tm: comparison thys/UTM.thy

equal deleted inserted replaced

-:1e89c65f844b
+:e995ae949731
 theory UTM
-imports Main uncomputable recursive abacus UF GCD
+imports Main recursive abacus UF GCD turing_hoare
 begin
 section {* Wang coding of input arguments *}
 text {*
 \newlength{\baseheight}
 \settoheight{\baseheight}{$B:R$}
 \newcommand{\vsep}{5\baseheight}
 The TM used to generate the Wang's code of input arguments is divided into three TMs
-executed sequentially, namely $prepare$, $mainwork$ and $adjust$¡£According to the
+executed sequentially, namely $prepare$, $mainwork$ and $adjust$\<exclamdown>\<pounds>According to the
 convention, start state of ever TM is fixed to state $1$ while the final state is
 fixed to $0$.
 The input and output of $prepare$ are illustrated respectively by Figure
 \ref{prepare_input} and \ref{prepare_output}.
 definition fourtimes_ly :: "nat list"
 where
 "fourtimes_ly = layout_of abc_fourtimes"
-definition t_twice :: "tprog"
+definition t_twice_compile :: "instr list"
 where
-"t_twice = change_termi_state (tm_of (abc_twice) @ (tMp 1 (start_of twice_ly (length abc_twice) - Suc 0)))"
+"t_twice_compile= (tm_of abc_twice @ (shift (mopup 1) (length (tm_of abc_twice) div 2)))"
-definition t_fourtimes :: "tprog"
+definition t_twice :: "instr list"
 where
-"t_fourtimes = change_termi_state (tm_of (abc_fourtimes) @
+"t_twice = adjust t_twice_compile"
-(tMp 1 (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)))"
+definition t_fourtimes_compile :: "instr list"
+where
+"t_fourtimes_compile= (tm_of abc_fourtimes @ (shift (mopup 1) (length (tm_of abc_fourtimes) div 2)))"
+definition t_fourtimes :: "instr list"
+where
+"t_fourtimes = adjust t_fourtimes_compile"
 definition t_twice_len :: "nat"
 where
 "t_twice_len = length t_twice div 2"
-definition t_wcode_main_first_part:: "tprog"
+definition t_wcode_main_first_part:: "instr list"
 where
 "t_wcode_main_first_part \<equiv>
 [(L, 1), (L, 2), (L, 7), (R, 3),
 (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"
+definition t_wcode_main :: "instr list"
 where
-"t_wcode_main = (t_wcode_main_first_part @ tshift t_twice 12 @ [(L, 1), (L, 1)]
+"t_wcode_main = (t_wcode_main_first_part @ shift t_twice 12 @ [(L, 1), (L, 1)]
-@ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])"
+@ shift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])"
-fun bl_bin :: "block list \<Rightarrow> nat"
+fun bl_bin :: "cell 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"
+type_synonym bin_inv_t = "cell list \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 fun wcode_before_double :: "bin_inv_t"
 where
 "wcode_before_double ires rs (l, r) =
-(\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+(\<exists> ln rn. l = Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and>
-r = Oc\<^bsup>(Suc (Suc rs))\<^esup> @ Bk\<^bsup>rn \<^esup>)"
+r = Oc\<up>((Suc (Suc rs))) @ Bk\<up>(rn ))"
 declare wcode_before_double.simps[simp del]
 fun wcode_after_double :: "bin_inv_t"
 where
 "wcode_after_double ires rs (l, r) =
-(\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+(\<exists> ln rn. l = Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and>
-r = Oc\<^bsup>Suc (Suc (Suc 2*rs))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+r = Oc\<up>(Suc (Suc (Suc 2*rs))) @ Bk\<up>(rn))"
 declare wcode_after_double.simps[simp del]
 fun wcode_on_left_moving_1_B :: "bin_inv_t"
 where
 "wcode_on_left_moving_1_B ires rs (l, r) =
-(\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Oc # ires \<and>
+(\<exists> ml mr rn. l = Bk\<up>(ml) @ Oc # Oc # ires \<and>
-r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Bk\<up>(mr) @ Oc\<up>(Suc rs) @ Bk\<up>(rn) \<and>
 ml + mr > Suc 0 \<and> mr > 0)"
 declare wcode_on_left_moving_1_B.simps[simp del]
 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>)"
+r = Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 declare wcode_on_left_moving_1_O.simps[simp del]
 fun wcode_on_left_moving_1 :: "bin_inv_t"
 where
 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>)"
+r = Oc # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 fun wcode_erase1 :: "bin_inv_t"
 where
 "wcode_erase1 ires rs (l, r) =
 (\<exists> ln rn. l = Oc # ires \<and>
-tl r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+tl r = Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 declare wcode_erase1.simps [simp del]
 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>
+l = Bk\<up>(ml) @ Oc # ires \<and>
-r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Bk\<up>(mr) @ Oc\<up>(Suc rs) @ Bk\<up>(rn) \<and>
 ml + mr > Suc 0)"
 declare wcode_on_right_moving_1.simps [simp del]
 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>
+l = Oc\<up>(ml) @ Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and>
-r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Oc\<up>(mr) @ Bk\<up>(rn) \<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>
+(\<exists> ln rn. l =  Bk # Bk\<up>(ln) @ Oc # ires \<and>
-r =  Bk # Oc\<^bsup>(Suc (Suc rs))\<^esup> @ Bk\<^bsup>rn \<^esup>)"
+r =  Bk # Oc\<up>((Suc (Suc rs))) @ Bk\<up>(rn ))"
 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>
+l = Oc\<up>(ml) @ Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and>
-r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Oc\<up>(mr) @ Bk\<up>(rn) \<and>
 ml + mr = Suc (Suc rs) \<and> mr > 0)"
 declare wcode_backto_standard_pos_O.simps[simp del]
 fun wcode_backto_standard_pos :: "bin_inv_t"
 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)
+apply(simp add: tape_of_nat_abv tape_of_nat_list.simps)
 done
 lemma tape_of_Suc_nat: "<Suc (a ::nat)> = replicate a Oc @ [Oc, Oc]"
-apply(simp add: tape_of_nat_abv exp_ind tape_of_nat_list.simps)
+apply(simp only: tape_of_nat_abv exp_ind, simp)
-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
+apply(simp add: tape_of_nat_abv tape_of_nl_abv
 tape_of_nat_list.simps)
 done
 lemma bin_wc_eq: "bl_bin xs = bl2wc xs"
 proof(induct xs)
 show " bl_bin [] = bl2wc []"
 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: tape_of_nat_abv bl_bin.simps)
-apply(simp add: bl2wc.simps)
+apply(induct a, auto simp: bl_bin.simps)
 done
-lemma [simp]: " rev (a\<^bsup>aa\<^esup>) = a\<^bsup>aa\<^esup>"
-apply(simp add: exponent_def)
+lemma [simp]: " rev (a\<up>(aa)) = a\<up>(aa)"
-done
+apply(simp)
+done
-declare tape_of_nl_abv_cons[simp del]
+lemma tape_of_nl_append_one: "lm \<noteq> [] \<Longrightarrow>  <lm @ [a]> = <lm> @ Bk # Oc\<up>Suc a"
+apply(induct lm, auto simp: tape_of_nl_cons split:if_splits)
+done
 lemma tape_of_nl_rev: "rev (<lm::nat list>) = (<rev lm>)"
-apply(induct lm rule: list_tl_induct, simp)
+apply(induct lm, simp, auto)
-apply(case_tac "list = []", simp add: tape_of_nl_abv tape_of_nat_list.simps)
+apply(auto simp: tape_of_nl_cons tape_of_nl_append_one split: if_splits)
-apply(simp add: tape_of_nat_list_butlast_last tape_of_nl_abv_cons)
+apply(simp add: exp_ind[THEN sym])
 done
-lemma [simp]: "a\<^bsup>Suc 0\<^esup> = [a]"
-by(simp add: exp_def)
+lemma [simp]: "a\<up>(Suc 0) = [a]"
-lemma tape_of_nl_cons_app1: "(<a # xs @ [b]>) = (Oc\<^bsup>Suc a\<^esup> @ Bk # (<xs@ [b]>))"
+by(simp)
+lemma tape_of_nl_cons_app1: "(<a # xs @ [b]>) = (Oc\<up>(Suc a) @ Bk # (<xs@ [b]>))"
 apply(case_tac xs, simp add: tape_of_nl_abv tape_of_nat_list.simps)
 apply(simp add: tape_of_nl_abv  tape_of_nat_list.simps)
 done
 lemma bl_bin_bk_oc[simp]:
 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>"
+lemma tape_of_nat[simp]: "(<a::nat>) = Oc\<up>(Suc a)"
 apply(simp add: tape_of_nat_abv)
 done
-lemma tape_of_nl_cons_app2: "(<c # xs @ [b]>) = (<c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>)"
+lemma tape_of_nl_cons_app2: "(<c # xs @ [b]>) = (<c # xs> @ Bk # Oc\<up>(Suc b))"
 proof(induct "length xs" arbitrary: xs c,
 simp add: tape_of_nl_abv  tape_of_nat_list.simps)
 fix x xs c
 assume ind: "\<And>xs c. x = length xs \<Longrightarrow> <c # xs @ [b]> =
-<c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>"
+<c # xs> @ Bk # Oc\<up>(Suc b)"
 and h: "Suc x = length (xs::nat list)"
-show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>"
+show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<up>(Suc b)"
 proof(case_tac xs, simp add: tape_of_nl_abv  tape_of_nat_list.simps)
 fix a list
 assume g: "xs = a # list"
-hence k: "<a # list @ [b]> =  <a # list> @ Bk # Oc\<^bsup>Suc b\<^esup>"
+hence k: "<a # list @ [b]> =  <a # list> @ Bk # Oc\<up>(Suc b)"
 apply(rule_tac ind)
 using h
 apply(simp)
 done
-from g and k show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>"
+from g and k show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<up>(Suc b)"
 apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
 done
 qed
 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]) =
+lemma [simp]: "bl_bin (Oc\<up>(Suc aa) @ Bk # tape_of_nat_list (a # lista) @ [Bk, Oc]) =
-bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)) +
+bl_bin (Oc\<up>(Suc aa) @ Bk # tape_of_nat_list (a # lista)) +
-2* 2^(length (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)))"
+2* 2^(length (Oc\<up>(Suc aa) @ Bk # tape_of_nat_list (a # lista)))"
-using bl_bin_bk_oc[of "Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)"]
+using bl_bin_bk_oc[of "Oc\<up>(Suc aa) @ Bk # tape_of_nat_list (a # lista)"]
 apply(simp)
 done
+declare replicate_Suc[simp del]
 lemma [simp]:
 "bl_bin (<aa # list>) + (4 * rs + 4) * 2 ^ (length (<aa # list>) - Suc 0)
-= bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [0]>) + rs * (2 * 2 ^ (aa + length (<list @ [0]>)))"
+= bl_bin (Oc\<up>(Suc aa) @ Bk # <list @ [0]>) + rs * (2 * 2 ^ (aa + length (<list @ [0]>)))"
-apply(case_tac "list", simp add: add_mult_distrib, simp)
+apply(case_tac "list", simp add: add_mult_distrib)
 apply(simp add: tape_of_nl_cons_app2 add_mult_distrib)
 apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
 done
 lemma tape_of_nl_app_Suc: "((<list @ [Suc ab]>)) = (<list @ [ab]>) @ [Oc]"
 apply(induct list)
 apply(simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind)
 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])
+lemma [simp]: "bl_bin (Oc # Oc\<up>(aa) @ Bk # <list @ [ab]> @ [Oc])
-= bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>) +
+= bl_bin (Oc # Oc\<up>(aa) @ Bk # <list @ [ab]>) +
-2^(length (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>))"
+2^(length (Oc # Oc\<up>(aa) @ Bk # <list @ [ab]>))"
 apply(simp add: bin_wc_eq)
 apply(simp add: bl2nat_cons_oc bl2wc.simps)
-using bl2nat_cons_oc[of "Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>"]
+using bl2nat_cons_oc[of "Oc # Oc\<up>(aa) @ Bk # <list @ [ab]>"]
 apply(simp)
 done
-lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>) + (4 * 2 ^ (aa + length (<list @ [ab]>)) +
+lemma [simp]: "bl_bin (Oc # Oc\<up>(aa) @ Bk # <list @ [ab]>) + (4 * 2 ^ (aa + length (<list @ [ab]>)) +
 4 * (rs * 2 ^ (aa + length (<list @ [ab]>)))) =
-bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [Suc ab]>) +
+bl_bin (Oc # Oc\<up>(aa) @ Bk # <list @ [Suc ab]>) +
 rs * (2 * 2 ^ (aa + length (<list @ [Suc ab]>)))"
 apply(simp add: tape_of_nl_app_Suc)
 done
 declare tape_of_nat[simp del]
 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"
+fun wcode_double_case_state :: "config \<Rightarrow> nat"
 where
 "wcode_double_case_state (st, l, r) =
 13 - st"
-fun wcode_double_case_step :: "t_conf \<Rightarrow> nat"
+fun wcode_double_case_step :: "config \<Rightarrow> nat"
 where
 "wcode_double_case_step (st, l, r) =
 (if st = Suc 0 then (length l)
 else if st = Suc (Suc 0) then (length r)
 else if st = 3 then
 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"
+fun wcode_double_case_measure :: "config \<Rightarrow> nat \<times> nat"
 where
 "wcode_double_case_measure (st, l, r) =
 (wcode_double_case_state (st, l, r),
 wcode_double_case_step (st, l, r))"
-definition wcode_double_case_le :: "(t_conf \<times> t_conf) set"
+definition wcode_double_case_le :: "(config \<times> config) set"
 where "wcode_double_case_le \<equiv> (inv_image lex_pair wcode_double_case_measure)"
 lemma [intro]: "wf lex_pair"
 by(auto intro:wf_lex_prod simp:lex_pair_def)
 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
+apply(subgoal_tac "4 = Suc 3")
-fetch.simps nth_of.simps)
+apply(simp only: t_wcode_main_def t_wcode_main_first_part_def
+fetch.simps nth_of.simps, auto)
 done
 lemma [simp]: "fetch t_wcode_main 4 Oc = (R, 5)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+apply(subgoal_tac "4 = Suc 3")
-fetch.simps nth_of.simps)
+apply(simp only: t_wcode_main_def t_wcode_main_first_part_def
+fetch.simps nth_of.simps, auto)
 done
 lemma [simp]: "fetch t_wcode_main 5 Oc = (R, 5)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+apply(subgoal_tac "5 = Suc 4")
-fetch.simps nth_of.simps)
+apply(simp only: t_wcode_main_def t_wcode_main_first_part_def
+fetch.simps nth_of.simps, auto)
 done
 lemma [simp]: "fetch t_wcode_main 5 Bk = (W1, 6)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+apply(subgoal_tac "5 = Suc 4")
-fetch.simps nth_of.simps)
+apply(simp only: t_wcode_main_def t_wcode_main_first_part_def
+fetch.simps nth_of.simps, auto)
 done
 lemma [simp]: "fetch t_wcode_main 6 Bk = (R, 13)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+apply(subgoal_tac "6 = Suc 5")
-fetch.simps nth_of.simps)
+apply(simp only: t_wcode_main_def t_wcode_main_first_part_def
-done
+fetch.simps nth_of.simps, auto)
+done
 lemma [simp]: "fetch t_wcode_main 6 Oc = (L, 6)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+apply(subgoal_tac "6 = Suc 5")
-fetch.simps nth_of.simps)
+apply(simp only: t_wcode_main_def t_wcode_main_first_part_def
-done
+fetch.simps nth_of.simps, auto)
-lemma [elim]: "Bk\<^bsup>mr\<^esup> = [] \<Longrightarrow> mr = 0"
+done
-apply(case_tac mr, auto simp: exponent_def)
+lemma [elim]: "Bk\<up>(mr) = [] \<Longrightarrow> mr = 0"
+apply(case_tac mr, auto)
 done
 lemma [simp]: "wcode_on_left_moving_1 ires rs (b, []) = False"
 apply(simp add: wcode_on_left_moving_1.simps wcode_on_left_moving_1_B.simps
-wcode_on_left_moving_1_O.simps, auto)
+wcode_on_left_moving_1_O.simps)
 done
 declare wcode_on_checking_1.simps[simp del]
 lemmas wcode_double_case_inv_simps =
 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)
+simp, simp add: replicate_Suc)
 apply(erule_tac exE)+
 apply(simp)
 done
+declare replicate_Suc[simp]
 lemma [elim]:
 "\<lbrakk>wcode_on_left_moving_1 ires rs (b, Oc # list); tl b = aa \<and> hd b # Oc # list = ba\<rbrakk>
 \<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(case_tac mr, simp, auto)
-apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
+done
-done
 lemma [simp]: "wcode_on_checking_1 ires rs (b, []) = False"
 apply(auto simp: 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)
+apply(simp add: wcode_double_case_inv_simps)
 done
 lemma [simp]: "wcode_on_right_moving_1 ires rs (b, []) = False"
-apply(simp add: wcode_double_case_inv_simps exp_ind_def)
+apply(simp add: wcode_double_case_inv_simps)
 done
 lemma [elim]: "\<lbrakk>wcode_on_right_moving_1 ires rs (b, Bk # ba);  Bk # b = aa \<and> list = b\<rbrakk> \<Longrightarrow>
 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(simp)
-apply(case_tac mr, simp, simp add: exp_ind_def)
+apply(case_tac mr, simp, simp)
 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 mr, simp_all)
 apply(case_tac ml, simp, case_tac nat, simp, simp)
-apply(simp add: exp_ind_def)
 done
 lemma [simp]:
 "wcode_on_right_moving_1 ires rs (b, []) \<Longrightarrow> False"
-apply(simp add: wcode_double_case_inv_simps exponent_def)
+apply(simp add: wcode_double_case_inv_simps)
 done
 lemma [elim]: "\<lbrakk>wcode_erase1 ires rs (b, Bk # ba); Bk # b = aa \<and> list = ba; c = Bk # ba\<rbrakk>
 \<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)
+rule_tac x = rn in exI, simp add: exp_ind del: replicate_Suc)
 done
 lemma [elim]: "\<lbrakk>wcode_erase1 ires rs (aa, Oc # list);  b = aa \<and> Bk # list = ba\<rbrakk> \<Longrightarrow>
 wcode_erase1 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(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)
+apply(case_tac mr, simp, case_tac rn, simp, simp_all)
 done
 lemma [elim]: "\<lbrakk>wcode_goon_right_moving_1 ires rs (b, Oc # ba);  Oc # b = aa \<and> list = ba\<rbrakk>
 \<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(simp)
-apply(case_tac mr, simp, case_tac rn, simp_all add: exp_ind_def)
+apply(case_tac mr, simp, case_tac rn, simp_all)
 done
 lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, []);  Bk # b = aa\<rbrakk> \<Longrightarrow> False"
 apply(auto simp: wcode_double_case_inv_simps wcode_backto_standard_pos_O.simps
 wcode_backto_standard_pos_B.simps)
-apply(case_tac mr, simp_all add: exp_ind_def)
 done
 lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, Bk # ba); Bk # b = aa \<and> list = ba\<rbrakk>
 \<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)
+apply(case_tac [!] mr, simp_all)
 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
 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
-done
+declare nth_of.simps[simp del] fetch.simps[simp del]
-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
+let f = (\<lambda> stp. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp) in
 \<exists> n .P (f n) \<and> Q (f (n::nat))"
 proof -
 let ?P = "(\<lambda> (st, l, r). st = 13)"
 let ?Q = "(\<lambda> (st, l, r). wcode_double_case_inv st ires rs (l, r))"
-let ?f = "(\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)"
+let ?f = "(\<lambda> stp. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp)"
 have "\<exists> n. ?P (?f n) \<and> ?Q (?f (n::nat))"
 proof(rule_tac halt_lemma2)
 show "wf wcode_double_case_le"
 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)
+proof(rule_tac allI, case_tac "?f na", simp add: step_red)
 fix na a b c
 show "a \<noteq> 13 \<and> wcode_double_case_inv a ires rs (b, c) \<longrightarrow>
-(case tstep (a, b, c) t_wcode_main of (st, x) \<Rightarrow>
+(case step0 (a, b, c) t_wcode_main of (st, x) \<Rightarrow>
 wcode_double_case_inv st ires rs x) \<and>
-(tstep (a, b, c) t_wcode_main, a, b, c) \<in> wcode_double_case_le"
+(step0 (a, b, c) t_wcode_main, a, b, c) \<in> wcode_double_case_le"
 apply(rule_tac impI, simp add: wcode_double_case_inv.simps)
-apply(auto split: if_splits simp: tstep.simps,
+apply(auto split: if_splits simp: step.simps,
-case_tac [!] c, simp_all, case_tac [!] "(c::block list)!0")
+case_tac [!] c, simp_all, case_tac [!] "(c::cell list)!0")
-apply(simp_all add: new_tape.simps wcode_double_case_inv.simps wcode_double_case_le_def
+apply(simp_all add: 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 m" in exI, simp)
-apply(rule_tac x = "Suc 0" in exI, simp add: exp_ind_def)
+apply(rule_tac x = "Suc 0" in exI, simp)
-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
+f = steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main
 in \<exists>n. P (f n) \<and> Q (f n)"
 apply(simp add: Let_def)
 done
 qed
-lemma [elim]: "t_ncorrect tp
+lemma tm_append_shift_append_steps:
-\<Longrightarrow> t_ncorrect (tshift tp a)"
+"\<lbrakk>steps0 (st, l, r) tp stp = (st', l', r');
-apply(simp add: t_ncorrect.simps shift_length)
+0 < st';
-done
+length tp1 mod 2 = 0
+\<rbrakk>
-lemma tshift_fetch: "\<lbrakk> fetch tp a b = (aa, st'); 0 < st'\<rbrakk>
+\<Longrightarrow> steps0 (st + length tp1 div 2, l, r) (tp1 @ shift tp (length tp1 div 2) @ tp2) stp
-\<Longrightarrow> fetch (tshift tp (length tp1 div 2)) a b
+= (st' + length tp1 div 2, l', r')"
-= (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 h:
-assume ind: "\<And>st' l' r'.
