--- a/utm/UTM.thy Sat Sep 29 12:38:12 2012 +0000
+++ b/utm/UTM.thy Mon Oct 15 13:23:52 2012 +0000
@@ -1,4704 +1,5165 @@
-theory UTM
-imports Main uncomputable recursive abacus UF GCD
-begin
-
-section {* Wang coding of input arguments *}
-
-text {*
- The direct compilation of the universal function @{text "rec_F"} can not give us UTM, because @{text "rec_F"} is of arity 2,
- where the first argument represents the Godel coding of the TM being simulated and the second argument represents the right number (in Wang's coding) of the TM tape.
- (Notice, left number is always @{text "0"} at the very beginning). However, UTM needs to simulate the execution of any TM which may
- very well take many input arguments. Therefore, a initialization TM needs to run before the TM compiled from @{text "rec_F"}, and the sequential
- composition of these two TMs will give rise to the UTM we are seeking. The purpose of this initialization TM is to transform the multiple
- input arguments of the TM being simulated into Wang's coding, so that it can be consumed by the TM compiled from @{text "rec_F"} as the second
- argument.
-
- However, this initialization TM (named @{text "t_wcode"}) can not be constructed by compiling from any resurve function, because every recursive
- function takes a fixed number of input arguments, while @{text "t_wcode"} needs to take varying number of arguments and tranform them into
- Wang's coding. Therefore, this section give a direct construction of @{text "t_wcode"} with just some parts being obtained from recursive functions.
-*}
-
-definition rec_twice :: "recf"
- where
- "rec_twice = Cn 1 rec_mult [id 1 0, constn 2]"
-
-definition rec_fourtimes :: "recf"
- where
- "rec_fourtimes = Cn 1 rec_mult [id 1 0, constn 4]"
-
-definition abc_twice :: "abc_prog"
- where
- "abc_twice = (let (aprog, ary, fp) = rec_ci rec_twice in
- aprog [+] dummy_abc ((Suc 0)))"
-
-definition abc_fourtimes :: "abc_prog"
- where
- "abc_fourtimes = (let (aprog, ary, fp) = rec_ci rec_fourtimes in
- aprog [+] dummy_abc ((Suc 0)))"
-
-definition twice_ly :: "nat list"
- where
- "twice_ly = layout_of abc_twice"
-
-definition fourtimes_ly :: "nat list"
- where
- "fourtimes_ly = layout_of abc_fourtimes"
-
-definition t_twice :: "tprog"
- where
- "t_twice = change_termi_state (tm_of (abc_twice) @ (tMp 1 (start_of twice_ly (length abc_twice) - Suc 0)))"
-
-definition t_fourtimes :: "tprog"
- where
- "t_fourtimes = change_termi_state (tm_of (abc_fourtimes) @
- (tMp 1 (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)))"
-
-
-definition t_twice_len :: "nat"
- where
- "t_twice_len = length t_twice div 2"
-
-definition t_wcode_main_first_part:: "tprog"
- where
- "t_wcode_main_first_part \<equiv>
- [(L, 1), (L, 2), (L, 7), (R, 3),
- (R, 4), (W0, 3), (R, 4), (R, 5),
- (W1, 6), (R, 5), (R, 13), (L, 6),
- (R, 0), (R, 8), (R, 9), (Nop, 8),
- (R, 10), (W0, 9), (R, 10), (R, 11),
- (W1, 12), (R, 11), (R, t_twice_len + 14), (L, 12)]"
-
-definition t_wcode_main :: "tprog"
- where
- "t_wcode_main = (t_wcode_main_first_part @ tshift t_twice 12 @ [(L, 1), (L, 1)]
- @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])"
-
-fun bl_bin :: "block list \<Rightarrow> nat"
- where
- "bl_bin [] = 0"
-| "bl_bin (Bk # xs) = 2 * bl_bin xs"
-| "bl_bin (Oc # xs) = Suc (2 * bl_bin xs)"
-
-declare bl_bin.simps[simp del]
-
-type_synonym bin_inv_t = "block list \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
-
-fun wcode_before_double :: "bin_inv_t"
- where
- "wcode_before_double ires rs (l, r) =
- (\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
- r = Oc\<^bsup>(Suc (Suc rs))\<^esup> @ Bk\<^bsup>rn \<^esup>)"
-
-declare wcode_before_double.simps[simp del]
-
-fun wcode_after_double :: "bin_inv_t"
- where
- "wcode_after_double ires rs (l, r) =
- (\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
- r = Oc\<^bsup>Suc (Suc (Suc 2*rs))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-declare wcode_after_double.simps[simp del]
-
-fun wcode_on_left_moving_1_B :: "bin_inv_t"
- where
- "wcode_on_left_moving_1_B ires rs (l, r) =
- (\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Oc # ires \<and>
- r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr > Suc 0 \<and> mr > 0)"
-
-declare wcode_on_left_moving_1_B.simps[simp del]
-
-fun wcode_on_left_moving_1_O :: "bin_inv_t"
- where
- "wcode_on_left_moving_1_O ires rs (l, r) =
- (\<exists> ln rn.
- l = Oc # ires \<and>
- r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-declare wcode_on_left_moving_1_O.simps[simp del]
-
-fun wcode_on_left_moving_1 :: "bin_inv_t"
- where
- "wcode_on_left_moving_1 ires rs (l, r) =
- (wcode_on_left_moving_1_B ires rs (l, r) \<or> wcode_on_left_moving_1_O ires rs (l, r))"
-
-declare wcode_on_left_moving_1.simps[simp del]
-
-fun wcode_on_checking_1 :: "bin_inv_t"
- where
- "wcode_on_checking_1 ires rs (l, r) =
- (\<exists> ln rn. l = ires \<and>
- r = Oc # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wcode_erase1 :: "bin_inv_t"
- where
-"wcode_erase1 ires rs (l, r) =
- (\<exists> ln rn. l = Oc # ires \<and>
- tl r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-declare wcode_erase1.simps [simp del]
-
-fun wcode_on_right_moving_1 :: "bin_inv_t"
- where
- "wcode_on_right_moving_1 ires rs (l, r) =
- (\<exists> ml mr rn.
- l = Bk\<^bsup>ml\<^esup> @ Oc # ires \<and>
- r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr > Suc 0)"
-
-declare wcode_on_right_moving_1.simps [simp del]
-
-declare wcode_on_right_moving_1.simps[simp del]
-
-fun wcode_goon_right_moving_1 :: "bin_inv_t"
- where
- "wcode_goon_right_moving_1 ires rs (l, r) =
- (\<exists> ml mr ln rn.
- l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
- r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr = Suc rs)"
-
-declare wcode_goon_right_moving_1.simps[simp del]
-
-fun wcode_backto_standard_pos_B :: "bin_inv_t"
- where
- "wcode_backto_standard_pos_B ires rs (l, r) =
- (\<exists> ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
- r = Bk # Oc\<^bsup>(Suc (Suc rs))\<^esup> @ Bk\<^bsup>rn \<^esup>)"
-
-declare wcode_backto_standard_pos_B.simps[simp del]
-
-fun wcode_backto_standard_pos_O :: "bin_inv_t"
- where
- "wcode_backto_standard_pos_O ires rs (l, r) =
- (\<exists> ml mr ln rn.
- l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
- r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr = Suc (Suc rs) \<and> mr > 0)"
-
-declare wcode_backto_standard_pos_O.simps[simp del]
-
-fun wcode_backto_standard_pos :: "bin_inv_t"
- where
- "wcode_backto_standard_pos ires rs (l, r) = (wcode_backto_standard_pos_B ires rs (l, r) \<or>
- wcode_backto_standard_pos_O ires rs (l, r))"
-
-declare wcode_backto_standard_pos.simps[simp del]
-
-lemma [simp]: "<0::nat> = [Oc]"
-apply(simp add: tape_of_nat_abv exponent_def tape_of_nat_list.simps)
-done
-
-lemma tape_of_Suc_nat: "<Suc (a ::nat)> = replicate a Oc @ [Oc, Oc]"
-apply(simp add: tape_of_nat_abv exp_ind tape_of_nat_list.simps)
-apply(simp only: exp_ind_def[THEN sym])
-apply(simp only: exp_ind, simp, simp add: exponent_def)
-done
-
-lemma [simp]: "length (<a::nat>) = Suc a"
-apply(simp add: tape_of_nat_abv tape_of_nat_list.simps)
-done
-
-lemma [simp]: "<[a::nat]> = <a>"
-apply(simp add: tape_of_nat_abv tape_of_nl_abv exponent_def
- tape_of_nat_list.simps)
-done
-
-lemma bin_wc_eq: "bl_bin xs = bl2wc xs"
-proof(induct xs)
- show " bl_bin [] = bl2wc []"
- apply(simp add: bl_bin.simps)
- done
-next
- fix a xs
- assume "bl_bin xs = bl2wc xs"
- thus " bl_bin (a # xs) = bl2wc (a # xs)"
- apply(case_tac a, simp_all add: bl_bin.simps bl2wc.simps)
- apply(simp_all add: bl2nat.simps bl2nat_double)
- done
-qed
-
-declare exp_def[simp del]
-
-lemma bl_bin_nat_Suc:
- "bl_bin (<Suc a>) = bl_bin (<a>) + 2^(Suc a)"
-apply(simp add: tape_of_nat_abv bin_wc_eq)
-apply(simp add: bl2wc.simps)
-done
-lemma [simp]: " rev (a\<^bsup>aa\<^esup>) = a\<^bsup>aa\<^esup>"
-apply(simp add: exponent_def)
-done
-
-declare tape_of_nl_abv_cons[simp del]
-
-lemma tape_of_nl_rev: "rev (<lm::nat list>) = (<rev lm>)"
-apply(induct lm rule: list_tl_induct, simp)
-apply(case_tac "list = []", simp add: tape_of_nl_abv tape_of_nat_list.simps)
-apply(simp add: tape_of_nat_list_butlast_last tape_of_nl_abv_cons)
-done
-lemma [simp]: "a\<^bsup>Suc 0\<^esup> = [a]"
-by(simp add: exp_def)
-lemma tape_of_nl_cons_app1: "(<a # xs @ [b]>) = (Oc\<^bsup>Suc a\<^esup> @ Bk # (<xs@ [b]>))"
-apply(case_tac xs, simp add: tape_of_nl_abv tape_of_nat_list.simps)
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
-done
-
-lemma bl_bin_bk_oc[simp]:
- "bl_bin (xs @ [Bk, Oc]) =
- bl_bin xs + 2*2^(length xs)"
-apply(simp add: bin_wc_eq)
-using bl2nat_cons_oc[of "xs @ [Bk]"]
-apply(simp add: bl2nat_cons_bk bl2wc.simps)
-done
-
-lemma tape_of_nat[simp]: "(<a::nat>) = Oc\<^bsup>Suc a\<^esup>"
-apply(simp add: tape_of_nat_abv)
-done
-lemma tape_of_nl_cons_app2: "(<c # xs @ [b]>) = (<c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>)"
-proof(induct "length xs" arbitrary: xs c,
- simp add: tape_of_nl_abv tape_of_nat_list.simps)
- fix x xs c
- assume ind: "\<And>xs c. x = length xs \<Longrightarrow> <c # xs @ [b]> =
- <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>"
- and h: "Suc x = length (xs::nat list)"
- show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>"
- proof(case_tac xs, simp add: tape_of_nl_abv tape_of_nat_list.simps)
- fix a list
- assume g: "xs = a # list"
- hence k: "<a # list @ [b]> = <a # list> @ Bk # Oc\<^bsup>Suc b\<^esup>"
- apply(rule_tac ind)
- using h
- apply(simp)
- done
- from g and k show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>"
- apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
- done
- qed
-qed
-
-lemma [simp]: "length (<aa # a # list>) = Suc (Suc aa) + length (<a # list>)"
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
-done
-
-lemma [simp]: "bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista) @ [Bk, Oc]) =
- bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)) +
- 2* 2^(length (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)))"
-using bl_bin_bk_oc[of "Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)"]
-apply(simp)
-done
-
-lemma [simp]:
- "bl_bin (<aa # list>) + (4 * rs + 4) * 2 ^ (length (<aa # list>) - Suc 0)
- = bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [0]>) + rs * (2 * 2 ^ (aa + length (<list @ [0]>)))"
-apply(case_tac "list", simp add: add_mult_distrib, simp)
-apply(simp add: tape_of_nl_cons_app2 add_mult_distrib)
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
-done
-
-lemma tape_of_nl_app_Suc: "((<list @ [Suc ab]>)) = (<list @ [ab]>) @ [Oc]"
-apply(induct list)
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind)
-apply(case_tac list)
-apply(simp_all add:tape_of_nl_abv tape_of_nat_list.simps exp_ind)
-done
-
-lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]> @ [Oc])
- = bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>) +
- 2^(length (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>))"
-apply(simp add: bin_wc_eq)
-apply(simp add: bl2nat_cons_oc bl2wc.simps)
-using bl2nat_cons_oc[of "Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>"]
-apply(simp)
-done
-lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>) + (4 * 2 ^ (aa + length (<list @ [ab]>)) +
- 4 * (rs * 2 ^ (aa + length (<list @ [ab]>)))) =
- bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [Suc ab]>) +
- rs * (2 * 2 ^ (aa + length (<list @ [Suc ab]>)))"
-apply(simp add: tape_of_nl_app_Suc)
-done
-
-declare tape_of_nat[simp del]
-
-text{* double case*}
-fun wcode_double_case_inv :: "nat \<Rightarrow> bin_inv_t"
- where
- "wcode_double_case_inv st ires rs (l, r) =
- (if st = Suc 0 then wcode_on_left_moving_1 ires rs (l, r)
- else if st = Suc (Suc 0) then wcode_on_checking_1 ires rs (l, r)
- else if st = 3 then wcode_erase1 ires rs (l, r)
- else if st = 4 then wcode_on_right_moving_1 ires rs (l, r)
- else if st = 5 then wcode_goon_right_moving_1 ires rs (l, r)
- else if st = 6 then wcode_backto_standard_pos ires rs (l, r)
- else if st = 13 then wcode_before_double ires rs (l, r)
- else False)"
-
-declare wcode_double_case_inv.simps[simp del]
-
-fun wcode_double_case_state :: "t_conf \<Rightarrow> nat"
- where
- "wcode_double_case_state (st, l, r) =
- 13 - st"
-
-fun wcode_double_case_step :: "t_conf \<Rightarrow> nat"
- where
- "wcode_double_case_step (st, l, r) =
- (if st = Suc 0 then (length l)
- else if st = Suc (Suc 0) then (length r)
- else if st = 3 then
- if hd r = Oc then 1 else 0
- else if st = 4 then (length r)
- else if st = 5 then (length r)
- else if st = 6 then (length l)
- else 0)"
-
-fun wcode_double_case_measure :: "t_conf \<Rightarrow> nat \<times> nat"
- where
- "wcode_double_case_measure (st, l, r) =
- (wcode_double_case_state (st, l, r),
- wcode_double_case_step (st, l, r))"
-
-definition wcode_double_case_le :: "(t_conf \<times> t_conf) set"
- where "wcode_double_case_le \<equiv> (inv_image lex_pair wcode_double_case_measure)"
-
-lemma [intro]: "wf lex_pair"
-by(auto intro:wf_lex_prod simp:lex_pair_def)
-
-lemma wf_wcode_double_case_le[intro]: "wf wcode_double_case_le"
-by(auto intro:wf_inv_image simp: wcode_double_case_le_def )
-term fetch
-
-lemma [simp]: "fetch t_wcode_main (Suc 0) Bk = (L, Suc 0)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
- fetch.simps nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main (Suc 0) Oc = (L, Suc (Suc 0))"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
- fetch.simps nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main (Suc (Suc 0)) Oc = (R, 3)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
- fetch.simps nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main (Suc (Suc (Suc 0))) Bk = (R, 4)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
- fetch.simps nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main (Suc (Suc (Suc 0))) Oc = (W0, 3)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
- fetch.simps nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 4 Bk = (R, 4)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
- fetch.simps nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 4 Oc = (R, 5)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
- fetch.simps nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 5 Oc = (R, 5)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
- fetch.simps nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 5 Bk = (W1, 6)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
- fetch.simps nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 6 Bk = (R, 13)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
- fetch.simps nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 6 Oc = (L, 6)"
-apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
- fetch.simps nth_of.simps)
-done
-lemma [elim]: "Bk\<^bsup>mr\<^esup> = [] \<Longrightarrow> mr = 0"
-apply(case_tac mr, auto simp: exponent_def)
-done
-
-lemma [simp]: "wcode_on_left_moving_1 ires rs (b, []) = False"
-apply(simp add: wcode_on_left_moving_1.simps wcode_on_left_moving_1_B.simps
- wcode_on_left_moving_1_O.simps, auto)
-done
-
-
-declare wcode_on_checking_1.simps[simp del]
-
-lemmas wcode_double_case_inv_simps =
- wcode_on_left_moving_1.simps wcode_on_left_moving_1_O.simps
- wcode_on_left_moving_1_B.simps wcode_on_checking_1.simps
- wcode_erase1.simps wcode_on_right_moving_1.simps
- wcode_goon_right_moving_1.simps wcode_backto_standard_pos.simps
-
-
-lemma [simp]: "wcode_on_left_moving_1 ires rs (b, r) \<Longrightarrow> b \<noteq> []"
-apply(simp add: wcode_double_case_inv_simps, auto)
-done
-
-
-lemma [elim]: "\<lbrakk>wcode_on_left_moving_1 ires rs (b, Bk # list);
- tl b = aa \<and> hd b # Bk # list = ba\<rbrakk> \<Longrightarrow>
- wcode_on_left_moving_1 ires rs (aa, ba)"
-apply(simp only: wcode_on_left_moving_1.simps wcode_on_left_moving_1_O.simps
- wcode_on_left_moving_1_B.simps)
-apply(erule_tac disjE)
-apply(erule_tac exE)+
-apply(case_tac ml, simp)
-apply(rule_tac x = "mr - Suc (Suc 0)" in exI, rule_tac x = rn in exI)
-apply(case_tac mr, simp, case_tac nat, simp, simp add: exp_ind)
-apply(rule_tac disjI1)
-apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = rn in exI,
- simp add: exp_ind_def)
-apply(erule_tac exE)+
-apply(simp)
-done
-
-
-lemma [elim]:
- "\<lbrakk>wcode_on_left_moving_1 ires rs (b, Oc # list); tl b = aa \<and> hd b # Oc # list = ba\<rbrakk>
- \<Longrightarrow> wcode_on_checking_1 ires rs (aa, ba)"
-apply(simp only: wcode_double_case_inv_simps)
-apply(erule_tac disjE)
-apply(erule_tac [!] exE)+
-apply(case_tac mr, simp, simp add: exp_ind_def)
-apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
-done
-
-
-lemma [simp]: "wcode_on_checking_1 ires rs (b, []) = False"
-apply(auto simp: wcode_double_case_inv_simps)
-done
-
-lemma [simp]: "wcode_on_checking_1 ires rs (b, Bk # list) = False"
-apply(auto simp: wcode_double_case_inv_simps)
-done
-
-lemma [elim]: "\<lbrakk>wcode_on_checking_1 ires rs (b, Oc # ba);Oc # b = aa \<and> list = ba\<rbrakk>
- \<Longrightarrow> wcode_erase1 ires rs (aa, ba)"
-apply(simp only: wcode_double_case_inv_simps)
-apply(erule_tac exE)+
-apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
-done
-
-
-lemma [simp]: "wcode_on_checking_1 ires rs (b, []) = False"
-apply(simp add: wcode_double_case_inv_simps)
-done
-
-lemma [simp]: "wcode_on_checking_1 ires rs ([], Bk # list) = False"
-apply(simp add: wcode_double_case_inv_simps)
-done
-
-lemma [simp]: "wcode_erase1 ires rs (b, []) = False"
-apply(simp add: wcode_double_case_inv_simps)
-done
-
-lemma [simp]: "wcode_on_right_moving_1 ires rs (b, []) = False"
-apply(simp add: wcode_double_case_inv_simps exp_ind_def)
-done
-
-lemma [simp]: "wcode_on_right_moving_1 ires rs (b, []) = False"
-apply(simp add: wcode_double_case_inv_simps exp_ind_def)
-done
-
-lemma [elim]: "\<lbrakk>wcode_on_right_moving_1 ires rs (b, Bk # ba); Bk # b = aa \<and> list = b\<rbrakk> \<Longrightarrow>
- wcode_on_right_moving_1 ires rs (aa, ba)"
-apply(simp only: wcode_double_case_inv_simps)
-apply(erule_tac exE)+
-apply(rule_tac x = "Suc ml" in exI, rule_tac x = "mr - Suc 0" in exI,
- rule_tac x = rn in exI)
-apply(simp add: exp_ind_def)
-apply(case_tac mr, simp, simp add: exp_ind_def)
-done
-
-lemma [elim]:
- "\<lbrakk>wcode_on_right_moving_1 ires rs (b, Oc # ba); Oc # b = aa \<and> list = ba\<rbrakk>
- \<Longrightarrow> wcode_goon_right_moving_1 ires rs (aa, ba)"
-apply(simp only: wcode_double_case_inv_simps)
-apply(erule_tac exE)+
-apply(rule_tac x = "Suc 0" in exI, rule_tac x = "rs" in exI,
- rule_tac x = "ml - Suc (Suc 0)" in exI, rule_tac x = rn in exI)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac ml, simp, case_tac nat, simp, simp)
-apply(simp add: exp_ind_def)
-done
-
-lemma [simp]:
- "wcode_on_right_moving_1 ires rs (b, []) \<Longrightarrow> False"
-apply(simp add: wcode_double_case_inv_simps exponent_def)
-done
-
-lemma [elim]: "\<lbrakk>wcode_erase1 ires rs (b, Bk # ba); Bk # b = aa \<and> list = ba; c = Bk # ba\<rbrakk>
- \<Longrightarrow> wcode_on_right_moving_1 ires rs (aa, ba)"
-apply(simp only: wcode_double_case_inv_simps)
-apply(erule_tac exE)+
-apply(rule_tac x = "Suc 0" in exI, rule_tac x = "Suc (Suc ln)" in exI,
- rule_tac x = rn in exI, simp add: exp_ind)
-done
-
-lemma [elim]: "\<lbrakk>wcode_erase1 ires rs (aa, Oc # list); b = aa \<and> Bk # list = ba\<rbrakk> \<Longrightarrow>
- wcode_erase1 ires rs (aa, ba)"
-apply(simp only: wcode_double_case_inv_simps)
-apply(erule_tac exE)+
-apply(rule_tac x = ln in exI, rule_tac x = rn in exI, auto)
-done
-
-lemma [elim]: "\<lbrakk>wcode_goon_right_moving_1 ires rs (aa, []); b = aa \<and> [Oc] = ba\<rbrakk>
- \<Longrightarrow> wcode_backto_standard_pos ires rs (aa, ba)"
-apply(simp only: wcode_double_case_inv_simps)
-apply(erule_tac exE)+
-apply(rule_tac disjI2)
-apply(simp only:wcode_backto_standard_pos_O.simps)
-apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI, rule_tac x = ln in exI,
- rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp_all add: exponent_def)
-done
-
-lemma [elim]:
- "\<lbrakk>wcode_goon_right_moving_1 ires rs (aa, Bk # list); b = aa \<and> Oc # list = ba\<rbrakk>
- \<Longrightarrow> wcode_backto_standard_pos ires rs (aa, ba)"
-apply(simp only: wcode_double_case_inv_simps)
-apply(erule_tac exE)+
-apply(rule_tac disjI2)
-apply(simp only:wcode_backto_standard_pos_O.simps)
-apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI, rule_tac x = ln in exI,
- rule_tac x = "rn - Suc 0" in exI, simp)
-apply(case_tac mr, simp, case_tac rn, simp, simp_all add: exp_ind_def)
-done
-
-lemma [elim]: "\<lbrakk>wcode_goon_right_moving_1 ires rs (b, Oc # ba); Oc # b = aa \<and> list = ba\<rbrakk>
- \<Longrightarrow> wcode_goon_right_moving_1 ires rs (aa, ba)"
-apply(simp only: wcode_double_case_inv_simps)
-apply(erule_tac exE)+
-apply(rule_tac x = "Suc ml" in exI, rule_tac x = "mr - Suc 0" in exI,
- rule_tac x = ln in exI, rule_tac x = rn in exI)
-apply(simp add: exp_ind_def)
-apply(case_tac mr, simp, case_tac rn, simp_all add: exp_ind_def)
-done
-
-lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, []); Bk # b = aa\<rbrakk> \<Longrightarrow> False"
-apply(auto simp: wcode_double_case_inv_simps wcode_backto_standard_pos_O.simps
- wcode_backto_standard_pos_B.simps)
-apply(case_tac mr, simp_all add: exp_ind_def)
-done
-
-lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, Bk # ba); Bk # b = aa \<and> list = ba\<rbrakk>
- \<Longrightarrow> wcode_before_double ires rs (aa, ba)"
-apply(simp only: wcode_double_case_inv_simps wcode_backto_standard_pos_B.simps
- wcode_backto_standard_pos_O.simps wcode_before_double.simps)
-apply(erule_tac disjE)
-apply(erule_tac exE)+
-apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
-apply(auto)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wcode_backto_standard_pos ires rs ([], Oc # list) = False"
-apply(auto simp: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps
- wcode_backto_standard_pos_O.simps)
-done
-
-lemma [simp]: "wcode_backto_standard_pos ires rs (b, []) = False"
-apply(auto simp: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps
- wcode_backto_standard_pos_O.simps)
-apply(case_tac mr, simp, simp add: exp_ind_def)
-done
-
-lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, Oc # list); tl b = aa; hd b # Oc # list = ba\<rbrakk>
- \<Longrightarrow> wcode_backto_standard_pos ires rs (aa, ba)"
-apply(simp only: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps
- wcode_backto_standard_pos_O.simps)
-apply(erule_tac disjE)
-apply(simp)
-apply(erule_tac exE)+
-apply(case_tac ml, simp)
-apply(rule_tac disjI1, rule_tac conjI)
-apply(rule_tac x = ln in exI, simp, rule_tac x = rn in exI, simp)
-apply(rule_tac disjI2)
-apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = ln in exI,
- rule_tac x = rn in exI, simp)
-apply(simp add: exp_ind_def)
-done
-
-declare new_tape.simps[simp del] nth_of.simps[simp del] fetch.simps[simp del]
-lemma wcode_double_case_first_correctness:
- "let P = (\<lambda> (st, l, r). st = 13) in
- let Q = (\<lambda> (st, l, r). wcode_double_case_inv st ires rs (l, r)) in
- let f = (\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp) in
- \<exists> n .P (f n) \<and> Q (f (n::nat))"
-proof -
- let ?P = "(\<lambda> (st, l, r). st = 13)"
- let ?Q = "(\<lambda> (st, l, r). wcode_double_case_inv st ires rs (l, r))"
- let ?f = "(\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)"
- have "\<exists> n. ?P (?f n) \<and> ?Q (?f (n::nat))"
- proof(rule_tac halt_lemma2)
- show "wf wcode_double_case_le"
- by auto
- next
- show "\<forall> na. \<not> ?P (?f na) \<and> ?Q (?f na) \<longrightarrow>
- ?Q (?f (Suc na)) \<and> (?f (Suc na), ?f na) \<in> wcode_double_case_le"
- proof(rule_tac allI, case_tac "?f na", simp add: tstep_red)
- fix na a b c
- show "a \<noteq> 13 \<and> wcode_double_case_inv a ires rs (b, c) \<longrightarrow>
- (case tstep (a, b, c) t_wcode_main of (st, x) \<Rightarrow>
- wcode_double_case_inv st ires rs x) \<and>
- (tstep (a, b, c) t_wcode_main, a, b, c) \<in> wcode_double_case_le"
- apply(rule_tac impI, simp add: wcode_double_case_inv.simps)
- apply(auto split: if_splits simp: tstep.simps,
- case_tac [!] c, simp_all, case_tac [!] "(c::block list)!0")
- apply(simp_all add: new_tape.simps wcode_double_case_inv.simps wcode_double_case_le_def
- lex_pair_def)
- apply(auto split: if_splits)
- done
- qed
- next
- show "?Q (?f 0)"
- apply(simp add: steps.simps wcode_double_case_inv.simps
- wcode_on_left_moving_1.simps
- wcode_on_left_moving_1_B.simps)
- apply(rule_tac disjI1)
- apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def)
- apply(rule_tac x = "Suc 0" in exI, simp add: exp_ind_def)
- apply(auto)
- done
- next
- show "\<not> ?P (?f 0)"
- apply(simp add: steps.simps)
- done
- qed
- thus "let P = \<lambda>(st, l, r). st = 13;
- Q = \<lambda>(st, l, r). wcode_double_case_inv st ires rs (l, r);
- f = steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main
- in \<exists>n. P (f n) \<and> Q (f n)"
- apply(simp add: Let_def)
- done
-qed
-
-lemma [elim]: "t_ncorrect tp
- \<Longrightarrow> t_ncorrect (abacus.tshift tp a)"
-apply(simp add: t_ncorrect.simps shift_length)
-done
-
-lemma tshift_fetch: "\<lbrakk> fetch tp a b = (aa, st'); 0 < st'\<rbrakk>
- \<Longrightarrow> fetch (abacus.tshift tp (length tp1 div 2)) a b
- = (aa, st' + length tp1 div 2)"
-apply(subgoal_tac "a > 0")
-apply(auto simp: fetch.simps nth_of.simps shift_length nth_map
- tshift.simps split: block.splits if_splits)
-done
-
-lemma t_steps_steps_eq: "\<lbrakk>steps (st, l, r) tp stp = (st', l', r');
- 0 < st';
- 0 < st \<and> st \<le> length tp div 2;
- t_ncorrect tp1;
- t_ncorrect tp\<rbrakk>
- \<Longrightarrow> t_steps (st + length tp1 div 2, l, r) (tshift tp (length tp1 div 2),
- length tp1 div 2) stp
- = (st' + length tp1 div 2, l', r')"
-apply(induct stp arbitrary: st' l' r', simp add: steps.simps t_steps.simps,
- simp add: tstep_red stepn)
-apply(case_tac "(steps (st, l, r) tp stp)", simp)
-proof -
- fix stp st' l' r' a b c
- assume ind: "\<And>st' l' r'.
- \<lbrakk>a = st' \<and> b = l' \<and> c = r'; 0 < st'\<rbrakk>
- \<Longrightarrow> t_steps (st + length tp1 div 2, l, r)
- (abacus.tshift tp (length tp1 div 2), length tp1 div 2) stp =
- (st' + length tp1 div 2, l', r')"
- and h: "tstep (a, b, c) tp = (st', l', r')" "0 < st'" "t_ncorrect tp1" "t_ncorrect tp"
- have k: "t_steps (st + length tp1 div 2, l, r) (abacus.tshift tp (length tp1 div 2),
- length tp1 div 2) stp = (a + length tp1 div 2, b, c)"
- apply(rule_tac ind, simp)
- using h
- apply(case_tac a, simp_all add: tstep.simps fetch.simps)
- done
- from h and this show "t_step (t_steps (st + length tp1 div 2, l, r) (abacus.tshift tp (length tp1 div 2), length tp1 div 2) stp)
- (abacus.tshift tp (length tp1 div 2), length tp1 div 2) =
- (st' + length tp1 div 2, l', r')"
- apply(simp add: k)
- apply(simp add: tstep.simps t_step.simps)
- apply(case_tac "fetch tp a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
- apply(subgoal_tac "fetch (abacus.tshift tp (length tp1 div 2)) a
- (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) = (aa, st' + length tp1 div 2)", simp)
- apply(simp add: tshift_fetch)
- done
-qed
-
-lemma t_tshift_lemma: "\<lbrakk> steps (st, l, r) tp stp = (st', l', r');
- st' \<noteq> 0;
- stp > 0;
- 0 < st \<and> st \<le> length tp div 2;
- t_ncorrect tp1;
- t_ncorrect tp;
- t_ncorrect tp2
- \<rbrakk>
- \<Longrightarrow> \<exists> stp>0. steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2) stp
- = (st' + length tp1 div 2, l', r')"
-proof -
- assume h: "steps (st, l, r) tp stp = (st', l', r')"
- "st' \<noteq> 0" "stp > 0"
- "0 < st \<and> st \<le> length tp div 2"
- "t_ncorrect tp1"
- "t_ncorrect tp"
- "t_ncorrect tp2"
- from h have
- "\<exists>stp>0. t_steps (st + length tp1 div 2, l, r) (tp1 @ abacus.tshift tp (length tp1 div 2) @ tp2, 0) stp =
- (st' + length tp1 div 2, l', r')"
- apply(rule_tac stp = stp in turing_shift, simp_all add: shift_length)
- apply(simp add: t_steps_steps_eq)
- apply(simp add: t_ncorrect.simps shift_length)
- done
- thus "\<exists> stp>0. steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2) stp
- = (st' + length tp1 div 2, l', r')"
- apply(erule_tac exE)
- apply(rule_tac x = stp in exI, simp)
- apply(subgoal_tac "length (tp1 @ abacus.tshift tp (length tp1 div 2) @ tp2) mod 2 = 0")
- apply(simp only: steps_eq)
- using h
- apply(auto simp: t_ncorrect.simps shift_length)
- apply arith
- done
-qed
-
-
-lemma t_twice_len_ge: "Suc 0 \<le> length t_twice div 2"
-apply(simp add: t_twice_def tMp.simps shift_length)
-done
-
-lemma [intro]: "rec_calc_rel (recf.id (Suc 0) 0) [rs] rs"
- apply(rule_tac calc_id, simp_all)
- done
-
-lemma [intro]: "rec_calc_rel (constn 2) [rs] 2"
-using prime_rel_exec_eq[of "constn 2" "[rs]" 2]
-apply(subgoal_tac "primerec (constn 2) 1", auto)
-done
-
-lemma [intro]: "rec_calc_rel rec_mult [rs, 2] (2 * rs)"
-using prime_rel_exec_eq[of "rec_mult" "[rs, 2]" "2*rs"]
-apply(subgoal_tac "primerec rec_mult (Suc (Suc 0))", auto)
-done
-lemma t_twice_correct: "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
- (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) stp =
- (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-proof(case_tac "rec_ci rec_twice")
- fix a b c
- assume h: "rec_ci rec_twice = (a, b, c)"
- have "\<exists>stp m l. steps (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<^bsup>n\<^esup>) (tm_of abc_twice @ tMp (Suc 0)
- (start_of twice_ly (length abc_twice) - 1)) stp = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2*rs)\<^esup> @ Bk\<^bsup>l\<^esup>)"
- proof(rule_tac t_compiled_by_rec)
- show "rec_ci rec_twice = (a, b, c)" by (simp add: h)
- next
- show "rec_calc_rel rec_twice [rs] (2 * rs)"
- apply(simp add: rec_twice_def)
- apply(rule_tac rs = "[rs, 2]" in calc_cn, simp_all)
- apply(rule_tac allI, case_tac k, auto)
- done
- next
- show "length [rs] = Suc 0" by simp
- next
- show "layout_of (a [+] dummy_abc (Suc 0)) = layout_of (a [+] dummy_abc (Suc 0))"
- by simp
- next
- show "start_of twice_ly (length abc_twice) =
- start_of (layout_of (a [+] dummy_abc (Suc 0))) (length (a [+] dummy_abc (Suc 0)))"
- using h
- apply(simp add: twice_ly_def abc_twice_def)
- done
- next
- show "tm_of abc_twice = tm_of (a [+] dummy_abc (Suc 0))"
- using h
- apply(simp add: abc_twice_def)
- done
- qed
- thus "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
- (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) stp =
- (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
- done
-qed
-
-lemma change_termi_state_fetch: "\<lbrakk>fetch ap a b = (aa, st); st > 0\<rbrakk>
- \<Longrightarrow> fetch (change_termi_state ap) a b = (aa, st)"
-apply(case_tac b, auto simp: fetch.simps nth_of.simps change_termi_state.simps nth_map
- split: if_splits block.splits)
-done
-
-lemma change_termi_state_exec_in_range:
- "\<lbrakk>steps (st, l, r) ap stp = (st', l', r'); st' \<noteq> 0\<rbrakk>
- \<Longrightarrow> steps (st, l, r) (change_termi_state ap) stp = (st', l', r')"
-proof(induct stp arbitrary: st l r st' l' r', simp add: steps.simps)
- fix stp st l r st' l' r'
- assume ind: "\<And>st l r st' l' r'.
