
diff -r cbb4ac6c8081 -r 1ce04eb1c8ad utm/UTM.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/utm/UTM.thy	Sat Sep 29 12:38:12 2012 +0000
@@ -0,0 +1,4704 @@
+theory UTM
+imports Main uncomputable recursive abacus UF GCD 
+begin
+
+section {* Wang coding of input arguments *}
+
+text {*
+  The direct compilation of the universal function @{text "rec_F"} can not give us UTM, because @{text "rec_F"} is of arity 2,
+  where the first argument represents the Godel coding of the TM being simulated and the second argument represents the right number (in Wang's coding) of the TM tape. 
+  (Notice, left number is always @{text "0"} at the very beginning). However, UTM needs to simulate the execution of any TM which may
+  very well take many input arguments. Therefore, a initialization TM needs to run before the TM compiled from @{text "rec_F"}, and the sequential 
+  composition of these two TMs will give rise to the UTM we are seeking. The purpose of this initialization TM is to transform the multiple 
+  input arguments of the TM being simulated into Wang's coding, so that it can be consumed by the TM compiled from @{text "rec_F"} as the second
+  argument. 
+
+  However, this initialization TM (named @{text "t_wcode"}) can not be constructed by compiling from any resurve function, because every recursive 
+  function takes a fixed number of input arguments, while @{text "t_wcode"} needs to take varying number of arguments and tranform them into 
+  Wang's coding. Therefore, this section give a direct construction of @{text "t_wcode"} with just some parts being obtained from recursive functions.
+*}
+
+definition rec_twice :: "recf"
+  where
+  "rec_twice = Cn 1 rec_mult [id 1 0, constn 2]"
+
+definition rec_fourtimes  :: "recf"
+  where
+  "rec_fourtimes = Cn 1 rec_mult [id 1 0, constn 4]"
+
+definition abc_twice :: "abc_prog"
+  where
+  "abc_twice = (let (aprog, ary, fp) = rec_ci rec_twice in 
+                       aprog [+] dummy_abc ((Suc 0)))"
+
+definition abc_fourtimes :: "abc_prog"
+  where
+  "abc_fourtimes = (let (aprog, ary, fp) = rec_ci rec_fourtimes in 
+                       aprog [+] dummy_abc ((Suc 0)))"
+
+definition twice_ly :: "nat list"
+  where
+  "twice_ly = layout_of abc_twice"
+
+definition fourtimes_ly :: "nat list"
+  where
+  "fourtimes_ly = layout_of abc_fourtimes"
+
+definition t_twice :: "tprog"
+  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"
+  where
+  "t_fourtimes = change_termi_state (tm_of (abc_fourtimes) @ 
+             (tMp 1 (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)))"
+
+
+definition t_twice_len :: "nat"
+  where
+  "t_twice_len = length t_twice div 2"
+
+definition t_wcode_main_first_part:: "tprog"
+  where
+  "t_wcode_main_first_part \<equiv> 
+                   [(L, 1), (L, 2), (L, 7), (R, 3),
+                    (R, 4), (W0, 3), (R, 4), (R, 5),
+                    (W1, 6), (R, 5), (R, 13), (L, 6),
+                    (R, 0), (R, 8), (R, 9), (Nop, 8),
+                    (R, 10), (W0, 9), (R, 10), (R, 11), 
+                    (W1, 12), (R, 11), (R, t_twice_len + 14), (L, 12)]"
+
+definition t_wcode_main :: "tprog"
+  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"
+  where
+  "bl_bin [] = 0" 
+| "bl_bin (Bk # xs) = 2 * bl_bin xs"
+| "bl_bin (Oc # xs) = Suc (2 * bl_bin xs)"
+
+declare bl_bin.simps[simp del]
+
+type_synonym bin_inv_t = "block 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>)"
+
+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>)"
+
+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>
+               ml + mr > Suc 0 \<and> mr > 0)"
+
+declare wcode_on_left_moving_1_B.simps[simp del]
+
+fun wcode_on_left_moving_1_O :: "bin_inv_t"
+  where
+  "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>)"
+
+declare wcode_on_left_moving_1_O.simps[simp del]
+
+fun wcode_on_left_moving_1 :: "bin_inv_t"
+  where
+  "wcode_on_left_moving_1 ires rs (l, r) = 
+          (wcode_on_left_moving_1_B ires rs (l, r) \<or> wcode_on_left_moving_1_O ires rs (l, r))"
+
+declare wcode_on_left_moving_1.simps[simp del]
+
+fun wcode_on_checking_1 :: "bin_inv_t"
+  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>)"
+
+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>)"
+
+declare wcode_erase1.simps [simp del]
+
+fun wcode_on_right_moving_1 :: "bin_inv_t"
+  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>
+             ml + mr > Suc 0)"
+
+declare wcode_on_right_moving_1.simps [simp del] 
+
+declare wcode_on_right_moving_1.simps[simp del]
+
+fun wcode_goon_right_moving_1 :: "bin_inv_t"
+  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>
+            ml + mr = Suc rs)"
+
+declare wcode_goon_right_moving_1.simps[simp del]
+
+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>)"
+
+declare wcode_backto_standard_pos_B.simps[simp del]
+
+fun wcode_backto_standard_pos_O :: "bin_inv_t"
+  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>
+            ml + mr = Suc (Suc rs) \<and> mr > 0)"
+
+declare wcode_backto_standard_pos_O.simps[simp del]
+
+fun wcode_backto_standard_pos :: "bin_inv_t"
+  where
+  "wcode_backto_standard_pos ires rs (l, r) = (wcode_backto_standard_pos_B ires rs (l, r) \<or>
+                                            wcode_backto_standard_pos_O ires rs (l, r))"
+
+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)
+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)
+done
+
+lemma [simp]: "length (<a::nat>) = Suc a"
+apply(simp add: tape_of_nat_abv tape_of_nat_list.simps)
+done
+
+lemma [simp]: "<[a::nat]> = <a>"
+apply(simp add: tape_of_nat_abv tape_of_nl_abv exponent_def
+                tape_of_nat_list.simps)
+done
+
+lemma bin_wc_eq: "bl_bin xs = bl2wc xs"
+proof(induct xs)
+  show " bl_bin [] = bl2wc []" 
+    apply(simp add: bl_bin.simps)
+    done
+next
+  fix a xs
+  assume "bl_bin xs = bl2wc xs"
+  thus " bl_bin (a # xs) = bl2wc (a # xs)"
+    apply(case_tac a, simp_all add: bl_bin.simps bl2wc.simps)
+    apply(simp_all add: bl2nat.simps bl2nat_double)
+    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]
+
+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(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
+
+lemma bl_bin_bk_oc[simp]:
+  "bl_bin (xs @ [Bk, Oc]) = 
+  bl_bin xs + 2*2^(length xs)"
+apply(simp add: bin_wc_eq)
+using bl2nat_cons_oc[of "xs @ [Bk]"]
+apply(simp add: bl2nat_cons_bk bl2wc.simps)
+done
+
+lemma tape_of_nat[simp]: "(<a::nat>) = Oc\<^bsup>Suc a\<^esup>"
+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>)"
+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>"
+    and h: "Suc x = length (xs::nat list)" 
+  show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>"
+  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>"
+      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>"
+      apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
+      done
+  qed
+qed
+
+lemma [simp]: "length (<aa # a # list>) = Suc (Suc aa) + length (<a # list>)"
+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)"]
+apply(simp)
+done
+
+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)
+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)
+apply(case_tac list)
+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]>))"
+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]>"]
+apply(simp)
+done
+lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ 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]>) +
+         rs * (2 * 2 ^ (aa + length (<list @ [Suc ab]>)))"
+apply(simp add: tape_of_nl_app_Suc)
+done
+
+declare tape_of_nat[simp del]
+
+text{* double case*}
+fun wcode_double_case_inv :: "nat \<Rightarrow> bin_inv_t"
+  where
+  "wcode_double_case_inv st ires rs (l, r) = 
+          (if st = Suc 0 then wcode_on_left_moving_1 ires rs (l, r)
+          else if st = Suc (Suc 0) then wcode_on_checking_1 ires rs (l, r)
+          else if st = 3 then wcode_erase1 ires rs (l, r)
+          else if st = 4 then wcode_on_right_moving_1 ires rs (l, r)
+          else if st = 5 then wcode_goon_right_moving_1 ires rs (l, r)
+          else if st = 6 then wcode_backto_standard_pos ires rs (l, r)
+          else if st = 13 then wcode_before_double ires rs (l, r)
+          else False)"
+
+declare wcode_double_case_inv.simps[simp del]
+
+fun wcode_double_case_state :: "t_conf \<Rightarrow> nat"
+  where
+  "wcode_double_case_state (st, l, r) = 
+   13 - st"
+
+fun wcode_double_case_step :: "t_conf \<Rightarrow> nat"
+  where
+  "wcode_double_case_step (st, l, r) = 
+      (if st = Suc 0 then (length l)
+      else if st = Suc (Suc 0) then (length r)
+      else if st = 3 then 
+                 if hd r = Oc then 1 else 0
+      else if st = 4 then (length r)
+      else if st = 5 then (length r)
+      else if st = 6 then (length l)
+      else 0)"
+
+fun wcode_double_case_measure :: "t_conf \<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"
+  where "wcode_double_case_le \<equiv> (inv_image lex_pair wcode_double_case_measure)"
+
+lemma [intro]: "wf lex_pair"
+by(auto intro:wf_lex_prod simp:lex_pair_def)
+
+lemma wf_wcode_double_case_le[intro]: "wf wcode_double_case_le"
+by(auto intro:wf_inv_image simp: wcode_double_case_le_def )
+term fetch
+
+lemma [simp]: "fetch t_wcode_main (Suc 0) Bk = (L, Suc 0)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+                fetch.simps nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main (Suc 0) Oc = (L, Suc (Suc 0))"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+                fetch.simps nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main (Suc (Suc 0)) Oc = (R, 3)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+                fetch.simps nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main (Suc (Suc (Suc 0))) Bk = (R, 4)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+                fetch.simps nth_of.simps)
+done 
+
+lemma [simp]: "fetch t_wcode_main (Suc (Suc (Suc 0))) Oc = (W0, 3)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+                fetch.simps nth_of.simps)
+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)
+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)
+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)
+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)
+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
+
+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)
+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
+
+
+declare wcode_on_checking_1.simps[simp del]
+
+lemmas wcode_double_case_inv_simps = 
+  wcode_on_left_moving_1.simps wcode_on_left_moving_1_O.simps
+  wcode_on_left_moving_1_B.simps wcode_on_checking_1.simps
+  wcode_erase1.simps wcode_on_right_moving_1.simps
+  wcode_goon_right_moving_1.simps wcode_backto_standard_pos.simps
+
+
+lemma [simp]: "wcode_on_left_moving_1 ires rs (b, r) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wcode_double_case_inv_simps, auto)