+"steps0 (st, l, r) tp stp = (st', l', r')"
-\<lbrakk>a = st' \<and> b = l' \<and> c = r'; 0 < st'\<rbrakk>
+"0 < st'"
-\<Longrightarrow> t_steps (st + length tp1 div 2, l, r)
+"length tp1 mod 2 = 0 "
-(tshift tp (length tp1 div 2), length tp1 div 2) stp =
+from h have
-(st' + length tp1 div 2, l', r')"
+"steps (st + length tp1 div 2, l, r) (tp1 @ shift tp (length tp1 div 2), 0) stp =
-and h: "tstep (a, b, c) tp = (st', l', r')" "0 < st'" "t_ncorrect tp1"  "t_ncorrect tp"
+(st' + length tp1 div 2, l', r')"
-have k: "t_steps (st + length tp1 div 2, l, r) (tshift tp (length tp1 div 2),
+by(rule_tac tm_append_second_steps_eq, simp_all)
-length tp1 div 2) stp = (a + length tp1 div 2, b, c)"
+then have "steps (st + length tp1 div 2, l, r) ((tp1 @ shift tp (length tp1 div 2)) @ tp2, 0) stp =
-apply(rule_tac ind, simp)
+(st' + length tp1 div 2, l', r')"
 using h
-apply(case_tac a, simp_all add: tstep.simps fetch.simps)
+apply(rule_tac tm_append_first_steps_eq, simp_all)
 done
-from h and this show "t_step (t_steps (st + length tp1 div 2, l, r) (tshift tp (length tp1 div 2), length tp1 div 2) stp)
+thus "?thesis"
-(tshift tp (length tp1 div 2), length tp1 div 2) =
+by simp
-(st' + length tp1 div 2, l', r')"
-apply(simp add: k)
-apply(simp add: tstep.simps t_step.simps)
-apply(case_tac "fetch tp a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
-apply(subgoal_tac "fetch (tshift tp (length tp1 div 2)) a
-(case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) = (aa, st' + length tp1 div 2)", simp)
-apply(simp add: tshift_fetch)
-done
 qed
-lemma t_tshift_lemma: "\<lbrakk> steps (st, l, r) tp stp = (st', l', r');
-st' \<noteq> 0;
-stp > 0;
-0 < st \<and> st \<le> length tp div 2;
-t_ncorrect tp1;
-t_ncorrect tp;
-t_ncorrect tp2
-\<rbrakk>
-\<Longrightarrow> \<exists> stp>0. steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2) stp
-= (st' + length tp1 div 2, l', r')"
-proof -
-assume h: "steps (st, l, r) tp stp = (st', l', r')"
-"st' \<noteq> 0" "stp > 0"
-"0 < st \<and> st \<le> length tp div 2"
-"t_ncorrect tp1"
-"t_ncorrect tp"
-"t_ncorrect tp2"
-from h have
-"\<exists>stp>0. t_steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2, 0) stp =
-(st' + length tp1 div 2, l', r')"
-apply(rule_tac stp = stp in turing_shift, simp_all add: shift_length)
-apply(simp add: t_steps_steps_eq)
-apply(simp add: t_ncorrect.simps shift_length)
-done
-thus "\<exists> stp>0. steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2) stp
-= (st' + length tp1 div 2, l', r')"
-apply(erule_tac exE)
-apply(rule_tac x = stp in exI, simp)
-apply(subgoal_tac "length (tp1 @ tshift tp (length tp1 div 2) @ tp2) mod 2 = 0")
-apply(simp only: steps_eq)
-using h
-apply(auto simp: t_ncorrect.simps shift_length)
-apply arith
-done
-qed
 lemma t_twice_len_ge: "Suc 0 \<le> length t_twice div 2"
-apply(simp add: t_twice_def tMp.simps shift_length)
+apply(simp add: t_twice_def mopup.simps t_twice_compile_def)
 done
 lemma [intro]: "rec_calc_rel (recf.id (Suc 0) 0) [rs] rs"
 apply(rule_tac calc_id, simp_all)
 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 =
+declare start_of.simps[simp del]
-(0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+lemma t_twice_correct:
+"\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
+(tm_of abc_twice @ shift (mopup (Suc 0)) ((length (tm_of abc_twice) div 2))) stp =
+(0, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
 proof(case_tac "rec_ci rec_twice")
 fix a b c
 assume h: "rec_ci rec_twice = (a, b, c)"
-have "\<exists>stp m l. steps (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<^bsup>n\<^esup>) (tm_of abc_twice @ tMp (Suc 0)
+have "\<exists>stp m l. steps0 (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<up>(n)) (tm_of abc_twice @ shift (mopup 1)
-(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>)"
+(length (tm_of abc_twice) div 2)) stp = (0, Bk\<up>(m) @ Bk # Bk # ires, Oc\<up>(Suc (2*rs)) @ Bk\<up>(l))"
-proof(rule_tac t_compiled_by_rec)
+proof(rule_tac recursive_compile_to_tm_correct)
 show "rec_ci rec_twice = (a, b, c)" by (simp add: h)
 next
 show "rec_calc_rel rec_twice [rs] (2 * rs)"
 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
+show "length [rs] = 1" by simp
+next
+show "layout_of (a [+] dummy_abc 1) = layout_of (a [+] dummy_abc 1)" by simp
 next
-show "layout_of (a [+] dummy_abc (Suc 0)) = layout_of (a [+] dummy_abc (Suc 0))"
+show "tm_of abc_twice = tm_of (a [+] dummy_abc 1)"
-by simp
-next
-show "start_of twice_ly (length abc_twice) =
-start_of (layout_of (a [+] dummy_abc (Suc 0))) (length (a [+] dummy_abc (Suc 0)))"
-using h
-apply(simp add: twice_ly_def abc_twice_def)
-done
-next
-show "tm_of abc_twice = tm_of (a [+] dummy_abc (Suc 0))"
 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>)
+thus "?thesis"
-(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)"
+lemma adjust_fetch0:
-apply(case_tac b, auto simp: fetch.simps nth_of.simps change_termi_state.simps nth_map
+"\<lbrakk>0 < a; a \<le> length ap div 2;  fetch ap a b = (aa, 0)\<rbrakk>
-split: if_splits block.splits)
+\<Longrightarrow> fetch (adjust ap) a b = (aa, Suc (length ap div 2))"
-done
+apply(case_tac b, auto simp: fetch.simps nth_of.simps nth_map
+split: if_splits)
-lemma change_termi_state_exec_in_range:
+apply(case_tac [!] a, auto simp: fetch.simps nth_of.simps)
-"\<lbrakk>steps (st, l, r) ap stp = (st', l', r'); st' \<noteq> 0\<rbrakk>
+done
-\<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)
+lemma adjust_fetch_norm:
-fix stp st l r st' l' r'
+"\<lbrakk>st > 0;  st \<le> length tp div 2; fetch ap st b = (aa, ns); ns \<noteq> 0\<rbrakk>
-assume ind: "\<And>st l r st' l' r'.
+\<Longrightarrow>  fetch (turing_basic.adjust ap) st b = (aa, ns)"
-\<lbrakk>steps (st, l, r) ap stp = (st', l', r'); st' \<noteq> 0\<rbrakk> \<Longrightarrow>
+apply(case_tac b, auto simp: fetch.simps nth_of.simps nth_map
-steps (st, l, r) (change_termi_state ap) stp = (st', l', r')"
+split: if_splits)
-and h: "steps (st, l, r) ap (Suc stp) = (st', l', r')" "st' \<noteq> 0"
+apply(case_tac [!] st, auto simp: fetch.simps nth_of.simps)
-from h show "steps (st, l, r) (change_termi_state ap) (Suc stp) = (st', l', r')"
+done
-proof(simp add: tstep_red, case_tac "steps (st, l, r) ap stp", simp)
-fix a b c
+lemma adjust_step_eq:
-assume g: "steps (st, l, r) ap stp = (a, b, c)"
+assumes exec: "step0 (st,l,r) ap = (st', l', r')"
-"tstep (a, b, c) ap = (st', l', r')" "0 < st'"
+and wf_tm: "tm_wf (ap, 0)"
-hence "steps (st, l, r) (change_termi_state ap) stp = (a, b, c)"
+and notfinal: "st' > 0"
-apply(rule_tac ind, simp)
+shows "step0 (st, l, r) (adjust ap) = (st', l', r')"
-apply(case_tac a, simp_all add: tstep_0)
+using assms
-done
+proof -
-from g and this show "tstep (steps (st, l, r) (change_termi_state ap) stp)
+have "st > 0"
-(change_termi_state ap) = (st', l', r')"
+using assms
-apply(simp add: tstep.simps)
+by(case_tac st, simp_all add: step.simps fetch.simps)
-apply(case_tac "fetch ap a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
+moreover hence "st \<le> (length ap) div 2"
-apply(subgoal_tac "fetch (change_termi_state ap) a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)
+using assms
-= (aa, st')", simp)
+apply(case_tac "st \<le> (length ap) div 2", simp)
-apply(simp add: change_termi_state_fetch)
+apply(case_tac st, auto simp: step.simps fetch.simps)
-done
+apply(case_tac "read r", simp_all add: fetch.simps nth_of.simps)
-qed
+done
+ultimately have "fetch (adjust ap) st (read r) = fetch ap st (read r)"
+using assms
+apply(case_tac "fetch ap st (read r)")
+apply(drule_tac adjust_fetch_norm, simp_all)
+apply(simp add: step.simps)
+done
+thus "?thesis"
+using exec
+by(simp add: step.simps)
 qed
-lemma change_termi_state_fetch0:
+declare adjust.simps[simp del]
-"\<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))"
+lemma adjust_steps_eq:
-apply(case_tac b, auto simp: fetch.simps nth_of.simps change_termi_state.simps nth_map
+assumes exec: "steps0 (st,l,r) ap stp = (st', l', r')"
-split: if_splits block.splits)
+and wf_tm: "tm_wf (ap, 0)"
-done
+and notfinal: "st' > 0"
+shows "steps0 (st, l, r) (adjust ap) stp = (st', l', r')"
-lemma turing_change_termi_state:
+using exec notfinal
-"\<lbrakk>steps (Suc 0, l, r) ap stp = (0, l', r'); t_correct ap\<rbrakk>
+proof(induct stp arbitrary: st' l' r')
-\<Longrightarrow> \<exists> stp. steps (Suc 0, l, r) (change_termi_state ap) stp =
+case 0
+thus "?case"
+by(simp add: steps.simps)
+next
+case (Suc stp st' l' r')
+have ind: "\<And>st' l' r'. \<lbrakk>steps0 (st, l, r) ap stp = (st', l', r'); 0 < st'\<rbrakk>
+\<Longrightarrow> steps0 (st, l, r) (turing_basic.adjust ap) stp = (st', l', r')" by fact
+have h: "steps0 (st, l, r) ap (Suc stp) = (st', l', r')" by fact
+have g:   "0 < st'" by fact
+obtain st'' l'' r'' where a: "steps0 (st, l, r) ap stp = (st'', l'', r'')"
+by (metis prod_cases3)
+hence c:"0 < st''"
+using h g
+apply(simp add: step_red)
+apply(case_tac st'', auto)
+done
+hence b: "steps0 (st, l, r) (turing_basic.adjust ap) stp = (st'', l'', r'')"
+using a
+by(rule_tac ind, simp_all)
+thus "?case"
+using assms a b h g
+apply(simp add: step_red)
+apply(rule_tac adjust_step_eq, simp_all)
+done
+qed
+lemma adjust_halt_eq:
+assumes exec: "steps0 (1, l, r) ap stp = (0, l', r')"
+and tm_wf: "tm_wf (ap, 0)"
+shows "\<exists> stp. steps0 (Suc 0, l, r) (adjust ap) stp =
 (Suc (length ap div 2), l', r')"
-apply(drule first_halt_point)
+proof -
-apply(erule_tac exE)
+have "\<exists> stp. \<not> is_final (steps0 (1, l, r) ap stp) \<and> (steps0 (1, l, r) ap (Suc stp) = (0, l', r'))"
-apply(rule_tac x = "Suc stp" in exI, simp add: tstep_red)
+thm before_final using exec
-apply(case_tac "steps (Suc 0, l, r) ap stp")
+by(erule_tac before_final)
-apply(simp add: isS0_def change_termi_state_exec_in_range)
+then obtain stpa where a:
-apply(subgoal_tac "steps (Suc 0, l, r) (change_termi_state ap) stp = (a, b, c)", simp)
+"\<not> is_final (steps0 (1, l, r) ap stpa) \<and> (steps0 (1, l, r) ap (Suc stpa) = (0, l', r'))" ..
-apply(simp add: tstep.simps)
+obtain sa la ra where b:"steps0 (1, l, r) ap stpa = (sa, la, ra)"  by (metis prod_cases3)
-apply(case_tac "fetch ap a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
+hence c: "steps0 (Suc 0, l, r) (adjust ap) stpa = (sa, la, ra)"
-apply(subgoal_tac "fetch (change_termi_state ap) a
+using assms a
-(case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) = (aa, Suc (length ap div 2))", simp)
+apply(rule_tac adjust_steps_eq, simp_all)
-apply(rule_tac ap = ap in change_termi_state_fetch0, simp_all)
+done
-apply(rule_tac tp = "(l, r)" and l = b and r = c  and stp = stp and A = ap in s_keep, simp_all)
+have d: "sa \<le> length ap div 2"
-apply(simp add: change_termi_state_exec_in_range)
+using steps_in_range[of  "(l, r)" ap stpa] a tm_wf b
+by(simp)
+obtain ac ns where e: "fetch ap sa (read ra) = (ac, ns)"
+by (metis prod.exhaust)
+hence f: "ns = 0"
+using b a
+apply(simp add: step_red step.simps)
+done
+have k: "fetch (adjust ap) sa (read ra) = (ac, Suc (length ap div 2))"
+using a b c d e f
+apply(rule_tac adjust_fetch0, simp_all)
+done
+from a b e f k and c show "?thesis"
+apply(rule_tac x = "Suc stpa" in exI)
+apply(simp add: step_red, auto)
+apply(simp add: step.simps)
+done
+qed
+declare tm_wf.simps[simp del]
+lemma [simp]: " tm_wf (t_twice_compile, 0)"
+apply(simp only: t_twice_compile_def)
+apply(rule_tac t_compiled_correct)
+apply(simp_all add: abc_twice_def)
 done
 lemma t_twice_change_term_state:
-"\<exists> stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp
+"\<exists> stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) t_twice stp
-= (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+= (Suc t_twice_len, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
-using t_twice_correct[of ires rs n]
+proof -
-apply(erule_tac exE)
+have "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
-apply(erule_tac exE)
+(tm_of abc_twice @ shift (mopup (Suc 0)) ((length (tm_of abc_twice) div 2))) stp =
-apply(erule_tac exE)
+(0, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
-proof(drule_tac turing_change_termi_state)
+by(rule_tac t_twice_correct)
-fix stp ln rn
+then obtain stp ln rn where " steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
-show "t_correct (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0))"
+(tm_of abc_twice @ shift (mopup (Suc 0)) ((length (tm_of abc_twice) div 2))) stp =
-apply(rule_tac t_compiled_correct, simp_all)
+(0, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))" by blast
-apply(simp add: twice_ly_def)
+hence "\<exists> stp. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
+(adjust t_twice_compile) stp
+= (Suc (length t_twice_compile div 2), Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
+thm adjust_halt_eq
+apply(rule_tac stp = stp in adjust_halt_eq)
+apply(simp add: t_twice_compile_def, auto)
 done
-next
+then obtain stpb where
-fix stp ln rn
+"steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
-show "\<exists>stp. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+(adjust t_twice_compile) stpb
-(change_termi_state (tm_of abc_twice @ tMp (Suc 0)
+= (Suc (length t_twice_compile div 2), Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))" ..
-(start_of twice_ly (length abc_twice) - Suc 0))) stp =
+thus "?thesis"
-(Suc (length (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) div 2),
+apply(simp add: t_twice_def t_twice_len_def)
-Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) \<Longrightarrow>
+by metis
-\<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 [intro]: "length t_wcode_main_first_part mod 2 = 0"
+apply(auto simp: t_wcode_main_first_part_def)
+done
 lemma t_twice_append_pre:
-"steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp
+"steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) t_twice stp
-= (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)
+= (Suc t_twice_len, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))
-\<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>)
+\<Longrightarrow> steps0 (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
-(t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @
+(t_wcode_main_first_part @ shift 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
+([(L, 1), (L, 1)] @ shift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp
-= (Suc (t_twice_len) + length t_wcode_main_first_part div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+= (Suc (t_twice_len) + length t_wcode_main_first_part div 2,
-proof(rule_tac t_tshift_lemma, simp_all add: t_twice_len_ge)
+Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
-assume "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp =
+by(rule_tac tm_append_shift_append_steps, auto)
-(Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-thus "0 < stp"
-apply(case_tac stp, simp add: steps.simps t_twice_len_ge t_twice_len_def)
-using t_twice_len_ge
-apply(simp, simp)
-done
-next
-show "t_ncorrect t_wcode_main_first_part"
-apply(simp add: t_ncorrect.simps t_wcode_main_first_part_def)
-done
-next
-show "t_ncorrect t_twice"
-using length_tm_even[of abc_twice]
-apply(auto simp: t_ncorrect.simps t_twice_def)
-apply(arith)
-done
-next
-show "t_ncorrect ((L, Suc 0) # (L, Suc 0) #
-tshift t_fourtimes (t_twice_len + 13) @ [(L, Suc 0), (L, Suc 0)])"
-using length_tm_even[of abc_fourtimes]
-apply(simp add: t_ncorrect.simps shift_length t_fourtimes_def)
-apply arith
-done
-qed
 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>)
+"\<exists> stp ln rn. steps0 (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
-(t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @
+(t_wcode_main_first_part @ shift 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
+([(L, 1), (L, 1)] @ shift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp
-= (Suc (t_twice_len) + length t_wcode_main_first_part div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+= (Suc (t_twice_len) + length t_wcode_main_first_part div 2, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
 using t_twice_change_term_state[of ires rs n]
 apply(erule_tac exE)
 apply(erule_tac exE)
 apply(erule_tac exE)
 apply(drule_tac t_twice_append_pre)
-apply(erule_tac exE)
+apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
-apply(rule_tac x = stpa in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
 apply(simp)
 done
+lemma mopup_mod2: "length (mopup k) mod 2  = 0"
+apply(auto simp: mopup.simps)
+by arith
 lemma [simp]: "fetch t_wcode_main (Suc (t_twice_len + length t_wcode_main_first_part div 2)) Oc
 = (L, Suc 0)"
 apply(subgoal_tac "length (t_twice) mod 2 = 0")
 apply(simp add: t_wcode_main_def nth_append fetch.simps t_wcode_main_first_part_def
-nth_of.simps shift_length t_twice_len_def, auto)
+nth_of.simps t_twice_len_def, auto)
-apply(simp add: t_twice_def)
+apply(simp add: t_twice_def t_twice_compile_def)
-apply(subgoal_tac "length (tm_of abc_twice) mod 2 = 0")
+using mopup_mod2[of 1]
-apply arith
+apply(simp)
-apply(rule_tac tm_even)
+by arith
-done
 lemma wcode_jump1:
-"\<exists> stp ln rn. steps (Suc (t_twice_len) + length t_wcode_main_first_part div 2,
+"\<exists> stp ln rn. steps0 (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>)
+Bk\<up>(m) @ Bk # Bk # ires, Oc\<up>(Suc (2 * rs)) @ Bk\<up>(n))
 t_wcode_main stp
-= (Suc 0, Bk\<^bsup>ln\<^esup> @ Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+= (Suc 0, Bk\<up>(ln) @ Bk # ires, Bk # Oc\<up>(Suc (2 * rs)) @ Bk\<up>(rn))"
 apply(rule_tac x = "Suc 0" in exI, rule_tac x = "m" in exI, rule_tac x = n in exI)
-apply(simp add: steps.simps tstep.simps exp_ind_def new_tape.simps)
+apply(simp add: steps.simps step.simps exp_ind)
-apply(case_tac m, simp, simp add: exp_ind_def)
+apply(case_tac m, simp_all)
-apply(simp add: exp_ind_def[THEN sym] exp_ind[THEN sym])
+apply(simp add: 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 =
+shows "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
-(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(Suc 0, Bk # Bk\<up>(ln) @ Oc # ires, Bk # Oc\<up>(Suc (2 * rs + 2)) @ Bk\<up>(rn))"
 proof -
-have "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+have "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
-(13,  Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(13,  Bk # Bk # Bk\<up>(ln) @ Oc # ires, Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 using wcode_double_case_first_correctness[of ires rs m n]
 apply(simp)
 apply(erule_tac exE)
-apply(case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires,
+apply(case_tac "steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires,
-Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na",
+Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main na",
 auto simp: wcode_double_case_inv.simps
 wcode_before_double.simps)
 apply(rule_tac x = na in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
 apply(simp)
 done
 from this obtain stpa lna rna where stp1:
-"steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
+"steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stpa =
-(13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
+(13, Bk # Bk # Bk\<up>(lna) @ Oc # ires, Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rna))" 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 =
+have "\<exists> stp ln rn. steps0 (13, Bk # Bk # Bk\<up>(lna) @ Oc # ires, Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rna)) t_wcode_main stp =
-(13 + t_twice_len, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(13 + t_twice_len, Bk # Bk # Bk\<up>(ln) @ Oc # ires, Oc\<up>(Suc (Suc (Suc (2 *rs)))) @ Bk\<up>(rn))"
-using t_twice_append[of "Bk\<^bsup>lna\<^esup> @ Oc # ires" "Suc rs" rna]
+using t_twice_append[of "Bk\<up>(lna) @ Oc # ires" "Suc rs" rna]
 apply(erule_tac exE)
 apply(erule_tac exE)
 apply(erule_tac exE)
 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])
+apply(simp add: replicate_Suc[THEN sym] exp_add[THEN sym] del: replicate_Suc)
 done
 from this obtain stpb lnb rnb where stp2:
-"steps (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb =
+"steps0 (13, Bk # Bk # Bk\<up>(lna) @ Oc # ires, Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rna)) t_wcode_main stpb =
-(13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>)" by blast
+(13 + t_twice_len, Bk # Bk # Bk\<up>(lnb) @ Oc # ires, Oc\<up>(Suc (Suc (Suc (2 *rs)))) @ Bk\<up>(rnb))" by blast
-have "\<exists>stp ln rn. steps (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires,
+have "\<exists>stp ln rn. steps0 (13 + t_twice_len, Bk # Bk # Bk\<up>(lnb) @ Oc # ires,
-Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stp =
+Oc\<up>(Suc (Suc (Suc (2 *rs)))) @ Bk\<up>(rnb)) 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>)"
+(Suc 0,  Bk # Bk\<up>(ln) @ Oc # ires, Bk # Oc\<up>(Suc (Suc (Suc (2 *rs)))) @ Bk\<up>(rn))"
 using wcode_jump1[of lnb "Oc # ires" "Suc rs" rnb]
 apply(erule_tac exE)
 apply(erule_tac exE)
 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(subgoal_tac "Bk\<up>(lnb) @ Bk # Bk # Oc # ires = Bk # Bk # Bk\<up>(lnb) @ Oc # ires", simp)
-apply(simp add: exp_ind_def[THEN sym] exp_ind[THEN sym])
+apply(simp add: replicate_Suc[THEN sym] exp_ind[THEN sym] del: replicate_Suc)
 apply(simp)
-apply(case_tac lnb, simp, simp add: exp_ind_def[THEN sym] exp_ind)
+apply(simp add: replicate_Suc[THEN sym] exp_ind del: replicate_Suc)
 done
 from this obtain stpc lnc rnc where stp3:
-"steps (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires,
+"steps0 (13 + t_twice_len, Bk # Bk # Bk\<up>(lnb) @ Oc # ires,
-Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stpc =
+Oc\<up>(Suc (Suc (Suc (2 *rs)))) @ Bk\<up>(rnb)) 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>)"
+(Suc 0,  Bk # Bk\<up>(lnc) @ Oc # ires, Bk # Oc\<up>(Suc (Suc (Suc (2 *rs)))) @ Bk\<up>(rnc))"
 by blast
 from stp1 stp2 stp3 show "?thesis"
 apply(rule_tac x = "stpa + stpb + stpc" in exI, rule_tac x = lnc in exI,
 rule_tac x = rnc in exI)
 apply(simp add: steps_add)
 (* 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>
+(\<exists> ml mr rn. l = Bk\<up>(ml) @ Oc # Bk # Oc # ires \<and>
-r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Bk\<up>(mr) @ Oc\<up>(Suc rs) @ Bk\<up>(rn) \<and>
 ml + mr > Suc 0 \<and> mr > 0)"
 fun wcode_on_left_moving_2_O :: "bin_inv_t"
 where
 "wcode_on_left_moving_2_O ires rs (l, r) =
 (\<exists> ln rn. l = Bk # Oc # ires \<and>
-r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+r = Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 fun wcode_on_left_moving_2 :: "bin_inv_t"
 where
 "wcode_on_left_moving_2 ires rs (l, r) =
 (wcode_on_left_moving_2_B ires rs (l, r) \<or>
 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>)"
+r = Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 fun wcode_goon_checking :: "bin_inv_t"
 where
 "wcode_goon_checking ires rs (l, r) =
 (\<exists> ln rn. l = ires \<and>
-r = Oc # Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+r = Oc # Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 fun wcode_right_move :: "bin_inv_t"
