply(rule_tac x = 0 in exI, simp)
-apply(rule_tac x = nat in exI, simp add: exp_ind_def)
-apply(simp)
+apply(case_tac n, simp, simp)
+apply(rule_tac x = nat in exI, simp)
 apply(rule_tac x = n in exI, simp)
 done
 
 lemma [simp]: "tinres r r' \<Longrightarrow> tinres (tl r) (tl r')"
 apply(auto simp: tinres_def)
-apply(case_tac r', simp_all)
-apply(case_tac n, simp_all add: exp_ind_def)
-apply(rule_tac x = 0 in exI, simp)
-apply(rule_tac x = nat in exI, simp_all)
+apply(case_tac r', simp)
+apply(case_tac n, simp_all)
+apply(rule_tac x = nat in exI, simp)
 apply(rule_tac x = n in exI, simp)
 done
 
 lemma [simp]: "\<lbrakk>tinres r [];  r \<noteq> []\<rbrakk> \<Longrightarrow> tinres (tl r) []"
 apply(case_tac r, auto simp: tinres_def)
-apply(case_tac n, simp_all add: exp_ind_def)
+apply(case_tac n, simp_all add: )
 apply(rule_tac x = nat in exI, simp)
 done
 
 lemma [simp]: "\<lbrakk>tinres [] r'\<rbrakk> \<Longrightarrow> tinres [] (tl r')"
 apply(case_tac r', auto simp: tinres_def)
-apply(case_tac n, simp_all add: exp_ind_def)
+apply(case_tac n, simp_all add: )
 apply(rule_tac x = nat in exI, simp)
 done
 
@@ -3503,32 +3572,38 @@
 apply(auto simp: tinres_def)
 done
 
+lemma [simp]: "tinres r [] \<Longrightarrow> tinres (Bk # tl r) [Bk]"
+apply(auto simp: tinres_def)
+done
+
+lemma [simp]: "tinres r [] \<Longrightarrow> tinres (Oc # tl r) [Oc]"
+apply(auto simp: tinres_def)
+done
+
 lemma tinres_step2: 
-  "\<lbrakk>tinres r r'; tstep (ss, l, r) t = (sa, la, ra); tstep (ss, l, r') t = (sb, lb, rb)\<rbrakk>
+  "\<lbrakk>tinres r r'; step0 (ss, l, r) t = (sa, la, ra); step0 (ss, l, r') t = (sb, lb, rb)\<rbrakk>
     \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb"
-apply(case_tac "ss = 0", simp add: tstep_0)
-apply(simp add: tstep.simps [simp del])
-apply(case_tac "fetch t ss (case r of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
-apply(auto simp: new_tape.simps)
-apply(simp_all split: taction.splits if_splits)
-apply(auto)
-done
-
+apply(case_tac "ss = 0", simp add: step_0)
+apply(simp add: step.simps [simp del], auto)
+apply(case_tac [!] "fetch t ss (read r')", simp)
+apply(auto simp: update.simps)
+apply(case_tac [!] a, auto split: if_splits)
+done
 
 lemma tinres_steps2: 
-  "\<lbrakk>tinres r r'; steps (ss, l, r) t stp = (sa, la, ra); steps (ss, l, r') t stp = (sb, lb, rb)\<rbrakk>
+  "\<lbrakk>tinres r r'; steps0 (ss, l, r) t stp = (sa, la, ra); steps0 (ss, l, r') t stp = (sb, lb, rb)\<rbrakk>
     \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb"
 apply(induct stp arbitrary: sa la ra sb lb rb, simp add: steps.simps)
-apply(simp add: tstep_red)
-apply(case_tac "(steps (ss, l, r) t stp)")
-apply(case_tac "(steps (ss, l, r') t stp)")
+apply(simp add: step_red)
+apply(case_tac "(steps0 (ss, l, r) t stp)")
+apply(case_tac "(steps0 (ss, l, r') t stp)")
 proof -
   fix stp sa la ra sb lb rb a b c aa ba ca
-  assume ind: "\<And>sa la ra sb lb rb. \<lbrakk>steps (ss, l, r) t stp = (sa, la, ra); 
-    steps (ss, l, r') t stp = (sb, lb, rb)\<rbrakk> \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb"
-  and h: " tinres r r'" "tstep (steps (ss, l, r) t stp) t = (sa, la, ra)"
-         "tstep (steps (ss, l, r') t stp) t = (sb, lb, rb)" "steps (ss, l, r) t stp = (a, b, c)" 
-         "steps (ss, l, r') t stp = (aa, ba, ca)"
+  assume ind: "\<And>sa la ra sb lb rb. \<lbrakk>steps0 (ss, l, r) t stp = (sa, la, ra); 
+    steps0 (ss, l, r') t stp = (sb, lb, rb)\<rbrakk> \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb"
+  and h: " tinres r r'" "step0 (steps0 (ss, l, r) t stp) t = (sa, la, ra)"
+         "step0 (steps0 (ss, l, r') t stp) t = (sb, lb, rb)" "steps0 (ss, l, r) t stp = (a, b, c)" 
+         "steps0 (ss, l, r') t stp = (aa, ba, ca)"
   have "b = ba \<and> tinres c ca \<and> a = aa"
     apply(rule_tac ind, simp_all add: h)
     done
@@ -3539,8 +3614,8 @@
     apply(simp, simp, simp)
     done
 qed
- 
-definition t_wcode_adjust :: "tprog"
+
+definition t_wcode_adjust :: "instr list"
   where
   "t_wcode_adjust = [(W1, 1), (R, 2), (Nop, 2), (R, 3), (R, 3), (R, 4), 
                    (L, 8), (L, 5), (L, 6), (W0, 5), (L, 6), (R, 7), 
@@ -3566,112 +3641,115 @@
 lemma [simp]: "fetch t_wcode_adjust  (Suc (Suc (Suc 0))) Bk = (R, 3)"
 apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
 done
-   
-lemma [simp]: "fetch t_wcode_adjust 4 Bk = (L, 8)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 4 Oc = (L, 5)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
+
+lemma [simp]: "fetch t_wcode_adjust (Suc (Suc (Suc (Suc 0)))) Bk = (L, 8)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_4_eq_4)
+done
+
+lemma [simp]: "fetch t_wcode_adjust (Suc (Suc (Suc (Suc 0)))) Oc = (L, 5)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_4_eq_4)
+done
+
+thm numeral_5_eq_5
 
 lemma [simp]: "fetch t_wcode_adjust 5 Oc = (W0, 5)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp only: fetch.simps t_wcode_adjust_def nth_of.simps numeral_5_eq_5, simp)
 done
 
 lemma [simp]: "fetch t_wcode_adjust 5 Bk = (L, 6)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
+apply(simp only: fetch.simps t_wcode_adjust_def nth_of.simps numeral_5_eq_5, auto)
+done
+
+thm numeral_6_eq_6
 lemma [simp]: "fetch t_wcode_adjust 6 Oc = (R, 7)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_6_eq_6)
 done
 
 lemma [simp]: "fetch t_wcode_adjust 6 Bk = (L, 6)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_6_eq_6)
 done
 
 lemma [simp]: "fetch t_wcode_adjust 7 Bk = (W1, 2)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_7_eq_7)
 done
 
 lemma [simp]: "fetch t_wcode_adjust 8 Bk = (L, 9)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_8_eq_8)
 done
 
 lemma [simp]: "fetch t_wcode_adjust 8 Oc = (W0, 8)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_8_eq_8)
 done
 
 lemma [simp]: "fetch t_wcode_adjust 9 Oc = (L, 10)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_9_eq_9)
 done
 
 lemma [simp]: "fetch t_wcode_adjust 9 Bk = (L, 9)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_9_eq_9)
 done
 
 lemma [simp]: "fetch t_wcode_adjust 10 Bk = (L, 11)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps  numeral_10_eq_10)
 done
 
 lemma [simp]: "fetch t_wcode_adjust 10 Oc = (L, 10)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps eval_nat_numeral)
 done
 
 lemma [simp]: "fetch t_wcode_adjust 11 Oc = (L, 11)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps eval_nat_numeral)
 done
 
 lemma [simp]: "fetch t_wcode_adjust 11 Bk = (R, 0)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps eval_nat_numeral)
 done
 
 fun wadjust_start :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wadjust_start m rs (l, r) = 
-         (\<exists> ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-                   tl r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+         (\<exists> ln rn. l = Bk # Oc\<up>(Suc m) \<and>
+                   tl r = Oc # Bk\<up>(ln) @ Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 
 fun wadjust_loop_start :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wadjust_loop_start m rs (l, r) = 
-          (\<exists> ln rn ml mr. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup>  \<and>
-                          r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+          (\<exists> ln rn ml mr. l = Oc\<up>(ml) @ Bk # Oc\<up>(Suc m)  \<and>
+                          r = Oc # Bk\<up>(ln) @ Bk # Oc\<up>(mr) @ Bk\<up>(rn) \<and>
                           ml + mr = Suc (Suc rs) \<and> mr > 0)"
 
 fun wadjust_loop_right_move :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wadjust_loop_right_move m rs (l, r) = 
-   (\<exists> ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-                      r = Bk\<^bsup>nr\<^esup> @ Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+   (\<exists> ml mr nl nr rn. l = Bk\<up>(nl) @ Oc # Oc\<up>(ml) @ Bk # Oc\<up>(Suc m) \<and>
+                      r = Bk\<up>(nr) @ Oc\<up>(mr) @ Bk\<up>(rn) \<and>
                       ml + mr = Suc (Suc rs) \<and> mr > 0 \<and>
                       nl + nr > 0)"
 
 fun wadjust_loop_check :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wadjust_loop_check m rs (l, r) = 
-  (\<exists> ml mr ln rn. l = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-                  r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = (Suc rs))"
+  (\<exists> ml mr ln rn. l = Oc # Bk\<up>(ln) @ Bk # Oc # Oc\<up>(ml) @ Bk # Oc\<up>(Suc m) \<and>
+                  r = Oc\<up>(mr) @ Bk\<up>(rn) \<and> ml + mr = (Suc rs))"
 
 fun wadjust_loop_erase :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wadjust_loop_erase m rs (l, r) = 
-    (\<exists> ml mr ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-                    tl r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = (Suc rs) \<and> mr > 0)"
+    (\<exists> ml mr ln rn. l = Bk\<up>(ln) @ Bk # Oc # Oc\<up>(ml) @ Bk # Oc\<up>(Suc m) \<and>
+                    tl r = Oc\<up>(mr) @ Bk\<up>(rn) \<and> ml + mr = (Suc rs) \<and> mr > 0)"
 
 fun wadjust_loop_on_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wadjust_loop_on_left_moving_O m rs (l, r) = 
-      (\<exists> ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m \<^esup>\<and>
-                      r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+      (\<exists> ml mr ln rn. l = Oc\<up>(ml) @ Bk # Oc\<up>(Suc m )\<and>
+                      r = Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(mr) @ Bk\<up>(rn) \<and>
                       ml + mr = Suc rs \<and> mr > 0)"
 
 fun wadjust_loop_on_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wadjust_loop_on_left_moving_B m rs (l, r) = 
-      (\<exists> ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-                         r = Bk\<^bsup>nr\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> 
+      (\<exists> ml mr nl nr rn. l = Bk\<up>(nl) @ Oc # Oc\<up>(ml) @ Bk # Oc\<up>(Suc m) \<and>
+                         r = Bk\<up>(nr) @ Bk # Bk # Oc\<up>(mr) @ Bk\<up>(rn) \<and> 
                          ml + mr = Suc rs \<and> mr > 0)"
 
 fun wadjust_loop_on_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
@@ -3683,27 +3761,27 @@
 fun wadjust_loop_right_move2 :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wadjust_loop_right_move2 m rs (l, r) = 
-        (\<exists> ml mr ln rn. l = Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-                        r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+        (\<exists> ml mr ln rn. l = Oc # Oc\<up>(ml) @ Bk # Oc\<up>(Suc m) \<and>
+                        r = Bk\<up>(ln) @ Bk # Bk # Oc\<up>(mr) @ Bk\<up>(rn) \<and>
                         ml + mr = Suc rs \<and> mr > 0)"
 