+done
+
+
+lemma [elim]: "\<lbrakk>wcode_on_left_moving_1 ires rs (b, Bk # list);
+                tl b = aa \<and> hd b # Bk # list = ba\<rbrakk> \<Longrightarrow> 
+               wcode_on_left_moving_1 ires rs (aa, ba)"
+apply(simp only: wcode_on_left_moving_1.simps wcode_on_left_moving_1_O.simps
+                wcode_on_left_moving_1_B.simps)
+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)
+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)
+apply(erule_tac exE)+
+apply(simp)
+done
+
+
+lemma [elim]: 
+  "\<lbrakk>wcode_on_left_moving_1 ires rs (b, Oc # list); tl b = aa \<and> hd b # Oc # list = ba\<rbrakk> 
+    \<Longrightarrow> wcode_on_checking_1 ires rs (aa, ba)"
+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
+
+
+lemma [simp]: "wcode_on_checking_1 ires rs (b, []) = False" 
+apply(auto simp: wcode_double_case_inv_simps)
+done         
+ 
+lemma [simp]: "wcode_on_checking_1 ires rs (b, Bk # list) = False"
+apply(auto simp: wcode_double_case_inv_simps)
+done         
+  
+lemma [elim]: "\<lbrakk>wcode_on_checking_1 ires rs (b, Oc # ba);Oc # b = aa \<and> list = ba\<rbrakk>
+  \<Longrightarrow> wcode_erase1 ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+apply(erule_tac exE)+
+apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
+done
+
+
+lemma [simp]: "wcode_on_checking_1 ires rs (b, []) = False"
+apply(simp add: wcode_double_case_inv_simps)
+done
+
+lemma [simp]: "wcode_on_checking_1 ires rs ([], Bk # list) = False"
+apply(simp add: wcode_double_case_inv_simps)
+done
+
+lemma [simp]: "wcode_erase1 ires rs (b, []) = False"
+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)
+done
+
+lemma [simp]: "wcode_on_right_moving_1 ires rs (b, []) = False"
+apply(simp add: wcode_double_case_inv_simps exp_ind_def)
+done
+
+lemma [elim]: "\<lbrakk>wcode_on_right_moving_1 ires rs (b, Bk # ba);  Bk # b = aa \<and> list = b\<rbrakk> \<Longrightarrow> 
+  wcode_on_right_moving_1 ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+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)
+done
+
+lemma [elim]: 
+  "\<lbrakk>wcode_on_right_moving_1 ires rs (b, Oc # ba); Oc # b = aa \<and> list = ba\<rbrakk> 
+  \<Longrightarrow> wcode_goon_right_moving_1 ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+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 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)
+done
+
+lemma [elim]: "\<lbrakk>wcode_erase1 ires rs (b, Bk # ba); Bk # b = aa \<and> list = ba; c = Bk # ba\<rbrakk> 
+  \<Longrightarrow> wcode_on_right_moving_1 ires rs (aa, ba)"
+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)
+done
+
+lemma [elim]: "\<lbrakk>wcode_erase1 ires rs (aa, Oc # list);  b = aa \<and> Bk # list = ba\<rbrakk> \<Longrightarrow> 
+  wcode_erase1 ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+apply(erule_tac exE)+
+apply(rule_tac x = ln in exI, rule_tac x = rn in exI, auto)
+done
+
+lemma [elim]: "\<lbrakk>wcode_goon_right_moving_1 ires rs (aa, []); b = aa \<and> [Oc] = ba\<rbrakk> 
+              \<Longrightarrow> wcode_backto_standard_pos ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+apply(erule_tac exE)+
+apply(rule_tac disjI2)
+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]: 
+  "\<lbrakk>wcode_goon_right_moving_1 ires rs (aa, Bk # list);  b = aa \<and> Oc # list = ba\<rbrakk>
+  \<Longrightarrow> wcode_backto_standard_pos ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+apply(erule_tac exE)+
+apply(rule_tac disjI2)
+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)
+done
+
+lemma [elim]: "\<lbrakk>wcode_goon_right_moving_1 ires rs (b, Oc # ba);  Oc # b = aa \<and> list = ba\<rbrakk> 
+  \<Longrightarrow> wcode_goon_right_moving_1 ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+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)
+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> 
+  \<Longrightarrow> wcode_before_double ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps wcode_backto_standard_pos_B.simps
+                 wcode_backto_standard_pos_O.simps wcode_before_double.simps)
+apply(erule_tac disjE)
+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)
+done
+
+lemma [simp]: "wcode_backto_standard_pos ires rs ([], Oc # list) = False"
+apply(auto simp: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps
+                 wcode_backto_standard_pos_O.simps)
+done
+
+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>
+       \<Longrightarrow> wcode_backto_standard_pos ires rs (aa, ba)"
+apply(simp only:  wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps
+                 wcode_backto_standard_pos_O.simps)
+apply(erule_tac disjE)
+apply(simp)
+apply(erule_tac exE)+
+apply(case_tac ml, simp)
+apply(rule_tac disjI1, rule_tac conjI)
+apply(rule_tac x = ln  in exI, simp, rule_tac x = rn in exI, simp)
+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]
+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
+       \<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)"
+  have "\<exists> n. ?P (?f n) \<and> ?Q (?f (n::nat))"
+  proof(rule_tac halt_lemma2)
+    show "wf wcode_double_case_le"
+      by auto
+  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)
+      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> 
+                   wcode_double_case_inv st ires rs x) \<and> 
+                (tstep (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
+                                        lex_pair_def)
+        apply(auto split: if_splits)
+        done
+    qed
+  next
+    show "?Q (?f 0)"
+      apply(simp add: steps.simps wcode_double_case_inv.simps 
+                                  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)
+      done
+  next
+    show "\<not> ?P (?f 0)"
+      apply(simp add: steps.simps)
+      done
+  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
+    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 (abacus.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 (abacus.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)
+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) 
+    (abacus.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) (abacus.tshift tp (length tp1 div 2),
+         length tp1 div 2) stp = (a + length tp1 div 2, b, c)"
+    apply(rule_tac ind, simp)
+    using h
+    apply(case_tac a, simp_all add: tstep.simps fetch.simps)
+    done
+  from h and this show "t_step (t_steps (st + length tp1 div 2, l, r) (abacus.tshift tp (length tp1 div 2), length tp1 div 2) stp)
+           (abacus.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 (abacus.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
+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 @ abacus.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 @ abacus.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)
+done
+
+lemma [intro]: "rec_calc_rel (recf.id (Suc 0) 0) [rs] rs"
+  apply(rule_tac calc_id, simp_all)
+  done
+  
+lemma [intro]: "rec_calc_rel (constn 2) [rs] 2"
+using prime_rel_exec_eq[of "constn 2" "[rs]" 2]
+apply(subgoal_tac "primerec (constn 2) 1", auto)
+done
+
+lemma  [intro]: "rec_calc_rel rec_mult [rs, 2] (2 * rs)"
+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>)"
+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)
+    show "rec_ci rec_twice = (a, b, c)" by (simp add: h)
+  next
+    show "rec_calc_rel rec_twice [rs] (2 * rs)"
+      apply(simp add: rec_twice_def)
+      apply(rule_tac rs =  "[rs, 2]" in calc_cn, simp_all)
+      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
+  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))"
+      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>)"
+    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
+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 = 
+        (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)
+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)
+    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
+qed
+
+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) #
+       abacus.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
+  
+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>)"
+  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(simp)
+  done
+  
+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
+
+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>)
+     t_wcode_main stp 
+    = (Suc 0, Bk\<^bsup>ln\<^esup> @ Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+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])
+done
+
+lemma wcode_main_first_part_len:
+  "length t_wcode_main_first_part = 24"
+  apply(simp add: t_wcode_main_first_part_def)
+  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>)"
+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>)"
+    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",
+          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]
+    apply(erule_tac exE)
+    apply(erule_tac exE)
+    apply(erule_tac exE)
+    apply(simp add: wcode_main_first_part_len)
+    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])
+    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>)"
+    using wcode_jump1[of lnb "Oc # ires" "Suc rs" rnb]
+    apply(erule_tac exE)
+    apply(erule_tac exE)
+    apply(erule_tac exE)
+    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(simp)
+    apply(case_tac lnb, simp, simp add: exp_ind_def[THEN sym] exp_ind)
+    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>)"
+    by blast
+  from stp1 stp2 stp3 show "?thesis"
+    apply(rule_tac x = "stpa + stpb + stpc" in exI, rule_tac x = lnc in exI,
+         rule_tac x = rnc in exI)
+    apply(simp add: steps_add)
+    done
+qed
+    
+
+(* Begin: fourtime_case*)
+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> 
+                 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>)"
+
+fun wcode_on_left_moving_2 :: "bin_inv_t"
+  where
+  "wcode_on_left_moving_2 ires rs (l, r) = 
+      (wcode_on_left_moving_2_B ires rs (l, r) \<or> 
+      wcode_on_left_moving_2_O ires rs (l, r))"
+
+fun wcode_on_checking_2 :: "bin_inv_t"
+  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>)"
+
+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>)"
+
+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>)"
+
+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>)"
+
+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)"
+
+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)"
+
+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>)"
+
+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> 
+                          ml + mr = (Suc (Suc rs)) \<and> mr > 0)"
+
+fun wcode_backto_standard_pos_2 :: "bin_inv_t"
+  where
+  "wcode_backto_standard_pos_2 ires rs (l, r) = 