- \<lbrakk>steps (st, l, r) ap stp = (st', l', r'); st' \<noteq> 0\<rbrakk> \<Longrightarrow>
- steps (st, l, r) (change_termi_state ap) stp = (st', l', r')"
- and h: "steps (st, l, r) ap (Suc stp) = (st', l', r')" "st' \<noteq> 0"
- from h show "steps (st, l, r) (change_termi_state ap) (Suc stp) = (st', l', r')"
- proof(simp add: tstep_red, case_tac "steps (st, l, r) ap stp", simp)
- fix a b c
- assume g: "steps (st, l, r) ap stp = (a, b, c)"
- "tstep (a, b, c) ap = (st', l', r')" "0 < st'"
- hence "steps (st, l, r) (change_termi_state ap) stp = (a, b, c)"
- apply(rule_tac ind, simp)
- apply(case_tac a, simp_all add: tstep_0)
- done
- from g and this show "tstep (steps (st, l, r) (change_termi_state ap) stp)
- (change_termi_state ap) = (st', l', r')"
- apply(simp add: tstep.simps)
- apply(case_tac "fetch ap a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
- apply(subgoal_tac "fetch (change_termi_state ap) a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)
- = (aa, st')", simp)
- apply(simp add: change_termi_state_fetch)
- done
- qed
-qed
-
-lemma change_termi_state_fetch0:
- "\<lbrakk>0 < a; a \<le> length ap div 2; t_correct ap; fetch ap a b = (aa, 0)\<rbrakk>
- \<Longrightarrow> fetch (change_termi_state ap) a b = (aa, Suc (length ap div 2))"
-apply(case_tac b, auto simp: fetch.simps nth_of.simps change_termi_state.simps nth_map
- split: if_splits block.splits)
-done
-
-lemma turing_change_termi_state:
- "\<lbrakk>steps (Suc 0, l, r) ap stp = (0, l', r'); t_correct ap\<rbrakk>
- \<Longrightarrow> \<exists> stp. steps (Suc 0, l, r) (change_termi_state ap) stp =
- (Suc (length ap div 2), l', r')"
-apply(drule first_halt_point)
-apply(erule_tac exE)
-apply(rule_tac x = "Suc stp" in exI, simp add: tstep_red)
-apply(case_tac "steps (Suc 0, l, r) ap stp")
-apply(simp add: isS0_def change_termi_state_exec_in_range)
-apply(subgoal_tac "steps (Suc 0, l, r) (change_termi_state ap) stp = (a, b, c)", simp)
-apply(simp add: tstep.simps)
-apply(case_tac "fetch ap a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
-apply(subgoal_tac "fetch (change_termi_state ap) a
- (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) = (aa, Suc (length ap div 2))", simp)
-apply(rule_tac ap = ap in change_termi_state_fetch0, simp_all)
-apply(rule_tac tp = "(l, r)" and l = b and r = c and stp = stp and A = ap in s_keep, simp_all)
-apply(simp add: change_termi_state_exec_in_range)
-done
-
-lemma t_twice_change_term_state:
- "\<exists> stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp
- = (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-using t_twice_correct[of ires rs n]
-apply(erule_tac exE)
-apply(erule_tac exE)
-apply(erule_tac exE)
-proof(drule_tac turing_change_termi_state)
- fix stp ln rn
- show "t_correct (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0))"
- apply(rule_tac t_compiled_correct, simp_all)
- apply(simp add: twice_ly_def)
- done
-next
- fix stp ln rn
- show "\<exists>stp. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
- (change_termi_state (tm_of abc_twice @ tMp (Suc 0)
- (start_of twice_ly (length abc_twice) - Suc 0))) stp =
- (Suc (length (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) div 2),
- Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) \<Longrightarrow>
- \<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp =
- (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- apply(erule_tac exE)
- apply(simp add: t_twice_len_def t_twice_def)
- apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
- done
-qed
-
-lemma t_twice_append_pre:
- "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp
- = (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)
- \<Longrightarrow> \<exists> stp>0. steps (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
- (t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @
- ([(L, 1), (L, 1)] @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp
- = (Suc (t_twice_len) + length t_wcode_main_first_part div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-proof(rule_tac t_tshift_lemma, simp_all add: t_twice_len_ge)
- assume "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp =
- (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- thus "0 < stp"
- apply(case_tac stp, simp add: steps.simps t_twice_len_ge t_twice_len_def)
- using t_twice_len_ge
- apply(simp, simp)
- done
-next
- show "t_ncorrect t_wcode_main_first_part"
- apply(simp add: t_ncorrect.simps t_wcode_main_first_part_def)
- done
-next
- show "t_ncorrect t_twice"
- using length_tm_even[of abc_twice]
- apply(auto simp: t_ncorrect.simps t_twice_def)
- apply(arith)
- done
-next
- show "t_ncorrect ((L, Suc 0) # (L, Suc 0) #
- abacus.tshift t_fourtimes (t_twice_len + 13) @ [(L, Suc 0), (L, Suc 0)])"
- using length_tm_even[of abc_fourtimes]
- apply(simp add: t_ncorrect.simps shift_length t_fourtimes_def)
- apply arith
- done
-qed
-
-lemma t_twice_append:
- "\<exists> stp ln rn. steps (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
- (t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @
- ([(L, 1), (L, 1)] @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp
- = (Suc (t_twice_len) + length t_wcode_main_first_part div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- using t_twice_change_term_state[of ires rs n]
- apply(erule_tac exE)
- apply(erule_tac exE)
- apply(erule_tac exE)
- apply(drule_tac t_twice_append_pre)
- apply(erule_tac exE)
- apply(rule_tac x = stpa in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
- apply(simp)
- done
-
-lemma [simp]: "fetch t_wcode_main (Suc (t_twice_len + length t_wcode_main_first_part div 2)) Oc
- = (L, Suc 0)"
-apply(subgoal_tac "length (t_twice) mod 2 = 0")
-apply(simp add: t_wcode_main_def nth_append fetch.simps t_wcode_main_first_part_def
- nth_of.simps shift_length t_twice_len_def, auto)
-apply(simp add: t_twice_def)
-apply(subgoal_tac "length (tm_of abc_twice) mod 2 = 0")
-apply arith
-apply(rule_tac tm_even)
-done
-
-lemma wcode_jump1:
- "\<exists> stp ln rn. steps (Suc (t_twice_len) + length t_wcode_main_first_part div 2,
- Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>n\<^esup>)
- t_wcode_main stp
- = (Suc 0, Bk\<^bsup>ln\<^esup> @ Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-apply(rule_tac x = "Suc 0" in exI, rule_tac x = "m" in exI, rule_tac x = n in exI)
-apply(simp add: steps.simps tstep.simps exp_ind_def new_tape.simps)
-apply(case_tac m, simp, simp add: exp_ind_def)
-apply(simp add: exp_ind_def[THEN sym] exp_ind[THEN sym])
-done
-
-lemma wcode_main_first_part_len:
- "length t_wcode_main_first_part = 24"
- apply(simp add: t_wcode_main_first_part_def)
- done
-
-lemma wcode_double_case:
- shows "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-proof -
- have "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (13, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- using wcode_double_case_first_correctness[of ires rs m n]
- apply(simp)
- apply(erule_tac exE)
- apply(case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires,
- Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na",
- auto simp: wcode_double_case_inv.simps
- wcode_before_double.simps)
- apply(rule_tac x = na in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
- apply(simp)
- done
- from this obtain stpa lna rna where stp1:
- "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
- (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
- have "\<exists> stp ln rn. steps (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp =
- (13 + t_twice_len, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- using t_twice_append[of "Bk\<^bsup>lna\<^esup> @ Oc # ires" "Suc rs" rna]
- apply(erule_tac exE)
- apply(erule_tac exE)
- apply(erule_tac exE)
- apply(simp add: wcode_main_first_part_len)
- apply(rule_tac x = stp in exI, rule_tac x = "ln + lna" in exI,
- rule_tac x = rn in exI)
- apply(simp add: t_wcode_main_def)
- apply(simp add: exp_ind_def[THEN sym] exp_add[THEN sym])
- done
- from this obtain stpb lnb rnb where stp2:
- "steps (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb =
- (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>)" by blast
- have "\<exists>stp ln rn. steps (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires,
- Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stp =
- (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- using wcode_jump1[of lnb "Oc # ires" "Suc rs" rnb]
- apply(erule_tac exE)
- apply(erule_tac exE)
- apply(erule_tac exE)
- apply(rule_tac x = stp in exI,
- rule_tac x = ln in exI,
- rule_tac x = rn in exI, simp add:wcode_main_first_part_len t_wcode_main_def)
- apply(subgoal_tac "Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc # ires = Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires", simp)
- apply(simp add: exp_ind_def[THEN sym] exp_ind[THEN sym])
- apply(simp)
- apply(case_tac lnb, simp, simp add: exp_ind_def[THEN sym] exp_ind)
- done
- from this obtain stpc lnc rnc where stp3:
- "steps (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires,
- Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stpc =
- (Suc 0, Bk # Bk\<^bsup>lnc\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnc\<^esup>)"
- by blast
- from stp1 stp2 stp3 show "?thesis"
- apply(rule_tac x = "stpa + stpb + stpc" in exI, rule_tac x = lnc in exI,
- rule_tac x = rnc in exI)
- apply(simp add: steps_add)
- done
-qed
-
-
-(* Begin: fourtime_case*)
-fun wcode_on_left_moving_2_B :: "bin_inv_t"
- where
- "wcode_on_left_moving_2_B ires rs (l, r) =
- (\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Oc # ires \<and>
- r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr > Suc 0 \<and> mr > 0)"
-
-fun wcode_on_left_moving_2_O :: "bin_inv_t"
- where
- "wcode_on_left_moving_2_O ires rs (l, r) =
- (\<exists> ln rn. l = Bk # Oc # ires \<and>
- r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wcode_on_left_moving_2 :: "bin_inv_t"
- where
- "wcode_on_left_moving_2 ires rs (l, r) =
- (wcode_on_left_moving_2_B ires rs (l, r) \<or>
- wcode_on_left_moving_2_O ires rs (l, r))"
-
-fun wcode_on_checking_2 :: "bin_inv_t"
- where
- "wcode_on_checking_2 ires rs (l, r) =
- (\<exists> ln rn. l = Oc#ires \<and>
- r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wcode_goon_checking :: "bin_inv_t"
- where
- "wcode_goon_checking ires rs (l, r) =
- (\<exists> ln rn. l = ires \<and>
- r = Oc # Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wcode_right_move :: "bin_inv_t"
- where
- "wcode_right_move ires rs (l, r) =
- (\<exists> ln rn. l = Oc # ires \<and>
- r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wcode_erase2 :: "bin_inv_t"
- where
- "wcode_erase2 ires rs (l, r) =
- (\<exists> ln rn. l = Bk # Oc # ires \<and>
- tl r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wcode_on_right_moving_2 :: "bin_inv_t"
- where
- "wcode_on_right_moving_2 ires rs (l, r) =
- (\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # ires \<and>
- r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr > Suc 0)"
-
-fun wcode_goon_right_moving_2 :: "bin_inv_t"
- where
- "wcode_goon_right_moving_2 ires rs (l, r) =
- (\<exists> ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
- r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = Suc rs)"
-
-fun wcode_backto_standard_pos_2_B :: "bin_inv_t"
- where
- "wcode_backto_standard_pos_2_B ires rs (l, r) =
- (\<exists> ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
- r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wcode_backto_standard_pos_2_O :: "bin_inv_t"
- where
- "wcode_backto_standard_pos_2_O ires rs (l, r) =
- (\<exists> ml mr ln rn. l = Oc\<^bsup>ml \<^esup>@ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
- r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr = (Suc (Suc rs)) \<and> mr > 0)"
-
-fun wcode_backto_standard_pos_2 :: "bin_inv_t"
- where
- "wcode_backto_standard_pos_2 ires rs (l, r) =
- (wcode_backto_standard_pos_2_O ires rs (l, r) \<or>
- wcode_backto_standard_pos_2_B ires rs (l, r))"
-
-fun wcode_before_fourtimes :: "bin_inv_t"
- where
- "wcode_before_fourtimes ires rs (l, r) =
- (\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
- r = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-declare wcode_on_left_moving_2_B.simps[simp del] wcode_on_left_moving_2.simps[simp del]
- wcode_on_left_moving_2_O.simps[simp del] wcode_on_checking_2.simps[simp del]
- wcode_goon_checking.simps[simp del] wcode_right_move.simps[simp del]
- wcode_erase2.simps[simp del]
- wcode_on_right_moving_2.simps[simp del] wcode_goon_right_moving_2.simps[simp del]
- wcode_backto_standard_pos_2_B.simps[simp del] wcode_backto_standard_pos_2_O.simps[simp del]
- wcode_backto_standard_pos_2.simps[simp del]
-
-lemmas wcode_fourtimes_invs =
- wcode_on_left_moving_2_B.simps wcode_on_left_moving_2.simps
- wcode_on_left_moving_2_O.simps wcode_on_checking_2.simps
- wcode_goon_checking.simps wcode_right_move.simps
- wcode_erase2.simps
- wcode_on_right_moving_2.simps wcode_goon_right_moving_2.simps
- wcode_backto_standard_pos_2_B.simps wcode_backto_standard_pos_2_O.simps
- wcode_backto_standard_pos_2.simps
-
-fun wcode_fourtimes_case_inv :: "nat \<Rightarrow> bin_inv_t"
- where
- "wcode_fourtimes_case_inv st ires rs (l, r) =
- (if st = Suc 0 then wcode_on_left_moving_2 ires rs (l, r)
- else if st = Suc (Suc 0) then wcode_on_checking_2 ires rs (l, r)
- else if st = 7 then wcode_goon_checking ires rs (l, r)
- else if st = 8 then wcode_right_move ires rs (l, r)
- else if st = 9 then wcode_erase2 ires rs (l, r)
- else if st = 10 then wcode_on_right_moving_2 ires rs (l, r)
- else if st = 11 then wcode_goon_right_moving_2 ires rs (l, r)
- else if st = 12 then wcode_backto_standard_pos_2 ires rs (l, r)
- else if st = t_twice_len + 14 then wcode_before_fourtimes ires rs (l, r)
- else False)"
-
-declare wcode_fourtimes_case_inv.simps[simp del]
-
-fun wcode_fourtimes_case_state :: "t_conf \<Rightarrow> nat"
- where
- "wcode_fourtimes_case_state (st, l, r) = 13 - st"
-
-fun wcode_fourtimes_case_step :: "t_conf \<Rightarrow> nat"
- where
- "wcode_fourtimes_case_step (st, l, r) =
- (if st = Suc 0 then length l
- else if st = 9 then
- (if hd r = Oc then 1
- else 0)
- else if st = 10 then length r
- else if st = 11 then length r
- else if st = 12 then length l
- else 0)"
-
-fun wcode_fourtimes_case_measure :: "t_conf \<Rightarrow> nat \<times> nat"
- where
- "wcode_fourtimes_case_measure (st, l, r) =
- (wcode_fourtimes_case_state (st, l, r),
- wcode_fourtimes_case_step (st, l, r))"
-
-definition wcode_fourtimes_case_le :: "(t_conf \<times> t_conf) set"
- where "wcode_fourtimes_case_le \<equiv> (inv_image lex_pair wcode_fourtimes_case_measure)"
-
-lemma wf_wcode_fourtimes_case_le[intro]: "wf wcode_fourtimes_case_le"
-by(auto intro:wf_inv_image simp: wcode_fourtimes_case_le_def)
-
-lemma [simp]: "fetch t_wcode_main (Suc (Suc 0)) Bk = (L, 7)"
-apply(simp add: t_wcode_main_def fetch.simps
- t_wcode_main_first_part_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 7 Oc = (R, 8)"
-apply(simp add: t_wcode_main_def fetch.simps
- t_wcode_main_first_part_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 8 Bk = (R, 9)"
-apply(simp add: t_wcode_main_def fetch.simps
- t_wcode_main_first_part_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 9 Bk = (R, 10)"
-apply(simp add: t_wcode_main_def fetch.simps
- t_wcode_main_first_part_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 9 Oc = (W0, 9)"
-apply(simp add: t_wcode_main_def fetch.simps
- t_wcode_main_first_part_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 10 Bk = (R, 10)"
-apply(simp add: t_wcode_main_def fetch.simps
- t_wcode_main_first_part_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 10 Oc = (R, 11)"
-apply(simp add: t_wcode_main_def fetch.simps
- t_wcode_main_first_part_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 11 Bk = (W1, 12)"
-apply(simp add: t_wcode_main_def fetch.simps
- t_wcode_main_first_part_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 11 Oc = (R, 11)"
-apply(simp add: t_wcode_main_def fetch.simps
- t_wcode_main_first_part_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 12 Oc = (L, 12)"
-apply(simp add: t_wcode_main_def fetch.simps
- t_wcode_main_first_part_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_main 12 Bk = (R, t_twice_len + 14)"
-apply(simp add: t_wcode_main_def fetch.simps
- t_wcode_main_first_part_def nth_of.simps)
-done
-
-
-lemma [simp]: "wcode_on_left_moving_2 ires rs (b, []) = False"
-apply(auto simp: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_on_checking_2 ires rs (b, []) = False"
-apply(auto simp: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_goon_checking ires rs (b, []) = False"
-apply(auto simp: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_right_move ires rs (b, []) = False"
-apply(auto simp: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_erase2 ires rs (b, []) = False"
-apply(auto simp: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_on_right_moving_2 ires rs (b, []) = False"
-apply(auto simp: wcode_fourtimes_invs exponent_def)
-done
-
-lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, []) = False"
-apply(auto simp: wcode_fourtimes_invs exponent_def)
-done
-
-lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(simp add: wcode_fourtimes_invs, auto)
-done
-
-lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \<Longrightarrow> wcode_on_left_moving_2 ires rs (tl b, hd b # Bk # list)"
-apply(simp only: wcode_fourtimes_invs)
-apply(erule_tac disjE)
-apply(erule_tac exE)+
-apply(case_tac ml, simp)
-apply(rule_tac x = "mr - (Suc (Suc 0))" in exI, rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp, case_tac nat, simp, simp add: exp_ind)
-apply(rule_tac disjI1)
-apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = rn in exI,
- simp add: exp_ind_def)
-apply(simp)
-done
-
-lemma [simp]: "wcode_on_checking_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(auto simp: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_on_checking_2 ires rs (b, Bk # list)
- \<Longrightarrow> wcode_goon_checking ires rs (tl b, hd b # Bk # list)"
-apply(simp only: wcode_fourtimes_invs)
-apply(auto)
-done
-
-lemma [simp]: "wcode_goon_checking ires rs (b, Bk # list) = False"
-apply(simp add: wcode_fourtimes_invs)
-done
-
-lemma [simp]: " wcode_right_move ires rs (b, Bk # list) \<Longrightarrow> b\<noteq> []"
-apply(simp add: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_right_move ires rs (b, Bk # list) \<Longrightarrow> wcode_erase2 ires rs (Bk # b, list)"
-apply(auto simp:wcode_fourtimes_invs )
-apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
-done
-
-lemma [simp]: "wcode_erase2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(auto simp: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_erase2 ires rs (b, Bk # list) \<Longrightarrow> wcode_on_right_moving_2 ires rs (Bk # b, list)"
-apply(auto simp:wcode_fourtimes_invs )
-apply(rule_tac x = "Suc (Suc 0)" in exI, simp add: exp_ind)
-apply(rule_tac x = "Suc (Suc ln)" in exI, simp add: exp_ind, auto)
-done
-
-lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(auto simp:wcode_fourtimes_invs )
-done
-
-lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list)
- \<Longrightarrow> wcode_on_right_moving_2 ires rs (Bk # b, list)"
-apply(auto simp: wcode_fourtimes_invs)
-apply(rule_tac x = "Suc ml" in exI, simp add: exp_ind_def)
-apply(rule_tac x = "mr - 1" in exI, case_tac mr, auto simp: exp_ind_def)
-done
-
-lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(auto simp: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow>
- wcode_backto_standard_pos_2 ires rs (b, Oc # list)"
-apply(simp add: wcode_fourtimes_invs, auto)
-apply(rule_tac x = ml in exI, auto)
-apply(rule_tac x = "Suc 0" in exI, simp)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(rule_tac x = "rn - 1" in exI, simp)
-apply(case_tac rn, simp, simp add: exp_ind_def)
-done
-
-lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(simp add: wcode_fourtimes_invs, auto)
-done
-
-lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
-apply(simp add: wcode_fourtimes_invs, auto)
-done
-
-lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \<Longrightarrow>
- wcode_on_checking_2 ires rs (tl b, hd b # Oc # list)"
-apply(auto simp: wcode_fourtimes_invs)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \<Longrightarrow> b \<noteq> []"
-apply(auto simp: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \<Longrightarrow>
- wcode_backto_standard_pos_2 ires rs (b, [Oc])"
-apply(simp only: wcode_fourtimes_invs)
-apply(erule_tac exE)+
-apply(rule_tac disjI1)
-apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI,
- rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp, simp add: exp_ind_def)
-done
-
-lemma "wcode_backto_standard_pos_2 ires rs (b, Bk # list)
- \<Longrightarrow> (\<exists>ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \<and> (\<exists>rn. list = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-apply(auto simp: wcode_fourtimes_invs)
-apply(case_tac [!] mr, auto simp: exp_ind_def)
-done
-
-
-lemma [simp]: "wcode_on_checking_2 ires rs (b, Oc # list) \<Longrightarrow> False"
-apply(simp add: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_goon_checking ires rs (b, Oc # list) \<Longrightarrow>
- (b = [] \<longrightarrow> wcode_right_move ires rs ([Oc], list)) \<and>
- (b \<noteq> [] \<longrightarrow> wcode_right_move ires rs (Oc # b, list))"
-apply(simp only: wcode_fourtimes_invs)
-apply(erule_tac exE)+
-apply(auto)
-done
-
-lemma [simp]: "wcode_right_move ires rs (b, Oc # list) = False"
-apply(auto simp: wcode_fourtimes_invs)
-done
-
-lemma [simp]: " wcode_erase2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
-apply(simp add: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_erase2 ires rs (b, Oc # list)
- \<Longrightarrow> wcode_erase2 ires rs (b, Bk # list)"
-apply(auto simp: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
-apply(simp only: wcode_fourtimes_invs)
-apply(auto)
-done
-
-lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Oc # list)
- \<Longrightarrow> wcode_goon_right_moving_2 ires rs (Oc # b, list)"
-apply(auto simp: wcode_fourtimes_invs)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(rule_tac x = "Suc 0" in exI, auto)
-apply(rule_tac x = "ml - 2" in exI)
-apply(case_tac ml, simp, case_tac nat, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
-apply(simp only:wcode_fourtimes_invs, auto)
-done
-
-lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list)
- \<Longrightarrow> (\<exists>ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \<and> (\<exists>rn. list = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-apply(simp add: wcode_fourtimes_invs, auto)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wcode_on_checking_2 ires rs (b, Oc # list) = False"
-apply(simp add: wcode_fourtimes_invs)
-done
-
-lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow>
- wcode_goon_right_moving_2 ires rs (Oc # b, list)"
-apply(simp only:wcode_fourtimes_invs, auto)
-apply(rule_tac x = "Suc ml" in exI, auto simp: exp_ind_def)
-apply(rule_tac x = "mr - 1" in exI)
-apply(case_tac mr, case_tac rn, auto simp: exp_ind_def)
-done
-
-lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
-apply(simp only: wcode_fourtimes_invs, auto)
-done
-
-lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list)
- \<Longrightarrow> wcode_backto_standard_pos_2 ires rs (tl b, hd b # Oc # list)"
-apply(simp only: wcode_fourtimes_invs)
-apply(erule_tac disjE)
-apply(erule_tac exE)+
-apply(case_tac ml, simp)
-apply(rule_tac disjI2)
-apply(rule_tac conjI, rule_tac x = ln in exI, simp)
-apply(rule_tac x = rn in exI, simp)
-apply(rule_tac disjI1)
-apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI,
- rule_tac x = ln in exI, rule_tac x = rn in exI, simp add: exp_ind_def)
-apply(simp)
-done
-
-lemma wcode_fourtimes_case_first_correctness:
- shows "let P = (\<lambda> (st, l, r). st = t_twice_len + 14) in
- let Q = (\<lambda> (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r)) in
- let f = (\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp) in
- \<exists> n .P (f n) \<and> Q (f (n::nat))"
-proof -
- let ?P = "(\<lambda> (st, l, r). st = t_twice_len + 14)"
- let ?Q = "(\<lambda> (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r))"
- let ?f = "(\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)"
- have "\<exists> n . ?P (?f n) \<and> ?Q (?f (n::nat))"
- proof(rule_tac halt_lemma2)
- show "wf wcode_fourtimes_case_le"
- by auto
- next
- show "\<forall> na. \<not> ?P (?f na) \<and> ?Q (?f na) \<longrightarrow>
- ?Q (?f (Suc na)) \<and> (?f (Suc na), ?f na) \<in> wcode_fourtimes_case_le"
- apply(rule_tac allI,
- case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na", simp,
- rule_tac impI)
- apply(simp add: tstep_red tstep.simps, case_tac c, simp, case_tac [2] aa, simp_all)
-
- apply(simp_all add: wcode_fourtimes_case_inv.simps new_tape.simps
- wcode_fourtimes_case_le_def lex_pair_def split: if_splits)
- done
- next
- show "?Q (?f 0)"
- apply(simp add: steps.simps wcode_fourtimes_case_inv.simps)
- apply(simp add: wcode_on_left_moving_2.simps wcode_on_left_moving_2_B.simps
- wcode_on_left_moving_2_O.simps)
- apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def)
- apply(rule_tac x ="Suc 0" in exI, auto)
- done
- next
- show "\<not> ?P (?f 0)"
- apply(simp add: steps.simps)
- done
- qed
- thus "?thesis"
- apply(erule_tac exE, simp)
- done
-qed
-
-definition t_fourtimes_len :: "nat"
- where
- "t_fourtimes_len = (length t_fourtimes div 2)"
-
-lemma t_fourtimes_len_gr: "t_fourtimes_len > 0"
-apply(simp add: t_fourtimes_len_def t_fourtimes_def)
-done
-
-lemma t_fourtimes_correct:
- "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
- (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) stp =
- (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-proof(case_tac "rec_ci rec_fourtimes")
- fix a b c
- assume h: "rec_ci rec_fourtimes = (a, b, c)"
- have "\<exists>stp m l. steps (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<^bsup>n\<^esup>) (tm_of abc_fourtimes @ tMp (Suc 0)
- (start_of fourtimes_ly (length abc_fourtimes) - 1)) stp = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4*rs)\<^esup> @ Bk\<^bsup>l\<^esup>)"
- proof(rule_tac t_compiled_by_rec)
- show "rec_ci rec_fourtimes = (a, b, c)" by (simp add: h)
- next
- show "rec_calc_rel rec_fourtimes [rs] (4 * rs)"
- using prime_rel_exec_eq [of rec_fourtimes "[rs]" "4 * rs"]
- apply(subgoal_tac "primerec rec_fourtimes (length [rs])")
- apply(simp_all add: rec_fourtimes_def rec_exec.simps)
- apply(auto)
- apply(simp only: Nat.One_nat_def[THEN sym], auto)
- done
- next
- show "length [rs] = Suc 0" by simp
- next
- show "layout_of (a [+] dummy_abc (Suc 0)) = layout_of (a [+] dummy_abc (Suc 0))"
- by simp
- next
- show "start_of fourtimes_ly (length abc_fourtimes) =
- start_of (layout_of (a [+] dummy_abc (Suc 0))) (length (a [+] dummy_abc (Suc 0)))"
- using h
- apply(simp add: fourtimes_ly_def abc_fourtimes_def)
- done
- next
- show "tm_of abc_fourtimes = tm_of (a [+] dummy_abc (Suc 0))"
- using h
- apply(simp add: abc_fourtimes_def)
- done
- qed
- thus "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
- (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) stp =
- (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
- done
-qed
-
-lemma t_fourtimes_change_term_state:
- "\<exists> stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp
- = (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-using t_fourtimes_correct[of ires rs n]
-apply(erule_tac exE)
-apply(erule_tac exE)
-apply(erule_tac exE)
-proof(drule_tac turing_change_termi_state)
- fix stp ln rn
- show "t_correct (tm_of abc_fourtimes @ tMp (Suc 0)
- (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))"
- apply(rule_tac t_compiled_correct, auto simp: fourtimes_ly_def)
- done
-next
- fix stp ln rn
- show "\<exists>stp. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
- (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0)
- (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) stp =
- (Suc (length (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly
- (length abc_fourtimes) - Suc 0)) div 2), Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) \<Longrightarrow>
- \<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp =
- (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- apply(erule_tac exE)
- apply(simp add: t_fourtimes_len_def t_fourtimes_def)
- apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
- done
-qed
-
-lemma t_fourtimes_append_pre:
- "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp
- = (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)
- \<Longrightarrow> \<exists> stp>0. steps (Suc 0 + length (t_wcode_main_first_part @
- tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2,
- Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
- ((t_wcode_main_first_part @
- tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) @
- tshift t_fourtimes (length (t_wcode_main_first_part @
- tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2) @ ([(L, 1), (L, 1)])) stp
- = (Suc t_fourtimes_len + length (t_wcode_main_first_part @
- tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2,
- Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-proof(rule_tac t_tshift_lemma, auto)
- assume "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp =
- (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- thus "0 < stp"
- using t_fourtimes_len_gr
- apply(case_tac stp, simp_all add: steps.simps)
- done
-next
- show "Suc 0 \<le> length t_fourtimes div 2"
- apply(simp add: t_fourtimes_def shift_length tMp.simps)
- done
-next
- show "t_ncorrect (t_wcode_main_first_part @
- abacus.tshift t_twice (length t_wcode_main_first_part div 2) @
- [(L, Suc 0), (L, Suc 0)])"
- apply(simp add: t_ncorrect.simps t_wcode_main_first_part_def shift_length
- t_twice_def)
- using tm_even[of abc_twice]
- by arith
-next
- show "t_ncorrect t_fourtimes"
- apply(simp add: t_fourtimes_def steps.simps t_ncorrect.simps)
- using tm_even[of abc_fourtimes]
- by arith
-next
- show "t_ncorrect [(L, Suc 0), (L, Suc 0)]"
- apply(simp add: t_ncorrect.simps)
- done
-qed
-
-lemma [simp]: "length t_wcode_main_first_part = 24"
-apply(simp add: t_wcode_main_first_part_def)
-done
-
-lemma [simp]: "(26 + length t_twice) div 2 = (length t_twice) div 2 + 13"
-using tm_even[of abc_twice]
-apply(simp add: t_twice_def)
-done
-
-lemma [simp]: "((26 + length (abacus.tshift t_twice 12)) div 2)
- = (length (abacus.tshift t_twice 12) div 2 + 13)"
-using tm_even[of abc_twice]
-apply(simp add: t_twice_def)
-done
-
-lemma [simp]: "t_twice_len + 14 = 14 + length (abacus.tshift t_twice 12) div 2"
-using tm_even[of abc_twice]
-apply(simp add: t_twice_def t_twice_len_def shift_length)
-done
-
-lemma t_fourtimes_append:
- "\<exists> stp ln rn.
- steps (Suc 0 + length (t_wcode_main_first_part @ tshift t_twice
- (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2,
- Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
- ((t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @
- [(L, 1), (L, 1)]) @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)]) stp
- = (Suc t_fourtimes_len + length (t_wcode_main_first_part @ tshift t_twice
- (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires,
- Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- using t_fourtimes_change_term_state[of ires rs n]
- apply(erule_tac exE)
- apply(erule_tac exE)
- apply(erule_tac exE)
- apply(drule_tac t_fourtimes_append_pre)
- apply(erule_tac exE)
- apply(rule_tac x = stpa in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
- apply(simp add: t_twice_len_def shift_length)
- done
-
-lemma t_wcode_main_len: "length t_wcode_main = length t_twice + length t_fourtimes + 28"
-apply(simp add: t_wcode_main_def shift_length)
-done
-
-lemma [simp]: "fetch t_wcode_main (14 + length t_twice div 2 + t_fourtimes_len) b
- = (L, Suc 0)"
-using tm_even[of "abc_twice"] tm_even[of "abc_fourtimes"]
-apply(case_tac b)
-apply(simp_all only: fetch.simps)
-apply(auto simp: nth_of.simps t_wcode_main_len t_twice_len_def
- t_fourtimes_def t_twice_def t_fourtimes_def t_fourtimes_len_def)
-apply(auto simp: t_wcode_main_def t_wcode_main_first_part_def shift_length t_twice_def nth_append
- t_fourtimes_def)
-done
-
-lemma wcode_jump2:
- "\<exists> stp ln rn. steps (t_twice_len + 14 + t_fourtimes_len
- , Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4 * rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stp =
- (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4 * rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-apply(rule_tac x = "Suc 0" in exI)
-apply(simp add: steps.simps shift_length)
-apply(rule_tac x = lnb in exI, rule_tac x = rnb in exI)
-apply(simp add: tstep.simps new_tape.simps)
-done
-
-lemma wcode_fourtimes_case:
- shows "\<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-proof -
- have "\<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (t_twice_len + 14, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- using wcode_fourtimes_case_first_correctness[of ires rs m n]
- apply(simp add: wcode_fourtimes_case_inv.simps, auto)
- apply(rule_tac x = na in exI, rule_tac x = ln in exI,
- rule_tac x = rn in exI)
- apply(simp)
- done
- from this obtain stpa lna rna where stp1:
- "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
- (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
- have "\<exists>stp ln rn. steps (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)
- t_wcode_main stp =
- (t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- using t_fourtimes_append[of " Bk\<^bsup>lna\<^esup> @ Oc # ires" "rs + 1" rna]
- apply(erule_tac exE)
- apply(erule_tac exE)
- apply(erule_tac exE)
- apply(simp add: t_wcode_main_def)
- apply(rule_tac x = stp in exI,
- rule_tac x = "ln + lna" in exI,
- rule_tac x = rn in exI, simp)
- apply(simp add: exp_ind_def[THEN sym] exp_add[THEN sym])
- done
- from this obtain stpb lnb rnb where stp2:
- "steps (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)
- t_wcode_main stpb =
- (t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>)"
- by blast
- have "\<exists>stp ln rn. steps (t_twice_len + 14 + t_fourtimes_len,
- Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>)
- t_wcode_main stp =
- (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- apply(rule wcode_jump2)
- done
- from this obtain stpc lnc rnc where stp3:
- "steps (t_twice_len + 14 + t_fourtimes_len,
- Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>)
- t_wcode_main stpc =
- (Suc 0, Bk # Bk\<^bsup>lnc\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnc\<^esup>)"
- by blast
- from stp1 stp2 stp3 show "?thesis"
- apply(rule_tac x = "stpa + stpb + stpc" in exI,
- rule_tac x = lnc in exI, rule_tac x = rnc in exI)
- apply(simp add: steps_add)
- done
-qed
-
-(**********************************************************)
-
-fun wcode_on_left_moving_3_B :: "bin_inv_t"
- where
- "wcode_on_left_moving_3_B ires rs (l, r) =
- (\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Bk # ires \<and>
- r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr > Suc 0 \<and> mr > 0 )"
-
-fun wcode_on_left_moving_3_O :: "bin_inv_t"
- where
- "wcode_on_left_moving_3_O ires rs (l, r) =
- (\<exists> ln rn. l = Bk # Bk # ires \<and>
- r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wcode_on_left_moving_3 :: "bin_inv_t"
- where
- "wcode_on_left_moving_3 ires rs (l, r) =
- (wcode_on_left_moving_3_B ires rs (l, r) \<or>
- wcode_on_left_moving_3_O ires rs (l, r))"
-
-fun wcode_on_checking_3 :: "bin_inv_t"
- where
- "wcode_on_checking_3 ires rs (l, r) =
- (\<exists> ln rn. l = Bk # ires \<and>
- r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wcode_goon_checking_3 :: "bin_inv_t"
- where
- "wcode_goon_checking_3 ires rs (l, r) =
- (\<exists> ln rn. l = ires \<and>
- r = Bk # Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wcode_stop :: "bin_inv_t"
- where
- "wcode_stop ires rs (l, r) =
- (\<exists> ln rn. l = Bk # ires \<and>
- r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wcode_halt_case_inv :: "nat \<Rightarrow> bin_inv_t"
- where
- "wcode_halt_case_inv st ires rs (l, r) =
- (if st = 0 then wcode_stop ires rs (l, r)
- else if st = Suc 0 then wcode_on_left_moving_3 ires rs (l, r)
- else if st = Suc (Suc 0) then wcode_on_checking_3 ires rs (l, r)
- else if st = 7 then wcode_goon_checking_3 ires rs (l, r)
- else False)"
-
-fun wcode_halt_case_state :: "t_conf \<Rightarrow> nat"
- where
- "wcode_halt_case_state (st, l, r) =
- (if st = 1 then 5
- else if st = Suc (Suc 0) then 4
- else if st = 7 then 3
- else 0)"
-
-fun wcode_halt_case_step :: "t_conf \<Rightarrow> nat"
- where
- "wcode_halt_case_step (st, l, r) =
- (if st = 1 then length l
- else 0)"
-
-fun wcode_halt_case_measure :: "t_conf \<Rightarrow> nat \<times> nat"
- where
- "wcode_halt_case_measure (st, l, r) =
- (wcode_halt_case_state (st, l, r),
- wcode_halt_case_step (st, l, r))"
-
-definition wcode_halt_case_le :: "(t_conf \<times> t_conf) set"
- where "wcode_halt_case_le \<equiv> (inv_image lex_pair wcode_halt_case_measure)"
-
-lemma wf_wcode_halt_case_le[intro]: "wf wcode_halt_case_le"
-by(auto intro:wf_inv_image simp: wcode_halt_case_le_def)
-
-declare wcode_on_left_moving_3_B.simps[simp del] wcode_on_left_moving_3_O.simps[simp del]
- wcode_on_checking_3.simps[simp del] wcode_goon_checking_3.simps[simp del]
- wcode_on_left_moving_3.simps[simp del] wcode_stop.simps[simp del]
-
-lemmas wcode_halt_invs =
- wcode_on_left_moving_3_B.simps wcode_on_left_moving_3_O.simps
- wcode_on_checking_3.simps wcode_goon_checking_3.simps
- wcode_on_left_moving_3.simps wcode_stop.simps
-
-lemma [simp]: "fetch t_wcode_main 7 Bk = (R, 0)"
-apply(simp add: fetch.simps t_wcode_main_def nth_append nth_of.simps
- t_wcode_main_first_part_def)
-done
-
-lemma [simp]: "wcode_on_left_moving_3 ires rs (b, []) = False"
-apply(simp only: wcode_halt_invs)
-apply(simp add: exp_ind_def)
-done
-
-lemma [simp]: "wcode_on_checking_3 ires rs (b, []) = False"
-apply(simp add: wcode_halt_invs)
-done
-
-lemma [simp]: "wcode_goon_checking_3 ires rs (b, []) = False"
-apply(simp add: wcode_halt_invs)
-done
-
-lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Bk # list)
- \<Longrightarrow> wcode_on_left_moving_3 ires rs (tl b, hd b # Bk # list)"
-apply(simp only: wcode_halt_invs)
-apply(erule_tac disjE)
-apply(erule_tac exE)+
-apply(case_tac ml, simp)
-apply(rule_tac x = "mr - 2" in exI, rule_tac x = rn in exI)
-apply(case_tac mr, simp, simp add: exp_ind, simp add: exp_ind[THEN sym])
-apply(rule_tac disjI1)
-apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI,
- rule_tac x = rn in exI, simp add: exp_ind_def)
-apply(simp)
-done
-
-lemma [simp]: "wcode_goon_checking_3 ires rs (b, Bk # list) \<Longrightarrow>
- (b = [] \<longrightarrow> wcode_stop ires rs ([Bk], list)) \<and>
- (b \<noteq> [] \<longrightarrow> wcode_stop ires rs (Bk # b, list))"
-apply(auto simp: wcode_halt_invs)
-done
-
-lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
-apply(auto simp: wcode_halt_invs)
-done
-
-lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Oc # list) \<Longrightarrow>
- wcode_on_checking_3 ires rs (tl b, hd b # Oc # list)"
-apply(simp add:wcode_halt_invs, auto)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wcode_on_checking_3 ires rs (b, Oc # list) = False"
-apply(auto simp: wcode_halt_invs)
-done
-
-lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(simp add: wcode_halt_invs, auto)
-done
-
-
-lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(auto simp: wcode_halt_invs)
-done
-
-lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \<Longrightarrow>
- wcode_goon_checking_3 ires rs (tl b, hd b # Bk # list)"
-apply(auto simp: wcode_halt_invs)
-done
-
-lemma [simp]: "wcode_goon_checking_3 ires rs (b, Oc # list) = False"
-apply(simp add: wcode_goon_checking_3.simps)
-done
-
-lemma t_halt_case_correctness:
-shows "let P = (\<lambda> (st, l, r). st = 0) in
- let Q = (\<lambda> (st, l, r). wcode_halt_case_inv st ires rs (l, r)) in
- let f = (\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp) in
- \<exists> n .P (f n) \<and> Q (f (n::nat))"
-proof -
- let ?P = "(\<lambda> (st, l, r). st = 0)"
- let ?Q = "(\<lambda> (st, l, r). wcode_halt_case_inv st ires rs (l, r))"
- let ?f = "(\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)"
- have "\<exists> n. ?P (?f n) \<and> ?Q (?f (n::nat))"
- proof(rule_tac halt_lemma2)
- show "wf wcode_halt_case_le" by auto
- next
- show "\<forall> na. \<not> ?P (?f na) \<and> ?Q (?f na) \<longrightarrow>
- ?Q (?f (Suc na)) \<and> (?f (Suc na), ?f na) \<in> wcode_halt_case_le"
- apply(rule_tac allI, rule_tac impI, case_tac "?f na")
- apply(simp add: tstep_red tstep.simps)
- apply(case_tac c, simp, case_tac [2] aa)
- apply(simp_all split: if_splits add: new_tape.simps wcode_halt_case_le_def lex_pair_def)
- done
- next
- show "?Q (?f 0)"
- apply(simp add: steps.simps wcode_halt_invs)
- apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def)
- apply(rule_tac x = "Suc 0" in exI, auto)
- done
- next
- show "\<not> ?P (?f 0)"
- apply(simp add: steps.simps)
- done
- qed
- thus "?thesis"
- apply(auto)
- done
-qed
-
-declare wcode_halt_case_inv.simps[simp del]
-lemma [intro]: "\<exists> xs. (<rev list @ [aa::nat]> :: block list) = Oc # xs"
-apply(case_tac "rev list", simp)
-apply(simp add: tape_of_nat_abv tape_of_nat_list.simps exp_ind_def)
-apply(case_tac list, simp, simp)
-done
-
-lemma wcode_halt_case:
- "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
- t_wcode_main stp = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- using t_halt_case_correctness[of ires rs m n]
-apply(simp)
-apply(erule_tac exE)
-apply(case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na")
-apply(auto simp: wcode_halt_case_inv.simps wcode_stop.simps)
-apply(rule_tac x = na in exI, rule_tac x = ln in exI,
- rule_tac x = rn in exI, simp)
-done
-
-lemma bl_bin_one: "bl_bin [Oc] = Suc 0"
-apply(simp add: bl_bin.simps)
-done
-
-lemma t_wcode_main_lemma_pre:
- "\<lbrakk>args \<noteq> []; lm = <args::nat list>\<rbrakk> \<Longrightarrow>
- \<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main
- stp
- = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2^(length lm - 1) \<^esup> @ Bk\<^bsup>rn\<^esup>)"
-proof(induct "length args" arbitrary: args lm rs m n, simp)
- fix x args lm rs m n
- assume ind:
- "\<And>args lm rs m n.
- \<lbrakk>x = length args; (args::nat list) \<noteq> []; lm = <args>\<rbrakk>
- \<Longrightarrow> \<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
- and h: "Suc x = length args" "(args::nat list) \<noteq> []" "lm = <args>"
- from h have "\<exists> (a::nat) xs. args = xs @ [a]"
- apply(rule_tac x = "last args" in exI)
- apply(rule_tac x = "butlast args" in exI, auto)
- done
- from this obtain a xs where "args = xs @ [a]" by blast
- from h and this show
- "\<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- proof(case_tac "xs::nat list", simp)
- show "\<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- proof(induct "a" arbitrary: m n rs ires, simp)
- fix m n rs ires
- show "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
- t_wcode_main stp = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin [Oc] + rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- apply(simp add: bl_bin_one)
- apply(rule_tac wcode_halt_case)
- done
- next
- fix a m n rs ires
- assume ind2:
- "\<And>m n rs ires.
- \<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- show "\<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<Suc a>) + rs * 2 ^ Suc a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- proof -
- have "\<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- apply(simp add: tape_of_nat)
- using wcode_double_case[of m "Oc\<^bsup>a\<^esup> @ Bk # Bk # ires" rs n]
- apply(simp add: exp_ind_def)
- done
- from this obtain stpa lna rna where stp1:
- "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
- (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
- moreover have
- "\<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp =
- (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + (2*rs + 2) * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- using ind2[of lna ires "2*rs + 2" rna] by simp
- from this obtain stpb lnb rnb where stp2:
- "steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb =
- (0, Bk # ires, Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + (2*rs + 2) * 2 ^ a\<^esup> @ Bk\<^bsup>rnb\<^esup>)"
- by blast
- from stp1 and stp2 show "?thesis"
- apply(rule_tac x = "stpa + stpb" in exI,
- rule_tac x = lnb in exI, rule_tac x = rnb in exI, simp)
- apply(simp add: steps_add bl_bin_nat_Suc exponent_def)
- done
- qed
- qed
- next
- fix aa list
- assume g: "Suc x = length args" "args \<noteq> []" "lm = <args>" "args = xs @ [a::nat]" "xs = (aa::nat) # list"
- thus "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- proof(induct a arbitrary: m n rs args lm, simp_all add: tape_of_nl_rev,
- simp only: tape_of_nl_cons_app1, simp)
- fix m n rs args lm
- have "\<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # rev (<(aa::nat) # list>) @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ rev (<aa # list>) @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- proof(simp add: tape_of_nl_rev)
- have "\<exists> xs. (<rev list @ [aa]>) = Oc # xs" by auto
- from this obtain xs where "(<rev list @ [aa]>) = Oc # xs" ..
- thus "\<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ <rev list @ [aa]> @ Bk # Bk # ires, Bk # Oc\<^bsup>5 + 4 * rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- apply(simp)
- using wcode_fourtimes_case[of m "xs @ Bk # Bk # ires" rs n]
- apply(simp)
- done
- qed
- from this obtain stpa lna rna where stp1:
- "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # rev (<aa # list>) @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
- (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<aa # list>) @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
- from g have
- "\<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<(aa::nat) # list>) @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp = (0, Bk # ires,
- Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<aa#list>)+ (4*rs + 4) * 2^(length (<aa#list>) - 1) \<^esup> @ Bk\<^bsup>rn\<^esup>)"
- apply(rule_tac args = "(aa::nat)#list" in ind, simp_all)
- done
- from this obtain stpb lnb rnb where stp2:
- "steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<(aa::nat) # list>) @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb = (0, Bk # ires,
- Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<aa#list>)+ (4*rs + 4) * 2^(length (<aa#list>) - 1) \<^esup> @ Bk\<^bsup>rnb\<^esup>)"
- by blast
- from stp1 and stp2 and h
- show "\<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
- Bk # Oc\<^bsup>bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [0]>) + rs * (2 * 2 ^ (aa + length (<list @ [0]>)))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- apply(rule_tac x = "stpa + stpb" in exI, rule_tac x = lnb in exI,
- rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_rev)
- done
- next
- fix ab m n rs args lm
- assume ind2:
- "\<And> m n rs args lm.
- \<lbrakk>lm = <aa # list @ [ab]>; args = aa # list @ [ab]\<rbrakk>
- \<Longrightarrow> \<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ <ab # rev list @ [aa]> @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
- Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]>) + rs * 2 ^ (length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- and k: "args = aa # list @ [Suc ab]" "lm = <aa # list @ [Suc ab]>"
- show "\<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ <Suc ab # rev list @ [aa]> @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (0, Bk # ires,Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
- Bk # Oc\<^bsup>bl_bin (<aa # list @ [Suc ab]>) + rs * 2 ^ (length (<aa # list @ [Suc ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- proof(simp add: tape_of_nl_cons_app1)
- have "\<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
- Bk # Oc # Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp
- = (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc\<^bsup>Suc ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- using wcode_double_case[of m "Oc\<^bsup>ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires"
- rs n]
- apply(simp add: exp_ind_def)
- done
- from this obtain stpa lna rna where stp1:
- "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
- Bk # Oc # Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa
- = (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ Oc\<^bsup>Suc ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
- from k have
- "\<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ <ab # rev list @ [aa]> @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp
- = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
- Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]> ) + (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- apply(rule_tac ind2, simp_all)
- done
- from this obtain stpb lnb rnb where stp2:
- "steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ <ab # rev list @ [aa]> @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb
- = (0, Bk # ires, Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk #
- Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]> ) + (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rnb\<^esup>)"
- by blast
- from stp1 and stp2 show
- "\<exists>stp ln rn.
- steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
- Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
- (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk #
- Oc\<^bsup>bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [Suc ab]>) + rs * (2 * 2 ^ (aa + length (<list @ [Suc ab]>)))\<^esup>
- @ Bk\<^bsup>rn\<^esup>)"
- apply(rule_tac x = "stpa + stpb" in exI, rule_tac x = lnb in exI,
- rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_cons_app1 exp_ind_def)
- done
- qed
- qed
- qed
- qed
-
-
-
-(* turing_shift can be used*)
-term t_wcode_main
-definition t_wcode_prepare :: "tprog"
- where
- "t_wcode_prepare \<equiv>
- [(W1, 2), (L, 1), (L, 3), (R, 2), (R, 4), (W0, 3),
- (R, 4), (R, 5), (R, 6), (R, 5), (R, 7), (R, 5),
- (W1, 7), (L, 0)]"
-
-fun wprepare_add_one :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_add_one m lm (l, r) =
- (\<exists> rn. l = [] \<and>
- (r = <m # lm> @ Bk\<^bsup>rn\<^esup> \<or>
- r = Bk # <m # lm> @ Bk\<^bsup>rn\<^esup>))"
-
-fun wprepare_goto_first_end :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_goto_first_end m lm (l, r) =
- (\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> \<and>
- r = Oc\<^bsup>mr\<^esup> @ Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr = Suc (Suc m))"
-
-fun wprepare_erase :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_erase m lm (l, r) =
- (\<exists> rn. l = Oc\<^bsup>Suc m\<^esup> \<and>
- tl r = Bk # <lm> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wprepare_goto_start_pos_B :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_goto_start_pos_B m lm (l, r) =
- (\<exists> rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- r = Bk # <lm> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wprepare_goto_start_pos_O :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_goto_start_pos_O m lm (l, r) =
- (\<exists> rn. l = Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- r = <lm> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wprepare_goto_start_pos :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_goto_start_pos m lm (l, r) =
- (wprepare_goto_start_pos_B m lm (l, r) \<or>
- wprepare_goto_start_pos_O m lm (l, r))"
-
-fun wprepare_loop_start_on_rightmost :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_loop_start_on_rightmost m lm (l, r) =
- (\<exists> rn mr. rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and>
- r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wprepare_loop_start_in_middle :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_loop_start_in_middle m lm (l, r) =
- (\<exists> rn (mr:: nat) (lm1::nat list).
- rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and>
- r = Oc\<^bsup>mr\<^esup> @ Bk # <lm1> @ Bk\<^bsup>rn\<^esup> \<and> lm1 \<noteq> [])"
-
-fun wprepare_loop_start :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_loop_start m lm (l, r) = (wprepare_loop_start_on_rightmost m lm (l, r) \<or>
- wprepare_loop_start_in_middle m lm (l, r))"
-
-fun wprepare_loop_goon_on_rightmost :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_loop_goon_on_rightmost m lm (l, r) =
- (\<exists> rn. l = Bk # <rev lm> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- r = Bk\<^bsup>rn\<^esup>)"
-
-fun wprepare_loop_goon_in_middle :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_loop_goon_in_middle m lm (l, r) =
- (\<exists> rn (mr:: nat) (lm1::nat list).
- rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and>
- (if lm1 = [] then r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>
- else r = Oc\<^bsup>mr\<^esup> @ Bk # <lm1> @ Bk\<^bsup>rn\<^esup>) \<and> mr > 0)"
-
-fun wprepare_loop_goon :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_loop_goon m lm (l, r) =
- (wprepare_loop_goon_in_middle m lm (l, r) \<or>
- wprepare_loop_goon_on_rightmost m lm (l, r))"
-
-fun wprepare_add_one2 :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_add_one2 m lm (l, r) =
- (\<exists> rn. l = Bk # Bk # <rev lm> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- (r = [] \<or> tl r = Bk\<^bsup>rn\<^esup>))"
-
-fun wprepare_stop :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_stop m lm (l, r) =
- (\<exists> rn. l = Bk # <rev lm> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- r = Bk # Oc # Bk\<^bsup>rn\<^esup>)"
-
-fun wprepare_inv :: "nat \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wprepare_inv st m lm (l, r) =
- (if st = 0 then wprepare_stop m lm (l, r)
- else if st = Suc 0 then wprepare_add_one m lm (l, r)
- else if st = Suc (Suc 0) then wprepare_goto_first_end m lm (l, r)
- else if st = Suc (Suc (Suc 0)) then wprepare_erase m lm (l, r)
- else if st = 4 then wprepare_goto_start_pos m lm (l, r)
- else if st = 5 then wprepare_loop_start m lm (l, r)
- else if st = 6 then wprepare_loop_goon m lm (l, r)
- else if st = 7 then wprepare_add_one2 m lm (l, r)
- else False)"
-
-fun wprepare_stage :: "t_conf \<Rightarrow> nat"
- where
- "wprepare_stage (st, l, r) =
- (if st \<ge> 1 \<and> st \<le> 4 then 3
- else if st = 5 \<or> st = 6 then 2
- else 1)"
-
-fun wprepare_state :: "t_conf \<Rightarrow> nat"
- where
- "wprepare_state (st, l, r) =
- (if st = 1 then 4
- else if st = Suc (Suc 0) then 3
- else if st = Suc (Suc (Suc 0)) then 2
- else if st = 4 then 1
- else if st = 7 then 2
- else 0)"
-
-fun wprepare_step :: "t_conf \<Rightarrow> nat"
- where
- "wprepare_step (st, l, r) =
- (if st = 1 then (if hd r = Oc then Suc (length l)
- else 0)
- else if st = Suc (Suc 0) then length r
- else if st = Suc (Suc (Suc 0)) then (if hd r = Oc then 1
- else 0)
- else if st = 4 then length r
- else if st = 5 then Suc (length r)
- else if st = 6 then (if r = [] then 0 else Suc (length r))
- else if st = 7 then (if (r \<noteq> [] \<and> hd r = Oc) then 0
- else 1)
- else 0)"
-
-fun wcode_prepare_measure :: "t_conf \<Rightarrow> nat \<times> nat \<times> nat"
- where
- "wcode_prepare_measure (st, l, r) =
- (wprepare_stage (st, l, r),
- wprepare_state (st, l, r),
- wprepare_step (st, l, r))"
-
-definition wcode_prepare_le :: "(t_conf \<times> t_conf) set"
- where "wcode_prepare_le \<equiv> (inv_image lex_triple wcode_prepare_measure)"
-
-lemma [intro]: "wf lex_pair"
-by(auto intro:wf_lex_prod simp:lex_pair_def)
-
-lemma wf_wcode_prepare_le[intro]: "wf wcode_prepare_le"
-by(auto intro:wf_inv_image simp: wcode_prepare_le_def
- recursive.lex_triple_def)
-
-declare wprepare_add_one.simps[simp del] wprepare_goto_first_end.simps[simp del]
- wprepare_erase.simps[simp del] wprepare_goto_start_pos.simps[simp del]
- wprepare_loop_start.simps[simp del] wprepare_loop_goon.simps[simp del]
- wprepare_add_one2.simps[simp del]
-
-lemmas wprepare_invs = wprepare_add_one.simps wprepare_goto_first_end.simps
- wprepare_erase.simps wprepare_goto_start_pos.simps
- wprepare_loop_start.simps wprepare_loop_goon.simps
- wprepare_add_one2.simps
-
-declare wprepare_inv.simps[simp del]
-lemma [simp]: "fetch t_wcode_prepare (Suc 0) Bk = (W1, 2)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_prepare (Suc 0) Oc = (L, 1)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_prepare (Suc (Suc 0)) Bk = (L, 3)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_prepare (Suc (Suc 0)) Oc = (R, 2)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_prepare (Suc (Suc (Suc 0))) Bk = (R, 4)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_prepare (Suc (Suc (Suc 0))) Oc = (W0, 3)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_prepare 4 Bk = (R, 4)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_prepare 4 Oc = (R, 5)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_prepare 5 Oc = (R, 5)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_prepare 5 Bk = (R, 6)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_prepare 6 Oc = (R, 5)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_prepare 6 Bk = (R, 7)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_prepare 7 Oc = (L, 0)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_prepare 7 Bk = (W1, 7)"
-apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
-done
-
-lemma tape_of_nl_not_null: "lm \<noteq> [] \<Longrightarrow> <lm::nat list> \<noteq> []"
-apply(case_tac lm, auto)
-apply(case_tac list, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
-done
-
-lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_add_one m lm (b, []) = False"
-apply(simp add: wprepare_invs)
-apply(simp add: tape_of_nl_not_null)
-done
-
-lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_goto_first_end m lm (b, []) = False"
-apply(simp add: wprepare_invs)
-done
-
-lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_erase m lm (b, []) = False"
-apply(simp add: wprepare_invs)
-done
-
-
-
-lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_goto_start_pos m lm (b, []) = False"
-apply(simp add: wprepare_invs tape_of_nl_not_null)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [])\<rbrakk> \<Longrightarrow> b \<noteq> []"
-apply(simp add: wprepare_invs tape_of_nl_not_null, auto)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [])\<rbrakk> \<Longrightarrow>
- wprepare_loop_goon m lm (Bk # b, [])"
-apply(simp only: wprepare_invs tape_of_nl_not_null)
-apply(erule_tac disjE)
-apply(rule_tac disjI2)
-apply(simp add: wprepare_loop_start_on_rightmost.simps
- wprepare_loop_goon_on_rightmost.simps, auto)
-apply(rule_tac rev_eq, simp add: tape_of_nl_rev)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, [])\<rbrakk> \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
-done
-
-lemma [simp]:"\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, [])\<rbrakk> \<Longrightarrow>
- wprepare_add_one2 m lm (Bk # b, [])"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto split: if_splits)
-apply(case_tac mr, simp, simp add: exp_ind_def)
-done
-
-lemma [simp]: "wprepare_add_one2 m lm (b, []) \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
-done
-
-lemma [simp]: "wprepare_add_one2 m lm (b, []) \<Longrightarrow> wprepare_add_one2 m lm (b, [Oc])"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
-done
-
-lemma [simp]: "Bk # list = <(m::nat) # lm> @ ys = False"
-apply(case_tac lm, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_add_one m lm (b, Bk # list)\<rbrakk>
- \<Longrightarrow> (b = [] \<longrightarrow> wprepare_goto_first_end m lm ([], Oc # list)) \<and>
- (b \<noteq> [] \<longrightarrow> wprepare_goto_first_end m lm (b, Oc # list))"
-apply(simp only: wprepare_invs, auto)
-apply(rule_tac x = 0 in exI, simp add: exp_ind_def)
-apply(case_tac lm, simp, simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
-apply(rule_tac x = rn in exI, simp)
-done
-
-lemma [simp]: "wprepare_goto_first_end m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wprepare_goto_first_end m lm (b, Bk # list) \<Longrightarrow>
- wprepare_erase m lm (tl b, hd b # Bk # list)"
-apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac mr, auto simp: exp_ind_def)
-done
-
-lemma [simp]: "wprepare_erase m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs exp_ind_def, auto)
-done
-
-lemma [simp]: "wprepare_erase m lm (b, Bk # list) \<Longrightarrow>
- wprepare_goto_start_pos m lm (Bk # b, list)"
-apply(simp only: wprepare_invs, auto)
-done
-
-lemma [simp]: "\<lbrakk>wprepare_add_one m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
-apply(simp only: wprepare_invs)
-apply(case_tac lm, simp_all add: tape_of_nl_abv
- tape_of_nat_list.simps exp_ind_def, auto)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_first_end m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
-apply(simp only: wprepare_invs, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(simp add: tape_of_nl_not_null)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_first_end m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs, auto)
-apply(case_tac mr, simp_all add: exp_ind_def tape_of_nl_not_null)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_erase m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
-apply(simp only: wprepare_invs, auto)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_erase m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs, auto simp: exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
-apply(simp only: wprepare_invs, auto)
-apply(simp add: tape_of_nl_not_null)
-apply(case_tac lm, simp, case_tac list)
-apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs)
-apply(auto)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs, auto)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, Bk # list)\<rbrakk> \<Longrightarrow>
- (list = [] \<longrightarrow> wprepare_add_one2 m lm (Bk # b, [])) \<and>
- (list \<noteq> [] \<longrightarrow> wprepare_add_one2 m lm (Bk # b, list))"
-apply(simp only: wprepare_invs, simp)
-apply(case_tac list, simp_all split: if_splits, auto)
-apply(case_tac [1-3] mr, simp_all add: exp_ind_def)
-apply(case_tac mr, simp_all add: exp_ind_def tape_of_nl_not_null)
-apply(case_tac [1-2] mr, simp_all add: exp_ind_def)
-apply(case_tac rn, simp, case_tac nat, auto simp: exp_ind_def)
-done
-
-lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs, simp)
-done
-
-lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \<Longrightarrow>
- (list = [] \<longrightarrow> wprepare_add_one2 m lm (b, [Oc])) \<and>
- (list \<noteq> [] \<longrightarrow> wprepare_add_one2 m lm (b, Oc # list))"
-apply(simp only: wprepare_invs, auto)
-done
-
-lemma [simp]: "wprepare_goto_first_end m lm (b, Oc # list)
- \<Longrightarrow> (b = [] \<longrightarrow> wprepare_goto_first_end m lm ([Oc], list)) \<and>
- (b \<noteq> [] \<longrightarrow> wprepare_goto_first_end m lm (Oc # b, list))"
-apply(simp only: wprepare_invs, auto)
-apply(rule_tac x = 1 in exI, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac ml, simp_all add: exp_ind_def)
-apply(rule_tac x = rn in exI, simp)
-apply(rule_tac x = "Suc ml" in exI, simp_all add: exp_ind_def)
-apply(rule_tac x = "mr - 1" in exI, simp)
-apply(case_tac mr, simp_all add: exp_ind_def, auto)
-done
-
-lemma [simp]: "wprepare_erase m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
-apply(simp only: wprepare_invs, auto simp: exp_ind_def)
-done
-
-lemma [simp]: "wprepare_erase m lm (b, Oc # list)
- \<Longrightarrow> wprepare_erase m lm (b, Bk # list)"
-apply(simp only:wprepare_invs, auto simp: exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk>
- \<Longrightarrow> wprepare_goto_start_pos m lm (Bk # b, list)"
-apply(simp only:wprepare_invs, auto)
-apply(case_tac [!] lm, simp, simp_all)
-done
-
-lemma [simp]: "wprepare_loop_start m lm (b, aa) \<Longrightarrow> b \<noteq> []"
-apply(simp only:wprepare_invs, auto)
-done
-lemma [elim]: "Bk # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<Longrightarrow> \<exists>rn. list = Bk\<^bsup>rn\<^esup>"
-apply(case_tac mr, simp_all)
-apply(case_tac rn, simp_all add: exp_ind_def, auto)
-done
-
-lemma rev_equal_iff: "x = y \<Longrightarrow> rev x = rev y"
-by simp
-
-lemma tape_of_nl_false1:
- "lm \<noteq> [] \<Longrightarrow> rev b @ [Bk] \<noteq> Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>m\<^esup> @ Bk # Bk # <lm::nat list>"
-apply(auto)
-apply(drule_tac rev_equal_iff, simp add: tape_of_nl_rev)
-apply(case_tac "rev lm")
-apply(case_tac [2] list, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
-done
-
-lemma [simp]: "wprepare_loop_start_in_middle m lm (b, [Bk]) = False"
-apply(simp add: wprepare_loop_start_in_middle.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac lm1, simp, simp add: tape_of_nl_not_null)
-done
-
-declare wprepare_loop_start_in_middle.simps[simp del]
-
-declare wprepare_loop_start_on_rightmost.simps[simp del]
- wprepare_loop_goon_in_middle.simps[simp del]
- wprepare_loop_goon_on_rightmost.simps[simp del]
-
-lemma [simp]: "wprepare_loop_goon_in_middle m lm (Bk # b, []) = False"
-apply(simp add: wprepare_loop_goon_in_middle.simps, auto)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [Bk])\<rbrakk> \<Longrightarrow>
- wprepare_loop_goon m lm (Bk # b, [])"
-apply(simp only: wprepare_invs, simp)
-apply(simp add: wprepare_loop_goon_on_rightmost.simps
- wprepare_loop_start_on_rightmost.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(rule_tac rev_eq)
-apply(simp add: tape_of_nl_rev)
-apply(simp add: exp_ind_def[THEN sym] exp_ind)
-done
-
-lemma [simp]: "wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista)
- \<Longrightarrow> wprepare_loop_goon_in_middle m lm (Bk # b, a # lista) = False"
-apply(auto simp: wprepare_loop_start_on_rightmost.simps
- wprepare_loop_goon_in_middle.simps)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista)\<rbrakk>
- \<Longrightarrow> wprepare_loop_goon_on_rightmost m lm (Bk # b, a # lista)"
-apply(simp only: wprepare_loop_start_on_rightmost.simps
- wprepare_loop_goon_on_rightmost.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(simp add: tape_of_nl_rev)
-apply(simp add: exp_ind_def[THEN sym] exp_ind)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\<rbrakk>
- \<Longrightarrow> wprepare_loop_goon_on_rightmost m lm (Bk # b, a # lista) = False"
-apply(simp add: wprepare_loop_start_in_middle.simps
- wprepare_loop_goon_on_rightmost.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac "lm1::nat list", simp_all, case_tac list, simp)
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps tape_of_nat_abv exp_ind_def)
-apply(case_tac [!] rna, simp_all add: exp_ind_def)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac lm1, simp, case_tac list, simp)
-apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def tape_of_nat_abv)
-done
-
-lemma [simp]:
- "\<lbrakk>lm \<noteq> []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\<rbrakk>
- \<Longrightarrow> wprepare_loop_goon_in_middle m lm (Bk # b, a # lista)"
-apply(simp add: wprepare_loop_start_in_middle.simps
- wprepare_loop_goon_in_middle.simps, auto)
-apply(rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac lm1, simp)
-apply(rule_tac x = "Suc aa" in exI, simp)
-apply(rule_tac x = list in exI)
-apply(case_tac list, simp_all add: tape_of_nl_abv tape_of_nat_list.simps)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, Bk # a # lista)\<rbrakk> \<Longrightarrow>
- wprepare_loop_goon m lm (Bk # b, a # lista)"
-apply(simp add: wprepare_loop_start.simps
- wprepare_loop_goon.simps)
-apply(erule_tac disjE, simp, auto)
-done
-
-lemma start_2_goon:
- "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, Bk # list)\<rbrakk> \<Longrightarrow>
- (list = [] \<longrightarrow> wprepare_loop_goon m lm (Bk # b, [])) \<and>
- (list \<noteq> [] \<longrightarrow> wprepare_loop_goon m lm (Bk # b, list))"
-apply(case_tac list, auto)
-done
-
-lemma add_one_2_add_one: "wprepare_add_one m lm (b, Oc # list)
- \<Longrightarrow> (hd b = Oc \<longrightarrow> (b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)) \<and>
- (b \<noteq> [] \<longrightarrow> wprepare_add_one m lm (tl b, Oc # Oc # list))) \<and>
- (hd b \<noteq> Oc \<longrightarrow> (b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)) \<and>
- (b \<noteq> [] \<longrightarrow> wprepare_add_one m lm (tl b, hd b # Oc # list)))"
-apply(simp only: wprepare_add_one.simps, auto)
-done
-
-lemma [simp]: "wprepare_loop_start m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
-apply(simp)
-done
-
-lemma [simp]: "wprepare_loop_start_on_rightmost m lm (b, Oc # list) \<Longrightarrow>
- wprepare_loop_start_on_rightmost m lm (Oc # b, list)"
-apply(simp add: wprepare_loop_start_on_rightmost.simps, auto)
-apply(rule_tac x = rn in exI, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac rn, auto simp: exp_ind_def)
-done
-
-lemma [simp]: "wprepare_loop_start_in_middle m lm (b, Oc # list) \<Longrightarrow>
- wprepare_loop_start_in_middle m lm (Oc # b, list)"
-apply(simp add: wprepare_loop_start_in_middle.simps, auto)
-apply(rule_tac x = rn in exI, auto)
-apply(case_tac mr, simp, simp add: exp_ind_def)
-apply(rule_tac x = nat in exI, simp)
-apply(rule_tac x = lm1 in exI, simp)
-done
-
-lemma start_2_start: "wprepare_loop_start m lm (b, Oc # list) \<Longrightarrow>
- wprepare_loop_start m lm (Oc # b, list)"
-apply(simp add: wprepare_loop_start.simps)
-apply(erule_tac disjE, simp_all )
-done
-
-lemma [simp]: "wprepare_loop_goon m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
-apply(simp add: wprepare_loop_goon.simps
- wprepare_loop_goon_in_middle.simps
- wprepare_loop_goon_on_rightmost.simps)
-apply(auto)
-done
-
-lemma [simp]: "wprepare_goto_start_pos m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
-apply(simp add: wprepare_goto_start_pos.simps)
-done
-
-lemma [simp]: "wprepare_loop_goon_on_rightmost m lm (b, Oc # list) = False"
-apply(simp add: wprepare_loop_goon_on_rightmost.simps)
-done
-lemma wprepare_loop1: "\<lbrakk>rev b @ Oc\<^bsup>mr\<^esup> = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm>;
- b \<noteq> []; 0 < mr; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>\<rbrakk>
- \<Longrightarrow> wprepare_loop_start_on_rightmost m lm (Oc # b, list)"
-apply(simp add: wprepare_loop_start_on_rightmost.simps)
-apply(rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp, simp add: exp_ind_def, auto)
-done
-
-lemma wprepare_loop2: "\<lbrakk>rev b @ Oc\<^bsup>mr\<^esup> @ Bk # <a # lista> = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm>;
- b \<noteq> []; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk # <(a::nat) # lista> @ Bk\<^bsup>rn\<^esup>\<rbrakk>
- \<Longrightarrow> wprepare_loop_start_in_middle m lm (Oc # b, list)"
-apply(simp add: wprepare_loop_start_in_middle.simps)
-apply(rule_tac x = rn in exI, simp)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(rule_tac x = nat in exI, simp)
-apply(rule_tac x = "a#lista" in exI, simp)
-done
-
-lemma [simp]: "wprepare_loop_goon_in_middle m lm (b, Oc # list) \<Longrightarrow>
- wprepare_loop_start_on_rightmost m lm (Oc # b, list) \<or>
- wprepare_loop_start_in_middle m lm (Oc # b, list)"
-apply(simp add: wprepare_loop_goon_in_middle.simps split: if_splits)
-apply(case_tac lm1, simp_all add: wprepare_loop1 wprepare_loop2)
-done
-
-lemma [simp]: "wprepare_loop_goon m lm (b, Oc # list)
- \<Longrightarrow> wprepare_loop_start m lm (Oc # b, list)"
-apply(simp add: wprepare_loop_goon.simps
- wprepare_loop_start.simps)
-done
-
-lemma [simp]: "wprepare_add_one m lm (b, Oc # list)
- \<Longrightarrow> b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)"
-apply(auto)
-apply(simp add: wprepare_add_one.simps)
-done
-
-lemma [simp]: "wprepare_goto_start_pos m [a] (b, Oc # list)
- \<Longrightarrow> wprepare_loop_start_on_rightmost m [a] (Oc # b, list) "
-apply(auto simp: wprepare_goto_start_pos.simps
- wprepare_loop_start_on_rightmost.simps)
-apply(rule_tac x = rn in exI, simp)
-apply(simp add: tape_of_nat_abv tape_of_nat_list.simps exp_ind_def, auto)
-done
-
-lemma [simp]: "wprepare_goto_start_pos m (a # aa # listaa) (b, Oc # list)
- \<Longrightarrow>wprepare_loop_start_in_middle m (a # aa # listaa) (Oc # b, list)"
-apply(auto simp: wprepare_goto_start_pos.simps
- wprepare_loop_start_in_middle.simps)
-apply(rule_tac x = rn in exI, simp)
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
-apply(rule_tac x = a in exI, rule_tac x = "aa#listaa" in exI, simp)
-done
-
-lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Oc # list)\<rbrakk>
- \<Longrightarrow> wprepare_loop_start m lm (Oc # b, list)"
-apply(case_tac lm, simp_all)
-apply(case_tac lista, simp_all add: wprepare_loop_start.simps)
-done
-
-lemma [simp]: "wprepare_add_one2 m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
-apply(auto simp: wprepare_add_one2.simps)
-done
-
-lemma add_one_2_stop:
- "wprepare_add_one2 m lm (b, Oc # list)
- \<Longrightarrow> wprepare_stop m lm (tl b, hd b # Oc # list)"
-apply(simp add: wprepare_stop.simps wprepare_add_one2.simps)
-done
-
-declare wprepare_stop.simps[simp del]
-
-lemma wprepare_correctness:
- assumes h: "lm \<noteq> []"
- shows "let P = (\<lambda> (st, l, r). st = 0) in
- let Q = (\<lambda> (st, l, r). wprepare_inv st m lm (l, r)) in
- let f = (\<lambda> stp. steps (Suc 0, [], (<m # lm>)) t_wcode_prepare stp) in
- \<exists> n .P (f n) \<and> Q (f n)"
-proof -
- let ?P = "(\<lambda> (st, l, r). st = 0)"
- let ?Q = "(\<lambda> (st, l, r). wprepare_inv st m lm (l, r))"
- let ?f = "(\<lambda> stp. steps (Suc 0, [], (<m # lm>)) t_wcode_prepare stp)"
- have "\<exists> n. ?P (?f n) \<and> ?Q (?f n)"
- proof(rule_tac halt_lemma2)
- show "wf wcode_prepare_le" by auto
- next
- show "\<forall> n. \<not> ?P (?f n) \<and> ?Q (?f n) \<longrightarrow>
- ?Q (?f (Suc n)) \<and> (?f (Suc n), ?f n) \<in> wcode_prepare_le"
- using h
- apply(rule_tac allI, rule_tac impI, case_tac "?f n",
- simp add: tstep_red tstep.simps)
- apply(case_tac c, simp, case_tac [2] aa)
- apply(simp_all add: wprepare_inv.simps wcode_prepare_le_def new_tape.simps
- lex_triple_def lex_pair_def
-
- split: if_splits)
- apply(simp_all add: start_2_goon start_2_start
- add_one_2_add_one add_one_2_stop)
- apply(auto simp: wprepare_add_one2.simps)
- done
- next
- show "?Q (?f 0)"
- apply(simp add: steps.simps wprepare_inv.simps wprepare_invs)
- done
- next
- show "\<not> ?P (?f 0)"
- apply(simp add: steps.simps)
- done
- qed
- thus "?thesis"
- apply(auto)
- done
-qed
-
-lemma [intro]: "t_correct t_wcode_prepare"
-apply(simp add: t_correct.simps t_wcode_prepare_def iseven_def)
-apply(rule_tac x = 7 in exI, simp)
-done
-
-lemma twice_len_even: "length (tm_of abc_twice) mod 2 = 0"
-apply(simp add: tm_even)
-done
-
-lemma fourtimes_len_even: "length (tm_of abc_fourtimes) mod 2 = 0"
-apply(simp add: tm_even)
-done
-
-lemma t_correct_termi: "t_correct tp \<Longrightarrow>
- list_all (\<lambda>(acn, st). (st \<le> Suc (length tp div 2))) (change_termi_state tp)"
-apply(auto simp: t_correct.simps List.list_all_length)
-apply(erule_tac x = n in allE, simp)
-apply(case_tac "tp!n", auto simp: change_termi_state.simps split: if_splits)
-done
-
-
-lemma t_correct_shift:
- "list_all (\<lambda>(acn, st). (st \<le> y)) tp \<Longrightarrow>
- list_all (\<lambda>(acn, st). (st \<le> y + off)) (tshift tp off) "
-apply(auto simp: t_correct.simps List.list_all_length)
-apply(erule_tac x = n in allE, simp add: shift_length)
-apply(case_tac "tp!n", auto simp: tshift.simps)
-done
-
-lemma [intro]:
- "t_correct (tm_of abc_twice @ tMp (Suc 0)
- (start_of twice_ly (length abc_twice) - Suc 0))"
-apply(rule_tac t_compiled_correct, simp_all)
-apply(simp add: twice_ly_def)
-done
-
-lemma [intro]: "t_correct (tm_of abc_fourtimes @ tMp (Suc 0)
- (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))"
-apply(rule_tac t_compiled_correct, simp_all)
-apply(simp add: fourtimes_ly_def)
-done
-
-
-lemma [intro]: "t_correct t_wcode_main"
-apply(auto simp: t_wcode_main_def t_correct.simps shift_length
- t_twice_def t_fourtimes_def)
-proof -
- show "iseven (60 + (length (tm_of abc_twice) +
- length (tm_of abc_fourtimes)))"
- using twice_len_even fourtimes_len_even
- apply(auto simp: iseven_def)
- apply(rule_tac x = "30 + q + qa" in exI, simp)
- done
-next
- show " list_all (\<lambda>(acn, s). s \<le> (60 + (length (tm_of abc_twice) +
- length (tm_of abc_fourtimes))) div 2) t_wcode_main_first_part"
- apply(auto simp: t_wcode_main_first_part_def shift_length t_twice_def)
- done
-next
- have "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_twice @ tMp (Suc 0)
- (start_of twice_ly (length abc_twice) - Suc 0)) div 2))
- (change_termi_state (tm_of abc_twice @ tMp (Suc 0)
- (start_of twice_ly (length abc_twice) - Suc 0)))"
- apply(rule_tac t_correct_termi, auto)
- done
- hence "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_twice @ tMp (Suc 0)
- (start_of twice_ly (length abc_twice) - Suc 0)) div 2) + 12)
- (abacus.tshift (change_termi_state (tm_of abc_twice @ tMp (Suc 0)
- (start_of twice_ly (length abc_twice) - Suc 0))) 12)"
- apply(rule_tac t_correct_shift, simp)
- done
- thus "list_all (\<lambda>(acn, s). s \<le>
- (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2)
- (abacus.tshift (change_termi_state (tm_of abc_twice @ tMp (Suc 0)
- (start_of twice_ly (length abc_twice) - Suc 0))) 12)"
- apply(simp)
- apply(simp add: list_all_length, auto)
- done
-next
- have "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_fourtimes @ tMp (Suc 0)
- (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) div 2))
- (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0)
- (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) "
- apply(rule_tac t_correct_termi, auto)
- done
- hence "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_fourtimes @ tMp (Suc 0)
- (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) div 2) + (t_twice_len + 13))
- (abacus.tshift (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0)
- (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) (t_twice_len + 13))"
- apply(rule_tac t_correct_shift, simp)
- done
- thus "list_all (\<lambda>(acn, s). s \<le> (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2)
- (abacus.tshift (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0)
- (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) (t_twice_len + 13))"
- apply(simp add: t_twice_len_def t_twice_def)
- using twice_len_even fourtimes_len_even
- apply(auto simp: list_all_length)
- done
-qed
-
-lemma [intro]: "t_correct (t_wcode_prepare |+| t_wcode_main)"
-apply(auto intro: t_correct_add)
-done
-
-lemma prepare_mainpart_lemma:
- "args \<noteq> [] \<Longrightarrow>
- \<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp
- = (0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-proof -
- let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <m # args>"
- let ?Q1 = "\<lambda> (l, r). wprepare_stop m args (l, r)"
- let ?P2 = ?Q1
- let ?Q2 = "\<lambda> (l, r). (\<exists> ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- let ?P3 = "\<lambda> tp. False"
- assume h: "args \<noteq> []"
- have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp)
- (t_wcode_prepare |+| t_wcode_main) stp = (0, tp') \<and> ?Q2 tp')"
- proof(rule_tac turing_merge.t_merge_halt[of t_wcode_prepare t_wcode_main ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2],
- auto simp: turing_merge_def)
- show "\<exists>stp. case steps (Suc 0, [], <m # args>) t_wcode_prepare stp of (st, tp')
- \<Rightarrow> st = 0 \<and> wprepare_stop m args tp'"
- using wprepare_correctness[of args m] h
- apply(simp, auto)
- apply(rule_tac x = n in exI, simp add: wprepare_inv.simps)
- done
- next
- fix a b
- assume "wprepare_stop m args (a, b)"
- thus "\<exists>stp. case steps (Suc 0, a, b) t_wcode_main stp of
- (st, tp') \<Rightarrow> (st = 0) \<and> (case tp' of (l, r) \<Rightarrow> l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- (\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>))"
- proof(simp only: wprepare_stop.simps, erule_tac exE)
- fix rn
- assume "a = Bk # <rev args> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- b = Bk # Oc # Bk\<^bsup>rn\<^esup>"
- thus "?thesis"
- using t_wcode_main_lemma_pre[of "args" "<args>" 0 "Oc\<^bsup>Suc m\<^esup>" 0 rn] h
- apply(simp)
- apply(erule_tac exE)+
- apply(rule_tac x = stp in exI, simp add: tape_of_nl_rev, auto)
- done
- qed
- next
- show "wprepare_stop m args \<turnstile>-> wprepare_stop m args"
- by(simp add: t_imply_def)
- qed
- thus "\<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp
- = (0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- apply(simp add: t_imply_def)
- apply(erule_tac exE)+
- apply(auto)
- done
-qed
-
-
-lemma [simp]: "tinres r r' \<Longrightarrow>
- fetch t ss (case r of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) =
- fetch t ss (case r' of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)"
-apply(simp add: fetch.simps, auto split: if_splits simp: tinres_def)
-apply(case_tac [!] r', simp_all)
-apply(case_tac [!] n, simp_all add: exp_ind_def)
-apply(case_tac [!] r, simp_all)
-done
-
-lemma [intro]: "\<exists> n. (a::block)\<^bsup>n\<^esup> = []"
-by auto
-
-lemma [simp]: "\<lbrakk>tinres r r'; r \<noteq> []; r' \<noteq> []\<rbrakk> \<Longrightarrow> hd r = hd r'"
-apply(auto simp: tinres_def)
-done
-
-lemma [intro]: "hd (Bk\<^bsup>Suc n\<^esup>) = Bk"
-apply(simp add: exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>tinres r []; r \<noteq> []\<rbrakk> \<Longrightarrow> hd r = Bk"
-apply(auto simp: tinres_def)
-apply(case_tac n, auto)
-done
-
-lemma [simp]: "\<lbrakk>tinres [] r'; r' \<noteq> []\<rbrakk> \<Longrightarrow> hd r' = Bk"
-apply(auto simp: tinres_def)
-done
-
-lemma [intro]: "\<exists>na. tl r = tl (r @ Bk\<^bsup>n\<^esup>) @ Bk\<^bsup>na\<^esup> \<or> tl (r @ Bk\<^bsup>n\<^esup>) = tl r @ Bk\<^bsup>na\<^esup>"
-apply(case_tac r, simp)
-apply(case_tac n, simp)
-apply(rule_tac x = 0 in exI, simp)
-apply(rule_tac x = nat in exI, simp add: exp_ind_def)
-apply(simp)
-apply(rule_tac x = n in exI, simp)
-done
-
-lemma [simp]: "tinres r r' \<Longrightarrow> tinres (tl r) (tl r')"
-apply(auto simp: tinres_def)
-apply(case_tac r', simp_all)
-apply(case_tac n, simp_all add: exp_ind_def)
-apply(rule_tac x = 0 in exI, simp)
-apply(rule_tac x = nat in exI, simp_all)
-apply(rule_tac x = n in exI, simp)
-done
-
-lemma [simp]: "\<lbrakk>tinres r []; r \<noteq> []\<rbrakk> \<Longrightarrow> tinres (tl r) []"
-apply(case_tac r, auto simp: tinres_def)
-apply(case_tac n, simp_all add: exp_ind_def)
-apply(rule_tac x = nat in exI, simp)
-done
-
-lemma [simp]: "\<lbrakk>tinres [] r'\<rbrakk> \<Longrightarrow> tinres [] (tl r')"
-apply(case_tac r', auto simp: tinres_def)
-apply(case_tac n, simp_all add: exp_ind_def)
-apply(rule_tac x = nat in exI, simp)
-done
-
-lemma [simp]: "tinres r r' \<Longrightarrow> tinres (b # r) (b # r')"
-apply(auto simp: tinres_def)
-done
-
-lemma tinres_step2:
- "\<lbrakk>tinres r r'; tstep (ss, l, r) t = (sa, la, ra); tstep (ss, l, r') t = (sb, lb, rb)\<rbrakk>
- \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb"
-apply(case_tac "ss = 0", simp add: tstep_0)
-apply(simp add: tstep.simps [simp del])
-apply(case_tac "fetch t ss (case r of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
-apply(auto simp: new_tape.simps)
-apply(simp_all split: taction.splits if_splits)
-apply(auto)
-done
-
-
-lemma tinres_steps2:
- "\<lbrakk>tinres r r'; steps (ss, l, r) t stp = (sa, la, ra); steps (ss, l, r') t stp = (sb, lb, rb)\<rbrakk>
- \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb"
-apply(induct stp arbitrary: sa la ra sb lb rb, simp add: steps.simps)
-apply(simp add: tstep_red)
-apply(case_tac "(steps (ss, l, r) t stp)")
-apply(case_tac "(steps (ss, l, r') t stp)")
-proof -
- fix stp sa la ra sb lb rb a b c aa ba ca
- assume ind: "\<And>sa la ra sb lb rb. \<lbrakk>steps (ss, l, r) t stp = (sa, la, ra);
- steps (ss, l, r') t stp = (sb, lb, rb)\<rbrakk> \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb"
- and h: " tinres r r'" "tstep (steps (ss, l, r) t stp) t = (sa, la, ra)"
- "tstep (steps (ss, l, r') t stp) t = (sb, lb, rb)" "steps (ss, l, r) t stp = (a, b, c)"
- "steps (ss, l, r') t stp = (aa, ba, ca)"
- have "b = ba \<and> tinres c ca \<and> a = aa"
- apply(rule_tac ind, simp_all add: h)
- done
- thus "la = lb \<and> tinres ra rb \<and> sa = sb"
- apply(rule_tac l = b and r = c and ss = a and r' = ca
- and t = t in tinres_step2)
- using h
- apply(simp, simp, simp)
- done
-qed
-
-
-text{**************Begin: adjust***************************}
-definition t_wcode_adjust :: "tprog"
- where
- "t_wcode_adjust = [(W1, 1), (R, 2), (Nop, 2), (R, 3), (R, 3), (R, 4),
- (L, 8), (L, 5), (L, 6), (W0, 5), (L, 6), (R, 7),
- (W1, 2), (Nop, 7), (L, 9), (W0, 8), (L, 9), (L, 10),
- (L, 11), (L, 10), (R, 0), (L, 11)]"
-
-lemma [simp]: "fetch t_wcode_adjust (Suc 0) Bk = (W1, 1)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust (Suc 0) Oc = (R, 2)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust (Suc (Suc 0)) Oc = (R, 3)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust (Suc (Suc (Suc 0))) Oc = (R, 4)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust (Suc (Suc (Suc 0))) Bk = (R, 3)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 4 Bk = (L, 8)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 4 Oc = (L, 5)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 5 Oc = (W0, 5)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 5 Bk = (L, 6)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 6 Oc = (R, 7)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 6 Bk = (L, 6)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 7 Bk = (W1, 2)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 8 Bk = (L, 9)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 8 Oc = (W0, 8)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 9 Oc = (L, 10)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 9 Bk = (L, 9)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 10 Bk = (L, 11)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 10 Oc = (L, 10)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 11 Oc = (L, 11)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-lemma [simp]: "fetch t_wcode_adjust 11 Bk = (R, 0)"
-apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
-done
-
-fun wadjust_start :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_start m rs (l, r) =
- (\<exists> ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- tl r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wadjust_loop_start :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_loop_start m rs (l, r) =
- (\<exists> ln rn ml mr. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr = Suc (Suc rs) \<and> mr > 0)"
-
-fun wadjust_loop_right_move :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_loop_right_move m rs (l, r) =
- (\<exists> ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- r = Bk\<^bsup>nr\<^esup> @ Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr = Suc (Suc rs) \<and> mr > 0 \<and>
- nl + nr > 0)"
-
-fun wadjust_loop_check :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_loop_check m rs (l, r) =
- (\<exists> ml mr ln rn. l = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = (Suc rs))"
-
-fun wadjust_loop_erase :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_loop_erase m rs (l, r) =
- (\<exists> ml mr ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- tl r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = (Suc rs) \<and> mr > 0)"
-
-fun wadjust_loop_on_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_loop_on_left_moving_O m rs (l, r) =
- (\<exists> ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m \<^esup>\<and>
- r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr = Suc rs \<and> mr > 0)"
-
-fun wadjust_loop_on_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_loop_on_left_moving_B m rs (l, r) =
- (\<exists> ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- r = Bk\<^bsup>nr\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr = Suc rs \<and> mr > 0)"
-
-fun wadjust_loop_on_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_loop_on_left_moving m rs (l, r) =
- (wadjust_loop_on_left_moving_O m rs (l, r) \<or>
- wadjust_loop_on_left_moving_B m rs (l, r))"
-
-fun wadjust_loop_right_move2 :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_loop_right_move2 m rs (l, r) =
- (\<exists> ml mr ln rn. l = Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr = Suc rs \<and> mr > 0)"
-
-fun wadjust_erase2 :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_erase2 m rs (l, r) =
- (\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- tl r = Bk\<^bsup>rn\<^esup>)"
-
-fun wadjust_on_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_on_left_moving_O m rs (l, r) =
- (\<exists> rn. l = Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- r = Oc # Bk\<^bsup>rn\<^esup>)"
-
-fun wadjust_on_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_on_left_moving_B m rs (l, r) =
- (\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- r = Bk\<^bsup>rn\<^esup>)"
-
-fun wadjust_on_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_on_left_moving m rs (l, r) =
- (wadjust_on_left_moving_O m rs (l, r) \<or>
- wadjust_on_left_moving_B m rs (l, r))"
-
-fun wadjust_goon_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_goon_left_moving_B m rs (l, r) =
- (\<exists> rn. l = Oc\<^bsup>Suc m\<^esup> \<and>
- r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wadjust_goon_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_goon_left_moving_O m rs (l, r) =
- (\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr = Suc (Suc rs) \<and> mr > 0)"
-
-fun wadjust_goon_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_goon_left_moving m rs (l, r) =
- (wadjust_goon_left_moving_B m rs (l, r) \<or>
- wadjust_goon_left_moving_O m rs (l, r))"
-
-fun wadjust_backto_standard_pos_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_backto_standard_pos_B m rs (l, r) =
- (\<exists> rn. l = [] \<and>
- r = Bk # Oc\<^bsup>Suc m \<^esup>@ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-fun wadjust_backto_standard_pos_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_backto_standard_pos_O m rs (l, r) =
- (\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> \<and>
- r = Oc\<^bsup>mr\<^esup> @ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
- ml + mr = Suc m \<and> mr > 0)"
-
-fun wadjust_backto_standard_pos :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_backto_standard_pos m rs (l, r) =
- (wadjust_backto_standard_pos_B m rs (l, r) \<or>
- wadjust_backto_standard_pos_O m rs (l, r))"
-
-fun wadjust_stop :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
-where
- "wadjust_stop m rs (l, r) =
- (\<exists> rn. l = [Bk] \<and>
- r = Oc\<^bsup>Suc m \<^esup>@ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-
-declare wadjust_start.simps[simp del] wadjust_loop_start.simps[simp del]
- wadjust_loop_right_move.simps[simp del] wadjust_loop_check.simps[simp del]
- wadjust_loop_erase.simps[simp del] wadjust_loop_on_left_moving.simps[simp del]
- wadjust_loop_right_move2.simps[simp del] wadjust_erase2.simps[simp del]
- wadjust_on_left_moving_O.simps[simp del] wadjust_on_left_moving_B.simps[simp del]
- wadjust_on_left_moving.simps[simp del] wadjust_goon_left_moving_B.simps[simp del]
- wadjust_goon_left_moving_O.simps[simp del] wadjust_goon_left_moving.simps[simp del]
- wadjust_backto_standard_pos.simps[simp del] wadjust_backto_standard_pos_B.simps[simp del]
- wadjust_backto_standard_pos_O.simps[simp del] wadjust_stop.simps[simp del]
-
-fun wadjust_inv :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
- where
- "wadjust_inv st m rs (l, r) =
- (if st = Suc 0 then wadjust_start m rs (l, r)
- else if st = Suc (Suc 0) then wadjust_loop_start m rs (l, r)
- else if st = Suc (Suc (Suc 0)) then wadjust_loop_right_move m rs (l, r)
- else if st = 4 then wadjust_loop_check m rs (l, r)
- else if st = 5 then wadjust_loop_erase m rs (l, r)
- else if st = 6 then wadjust_loop_on_left_moving m rs (l, r)
- else if st = 7 then wadjust_loop_right_move2 m rs (l, r)
- else if st = 8 then wadjust_erase2 m rs (l, r)
- else if st = 9 then wadjust_on_left_moving m rs (l, r)
- else if st = 10 then wadjust_goon_left_moving m rs (l, r)
- else if st = 11 then wadjust_backto_standard_pos m rs (l, r)
- else if st = 0 then wadjust_stop m rs (l, r)
- else False
-)"
-
-declare wadjust_inv.simps[simp del]
-
-fun wadjust_phase :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
- where
- "wadjust_phase rs (st, l, r) =
- (if st = 1 then 3
- else if st \<ge> 2 \<and> st \<le> 7 then 2
- else if st \<ge> 8 \<and> st \<le> 11 then 1
- else 0)"
-
-thm dropWhile.simps
-
-fun wadjust_stage :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
- where
- "wadjust_stage rs (st, l, r) =
- (if st \<ge> 2 \<and> st \<le> 7 then
- rs - length (takeWhile (\<lambda> a. a = Oc)
- (tl (dropWhile (\<lambda> a. a = Oc) (rev l @ r))))
- else 0)"
-
-fun wadjust_state :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
- where
- "wadjust_state rs (st, l, r) =
- (if st \<ge> 2 \<and> st \<le> 7 then 8 - st
- else if st \<ge> 8 \<and> st \<le> 11 then 12 - st
- else 0)"
-
-fun wadjust_step :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
- where
- "wadjust_step rs (st, l, r) =
- (if st = 1 then (if hd r = Bk then 1
- else 0)
- else if st = 3 then length r
- else if st = 5 then (if hd r = Oc then 1
- else 0)
- else if st = 6 then length l
- else if st = 8 then (if hd r = Oc then 1
- else 0)
- else if st = 9 then length l
- else if st = 10 then length l
- else if st = 11 then (if hd r = Bk then 0
- else Suc (length l))
- else 0)"
-
-fun wadjust_measure :: "(nat \<times> t_conf) \<Rightarrow> nat \<times> nat \<times> nat \<times> nat"
- where
- "wadjust_measure (rs, (st, l, r)) =
- (wadjust_phase rs (st, l, r),
- wadjust_stage rs (st, l, r),
- wadjust_state rs (st, l, r),
- wadjust_step rs (st, l, r))"
-
-definition wadjust_le :: "((nat \<times> t_conf) \<times> nat \<times> t_conf) set"
- where "wadjust_le \<equiv> (inv_image lex_square wadjust_measure)"
-
-lemma [intro]: "wf lex_square"
-by(auto intro:wf_lex_prod simp: abacus.lex_pair_def lex_square_def
- abacus.lex_triple_def)
-
-lemma wf_wadjust_le[intro]: "wf wadjust_le"
-by(auto intro:wf_inv_image simp: wadjust_le_def
- abacus.lex_triple_def abacus.lex_pair_def)
-
-lemma [simp]: "wadjust_start m rs (c, []) = False"
-apply(auto simp: wadjust_start.simps)
-done
-
-lemma [simp]: "wadjust_loop_right_move m rs (c, []) \<Longrightarrow> c \<noteq> []"
-apply(auto simp: wadjust_loop_right_move.simps)
-done
-
-lemma [simp]: "wadjust_loop_right_move m rs (c, [])
- \<Longrightarrow> wadjust_loop_check m rs (Bk # c, [])"
-apply(simp only: wadjust_loop_right_move.simps wadjust_loop_check.simps)
-apply(auto)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_loop_check m rs (c, []) \<Longrightarrow> c \<noteq> []"
-apply(simp only: wadjust_loop_check.simps, auto)
-done
-
-lemma [simp]: "wadjust_loop_start m rs (c, []) = False"
-apply(simp add: wadjust_loop_start.simps)
-done
-
-lemma [simp]: "wadjust_loop_right_move m rs (c, []) \<Longrightarrow>
- wadjust_loop_right_move m rs (Bk # c, [])"
-apply(simp only: wadjust_loop_right_move.simps)
-apply(erule_tac exE)+
-apply(auto)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_loop_check m rs (c, []) \<Longrightarrow> wadjust_erase2 m rs (tl c, [hd c])"
-apply(simp only: wadjust_loop_check.simps wadjust_erase2.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def, auto)
-done
-
-lemma [simp]: " wadjust_loop_erase m rs (c, [])
- \<Longrightarrow> (c = [] \<longrightarrow> wadjust_loop_on_left_moving m rs ([], [Bk])) \<and>
- (c \<noteq> [] \<longrightarrow> wadjust_loop_on_left_moving m rs (tl c, [hd c]))"
-apply(simp add: wadjust_loop_erase.simps, auto)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_loop_on_left_moving m rs (c, []) = False"
-apply(auto simp: wadjust_loop_on_left_moving.simps)
-done
-
-
-lemma [simp]: "wadjust_loop_right_move2 m rs (c, []) = False"
-apply(auto simp: wadjust_loop_right_move2.simps)
-done
-
-lemma [simp]: "wadjust_erase2 m rs ([], []) = False"
-apply(auto simp: wadjust_erase2.simps)
-done
-
-lemma [simp]: "wadjust_on_left_moving_B m rs
- (Oc # Oc # Oc\<^bsup>rs\<^esup> @ Bk # Oc # Oc\<^bsup>m\<^esup>, [Bk])"
-apply(simp add: wadjust_on_left_moving_B.simps, auto)
-apply(rule_tac x = 0 in exI, simp add: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_on_left_moving_B m rs
- (Bk\<^bsup>n\<^esup> @ Bk # Oc # Oc # Oc\<^bsup>rs\<^esup> @ Bk # Oc # Oc\<^bsup>m\<^esup>, [Bk])"
-apply(simp add: wadjust_on_left_moving_B.simps exp_ind_def, auto)
-apply(rule_tac x = "Suc n" in exI, simp add: exp_ind)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_erase2 m rs (c, []); c \<noteq> []\<rbrakk> \<Longrightarrow>
- wadjust_on_left_moving m rs (tl c, [hd c])"
-apply(simp only: wadjust_erase2.simps)
-apply(erule_tac exE)+
-apply(case_tac ln, simp_all add: exp_ind_def wadjust_on_left_moving.simps)
-done
-
-lemma [simp]: "wadjust_erase2 m rs (c, [])
- \<Longrightarrow> (c = [] \<longrightarrow> wadjust_on_left_moving m rs ([], [Bk])) \<and>
- (c \<noteq> [] \<longrightarrow> wadjust_on_left_moving m rs (tl c, [hd c]))"
-apply(auto)
-done
-
-lemma [simp]: "wadjust_on_left_moving m rs ([], []) = False"
-apply(simp add: wadjust_on_left_moving.simps
- wadjust_on_left_moving_O.simps wadjust_on_left_moving_B.simps)
-done
-
-lemma [simp]: "wadjust_on_left_moving_O m rs (c, []) = False"
-apply(simp add: wadjust_on_left_moving_O.simps)
-done
-
-lemma [simp]: " \<lbrakk>wadjust_on_left_moving_B m rs (c, []); c \<noteq> []; hd c = Bk\<rbrakk> \<Longrightarrow>
- wadjust_on_left_moving_B m rs (tl c, [Bk])"
-apply(simp add: wadjust_on_left_moving_B.simps, auto)
-apply(case_tac [!] ln, simp_all add: exp_ind_def, auto)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, []); c \<noteq> []; hd c = Oc\<rbrakk> \<Longrightarrow>
- wadjust_on_left_moving_O m rs (tl c, [Oc])"
-apply(simp add: wadjust_on_left_moving_B.simps wadjust_on_left_moving_O.simps, auto)
-apply(case_tac [!] ln, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_on_left_moving m rs (c, []); c \<noteq> []\<rbrakk> \<Longrightarrow>
- wadjust_on_left_moving m rs (tl c, [hd c])"
-apply(simp add: wadjust_on_left_moving.simps)
-apply(case_tac "hd c", simp_all)
-done
-
-lemma [simp]: "wadjust_on_left_moving m rs (c, [])
- \<Longrightarrow> (c = [] \<longrightarrow> wadjust_on_left_moving m rs ([], [Bk])) \<and>
- (c \<noteq> [] \<longrightarrow> wadjust_on_left_moving m rs (tl c, [hd c]))"
-apply(auto)
-done
-
-lemma [simp]: "wadjust_goon_left_moving m rs (c, []) = False"
-apply(auto simp: wadjust_goon_left_moving.simps wadjust_goon_left_moving_B.simps
- wadjust_goon_left_moving_O.simps)
-done
-
-lemma [simp]: "wadjust_backto_standard_pos m rs (c, []) = False"
-apply(auto simp: wadjust_backto_standard_pos.simps
- wadjust_backto_standard_pos_B.simps wadjust_backto_standard_pos_O.simps)
-done
-
-lemma [simp]:
- "wadjust_start m rs (c, Bk # list) \<Longrightarrow>
- (c = [] \<longrightarrow> wadjust_start m rs ([], Oc # list)) \<and>
- (c \<noteq> [] \<longrightarrow> wadjust_start m rs (c, Oc # list))"
-apply(auto simp: wadjust_start.simps)
-done
-
-lemma [simp]: "wadjust_loop_start m rs (c, Bk # list) = False"
-apply(auto simp: wadjust_loop_start.simps)
-done
-
-lemma [simp]: "wadjust_loop_right_move m rs (c, b) \<Longrightarrow> c \<noteq> []"
-apply(simp only: wadjust_loop_right_move.simps, auto)
-done
-
-lemma [simp]: "wadjust_loop_right_move m rs (c, Bk # list)
- \<Longrightarrow> wadjust_loop_right_move m rs (Bk # c, list)"
-apply(simp only: wadjust_loop_right_move.simps)
-apply(erule_tac exE)+
-apply(rule_tac x = ml in exI, simp)
-apply(rule_tac x = mr in exI, simp)
-apply(rule_tac x = "Suc nl" in exI, simp add: exp_ind_def)
-apply(case_tac nr, simp, case_tac mr, simp_all add: exp_ind_def)
-apply(rule_tac x = nat in exI, auto)
-done
-
-lemma [simp]: "wadjust_loop_check m rs (c, b) \<Longrightarrow> c \<noteq> []"
-apply(simp only: wadjust_loop_check.simps, auto)
-done
-
-lemma [simp]: "wadjust_loop_check m rs (c, Bk # list)
- \<Longrightarrow> wadjust_erase2 m rs (tl c, hd c # Bk # list)"
-apply(auto simp: wadjust_loop_check.simps wadjust_erase2.simps)
-apply(case_tac [!] mr, simp_all add: exp_ind_def, auto)
-done
-
-lemma [simp]: "wadjust_loop_erase m rs (c, b) \<Longrightarrow> c \<noteq> []"
-apply(simp only: wadjust_loop_erase.simps, auto)
-done
-
-declare wadjust_loop_on_left_moving_O.simps[simp del]
- wadjust_loop_on_left_moving_B.simps[simp del]
-
-lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); hd c = Bk\<rbrakk>
- \<Longrightarrow> wadjust_loop_on_left_moving_B m rs (tl c, Bk # Bk # list)"
-apply(simp only: wadjust_loop_erase.simps
- wadjust_loop_on_left_moving_B.simps)
-apply(erule_tac exE)+
-apply(rule_tac x = ml in exI, rule_tac x = mr in exI,
- rule_tac x = ln in exI, rule_tac x = 0 in exI, simp)
-apply(case_tac ln, simp_all add: exp_ind_def, auto)
-apply(simp add: exp_ind exp_ind_def[THEN sym])
-done
-
-lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); c \<noteq> []; hd c = Oc\<rbrakk> \<Longrightarrow>
- wadjust_loop_on_left_moving_O m rs (tl c, Oc # Bk # list)"
-apply(simp only: wadjust_loop_erase.simps wadjust_loop_on_left_moving_O.simps,
- auto)
-apply(case_tac [!] ln, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); c \<noteq> []\<rbrakk> \<Longrightarrow>
- wadjust_loop_on_left_moving m rs (tl c, hd c # Bk # list)"
-apply(case_tac "hd c", simp_all add:wadjust_loop_on_left_moving.simps)
-done
-
-lemma [simp]: "wadjust_loop_on_left_moving m rs (c, b) \<Longrightarrow> c \<noteq> []"
-apply(simp add: wadjust_loop_on_left_moving.simps
-wadjust_loop_on_left_moving_O.simps wadjust_loop_on_left_moving_B.simps, auto)
-done
-
-lemma [simp]: "wadjust_loop_on_left_moving_O m rs (c, Bk # list) = False"
-apply(simp add: wadjust_loop_on_left_moving_O.simps)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Bk\<rbrakk>
- \<Longrightarrow> wadjust_loop_on_left_moving_B m rs (tl c, Bk # Bk # list)"
-apply(simp only: wadjust_loop_on_left_moving_B.simps)
-apply(erule_tac exE)+
-apply(rule_tac x = ml in exI, rule_tac x = mr in exI)
-apply(case_tac nl, simp_all add: exp_ind_def, auto)
-apply(rule_tac x = "Suc nr" in exI, auto simp: exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk>
- \<Longrightarrow> wadjust_loop_on_left_moving_O m rs (tl c, Oc # Bk # list)"
-apply(simp only: wadjust_loop_on_left_moving_O.simps
- wadjust_loop_on_left_moving_B.simps)
-apply(erule_tac exE)+
-apply(rule_tac x = ml in exI, rule_tac x = mr in exI)
-apply(case_tac nl, simp_all add: exp_ind_def, auto)
-done
-
-lemma [simp]: "wadjust_loop_on_left_moving m rs (c, Bk # list)
- \<Longrightarrow> wadjust_loop_on_left_moving m rs (tl c, hd c # Bk # list)"
-apply(simp add: wadjust_loop_on_left_moving.simps)
-apply(case_tac "hd c", simp_all)
-done
-
-lemma [simp]: "wadjust_loop_right_move2 m rs (c, b) \<Longrightarrow> c \<noteq> []"
-apply(simp only: wadjust_loop_right_move2.simps, auto)
-done
-
-lemma [simp]: "wadjust_loop_right_move2 m rs (c, Bk # list) \<Longrightarrow> wadjust_loop_start m rs (c, Oc # list)"
-apply(auto simp: wadjust_loop_right_move2.simps wadjust_loop_start.simps)
-apply(case_tac ln, simp_all add: exp_ind_def)
-apply(rule_tac x = 0 in exI, simp)
-apply(rule_tac x = rn in exI, simp)
-apply(rule_tac x = "Suc ml" in exI, simp add: exp_ind_def, auto)
-apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind)
-apply(rule_tac x = rn in exI, auto)
-apply(rule_tac x = "Suc ml" in exI, auto simp: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \<Longrightarrow> c \<noteq> []"
-apply(auto simp:wadjust_erase2.simps )
-done
-
-lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \<Longrightarrow>
- wadjust_on_left_moving m rs (tl c, hd c # Bk # list)"
-apply(auto simp: wadjust_erase2.simps)
-apply(case_tac ln, simp_all add: exp_ind_def wadjust_on_left_moving.simps
- wadjust_on_left_moving_O.simps wadjust_on_left_moving_B.simps)
-apply(auto)
-apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: exp_ind_def)
-apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind)
-apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_on_left_moving m rs (c,b) \<Longrightarrow> c \<noteq> []"
-apply(simp only:wadjust_on_left_moving.simps
- wadjust_on_left_moving_O.simps
- wadjust_on_left_moving_B.simps
- , auto)
-done
-
-lemma [simp]: "wadjust_on_left_moving_O m rs (c, Bk # list) = False"
-apply(simp add: wadjust_on_left_moving_O.simps)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Bk\<rbrakk>
- \<Longrightarrow> wadjust_on_left_moving_B m rs (tl c, Bk # Bk # list)"
-apply(auto simp: wadjust_on_left_moving_B.simps)
-apply(case_tac ln, simp_all add: exp_ind_def, auto)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk>
- \<Longrightarrow> wadjust_on_left_moving_O m rs (tl c, Oc # Bk # list)"
-apply(auto simp: wadjust_on_left_moving_O.simps
- wadjust_on_left_moving_B.simps)
-apply(case_tac ln, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_on_left_moving m rs (c, Bk # list) \<Longrightarrow>
- wadjust_on_left_moving m rs (tl c, hd c # Bk # list)"
-apply(simp add: wadjust_on_left_moving.simps)
-apply(case_tac "hd c", simp_all)
-done
-
-lemma [simp]: "wadjust_goon_left_moving m rs (c, b) \<Longrightarrow> c \<noteq> []"
-apply(simp add: wadjust_goon_left_moving.simps
- wadjust_goon_left_moving_B.simps
- wadjust_goon_left_moving_O.simps exp_ind_def, auto)
-done
-
-lemma [simp]: "wadjust_goon_left_moving_O m rs (c, Bk # list) = False"
-apply(simp add: wadjust_goon_left_moving_O.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Bk\<rbrakk>
- \<Longrightarrow> wadjust_backto_standard_pos_B m rs (tl c, Bk # Bk # list)"
-apply(auto simp: wadjust_goon_left_moving_B.simps
- wadjust_backto_standard_pos_B.simps exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk>
- \<Longrightarrow> wadjust_backto_standard_pos_O m rs (tl c, Oc # Bk # list)"
-apply(auto simp: wadjust_goon_left_moving_B.simps
- wadjust_backto_standard_pos_O.simps exp_ind_def)
-apply(rule_tac x = m in exI, simp, auto)
-done
-
-lemma [simp]: "wadjust_goon_left_moving m rs (c, Bk # list) \<Longrightarrow>
- wadjust_backto_standard_pos m rs (tl c, hd c # Bk # list)"
-apply(case_tac "hd c", simp_all add: wadjust_backto_standard_pos.simps
- wadjust_goon_left_moving.simps)
-done
-
-lemma [simp]: "wadjust_backto_standard_pos m rs (c, Bk # list) \<Longrightarrow>
- (c = [] \<longrightarrow> wadjust_stop m rs ([Bk], list)) \<and> (c \<noteq> [] \<longrightarrow> wadjust_stop m rs (Bk # c, list))"
-apply(auto simp: wadjust_backto_standard_pos.simps
- wadjust_backto_standard_pos_B.simps
- wadjust_backto_standard_pos_O.simps wadjust_stop.simps)
-apply(case_tac [!] mr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_start m rs (c, Oc # list)
- \<Longrightarrow> (c = [] \<longrightarrow> wadjust_loop_start m rs ([Oc], list)) \<and>
- (c \<noteq> [] \<longrightarrow> wadjust_loop_start m rs (Oc # c, list))"
-apply(auto simp:wadjust_loop_start.simps wadjust_start.simps )
-apply(rule_tac x = ln in exI, rule_tac x = rn in exI,
- rule_tac x = "Suc 0" in exI, simp)
-done
-
-lemma [simp]: "wadjust_loop_start m rs (c, b) \<Longrightarrow> c \<noteq> []"
-apply(simp add: wadjust_loop_start.simps, auto)
-done
-
-lemma [simp]: "wadjust_loop_start m rs (c, Oc # list)
- \<Longrightarrow> wadjust_loop_right_move m rs (Oc # c, list)"
-apply(simp add: wadjust_loop_start.simps wadjust_loop_right_move.simps, auto)
-apply(rule_tac x = ml in exI, rule_tac x = mr in exI,
- rule_tac x = 0 in exI, simp)
-apply(rule_tac x = "Suc ln" in exI, simp add: exp_ind, auto)
-done
-
-lemma [simp]: "wadjust_loop_right_move m rs (c, Oc # list) \<Longrightarrow>
- wadjust_loop_check m rs (Oc # c, list)"
-apply(simp add: wadjust_loop_right_move.simps
- wadjust_loop_check.simps, auto)
-apply(rule_tac [!] x = ml in exI, simp_all, auto)
-apply(case_tac nl, auto simp: exp_ind_def)
-apply(rule_tac x = "mr - 1" in exI, case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac [!] nr, simp_all add: exp_ind_def, auto)
-done
-
-lemma [simp]: "wadjust_loop_check m rs (c, Oc # list) \<Longrightarrow>
- wadjust_loop_erase m rs (tl c, hd c # Oc # list)"
-apply(simp only: wadjust_loop_check.simps wadjust_loop_erase.simps)
-apply(erule_tac exE)+
-apply(rule_tac x = ml in exI, rule_tac x = mr in exI, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-apply(case_tac rn, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_loop_erase m rs (c, Oc # list) \<Longrightarrow>
- wadjust_loop_erase m rs (c, Bk # list)"
-apply(auto simp: wadjust_loop_erase.simps)
-done
-
-lemma [simp]: "wadjust_loop_on_left_moving_B m rs (c, Oc # list) = False"
-apply(auto simp: wadjust_loop_on_left_moving_B.simps)
-apply(case_tac nr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_loop_on_left_moving m rs (c, Oc # list)
- \<Longrightarrow> wadjust_loop_right_move2 m rs (Oc # c, list)"
-apply(simp add:wadjust_loop_on_left_moving.simps)
-apply(auto simp: wadjust_loop_on_left_moving_O.simps
- wadjust_loop_right_move2.simps)
-done
-
-lemma [simp]: "wadjust_loop_right_move2 m rs (c, Oc # list) = False"
-apply(auto simp: wadjust_loop_right_move2.simps )
-apply(case_tac ln, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_erase2 m rs (c, Oc # list)
- \<Longrightarrow> (c = [] \<longrightarrow> wadjust_erase2 m rs ([], Bk # list))
- \<and> (c \<noteq> [] \<longrightarrow> wadjust_erase2 m rs (c, Bk # list))"
-apply(auto simp: wadjust_erase2.simps )
-done
-
-lemma [simp]: "wadjust_on_left_moving_B m rs (c, Oc # list) = False"
-apply(auto simp: wadjust_on_left_moving_B.simps)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Bk\<rbrakk> \<Longrightarrow>
- wadjust_goon_left_moving_B m rs (tl c, Bk # Oc # list)"
-apply(auto simp: wadjust_on_left_moving_O.simps
- wadjust_goon_left_moving_B.simps exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Oc\<rbrakk>
- \<Longrightarrow> wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)"
-apply(auto simp: wadjust_on_left_moving_O.simps
- wadjust_goon_left_moving_O.simps exp_ind_def)
-apply(rule_tac x = rs in exI, simp)
-apply(auto simp: exp_ind_def numeral_2_eq_2)
-done
-
-
-lemma [simp]: "wadjust_on_left_moving m rs (c, Oc # list) \<Longrightarrow>
- wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)"
-apply(simp add: wadjust_on_left_moving.simps
- wadjust_goon_left_moving.simps)
-apply(case_tac "hd c", simp_all)
-done
-
-lemma [simp]: "wadjust_on_left_moving m rs (c, Oc # list) \<Longrightarrow>
- wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)"
-apply(simp add: wadjust_on_left_moving.simps
- wadjust_goon_left_moving.simps)
-apply(case_tac "hd c", simp_all)
-done
-
-lemma [simp]: "wadjust_goon_left_moving_B m rs (c, Oc # list) = False"
-apply(auto simp: wadjust_goon_left_moving_B.simps)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Bk\<rbrakk>
- \<Longrightarrow> wadjust_goon_left_moving_B m rs (tl c, Bk # Oc # list)"
-apply(auto simp: wadjust_goon_left_moving_O.simps wadjust_goon_left_moving_B.simps)
-apply(case_tac [!] ml, auto simp: exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Oc\<rbrakk> \<Longrightarrow>
- wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)"
-apply(auto simp: wadjust_goon_left_moving_O.simps wadjust_goon_left_moving_B.simps)
-apply(rule_tac x = "ml - 1" in exI, simp)
-apply(case_tac ml, simp_all add: exp_ind_def)
-apply(rule_tac x = "Suc mr" in exI, auto simp: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_goon_left_moving m rs (c, Oc # list) \<Longrightarrow>
- wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)"
-apply(simp add: wadjust_goon_left_moving.simps)
-apply(case_tac "hd c", simp_all)
-done
-
-lemma [simp]: "wadjust_backto_standard_pos_B m rs (c, Oc # list) = False"
-apply(simp add: wadjust_backto_standard_pos_B.simps)
-done
-
-lemma [simp]: "wadjust_backto_standard_pos_O m rs (c, Bk # xs) = False"
-apply(simp add: wadjust_backto_standard_pos_O.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-done
-
-
-
-lemma [simp]: "wadjust_backto_standard_pos_O m rs ([], Oc # list) \<Longrightarrow>
- wadjust_backto_standard_pos_B m rs ([], Bk # Oc # list)"
-apply(auto simp: wadjust_backto_standard_pos_O.simps
- wadjust_backto_standard_pos_B.simps)
-apply(rule_tac x = rn in exI, simp)
-apply(case_tac ml, simp_all add: exp_ind_def)
-done
-
-
-lemma [simp]:
- "\<lbrakk>wadjust_backto_standard_pos_O m rs (c, Oc # list); c \<noteq> []; hd c = Bk\<rbrakk>
- \<Longrightarrow> wadjust_backto_standard_pos_B m rs (tl c, Bk # Oc # list)"
-apply(simp add:wadjust_backto_standard_pos_O.simps
- wadjust_backto_standard_pos_B.simps, auto)
-apply(case_tac [!] ml, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>wadjust_backto_standard_pos_O m rs (c, Oc # list); c \<noteq> []; hd c = Oc\<rbrakk>
- \<Longrightarrow> wadjust_backto_standard_pos_O m rs (tl c, Oc # Oc # list)"
-apply(simp add: wadjust_backto_standard_pos_O.simps, auto)
-apply(case_tac ml, simp_all add: exp_ind_def, auto)
-apply(rule_tac x = nat in exI, auto simp: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_backto_standard_pos m rs (c, Oc # list)
- \<Longrightarrow> (c = [] \<longrightarrow> wadjust_backto_standard_pos m rs ([], Bk # Oc # list)) \<and>
- (c \<noteq> [] \<longrightarrow> wadjust_backto_standard_pos m rs (tl c, hd c # Oc # list))"
-apply(auto simp: wadjust_backto_standard_pos.simps)
-apply(case_tac "hd c", simp_all)
-done
-thm wadjust_loop_right_move.simps
-
-lemma [simp]: "wadjust_loop_right_move m rs (c, []) = False"
-apply(simp only: wadjust_loop_right_move.simps)
-apply(rule_tac iffI)
-apply(erule_tac exE)+
-apply(case_tac nr, simp_all add: exp_ind_def)
-apply(case_tac mr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_loop_erase m rs (c, []) = False"
-apply(simp only: wadjust_loop_erase.simps, auto)
-apply(case_tac mr, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>Suc (Suc rs) = a; wadjust_loop_erase m rs (c, Bk # list)\<rbrakk>
- \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list))))
- < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list)))) \<or>
- a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) =
- a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))"
-apply(simp only: wadjust_loop_erase.simps)
-apply(rule_tac disjI2)
-apply(case_tac c, simp, simp)
-done
-
-lemma [simp]:
- "\<lbrakk>Suc (Suc rs) = a; wadjust_loop_on_left_moving m rs (c, Bk # list)\<rbrakk>
- \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list))))
- < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list)))) \<or>
- a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) =
- a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))"
-apply(subgoal_tac "c \<noteq> []")
-apply(case_tac c, simp_all)
-done
-
-lemma dropWhile_exp1: "dropWhile (\<lambda>a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = dropWhile (\<lambda>a. a = Oc) xs"
-apply(induct n, simp_all add: exp_ind_def)
-done
-lemma takeWhile_exp1: "takeWhile (\<lambda>a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = Oc\<^bsup>n\<^esup> @ takeWhile (\<lambda>a. a = Oc) xs"
-apply(induct n, simp_all add: exp_ind_def)
-done
-
-lemma [simp]: "\<lbrakk>Suc (Suc rs) = a; wadjust_loop_right_move2 m rs (c, Bk # list)\<rbrakk>
- \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list))))
- < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))"
-apply(simp add: wadjust_loop_right_move2.simps, auto)
-apply(simp add: dropWhile_exp1 takeWhile_exp1)
-apply(case_tac ln, simp, simp add: exp_ind_def)
-done
-
-lemma [simp]: "wadjust_loop_check m rs ([], b) = False"
-apply(simp add: wadjust_loop_check.simps)
-done
-
-lemma [simp]: "\<lbrakk>Suc (Suc rs) = a; wadjust_loop_check m rs (c, Oc # list)\<rbrakk>
- \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Oc # list))))
- < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list)))) \<or>
- a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Oc # list)))) =
- a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list))))"
-apply(case_tac "c", simp_all)
-done
-
-lemma [simp]:
- "\<lbrakk>Suc (Suc rs) = a; wadjust_loop_erase m rs (c, Oc # list)\<rbrakk>
- \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))
- < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list)))) \<or>
- a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list)))) =
- a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list))))"
-apply(simp add: wadjust_loop_erase.simps)
-apply(rule_tac disjI2)
-apply(auto)
-apply(simp add: dropWhile_exp1 takeWhile_exp1)
-done
-
-declare numeral_2_eq_2[simp del]
-
-lemma wadjust_correctness:
- shows "let P = (\<lambda> (len, st, l, r). st = 0) in
- let Q = (\<lambda> (len, st, l, r). wadjust_inv st m rs (l, r)) in
- let f = (\<lambda> stp. (Suc (Suc rs), steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>,
- Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp)) in
- \<exists> n .P (f n) \<and> Q (f n)"
-proof -
- let ?P = "(\<lambda> (len, st, l, r). st = 0)"
- let ?Q = "\<lambda> (len, st, l, r). wadjust_inv st m rs (l, r)"
- let ?f = "\<lambda> stp. (Suc (Suc rs), steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>,
- Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp)"
- have "\<exists> n. ?P (?f n) \<and> ?Q (?f n)"
- proof(rule_tac halt_lemma2)
- show "wf wadjust_le" by auto
- next
- show "\<forall> n. \<not> ?P (?f n) \<and> ?Q (?f n) \<longrightarrow>
- ?Q (?f (Suc n)) \<and> (?f (Suc n), ?f n) \<in> wadjust_le"
- proof(rule_tac allI, rule_tac impI, case_tac "?f n",
- simp add: tstep_red tstep.simps, rule_tac conjI, erule_tac conjE,
- erule_tac conjE)
- fix n a b c d
- assume "0 < b" "wadjust_inv b m rs (c, d)" "Suc (Suc rs) = a"
- thus "case case fetch t_wcode_adjust b (case d of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)
- of (ac, ns) \<Rightarrow> (ns, new_tape ac (c, d)) of (st, x) \<Rightarrow> wadjust_inv st m rs x"
- apply(case_tac d, simp, case_tac [2] aa)
- apply(simp_all add: wadjust_inv.simps wadjust_le_def new_tape.simps
- abacus.lex_triple_def abacus.lex_pair_def lex_square_def
- split: if_splits)
- done
- next
- fix n a b c d
- assume "0 < b \<and> wadjust_inv b m rs (c, d)"
- "Suc (Suc rs) = a \<and> steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>,
- Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust n = (b, c, d)"
- thus "((a, case fetch t_wcode_adjust b (case d of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)
- of (ac, ns) \<Rightarrow> (ns, new_tape ac (c, d))), a, b, c, d) \<in> wadjust_le"
- proof(erule_tac conjE, erule_tac conjE, erule_tac conjE)
- assume "0 < b" "wadjust_inv b m rs (c, d)" "Suc (Suc rs) = a"
- thus "?thesis"
- apply(case_tac d, case_tac [2] aa)
- apply(simp_all add: wadjust_inv.simps wadjust_le_def new_tape.simps
- abacus.lex_triple_def abacus.lex_pair_def lex_square_def
- split: if_splits)
- done
- qed
- qed
- next
- show "?Q (?f 0)"
- apply(simp add: steps.simps wadjust_inv.simps wadjust_start.simps)
- apply(rule_tac x = ln in exI,auto)
- done
- next
- show "\<not> ?P (?f 0)"
- apply(simp add: steps.simps)
- done
- qed
- thus "?thesis"
- apply(auto)
- done
-qed
-
-lemma [intro]: "t_correct t_wcode_adjust"
-apply(auto simp: t_wcode_adjust_def t_correct.simps iseven_def)
-apply(rule_tac x = 11 in exI, simp)
-done
-
-lemma wcode_lemma_pre':
- "args \<noteq> [] \<Longrightarrow>
- \<exists> stp rn. steps (Suc 0, [], <m # args>)
- ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp
- = (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-proof -
- let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <m # args>"
- let ?Q1 = "\<lambda>(l, r). l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- (\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- let ?P2 = ?Q1
- let ?Q2 = "\<lambda> (l, r). (wadjust_stop m (bl_bin (<args>) - 1) (l, r))"
- let ?P3 = "\<lambda> tp. False"
- assume h: "args \<noteq> []"
- have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp)
- ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp = (0, tp') \<and> ?Q2 tp')"
- proof(rule_tac turing_merge.t_merge_halt[of "t_wcode_prepare |+| t_wcode_main"
- t_wcode_adjust ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2],
- auto simp: turing_merge_def)
-
- show "\<exists>stp. case steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp of
- (st, tp') \<Rightarrow> st = 0 \<and> (case tp' of (l, r) \<Rightarrow> l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
- (\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>))"
- using h prepare_mainpart_lemma[of args m]
- apply(auto)
- apply(rule_tac x = stp in exI, simp)
- apply(rule_tac x = ln in exI, auto)
- done
- next
- fix ln rn
- show "\<exists>stp. case steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk #
- Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp of
- (st, tp') \<Rightarrow> st = 0 \<and> wadjust_stop m (bl_bin (<args>) - Suc 0) tp'"
- using wadjust_correctness[of m "bl_bin (<args>) - 1" "Suc ln" rn]
- apply(subgoal_tac "bl_bin (<args>) > 0", auto simp: wadjust_inv.simps)
- apply(rule_tac x = n in exI, simp add: exp_ind)
- using h
- apply(case_tac args, simp_all, case_tac list,
- simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def
- bl_bin.simps)
- done
- next
- show "?Q1 \<turnstile>-> ?P2"
- by(simp add: t_imply_def)
- qed
- thus "\<exists>stp rn. steps (Suc 0, [], <m # args>) ((t_wcode_prepare |+| t_wcode_main) |+|
- t_wcode_adjust) stp = (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- apply(simp add: t_imply_def)
- apply(erule_tac exE)+
- apply(subgoal_tac "bl_bin (<args>) > 0", auto simp: wadjust_stop.simps)
- using h
- apply(case_tac args, simp_all, case_tac list,
- simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def
- bl_bin.simps)
- done
-qed
-
-text {*
- The initialization TM @{text "t_wcode"}.