 where
 "wcode_right_move ires rs (l, r) =
 (\<exists> ln rn. l = Oc # ires \<and>
-r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+r = Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 fun wcode_erase2 :: "bin_inv_t"
 where
 "wcode_erase2 ires rs (l, r) =
 (\<exists> ln rn. l = Bk # Oc # ires \<and>
-tl r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+tl r = Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 fun wcode_on_right_moving_2 :: "bin_inv_t"
 where
 "wcode_on_right_moving_2 ires rs (l, r) =
-(\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # ires \<and>
+(\<exists> ml mr rn. l = Bk\<up>(ml) @ Oc # ires \<and>
-r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr > Suc 0)"
+r = Bk\<up>(mr) @ Oc\<up>(Suc rs) @ Bk\<up>(rn) \<and> ml + mr > Suc 0)"
 fun wcode_goon_right_moving_2 :: "bin_inv_t"
 where
 "wcode_goon_right_moving_2 ires rs (l, r) =
-(\<exists> ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+(\<exists> ml mr ln rn. l = Oc\<up>(ml) @ Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and>
-r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = Suc rs)"
+r = Oc\<up>(mr) @ Bk\<up>(rn) \<and> ml + mr = Suc rs)"
 fun wcode_backto_standard_pos_2_B :: "bin_inv_t"
 where
 "wcode_backto_standard_pos_2_B ires rs (l, r) =
-(\<exists> ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+(\<exists> ln rn. l = Bk # Bk\<up>(ln) @ Oc # ires \<and>
-r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+r = Bk # Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 fun wcode_backto_standard_pos_2_O :: "bin_inv_t"
 where
 "wcode_backto_standard_pos_2_O ires rs (l, r) =
-(\<exists> ml mr ln rn. l = Oc\<^bsup>ml \<^esup>@ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+(\<exists> ml mr ln rn. l = Oc\<up>(ml )@ Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and>
-r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Oc\<up>(mr) @ Bk\<up>(rn) \<and>
 ml + mr = (Suc (Suc rs)) \<and> mr > 0)"
 fun wcode_backto_standard_pos_2 :: "bin_inv_t"
 where
 "wcode_backto_standard_pos_2 ires rs (l, r) =
 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>
+(\<exists> ln rn. l = Bk # Bk # Bk\<up>(ln) @ Oc # ires \<and>
-r = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+r = Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 declare wcode_on_left_moving_2_B.simps[simp del] wcode_on_left_moving_2.simps[simp del]
 wcode_on_left_moving_2_O.simps[simp del] wcode_on_checking_2.simps[simp del]
 wcode_goon_checking.simps[simp del] wcode_right_move.simps[simp del]
 wcode_erase2.simps[simp del]
 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"
+fun wcode_fourtimes_case_state :: "config \<Rightarrow> nat"
 where
 "wcode_fourtimes_case_state (st, l, r) = 13 - st"
-fun wcode_fourtimes_case_step :: "t_conf \<Rightarrow> nat"
+fun wcode_fourtimes_case_step :: "config \<Rightarrow> nat"
 where
 "wcode_fourtimes_case_step (st, l, r) =
 (if st = Suc 0 then length l
 else if st = 9 then
 (if hd r = Oc then 1
 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"
+fun wcode_fourtimes_case_measure :: "config \<Rightarrow> nat \<times> nat"
 where
 "wcode_fourtimes_case_measure (st, l, r) =
 (wcode_fourtimes_case_state (st, l, r),
 wcode_fourtimes_case_step (st, l, r))"
-definition wcode_fourtimes_case_le :: "(t_conf \<times> t_conf) set"
+definition wcode_fourtimes_case_le :: "(config \<times> config) set"
 where "wcode_fourtimes_case_le \<equiv> (inv_image lex_pair wcode_fourtimes_case_measure)"
 lemma wf_wcode_fourtimes_case_le[intro]: "wf wcode_fourtimes_case_le"
 by(auto intro:wf_inv_image simp: wcode_fourtimes_case_le_def)
 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
+apply(subgoal_tac "7 = Suc 6")
+apply(simp only: t_wcode_main_def fetch.simps
 t_wcode_main_first_part_def nth_of.simps)
+apply(auto)
 done
 lemma [simp]: "fetch t_wcode_main 8 Bk = (R, 9)"
-apply(simp add: t_wcode_main_def fetch.simps
+apply(subgoal_tac "8 = Suc 7")
+apply(simp only: t_wcode_main_def fetch.simps
 t_wcode_main_first_part_def nth_of.simps)
-done
+apply(auto)
+done
 lemma [simp]: "fetch t_wcode_main 9 Bk = (R, 10)"
-apply(simp add: t_wcode_main_def fetch.simps
+apply(subgoal_tac "9 = Suc 8")
+apply(simp only: t_wcode_main_def fetch.simps
 t_wcode_main_first_part_def nth_of.simps)
+apply(auto)
 done
 lemma [simp]: "fetch t_wcode_main 9 Oc = (W0, 9)"
-apply(simp add: t_wcode_main_def fetch.simps
+apply(subgoal_tac "9 = Suc 8")
+apply(simp only: t_wcode_main_def fetch.simps
 t_wcode_main_first_part_def nth_of.simps)
+apply(auto)
 done
 lemma [simp]: "fetch t_wcode_main 10 Bk = (R, 10)"
-apply(simp add: t_wcode_main_def fetch.simps
+apply(subgoal_tac "10 = Suc 9")
+apply(simp only: t_wcode_main_def fetch.simps
 t_wcode_main_first_part_def nth_of.simps)
+apply(auto)
 done
 lemma [simp]: "fetch t_wcode_main 10 Oc = (R, 11)"
-apply(simp add: t_wcode_main_def fetch.simps
+apply(subgoal_tac "10 = Suc 9")
+apply(simp only: t_wcode_main_def fetch.simps
 t_wcode_main_first_part_def nth_of.simps)
-done
+apply(auto)
+done
 lemma [simp]: "fetch t_wcode_main 11 Bk = (W1, 12)"
-apply(simp add: t_wcode_main_def fetch.simps
+apply(subgoal_tac "11 = Suc 10")
+apply(simp only: t_wcode_main_def fetch.simps
 t_wcode_main_first_part_def nth_of.simps)
+apply(auto)
 done
 lemma [simp]: "fetch t_wcode_main 11 Oc = (R, 11)"
-apply(simp add: t_wcode_main_def fetch.simps
+apply(subgoal_tac "11 = Suc 10")
+apply(simp only: t_wcode_main_def fetch.simps
 t_wcode_main_first_part_def nth_of.simps)
-done
+apply(auto)
+done
 lemma [simp]: "fetch t_wcode_main 12 Oc = (L, 12)"
-apply(simp add: t_wcode_main_def fetch.simps
+apply(subgoal_tac "12 = Suc 11")
+apply(simp only: t_wcode_main_def fetch.simps
 t_wcode_main_first_part_def nth_of.simps)
-done
+apply(auto)
+done
 lemma [simp]: "fetch t_wcode_main 12 Bk = (R, t_twice_len + 14)"
-apply(simp add: t_wcode_main_def fetch.simps
+apply(subgoal_tac "12 = Suc 11")
+apply(simp only: t_wcode_main_def fetch.simps
 t_wcode_main_first_part_def nth_of.simps)
-done
+apply(auto)
+done
 lemma [simp]: "wcode_on_left_moving_2 ires rs (b, []) = False"
 apply(auto simp: wcode_fourtimes_invs)
 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)
+apply(auto simp: wcode_fourtimes_invs)
 done
 lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, []) = False"
-apply(auto simp: wcode_fourtimes_invs exponent_def)
+apply(auto simp: wcode_fourtimes_invs)
 done
 lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
 apply(simp add: wcode_fourtimes_invs, auto)
 done
 lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \<Longrightarrow>  wcode_on_left_moving_2 ires rs (tl b, hd b # Bk # list)"
 apply(simp only: wcode_fourtimes_invs)
 apply(erule_tac disjE)
 apply(erule_tac exE)+
 apply(case_tac ml, simp)
 apply(rule_tac x = "mr - (Suc (Suc 0))" in exI, rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp, case_tac nat, simp, simp add: exp_ind)
+apply(case_tac mr, simp, case_tac nat, simp, simp add: exp_ind del: replicate_Suc)
 apply(rule_tac disjI1)
 apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = rn in exI,
-simp add: exp_ind_def)
+simp add: replicate_Suc)
 apply(simp)
 done
 lemma [simp]: "wcode_on_checking_2 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)
+apply(rule_tac x =  "Suc (Suc ln)" in exI, simp add: exp_ind del: replicate_Suc)
 done
 lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
 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 = "Suc ml" in exI, simp)
-apply(rule_tac x = "mr - 1" in exI, case_tac mr, auto simp: exp_ind_def)
+apply(rule_tac x = "mr - 1" in exI, case_tac mr,auto)
 done
 lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
 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(case_tac mr, simp_all)
 apply(rule_tac x = "rn - 1" in exI, simp)
-apply(case_tac rn, simp, simp add: exp_ind_def)
+apply(case_tac rn, simp, simp)
 done
 lemma  [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list) \<Longrightarrow>  b \<noteq> []"
 apply(simp add: wcode_fourtimes_invs, auto)
 done
 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)
+apply(case_tac [!] mr, simp_all)
 done
 lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \<Longrightarrow> b \<noteq> []"
 apply(auto simp: wcode_fourtimes_invs)
 done
 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>)"
+\<Longrightarrow> (\<exists>ln. b = Bk # Bk\<up>(ln) @ Oc # ires) \<and> (\<exists>rn. list = Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 apply(auto simp: wcode_fourtimes_invs)
-apply(case_tac [!] mr, auto simp: exp_ind_def)
+apply(case_tac [!] mr, auto)
 done
 lemma [simp]: "wcode_on_checking_2 ires rs (b, Oc # list) \<Longrightarrow> False"
 apply(simp add: wcode_fourtimes_invs)
 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(case_tac mr, simp_all)
 apply(rule_tac x = "Suc 0" in exI, auto)
 apply(rule_tac x = "ml - 2" in exI)
-apply(case_tac ml, simp, case_tac nat, simp_all add: exp_ind_def)
+apply(case_tac ml, simp, case_tac nat, simp_all)
 done
 lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
 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>)"
+\<Longrightarrow> (\<exists>ln. b = Bk # Bk\<up>(ln) @ Oc # ires) \<and> (\<exists>rn. list = Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 apply(simp add: wcode_fourtimes_invs, auto)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
+apply(case_tac [!] mr, simp_all)
 done
 lemma [simp]: "wcode_on_checking_2 ires rs (b, Oc # list) = False"
 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 = "Suc ml" in exI, simp)
 apply(rule_tac x = "mr - 1" in exI)
-apply(case_tac mr, case_tac rn, auto simp: exp_ind_def)
+apply(case_tac mr, case_tac rn, auto)
 done
 lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
 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(case_tac ml, auto)
-apply(rule_tac disjI2)
+apply(rule_tac x = nat in exI, auto)
-apply(rule_tac conjI, rule_tac x = ln in exI, simp)
+apply(rule_tac x = "Suc mr" 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
+let f = (\<lambda> stp. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp) in
 \<exists> n .P (f n) \<and> Q (f (n::nat))"
 proof -
 let ?P = "(\<lambda> (st, l, r). st = t_twice_len + 14)"
 let ?Q = "(\<lambda> (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r))"
-let ?f = "(\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)"
+let ?f = "(\<lambda> stp. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp)"
 have "\<exists> n . ?P (?f n) \<and> ?Q (?f (n::nat))"
 proof(rule_tac halt_lemma2)
 show "wf wcode_fourtimes_case_le"
 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,
+case_tac "steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main na", simp,
 rule_tac impI)
-apply(simp add: tstep_red tstep.simps, case_tac c, simp, case_tac [2] aa, simp_all)
+apply(simp add: step_red step.simps, case_tac c, simp, case_tac [2] aa, simp_all)
+apply(simp_all add: wcode_fourtimes_case_inv.simps
-apply(simp_all add: wcode_fourtimes_case_inv.simps new_tape.simps
 wcode_fourtimes_case_le_def lex_pair_def split: if_splits)
+apply(auto simp: wcode_backto_standard_pos_2.simps wcode_backto_standard_pos_2_O.simps
+wcode_backto_standard_pos_2_B.simps)
+apply(case_tac mr, simp_all)
 done
 next
 show "?Q (?f 0)"
 apply(simp add: steps.simps wcode_fourtimes_case_inv.simps)
 apply(simp add: wcode_on_left_moving_2.simps wcode_on_left_moving_2_B.simps
 wcode_on_left_moving_2_O.simps)
-apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def)
+apply(rule_tac x = "Suc m" in exI, simp )
 apply(rule_tac x ="Suc 0" in exI, auto)
 done
 next
 show "\<not> ?P (?f 0)"
 apply(simp add: steps.simps)
 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)
+apply(simp add: t_fourtimes_len_def t_fourtimes_def mopup.simps t_fourtimes_compile_def)
+done
+lemma [intro]: "rec_calc_rel (constn 4) [rs] 4"
+using prime_rel_exec_eq[of "constn 4" "[rs]" 4]
+apply(subgoal_tac "primerec (constn 4) 1", auto)
+done
+lemma [intro]: "rec_calc_rel rec_mult [rs, 4] (4 * rs)"
+using prime_rel_exec_eq[of "rec_mult" "[rs, 4]"  "4*rs"]
+apply(subgoal_tac "primerec rec_mult 2", auto simp: numeral_2_eq_2)
 done
 lemma t_fourtimes_correct:
-"\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+"\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
-(tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) stp =
+(tm_of abc_fourtimes @ shift (mopup 1) (length (tm_of abc_fourtimes) div 2)) stp =
-(0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(0, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))"
 proof(case_tac "rec_ci rec_fourtimes")
 fix a b c
 assume h: "rec_ci rec_fourtimes = (a, b, c)"
-have "\<exists>stp m l. steps (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<^bsup>n\<^esup>) (tm_of abc_fourtimes @ tMp (Suc 0)
+have "\<exists>stp m l. steps0 (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<up>(n)) (tm_of abc_fourtimes @ shift (mopup 1)
-(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>)"
+(length (tm_of abc_fourtimes) div 2)) stp = (0, Bk\<up>(m) @ Bk # Bk # ires, Oc\<up>(Suc (4*rs)) @ Bk\<up>(l))"
-proof(rule_tac t_compiled_by_rec)
+proof(rule_tac recursive_compile_to_tm_correct)
 show "rec_ci rec_fourtimes = (a, b, c)" by (simp add: h)
 next
 show "rec_calc_rel rec_fourtimes [rs] (4 * rs)"
-using prime_rel_exec_eq [of rec_fourtimes "[rs]" "4 * rs"]
+apply(simp add: rec_fourtimes_def)
-apply(subgoal_tac "primerec rec_fourtimes (length [rs])")
+apply(rule_tac rs =  "[rs, 4]" in calc_cn, simp_all)
-apply(simp_all add: rec_fourtimes_def rec_exec.simps)
+apply(rule_tac allI, case_tac k, auto simp: mult_lemma)
-apply(auto)
-apply(simp only: Nat.One_nat_def[THEN sym], auto)
 done
 next
-show "length [rs] = Suc 0" by simp
+show "length [rs] = 1" by simp
+next
+show "layout_of (a [+] dummy_abc 1) = layout_of (a [+] dummy_abc 1)" by simp
 next
-show "layout_of (a [+] dummy_abc (Suc 0)) = layout_of (a [+] dummy_abc (Suc 0))"
+show "tm_of abc_fourtimes = tm_of (a [+] dummy_abc 1)"
-by simp
-next
-show "start_of fourtimes_ly (length abc_fourtimes) =
-start_of (layout_of (a [+] dummy_abc (Suc 0))) (length (a [+] dummy_abc (Suc 0)))"
-using h
-apply(simp add: fourtimes_ly_def abc_fourtimes_def)
-done
-next
-show "tm_of abc_fourtimes = tm_of (a [+] dummy_abc (Suc 0))"
 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>)
+thus "?thesis"
-(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 wf_fourtimes[intro]: "tm_wf (t_fourtimes_compile, 0)"
+apply(simp only: t_fourtimes_compile_def)
+apply(rule_tac t_compiled_correct)
+apply(simp_all add: abc_twice_def)
+done
 lemma t_fourtimes_change_term_state:
-"\<exists> stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp
+"\<exists> stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) t_fourtimes stp
-= (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+= (Suc t_fourtimes_len, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))"
-using t_fourtimes_correct[of ires rs n]
+proof -
-apply(erule_tac exE)
+have "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
-apply(erule_tac exE)
+(tm_of abc_fourtimes @ shift (mopup 1) ((length (tm_of abc_fourtimes) div 2))) stp =
-apply(erule_tac exE)
+(0, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))"
-proof(drule_tac turing_change_termi_state)
+by(rule_tac t_fourtimes_correct)
-fix stp ln rn
+then obtain stp ln rn where
-show "t_correct (tm_of abc_fourtimes @ tMp (Suc 0)
+"steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
-(start_of fourtimes_ly (length abc_fourtimes) - Suc 0))"
+(tm_of abc_fourtimes @ shift (mopup 1) ((length (tm_of abc_fourtimes) div 2))) stp =
-apply(rule_tac t_compiled_correct, auto simp: fourtimes_ly_def)
+(0, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))" by blast
+hence "\<exists> stp. steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
+(adjust t_fourtimes_compile) stp
+= (Suc (length t_fourtimes_compile div 2), Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))"
+apply(rule_tac stp = stp in adjust_halt_eq)
+apply(simp add: t_fourtimes_compile_def, auto)
 done
-next
+then obtain stpb where
-fix stp ln rn
+"steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
-show "\<exists>stp. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+(adjust t_fourtimes_compile) stpb
-(change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0)
+= (Suc (length t_fourtimes_compile div 2), Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))" ..
-(start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) stp =
+thus "?thesis"
-(Suc (length (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly
+apply(simp add: t_fourtimes_def t_fourtimes_len_def)
-(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>
+by metis
-\<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 [intro]: "length t_twice mod 2 = 0"
+apply(auto simp: t_twice_def t_twice_compile_def)
+done
 lemma t_fourtimes_append_pre:
-"steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp
+"steps0 (Suc 0, Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n)) t_fourtimes stp
-= (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)
+= (Suc t_fourtimes_len, Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))
-\<Longrightarrow> \<exists> stp>0. steps (Suc 0 + length (t_wcode_main_first_part @
+\<Longrightarrow> steps0 (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,
+shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2,
-Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
 ((t_wcode_main_first_part @
-tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) @
+shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) @
-tshift t_fourtimes (length (t_wcode_main_first_part @
+shift 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
+shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2) @ ([(L, 1), (L, 1)])) stp
-= (Suc t_fourtimes_len + length (t_wcode_main_first_part @
+= ((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,
+shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2,
-Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+Bk\<up>(ln) @ Bk # Bk # ires, Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))"
-proof(rule_tac t_tshift_lemma, auto)
+apply(rule_tac tm_append_shift_append_steps, simp_all)
-assume "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp =
+apply(auto simp: t_wcode_main_first_part_def)
-(Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+done
-thus "0 < stp"
-using t_fourtimes_len_gr
-apply(case_tac stp, simp_all add: steps.simps)
-done
-next
-show "Suc 0 \<le> length t_fourtimes div 2"
-apply(simp add: t_fourtimes_def shift_length tMp.simps)
-done
-next
-show "t_ncorrect (t_wcode_main_first_part @
-tshift t_twice (length t_wcode_main_first_part div 2) @
-[(L, Suc 0), (L, Suc 0)])"
-apply(simp add: t_ncorrect.simps t_wcode_main_first_part_def shift_length
-t_twice_def)
-using tm_even[of abc_twice]
-by arith
-next
-show "t_ncorrect t_fourtimes"
-apply(simp add: t_fourtimes_def steps.simps t_ncorrect.simps)
-using tm_even[of abc_fourtimes]
-by arith
-next
-show "t_ncorrect [(L, Suc 0), (L, Suc 0)]"
-apply(simp add: t_ncorrect.simps)
-done
-qed
 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 t_twice_def)
-apply(simp add: t_twice_def)
+done
-done
+lemma [simp]: "((26 + length (shift t_twice 12)) div 2)
-lemma [simp]: "((26 + length (tshift t_twice 12)) div 2)
+= (length (shift t_twice 12) div 2 + 13)"
-= (length (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 (tshift t_twice 12) div 2"
+lemma [simp]: "t_twice_len + 14 =  14 + length (shift t_twice 12) div 2"
-using tm_even[of abc_twice]
+apply(simp add: t_twice_def t_twice_len_def)
-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
+steps0 (Suc 0 + length (t_wcode_main_first_part @ shift t_twice
 (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2,
-Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+Bk # Bk # ires, Oc\<up>(Suc rs) @ Bk\<up>(n))
-((t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @
+((t_wcode_main_first_part @ shift 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
+[(L, 1), (L, 1)]) @ shift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)]) stp
-= (Suc t_fourtimes_len + length (t_wcode_main_first_part @ tshift t_twice
+= (Suc t_fourtimes_len + length (t_wcode_main_first_part @ shift t_twice
-(length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires,
+(length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, Bk\<up>(ln) @ Bk # Bk # ires,
-Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+Oc\<up>(Suc (4 * rs)) @ Bk\<up>(rn))"
 using t_fourtimes_change_term_state[of ires rs n]
 apply(erule_tac exE)
 apply(erule_tac exE)
 apply(erule_tac exE)
 apply(drule_tac t_fourtimes_append_pre)
-apply(erule_tac exE)
+apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
-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)
-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)
+apply(simp add: t_wcode_main_def)
 done
+lemma even_twice_len: "length t_twice mod 2 = 0"
+apply(auto simp: t_twice_def t_twice_compile_def)
+done
+lemma even_fourtimes_len: "length t_fourtimes mod 2 = 0"
+apply(auto simp: t_fourtimes_def t_fourtimes_compile_def)
+done
+lemma [simp]: "2 * (length t_twice div 2) = length t_twice"
+using even_twice_len
+by arith
+lemma [simp]: "2 * (length t_fourtimes div 2) = length t_fourtimes"
+using even_fourtimes_len
+by arith
+lemma [simp]: "fetch t_wcode_main (14 + length t_twice div 2 + t_fourtimes_len) Oc
+= (L, Suc 0)"
+apply(subgoal_tac "14 = Suc 13")
+apply(simp only: fetch.simps add_Suc nth_of.simps t_wcode_main_def)
+apply(simp add:length_append length_shift Parity.two_times_even_div_two even_twice_len t_fourtimes_len_def)
+by arith
+lemma [simp]: "fetch t_wcode_main (14 + length t_twice div 2 + t_fourtimes_len) Bk
+= (L, Suc 0)"
+apply(subgoal_tac "14 = Suc 13")
+apply(simp only: fetch.simps add_Suc nth_of.simps t_wcode_main_def)
+apply(simp add:length_append length_shift Parity.two_times_even_div_two even_twice_len t_fourtimes_len_def nth_append)
+by arith
 lemma [simp]: "fetch t_wcode_main (14 + length t_twice div 2 + t_fourtimes_len) b
 = (L, Suc 0)"
-using tm_even[of "abc_twice"] tm_even[of "abc_fourtimes"]
+apply(case_tac b, simp_all)
-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
+"\<exists> stp ln rn. steps0 (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 =
+, Bk # Bk # Bk\<up>(lnb) @ Oc # ires, Oc\<up>(Suc (4 * rs + 4)) @ Bk\<up>(rnb)) t_wcode_main stp =
-(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4 * rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(Suc 0, Bk # Bk\<up>(ln) @ Oc # ires, Bk # Oc\<up>(Suc (4 * rs + 4)) @ Bk\<up>(rn))"
 apply(rule_tac x = "Suc 0" in exI)
-apply(simp add: steps.simps shift_length)
+apply(simp add: steps.simps)
 apply(rule_tac x = lnb in exI, rule_tac x = rnb in exI)
-apply(simp add: tstep.simps new_tape.simps)
+apply(simp add: step.simps)
 done
 lemma wcode_fourtimes_case:
 shows "\<exists>stp ln rn.
-steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
-(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(Suc 0, Bk # Bk\<up>(ln) @ Oc # ires, Bk # Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rn))"
 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 =
+steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
-(t_twice_len + 14, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(t_twice_len + 14, Bk # Bk # Bk\<up>(ln) @ Oc # ires, Oc\<up>(Suc (rs + 1)) @ Bk\<up>(rn))"
 using wcode_fourtimes_case_first_correctness[of ires rs m n]
 apply(simp add: wcode_fourtimes_case_inv.simps, auto)
 apply(rule_tac x = na in exI, rule_tac x = ln in exI,
 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 =
+"steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Oc # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stpa =
-(t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
+(t_twice_len + 14, Bk # Bk # Bk\<up>(lna) @ Oc # ires, Oc\<up>(Suc (rs + 1)) @ Bk\<up>(rna))" by blast
-have "\<exists>stp ln rn. steps (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)
+have "\<exists>stp ln rn. steps0 (t_twice_len + 14, Bk # Bk # Bk\<up>(lna) @ Oc # ires, Oc\<up>(Suc (rs + 1)) @ Bk\<up>(rna))
 t_wcode_main stp =
-(t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires,  Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<up>(ln) @ Oc # ires,  Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rn))"
-using t_fourtimes_append[of " Bk\<^bsup>lna\<^esup> @ Oc # ires" "rs + 1" rna]
+using t_fourtimes_append[of " Bk\<up>(lna) @ Oc # ires" "rs + 1" rna]
 apply(erule_tac exE)
 apply(erule_tac exE)
 apply(erule_tac exE)
 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])
+apply(simp add: replicate_Suc[THEN sym] exp_add[THEN sym] del: replicate_Suc)
 done
 from this obtain stpb lnb rnb where stp2:
-"steps (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)
+"steps0 (t_twice_len + 14, Bk # Bk # Bk\<up>(lna) @ Oc # ires, Oc\<up>(Suc (rs + 1)) @ Bk\<up>(rna))
 t_wcode_main stpb =
-(t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires,  Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>)"
+(t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<up>(lnb) @ Oc # ires,  Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rnb))"
 by blast
-have "\<exists>stp ln rn. steps (t_twice_len + 14 + t_fourtimes_len,
+have "\<exists>stp ln rn. steps0 (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>)
+Bk # Bk # Bk\<up>(lnb) @ Oc # ires,  Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rnb))
 t_wcode_main stp =
-(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(Suc 0, Bk # Bk\<up>(ln) @ Oc # ires, Bk # Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rn))"
 apply(rule wcode_jump2)
 done
 from this obtain stpc lnc rnc where stp3:
-"steps (t_twice_len + 14 + t_fourtimes_len,
+"steps0 (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>)
+Bk # Bk # Bk\<up>(lnb) @ Oc # ires,  Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rnb))
 t_wcode_main stpc =
-(Suc 0, Bk # Bk\<^bsup>lnc\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnc\<^esup>)"
+(Suc 0, Bk # Bk\<up>(lnc) @ Oc # ires, Bk # Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rnc))"
 by blast
 from stp1 stp2 stp3 show "?thesis"
 apply(rule_tac x = "stpa + stpb + stpc" in exI,
 rule_tac x = lnc in exI, rule_tac x = rnc in exI)
 apply(simp add: steps_add)
 (**********************************************************)
 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>
+(\<exists> ml mr rn. l = Bk\<up>(ml) @ Oc # Bk # Bk # ires \<and>
-r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Bk\<up>(mr) @ Oc\<up>(Suc rs) @ Bk\<up>(rn) \<and>
 ml + mr > Suc 0 \<and> mr > 0 )"
 fun wcode_on_left_moving_3_O :: "bin_inv_t"
 where
 "wcode_on_left_moving_3_O ires rs (l, r) =
 (\<exists> ln rn. l = Bk # Bk # ires \<and>
-r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+r = Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 fun wcode_on_left_moving_3 :: "bin_inv_t"
 where
 "wcode_on_left_moving_3 ires rs (l, r) =
 (wcode_on_left_moving_3_B ires rs (l, r) \<or>
 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>)"
+r = Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 fun wcode_goon_checking_3 :: "bin_inv_t"
 where
 "wcode_goon_checking_3 ires rs (l, r) =
 (\<exists> ln rn. l = ires \<and>
-r = Bk # Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+r = Bk # Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 fun wcode_stop :: "bin_inv_t"
 where
 "wcode_stop ires rs (l, r) =
 (\<exists> ln rn. l = Bk # ires \<and>
-r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+r = Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 fun wcode_halt_case_inv :: "nat \<Rightarrow> bin_inv_t"
 where
 "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"
+fun wcode_halt_case_state :: "config \<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"
+fun wcode_halt_case_step :: "config \<Rightarrow> nat"
 where
 "wcode_halt_case_step (st, l, r) =
 (if st = 1 then length l
 else 0)"
-fun wcode_halt_case_measure :: "t_conf \<Rightarrow> nat \<times> nat"
+fun wcode_halt_case_measure :: "config \<Rightarrow> nat \<times> nat"
 where
 "wcode_halt_case_measure (st, l, r) =
 (wcode_halt_case_state (st, l, r),
 wcode_halt_case_step (st, l, r))"
-definition wcode_halt_case_le :: "(t_conf \<times> t_conf) set"
+definition wcode_halt_case_le :: "(config \<times> config) set"
 where "wcode_halt_case_le \<equiv> (inv_image lex_pair wcode_halt_case_measure)"
 lemma wf_wcode_halt_case_le[intro]: "wf wcode_halt_case_le"
 by(auto intro:wf_inv_image simp: wcode_halt_case_le_def)
 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
+apply(subgoal_tac "7 = Suc 6")
+apply(simp only: fetch.simps t_wcode_main_def nth_append nth_of.simps
 t_wcode_main_first_part_def)
+apply(auto)
 done
 lemma [simp]: "wcode_on_left_moving_3 ires rs (b, [])  = False"
 apply(simp only: wcode_halt_invs)
-apply(simp add: exp_ind_def)
+apply(simp)
 done
 lemma [simp]: "wcode_on_checking_3 ires rs (b, []) = False"
 apply(simp add: wcode_halt_invs)
 done
 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(case_tac mr, simp, simp add: exp_ind del: replicate_Suc)
+apply(case_tac nat, simp, simp add: exp_ind del: replicate_Suc)
 apply(rule_tac disjI1)
 apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI,
-rule_tac x = rn in exI, simp add: exp_ind_def)
+rule_tac x = rn in exI, simp)
 apply(simp)
 done
 lemma [simp]: "wcode_goon_checking_3 ires rs (b, Bk # list) \<Longrightarrow>
 (b = [] \<longrightarrow> wcode_stop ires rs ([Bk], list)) \<and>
 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)
+apply(case_tac [!] mr, simp_all)
 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
 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
+let f = (\<lambda> stp. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp) in
 \<exists> n .P (f n) \<and> Q (f (n::nat))"
 proof -
 let ?P = "(\<lambda> (st, l, r). st = 0)"
 let ?Q = "(\<lambda> (st, l, r). wcode_halt_case_inv st ires rs (l, r))"
-let ?f = "(\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)"
+let ?f = "(\<lambda> stp. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp)"
 have "\<exists> n. ?P (?f n) \<and> ?Q (?f (n::nat))"
 proof(rule_tac halt_lemma2)
 show "wf wcode_halt_case_le" by auto
 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(simp add: step_red step.simps)
 apply(case_tac c, simp, case_tac [2] aa)
-apply(simp_all split: if_splits add: new_tape.simps wcode_halt_case_le_def lex_pair_def)
+apply(simp_all split: if_splits add: wcode_halt_case_le_def lex_pair_def)
 done
 next
 show "?Q (?f 0)"
 apply(simp add: steps.simps wcode_halt_invs)
-apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def)
+apply(rule_tac x = "Suc m" in exI, simp)
 apply(rule_tac x = "Suc 0" in exI, auto)
 done
 next
 show "\<not> ?P (?f 0)"
 apply(simp add: steps.simps)
 apply(auto)
 done
 qed
 declare wcode_halt_case_inv.simps[simp del]
-lemma [intro]: "\<exists> xs. (<rev list @ [aa::nat]> :: block list) = Oc # xs"
+lemma [intro]: "\<exists> xs. (<rev list @ [aa::nat]> :: cell list) = Oc # xs"
 apply(case_tac "rev list", simp)
-apply(simp add: tape_of_nat_abv tape_of_nat_list.simps exp_ind_def)
+apply(simp add: tape_of_nl_cons)
-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>)
+"\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n))
-t_wcode_main stp  = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+t_wcode_main stp  = (0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 using t_halt_case_correctness[of ires rs m n]
 apply(simp)
 apply(erule_tac exE)
-apply(case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires,
+apply(case_tac "steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na")
+Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main na")
 apply(auto simp: wcode_halt_case_inv.simps wcode_stop.simps)
 apply(rule_tac x = na in exI, rule_tac x = ln in exI,
 rule_tac x = rn in exI, simp)
 done
 lemma bl_bin_one: "bl_bin [Oc] =  Suc 0"
 apply(simp add: bl_bin.simps)
 done
+lemma [simp]: "bl_bin [Oc] = 1"
+apply(simp add: bl_bin.simps)
+done
+lemma [intro]: "2 * 2 ^ a = Suc (Suc (2 * bl_bin (Oc \<up> a)))"
+apply(induct a, auto simp: bl_bin.simps)
+done
+declare replicate_Suc[simp del]
 lemma t_wcode_main_lemma_pre:
 "\<lbrakk>args \<noteq> []; lm = <args::nat list>\<rbrakk> \<Longrightarrow>
-\<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main
+\<exists> stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(m) @ rev lm @ Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main
 stp
-= (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2^(length lm - 1) \<^esup> @ Bk\<^bsup>rn\<^esup>)"
+= (0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin lm + rs * 2^(length lm - 1) ) @ Bk\<up>(rn))"
 proof(induct "length args" arbitrary: args lm rs m n, simp)
 fix x args lm rs m n
 assume ind:
 "\<And>args lm rs m n.
 \<lbrakk>x = length args; (args::nat list) \<noteq> []; lm = <args>\<rbrakk>
 \<Longrightarrow> \<exists>stp ln rn.
-steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+steps0 (Suc 0, Bk # Bk\<up>(m) @ rev lm @ Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
-(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin lm + rs * 2 ^ (length lm - 1)) @ Bk\<up>(rn))"
 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 =
+steps0 (Suc 0, Bk # Bk\<up>(m) @ rev lm @ Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
-(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin lm + rs * 2 ^ (length lm - 1)) @ Bk\<up>(rn))"
 proof(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 =
+steps0 (Suc 0, Bk # Bk \<up> m @ Oc \<up> Suc a @ Bk # Bk # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main stp =
-(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(0, Bk # ires, Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> (bl_bin (Oc \<up> Suc a) + rs * 2 ^ a) @ Bk \<up> rn)"
 proof(induct "a" arbitrary: m n rs ires, simp)
 fix m n rs ires
-show "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+show "\<exists>stp ln rn.
-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>)"
+steps0 (Suc 0, Bk # Bk \<up> m @ Oc # Bk # Bk # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main stp =
-apply(simp add: bl_bin_one)
+(0, Bk # ires, Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> Suc rs @ Bk \<up> rn)"
 apply(rule_tac wcode_halt_case)
 done
 next
 fix a m n rs ires
 assume ind2:
 "\<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 =
+steps0 (Suc 0, Bk # Bk \<up> m @ Oc \<up> Suc a @ Bk # Bk # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main stp =
-(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(0, Bk # ires, Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> (bl_bin (Oc \<up> Suc a) + rs * 2 ^ a) @ Bk \<up> rn)"
-show "\<exists>stp ln rn.
+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 =
+steps0 (Suc 0, Bk # Bk \<up> m @ Oc \<up> Suc (Suc a) @ Bk # Bk # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main stp =
-(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<Suc a>) + rs * 2 ^ Suc a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(0, Bk # ires, Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> (bl_bin (Oc \<up> Suc (Suc a)) + rs * 2 ^ Suc a) @ Bk \<up> rn)"
 proof -
 have "\<exists>stp ln rn.
-steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+steps0 (Suc 0, Bk # Bk\<up>(m) @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
-(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(Suc 0, Bk # Bk\<up>(ln) @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<up>(Suc (2 * rs + 2)) @ Bk\<up>(rn))"
 apply(simp add: tape_of_nat)
-using wcode_double_case[of m "Oc\<^bsup>a\<^esup> @ Bk # Bk # ires" rs n]
+using wcode_double_case[of m "Oc\<up>(a) @ Bk # Bk # ires" rs n]
-apply(simp add: exp_ind_def)
+apply(simp add: replicate_Suc)
 done
 from this obtain stpa lna rna where stp1:
-"steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
+"steps0 (Suc 0, Bk # Bk\<up>(m) @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stpa =
-(Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
+(Suc 0, Bk # Bk\<up>(lna) @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<up>(Suc (2 * rs + 2)) @ Bk\<up>(rna))" by blast
 moreover have
 "\<exists>stp ln rn.
-steps (Suc 0,  Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp =
+steps0 (Suc 0,  Bk # Bk\<up>(lna) @ rev (<a::nat>) @ Bk # Bk # ires, Bk # Oc\<up>(Suc (2 * rs + 2)) @ Bk\<up>(rna)) t_wcode_main stp =
-(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + (2*rs + 2)  * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin (<a>) + (2*rs + 2)  * 2 ^ a) @ Bk\<up>(rn))"
-using ind2[of lna ires "2*rs + 2" rna] by simp
+using ind2[of lna ires "2*rs + 2" rna] by(simp add: tape_of_nl_abv tape_of_nat_abv)
 from this obtain stpb lnb rnb where stp2:
-"steps (Suc 0,  Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb =
+"steps0 (Suc 0,  Bk # Bk\<up>(lna) @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<up>(Suc (2 * rs + 2)) @ Bk\<up>(rna)) t_wcode_main stpb =
-(0, Bk # ires, Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + (2*rs + 2)  * 2 ^ a\<^esup> @ Bk\<^bsup>rnb\<^esup>)"
+(0, Bk # ires, Bk # Oc # Bk\<up>(lnb) @ Bk # Bk # Oc\<up>(bl_bin (<a>) + (2*rs + 2)  * 2 ^ a) @ Bk\<up>(rnb))"
 by blast
 from stp1 and stp2 show "?thesis"
 apply(rule_tac x = "stpa + stpb" in exI,
-rule_tac x = lnb in exI, rule_tac x = rnb in exI, simp)
+rule_tac x = lnb in exI, rule_tac x = rnb in exI, simp add: tape_of_nat_abv)
-apply(simp add: steps_add bl_bin_nat_Suc exponent_def)
+apply(simp add:  bl_bin.simps replicate_Suc)
+apply(auto)
 done
 qed
 qed
 next
 fix aa list
 assume g: "Suc x = length args" "args \<noteq> []" "lm = <args>" "args = xs @ [a::nat]" "xs = (aa::nat) # list"
-thus "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+thus "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(m) @ rev lm @ Bk # Bk # ires, Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
-(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin lm + rs * 2 ^ (length lm - 1)) @ Bk\<up>(rn))"
 proof(induct 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,
+steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # rev (<(aa::nat) # list>) @ Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
-(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ rev (<aa # list>) @ Bk # Bk # ires,
+(Suc 0, Bk # Bk\<up>(ln) @ rev (<aa # list>) @ Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+Bk # Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rn))"
 proof(simp add: tape_of_nl_rev)
 have "\<exists> xs. (<rev list @ [aa]>) = Oc # xs" by auto
 from this obtain xs where "(<rev list @ [aa]>) = Oc # xs" ..
 thus "\<exists>stp ln rn.
-steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) 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>)"
+(Suc 0, Bk # Bk\<up>(ln) @ <rev list @ [aa]> @ Bk # Bk # ires, Bk # Oc\<up>(5 + 4 * rs) @ Bk\<up>(rn))"
 apply(simp)
 using wcode_fourtimes_case[of m "xs @ Bk # Bk # ires" rs n]
 apply(simp)
 done
 qed
 from this obtain stpa lna rna where stp1:
-"steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # rev (<aa # list>) @ Bk # Bk # ires,
+"steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # rev (<aa # list>) @ Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
+Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stpa =
-(Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<aa # list>) @ Bk # Bk # ires,
+(Suc 0, Bk # Bk\<up>(lna) @ rev (<aa # list>) @ Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
+Bk # Oc\<up>(Suc (4*rs + 4)) @ Bk\<up>(rna))" by blast
 from g have
-"\<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<(aa::nat) # list>) @ Bk # Bk # ires,
+"\<exists> stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(lna) @ 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\<up>(Suc (4*rs + 4)) @ Bk\<up>(rna)) 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>)"
+Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin (<aa#list>)+ (4*rs + 4) * 2^(length (<aa#list>) - 1) ) @ Bk\<up>(rn))"
 apply(rule_tac args = "(aa::nat)#list" in ind, simp_all)
 done
 from this obtain stpb lnb rnb where stp2:
-"steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<(aa::nat) # list>) @ Bk # Bk # ires,
+"steps0 (Suc 0, Bk # Bk\<up>(lna) @ 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\<up>(Suc (4*rs + 4)) @ Bk\<up>(rna)) 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>)"
+Bk # Oc # Bk\<up>(lnb) @ Bk # Bk # Oc\<up>(bl_bin (<aa#list>)+ (4*rs + 4) * 2^(length (<aa#list>) - 1) ) @ Bk\<up>(rnb))"
 by blast
 from stp1 and stp2 and h
 show "\<exists>stp ln rn.
-steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
-(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
+(0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ 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>)"
+Bk # Oc\<up>(bl_bin (Oc\<up>(Suc aa) @ Bk # <list @ [0]>) + rs * (2 * 2 ^ (aa + length (<list @ [0]>)))) @ Bk\<up>(rn))"
 apply(rule_tac x = "stpa + stpb" in exI, rule_tac x = lnb in exI,
 rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_rev)
 done
 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,
+steps0 (Suc 0, Bk # Bk\<up>(m) @ <ab # rev list @ [aa]> @ Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
-(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
+(0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk #
-Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]>) + rs * 2 ^ (length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+Bk # Oc\<up>(bl_bin (<aa # list @ [ab]>) + rs * 2 ^ (length (<aa # list @ [ab]>) - Suc 0)) @ Bk\<up>(rn))"
 and k: "args = aa # list @ [Suc ab]" "lm = <aa # list @ [Suc ab]>"
 show "\<exists>stp ln rn.
-steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ <Suc ab # rev list @ [aa]> @ Bk # Bk # ires,
+steps0 (Suc 0, Bk # Bk\<up>(m) @ <Suc ab # rev list @ [aa]> @ Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
-(0, Bk # ires,Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
+(0, Bk # ires,Bk # Oc # Bk\<up>(ln) @ Bk #
-Bk # Oc\<^bsup>bl_bin (<aa # list @ [Suc ab]>) + rs * 2 ^ (length (<aa # list @ [Suc ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+Bk # Oc\<up>(bl_bin (<aa # list @ [Suc ab]>) + rs * 2 ^ (length (<aa # list @ [Suc ab]>) - Suc 0)) @ Bk\<up>(rn))"
 proof(simp add: tape_of_nl_cons_app1)
 have "\<exists>stp ln rn.
-steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc\<up>(Suc (Suc ab)) @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
-Bk # Oc # Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp
+Bk # Oc # Oc\<up>(rs) @ Bk\<up>(n)) t_wcode_main stp
-= (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc\<^bsup>Suc ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+= (Suc 0, Bk # Bk\<up>(ln) @ Oc\<up>(Suc ab) @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+Bk # Oc\<up>(Suc (2*rs + 2)) @ Bk\<up>(rn))"
-using wcode_double_case[of m "Oc\<^bsup>ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires"
+using wcode_double_case[of m "Oc\<up>(ab) @ Bk # <rev list @ [aa]> @ Bk # Bk # ires"
 rs n]
-apply(simp add: exp_ind_def)
+apply(simp add: replicate_Suc)
 done
 from this obtain stpa lna rna where stp1:
-"steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+"steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc\<up>(Suc (Suc ab)) @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
-Bk # Oc # Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa
+Bk # Oc # Oc\<up>(rs) @ Bk\<up>(n)) t_wcode_main stpa
-= (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ Oc\<^bsup>Suc ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+= (Suc 0, Bk # Bk\<up>(lna) @ Oc\<up>(Suc ab) @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
+Bk # Oc\<up>(Suc (2*rs + 2)) @ Bk\<up>(rna))" by blast
 from k have
-"\<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ <ab # rev list @ [aa]> @ Bk # Bk # ires,
+"\<exists> stp ln rn. steps0 (Suc 0, Bk # Bk\<up>(lna) @ <ab # rev list @ [aa]> @ Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp
+Bk # Oc\<up>(Suc (2*rs + 2)) @ Bk\<up>(rna)) t_wcode_main stp
-= (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
+= (0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk #
-Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]> ) +  (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+Bk # Oc\<up>(bl_bin (<aa # list @ [ab]> ) +  (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)) @ Bk\<up>(rn))"
 apply(rule_tac ind2, simp_all)
 done
 from this obtain stpb lnb rnb where stp2:
-"steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @  <ab # rev list @ [aa]> @ Bk # Bk # ires,
+"steps0 (Suc 0, Bk # Bk\<up>(lna) @  <ab # rev list @ [aa]> @ Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb
+Bk # Oc\<up>(Suc (2*rs + 2)) @ Bk\<up>(rna)) t_wcode_main stpb
-= (0, Bk # ires, Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk #
+= (0, Bk # ires, Bk # Oc # Bk\<up>(lnb) @ Bk #
-Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]> ) +  (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rnb\<^esup>)"
+Bk # Oc\<up>(bl_bin (<aa # list @ [ab]> ) +  (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)) @ Bk\<up>(rnb))"
 by blast
 from stp1 and stp2 show
 "\<exists>stp ln rn.
-steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+steps0 (Suc 0, Bk # Bk\<up>(m) @ Oc\<up>(Suc (Suc ab)) @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
-Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+Bk # Oc\<up>(Suc rs) @ Bk\<up>(n)) t_wcode_main stp =
-(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk #
+(0, Bk # ires, Bk # Oc # Bk\<up>(ln) @ Bk # Bk #
-Oc\<^bsup>bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [Suc ab]>) + rs * (2 * 2 ^ (aa + length (<list @ [Suc ab]>)))\<^esup>
+Oc\<up>(bl_bin (Oc\<up>(Suc aa) @ Bk # <list @ [Suc ab]>) + rs * (2 * 2 ^ (aa + length (<list @ [Suc ab]>))))
-@ Bk\<^bsup>rn\<^esup>)"
+@ Bk\<up>(rn))"
 apply(rule_tac x = "stpa + stpb" in exI, rule_tac x = lnb in exI,
-rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_cons_app1 exp_ind_def)
+rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_cons_app1 replicate_Suc)
 done
 qed
 qed
 qed
 qed
+definition t_wcode_prepare :: "instr list"
-(* 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 = <m # lm> @ Bk\<up>(rn) \<or>
-r = Bk # <m # lm> @ Bk\<^bsup>rn\<^esup>))"
+r = Bk # <m # lm> @ Bk\<up>(rn)))"
 fun wprepare_goto_first_end :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wprepare_goto_first_end m lm (l, r) =
-(\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> \<and>
+(\<exists> ml mr rn. l = Oc\<up>(ml) \<and>
-r = Oc\<^bsup>mr\<^esup> @ Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Oc\<up>(mr) @ Bk # <lm> @ Bk\<up>(rn) \<and>
 ml + mr = Suc (Suc m))"
 fun wprepare_erase :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow>  bool"
 where
 "wprepare_erase m lm (l, r) =
-(\<exists> rn. l = Oc\<^bsup>Suc m\<^esup> \<and>
+(\<exists> rn. l = Oc\<up>(Suc m) \<and>
-tl r = Bk # <lm> @ Bk\<^bsup>rn\<^esup>)"
+tl r = Bk # <lm> @ Bk\<up>(rn))"
 fun wprepare_goto_start_pos_B :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wprepare_goto_start_pos_B m lm (l, r) =
-(\<exists> rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+(\<exists> rn. l = Bk # Oc\<up>(Suc m) \<and>
-r = Bk # <lm> @ Bk\<^bsup>rn\<^esup>)"
+r = Bk # <lm> @ Bk\<up>(rn))"
 fun wprepare_goto_start_pos_O :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wprepare_goto_start_pos_O m lm (l, r) =
-(\<exists> rn. l = Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+(\<exists> rn. l = Bk # Bk # Oc\<up>(Suc m) \<and>
-r = <lm> @ Bk\<^bsup>rn\<^esup>)"
+r = <lm> @ Bk\<up>(rn))"
 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>
+(\<exists> rn mr. rev l @ r = Oc\<up>(Suc m) @ Bk # Bk # <lm> @ Bk\<up>(rn) \<and> l \<noteq> [] \<and>
-r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+r = Oc\<up>(mr) @ Bk\<up>(rn))"
 fun wprepare_loop_start_in_middle :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wprepare_loop_start_in_middle m lm (l, r) =
 (\<exists> rn (mr:: nat) (lm1::nat list).
-rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and>
+rev l @ r = Oc\<up>(Suc m) @ Bk # Bk # <lm> @ Bk\<up>(rn) \<and> l \<noteq> [] \<and>
-r = Oc\<^bsup>mr\<^esup> @ Bk # <lm1> @ Bk\<^bsup>rn\<^esup> \<and> lm1 \<noteq> [])"
+r = Oc\<up>(mr) @ Bk # <lm1> @ Bk\<up>(rn) \<and> lm1 \<noteq> [])"
 fun wprepare_loop_start :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
 where
 "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>
+(\<exists> rn. l = Bk # <rev lm> @ Bk # Bk # Oc\<up>(Suc m) \<and>
-r = Bk\<^bsup>rn\<^esup>)"
+r = Bk\<up>(rn))"
 fun wprepare_loop_goon_in_middle :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wprepare_loop_goon_in_middle m lm (l, r) =
 (\<exists> rn (mr:: nat) (lm1::nat list).
-rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and>
+rev l @ r = Oc\<up>(Suc m) @ Bk # Bk # <lm> @ Bk\<up>(rn) \<and> l \<noteq> [] \<and>
-(if lm1 = [] then r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>
+(if lm1 = [] then r = Oc\<up>(mr) @ Bk\<up>(rn)
-else r = Oc\<^bsup>mr\<^esup> @ Bk # <lm1> @ Bk\<^bsup>rn\<^esup>) \<and> mr > 0)"
+else r = Oc\<up>(mr) @ Bk # <lm1> @ Bk\<up>(rn)) \<and> mr > 0)"
 fun wprepare_loop_goon :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
 where
 "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>
+(\<exists> rn. l = Bk # Bk # <rev lm> @ Bk # Bk # Oc\<up>(Suc m) \<and>
-(r = [] \<or> tl r = Bk\<^bsup>rn\<^esup>))"
+(r = [] \<or> tl r = Bk\<up>(rn)))"
 fun wprepare_stop :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wprepare_stop m lm (l, r) =
-(\<exists> rn. l = Bk # <rev lm> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+(\<exists> rn. l = Bk # <rev lm> @ Bk # Bk # Oc\<up>(Suc m) \<and>
-r = Bk # Oc # Bk\<^bsup>rn\<^esup>)"
+r = Bk # Oc # Bk\<up>(rn))"
 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 = 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"
+fun wprepare_stage :: "config \<Rightarrow> nat"
 where
 "wprepare_stage (st, l, r) =
 (if st \<ge> 1 \<and> st \<le> 4 then 3
 else if st = 5 \<or> st = 6 then 2
 else 1)"
-fun wprepare_state :: "t_conf \<Rightarrow> nat"
+fun wprepare_state :: "config \<Rightarrow> nat"
 where
 "wprepare_state (st, l, r) =
 (if st = 1 then 4
 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"
+fun wprepare_step :: "config \<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 = 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"
+fun wcode_prepare_measure :: "config \<Rightarrow> nat \<times> nat \<times> nat"
 where
 "wcode_prepare_measure (st, l, r) =
 (wprepare_stage (st, l, r),
 wprepare_state (st, l, r),
 wprepare_step (st, l, r))"
-definition wcode_prepare_le :: "(t_conf \<times> t_conf) set"
+definition wcode_prepare_le :: "(config \<times> config) set"
 where "wcode_prepare_le \<equiv> (inv_image lex_triple wcode_prepare_measure)"
 lemma [intro]: "wf lex_pair"
 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)
+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]
 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)
+apply(subgoal_tac "4 = Suc 3")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
 done
 lemma [simp]: "fetch t_wcode_prepare 4 Oc = (R, 5)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(subgoal_tac "4 = Suc 3")
-done
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
+done
 lemma [simp]: "fetch t_wcode_prepare 5 Oc = (R, 5)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(subgoal_tac "5 = Suc 4")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
 done
 lemma [simp]: "fetch t_wcode_prepare 5 Bk = (R, 6)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(subgoal_tac "5 = Suc 4")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
 done
 lemma [simp]: "fetch t_wcode_prepare 6 Oc = (R, 5)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(subgoal_tac "6 = Suc 5")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
 done
 lemma [simp]: "fetch t_wcode_prepare 6 Bk = (R, 7)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(subgoal_tac "6 = Suc 5")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
 done
 lemma [simp]: "fetch t_wcode_prepare 7 Oc = (L, 0)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(subgoal_tac "7 = Suc 6")
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
 done
 lemma [simp]: "fetch t_wcode_prepare 7 Bk = (W1, 7)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(subgoal_tac "7 = Suc 6")
-done
+apply(simp_all only: fetch.simps t_wcode_prepare_def nth_of.simps)
+apply(auto)
-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)
+apply(simp add: wprepare_invs)
 done
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [])\<rbrakk> \<Longrightarrow> b \<noteq> []"
-apply(simp add: wprepare_invs tape_of_nl_not_null, auto)
+apply(simp add: wprepare_invs)
 done
+lemma rev_eq: "rev xs = rev ys \<Longrightarrow> xs = ys"
+by auto
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [])\<rbrakk> \<Longrightarrow>
 wprepare_loop_goon m lm (Bk # b, [])"
-apply(simp only: wprepare_invs tape_of_nl_not_null)
+apply(simp only: wprepare_invs)
 apply(erule_tac disjE)
 apply(rule_tac disjI2)
 apply(simp add: wprepare_loop_start_on_rightmost.simps
 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)
+apply(simp only: wprepare_invs, auto)
 done
 lemma [simp]:"\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, [])\<rbrakk> \<Longrightarrow>
 wprepare_add_one2 m lm (Bk # b, [])"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto split: if_splits)
+apply(simp only: wprepare_invs, 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)
+apply(simp only: wprepare_invs, auto)
 done
 lemma [simp]: "wprepare_add_one2 m lm (b, []) \<Longrightarrow> wprepare_add_one2 m lm (b, [Oc])"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
+apply(simp only: wprepare_invs, auto)
 done
 lemma [simp]: "Bk # list = <(m::nat) # lm> @ ys = False"
-apply(case_tac lm, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+apply(case_tac lm, auto simp: tape_of_nl_cons replicate_Suc)
 done
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_add_one m lm (b, Bk # list)\<rbrakk>
 \<Longrightarrow> (b = [] \<longrightarrow> wprepare_goto_first_end m lm ([], Oc # list)) \<and>
 (b \<noteq> [] \<longrightarrow> wprepare_goto_first_end m lm (b, Oc # list))"
-apply(simp only: wprepare_invs, auto)
+apply(simp only: wprepare_invs)
-apply(rule_tac x = 0 in exI, simp add: exp_ind_def)
+apply(auto simp: tape_of_nl_cons split: if_splits)
-apply(case_tac lm, simp, simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+apply(rule_tac x = 0 in exI, simp add: replicate_Suc)
-apply(rule_tac x = rn in exI, simp)
+apply(case_tac lm, simp, simp add: tape_of_nl_abv tape_of_nat_list.simps replicate_Suc)
 done
 lemma [simp]: "wprepare_goto_first_end m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
+apply(simp only: wprepare_invs , auto simp: replicate_Suc)
-apply(case_tac mr, simp_all add: exp_ind_def)
+done
-done
+declare replicate_Suc[simp]
 lemma [simp]: "wprepare_goto_first_end m lm (b, Bk # list) \<Longrightarrow>
 wprepare_erase m lm (tl b, hd b # Bk # list)"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
+apply(simp only: wprepare_invs, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all)
-apply(case_tac mr, auto simp: exp_ind_def)
+apply(case_tac mr, auto)
 done
 lemma [simp]: "wprepare_erase m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs exp_ind_def, auto)
+apply(simp only: wprepare_invs, auto)
 done
 lemma [simp]: "wprepare_erase m lm (b, Bk # list) \<Longrightarrow>
 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)
+tape_of_nat_list.simps, auto)
 done
 lemma [simp]: "\<lbrakk>lm \<noteq> [];  wprepare_goto_first_end m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
 apply(simp only: wprepare_invs, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all)
-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)
+apply(simp only: wprepare_invs, auto)
 done
 lemma [simp]: "\<lbrakk>lm \<noteq> [];  wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
 apply(simp only: wprepare_invs, auto)
-apply(simp add: tape_of_nl_not_null)
 apply(case_tac lm, simp, case_tac list)
-apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps)
 done
 lemma [simp]: "\<lbrakk>lm \<noteq> [];  wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
 apply(simp only: wprepare_invs)
 apply(auto)
 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 [1-3] mr, simp_all add: )
-apply(case_tac mr, simp_all add: exp_ind_def tape_of_nl_not_null)
+apply(case_tac mr, simp_all)
-apply(case_tac [1-2] mr, simp_all add: exp_ind_def)
+apply(case_tac [1-2] mr, simp_all add: )
-apply(case_tac rn, simp, case_tac nat, auto simp: exp_ind_def)
+apply(case_tac rn, simp, case_tac nat, auto simp: )
 done
 lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
 apply(simp only: wprepare_invs, simp)
 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 mr, simp_all add: )
-apply(case_tac ml, simp_all add: exp_ind_def)
+apply(case_tac ml, simp_all add: )
-apply(rule_tac x = rn in exI, simp)
+apply(rule_tac x = "Suc ml" in exI, simp_all add: )
-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)
+apply(simp only: wprepare_invs, auto simp: )
 done
 lemma [simp]: "wprepare_erase m lm (b, Oc # list)
 \<Longrightarrow> wprepare_erase m lm (b, Bk # list)"
-apply(simp  only:wprepare_invs, auto simp: exp_ind_def)
+apply(simp  only:wprepare_invs, auto simp: )
 done
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk>
 \<Longrightarrow> wprepare_goto_start_pos m lm (Bk # b, list)"
 apply(simp only:wprepare_invs, auto)
 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>"
+lemma [elim]: "Bk # list = Oc\<up>(mr) @ Bk\<up>(rn)  \<Longrightarrow> \<exists>rn. list = Bk\<up>(rn)"
 apply(case_tac mr, simp_all)
-apply(case_tac rn, simp_all add: exp_ind_def, auto)
+apply(case_tac rn, simp_all)
 done
 lemma rev_equal_iff: "x = y \<Longrightarrow> rev x = rev y"
 by simp
 lemma tape_of_nl_false1:
-"lm \<noteq> [] \<Longrightarrow> rev b @ [Bk] \<noteq> Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>m\<^esup> @ Bk # Bk # <lm::nat list>"
+"lm \<noteq> [] \<Longrightarrow> rev b @ [Bk] \<noteq> Bk\<up>(ln) @ Oc # Oc\<up>(m) @ Bk # Bk # <lm::nat list>"
 apply(auto)
 apply(drule_tac rev_equal_iff, simp add: tape_of_nl_rev)
 apply(case_tac "rev lm")
-apply(case_tac [2] list, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+apply(case_tac [2] list, auto simp: tape_of_nl_abv tape_of_nat_list.simps )
 done
 lemma [simp]: "wprepare_loop_start_in_middle m lm (b, [Bk]) = False"
 apply(simp add: wprepare_loop_start_in_middle.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
-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]
 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(case_tac mr, simp_all add: )
 apply(rule_tac rev_eq)
 apply(simp add: tape_of_nl_rev)
-apply(simp add: exp_ind_def[THEN sym] exp_ind)
+apply(simp add: exp_ind replicate_Suc[THEN sym] del: replicate_Suc)
 done
 lemma [simp]: "wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista)
 \<Longrightarrow> wprepare_loop_goon_in_middle m lm (Bk # b, a # lista) = False"
 apply(auto simp: wprepare_loop_start_on_rightmost.simps
 wprepare_loop_goon_in_middle.simps)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
+apply(case_tac [!] mr, simp_all)
 done
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista)\<rbrakk>
 \<Longrightarrow> wprepare_loop_goon_on_rightmost m lm (Bk # b, a # lista)"
 apply(simp only: wprepare_loop_start_on_rightmost.simps
 wprepare_loop_goon_on_rightmost.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 apply(simp add: tape_of_nl_rev)
-apply(simp add: exp_ind_def[THEN sym] exp_ind)
+apply(simp add: replicate_Suc[THEN sym] exp_ind del: replicate_Suc)
 done
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\<rbrakk>
 \<Longrightarrow> wprepare_loop_goon_on_rightmost m lm (Bk # b, a # lista) = False"
 apply(simp add: wprepare_loop_start_in_middle.simps
 wprepare_loop_goon_on_rightmost.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 apply(case_tac  "lm1::nat list", simp_all, case_tac  list, simp)
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps tape_of_nat_abv exp_ind_def)
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps tape_of_nat_abv )
-apply(case_tac [!] rna, simp_all add: exp_ind_def)
+apply(case_tac [!] rna, simp_all add: )
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 apply(case_tac lm1, simp, case_tac list, simp)
-apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def tape_of_nat_abv)
+apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps  tape_of_nat_abv)
 done
 lemma [simp]:
 "\<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 mr, simp_all add: )
 apply(case_tac lm1, simp)
 apply(rule_tac x = "Suc aa" in exI, simp)
 apply(rule_tac x = list in exI)
 apply(case_tac list, simp_all add: tape_of_nl_abv tape_of_nat_list.simps)
 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 mr, simp_all add: )
-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(case_tac mr, simp, simp add: )
 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>
 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>;
+lemma wprepare_loop1: "\<lbrakk>rev b @ Oc\<up>(mr) =  Oc\<up>(Suc m) @ Bk # Bk # <lm>;
-b \<noteq> []; 0 < mr; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>\<rbrakk>
+b \<noteq> []; 0 < mr; Oc # list = Oc\<up>(mr) @ Bk\<up>(rn)\<rbrakk>
 \<Longrightarrow> wprepare_loop_start_on_rightmost m lm (Oc # b, list)"
 apply(simp add: wprepare_loop_start_on_rightmost.simps)
 apply(rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp, simp add: exp_ind_def, auto)
+apply(case_tac mr, simp, simp)
 done
-lemma wprepare_loop2: "\<lbrakk>rev b @ Oc\<^bsup>mr\<^esup> @ Bk # <a # lista> = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm>;
+lemma wprepare_loop2: "\<lbrakk>rev b @ Oc\<up>(mr) @ Bk # <a # lista> = Oc\<up>(Suc m) @ Bk # Bk # <lm>;
-b \<noteq> []; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk # <(a::nat) # lista> @ Bk\<^bsup>rn\<^esup>\<rbrakk>
+b \<noteq> []; Oc # list = Oc\<up>(mr) @ Bk # <(a::nat) # lista> @ Bk\<up>(rn)\<rbrakk>
 \<Longrightarrow>  wprepare_loop_start_in_middle m lm (Oc # b, list)"
 apply(simp add: wprepare_loop_start_in_middle.simps)
 apply(rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 apply(rule_tac x = nat in exI, simp)
 apply(rule_tac x = "a#lista" in exI, simp)
 done
 lemma [simp]: "wprepare_loop_goon_in_middle m lm (b, Oc # list) \<Longrightarrow>
 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)
+apply(simp add: replicate_Suc[THEN sym] exp_ind del: replicate_Suc)
 done
 lemma [simp]:  "wprepare_goto_start_pos m (a # aa # listaa) (b, Oc # list)
 \<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(simp add: exp_ind[THEN sym])
 apply(rule_tac x = a in exI, rule_tac x = "aa#listaa" in exI, simp)
+apply(simp add: tape_of_nl_cons)
 done
 lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Oc # list)\<rbrakk>
 \<Longrightarrow> wprepare_loop_start m lm (Oc # b, list)"
 apply(case_tac lm, simp_all)
 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
+let f = (\<lambda> stp. steps0 (Suc 0, [], (<m # lm>)) t_wcode_prepare stp) in
 \<exists> n .P (f n) \<and> Q (f n)"
 proof -
 let ?P = "(\<lambda> (st, l, r). st = 0)"
 let ?Q = "(\<lambda> (st, l, r). wprepare_inv st m lm (l, r))"
-let ?f = "(\<lambda> stp. steps (Suc 0, [], (<m # lm>)) t_wcode_prepare stp)"
+let ?f = "(\<lambda> stp. steps0 (Suc 0, [], (<m # lm>)) t_wcode_prepare stp)"
 have "\<exists> n. ?P (?f n) \<and> ?Q (?f n)"
 proof(rule_tac halt_lemma2)
 show "wf wcode_prepare_le" by auto
 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)
+simp add: step_red step.simps)
 apply(case_tac c, simp, case_tac [2] aa)
-apply(simp_all add: wprepare_inv.simps wcode_prepare_le_def new_tape.simps
+apply(simp_all add: wprepare_inv.simps wcode_prepare_le_def lex_triple_def lex_pair_def
-lex_triple_def lex_pair_def
 split: if_splits)
 apply(simp_all add: start_2_goon  start_2_start
 add_one_2_add_one add_one_2_stop)
 apply(auto simp: wprepare_add_one2.simps)
 done
 thus "?thesis"
 apply(auto)
 done
 qed
-lemma [intro]: "t_correct t_wcode_prepare"
+lemma [intro]: "tm_wf (t_wcode_prepare, 0)"
-apply(simp add: t_correct.simps t_wcode_prepare_def iseven_def)
+apply(simp add:tm_wf.simps t_wcode_prepare_def)
-apply(rule_tac x = 7 in exI, simp)
+done
-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) "
+list_all (\<lambda>(acn, st). (st \<le> y + off)) (shift tp off) "
-apply(auto simp: t_correct.simps List.list_all_length)
+apply(auto simp: List.list_all_length)
-apply(erule_tac x = n in allE, simp add: shift_length)
+apply(erule_tac x = n in allE, simp add: length_shift)
-apply(case_tac "tp!n", auto simp: tshift.simps)
+apply(case_tac "tp!n", auto simp: shift.simps)
 done
+(*
 lemma [intro]:
 "t_correct (tm_of abc_twice @ tMp (Suc 0)
 (start_of twice_ly (length abc_twice) - Suc 0))"
 apply(rule_tac t_compiled_correct, simp_all)
 apply(simp add: twice_ly_def)
 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"
+lemma [intro]: "(28 + (length t_twice_compile + length t_fourtimes_compile)) mod 2 = 0"
-apply(auto simp: t_wcode_main_def t_correct.simps shift_length
+apply(auto simp: t_twice_compile_def t_fourtimes_compile_def)
-t_twice_def t_fourtimes_def)
+by arith
+lemma [elim]: "(a, b) \<in> set t_wcode_main_first_part \<Longrightarrow>
+b \<le> (28 + (length t_twice_compile + length t_fourtimes_compile)) div 2"
+apply(auto simp: t_wcode_main_first_part_def t_twice_def)
+done
+lemma tm_wf_change_termi: "tm_wf (tp, 0) \<Longrightarrow>
+list_all (\<lambda>(acn, st). (st \<le> Suc (length tp div 2))) (adjust tp)"
+apply(auto simp: tm_wf.simps List.list_all_length)
+apply(case_tac "tp!n", auto simp: adjust.simps split: if_splits)
+apply(erule_tac x = "(a, b)" in ballE, auto)
+by (metis in_set_conv_nth)
+lemma tm_wf_shift:
+"list_all (\<lambda>(acn, st). (st \<le> y)) tp \<Longrightarrow>
+list_all (\<lambda>(acn, st). (st \<le> y + off)) (shift tp off) "
+apply(auto simp: tm_wf.simps List.list_all_length)
+apply(erule_tac x = n in allE, simp add: length_shift)
+apply(case_tac "tp!n", auto simp: shift.simps)
+done
+declare length_tp'[simp del]
+lemma [simp]: "length (mopup (Suc 0)) = 16"
+apply(auto simp: mopup.simps)
+done
+lemma [elim]: "(a, b) \<in> set (shift (turing_basic.adjust t_twice_compile) 12) \<Longrightarrow>
+b \<le> (28 + (length t_twice_compile + length t_fourtimes_compile)) div 2"
+apply(simp add: t_twice_compile_def t_fourtimes_compile_def)
 proof -
-show "iseven (60 + (length (tm_of abc_twice) +
+assume g: "(a, b) \<in> set (shift (turing_basic.adjust (tm_of abc_twice @ shift (mopup (Suc 0)) (length (tm_of abc_twice) div 2))) 12)"
-length (tm_of abc_fourtimes)))"
+moreover have "length (tm_of abc_twice) mod 2 = 0" by auto
-using twice_len_even fourtimes_len_even
+moreover have "length (tm_of abc_fourtimes) mod 2 = 0" by auto
-apply(auto simp: iseven_def)
+ultimately have "list_all (\<lambda>(acn, st). (st \<le> (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2))
-apply(rule_tac x = "30 + q + qa" in exI, simp)
+(shift (turing_basic.adjust t_twice_compile) 12)"
+proof(auto simp: mod_ex1)
+fix q qa
+assume h: "length (tm_of abc_twice) = 2 * q" "length (tm_of abc_fourtimes) = 2 * qa"
+hence "list_all (\<lambda>(acn, st). st \<le> (18 + (q + qa)) + 12) (shift (turing_basic.adjust t_twice_compile) 12)"
+proof(rule_tac tm_wf_shift t_twice_compile_def)
+have "list_all (\<lambda>(acn, st). st \<le> Suc (length t_twice_compile div 2)) (adjust t_twice_compile)"
+by(rule_tac tm_wf_change_termi, auto)
+thus "list_all (\<lambda>(acn, st). st \<le> 18 + (q + qa)) (turing_basic.adjust t_twice_compile)"
+using h
+apply(simp add: t_twice_compile_def, auto simp: List.list_all_length)
+done
+qed
+thus "list_all (\<lambda>(acn, st). st \<le> 30 + (q + qa)) (shift (turing_basic.adjust t_twice_compile) 12)"
+by simp
+qed
+thus "b \<le> (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2"
+using g
+apply(auto simp:t_twice_compile_def)
+apply(simp add: Ball_set[THEN sym])
+apply(erule_tac x = "(a, b)" in ballE, simp, simp)
 done
-next
+qed
-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"
+lemma [elim]: "(a, b) \<in> set (shift (turing_basic.adjust t_fourtimes_compile) (t_twice_len + 13))
-apply(auto simp: t_wcode_main_first_part_def shift_length t_twice_def)
+\<Longrightarrow> b \<le> (28 + (length t_twice_compile + length t_fourtimes_compile)) div 2"
-done
+apply(simp add: t_twice_compile_def t_fourtimes_compile_def t_twice_len_def)
-next
+proof -
-have "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_twice @ tMp (Suc 0)
+assume g: "(a, b) \<in> set (shift (turing_basic.adjust (tm_of abc_fourtimes @ shift (mopup (Suc 0))
-(start_of twice_ly (length abc_twice) - Suc 0)) div 2))
+(length (tm_of abc_fourtimes) div 2))) (length t_twice div 2 + 13))"
-(change_termi_state (tm_of abc_twice @ tMp (Suc 0)
+moreover have "length (tm_of abc_twice) mod 2 = 0" by auto
-(start_of twice_ly (length abc_twice) - Suc 0)))"
+moreover have "length (tm_of abc_fourtimes) mod 2 = 0" by auto
-apply(rule_tac t_correct_termi, auto)
+ultimately have "list_all (\<lambda>(acn, st). (st \<le> (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2))
-done
+(shift (turing_basic.adjust (tm_of abc_fourtimes @ shift (mopup (Suc 0))
-hence "list_all (\<lambda>(acn, s). s \<le>  Suc (length (tm_of abc_twice @ tMp (Suc 0)
+(length (tm_of abc_fourtimes) div 2))) (length t_twice div 2 + 13))"
-(start_of twice_ly (length abc_twice) - Suc 0)) div 2) + 12)
+proof(auto simp: mod_ex1 t_twice_def t_twice_compile_def)
-(tshift (change_termi_state (tm_of abc_twice @ tMp (Suc 0)
+fix q qa
-(start_of twice_ly (length abc_twice) - Suc 0))) 12)"
+assume h: "length (tm_of abc_twice) = 2 * q" "length (tm_of abc_fourtimes) = 2 * qa"
-apply(rule_tac t_correct_shift, simp)
+hence "list_all (\<lambda>(acn, st). st \<le> (9 + qa + (21 + q)))
-done
+(shift (turing_basic.adjust (tm_of abc_fourtimes @ shift (mopup (Suc 0)) qa)) (21 + q))"
-thus  "list_all (\<lambda>(acn, s). s \<le>
+proof(rule_tac tm_wf_shift t_twice_compile_def)
-(60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2)
+have "list_all (\<lambda>(acn, st). st \<le> Suc (length (tm_of abc_fourtimes @ shift
-(tshift (change_termi_state (tm_of abc_twice @ tMp (Suc 0)
+(mopup (Suc 0)) qa) div 2)) (adjust (tm_of abc_fourtimes @ shift (mopup (Suc 0)) qa))"
-(start_of twice_ly (length abc_twice) - Suc 0))) 12)"
+apply(rule_tac tm_wf_change_termi)
-apply(simp)
+using wf_fourtimes h
-apply(simp add: list_all_length, auto)
+apply(simp add: t_fourtimes_compile_def)
 done
-next
+thus "list_all (\<lambda>(acn, st). st \<le> 9 + qa) ((turing_basic.adjust (tm_of abc_fourtimes @ shift (mopup (Suc 0)) qa)))"
-have "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_fourtimes @ tMp (Suc 0)
+using h
-(start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) div 2))
+apply(simp)
-(change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0)
+done
-(start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) "
+qed
-apply(rule_tac t_correct_termi, auto)
+thus "list_all (\<lambda>(acn, st). st \<le> 30 + (q + qa)) (shift (turing_basic.adjust (tm_of abc_fourtimes @ shift (mopup (Suc 0)) qa)) (21 + q))"
-done
+apply(subgoal_tac "qa + q = q + qa")
-hence "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_fourtimes @ tMp (Suc 0)
+apply(simp, simp)
-(start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) div 2) + (t_twice_len + 13))
+done
-(tshift (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0)
+qed
-(start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) (t_twice_len + 13))"
+thus "b \<le> (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2"
-apply(rule_tac t_correct_shift, simp)
+using g
-done
+apply(simp add: Ball_set[THEN sym])
-thus "list_all (\<lambda>(acn, s). s \<le> (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2)
+apply(erule_tac x = "(a, b)" in ballE, simp, simp)
-(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)"
+lemma [intro]: "tm_wf (t_wcode_main, 0)"
-apply(auto intro: t_correct_add)
+apply(auto simp: t_wcode_main_def tm_wf.simps
+t_twice_def t_fourtimes_def del: List.list_all_iff)
+done
+declare tm_comp.simps[simp del]
+lemma tm_wf_comp: "\<lbrakk>tm_wf (A, 0); tm_wf (B, 0)\<rbrakk> \<Longrightarrow> tm_wf (A |+| B, 0)"
+apply(auto simp: tm_wf.simps shift.simps adjust.simps tm_comp_length
+tm_comp.simps)
+done
+lemma [intro]: "tm_wf (t_wcode_prepare |+| t_wcode_main, 0)"
+apply(rule_tac tm_wf_comp, auto)
 done
 lemma prepare_mainpart_lemma:
 "args \<noteq> [] \<Longrightarrow>
-\<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp
+\<exists> stp ln rn. steps0 (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>)"
+= (0,  Bk # Oc\<up>(Suc m), Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin (<args>)) @ Bk\<up>(rn))"
 proof -
-let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <m # args>"
+let ?P1 = "(\<lambda> (l, r). (l::cell list) = [] \<and> r = <m # args>)"
-let ?Q1 = "\<lambda> (l, r). wprepare_stop m args (l, r)"
+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>
+let ?Q2 = "(\<lambda> (l, r). (\<exists> ln rn. l = Bk # Oc\<up>(Suc m) \<and>
-r =  Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+r =  Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin (<args>)) @ Bk\<up>(rn)))"
 let ?P3 = "\<lambda> tp. False"
 assume h: "args \<noteq> []"
-have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp)
+have "{?P1} t_wcode_prepare |+| t_wcode_main {?Q2}"
-(t_wcode_prepare |+| t_wcode_main) stp = (0, tp') \<and> ?Q2 tp')"
+proof(rule_tac Hoare_plus_halt)
-proof(rule_tac turing_merge.t_merge_halt[of t_wcode_prepare t_wcode_main ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2],