+           (wcode_backto_standard_pos_2_O ires rs (l, r) \<or> 
+           wcode_backto_standard_pos_2_B ires rs (l, r))"
+
+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>)"
+
+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]
+        wcode_goon_checking.simps[simp del] wcode_right_move.simps[simp del]
+        wcode_erase2.simps[simp del]
+        wcode_on_right_moving_2.simps[simp del] wcode_goon_right_moving_2.simps[simp del]
+        wcode_backto_standard_pos_2_B.simps[simp del] wcode_backto_standard_pos_2_O.simps[simp del]
+        wcode_backto_standard_pos_2.simps[simp del]
+
+lemmas wcode_fourtimes_invs = 
+       wcode_on_left_moving_2_B.simps wcode_on_left_moving_2.simps
+        wcode_on_left_moving_2_O.simps wcode_on_checking_2.simps
+        wcode_goon_checking.simps wcode_right_move.simps
+        wcode_erase2.simps
+        wcode_on_right_moving_2.simps wcode_goon_right_moving_2.simps
+        wcode_backto_standard_pos_2_B.simps wcode_backto_standard_pos_2_O.simps
+        wcode_backto_standard_pos_2.simps
+
+fun wcode_fourtimes_case_inv :: "nat \<Rightarrow> bin_inv_t"
+  where
+  "wcode_fourtimes_case_inv st ires rs (l, r) = 
+           (if st = Suc 0 then wcode_on_left_moving_2 ires rs (l, r)
+            else if st = Suc (Suc 0) then wcode_on_checking_2 ires rs (l, r)
+            else if st = 7 then wcode_goon_checking ires rs (l, r)
+            else if st = 8 then wcode_right_move ires rs (l, r)
+            else if st = 9 then wcode_erase2 ires rs (l, r)
+            else if st = 10 then wcode_on_right_moving_2 ires rs (l, r)
+            else if st = 11 then wcode_goon_right_moving_2 ires rs (l, r)
+            else if st = 12 then wcode_backto_standard_pos_2 ires rs (l, r)
+            else if st = t_twice_len + 14 then wcode_before_fourtimes ires rs (l, r)
+            else False)"
+
+declare wcode_fourtimes_case_inv.simps[simp del]
+
+fun wcode_fourtimes_case_state :: "t_conf \<Rightarrow> nat"
+  where
+  "wcode_fourtimes_case_state (st, l, r) = 13 - st"
+
+fun wcode_fourtimes_case_step :: "t_conf \<Rightarrow> nat"
+  where
+  "wcode_fourtimes_case_step (st, l, r) = 
+         (if st = Suc 0 then length l
+          else if st = 9 then 
+           (if hd r = Oc then 1
+            else 0)
+          else if st = 10 then length r
+          else if st = 11 then length r
+          else if st = 12 then length l
+          else 0)"
+
+fun wcode_fourtimes_case_measure :: "t_conf \<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"
+  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"
+by(auto intro:wf_inv_image simp: wcode_fourtimes_case_le_def)
+
+lemma [simp]: "fetch t_wcode_main (Suc (Suc 0)) Bk = (L, 7)"
+apply(simp add: t_wcode_main_def fetch.simps 
+  t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 7 Oc = (R, 8)"
+apply(simp add: t_wcode_main_def fetch.simps 
+  t_wcode_main_first_part_def nth_of.simps)
+done
+ 
+lemma [simp]: "fetch t_wcode_main 8 Bk = (R, 9)"
+apply(simp add: t_wcode_main_def fetch.simps 
+  t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 9 Bk = (R, 10)"
+apply(simp add: t_wcode_main_def fetch.simps 
+  t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 9 Oc = (W0, 9)"
+apply(simp add: t_wcode_main_def fetch.simps 
+  t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 10 Bk = (R, 10)"
+apply(simp add: t_wcode_main_def fetch.simps 
+  t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 10 Oc = (R, 11)"
+apply(simp add: t_wcode_main_def fetch.simps 
+  t_wcode_main_first_part_def nth_of.simps)
+done 
+
+lemma [simp]: "fetch t_wcode_main 11 Bk = (W1, 12)"
+apply(simp add: t_wcode_main_def fetch.simps 
+  t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 11 Oc = (R, 11)"
+apply(simp add: t_wcode_main_def fetch.simps 
+  t_wcode_main_first_part_def nth_of.simps)
+done 
+
+lemma [simp]: "fetch t_wcode_main 12 Oc = (L, 12)"
+apply(simp add: t_wcode_main_def fetch.simps 
+  t_wcode_main_first_part_def nth_of.simps)
+done 
+
+lemma [simp]: "fetch t_wcode_main 12 Bk = (R, t_twice_len + 14)"
+apply(simp add: t_wcode_main_def fetch.simps 
+  t_wcode_main_first_part_def nth_of.simps)
+done
+
+
+lemma [simp]: "wcode_on_left_moving_2 ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_on_checking_2 ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs)
+done          
+
+lemma [simp]: "wcode_goon_checking ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_right_move ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_erase2 ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_on_right_moving_2 ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs exponent_def)
+done
+
+lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs exponent_def)
+done
+    
+lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wcode_fourtimes_invs, auto)
+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(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)
+apply(simp)
+done
+
+lemma [simp]: "wcode_on_checking_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma  [simp]: "wcode_on_checking_2 ires rs (b, Bk # list)
+       \<Longrightarrow>   wcode_goon_checking ires rs (tl b, hd b # Bk # list)"
+apply(simp only: wcode_fourtimes_invs)
+apply(auto)
+done
+
+lemma [simp]: "wcode_goon_checking ires rs (b, Bk # list) = False"
+apply(simp add: wcode_fourtimes_invs)
+done
+
+lemma [simp]: " wcode_right_move ires rs (b, Bk # list) \<Longrightarrow> b\<noteq> []" 
+apply(simp add: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_right_move ires rs (b, Bk # list) \<Longrightarrow>  wcode_erase2 ires rs (Bk # b, list)"
+apply(auto simp:wcode_fourtimes_invs )
+apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
+done
+
+lemma [simp]: "wcode_erase2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+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)
+done
+
+lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(auto simp:wcode_fourtimes_invs )
+done
+
+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)
+done
+
+lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> 
+                 wcode_backto_standard_pos_2 ires rs (b, Oc # list)"
+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(rule_tac x = "rn - 1" in exI, simp)
+apply(case_tac rn, simp, simp add: exp_ind_def)
+done
+   
+lemma  [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list) \<Longrightarrow>  b \<noteq> []"
+apply(simp add: wcode_fourtimes_invs, auto)
+done
+
+lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wcode_fourtimes_invs, auto)
+done
+
+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)
+done
+
+lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \<Longrightarrow> b \<noteq> []"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \<Longrightarrow>
+              wcode_backto_standard_pos_2 ires rs (b, [Oc])"
+apply(simp only: wcode_fourtimes_invs)
+apply(erule_tac exE)+
+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>)"
+apply(auto simp: wcode_fourtimes_invs)
+apply(case_tac [!] mr, auto simp: exp_ind_def)
+done
+
+
+lemma [simp]: "wcode_on_checking_2 ires rs (b, Oc # list) \<Longrightarrow> False"
+apply(simp add: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_goon_checking ires rs (b, Oc # list) \<Longrightarrow>
+  (b = [] \<longrightarrow> wcode_right_move ires rs ([Oc], list)) \<and>
+  (b \<noteq> [] \<longrightarrow> wcode_right_move ires rs (Oc # b, list))"
+apply(simp only: wcode_fourtimes_invs)
+apply(erule_tac exE)+
+apply(auto)
+done
+
+lemma [simp]: "wcode_right_move ires rs (b, Oc # list) = False"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: " wcode_erase2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_erase2 ires rs (b, Oc # list)
+       \<Longrightarrow> wcode_erase2 ires rs (b, Bk # list)"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp only: wcode_fourtimes_invs)
+apply(auto)
+done
+
+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(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)
+done
+
+lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp only:wcode_fourtimes_invs, auto)
+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>)"
+apply(simp add: wcode_fourtimes_invs, auto)
+apply(case_tac [!] mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wcode_on_checking_2 ires rs (b, Oc # list) = False"
+apply(simp add: wcode_fourtimes_invs)
+done
+
+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 = "mr - 1" in exI)
+apply(case_tac mr, case_tac rn, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp only: wcode_fourtimes_invs, auto)
+done
+ 
+lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list)    
+            \<Longrightarrow> wcode_backto_standard_pos_2 ires rs (tl b, hd b # Oc # list)"
+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)
+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
+  \<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)"
+  have "\<exists> n . ?P (?f n) \<and> ?Q (?f (n::nat))"
+  proof(rule_tac halt_lemma2)
+    show "wf wcode_fourtimes_case_le"
+      by auto
+  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_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,
+     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 
+                        wcode_fourtimes_case_le_def lex_pair_def split: if_splits)
+    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 0" in exI, auto)
+      done
+  next
+    show "\<not> ?P (?f 0)"
+      apply(simp add: steps.simps)
+      done
+  qed
+  thus "?thesis"
+    apply(erule_tac exE, simp)
+    done
+qed
+
+definition t_fourtimes_len :: "nat"
+  where
+  "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)
+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>)"
+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)
+    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)
+      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
+  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))"
+      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>)"
+    apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
+    done
+qed
+
+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)
+    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
+qed
+
+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>)
+     ((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 @ 
+    abacus.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
+
+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 (abacus.tshift t_twice 12)) div 2)
+             = (length (abacus.tshift t_twice 12) div 2 + 13)"
+using tm_even[of abc_twice]
+apply(simp add: t_twice_def)
+done 
+
+lemma [simp]: "t_twice_len + 14 =  14 + length (abacus.tshift t_twice 12) div 2"
+using tm_even[of abc_twice]
+apply(simp add: t_twice_def t_twice_len_def shift_length)
+done
+
+lemma t_fourtimes_append:
+  "\<exists> stp ln rn. 