- *}
-definition t_wcode :: "tprog"
- where
- "t_wcode = (t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust"
-
-
-text {*
- The correctness of @{text "t_wcode"}.
- *}
-lemma wcode_lemma_1:
- "args \<noteq> [] \<Longrightarrow>
- \<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode) stp =
- (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
-apply(simp add: wcode_lemma_pre' t_wcode_def)
-done
-
-lemma wcode_lemma:
- "args \<noteq> [] \<Longrightarrow>
- \<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode) stp =
- (0, [Bk], <[m ,bl_bin (<args>)]> @ Bk\<^bsup>rn\<^esup>)"
-using wcode_lemma_1[of args m]
-apply(simp add: t_wcode_def tape_of_nl_abv tape_of_nat_list.simps)
-done
-
-section {* The universal TM @{text "UTM"} *}
-
-text {*
- This section gives the explicit construction of {\em Universal Turing Machine}, defined as @{text "UTM"} and proves its
- correctness. It is pretty easy by composing the partial results we have got so far.
- *}
-
-
-definition UTM :: "tprog"
- where
- "UTM = (let (aprog, rs_pos, a_md) = rec_ci rec_F in
- let abc_F = aprog [+] dummy_abc (Suc (Suc 0)) in
- (t_wcode |+| (tm_of abc_F @ tMp (Suc (Suc 0)) (start_of (layout_of abc_F)
- (length abc_F) - Suc 0))))"
-
-definition F_aprog :: "abc_prog"
- where
- "F_aprog \<equiv> (let (aprog, rs_pos, a_md) = rec_ci rec_F in
- aprog [+] dummy_abc (Suc (Suc 0)))"
-
-definition F_tprog :: "tprog"
- where
- "F_tprog = tm_of (F_aprog)"
-
-definition t_utm :: "tprog"
- where
- "t_utm \<equiv>
- (F_tprog) @ tMp (Suc (Suc 0)) (start_of (layout_of (F_aprog))
- (length (F_aprog)) - Suc 0)"
-
-definition UTM_pre :: "tprog"
- where
- "UTM_pre = t_wcode |+| t_utm"
-
-lemma F_abc_halt_eq:
- "\<lbrakk>turing_basic.t_correct tp;
- length lm = k;
- steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>);
- rs > 0\<rbrakk>
- \<Longrightarrow> \<exists> stp m. abc_steps_l (0, [code tp, bl2wc (<lm>)]) (F_aprog) stp =
- (length (F_aprog), code tp # bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>)"
-apply(drule_tac F_t_halt_eq, simp, simp, simp)
-apply(case_tac "rec_ci rec_F")
-apply(frule_tac abc_append_dummy_complie, simp, simp, erule_tac exE,
- erule_tac exE)
-apply(rule_tac x = stp in exI, rule_tac x = m in exI)
-apply(simp add: F_aprog_def dummy_abc_def)
-done
-
-lemma F_abc_utm_halt_eq:
- "\<lbrakk>rs > 0;
- abc_steps_l (0, [code tp, bl2wc (<lm>)]) F_aprog stp =
- (length F_aprog, code tp # bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>)\<rbrakk>
- \<Longrightarrow> \<exists>stp m n.(steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp =
- (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>))"
- thm abacus_turing_eq_halt
- using abacus_turing_eq_halt
- [of "layout_of F_aprog" "F_aprog" "F_tprog" "length (F_aprog)"
- "[code tp, bl2wc (<lm>)]" stp "code tp # bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>" "Suc (Suc 0)"
- "start_of (layout_of (F_aprog)) (length (F_aprog))" "[]" 0]
-apply(simp add: F_tprog_def t_utm_def abc_lm_v.simps nth_append)
-apply(erule_tac exE)+
-apply(rule_tac x = stpa in exI, rule_tac x = "Suc (Suc ma)" in exI,
- rule_tac x = l in exI, simp add: exp_ind)
-done
-
-declare tape_of_nl_abv_cons[simp del]
-
-lemma t_utm_halt_eq':
- "\<lbrakk>turing_basic.t_correct tp;
- 0 < rs;
- steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\<rbrakk>
- \<Longrightarrow> \<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp =
- (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
-apply(drule_tac l = l in F_abc_halt_eq, simp, simp, simp)
-apply(erule_tac exE, erule_tac exE)
-apply(rule_tac F_abc_utm_halt_eq, simp_all)
-done
-
-lemma [simp]: "tinres xs (xs @ Bk\<^bsup>i\<^esup>)"
-apply(auto simp: tinres_def)
-done
-
-lemma [elim]: "\<lbrakk>rs > 0; Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup> = c @ Bk\<^bsup>n\<^esup>\<rbrakk>
- \<Longrightarrow> \<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
-apply(case_tac "na > n")
-apply(subgoal_tac "\<exists> d. na = d + n", auto simp: exp_add)
-apply(rule_tac x = "na - n" in exI, simp)
-apply(subgoal_tac "\<exists> d. n = d + na", auto simp: exp_add)
-apply(case_tac rs, simp_all add: exp_ind, case_tac d,
- simp_all add: exp_ind)
-apply(rule_tac x = "n - na" in exI, simp)
-done
-
-
-lemma t_utm_halt_eq'':
- "\<lbrakk>turing_basic.t_correct tp;
- 0 < rs;
- steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\<rbrakk>
- \<Longrightarrow> \<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp =
- (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
-apply(drule_tac t_utm_halt_eq', simp_all)
-apply(erule_tac exE)+
-proof -
- fix stpa ma na
- assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
- and gr: "rs > 0"
- thus "\<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
- apply(rule_tac x = stpa in exI, rule_tac x = ma in exI, simp)
- proof(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp)
- fix a b c
- assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
- "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)"
- thus " a = 0 \<and> b = Bk\<^bsup>ma\<^esup> \<and> (\<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
- using tinres_steps2[of "<[code tp, bl2wc (<lm>)]>" "<[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>"
- "Suc 0" " [Bk, Bk]" t_utm stpa 0 "Bk\<^bsup>ma\<^esup>" "Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>" a b c]
- apply(simp)
- using gr
- apply(simp only: tinres_def, auto)
- apply(rule_tac x = "na + n" in exI, simp add: exp_add)
- done
- qed
-qed
-
-lemma [simp]: "tinres [Bk, Bk] [Bk]"
-apply(auto simp: tinres_def)
-done
-
-lemma [elim]: "Bk\<^bsup>ma\<^esup> = b @ Bk\<^bsup>n\<^esup> \<Longrightarrow> \<exists>m. b = Bk\<^bsup>m\<^esup>"
-apply(subgoal_tac "ma = length b + n")
-apply(rule_tac x = "ma - n" in exI, simp add: exp_add)
-apply(drule_tac length_equal)
-apply(simp)
-done
-
-lemma t_utm_halt_eq:
- "\<lbrakk>turing_basic.t_correct tp;
- 0 < rs;
- steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\<rbrakk>
- \<Longrightarrow> \<exists>stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp =
- (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
-apply(drule_tac i = i in t_utm_halt_eq'', simp_all)
-apply(erule_tac exE)+
-proof -
- fix stpa ma na
- assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
- and gr: "rs > 0"
- thus "\<exists>stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
- apply(rule_tac x = stpa in exI)
- proof(case_tac "steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp)
- fix a b c
- assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
- "steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)"
- thus "a = 0 \<and> (\<exists>m. b = Bk\<^bsup>m\<^esup>) \<and> (\<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
- using tinres_steps[of "[Bk, Bk]" "[Bk]" "Suc 0" "<[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>" t_utm stpa 0
- "Bk\<^bsup>ma\<^esup>" "Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>" a b c]
- apply(simp)
- apply(auto simp: tinres_def)
- apply(rule_tac x = "ma + n" in exI, simp add: exp_add)
- done
- qed
-qed
-
-lemma [intro]: "t_correct t_wcode"
-apply(simp add: t_wcode_def)
-apply(auto)
-done
-
-lemma [intro]: "t_correct t_utm"
-apply(simp add: t_utm_def F_tprog_def)
-apply(rule_tac t_compiled_correct, auto)
-done
-
-lemma UTM_halt_lemma_pre:
- "\<lbrakk>turing_basic.t_correct tp;
- 0 < rs;
- args \<noteq> [];
- steps (Suc 0, Bk\<^bsup>i\<^esup>, <args::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\<rbrakk>
- \<Longrightarrow> \<exists>stp m n. steps (Suc 0, [], <code tp # args>) UTM_pre stp =
- (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
-proof -
- let ?Q2 = "\<lambda> (l, r). (\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> \<and> r = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
- term ?Q2
- let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <code tp # args>"
- let ?Q1 = "\<lambda> (l, r). (l = [Bk] \<and>
- (\<exists> rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))"
- let ?P2 = ?Q1
- let ?P3 = "\<lambda> (l, r). False"
- assume h: "turing_basic.t_correct tp" "0 < rs"
- "args \<noteq> []" "steps (Suc 0, Bk\<^bsup>i\<^esup>, <args::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)"
- have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp)
- (t_wcode |+| t_utm) stp = (0, tp') \<and> ?Q2 tp')"
- proof(rule_tac turing_merge.t_merge_halt [of "t_wcode" "t_utm"
- ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2], auto simp: turing_merge_def)
- show "\<exists>stp. case steps (Suc 0, [], <code tp # args>) t_wcode stp of (st, tp') \<Rightarrow>
- st = 0 \<and> (case tp' of (l, r) \<Rightarrow> l = [Bk] \<and>
- (\<exists>rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))"
- using wcode_lemma_1[of args "code tp"] h
- apply(simp, auto)
- apply(rule_tac x = stpa in exI, auto)
- done
- next
- fix rn
- show "\<exists>stp. case steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @
- Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp of
- (st, tp') \<Rightarrow> st = 0 \<and> (case tp' of (l, r) \<Rightarrow>
- (\<exists>ln. l = Bk\<^bsup>ln\<^esup>) \<and> (\<exists>rn. r = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>rn\<^esup>))"
- using t_utm_halt_eq[of tp rs i args stp m k rn] h
- apply(auto)
- apply(rule_tac x = stpa in exI, simp add: bin_wc_eq
- tape_of_nat_list.simps tape_of_nl_abv)
- apply(auto)
- done
- next
- show "?Q1 \<turnstile>-> ?P2"
- apply(simp add: t_imply_def)
- done
- qed
- thus "?thesis"
- apply(simp add: t_imply_def)
- apply(auto simp: UTM_pre_def)
- done
-qed
-
-text {*
- The correctness of @{text "UTM"}, the halt case.
-*}
-theorem UTM_halt_lemma:
- "\<lbrakk>turing_basic.t_correct tp;
- 0 < rs;
- args \<noteq> [];
- steps (Suc 0, Bk\<^bsup>i\<^esup>, <args::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\<rbrakk>
- \<Longrightarrow> \<exists>stp m n. steps (Suc 0, [], <code tp # args>) UTM stp =
- (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
-using UTM_halt_lemma_pre[of tp rs args i stp m k]
-apply(simp add: UTM_pre_def t_utm_def UTM_def F_aprog_def F_tprog_def)
-apply(case_tac "rec_ci rec_F", simp)
-done
-
-definition TSTD:: "t_conf \<Rightarrow> bool"
- where
- "TSTD c = (let (st, l, r) = c in
- st = 0 \<and> (\<exists> m. l = Bk\<^bsup>m\<^esup>) \<and> (\<exists> rs n. r = Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>))"
-
-thm abacus_turing_eq_uhalt
-
-lemma nstd_case1: "0 < a \<Longrightarrow> NSTD (trpl_code (a, b, c))"
-apply(simp add: NSTD.simps trpl_code.simps)
-done
-
-lemma [simp]: "\<forall>m. b \<noteq> Bk\<^bsup>m\<^esup> \<Longrightarrow> 0 < bl2wc b"
-apply(rule classical, simp)
-apply(induct b, erule_tac x = 0 in allE, simp)
-apply(simp add: bl2wc.simps, case_tac a, simp_all
- add: bl2nat.simps bl2nat_double)
-apply(case_tac "\<exists> m. b = Bk\<^bsup>m\<^esup>", erule exE)
-apply(erule_tac x = "Suc m" in allE, simp add: exp_ind_def, simp)
-done
-lemma nstd_case2: "\<forall>m. b \<noteq> Bk\<^bsup>m\<^esup> \<Longrightarrow> NSTD (trpl_code (a, b, c))"
-apply(simp add: NSTD.simps trpl_code.simps)
-done
-
-thm lg.simps
-thm lgR.simps
-
-lemma [elim]: "Suc (2 * x) = 2 * y \<Longrightarrow> RR"
-apply(induct x arbitrary: y, simp, simp)
-apply(case_tac y, simp, simp)
-done
-
-lemma bl2nat_zero_eq[simp]: "(bl2nat c 0 = 0) = (\<exists>n. c = Bk\<^bsup>n\<^esup>)"
-apply(auto)
-apply(induct c, simp add: bl2nat.simps)
-apply(rule_tac x = 0 in exI, simp)
-apply(case_tac a, auto simp: bl2nat.simps bl2nat_double)
-done
-
-lemma bl2wc_exp_ex:
- "\<lbrakk>Suc (bl2wc c) = 2 ^ m\<rbrakk> \<Longrightarrow> \<exists> rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
-apply(induct c arbitrary: m, simp add: bl2wc.simps bl2nat.simps)
-apply(case_tac a, auto)
-apply(case_tac m, simp_all add: bl2wc.simps, auto)
-apply(rule_tac x = 0 in exI, rule_tac x = "Suc n" in exI,
- simp add: exp_ind_def)
-apply(simp add: bl2wc.simps bl2nat.simps bl2nat_double)
-apply(case_tac m, simp, simp)
-proof -
- fix c m nat
- assume ind:
- "\<And>m. Suc (bl2nat c 0) = 2 ^ m \<Longrightarrow> \<exists>rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
- and h:
- "Suc (Suc (2 * bl2nat c 0)) = 2 * 2 ^ nat"
- have "\<exists>rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
- apply(rule_tac m = nat in ind)
- using h
- apply(simp)
- done
- from this obtain rs n where " c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" by blast
- thus "\<exists>rs n. Oc # c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
- apply(rule_tac x = "Suc rs" in exI, simp add: exp_ind_def)
- apply(rule_tac x = n in exI, simp)
- done
-qed
-
-lemma [elim]:
- "\<lbrakk>\<forall>rs n. c \<noteq> Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>;
- bl2wc c = 2 ^ lg (Suc (bl2wc c)) 2 - Suc 0\<rbrakk> \<Longrightarrow> bl2wc c = 0"
-apply(subgoal_tac "\<exists> m. Suc (bl2wc c) = 2^m", erule_tac exE)
-apply(drule_tac bl2wc_exp_ex, simp, erule_tac exE, erule_tac exE)
-apply(case_tac rs, simp, simp, erule_tac x = nat in allE,
- erule_tac x = n in allE, simp)
-using bl2wc_exp_ex[of c "lg (Suc (bl2wc c)) 2"]
-apply(case_tac "(2::nat) ^ lg (Suc (bl2wc c)) 2",
- simp, simp, erule_tac exE, erule_tac exE, simp)
-apply(simp add: bl2wc.simps)
-apply(rule_tac x = rs in exI)
-apply(case_tac "(2::nat)^rs", simp, simp)
-done
-
-lemma nstd_case3:
- "\<forall>rs n. c \<noteq> Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup> \<Longrightarrow> NSTD (trpl_code (a, b, c))"
-apply(simp add: NSTD.simps trpl_code.simps)
-apply(rule_tac impI)
-apply(rule_tac disjI2, rule_tac disjI2, auto)
-done
-
-lemma NSTD_1: "\<not> TSTD (a, b, c)
- \<Longrightarrow> rec_exec rec_NSTD [trpl_code (a, b, c)] = Suc 0"
- using NSTD_lemma1[of "trpl_code (a, b, c)"]
- NSTD_lemma2[of "trpl_code (a, b, c)"]
- apply(simp add: TSTD_def)
- apply(erule_tac disjE, erule_tac nstd_case1)
- apply(erule_tac disjE, erule_tac nstd_case2)
- apply(erule_tac nstd_case3)
- done
-
-lemma nonstop_t_uhalt_eq:
- "\<lbrakk>turing_basic.t_correct tp;
- steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp = (a, b, c);
- \<not> TSTD (a, b, c)\<rbrakk>
- \<Longrightarrow> rec_exec rec_nonstop [code tp, bl2wc (<lm>), stp] = Suc 0"
-apply(simp add: rec_nonstop_def rec_exec.simps)
-apply(subgoal_tac
- "rec_exec rec_conf [code tp, bl2wc (<lm>), stp] =
- trpl_code (a, b, c)", simp)
-apply(erule_tac NSTD_1)
-using rec_t_eq_steps[of tp l lm stp]
-apply(simp)
-done
-
-lemma nonstop_true:
- "\<lbrakk>turing_basic.t_correct tp;
- \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk>
- \<Longrightarrow> \<forall>y. rec_calc_rel rec_nonstop
- ([code tp, bl2wc (<lm>), y]) (Suc 0)"
-apply(rule_tac allI, erule_tac x = y in allE)
-apply(case_tac "steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp y", simp)
-apply(rule_tac nonstop_t_uhalt_eq, simp_all)
-done
-
-(*
-lemma [simp]:
- "\<forall>j<Suc k. Ex (rec_calc_rel (get_fstn_args (Suc k) (Suc k) ! j)
- (code tp # lm))"
-apply(auto simp: get_fstn_args_nth)
-apply(rule_tac x = "(code tp # lm) ! j" in exI)
-apply(rule_tac calc_id, simp_all)
-done
-*)
-declare ci_cn_para_eq[simp]
-
-lemma F_aprog_uhalt:
- "\<lbrakk>turing_basic.t_correct tp;
- \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp));
- rec_ci rec_F = (F_ap, rs_pos, a_md)\<rbrakk>
- \<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)] @ 0\<^bsup>a_md - rs_pos \<^esup>
- @ suflm) (F_ap) stp of (ss, e) \<Rightarrow> ss < length (F_ap)"
-apply(case_tac "rec_ci (Cn (Suc (Suc 0)) rec_right [Cn (Suc (Suc 0)) rec_conf
- ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])])")
-apply(simp only: rec_F_def, rule_tac i = 0 and ga = a and gb = b and
- gc = c in cn_gi_uhalt, simp, simp, simp, simp, simp, simp, simp)
-apply(simp add: ci_cn_para_eq)
-apply(case_tac "rec_ci (Cn (Suc (Suc 0)) rec_conf
- ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt]))")
-apply(rule_tac rf = "(Cn (Suc (Suc 0)) rec_right [Cn (Suc (Suc 0)) rec_conf
- ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])])"
- and n = "Suc (Suc 0)" and f = rec_right and
- gs = "[Cn (Suc (Suc 0)) rec_conf
- ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])]"
- and i = 0 and ga = aa and gb = ba and gc = ca in
- cn_gi_uhalt)
-apply(simp, simp, simp, simp, simp, simp, simp,
- simp add: ci_cn_para_eq)
-apply(case_tac "rec_ci rec_halt")
-apply(rule_tac rf = "(Cn (Suc (Suc 0)) rec_conf
- ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt]))"
- and n = "Suc (Suc 0)" and f = "rec_conf" and
- gs = "([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])" and
- i = "Suc (Suc 0)" and gi = "rec_halt" and ga = ab and gb = bb and
- gc = cb in cn_gi_uhalt)
-apply(simp, simp, simp, simp, simp add: nth_append, simp,
- simp add: nth_append, simp add: rec_halt_def)
-apply(simp only: rec_halt_def)
-apply(case_tac [!] "rec_ci ((rec_nonstop))")
-apply(rule_tac allI, rule_tac impI, simp)
-apply(case_tac j, simp)
-apply(rule_tac x = "code tp" in exI, rule_tac calc_id, simp, simp, simp, simp)
-apply(rule_tac x = "bl2wc (<lm>)" in exI, rule_tac calc_id, simp, simp, simp)
-apply(rule_tac rf = "Mn (Suc (Suc 0)) (rec_nonstop)"
- and f = "(rec_nonstop)" and n = "Suc (Suc 0)"
- and aprog' = ac and rs_pos' = bc and a_md' = cc in Mn_unhalt)
-apply(simp, simp add: rec_halt_def , simp, simp)
-apply(drule_tac nonstop_true, simp_all)
-apply(rule_tac allI)
-apply(erule_tac x = y in allE)+
-apply(simp)
-done
-
-thm abc_list_crsp_steps
-
-lemma uabc_uhalt':
- "\<lbrakk>turing_basic.t_correct tp;
- \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp));
- rec_ci rec_F = (ap, pos, md)\<rbrakk>
- \<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) ap stp of (ss, e)
- \<Rightarrow> ss < length ap"
-proof(frule_tac F_ap = ap and rs_pos = pos and a_md = md
- and suflm = "[]" in F_aprog_uhalt, auto)
- fix stp a b
- assume h:
- "\<forall>stp. case abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp of
- (ss, e) \<Rightarrow> ss < length ap"
- "abc_steps_l (0, [code tp, bl2wc (<lm>)]) ap stp = (a, b)"
- "turing_basic.t_correct tp"
- "rec_ci rec_F = (ap, pos, md)"
- moreover have "ap \<noteq> []"
- using h apply(rule_tac rec_ci_not_null, simp)
- done
- ultimately show "a < length ap"
- proof(erule_tac x = stp in allE,
- case_tac "abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp", simp)
- fix aa ba
- assume g: "aa < length ap"
- "abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp = (aa, ba)"
- "ap \<noteq> []"
- thus "?thesis"
- using abc_list_crsp_steps[of "[code tp, bl2wc (<lm>)]"
- "md - pos" ap stp aa ba] h
- apply(simp)
- done
- qed
-qed
-
-lemma uabc_uhalt:
- "\<lbrakk>turing_basic.t_correct tp;
- \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk>
- \<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) F_aprog
- stp of (ss, e) \<Rightarrow> ss < length F_aprog"
-apply(case_tac "rec_ci rec_F", simp add: F_aprog_def)
-thm uabc_uhalt'
-apply(drule_tac ap = a and pos = b and md = c in uabc_uhalt', simp_all)
-proof -
- fix a b c
- assume
- "\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) a stp of (ss, e)
- \<Rightarrow> ss < length a"
- "rec_ci rec_F = (a, b, c)"
- thus
- "\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)])
- (a [+] dummy_abc (Suc (Suc 0))) stp of (ss, e) \<Rightarrow>
- ss < Suc (Suc (Suc (length a)))"
- using abc_append_uhalt1[of a "[code tp, bl2wc (<lm>)]"
- "a [+] dummy_abc (Suc (Suc 0))" "[]" "dummy_abc (Suc (Suc 0))"]
- apply(simp)
- done
-qed
-
-thm abacus_turing_eq_uhalt
-lemma tutm_uhalt':
- "\<lbrakk>turing_basic.t_correct tp;
- \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk>
- \<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp)"
- using abacus_turing_eq_uhalt[of "layout_of (F_aprog)"
- "F_aprog" "F_tprog" "[code tp, bl2wc (<lm>)]"
- "start_of (layout_of (F_aprog )) (length (F_aprog))"
- "Suc (Suc 0)"]
-apply(simp add: F_tprog_def)
-apply(subgoal_tac "\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)])
- (F_aprog) stp of (as, am) \<Rightarrow> as < length (F_aprog)", simp)
-thm abacus_turing_eq_uhalt
-apply(simp add: t_utm_def F_tprog_def)
-apply(rule_tac uabc_uhalt, simp_all)
-done
-
-lemma tinres_commute: "tinres r r' \<Longrightarrow> tinres r' r"
-apply(auto simp: tinres_def)
-done
-
-lemma inres_tape:
- "\<lbrakk>steps (st, l, r) tp stp = (a, b, c); steps (st, l', r') tp stp = (a', b', c');
- tinres l l'; tinres r r'\<rbrakk>
- \<Longrightarrow> a = a' \<and> tinres b b' \<and> tinres c c'"
-proof(case_tac "steps (st, l', r) tp stp")
- fix aa ba ca
- assume h: "steps (st, l, r) tp stp = (a, b, c)"
- "steps (st, l', r') tp stp = (a', b', c')"
- "tinres l l'" "tinres r r'"
- "steps (st, l', r) tp stp = (aa, ba, ca)"
- have "tinres b ba \<and> c = ca \<and> a = aa"
- using h
- apply(rule_tac tinres_steps, auto)
- done
-
- thm tinres_steps2
- moreover have "b' = ba \<and> tinres c' ca \<and> a' = aa"
- using h
- apply(rule_tac tinres_steps2, auto intro: tinres_commute)
- done
- ultimately show "?thesis"
- apply(auto intro: tinres_commute)
- done
-qed
-
-lemma tape_normalize: "\<forall> stp. \<not> isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp)
- \<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>n\<^esup>) t_utm stp)"
-apply(rule_tac allI, case_tac "(steps (Suc 0, Bk\<^bsup>m\<^esup>,
- <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>n\<^esup>) t_utm stp)", simp add: isS0_def)
-apply(erule_tac x = stp in allE)
-apply(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp", simp)
-apply(drule_tac inres_tape, auto)
-apply(auto simp: tinres_def)
-apply(case_tac "m > Suc (Suc 0)")
-apply(rule_tac x = "m - Suc (Suc 0)" in exI)
-apply(case_tac m, simp_all add: exp_ind_def, case_tac nat, simp_all add: exp_ind_def)
-apply(rule_tac x = "2 - m" in exI, simp add: exp_ind_def[THEN sym] exp_add[THEN sym])
-apply(simp only: numeral_2_eq_2, simp add: exp_ind_def)
-done
-
-lemma tutm_uhalt:
- "\<lbrakk>turing_basic.t_correct tp;
- \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp))\<rbrakk>
- \<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc (<args>)]> @ Bk\<^bsup>n\<^esup>) t_utm stp)"
-apply(rule_tac tape_normalize)
-apply(rule_tac tutm_uhalt', simp_all)
-done
-
-lemma UTM_uhalt_lemma_pre:
- "\<lbrakk>turing_basic.t_correct tp;
- \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp));
- args \<noteq> []\<rbrakk>
- \<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, [], <code tp # args>) UTM_pre stp)"
-proof -
- let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <code tp # args>"
- let ?Q1 = "\<lambda> (l, r). (l = [Bk] \<and>
- (\<exists> rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))"
- let ?P4 = ?Q1
- let ?P3 = "\<lambda> (l, r). False"
- assume h: "turing_basic.t_correct tp" "\<forall>stp. \<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp)"
- "args \<noteq> []"
- have "?P1 \<turnstile>-> \<lambda> tp. \<not> (\<exists> stp. isS0 (steps (Suc 0, tp) (t_wcode |+| t_utm) stp))"
- proof(rule_tac turing_merge.t_merge_uhalt [of "t_wcode" "t_utm"
- ?P1 ?P3 ?P3 ?P4 ?Q1 ?P3], auto simp: turing_merge_def)
- show "\<exists>stp. case steps (Suc 0, [], <code tp # args>) t_wcode stp of (st, tp') \<Rightarrow>
- st = 0 \<and> (case tp' of (l, r) \<Rightarrow> l = [Bk] \<and>
- (\<exists>rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))"
- using wcode_lemma_1[of args "code tp"] h
- apply(simp, auto)
- apply(rule_tac x = stp in exI, auto)
- done
- next
- fix rn stp
- show " isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp)
- \<Longrightarrow> False"
- using tutm_uhalt[of tp l args "Suc 0" rn] h
- apply(simp)
- apply(erule_tac x = stp in allE)
- apply(simp add: tape_of_nl_abv tape_of_nat_list.simps bin_wc_eq)
- done
- next
- fix rn stp
- show "isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp) \<Longrightarrow>
- isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp)"
- by simp
- next
- show "?Q1 \<turnstile>-> ?P4"
- apply(simp add: t_imply_def)
- done
- qed
- thus "?thesis"
- apply(simp add: t_imply_def UTM_pre_def)
- done
-qed
-
-text {*
- The correctness of @{text "UTM"}, the unhalt case.
- *}
-
-theorem UTM_uhalt_lemma:
- "\<lbrakk>turing_basic.t_correct tp;
- \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp));
- args \<noteq> []\<rbrakk>
- \<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, [], <code tp # args>) UTM stp)"
-using UTM_uhalt_lemma_pre[of tp l args]
-apply(simp add: UTM_pre_def t_utm_def UTM_def F_aprog_def F_tprog_def)
-apply(case_tac "rec_ci rec_F", simp)
-done
-
+theory UTM
+imports Main uncomputable recursive abacus UF GCD
+begin
+
+section {* Wang coding of input arguments *}
+
+text {*
+ The direct compilation of the universal function @{text "rec_F"} can not give us UTM, because @{text "rec_F"} is of arity 2,
+ where the first argument represents the Godel coding of the TM being simulated and the second argument represents the right number (in Wang's coding) of the TM tape.
+ (Notice, left number is always @{text "0"} at the very beginning). However, UTM needs to simulate the execution of any TM which may
+ very well take many input arguments. Therefore, a initialization TM needs to run before the TM compiled from @{text "rec_F"}, and the sequential
+ composition of these two TMs will give rise to the UTM we are seeking. The purpose of this initialization TM is to transform the multiple
+ input arguments of the TM being simulated into Wang's coding, so that it can be consumed by the TM compiled from @{text "rec_F"} as the second
+ argument.
+
+ However, this initialization TM (named @{text "t_wcode"}) can not be constructed by compiling from any resurve function, because every recursive
+ function takes a fixed number of input arguments, while @{text "t_wcode"} needs to take varying number of arguments and tranform them into
+ Wang's coding. Therefore, this section give a direct construction of @{text "t_wcode"} with just some parts being obtained from recursive functions.
+
+\newlength{\basewidth}
+\settowidth{\basewidth}{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}
+\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
+ 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}.
+
+
+\begin{figure}[h!]
+\centering
+\scalebox{1.2}{
+\begin{tikzpicture}
+ [tbox/.style = {draw, thick, inner sep = 5pt}]
+ \node (0) {};
+ \node (1) [tbox, text height = 3.5pt, right = -0.9pt of 0] {\wuhao $m$};
+ \node (2) [tbox, right = -0.9pt of 1] {\wuhao $0$};
+ \node (3) [tbox, right = -0.9pt of 2] {\wuhao $a_1$};
+ \node (4) [tbox, right = -0.9pt of 3] {\wuhao $0$};
+ \node (5) [tbox, right = -0.9pt of 4] {\wuhao $a_2$};
+ \node (6) [right = -0.9pt of 5] {\ldots \ldots};
+ \node (7) [tbox, right = -0.9pt of 6] {\wuhao $a_n$};
+ \draw [->, >=latex, thick] (1)+(0, -4\baseheight) -- (1);
+\end{tikzpicture}}
+\caption{The input of TM $prepare$} \label{prepare_input}
+\end{figure}
+
+\begin{figure}[h!]
+\centering
+\scalebox{1.2}{
+\begin{tikzpicture}
+ \node (0) {};
+ \node (1) [draw, text height = 3.5pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$};
+ \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$};
+ \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$};
+ \node (7) [right = -0.9pt of 6] {\ldots \ldots};
+ \node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_n$};
+ \node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $1$};
+ \draw [->, >=latex, thick] (10)+(0, -4\baseheight) -- (10);
+\end{tikzpicture}}
+\caption{The output of TM $prepare$} \label{prepare_output}
+\end{figure}
+
+As shown in Figure \ref{prepare_input}, the input of $prepare$ is the same as the the input
+of UTM, where $m$ is the Godel coding of the TM being interpreted and $a_1$ through $a_n$ are the $n$ input arguments of the TM under interpretation. The purpose of $purpose$ is to transform this initial tape layout to the one shown in Figure \ref{prepare_output},
+which is convenient for the generation of Wang's codding of $a_1, \ldots, a_n$. The coding procedure starts from $a_n$ and ends after $a_1$ is encoded. The coding result is stored in an accumulator at the end of the tape (initially represented by the $1$ two blanks right to $a_n$ in Figure \ref{prepare_output}). In Figure \ref{prepare_output}, arguments $a_1, \ldots, a_n$ are separated by two blanks on both ends with the rest so that movement conditions can be implemented conveniently in subsequent TMs, because, by convention,
+two consecutive blanks are usually used to signal the end or start of a large chunk of data. The diagram of $prepare$ is given in Figure \ref{prepare_diag}.
+
+
+\begin{figure}[h!]
+\centering
+\scalebox{0.9}{
+\begin{tikzpicture}
+ \node[circle,draw] (1) {$1$};
+ \node[circle,draw] (2) at ($(1)+(0.3\basewidth, 0)$) {$2$};
+ \node[circle,draw] (3) at ($(2)+(0.3\basewidth, 0)$) {$3$};
+ \node[circle,draw] (4) at ($(3)+(0.3\basewidth, 0)$) {$4$};
+ \node[circle,draw] (5) at ($(4)+(0.3\basewidth, 0)$) {$5$};
+ \node[circle,draw] (6) at ($(5)+(0.3\basewidth, 0)$) {$6$};
+ \node[circle,draw] (7) at ($(6)+(0.3\basewidth, 0)$) {$7$};
+ \node[circle,draw] (8) at ($(7)+(0.3\basewidth, 0)$) {$0$};
+
+
+ \draw [->, >=latex] (1) edge [loop above] node[above] {$S_1:L$} (1)
+ ;
+ \draw [->, >=latex] (1) -- node[above] {$S_0:S_1$} (2)
+ ;
+ \draw [->, >=latex] (2) edge [loop above] node[above] {$S_1:R$} (2)
+ ;
+ \draw [->, >=latex] (2) -- node[above] {$S_0:L$} (3)
+ ;
+ \draw [->, >=latex] (3) edge[loop above] node[above] {$S_1:S_0$} (3)
+ ;
+ \draw [->, >=latex] (3) -- node[above] {$S_0:R$} (4)
+ ;
+ \draw [->, >=latex] (4) edge[loop above] node[above] {$S_0:R$} (4)
+ ;
+ \draw [->, >=latex] (4) -- node[above] {$S_0:R$} (5)
+ ;
+ \draw [->, >=latex] (5) edge[loop above] node[above] {$S_1:R$} (5)
+ ;
+ \draw [->, >=latex] (5) -- node[above] {$S_0:R$} (6)
+ ;
+ \draw [->, >=latex] (6) edge[bend left = 50] node[below] {$S_1:R$} (5)
+ ;
+ \draw [->, >=latex] (6) -- node[above] {$S_0:R$} (7)
+ ;
+ \draw [->, >=latex] (7) edge[loop above] node[above] {$S_0:S_1$} (7)
+ ;
+ \draw [->, >=latex] (7) -- node[above] {$S_1:L$} (8)
+ ;
+ \end{tikzpicture}}
+\caption{The diagram of TM $prepare$} \label{prepare_diag}
+\end{figure}
+
+The purpose of TM $mainwork$ is to compute the Wang's encoding of $a_1, \ldots, a_n$. Every bit of $a_1, \ldots, a_n$, including the separating bits, is processed from left to right.
+In order to detect the termination condition when the left most bit of $a_1$ is reached,
+TM $mainwork$ needs to look ahead and consider three different situations at the start of
+every iteration:
+\begin{enumerate}
+ \item The TM configuration for the first situation is shown in Figure \ref{mainwork_case_one_input},
+ where the accumulator is stored in $r$, both of the next two bits
+ to be encoded are $1$. The configuration at the end of the iteration
+ is shown in Figure \ref{mainwork_case_one_output}, where the first 1-bit has been
+ encoded and cleared. Notice that the accumulator has been changed to
+ $(r+1) \times 2$ to reflect the encoded bit.
+ \item The TM configuration for the second situation is shown in Figure
+ \ref{mainwork_case_two_input},
+ where the accumulator is stored in $r$, the next two bits
+ to be encoded are $1$ and $0$. After the first
+ $1$-bit was encoded and cleared, the second $0$-bit is difficult to detect
+ and process. To solve this problem, these two consecutive bits are
+ encoded in one iteration. In this situation, only the first $1$-bit needs
+ to be cleared since the second one is cleared by definition.
+ The configuration at the end of the iteration
+ is shown in Figure \ref{mainwork_case_two_output}.
+ Notice that the accumulator has been changed to
+ $(r+1) \times 4$ to reflect the two encoded bits.
+ \item The third situation corresponds to the case when the last bit of $a_1$ is reached.
+ The TM configurations at the start and end of the iteration are shown in
+ Figure \ref{mainwork_case_three_input} and \ref{mainwork_case_three_output}
+ respectively. For this situation, only the read write head needs to be moved to
+ the left to prepare a initial configuration for TM $adjust$ to start with.
+\end{enumerate}
+The diagram of $mainwork$ is given in Figure \ref{mainwork_diag}. The two rectangular nodes
+labeled with $2 \times x$ and $4 \times x$ are two TMs compiling from recursive functions
+so that we do not have to design and verify two quite complicated TMs.
+
+
+\begin{figure}[h!]
+\centering
+\scalebox{1.2}{
+\begin{tikzpicture}
+ \node (0) {};
+ \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$};
+ \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$};
+ \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$};
+ \node (7) [right = -0.9pt of 6] {\ldots \ldots};
+ \node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_i$};
+ \node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $1$};
+ \node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $1$};
+ \node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (12) [right = -0.9pt of 11] {\ldots \ldots};
+ \node (13) [draw, right = -0.9pt of 12, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (14) [draw, text height = 3.9pt, right = -0.9pt of 13, thick, inner sep = 5pt] {\wuhao $r$};
+ \draw [->, >=latex, thick] (13)+(0, -4\baseheight) -- (13);
+\end{tikzpicture}}
+\caption{The first situation for TM $mainwork$ to consider} \label{mainwork_case_one_input}
+\end{figure}
+
+
+\begin{figure}[h!]
+\centering
+\scalebox{1.2}{
+\begin{tikzpicture}
+ \node (0) {};
+ \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$};
+ \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$};
+ \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$};
+ \node (7) [right = -0.9pt of 6] {\ldots \ldots};
+ \node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_i$};
+ \node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $1$};
+ \node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (12) [right = -0.9pt of 11] {\ldots \ldots};
+ \node (13) [draw, right = -0.9pt of 12, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (14) [draw, text height = 2.7pt, right = -0.9pt of 13, thick, inner sep = 5pt] {\wuhao $(r+1) \times 2$};
+ \draw [->, >=latex, thick] (13)+(0, -4\baseheight) -- (13);
+\end{tikzpicture}}
+\caption{The output for the first case of TM $mainwork$'s processing}
+\label{mainwork_case_one_output}
+\end{figure}
+
+\begin{figure}[h!]
+\centering
+\scalebox{1.2}{
+\begin{tikzpicture}
+ \node (0) {};
+ \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$};
+ \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$};
+ \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$};
+ \node (7) [right = -0.9pt of 6] {\ldots \ldots};
+ \node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_i$};
+ \node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $1$};
+ \node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $1$};
+ \node (12) [draw, right = -0.9pt of 11, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (13) [right = -0.9pt of 12] {\ldots \ldots};
+ \node (14) [draw, right = -0.9pt of 13, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (15) [draw, text height = 3.9pt, right = -0.9pt of 14, thick, inner sep = 5pt] {\wuhao $r$};
+ \draw [->, >=latex, thick] (14)+(0, -4\baseheight) -- (14);
+\end{tikzpicture}}
+\caption{The second situation for TM $mainwork$ to consider} \label{mainwork_case_two_input}
+\end{figure}
+
+\begin{figure}[h!]
+\centering
+\scalebox{1.2}{
+\begin{tikzpicture}
+ \node (0) {};
+ \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$};
+ \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$};
+ \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$};
+ \node (7) [right = -0.9pt of 6] {\ldots \ldots};
+ \node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_i$};
+ \node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $1$};
+ \node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (12) [draw, right = -0.9pt of 11, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (13) [right = -0.9pt of 12] {\ldots \ldots};
+ \node (14) [draw, right = -0.9pt of 13, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (15) [draw, text height = 2.7pt, right = -0.9pt of 14, thick, inner sep = 5pt] {\wuhao $(r+1) \times 4$};
+ \draw [->, >=latex, thick] (14)+(0, -4\baseheight) -- (14);
+\end{tikzpicture}}
+\caption{The output for the second case of TM $mainwork$'s processing}
+\label{mainwork_case_two_output}
+\end{figure}
+
+\begin{figure}[h!]