 fun wadjust_erase2 :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wadjust_erase2 m rs (l, r) = 
-     (\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-                     tl r = Bk\<^bsup>rn\<^esup>)"
+     (\<exists> ln rn. l = Bk\<up>(ln) @ Bk # Oc # Oc\<up>(Suc rs) @ Bk # Oc\<up>(Suc m) \<and>
+                     tl r = Bk\<up>(rn))"
 
 fun wadjust_on_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wadjust_on_left_moving_O m rs (l, r) = 
-        (\<exists> rn. l = Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-                  r = Oc # Bk\<^bsup>rn\<^esup>)"
+        (\<exists> rn. l = Oc\<up>(Suc rs) @ Bk # Oc\<up>(Suc m) \<and>
+                  r = Oc # Bk\<up>(rn))"
 
 fun wadjust_on_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wadjust_on_left_moving_B m rs (l, r) = 
-         (\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-                   r = Bk\<^bsup>rn\<^esup>)"
+         (\<exists> ln rn. l = Bk\<up>(ln) @ Oc # Oc\<up>(Suc rs) @ Bk # Oc\<up>(Suc m) \<and>
+                   r = Bk\<up>(rn))"
 
 fun wadjust_on_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
@@ -3714,14 +3792,14 @@
 fun wadjust_goon_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where 
   "wadjust_goon_left_moving_B m rs (l, r) = 
-        (\<exists> rn. l = Oc\<^bsup>Suc m\<^esup> \<and> 
-               r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+        (\<exists> rn. l = Oc\<up>(Suc m) \<and> 
+               r = Bk # Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 
 fun wadjust_goon_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wadjust_goon_left_moving_O m rs (l, r) = 
-      (\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-                      r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> 
+      (\<exists> ml mr rn. l = Oc\<up>(ml) @ Bk # Oc\<up>(Suc m) \<and>
+                      r = Oc\<up>(mr) @ Bk\<up>(rn) \<and> 
                       ml + mr = Suc (Suc rs) \<and> mr > 0)"
 
 fun wadjust_goon_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
@@ -3734,13 +3812,13 @@
   where
   "wadjust_backto_standard_pos_B m rs (l, r) =
         (\<exists> rn. l = [] \<and> 
-               r = Bk # Oc\<^bsup>Suc m \<^esup>@ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+               r = Bk # Oc\<up>(Suc m )@ Bk # Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 
 fun wadjust_backto_standard_pos_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
   where
   "wadjust_backto_standard_pos_O m rs (l, r) = 
-      (\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> \<and>
-                      r = Oc\<^bsup>mr\<^esup> @ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> 
+      (\<exists> ml mr rn. l = Oc\<up>(ml) \<and>
+                      r = Oc\<up>(mr) @ Bk # Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn) \<and> 
                       ml + mr = Suc m \<and> mr > 0)"
 
 fun wadjust_backto_standard_pos :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
@@ -3753,7 +3831,7 @@
 where
   "wadjust_stop m rs (l, r) =
         (\<exists> rn. l = [Bk] \<and> 
-               r = Oc\<^bsup>Suc m \<^esup>@ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+               r = Oc\<up>(Suc m )@ Bk # Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 
 declare wadjust_start.simps[simp del]  wadjust_loop_start.simps[simp del]
         wadjust_loop_right_move.simps[simp del]  wadjust_loop_check.simps[simp del]
@@ -3785,7 +3863,7 @@
 
 declare wadjust_inv.simps[simp del]
 
-fun wadjust_phase :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
+fun wadjust_phase :: "nat \<Rightarrow> config \<Rightarrow> nat"
   where
   "wadjust_phase rs (st, l, r) = 
          (if st = 1 then 3 
@@ -3793,9 +3871,7 @@
           else if st \<ge> 8 \<and> st \<le> 11 then 1
           else 0)"
 
-thm dropWhile.simps
-
-fun wadjust_stage :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
+fun wadjust_stage :: "nat \<Rightarrow> config \<Rightarrow> nat"
   where
   "wadjust_stage rs (st, l, r) = 
            (if st \<ge> 2 \<and> st \<le> 7 then 
@@ -3803,14 +3879,14 @@
                           (tl (dropWhile (\<lambda> a. a = Oc) (rev l @ r))))
             else 0)"
 
-fun wadjust_state :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
+fun wadjust_state :: "nat \<Rightarrow> config \<Rightarrow> nat"
   where
   "wadjust_state rs (st, l, r) = 
        (if st \<ge> 2 \<and> st \<le> 7 then 8 - st
         else if st \<ge> 8 \<and> st \<le> 11 then 12 - st
         else 0)"
 
-fun wadjust_step :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
+fun wadjust_step :: "nat \<Rightarrow> config \<Rightarrow> nat"
   where
   "wadjust_step rs (st, l, r) = 
        (if st = 1 then (if hd r = Bk then 1
@@ -3827,7 +3903,7 @@
                               else Suc (length l))
         else 0)"
 
-fun wadjust_measure :: "(nat \<times> t_conf) \<Rightarrow> nat \<times> nat \<times> nat \<times> nat"
+fun wadjust_measure :: "(nat \<times> config) \<Rightarrow> nat \<times> nat \<times> nat \<times> nat"
   where
   "wadjust_measure (rs, (st, l, r)) = 
      (wadjust_phase rs (st, l, r), 
@@ -3835,7 +3911,7 @@
       wadjust_state rs (st, l, r), 
       wadjust_step rs (st, l, r))"
 
-definition wadjust_le :: "((nat \<times> t_conf) \<times> nat \<times> t_conf) set"
+definition wadjust_le :: "((nat \<times> config) \<times> nat \<times> config) set"
   where "wadjust_le \<equiv> (inv_image lex_square wadjust_measure)"
 
 lemma [intro]: "wf lex_square"
@@ -3858,7 +3934,6 @@
         \<Longrightarrow>  wadjust_loop_check m rs (Bk # c, [])"
 apply(simp only: wadjust_loop_right_move.simps wadjust_loop_check.simps)
 apply(auto)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
 done
 
 lemma [simp]: "wadjust_loop_check m rs (c, []) \<Longrightarrow> c \<noteq> []"
@@ -3874,19 +3949,16 @@
 apply(simp only: wadjust_loop_right_move.simps)
 apply(erule_tac exE)+
 apply(auto)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
 done
 
 lemma [simp]: "wadjust_loop_check m rs (c, []) \<Longrightarrow> wadjust_erase2 m rs (tl c, [hd c])"
 apply(simp only: wadjust_loop_check.simps wadjust_erase2.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def, auto)
 done
 
 lemma [simp]: " wadjust_loop_erase m rs (c, [])
     \<Longrightarrow> (c = [] \<longrightarrow> wadjust_loop_on_left_moving m rs ([], [Bk])) \<and>
         (c \<noteq> [] \<longrightarrow> wadjust_loop_on_left_moving m rs (tl c, [hd c]))"
-apply(simp add: wadjust_loop_erase.simps, auto)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
+apply(simp add: wadjust_loop_erase.simps)
 done
 
 lemma [simp]: "wadjust_loop_on_left_moving m rs (c, []) = False"
@@ -3903,22 +3975,21 @@
 done
 
 lemma [simp]: "wadjust_on_left_moving_B m rs 
-                 (Oc # Oc # Oc\<^bsup>rs\<^esup> @ Bk # Oc # Oc\<^bsup>m\<^esup>, [Bk])"
+                 (Oc # Oc # Oc\<up>(rs) @ Bk # Oc # Oc\<up>(m), [Bk])"
 apply(simp add: wadjust_on_left_moving_B.simps, auto)
-apply(rule_tac x = 0 in exI, simp add: exp_ind_def)
 done
 
 lemma [simp]: "wadjust_on_left_moving_B m rs 
-                 (Bk\<^bsup>n\<^esup> @ Bk # Oc # Oc # Oc\<^bsup>rs\<^esup> @ Bk # Oc # Oc\<^bsup>m\<^esup>, [Bk])"
-apply(simp add: wadjust_on_left_moving_B.simps exp_ind_def, auto)
-apply(rule_tac x = "Suc n" in exI, simp add: exp_ind)
+                 (Bk\<up>(n) @ Bk # Oc # Oc # Oc\<up>(rs) @ Bk # Oc # Oc\<up>(m), [Bk])"
+apply(simp add: wadjust_on_left_moving_B.simps , auto)
+apply(rule_tac x = "Suc n" in exI, simp add: exp_ind del: replicate_Suc)
 done
 
 lemma [simp]: "\<lbrakk>wadjust_erase2 m rs (c, []); c \<noteq> []\<rbrakk> \<Longrightarrow>
             wadjust_on_left_moving m rs (tl c, [hd c])"
 apply(simp only: wadjust_erase2.simps)
 apply(erule_tac exE)+
-apply(case_tac ln, simp_all add: exp_ind_def wadjust_on_left_moving.simps)
+apply(case_tac ln, simp_all add:  wadjust_on_left_moving.simps)
 done
 
 lemma [simp]: "wadjust_erase2 m rs (c, [])
@@ -3939,13 +4010,13 @@
 lemma [simp]: " \<lbrakk>wadjust_on_left_moving_B m rs (c, []); c \<noteq> []; hd c = Bk\<rbrakk> \<Longrightarrow>
                                       wadjust_on_left_moving_B m rs (tl c, [Bk])"
 apply(simp add: wadjust_on_left_moving_B.simps, auto)
-apply(case_tac [!] ln, simp_all add: exp_ind_def, auto)
+apply(case_tac [!] ln, simp_all)
 done
 
 lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, []); c \<noteq> []; hd c = Oc\<rbrakk> \<Longrightarrow>
                                   wadjust_on_left_moving_O m rs (tl c, [Oc])"
 apply(simp add: wadjust_on_left_moving_B.simps wadjust_on_left_moving_O.simps, auto)
-apply(case_tac [!] ln, simp_all add: exp_ind_def)
+apply(case_tac [!] ln, simp_all add: )
 done
 
 lemma [simp]: "\<lbrakk>wadjust_on_left_moving m rs (c, []); c \<noteq> []\<rbrakk> \<Longrightarrow> 
@@ -3991,8 +4062,8 @@
 apply(erule_tac exE)+
 apply(rule_tac x = ml in exI, simp)
 apply(rule_tac x = mr in exI, simp)
-apply(rule_tac x = "Suc nl" in exI, simp add: exp_ind_def)
-apply(case_tac nr, simp, case_tac mr, simp_all add: exp_ind_def)
+apply(rule_tac x = "Suc nl" in exI, simp add: )
+apply(case_tac nr, simp, case_tac mr, simp_all add: )
 apply(rule_tac x = nat in exI, auto)
 done
 