+  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>)
+  ((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>)"
+  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)
+  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
+ 
+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)
+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>)"
+apply(rule_tac x = "Suc 0" in exI)
+apply(simp add: steps.simps shift_length)
+apply(rule_tac x = lnb in exI, rule_tac x = rnb in exI)
+apply(simp add: tstep.simps new_tape.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>)"
+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>)"
+    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,
+          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 # 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>)
+                     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]
+    apply(erule_tac exE)
+    apply(erule_tac exE)
+    apply(erule_tac exE)
+    apply(simp add: t_wcode_main_def)
+    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])
+    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>)
+                     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>)"
+    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>)
+    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>)"
+    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>)
+    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>)"
+    by blast
+  from stp1 stp2 stp3 show "?thesis"
+    apply(rule_tac x = "stpa + stpb + stpc" in exI,
+          rule_tac x = lnc in exI, rule_tac x = rnc in exI)
+    apply(simp add: steps_add)
+    done
+qed
+
+(**********************************************************)
+
+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> 
+                    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>)"
+
+fun wcode_on_left_moving_3 :: "bin_inv_t"
+  where
+  "wcode_on_left_moving_3 ires rs (l, r) = 
+       (wcode_on_left_moving_3_B ires rs (l, r) \<or>  
+        wcode_on_left_moving_3_O ires rs (l, r))"
+
+fun wcode_on_checking_3 :: "bin_inv_t"
+  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>)"
+
+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>)"
+
+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>)"
+
+fun wcode_halt_case_inv :: "nat \<Rightarrow> bin_inv_t"
+  where
+  "wcode_halt_case_inv st ires rs (l, r) = 
+          (if st = 0 then wcode_stop ires rs (l, r)
+           else if st = Suc 0 then wcode_on_left_moving_3 ires rs (l, r)
+           else if st = Suc (Suc 0) then wcode_on_checking_3 ires rs (l, r)
+           else if st = 7 then wcode_goon_checking_3 ires rs (l, r)
+           else False)"
+
+fun wcode_halt_case_state :: "t_conf \<Rightarrow> nat"
+  where
+  "wcode_halt_case_state (st, l, r) = 
+           (if st = 1 then 5
+            else if st = Suc (Suc 0) then 4
+            else if st = 7 then 3
+            else 0)"
+
+fun wcode_halt_case_step :: "t_conf \<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"
+  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"
+  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"
+by(auto intro:wf_inv_image simp: wcode_halt_case_le_def)
+
+declare wcode_on_left_moving_3_B.simps[simp del] wcode_on_left_moving_3_O.simps[simp del]  
+        wcode_on_checking_3.simps[simp del] wcode_goon_checking_3.simps[simp del] 
+        wcode_on_left_moving_3.simps[simp del] wcode_stop.simps[simp del]
+
+lemmas wcode_halt_invs = 
+  wcode_on_left_moving_3_B.simps wcode_on_left_moving_3_O.simps
+  wcode_on_checking_3.simps wcode_goon_checking_3.simps 
+  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
+                t_wcode_main_first_part_def)
+done
+
+lemma [simp]: "wcode_on_left_moving_3 ires rs (b, [])  = False"
+apply(simp only: wcode_halt_invs)
+apply(simp add: exp_ind_def)
+done    
+
+lemma [simp]: "wcode_on_checking_3 ires rs (b, []) = False"
+apply(simp add: wcode_halt_invs)
+done
+              
+lemma [simp]: "wcode_goon_checking_3 ires rs (b, []) = False"
+apply(simp add: wcode_halt_invs)
+done 
+
+lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Bk # list)
+ \<Longrightarrow> wcode_on_left_moving_3 ires rs (tl b, hd b # Bk # list)"
+apply(simp only: wcode_halt_invs)
+apply(erule_tac disjE)
+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(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)
+apply(simp)
+done
+
+lemma [simp]: "wcode_goon_checking_3 ires rs (b, Bk # list) \<Longrightarrow> 
+  (b = [] \<longrightarrow> wcode_stop ires rs ([Bk], list)) \<and>
+  (b \<noteq> [] \<longrightarrow> wcode_stop ires rs (Bk # b, list))"
+apply(auto simp: wcode_halt_invs)
+done
+
+lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(auto simp: wcode_halt_invs)
+done
+
+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)
+done     
+
+lemma [simp]: "wcode_on_checking_3 ires rs (b, Oc # list) = False"
+apply(auto simp: wcode_halt_invs)
+done
+
+lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+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
+
+lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \<Longrightarrow> 
+  wcode_goon_checking_3 ires rs (tl b, hd b # Bk # list)"
+apply(auto simp: wcode_halt_invs)
+done
+
+lemma [simp]: "wcode_goon_checking_3 ires rs (b, Oc # list) = False"
+apply(simp add: wcode_goon_checking_3.simps)
+done
+
+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
+       \<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)"
+  have "\<exists> n. ?P (?f n) \<and> ?Q (?f (n::nat))"
+  proof(rule_tac halt_lemma2)
+    show "wf wcode_halt_case_le" by auto
+  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_halt_case_le"
+      apply(rule_tac allI, rule_tac impI, case_tac "?f na")
+      apply(simp add: tstep_red tstep.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)
+      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 0" in exI, auto)
+      done
+  next
+    show "\<not> ?P (?f 0)"
+      apply(simp add: steps.simps)
+      done
+  qed
+  thus "?thesis"
+    apply(auto)
+    done
+qed
+
+declare wcode_halt_case_inv.simps[simp del]
+lemma [intro]: "\<exists> xs. (<rev list @ [aa::nat]> :: block 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)
+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>)"
+  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(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)
+done
+
+lemma bl_bin_one: "bl_bin [Oc] =  Suc 0"
+apply(simp add: bl_bin.simps)
+done
+
+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
+                    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>)"
+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>)"
+  
+    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)
+    apply(rule_tac x = "butlast args" in exI, auto)
+    done    
+  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>)"
+  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>)"
+    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)
+        done
+    next
+      fix a m n rs ires
+      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>)"
+      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>)"
+          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)
+          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
+        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   
+        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>)"
+          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)
+          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>)"
+    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>)"
+        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>)"
+            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
+      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>)"
+         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>)"
+         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>)"
+         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
+     next
+       fix ab m n rs args lm
+       assume ind2:
+         "\<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>)"
+         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>)"
+       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"
+                                      rs n]
+           apply(simp add: exp_ind_def)
+           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
+         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>)"
+           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>)" 
+           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>)"
+           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)
+           done
+       qed
+     qed
+   qed
+ qed
+
+
+         
+(* turing_shift can be used*)
+term t_wcode_main
+definition t_wcode_prepare :: "tprog"
+  where
+  "t_wcode_prepare \<equiv> 
+         [(W1, 2), (L, 1), (L, 3), (R, 2), (R, 4), (W0, 3),
+          (R, 4), (R, 5), (R, 6), (R, 5), (R, 7), (R, 5),
+          (W1, 7), (L, 0)]"
+
+fun wprepare_add_one :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+  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>))"
+
+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>
+                      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>)"
+
+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>)"
+
+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>)"
+
+fun wprepare_goto_start_pos :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+  where
+  "wprepare_goto_start_pos m lm (l, r) = 
+       (wprepare_goto_start_pos_B m lm (l, r) \<or>
+        wprepare_goto_start_pos_O m lm (l, r))"
+
+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>)"
+
+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> [])"
+
+fun wprepare_loop_start :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+  where
+  "wprepare_loop_start m lm (l, r) = (wprepare_loop_start_on_rightmost m lm (l, r) \<or> 
+                                      wprepare_loop_start_in_middle m lm (l, r))"
+
+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>)"
+
+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)"
+
+fun wprepare_loop_goon :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+  where
+  "wprepare_loop_goon m lm (l, r) = 
+              (wprepare_loop_goon_in_middle m lm (l, r) \<or> 
+               wprepare_loop_goon_on_rightmost m lm (l, r))"
+
+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>))"
+
+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>)"
+
+fun wprepare_inv :: "nat \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+  where
+  "wprepare_inv st m lm (l, r) = 
+        (if st = 0 then wprepare_stop m lm (l, r) 
+         else if st = Suc 0 then wprepare_add_one m lm (l, r)
+         else if st = Suc (Suc 0) then wprepare_goto_first_end m lm (l, r)
+         else if st = Suc (Suc (Suc 0)) then wprepare_erase m lm (l, r)
+         else if st = 4 then wprepare_goto_start_pos m lm (l, r)
+         else if st = 5 then wprepare_loop_start m lm (l, r)
+         else if st = 6 then wprepare_loop_goon m lm (l, r)
+         else if st = 7 then wprepare_add_one2 m lm (l, r)
+         else False)"
+
+fun wprepare_stage :: "t_conf \<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"
+  where
+  "wprepare_state (st, l, r) = 
+       (if st = 1 then 4
+        else if st = Suc (Suc 0) then 3
+        else if st = Suc (Suc (Suc 0)) then 2
+        else if st = 4 then 1
+        else if st = 7 then 2
+        else 0)"
+
+fun wprepare_step :: "t_conf \<Rightarrow> nat"
+  where
+  "wprepare_step (st, l, r) = 
+      (if st = 1 then (if hd r = Oc then Suc (length l)
+                       else 0)
+       else if st = Suc (Suc 0) then length r
+       else if st = Suc (Suc (Suc 0)) then (if hd r = Oc then 1
+                            else 0)
+       else if st = 4 then length r
+       else if st = 5 then Suc (length r)
+       else if st = 6 then (if r = [] then 0 else Suc (length r))
+       else if st = 7 then (if (r \<noteq> [] \<and> hd r = Oc) then 0
+                            else 1)
+       else 0)"
+
+fun wcode_prepare_measure :: "t_conf \<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"