+show "?Q1 \<mapsto> ?P2"
-auto simp: turing_merge_def)
+by(simp add: assert_imp_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
+show "tm_wf (t_wcode_prepare, 0)"
-assume "wprepare_stop m args (a, b)"
+by auto
-thus "\<exists>stp. case steps (Suc 0, a, b) t_wcode_main stp of
+next
-(st, tp') \<Rightarrow> (st = 0) \<and> (case tp' of (l, r) \<Rightarrow> l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+show "{?P1} t_wcode_prepare {?Q1}"
-(\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>))"
+proof(rule_tac HoareI, auto)
-proof(simp only: wprepare_stop.simps, erule_tac exE)
+show "\<exists>n. is_final (steps0 (Suc 0, [], <m # args>) t_wcode_prepare n) \<and>
+wprepare_stop m args holds_for steps0 (Suc 0, [], <m # args>) t_wcode_prepare n"
+using wprepare_correctness[of args m] h
+apply(auto)
+apply(rule_tac x = n in exI, simp add: wprepare_inv.simps)
+done
+qed
+next
+show "{?P2} t_wcode_main {?Q2}"
+proof(rule_tac HoareI, auto)
+fix l r
+assume "wprepare_stop m args (l, r)"
+thus "\<exists>n. is_final (steps0 (Suc 0, l, r) t_wcode_main n) \<and>
+(\<lambda>(l, r). l = Bk # Oc # Oc \<up> m \<and> (\<exists>ln rn. r = Bk # Oc # Bk \<up> ln @
+Bk # Bk # Oc \<up> bl_bin (<args>) @ Bk \<up> rn)) holds_for steps0 (Suc 0, l, r) t_wcode_main n"
+proof(auto simp: wprepare_stop.simps)
 fix rn
-assume "a = Bk # <rev args> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+show " \<exists>n. is_final (steps0 (Suc 0, Bk # <rev args> @ Bk # Bk # Oc # Oc \<up> m, Bk # Oc # Bk \<up> rn) t_wcode_main n) \<and>
-b = Bk # Oc # Bk\<^bsup>rn\<^esup>"
+(\<lambda>(l, r). l = Bk # Oc # Oc \<up> m \<and>
-thus "?thesis"
+(\<exists>ln rn. r = Bk # Oc # Bk \<up> ln @
-using t_wcode_main_lemma_pre[of "args" "<args>" 0 "Oc\<^bsup>Suc m\<^esup>" 0 rn] h
+Bk # Bk # Oc \<up> bl_bin (<args>) @
-apply(simp)
+Bk \<up> rn)) holds_for steps0 (Suc 0, Bk # <rev args> @ Bk # Bk # Oc # Oc \<up> m, Bk # Oc # Bk \<up> rn) t_wcode_main n"
-apply(erule_tac exE)+
+using t_wcode_main_lemma_pre[of "args" "<args>" 0 "Oc\<up>(Suc m)" 0 rn] h
-apply(rule_tac x = stp in exI, simp add: tape_of_nl_rev, auto)
+apply(auto simp: tape_of_nl_rev)
+apply(rule_tac x = stp in exI, auto)
 done
 qed
-next
+qed
-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
+thus "?thesis"
-= (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(auto simp: Hoare_def)
-apply(simp add: t_imply_def)
+apply(rule_tac x = n in exI)
-apply(erule_tac exE)+
+apply(case_tac "(steps0 (Suc 0, [], <m # args>)
-apply(auto)
+(turing_basic.adjust t_wcode_prepare @ shift t_wcode_main (length t_wcode_prepare div 2)) n)")
+apply(auto simp: tm_comp.simps)
 done
 qed
 lemma [simp]:  "tinres r r' \<Longrightarrow>
-fetch t ss (case r of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) =
+fetch t ss (read r) =
-fetch t ss (case r' of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)"
+fetch t ss (read r')"
 apply(simp add: fetch.simps, auto split: if_splits simp: tinres_def)
-apply(case_tac [!] r', simp_all)
+apply(case_tac [!] n, simp_all)
-apply(case_tac [!] n, simp_all add: exp_ind_def)
+done
-apply(case_tac [!] r, simp_all)
-done
+lemma [intro]: "\<exists> n. (a::cell)\<up>(n) = []"
-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"
+lemma [intro]: "hd (Bk\<up>(Suc n)) = Bk"
-apply(simp add: exp_ind_def)
+apply(simp add: )
 done
 lemma [simp]: "\<lbrakk>tinres r []; r \<noteq> []\<rbrakk> \<Longrightarrow> hd r = Bk"
 apply(auto simp: tinres_def)
-apply(case_tac n, auto)
 done
 lemma [simp]: "\<lbrakk>tinres [] r'; r' \<noteq> []\<rbrakk> \<Longrightarrow> hd r' = Bk"
 apply(auto simp: tinres_def)
 done
-lemma [intro]: "\<exists>na. tl r = tl (r @ Bk\<^bsup>n\<^esup>) @ Bk\<^bsup>na\<^esup> \<or> tl (r @ Bk\<^bsup>n\<^esup>) = tl r @ Bk\<^bsup>na\<^esup>"
+lemma [intro]: "\<exists>na. tl r = tl (r @ Bk\<up>(n)) @ Bk\<up>(na) \<or> tl (r @ Bk\<up>(n)) = tl r @ Bk\<up>(na)"
 apply(case_tac r, simp)
-apply(case_tac n, simp)
+apply(case_tac n, simp, simp)
-apply(rule_tac x = 0 in exI, simp)
+apply(rule_tac x = nat 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 r', simp)
-apply(case_tac n, simp_all add: exp_ind_def)
+apply(case_tac n, simp_all)
-apply(rule_tac x = 0 in exI, simp)
+apply(rule_tac x = nat 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(case_tac n, simp_all add: )
 apply(rule_tac x = nat in exI, simp)
 done
 lemma [simp]: "\<lbrakk>tinres [] r'\<rbrakk> \<Longrightarrow> tinres [] (tl r')"
 apply(case_tac r', auto simp: tinres_def)
-apply(case_tac n, simp_all add: exp_ind_def)
+apply(case_tac n, simp_all add: )
 apply(rule_tac x = nat in exI, simp)
 done
 lemma [simp]: "tinres r r' \<Longrightarrow> tinres (b # r) (b # r')"
 apply(auto simp: tinres_def)
 done
+lemma [simp]: "tinres r [] \<Longrightarrow> tinres (Bk # tl r) [Bk]"
+apply(auto simp: tinres_def)
+done
+lemma [simp]: "tinres r [] \<Longrightarrow> tinres (Oc # tl r) [Oc]"
+apply(auto simp: tinres_def)
+done
 lemma tinres_step2:
-"\<lbrakk>tinres r r'; tstep (ss, l, r) t = (sa, la, ra); tstep (ss, l, r') t = (sb, lb, rb)\<rbrakk>
+"\<lbrakk>tinres r r'; step0 (ss, l, r) t = (sa, la, ra); step0 (ss, l, r') t = (sb, lb, rb)\<rbrakk>
 \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb"
-apply(case_tac "ss = 0", simp add: tstep_0)
+apply(case_tac "ss = 0", simp add: step_0)
-apply(simp add: tstep.simps [simp del])
+apply(simp add: step.simps [simp del], auto)
-apply(case_tac "fetch t ss (case r of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
+apply(case_tac [!] "fetch t ss (read r')", simp)
-apply(auto simp: new_tape.simps)
+apply(auto simp: update.simps)
-apply(simp_all split: taction.splits if_splits)
+apply(case_tac [!] a, auto split: if_splits)
-apply(auto)
+done
-done
 lemma tinres_steps2:
-"\<lbrakk>tinres r r'; steps (ss, l, r) t stp = (sa, la, ra); steps (ss, l, r') t stp = (sb, lb, rb)\<rbrakk>
+"\<lbrakk>tinres r r'; steps0 (ss, l, r) t stp = (sa, la, ra); steps0 (ss, l, r') t stp = (sb, lb, rb)\<rbrakk>
 \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb"
 apply(induct stp arbitrary: sa la ra sb lb rb, simp add: steps.simps)
-apply(simp add: tstep_red)
+apply(simp add: step_red)
-apply(case_tac "(steps (ss, l, r) t stp)")
+apply(case_tac "(steps0 (ss, l, r) t stp)")
-apply(case_tac "(steps (ss, l, r') t stp)")
+apply(case_tac "(steps0 (ss, l, r') t stp)")
 proof -
 fix stp sa la ra sb lb rb a b c aa ba ca
-assume ind: "\<And>sa la ra sb lb rb. \<lbrakk>steps (ss, l, r) t stp = (sa, la, ra);
+assume ind: "\<And>sa la ra sb lb rb. \<lbrakk>steps0 (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"
+steps0 (ss, l, r') t stp = (sb, lb, rb)\<rbrakk> \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb"
-and h: " tinres r r'" "tstep (steps (ss, l, r) t stp) t = (sa, la, ra)"
+and h: " tinres r r'" "step0 (steps0 (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)"
+"step0 (steps0 (ss, l, r') t stp) t = (sb, lb, rb)" "steps0 (ss, l, r) t stp = (a, b, c)"
-"steps (ss, l, r') t stp = (aa, ba, ca)"
+"steps0 (ss, l, r') t stp = (aa, ba, ca)"
 have "b = ba \<and> tinres c ca \<and> a = aa"
 apply(rule_tac ind, simp_all add: h)
 done
 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
-definition t_wcode_adjust :: "tprog"
+definition t_wcode_adjust :: "instr list"
 where
 "t_wcode_adjust = [(W1, 1), (R, 2), (Nop, 2), (R, 3), (R, 3), (R, 4),
 (L, 8), (L, 5), (L, 6), (W0, 5), (L, 6), (R, 7),
 (W1, 2), (Nop, 7), (L, 9), (W0, 8), (L, 9), (L, 10),
 (L, 11), (L, 10), (R, 0), (L, 11)]"
 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)"
+lemma [simp]: "fetch t_wcode_adjust (Suc (Suc (Suc (Suc 0)))) Bk = (L, 8)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_4_eq_4)
 done
-lemma [simp]: "fetch t_wcode_adjust 4 Oc = (L, 5)"
+lemma [simp]: "fetch t_wcode_adjust (Suc (Suc (Suc (Suc 0)))) Oc = (L, 5)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_4_eq_4)
 done
+thm numeral_5_eq_5
 lemma [simp]: "fetch t_wcode_adjust 5 Oc = (W0, 5)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp only: fetch.simps t_wcode_adjust_def nth_of.simps numeral_5_eq_5, simp)
 done
 lemma [simp]: "fetch t_wcode_adjust 5 Bk = (L, 6)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp only: fetch.simps t_wcode_adjust_def nth_of.simps numeral_5_eq_5, auto)
 done
+thm numeral_6_eq_6
 lemma [simp]: "fetch t_wcode_adjust 6 Oc = (R, 7)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_6_eq_6)
 done
 lemma [simp]: "fetch t_wcode_adjust 6 Bk = (L, 6)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_6_eq_6)
 done
 lemma [simp]: "fetch t_wcode_adjust 7 Bk = (W1, 2)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_7_eq_7)
 done
 lemma [simp]: "fetch t_wcode_adjust 8 Bk = (L, 9)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_8_eq_8)
 done
 lemma [simp]: "fetch t_wcode_adjust 8 Oc = (W0, 8)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_8_eq_8)
 done
 lemma [simp]: "fetch t_wcode_adjust 9 Oc = (L, 10)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_9_eq_9)
 done
 lemma [simp]: "fetch t_wcode_adjust 9 Bk = (L, 9)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps numeral_9_eq_9)
 done
 lemma [simp]: "fetch t_wcode_adjust 10 Bk = (L, 11)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps  numeral_10_eq_10)
 done
 lemma [simp]: "fetch t_wcode_adjust 10 Oc = (L, 10)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps eval_nat_numeral)
 done
 lemma [simp]: "fetch t_wcode_adjust 11 Oc = (L, 11)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps eval_nat_numeral)
 done
 lemma [simp]: "fetch t_wcode_adjust 11 Bk = (R, 0)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps eval_nat_numeral)
 done
 fun wadjust_start :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_start m rs (l, r) =
-(\<exists> ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+(\<exists> ln rn. l = Bk # Oc\<up>(Suc m) \<and>
-tl r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+tl r = Oc # Bk\<up>(ln) @ Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn))"
 fun wadjust_loop_start :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_loop_start m rs (l, r) =
-(\<exists> ln rn ml mr. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup>  \<and>
+(\<exists> ln rn ml mr. l = Oc\<up>(ml) @ Bk # Oc\<up>(Suc m)  \<and>
-r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Oc # Bk\<up>(ln) @ Bk # Oc\<up>(mr) @ Bk\<up>(rn) \<and>
 ml + mr = Suc (Suc rs) \<and> mr > 0)"
 fun wadjust_loop_right_move :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_loop_right_move m rs (l, r) =
-(\<exists> ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+(\<exists> ml mr nl nr rn. l = Bk\<up>(nl) @ Oc # Oc\<up>(ml) @ Bk # Oc\<up>(Suc m) \<and>
-r = Bk\<^bsup>nr\<^esup> @ Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Bk\<up>(nr) @ Oc\<up>(mr) @ Bk\<up>(rn) \<and>
 ml + mr = Suc (Suc rs) \<and> mr > 0 \<and>
 nl + nr > 0)"
 fun wadjust_loop_check :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_loop_check m rs (l, r) =
-(\<exists> ml mr ln rn. l = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+(\<exists> ml mr ln rn. l = Oc # Bk\<up>(ln) @ Bk # Oc # Oc\<up>(ml) @ Bk # Oc\<up>(Suc m) \<and>
-r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = (Suc rs))"
+r = Oc\<up>(mr) @ Bk\<up>(rn) \<and> ml + mr = (Suc rs))"
 fun wadjust_loop_erase :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_loop_erase m rs (l, r) =
-(\<exists> ml mr ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+(\<exists> ml mr ln rn. l = Bk\<up>(ln) @ Bk # Oc # Oc\<up>(ml) @ Bk # Oc\<up>(Suc m) \<and>
-tl r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = (Suc rs) \<and> mr > 0)"
+tl r = Oc\<up>(mr) @ Bk\<up>(rn) \<and> ml + mr = (Suc rs) \<and> mr > 0)"
 fun wadjust_loop_on_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_loop_on_left_moving_O m rs (l, r) =
-(\<exists> ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m \<^esup>\<and>
+(\<exists> ml mr ln rn. l = Oc\<up>(ml) @ Bk # Oc\<up>(Suc m )\<and>
-r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(mr) @ Bk\<up>(rn) \<and>
 ml + mr = Suc rs \<and> mr > 0)"
 fun wadjust_loop_on_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_loop_on_left_moving_B m rs (l, r) =
-(\<exists> ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+(\<exists> ml mr nl nr rn. l = Bk\<up>(nl) @ Oc # Oc\<up>(ml) @ Bk # Oc\<up>(Suc m) \<and>
-r = Bk\<^bsup>nr\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Bk\<up>(nr) @ Bk # Bk # Oc\<up>(mr) @ Bk\<up>(rn) \<and>
 ml + mr = Suc rs \<and> mr > 0)"
 fun wadjust_loop_on_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_loop_on_left_moving m rs (l, r) =
 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>
+(\<exists> ml mr ln rn. l = Oc # Oc\<up>(ml) @ Bk # Oc\<up>(Suc m) \<and>
-r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Bk\<up>(ln) @ Bk # Bk # Oc\<up>(mr) @ Bk\<up>(rn) \<and>
 ml + mr = Suc rs \<and> mr > 0)"
 fun wadjust_erase2 :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_erase2 m rs (l, r) =
-(\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+(\<exists> ln rn. l = Bk\<up>(ln) @ Bk # Oc # Oc\<up>(Suc rs) @ Bk # Oc\<up>(Suc m) \<and>
-tl r = Bk\<^bsup>rn\<^esup>)"
+tl r = Bk\<up>(rn))"
 fun wadjust_on_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_on_left_moving_O m rs (l, r) =
-(\<exists> rn. l = Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+(\<exists> rn. l = Oc\<up>(Suc rs) @ Bk # Oc\<up>(Suc m) \<and>
-r = Oc # Bk\<^bsup>rn\<^esup>)"
+r = Oc # Bk\<up>(rn))"
 fun wadjust_on_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_on_left_moving_B m rs (l, r) =
-(\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+(\<exists> ln rn. l = Bk\<up>(ln) @ Oc # Oc\<up>(Suc rs) @ Bk # Oc\<up>(Suc m) \<and>
-r = Bk\<^bsup>rn\<^esup>)"
+r = Bk\<up>(rn))"
 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>
+(\<exists> rn. l = Oc\<up>(Suc m) \<and>
-r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+r = Bk # Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 fun wadjust_goon_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_goon_left_moving_O m rs (l, r) =
-(\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+(\<exists> ml mr rn. l = Oc\<up>(ml) @ Bk # Oc\<up>(Suc m) \<and>
-r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Oc\<up>(mr) @ Bk\<up>(rn) \<and>
 ml + mr = Suc (Suc rs) \<and> mr > 0)"
 fun wadjust_goon_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_goon_left_moving 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>)"
+r = Bk # Oc\<up>(Suc m )@ Bk # Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 fun wadjust_backto_standard_pos_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_backto_standard_pos_O m rs (l, r) =
-(\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> \<and>
+(\<exists> ml mr rn. l = Oc\<up>(ml) \<and>
-r = Oc\<^bsup>mr\<^esup> @ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+r = Oc\<up>(mr) @ Bk # Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn) \<and>
 ml + mr = Suc m \<and> mr > 0)"
 fun wadjust_backto_standard_pos :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
 where
 "wadjust_backto_standard_pos 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>)"
+r = Oc\<up>(Suc m )@ Bk # Oc\<up>(Suc (Suc rs)) @ Bk\<up>(rn))"
 declare wadjust_start.simps[simp del]  wadjust_loop_start.simps[simp del]
 wadjust_loop_right_move.simps[simp del]  wadjust_loop_check.simps[simp del]
 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]
 else False
 )"
 declare wadjust_inv.simps[simp del]
-fun wadjust_phase :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
+fun wadjust_phase :: "nat \<Rightarrow> config \<Rightarrow> nat"
 where
 "wadjust_phase rs (st, l, r) =
 (if st = 1 then 3
 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> config \<Rightarrow> nat"
-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"
+fun wadjust_state :: "nat \<Rightarrow> config \<Rightarrow> nat"
 where
 "wadjust_state rs (st, l, r) =
 (if st \<ge> 2 \<and> st \<le> 7 then 8 - st
 else if st \<ge> 8 \<and> st \<le> 11 then 12 - st
 else 0)"
-fun wadjust_step :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
+fun wadjust_step :: "nat \<Rightarrow> config \<Rightarrow> nat"
 where
 "wadjust_step rs (st, l, r) =
 (if st = 1 then (if hd r = Bk then 1
 else 0)
 else if st = 3 then length r
 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"
+fun wadjust_measure :: "(nat \<times> config) \<Rightarrow> nat \<times> nat \<times> nat \<times> nat"
 where
 "wadjust_measure (rs, (st, l, r)) =
 (wadjust_phase rs (st, l, r),
 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"
+definition wadjust_le :: "((nat \<times> config) \<times> nat \<times> config) set"
 where "wadjust_le \<equiv> (inv_image lex_square wadjust_measure)"
 lemma [intro]: "wf lex_square"
 by(auto intro:wf_lex_prod simp: abacus.lex_pair_def lex_square_def
 abacus.lex_triple_def)
 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_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(simp add: wadjust_loop_erase.simps)
-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_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])"
+(Oc # Oc # Oc\<up>(rs) @ Bk # Oc # Oc\<up>(m), [Bk])"
 apply(simp add: wadjust_on_left_moving_B.simps, auto)
-apply(rule_tac x = 0 in exI, simp add: exp_ind_def)
 done
 lemma [simp]: "wadjust_on_left_moving_B m rs
-(Bk\<^bsup>n\<^esup> @ Bk # Oc # Oc # Oc\<^bsup>rs\<^esup> @ Bk # Oc # Oc\<^bsup>m\<^esup>, [Bk])"
+(Bk\<up>(n) @ Bk # Oc # Oc # Oc\<up>(rs) @ Bk # Oc # Oc\<up>(m), [Bk])"
-apply(simp add: wadjust_on_left_moving_B.simps exp_ind_def, auto)
+apply(simp add: wadjust_on_left_moving_B.simps , auto)
-apply(rule_tac x = "Suc n" in exI, simp add: exp_ind)
+apply(rule_tac x = "Suc n" in exI, simp add: exp_ind del: replicate_Suc)
 done
 lemma [simp]: "\<lbrakk>wadjust_erase2 m rs (c, []); c \<noteq> []\<rbrakk> \<Longrightarrow>
 wadjust_on_left_moving m rs (tl c, [hd c])"
 apply(simp only: wadjust_erase2.simps)
 apply(erule_tac exE)+
-apply(case_tac ln, simp_all add: exp_ind_def wadjust_on_left_moving.simps)
+apply(case_tac ln, simp_all add:  wadjust_on_left_moving.simps)
 done
 lemma [simp]: "wadjust_erase2 m rs (c, [])
 \<Longrightarrow> (c = [] \<longrightarrow> wadjust_on_left_moving m rs ([], [Bk])) \<and>
 (c \<noteq> [] \<longrightarrow> wadjust_on_left_moving m rs (tl c, [hd c]))"
 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)
+apply(case_tac [!] ln, simp_all)
 done
 lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, []); c \<noteq> []; hd c = Oc\<rbrakk> \<Longrightarrow>
 wadjust_on_left_moving_O m rs (tl c, [Oc])"
 apply(simp add: wadjust_on_left_moving_B.simps wadjust_on_left_moving_O.simps, auto)
-apply(case_tac [!] ln, simp_all add: exp_ind_def)
+apply(case_tac [!] ln, simp_all add: )
 done
 lemma [simp]: "\<lbrakk>wadjust_on_left_moving m rs (c, []); c \<noteq> []\<rbrakk> \<Longrightarrow>
 wadjust_on_left_moving m rs (tl c, [hd c])"
 apply(simp add: wadjust_on_left_moving.simps)
 \<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(rule_tac x = "Suc nl" in exI, simp add: )
-apply(case_tac nr, simp, case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac nr, simp, case_tac mr, simp_all add: )
 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)
+apply(case_tac [!] mr, simp_all)
 done
 lemma [simp]: "wadjust_loop_erase m rs (c, b) \<Longrightarrow> c \<noteq> []"
 apply(simp only: wadjust_loop_erase.simps, auto)
 done
 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(case_tac ln, simp_all add: , auto)
-apply(simp add: exp_ind exp_ind_def[THEN sym])
+apply(simp add: exp_ind [THEN sym])
 done
 lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); c \<noteq> []; hd c = Oc\<rbrakk> \<Longrightarrow>
 wadjust_loop_on_left_moving_O m rs (tl c, Oc # Bk # list)"
 apply(simp only: wadjust_loop_erase.simps wadjust_loop_on_left_moving_O.simps,
 auto)
-apply(case_tac [!] ln, simp_all add: exp_ind_def)
+apply(case_tac [!] ln, simp_all add: )
 done
 lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); c \<noteq> []\<rbrakk> \<Longrightarrow>
 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)
 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(case_tac nl, simp_all add: , auto)
-apply(rule_tac x = "Suc nr" in exI, auto simp: exp_ind_def)
+apply(rule_tac x = "Suc nr" in exI, auto simp: )
 done
 lemma [simp]: "\<lbrakk>wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk>
 \<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)
+apply(case_tac nl, simp_all add: , 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(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(case_tac ln, simp_all add: )
 apply(rule_tac x = 0 in exI, simp)
 apply(rule_tac x = rn in exI, simp)
-apply(rule_tac x = "Suc ml" in exI, simp add: exp_ind_def, auto)
+apply(rule_tac x = "Suc ml" in exI, simp add: , auto)
-apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind)
+apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind del: replicate_Suc)
 apply(rule_tac x = rn in exI, auto)
-apply(rule_tac x = "Suc ml" in exI, auto simp: exp_ind_def)
+apply(rule_tac x = "Suc ml" in exI, auto )
 done
 lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \<Longrightarrow> c \<noteq> []"
 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
+apply(case_tac ln, simp_all add:  wadjust_on_left_moving.simps
 wadjust_on_left_moving_O.simps wadjust_on_left_moving_B.simps)
 apply(auto)
-apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: exp_ind_def)
+apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: )
-apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind)
+apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind del: replicate_Suc)
-apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: exp_ind_def)
+apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: )
 done
 lemma [simp]: "wadjust_on_left_moving m rs (c,b) \<Longrightarrow> c \<noteq> []"
 apply(simp only:wadjust_on_left_moving.simps
 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)
+apply(case_tac ln, simp_all)
 done
 lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk>
 \<Longrightarrow> wadjust_on_left_moving_O m rs (tl c, Oc # Bk # list)"
 apply(auto simp: wadjust_on_left_moving_O.simps
 wadjust_on_left_moving_B.simps)
-apply(case_tac ln, simp_all add: exp_ind_def)
+apply(case_tac ln, simp_all add: )
 done
 lemma [simp]: "wadjust_on_left_moving  m rs (c, Bk # list) \<Longrightarrow>
 wadjust_on_left_moving m rs (tl c, hd c # Bk # list)"
 apply(simp add: wadjust_on_left_moving.simps)
 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)
+wadjust_goon_left_moving_O.simps , auto)
 done
 lemma [simp]: "wadjust_goon_left_moving_O m rs (c, Bk # list) = False"
 apply(simp add: wadjust_goon_left_moving_O.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 done
 lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Bk\<rbrakk>
 \<Longrightarrow> wadjust_backto_standard_pos_B m rs (tl c, Bk # Bk # list)"
 apply(auto simp: wadjust_goon_left_moving_B.simps
-wadjust_backto_standard_pos_B.simps exp_ind_def)
+wadjust_backto_standard_pos_B.simps )
 done
 lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk>
 \<Longrightarrow> wadjust_backto_standard_pos_O m rs (tl c, Oc # Bk # list)"
 apply(auto simp: wadjust_goon_left_moving_B.simps
-wadjust_backto_standard_pos_O.simps exp_ind_def)
+wadjust_backto_standard_pos_O.simps)
-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
 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)
+apply(case_tac [!] mr, simp_all add: )
 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))"
 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)
+apply(rule_tac x = "Suc ln" in exI, simp add: exp_ind del: replicate_Suc)
 done
 lemma [simp]: "wadjust_loop_right_move m rs (c, Oc # list) \<Longrightarrow>
 wadjust_loop_check m rs (Oc # c, list)"
 apply(simp add: wadjust_loop_right_move.simps
 wadjust_loop_check.simps, auto)
-apply(rule_tac [!] x = ml in exI, simp_all, auto)
+apply(rule_tac [!] x = ml in exI, simp_all add: exp_ind del: replicate_Suc, auto)
-apply(case_tac nl, auto simp: exp_ind_def)
+apply(case_tac nl, simp_all add: exp_ind del: replicate_Suc)
-apply(rule_tac x = "mr - 1" in exI, case_tac mr, simp_all add: exp_ind_def)
+apply(rule_tac x = "mr - 1" in exI, case_tac mr, simp_all add: )
-apply(case_tac [!] nr, simp_all add: exp_ind_def, auto)
+apply(case_tac [!] nr, simp_all)
 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 mr, simp_all add: )
-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)
+apply(case_tac nr, simp_all add: )
 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)
 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)
+apply(case_tac ln, simp_all add: )
 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))"
 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)
+wadjust_goon_left_moving_B.simps )
 done
 lemma [simp]: "\<lbrakk>wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Oc\<rbrakk>
 \<Longrightarrow> wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)"