@@ -4003,7 +4074,7 @@
 lemma [simp]: "wadjust_loop_check m rs (c, Bk # list)
               \<Longrightarrow>  wadjust_erase2 m rs (tl c, hd c # Bk # list)"
 apply(auto simp: wadjust_loop_check.simps wadjust_erase2.simps)
-apply(case_tac [!] mr, simp_all add: exp_ind_def, auto)
+apply(case_tac [!] mr, simp_all)
 done
 
 lemma [simp]: "wadjust_loop_erase m rs (c, b) \<Longrightarrow> c \<noteq> []"
@@ -4020,15 +4091,15 @@
 apply(erule_tac exE)+
 apply(rule_tac x = ml in exI, rule_tac x = mr in exI, 
       rule_tac x = ln in exI, rule_tac x = 0 in exI, simp)
-apply(case_tac ln, simp_all add: exp_ind_def, auto)
-apply(simp add: exp_ind exp_ind_def[THEN sym])
+apply(case_tac ln, simp_all add: , auto)
+apply(simp add: exp_ind [THEN sym])
 done
 
 lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); c \<noteq> []; hd c = Oc\<rbrakk> \<Longrightarrow>
              wadjust_loop_on_left_moving_O m rs (tl c, Oc # Bk # list)"
 apply(simp only: wadjust_loop_erase.simps wadjust_loop_on_left_moving_O.simps,
        auto)
-apply(case_tac [!] ln, simp_all add: exp_ind_def)
+apply(case_tac [!] ln, simp_all add: )
 done
 
 lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); c \<noteq> []\<rbrakk> \<Longrightarrow> 
@@ -4050,8 +4121,8 @@
 apply(simp only: wadjust_loop_on_left_moving_B.simps)
 apply(erule_tac exE)+
 apply(rule_tac x = ml in exI, rule_tac x = mr in exI)
-apply(case_tac nl, simp_all add: exp_ind_def, auto)
-apply(rule_tac x = "Suc nr" in exI, auto simp: exp_ind_def)
+apply(case_tac nl, simp_all add: , auto)
+apply(rule_tac x = "Suc nr" in exI, auto simp: )
 done
 
 lemma [simp]: "\<lbrakk>wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk>
@@ -4060,7 +4131,7 @@
                  wadjust_loop_on_left_moving_B.simps)
 apply(erule_tac exE)+
 apply(rule_tac x = ml in exI, rule_tac x = mr in exI)
-apply(case_tac nl, simp_all add: exp_ind_def, auto)
+apply(case_tac nl, simp_all add: , auto)
 done
 
 lemma [simp]: "wadjust_loop_on_left_moving m rs (c, Bk # list)
@@ -4075,13 +4146,13 @@
 
 lemma [simp]: "wadjust_loop_right_move2 m rs (c, Bk # list) \<Longrightarrow>  wadjust_loop_start m rs (c, Oc # list)"
 apply(auto simp: wadjust_loop_right_move2.simps wadjust_loop_start.simps)
-apply(case_tac ln, simp_all add: exp_ind_def)
+apply(case_tac ln, simp_all add: )
 apply(rule_tac x = 0 in exI, simp)
 apply(rule_tac x = rn in exI, simp)
-apply(rule_tac x = "Suc ml" in exI, simp add: exp_ind_def, auto)
-apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind)
+apply(rule_tac x = "Suc ml" in exI, simp add: , auto)
+apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind del: replicate_Suc)
 apply(rule_tac x = rn in exI, auto)
-apply(rule_tac x = "Suc ml" in exI, auto simp: exp_ind_def)
+apply(rule_tac x = "Suc ml" in exI, auto )
 done
 
 lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \<Longrightarrow> c \<noteq> []"
@@ -4091,12 +4162,12 @@
 lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \<Longrightarrow> 
                  wadjust_on_left_moving m rs (tl c, hd c # Bk # list)"
 apply(auto simp: wadjust_erase2.simps)
-apply(case_tac ln, simp_all add: exp_ind_def wadjust_on_left_moving.simps 
+apply(case_tac ln, simp_all add:  wadjust_on_left_moving.simps 
         wadjust_on_left_moving_O.simps wadjust_on_left_moving_B.simps)
 apply(auto)
-apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: exp_ind_def)
-apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind)
-apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: exp_ind_def)
+apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: )
+apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind del: replicate_Suc)
+apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: )
 done
 
 lemma [simp]: "wadjust_on_left_moving m rs (c,b) \<Longrightarrow> c \<noteq> []"
@@ -4113,14 +4184,14 @@
 lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Bk\<rbrakk>
     \<Longrightarrow> wadjust_on_left_moving_B m rs (tl c, Bk # Bk # list)"
 apply(auto simp: wadjust_on_left_moving_B.simps)
-apply(case_tac ln, simp_all add: exp_ind_def, auto)
+apply(case_tac ln, simp_all)
 done
 
 lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk>
     \<Longrightarrow> wadjust_on_left_moving_O m rs (tl c, Oc # Bk # list)"
 apply(auto simp: wadjust_on_left_moving_O.simps
                  wadjust_on_left_moving_B.simps)
-apply(case_tac ln, simp_all add: exp_ind_def)
+apply(case_tac ln, simp_all add: )
 done
 
 lemma [simp]: "wadjust_on_left_moving  m rs (c, Bk # list) \<Longrightarrow>  
@@ -4132,25 +4203,24 @@
 lemma [simp]: "wadjust_goon_left_moving m rs (c, b) \<Longrightarrow> c \<noteq> []"
 apply(simp add: wadjust_goon_left_moving.simps
                 wadjust_goon_left_moving_B.simps
-                wadjust_goon_left_moving_O.simps exp_ind_def, auto)
+                wadjust_goon_left_moving_O.simps , auto)
 done
 
 lemma [simp]: "wadjust_goon_left_moving_O m rs (c, Bk # list) = False"
 apply(simp add: wadjust_goon_left_moving_O.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 done
 
 lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Bk\<rbrakk>
     \<Longrightarrow> wadjust_backto_standard_pos_B m rs (tl c, Bk # Bk # list)"
 apply(auto simp: wadjust_goon_left_moving_B.simps 
-                 wadjust_backto_standard_pos_B.simps exp_ind_def)
+                 wadjust_backto_standard_pos_B.simps )
 done
 
 lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk>
     \<Longrightarrow> wadjust_backto_standard_pos_O m rs (tl c, Oc # Bk # list)"
 apply(auto simp: wadjust_goon_left_moving_B.simps 
-                 wadjust_backto_standard_pos_O.simps exp_ind_def)
-apply(rule_tac x = m in exI, simp, auto)
+                 wadjust_backto_standard_pos_O.simps)
 done
 
 lemma [simp]: "wadjust_goon_left_moving m rs (c, Bk # list) \<Longrightarrow>
@@ -4164,7 +4234,7 @@
 apply(auto simp: wadjust_backto_standard_pos.simps 
                  wadjust_backto_standard_pos_B.simps
                  wadjust_backto_standard_pos_O.simps wadjust_stop.simps)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
+apply(case_tac [!] mr, simp_all add: )
 done
 
 lemma [simp]: "wadjust_start m rs (c, Oc # list)
@@ -4184,17 +4254,17 @@
 apply(simp add: wadjust_loop_start.simps wadjust_loop_right_move.simps, auto)
 apply(rule_tac x = ml in exI, rule_tac x = mr in exI, 
       rule_tac x = 0 in exI, simp)
-apply(rule_tac x = "Suc ln" in exI, simp add: exp_ind, auto)
+apply(rule_tac x = "Suc ln" in exI, simp add: exp_ind del: replicate_Suc)
 done
 
 lemma [simp]: "wadjust_loop_right_move m rs (c, Oc # list) \<Longrightarrow> 
                        wadjust_loop_check m rs (Oc # c, list)"
 apply(simp add: wadjust_loop_right_move.simps  
                  wadjust_loop_check.simps, auto)
-apply(rule_tac [!] x = ml in exI, simp_all, auto)
-apply(case_tac nl, auto simp: exp_ind_def)
-apply(rule_tac x = "mr - 1" in exI, case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac [!] nr, simp_all add: exp_ind_def, auto)
+apply(rule_tac [!] x = ml in exI, simp_all add: exp_ind del: replicate_Suc, auto)
+apply(case_tac nl, simp_all add: exp_ind del: replicate_Suc)
+apply(rule_tac x = "mr - 1" in exI, case_tac mr, simp_all add: )
+apply(case_tac [!] nr, simp_all)
 done
 
 lemma [simp]: "wadjust_loop_check m rs (c, Oc # list) \<Longrightarrow> 
@@ -4202,8 +4272,7 @@
 apply(simp only: wadjust_loop_check.simps wadjust_loop_erase.simps)
 apply(erule_tac exE)+
 apply(rule_tac x = ml in exI, rule_tac x = mr in exI, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac rn, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 done
 
 lemma [simp]: "wadjust_loop_erase m rs (c, Oc # list) \<Longrightarrow> 
@@ -4213,7 +4282,7 @@
 
 lemma [simp]: "wadjust_loop_on_left_moving_B m rs (c, Oc # list) = False"
 apply(auto simp: wadjust_loop_on_left_moving_B.simps)
-apply(case_tac nr, simp_all add: exp_ind_def)
+apply(case_tac nr, simp_all add: )
 done
 
 lemma [simp]: "wadjust_loop_on_left_moving m rs (c, Oc # list)
@@ -4225,7 +4294,7 @@
 
 lemma [simp]: "wadjust_loop_right_move2 m rs (c, Oc # list) = False"
 apply(auto simp: wadjust_loop_right_move2.simps )
-apply(case_tac ln, simp_all add: exp_ind_def)
+apply(case_tac ln, simp_all add: )
 done
 
 lemma [simp]: "wadjust_erase2 m rs (c, Oc # list)
@@ -4241,15 +4310,14 @@
 lemma [simp]: "\<lbrakk>wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Bk\<rbrakk> \<Longrightarrow> 
          wadjust_goon_left_moving_B m rs (tl c, Bk # Oc # list)"
 apply(auto simp: wadjust_on_left_moving_O.simps 
-     wadjust_goon_left_moving_B.simps exp_ind_def)
+     wadjust_goon_left_moving_B.simps )
 done
 
 lemma [simp]: "\<lbrakk>wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Oc\<rbrakk>
     \<Longrightarrow> wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)"
 apply(auto simp: wadjust_on_left_moving_O.simps 
-                 wadjust_goon_left_moving_O.simps exp_ind_def)
-apply(rule_tac x = rs in exI, simp)
-apply(auto simp: exp_ind_def numeral_2_eq_2)
+                 wadjust_goon_left_moving_O.simps )
+apply(auto simp:  numeral_2_eq_2)
 done
 
 
@@ -4274,15 +4342,15 @@
 lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Bk\<rbrakk> 
                \<Longrightarrow> wadjust_goon_left_moving_B m rs (tl c, Bk # Oc # list)"
 apply(auto simp: wadjust_goon_left_moving_O.simps wadjust_goon_left_moving_B.simps)
-apply(case_tac [!] ml, auto simp: exp_ind_def)
+apply(case_tac [!] ml, auto simp: )
 done
 
 lemma  [simp]: "\<lbrakk>wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Oc\<rbrakk> \<Longrightarrow> 
   wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)"
 apply(auto simp: wadjust_goon_left_moving_O.simps wadjust_goon_left_moving_B.simps)
 apply(rule_tac x = "ml - 1" in exI, simp)
-apply(case_tac ml, simp_all add: exp_ind_def)
-apply(rule_tac x = "Suc mr" in exI, auto simp: exp_ind_def)
+apply(case_tac ml, simp_all add: )
+apply(rule_tac x = "Suc mr" in exI, auto simp: )
 done
 
 lemma [simp]: "wadjust_goon_left_moving m rs (c, Oc # list) \<Longrightarrow> 
@@ -4297,33 +4365,26 @@
 
 lemma [simp]: "wadjust_backto_standard_pos_O m rs (c, Bk # xs) = False"
 apply(simp add: wadjust_backto_standard_pos_O.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-done
-
-
+apply(case_tac mr, simp_all add: )
+done
 
 lemma [simp]: "wadjust_backto_standard_pos_O m rs ([], Oc # list) \<Longrightarrow> 
   wadjust_backto_standard_pos_B m rs ([], Bk # Oc # list)"
 apply(auto simp: wadjust_backto_standard_pos_O.simps
                  wadjust_backto_standard_pos_B.simps)
-apply(rule_tac x = rn in exI, simp)
-apply(case_tac ml, simp_all add: exp_ind_def)
-done
-
+done
 
 lemma [simp]: 
   "\<lbrakk>wadjust_backto_standard_pos_O m rs (c, Oc # list); c \<noteq> []; hd c = Bk\<rbrakk>
   \<Longrightarrow> wadjust_backto_standard_pos_B m rs (tl c, Bk # Oc # list)"
 apply(simp add:wadjust_backto_standard_pos_O.simps 
         wadjust_backto_standard_pos_B.simps, auto)
-apply(case_tac [!] ml, simp_all add: exp_ind_def)
 done 
 
 lemma [simp]: "\<lbrakk>wadjust_backto_standard_pos_O m rs (c, Oc # list); c \<noteq> []; hd c = Oc\<rbrakk>
           \<Longrightarrow>  wadjust_backto_standard_pos_O m rs (tl c, Oc # Oc # list)"
 apply(simp add: wadjust_backto_standard_pos_O.simps, auto)
-apply(case_tac ml, simp_all add: exp_ind_def, auto)
-apply(rule_tac x = nat in exI, auto simp: exp_ind_def)
+apply(case_tac ml, simp_all add: , auto)
 done
 
 lemma [simp]: "wadjust_backto_standard_pos m rs (c, Oc # list)
@@ -4332,19 +4393,17 @@
 apply(auto simp: wadjust_backto_standard_pos.simps)
 apply(case_tac "hd c", simp_all)
 done
-thm wadjust_loop_right_move.simps
 
 lemma [simp]: "wadjust_loop_right_move m rs (c, []) = False"
 apply(simp only: wadjust_loop_right_move.simps)
 apply(rule_tac iffI)
 apply(erule_tac exE)+
-apply(case_tac nr, simp_all add: exp_ind_def)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac nr, simp_all add: )
+apply(case_tac mr, simp_all add: )
 done
 
 lemma [simp]: "wadjust_loop_erase m rs (c, []) = False"
 apply(simp only: wadjust_loop_erase.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
 done
 
 lemma [simp]: "\<lbrakk>Suc (Suc rs) = a;  wadjust_loop_erase m rs (c, Bk # list)\<rbrakk>
@@ -4367,11 +4426,11 @@
 apply(case_tac c, simp_all)
 done
 