+\centering
+\scalebox{1.2}{
+\begin{tikzpicture}
+ \node (0) {};
+ \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$};
+ \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $1$};
+ \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (6) [right = -0.9pt of 5] {\ldots \ldots};
+ \node (7) [draw, right = -0.9pt of 6, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (8) [draw, text height = 3.9pt, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $r$};
+ \draw [->, >=latex, thick] (7)+(0, -4\baseheight) -- (7);
+\end{tikzpicture}}
+\caption{The third situation for TM $mainwork$ to consider} \label{mainwork_case_three_input}
+\end{figure}
+
+\begin{figure}[h!]
+\centering
+\scalebox{1.2}{
+\begin{tikzpicture}
+ \node (0) {};
+ \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$};
+ \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $1$};
+ \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (6) [right = -0.9pt of 5] {\ldots \ldots};
+ \node (7) [draw, right = -0.9pt of 6, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (8) [draw, text height = 3.9pt, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $r$};
+ \draw [->, >=latex, thick] (3)+(0, -4\baseheight) -- (3);
+\end{tikzpicture}}
+\caption{The output for the third case of TM $mainwork$'s processing}
+\label{mainwork_case_three_output}
+\end{figure}
+
+\begin{figure}[h!]
+\centering
+\scalebox{0.9}{
+\begin{tikzpicture}
+ \node[circle,draw] (1) {$1$};
+ \node[circle,draw] (2) at ($(1)+(0.3\basewidth, 0)$) {$2$};
+ \node[circle,draw] (3) at ($(2)+(0.3\basewidth, 0)$) {$3$};
+ \node[circle,draw] (4) at ($(3)+(0.3\basewidth, 0)$) {$4$};
+ \node[circle,draw] (5) at ($(4)+(0.3\basewidth, 0)$) {$5$};
+ \node[circle,draw] (6) at ($(5)+(0.3\basewidth, 0)$) {$6$};
+ \node[circle,draw] (7) at ($(2)+(0, -7\baseheight)$) {$7$};
+ \node[circle,draw] (8) at ($(7)+(0, -7\baseheight)$) {$8$};
+ \node[circle,draw] (9) at ($(8)+(0.3\basewidth, 0)$) {$9$};
+ \node[circle,draw] (10) at ($(9)+(0.3\basewidth, 0)$) {$10$};
+ \node[circle,draw] (11) at ($(10)+(0.3\basewidth, 0)$) {$11$};
+ \node[circle,draw] (12) at ($(11)+(0.3\basewidth, 0)$) {$12$};
+ \node[draw] (13) at ($(6)+(0.3\basewidth, 0)$) {$2 \times x$};
+ \node[circle,draw] (14) at ($(13)+(0.3\basewidth, 0)$) {$j_1$};
+ \node[draw] (15) at ($(12)+(0.3\basewidth, 0)$) {$4 \times x$};
+ \node[draw] (16) at ($(15)+(0.3\basewidth, 0)$) {$j_2$};
+ \node[draw] (17) at ($(7)+(0.3\basewidth, 0)$) {$0$};
+
+ \draw [->, >=latex] (1) edge[loop left] node[above] {$S_0:L$} (1)
+ ;
+ \draw [->, >=latex] (1) -- node[above] {$S_1:L$} (2)
+ ;
+ \draw [->, >=latex] (2) -- node[above] {$S_1:R$} (3)
+ ;
+ \draw [->, >=latex] (2) -- node[left] {$S_1:L$} (7)
+ ;
+ \draw [->, >=latex] (3) edge[loop above] node[above] {$S_1:S_0$} (3)
+ ;
+ \draw [->, >=latex] (3) -- node[above] {$S_0:R$} (4)
+ ;
+ \draw [->, >=latex] (4) edge[loop above] node[above] {$S_0:R$} (4)
+ ;
+ \draw [->, >=latex] (4) -- node[above] {$S_1:R$} (5)
+ ;
+ \draw [->, >=latex] (5) edge[loop above] node[above] {$S_1:R$} (5)
+ ;
+ \draw [->, >=latex] (5) -- node[above] {$S_0:S_1$} (6)
+ ;
+ \draw [->, >=latex] (6) edge[loop above] node[above] {$S_1:L$} (6)
+ ;
+ \draw [->, >=latex] (6) -- node[above] {$S_0:R$} (13)
+ ;
+ \draw [->, >=latex] (13) -- (14)
+ ;
+ \draw (14) -- ($(14)+(0, 6\baseheight)$) -- ($(1) + (0, 6\baseheight)$) node [above,midway] {$S_1:L$}
+ ;
+ \draw [->, >=latex] ($(1) + (0, 6\baseheight)$) -- (1)
+ ;
+ \draw [->, >=latex] (7) -- node[above] {$S_0:R$} (17)
+ ;
+ \draw [->, >=latex] (7) -- node[left] {$S_1:R$} (8)
+ ;
+ \draw [->, >=latex] (8) -- node[above] {$S_0:R$} (9)
+ ;
+ \draw [->, >=latex] (9) -- node[above] {$S_0:R$} (10)
+ ;
+ \draw [->, >=latex] (10) -- node[above] {$S_1:R$} (11)
+ ;
+ \draw [->, >=latex] (10) edge[loop above] node[above] {$S_0:R$} (10)
+ ;
+ \draw [->, >=latex] (11) edge[loop above] node[above] {$S_1:R$} (11)
+ ;
+ \draw [->, >=latex] (11) -- node[above] {$S_0:S_1$} (12)
+ ;
+ \draw [->, >=latex] (12) -- node[above] {$S_0:R$} (15)
+ ;
+ \draw [->, >=latex] (12) edge[loop above] node[above] {$S_1:L$} (12)
+ ;
+ \draw [->, >=latex] (15) -- (16)
+ ;
+ \draw (16) -- ($(16)+(0, -4\baseheight)$) -- ($(1) + (0, -18\baseheight)$) node [below,midway] {$S_1:L$}
+ ;
+ \draw [->, >=latex] ($(1) + (0, -18\baseheight)$) -- (1)
+ ;
+ \end{tikzpicture}}
+\caption{The diagram of TM $mainwork$} \label{mainwork_diag}
+\end{figure}
+
+The purpose of TM $adjust$ is to encode the last bit of $a_1$. The initial and final configuration
+of this TM are shown in Figure \ref{adjust_initial} and \ref{adjust_final} respectively.
+The diagram of TM $adjust$ is shown in Figure \ref{adjust_diag}.
+
+
+\begin{figure}[h!]
+\centering
+\scalebox{1.2}{
+\begin{tikzpicture}
+ \node (0) {};
+ \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$};
+ \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $1$};
+ \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (6) [right = -0.9pt of 5] {\ldots \ldots};
+ \node (7) [draw, right = -0.9pt of 6, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (8) [draw, text height = 3.9pt, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $r$};
+ \draw [->, >=latex, thick] (3)+(0, -4\baseheight) -- (3);
+\end{tikzpicture}}
+\caption{Initial configuration of TM $adjust$} \label{adjust_initial}
+\end{figure}
+
+\begin{figure}[h!]
+\centering
+\scalebox{1.2}{
+\begin{tikzpicture}
+ \node (0) {};
+ \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$};
+ \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (3) [draw, text height = 2.9pt, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $r+1$};
+ \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$};
+ \node (6) [right = -0.9pt of 5] {\ldots \ldots};
+ \draw [->, >=latex, thick] (1)+(0, -4\baseheight) -- (1);
+\end{tikzpicture}}
+\caption{Final configuration of TM $adjust$} \label{adjust_final}
+\end{figure}
+
+
+\begin{figure}[h!]
+\centering
+\scalebox{0.9}{
+\begin{tikzpicture}
+ \node[circle,draw] (1) {$1$};
+ \node[circle,draw] (2) at ($(1)+(0.3\basewidth, 0)$) {$2$};
+ \node[circle,draw] (3) at ($(2)+(0.3\basewidth, 0)$) {$3$};
+ \node[circle,draw] (4) at ($(3)+(0.3\basewidth, 0)$) {$4$};
+ \node[circle,draw] (5) at ($(4)+(0.3\basewidth, 0)$) {$5$};
+ \node[circle,draw] (6) at ($(5)+(0.3\basewidth, 0)$) {$6$};
+ \node[circle,draw] (7) at ($(6)+(0.3\basewidth, 0)$) {$7$};
+ \node[circle,draw] (8) at ($(4)+(0, -7\baseheight)$) {$8$};
+ \node[circle,draw] (9) at ($(8)+(0.3\basewidth, 0)$) {$9$};
+ \node[circle,draw] (10) at ($(9)+(0.3\basewidth, 0)$) {$10$};
+ \node[circle,draw] (11) at ($(10)+(0.3\basewidth, 0)$) {$11$};
+ \node[circle,draw] (12) at ($(11)+(0.3\basewidth, 0)$) {$0$};
+
+
+ \draw [->, >=latex] (1) -- node[above] {$S_1:R$} (2)
+ ;
+ \draw [->, >=latex] (1) edge[loop above] node[above] {$S_0:S_1$} (1)
+ ;
+ \draw [->, >=latex] (2) -- node[above] {$S_1:R$} (3)
+ ;
+ \draw [->, >=latex] (3) edge[loop above] node[above] {$S_0:R$} (3)
+ ;
+ \draw [->, >=latex] (3) -- node[above] {$S_1:R$} (4)
+ ;
+ \draw [->, >=latex] (4) -- node[above] {$S_1:L$} (5)
+ ;
+ \draw [->, >=latex] (4) -- node[right] {$S_0:L$} (8)
+ ;
+ \draw [->, >=latex] (5) -- node[above] {$S_0:L$} (6)
+ ;
+ \draw [->, >=latex] (5) edge[loop above] node[above] {$S_1:S_0$} (5)
+ ;
+ \draw [->, >=latex] (6) -- node[above] {$S_1:R$} (7)
+ ;
+ \draw [->, >=latex] (6) edge[loop above] node[above] {$S_0:L$} (6)
+ ;
+ \draw (7) -- ($(7)+(0, 6\baseheight)$) -- ($(2) + (0, 6\baseheight)$) node [above,midway] {$S_0:S_1$}
+ ;
+ \draw [->, >=latex] ($(2) + (0, 6\baseheight)$) -- (2)
+ ;
+ \draw [->, >=latex] (8) edge[loop left] node[left] {$S_1:S_0$} (8)
+ ;
+ \draw [->, >=latex] (8) -- node[above] {$S_0:L$} (9)
+ ;
+ \draw [->, >=latex] (9) edge[loop above] node[above] {$S_0:L$} (9)
+ ;
+ \draw [->, >=latex] (9) -- node[above] {$S_1:L$} (10)
+ ;
+ \draw [->, >=latex] (10) edge[loop above] node[above] {$S_0:L$} (10)
+ ;
+ \draw [->, >=latex] (10) -- node[above] {$S_0:L$} (11)
+ ;
+ \draw [->, >=latex] (11) edge[loop above] node[above] {$S_1:L$} (11)
+ ;
+ \draw [->, >=latex] (11) -- node[above] {$S_0:R$} (12)
+ ;
+ \end{tikzpicture}}
+\caption{Diagram of TM $adjust$} \label{adjust_diag}
+\end{figure}
+*}
+
+
+definition rec_twice :: "recf"
+ where
+ "rec_twice = Cn 1 rec_mult [id 1 0, constn 2]"
+
+definition rec_fourtimes :: "recf"
+ where
+ "rec_fourtimes = Cn 1 rec_mult [id 1 0, constn 4]"
+
+definition abc_twice :: "abc_prog"
+ where
+ "abc_twice = (let (aprog, ary, fp) = rec_ci rec_twice in
+ aprog [+] dummy_abc ((Suc 0)))"
+
+definition abc_fourtimes :: "abc_prog"
+ where
+ "abc_fourtimes = (let (aprog, ary, fp) = rec_ci rec_fourtimes in
+ aprog [+] dummy_abc ((Suc 0)))"
+
+definition twice_ly :: "nat list"
+ where
+ "twice_ly = layout_of abc_twice"
+
+definition fourtimes_ly :: "nat list"
+ where
+ "fourtimes_ly = layout_of abc_fourtimes"
+
+definition t_twice :: "tprog"
+ where
+ "t_twice = change_termi_state (tm_of (abc_twice) @ (tMp 1 (start_of twice_ly (length abc_twice) - Suc 0)))"
+
+definition t_fourtimes :: "tprog"
+ where
+ "t_fourtimes = change_termi_state (tm_of (abc_fourtimes) @
+ (tMp 1 (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)))"
+
+
+definition t_twice_len :: "nat"
+ where
+ "t_twice_len = length t_twice div 2"
+
+definition t_wcode_main_first_part:: "tprog"
+ where
+ "t_wcode_main_first_part \<equiv>
+ [(L, 1), (L, 2), (L, 7), (R, 3),
+ (R, 4), (W0, 3), (R, 4), (R, 5),
+ (W1, 6), (R, 5), (R, 13), (L, 6),
+ (R, 0), (R, 8), (R, 9), (Nop, 8),
+ (R, 10), (W0, 9), (R, 10), (R, 11),
+ (W1, 12), (R, 11), (R, t_twice_len + 14), (L, 12)]"
+
+definition t_wcode_main :: "tprog"
+ where
+ "t_wcode_main = (t_wcode_main_first_part @ tshift t_twice 12 @ [(L, 1), (L, 1)]
+ @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])"
+
+fun bl_bin :: "block list \<Rightarrow> nat"
+ where
+ "bl_bin [] = 0"
+| "bl_bin (Bk # xs) = 2 * bl_bin xs"
+| "bl_bin (Oc # xs) = Suc (2 * bl_bin xs)"
+
+declare bl_bin.simps[simp del]
+
+type_synonym bin_inv_t = "block list \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+
+fun wcode_before_double :: "bin_inv_t"
+ where
+ "wcode_before_double ires rs (l, r) =
+ (\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+ r = Oc\<^bsup>(Suc (Suc rs))\<^esup> @ Bk\<^bsup>rn \<^esup>)"
+
+declare wcode_before_double.simps[simp del]
+
+fun wcode_after_double :: "bin_inv_t"
+ where
+ "wcode_after_double ires rs (l, r) =
+ (\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+ r = Oc\<^bsup>Suc (Suc (Suc 2*rs))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+declare wcode_after_double.simps[simp del]
+
+fun wcode_on_left_moving_1_B :: "bin_inv_t"
+ where
+ "wcode_on_left_moving_1_B ires rs (l, r) =
+ (\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Oc # ires \<and>
+ r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr > Suc 0 \<and> mr > 0)"
+
+declare wcode_on_left_moving_1_B.simps[simp del]
+
+fun wcode_on_left_moving_1_O :: "bin_inv_t"
+ where
+ "wcode_on_left_moving_1_O ires rs (l, r) =
+ (\<exists> ln rn.
+ l = Oc # ires \<and>
+ r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+declare wcode_on_left_moving_1_O.simps[simp del]
+
+fun wcode_on_left_moving_1 :: "bin_inv_t"
+ where
+ "wcode_on_left_moving_1 ires rs (l, r) =
+ (wcode_on_left_moving_1_B ires rs (l, r) \<or> wcode_on_left_moving_1_O ires rs (l, r))"
+
+declare wcode_on_left_moving_1.simps[simp del]
+
+fun wcode_on_checking_1 :: "bin_inv_t"
+ where
+ "wcode_on_checking_1 ires rs (l, r) =
+ (\<exists> ln rn. l = ires \<and>
+ r = Oc # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wcode_erase1 :: "bin_inv_t"
+ where
+"wcode_erase1 ires rs (l, r) =
+ (\<exists> ln rn. l = Oc # ires \<and>
+ tl r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+declare wcode_erase1.simps [simp del]
+
+fun wcode_on_right_moving_1 :: "bin_inv_t"
+ where
+ "wcode_on_right_moving_1 ires rs (l, r) =
+ (\<exists> ml mr rn.
+ l = Bk\<^bsup>ml\<^esup> @ Oc # ires \<and>
+ r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr > Suc 0)"
+
+declare wcode_on_right_moving_1.simps [simp del]
+
+declare wcode_on_right_moving_1.simps[simp del]
+
+fun wcode_goon_right_moving_1 :: "bin_inv_t"
+ where
+ "wcode_goon_right_moving_1 ires rs (l, r) =
+ (\<exists> ml mr ln rn.
+ l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+ r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr = Suc rs)"
+
+declare wcode_goon_right_moving_1.simps[simp del]
+
+fun wcode_backto_standard_pos_B :: "bin_inv_t"
+ where
+ "wcode_backto_standard_pos_B ires rs (l, r) =
+ (\<exists> ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+ r = Bk # Oc\<^bsup>(Suc (Suc rs))\<^esup> @ Bk\<^bsup>rn \<^esup>)"
+
+declare wcode_backto_standard_pos_B.simps[simp del]
+
+fun wcode_backto_standard_pos_O :: "bin_inv_t"
+ where
+ "wcode_backto_standard_pos_O ires rs (l, r) =
+ (\<exists> ml mr ln rn.
+ l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+ r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr = Suc (Suc rs) \<and> mr > 0)"
+
+declare wcode_backto_standard_pos_O.simps[simp del]
+
+fun wcode_backto_standard_pos :: "bin_inv_t"
+ where
+ "wcode_backto_standard_pos ires rs (l, r) = (wcode_backto_standard_pos_B ires rs (l, r) \<or>
+ wcode_backto_standard_pos_O ires rs (l, r))"
+
+declare wcode_backto_standard_pos.simps[simp del]
+
+lemma [simp]: "<0::nat> = [Oc]"
+apply(simp add: tape_of_nat_abv exponent_def tape_of_nat_list.simps)
+done
+
+lemma tape_of_Suc_nat: "<Suc (a ::nat)> = replicate a Oc @ [Oc, Oc]"
+apply(simp add: tape_of_nat_abv exp_ind tape_of_nat_list.simps)
+apply(simp only: exp_ind_def[THEN sym])
+apply(simp only: exp_ind, simp, simp add: exponent_def)
+done
+
+lemma [simp]: "length (<a::nat>) = Suc a"
+apply(simp add: tape_of_nat_abv tape_of_nat_list.simps)
+done
+
+lemma [simp]: "<[a::nat]> = <a>"
+apply(simp add: tape_of_nat_abv tape_of_nl_abv exponent_def
+ tape_of_nat_list.simps)
+done
+
+lemma bin_wc_eq: "bl_bin xs = bl2wc xs"
+proof(induct xs)
+ show " bl_bin [] = bl2wc []"
+ apply(simp add: bl_bin.simps)
+ done
+next
+ fix a xs
+ assume "bl_bin xs = bl2wc xs"
+ thus " bl_bin (a # xs) = bl2wc (a # xs)"
+ apply(case_tac a, simp_all add: bl_bin.simps bl2wc.simps)
+ apply(simp_all add: bl2nat.simps bl2nat_double)
+ done
+qed
+
+declare exp_def[simp del]
+
+lemma bl_bin_nat_Suc:
+ "bl_bin (<Suc a>) = bl_bin (<a>) + 2^(Suc a)"
+apply(simp add: tape_of_nat_abv bin_wc_eq)
+apply(simp add: bl2wc.simps)
+done
+lemma [simp]: " rev (a\<^bsup>aa\<^esup>) = a\<^bsup>aa\<^esup>"
+apply(simp add: exponent_def)
+done
+
+declare tape_of_nl_abv_cons[simp del]
+
+lemma tape_of_nl_rev: "rev (<lm::nat list>) = (<rev lm>)"
+apply(induct lm rule: list_tl_induct, simp)
+apply(case_tac "list = []", simp add: tape_of_nl_abv tape_of_nat_list.simps)
+apply(simp add: tape_of_nat_list_butlast_last tape_of_nl_abv_cons)
+done
+lemma [simp]: "a\<^bsup>Suc 0\<^esup> = [a]"
+by(simp add: exp_def)
+lemma tape_of_nl_cons_app1: "(<a # xs @ [b]>) = (Oc\<^bsup>Suc a\<^esup> @ Bk # (<xs@ [b]>))"
+apply(case_tac xs, simp add: tape_of_nl_abv tape_of_nat_list.simps)
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
+done
+
+lemma bl_bin_bk_oc[simp]:
+ "bl_bin (xs @ [Bk, Oc]) =
+ bl_bin xs + 2*2^(length xs)"
+apply(simp add: bin_wc_eq)
+using bl2nat_cons_oc[of "xs @ [Bk]"]
+apply(simp add: bl2nat_cons_bk bl2wc.simps)
+done
+
+lemma tape_of_nat[simp]: "(<a::nat>) = Oc\<^bsup>Suc a\<^esup>"
+apply(simp add: tape_of_nat_abv)
+done
+lemma tape_of_nl_cons_app2: "(<c # xs @ [b]>) = (<c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>)"
+proof(induct "length xs" arbitrary: xs c,
+ simp add: tape_of_nl_abv tape_of_nat_list.simps)
+ fix x xs c
+ assume ind: "\<And>xs c. x = length xs \<Longrightarrow> <c # xs @ [b]> =
+ <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>"
+ and h: "Suc x = length (xs::nat list)"
+ show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>"
+ proof(case_tac xs, simp add: tape_of_nl_abv tape_of_nat_list.simps)
+ fix a list
+ assume g: "xs = a # list"
+ hence k: "<a # list @ [b]> = <a # list> @ Bk # Oc\<^bsup>Suc b\<^esup>"
+ apply(rule_tac ind)
+ using h
+ apply(simp)
+ done
+ from g and k show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>"
+ apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
+ done
+ qed
+qed
+
+lemma [simp]: "length (<aa # a # list>) = Suc (Suc aa) + length (<a # list>)"
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
+done
+
+lemma [simp]: "bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista) @ [Bk, Oc]) =
+ bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)) +
+ 2* 2^(length (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)))"
+using bl_bin_bk_oc[of "Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)"]
+apply(simp)
+done
+
+lemma [simp]:
+ "bl_bin (<aa # list>) + (4 * rs + 4) * 2 ^ (length (<aa # list>) - Suc 0)
+ = bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [0]>) + rs * (2 * 2 ^ (aa + length (<list @ [0]>)))"
+apply(case_tac "list", simp add: add_mult_distrib, simp)
+apply(simp add: tape_of_nl_cons_app2 add_mult_distrib)
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
+done
+
+lemma tape_of_nl_app_Suc: "((<list @ [Suc ab]>)) = (<list @ [ab]>) @ [Oc]"
+apply(induct list)
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind)
+apply(case_tac list)
+apply(simp_all add:tape_of_nl_abv tape_of_nat_list.simps exp_ind)
+done
+
+lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]> @ [Oc])
+ = bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>) +
+ 2^(length (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>))"
+apply(simp add: bin_wc_eq)
+apply(simp add: bl2nat_cons_oc bl2wc.simps)
+using bl2nat_cons_oc[of "Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>"]
+apply(simp)
+done
+lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>) + (4 * 2 ^ (aa + length (<list @ [ab]>)) +
+ 4 * (rs * 2 ^ (aa + length (<list @ [ab]>)))) =
+ bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [Suc ab]>) +
+ rs * (2 * 2 ^ (aa + length (<list @ [Suc ab]>)))"
+apply(simp add: tape_of_nl_app_Suc)
+done
+
+declare tape_of_nat[simp del]
+
+fun wcode_double_case_inv :: "nat \<Rightarrow> bin_inv_t"
+ where
+ "wcode_double_case_inv st ires rs (l, r) =
+ (if st = Suc 0 then wcode_on_left_moving_1 ires rs (l, r)
+ else if st = Suc (Suc 0) then wcode_on_checking_1 ires rs (l, r)
+ else if st = 3 then wcode_erase1 ires rs (l, r)
+ else if st = 4 then wcode_on_right_moving_1 ires rs (l, r)
+ else if st = 5 then wcode_goon_right_moving_1 ires rs (l, r)
+ else if st = 6 then wcode_backto_standard_pos ires rs (l, r)
+ else if st = 13 then wcode_before_double ires rs (l, r)
+ else False)"
+
+declare wcode_double_case_inv.simps[simp del]
+
+fun wcode_double_case_state :: "t_conf \<Rightarrow> nat"
+ where
+ "wcode_double_case_state (st, l, r) =
+ 13 - st"
+
+fun wcode_double_case_step :: "t_conf \<Rightarrow> nat"
+ where
+ "wcode_double_case_step (st, l, r) =
+ (if st = Suc 0 then (length l)
+ else if st = Suc (Suc 0) then (length r)
+ else if st = 3 then
+ if hd r = Oc then 1 else 0
+ else if st = 4 then (length r)
+ else if st = 5 then (length r)
+ else if st = 6 then (length l)
+ else 0)"
+
+fun wcode_double_case_measure :: "t_conf \<Rightarrow> nat \<times> nat"
+ where
+ "wcode_double_case_measure (st, l, r) =
+ (wcode_double_case_state (st, l, r),
+ wcode_double_case_step (st, l, r))"
+
+definition wcode_double_case_le :: "(t_conf \<times> t_conf) set"
+ where "wcode_double_case_le \<equiv> (inv_image lex_pair wcode_double_case_measure)"
+
+lemma [intro]: "wf lex_pair"
+by(auto intro:wf_lex_prod simp:lex_pair_def)
+
+lemma wf_wcode_double_case_le[intro]: "wf wcode_double_case_le"
+by(auto intro:wf_inv_image simp: wcode_double_case_le_def )
+term fetch
+
+lemma [simp]: "fetch t_wcode_main (Suc 0) Bk = (L, Suc 0)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+ fetch.simps nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main (Suc 0) Oc = (L, Suc (Suc 0))"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+ fetch.simps nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main (Suc (Suc 0)) Oc = (R, 3)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+ fetch.simps nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main (Suc (Suc (Suc 0))) Bk = (R, 4)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+ fetch.simps nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main (Suc (Suc (Suc 0))) Oc = (W0, 3)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+ fetch.simps nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 4 Bk = (R, 4)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+ fetch.simps nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 4 Oc = (R, 5)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+ fetch.simps nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 5 Oc = (R, 5)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+ fetch.simps nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 5 Bk = (W1, 6)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+ fetch.simps nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 6 Bk = (R, 13)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+ fetch.simps nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 6 Oc = (L, 6)"
+apply(simp add: t_wcode_main_def t_wcode_main_first_part_def
+ fetch.simps nth_of.simps)
+done
+lemma [elim]: "Bk\<^bsup>mr\<^esup> = [] \<Longrightarrow> mr = 0"
+apply(case_tac mr, auto simp: exponent_def)
+done
+
+lemma [simp]: "wcode_on_left_moving_1 ires rs (b, []) = False"
+apply(simp add: wcode_on_left_moving_1.simps wcode_on_left_moving_1_B.simps
+ wcode_on_left_moving_1_O.simps, auto)
+done
+
+
+declare wcode_on_checking_1.simps[simp del]
+
+lemmas wcode_double_case_inv_simps =
+ wcode_on_left_moving_1.simps wcode_on_left_moving_1_O.simps
+ wcode_on_left_moving_1_B.simps wcode_on_checking_1.simps
+ wcode_erase1.simps wcode_on_right_moving_1.simps
+ wcode_goon_right_moving_1.simps wcode_backto_standard_pos.simps
+
+
+lemma [simp]: "wcode_on_left_moving_1 ires rs (b, r) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wcode_double_case_inv_simps, auto)
+done
+
+
+lemma [elim]: "\<lbrakk>wcode_on_left_moving_1 ires rs (b, Bk # list);
+ tl b = aa \<and> hd b # Bk # list = ba\<rbrakk> \<Longrightarrow>
+ wcode_on_left_moving_1 ires rs (aa, ba)"
+apply(simp only: wcode_on_left_moving_1.simps wcode_on_left_moving_1_O.simps
+ wcode_on_left_moving_1_B.simps)
+apply(erule_tac disjE)
+apply(erule_tac exE)+
+apply(case_tac ml, simp)
+apply(rule_tac x = "mr - Suc (Suc 0)" in exI, rule_tac x = rn in exI)
+apply(case_tac mr, simp, case_tac nat, simp, simp add: exp_ind)
+apply(rule_tac disjI1)
+apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = rn in exI,
+ simp add: exp_ind_def)
+apply(erule_tac exE)+
+apply(simp)
+done
+
+
+lemma [elim]:
+ "\<lbrakk>wcode_on_left_moving_1 ires rs (b, Oc # list); tl b = aa \<and> hd b # Oc # list = ba\<rbrakk>
+ \<Longrightarrow> wcode_on_checking_1 ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+apply(erule_tac disjE)
+apply(erule_tac [!] exE)+
+apply(case_tac mr, simp, simp add: exp_ind_def)
+apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
+done
+
+
+lemma [simp]: "wcode_on_checking_1 ires rs (b, []) = False"
+apply(auto simp: wcode_double_case_inv_simps)
+done
+
+lemma [simp]: "wcode_on_checking_1 ires rs (b, Bk # list) = False"
+apply(auto simp: wcode_double_case_inv_simps)
+done
+
+lemma [elim]: "\<lbrakk>wcode_on_checking_1 ires rs (b, Oc # ba);Oc # b = aa \<and> list = ba\<rbrakk>
+ \<Longrightarrow> wcode_erase1 ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+apply(erule_tac exE)+
+apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
+done
+
+
+lemma [simp]: "wcode_on_checking_1 ires rs (b, []) = False"
+apply(simp add: wcode_double_case_inv_simps)
+done
+
+lemma [simp]: "wcode_on_checking_1 ires rs ([], Bk # list) = False"
+apply(simp add: wcode_double_case_inv_simps)
+done
+
+lemma [simp]: "wcode_erase1 ires rs (b, []) = False"
+apply(simp add: wcode_double_case_inv_simps)
+done
+
+lemma [simp]: "wcode_on_right_moving_1 ires rs (b, []) = False"
+apply(simp add: wcode_double_case_inv_simps exp_ind_def)
+done
+
+lemma [simp]: "wcode_on_right_moving_1 ires rs (b, []) = False"
+apply(simp add: wcode_double_case_inv_simps exp_ind_def)
+done
+
+lemma [elim]: "\<lbrakk>wcode_on_right_moving_1 ires rs (b, Bk # ba); Bk # b = aa \<and> list = b\<rbrakk> \<Longrightarrow>
+ wcode_on_right_moving_1 ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+apply(erule_tac exE)+
+apply(rule_tac x = "Suc ml" in exI, rule_tac x = "mr - Suc 0" in exI,
+ rule_tac x = rn in exI)
+apply(simp add: exp_ind_def)
+apply(case_tac mr, simp, simp add: exp_ind_def)
+done
+
+lemma [elim]:
+ "\<lbrakk>wcode_on_right_moving_1 ires rs (b, Oc # ba); Oc # b = aa \<and> list = ba\<rbrakk>
+ \<Longrightarrow> wcode_goon_right_moving_1 ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+apply(erule_tac exE)+
+apply(rule_tac x = "Suc 0" in exI, rule_tac x = "rs" in exI,
+ rule_tac x = "ml - Suc (Suc 0)" in exI, rule_tac x = rn in exI)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac ml, simp, case_tac nat, simp, simp)
+apply(simp add: exp_ind_def)
+done
+
+lemma [simp]:
+ "wcode_on_right_moving_1 ires rs (b, []) \<Longrightarrow> False"
+apply(simp add: wcode_double_case_inv_simps exponent_def)
+done
+
+lemma [elim]: "\<lbrakk>wcode_erase1 ires rs (b, Bk # ba); Bk # b = aa \<and> list = ba; c = Bk # ba\<rbrakk>
+ \<Longrightarrow> wcode_on_right_moving_1 ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+apply(erule_tac exE)+
+apply(rule_tac x = "Suc 0" in exI, rule_tac x = "Suc (Suc ln)" in exI,
+ rule_tac x = rn in exI, simp add: exp_ind)
+done
+
+lemma [elim]: "\<lbrakk>wcode_erase1 ires rs (aa, Oc # list); b = aa \<and> Bk # list = ba\<rbrakk> \<Longrightarrow>
+ wcode_erase1 ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+apply(erule_tac exE)+
+apply(rule_tac x = ln in exI, rule_tac x = rn in exI, auto)
+done
+
+lemma [elim]: "\<lbrakk>wcode_goon_right_moving_1 ires rs (aa, []); b = aa \<and> [Oc] = ba\<rbrakk>
+ \<Longrightarrow> wcode_backto_standard_pos ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+apply(erule_tac exE)+
+apply(rule_tac disjI2)
+apply(simp only:wcode_backto_standard_pos_O.simps)
+apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI, rule_tac x = ln in exI,
+ rule_tac x = rn in exI, simp)
+apply(case_tac mr, simp_all add: exponent_def)
+done
+
+lemma [elim]:
+ "\<lbrakk>wcode_goon_right_moving_1 ires rs (aa, Bk # list); b = aa \<and> Oc # list = ba\<rbrakk>
+ \<Longrightarrow> wcode_backto_standard_pos ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+apply(erule_tac exE)+
+apply(rule_tac disjI2)
+apply(simp only:wcode_backto_standard_pos_O.simps)
+apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI, rule_tac x = ln in exI,
+ rule_tac x = "rn - Suc 0" in exI, simp)
+apply(case_tac mr, simp, case_tac rn, simp, simp_all add: exp_ind_def)
+done
+
+lemma [elim]: "\<lbrakk>wcode_goon_right_moving_1 ires rs (b, Oc # ba); Oc # b = aa \<and> list = ba\<rbrakk>
+ \<Longrightarrow> wcode_goon_right_moving_1 ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps)
+apply(erule_tac exE)+
+apply(rule_tac x = "Suc ml" in exI, rule_tac x = "mr - Suc 0" in exI,
+ rule_tac x = ln in exI, rule_tac x = rn in exI)
+apply(simp add: exp_ind_def)
+apply(case_tac mr, simp, case_tac rn, simp_all add: exp_ind_def)
+done
+
+lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, []); Bk # b = aa\<rbrakk> \<Longrightarrow> False"
+apply(auto simp: wcode_double_case_inv_simps wcode_backto_standard_pos_O.simps
+ wcode_backto_standard_pos_B.simps)
+apply(case_tac mr, simp_all add: exp_ind_def)
+done
+
+lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, Bk # ba); Bk # b = aa \<and> list = ba\<rbrakk>
+ \<Longrightarrow> wcode_before_double ires rs (aa, ba)"
+apply(simp only: wcode_double_case_inv_simps wcode_backto_standard_pos_B.simps
+ wcode_backto_standard_pos_O.simps wcode_before_double.simps)
+apply(erule_tac disjE)
+apply(erule_tac exE)+
+apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
+apply(auto)
+apply(case_tac [!] mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wcode_backto_standard_pos ires rs ([], Oc # list) = False"
+apply(auto simp: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps
+ wcode_backto_standard_pos_O.simps)
+done
+
+lemma [simp]: "wcode_backto_standard_pos ires rs (b, []) = False"
+apply(auto simp: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps
+ wcode_backto_standard_pos_O.simps)
+apply(case_tac mr, simp, simp add: exp_ind_def)
+done
+
+lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, Oc # list); tl b = aa; hd b # Oc # list = ba\<rbrakk>
+ \<Longrightarrow> wcode_backto_standard_pos ires rs (aa, ba)"
+apply(simp only: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps
+ wcode_backto_standard_pos_O.simps)
+apply(erule_tac disjE)
+apply(simp)
+apply(erule_tac exE)+
+apply(case_tac ml, simp)
+apply(rule_tac disjI1, rule_tac conjI)
+apply(rule_tac x = ln in exI, simp, rule_tac x = rn in exI, simp)
+apply(rule_tac disjI2)
+apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = ln in exI,
+ rule_tac x = rn in exI, simp)
+apply(simp add: exp_ind_def)
+done
+
+declare new_tape.simps[simp del] nth_of.simps[simp del] fetch.simps[simp del]
+lemma wcode_double_case_first_correctness:
+ "let P = (\<lambda> (st, l, r). st = 13) in
+ let Q = (\<lambda> (st, l, r). wcode_double_case_inv st ires rs (l, r)) in
+ let f = (\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp) in
+ \<exists> n .P (f n) \<and> Q (f (n::nat))"
+proof -
+ let ?P = "(\<lambda> (st, l, r). st = 13)"
+ let ?Q = "(\<lambda> (st, l, r). wcode_double_case_inv st ires rs (l, r))"
+ let ?f = "(\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)"
+ have "\<exists> n. ?P (?f n) \<and> ?Q (?f (n::nat))"
+ proof(rule_tac halt_lemma2)
+ show "wf wcode_double_case_le"
+ by auto
+ next
+ show "\<forall> na. \<not> ?P (?f na) \<and> ?Q (?f na) \<longrightarrow>
+ ?Q (?f (Suc na)) \<and> (?f (Suc na), ?f na) \<in> wcode_double_case_le"
+ proof(rule_tac allI, case_tac "?f na", simp add: tstep_red)
+ fix na a b c
+ show "a \<noteq> 13 \<and> wcode_double_case_inv a ires rs (b, c) \<longrightarrow>
+ (case tstep (a, b, c) t_wcode_main of (st, x) \<Rightarrow>
+ wcode_double_case_inv st ires rs x) \<and>
+ (tstep (a, b, c) t_wcode_main, a, b, c) \<in> wcode_double_case_le"
+ apply(rule_tac impI, simp add: wcode_double_case_inv.simps)
+ apply(auto split: if_splits simp: tstep.simps,
+ case_tac [!] c, simp_all, case_tac [!] "(c::block list)!0")
+ apply(simp_all add: new_tape.simps wcode_double_case_inv.simps wcode_double_case_le_def
+ lex_pair_def)
+ apply(auto split: if_splits)
+ done
+ qed
+ next
+ show "?Q (?f 0)"
+ apply(simp add: steps.simps wcode_double_case_inv.simps
+ wcode_on_left_moving_1.simps
+ wcode_on_left_moving_1_B.simps)
+ apply(rule_tac disjI1)
+ apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def)
+ apply(rule_tac x = "Suc 0" in exI, simp add: exp_ind_def)
+ apply(auto)
+ done
+ next
+ show "\<not> ?P (?f 0)"
+ apply(simp add: steps.simps)
+ done
+ qed
+ thus "let P = \<lambda>(st, l, r). st = 13;
+ Q = \<lambda>(st, l, r). wcode_double_case_inv st ires rs (l, r);
+ f = steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main
+ in \<exists>n. P (f n) \<and> Q (f n)"
+ apply(simp add: Let_def)
+ done
+qed
+
+lemma [elim]: "t_ncorrect tp
+ \<Longrightarrow> t_ncorrect (abacus.tshift tp a)"
+apply(simp add: t_ncorrect.simps shift_length)
+done
+
+lemma tshift_fetch: "\<lbrakk> fetch tp a b = (aa, st'); 0 < st'\<rbrakk>
+ \<Longrightarrow> fetch (abacus.tshift tp (length tp1 div 2)) a b
+ = (aa, st' + length tp1 div 2)"
+apply(subgoal_tac "a > 0")
+apply(auto simp: fetch.simps nth_of.simps shift_length nth_map
+ tshift.simps split: block.splits if_splits)
+done
+
+lemma t_steps_steps_eq: "\<lbrakk>steps (st, l, r) tp stp = (st', l', r');
+ 0 < st';
+ 0 < st \<and> st \<le> length tp div 2;
+ t_ncorrect tp1;
+ t_ncorrect tp\<rbrakk>
+ \<Longrightarrow> t_steps (st + length tp1 div 2, l, r) (tshift tp (length tp1 div 2),
+ length tp1 div 2) stp
+ = (st' + length tp1 div 2, l', r')"
+apply(induct stp arbitrary: st' l' r', simp add: steps.simps t_steps.simps,
+ simp add: tstep_red stepn)
+apply(case_tac "(steps (st, l, r) tp stp)", simp)
+proof -
+ fix stp st' l' r' a b c
+ assume ind: "\<And>st' l' r'.
+ \<lbrakk>a = st' \<and> b = l' \<and> c = r'; 0 < st'\<rbrakk>
+ \<Longrightarrow> t_steps (st + length tp1 div 2, l, r)
+ (abacus.tshift tp (length tp1 div 2), length tp1 div 2) stp =
+ (st' + length tp1 div 2, l', r')"
+ and h: "tstep (a, b, c) tp = (st', l', r')" "0 < st'" "t_ncorrect tp1" "t_ncorrect tp"
+ have k: "t_steps (st + length tp1 div 2, l, r) (abacus.tshift tp (length tp1 div 2),
+ length tp1 div 2) stp = (a + length tp1 div 2, b, c)"
+ apply(rule_tac ind, simp)
+ using h
+ apply(case_tac a, simp_all add: tstep.simps fetch.simps)
+ done
+ from h and this show "t_step (t_steps (st + length tp1 div 2, l, r) (abacus.tshift tp (length tp1 div 2), length tp1 div 2) stp)
+ (abacus.tshift tp (length tp1 div 2), length tp1 div 2) =
+ (st' + length tp1 div 2, l', r')"
+ apply(simp add: k)
+ apply(simp add: tstep.simps t_step.simps)
+ apply(case_tac "fetch tp a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
+ apply(subgoal_tac "fetch (abacus.tshift tp (length tp1 div 2)) a
+ (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) = (aa, st' + length tp1 div 2)", simp)
+ apply(simp add: tshift_fetch)
+ done
+qed
+
+lemma t_tshift_lemma: "\<lbrakk> steps (st, l, r) tp stp = (st', l', r');
+ st' \<noteq> 0;
+ stp > 0;
+ 0 < st \<and> st \<le> length tp div 2;
+ t_ncorrect tp1;
+ t_ncorrect tp;
+ t_ncorrect tp2
+ \<rbrakk>
+ \<Longrightarrow> \<exists> stp>0. steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2) stp
+ = (st' + length tp1 div 2, l', r')"
+proof -
+ assume h: "steps (st, l, r) tp stp = (st', l', r')"
+ "st' \<noteq> 0" "stp > 0"
+ "0 < st \<and> st \<le> length tp div 2"
+ "t_ncorrect tp1"
+ "t_ncorrect tp"
+ "t_ncorrect tp2"
+ from h have
+ "\<exists>stp>0. t_steps (st + length tp1 div 2, l, r) (tp1 @ abacus.tshift tp (length tp1 div 2) @ tp2, 0) stp =
+ (st' + length tp1 div 2, l', r')"
+ apply(rule_tac stp = stp in turing_shift, simp_all add: shift_length)
+ apply(simp add: t_steps_steps_eq)
+ apply(simp add: t_ncorrect.simps shift_length)
+ done
+ thus "\<exists> stp>0. steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2) stp
+ = (st' + length tp1 div 2, l', r')"
+ apply(erule_tac exE)
+ apply(rule_tac x = stp in exI, simp)
+ apply(subgoal_tac "length (tp1 @ abacus.tshift tp (length tp1 div 2) @ tp2) mod 2 = 0")
+ apply(simp only: steps_eq)
+ using h
+ apply(auto simp: t_ncorrect.simps shift_length)
+ apply arith
+ done
+qed
+
+
+lemma t_twice_len_ge: "Suc 0 \<le> length t_twice div 2"
+apply(simp add: t_twice_def tMp.simps shift_length)
+done
+
+lemma [intro]: "rec_calc_rel (recf.id (Suc 0) 0) [rs] rs"
+ apply(rule_tac calc_id, simp_all)
+ done
+
+lemma [intro]: "rec_calc_rel (constn 2) [rs] 2"
+using prime_rel_exec_eq[of "constn 2" "[rs]" 2]
+apply(subgoal_tac "primerec (constn 2) 1", auto)
+done
+
+lemma [intro]: "rec_calc_rel rec_mult [rs, 2] (2 * rs)"
+using prime_rel_exec_eq[of "rec_mult" "[rs, 2]" "2*rs"]
+apply(subgoal_tac "primerec rec_mult (Suc (Suc 0))", auto)
+done
+lemma t_twice_correct: "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+ (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) stp =
+ (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+proof(case_tac "rec_ci rec_twice")
+ fix a b c
+ assume h: "rec_ci rec_twice = (a, b, c)"
+ have "\<exists>stp m l. steps (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<^bsup>n\<^esup>) (tm_of abc_twice @ tMp (Suc 0)
+ (start_of twice_ly (length abc_twice) - 1)) stp = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2*rs)\<^esup> @ Bk\<^bsup>l\<^esup>)"
+ proof(rule_tac t_compiled_by_rec)
+ show "rec_ci rec_twice = (a, b, c)" by (simp add: h)
+ next
+ show "rec_calc_rel rec_twice [rs] (2 * rs)"
+ apply(simp add: rec_twice_def)
+ apply(rule_tac rs = "[rs, 2]" in calc_cn, simp_all)
+ apply(rule_tac allI, case_tac k, auto)
+ done
+ next
+ show "length [rs] = Suc 0" by simp
+ next
+ show "layout_of (a [+] dummy_abc (Suc 0)) = layout_of (a [+] dummy_abc (Suc 0))"
+ by simp
+ next
+ show "start_of twice_ly (length abc_twice) =
+ start_of (layout_of (a [+] dummy_abc (Suc 0))) (length (a [+] dummy_abc (Suc 0)))"
+ using h
+ apply(simp add: twice_ly_def abc_twice_def)
+ done
+ next
+ show "tm_of abc_twice = tm_of (a [+] dummy_abc (Suc 0))"
+ using h
+ apply(simp add: abc_twice_def)
+ done
+ qed
+ thus "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+ (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) stp =
+ (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
+ done
+qed
+
+lemma change_termi_state_fetch: "\<lbrakk>fetch ap a b = (aa, st); st > 0\<rbrakk>
+ \<Longrightarrow> fetch (change_termi_state ap) a b = (aa, st)"
+apply(case_tac b, auto simp: fetch.simps nth_of.simps change_termi_state.simps nth_map
+ split: if_splits block.splits)
+done
+
+lemma change_termi_state_exec_in_range:
+ "\<lbrakk>steps (st, l, r) ap stp = (st', l', r'); st' \<noteq> 0\<rbrakk>
+ \<Longrightarrow> steps (st, l, r) (change_termi_state ap) stp = (st', l', r')"
+proof(induct stp arbitrary: st l r st' l' r', simp add: steps.simps)
+ fix stp st l r st' l' r'
+ assume ind: "\<And>st l r st' l' r'.