+  where "wcode_prepare_le \<equiv> (inv_image lex_triple wcode_prepare_measure)"
+
+lemma [intro]: "wf lex_pair"
+by(auto intro:wf_lex_prod simp:lex_pair_def)
+
+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)
+
+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]
+        wprepare_loop_start.simps[simp del] wprepare_loop_goon.simps[simp del]
+        wprepare_add_one2.simps[simp del]
+
+lemmas wprepare_invs = wprepare_add_one.simps wprepare_goto_first_end.simps
+        wprepare_erase.simps wprepare_goto_start_pos.simps
+        wprepare_loop_start.simps wprepare_loop_goon.simps
+        wprepare_add_one2.simps
+
+declare wprepare_inv.simps[simp del]
+lemma [simp]: "fetch t_wcode_prepare (Suc 0) Bk = (W1, 2)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare (Suc 0) Oc = (L, 1)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare (Suc (Suc 0)) Bk = (L, 3)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare (Suc (Suc 0)) Oc = (R, 2)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare (Suc (Suc (Suc 0))) Bk = (R, 4)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare (Suc (Suc (Suc 0))) Oc = (W0, 3)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 4 Bk = (R, 4)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 4 Oc = (R, 5)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 5 Oc = (R, 5)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 5 Bk = (R, 6)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 6 Oc = (R, 5)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 6 Bk = (R, 7)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 7 Oc = (L, 0)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+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)
+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"
+apply(simp add: wprepare_invs)
+done
+
+lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_erase m lm (b, []) = False"
+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)
+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
+
+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(erule_tac disjE)
+apply(rule_tac disjI2)
+apply(simp add: wprepare_loop_start_on_rightmost.simps
+                wprepare_loop_goon_on_rightmost.simps, auto)
+apply(rule_tac rev_eq, simp add: tape_of_nl_rev)
+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)
+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)
+done
+
+lemma [simp]: "wprepare_add_one2 m lm (b, []) \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs tape_of_nl_not_null, 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)
+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)
+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)
+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
+
+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)
+done
+
+lemma [simp]: "wprepare_erase m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs exp_ind_def, auto)
+done
+
+lemma [simp]: "wprepare_erase m lm (b, Bk # list) \<Longrightarrow> 
+                           wprepare_goto_start_pos m lm (Bk # b, list)"
+apply(simp only: wprepare_invs, auto)
+done
+
+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)
+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)
+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> []"
+apply(simp only: wprepare_invs, auto)
+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)
+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)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> [];  wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs)
+apply(auto)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs, auto)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> 
+  (list = [] \<longrightarrow> wprepare_add_one2 m lm (Bk # b, [])) \<and> 
+  (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)
+done
+
+lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs, simp)
+done
+
+lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \<Longrightarrow> 
+      (list = [] \<longrightarrow> wprepare_add_one2 m lm (b, [Oc])) \<and> 
+      (list \<noteq> [] \<longrightarrow> wprepare_add_one2 m lm (b, Oc # list))"
+apply(simp only:  wprepare_invs, auto)
+done
+
+lemma [simp]: "wprepare_goto_first_end m lm (b, Oc # list)
+       \<Longrightarrow> (b = [] \<longrightarrow> wprepare_goto_first_end m lm ([Oc], list)) \<and> 
+           (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(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)
+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)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk>
+       \<Longrightarrow> wprepare_goto_start_pos m lm (Bk # b, list)"
+apply(simp only:wprepare_invs, auto)
+apply(case_tac [!] lm, simp, simp_all)
+done
+
+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>"
+apply(case_tac mr, simp_all)
+apply(case_tac rn, simp_all add: exp_ind_def, auto)
+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>"
+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)
+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)
+done
+
+declare wprepare_loop_start_in_middle.simps[simp del]
+
+declare wprepare_loop_start_on_rightmost.simps[simp del] 
+        wprepare_loop_goon_in_middle.simps[simp del]
+        wprepare_loop_goon_on_rightmost.simps[simp del]
+
+lemma [simp]: "wprepare_loop_goon_in_middle m lm (Bk # b, []) = False"
+apply(simp add: wprepare_loop_goon_in_middle.simps, auto)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [Bk])\<rbrakk> \<Longrightarrow>
+  wprepare_loop_goon m lm (Bk # b, [])"
+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(rule_tac rev_eq)
+apply(simp add: tape_of_nl_rev)
+apply(simp add: exp_ind_def[THEN sym] exp_ind)
+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)
+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(simp add: tape_of_nl_rev)
+apply(simp add: exp_ind_def[THEN sym] exp_ind)
+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  "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(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)
+done
+
+lemma [simp]: 
+  "\<lbrakk>lm \<noteq> []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\<rbrakk> 
+  \<Longrightarrow> wprepare_loop_goon_in_middle m lm (Bk # b, a # lista)"
+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 lm1, simp)
+apply(rule_tac x = "Suc aa" in exI, simp)
+apply(rule_tac x = list in exI)
+apply(case_tac list, simp_all add: tape_of_nl_abv tape_of_nat_list.simps)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, Bk # a # lista)\<rbrakk> \<Longrightarrow> 
+  wprepare_loop_goon m lm (Bk # b, a # lista)"
+apply(simp add: wprepare_loop_start.simps 
+                wprepare_loop_goon.simps)
+apply(erule_tac disjE, simp, auto)
+done
+
+lemma start_2_goon:
+  "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, Bk # list)\<rbrakk> \<Longrightarrow>
+   (list = [] \<longrightarrow> wprepare_loop_goon m lm (Bk # b, [])) \<and>
+  (list \<noteq> [] \<longrightarrow> wprepare_loop_goon m lm (Bk # b, list))"
+apply(case_tac list, auto)
+done
+
+lemma add_one_2_add_one: "wprepare_add_one m lm (b, Oc # list)
+  \<Longrightarrow> (hd b = Oc \<longrightarrow> (b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)) \<and>
+                     (b \<noteq> [] \<longrightarrow> wprepare_add_one m lm (tl b, Oc # Oc # list))) \<and>
+  (hd b \<noteq> Oc \<longrightarrow> (b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)) \<and>
+                 (b \<noteq> [] \<longrightarrow> wprepare_add_one m lm (tl b, hd b # Oc # list)))"
+apply(simp only: wprepare_add_one.simps, auto)
+done
+
+lemma [simp]: "wprepare_loop_start m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp)
+done
+
+lemma [simp]: "wprepare_loop_start_on_rightmost m lm (b, Oc # list) \<Longrightarrow> 
+  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)
+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(rule_tac x = nat in exI, simp)
+apply(rule_tac x = lm1 in exI, simp)
+done
+
+lemma start_2_start: "wprepare_loop_start m lm (b, Oc # list) \<Longrightarrow> 
+       wprepare_loop_start m lm (Oc # b, list)"
+apply(simp add: wprepare_loop_start.simps)
+apply(erule_tac disjE, simp_all )
+done
+
+lemma [simp]: "wprepare_loop_goon m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wprepare_loop_goon.simps     
+                wprepare_loop_goon_in_middle.simps 
+                wprepare_loop_goon_on_rightmost.simps)
+apply(auto)
+done
+
+lemma [simp]: "wprepare_goto_start_pos m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wprepare_goto_start_pos.simps)
+done
+
+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>
+       \<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>
+       \<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(rule_tac x = nat in exI, simp)
+apply(rule_tac x = "a#lista" in exI, simp)
+done
+
+lemma [simp]: "wprepare_loop_goon_in_middle m lm (b, Oc # list) \<Longrightarrow>
+                wprepare_loop_start_on_rightmost m lm (Oc # b, list) \<or>
+                wprepare_loop_start_in_middle m lm (Oc # b, list)"
+apply(simp add: wprepare_loop_goon_in_middle.simps split: if_splits)
+apply(case_tac lm1, simp_all add: wprepare_loop1 wprepare_loop2)
+done
+
+lemma [simp]: "wprepare_loop_goon m lm (b, Oc # list)
+  \<Longrightarrow>  wprepare_loop_start m lm (Oc # b, list)"
+apply(simp add: wprepare_loop_goon.simps
+                wprepare_loop_start.simps)
+done
+
+lemma [simp]: "wprepare_add_one m lm (b, Oc # list)
+       \<Longrightarrow> b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)"
+apply(auto)
+apply(simp add: wprepare_add_one.simps)
+done
+
+lemma [simp]: "wprepare_goto_start_pos m [a] (b, Oc # list)
+              \<Longrightarrow> wprepare_loop_start_on_rightmost m [a] (Oc # b, list) "
+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)
+done
+
+lemma [simp]:  "wprepare_goto_start_pos m (a # aa # listaa) (b, Oc # list)
+       \<Longrightarrow>wprepare_loop_start_in_middle m (a # aa # listaa) (Oc # b, list)"
+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(rule_tac x = a in exI, rule_tac x = "aa#listaa" in exI, simp)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Oc # list)\<rbrakk>
+       \<Longrightarrow> wprepare_loop_start m lm (Oc # b, list)"
+apply(case_tac lm, simp_all)
+apply(case_tac lista, simp_all add: wprepare_loop_start.simps)
+done
+
+lemma [simp]: "wprepare_add_one2 m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(auto simp: wprepare_add_one2.simps)
+done
+
+lemma add_one_2_stop:
+  "wprepare_add_one2 m lm (b, Oc # list)      
+  \<Longrightarrow>  wprepare_stop m lm (tl b, hd b # Oc # list)"
+apply(simp add: wprepare_stop.simps wprepare_add_one2.simps)
+done
+
+declare wprepare_stop.simps[simp del]
+
+lemma wprepare_correctness:
+  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
+    \<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)"
+  have "\<exists> n. ?P (?f n) \<and> ?Q (?f n)"
+  proof(rule_tac halt_lemma2)
+    show "wf wcode_prepare_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> wcode_prepare_le"
+      using h
+      apply(rule_tac allI, rule_tac impI, case_tac "?f n", 
+            simp add: tstep_red tstep.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
+
+                 split: if_splits)
+      apply(simp_all add: start_2_goon  start_2_start
+                           add_one_2_add_one add_one_2_stop)
+      apply(auto simp: wprepare_add_one2.simps)
+      done   
+  next
+    show "?Q (?f 0)"
+      apply(simp add: steps.simps wprepare_inv.simps wprepare_invs)
+      done
+  next
+    show "\<not> ?P (?f 0)"
+      apply(simp add: steps.simps)
+      done
+  qed
+  thus "?thesis"
+    apply(auto)
+    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 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
+
+lemma [intro]: 
+  "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)
+done
+
+lemma [intro]: "t_correct (tm_of abc_fourtimes @ tMp (Suc 0) 
+   (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))"
+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)
+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)
+    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)
+     (abacus.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)