-lemma dropWhile_exp1: "dropWhile (\<lambda>a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = dropWhile (\<lambda>a. a = Oc) xs"
-apply(induct n, simp_all add: exp_ind_def)
-done
-lemma takeWhile_exp1: "takeWhile (\<lambda>a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = Oc\<^bsup>n\<^esup> @ takeWhile (\<lambda>a. a = Oc) xs"
-apply(induct n, simp_all add: exp_ind_def)
+lemma dropWhile_exp1: "dropWhile (\<lambda>a. a = Oc) (Oc\<up>(n) @ xs) = dropWhile (\<lambda>a. a = Oc) xs"
+apply(induct n, simp_all add: )
+done
+lemma takeWhile_exp1: "takeWhile (\<lambda>a. a = Oc) (Oc\<up>(n) @ xs) = Oc\<up>(n) @ takeWhile (\<lambda>a. a = Oc) xs"
+apply(induct n, simp_all add: )
 done
 
 lemma [simp]: "\<lbrakk>Suc (Suc rs) = a;  wadjust_loop_right_move2 m rs (c, Bk # list)\<rbrakk>
@@ -4379,7 +4438,7 @@
                  < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))"
 apply(simp add: wadjust_loop_right_move2.simps, auto)
 apply(simp add: dropWhile_exp1 takeWhile_exp1)
-apply(case_tac ln, simp, simp add: exp_ind_def)
+apply(case_tac ln, simp, simp add: )
 done
 
 lemma [simp]: "wadjust_loop_check m rs ([], b) = False"
@@ -4411,129 +4470,119 @@
 lemma wadjust_correctness:
   shows "let P = (\<lambda> (len, st, l, r). st = 0) in 
   let Q = (\<lambda> (len, st, l, r). wadjust_inv st m rs (l, r)) in 
-  let f = (\<lambda> stp. (Suc (Suc rs),  steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, 
-                Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #  Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp)) in
+  let f = (\<lambda> stp. (Suc (Suc rs),  steps0 (Suc 0, Bk # Oc\<up>(Suc m), 
+                Bk # Oc # Bk\<up>(ln) @ Bk #  Oc\<up>(Suc rs) @ Bk\<up>(rn)) t_wcode_adjust stp)) in
     \<exists> n .P (f n) \<and> Q (f n)"
 proof -
   let ?P = "(\<lambda> (len, st, l, r). st = 0)"
   let ?Q = "\<lambda> (len, st, l, r). wadjust_inv st m rs (l, r)"
-  let ?f = "\<lambda> stp. (Suc (Suc rs),  steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, 
-                Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp)"
+  let ?f = "\<lambda> stp. (Suc (Suc rs),  steps0 (Suc 0, Bk # Oc\<up>(Suc m), 
+                Bk # Oc # Bk\<up>(ln) @ Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn)) t_wcode_adjust stp)"
   have "\<exists> n. ?P (?f n) \<and> ?Q (?f n)"
   proof(rule_tac halt_lemma2)
     show "wf wadjust_le" by auto
   next
     show "\<forall> n. \<not> ?P (?f n) \<and> ?Q (?f n) \<longrightarrow> 
                  ?Q (?f (Suc n)) \<and> (?f (Suc n), ?f n) \<in> wadjust_le"
-    proof(rule_tac allI, rule_tac impI, case_tac "?f n", 
-            simp add: tstep_red tstep.simps, rule_tac conjI, erule_tac conjE,
-          erule_tac conjE)      
-      fix n a b c d
-      assume "0 < b" "wadjust_inv b m rs (c, d)" "Suc (Suc rs) = a"
-      thus "case case fetch t_wcode_adjust b (case d of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)
-        of (ac, ns) \<Rightarrow> (ns, new_tape ac (c, d)) of (st, x) \<Rightarrow> wadjust_inv st m rs x"
-        apply(case_tac d, simp, case_tac [2] aa)
-        apply(simp_all add: wadjust_inv.simps wadjust_le_def new_tape.simps
-          abacus.lex_triple_def abacus.lex_pair_def lex_square_def
-          split: if_splits)
-        done
-    next
-      fix n a b c d
-      assume "0 < b \<and> wadjust_inv b m rs (c, d)"
-        "Suc (Suc rs) = a \<and> steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>,
-         Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust n = (b, c, d)"
-      thus "((a, case fetch t_wcode_adjust b (case d of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)
-        of (ac, ns) \<Rightarrow> (ns, new_tape ac (c, d))), a, b, c, d) \<in> wadjust_le"
-      proof(erule_tac conjE, erule_tac conjE, erule_tac conjE)
-        assume "0 < b" "wadjust_inv b m rs (c, d)" "Suc (Suc rs) = a"
-        thus "?thesis"
-          apply(case_tac d, case_tac [2] aa)
-          apply(simp_all add: wadjust_inv.simps wadjust_le_def new_tape.simps
-            abacus.lex_triple_def abacus.lex_pair_def lex_square_def
+      apply(rule_tac allI, rule_tac impI, case_tac "?f n", simp)
+      apply(simp add: step.simps)
+      apply(case_tac d, case_tac [2] aa, simp_all)
+      apply(simp_all add: wadjust_inv.simps wadjust_le_def
+            abacus.lex_triple_def abacus.lex_pair_def lex_square_def numeral_4_eq_4
             split: if_splits)
-          done
-      qed
-    qed
+      done
   next
     show "?Q (?f 0)"
-      apply(simp add: steps.simps wadjust_inv.simps wadjust_start.simps)
-      apply(rule_tac x = ln in exI,auto)
+      apply(simp add: steps.simps wadjust_inv.simps wadjust_start.simps, auto)
       done
   next
     show "\<not> ?P (?f 0)"
       apply(simp add: steps.simps)
       done
   qed
-  thus "?thesis"
-    apply(auto)
+  thus"?thesis"
+    apply(simp)
     done
 qed
 
-lemma [intro]: "t_correct t_wcode_adjust"
-apply(auto simp: t_wcode_adjust_def t_correct.simps iseven_def)
-apply(rule_tac x = 11 in exI, simp)
+lemma [intro]: "tm_wf (t_wcode_adjust, 0)"
+apply(auto simp: t_wcode_adjust_def tm_wf.simps)
+done
+
+declare tm_comp.simps[simp del]
+
+lemma [simp]: "args \<noteq> [] \<Longrightarrow> bl_bin (<args::nat list>) > 0"
+apply(case_tac args)
+apply(auto simp: tape_of_nl_cons bl_bin.simps split: if_splits)
 done
 
 lemma wcode_lemma_pre':
   "args \<noteq> [] \<Longrightarrow> 
-  \<exists> stp rn. steps (Suc 0, [], <m # args>) 
+  \<exists> stp rn. steps0 (Suc 0, [], <m # args>) 
               ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp
-  = (0,  [Bk],  Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>)" 
+  = (0,  [Bk],  Oc\<up>(Suc m) @ Bk # Oc\<up>(Suc (bl_bin (<args>))) @ Bk\<up>(rn))" 
 proof -
   let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <m # args>"
-  let ?Q1 = "\<lambda>(l, r). l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-    (\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+  let ?Q1 = "\<lambda>(l, r). l = Bk # Oc\<up>(Suc m) \<and>
+    (\<exists>ln rn. r = Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin (<args>)) @ Bk\<up>(rn))"
   let ?P2 = ?Q1
   let ?Q2 = "\<lambda> (l, r). (wadjust_stop m (bl_bin (<args>) - 1) (l, r))"
   let ?P3 = "\<lambda> tp. False"
   assume h: "args \<noteq> []"
-  have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp)
-                      ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp = (0, tp') \<and> ?Q2 tp')"
-  proof(rule_tac turing_merge.t_merge_halt[of "t_wcode_prepare |+| t_wcode_main" 
-               t_wcode_adjust ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2], 
-        auto simp: turing_merge_def)
-
-    show "\<exists>stp. case steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp of
-          (st, tp') \<Rightarrow> st = 0 \<and> (case tp' of (l, r) \<Rightarrow> l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
-                (\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>))"
-      using h prepare_mainpart_lemma[of args m]
-      apply(auto)
-      apply(rule_tac x = stp in exI, simp)
-      apply(rule_tac x = ln in exI, auto)
+  hence a: "bl_bin (<args>) > 0"
+    using h by simp
+  hence "{?P1} (t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust {?Q2}"
+  proof(rule_tac Hoare_plus_halt)
+    show "?Q1 \<mapsto> ?P2"
+      by(simp add: assert_imp_def)
+  next
+    show "tm_wf (t_wcode_prepare |+| t_wcode_main, 0)"
+      apply(rule_tac tm_wf_comp, auto)
       done
   next
-    fix ln rn
-    show "\<exists>stp. case steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # 
-                               Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp of
-      (st, tp') \<Rightarrow> st = 0 \<and> wadjust_stop m (bl_bin (<args>) - Suc 0) tp'"
-      using wadjust_correctness[of m "bl_bin (<args>) - 1" "Suc ln" rn]
-      apply(subgoal_tac "bl_bin (<args>) > 0", auto simp: wadjust_inv.simps)
-      apply(rule_tac x = n in exI, simp add: exp_ind)
-      using h
-      apply(case_tac args, simp_all, case_tac list,
-            simp_all add: tape_of_nl_abv  tape_of_nat_list.simps exp_ind_def
-            bl_bin.simps)
-      done     
+    show "{?P1} t_wcode_prepare |+| t_wcode_main {?Q1}"
+    proof(rule_tac HoareI, auto)
+      show 
+        "\<exists>n. is_final (steps0 (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) n) \<and>
+        (\<lambda>(l, r). l = Bk # Oc # Oc \<up> m \<and>
+        (\<exists>ln rn. r = Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> bl_bin (<args>) @ Bk \<up> rn))
+        holds_for steps0 (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) n"
+        using h prepare_mainpart_lemma[of args m]
+        apply(auto)
+        apply(rule_tac x = stp in exI, simp)
+        apply(rule_tac x = ln in exI, auto)
+        done
+    qed
   next
-    show "?Q1 \<turnstile>-> ?P2"
-      by(simp add: t_imply_def)
+    show "{?P2} t_wcode_adjust {?Q2}"
+    proof(rule_tac HoareI, auto del: replicate_Suc)
+      fix ln rn
+      show "\<exists>n. is_final (steps0 (Suc 0, Bk # Oc # Oc \<up> m, 
+        Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> bl_bin (<args>) @ Bk \<up> rn) t_wcode_adjust n) \<and>
+        wadjust_stop m (bl_bin (<args>) - Suc 0) holds_for steps0
+        (Suc 0, Bk # Oc # Oc \<up> m, Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> bl_bin (<args>) @ Bk \<up> rn) t_wcode_adjust n"
+        using wadjust_correctness[of m "bl_bin (<args>) - 1" "Suc ln" rn]
+        apply(simp del: replicate_Suc add: replicate_Suc[THEN sym] exp_ind, auto)
+        apply(rule_tac x = n in exI)
+        using a
+        apply(case_tac "bl_bin (<args>)", simp, simp del: replicate_Suc add: exp_ind wadjust_inv.simps)
+        done
+    qed
   qed
-  thus "\<exists>stp rn. steps (Suc 0, [], <m # args>) ((t_wcode_prepare |+| t_wcode_main) |+| 
-        t_wcode_adjust) stp = (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-    apply(simp add: t_imply_def)
-    apply(erule_tac exE)+
-    apply(subgoal_tac "bl_bin (<args>) > 0", auto simp: wadjust_stop.simps)
-    using h
-    apply(case_tac args, simp_all, case_tac list,  
-          simp_all add: tape_of_nl_abv  tape_of_nat_list.simps exp_ind_def
-            bl_bin.simps)
+  thus "?thesis"
+    apply(simp add: Hoare_def, auto)
+    apply(case_tac "(steps0 (Suc 0, [], <(m::nat) # args>) 
+      ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) n)")
+    apply(rule_tac x = n in exI, auto simp: wadjust_stop.simps)
+    using a
+    apply(case_tac "bl_bin (<args>)", simp_all)
     done
 qed
-
+    
 text {*
   The initialization TM @{text "t_wcode"}.
   *}
-definition t_wcode :: "tprog"
+definition t_wcode :: "instr list"
   where
   "t_wcode = (t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust"
 