+ \<lbrakk>steps (st, l, r) ap stp = (st', l', r'); st' \<noteq> 0\<rbrakk> \<Longrightarrow>
+ steps (st, l, r) (change_termi_state ap) stp = (st', l', r')"
+ and h: "steps (st, l, r) ap (Suc stp) = (st', l', r')" "st' \<noteq> 0"
+ from h show "steps (st, l, r) (change_termi_state ap) (Suc stp) = (st', l', r')"
+ proof(simp add: tstep_red, case_tac "steps (st, l, r) ap stp", simp)
+ fix a b c
+ assume g: "steps (st, l, r) ap stp = (a, b, c)"
+ "tstep (a, b, c) ap = (st', l', r')" "0 < st'"
+ hence "steps (st, l, r) (change_termi_state ap) stp = (a, b, c)"
+ apply(rule_tac ind, simp)
+ apply(case_tac a, simp_all add: tstep_0)
+ done
+ from g and this show "tstep (steps (st, l, r) (change_termi_state ap) stp)
+ (change_termi_state ap) = (st', l', r')"
+ apply(simp add: tstep.simps)
+ apply(case_tac "fetch ap a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
+ apply(subgoal_tac "fetch (change_termi_state ap) a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)
+ = (aa, st')", simp)
+ apply(simp add: change_termi_state_fetch)
+ done
+ qed
+qed
+
+lemma change_termi_state_fetch0:
+ "\<lbrakk>0 < a; a \<le> length ap div 2; t_correct ap; fetch ap a b = (aa, 0)\<rbrakk>
+ \<Longrightarrow> fetch (change_termi_state ap) a b = (aa, Suc (length ap div 2))"
+apply(case_tac b, auto simp: fetch.simps nth_of.simps change_termi_state.simps nth_map
+ split: if_splits block.splits)
+done
+
+lemma turing_change_termi_state:
+ "\<lbrakk>steps (Suc 0, l, r) ap stp = (0, l', r'); t_correct ap\<rbrakk>
+ \<Longrightarrow> \<exists> stp. steps (Suc 0, l, r) (change_termi_state ap) stp =
+ (Suc (length ap div 2), l', r')"
+apply(drule first_halt_point)
+apply(erule_tac exE)
+apply(rule_tac x = "Suc stp" in exI, simp add: tstep_red)
+apply(case_tac "steps (Suc 0, l, r) ap stp")
+apply(simp add: isS0_def change_termi_state_exec_in_range)
+apply(subgoal_tac "steps (Suc 0, l, r) (change_termi_state ap) stp = (a, b, c)", simp)
+apply(simp add: tstep.simps)
+apply(case_tac "fetch ap a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
+apply(subgoal_tac "fetch (change_termi_state ap) a
+ (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) = (aa, Suc (length ap div 2))", simp)
+apply(rule_tac ap = ap in change_termi_state_fetch0, simp_all)
+apply(rule_tac tp = "(l, r)" and l = b and r = c and stp = stp and A = ap in s_keep, simp_all)
+apply(simp add: change_termi_state_exec_in_range)
+done
+
+lemma t_twice_change_term_state:
+ "\<exists> stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp
+ = (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+using t_twice_correct[of ires rs n]
+apply(erule_tac exE)
+apply(erule_tac exE)
+apply(erule_tac exE)
+proof(drule_tac turing_change_termi_state)
+ fix stp ln rn
+ show "t_correct (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0))"
+ apply(rule_tac t_compiled_correct, simp_all)
+ apply(simp add: twice_ly_def)
+ done
+next
+ fix stp ln rn
+ show "\<exists>stp. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+ (change_termi_state (tm_of abc_twice @ tMp (Suc 0)
+ (start_of twice_ly (length abc_twice) - Suc 0))) stp =
+ (Suc (length (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) div 2),
+ Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) \<Longrightarrow>
+ \<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp =
+ (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ apply(erule_tac exE)
+ apply(simp add: t_twice_len_def t_twice_def)
+ apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
+ done
+qed
+
+lemma t_twice_append_pre:
+ "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp
+ = (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)
+ \<Longrightarrow> \<exists> stp>0. steps (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+ (t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @
+ ([(L, 1), (L, 1)] @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp
+ = (Suc (t_twice_len) + length t_wcode_main_first_part div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+proof(rule_tac t_tshift_lemma, simp_all add: t_twice_len_ge)
+ assume "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp =
+ (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ thus "0 < stp"
+ apply(case_tac stp, simp add: steps.simps t_twice_len_ge t_twice_len_def)
+ using t_twice_len_ge
+ apply(simp, simp)
+ done
+next
+ show "t_ncorrect t_wcode_main_first_part"
+ apply(simp add: t_ncorrect.simps t_wcode_main_first_part_def)
+ done
+next
+ show "t_ncorrect t_twice"
+ using length_tm_even[of abc_twice]
+ apply(auto simp: t_ncorrect.simps t_twice_def)
+ apply(arith)
+ done
+next
+ show "t_ncorrect ((L, Suc 0) # (L, Suc 0) #
+ abacus.tshift t_fourtimes (t_twice_len + 13) @ [(L, Suc 0), (L, Suc 0)])"
+ using length_tm_even[of abc_fourtimes]
+ apply(simp add: t_ncorrect.simps shift_length t_fourtimes_def)
+ apply arith
+ done
+qed
+
+lemma t_twice_append:
+ "\<exists> stp ln rn. steps (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+ (t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @
+ ([(L, 1), (L, 1)] @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp
+ = (Suc (t_twice_len) + length t_wcode_main_first_part div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ using t_twice_change_term_state[of ires rs n]
+ apply(erule_tac exE)
+ apply(erule_tac exE)
+ apply(erule_tac exE)
+ apply(drule_tac t_twice_append_pre)
+ apply(erule_tac exE)
+ apply(rule_tac x = stpa in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
+ apply(simp)
+ done
+
+lemma [simp]: "fetch t_wcode_main (Suc (t_twice_len + length t_wcode_main_first_part div 2)) Oc
+ = (L, Suc 0)"
+apply(subgoal_tac "length (t_twice) mod 2 = 0")
+apply(simp add: t_wcode_main_def nth_append fetch.simps t_wcode_main_first_part_def
+ nth_of.simps shift_length t_twice_len_def, auto)
+apply(simp add: t_twice_def)
+apply(subgoal_tac "length (tm_of abc_twice) mod 2 = 0")
+apply arith
+apply(rule_tac tm_even)
+done
+
+lemma wcode_jump1:
+ "\<exists> stp ln rn. steps (Suc (t_twice_len) + length t_wcode_main_first_part div 2,
+ Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>n\<^esup>)
+ t_wcode_main stp
+ = (Suc 0, Bk\<^bsup>ln\<^esup> @ Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+apply(rule_tac x = "Suc 0" in exI, rule_tac x = "m" in exI, rule_tac x = n in exI)
+apply(simp add: steps.simps tstep.simps exp_ind_def new_tape.simps)
+apply(case_tac m, simp, simp add: exp_ind_def)
+apply(simp add: exp_ind_def[THEN sym] exp_ind[THEN sym])
+done
+
+lemma wcode_main_first_part_len:
+ "length t_wcode_main_first_part = 24"
+ apply(simp add: t_wcode_main_first_part_def)
+ done
+
+lemma wcode_double_case:
+ shows "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+proof -
+ have "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (13, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ using wcode_double_case_first_correctness[of ires rs m n]
+ apply(simp)
+ apply(erule_tac exE)
+ apply(case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires,
+ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na",
+ auto simp: wcode_double_case_inv.simps
+ wcode_before_double.simps)
+ apply(rule_tac x = na in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
+ apply(simp)
+ done
+ from this obtain stpa lna rna where stp1:
+ "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
+ (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
+ have "\<exists> stp ln rn. steps (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp =
+ (13 + t_twice_len, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ using t_twice_append[of "Bk\<^bsup>lna\<^esup> @ Oc # ires" "Suc rs" rna]
+ apply(erule_tac exE)
+ apply(erule_tac exE)
+ apply(erule_tac exE)
+ apply(simp add: wcode_main_first_part_len)
+ apply(rule_tac x = stp in exI, rule_tac x = "ln + lna" in exI,
+ rule_tac x = rn in exI)
+ apply(simp add: t_wcode_main_def)
+ apply(simp add: exp_ind_def[THEN sym] exp_add[THEN sym])
+ done
+ from this obtain stpb lnb rnb where stp2:
+ "steps (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb =
+ (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>)" by blast
+ have "\<exists>stp ln rn. steps (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires,
+ Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stp =
+ (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ using wcode_jump1[of lnb "Oc # ires" "Suc rs" rnb]
+ apply(erule_tac exE)
+ apply(erule_tac exE)
+ apply(erule_tac exE)
+ apply(rule_tac x = stp in exI,
+ rule_tac x = ln in exI,
+ rule_tac x = rn in exI, simp add:wcode_main_first_part_len t_wcode_main_def)
+ apply(subgoal_tac "Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc # ires = Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires", simp)
+ apply(simp add: exp_ind_def[THEN sym] exp_ind[THEN sym])
+ apply(simp)
+ apply(case_tac lnb, simp, simp add: exp_ind_def[THEN sym] exp_ind)
+ done
+ from this obtain stpc lnc rnc where stp3:
+ "steps (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires,
+ Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stpc =
+ (Suc 0, Bk # Bk\<^bsup>lnc\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnc\<^esup>)"
+ by blast
+ from stp1 stp2 stp3 show "?thesis"
+ apply(rule_tac x = "stpa + stpb + stpc" in exI, rule_tac x = lnc in exI,
+ rule_tac x = rnc in exI)
+ apply(simp add: steps_add)
+ done
+qed
+
+
+(* Begin: fourtime_case*)
+fun wcode_on_left_moving_2_B :: "bin_inv_t"
+ where
+ "wcode_on_left_moving_2_B ires rs (l, r) =
+ (\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Oc # ires \<and>
+ r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr > Suc 0 \<and> mr > 0)"
+
+fun wcode_on_left_moving_2_O :: "bin_inv_t"
+ where
+ "wcode_on_left_moving_2_O ires rs (l, r) =
+ (\<exists> ln rn. l = Bk # Oc # ires \<and>
+ r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wcode_on_left_moving_2 :: "bin_inv_t"
+ where
+ "wcode_on_left_moving_2 ires rs (l, r) =
+ (wcode_on_left_moving_2_B ires rs (l, r) \<or>
+ wcode_on_left_moving_2_O ires rs (l, r))"
+
+fun wcode_on_checking_2 :: "bin_inv_t"
+ where
+ "wcode_on_checking_2 ires rs (l, r) =
+ (\<exists> ln rn. l = Oc#ires \<and>
+ r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wcode_goon_checking :: "bin_inv_t"
+ where
+ "wcode_goon_checking ires rs (l, r) =
+ (\<exists> ln rn. l = ires \<and>
+ r = Oc # Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wcode_right_move :: "bin_inv_t"
+ where
+ "wcode_right_move ires rs (l, r) =
+ (\<exists> ln rn. l = Oc # ires \<and>
+ r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wcode_erase2 :: "bin_inv_t"
+ where
+ "wcode_erase2 ires rs (l, r) =
+ (\<exists> ln rn. l = Bk # Oc # ires \<and>
+ tl r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wcode_on_right_moving_2 :: "bin_inv_t"
+ where
+ "wcode_on_right_moving_2 ires rs (l, r) =
+ (\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # ires \<and>
+ r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr > Suc 0)"
+
+fun wcode_goon_right_moving_2 :: "bin_inv_t"
+ where
+ "wcode_goon_right_moving_2 ires rs (l, r) =
+ (\<exists> ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+ r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = Suc rs)"
+
+fun wcode_backto_standard_pos_2_B :: "bin_inv_t"
+ where
+ "wcode_backto_standard_pos_2_B ires rs (l, r) =
+ (\<exists> ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+ r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wcode_backto_standard_pos_2_O :: "bin_inv_t"
+ where
+ "wcode_backto_standard_pos_2_O ires rs (l, r) =
+ (\<exists> ml mr ln rn. l = Oc\<^bsup>ml \<^esup>@ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+ r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr = (Suc (Suc rs)) \<and> mr > 0)"
+
+fun wcode_backto_standard_pos_2 :: "bin_inv_t"
+ where
+ "wcode_backto_standard_pos_2 ires rs (l, r) =
+ (wcode_backto_standard_pos_2_O ires rs (l, r) \<or>
+ wcode_backto_standard_pos_2_B ires rs (l, r))"
+
+fun wcode_before_fourtimes :: "bin_inv_t"
+ where
+ "wcode_before_fourtimes ires rs (l, r) =
+ (\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and>
+ r = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+declare wcode_on_left_moving_2_B.simps[simp del] wcode_on_left_moving_2.simps[simp del]
+ wcode_on_left_moving_2_O.simps[simp del] wcode_on_checking_2.simps[simp del]
+ wcode_goon_checking.simps[simp del] wcode_right_move.simps[simp del]
+ wcode_erase2.simps[simp del]
+ wcode_on_right_moving_2.simps[simp del] wcode_goon_right_moving_2.simps[simp del]
+ wcode_backto_standard_pos_2_B.simps[simp del] wcode_backto_standard_pos_2_O.simps[simp del]
+ wcode_backto_standard_pos_2.simps[simp del]
+
+lemmas wcode_fourtimes_invs =
+ wcode_on_left_moving_2_B.simps wcode_on_left_moving_2.simps
+ wcode_on_left_moving_2_O.simps wcode_on_checking_2.simps
+ wcode_goon_checking.simps wcode_right_move.simps
+ wcode_erase2.simps
+ wcode_on_right_moving_2.simps wcode_goon_right_moving_2.simps
+ wcode_backto_standard_pos_2_B.simps wcode_backto_standard_pos_2_O.simps
+ wcode_backto_standard_pos_2.simps
+
+fun wcode_fourtimes_case_inv :: "nat \<Rightarrow> bin_inv_t"
+ where
+ "wcode_fourtimes_case_inv st ires rs (l, r) =
+ (if st = Suc 0 then wcode_on_left_moving_2 ires rs (l, r)
+ else if st = Suc (Suc 0) then wcode_on_checking_2 ires rs (l, r)
+ else if st = 7 then wcode_goon_checking ires rs (l, r)
+ else if st = 8 then wcode_right_move ires rs (l, r)
+ else if st = 9 then wcode_erase2 ires rs (l, r)
+ else if st = 10 then wcode_on_right_moving_2 ires rs (l, r)
+ else if st = 11 then wcode_goon_right_moving_2 ires rs (l, r)
+ else if st = 12 then wcode_backto_standard_pos_2 ires rs (l, r)
+ else if st = t_twice_len + 14 then wcode_before_fourtimes ires rs (l, r)
+ else False)"
+
+declare wcode_fourtimes_case_inv.simps[simp del]
+
+fun wcode_fourtimes_case_state :: "t_conf \<Rightarrow> nat"
+ where
+ "wcode_fourtimes_case_state (st, l, r) = 13 - st"
+
+fun wcode_fourtimes_case_step :: "t_conf \<Rightarrow> nat"
+ where
+ "wcode_fourtimes_case_step (st, l, r) =
+ (if st = Suc 0 then length l
+ else if st = 9 then
+ (if hd r = Oc then 1
+ else 0)
+ else if st = 10 then length r
+ else if st = 11 then length r
+ else if st = 12 then length l
+ else 0)"
+
+fun wcode_fourtimes_case_measure :: "t_conf \<Rightarrow> nat \<times> nat"
+ where
+ "wcode_fourtimes_case_measure (st, l, r) =
+ (wcode_fourtimes_case_state (st, l, r),
+ wcode_fourtimes_case_step (st, l, r))"
+
+definition wcode_fourtimes_case_le :: "(t_conf \<times> t_conf) set"
+ where "wcode_fourtimes_case_le \<equiv> (inv_image lex_pair wcode_fourtimes_case_measure)"
+
+lemma wf_wcode_fourtimes_case_le[intro]: "wf wcode_fourtimes_case_le"
+by(auto intro:wf_inv_image simp: wcode_fourtimes_case_le_def)
+
+lemma [simp]: "fetch t_wcode_main (Suc (Suc 0)) Bk = (L, 7)"
+apply(simp add: t_wcode_main_def fetch.simps
+ t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 7 Oc = (R, 8)"
+apply(simp add: t_wcode_main_def fetch.simps
+ t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 8 Bk = (R, 9)"
+apply(simp add: t_wcode_main_def fetch.simps
+ t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 9 Bk = (R, 10)"
+apply(simp add: t_wcode_main_def fetch.simps
+ t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 9 Oc = (W0, 9)"
+apply(simp add: t_wcode_main_def fetch.simps
+ t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 10 Bk = (R, 10)"
+apply(simp add: t_wcode_main_def fetch.simps
+ t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 10 Oc = (R, 11)"
+apply(simp add: t_wcode_main_def fetch.simps
+ t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 11 Bk = (W1, 12)"
+apply(simp add: t_wcode_main_def fetch.simps
+ t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 11 Oc = (R, 11)"
+apply(simp add: t_wcode_main_def fetch.simps
+ t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 12 Oc = (L, 12)"
+apply(simp add: t_wcode_main_def fetch.simps
+ t_wcode_main_first_part_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_main 12 Bk = (R, t_twice_len + 14)"
+apply(simp add: t_wcode_main_def fetch.simps
+ t_wcode_main_first_part_def nth_of.simps)
+done
+
+
+lemma [simp]: "wcode_on_left_moving_2 ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_on_checking_2 ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_goon_checking ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_right_move ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_erase2 ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_on_right_moving_2 ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs exponent_def)
+done
+
+lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, []) = False"
+apply(auto simp: wcode_fourtimes_invs exponent_def)
+done
+
+lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wcode_fourtimes_invs, auto)
+done
+
+lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \<Longrightarrow> wcode_on_left_moving_2 ires rs (tl b, hd b # Bk # list)"
+apply(simp only: wcode_fourtimes_invs)
+apply(erule_tac disjE)
+apply(erule_tac exE)+
+apply(case_tac ml, simp)
+apply(rule_tac x = "mr - (Suc (Suc 0))" in exI, rule_tac x = rn in exI, simp)
+apply(case_tac mr, simp, case_tac nat, simp, simp add: exp_ind)
+apply(rule_tac disjI1)
+apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = rn in exI,
+ simp add: exp_ind_def)
+apply(simp)
+done
+
+lemma [simp]: "wcode_on_checking_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_on_checking_2 ires rs (b, Bk # list)
+ \<Longrightarrow> wcode_goon_checking ires rs (tl b, hd b # Bk # list)"
+apply(simp only: wcode_fourtimes_invs)
+apply(auto)
+done
+
+lemma [simp]: "wcode_goon_checking ires rs (b, Bk # list) = False"
+apply(simp add: wcode_fourtimes_invs)
+done
+
+lemma [simp]: " wcode_right_move ires rs (b, Bk # list) \<Longrightarrow> b\<noteq> []"
+apply(simp add: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_right_move ires rs (b, Bk # list) \<Longrightarrow> wcode_erase2 ires rs (Bk # b, list)"
+apply(auto simp:wcode_fourtimes_invs )
+apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
+done
+
+lemma [simp]: "wcode_erase2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_erase2 ires rs (b, Bk # list) \<Longrightarrow> wcode_on_right_moving_2 ires rs (Bk # b, list)"
+apply(auto simp:wcode_fourtimes_invs )
+apply(rule_tac x = "Suc (Suc 0)" in exI, simp add: exp_ind)
+apply(rule_tac x = "Suc (Suc ln)" in exI, simp add: exp_ind, auto)
+done
+
+lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(auto simp:wcode_fourtimes_invs )
+done
+
+lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list)
+ \<Longrightarrow> wcode_on_right_moving_2 ires rs (Bk # b, list)"
+apply(auto simp: wcode_fourtimes_invs)
+apply(rule_tac x = "Suc ml" in exI, simp add: exp_ind_def)
+apply(rule_tac x = "mr - 1" in exI, case_tac mr, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow>
+ wcode_backto_standard_pos_2 ires rs (b, Oc # list)"
+apply(simp add: wcode_fourtimes_invs, auto)
+apply(rule_tac x = ml in exI, auto)
+apply(rule_tac x = "Suc 0" in exI, simp)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(rule_tac x = "rn - 1" in exI, simp)
+apply(case_tac rn, simp, simp add: exp_ind_def)
+done
+
+lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wcode_fourtimes_invs, auto)
+done
+
+lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wcode_fourtimes_invs, auto)
+done
+
+lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \<Longrightarrow>
+ wcode_on_checking_2 ires rs (tl b, hd b # Oc # list)"
+apply(auto simp: wcode_fourtimes_invs)
+apply(case_tac [!] mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \<Longrightarrow> b \<noteq> []"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \<Longrightarrow>
+ wcode_backto_standard_pos_2 ires rs (b, [Oc])"
+apply(simp only: wcode_fourtimes_invs)
+apply(erule_tac exE)+
+apply(rule_tac disjI1)
+apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI,
+ rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
+apply(case_tac mr, simp, simp add: exp_ind_def)
+done
+
+lemma "wcode_backto_standard_pos_2 ires rs (b, Bk # list)
+ \<Longrightarrow> (\<exists>ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \<and> (\<exists>rn. list = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+apply(auto simp: wcode_fourtimes_invs)
+apply(case_tac [!] mr, auto simp: exp_ind_def)
+done
+
+
+lemma [simp]: "wcode_on_checking_2 ires rs (b, Oc # list) \<Longrightarrow> False"
+apply(simp add: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_goon_checking ires rs (b, Oc # list) \<Longrightarrow>
+ (b = [] \<longrightarrow> wcode_right_move ires rs ([Oc], list)) \<and>
+ (b \<noteq> [] \<longrightarrow> wcode_right_move ires rs (Oc # b, list))"
+apply(simp only: wcode_fourtimes_invs)
+apply(erule_tac exE)+
+apply(auto)
+done
+
+lemma [simp]: "wcode_right_move ires rs (b, Oc # list) = False"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: " wcode_erase2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_erase2 ires rs (b, Oc # list)
+ \<Longrightarrow> wcode_erase2 ires rs (b, Bk # list)"
+apply(auto simp: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp only: wcode_fourtimes_invs)
+apply(auto)
+done
+
+lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Oc # list)
+ \<Longrightarrow> wcode_goon_right_moving_2 ires rs (Oc # b, list)"
+apply(auto simp: wcode_fourtimes_invs)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(rule_tac x = "Suc 0" in exI, auto)
+apply(rule_tac x = "ml - 2" in exI)
+apply(case_tac ml, simp, case_tac nat, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp only:wcode_fourtimes_invs, auto)
+done
+
+lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list)
+ \<Longrightarrow> (\<exists>ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \<and> (\<exists>rn. list = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+apply(simp add: wcode_fourtimes_invs, auto)
+apply(case_tac [!] mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wcode_on_checking_2 ires rs (b, Oc # list) = False"
+apply(simp add: wcode_fourtimes_invs)
+done
+
+lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow>
+ wcode_goon_right_moving_2 ires rs (Oc # b, list)"
+apply(simp only:wcode_fourtimes_invs, auto)
+apply(rule_tac x = "Suc ml" in exI, auto simp: exp_ind_def)
+apply(rule_tac x = "mr - 1" in exI)
+apply(case_tac mr, case_tac rn, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp only: wcode_fourtimes_invs, auto)
+done
+
+lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list)
+ \<Longrightarrow> wcode_backto_standard_pos_2 ires rs (tl b, hd b # Oc # list)"
+apply(simp only: wcode_fourtimes_invs)
+apply(erule_tac disjE)
+apply(erule_tac exE)+
+apply(case_tac ml, simp)
+apply(rule_tac disjI2)
+apply(rule_tac conjI, rule_tac x = ln in exI, simp)
+apply(rule_tac x = rn in exI, simp)
+apply(rule_tac disjI1)
+apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI,
+ rule_tac x = ln in exI, rule_tac x = rn in exI, simp add: exp_ind_def)
+apply(simp)
+done
+
+lemma wcode_fourtimes_case_first_correctness:
+ shows "let P = (\<lambda> (st, l, r). st = t_twice_len + 14) in
+ let Q = (\<lambda> (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r)) in
+ let f = (\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp) in
+ \<exists> n .P (f n) \<and> Q (f (n::nat))"
+proof -
+ let ?P = "(\<lambda> (st, l, r). st = t_twice_len + 14)"
+ let ?Q = "(\<lambda> (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r))"
+ let ?f = "(\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)"
+ have "\<exists> n . ?P (?f n) \<and> ?Q (?f (n::nat))"
+ proof(rule_tac halt_lemma2)
+ show "wf wcode_fourtimes_case_le"
+ by auto
+ next
+ show "\<forall> na. \<not> ?P (?f na) \<and> ?Q (?f na) \<longrightarrow>
+ ?Q (?f (Suc na)) \<and> (?f (Suc na), ?f na) \<in> wcode_fourtimes_case_le"
+ apply(rule_tac allI,
+ case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na", simp,
+ rule_tac impI)
+ apply(simp add: tstep_red tstep.simps, case_tac c, simp, case_tac [2] aa, simp_all)
+
+ apply(simp_all add: wcode_fourtimes_case_inv.simps new_tape.simps
+ wcode_fourtimes_case_le_def lex_pair_def split: if_splits)
+ done
+ next
+ show "?Q (?f 0)"
+ apply(simp add: steps.simps wcode_fourtimes_case_inv.simps)
+ apply(simp add: wcode_on_left_moving_2.simps wcode_on_left_moving_2_B.simps
+ wcode_on_left_moving_2_O.simps)
+ apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def)
+ apply(rule_tac x ="Suc 0" in exI, auto)
+ done
+ next
+ show "\<not> ?P (?f 0)"
+ apply(simp add: steps.simps)
+ done
+ qed
+ thus "?thesis"
+ apply(erule_tac exE, simp)
+ done
+qed
+
+definition t_fourtimes_len :: "nat"
+ where
+ "t_fourtimes_len = (length t_fourtimes div 2)"
+
+lemma t_fourtimes_len_gr: "t_fourtimes_len > 0"
+apply(simp add: t_fourtimes_len_def t_fourtimes_def)
+done
+
+lemma t_fourtimes_correct:
+ "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+ (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) stp =
+ (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+proof(case_tac "rec_ci rec_fourtimes")
+ fix a b c
+ assume h: "rec_ci rec_fourtimes = (a, b, c)"
+ have "\<exists>stp m l. steps (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<^bsup>n\<^esup>) (tm_of abc_fourtimes @ tMp (Suc 0)
+ (start_of fourtimes_ly (length abc_fourtimes) - 1)) stp = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4*rs)\<^esup> @ Bk\<^bsup>l\<^esup>)"
+ proof(rule_tac t_compiled_by_rec)
+ show "rec_ci rec_fourtimes = (a, b, c)" by (simp add: h)
+ next
+ show "rec_calc_rel rec_fourtimes [rs] (4 * rs)"
+ using prime_rel_exec_eq [of rec_fourtimes "[rs]" "4 * rs"]
+ apply(subgoal_tac "primerec rec_fourtimes (length [rs])")
+ apply(simp_all add: rec_fourtimes_def rec_exec.simps)
+ apply(auto)
+ apply(simp only: Nat.One_nat_def[THEN sym], auto)
+ done
+ next
+ show "length [rs] = Suc 0" by simp
+ next
+ show "layout_of (a [+] dummy_abc (Suc 0)) = layout_of (a [+] dummy_abc (Suc 0))"
+ by simp
+ next
+ show "start_of fourtimes_ly (length abc_fourtimes) =
+ start_of (layout_of (a [+] dummy_abc (Suc 0))) (length (a [+] dummy_abc (Suc 0)))"
+ using h
+ apply(simp add: fourtimes_ly_def abc_fourtimes_def)
+ done
+ next
+ show "tm_of abc_fourtimes = tm_of (a [+] dummy_abc (Suc 0))"
+ using h
+ apply(simp add: abc_fourtimes_def)
+ done
+ qed
+ thus "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+ (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) stp =
+ (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
+ done
+qed
+
+lemma t_fourtimes_change_term_state:
+ "\<exists> stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp
+ = (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+using t_fourtimes_correct[of ires rs n]
+apply(erule_tac exE)
+apply(erule_tac exE)
+apply(erule_tac exE)
+proof(drule_tac turing_change_termi_state)
+ fix stp ln rn
+ show "t_correct (tm_of abc_fourtimes @ tMp (Suc 0)
+ (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))"
+ apply(rule_tac t_compiled_correct, auto simp: fourtimes_ly_def)
+ done
+next
+ fix stp ln rn
+ show "\<exists>stp. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+ (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0)
+ (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) stp =
+ (Suc (length (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly
+ (length abc_fourtimes) - Suc 0)) div 2), Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) \<Longrightarrow>
+ \<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp =
+ (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ apply(erule_tac exE)
+ apply(simp add: t_fourtimes_len_def t_fourtimes_def)
+ apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI, simp)
+ done
+qed
+
+lemma t_fourtimes_append_pre:
+ "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp
+ = (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)
+ \<Longrightarrow> \<exists> stp>0. steps (Suc 0 + length (t_wcode_main_first_part @
+ tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2,
+ Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+ ((t_wcode_main_first_part @
+ tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) @
+ tshift t_fourtimes (length (t_wcode_main_first_part @
+ tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2) @ ([(L, 1), (L, 1)])) stp
+ = (Suc t_fourtimes_len + length (t_wcode_main_first_part @
+ tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2,
+ Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+proof(rule_tac t_tshift_lemma, auto)
+ assume "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp =
+ (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ thus "0 < stp"
+ using t_fourtimes_len_gr
+ apply(case_tac stp, simp_all add: steps.simps)
+ done
+next
+ show "Suc 0 \<le> length t_fourtimes div 2"
+ apply(simp add: t_fourtimes_def shift_length tMp.simps)
+ done
+next
+ show "t_ncorrect (t_wcode_main_first_part @
+ abacus.tshift t_twice (length t_wcode_main_first_part div 2) @
+ [(L, Suc 0), (L, Suc 0)])"
+ apply(simp add: t_ncorrect.simps t_wcode_main_first_part_def shift_length
+ t_twice_def)
+ using tm_even[of abc_twice]
+ by arith
+next
+ show "t_ncorrect t_fourtimes"
+ apply(simp add: t_fourtimes_def steps.simps t_ncorrect.simps)
+ using tm_even[of abc_fourtimes]
+ by arith
+next
+ show "t_ncorrect [(L, Suc 0), (L, Suc 0)]"
+ apply(simp add: t_ncorrect.simps)
+ done
+qed
+
+lemma [simp]: "length t_wcode_main_first_part = 24"
+apply(simp add: t_wcode_main_first_part_def)
+done
+
+lemma [simp]: "(26 + length t_twice) div 2 = (length t_twice) div 2 + 13"
+using tm_even[of abc_twice]
+apply(simp add: t_twice_def)
+done
+
+lemma [simp]: "((26 + length (abacus.tshift t_twice 12)) div 2)
+ = (length (abacus.tshift t_twice 12) div 2 + 13)"
+using tm_even[of abc_twice]
+apply(simp add: t_twice_def)
+done
+
+lemma [simp]: "t_twice_len + 14 = 14 + length (abacus.tshift t_twice 12) div 2"
+using tm_even[of abc_twice]
+apply(simp add: t_twice_def t_twice_len_def shift_length)
+done
+
+lemma t_fourtimes_append:
+ "\<exists> stp ln rn.
+ steps (Suc 0 + length (t_wcode_main_first_part @ tshift t_twice
+ (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2,
+ Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+ ((t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @
+ [(L, 1), (L, 1)]) @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)]) stp
+ = (Suc t_fourtimes_len + length (t_wcode_main_first_part @ tshift t_twice
+ (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires,
+ Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ using t_fourtimes_change_term_state[of ires rs n]
+ apply(erule_tac exE)
+ apply(erule_tac exE)
+ apply(erule_tac exE)
+ apply(drule_tac t_fourtimes_append_pre)
+ apply(erule_tac exE)
+ apply(rule_tac x = stpa in exI, rule_tac x = ln in exI, rule_tac x = rn in exI)
+ apply(simp add: t_twice_len_def shift_length)
+ done
+
+lemma t_wcode_main_len: "length t_wcode_main = length t_twice + length t_fourtimes + 28"
+apply(simp add: t_wcode_main_def shift_length)
+done
+
+lemma [simp]: "fetch t_wcode_main (14 + length t_twice div 2 + t_fourtimes_len) b
+ = (L, Suc 0)"
+using tm_even[of "abc_twice"] tm_even[of "abc_fourtimes"]
+apply(case_tac b)
+apply(simp_all only: fetch.simps)
+apply(auto simp: nth_of.simps t_wcode_main_len t_twice_len_def
+ t_fourtimes_def t_twice_def t_fourtimes_def t_fourtimes_len_def)
+apply(auto simp: t_wcode_main_def t_wcode_main_first_part_def shift_length t_twice_def nth_append
+ t_fourtimes_def)
+done
+
+lemma wcode_jump2:
+ "\<exists> stp ln rn. steps (t_twice_len + 14 + t_fourtimes_len
+ , Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4 * rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stp =
+ (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4 * rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+apply(rule_tac x = "Suc 0" in exI)
+apply(simp add: steps.simps shift_length)
+apply(rule_tac x = lnb in exI, rule_tac x = rnb in exI)
+apply(simp add: tstep.simps new_tape.simps)
+done
+
+lemma wcode_fourtimes_case:
+ shows "\<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+proof -
+ have "\<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (t_twice_len + 14, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ using wcode_fourtimes_case_first_correctness[of ires rs m n]
+ apply(simp add: wcode_fourtimes_case_inv.simps, auto)
+ apply(rule_tac x = na in exI, rule_tac x = ln in exI,
+ rule_tac x = rn in exI)
+ apply(simp)
+ done
+ from this obtain stpa lna rna where stp1:
+ "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
+ (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
+ have "\<exists>stp ln rn. steps (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)
+ t_wcode_main stp =
+ (t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ using t_fourtimes_append[of " Bk\<^bsup>lna\<^esup> @ Oc # ires" "rs + 1" rna]
+ apply(erule_tac exE)
+ apply(erule_tac exE)
+ apply(erule_tac exE)
+ apply(simp add: t_wcode_main_def)
+ apply(rule_tac x = stp in exI,
+ rule_tac x = "ln + lna" in exI,
+ rule_tac x = rn in exI, simp)
+ apply(simp add: exp_ind_def[THEN sym] exp_add[THEN sym])
+ done
+ from this obtain stpb lnb rnb where stp2:
+ "steps (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)
+ t_wcode_main stpb =
+ (t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>)"
+ by blast
+ have "\<exists>stp ln rn. steps (t_twice_len + 14 + t_fourtimes_len,
+ Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>)
+ t_wcode_main stp =
+ (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ apply(rule wcode_jump2)
+ done
+ from this obtain stpc lnc rnc where stp3:
+ "steps (t_twice_len + 14 + t_fourtimes_len,
+ Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>)
+ t_wcode_main stpc =
+ (Suc 0, Bk # Bk\<^bsup>lnc\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnc\<^esup>)"
+ by blast
+ from stp1 stp2 stp3 show "?thesis"
+ apply(rule_tac x = "stpa + stpb + stpc" in exI,
+ rule_tac x = lnc in exI, rule_tac x = rnc in exI)
+ apply(simp add: steps_add)
+ done
+qed
+
+(**********************************************************)
+
+fun wcode_on_left_moving_3_B :: "bin_inv_t"
+ where
+ "wcode_on_left_moving_3_B ires rs (l, r) =
+ (\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Bk # ires \<and>
+ r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr > Suc 0 \<and> mr > 0 )"
+
+fun wcode_on_left_moving_3_O :: "bin_inv_t"
+ where
+ "wcode_on_left_moving_3_O ires rs (l, r) =
+ (\<exists> ln rn. l = Bk # Bk # ires \<and>
+ r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wcode_on_left_moving_3 :: "bin_inv_t"
+ where
+ "wcode_on_left_moving_3 ires rs (l, r) =
+ (wcode_on_left_moving_3_B ires rs (l, r) \<or>
+ wcode_on_left_moving_3_O ires rs (l, r))"
+
+fun wcode_on_checking_3 :: "bin_inv_t"
+ where
+ "wcode_on_checking_3 ires rs (l, r) =
+ (\<exists> ln rn. l = Bk # ires \<and>
+ r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wcode_goon_checking_3 :: "bin_inv_t"
+ where
+ "wcode_goon_checking_3 ires rs (l, r) =
+ (\<exists> ln rn. l = ires \<and>
+ r = Bk # Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wcode_stop :: "bin_inv_t"
+ where
+ "wcode_stop ires rs (l, r) =
+ (\<exists> ln rn. l = Bk # ires \<and>
+ r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wcode_halt_case_inv :: "nat \<Rightarrow> bin_inv_t"
+ where
+ "wcode_halt_case_inv st ires rs (l, r) =
+ (if st = 0 then wcode_stop ires rs (l, r)
+ else if st = Suc 0 then wcode_on_left_moving_3 ires rs (l, r)
+ else if st = Suc (Suc 0) then wcode_on_checking_3 ires rs (l, r)
+ else if st = 7 then wcode_goon_checking_3 ires rs (l, r)
+ else False)"
+
+fun wcode_halt_case_state :: "t_conf \<Rightarrow> nat"
+ where
+ "wcode_halt_case_state (st, l, r) =
+ (if st = 1 then 5
+ else if st = Suc (Suc 0) then 4
+ else if st = 7 then 3
+ else 0)"
+
+fun wcode_halt_case_step :: "t_conf \<Rightarrow> nat"
+ where
+ "wcode_halt_case_step (st, l, r) =
+ (if st = 1 then length l
+ else 0)"
+
+fun wcode_halt_case_measure :: "t_conf \<Rightarrow> nat \<times> nat"
+ where
+ "wcode_halt_case_measure (st, l, r) =
+ (wcode_halt_case_state (st, l, r),
+ wcode_halt_case_step (st, l, r))"
+
+definition wcode_halt_case_le :: "(t_conf \<times> t_conf) set"
+ where "wcode_halt_case_le \<equiv> (inv_image lex_pair wcode_halt_case_measure)"
+
+lemma wf_wcode_halt_case_le[intro]: "wf wcode_halt_case_le"
+by(auto intro:wf_inv_image simp: wcode_halt_case_le_def)
+
+declare wcode_on_left_moving_3_B.simps[simp del] wcode_on_left_moving_3_O.simps[simp del]
+ wcode_on_checking_3.simps[simp del] wcode_goon_checking_3.simps[simp del]
+ wcode_on_left_moving_3.simps[simp del] wcode_stop.simps[simp del]
+
+lemmas wcode_halt_invs =
+ wcode_on_left_moving_3_B.simps wcode_on_left_moving_3_O.simps
+ wcode_on_checking_3.simps wcode_goon_checking_3.simps
+ wcode_on_left_moving_3.simps wcode_stop.simps
+
+lemma [simp]: "fetch t_wcode_main 7 Bk = (R, 0)"
+apply(simp add: fetch.simps t_wcode_main_def nth_append nth_of.simps
+ t_wcode_main_first_part_def)
+done
+
+lemma [simp]: "wcode_on_left_moving_3 ires rs (b, []) = False"
+apply(simp only: wcode_halt_invs)
+apply(simp add: exp_ind_def)
+done
+
+lemma [simp]: "wcode_on_checking_3 ires rs (b, []) = False"
+apply(simp add: wcode_halt_invs)
+done
+
+lemma [simp]: "wcode_goon_checking_3 ires rs (b, []) = False"
+apply(simp add: wcode_halt_invs)
+done
+
+lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Bk # list)
+ \<Longrightarrow> wcode_on_left_moving_3 ires rs (tl b, hd b # Bk # list)"
+apply(simp only: wcode_halt_invs)
+apply(erule_tac disjE)
+apply(erule_tac exE)+
+apply(case_tac ml, simp)
+apply(rule_tac x = "mr - 2" in exI, rule_tac x = rn in exI)
+apply(case_tac mr, simp, simp add: exp_ind, simp add: exp_ind[THEN sym])
+apply(rule_tac disjI1)
+apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI,
+ rule_tac x = rn in exI, simp add: exp_ind_def)
+apply(simp)
+done
+
+lemma [simp]: "wcode_goon_checking_3 ires rs (b, Bk # list) \<Longrightarrow>
+ (b = [] \<longrightarrow> wcode_stop ires rs ([Bk], list)) \<and>
+ (b \<noteq> [] \<longrightarrow> wcode_stop ires rs (Bk # b, list))"
+apply(auto simp: wcode_halt_invs)
+done
+
+lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(auto simp: wcode_halt_invs)
+done
+
+lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Oc # list) \<Longrightarrow>
+ wcode_on_checking_3 ires rs (tl b, hd b # Oc # list)"
+apply(simp add:wcode_halt_invs, auto)
+apply(case_tac [!] mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wcode_on_checking_3 ires rs (b, Oc # list) = False"
+apply(auto simp: wcode_halt_invs)
+done
+
+lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wcode_halt_invs, auto)
+done
+
+
+lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(auto simp: wcode_halt_invs)
+done
+
+lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \<Longrightarrow>
+ wcode_goon_checking_3 ires rs (tl b, hd b # Bk # list)"
+apply(auto simp: wcode_halt_invs)
+done
+
+lemma [simp]: "wcode_goon_checking_3 ires rs (b, Oc # list) = False"
+apply(simp add: wcode_goon_checking_3.simps)
+done
+
+lemma t_halt_case_correctness:
+shows "let P = (\<lambda> (st, l, r). st = 0) in
+ let Q = (\<lambda> (st, l, r). wcode_halt_case_inv st ires rs (l, r)) in
+ let f = (\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp) in
+ \<exists> n .P (f n) \<and> Q (f (n::nat))"
+proof -
+ let ?P = "(\<lambda> (st, l, r). st = 0)"
+ let ?Q = "(\<lambda> (st, l, r). wcode_halt_case_inv st ires rs (l, r))"
+ let ?f = "(\<lambda> stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)"
+ have "\<exists> n. ?P (?f n) \<and> ?Q (?f (n::nat))"
+ proof(rule_tac halt_lemma2)
+ show "wf wcode_halt_case_le" by auto
+ next
+ show "\<forall> na. \<not> ?P (?f na) \<and> ?Q (?f na) \<longrightarrow>
+ ?Q (?f (Suc na)) \<and> (?f (Suc na), ?f na) \<in> wcode_halt_case_le"
+ apply(rule_tac allI, rule_tac impI, case_tac "?f na")
+ apply(simp add: tstep_red tstep.simps)
+ apply(case_tac c, simp, case_tac [2] aa)
+ apply(simp_all split: if_splits add: new_tape.simps wcode_halt_case_le_def lex_pair_def)
+ done
+ next
+ show "?Q (?f 0)"
+ apply(simp add: steps.simps wcode_halt_invs)
+ apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def)
+ apply(rule_tac x = "Suc 0" in exI, auto)
+ done
+ next
+ show "\<not> ?P (?f 0)"
+ apply(simp add: steps.simps)
+ done
+ qed
+ thus "?thesis"
+ apply(auto)
+ done
+qed
+
+declare wcode_halt_case_inv.simps[simp del]
+lemma [intro]: "\<exists> xs. (<rev list @ [aa::nat]> :: block list) = Oc # xs"
+apply(case_tac "rev list", simp)
+apply(simp add: tape_of_nat_abv tape_of_nat_list.simps exp_ind_def)
+apply(case_tac list, simp, simp)
+done
+
+lemma wcode_halt_case:
+ "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+ t_wcode_main stp = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ using t_halt_case_correctness[of ires rs m n]
+apply(simp)
+apply(erule_tac exE)
+apply(case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na")
+apply(auto simp: wcode_halt_case_inv.simps wcode_stop.simps)
+apply(rule_tac x = na in exI, rule_tac x = ln in exI,
+ rule_tac x = rn in exI, simp)
+done
+
+lemma bl_bin_one: "bl_bin [Oc] = Suc 0"
+apply(simp add: bl_bin.simps)
+done
+
+lemma t_wcode_main_lemma_pre:
+ "\<lbrakk>args \<noteq> []; lm = <args::nat list>\<rbrakk> \<Longrightarrow>
+ \<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main
+ stp
+ = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2^(length lm - 1) \<^esup> @ Bk\<^bsup>rn\<^esup>)"
+proof(induct "length args" arbitrary: args lm rs m n, simp)
+ fix x args lm rs m n
+ assume ind:
+ "\<And>args lm rs m n.
+ \<lbrakk>x = length args; (args::nat list) \<noteq> []; lm = <args>\<rbrakk>
+ \<Longrightarrow> \<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+ and h: "Suc x = length args" "(args::nat list) \<noteq> []" "lm = <args>"
+ from h have "\<exists> (a::nat) xs. args = xs @ [a]"
+ apply(rule_tac x = "last args" in exI)
+ apply(rule_tac x = "butlast args" in exI, auto)
+ done
+ from this obtain a xs where "args = xs @ [a]" by blast
+ from h and this show
+ "\<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ proof(case_tac "xs::nat list", simp)
+ show "\<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ proof(induct "a" arbitrary: m n rs ires, simp)
+ fix m n rs ires
+ show "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>)
+ t_wcode_main stp = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin [Oc] + rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ apply(simp add: bl_bin_one)
+ apply(rule_tac wcode_halt_case)
+ done
+ next
+ fix a m n rs ires
+ assume ind2:
+ "\<And>m n rs ires.
+ \<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ show "\<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<Suc a>) + rs * 2 ^ Suc a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ proof -
+ have "\<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ apply(simp add: tape_of_nat)
+ using wcode_double_case[of m "Oc\<^bsup>a\<^esup> @ Bk # Bk # ires" rs n]
+ apply(simp add: exp_ind_def)
+ done
+ from this obtain stpa lna rna where stp1:
+ "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
+ (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
+ moreover have
+ "\<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp =
+ (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + (2*rs + 2) * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ using ind2[of lna ires "2*rs + 2" rna] by simp
+ from this obtain stpb lnb rnb where stp2:
+ "steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb =
+ (0, Bk # ires, Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + (2*rs + 2) * 2 ^ a\<^esup> @ Bk\<^bsup>rnb\<^esup>)"
+ by blast
+ from stp1 and stp2 show "?thesis"
+ apply(rule_tac x = "stpa + stpb" in exI,
+ rule_tac x = lnb in exI, rule_tac x = rnb in exI, simp)
+ apply(simp add: steps_add bl_bin_nat_Suc exponent_def)
+ done
+ qed
+ qed
+ next
+ fix aa list
+ assume g: "Suc x = length args" "args \<noteq> []" "lm = <args>" "args = xs @ [a::nat]" "xs = (aa::nat) # list"
+ thus "\<exists>stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ proof(induct a arbitrary: m n rs args lm, simp_all add: tape_of_nl_rev,
+ simp only: tape_of_nl_cons_app1, simp)
+ fix m n rs args lm
+ have "\<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # rev (<(aa::nat) # list>) @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ rev (<aa # list>) @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ proof(simp add: tape_of_nl_rev)
+ have "\<exists> xs. (<rev list @ [aa]>) = Oc # xs" by auto
+ from this obtain xs where "(<rev list @ [aa]>) = Oc # xs" ..
+ thus "\<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ <rev list @ [aa]> @ Bk # Bk # ires, Bk # Oc\<^bsup>5 + 4 * rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ apply(simp)
+ using wcode_fourtimes_case[of m "xs @ Bk # Bk # ires" rs n]
+ apply(simp)
+ done
+ qed
+ from this obtain stpa lna rna where stp1:
+ "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # rev (<aa # list>) @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa =
+ (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<aa # list>) @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
+ from g have
+ "\<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<(aa::nat) # list>) @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp = (0, Bk # ires,
+ Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<aa#list>)+ (4*rs + 4) * 2^(length (<aa#list>) - 1) \<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ apply(rule_tac args = "(aa::nat)#list" in ind, simp_all)
+ done
+ from this obtain stpb lnb rnb where stp2:
+ "steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<(aa::nat) # list>) @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb = (0, Bk # ires,
+ Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<aa#list>)+ (4*rs + 4) * 2^(length (<aa#list>) - 1) \<^esup> @ Bk\<^bsup>rnb\<^esup>)"
+ by blast
+ from stp1 and stp2 and h
+ show "\<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
+ Bk # Oc\<^bsup>bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [0]>) + rs * (2 * 2 ^ (aa + length (<list @ [0]>)))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ apply(rule_tac x = "stpa + stpb" in exI, rule_tac x = lnb in exI,
+ rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_rev)
+ done
+ next
+ fix ab m n rs args lm
+ assume ind2:
+ "\<And> m n rs args lm.
+ \<lbrakk>lm = <aa # list @ [ab]>; args = aa # list @ [ab]\<rbrakk>
+ \<Longrightarrow> \<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ <ab # rev list @ [aa]> @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
+ Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]>) + rs * 2 ^ (length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ and k: "args = aa # list @ [Suc ab]" "lm = <aa # list @ [Suc ab]>"
+ show "\<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ <Suc ab # rev list @ [aa]> @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (0, Bk # ires,Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
+ Bk # Oc\<^bsup>bl_bin (<aa # list @ [Suc ab]>) + rs * 2 ^ (length (<aa # list @ [Suc ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ proof(simp add: tape_of_nl_cons_app1)
+ have "\<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+ Bk # Oc # Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp
+ = (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc\<^bsup>Suc ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ using wcode_double_case[of m "Oc\<^bsup>ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires"
+ rs n]
+ apply(simp add: exp_ind_def)
+ done
+ from this obtain stpa lna rna where stp1:
+ "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+ Bk # Oc # Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa
+ = (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ Oc\<^bsup>Suc ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast
+ from k have
+ "\<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ <ab # rev list @ [aa]> @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp
+ = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk #
+ Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]> ) + (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ apply(rule_tac ind2, simp_all)
+ done
+ from this obtain stpb lnb rnb where stp2:
+ "steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ <ab # rev list @ [aa]> @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb
+ = (0, Bk # ires, Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk #
+ Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]> ) + (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rnb\<^esup>)"
+ by blast
+ from stp1 and stp2 show
+ "\<exists>stp ln rn.
+ steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires,
+ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp =
+ (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk #
+ Oc\<^bsup>bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [Suc ab]>) + rs * (2 * 2 ^ (aa + length (<list @ [Suc ab]>)))\<^esup>
+ @ Bk\<^bsup>rn\<^esup>)"
+ apply(rule_tac x = "stpa + stpb" in exI, rule_tac x = lnb in exI,
+ rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_cons_app1 exp_ind_def)
+ done
+ qed
+ qed
+ qed
+ qed
+
+
+
+(* turing_shift can be used*)
+term t_wcode_main
+definition t_wcode_prepare :: "tprog"
+ where
+ "t_wcode_prepare \<equiv>
+ [(W1, 2), (L, 1), (L, 3), (R, 2), (R, 4), (W0, 3),
+ (R, 4), (R, 5), (R, 6), (R, 5), (R, 7), (R, 5),
+ (W1, 7), (L, 0)]"
+
+fun wprepare_add_one :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_add_one m lm (l, r) =
+ (\<exists> rn. l = [] \<and>
+ (r = <m # lm> @ Bk\<^bsup>rn\<^esup> \<or>
+ r = Bk # <m # lm> @ Bk\<^bsup>rn\<^esup>))"
+
+fun wprepare_goto_first_end :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_goto_first_end m lm (l, r) =
+ (\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> \<and>
+ r = Oc\<^bsup>mr\<^esup> @ Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr = Suc (Suc m))"
+
+fun wprepare_erase :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_erase m lm (l, r) =
+ (\<exists> rn. l = Oc\<^bsup>Suc m\<^esup> \<and>
+ tl r = Bk # <lm> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wprepare_goto_start_pos_B :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_goto_start_pos_B m lm (l, r) =
+ (\<exists> rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ r = Bk # <lm> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wprepare_goto_start_pos_O :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_goto_start_pos_O m lm (l, r) =
+ (\<exists> rn. l = Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ r = <lm> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wprepare_goto_start_pos :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_goto_start_pos m lm (l, r) =
+ (wprepare_goto_start_pos_B m lm (l, r) \<or>
+ wprepare_goto_start_pos_O m lm (l, r))"
+
+fun wprepare_loop_start_on_rightmost :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_loop_start_on_rightmost m lm (l, r) =
+ (\<exists> rn mr. rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and>
+ r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wprepare_loop_start_in_middle :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_loop_start_in_middle m lm (l, r) =
+ (\<exists> rn (mr:: nat) (lm1::nat list).
+ rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and>
+ r = Oc\<^bsup>mr\<^esup> @ Bk # <lm1> @ Bk\<^bsup>rn\<^esup> \<and> lm1 \<noteq> [])"
+
+fun wprepare_loop_start :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_loop_start m lm (l, r) = (wprepare_loop_start_on_rightmost m lm (l, r) \<or>
+ wprepare_loop_start_in_middle m lm (l, r))"
+
+fun wprepare_loop_goon_on_rightmost :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_loop_goon_on_rightmost m lm (l, r) =
+ (\<exists> rn. l = Bk # <rev lm> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ r = Bk\<^bsup>rn\<^esup>)"
+
+fun wprepare_loop_goon_in_middle :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_loop_goon_in_middle m lm (l, r) =
+ (\<exists> rn (mr:: nat) (lm1::nat list).
+ rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and>
+ (if lm1 = [] then r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>
+ else r = Oc\<^bsup>mr\<^esup> @ Bk # <lm1> @ Bk\<^bsup>rn\<^esup>) \<and> mr > 0)"
+
+fun wprepare_loop_goon :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_loop_goon m lm (l, r) =
+ (wprepare_loop_goon_in_middle m lm (l, r) \<or>
+ wprepare_loop_goon_on_rightmost m lm (l, r))"
+
+fun wprepare_add_one2 :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_add_one2 m lm (l, r) =
+ (\<exists> rn. l = Bk # Bk # <rev lm> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ (r = [] \<or> tl r = Bk\<^bsup>rn\<^esup>))"
+
+fun wprepare_stop :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_stop m lm (l, r) =
+ (\<exists> rn. l = Bk # <rev lm> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ r = Bk # Oc # Bk\<^bsup>rn\<^esup>)"
+
+fun wprepare_inv :: "nat \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wprepare_inv st m lm (l, r) =
+ (if st = 0 then wprepare_stop m lm (l, r)
+ else if st = Suc 0 then wprepare_add_one m lm (l, r)
+ else if st = Suc (Suc 0) then wprepare_goto_first_end m lm (l, r)
+ else if st = Suc (Suc (Suc 0)) then wprepare_erase m lm (l, r)
+ else if st = 4 then wprepare_goto_start_pos m lm (l, r)
+ else if st = 5 then wprepare_loop_start m lm (l, r)
+ else if st = 6 then wprepare_loop_goon m lm (l, r)
+ else if st = 7 then wprepare_add_one2 m lm (l, r)
+ else False)"
+
+fun wprepare_stage :: "t_conf \<Rightarrow> nat"
+ where
+ "wprepare_stage (st, l, r) =
+ (if st \<ge> 1 \<and> st \<le> 4 then 3
+ else if st = 5 \<or> st = 6 then 2
+ else 1)"
+
+fun wprepare_state :: "t_conf \<Rightarrow> nat"
+ where
+ "wprepare_state (st, l, r) =
+ (if st = 1 then 4
+ else if st = Suc (Suc 0) then 3
+ else if st = Suc (Suc (Suc 0)) then 2
+ else if st = 4 then 1
+ else if st = 7 then 2
+ else 0)"
+
+fun wprepare_step :: "t_conf \<Rightarrow> nat"
+ where
+ "wprepare_step (st, l, r) =
+ (if st = 1 then (if hd r = Oc then Suc (length l)
+ else 0)
+ else if st = Suc (Suc 0) then length r
+ else if st = Suc (Suc (Suc 0)) then (if hd r = Oc then 1
+ else 0)
+ else if st = 4 then length r
+ else if st = 5 then Suc (length r)
+ else if st = 6 then (if r = [] then 0 else Suc (length r))
+ else if st = 7 then (if (r \<noteq> [] \<and> hd r = Oc) then 0
+ else 1)
+ else 0)"
+
+fun wcode_prepare_measure :: "t_conf \<Rightarrow> nat \<times> nat \<times> nat"
+ where
+ "wcode_prepare_measure (st, l, r) =
+ (wprepare_stage (st, l, r),
+ wprepare_state (st, l, r),
+ wprepare_step (st, l, r))"
+
+definition wcode_prepare_le :: "(t_conf \<times> t_conf) set"
+ where "wcode_prepare_le \<equiv> (inv_image lex_triple wcode_prepare_measure)"
+
+lemma [intro]: "wf lex_pair"
+by(auto intro:wf_lex_prod simp:lex_pair_def)
+
+lemma wf_wcode_prepare_le[intro]: "wf wcode_prepare_le"
+by(auto intro:wf_inv_image simp: wcode_prepare_le_def
+ recursive.lex_triple_def)
+
+declare wprepare_add_one.simps[simp del] wprepare_goto_first_end.simps[simp del]
+ wprepare_erase.simps[simp del] wprepare_goto_start_pos.simps[simp del]
+ wprepare_loop_start.simps[simp del] wprepare_loop_goon.simps[simp del]
+ wprepare_add_one2.simps[simp del]
+
+lemmas wprepare_invs = wprepare_add_one.simps wprepare_goto_first_end.simps
+ wprepare_erase.simps wprepare_goto_start_pos.simps
+ wprepare_loop_start.simps wprepare_loop_goon.simps
+ wprepare_add_one2.simps
+
+declare wprepare_inv.simps[simp del]
+lemma [simp]: "fetch t_wcode_prepare (Suc 0) Bk = (W1, 2)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare (Suc 0) Oc = (L, 1)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare (Suc (Suc 0)) Bk = (L, 3)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare (Suc (Suc 0)) Oc = (R, 2)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare (Suc (Suc (Suc 0))) Bk = (R, 4)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare (Suc (Suc (Suc 0))) Oc = (W0, 3)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 4 Bk = (R, 4)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 4 Oc = (R, 5)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 5 Oc = (R, 5)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 5 Bk = (R, 6)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 6 Oc = (R, 5)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 6 Bk = (R, 7)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 7 Oc = (L, 0)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_prepare 7 Bk = (W1, 7)"
+apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps)
+done
+
+lemma tape_of_nl_not_null: "lm \<noteq> [] \<Longrightarrow> <lm::nat list> \<noteq> []"
+apply(case_tac lm, auto)
+apply(case_tac list, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+done
+
+lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_add_one m lm (b, []) = False"
+apply(simp add: wprepare_invs)
+apply(simp add: tape_of_nl_not_null)
+done
+
+lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_goto_first_end m lm (b, []) = False"
+apply(simp add: wprepare_invs)
+done
+
+lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_erase m lm (b, []) = False"
+apply(simp add: wprepare_invs)
+done
+
+
+
+lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_goto_start_pos m lm (b, []) = False"
+apply(simp add: wprepare_invs tape_of_nl_not_null)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [])\<rbrakk> \<Longrightarrow> b \<noteq> []"
+apply(simp add: wprepare_invs tape_of_nl_not_null, auto)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [])\<rbrakk> \<Longrightarrow>
+ wprepare_loop_goon m lm (Bk # b, [])"
+apply(simp only: wprepare_invs tape_of_nl_not_null)
+apply(erule_tac disjE)
+apply(rule_tac disjI2)
+apply(simp add: wprepare_loop_start_on_rightmost.simps
+ wprepare_loop_goon_on_rightmost.simps, auto)
+apply(rule_tac rev_eq, simp add: tape_of_nl_rev)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, [])\<rbrakk> \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
+done
+
+lemma [simp]:"\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, [])\<rbrakk> \<Longrightarrow>
+ wprepare_add_one2 m lm (Bk # b, [])"
+apply(simp only: wprepare_invs tape_of_nl_not_null, auto split: if_splits)
+apply(case_tac mr, simp, simp add: exp_ind_def)
+done
+
+lemma [simp]: "wprepare_add_one2 m lm (b, []) \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
+done
+
+lemma [simp]: "wprepare_add_one2 m lm (b, []) \<Longrightarrow> wprepare_add_one2 m lm (b, [Oc])"
+apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
+done
+
+lemma [simp]: "Bk # list = <(m::nat) # lm> @ ys = False"
+apply(case_tac lm, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_add_one m lm (b, Bk # list)\<rbrakk>
+ \<Longrightarrow> (b = [] \<longrightarrow> wprepare_goto_first_end m lm ([], Oc # list)) \<and>
+ (b \<noteq> [] \<longrightarrow> wprepare_goto_first_end m lm (b, Oc # list))"
+apply(simp only: wprepare_invs, auto)
+apply(rule_tac x = 0 in exI, simp add: exp_ind_def)
+apply(case_tac lm, simp, simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+apply(rule_tac x = rn in exI, simp)
+done
+
+lemma [simp]: "wprepare_goto_first_end m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
+apply(case_tac mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wprepare_goto_first_end m lm (b, Bk # list) \<Longrightarrow>
+ wprepare_erase m lm (tl b, hd b # Bk # list)"
+apply(simp only: wprepare_invs tape_of_nl_not_null, auto)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac mr, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "wprepare_erase m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs exp_ind_def, auto)
+done
+
+lemma [simp]: "wprepare_erase m lm (b, Bk # list) \<Longrightarrow>
+ wprepare_goto_start_pos m lm (Bk # b, list)"
+apply(simp only: wprepare_invs, auto)
+done
+
+lemma [simp]: "\<lbrakk>wprepare_add_one m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
+apply(simp only: wprepare_invs)
+apply(case_tac lm, simp_all add: tape_of_nl_abv
+ tape_of_nat_list.simps exp_ind_def, auto)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_first_end m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
+apply(simp only: wprepare_invs, auto)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(simp add: tape_of_nl_not_null)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_first_end m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs, auto)
+apply(case_tac mr, simp_all add: exp_ind_def tape_of_nl_not_null)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_erase m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
+apply(simp only: wprepare_invs, auto)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_erase m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []"
+apply(simp only: wprepare_invs, auto)
+apply(simp add: tape_of_nl_not_null)
+apply(case_tac lm, simp, case_tac list)
+apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs)
+apply(auto)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs, auto)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, Bk # list)\<rbrakk> \<Longrightarrow>
+ (list = [] \<longrightarrow> wprepare_add_one2 m lm (Bk # b, [])) \<and>
+ (list \<noteq> [] \<longrightarrow> wprepare_add_one2 m lm (Bk # b, list))"
+apply(simp only: wprepare_invs, simp)
+apply(case_tac list, simp_all split: if_splits, auto)
+apply(case_tac [1-3] mr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: exp_ind_def tape_of_nl_not_null)
+apply(case_tac [1-2] mr, simp_all add: exp_ind_def)
+apply(case_tac rn, simp, case_tac nat, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs, simp)
+done
+
+lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \<Longrightarrow>
+ (list = [] \<longrightarrow> wprepare_add_one2 m lm (b, [Oc])) \<and>
+ (list \<noteq> [] \<longrightarrow> wprepare_add_one2 m lm (b, Oc # list))"
+apply(simp only: wprepare_invs, auto)
+done
+
+lemma [simp]: "wprepare_goto_first_end m lm (b, Oc # list)
+ \<Longrightarrow> (b = [] \<longrightarrow> wprepare_goto_first_end m lm ([Oc], list)) \<and>
+ (b \<noteq> [] \<longrightarrow> wprepare_goto_first_end m lm (Oc # b, list))"
+apply(simp only: wprepare_invs, auto)
+apply(rule_tac x = 1 in exI, auto)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac ml, simp_all add: exp_ind_def)
+apply(rule_tac x = rn in exI, simp)
+apply(rule_tac x = "Suc ml" in exI, simp_all add: exp_ind_def)
+apply(rule_tac x = "mr - 1" in exI, simp)
+apply(case_tac mr, simp_all add: exp_ind_def, auto)
+done
+
+lemma [simp]: "wprepare_erase m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp only: wprepare_invs, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "wprepare_erase m lm (b, Oc # list)
+ \<Longrightarrow> wprepare_erase m lm (b, Bk # list)"
+apply(simp only:wprepare_invs, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk>
+ \<Longrightarrow> wprepare_goto_start_pos m lm (Bk # b, list)"
+apply(simp only:wprepare_invs, auto)
+apply(case_tac [!] lm, simp, simp_all)
+done
+
+lemma [simp]: "wprepare_loop_start m lm (b, aa) \<Longrightarrow> b \<noteq> []"
+apply(simp only:wprepare_invs, auto)
+done
+lemma [elim]: "Bk # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<Longrightarrow> \<exists>rn. list = Bk\<^bsup>rn\<^esup>"
+apply(case_tac mr, simp_all)
+apply(case_tac rn, simp_all add: exp_ind_def, auto)
+done
+
+lemma rev_equal_iff: "x = y \<Longrightarrow> rev x = rev y"
+by simp
+
+lemma tape_of_nl_false1:
+ "lm \<noteq> [] \<Longrightarrow> rev b @ [Bk] \<noteq> Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>m\<^esup> @ Bk # Bk # <lm::nat list>"
+apply(auto)
+apply(drule_tac rev_equal_iff, simp add: tape_of_nl_rev)
+apply(case_tac "rev lm")
+apply(case_tac [2] list, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+done
+
+lemma [simp]: "wprepare_loop_start_in_middle m lm (b, [Bk]) = False"
+apply(simp add: wprepare_loop_start_in_middle.simps, auto)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac lm1, simp, simp add: tape_of_nl_not_null)
+done
+
+declare wprepare_loop_start_in_middle.simps[simp del]
+
+declare wprepare_loop_start_on_rightmost.simps[simp del]
+ wprepare_loop_goon_in_middle.simps[simp del]
+ wprepare_loop_goon_on_rightmost.simps[simp del]
+
+lemma [simp]: "wprepare_loop_goon_in_middle m lm (Bk # b, []) = False"
+apply(simp add: wprepare_loop_goon_in_middle.simps, auto)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [Bk])\<rbrakk> \<Longrightarrow>
+ wprepare_loop_goon m lm (Bk # b, [])"
+apply(simp only: wprepare_invs, simp)
+apply(simp add: wprepare_loop_goon_on_rightmost.simps
+ wprepare_loop_start_on_rightmost.simps, auto)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(rule_tac rev_eq)
+apply(simp add: tape_of_nl_rev)
+apply(simp add: exp_ind_def[THEN sym] exp_ind)
+done
+
+lemma [simp]: "wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista)
+ \<Longrightarrow> wprepare_loop_goon_in_middle m lm (Bk # b, a # lista) = False"
+apply(auto simp: wprepare_loop_start_on_rightmost.simps
+ wprepare_loop_goon_in_middle.simps)
+apply(case_tac [!] mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista)\<rbrakk>
+ \<Longrightarrow> wprepare_loop_goon_on_rightmost m lm (Bk # b, a # lista)"
+apply(simp only: wprepare_loop_start_on_rightmost.simps
+ wprepare_loop_goon_on_rightmost.simps, auto)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(simp add: tape_of_nl_rev)
+apply(simp add: exp_ind_def[THEN sym] exp_ind)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\<rbrakk>
+ \<Longrightarrow> wprepare_loop_goon_on_rightmost m lm (Bk # b, a # lista) = False"
+apply(simp add: wprepare_loop_start_in_middle.simps
+ wprepare_loop_goon_on_rightmost.simps, auto)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac "lm1::nat list", simp_all, case_tac list, simp)
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps tape_of_nat_abv exp_ind_def)
+apply(case_tac [!] rna, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac lm1, simp, case_tac list, simp)
+apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def tape_of_nat_abv)
+done
+
+lemma [simp]:
+ "\<lbrakk>lm \<noteq> []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\<rbrakk>
+ \<Longrightarrow> wprepare_loop_goon_in_middle m lm (Bk # b, a # lista)"
+apply(simp add: wprepare_loop_start_in_middle.simps
+ wprepare_loop_goon_in_middle.simps, auto)
+apply(rule_tac x = rn in exI, simp)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac lm1, simp)
+apply(rule_tac x = "Suc aa" in exI, simp)
+apply(rule_tac x = list in exI)
+apply(case_tac list, simp_all add: tape_of_nl_abv tape_of_nat_list.simps)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, Bk # a # lista)\<rbrakk> \<Longrightarrow>
+ wprepare_loop_goon m lm (Bk # b, a # lista)"
+apply(simp add: wprepare_loop_start.simps
+ wprepare_loop_goon.simps)
+apply(erule_tac disjE, simp, auto)
+done
+
+lemma start_2_goon:
+ "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, Bk # list)\<rbrakk> \<Longrightarrow>
+ (list = [] \<longrightarrow> wprepare_loop_goon m lm (Bk # b, [])) \<and>
+ (list \<noteq> [] \<longrightarrow> wprepare_loop_goon m lm (Bk # b, list))"
+apply(case_tac list, auto)
+done
+
+lemma add_one_2_add_one: "wprepare_add_one m lm (b, Oc # list)
+ \<Longrightarrow> (hd b = Oc \<longrightarrow> (b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)) \<and>
+ (b \<noteq> [] \<longrightarrow> wprepare_add_one m lm (tl b, Oc # Oc # list))) \<and>
+ (hd b \<noteq> Oc \<longrightarrow> (b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)) \<and>
+ (b \<noteq> [] \<longrightarrow> wprepare_add_one m lm (tl b, hd b # Oc # list)))"
+apply(simp only: wprepare_add_one.simps, auto)
+done
+
+lemma [simp]: "wprepare_loop_start m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp)
+done
+
+lemma [simp]: "wprepare_loop_start_on_rightmost m lm (b, Oc # list) \<Longrightarrow>
+ wprepare_loop_start_on_rightmost m lm (Oc # b, list)"
+apply(simp add: wprepare_loop_start_on_rightmost.simps, auto)
+apply(rule_tac x = rn in exI, auto)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac rn, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "wprepare_loop_start_in_middle m lm (b, Oc # list) \<Longrightarrow>
+ wprepare_loop_start_in_middle m lm (Oc # b, list)"
+apply(simp add: wprepare_loop_start_in_middle.simps, auto)
+apply(rule_tac x = rn in exI, auto)
+apply(case_tac mr, simp, simp add: exp_ind_def)
+apply(rule_tac x = nat in exI, simp)
+apply(rule_tac x = lm1 in exI, simp)
+done
+
+lemma start_2_start: "wprepare_loop_start m lm (b, Oc # list) \<Longrightarrow>
+ wprepare_loop_start m lm (Oc # b, list)"
+apply(simp add: wprepare_loop_start.simps)
+apply(erule_tac disjE, simp_all )
+done
+
+lemma [simp]: "wprepare_loop_goon m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wprepare_loop_goon.simps
+ wprepare_loop_goon_in_middle.simps
+ wprepare_loop_goon_on_rightmost.simps)
+apply(auto)
+done
+
+lemma [simp]: "wprepare_goto_start_pos m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(simp add: wprepare_goto_start_pos.simps)
+done
+
+lemma [simp]: "wprepare_loop_goon_on_rightmost m lm (b, Oc # list) = False"
+apply(simp add: wprepare_loop_goon_on_rightmost.simps)
+done
+lemma wprepare_loop1: "\<lbrakk>rev b @ Oc\<^bsup>mr\<^esup> = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm>;
+ b \<noteq> []; 0 < mr; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>\<rbrakk>
+ \<Longrightarrow> wprepare_loop_start_on_rightmost m lm (Oc # b, list)"
+apply(simp add: wprepare_loop_start_on_rightmost.simps)
+apply(rule_tac x = rn in exI, simp)
+apply(case_tac mr, simp, simp add: exp_ind_def, auto)
+done
+
+lemma wprepare_loop2: "\<lbrakk>rev b @ Oc\<^bsup>mr\<^esup> @ Bk # <a # lista> = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm>;
+ b \<noteq> []; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk # <(a::nat) # lista> @ Bk\<^bsup>rn\<^esup>\<rbrakk>
+ \<Longrightarrow> wprepare_loop_start_in_middle m lm (Oc # b, list)"
+apply(simp add: wprepare_loop_start_in_middle.simps)
+apply(rule_tac x = rn in exI, simp)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(rule_tac x = nat in exI, simp)
+apply(rule_tac x = "a#lista" in exI, simp)
+done
+
+lemma [simp]: "wprepare_loop_goon_in_middle m lm (b, Oc # list) \<Longrightarrow>
+ wprepare_loop_start_on_rightmost m lm (Oc # b, list) \<or>
+ wprepare_loop_start_in_middle m lm (Oc # b, list)"
+apply(simp add: wprepare_loop_goon_in_middle.simps split: if_splits)
+apply(case_tac lm1, simp_all add: wprepare_loop1 wprepare_loop2)
+done
+
+lemma [simp]: "wprepare_loop_goon m lm (b, Oc # list)
+ \<Longrightarrow> wprepare_loop_start m lm (Oc # b, list)"
+apply(simp add: wprepare_loop_goon.simps
+ wprepare_loop_start.simps)
+done
+
+lemma [simp]: "wprepare_add_one m lm (b, Oc # list)
+ \<Longrightarrow> b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)"
+apply(auto)
+apply(simp add: wprepare_add_one.simps)
+done
+
+lemma [simp]: "wprepare_goto_start_pos m [a] (b, Oc # list)
+ \<Longrightarrow> wprepare_loop_start_on_rightmost m [a] (Oc # b, list) "
+apply(auto simp: wprepare_goto_start_pos.simps
+ wprepare_loop_start_on_rightmost.simps)
+apply(rule_tac x = rn in exI, simp)
+apply(simp add: tape_of_nat_abv tape_of_nat_list.simps exp_ind_def, auto)
+done
+
+lemma [simp]: "wprepare_goto_start_pos m (a # aa # listaa) (b, Oc # list)
+ \<Longrightarrow>wprepare_loop_start_in_middle m (a # aa # listaa) (Oc # b, list)"
+apply(auto simp: wprepare_goto_start_pos.simps
+ wprepare_loop_start_in_middle.simps)
+apply(rule_tac x = rn in exI, simp)
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def)
+apply(rule_tac x = a in exI, rule_tac x = "aa#listaa" in exI, simp)
+done
+
+lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Oc # list)\<rbrakk>
+ \<Longrightarrow> wprepare_loop_start m lm (Oc # b, list)"
+apply(case_tac lm, simp_all)
+apply(case_tac lista, simp_all add: wprepare_loop_start.simps)
+done
+
+lemma [simp]: "wprepare_add_one2 m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []"
+apply(auto simp: wprepare_add_one2.simps)
+done
+
+lemma add_one_2_stop:
+ "wprepare_add_one2 m lm (b, Oc # list)
+ \<Longrightarrow> wprepare_stop m lm (tl b, hd b # Oc # list)"
+apply(simp add: wprepare_stop.simps wprepare_add_one2.simps)
+done
+
+declare wprepare_stop.simps[simp del]
+
+lemma wprepare_correctness:
+ assumes h: "lm \<noteq> []"
+ shows "let P = (\<lambda> (st, l, r). st = 0) in
+ let Q = (\<lambda> (st, l, r). wprepare_inv st m lm (l, r)) in
+ let f = (\<lambda> stp. steps (Suc 0, [], (<m # lm>)) t_wcode_prepare stp) in
+ \<exists> n .P (f n) \<and> Q (f n)"
+proof -
+ let ?P = "(\<lambda> (st, l, r). st = 0)"
+ let ?Q = "(\<lambda> (st, l, r). wprepare_inv st m lm (l, r))"
+ let ?f = "(\<lambda> stp. steps (Suc 0, [], (<m # lm>)) t_wcode_prepare stp)"
+ have "\<exists> n. ?P (?f n) \<and> ?Q (?f n)"
+ proof(rule_tac halt_lemma2)
+ show "wf wcode_prepare_le" by auto
+ next
+ show "\<forall> n. \<not> ?P (?f n) \<and> ?Q (?f n) \<longrightarrow>
+ ?Q (?f (Suc n)) \<and> (?f (Suc n), ?f n) \<in> wcode_prepare_le"
+ using h
+ apply(rule_tac allI, rule_tac impI, case_tac "?f n",
+ simp add: tstep_red tstep.simps)
+ apply(case_tac c, simp, case_tac [2] aa)
+ apply(simp_all add: wprepare_inv.simps wcode_prepare_le_def new_tape.simps
+ lex_triple_def lex_pair_def
+
+ split: if_splits)
+ apply(simp_all add: start_2_goon start_2_start
+ add_one_2_add_one add_one_2_stop)
+ apply(auto simp: wprepare_add_one2.simps)
+ done
+ next
+ show "?Q (?f 0)"
+ apply(simp add: steps.simps wprepare_inv.simps wprepare_invs)
+ done
+ next
+ show "\<not> ?P (?f 0)"
+ apply(simp add: steps.simps)
+ done
+ qed
+ thus "?thesis"
+ apply(auto)
+ done
+qed
+
+lemma [intro]: "t_correct t_wcode_prepare"
+apply(simp add: t_correct.simps t_wcode_prepare_def iseven_def)
+apply(rule_tac x = 7 in exI, simp)
+done
+
+lemma twice_len_even: "length (tm_of abc_twice) mod 2 = 0"
+apply(simp add: tm_even)
+done
+
+lemma fourtimes_len_even: "length (tm_of abc_fourtimes) mod 2 = 0"
+apply(simp add: tm_even)
+done
+
+lemma t_correct_termi: "t_correct tp \<Longrightarrow>
+ list_all (\<lambda>(acn, st). (st \<le> Suc (length tp div 2))) (change_termi_state tp)"
+apply(auto simp: t_correct.simps List.list_all_length)
+apply(erule_tac x = n in allE, simp)
+apply(case_tac "tp!n", auto simp: change_termi_state.simps split: if_splits)
+done
+
+
+lemma t_correct_shift:
+ "list_all (\<lambda>(acn, st). (st \<le> y)) tp \<Longrightarrow>
+ list_all (\<lambda>(acn, st). (st \<le> y + off)) (tshift tp off) "
+apply(auto simp: t_correct.simps List.list_all_length)
+apply(erule_tac x = n in allE, simp add: shift_length)
+apply(case_tac "tp!n", auto simp: tshift.simps)
+done
+
+lemma [intro]:
+ "t_correct (tm_of abc_twice @ tMp (Suc 0)
+ (start_of twice_ly (length abc_twice) - Suc 0))"
+apply(rule_tac t_compiled_correct, simp_all)
+apply(simp add: twice_ly_def)
+done
+
+lemma [intro]: "t_correct (tm_of abc_fourtimes @ tMp (Suc 0)
+ (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))"
+apply(rule_tac t_compiled_correct, simp_all)
+apply(simp add: fourtimes_ly_def)
+done
+
+
+lemma [intro]: "t_correct t_wcode_main"
+apply(auto simp: t_wcode_main_def t_correct.simps shift_length
+ t_twice_def t_fourtimes_def)
+proof -
+ show "iseven (60 + (length (tm_of abc_twice) +
+ length (tm_of abc_fourtimes)))"
+ using twice_len_even fourtimes_len_even
+ apply(auto simp: iseven_def)
+ apply(rule_tac x = "30 + q + qa" in exI, simp)
+ done
+next
+ show " list_all (\<lambda>(acn, s). s \<le> (60 + (length (tm_of abc_twice) +
+ length (tm_of abc_fourtimes))) div 2) t_wcode_main_first_part"
+ apply(auto simp: t_wcode_main_first_part_def shift_length t_twice_def)
+ done
+next
+ have "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_twice @ tMp (Suc 0)
+ (start_of twice_ly (length abc_twice) - Suc 0)) div 2))
+ (change_termi_state (tm_of abc_twice @ tMp (Suc 0)
+ (start_of twice_ly (length abc_twice) - Suc 0)))"
+ apply(rule_tac t_correct_termi, auto)
+ done
+ hence "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_twice @ tMp (Suc 0)
+ (start_of twice_ly (length abc_twice) - Suc 0)) div 2) + 12)
+ (abacus.tshift (change_termi_state (tm_of abc_twice @ tMp (Suc 0)
+ (start_of twice_ly (length abc_twice) - Suc 0))) 12)"
+ apply(rule_tac t_correct_shift, simp)
+ done
+ thus "list_all (\<lambda>(acn, s). s \<le>
+ (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2)
+ (abacus.tshift (change_termi_state (tm_of abc_twice @ tMp (Suc 0)
+ (start_of twice_ly (length abc_twice) - Suc 0))) 12)"
+ apply(simp)
+ apply(simp add: list_all_length, auto)
+ done
+next
+ have "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_fourtimes @ tMp (Suc 0)
+ (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) div 2))
+ (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0)
+ (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) "
+ apply(rule_tac t_correct_termi, auto)
+ done
+ hence "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_fourtimes @ tMp (Suc 0)
+ (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) div 2) + (t_twice_len + 13))
+ (abacus.tshift (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0)
+ (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) (t_twice_len + 13))"
+ apply(rule_tac t_correct_shift, simp)
+ done
+ thus "list_all (\<lambda>(acn, s). s \<le> (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2)
+ (abacus.tshift (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0)
+ (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) (t_twice_len + 13))"
+ apply(simp add: t_twice_len_def t_twice_def)
+ using twice_len_even fourtimes_len_even
+ apply(auto simp: list_all_length)
+ done
+qed
+
+lemma [intro]: "t_correct (t_wcode_prepare |+| t_wcode_main)"
+apply(auto intro: t_correct_add)
+done
+
+lemma prepare_mainpart_lemma:
+ "args \<noteq> [] \<Longrightarrow>
+ \<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp
+ = (0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+proof -
+ let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <m # args>"
+ let ?Q1 = "\<lambda> (l, r). wprepare_stop m args (l, r)"
+ let ?P2 = ?Q1
+ let ?Q2 = "\<lambda> (l, r). (\<exists> ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ let ?P3 = "\<lambda> tp. False"
+ assume h: "args \<noteq> []"
+ have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp)
+ (t_wcode_prepare |+| t_wcode_main) stp = (0, tp') \<and> ?Q2 tp')"
+ proof(rule_tac turing_merge.t_merge_halt[of t_wcode_prepare t_wcode_main ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2],
+ auto simp: turing_merge_def)
+ show "\<exists>stp. case steps (Suc 0, [], <m # args>) t_wcode_prepare stp of (st, tp')
+ \<Rightarrow> st = 0 \<and> wprepare_stop m args tp'"
+ using wprepare_correctness[of args m] h
+ apply(simp, auto)
+ apply(rule_tac x = n in exI, simp add: wprepare_inv.simps)
+ done
+ next
+ fix a b
+ assume "wprepare_stop m args (a, b)"
+ thus "\<exists>stp. case steps (Suc 0, a, b) t_wcode_main stp of
+ (st, tp') \<Rightarrow> (st = 0) \<and> (case tp' of (l, r) \<Rightarrow> l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ (\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>))"
+ proof(simp only: wprepare_stop.simps, erule_tac exE)
+ fix rn
+ assume "a = Bk # <rev args> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ b = Bk # Oc # Bk\<^bsup>rn\<^esup>"
+ thus "?thesis"
+ using t_wcode_main_lemma_pre[of "args" "<args>" 0 "Oc\<^bsup>Suc m\<^esup>" 0 rn] h
+ apply(simp)
+ apply(erule_tac exE)+
+ apply(rule_tac x = stp in exI, simp add: tape_of_nl_rev, auto)
+ done
+ qed
+ next
+ show "wprepare_stop m args \<turnstile>-> wprepare_stop m args"
+ by(simp add: t_imply_def)
+ qed
+ thus "\<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp
+ = (0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ apply(simp add: t_imply_def)
+ apply(erule_tac exE)+
+ apply(auto)
+ done
+qed
+
+
+lemma [simp]: "tinres r r' \<Longrightarrow>
+ fetch t ss (case r of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) =
+ fetch t ss (case r' of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)"
+apply(simp add: fetch.simps, auto split: if_splits simp: tinres_def)
+apply(case_tac [!] r', simp_all)
+apply(case_tac [!] n, simp_all add: exp_ind_def)
+apply(case_tac [!] r, simp_all)
+done
+
+lemma [intro]: "\<exists> n. (a::block)\<^bsup>n\<^esup> = []"
+by auto
+
+lemma [simp]: "\<lbrakk>tinres r r'; r \<noteq> []; r' \<noteq> []\<rbrakk> \<Longrightarrow> hd r = hd r'"
+apply(auto simp: tinres_def)
+done
+
+lemma [intro]: "hd (Bk\<^bsup>Suc n\<^esup>) = Bk"
+apply(simp add: exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>tinres r []; r \<noteq> []\<rbrakk> \<Longrightarrow> hd r = Bk"
+apply(auto simp: tinres_def)
+apply(case_tac n, auto)
+done
+
+lemma [simp]: "\<lbrakk>tinres [] r'; r' \<noteq> []\<rbrakk> \<Longrightarrow> hd r' = Bk"
+apply(auto simp: tinres_def)
+done
+
+lemma [intro]: "\<exists>na. tl r = tl (r @ Bk\<^bsup>n\<^esup>) @ Bk\<^bsup>na\<^esup> \<or> tl (r @ Bk\<^bsup>n\<^esup>) = tl r @ Bk\<^bsup>na\<^esup>"
+apply(case_tac r, simp)
+apply(case_tac n, simp)
+apply(rule_tac x = 0 in exI, simp)
+apply(rule_tac x = nat in exI, simp add: exp_ind_def)
+apply(simp)
+apply(rule_tac x = n in exI, simp)
+done
+
+lemma [simp]: "tinres r r' \<Longrightarrow> tinres (tl r) (tl r')"
+apply(auto simp: tinres_def)
+apply(case_tac r', simp_all)
+apply(case_tac n, simp_all add: exp_ind_def)
+apply(rule_tac x = 0 in exI, simp)
+apply(rule_tac x = nat in exI, simp_all)
+apply(rule_tac x = n in exI, simp)
+done
+
+lemma [simp]: "\<lbrakk>tinres r []; r \<noteq> []\<rbrakk> \<Longrightarrow> tinres (tl r) []"
+apply(case_tac r, auto simp: tinres_def)
+apply(case_tac n, simp_all add: exp_ind_def)
+apply(rule_tac x = nat in exI, simp)
+done
+
+lemma [simp]: "\<lbrakk>tinres [] r'\<rbrakk> \<Longrightarrow> tinres [] (tl r')"
+apply(case_tac r', auto simp: tinres_def)
+apply(case_tac n, simp_all add: exp_ind_def)
+apply(rule_tac x = nat in exI, simp)
+done
+
+lemma [simp]: "tinres r r' \<Longrightarrow> tinres (b # r) (b # r')"
+apply(auto simp: tinres_def)
+done
+
+lemma tinres_step2:
+ "\<lbrakk>tinres r r'; tstep (ss, l, r) t = (sa, la, ra); tstep (ss, l, r') t = (sb, lb, rb)\<rbrakk>
+ \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb"
+apply(case_tac "ss = 0", simp add: tstep_0)
+apply(simp add: tstep.simps [simp del])
+apply(case_tac "fetch t ss (case r of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)
+apply(auto simp: new_tape.simps)
+apply(simp_all split: taction.splits if_splits)
+apply(auto)
+done
+
+
+lemma tinres_steps2:
+ "\<lbrakk>tinres r r'; steps (ss, l, r) t stp = (sa, la, ra); steps (ss, l, r') t stp = (sb, lb, rb)\<rbrakk>
+ \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb"
+apply(induct stp arbitrary: sa la ra sb lb rb, simp add: steps.simps)
+apply(simp add: tstep_red)
+apply(case_tac "(steps (ss, l, r) t stp)")
+apply(case_tac "(steps (ss, l, r') t stp)")
+proof -
+ fix stp sa la ra sb lb rb a b c aa ba ca
+ assume ind: "\<And>sa la ra sb lb rb. \<lbrakk>steps (ss, l, r) t stp = (sa, la, ra);
+ steps (ss, l, r') t stp = (sb, lb, rb)\<rbrakk> \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb"
+ and h: " tinres r r'" "tstep (steps (ss, l, r) t stp) t = (sa, la, ra)"
+ "tstep (steps (ss, l, r') t stp) t = (sb, lb, rb)" "steps (ss, l, r) t stp = (a, b, c)"
+ "steps (ss, l, r') t stp = (aa, ba, ca)"
+ have "b = ba \<and> tinres c ca \<and> a = aa"
+ apply(rule_tac ind, simp_all add: h)
+ done
+ thus "la = lb \<and> tinres ra rb \<and> sa = sb"
+ apply(rule_tac l = b and r = c and ss = a and r' = ca
+ and t = t in tinres_step2)
+ using h
+ apply(simp, simp, simp)
+ done
+qed
+
+definition t_wcode_adjust :: "tprog"
+ where
+ "t_wcode_adjust = [(W1, 1), (R, 2), (Nop, 2), (R, 3), (R, 3), (R, 4),
+ (L, 8), (L, 5), (L, 6), (W0, 5), (L, 6), (R, 7),
+ (W1, 2), (Nop, 7), (L, 9), (W0, 8), (L, 9), (L, 10),
+ (L, 11), (L, 10), (R, 0), (L, 11)]"
+
+lemma [simp]: "fetch t_wcode_adjust (Suc 0) Bk = (W1, 1)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust (Suc 0) Oc = (R, 2)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust (Suc (Suc 0)) Oc = (R, 3)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust (Suc (Suc (Suc 0))) Oc = (R, 4)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust (Suc (Suc (Suc 0))) Bk = (R, 3)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 4 Bk = (L, 8)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 4 Oc = (L, 5)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 5 Oc = (W0, 5)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 5 Bk = (L, 6)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 6 Oc = (R, 7)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 6 Bk = (L, 6)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 7 Bk = (W1, 2)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 8 Bk = (L, 9)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 8 Oc = (W0, 8)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 9 Oc = (L, 10)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 9 Bk = (L, 9)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 10 Bk = (L, 11)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 10 Oc = (L, 10)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 11 Oc = (L, 11)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+lemma [simp]: "fetch t_wcode_adjust 11 Bk = (R, 0)"
+apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps)
+done
+
+fun wadjust_start :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_start m rs (l, r) =
+ (\<exists> ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ tl r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wadjust_loop_start :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_loop_start m rs (l, r) =
+ (\<exists> ln rn ml mr. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr = Suc (Suc rs) \<and> mr > 0)"
+
+fun wadjust_loop_right_move :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_loop_right_move m rs (l, r) =
+ (\<exists> ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ r = Bk\<^bsup>nr\<^esup> @ Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr = Suc (Suc rs) \<and> mr > 0 \<and>
+ nl + nr > 0)"
+
+fun wadjust_loop_check :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_loop_check m rs (l, r) =
+ (\<exists> ml mr ln rn. l = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = (Suc rs))"
+
+fun wadjust_loop_erase :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_loop_erase m rs (l, r) =
+ (\<exists> ml mr ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ tl r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = (Suc rs) \<and> mr > 0)"
+
+fun wadjust_loop_on_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_loop_on_left_moving_O m rs (l, r) =
+ (\<exists> ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m \<^esup>\<and>
+ r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr = Suc rs \<and> mr > 0)"
+
+fun wadjust_loop_on_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_loop_on_left_moving_B m rs (l, r) =
+ (\<exists> ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ r = Bk\<^bsup>nr\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr = Suc rs \<and> mr > 0)"
+
+fun wadjust_loop_on_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_loop_on_left_moving m rs (l, r) =
+ (wadjust_loop_on_left_moving_O m rs (l, r) \<or>
+ wadjust_loop_on_left_moving_B m rs (l, r))"
+
+fun wadjust_loop_right_move2 :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_loop_right_move2 m rs (l, r) =
+ (\<exists> ml mr ln rn. l = Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr = Suc rs \<and> mr > 0)"
+
+fun wadjust_erase2 :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_erase2 m rs (l, r) =
+ (\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ tl r = Bk\<^bsup>rn\<^esup>)"
+
+fun wadjust_on_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_on_left_moving_O m rs (l, r) =
+ (\<exists> rn. l = Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ r = Oc # Bk\<^bsup>rn\<^esup>)"
+
+fun wadjust_on_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_on_left_moving_B m rs (l, r) =
+ (\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ r = Bk\<^bsup>rn\<^esup>)"
+
+fun wadjust_on_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_on_left_moving m rs (l, r) =
+ (wadjust_on_left_moving_O m rs (l, r) \<or>
+ wadjust_on_left_moving_B m rs (l, r))"
+
+fun wadjust_goon_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_goon_left_moving_B m rs (l, r) =
+ (\<exists> rn. l = Oc\<^bsup>Suc m\<^esup> \<and>
+ r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wadjust_goon_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_goon_left_moving_O m rs (l, r) =
+ (\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr = Suc (Suc rs) \<and> mr > 0)"
+
+fun wadjust_goon_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_goon_left_moving m rs (l, r) =
+ (wadjust_goon_left_moving_B m rs (l, r) \<or>
+ wadjust_goon_left_moving_O m rs (l, r))"
+
+fun wadjust_backto_standard_pos_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_backto_standard_pos_B m rs (l, r) =
+ (\<exists> rn. l = [] \<and>
+ r = Bk # Oc\<^bsup>Suc m \<^esup>@ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+fun wadjust_backto_standard_pos_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_backto_standard_pos_O m rs (l, r) =
+ (\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> \<and>
+ r = Oc\<^bsup>mr\<^esup> @ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup> \<and>
+ ml + mr = Suc m \<and> mr > 0)"
+
+fun wadjust_backto_standard_pos :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_backto_standard_pos m rs (l, r) =
+ (wadjust_backto_standard_pos_B m rs (l, r) \<or>
+ wadjust_backto_standard_pos_O m rs (l, r))"
+
+fun wadjust_stop :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+where
+ "wadjust_stop m rs (l, r) =
+ (\<exists> rn. l = [Bk] \<and>
+ r = Oc\<^bsup>Suc m \<^esup>@ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+
+declare wadjust_start.simps[simp del] wadjust_loop_start.simps[simp del]
+ wadjust_loop_right_move.simps[simp del] wadjust_loop_check.simps[simp del]
+ wadjust_loop_erase.simps[simp del] wadjust_loop_on_left_moving.simps[simp del]
+ wadjust_loop_right_move2.simps[simp del] wadjust_erase2.simps[simp del]
+ wadjust_on_left_moving_O.simps[simp del] wadjust_on_left_moving_B.simps[simp del]
+ wadjust_on_left_moving.simps[simp del] wadjust_goon_left_moving_B.simps[simp del]
+ wadjust_goon_left_moving_O.simps[simp del] wadjust_goon_left_moving.simps[simp del]
+ wadjust_backto_standard_pos.simps[simp del] wadjust_backto_standard_pos_B.simps[simp del]
+ wadjust_backto_standard_pos_O.simps[simp del] wadjust_stop.simps[simp del]
+
+fun wadjust_inv :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "wadjust_inv st m rs (l, r) =
+ (if st = Suc 0 then wadjust_start m rs (l, r)
+ else if st = Suc (Suc 0) then wadjust_loop_start m rs (l, r)
+ else if st = Suc (Suc (Suc 0)) then wadjust_loop_right_move m rs (l, r)
+ else if st = 4 then wadjust_loop_check m rs (l, r)
+ else if st = 5 then wadjust_loop_erase m rs (l, r)
+ else if st = 6 then wadjust_loop_on_left_moving m rs (l, r)
+ else if st = 7 then wadjust_loop_right_move2 m rs (l, r)
+ else if st = 8 then wadjust_erase2 m rs (l, r)
+ else if st = 9 then wadjust_on_left_moving m rs (l, r)
+ else if st = 10 then wadjust_goon_left_moving m rs (l, r)
+ else if st = 11 then wadjust_backto_standard_pos m rs (l, r)
+ else if st = 0 then wadjust_stop m rs (l, r)
+ else False
+)"
+
+declare wadjust_inv.simps[simp del]
+
+fun wadjust_phase :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
+ where
+ "wadjust_phase rs (st, l, r) =
+ (if st = 1 then 3
+ else if st \<ge> 2 \<and> st \<le> 7 then 2
+ else if st \<ge> 8 \<and> st \<le> 11 then 1
+ else 0)"
+
+thm dropWhile.simps
+
+fun wadjust_stage :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
+ where
+ "wadjust_stage rs (st, l, r) =
+ (if st \<ge> 2 \<and> st \<le> 7 then
+ rs - length (takeWhile (\<lambda> a. a = Oc)
+ (tl (dropWhile (\<lambda> a. a = Oc) (rev l @ r))))
+ else 0)"
+
+fun wadjust_state :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
+ where
+ "wadjust_state rs (st, l, r) =
+ (if st \<ge> 2 \<and> st \<le> 7 then 8 - st
+ else if st \<ge> 8 \<and> st \<le> 11 then 12 - st
+ else 0)"
+
+fun wadjust_step :: "nat \<Rightarrow> t_conf \<Rightarrow> nat"
+ where
+ "wadjust_step rs (st, l, r) =
+ (if st = 1 then (if hd r = Bk then 1
+ else 0)
+ else if st = 3 then length r
+ else if st = 5 then (if hd r = Oc then 1
+ else 0)
+ else if st = 6 then length l
+ else if st = 8 then (if hd r = Oc then 1
+ else 0)
+ else if st = 9 then length l
+ else if st = 10 then length l
+ else if st = 11 then (if hd r = Bk then 0
+ else Suc (length l))
+ else 0)"
+
+fun wadjust_measure :: "(nat \<times> t_conf) \<Rightarrow> nat \<times> nat \<times> nat \<times> nat"
+ where
+ "wadjust_measure (rs, (st, l, r)) =
+ (wadjust_phase rs (st, l, r),
+ wadjust_stage rs (st, l, r),
+ wadjust_state rs (st, l, r),
+ wadjust_step rs (st, l, r))"
+
+definition wadjust_le :: "((nat \<times> t_conf) \<times> nat \<times> t_conf) set"
+ where "wadjust_le \<equiv> (inv_image lex_square wadjust_measure)"
+
+lemma [intro]: "wf lex_square"
+by(auto intro:wf_lex_prod simp: abacus.lex_pair_def lex_square_def
+ abacus.lex_triple_def)
+
+lemma wf_wadjust_le[intro]: "wf wadjust_le"
+by(auto intro:wf_inv_image simp: wadjust_le_def
+ abacus.lex_triple_def abacus.lex_pair_def)
+
+lemma [simp]: "wadjust_start m rs (c, []) = False"
+apply(auto simp: wadjust_start.simps)
+done
+
+lemma [simp]: "wadjust_loop_right_move m rs (c, []) \<Longrightarrow> c \<noteq> []"
+apply(auto simp: wadjust_loop_right_move.simps)
+done
+
+lemma [simp]: "wadjust_loop_right_move m rs (c, [])
+ \<Longrightarrow> wadjust_loop_check m rs (Bk # c, [])"
+apply(simp only: wadjust_loop_right_move.simps wadjust_loop_check.simps)
+apply(auto)
+apply(case_tac [!] mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_loop_check m rs (c, []) \<Longrightarrow> c \<noteq> []"
+apply(simp only: wadjust_loop_check.simps, auto)
+done
+
+lemma [simp]: "wadjust_loop_start m rs (c, []) = False"
+apply(simp add: wadjust_loop_start.simps)
+done
+
+lemma [simp]: "wadjust_loop_right_move m rs (c, []) \<Longrightarrow>
+ wadjust_loop_right_move m rs (Bk # c, [])"
+apply(simp only: wadjust_loop_right_move.simps)
+apply(erule_tac exE)+
+apply(auto)
+apply(case_tac [!] mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_loop_check m rs (c, []) \<Longrightarrow> wadjust_erase2 m rs (tl c, [hd c])"
+apply(simp only: wadjust_loop_check.simps wadjust_erase2.simps, auto)
+apply(case_tac mr, simp_all add: exp_ind_def, auto)
+done
+
+lemma [simp]: " wadjust_loop_erase m rs (c, [])
+ \<Longrightarrow> (c = [] \<longrightarrow> wadjust_loop_on_left_moving m rs ([], [Bk])) \<and>
+ (c \<noteq> [] \<longrightarrow> wadjust_loop_on_left_moving m rs (tl c, [hd c]))"
+apply(simp add: wadjust_loop_erase.simps, auto)
+apply(case_tac [!] mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_loop_on_left_moving m rs (c, []) = False"
+apply(auto simp: wadjust_loop_on_left_moving.simps)
+done
+
+
+lemma [simp]: "wadjust_loop_right_move2 m rs (c, []) = False"
+apply(auto simp: wadjust_loop_right_move2.simps)
+done
+
+lemma [simp]: "wadjust_erase2 m rs ([], []) = False"
+apply(auto simp: wadjust_erase2.simps)
+done
+
+lemma [simp]: "wadjust_on_left_moving_B m rs
+ (Oc # Oc # Oc\<^bsup>rs\<^esup> @ Bk # Oc # Oc\<^bsup>m\<^esup>, [Bk])"
+apply(simp add: wadjust_on_left_moving_B.simps, auto)
+apply(rule_tac x = 0 in exI, simp add: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_on_left_moving_B m rs
+ (Bk\<^bsup>n\<^esup> @ Bk # Oc # Oc # Oc\<^bsup>rs\<^esup> @ Bk # Oc # Oc\<^bsup>m\<^esup>, [Bk])"
+apply(simp add: wadjust_on_left_moving_B.simps exp_ind_def, auto)
+apply(rule_tac x = "Suc n" in exI, simp add: exp_ind)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_erase2 m rs (c, []); c \<noteq> []\<rbrakk> \<Longrightarrow>
+ wadjust_on_left_moving m rs (tl c, [hd c])"
+apply(simp only: wadjust_erase2.simps)
+apply(erule_tac exE)+
+apply(case_tac ln, simp_all add: exp_ind_def wadjust_on_left_moving.simps)
+done
+
+lemma [simp]: "wadjust_erase2 m rs (c, [])
+ \<Longrightarrow> (c = [] \<longrightarrow> wadjust_on_left_moving m rs ([], [Bk])) \<and>
+ (c \<noteq> [] \<longrightarrow> wadjust_on_left_moving m rs (tl c, [hd c]))"
+apply(auto)
+done
+
+lemma [simp]: "wadjust_on_left_moving m rs ([], []) = False"
+apply(simp add: wadjust_on_left_moving.simps
+ wadjust_on_left_moving_O.simps wadjust_on_left_moving_B.simps)
+done
+
+lemma [simp]: "wadjust_on_left_moving_O m rs (c, []) = False"
+apply(simp add: wadjust_on_left_moving_O.simps)
+done
+
+lemma [simp]: " \<lbrakk>wadjust_on_left_moving_B m rs (c, []); c \<noteq> []; hd c = Bk\<rbrakk> \<Longrightarrow>
+ wadjust_on_left_moving_B m rs (tl c, [Bk])"
+apply(simp add: wadjust_on_left_moving_B.simps, auto)
+apply(case_tac [!] ln, simp_all add: exp_ind_def, auto)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, []); c \<noteq> []; hd c = Oc\<rbrakk> \<Longrightarrow>
+ wadjust_on_left_moving_O m rs (tl c, [Oc])"
+apply(simp add: wadjust_on_left_moving_B.simps wadjust_on_left_moving_O.simps, auto)
+apply(case_tac [!] ln, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_on_left_moving m rs (c, []); c \<noteq> []\<rbrakk> \<Longrightarrow>
+ wadjust_on_left_moving m rs (tl c, [hd c])"
+apply(simp add: wadjust_on_left_moving.simps)
+apply(case_tac "hd c", simp_all)
+done
+
+lemma [simp]: "wadjust_on_left_moving m rs (c, [])
+ \<Longrightarrow> (c = [] \<longrightarrow> wadjust_on_left_moving m rs ([], [Bk])) \<and>
+ (c \<noteq> [] \<longrightarrow> wadjust_on_left_moving m rs (tl c, [hd c]))"
+apply(auto)
+done
+
+lemma [simp]: "wadjust_goon_left_moving m rs (c, []) = False"
+apply(auto simp: wadjust_goon_left_moving.simps wadjust_goon_left_moving_B.simps
+ wadjust_goon_left_moving_O.simps)
+done
+
+lemma [simp]: "wadjust_backto_standard_pos m rs (c, []) = False"
+apply(auto simp: wadjust_backto_standard_pos.simps
+ wadjust_backto_standard_pos_B.simps wadjust_backto_standard_pos_O.simps)
+done
+
+lemma [simp]:
+ "wadjust_start m rs (c, Bk # list) \<Longrightarrow>
+ (c = [] \<longrightarrow> wadjust_start m rs ([], Oc # list)) \<and>
+ (c \<noteq> [] \<longrightarrow> wadjust_start m rs (c, Oc # list))"