+     (abacus.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))
+    (abacus.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)
+    (abacus.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)
+    done
+qed
+
+lemma [intro]: "t_correct (t_wcode_prepare |+| t_wcode_main)"
+apply(auto intro: t_correct_add)
+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>)"
+proof -
+  let ?P1 = "\<lambda> (l, r). l = [] \<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 ?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
+  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)
+        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)
+          done
+      qed
+  next
+    show "wprepare_stop m args \<turnstile>-> wprepare_stop m args"
+      by(simp add: t_imply_def)
+  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)
+    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)"
+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> = []"
+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)
+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>"
+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(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(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(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(rule_tac x = nat in exI, simp)
+done
+
+lemma [simp]: "tinres r r' \<Longrightarrow> tinres (b # r) (b # r')"
+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>
+    \<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
+
+
+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>
+    \<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)")
+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)"
+  have "b = ba \<and> tinres c ca \<and> a = aa"
+    apply(rule_tac ind, simp_all add: h)
+    done
+  thus "la = lb \<and> tinres ra rb \<and> sa = sb"
+    apply(rule_tac l = b  and r = c  and ss = a and r' = ca   
+            and t = t in tinres_step2)
+    using h
+    apply(simp, simp, simp)
+    done
+qed
+
+
+text{**************Begin: adjust***************************}   
+definition t_wcode_adjust :: "tprog"
+  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), 
+                   (W1, 2), (Nop, 7), (L, 9), (W0, 8), (L, 9), (L, 10), 
+                    (L, 11), (L, 10), (R, 0), (L, 11)]"
+                 
+lemma [simp]: "fetch t_wcode_adjust (Suc 0) Bk = (W1, 1)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust (Suc 0) Oc = (R, 2)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust (Suc (Suc 0)) Oc = (R, 3)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust (Suc (Suc (Suc 0))) Oc = (R, 4)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+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 5 Oc = (W0, 5)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 5 Bk = (L, 6)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 6 Oc = (R, 7)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 6 Bk = (L, 6)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 7 Bk = (W1, 2)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 8 Bk = (L, 9)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 8 Oc = (W0, 8)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 9 Oc = (L, 10)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 9 Bk = (L, 9)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 10 Bk = (L, 11)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 10 Oc = (L, 10)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 11 Oc = (L, 11)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 11 Bk = (R, 0)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+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>)"
+
+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>
+                          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>
+                      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))"
+
+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)"
+
+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>
+                      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> 
+                         ml + mr = Suc rs \<and> mr > 0)"
+
+fun wadjust_loop_on_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+  where
+  "wadjust_loop_on_left_moving m rs (l, r) = 
+       (wadjust_loop_on_left_moving_O m rs (l, r) \<or>
+       wadjust_loop_on_left_moving_B m rs (l, r))"
+
+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>
+                        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>)"
+
+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>)"
+
+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>)"
+
+fun wadjust_on_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+  where
+  "wadjust_on_left_moving m rs (l, r) = 
+      (wadjust_on_left_moving_O m rs (l, r) \<or>
+       wadjust_on_left_moving_B m rs (l, r))"
+
+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>)"
+
+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> 
+                      ml + mr = Suc (Suc rs) \<and> mr > 0)"
+
+fun wadjust_goon_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+  where
+  "wadjust_goon_left_moving m rs (l, r) = 
+            (wadjust_goon_left_moving_B m rs (l, r) \<or>
+             wadjust_goon_left_moving_O m rs (l, r))"
+
+fun wadjust_backto_standard_pos_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+  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>)"
+
+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> 
+                      ml + mr = Suc m \<and> mr > 0)"
+
+fun wadjust_backto_standard_pos :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+  where
+  "wadjust_backto_standard_pos m rs (l, r) = 
+        (wadjust_backto_standard_pos_B m rs (l, r) \<or> 
+        wadjust_backto_standard_pos_O m rs (l, r))"
+
+fun wadjust_stop :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+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>)"
+
+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]
+        wadjust_loop_erase.simps[simp del] wadjust_loop_on_left_moving.simps[simp del]
+        wadjust_loop_right_move2.simps[simp del] wadjust_erase2.simps[simp del]
+        wadjust_on_left_moving_O.simps[simp del] wadjust_on_left_moving_B.simps[simp del]
+        wadjust_on_left_moving.simps[simp del] wadjust_goon_left_moving_B.simps[simp del]
+        wadjust_goon_left_moving_O.simps[simp del] wadjust_goon_left_moving.simps[simp del]
+        wadjust_backto_standard_pos.simps[simp del] wadjust_backto_standard_pos_B.simps[simp del]
+        wadjust_backto_standard_pos_O.simps[simp del] wadjust_stop.simps[simp del]
+
+fun wadjust_inv :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+  where
+  "wadjust_inv st m rs (l, r) = 
+       (if st = Suc 0 then wadjust_start m rs (l, r) 
+        else if st = Suc (Suc 0) then wadjust_loop_start m rs (l, r)
+        else if st = Suc (Suc (Suc 0)) then wadjust_loop_right_move m rs (l, r)
+        else if st = 4 then wadjust_loop_check m rs (l, r)
+        else if st = 5 then wadjust_loop_erase m rs (l, r)
+        else if st = 6 then wadjust_loop_on_left_moving m rs (l, r)
+        else if st = 7 then wadjust_loop_right_move2 m rs (l, r)
+        else if st = 8 then wadjust_erase2 m rs (l, r)
+        else if st = 9 then wadjust_on_left_moving m rs (l, r)
+        else if st = 10 then wadjust_goon_left_moving m rs (l, r)
+        else if st = 11 then wadjust_backto_standard_pos m rs (l, r)
+        else if st = 0 then wadjust_stop m rs (l, r)
+        else False
+)"
+
+declare wadjust_inv.simps[simp del]
+
+fun wadjust_phase :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
+  where
+  "wadjust_phase rs (st, l, r) = 
+         (if st = 1 then 3 
+          else if st \<ge> 2 \<and> st \<le> 7 then 2
+          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"
+  where
+  "wadjust_stage rs (st, l, r) = 
+           (if st \<ge> 2 \<and> st \<le> 7 then 
+                  rs - length (takeWhile (\<lambda> a. a = Oc) 
+                          (tl (dropWhile (\<lambda> a. a = Oc) (rev l @ r))))
+            else 0)"
+
+fun wadjust_state :: "nat \<Rightarrow> t_conf \<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"
+  where
+  "wadjust_step rs (st, l, r) = 
+       (if st = 1 then (if hd r = Bk then 1
+                        else 0) 
+        else if st = 3 then length r
+        else if st = 5 then (if hd r = Oc then 1
+                             else 0)
+        else if st = 6 then length l
+        else if st = 8 then (if hd r = Oc then 1
+                             else 0)
+        else if st = 9 then length l
+        else if st = 10 then length l
+        else if st = 11 then (if hd r = Bk then 0
+                              else Suc (length l))
+        else 0)"
+
+fun wadjust_measure :: "(nat \<times> t_conf) \<Rightarrow> nat \<times> nat \<times> nat \<times> nat"
+  where
+  "wadjust_measure (rs, (st, l, r)) = 
+     (wadjust_phase rs (st, l, r), 
+      wadjust_stage rs (st, l, r),
+      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"
+  where "wadjust_le \<equiv> (inv_image lex_square wadjust_measure)"
+
+lemma [intro]: "wf lex_square"
+by(auto intro:wf_lex_prod simp: abacus.lex_pair_def lex_square_def 
+  abacus.lex_triple_def)
+
+lemma wf_wadjust_le[intro]: "wf wadjust_le"
+by(auto intro:wf_inv_image simp: wadjust_le_def
+           abacus.lex_triple_def abacus.lex_pair_def)
+
+lemma [simp]: "wadjust_start m rs (c, []) = False"
+apply(auto simp: wadjust_start.simps)
+done
+
+lemma [simp]: "wadjust_loop_right_move m rs (c, []) \<Longrightarrow> c \<noteq> []"
+apply(auto simp: wadjust_loop_right_move.simps)
+done
+
+lemma [simp]: "wadjust_loop_right_move m rs (c, [])
+        \<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> []"
+apply(simp only: wadjust_loop_check.simps, auto)
+done
+ 
+lemma [simp]: "wadjust_loop_start m rs (c, []) = False"
+apply(simp add: wadjust_loop_start.simps)
+done
+
+lemma [simp]: "wadjust_loop_right_move m rs (c, []) \<Longrightarrow> 
+  wadjust_loop_right_move m rs (Bk # c, [])"
+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)
+done
+
+lemma [simp]: "wadjust_loop_on_left_moving m rs (c, []) = False"
+apply(auto simp: wadjust_loop_on_left_moving.simps)
+done
+
+
+lemma [simp]: "wadjust_loop_right_move2 m rs (c, []) = False"
+apply(auto simp: wadjust_loop_right_move2.simps)
+done
+   
+lemma [simp]: "wadjust_erase2 m rs ([], []) = False"
+apply(auto simp: wadjust_erase2.simps)
+done
+
+lemma [simp]: "wadjust_on_left_moving_B m rs 
+                 (Oc # Oc # Oc\<^bsup>rs\<^esup> @ Bk # Oc # Oc\<^bsup>m\<^esup>, [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)
+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)
+done
+
+lemma [simp]: "wadjust_erase2 m rs (c, [])
+    \<Longrightarrow> (c = [] \<longrightarrow> wadjust_on_left_moving m rs ([], [Bk])) \<and> 
+       (c \<noteq> [] \<longrightarrow> wadjust_on_left_moving m rs (tl c, [hd c]))"
+apply(auto)
+done
+
+lemma [simp]: "wadjust_on_left_moving m rs ([], []) = False"
+apply(simp add: wadjust_on_left_moving.simps 
+  wadjust_on_left_moving_O.simps wadjust_on_left_moving_B.simps)
+done
+
+lemma [simp]: "wadjust_on_left_moving_O m rs (c, []) = False"
+apply(simp add: wadjust_on_left_moving_O.simps)
+done
+
+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)
+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)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_on_left_moving m rs (c, []); c \<noteq> []\<rbrakk> \<Longrightarrow> 
+  wadjust_on_left_moving m rs (tl c, [hd c])"
+apply(simp add: wadjust_on_left_moving.simps)
+apply(case_tac "hd c", simp_all)
+done
+
+lemma [simp]: "wadjust_on_left_moving m rs (c, [])
+    \<Longrightarrow> (c = [] \<longrightarrow> wadjust_on_left_moving m rs ([], [Bk])) \<and> 
+       (c \<noteq> [] \<longrightarrow> wadjust_on_left_moving m rs (tl c, [hd c]))"
+apply(auto)
+done
+
+lemma [simp]: "wadjust_goon_left_moving m rs (c, []) = False"
+apply(auto simp: wadjust_goon_left_moving.simps wadjust_goon_left_moving_B.simps
+                 wadjust_goon_left_moving_O.simps)
+done
+
+lemma [simp]: "wadjust_backto_standard_pos m rs (c, []) = False"
+apply(auto simp: wadjust_backto_standard_pos.simps