@@ -4541,17 +4590,18 @@
 text {*
   The correctness of @{text "t_wcode"}.
   *}
+
 lemma wcode_lemma_1:
   "args \<noteq> [] \<Longrightarrow> 
-  \<exists> stp ln rn. steps (Suc 0, [], <m # args>)  (t_wcode) stp = 
-              (0,  [Bk],  Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-apply(simp add: wcode_lemma_pre' t_wcode_def)
+  \<exists> stp ln rn. steps0 (Suc 0, [], <m # args>)  (t_wcode) stp = 
+              (0,  [Bk],  Oc\<up>(Suc m) @ Bk # Oc\<up>(Suc (bl_bin (<args>))) @ Bk\<up>(rn))"
+apply(simp add: wcode_lemma_pre' t_wcode_def del: replicate_Suc)
 done
 
 lemma wcode_lemma: 
   "args \<noteq> [] \<Longrightarrow> 
-  \<exists> stp ln rn. steps (Suc 0, [], <m # args>)  (t_wcode) stp = 
-              (0,  [Bk],  <[m ,bl_bin (<args>)]> @ Bk\<^bsup>rn\<^esup>)"
+  \<exists> stp ln rn. steps0 (Suc 0, [], <m # args>)  (t_wcode) stp = 
+              (0,  [Bk],  <[m ,bl_bin (<args>)]> @ Bk\<up>(rn))"
 using wcode_lemma_1[of args m]
 apply(simp add: t_wcode_def tape_of_nl_abv tape_of_nat_list.simps)
 done
@@ -4564,39 +4614,38 @@
   *}
 
 
-definition UTM :: "tprog"
+definition UTM :: "instr list"
   where
   "UTM = (let (aprog, rs_pos, a_md) = rec_ci rec_F in 
           let abc_F = aprog [+] dummy_abc (Suc (Suc 0)) in 
-          (t_wcode |+| (tm_of abc_F @ tMp (Suc (Suc 0)) (start_of (layout_of abc_F) 
-                                                   (length abc_F) - Suc 0))))"
+          (t_wcode |+| (tm_of abc_F @ shift (mopup (Suc (Suc 0))) (length (tm_of abc_F) div 2))))"
 
 definition F_aprog :: "abc_prog"
   where
   "F_aprog \<equiv> (let (aprog, rs_pos, a_md) = rec_ci rec_F in 
                        aprog [+] dummy_abc (Suc (Suc 0)))"
 
-definition F_tprog :: "tprog"
+definition F_tprog :: "instr list"
   where
   "F_tprog = tm_of (F_aprog)"
 
-definition t_utm :: "tprog"
+definition t_utm :: "instr list"
   where
   "t_utm \<equiv>
-     (F_tprog) @ tMp (Suc (Suc 0)) (start_of (layout_of (F_aprog)) 
-                                  (length (F_aprog)) - Suc 0)"
-
-definition UTM_pre :: "tprog"
+     F_tprog @ shift (mopup (Suc (Suc 0))) (length F_tprog div 2)"
+
+definition UTM_pre :: "instr list"
   where
   "UTM_pre = t_wcode |+| t_utm"
 
+(*
 lemma F_abc_halt_eq:
   "\<lbrakk>turing_basic.t_correct tp; 
     length lm = k;
-    steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>);
+    steps (Suc 0, Bk\<up>(l), <lm>) tp stp = (0, Bk\<up>(m), Oc\<up>(rs)@Bk\<up>(n));
     rs > 0\<rbrakk>
     \<Longrightarrow> \<exists> stp m. abc_steps_l (0, [code tp, bl2wc (<lm>)]) (F_aprog) stp =
-                       (length (F_aprog), code tp # bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>)"
+                       (length (F_aprog), code tp # bl2wc (<lm>) # (rs - 1) # 0\<up>(m))"
 apply(drule_tac  F_t_halt_eq, simp, simp, simp)
 apply(case_tac "rec_ci rec_F")
 apply(frule_tac abc_append_dummy_complie, simp, simp, erule_tac exE,
@@ -4608,13 +4657,13 @@
 lemma F_abc_utm_halt_eq: 
   "\<lbrakk>rs > 0; 
   abc_steps_l (0, [code tp, bl2wc (<lm>)]) F_aprog stp =
-        (length F_aprog, code tp #  bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>)\<rbrakk>
+        (length F_aprog, code tp #  bl2wc (<lm>) # (rs - 1) # 0\<up>(m))\<rbrakk>
   \<Longrightarrow> \<exists>stp m n.(steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp =
-                                             (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>))"
+                                             (0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n)))"
   thm abacus_turing_eq_halt
   using abacus_turing_eq_halt
   [of "layout_of F_aprog" "F_aprog" "F_tprog" "length (F_aprog)" 
-    "[code tp, bl2wc (<lm>)]" stp "code tp # bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>" "Suc (Suc 0)"
+    "[code tp, bl2wc (<lm>)]" stp "code tp # bl2wc (<lm>) # (rs - 1) # 0\<up>(m)" "Suc (Suc 0)"
     "start_of (layout_of (F_aprog)) (length (F_aprog))" "[]" 0]
 apply(simp add: F_tprog_def t_utm_def abc_lm_v.simps nth_append)
 apply(erule_tac exE)+
@@ -4627,20 +4676,21 @@
 lemma t_utm_halt_eq': 
   "\<lbrakk>turing_basic.t_correct tp;
    0 < rs;
-  steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\<rbrakk>
+  steps (Suc 0, Bk\<up>(l), <lm::nat list>) tp stp = (0, Bk\<up>(m), Oc\<up>(rs)@Bk\<up>(n))\<rbrakk>
   \<Longrightarrow>  \<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp = 
-                                                (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+                                                (0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n))"
 apply(drule_tac  l = l in F_abc_halt_eq, simp, simp, simp)
 apply(erule_tac exE, erule_tac exE)
 apply(rule_tac F_abc_utm_halt_eq, simp_all)
 done
-
-lemma [simp]: "tinres xs (xs @ Bk\<^bsup>i\<^esup>)"
+*)
+(*
+lemma [simp]: "tinres xs (xs @ Bk\<up>(i))"
 apply(auto simp: tinres_def)
 done
 
-lemma [elim]: "\<lbrakk>rs > 0; Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup> = c @ Bk\<^bsup>n\<^esup>\<rbrakk>
-        \<Longrightarrow> \<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+lemma [elim]: "\<lbrakk>rs > 0; Oc\<up>(rs) @ Bk\<up>(na) = c @ Bk\<up>(n)\<rbrakk>
+        \<Longrightarrow> \<exists>n. c = Oc\<up>(rs) @ Bk\<up>(n)"
 apply(case_tac "na > n")
 apply(subgoal_tac "\<exists> d. na = d + n", auto simp: exp_add)
 apply(rule_tac x = "na - n" in exI, simp)
@@ -4649,29 +4699,29 @@
            simp_all add: exp_ind)
 apply(rule_tac x = "n - na" in exI, simp)
 done
-
-
+*)
+(*
 lemma t_utm_halt_eq'': 
   "\<lbrakk>turing_basic.t_correct tp;
    0 < rs;
-   steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\<rbrakk>
-  \<Longrightarrow>  \<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp = 
-                                                (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+   steps (Suc 0, Bk\<up>(l), <lm::nat list>) tp stp = (0, Bk\<up>(m), Oc\<up>(rs)@Bk\<up>(n))\<rbrakk>
+  \<Longrightarrow>  \<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>(i)) t_utm stp = 
+                                                (0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n))"
 apply(drule_tac t_utm_halt_eq', simp_all)
 apply(erule_tac exE)+
 proof -
   fix stpa ma na
-  assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
+  assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<up>(ma), Oc\<up>(rs) @ Bk\<up>(na))"
   and gr: "rs > 0"
-  thus "\<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+  thus "\<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>(i)) t_utm stp = (0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n))"
     apply(rule_tac x = stpa in exI, rule_tac x = ma in exI,  simp)
-  proof(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp)
+  proof(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>(i)) t_utm stpa", simp)
     fix a b c
-    assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
-            "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)"
-    thus " a = 0 \<and> b = Bk\<^bsup>ma\<^esup> \<and> (\<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
-      using tinres_steps2[of "<[code tp, bl2wc (<lm>)]>" "<[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>" 
-                           "Suc 0" " [Bk, Bk]" t_utm stpa 0 "Bk\<^bsup>ma\<^esup>" "Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>" a b c]
+    assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<up>(ma), Oc\<up>(rs) @ Bk\<up>(na))"
+            "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>(i)) t_utm stpa = (a, b, c)"
+    thus " a = 0 \<and> b = Bk\<up>(ma) \<and> (\<exists>n. c = Oc\<up>(rs) @ Bk\<up>(n))"
+      using tinres_steps2[of "<[code tp, bl2wc (<lm>)]>" "<[code tp, bl2wc (<lm>)]> @ Bk\<up>(i)" 
+                           "Suc 0" " [Bk, Bk]" t_utm stpa 0 "Bk\<up>(ma)" "Oc\<up>(rs) @ Bk\<up>(na)" a b c]
       apply(simp)
       using gr
       apply(simp only: tinres_def, auto)
@@ -4684,99 +4734,195 @@
 apply(auto simp: tinres_def)
 done
 
-lemma [elim]: "Bk\<^bsup>ma\<^esup> = b @ Bk\<^bsup>n\<^esup>  \<Longrightarrow> \<exists>m. b = Bk\<^bsup>m\<^esup>"
+lemma [elim]: "Bk\<up>(ma) = b @ Bk\<up>(n)  \<Longrightarrow> \<exists>m. b = Bk\<up>(m)"
 apply(subgoal_tac "ma = length b + n")
 apply(rule_tac x = "ma - n" in exI, simp add: exp_add)
 apply(drule_tac length_equal)
 apply(simp)
 done
+*)
+
+
+
+lemma tinres_step1: 
+  "\<lbrakk>tinres l l'; step (ss, l, r) (t, 0) = (sa, la, ra); 
+                 step (ss, l', r) (t, 0) = (sb, lb, rb)\<rbrakk>
+    \<Longrightarrow> tinres la lb \<and> ra = rb \<and> sa = sb"
+apply(case_tac ss, case_tac [!]r, case_tac [!] "a::cell")
+apply(auto simp: step.simps fetch.simps nth_of.simps
+        split: if_splits )
+apply(case_tac [!] "t ! (2 * nat)", 
+     auto simp: tinres_def split: if_splits)
+apply(case_tac [1-8] a, auto split: if_splits)
+apply(case_tac [!] "t ! (2 * nat)", 
+     auto simp: tinres_def split: if_splits)
+apply(case_tac [1-4] a, auto split: if_splits)
+apply(case_tac [!] "t ! Suc (2 * nat)", 
+     auto simp: if_splits)
+apply(case_tac [!] aa, auto split: if_splits)
+done
+
+lemma tinres_steps1: 
+  "\<lbrakk>tinres l l'; steps (ss, l, r) (t, 0) stp = (sa, la, ra); 
+                 steps (ss, l', r) (t, 0) stp = (sb, lb, rb)\<rbrakk>
+    \<Longrightarrow> tinres la lb \<and> ra = rb \<and> sa = sb"
+apply(induct stp arbitrary: sa la ra sb lb rb, simp add: steps.simps)
+apply(simp add: step_red)
+apply(case_tac "(steps (ss, l, r) (t, 0) stp)")
+apply(case_tac "(steps (ss, l', r) (t, 0) stp)")
+proof -
+  fix stp sa la ra sb lb rb a b c aa ba ca
+  assume ind: "\<And>sa la ra sb lb rb. \<lbrakk>steps (ss, l, r) (t, 0) stp = (sa, (la::cell list), ra); 
+          steps (ss, l', r) (t, 0) stp = (sb, lb, rb)\<rbrakk> \<Longrightarrow> tinres la lb \<and> ra = rb \<and> sa = sb"
+  and h: " tinres l l'" "step (steps (ss, l, r) (t, 0) stp) (t, 0) = (sa, la, ra)"
+         "step (steps (ss, l', r) (t, 0) stp) (t, 0) = (sb, lb, rb)" "steps (ss, l, r) (t, 0) stp = (a, b, c)" 
+         "steps (ss, l', r) (t, 0) stp = (aa, ba, ca)"
+  have "tinres b ba \<and> c = ca \<and> a = aa"
+    apply(rule_tac ind, simp_all add: h)
+    done
+  thus "tinres la lb \<and> ra = rb \<and> sa = sb"
+    apply(rule_tac l = b and l' = ba and r = c  and ss = a   
+            and t = t in tinres_step1)
+    using h
+    apply(simp, simp, simp)
+    done
+qed
+
+lemma [simp]: 
+  "tinres (Bk \<up> m @ [Bk, Bk]) la \<Longrightarrow> \<exists>m. la = Bk \<up> m"
+apply(auto simp: tinres_def)
+apply(case_tac n, simp add: exp_ind)
+apply(rule_tac  x ="Suc (Suc m)" in exI, simp only: exp_ind, simp)
+apply(simp add: exp_ind del: replicate_Suc)
+apply(case_tac nat, simp add: exp_ind)
+apply(rule_tac x = "Suc m" in exI, simp only: exp_ind)
+apply(simp only: exp_ind, simp)
+apply(subgoal_tac "m = length la + nata")
+apply(rule_tac x = "m - nata" in exI, simp add: exp_add)
+apply(drule_tac length_equal, simp)
+apply(simp only: exp_ind[THEN sym] replicate_Suc[THEN sym] replicate_add[THEN sym])
+apply(rule_tac x = "m + Suc (Suc n)" in exI, simp)
+done
 