 apply(auto simp: wadjust_on_left_moving_O.simps
-wadjust_goon_left_moving_O.simps exp_ind_def)
+wadjust_goon_left_moving_O.simps )
-apply(rule_tac x = rs in exI, simp)
+apply(auto simp:  numeral_2_eq_2)
-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)"
 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)
+apply(case_tac [!] ml, auto simp: )
 done
 lemma  [simp]: "\<lbrakk>wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Oc\<rbrakk> \<Longrightarrow>
 wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)"
 apply(auto simp: wadjust_goon_left_moving_O.simps wadjust_goon_left_moving_B.simps)
 apply(rule_tac x = "ml - 1" in exI, simp)
-apply(case_tac ml, simp_all add: exp_ind_def)
+apply(case_tac ml, simp_all add: )
-apply(rule_tac x = "Suc mr" in exI, auto simp: exp_ind_def)
+apply(rule_tac x = "Suc mr" in exI, auto simp: )
 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(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)
+apply(case_tac mr, simp_all add: )
 done
 lemma [simp]: "wadjust_backto_standard_pos_O m rs ([], Oc # list) \<Longrightarrow>
 wadjust_backto_standard_pos_B m rs ([], Bk # Oc # list)"
 apply(auto simp: wadjust_backto_standard_pos_O.simps
 wadjust_backto_standard_pos_B.simps)
-apply(rule_tac x = rn in exI, simp)
+done
-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(case_tac ml, simp_all add: , 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 nr, simp_all add: )
-apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: )
 done
 lemma [simp]: "wadjust_loop_erase m rs (c, []) = False"
 apply(simp only: wadjust_loop_erase.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
 done
 lemma [simp]: "\<lbrakk>Suc (Suc rs) = a;  wadjust_loop_erase m rs (c, Bk # list)\<rbrakk>
 \<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 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"
+lemma dropWhile_exp1: "dropWhile (\<lambda>a. a = Oc) (Oc\<up>(n) @ xs) = dropWhile (\<lambda>a. a = Oc) xs"
-apply(induct n, simp_all add: exp_ind_def)
+apply(induct n, simp_all add: )
 done
-lemma takeWhile_exp1: "takeWhile (\<lambda>a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = Oc\<^bsup>n\<^esup> @ takeWhile (\<lambda>a. a = Oc) xs"
+lemma takeWhile_exp1: "takeWhile (\<lambda>a. a = Oc) (Oc\<up>(n) @ xs) = Oc\<up>(n) @ takeWhile (\<lambda>a. a = Oc) xs"
-apply(induct n, simp_all add: exp_ind_def)
+apply(induct n, simp_all add: )
 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)
+apply(case_tac ln, simp, simp add: )
 done
 lemma [simp]: "wadjust_loop_check m rs ([], b) = False"
 apply(simp add: wadjust_loop_check.simps)
 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>,
+let f = (\<lambda> stp. (Suc (Suc rs),  steps0 (Suc 0, Bk # Oc\<up>(Suc m),
-Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #  Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp)) in
+Bk # Oc # Bk\<up>(ln) @ Bk #  Oc\<up>(Suc rs) @ Bk\<up>(rn)) t_wcode_adjust stp)) in
 \<exists> n .P (f n) \<and> Q (f n)"
 proof -
 let ?P = "(\<lambda> (len, st, l, r). st = 0)"
 let ?Q = "\<lambda> (len, st, l, r). wadjust_inv st m rs (l, r)"
-let ?f = "\<lambda> stp. (Suc (Suc rs),  steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>,
+let ?f = "\<lambda> stp. (Suc (Suc rs),  steps0 (Suc 0, Bk # Oc\<up>(Suc m),
-Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp)"
+Bk # Oc # Bk\<up>(ln) @ Bk # Oc\<up>(Suc rs) @ Bk\<up>(rn)) t_wcode_adjust stp)"
 have "\<exists> n. ?P (?f n) \<and> ?Q (?f n)"
 proof(rule_tac halt_lemma2)
 show "wf wadjust_le" by auto
 next
 show "\<forall> n. \<not> ?P (?f n) \<and> ?Q (?f n) \<longrightarrow>
 ?Q (?f (Suc n)) \<and> (?f (Suc n), ?f n) \<in> wadjust_le"
-proof(rule_tac allI, rule_tac impI, case_tac "?f n",
+apply(rule_tac allI, rule_tac impI, case_tac "?f n", simp)
-simp add: tstep_red tstep.simps, rule_tac conjI, erule_tac conjE,
+apply(simp add: step.simps)
-erule_tac conjE)
+apply(case_tac d, case_tac [2] aa, simp_all)
-fix n a b c d
+apply(simp_all add: wadjust_inv.simps wadjust_le_def
-assume "0 < b" "wadjust_inv b m rs (c, d)" "Suc (Suc rs) = a"
+abacus.lex_triple_def abacus.lex_pair_def lex_square_def numeral_4_eq_4
-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(simp add: steps.simps wadjust_inv.simps wadjust_start.simps, auto)
-apply(rule_tac x = ln in exI,auto)
 done
 next
 show "\<not> ?P (?f 0)"
 apply(simp add: steps.simps)
 done
 qed
-thus "?thesis"
+thus"?thesis"
-apply(auto)
+apply(simp)
 done
 qed
-lemma [intro]: "t_correct t_wcode_adjust"
+lemma [intro]: "tm_wf (t_wcode_adjust, 0)"
-apply(auto simp: t_wcode_adjust_def t_correct.simps iseven_def)
+apply(auto simp: t_wcode_adjust_def tm_wf.simps)
-apply(rule_tac x = 11 in exI, simp)
+done
+declare tm_comp.simps[simp del]
+lemma [simp]: "args \<noteq> [] \<Longrightarrow> bl_bin (<args::nat list>) > 0"
+apply(case_tac args)
+apply(auto simp: tape_of_nl_cons bl_bin.simps split: if_splits)
 done
 lemma wcode_lemma_pre':
 "args \<noteq> [] \<Longrightarrow>
-\<exists> stp rn. steps (Suc 0, [], <m # args>)
+\<exists> stp rn. steps0 (Suc 0, [], <m # args>)
 ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp
-= (0,  [Bk],  Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+= (0,  [Bk],  Oc\<up>(Suc m) @ Bk # Oc\<up>(Suc (bl_bin (<args>))) @ Bk\<up>(rn))"
 proof -
 let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <m # args>"
-let ?Q1 = "\<lambda>(l, r). l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+let ?Q1 = "\<lambda>(l, r). l = Bk # Oc\<up>(Suc m) \<and>
-(\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+(\<exists>ln rn. r = Bk # Oc # Bk\<up>(ln) @ Bk # Bk # Oc\<up>(bl_bin (<args>)) @ Bk\<up>(rn))"
 let ?P2 = ?Q1
 let ?Q2 = "\<lambda> (l, r). (wadjust_stop m (bl_bin (<args>) - 1) (l, r))"
 let ?P3 = "\<lambda> tp. False"
 assume h: "args \<noteq> []"
-have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp)
+hence a: "bl_bin (<args>) > 0"
-((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp = (0, tp') \<and> ?Q2 tp')"
+using h by simp
-proof(rule_tac turing_merge.t_merge_halt[of "t_wcode_prepare |+| t_wcode_main"
+hence "{?P1} (t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust {?Q2}"
-t_wcode_adjust ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2],
+proof(rule_tac Hoare_plus_halt)
-auto simp: turing_merge_def)
+show "?Q1 \<mapsto> ?P2"
+by(simp add: assert_imp_def)
-show "\<exists>stp. case steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp of
+next
-(st, tp') \<Rightarrow> st = 0 \<and> (case tp' of (l, r) \<Rightarrow> l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+show "tm_wf (t_wcode_prepare |+| t_wcode_main, 0)"
-(\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>))"
+apply(rule_tac tm_wf_comp, auto)
-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 "{?P1} t_wcode_prepare |+| t_wcode_main {?Q1}"
-show "\<exists>stp. case steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk #
+proof(rule_tac HoareI, auto)
-Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp of
+show
-(st, tp') \<Rightarrow> st = 0 \<and> wadjust_stop m (bl_bin (<args>) - Suc 0) tp'"
+"\<exists>n. is_final (steps0 (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) n) \<and>
-using wadjust_correctness[of m "bl_bin (<args>) - 1" "Suc ln" rn]
+(\<lambda>(l, r). l = Bk # Oc # Oc \<up> m \<and>
-apply(subgoal_tac "bl_bin (<args>) > 0", auto simp: wadjust_inv.simps)
+(\<exists>ln rn. r = Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> bl_bin (<args>) @ Bk \<up> rn))
-apply(rule_tac x = n in exI, simp add: exp_ind)
+holds_for steps0 (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) n"
-using h
+using h prepare_mainpart_lemma[of args m]
-apply(case_tac args, simp_all, case_tac list,
+apply(auto)
-simp_all add: tape_of_nl_abv  tape_of_nat_list.simps exp_ind_def
+apply(rule_tac x = stp in exI, simp)
-bl_bin.simps)
+apply(rule_tac x = ln in exI, auto)
 done
+qed
 next
-show "?Q1 \<turnstile>-> ?P2"
+show "{?P2} t_wcode_adjust {?Q2}"
-by(simp add: t_imply_def)
+proof(rule_tac HoareI, auto del: replicate_Suc)
+fix ln rn
+show "\<exists>n. is_final (steps0 (Suc 0, Bk # Oc # Oc \<up> m,
+Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> bl_bin (<args>) @ Bk \<up> rn) t_wcode_adjust n) \<and>
+wadjust_stop m (bl_bin (<args>) - Suc 0) holds_for steps0
+(Suc 0, Bk # Oc # Oc \<up> m, Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> bl_bin (<args>) @ Bk \<up> rn) t_wcode_adjust n"
+using wadjust_correctness[of m "bl_bin (<args>) - 1" "Suc ln" rn]
+apply(simp del: replicate_Suc add: replicate_Suc[THEN sym] exp_ind, auto)
+apply(rule_tac x = n in exI)
+using a
+apply(case_tac "bl_bin (<args>)", simp, simp del: replicate_Suc add: exp_ind wadjust_inv.simps)
+done
+qed
 qed
-thus "\<exists>stp rn. steps (Suc 0, [], <m # args>) ((t_wcode_prepare |+| t_wcode_main) |+|
+thus "?thesis"
-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: Hoare_def, auto)
-apply(simp add: t_imply_def)
+apply(case_tac "(steps0 (Suc 0, [], <(m::nat) # args>)
-apply(erule_tac exE)+
+((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) n)")
-apply(subgoal_tac "bl_bin (<args>) > 0", auto simp: wadjust_stop.simps)
+apply(rule_tac x = n in exI, auto simp: wadjust_stop.simps)
-using h
+using a
-apply(case_tac args, simp_all, case_tac list,
+apply(case_tac "bl_bin (<args>)", simp_all)
-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"
+definition t_wcode :: "instr list"
 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 =
+\<exists> stp ln rn. steps0 (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>)"
+(0,  [Bk],  Oc\<up>(Suc m) @ Bk # Oc\<up>(Suc (bl_bin (<args>))) @ Bk\<up>(rn))"
-apply(simp add: wcode_lemma_pre' t_wcode_def)
+apply(simp add: wcode_lemma_pre' t_wcode_def del: replicate_Suc)
 done
 lemma wcode_lemma:
 "args \<noteq> [] \<Longrightarrow>
-\<exists> stp ln rn. steps (Suc 0, [], <m # args>)  (t_wcode) stp =
+\<exists> stp ln rn. steps0 (Suc 0, [], <m # args>)  (t_wcode) stp =
-(0,  [Bk],  <[m ,bl_bin (<args>)]> @ Bk\<^bsup>rn\<^esup>)"
+(0,  [Bk],  <[m ,bl_bin (<args>)]> @ Bk\<up>(rn))"
 using wcode_lemma_1[of args m]
 apply(simp add: t_wcode_def tape_of_nl_abv tape_of_nat_list.simps)
 done
 section {* The universal TM *}
 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"
+definition UTM :: "instr list"
 where
 "UTM = (let (aprog, rs_pos, a_md) = rec_ci rec_F in
 let abc_F = aprog [+] dummy_abc (Suc (Suc 0)) in
-(t_wcode |+| (tm_of abc_F @ tMp (Suc (Suc 0)) (start_of (layout_of abc_F)
+(t_wcode |+| (tm_of abc_F @ shift (mopup (Suc (Suc 0))) (length (tm_of abc_F) div 2))))"
-(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"
+definition F_tprog :: "instr list"
 where
 "F_tprog = tm_of (F_aprog)"
-definition t_utm :: "tprog"
+definition t_utm :: "instr list"
 where
 "t_utm \<equiv>
-(F_tprog) @ tMp (Suc (Suc 0)) (start_of (layout_of (F_aprog))
+F_tprog @ shift (mopup (Suc (Suc 0))) (length F_tprog div 2)"
-(length (F_aprog)) - Suc 0)"
+definition UTM_pre :: "instr list"
-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>);
+steps (Suc 0, Bk\<up>(l), <lm>) tp stp = (0, Bk\<up>(m), Oc\<up>(rs)@Bk\<up>(n));
 rs > 0\<rbrakk>
 \<Longrightarrow> \<exists> stp m. abc_steps_l (0, [code tp, bl2wc (<lm>)]) (F_aprog) stp =
-(length (F_aprog), code tp # bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>)"
+(length (F_aprog), code tp # bl2wc (<lm>) # (rs - 1) # 0\<up>(m))"
 apply(drule_tac  F_t_halt_eq, simp, simp, simp)
 apply(case_tac "rec_ci rec_F")
 apply(frule_tac abc_append_dummy_complie, simp, simp, erule_tac exE,
 erule_tac exE)
 apply(rule_tac x = stp in exI, rule_tac x = m in exI)
 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>
+(length F_aprog, code tp #  bl2wc (<lm>) # (rs - 1) # 0\<up>(m))\<rbrakk>
 \<Longrightarrow> \<exists>stp m n.(steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp =
-(0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>))"
+(0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n)))"
 thm abacus_turing_eq_halt
 using abacus_turing_eq_halt
 [of "layout_of F_aprog" "F_aprog" "F_tprog" "length (F_aprog)"
-"[code tp, bl2wc (<lm>)]" stp "code tp # bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>" "Suc (Suc 0)"
+"[code tp, bl2wc (<lm>)]" stp "code tp # bl2wc (<lm>) # (rs - 1) # 0\<up>(m)" "Suc (Suc 0)"
 "start_of (layout_of (F_aprog)) (length (F_aprog))" "[]" 0]
 apply(simp add: F_tprog_def t_utm_def abc_lm_v.simps nth_append)
 apply(erule_tac exE)+
 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)
 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>
+steps (Suc 0, Bk\<up>(l), <lm::nat list>) tp stp = (0, Bk\<up>(m), Oc\<up>(rs)@Bk\<up>(n))\<rbrakk>
 \<Longrightarrow>  \<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp =
-(0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+(0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n))"
 apply(drule_tac  l = l in F_abc_halt_eq, simp, simp, simp)
 apply(erule_tac exE, erule_tac exE)
 apply(rule_tac F_abc_utm_halt_eq, simp_all)
 done
+*)
-lemma [simp]: "tinres xs (xs @ Bk\<^bsup>i\<^esup>)"
+(*
+lemma [simp]: "tinres xs (xs @ Bk\<up>(i))"
 apply(auto simp: tinres_def)
 done
-lemma [elim]: "\<lbrakk>rs > 0; Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup> = c @ Bk\<^bsup>n\<^esup>\<rbrakk>
+lemma [elim]: "\<lbrakk>rs > 0; Oc\<up>(rs) @ Bk\<up>(na) = c @ Bk\<up>(n)\<rbrakk>
-\<Longrightarrow> \<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+\<Longrightarrow> \<exists>n. c = Oc\<up>(rs) @ Bk\<up>(n)"
 apply(case_tac "na > n")
 apply(subgoal_tac "\<exists> d. na = d + n", auto simp: exp_add)
 apply(rule_tac x = "na - n" in exI, simp)
 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>
+steps (Suc 0, Bk\<up>(l), <lm::nat list>) tp stp = (0, Bk\<up>(m), Oc\<up>(rs)@Bk\<up>(n))\<rbrakk>
-\<Longrightarrow>  \<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp =
+\<Longrightarrow>  \<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>(i)) t_utm stp =
-(0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+(0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n))"
 apply(drule_tac t_utm_halt_eq', simp_all)
 apply(erule_tac exE)+
 proof -
 fix stpa ma na
-assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
+assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<up>(ma), Oc\<up>(rs) @ Bk\<up>(na))"
 and gr: "rs > 0"
-thus "\<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+thus "\<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>(i)) t_utm stp = (0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n))"
 apply(rule_tac x = stpa in exI, rule_tac x = ma in exI,  simp)
-proof(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp)
+proof(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>(i)) t_utm stpa", simp)
 fix a b c
-assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
+assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<up>(ma), Oc\<up>(rs) @ Bk\<up>(na))"
-"steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)"
+"steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>(i)) 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>)"
+thus " a = 0 \<and> b = Bk\<up>(ma) \<and> (\<exists>n. c = Oc\<up>(rs) @ Bk\<up>(n))"
-using tinres_steps2[of "<[code tp, bl2wc (<lm>)]>" "<[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>"
+using tinres_steps2[of "<[code tp, bl2wc (<lm>)]>" "<[code tp, bl2wc (<lm>)]> @ Bk\<up>(i)"
-"Suc 0" " [Bk, Bk]" t_utm stpa 0 "Bk\<^bsup>ma\<^esup>" "Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>" a b c]
+"Suc 0" " [Bk, Bk]" t_utm stpa 0 "Bk\<up>(ma)" "Oc\<up>(rs) @ Bk\<up>(na)" a b c]
 apply(simp)
 using gr
 apply(simp only: tinres_def, auto)
 apply(rule_tac x = "na + n" in exI, simp add: exp_add)
 done
 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>"
+lemma [elim]: "Bk\<up>(ma) = b @ Bk\<up>(n)  \<Longrightarrow> \<exists>m. b = Bk\<up>(m)"
 apply(subgoal_tac "ma = length b + n")
 apply(rule_tac x = "ma - n" in exI, simp add: exp_add)
 apply(drule_tac length_equal)
 apply(simp)
 done
+*)
+lemma tinres_step1:
+"\<lbrakk>tinres l l'; step (ss, l, r) (t, 0) = (sa, la, ra);
+step (ss, l', r) (t, 0) = (sb, lb, rb)\<rbrakk>
+\<Longrightarrow> tinres la lb \<and> ra = rb \<and> sa = sb"
+apply(case_tac ss, case_tac [!]r, case_tac [!] "a::cell")
+apply(auto simp: step.simps fetch.simps nth_of.simps
+split: if_splits )
+apply(case_tac [!] "t ! (2 * nat)",
+auto simp: tinres_def split: if_splits)
+apply(case_tac [1-8] a, auto split: if_splits)
+apply(case_tac [!] "t ! (2 * nat)",
+auto simp: tinres_def split: if_splits)
+apply(case_tac [1-4] a, auto split: if_splits)
+apply(case_tac [!] "t ! Suc (2 * nat)",
+auto simp: if_splits)
+apply(case_tac [!] aa, auto split: if_splits)
+done
+lemma tinres_steps1:
+"\<lbrakk>tinres l l'; steps (ss, l, r) (t, 0) stp = (sa, la, ra);
+steps (ss, l', r) (t, 0) stp = (sb, lb, rb)\<rbrakk>
+\<Longrightarrow> tinres la lb \<and> ra = rb \<and> sa = sb"
+apply(induct stp arbitrary: sa la ra sb lb rb, simp add: steps.simps)
+apply(simp add: step_red)
+apply(case_tac "(steps (ss, l, r) (t, 0) stp)")
+apply(case_tac "(steps (ss, l', r) (t, 0) stp)")
+proof -
+fix stp sa la ra sb lb rb a b c aa ba ca
+assume ind: "\<And>sa la ra sb lb rb. \<lbrakk>steps (ss, l, r) (t, 0) stp = (sa, (la::cell list), ra);
+steps (ss, l', r) (t, 0) stp = (sb, lb, rb)\<rbrakk> \<Longrightarrow> tinres la lb \<and> ra = rb \<and> sa = sb"
+and h: " tinres l l'" "step (steps (ss, l, r) (t, 0) stp) (t, 0) = (sa, la, ra)"
+"step (steps (ss, l', r) (t, 0) stp) (t, 0) = (sb, lb, rb)" "steps (ss, l, r) (t, 0) stp = (a, b, c)"
+"steps (ss, l', r) (t, 0) stp = (aa, ba, ca)"
+have "tinres b ba \<and> c = ca \<and> a = aa"
+apply(rule_tac ind, simp_all add: h)
+done
+thus "tinres la lb \<and> ra = rb \<and> sa = sb"
+apply(rule_tac l = b and l' = ba and r = c  and ss = a
+and t = t in tinres_step1)
+using h
+apply(simp, simp, simp)
+done
+qed
+lemma [simp]:
+"tinres (Bk \<up> m @ [Bk, Bk]) la \<Longrightarrow> \<exists>m. la = Bk \<up> m"
+apply(auto simp: tinres_def)
+apply(case_tac n, simp add: exp_ind)
+apply(rule_tac  x ="Suc (Suc m)" in exI, simp only: exp_ind, simp)
+apply(simp add: exp_ind del: replicate_Suc)
+apply(case_tac nat, simp add: exp_ind)
+apply(rule_tac x = "Suc m" in exI, simp only: exp_ind)
+apply(simp only: exp_ind, simp)
+apply(subgoal_tac "m = length la + nata")
+apply(rule_tac x = "m - nata" in exI, simp add: exp_add)
+apply(drule_tac length_equal, simp)
+apply(simp only: exp_ind[THEN sym] replicate_Suc[THEN sym] replicate_add[THEN sym])
+apply(rule_tac x = "m + Suc (Suc n)" in exI, simp)
+done
 lemma t_utm_halt_eq:
-"\<lbrakk>turing_basic.t_correct tp;
+assumes tm_wf: "tm_wf (tp, 0)"
-0 < rs;
+and exec: "steps0 (Suc 0, Bk\<up>(l), <lm::nat list>) tp stp = (0, Bk\<up>(m), Oc\<up>(rs)@Bk\<up>(n))"
-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>
+and resutl: "0 < rs"
-\<Longrightarrow>  \<exists>stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp =
+shows "\<exists>stp m n. steps0 (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>(i)) t_utm stp =
-(0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+(0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n))"
-apply(drule_tac i = i in t_utm_halt_eq'', simp_all)
-apply(erule_tac exE)+
 proof -
-fix stpa ma na
+obtain ap arity fp where a: "rec_ci rec_F = (ap, arity, fp)"
-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>)"
+by (metis prod_cases3)
-and gr: "rs > 0"
+moreover have b: "rec_calc_rel  rec_F [code tp, (bl2wc (<lm>))] (rs - Suc 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>)"
+using assms
-apply(rule_tac x = stpa in exI)
+apply(rule_tac F_correct, simp_all)
-proof(case_tac "steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp)
+done
-fix a b c
+have "\<exists> stp m l. steps0 (Suc 0, Bk # Bk # [], <[code tp, bl2wc (<lm>)]> @ Bk\<up>i)
-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>)"
+(F_tprog @ shift (mopup 2) (length F_tprog div 2)) stp
-"steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)"
+= (0, Bk\<up>m @ Bk # Bk # [], Oc\<up>Suc (rs - 1) @ Bk\<up>l)"
-thus "a = 0 \<and> (\<exists>m. b = Bk\<^bsup>m\<^esup>) \<and> (\<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+proof(rule_tac recursive_compile_to_tm_correct)
-using tinres_steps[of "[Bk, Bk]" "[Bk]" "Suc 0" "<[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>" t_utm stpa 0
+show "rec_ci rec_F = (ap, arity, fp)" using a by simp
-"Bk\<^bsup>ma\<^esup>" "Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>" a b c]
+next
-apply(simp)
+show "rec_calc_rel rec_F [code tp, bl2wc (<lm>)] (rs - 1)"
-apply(auto simp: tinres_def)
+using b by simp
-apply(rule_tac x = "ma + n" in exI, simp add: exp_add)
+next
+show "length [code tp, bl2wc (<lm>)] = 2" by simp
+next
+show "layout_of (ap [+] dummy_abc 2) = layout_of (ap [+] dummy_abc 2)"
+by simp
+next
+show "F_tprog = tm_of (ap [+] dummy_abc 2)"
+using a
+apply(simp add: F_tprog_def F_aprog_def numeral_2_eq_2)
 done
 qed
-qed
+then obtain stp m l where
+"steps0 (Suc 0, Bk # Bk # [], <[code tp, bl2wc (<lm>)]> @ Bk\<up>i)
-lemma [intro]: "t_correct t_wcode"
+(F_tprog @ shift (mopup 2) (length F_tprog div 2)) stp
-apply(simp add: t_wcode_def)
+= (0, Bk\<up>m @ Bk # Bk # [], Oc\<up>Suc (rs - 1) @ Bk\<up>l)" by blast
-apply(auto)
+hence "\<exists> m. steps0 (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>i)
-done
+(F_tprog @ shift (mopup 2) (length F_tprog div 2)) stp
+= (0, Bk\<up>m, Oc\<up>Suc (rs - 1) @ Bk\<up>l)"
-lemma [intro]: "t_correct t_utm"
+proof -
-apply(simp add: t_utm_def F_tprog_def)
+assume g: "steps0 (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk \<up> i)
-apply(rule_tac t_compiled_correct, auto)
+(F_tprog @ shift (mopup 2) (length F_tprog div 2)) stp =
-done
+(0, Bk \<up> m @ [Bk, Bk], Oc \<up> Suc (rs - 1) @ Bk \<up> l)"
+moreover have "tinres [Bk, Bk] [Bk]"
-lemma UTM_halt_lemma_pre:
+apply(auto simp: tinres_def)
-"\<lbrakk>turing_basic.t_correct tp;
+done
-0 < rs;
+moreover obtain sa la ra where "steps0 (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>i)
-args \<noteq> [];
+(F_tprog @ shift (mopup 2) (length F_tprog div 2)) stp = (sa, la, ra)"
-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>
+apply(case_tac "steps0 (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<up>i)
-\<Longrightarrow>  \<exists>stp m n. steps (Suc 0, [], <code tp # args>) UTM_pre stp =
+(F_tprog @ shift (mopup 2) (length F_tprog div 2)) stp", auto)
-(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
+ultimately show "?thesis"
-show "?Q1 \<turnstile>-> ?P2"
+apply(drule_tac tinres_steps1, auto)
-apply(simp add: t_imply_def)
 done
 qed
 thus "?thesis"
-apply(simp add: t_imply_def)
+apply(auto)
-apply(auto simp: UTM_pre_def)
+apply(rule_tac x = stp in exI, simp add: t_utm_def)
+using assms
+apply(case_tac rs, simp_all add: numeral_2_eq_2)
+done
+qed
+lemma [intro]: "tm_wf (t_wcode, 0)"
+apply(simp add: t_wcode_def)
+apply(rule_tac tm_wf_comp)
+apply(rule_tac tm_wf_comp, auto)
+done
+lemma [intro]: "tm_wf (t_utm, 0)"
+apply(simp only: t_utm_def F_tprog_def)
+apply(rule_tac t_compiled_correct, auto)
+done
+lemma UTM_halt_lemma_pre:
+assumes wf_tm: "tm_wf (tp, 0)"
+and result: "0 < rs"
+and args: "args \<noteq> []"
+and exec: "steps0 (Suc 0, Bk\<up>(i), <args::nat list>) tp stp = (0, Bk\<up>(m), Oc\<up>(rs)@Bk\<up>(k))"
+shows "\<exists>stp m n. steps0 (Suc 0, [], <code tp # args>) UTM_pre stp =
+(0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n))"
+proof -
+let ?Q2 = "\<lambda> (l, r). (\<exists> ln rn. l = Bk\<up>(ln) \<and> r = Oc\<up>(rs) @ Bk\<up>(rn))"
+let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <code tp # args>"
+let ?Q1 = "\<lambda> (l, r). (l = [Bk] \<and>
+(\<exists> rn. r = Oc\<up>(Suc (code tp)) @ Bk # Oc\<up>(Suc (bl_bin (<args>))) @ Bk\<up>(rn)))"
+let ?P2 = ?Q1
+let ?P3 = "\<lambda> (l, r). False"
+have "{?P1} (t_wcode |+| t_utm) {?Q2}"
+proof(rule_tac Hoare_plus_halt)
+show "?Q1 \<mapsto> ?P2"
+by(simp add: assert_imp_def)
+next
+show "tm_wf (t_wcode, 0)" by auto
+next
+show "{?P1} t_wcode {?Q1}"
+apply(rule_tac HoareI, auto)
+using wcode_lemma_1[of args "code tp"] args
+apply(auto)
+apply(rule_tac x = stp in exI, simp)
+done
+next
+show "{?P2} t_utm {?Q2}"
+proof(rule_tac HoareI, auto)
+fix rn
+show "\<exists>n. is_final (steps0 (Suc 0, [Bk], Oc # Oc \<up> code tp @ Bk # Oc # Oc \<up> bl_bin (<args>) @ Bk \<up> rn) t_utm n) \<and>
+(\<lambda>(l, r). (\<exists>ln. l = Bk \<up> ln) \<and>
+(\<exists>rn. r = Oc \<up> rs @ Bk \<up> rn)) holds_for steps0 (Suc 0, [Bk],
+Oc # Oc \<up> code tp @ Bk # Oc # Oc \<up> bl_bin (<args>) @ Bk \<up> rn) t_utm n"
+using t_utm_halt_eq[of tp i "args" stp m rs k rn] assms
+apply(auto simp: bin_wc_eq)
+apply(rule_tac x = stpa in exI, simp add: tape_of_nl_abv)
+done
+qed
+qed
+thus "?thesis"
+apply(auto simp: Hoare_def UTM_pre_def)
+apply(case_tac "steps0 (Suc 0, [], <code tp # args>) (t_wcode |+| t_utm) n")
+apply(rule_tac x = n in exI, simp)
 done
 qed
 text {*
 The correctness of @{text "UTM"}, the halt case.