+ wadjust_backto_standard_pos_B.simps wadjust_backto_standard_pos_O.simps)
+done
+
+lemma [simp]:
+  "wadjust_start m rs (c, Bk # list) \<Longrightarrow> 
+  (c = [] \<longrightarrow> wadjust_start m rs ([], Oc # list)) \<and> 
+  (c \<noteq> [] \<longrightarrow> wadjust_start m rs (c, Oc # list))"
+apply(auto simp: wadjust_start.simps)
+done
+
+lemma [simp]: "wadjust_loop_start m rs (c, Bk # list) = False"
+apply(auto simp: wadjust_loop_start.simps)
+done
+
+lemma [simp]: "wadjust_loop_right_move m rs (c, b) \<Longrightarrow> c \<noteq> []"
+apply(simp only: wadjust_loop_right_move.simps, auto)
+done
+
+lemma [simp]: "wadjust_loop_right_move m rs (c, Bk # list)
+    \<Longrightarrow> wadjust_loop_right_move m rs (Bk # c, list)"
+apply(simp only: wadjust_loop_right_move.simps)
+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 = nat in exI, auto)
+done
+
+lemma [simp]: "wadjust_loop_check m rs (c, b) \<Longrightarrow> c \<noteq> []"
+apply(simp only: wadjust_loop_check.simps, auto)
+done
+
+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)
+done
+
+lemma [simp]: "wadjust_loop_erase m rs (c, b) \<Longrightarrow> c \<noteq> []"
+apply(simp only: wadjust_loop_erase.simps, auto)
+done
+
+declare wadjust_loop_on_left_moving_O.simps[simp del]
+        wadjust_loop_on_left_moving_B.simps[simp del]
+
+lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); hd c = Bk\<rbrakk>
+    \<Longrightarrow> wadjust_loop_on_left_moving_B m rs (tl c, Bk # Bk # list)"
+apply(simp only: wadjust_loop_erase.simps 
+  wadjust_loop_on_left_moving_B.simps)
+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])
+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)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); c \<noteq> []\<rbrakk> \<Longrightarrow> 
+                wadjust_loop_on_left_moving m rs (tl c, hd c # Bk # list)"
+apply(case_tac "hd c", simp_all add:wadjust_loop_on_left_moving.simps)
+done
+
+lemma [simp]: "wadjust_loop_on_left_moving m rs (c, b) \<Longrightarrow> c \<noteq> []"
+apply(simp add: wadjust_loop_on_left_moving.simps 
+wadjust_loop_on_left_moving_O.simps wadjust_loop_on_left_moving_B.simps, auto)
+done
+
+lemma [simp]: "wadjust_loop_on_left_moving_O m rs (c, Bk # list) = False"
+apply(simp add: wadjust_loop_on_left_moving_O.simps)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Bk\<rbrakk>
+    \<Longrightarrow>  wadjust_loop_on_left_moving_B m rs (tl c, Bk # Bk # list)"
+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)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk>
+    \<Longrightarrow> wadjust_loop_on_left_moving_O m rs (tl c, Oc # Bk # list)"
+apply(simp only: wadjust_loop_on_left_moving_O.simps 
+                 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)
+done
+
+lemma [simp]: "wadjust_loop_on_left_moving m rs (c, Bk # list)
+            \<Longrightarrow> wadjust_loop_on_left_moving m rs (tl c, hd c # Bk # list)"
+apply(simp add: wadjust_loop_on_left_moving.simps)
+apply(case_tac "hd c", simp_all)
+done
+
+lemma [simp]: "wadjust_loop_right_move2 m rs (c, b) \<Longrightarrow> c \<noteq> []"
+apply(simp only: wadjust_loop_right_move2.simps, auto)
+done
+
+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(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 = rn in exI, auto)
+apply(rule_tac x = "Suc ml" in exI, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \<Longrightarrow> c \<noteq> []"
+apply(auto simp:wadjust_erase2.simps )
+done
+
+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 
+        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)
+done
+
+lemma [simp]: "wadjust_on_left_moving m rs (c,b) \<Longrightarrow> c \<noteq> []"
+apply(simp only:wadjust_on_left_moving.simps
+                wadjust_on_left_moving_O.simps
+                wadjust_on_left_moving_B.simps
+             , auto)
+done
+
+lemma [simp]: "wadjust_on_left_moving_O m rs (c, Bk # list) = False"
+apply(simp add: wadjust_on_left_moving_O.simps)
+done
+
+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)
+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)
+done
+
+lemma [simp]: "wadjust_on_left_moving  m rs (c, Bk # list) \<Longrightarrow>  
+                  wadjust_on_left_moving m rs (tl c, hd c # Bk # list)"
+apply(simp add: wadjust_on_left_moving.simps)
+apply(case_tac "hd c", simp_all)
+done
+
+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)
+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)
+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)
+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)
+done
+
+lemma [simp]: "wadjust_goon_left_moving m rs (c, Bk # list) \<Longrightarrow>
+  wadjust_backto_standard_pos m rs (tl c, hd c # Bk # list)"
+apply(case_tac "hd c", simp_all add: wadjust_backto_standard_pos.simps 
+                                     wadjust_goon_left_moving.simps)
+done
+
+lemma [simp]: "wadjust_backto_standard_pos m rs (c, Bk # list) \<Longrightarrow>
+  (c = [] \<longrightarrow> wadjust_stop m rs ([Bk], list)) \<and> (c \<noteq> [] \<longrightarrow> wadjust_stop m rs (Bk # c, list))"
+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)
+done
+
+lemma [simp]: "wadjust_start m rs (c, Oc # list)
+              \<Longrightarrow> (c = [] \<longrightarrow> wadjust_loop_start m rs ([Oc], list)) \<and>
+                (c \<noteq> [] \<longrightarrow> wadjust_loop_start m rs (Oc # c, list))"
+apply(auto simp:wadjust_loop_start.simps wadjust_start.simps )
+apply(rule_tac x = ln in exI, rule_tac x = rn in exI,
+      rule_tac x = "Suc 0" in exI, simp)
+done
+
+lemma [simp]: "wadjust_loop_start m rs (c, b) \<Longrightarrow> c \<noteq> []"
+apply(simp add: wadjust_loop_start.simps, auto)
+done
+
+lemma [simp]: "wadjust_loop_start m rs (c, Oc # list)
+              \<Longrightarrow> wadjust_loop_right_move m rs (Oc # c, list)"
+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)
+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)
+done
+
+lemma [simp]: "wadjust_loop_check m rs (c, Oc # list) \<Longrightarrow> 
+               wadjust_loop_erase m rs (tl c, hd c # Oc # list)"
+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)
+done
+
+lemma [simp]: "wadjust_loop_erase m rs (c, Oc # list) \<Longrightarrow> 
+                wadjust_loop_erase m rs (c, Bk # list)"
+apply(auto simp: wadjust_loop_erase.simps)
+done
+
+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)
+done
+
+lemma [simp]: "wadjust_loop_on_left_moving m rs (c, Oc # list)
+           \<Longrightarrow> wadjust_loop_right_move2 m rs (Oc # c, list)"
+apply(simp add:wadjust_loop_on_left_moving.simps)
+apply(auto simp: wadjust_loop_on_left_moving_O.simps
+                 wadjust_loop_right_move2.simps)
+done
+
+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)
+done
+
+lemma [simp]: "wadjust_erase2 m rs (c, Oc # list)
+              \<Longrightarrow> (c = [] \<longrightarrow> wadjust_erase2 m rs ([], Bk # list))
+               \<and> (c \<noteq> [] \<longrightarrow> wadjust_erase2 m rs (c, Bk # list))"
+apply(auto simp: wadjust_erase2.simps )
+done
+
+lemma [simp]: "wadjust_on_left_moving_B m rs (c, Oc # list) = False"
+apply(auto simp: wadjust_on_left_moving_B.simps)
+done
+
+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)
+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)
+done
+
+
+lemma [simp]: "wadjust_on_left_moving m rs (c, Oc # list) \<Longrightarrow> 
+              wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)"
+apply(simp add: wadjust_on_left_moving.simps   
+                 wadjust_goon_left_moving.simps)
+apply(case_tac "hd c", simp_all)
+done
+
+lemma [simp]: "wadjust_on_left_moving m rs (c, Oc # list) \<Longrightarrow> 
+  wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)"
+apply(simp add: wadjust_on_left_moving.simps 
+  wadjust_goon_left_moving.simps)
+apply(case_tac "hd c", simp_all)
+done
+
+lemma [simp]: "wadjust_goon_left_moving_B m rs (c, Oc # list) = False"
+apply(auto simp: wadjust_goon_left_moving_B.simps)
+done
+
+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)
+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)
+done
+
+lemma [simp]: "wadjust_goon_left_moving m rs (c, Oc # list) \<Longrightarrow> 
+  wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)"
+apply(simp add: wadjust_goon_left_moving.simps)
+apply(case_tac "hd c", simp_all)
+done
+
+lemma [simp]: "wadjust_backto_standard_pos_B m rs (c, Oc # list) = False"
+apply(simp add: wadjust_backto_standard_pos_B.simps)
+done
+
+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
+
+
+
+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
+
+
+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)
+done
+
+lemma [simp]: "wadjust_backto_standard_pos m rs (c, Oc # list)
+  \<Longrightarrow> (c = [] \<longrightarrow> wadjust_backto_standard_pos m rs ([], Bk # Oc # list)) \<and> 
+ (c \<noteq> [] \<longrightarrow> wadjust_backto_standard_pos m rs (tl c, hd c # Oc # list))"
+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)
+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>
+  \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list))))
+  < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list)))) \<or>
+  a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) =
+  a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))"
+apply(simp only: wadjust_loop_erase.simps)
+apply(rule_tac disjI2)
+apply(case_tac c, simp, simp)
+done
+
+lemma [simp]:
+  "\<lbrakk>Suc (Suc rs) = a;  wadjust_loop_on_left_moving m rs (c, Bk # list)\<rbrakk>
+  \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list))))
+  < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list)))) \<or>
+  a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) =
+  a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))"
+apply(subgoal_tac "c \<noteq> []")
+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)
+done
+
+lemma [simp]: "\<lbrakk>Suc (Suc rs) = a;  wadjust_loop_right_move2 m rs (c, Bk # list)\<rbrakk>
+              \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list))))
+                 < 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)
+done
+
+lemma [simp]: "wadjust_loop_check m rs ([], b) = False"
+apply(simp add: wadjust_loop_check.simps)
+done
+
+lemma [simp]: "\<lbrakk>Suc (Suc rs) = a;  wadjust_loop_check m rs (c, Oc # list)\<rbrakk>
+  \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Oc # list))))
+  < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list)))) \<or>
+  a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Oc # list)))) =
+  a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list))))"
+apply(case_tac "c", simp_all)
+done
+
+lemma [simp]: 
+  "\<lbrakk>Suc (Suc rs) = a;  wadjust_loop_erase m rs (c, Oc # list)\<rbrakk>
+  \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))
+  < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list)))) \<or>
+  a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list)))) =
+  a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list))))"
+apply(simp add: wadjust_loop_erase.simps)
+apply(rule_tac disjI2)
+apply(auto)
+apply(simp add: dropWhile_exp1 takeWhile_exp1)
+done
+
+declare numeral_2_eq_2[simp del]
+
+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
+    \<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)"
+  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
+            split: if_splits)
+          done
+      qed
+    qed
+  next
+    show "?Q (?f 0)"
+      apply(simp add: steps.simps wadjust_inv.simps wadjust_start.simps)
+      apply(rule_tac x = ln in exI,auto)
+      done
+  next
+    show "\<not> ?P (?f 0)"
+      apply(simp add: steps.simps)
+      done
+  qed
+  thus "?thesis"
+    apply(auto)
+    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)
+done
+
+lemma wcode_lemma_pre':
+  "args \<noteq> [] \<Longrightarrow> 
+  \<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>)" 
+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 ?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)
+      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     
+  next
+    show "?Q1 \<turnstile>-> ?P2"
+      by(simp add: t_imply_def)
+  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)
+    done
+qed
+
+text {*
+  The initialization TM @{text "t_wcode"}.