 lemma t_utm_halt_eq: 
-  "\<lbrakk>turing_basic.t_correct tp;
-   0 < rs;
-   steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\<rbrakk>
-  \<Longrightarrow>  \<exists>stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp = 
-                                                (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
-apply(drule_tac i = i in t_utm_halt_eq'', simp_all)
-apply(erule_tac exE)+
+  assumes tm_wf: "tm_wf (tp, 0)"
+  and exec: "steps0 (Suc 0, Bk\<up>(l), <lm::nat list>) tp stp = (0, Bk\<up>(m), Oc\<up>(rs)@Bk\<up>(n))"
+  and resutl: "0 < rs"
+  shows "\<exists>stp m n. steps0 (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>(i)) t_utm stp = 
+                                                (0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n))"
 proof -
-  fix stpa ma na
-  assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
-  and gr: "rs > 0"
-  thus "\<exists>stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
-    apply(rule_tac x = stpa in exI)
-  proof(case_tac "steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp)
-    fix a b c
-    assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
-            "steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)"
-    thus "a = 0 \<and> (\<exists>m. b = Bk\<^bsup>m\<^esup>) \<and> (\<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
-      using tinres_steps[of "[Bk, Bk]" "[Bk]" "Suc 0" "<[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>" t_utm stpa 0
-                             "Bk\<^bsup>ma\<^esup>" "Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>" a b c]
-      apply(simp)
-      apply(auto simp: tinres_def)
-      apply(rule_tac x = "ma + n" in exI, simp add: exp_add)
+  obtain ap arity fp where a: "rec_ci rec_F = (ap, arity, fp)"
+    by (metis prod_cases3) 
+  moreover have b: "rec_calc_rel  rec_F [code tp, (bl2wc (<lm>))] (rs - Suc 0)"
+    using assms
+    apply(rule_tac F_correct, simp_all)
+    done 
+  have "\<exists> stp m l. steps0 (Suc 0, Bk # Bk # [], <[code tp, bl2wc (<lm>)]> @ Bk\<up>i)
+    (F_tprog @ shift (mopup 2) (length F_tprog div 2)) stp
+    = (0, Bk\<up>m @ Bk # Bk # [], Oc\<up>Suc (rs - 1) @ Bk\<up>l)"  
+  proof(rule_tac recursive_compile_to_tm_correct)
+    show "rec_ci rec_F = (ap, arity, fp)" using a by simp
+  next
+    show "rec_calc_rel rec_F [code tp, bl2wc (<lm>)] (rs - 1)"
+      using b by simp
+  next
+    show "length [code tp, bl2wc (<lm>)] = 2" by simp
+  next
+    show "layout_of (ap [+] dummy_abc 2) = layout_of (ap [+] dummy_abc 2)"
+      by simp
+  next
+    show "F_tprog = tm_of (ap [+] dummy_abc 2)"
+      using a
+      apply(simp add: F_tprog_def F_aprog_def numeral_2_eq_2)
       done
   qed
+  then obtain stp m l where 
+    "steps0 (Suc 0, Bk # Bk # [], <[code tp, bl2wc (<lm>)]> @ Bk\<up>i)
+    (F_tprog @ shift (mopup 2) (length F_tprog div 2)) stp
+    = (0, Bk\<up>m @ Bk # Bk # [], Oc\<up>Suc (rs - 1) @ Bk\<up>l)" by blast
+  hence "\<exists> m. steps0 (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>i)
+    (F_tprog @ shift (mopup 2) (length F_tprog div 2)) stp
+    = (0, Bk\<up>m, Oc\<up>Suc (rs - 1) @ Bk\<up>l)"
+  proof -
+    assume g: "steps0 (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk \<up> i)
+      (F_tprog @ shift (mopup 2) (length F_tprog div 2)) stp =
+      (0, Bk \<up> m @ [Bk, Bk], Oc \<up> Suc (rs - 1) @ Bk \<up> l)"
+   moreover have "tinres [Bk, Bk] [Bk]"
+     apply(auto simp: tinres_def)
+     done
+    moreover obtain sa la ra where "steps0 (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>i)
+    (F_tprog @ shift (mopup 2) (length F_tprog div 2)) stp = (sa, la, ra)"
+      apply(case_tac "steps0 (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>i)
+    (F_tprog @ shift (mopup 2) (length F_tprog div 2)) stp", auto)
+      done
+    ultimately show "?thesis"
+      apply(drule_tac tinres_steps1, auto)
+      done
+  qed
+  thus "?thesis"
+    apply(auto)
+    apply(rule_tac x = stp in exI, simp add: t_utm_def)
+    using assms
+    apply(case_tac rs, simp_all add: numeral_2_eq_2)
+    done
 qed
 
-lemma [intro]: "t_correct t_wcode"
+lemma [intro]: "tm_wf (t_wcode, 0)"
 apply(simp add: t_wcode_def)
-apply(auto)
+apply(rule_tac tm_wf_comp)
+apply(rule_tac tm_wf_comp, auto)
 done
       
-lemma [intro]: "t_correct t_utm"
-apply(simp add: t_utm_def F_tprog_def)
+lemma [intro]: "tm_wf (t_utm, 0)"
+apply(simp only: t_utm_def F_tprog_def)
 apply(rule_tac t_compiled_correct, auto)
 done   
 
 lemma UTM_halt_lemma_pre: 
-  "\<lbrakk>turing_basic.t_correct tp;
-   0 < rs;
-   args \<noteq> [];
-   steps (Suc 0, Bk\<^bsup>i\<^esup>, <args::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\<rbrakk>
-  \<Longrightarrow>  \<exists>stp m n. steps (Suc 0, [], <code tp # args>) UTM_pre stp = 
-                                                (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+  assumes wf_tm: "tm_wf (tp, 0)"
+  and result: "0 < rs"
+  and args: "args \<noteq> []"
+  and exec: "steps0 (Suc 0, Bk\<up>(i), <args::nat list>) tp stp = (0, Bk\<up>(m), Oc\<up>(rs)@Bk\<up>(k))"
+  shows "\<exists>stp m n. steps0 (Suc 0, [], <code tp # args>) UTM_pre stp = 
+                                                (0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n))"
 proof -
-  let ?Q2 = "\<lambda> (l, r). (\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> \<and> r = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-  term ?Q2
+  let ?Q2 = "\<lambda> (l, r). (\<exists> ln rn. l = Bk\<up>(ln) \<and> r = Oc\<up>(rs) @ Bk\<up>(rn))"
   let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <code tp # args>"
   let ?Q1 = "\<lambda> (l, r). (l = [Bk] \<and>
-             (\<exists> rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))"
+    (\<exists> rn. r = Oc\<up>(Suc (code tp)) @ Bk # Oc\<up>(Suc (bl_bin (<args>))) @ Bk\<up>(rn)))"
   let ?P2 = ?Q1
   let ?P3 = "\<lambda> (l, r). False"
-  assume h: "turing_basic.t_correct tp" "0 < rs"
-            "args \<noteq> []" "steps (Suc 0, Bk\<^bsup>i\<^esup>, <args::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)"
-  have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp)
-                    (t_wcode |+| t_utm) stp = (0, tp') \<and> ?Q2 tp')"
-  proof(rule_tac turing_merge.t_merge_halt [of "t_wcode" "t_utm"
-          ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2], auto simp: turing_merge_def)
-    show "\<exists>stp. case steps (Suc 0, [], <code tp # args>) t_wcode stp of (st, tp') \<Rightarrow> 
-       st = 0 \<and> (case tp' of (l, r) \<Rightarrow> l = [Bk] \<and>
-                   (\<exists>rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))"
-      using wcode_lemma_1[of args "code tp"] h
-      apply(simp, auto)
-      apply(rule_tac x = stpa in exI, auto)
-      done      
+  have "{?P1} (t_wcode |+| t_utm) {?Q2}"
+  proof(rule_tac Hoare_plus_halt)
+    show "?Q1 \<mapsto> ?P2"
+      by(simp add: assert_imp_def)
   next
-    fix rn 
-    show "\<exists>stp. case steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @
-      Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp of
-      (st, tp') \<Rightarrow> st = 0 \<and> (case tp' of (l, r) \<Rightarrow>
-      (\<exists>ln. l = Bk\<^bsup>ln\<^esup>) \<and> (\<exists>rn. r = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>rn\<^esup>))"
-      using t_utm_halt_eq[of tp rs i args stp m k rn] h
+    show "tm_wf (t_wcode, 0)" by auto
+  next
+    show "{?P1} t_wcode {?Q1}"
+      apply(rule_tac HoareI, auto)
+      using wcode_lemma_1[of args "code tp"] args
       apply(auto)
-      apply(rule_tac x = stpa in exI, simp add: bin_wc_eq 
-        tape_of_nat_list.simps tape_of_nl_abv)
-      apply(auto)
+      apply(rule_tac x = stp in exI, simp)
       done
   next
-    show "?Q1 \<turnstile>-> ?P2"
-      apply(simp add: t_imply_def)
+    show "{?P2} t_utm {?Q2}"
+    proof(rule_tac HoareI, auto)
+      fix rn
+      show "\<exists>n. is_final (steps0 (Suc 0, [Bk], Oc # Oc \<up> code tp @ Bk # Oc # Oc \<up> bl_bin (<args>) @ Bk \<up> rn) t_utm n) \<and>
+        (\<lambda>(l, r). (\<exists>ln. l = Bk \<up> ln) \<and>
+        (\<exists>rn. r = Oc \<up> rs @ Bk \<up> rn)) holds_for steps0 (Suc 0, [Bk],
+        Oc # Oc \<up> code tp @ Bk # Oc # Oc \<up> bl_bin (<args>) @ Bk \<up> rn) t_utm n"
+        using t_utm_halt_eq[of tp i "args" stp m rs k rn] assms
+      apply(auto simp: bin_wc_eq)
+      apply(rule_tac x = stpa in exI, simp add: tape_of_nl_abv)
       done
+    qed
   qed
   thus "?thesis"
-    apply(simp add: t_imply_def)
-    apply(auto simp: UTM_pre_def)
+    apply(auto simp: Hoare_def UTM_pre_def)
+    apply(case_tac "steps0 (Suc 0, [], <code tp # args>) (t_wcode |+| t_utm) n")
+    apply(rule_tac x = n in exI, simp)
     done
 qed
 