 *}
 lemma UTM_halt_lemma:
-"\<lbrakk>turing_basic.t_correct tp;
+assumes tm_wf: "tm_wf (tp, 0)"
-0 < rs;
+and result: "0 < rs"
-args \<noteq> [];
+and args: "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>
+and exec: "steps0 (Suc 0, Bk\<up>(i), <args::nat list>) tp stp = (0, Bk\<up>(m), Oc\<up>(rs)@Bk\<up>(k))"
-\<Longrightarrow>  \<exists>stp m n. steps (Suc 0, [], <code tp # args>) UTM stp =
+shows "\<exists>stp m n. steps0 (Suc 0, [], <code tp # args>) UTM stp =
-(0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+(0, Bk\<up>(m), Oc\<up>(rs) @ Bk\<up>(n))"
-using UTM_halt_lemma_pre[of tp rs args i stp m k]
+using UTM_halt_lemma_pre[of tp rs args i stp m k] assms
 apply(simp add: UTM_pre_def t_utm_def UTM_def F_aprog_def F_tprog_def)
 apply(case_tac "rec_ci rec_F", simp)
 done
-definition TSTD:: "t_conf \<Rightarrow> bool"
+definition TSTD:: "config \<Rightarrow> bool"
 where
 "TSTD c = (let (st, l, r) = c in
-st = 0 \<and> (\<exists> m. l = Bk\<^bsup>m\<^esup>) \<and> (\<exists> rs n. r = Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>))"
+st = 0 \<and> (\<exists> m. l = Bk\<up>(m)) \<and> (\<exists> rs n. r = Oc\<up>(Suc rs) @ Bk\<up>(n)))"
-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"
+lemma [simp]: "\<forall>m. b \<noteq> Bk\<up>(m) \<Longrightarrow> 0 < bl2wc b"
 apply(rule classical, simp)
 apply(induct b, erule_tac x = 0 in allE, simp)
 apply(simp add: bl2wc.simps, case_tac a, simp_all
 add: bl2nat.simps bl2nat_double)
-apply(case_tac "\<exists> m. b = Bk\<^bsup>m\<^esup>",  erule exE)
+apply(case_tac "\<exists> m. b = Bk\<up>(m)",  erule exE)
-apply(erule_tac x = "Suc m" in allE, simp add: exp_ind_def, simp)
+apply(erule_tac x = "Suc m" in allE, simp add: , simp)
 done
-lemma nstd_case2: "\<forall>m. b \<noteq> Bk\<^bsup>m\<^esup> \<Longrightarrow> NSTD (trpl_code (a, b, c))"
+lemma nstd_case2: "\<forall>m. b \<noteq> Bk\<up>(m) \<Longrightarrow> NSTD (trpl_code (a, b, c))"
 apply(simp add: NSTD.simps trpl_code.simps)
 done
-thm lg.simps
-thm lgR.simps
 lemma [elim]: "Suc (2 * x) = 2 * y \<Longrightarrow> RR"
 apply(induct x arbitrary: y, simp, simp)
 apply(case_tac y, simp, simp)
 done
-lemma bl2nat_zero_eq[simp]: "(bl2nat c 0 = 0) = (\<exists>n. c = Bk\<^bsup>n\<^esup>)"
+declare replicate_Suc[simp del]
+lemma bl2nat_zero_eq[simp]: "(bl2nat c 0 = 0) = (\<exists>n. c = Bk\<up>(n))"
 apply(auto)
-apply(induct c, simp add: bl2nat.simps)
+apply(induct c, simp_all 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>"
+"\<lbrakk>Suc (bl2wc c) = 2 ^  m\<rbrakk> \<Longrightarrow> \<exists> rs n. c = Oc\<up>(rs) @ Bk\<up>(n)"
 apply(induct c arbitrary: m, simp add: bl2wc.simps bl2nat.simps)
 apply(case_tac a, auto)
 apply(case_tac m, simp_all add: bl2wc.simps, auto)
 apply(rule_tac x = 0 in exI, rule_tac x = "Suc n" in exI,
-simp add: exp_ind_def)
+simp add: replicate_Suc)
 apply(simp add: bl2wc.simps bl2nat.simps bl2nat_double)
 apply(case_tac m, simp, simp)
 proof -
 fix c m nat
 assume ind:
-"\<And>m. Suc (bl2nat c 0) = 2 ^ m \<Longrightarrow> \<exists>rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+"\<And>m. Suc (bl2nat c 0) = 2 ^ m \<Longrightarrow> \<exists>rs n. c = Oc\<up>(rs) @ Bk\<up>(n)"
 and h:
 "Suc (Suc (2 * bl2nat c 0)) = 2 * 2 ^ nat"
-have "\<exists>rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+have "\<exists>rs n. c = Oc\<up>(rs) @ Bk\<up>(n)"
 apply(rule_tac m = nat in ind)
 using h
 apply(simp)
 done
-from this obtain rs n where " c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" by blast
+from this obtain rs n where " c = Oc\<up>(rs) @ Bk\<up>(n)" by blast
-thus "\<exists>rs n. Oc # c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+thus "\<exists>rs n. Oc # c = Oc\<up>(rs) @ Bk\<up>(n)"
-apply(rule_tac x = "Suc rs" in exI, simp add: exp_ind_def)
+apply(rule_tac x = "Suc rs" in exI, simp add: )
-apply(rule_tac x = n in exI, simp)
+apply(rule_tac x = n in exI, simp add: replicate_Suc)
 done
 qed
-lemma [elim]:
+lemma lg_bin:
-"\<lbrakk>\<forall>rs n. c \<noteq> Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>;
+"\<lbrakk>\<forall>rs n. c \<noteq> Oc\<up>(Suc rs) @ Bk\<up>(n);
 bl2wc c = 2 ^ lg (Suc (bl2wc c)) 2 - Suc 0\<rbrakk> \<Longrightarrow> bl2wc c = 0"
 apply(subgoal_tac "\<exists> m. Suc (bl2wc c) = 2^m", erule_tac exE)
 apply(drule_tac bl2wc_exp_ex, simp, erule_tac exE, erule_tac exE)
 apply(case_tac rs, simp, simp, erule_tac x = nat in allE,
 erule_tac x = n in allE, simp)
 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))"
+"\<forall>rs n. c \<noteq> Oc\<up>(Suc rs) @ Bk\<up>(n) \<Longrightarrow>  NSTD (trpl_code (a, b, c))"
 apply(simp add: NSTD.simps trpl_code.simps)
-apply(rule_tac impI)
+apply(auto)
-apply(rule_tac disjI2, rule_tac disjI2, auto)
+apply(drule_tac lg_bin, simp_all)
 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)"]
 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;
+"\<lbrakk>tm_wf (tp, 0);
-steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp = (a, b, c);
+steps0 (Suc 0, Bk\<up>(l), <lm>) tp stp = (a, b, c);
 \<not> TSTD (a, b, c)\<rbrakk>
 \<Longrightarrow> rec_exec rec_nonstop [code tp, bl2wc (<lm>), stp] = Suc 0"
 apply(simp add: rec_nonstop_def rec_exec.simps)
 apply(subgoal_tac
 "rec_exec rec_conf [code tp, bl2wc (<lm>), stp] =
 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;
+"\<lbrakk>tm_wf (tp, 0);
-\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk>
+\<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <lm>) tp stp))\<rbrakk>
 \<Longrightarrow> \<forall>y. rec_calc_rel rec_nonstop
 ([code tp, bl2wc (<lm>), y]) (Suc 0)"
 apply(rule_tac allI, erule_tac x = y in allE)
-apply(case_tac "steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp y", simp)
+apply(case_tac "steps0 (Suc 0, Bk\<up>(l), <lm>) tp y", simp)
 apply(rule_tac nonstop_t_uhalt_eq, simp_all)
 done
 (*
 lemma [simp]:
 done
 *)
 declare ci_cn_para_eq[simp]
 lemma F_aprog_uhalt:
-"\<lbrakk>turing_basic.t_correct tp;
+"\<lbrakk>tm_wf (tp,0);
-\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp));
+\<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <lm>) tp stp));
 rec_ci rec_F = (F_ap, rs_pos, a_md)\<rbrakk>
-\<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)] @ 0\<^bsup>a_md - rs_pos \<^esup>
+\<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)] @ 0\<up>(a_md - rs_pos )
 @ suflm) (F_ap) stp of (ss, e) \<Rightarrow> ss < length (F_ap)"
 apply(case_tac "rec_ci (Cn (Suc (Suc 0)) rec_right [Cn (Suc (Suc 0)) rec_conf
 ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])])")
 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(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;
+"\<lbrakk>tm_wf (tp, 0);
-\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp));
+\<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <lm>) tp stp));
 rec_ci rec_F = (ap, pos, md)\<rbrakk>
 \<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) ap stp of (ss, e)
 \<Rightarrow>  ss < length ap"
 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
+"\<forall>stp. case abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<up>(md - pos)) ap stp of
 (ss, e) \<Rightarrow> ss < length ap"
 "abc_steps_l (0, [code tp, bl2wc (<lm>)]) ap stp = (a, b)"
-"turing_basic.t_correct tp"
+"tm_wf (tp, 0)"
 "rec_ci rec_F = (ap, pos, md)"
 moreover have "ap \<noteq> []"
 using h apply(rule_tac rec_ci_not_null, simp)
 done
 ultimately show "a < length ap"
 proof(erule_tac x = stp in allE,
-case_tac "abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp", simp)
+case_tac "abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<up>(md - pos)) ap stp", simp)
 fix aa ba
 assume g: "aa < length ap"
-"abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp = (aa, ba)"
+"abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<up>(md - pos)) ap stp = (aa, ba)"
 "ap \<noteq> []"
 thus "?thesis"
 using abc_list_crsp_steps[of "[code tp, bl2wc (<lm>)]"
 "md - pos" ap stp aa ba] h
 apply(simp)
 done
 qed
 qed
 lemma uabc_uhalt:
-"\<lbrakk>turing_basic.t_correct tp;
+"\<lbrakk>tm_wf (tp, 0);
-\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk>
+\<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <lm>) tp stp))\<rbrakk>
 \<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) F_aprog
 stp of (ss, e) \<Rightarrow> ss < length F_aprog"
 apply(case_tac "rec_ci rec_F", simp add: F_aprog_def)
 thm uabc_uhalt'
 apply(drule_tac ap = a and pos = b and md = c in uabc_uhalt', simp_all)
 apply(simp)
 done
 qed
 lemma tutm_uhalt':
-"\<lbrakk>turing_basic.t_correct tp;
+assumes tm_wf:  "tm_wf (tp,0)"
-\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk>
+and unhalt: "\<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <lm>) tp stp))"
-\<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp)"
+shows "\<forall> stp. \<not> is_final (steps0 (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp)"
-using abacus_turing_eq_uhalt[of "layout_of (F_aprog)"
+apply(simp add: t_utm_def)
-"F_aprog" "F_tprog" "[code tp, bl2wc (<lm>)]"
+proof(rule_tac compile_correct_unhalt)
-"start_of (layout_of (F_aprog )) (length (F_aprog))"
+show "layout_of F_aprog = layout_of F_aprog" by simp
-"Suc (Suc 0)"]
+next
-apply(simp add: F_tprog_def)
+show "F_tprog = tm_of F_aprog"
-apply(subgoal_tac "\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)])
+by(simp add:  F_tprog_def)
-(F_aprog) stp of (as, am) \<Rightarrow> as < length (F_aprog)", simp)
+next
-thm abacus_turing_eq_uhalt
+show "crsp (layout_of F_aprog) (0, [code tp, bl2wc (<lm>)]) (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>)  []"
-apply(simp add: t_utm_def F_tprog_def)
+by(auto simp: crsp.simps start_of.simps)
-apply(rule_tac uabc_uhalt, simp_all)
+next
-done
+show "length F_tprog div 2 = length F_tprog div 2" by simp
+next
+show "\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) F_aprog stp of (as, am) \<Rightarrow> as < length F_aprog"
+using assms
+apply(erule_tac uabc_uhalt, simp)
+done
+qed
 lemma tinres_commute: "tinres r r' \<Longrightarrow> tinres r' r"
 apply(auto simp: tinres_def)
 done
 lemma inres_tape:
-"\<lbrakk>steps (st, l, r) tp stp = (a, b, c); steps (st, l', r') tp stp = (a', b', c');
+"\<lbrakk>steps0 (st, l, r) tp stp = (a, b, c); steps0 (st, l', r') tp stp = (a', b', c');
 tinres l l'; tinres r r'\<rbrakk>
 \<Longrightarrow> a = a' \<and> tinres b b' \<and> tinres c c'"
-proof(case_tac "steps (st, l', r) tp stp")
+proof(case_tac "steps0 (st, l', r) tp stp")
 fix aa ba ca
-assume h: "steps (st, l, r) tp stp = (a, b, c)"
+assume h: "steps0 (st, l, r) tp stp = (a, b, c)"
-"steps (st, l', r') tp stp = (a', b', c')"
+"steps0 (st, l', r') tp stp = (a', b', c')"
 "tinres l l'" "tinres r r'"
-"steps (st, l', r) tp stp = (aa, ba, ca)"
+"steps0 (st, l', r) tp stp = (aa, ba, ca)"
 have "tinres b ba \<and> c = ca \<and> a = aa"
 using h
-apply(rule_tac tinres_steps, auto)
+apply(rule_tac tinres_steps1, auto)
 done
-thm tinres_steps2
 moreover have "b' = ba \<and> tinres c' ca \<and> a' =  aa"
 using h
 apply(rule_tac tinres_steps2, auto intro: tinres_commute)
 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)
+lemma tape_normalize: "\<forall> stp. \<not> is_final(steps0 (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)"
+\<Longrightarrow> \<forall> stp. \<not> is_final (steps0 (Suc 0, Bk\<up>(m), <[code tp, bl2wc (<lm>)]> @ Bk\<up>(n)) t_utm stp)"
-apply(rule_tac allI, case_tac "(steps (Suc 0, Bk\<^bsup>m\<^esup>,
+apply(rule_tac allI, case_tac "(steps0 (Suc 0, Bk\<up>(m),
-<[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>n\<^esup>) t_utm stp)", simp add: isS0_def)
+<[code tp, bl2wc (<lm>)]> @ Bk\<up>(n)) t_utm stp)", simp)
 apply(erule_tac x = stp in allE)
-apply(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp", simp)
+apply(case_tac "steps0 (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp", simp)
 apply(drule_tac inres_tape, auto)
 apply(auto simp: tinres_def)
 apply(case_tac "m > Suc (Suc 0)")
 apply(rule_tac x = "m - Suc (Suc 0)" in exI)
-apply(case_tac m, simp_all add: exp_ind_def, case_tac nat, simp_all add: exp_ind_def)
+apply(case_tac m, simp_all add: , case_tac nat, simp_all add: replicate_Suc)
-apply(rule_tac x = "2 - m" in exI, simp add: exp_ind_def[THEN sym] exp_add[THEN sym])
+apply(rule_tac x = "2 - m" in exI, simp add: exp_add[THEN sym])
-apply(simp only: numeral_2_eq_2, simp add: exp_ind_def)
+apply(simp only: numeral_2_eq_2, simp add: replicate_Suc)
 done
 lemma tutm_uhalt:
-"\<lbrakk>turing_basic.t_correct tp;
+"\<lbrakk>tm_wf (tp,0);
-\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp))\<rbrakk>
+\<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <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)"
+\<Longrightarrow> \<forall> stp. \<not> is_final (steps0 (Suc 0, Bk\<up>(m), <[code tp, bl2wc (<args>)]> @ Bk\<up>(n)) t_utm stp)"
 apply(rule_tac tape_normalize)
 apply(rule_tac tutm_uhalt', simp_all)
 done
 lemma UTM_uhalt_lemma_pre:
-"\<lbrakk>turing_basic.t_correct tp;
+assumes tm_wf: "tm_wf (tp, 0)"
-\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp));
+and exec: "\<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <args>) tp stp))"
-args \<noteq> []\<rbrakk>
+and args: "args \<noteq> []"
-\<Longrightarrow>  \<forall> stp. \<not> isS0 (steps (Suc 0, [], <code tp # args>)  UTM_pre stp)"
+shows "\<forall> stp. \<not> is_final (steps0 (Suc 0, [], <code tp # args>)  UTM_pre stp)"
 proof -
 let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <code tp # args>"
 let ?Q1 = "\<lambda> (l, r). (l = [Bk] \<and>
-(\<exists> rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))"
+(\<exists> rn. r = Oc\<up>(Suc (code tp)) @ Bk # Oc\<up>(Suc (bl_bin (<args>))) @ Bk\<up>(rn)))"
-let ?P4 = ?Q1
+let ?P2 = ?Q1
-let ?P3 = "\<lambda> (l, r). False"
+have "{?P1} (t_wcode |+| t_utm) \<up>"
-assume h: "turing_basic.t_correct tp" "\<forall>stp. \<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp)"
+proof(rule_tac Hoare_plus_unhalt)
-"args \<noteq> []"
+show "?Q1 \<mapsto> ?P2"
-have "?P1 \<turnstile>-> \<lambda> tp. \<not> (\<exists> stp. isS0 (steps (Suc 0, tp) (t_wcode |+| t_utm) stp))"
+by(simp add: assert_imp_def)
-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 "tm_wf (t_wcode, 0)" by auto
-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)
+next
-\<Longrightarrow> False"
+show "{?P1} t_wcode {?Q1}"
-using tutm_uhalt[of tp l args "Suc 0" rn] h
+apply(rule_tac HoareI, auto)
-apply(simp)
+using wcode_lemma_1[of args "code tp"] args
-apply(erule_tac x = stp in allE)
+apply(auto)
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps bin_wc_eq)
+apply(rule_tac x = stp in exI, simp)
 done
 next
-fix rn stp
+show "{?P2} t_utm \<up>"
-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>
+proof(rule_tac Hoare_unhalt_I, auto)
-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)"
+fix n rn
-by simp
+assume h: "is_final (steps0 (Suc 0, [Bk], Oc \<up> Suc (code tp) @ Bk # Oc \<up> Suc (bl_bin (<args>)) @ Bk \<up> rn) t_utm n)"
-next
+have "\<forall> stp. \<not> is_final (steps0 (Suc 0, Bk\<up>(Suc 0), <[code tp, bl2wc (<args>)]> @ Bk\<up>(rn)) t_utm stp)"
-show "?Q1 \<turnstile>-> ?P4"
+using assms
-apply(simp add: t_imply_def)
+apply(rule_tac tutm_uhalt, simp_all)
 done
+thus "False"
+using h
+apply(erule_tac x = n in allE)
+apply(simp add: tape_of_nl_abv bin_wc_eq)
+done
+qed
 qed
 thus "?thesis"
-apply(simp add: t_imply_def UTM_pre_def)
+apply(simp add: Hoare_unhalt_def UTM_pre_def)
 done
 qed
 text {*
 The correctness of @{text "UTM"}, the unhalt case.
 *}
 lemma UTM_uhalt_lemma:
-"\<lbrakk>turing_basic.t_correct tp;
+assumes tm_wf: "tm_wf (tp, 0)"
-\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp));
+and unhalt: "\<forall> stp. (\<not> TSTD (steps0 (Suc 0, Bk\<up>(l), <args>) tp stp))"
-args \<noteq> []\<rbrakk>
+and args: "args \<noteq> []"
-\<Longrightarrow>  \<forall> stp. \<not> isS0 (steps (Suc 0, [], <code tp # args>)  UTM stp)"
+shows " \<forall> stp. \<not> is_final (steps0 (Suc 0, [], <code tp # args>)  UTM stp)"
-using UTM_uhalt_lemma_pre[of tp l args]
+using UTM_uhalt_lemma_pre[of tp l args] assms
 apply(simp add: UTM_pre_def t_utm_def UTM_def F_aprog_def F_tprog_def)
 apply(case_tac "rec_ci rec_F", simp)
 done
 end

changeset 131	e995ae949731
parent 130	1e89c65f844b
child 133	ca7fb6848715