+  *}
+definition t_wcode :: "tprog"
+  where
+  "t_wcode = (t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust"
+
+
+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)
+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>)"
+using wcode_lemma_1[of args m]
+apply(simp add: t_wcode_def tape_of_nl_abv tape_of_nat_list.simps)
+done
+
+section {* The universal TM @{text "UTM"} *}
+
+text {*
+  This section gives the explicit construction of {\em Universal Turing Machine}, defined as @{text "UTM"} and proves its 
+  correctness. It is pretty easy by composing the partial results we have got so far.
+  *}
+
+
+definition UTM :: "tprog"
+  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))))"
+
+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"
+  where
+  "F_tprog = tm_of (F_aprog)"
+
+definition t_utm :: "tprog"
+  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"
+  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>);
+    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>)"
+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,
+      erule_tac exE)
+apply(rule_tac x = stp in exI, rule_tac x = m in exI)
+apply(simp add: F_aprog_def dummy_abc_def)
+done
+
+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>
+  \<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>))"
+  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)"
+    "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)+
+apply(rule_tac x = stpa in exI, rule_tac x = "Suc (Suc ma)" in exI, 
+       rule_tac x = l in exI, simp add: exp_ind)
+done
+
+declare tape_of_nl_abv_cons[simp del]
+
+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>)]>) t_utm stp = 
+                                                (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+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>)"
+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>"
+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)
+apply(subgoal_tac "\<exists> d. n = d + na", auto simp: exp_add)
+apply(case_tac rs, simp_all add: exp_ind, case_tac d, 
+           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>)"
+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>)"
+  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>)"
+    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)
+    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]
+      apply(simp)
+      using gr
+      apply(simp only: tinres_def, auto)
+      apply(rule_tac x = "na + n" in exI, simp add: exp_add)
+      done
+  qed
+qed
+
+lemma [simp]: "tinres [Bk, Bk] [Bk]"
+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>"
+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 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)+
+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)
+      done
+  qed
+qed
+
+lemma [intro]: "t_correct t_wcode"
+apply(simp add: t_wcode_def)
+apply(auto)
+done
+      
+lemma [intro]: "t_correct t_utm"
+apply(simp add: 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>)"
+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 ?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 ?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      
+  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
+      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)
+      done
+  next
+    show "?Q1 \<turnstile>-> ?P2"
+      apply(simp add: t_imply_def)
+      done
+  qed
+  thus "?thesis"
+    apply(simp add: t_imply_def)
+    apply(auto simp: UTM_pre_def)
+    done
+qed
+
+text {*
+  The correctness of @{text "UTM"}, the halt case.
+*}
+theorem 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]
+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"
+  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
+
+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"
+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(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>)"
+apply(auto)
+apply(induct c, simp add: bl2nat.simps)
+apply(rule_tac x = 0 in exI, simp)
+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>"
+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)
+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 h: 
+    "Suc (Suc (2 * bl2nat c 0)) = 2 * 2 ^ nat"
+  have "\<exists>rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+    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)
+    done
+qed
+
+lemma [elim]: 
+  "\<lbrakk>\<forall>rs n. c \<noteq> Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>; 
+  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)
+apply(case_tac rs, simp, simp, erule_tac x = nat in allE,
+  erule_tac x = n in allE, simp)
+using bl2wc_exp_ex[of c "lg (Suc (bl2wc c)) 2"]
+apply(case_tac "(2::nat) ^ lg (Suc (bl2wc c)) 2", 
+  simp, simp, erule_tac exE, erule_tac exE, simp)
+apply(simp add: bl2wc.simps)
+apply(rule_tac x = rs in exI)
+apply(case_tac "(2::nat)^rs", simp, simp)
+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))"
+apply(simp add: NSTD.simps trpl_code.simps)
+apply(rule_tac impI)
+apply(rule_tac disjI2, rule_tac disjI2, auto)
+done
+
+lemma NSTD_1: "\<not> TSTD (a, b, c)
+    \<Longrightarrow> rec_exec rec_NSTD [trpl_code (a, b, c)] = Suc 0"
+  using NSTD_lemma1[of "trpl_code (a, b, c)"]
+       NSTD_lemma2[of "trpl_code (a, b, c)"]
+  apply(simp add: TSTD_def)
+  apply(erule_tac disjE, erule_tac nstd_case1)
+  apply(erule_tac disjE, erule_tac nstd_case2)
+  apply(erule_tac nstd_case3)
+  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"
+apply(simp add: rec_nonstop_def rec_exec.simps)
+apply(subgoal_tac 
+  "rec_exec rec_conf [code tp, bl2wc (<lm>), stp] =
+  trpl_code (a, b, c)", simp)
+apply(erule_tac NSTD_1)
+using rec_t_eq_steps[of tp l lm stp]
+apply(simp)
+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)"
+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(rule_tac nonstop_t_uhalt_eq, simp_all)
+done
+
+(*
+lemma [simp]: 
+  "\<forall>j<Suc k. Ex (rec_calc_rel (get_fstn_args (Suc k) (Suc k) ! j)
+                                                     (code tp # lm))"
+apply(auto simp: get_fstn_args_nth)
+apply(rule_tac x = "(code tp # lm) ! j" in exI)
+apply(rule_tac calc_id, simp_all)
+done
+*)
+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)); 
+    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>
+               @ 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])])")
+apply(simp only: rec_F_def, rule_tac i = 0  and ga = a and gb = b and 
+  gc = c in cn_gi_uhalt, simp, simp, simp, simp, simp, simp, simp)
+apply(simp add: ci_cn_para_eq)
+apply(case_tac "rec_ci (Cn (Suc (Suc 0)) rec_conf 
+  ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt]))")
+apply(rule_tac rf = "(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])])" 
+           and n = "Suc (Suc 0)" and f = rec_right and 
+          gs = "[Cn (Suc (Suc 0)) rec_conf 
+           ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])]"
+           and i = 0 and ga = aa and gb = ba and gc = ca in 
+          cn_gi_uhalt)
+apply(simp, simp, simp, simp, simp, simp, simp, 
+     simp add: ci_cn_para_eq)
+apply(case_tac "rec_ci rec_halt")
+apply(rule_tac rf = "(Cn (Suc (Suc 0)) rec_conf 
+  ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt]))" 
+  and n = "Suc (Suc 0)" and f = "rec_conf" and 
+  gs = "([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])"  and 
+  i = "Suc (Suc 0)" and gi = "rec_halt" and ga = ab and gb = bb and
+  gc = cb in cn_gi_uhalt)
+apply(simp, simp, simp, simp, simp add: nth_append, simp, 
+  simp add: nth_append, simp add: rec_halt_def)
+apply(simp only: rec_halt_def)
+apply(case_tac [!] "rec_ci ((rec_nonstop))")
+apply(rule_tac allI, rule_tac impI, simp)
+apply(case_tac j, simp)
+apply(rule_tac x = "code tp" in exI, rule_tac calc_id, simp, simp, simp, simp)
+apply(rule_tac x = "bl2wc (<lm>)" in exI, rule_tac calc_id, simp, simp, simp)
+apply(rule_tac rf = "Mn (Suc (Suc 0)) (rec_nonstop)"
+  and f = "(rec_nonstop)" and n = "Suc (Suc 0)"
+  and  aprog' = ac and rs_pos' =  bc and a_md' = cc in Mn_unhalt)
+apply(simp, simp add: rec_halt_def , simp, simp)
+apply(drule_tac  nonstop_true, simp_all)
+apply(rule_tac allI)
+apply(erule_tac x = y in allE)+
+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));
+  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"
+proof(frule_tac F_ap = ap and rs_pos = pos and a_md = md
+    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 
+    (ss, e) \<Rightarrow> ss < length ap"
+    "abc_steps_l (0, [code tp, bl2wc (<lm>)]) ap stp = (a, b)" 
+    "turing_basic.t_correct tp" 
+    "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)
+    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)" 
+      "ap \<noteq> []"
+    thus "?thesis"
+      using abc_list_crsp_steps[of "[code tp, bl2wc (<lm>)]"
+                                   "md - pos" ap stp aa ba] h
+      apply(simp)
+      done
+  qed
+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>
+  \<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)
+thm uabc_uhalt'
+apply(drule_tac ap = a and pos = b and md = c in uabc_uhalt', simp_all)
+proof -
+  fix a b c
+  assume 
+    "\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) a stp of (ss, e) 
+                                                   \<Rightarrow> ss < length a"
+    "rec_ci rec_F = (a, b, c)"
+  thus 
+    "\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) 
+    (a [+] dummy_abc (Suc (Suc 0))) stp of (ss, e) \<Rightarrow> 
+           ss < Suc (Suc (Suc (length a)))"
+    using abc_append_uhalt1[of a "[code tp, bl2wc (<lm>)]" 
+      "a [+] dummy_abc (Suc (Suc 0))" "[]" "dummy_abc (Suc (Suc 0))"]  
+    apply(simp)
+    done
+qed
+
+thm abacus_turing_eq_uhalt
+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
+
+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'); 
+  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")
+  fix aa ba ca
+  assume h: "steps (st, l, r) tp stp = (a, b, c)" 
+            "steps (st, l', r') tp stp = (a', b', c')"
+            "tinres l l'" "tinres r r'"
+            "steps (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)
+    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)
+    done
+  ultimately show "?thesis"
+    apply(auto intro: tinres_commute)
+    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)
+apply(erule_tac x = stp in allE)
+apply(case_tac "steps (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)
+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)"
+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)"
+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      
+  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)
+      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
+  qed
+  thus "?thesis"
+    apply(simp add: t_imply_def UTM_pre_def)
+    done
+qed
+
+text {*
+  The correctness of @{text "UTM"}, the unhalt case.
+  *}
+
+theorem 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]
+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
+
+end                               
\ No newline at end of file