@@ -4784,84 +4930,81 @@
   The correctness of @{text "UTM"}, the halt case.
 *}
 lemma UTM_halt_lemma: 
-  "\<lbrakk>turing_basic.t_correct tp;
-   0 < rs;
-   args \<noteq> [];
-   steps (Suc 0, Bk\<^bsup>i\<^esup>, <args::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\<rbrakk>
-  \<Longrightarrow>  \<exists>stp m n. steps (Suc 0, [], <code tp # args>) UTM stp = 
-                                                (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
-using UTM_halt_lemma_pre[of tp rs args i stp m k]
+  assumes tm_wf: "tm_wf (tp, 0)"
+  and result: "0 < rs"
+  and args: "args \<noteq> []"
+  and exec: "steps0 (Suc 0, Bk\<up>(i), <args::nat list>) tp stp = (0, Bk\<up>(m), Oc\<up>(rs)@Bk\<up>(k))"
+  shows "\<exists>stp m n. steps0 (Suc 0, [], <code tp # args>) UTM stp = 
+                                                (0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n))"
+using UTM_halt_lemma_pre[of tp rs args i stp m k] assms
 apply(simp add: UTM_pre_def t_utm_def UTM_def F_aprog_def F_tprog_def)
 apply(case_tac "rec_ci rec_F", simp)
 done
 
-definition TSTD:: "t_conf \<Rightarrow> bool"
+definition TSTD:: "config \<Rightarrow> bool"
   where
   "TSTD c = (let (st, l, r) = c in 
-             st = 0 \<and> (\<exists> m. l = Bk\<^bsup>m\<^esup>) \<and> (\<exists> rs n. r = Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>))"
-
-thm abacus_turing_eq_uhalt
+             st = 0 \<and> (\<exists> m. l = Bk\<up>(m)) \<and> (\<exists> rs n. r = Oc\<up>(Suc rs) @ Bk\<up>(n)))"
 
 lemma nstd_case1: "0 < a \<Longrightarrow> NSTD (trpl_code (a, b, c))"
 apply(simp add: NSTD.simps trpl_code.simps)
 done
 
-lemma [simp]: "\<forall>m. b \<noteq> Bk\<^bsup>m\<^esup> \<Longrightarrow> 0 < bl2wc b"
+lemma [simp]: "\<forall>m. b \<noteq> Bk\<up>(m) \<Longrightarrow> 0 < bl2wc b"
 apply(rule classical, simp)
 apply(induct b, erule_tac x = 0 in allE, simp)
 apply(simp add: bl2wc.simps, case_tac a, simp_all 
   add: bl2nat.simps bl2nat_double)
-apply(case_tac "\<exists> m. b = Bk\<^bsup>m\<^esup>",  erule exE)
-apply(erule_tac x = "Suc m" in allE, simp add: exp_ind_def, simp)
-done
-lemma nstd_case2: "\<forall>m. b \<noteq> Bk\<^bsup>m\<^esup> \<Longrightarrow> NSTD (trpl_code (a, b, c))"
+apply(case_tac "\<exists> m. b = Bk\<up>(m)",  erule exE)
+apply(erule_tac x = "Suc m" in allE, simp add: , simp)
+done
+
+lemma nstd_case2: "\<forall>m. b \<noteq> Bk\<up>(m) \<Longrightarrow> NSTD (trpl_code (a, b, c))"
 apply(simp add: NSTD.simps trpl_code.simps)
 done
 
-thm lg.simps
-thm lgR.simps
-
 lemma [elim]: "Suc (2 * x) = 2 * y \<Longrightarrow> RR"
 apply(induct x arbitrary: y, simp, simp)
 apply(case_tac y, simp, simp)
 done
 
-lemma bl2nat_zero_eq[simp]: "(bl2nat c 0 = 0) = (\<exists>n. c = Bk\<^bsup>n\<^esup>)"
+declare replicate_Suc[simp del]
+
+lemma bl2nat_zero_eq[simp]: "(bl2nat c 0 = 0) = (\<exists>n. c = Bk\<up>(n))"
 apply(auto)
-apply(induct c, simp add: bl2nat.simps)
-apply(rule_tac x = 0 in exI, simp)
+apply(induct c, simp_all add: bl2nat.simps)
 apply(case_tac a, auto simp: bl2nat.simps bl2nat_double)
 done
 
 lemma bl2wc_exp_ex: 
-  "\<lbrakk>Suc (bl2wc c) = 2 ^  m\<rbrakk> \<Longrightarrow> \<exists> rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+  "\<lbrakk>Suc (bl2wc c) = 2 ^  m\<rbrakk> \<Longrightarrow> \<exists> rs n. c = Oc\<up>(rs) @ Bk\<up>(n)"
 apply(induct c arbitrary: m, simp add: bl2wc.simps bl2nat.simps)
 apply(case_tac a, auto)
 apply(case_tac m, simp_all add: bl2wc.simps, auto)
 apply(rule_tac x = 0 in exI, rule_tac x = "Suc n" in exI, 
-  simp add: exp_ind_def)
+  simp add: replicate_Suc)
 apply(simp add: bl2wc.simps bl2nat.simps bl2nat_double)
 apply(case_tac m, simp, simp)
 proof -
   fix c m nat
   assume ind: 
-    "\<And>m. Suc (bl2nat c 0) = 2 ^ m \<Longrightarrow> \<exists>rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+    "\<And>m. Suc (bl2nat c 0) = 2 ^ m \<Longrightarrow> \<exists>rs n. c = Oc\<up>(rs) @ Bk\<up>(n)"
   and h: 
     "Suc (Suc (2 * bl2nat c 0)) = 2 * 2 ^ nat"
-  have "\<exists>rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+  have "\<exists>rs n. c = Oc\<up>(rs) @ Bk\<up>(n)"
     apply(rule_tac m = nat in ind)
     using h
     apply(simp)
     done
-  from this obtain rs n where " c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" by blast 
-  thus "\<exists>rs n. Oc # c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
-    apply(rule_tac x = "Suc rs" in exI, simp add: exp_ind_def)
-    apply(rule_tac x = n in exI, simp)
+  from this obtain rs n where " c = Oc\<up>(rs) @ Bk\<up>(n)" by blast 
+  thus "\<exists>rs n. Oc # c = Oc\<up>(rs) @ Bk\<up>(n)"
+    apply(rule_tac x = "Suc rs" in exI, simp add: )
+    apply(rule_tac x = n in exI, simp add: replicate_Suc)
     done
 qed
 
-lemma [elim]: 
-  "\<lbrakk>\<forall>rs n. c \<noteq> Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>; 
+lemma lg_bin: 
+  "\<lbrakk>\<forall>rs n. c \<noteq> Oc\<up>(Suc rs) @ Bk\<up>(n); 
   bl2wc c = 2 ^ lg (Suc (bl2wc c)) 2 - Suc 0\<rbrakk> \<Longrightarrow> bl2wc c = 0"
 apply(subgoal_tac "\<exists> m. Suc (bl2wc c) = 2^m", erule_tac exE)
 apply(drule_tac bl2wc_exp_ex, simp, erule_tac exE, erule_tac exE)
@@ -4876,10 +5019,10 @@
 done
 
 lemma nstd_case3: 
-  "\<forall>rs n. c \<noteq> Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup> \<Longrightarrow>  NSTD (trpl_code (a, b, c))"
+  "\<forall>rs n. c \<noteq> Oc\<up>(Suc rs) @ Bk\<up>(n) \<Longrightarrow>  NSTD (trpl_code (a, b, c))"
 apply(simp add: NSTD.simps trpl_code.simps)
-apply(rule_tac impI)
-apply(rule_tac disjI2, rule_tac disjI2, auto)
+apply(auto)
+apply(drule_tac lg_bin, simp_all)
 done
 
 lemma NSTD_1: "\<not> TSTD (a, b, c)
@@ -4893,10 +5036,10 @@
   done
  
 lemma nonstop_t_uhalt_eq:
-      "\<lbrakk>turing_basic.t_correct tp;
-        steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp = (a, b, c);
-       \<not> TSTD (a, b, c)\<rbrakk>
-       \<Longrightarrow> rec_exec rec_nonstop [code tp, bl2wc (<lm>), stp] = Suc 0"
+  "\<lbrakk>tm_wf (tp, 0);
+  steps0 (Suc 0, Bk\<up>(l), <lm>) tp stp = (a, b, c);
+  \<not> TSTD (a, b, c)\<rbrakk>
+  \<Longrightarrow> rec_exec rec_nonstop [code tp, bl2wc (<lm>), stp] = Suc 0"
 apply(simp add: rec_nonstop_def rec_exec.simps)
 apply(subgoal_tac 
   "rec_exec rec_conf [code tp, bl2wc (<lm>), stp] =
@@ -4907,12 +5050,12 @@
 done
 
 lemma nonstop_true:
-  "\<lbrakk>turing_basic.t_correct tp;
-  \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk>
-     \<Longrightarrow> \<forall>y. rec_calc_rel rec_nonstop 
-                        ([code tp, bl2wc (<lm>), y]) (Suc 0)"
+  "\<lbrakk>tm_wf (tp, 0);
+  \<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <lm>) tp stp))\<rbrakk>
+  \<Longrightarrow> \<forall>y. rec_calc_rel rec_nonstop 
+  ([code tp, bl2wc (<lm>), y]) (Suc 0)"
 apply(rule_tac allI, erule_tac x = y in allE)
-apply(case_tac "steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp y", simp)
+apply(case_tac "steps0 (Suc 0, Bk\<up>(l), <lm>) tp y", simp)
 apply(rule_tac nonstop_t_uhalt_eq, simp_all)
 done
 
@@ -4928,10 +5071,10 @@
 declare ci_cn_para_eq[simp]
 
 lemma F_aprog_uhalt: 
-  "\<lbrakk>turing_basic.t_correct tp; 
-    \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp)); 
+  "\<lbrakk>tm_wf (tp,0); 
+    \<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <lm>) tp stp)); 
     rec_ci rec_F = (F_ap, rs_pos, a_md)\<rbrakk>
-  \<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)] @ 0\<^bsup>a_md - rs_pos \<^esup>
+  \<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)] @ 0\<up>(a_md - rs_pos )
                @ suflm) (F_ap) stp of (ss, e) \<Rightarrow> ss < length (F_ap)"
 apply(case_tac "rec_ci (Cn (Suc (Suc 0)) rec_right [Cn (Suc (Suc 0)) rec_conf 
                ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])])")
@@ -4974,11 +5117,9 @@
 apply(simp)
 done
 
-thm abc_list_crsp_steps
-
 lemma uabc_uhalt': 
-  "\<lbrakk>turing_basic.t_correct tp;
-  \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp));
+  "\<lbrakk>tm_wf (tp, 0);
+  \<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <lm>) tp stp));
   rec_ci rec_F = (ap, pos, md)\<rbrakk>
   \<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) ap stp of (ss, e)
            \<Rightarrow>  ss < length ap"
@@ -4986,20 +5127,20 @@
     and suflm = "[]" in F_aprog_uhalt, auto)
   fix stp a b
   assume h: 
-    "\<forall>stp. case abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp of 
+    "\<forall>stp. case abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<up>(md - pos)) ap stp of 
     (ss, e) \<Rightarrow> ss < length ap"
     "abc_steps_l (0, [code tp, bl2wc (<lm>)]) ap stp = (a, b)" 
-    "turing_basic.t_correct tp" 
+    "tm_wf (tp, 0)" 
     "rec_ci rec_F = (ap, pos, md)"
   moreover have "ap \<noteq> []"
     using h apply(rule_tac rec_ci_not_null, simp)
     done
   ultimately show "a < length ap"
   proof(erule_tac x = stp in allE,
-  case_tac "abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp", simp)
+  case_tac "abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<up>(md - pos)) ap stp", simp)
     fix aa ba
     assume g: "aa < length ap" 
-      "abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp = (aa, ba)" 
+      "abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<up>(md - pos)) ap stp = (aa, ba)" 
       "ap \<noteq> []"
     thus "?thesis"
       using abc_list_crsp_steps[of "[code tp, bl2wc (<lm>)]"
@@ -5010,8 +5151,8 @@
 qed
 
 lemma uabc_uhalt: 
-  "\<lbrakk>turing_basic.t_correct tp; 
-  \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk>
+  "\<lbrakk>tm_wf (tp, 0); 
+  \<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <lm>) tp stp))\<rbrakk>
   \<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) F_aprog 
        stp of (ss, e) \<Rightarrow> ss < length F_aprog"
 apply(case_tac "rec_ci rec_F", simp add: F_aprog_def)
@@ -5034,41 +5175,46 @@
 qed
 
 lemma tutm_uhalt': 
-  "\<lbrakk>turing_basic.t_correct tp;
-    \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk>
-  \<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp)"
-  using abacus_turing_eq_uhalt[of "layout_of (F_aprog)" 
-               "F_aprog" "F_tprog" "[code tp, bl2wc (<lm>)]" 
-               "start_of (layout_of (F_aprog )) (length (F_aprog))" 
-               "Suc (Suc 0)"]
-apply(simp add: F_tprog_def)
-apply(subgoal_tac "\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)])
-  (F_aprog) stp of (as, am) \<Rightarrow> as < length (F_aprog)", simp)
-thm abacus_turing_eq_uhalt
-apply(simp add: t_utm_def F_tprog_def)
-apply(rule_tac uabc_uhalt, simp_all)
-done
-
+assumes tm_wf:  "tm_wf (tp,0)"
+  and unhalt: "\<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <lm>) tp stp))"
+  shows "\<forall> stp. \<not> is_final (steps0 (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp)"
+apply(simp add: t_utm_def)
+proof(rule_tac compile_correct_unhalt)
+  show "layout_of F_aprog = layout_of F_aprog" by simp
+next
+  show "F_tprog = tm_of F_aprog"
+    by(simp add:  F_tprog_def)
+next
+  show "crsp (layout_of F_aprog) (0, [code tp, bl2wc (<lm>)]) (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>)  []"
+    by(auto simp: crsp.simps start_of.simps)
+next
+  show "length F_tprog div 2 = length F_tprog div 2" by simp
+next
+  show "\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) F_aprog stp of (as, am) \<Rightarrow> as < length F_aprog"
+    using assms
+    apply(erule_tac uabc_uhalt, simp)
+    done
+qed
+
+ 
 lemma tinres_commute: "tinres r r' \<Longrightarrow> tinres r' r"
 apply(auto simp: tinres_def)
 done
 
 lemma inres_tape:
-  "\<lbrakk>steps (st, l, r) tp stp = (a, b, c); steps (st, l', r') tp stp = (a', b', c'); 
+  "\<lbrakk>steps0 (st, l, r) tp stp = (a, b, c); steps0 (st, l', r') tp stp = (a', b', c'); 
   tinres l l'; tinres r r'\<rbrakk>
   \<Longrightarrow> a = a' \<and> tinres b b' \<and> tinres c c'"
-proof(case_tac "steps (st, l', r) tp stp")
+proof(case_tac "steps0 (st, l', r) tp stp")
   fix aa ba ca
-  assume h: "steps (st, l, r) tp stp = (a, b, c)" 
-            "steps (st, l', r') tp stp = (a', b', c')"
+  assume h: "steps0 (st, l, r) tp stp = (a, b, c)" 
+            "steps0 (st, l', r') tp stp = (a', b', c')"
             "tinres l l'" "tinres r r'"
-            "steps (st, l', r) tp stp = (aa, ba, ca)"
+            "steps0 (st, l', r) tp stp = (aa, ba, ca)"
   have "tinres b ba \<and> c = ca \<and> a = aa"
     using h
-    apply(rule_tac tinres_steps, auto)
+    apply(rule_tac tinres_steps1, auto)
     done
-
-  thm tinres_steps2
   moreover have "b' = ba \<and> tinres c' ca \<and> a' =  aa"
     using h
     apply(rule_tac tinres_steps2, auto intro: tinres_commute)
@@ -5078,73 +5224,70 @@
     done
 qed
 
-lemma tape_normalize: "\<forall> stp. \<not> isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp)
-      \<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>n\<^esup>) t_utm stp)"
-apply(rule_tac allI, case_tac "(steps (Suc 0, Bk\<^bsup>m\<^esup>, 
-               <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>n\<^esup>) t_utm stp)", simp add: isS0_def)
+lemma tape_normalize: "\<forall> stp. \<not> is_final(steps0 (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp)
+      \<Longrightarrow> \<forall> stp. \<not> is_final (steps0 (Suc 0, Bk\<up>(m), <[code tp, bl2wc (<lm>)]> @ Bk\<up>(n)) t_utm stp)"
+apply(rule_tac allI, case_tac "(steps0 (Suc 0, Bk\<up>(m), 
+               <[code tp, bl2wc (<lm>)]> @ Bk\<up>(n)) t_utm stp)", simp)
 apply(erule_tac x = stp in allE)
-apply(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp", simp)
+apply(case_tac "steps0 (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp", simp)
 apply(drule_tac inres_tape, auto)
 apply(auto simp: tinres_def)
 apply(case_tac "m > Suc (Suc 0)")
 apply(rule_tac x = "m - Suc (Suc 0)" in exI) 
-apply(case_tac m, simp_all add: exp_ind_def, case_tac nat, simp_all add: exp_ind_def)
-apply(rule_tac x = "2 - m" in exI, simp add: exp_ind_def[THEN sym] exp_add[THEN sym])
-apply(simp only: numeral_2_eq_2, simp add: exp_ind_def)
+apply(case_tac m, simp_all add: , case_tac nat, simp_all add: replicate_Suc)
+apply(rule_tac x = "2 - m" in exI, simp add: exp_add[THEN sym])
+apply(simp only: numeral_2_eq_2, simp add: replicate_Suc)
 done
 
 lemma tutm_uhalt: 
-  "\<lbrakk>turing_basic.t_correct tp;
-    \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp))\<rbrakk>
-  \<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc (<args>)]> @ Bk\<^bsup>n\<^esup>) t_utm stp)"
+  "\<lbrakk>tm_wf (tp,0);
+    \<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <args>) tp stp))\<rbrakk>
+  \<Longrightarrow> \<forall> stp. \<not> is_final (steps0 (Suc 0, Bk\<up>(m), <[code tp, bl2wc (<args>)]> @ Bk\<up>(n)) t_utm stp)"
 apply(rule_tac tape_normalize)
 apply(rule_tac tutm_uhalt', simp_all)
 done
 
 lemma UTM_uhalt_lemma_pre:
-  "\<lbrakk>turing_basic.t_correct tp;
-   \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp));
-   args \<noteq> []\<rbrakk>
-  \<Longrightarrow>  \<forall> stp. \<not> isS0 (steps (Suc 0, [], <code tp # args>)  UTM_pre stp)"
+  assumes tm_wf: "tm_wf (tp, 0)"
+  and exec: "\<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <args>) tp stp))"
+  and args: "args \<noteq> []"
+  shows "\<forall> stp. \<not> is_final (steps0 (Suc 0, [], <code tp # args>)  UTM_pre stp)"
 proof -
   let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <code tp # args>"
   let ?Q1 = "\<lambda> (l, r). (l = [Bk] \<and>
-             (\<exists> rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))"
-  let ?P4 = ?Q1
-  let ?P3 = "\<lambda> (l, r). False"
-  assume h: "turing_basic.t_correct tp" "\<forall>stp. \<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp)"
-            "args \<noteq> []"
-  have "?P1 \<turnstile>-> \<lambda> tp. \<not> (\<exists> stp. isS0 (steps (Suc 0, tp) (t_wcode |+| t_utm) stp))"
-  proof(rule_tac turing_merge.t_merge_uhalt [of "t_wcode" "t_utm"
-          ?P1 ?P3 ?P3 ?P4 ?Q1 ?P3], auto simp: turing_merge_def)
-    show "\<exists>stp. case steps (Suc 0, [], <code tp # args>) t_wcode stp of (st, tp') \<Rightarrow> 
-       st = 0 \<and> (case tp' of (l, r) \<Rightarrow> l = [Bk] \<and>
-                   (\<exists>rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))"
-      using wcode_lemma_1[of args "code tp"] h
-      apply(simp, auto)
-      apply(rule_tac x = stp in exI, auto)
-      done      
+             (\<exists> rn. r = Oc\<up>(Suc (code tp)) @ Bk # Oc\<up>(Suc (bl_bin (<args>))) @ Bk\<up>(rn)))"
+  let ?P2 = ?Q1
+  have "{?P1} (t_wcode |+| t_utm) \<up>"
+  proof(rule_tac Hoare_plus_unhalt)
+    show "?Q1 \<mapsto> ?P2"
+      by(simp add: assert_imp_def)
   next
-    fix rn  stp
-    show " isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp)
-          \<Longrightarrow> False"
-      using tutm_uhalt[of tp l args "Suc 0" rn] h
-      apply(simp)
-      apply(erule_tac x = stp in allE)
-      apply(simp add: tape_of_nl_abv tape_of_nat_list.simps bin_wc_eq)
+    show "tm_wf (t_wcode, 0)" by auto
+  next
+    show "{?P1} t_wcode {?Q1}"
+      apply(rule_tac HoareI, auto)
+      using wcode_lemma_1[of args "code tp"] args
+      apply(auto)
+      apply(rule_tac x = stp in exI, simp)
       done
   next
-    fix rn stp
-    show "isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp) \<Longrightarrow>
-      isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp)"
-      by simp
-  next
-    show "?Q1 \<turnstile>-> ?P4"
-      apply(simp add: t_imply_def)
-      done
+    show "{?P2} t_utm \<up>"
+    proof(rule_tac Hoare_unhalt_I, auto)
+      fix n rn
+      assume h: "is_final (steps0 (Suc 0, [Bk], Oc \<up> Suc (code tp) @ Bk # Oc \<up> Suc (bl_bin (<args>)) @ Bk \<up> rn) t_utm n)"
+      have "\<forall> stp. \<not> is_final (steps0 (Suc 0, Bk\<up>(Suc 0), <[code tp, bl2wc (<args>)]> @ Bk\<up>(rn)) t_utm stp)"
+        using assms
+        apply(rule_tac tutm_uhalt, simp_all)
+        done
+      thus "False"
+        using h
+        apply(erule_tac x = n in allE)
+        apply(simp add: tape_of_nl_abv bin_wc_eq)
+        done
+    qed
   qed
   thus "?thesis"
-    apply(simp add: t_imply_def UTM_pre_def)
+    apply(simp add: Hoare_unhalt_def UTM_pre_def)
     done
 qed
 
@@ -5153,11 +5296,11 @@
   *}
 
 lemma UTM_uhalt_lemma:
-  "\<lbrakk>turing_basic.t_correct tp;
-   \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp));
-   args \<noteq> []\<rbrakk>
-  \<Longrightarrow>  \<forall> stp. \<not> isS0 (steps (Suc 0, [], <code tp # args>)  UTM stp)"
-using UTM_uhalt_lemma_pre[of tp l args]
+  assumes tm_wf: "tm_wf (tp, 0)"
+  and unhalt: "\<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <args>) tp stp))"
+  and args: "args \<noteq> []"
+  shows " \<forall> stp. \<not> is_final (steps0 (Suc 0, [], <code tp # args>)  UTM stp)"
+  using UTM_uhalt_lemma_pre[of tp l args] assms
 apply(simp add: UTM_pre_def t_utm_def UTM_def F_aprog_def F_tprog_def)
 apply(case_tac "rec_ci rec_F", simp)
 done
diff -r 1e89c65f844b -r e995ae949731 thys/recursive.thy
--- a/thys/recursive.thy	Wed Feb 06 04:11:06 2013 +0000
+++ b/thys/recursive.thy	Wed Feb 06 04:27:03 2013 +0000
@@ -4879,10 +4879,18 @@
 apply(drule_tac x="length args" in meta_spec)
 apply(drule_tac x="tm_of (a [+] dummy_abc (length args))" in meta_spec)
 apply(auto)
-apply(rule_tac x="m" in exI)
-apply(rule_tac x="n" in exI)
 apply(simp add: tape_of_nat_abv)
 apply(subgoal_tac "b = length args")
+apply(simp add: Hoare_halt_def)
+apply(auto)[1]
+apply(rule_tac x="na" in exI)
+apply(auto)[1]
+apply(case_tac "steps0 (Suc 0, [Bk, Bk], <args>)
+                                   (tm_of (a [+] dummy_abc (length args)) @
+                                    shift (mopup (length args))
+                                     (listsum
+ (layout_of (a [+] dummy_abc (length args)))))
+                                   na")
 apply(simp)
 by (metis assms para_pattern)