+apply(auto simp: wadjust_start.simps)
+done
+
+lemma [simp]: "wadjust_loop_start m rs (c, Bk # list) = False"
+apply(auto simp: wadjust_loop_start.simps)
+done
+
+lemma [simp]: "wadjust_loop_right_move m rs (c, b) \<Longrightarrow> c \<noteq> []"
+apply(simp only: wadjust_loop_right_move.simps, auto)
+done
+
+lemma [simp]: "wadjust_loop_right_move m rs (c, Bk # list)
+ \<Longrightarrow> wadjust_loop_right_move m rs (Bk # c, list)"
+apply(simp only: wadjust_loop_right_move.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = ml in exI, simp)
+apply(rule_tac x = mr in exI, simp)
+apply(rule_tac x = "Suc nl" in exI, simp add: exp_ind_def)
+apply(case_tac nr, simp, case_tac mr, simp_all add: exp_ind_def)
+apply(rule_tac x = nat in exI, auto)
+done
+
+lemma [simp]: "wadjust_loop_check m rs (c, b) \<Longrightarrow> c \<noteq> []"
+apply(simp only: wadjust_loop_check.simps, auto)
+done
+
+lemma [simp]: "wadjust_loop_check m rs (c, Bk # list)
+ \<Longrightarrow> wadjust_erase2 m rs (tl c, hd c # Bk # list)"
+apply(auto simp: wadjust_loop_check.simps wadjust_erase2.simps)
+apply(case_tac [!] mr, simp_all add: exp_ind_def, auto)
+done
+
+lemma [simp]: "wadjust_loop_erase m rs (c, b) \<Longrightarrow> c \<noteq> []"
+apply(simp only: wadjust_loop_erase.simps, auto)
+done
+
+declare wadjust_loop_on_left_moving_O.simps[simp del]
+ wadjust_loop_on_left_moving_B.simps[simp del]
+
+lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); hd c = Bk\<rbrakk>
+ \<Longrightarrow> wadjust_loop_on_left_moving_B m rs (tl c, Bk # Bk # list)"
+apply(simp only: wadjust_loop_erase.simps
+ wadjust_loop_on_left_moving_B.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = ml in exI, rule_tac x = mr in exI,
+ rule_tac x = ln in exI, rule_tac x = 0 in exI, simp)
+apply(case_tac ln, simp_all add: exp_ind_def, auto)
+apply(simp add: exp_ind exp_ind_def[THEN sym])
+done
+
+lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); c \<noteq> []; hd c = Oc\<rbrakk> \<Longrightarrow>
+ wadjust_loop_on_left_moving_O m rs (tl c, Oc # Bk # list)"
+apply(simp only: wadjust_loop_erase.simps wadjust_loop_on_left_moving_O.simps,
+ auto)
+apply(case_tac [!] ln, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); c \<noteq> []\<rbrakk> \<Longrightarrow>
+ wadjust_loop_on_left_moving m rs (tl c, hd c # Bk # list)"
+apply(case_tac "hd c", simp_all add:wadjust_loop_on_left_moving.simps)
+done
+
+lemma [simp]: "wadjust_loop_on_left_moving m rs (c, b) \<Longrightarrow> c \<noteq> []"
+apply(simp add: wadjust_loop_on_left_moving.simps
+wadjust_loop_on_left_moving_O.simps wadjust_loop_on_left_moving_B.simps, auto)
+done
+
+lemma [simp]: "wadjust_loop_on_left_moving_O m rs (c, Bk # list) = False"
+apply(simp add: wadjust_loop_on_left_moving_O.simps)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Bk\<rbrakk>
+ \<Longrightarrow> wadjust_loop_on_left_moving_B m rs (tl c, Bk # Bk # list)"
+apply(simp only: wadjust_loop_on_left_moving_B.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = ml in exI, rule_tac x = mr in exI)
+apply(case_tac nl, simp_all add: exp_ind_def, auto)
+apply(rule_tac x = "Suc nr" in exI, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk>
+ \<Longrightarrow> wadjust_loop_on_left_moving_O m rs (tl c, Oc # Bk # list)"
+apply(simp only: wadjust_loop_on_left_moving_O.simps
+ wadjust_loop_on_left_moving_B.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = ml in exI, rule_tac x = mr in exI)
+apply(case_tac nl, simp_all add: exp_ind_def, auto)
+done
+
+lemma [simp]: "wadjust_loop_on_left_moving m rs (c, Bk # list)
+ \<Longrightarrow> wadjust_loop_on_left_moving m rs (tl c, hd c # Bk # list)"
+apply(simp add: wadjust_loop_on_left_moving.simps)
+apply(case_tac "hd c", simp_all)
+done
+
+lemma [simp]: "wadjust_loop_right_move2 m rs (c, b) \<Longrightarrow> c \<noteq> []"
+apply(simp only: wadjust_loop_right_move2.simps, auto)
+done
+
+lemma [simp]: "wadjust_loop_right_move2 m rs (c, Bk # list) \<Longrightarrow> wadjust_loop_start m rs (c, Oc # list)"
+apply(auto simp: wadjust_loop_right_move2.simps wadjust_loop_start.simps)
+apply(case_tac ln, simp_all add: exp_ind_def)
+apply(rule_tac x = 0 in exI, simp)
+apply(rule_tac x = rn in exI, simp)
+apply(rule_tac x = "Suc ml" in exI, simp add: exp_ind_def, auto)
+apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind)
+apply(rule_tac x = rn in exI, auto)
+apply(rule_tac x = "Suc ml" in exI, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \<Longrightarrow> c \<noteq> []"
+apply(auto simp:wadjust_erase2.simps )
+done
+
+lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \<Longrightarrow>
+ wadjust_on_left_moving m rs (tl c, hd c # Bk # list)"
+apply(auto simp: wadjust_erase2.simps)
+apply(case_tac ln, simp_all add: exp_ind_def wadjust_on_left_moving.simps
+ wadjust_on_left_moving_O.simps wadjust_on_left_moving_B.simps)
+apply(auto)
+apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: exp_ind_def)
+apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind)
+apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_on_left_moving m rs (c,b) \<Longrightarrow> c \<noteq> []"
+apply(simp only:wadjust_on_left_moving.simps
+ wadjust_on_left_moving_O.simps
+ wadjust_on_left_moving_B.simps
+ , auto)
+done
+
+lemma [simp]: "wadjust_on_left_moving_O m rs (c, Bk # list) = False"
+apply(simp add: wadjust_on_left_moving_O.simps)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Bk\<rbrakk>
+ \<Longrightarrow> wadjust_on_left_moving_B m rs (tl c, Bk # Bk # list)"
+apply(auto simp: wadjust_on_left_moving_B.simps)
+apply(case_tac ln, simp_all add: exp_ind_def, auto)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk>
+ \<Longrightarrow> wadjust_on_left_moving_O m rs (tl c, Oc # Bk # list)"
+apply(auto simp: wadjust_on_left_moving_O.simps
+ wadjust_on_left_moving_B.simps)
+apply(case_tac ln, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_on_left_moving m rs (c, Bk # list) \<Longrightarrow>
+ wadjust_on_left_moving m rs (tl c, hd c # Bk # list)"
+apply(simp add: wadjust_on_left_moving.simps)
+apply(case_tac "hd c", simp_all)
+done
+
+lemma [simp]: "wadjust_goon_left_moving m rs (c, b) \<Longrightarrow> c \<noteq> []"
+apply(simp add: wadjust_goon_left_moving.simps
+ wadjust_goon_left_moving_B.simps
+ wadjust_goon_left_moving_O.simps exp_ind_def, auto)
+done
+
+lemma [simp]: "wadjust_goon_left_moving_O m rs (c, Bk # list) = False"
+apply(simp add: wadjust_goon_left_moving_O.simps, auto)
+apply(case_tac mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Bk\<rbrakk>
+ \<Longrightarrow> wadjust_backto_standard_pos_B m rs (tl c, Bk # Bk # list)"
+apply(auto simp: wadjust_goon_left_moving_B.simps
+ wadjust_backto_standard_pos_B.simps exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk>
+ \<Longrightarrow> wadjust_backto_standard_pos_O m rs (tl c, Oc # Bk # list)"
+apply(auto simp: wadjust_goon_left_moving_B.simps
+ wadjust_backto_standard_pos_O.simps exp_ind_def)
+apply(rule_tac x = m in exI, simp, auto)
+done
+
+lemma [simp]: "wadjust_goon_left_moving m rs (c, Bk # list) \<Longrightarrow>
+ wadjust_backto_standard_pos m rs (tl c, hd c # Bk # list)"
+apply(case_tac "hd c", simp_all add: wadjust_backto_standard_pos.simps
+ wadjust_goon_left_moving.simps)
+done
+
+lemma [simp]: "wadjust_backto_standard_pos m rs (c, Bk # list) \<Longrightarrow>
+ (c = [] \<longrightarrow> wadjust_stop m rs ([Bk], list)) \<and> (c \<noteq> [] \<longrightarrow> wadjust_stop m rs (Bk # c, list))"
+apply(auto simp: wadjust_backto_standard_pos.simps
+ wadjust_backto_standard_pos_B.simps
+ wadjust_backto_standard_pos_O.simps wadjust_stop.simps)
+apply(case_tac [!] mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_start m rs (c, Oc # list)
+ \<Longrightarrow> (c = [] \<longrightarrow> wadjust_loop_start m rs ([Oc], list)) \<and>
+ (c \<noteq> [] \<longrightarrow> wadjust_loop_start m rs (Oc # c, list))"
+apply(auto simp:wadjust_loop_start.simps wadjust_start.simps )
+apply(rule_tac x = ln in exI, rule_tac x = rn in exI,
+ rule_tac x = "Suc 0" in exI, simp)
+done
+
+lemma [simp]: "wadjust_loop_start m rs (c, b) \<Longrightarrow> c \<noteq> []"
+apply(simp add: wadjust_loop_start.simps, auto)
+done
+
+lemma [simp]: "wadjust_loop_start m rs (c, Oc # list)
+ \<Longrightarrow> wadjust_loop_right_move m rs (Oc # c, list)"
+apply(simp add: wadjust_loop_start.simps wadjust_loop_right_move.simps, auto)
+apply(rule_tac x = ml in exI, rule_tac x = mr in exI,
+ rule_tac x = 0 in exI, simp)
+apply(rule_tac x = "Suc ln" in exI, simp add: exp_ind, auto)
+done
+
+lemma [simp]: "wadjust_loop_right_move m rs (c, Oc # list) \<Longrightarrow>
+ wadjust_loop_check m rs (Oc # c, list)"
+apply(simp add: wadjust_loop_right_move.simps
+ wadjust_loop_check.simps, auto)
+apply(rule_tac [!] x = ml in exI, simp_all, auto)
+apply(case_tac nl, auto simp: exp_ind_def)
+apply(rule_tac x = "mr - 1" in exI, case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac [!] nr, simp_all add: exp_ind_def, auto)
+done
+
+lemma [simp]: "wadjust_loop_check m rs (c, Oc # list) \<Longrightarrow>
+ wadjust_loop_erase m rs (tl c, hd c # Oc # list)"
+apply(simp only: wadjust_loop_check.simps wadjust_loop_erase.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = ml in exI, rule_tac x = mr in exI, auto)
+apply(case_tac mr, simp_all add: exp_ind_def)
+apply(case_tac rn, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_loop_erase m rs (c, Oc # list) \<Longrightarrow>
+ wadjust_loop_erase m rs (c, Bk # list)"
+apply(auto simp: wadjust_loop_erase.simps)
+done
+
+lemma [simp]: "wadjust_loop_on_left_moving_B m rs (c, Oc # list) = False"
+apply(auto simp: wadjust_loop_on_left_moving_B.simps)
+apply(case_tac nr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_loop_on_left_moving m rs (c, Oc # list)
+ \<Longrightarrow> wadjust_loop_right_move2 m rs (Oc # c, list)"
+apply(simp add:wadjust_loop_on_left_moving.simps)
+apply(auto simp: wadjust_loop_on_left_moving_O.simps
+ wadjust_loop_right_move2.simps)
+done
+
+lemma [simp]: "wadjust_loop_right_move2 m rs (c, Oc # list) = False"
+apply(auto simp: wadjust_loop_right_move2.simps )
+apply(case_tac ln, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_erase2 m rs (c, Oc # list)
+ \<Longrightarrow> (c = [] \<longrightarrow> wadjust_erase2 m rs ([], Bk # list))
+ \<and> (c \<noteq> [] \<longrightarrow> wadjust_erase2 m rs (c, Bk # list))"
+apply(auto simp: wadjust_erase2.simps )
+done
+
+lemma [simp]: "wadjust_on_left_moving_B m rs (c, Oc # list) = False"
+apply(auto simp: wadjust_on_left_moving_B.simps)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Bk\<rbrakk> \<Longrightarrow>
+ wadjust_goon_left_moving_B m rs (tl c, Bk # Oc # list)"
+apply(auto simp: wadjust_on_left_moving_O.simps
+ wadjust_goon_left_moving_B.simps exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Oc\<rbrakk>
+ \<Longrightarrow> wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)"
+apply(auto simp: wadjust_on_left_moving_O.simps
+ wadjust_goon_left_moving_O.simps exp_ind_def)
+apply(rule_tac x = rs in exI, simp)
+apply(auto simp: exp_ind_def numeral_2_eq_2)
+done
+
+
+lemma [simp]: "wadjust_on_left_moving m rs (c, Oc # list) \<Longrightarrow>
+ wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)"
+apply(simp add: wadjust_on_left_moving.simps
+ wadjust_goon_left_moving.simps)
+apply(case_tac "hd c", simp_all)
+done
+
+lemma [simp]: "wadjust_on_left_moving m rs (c, Oc # list) \<Longrightarrow>
+ wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)"
+apply(simp add: wadjust_on_left_moving.simps
+ wadjust_goon_left_moving.simps)
+apply(case_tac "hd c", simp_all)
+done
+
+lemma [simp]: "wadjust_goon_left_moving_B m rs (c, Oc # list) = False"
+apply(auto simp: wadjust_goon_left_moving_B.simps)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Bk\<rbrakk>
+ \<Longrightarrow> wadjust_goon_left_moving_B m rs (tl c, Bk # Oc # list)"
+apply(auto simp: wadjust_goon_left_moving_O.simps wadjust_goon_left_moving_B.simps)
+apply(case_tac [!] ml, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Oc\<rbrakk> \<Longrightarrow>
+ wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)"
+apply(auto simp: wadjust_goon_left_moving_O.simps wadjust_goon_left_moving_B.simps)
+apply(rule_tac x = "ml - 1" in exI, simp)
+apply(case_tac ml, simp_all add: exp_ind_def)
+apply(rule_tac x = "Suc mr" in exI, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_goon_left_moving m rs (c, Oc # list) \<Longrightarrow>
+ wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)"
+apply(simp add: wadjust_goon_left_moving.simps)
+apply(case_tac "hd c", simp_all)
+done
+
+lemma [simp]: "wadjust_backto_standard_pos_B m rs (c, Oc # list) = False"
+apply(simp add: wadjust_backto_standard_pos_B.simps)
+done
+
+lemma [simp]: "wadjust_backto_standard_pos_O m rs (c, Bk # xs) = False"
+apply(simp add: wadjust_backto_standard_pos_O.simps, auto)
+apply(case_tac mr, simp_all add: exp_ind_def)
+done
+
+
+
+lemma [simp]: "wadjust_backto_standard_pos_O m rs ([], Oc # list) \<Longrightarrow>
+ wadjust_backto_standard_pos_B m rs ([], Bk # Oc # list)"
+apply(auto simp: wadjust_backto_standard_pos_O.simps
+ wadjust_backto_standard_pos_B.simps)
+apply(rule_tac x = rn in exI, simp)
+apply(case_tac ml, simp_all add: exp_ind_def)
+done
+
+
+lemma [simp]:
+ "\<lbrakk>wadjust_backto_standard_pos_O m rs (c, Oc # list); c \<noteq> []; hd c = Bk\<rbrakk>
+ \<Longrightarrow> wadjust_backto_standard_pos_B m rs (tl c, Bk # Oc # list)"
+apply(simp add:wadjust_backto_standard_pos_O.simps
+ wadjust_backto_standard_pos_B.simps, auto)
+apply(case_tac [!] ml, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>wadjust_backto_standard_pos_O m rs (c, Oc # list); c \<noteq> []; hd c = Oc\<rbrakk>
+ \<Longrightarrow> wadjust_backto_standard_pos_O m rs (tl c, Oc # Oc # list)"
+apply(simp add: wadjust_backto_standard_pos_O.simps, auto)
+apply(case_tac ml, simp_all add: exp_ind_def, auto)
+apply(rule_tac x = nat in exI, auto simp: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_backto_standard_pos m rs (c, Oc # list)
+ \<Longrightarrow> (c = [] \<longrightarrow> wadjust_backto_standard_pos m rs ([], Bk # Oc # list)) \<and>
+ (c \<noteq> [] \<longrightarrow> wadjust_backto_standard_pos m rs (tl c, hd c # Oc # list))"
+apply(auto simp: wadjust_backto_standard_pos.simps)
+apply(case_tac "hd c", simp_all)
+done
+thm wadjust_loop_right_move.simps
+
+lemma [simp]: "wadjust_loop_right_move m rs (c, []) = False"
+apply(simp only: wadjust_loop_right_move.simps)
+apply(rule_tac iffI)
+apply(erule_tac exE)+
+apply(case_tac nr, simp_all add: exp_ind_def)
+apply(case_tac mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_loop_erase m rs (c, []) = False"
+apply(simp only: wadjust_loop_erase.simps, auto)
+apply(case_tac mr, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>Suc (Suc rs) = a; wadjust_loop_erase m rs (c, Bk # list)\<rbrakk>
+ \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list))))
+ < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list)))) \<or>
+ a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) =
+ a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))"
+apply(simp only: wadjust_loop_erase.simps)
+apply(rule_tac disjI2)
+apply(case_tac c, simp, simp)
+done
+
+lemma [simp]:
+ "\<lbrakk>Suc (Suc rs) = a; wadjust_loop_on_left_moving m rs (c, Bk # list)\<rbrakk>
+ \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list))))
+ < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list)))) \<or>
+ a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) =
+ a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))"
+apply(subgoal_tac "c \<noteq> []")
+apply(case_tac c, simp_all)
+done
+
+lemma dropWhile_exp1: "dropWhile (\<lambda>a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = dropWhile (\<lambda>a. a = Oc) xs"
+apply(induct n, simp_all add: exp_ind_def)
+done
+lemma takeWhile_exp1: "takeWhile (\<lambda>a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = Oc\<^bsup>n\<^esup> @ takeWhile (\<lambda>a. a = Oc) xs"
+apply(induct n, simp_all add: exp_ind_def)
+done
+
+lemma [simp]: "\<lbrakk>Suc (Suc rs) = a; wadjust_loop_right_move2 m rs (c, Bk # list)\<rbrakk>
+ \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list))))
+ < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))"
+apply(simp add: wadjust_loop_right_move2.simps, auto)
+apply(simp add: dropWhile_exp1 takeWhile_exp1)
+apply(case_tac ln, simp, simp add: exp_ind_def)
+done
+
+lemma [simp]: "wadjust_loop_check m rs ([], b) = False"
+apply(simp add: wadjust_loop_check.simps)
+done
+
+lemma [simp]: "\<lbrakk>Suc (Suc rs) = a; wadjust_loop_check m rs (c, Oc # list)\<rbrakk>
+ \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Oc # list))))
+ < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list)))) \<or>
+ a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Oc # list)))) =
+ a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list))))"
+apply(case_tac "c", simp_all)
+done
+
+lemma [simp]:
+ "\<lbrakk>Suc (Suc rs) = a; wadjust_loop_erase m rs (c, Oc # list)\<rbrakk>
+ \<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))
+ < a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list)))) \<or>
+ a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list)))) =
+ a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list))))"
+apply(simp add: wadjust_loop_erase.simps)
+apply(rule_tac disjI2)
+apply(auto)
+apply(simp add: dropWhile_exp1 takeWhile_exp1)
+done
+
+declare numeral_2_eq_2[simp del]
+
+lemma wadjust_correctness:
+ shows "let P = (\<lambda> (len, st, l, r). st = 0) in
+ let Q = (\<lambda> (len, st, l, r). wadjust_inv st m rs (l, r)) in
+ let f = (\<lambda> stp. (Suc (Suc rs), steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>,
+ Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp)) in
+ \<exists> n .P (f n) \<and> Q (f n)"
+proof -
+ let ?P = "(\<lambda> (len, st, l, r). st = 0)"
+ let ?Q = "\<lambda> (len, st, l, r). wadjust_inv st m rs (l, r)"
+ let ?f = "\<lambda> stp. (Suc (Suc rs), steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>,
+ Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp)"
+ have "\<exists> n. ?P (?f n) \<and> ?Q (?f n)"
+ proof(rule_tac halt_lemma2)
+ show "wf wadjust_le" by auto
+ next
+ show "\<forall> n. \<not> ?P (?f n) \<and> ?Q (?f n) \<longrightarrow>
+ ?Q (?f (Suc n)) \<and> (?f (Suc n), ?f n) \<in> wadjust_le"
+ proof(rule_tac allI, rule_tac impI, case_tac "?f n",
+ simp add: tstep_red tstep.simps, rule_tac conjI, erule_tac conjE,
+ erule_tac conjE)
+ fix n a b c d
+ assume "0 < b" "wadjust_inv b m rs (c, d)" "Suc (Suc rs) = a"
+ thus "case case fetch t_wcode_adjust b (case d of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)
+ of (ac, ns) \<Rightarrow> (ns, new_tape ac (c, d)) of (st, x) \<Rightarrow> wadjust_inv st m rs x"
+ apply(case_tac d, simp, case_tac [2] aa)
+ apply(simp_all add: wadjust_inv.simps wadjust_le_def new_tape.simps
+ abacus.lex_triple_def abacus.lex_pair_def lex_square_def
+ split: if_splits)
+ done
+ next
+ fix n a b c d
+ assume "0 < b \<and> wadjust_inv b m rs (c, d)"
+ "Suc (Suc rs) = a \<and> steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>,
+ Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust n = (b, c, d)"
+ thus "((a, case fetch t_wcode_adjust b (case d of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)
+ of (ac, ns) \<Rightarrow> (ns, new_tape ac (c, d))), a, b, c, d) \<in> wadjust_le"
+ proof(erule_tac conjE, erule_tac conjE, erule_tac conjE)
+ assume "0 < b" "wadjust_inv b m rs (c, d)" "Suc (Suc rs) = a"
+ thus "?thesis"
+ apply(case_tac d, case_tac [2] aa)
+ apply(simp_all add: wadjust_inv.simps wadjust_le_def new_tape.simps
+ abacus.lex_triple_def abacus.lex_pair_def lex_square_def
+ split: if_splits)
+ done
+ qed
+ qed
+ next
+ show "?Q (?f 0)"
+ apply(simp add: steps.simps wadjust_inv.simps wadjust_start.simps)
+ apply(rule_tac x = ln in exI,auto)
+ done
+ next
+ show "\<not> ?P (?f 0)"
+ apply(simp add: steps.simps)
+ done
+ qed
+ thus "?thesis"
+ apply(auto)
+ done
+qed
+
+lemma [intro]: "t_correct t_wcode_adjust"
+apply(auto simp: t_wcode_adjust_def t_correct.simps iseven_def)
+apply(rule_tac x = 11 in exI, simp)
+done
+
+lemma wcode_lemma_pre':
+ "args \<noteq> [] \<Longrightarrow>
+ \<exists> stp rn. steps (Suc 0, [], <m # args>)
+ ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp
+ = (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+proof -
+ let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <m # args>"
+ let ?Q1 = "\<lambda>(l, r). l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ (\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ let ?P2 = ?Q1
+ let ?Q2 = "\<lambda> (l, r). (wadjust_stop m (bl_bin (<args>) - 1) (l, r))"
+ let ?P3 = "\<lambda> tp. False"
+ assume h: "args \<noteq> []"
+ have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp)
+ ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp = (0, tp') \<and> ?Q2 tp')"
+ proof(rule_tac turing_merge.t_merge_halt[of "t_wcode_prepare |+| t_wcode_main"
+ t_wcode_adjust ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2],
+ auto simp: turing_merge_def)
+
+ show "\<exists>stp. case steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp of
+ (st, tp') \<Rightarrow> st = 0 \<and> (case tp' of (l, r) \<Rightarrow> l = Bk # Oc\<^bsup>Suc m\<^esup> \<and>
+ (\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>))"
+ using h prepare_mainpart_lemma[of args m]
+ apply(auto)
+ apply(rule_tac x = stp in exI, simp)
+ apply(rule_tac x = ln in exI, auto)
+ done
+ next
+ fix ln rn
+ show "\<exists>stp. case steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk #
+ Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp of
+ (st, tp') \<Rightarrow> st = 0 \<and> wadjust_stop m (bl_bin (<args>) - Suc 0) tp'"
+ using wadjust_correctness[of m "bl_bin (<args>) - 1" "Suc ln" rn]
+ apply(subgoal_tac "bl_bin (<args>) > 0", auto simp: wadjust_inv.simps)
+ apply(rule_tac x = n in exI, simp add: exp_ind)
+ using h
+ apply(case_tac args, simp_all, case_tac list,
+ simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def
+ bl_bin.simps)
+ done
+ next
+ show "?Q1 \<turnstile>-> ?P2"
+ by(simp add: t_imply_def)
+ qed
+ thus "\<exists>stp rn. steps (Suc 0, [], <m # args>) ((t_wcode_prepare |+| t_wcode_main) |+|
+ t_wcode_adjust) stp = (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ apply(simp add: t_imply_def)
+ apply(erule_tac exE)+
+ apply(subgoal_tac "bl_bin (<args>) > 0", auto simp: wadjust_stop.simps)
+ using h
+ apply(case_tac args, simp_all, case_tac list,
+ simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def
+ bl_bin.simps)
+ done
+qed
+
+text {*
+ The initialization TM @{text "t_wcode"}.
+ *}
+definition t_wcode :: "tprog"
+ where
+ "t_wcode = (t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust"
+
+
+text {*
+ The correctness of @{text "t_wcode"}.
+ *}
+lemma wcode_lemma_1:
+ "args \<noteq> [] \<Longrightarrow>
+ \<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode) stp =
+ (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+apply(simp add: wcode_lemma_pre' t_wcode_def)
+done
+
+lemma wcode_lemma:
+ "args \<noteq> [] \<Longrightarrow>
+ \<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode) stp =
+ (0, [Bk], <[m ,bl_bin (<args>)]> @ Bk\<^bsup>rn\<^esup>)"
+using wcode_lemma_1[of args m]
+apply(simp add: t_wcode_def tape_of_nl_abv tape_of_nat_list.simps)
+done
+
+section {* The universal TM *}
+
+text {*
+ This section gives the explicit construction of {\em Universal Turing Machine}, defined as @{text "UTM"} and proves its
+ correctness. It is pretty easy by composing the partial results we have got so far.
+ *}
+
+
+definition UTM :: "tprog"
+ where
+ "UTM = (let (aprog, rs_pos, a_md) = rec_ci rec_F in
+ let abc_F = aprog [+] dummy_abc (Suc (Suc 0)) in
+ (t_wcode |+| (tm_of abc_F @ tMp (Suc (Suc 0)) (start_of (layout_of abc_F)
+ (length abc_F) - Suc 0))))"
+
+definition F_aprog :: "abc_prog"
+ where
+ "F_aprog \<equiv> (let (aprog, rs_pos, a_md) = rec_ci rec_F in
+ aprog [+] dummy_abc (Suc (Suc 0)))"
+
+definition F_tprog :: "tprog"
+ where
+ "F_tprog = tm_of (F_aprog)"
+
+definition t_utm :: "tprog"
+ where
+ "t_utm \<equiv>
+ (F_tprog) @ tMp (Suc (Suc 0)) (start_of (layout_of (F_aprog))
+ (length (F_aprog)) - Suc 0)"
+
+definition UTM_pre :: "tprog"
+ where
+ "UTM_pre = t_wcode |+| t_utm"
+
+lemma F_abc_halt_eq:
+ "\<lbrakk>turing_basic.t_correct tp;
+ length lm = k;
+ steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>);
+ rs > 0\<rbrakk>
+ \<Longrightarrow> \<exists> stp m. abc_steps_l (0, [code tp, bl2wc (<lm>)]) (F_aprog) stp =
+ (length (F_aprog), code tp # bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>)"
+apply(drule_tac F_t_halt_eq, simp, simp, simp)
+apply(case_tac "rec_ci rec_F")
+apply(frule_tac abc_append_dummy_complie, simp, simp, erule_tac exE,
+ erule_tac exE)
+apply(rule_tac x = stp in exI, rule_tac x = m in exI)
+apply(simp add: F_aprog_def dummy_abc_def)
+done
+
+lemma F_abc_utm_halt_eq:
+ "\<lbrakk>rs > 0;
+ abc_steps_l (0, [code tp, bl2wc (<lm>)]) F_aprog stp =
+ (length F_aprog, code tp # bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>)\<rbrakk>
+ \<Longrightarrow> \<exists>stp m n.(steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp =
+ (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>))"
+ thm abacus_turing_eq_halt
+ using abacus_turing_eq_halt
+ [of "layout_of F_aprog" "F_aprog" "F_tprog" "length (F_aprog)"
+ "[code tp, bl2wc (<lm>)]" stp "code tp # bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>" "Suc (Suc 0)"
+ "start_of (layout_of (F_aprog)) (length (F_aprog))" "[]" 0]
+apply(simp add: F_tprog_def t_utm_def abc_lm_v.simps nth_append)
+apply(erule_tac exE)+
+apply(rule_tac x = stpa in exI, rule_tac x = "Suc (Suc ma)" in exI,
+ rule_tac x = l in exI, simp add: exp_ind)
+done
+
+declare tape_of_nl_abv_cons[simp del]
+
+lemma t_utm_halt_eq':
+ "\<lbrakk>turing_basic.t_correct tp;
+ 0 < rs;
+ steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\<rbrakk>
+ \<Longrightarrow> \<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp =
+ (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+apply(drule_tac l = l in F_abc_halt_eq, simp, simp, simp)
+apply(erule_tac exE, erule_tac exE)
+apply(rule_tac F_abc_utm_halt_eq, simp_all)
+done
+
+lemma [simp]: "tinres xs (xs @ Bk\<^bsup>i\<^esup>)"
+apply(auto simp: tinres_def)
+done
+
+lemma [elim]: "\<lbrakk>rs > 0; Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup> = c @ Bk\<^bsup>n\<^esup>\<rbrakk>
+ \<Longrightarrow> \<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+apply(case_tac "na > n")
+apply(subgoal_tac "\<exists> d. na = d + n", auto simp: exp_add)
+apply(rule_tac x = "na - n" in exI, simp)
+apply(subgoal_tac "\<exists> d. n = d + na", auto simp: exp_add)
+apply(case_tac rs, simp_all add: exp_ind, case_tac d,
+ simp_all add: exp_ind)
+apply(rule_tac x = "n - na" in exI, simp)
+done
+
+
+lemma t_utm_halt_eq'':
+ "\<lbrakk>turing_basic.t_correct tp;
+ 0 < rs;
+ steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\<rbrakk>
+ \<Longrightarrow> \<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp =
+ (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+apply(drule_tac t_utm_halt_eq', simp_all)
+apply(erule_tac exE)+
+proof -
+ fix stpa ma na
+ assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
+ and gr: "rs > 0"
+ thus "\<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+ apply(rule_tac x = stpa in exI, rule_tac x = ma in exI, simp)
+ proof(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp)
+ fix a b c
+ assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
+ "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)"
+ thus " a = 0 \<and> b = Bk\<^bsup>ma\<^esup> \<and> (\<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+ using tinres_steps2[of "<[code tp, bl2wc (<lm>)]>" "<[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>"
+ "Suc 0" " [Bk, Bk]" t_utm stpa 0 "Bk\<^bsup>ma\<^esup>" "Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>" a b c]
+ apply(simp)
+ using gr
+ apply(simp only: tinres_def, auto)
+ apply(rule_tac x = "na + n" in exI, simp add: exp_add)
+ done
+ qed
+qed
+
+lemma [simp]: "tinres [Bk, Bk] [Bk]"
+apply(auto simp: tinres_def)
+done
+
+lemma [elim]: "Bk\<^bsup>ma\<^esup> = b @ Bk\<^bsup>n\<^esup> \<Longrightarrow> \<exists>m. b = Bk\<^bsup>m\<^esup>"
+apply(subgoal_tac "ma = length b + n")
+apply(rule_tac x = "ma - n" in exI, simp add: exp_add)
+apply(drule_tac length_equal)
+apply(simp)
+done
+
+lemma t_utm_halt_eq:
+ "\<lbrakk>turing_basic.t_correct tp;
+ 0 < rs;
+ steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\<rbrakk>
+ \<Longrightarrow> \<exists>stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp =
+ (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+apply(drule_tac i = i in t_utm_halt_eq'', simp_all)
+apply(erule_tac exE)+
+proof -
+ fix stpa ma na
+ assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
+ and gr: "rs > 0"
+ thus "\<exists>stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+ apply(rule_tac x = stpa in exI)
+ proof(case_tac "steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp)
+ fix a b c
+ assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)"
+ "steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)"
+ thus "a = 0 \<and> (\<exists>m. b = Bk\<^bsup>m\<^esup>) \<and> (\<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+ using tinres_steps[of "[Bk, Bk]" "[Bk]" "Suc 0" "<[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>" t_utm stpa 0
+ "Bk\<^bsup>ma\<^esup>" "Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>" a b c]
+ apply(simp)
+ apply(auto simp: tinres_def)
+ apply(rule_tac x = "ma + n" in exI, simp add: exp_add)
+ done
+ qed
+qed
+
+lemma [intro]: "t_correct t_wcode"
+apply(simp add: t_wcode_def)
+apply(auto)
+done
+
+lemma [intro]: "t_correct t_utm"
+apply(simp add: t_utm_def F_tprog_def)
+apply(rule_tac t_compiled_correct, auto)
+done
+
+lemma UTM_halt_lemma_pre:
+ "\<lbrakk>turing_basic.t_correct tp;
+ 0 < rs;
+ args \<noteq> [];
+ steps (Suc 0, Bk\<^bsup>i\<^esup>, <args::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\<rbrakk>
+ \<Longrightarrow> \<exists>stp m n. steps (Suc 0, [], <code tp # args>) UTM_pre stp =
+ (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+proof -
+ let ?Q2 = "\<lambda> (l, r). (\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> \<and> r = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>rn\<^esup>)"
+ term ?Q2
+ let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <code tp # args>"
+ let ?Q1 = "\<lambda> (l, r). (l = [Bk] \<and>
+ (\<exists> rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))"
+ let ?P2 = ?Q1
+ let ?P3 = "\<lambda> (l, r). False"
+ assume h: "turing_basic.t_correct tp" "0 < rs"
+ "args \<noteq> []" "steps (Suc 0, Bk\<^bsup>i\<^esup>, <args::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)"
+ have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp)
+ (t_wcode |+| t_utm) stp = (0, tp') \<and> ?Q2 tp')"
+ proof(rule_tac turing_merge.t_merge_halt [of "t_wcode" "t_utm"
+ ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2], auto simp: turing_merge_def)
+ show "\<exists>stp. case steps (Suc 0, [], <code tp # args>) t_wcode stp of (st, tp') \<Rightarrow>
+ st = 0 \<and> (case tp' of (l, r) \<Rightarrow> l = [Bk] \<and>
+ (\<exists>rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))"
+ using wcode_lemma_1[of args "code tp"] h
+ apply(simp, auto)
+ apply(rule_tac x = stpa in exI, auto)
+ done
+ next
+ fix rn
+ show "\<exists>stp. case steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @
+ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp of
+ (st, tp') \<Rightarrow> st = 0 \<and> (case tp' of (l, r) \<Rightarrow>
+ (\<exists>ln. l = Bk\<^bsup>ln\<^esup>) \<and> (\<exists>rn. r = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>rn\<^esup>))"
+ using t_utm_halt_eq[of tp rs i args stp m k rn] h
+ apply(auto)
+ apply(rule_tac x = stpa in exI, simp add: bin_wc_eq
+ tape_of_nat_list.simps tape_of_nl_abv)
+ apply(auto)
+ done
+ next
+ show "?Q1 \<turnstile>-> ?P2"
+ apply(simp add: t_imply_def)
+ done
+ qed
+ thus "?thesis"
+ apply(simp add: t_imply_def)
+ apply(auto simp: UTM_pre_def)
+ done
+qed
+
+text {*
+ The correctness of @{text "UTM"}, the halt case.
+*}
+lemma UTM_halt_lemma:
+ "\<lbrakk>turing_basic.t_correct tp;
+ 0 < rs;
+ args \<noteq> [];
+ steps (Suc 0, Bk\<^bsup>i\<^esup>, <args::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\<rbrakk>
+ \<Longrightarrow> \<exists>stp m n. steps (Suc 0, [], <code tp # args>) UTM stp =
+ (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)"
+using UTM_halt_lemma_pre[of tp rs args i stp m k]
+apply(simp add: UTM_pre_def t_utm_def UTM_def F_aprog_def F_tprog_def)
+apply(case_tac "rec_ci rec_F", simp)
+done
+
+definition TSTD:: "t_conf \<Rightarrow> bool"
+ where
+ "TSTD c = (let (st, l, r) = c in
+ st = 0 \<and> (\<exists> m. l = Bk\<^bsup>m\<^esup>) \<and> (\<exists> rs n. r = Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>))"
+
+thm abacus_turing_eq_uhalt
+
+lemma nstd_case1: "0 < a \<Longrightarrow> NSTD (trpl_code (a, b, c))"
+apply(simp add: NSTD.simps trpl_code.simps)
+done
+
+lemma [simp]: "\<forall>m. b \<noteq> Bk\<^bsup>m\<^esup> \<Longrightarrow> 0 < bl2wc b"
+apply(rule classical, simp)
+apply(induct b, erule_tac x = 0 in allE, simp)
+apply(simp add: bl2wc.simps, case_tac a, simp_all
+ add: bl2nat.simps bl2nat_double)
+apply(case_tac "\<exists> m. b = Bk\<^bsup>m\<^esup>", erule exE)
+apply(erule_tac x = "Suc m" in allE, simp add: exp_ind_def, simp)
+done
+lemma nstd_case2: "\<forall>m. b \<noteq> Bk\<^bsup>m\<^esup> \<Longrightarrow> NSTD (trpl_code (a, b, c))"
+apply(simp add: NSTD.simps trpl_code.simps)
+done
+
+thm lg.simps
+thm lgR.simps
+
+lemma [elim]: "Suc (2 * x) = 2 * y \<Longrightarrow> RR"
+apply(induct x arbitrary: y, simp, simp)
+apply(case_tac y, simp, simp)
+done
+
+lemma bl2nat_zero_eq[simp]: "(bl2nat c 0 = 0) = (\<exists>n. c = Bk\<^bsup>n\<^esup>)"
+apply(auto)
+apply(induct c, simp add: bl2nat.simps)
+apply(rule_tac x = 0 in exI, simp)
+apply(case_tac a, auto simp: bl2nat.simps bl2nat_double)
+done
+
+lemma bl2wc_exp_ex:
+ "\<lbrakk>Suc (bl2wc c) = 2 ^ m\<rbrakk> \<Longrightarrow> \<exists> rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+apply(induct c arbitrary: m, simp add: bl2wc.simps bl2nat.simps)
+apply(case_tac a, auto)
+apply(case_tac m, simp_all add: bl2wc.simps, auto)
+apply(rule_tac x = 0 in exI, rule_tac x = "Suc n" in exI,
+ simp add: exp_ind_def)
+apply(simp add: bl2wc.simps bl2nat.simps bl2nat_double)
+apply(case_tac m, simp, simp)
+proof -
+ fix c m nat
+ assume ind:
+ "\<And>m. Suc (bl2nat c 0) = 2 ^ m \<Longrightarrow> \<exists>rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+ and h:
+ "Suc (Suc (2 * bl2nat c 0)) = 2 * 2 ^ nat"
+ have "\<exists>rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+ apply(rule_tac m = nat in ind)
+ using h
+ apply(simp)
+ done
+ from this obtain rs n where " c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" by blast
+ thus "\<exists>rs n. Oc # c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>"
+ apply(rule_tac x = "Suc rs" in exI, simp add: exp_ind_def)
+ apply(rule_tac x = n in exI, simp)
+ done
+qed
+
+lemma [elim]:
+ "\<lbrakk>\<forall>rs n. c \<noteq> Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>;
+ bl2wc c = 2 ^ lg (Suc (bl2wc c)) 2 - Suc 0\<rbrakk> \<Longrightarrow> bl2wc c = 0"
+apply(subgoal_tac "\<exists> m. Suc (bl2wc c) = 2^m", erule_tac exE)
+apply(drule_tac bl2wc_exp_ex, simp, erule_tac exE, erule_tac exE)
+apply(case_tac rs, simp, simp, erule_tac x = nat in allE,
+ erule_tac x = n in allE, simp)
+using bl2wc_exp_ex[of c "lg (Suc (bl2wc c)) 2"]
+apply(case_tac "(2::nat) ^ lg (Suc (bl2wc c)) 2",
+ simp, simp, erule_tac exE, erule_tac exE, simp)
+apply(simp add: bl2wc.simps)
+apply(rule_tac x = rs in exI)
+apply(case_tac "(2::nat)^rs", simp, simp)
+done
+
+lemma nstd_case3:
+ "\<forall>rs n. c \<noteq> Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup> \<Longrightarrow> NSTD (trpl_code (a, b, c))"
+apply(simp add: NSTD.simps trpl_code.simps)
+apply(rule_tac impI)
+apply(rule_tac disjI2, rule_tac disjI2, auto)
+done
+
+lemma NSTD_1: "\<not> TSTD (a, b, c)
+ \<Longrightarrow> rec_exec rec_NSTD [trpl_code (a, b, c)] = Suc 0"
+ using NSTD_lemma1[of "trpl_code (a, b, c)"]
+ NSTD_lemma2[of "trpl_code (a, b, c)"]
+ apply(simp add: TSTD_def)
+ apply(erule_tac disjE, erule_tac nstd_case1)
+ apply(erule_tac disjE, erule_tac nstd_case2)
+ apply(erule_tac nstd_case3)
+ done
+
+lemma nonstop_t_uhalt_eq:
+ "\<lbrakk>turing_basic.t_correct tp;
+ steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp = (a, b, c);
+ \<not> TSTD (a, b, c)\<rbrakk>
+ \<Longrightarrow> rec_exec rec_nonstop [code tp, bl2wc (<lm>), stp] = Suc 0"
+apply(simp add: rec_nonstop_def rec_exec.simps)
+apply(subgoal_tac
+ "rec_exec rec_conf [code tp, bl2wc (<lm>), stp] =
+ trpl_code (a, b, c)", simp)
+apply(erule_tac NSTD_1)
+using rec_t_eq_steps[of tp l lm stp]
+apply(simp)
+done
+
+lemma nonstop_true:
+ "\<lbrakk>turing_basic.t_correct tp;
+ \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk>
+ \<Longrightarrow> \<forall>y. rec_calc_rel rec_nonstop
+ ([code tp, bl2wc (<lm>), y]) (Suc 0)"
+apply(rule_tac allI, erule_tac x = y in allE)
+apply(case_tac "steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp y", simp)
+apply(rule_tac nonstop_t_uhalt_eq, simp_all)
+done
+
+(*
+lemma [simp]:
+ "\<forall>j<Suc k. Ex (rec_calc_rel (get_fstn_args (Suc k) (Suc k) ! j)
+ (code tp # lm))"
+apply(auto simp: get_fstn_args_nth)
+apply(rule_tac x = "(code tp # lm) ! j" in exI)
+apply(rule_tac calc_id, simp_all)
+done
+*)
+declare ci_cn_para_eq[simp]
+
+lemma F_aprog_uhalt:
+ "\<lbrakk>turing_basic.t_correct tp;
+ \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp));
+ rec_ci rec_F = (F_ap, rs_pos, a_md)\<rbrakk>
+ \<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)] @ 0\<^bsup>a_md - rs_pos \<^esup>
+ @ suflm) (F_ap) stp of (ss, e) \<Rightarrow> ss < length (F_ap)"
+apply(case_tac "rec_ci (Cn (Suc (Suc 0)) rec_right [Cn (Suc (Suc 0)) rec_conf
+ ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])])")
+apply(simp only: rec_F_def, rule_tac i = 0 and ga = a and gb = b and
+ gc = c in cn_gi_uhalt, simp, simp, simp, simp, simp, simp, simp)
+apply(simp add: ci_cn_para_eq)
+apply(case_tac "rec_ci (Cn (Suc (Suc 0)) rec_conf
+ ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt]))")
+apply(rule_tac rf = "(Cn (Suc (Suc 0)) rec_right [Cn (Suc (Suc 0)) rec_conf
+ ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])])"
+ and n = "Suc (Suc 0)" and f = rec_right and
+ gs = "[Cn (Suc (Suc 0)) rec_conf
+ ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])]"
+ and i = 0 and ga = aa and gb = ba and gc = ca in
+ cn_gi_uhalt)
+apply(simp, simp, simp, simp, simp, simp, simp,
+ simp add: ci_cn_para_eq)
+apply(case_tac "rec_ci rec_halt")
+apply(rule_tac rf = "(Cn (Suc (Suc 0)) rec_conf
+ ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt]))"
+ and n = "Suc (Suc 0)" and f = "rec_conf" and
+ gs = "([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])" and
+ i = "Suc (Suc 0)" and gi = "rec_halt" and ga = ab and gb = bb and
+ gc = cb in cn_gi_uhalt)
+apply(simp, simp, simp, simp, simp add: nth_append, simp,
+ simp add: nth_append, simp add: rec_halt_def)
+apply(simp only: rec_halt_def)
+apply(case_tac [!] "rec_ci ((rec_nonstop))")
+apply(rule_tac allI, rule_tac impI, simp)
+apply(case_tac j, simp)
+apply(rule_tac x = "code tp" in exI, rule_tac calc_id, simp, simp, simp, simp)
+apply(rule_tac x = "bl2wc (<lm>)" in exI, rule_tac calc_id, simp, simp, simp)
+apply(rule_tac rf = "Mn (Suc (Suc 0)) (rec_nonstop)"
+ and f = "(rec_nonstop)" and n = "Suc (Suc 0)"
+ and aprog' = ac and rs_pos' = bc and a_md' = cc in Mn_unhalt)
+apply(simp, simp add: rec_halt_def , simp, simp)
+apply(drule_tac nonstop_true, simp_all)
+apply(rule_tac allI)
+apply(erule_tac x = y in allE)+
+apply(simp)
+done
+
+thm abc_list_crsp_steps
+
+lemma uabc_uhalt':
+ "\<lbrakk>turing_basic.t_correct tp;
+ \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp));
+ rec_ci rec_F = (ap, pos, md)\<rbrakk>
+ \<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) ap stp of (ss, e)
+ \<Rightarrow> ss < length ap"
+proof(frule_tac F_ap = ap and rs_pos = pos and a_md = md
+ and suflm = "[]" in F_aprog_uhalt, auto)
+ fix stp a b
+ assume h:
+ "\<forall>stp. case abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp of
+ (ss, e) \<Rightarrow> ss < length ap"
+ "abc_steps_l (0, [code tp, bl2wc (<lm>)]) ap stp = (a, b)"
+ "turing_basic.t_correct tp"
+ "rec_ci rec_F = (ap, pos, md)"
+ moreover have "ap \<noteq> []"
+ using h apply(rule_tac rec_ci_not_null, simp)
+ done
+ ultimately show "a < length ap"
+ proof(erule_tac x = stp in allE,
+ case_tac "abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp", simp)
+ fix aa ba
+ assume g: "aa < length ap"
+ "abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp = (aa, ba)"
+ "ap \<noteq> []"
+ thus "?thesis"
+ using abc_list_crsp_steps[of "[code tp, bl2wc (<lm>)]"
+ "md - pos" ap stp aa ba] h
+ apply(simp)
+ done
+ qed
+qed
+
+lemma uabc_uhalt:
+ "\<lbrakk>turing_basic.t_correct tp;
+ \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk>
+ \<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) F_aprog
+ stp of (ss, e) \<Rightarrow> ss < length F_aprog"
+apply(case_tac "rec_ci rec_F", simp add: F_aprog_def)
+thm uabc_uhalt'
+apply(drule_tac ap = a and pos = b and md = c in uabc_uhalt', simp_all)
+proof -
+ fix a b c
+ assume
+ "\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) a stp of (ss, e)
+ \<Rightarrow> ss < length a"
+ "rec_ci rec_F = (a, b, c)"
+ thus
+ "\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)])
+ (a [+] dummy_abc (Suc (Suc 0))) stp of (ss, e) \<Rightarrow>
+ ss < Suc (Suc (Suc (length a)))"
+ using abc_append_uhalt1[of a "[code tp, bl2wc (<lm>)]"
+ "a [+] dummy_abc (Suc (Suc 0))" "[]" "dummy_abc (Suc (Suc 0))"]
+ apply(simp)
+ done
+qed
+
+lemma tutm_uhalt':
+ "\<lbrakk>turing_basic.t_correct tp;
+ \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk>
+ \<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp)"
+ using abacus_turing_eq_uhalt[of "layout_of (F_aprog)"
+ "F_aprog" "F_tprog" "[code tp, bl2wc (<lm>)]"
+ "start_of (layout_of (F_aprog )) (length (F_aprog))"
+ "Suc (Suc 0)"]
+apply(simp add: F_tprog_def)
+apply(subgoal_tac "\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)])
+ (F_aprog) stp of (as, am) \<Rightarrow> as < length (F_aprog)", simp)
+thm abacus_turing_eq_uhalt
+apply(simp add: t_utm_def F_tprog_def)
+apply(rule_tac uabc_uhalt, simp_all)
+done
+
+lemma tinres_commute: "tinres r r' \<Longrightarrow> tinres r' r"
+apply(auto simp: tinres_def)
+done
+
+lemma inres_tape:
+ "\<lbrakk>steps (st, l, r) tp stp = (a, b, c); steps (st, l', r') tp stp = (a', b', c');
+ tinres l l'; tinres r r'\<rbrakk>
+ \<Longrightarrow> a = a' \<and> tinres b b' \<and> tinres c c'"
+proof(case_tac "steps (st, l', r) tp stp")
+ fix aa ba ca
+ assume h: "steps (st, l, r) tp stp = (a, b, c)"
+ "steps (st, l', r') tp stp = (a', b', c')"
+ "tinres l l'" "tinres r r'"
+ "steps (st, l', r) tp stp = (aa, ba, ca)"
+ have "tinres b ba \<and> c = ca \<and> a = aa"
+ using h
+ apply(rule_tac tinres_steps, auto)
+ done
+
+ thm tinres_steps2
+ moreover have "b' = ba \<and> tinres c' ca \<and> a' = aa"
+ using h
+ apply(rule_tac tinres_steps2, auto intro: tinres_commute)
+ done
+ ultimately show "?thesis"
+ apply(auto intro: tinres_commute)
+ done
+qed
+
+lemma tape_normalize: "\<forall> stp. \<not> isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp)
+ \<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>n\<^esup>) t_utm stp)"
+apply(rule_tac allI, case_tac "(steps (Suc 0, Bk\<^bsup>m\<^esup>,
+ <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>n\<^esup>) t_utm stp)", simp add: isS0_def)
+apply(erule_tac x = stp in allE)
+apply(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp", simp)
+apply(drule_tac inres_tape, auto)
+apply(auto simp: tinres_def)
+apply(case_tac "m > Suc (Suc 0)")
+apply(rule_tac x = "m - Suc (Suc 0)" in exI)
+apply(case_tac m, simp_all add: exp_ind_def, case_tac nat, simp_all add: exp_ind_def)
+apply(rule_tac x = "2 - m" in exI, simp add: exp_ind_def[THEN sym] exp_add[THEN sym])
+apply(simp only: numeral_2_eq_2, simp add: exp_ind_def)
+done
+
+lemma tutm_uhalt:
+ "\<lbrakk>turing_basic.t_correct tp;
+ \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp))\<rbrakk>
+ \<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc (<args>)]> @ Bk\<^bsup>n\<^esup>) t_utm stp)"
+apply(rule_tac tape_normalize)
+apply(rule_tac tutm_uhalt', simp_all)
+done
+
+lemma UTM_uhalt_lemma_pre:
+ "\<lbrakk>turing_basic.t_correct tp;
+ \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp));
+ args \<noteq> []\<rbrakk>
+ \<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, [], <code tp # args>) UTM_pre stp)"
+proof -
+ let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <code tp # args>"
+ let ?Q1 = "\<lambda> (l, r). (l = [Bk] \<and>
+ (\<exists> rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))"
+ let ?P4 = ?Q1
+ let ?P3 = "\<lambda> (l, r). False"
+ assume h: "turing_basic.t_correct tp" "\<forall>stp. \<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp)"
+ "args \<noteq> []"
+ have "?P1 \<turnstile>-> \<lambda> tp. \<not> (\<exists> stp. isS0 (steps (Suc 0, tp) (t_wcode |+| t_utm) stp))"
+ proof(rule_tac turing_merge.t_merge_uhalt [of "t_wcode" "t_utm"
+ ?P1 ?P3 ?P3 ?P4 ?Q1 ?P3], auto simp: turing_merge_def)
+ show "\<exists>stp. case steps (Suc 0, [], <code tp # args>) t_wcode stp of (st, tp') \<Rightarrow>
+ st = 0 \<and> (case tp' of (l, r) \<Rightarrow> l = [Bk] \<and>
+ (\<exists>rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))"
+ using wcode_lemma_1[of args "code tp"] h
+ apply(simp, auto)
+ apply(rule_tac x = stp in exI, auto)
+ done
+ next
+ fix rn stp
+ show " isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp)
+ \<Longrightarrow> False"
+ using tutm_uhalt[of tp l args "Suc 0" rn] h
+ apply(simp)
+ apply(erule_tac x = stp in allE)
+ apply(simp add: tape_of_nl_abv tape_of_nat_list.simps bin_wc_eq)
+ done
+ next
+ fix rn stp
+ show "isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp) \<Longrightarrow>
+ isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp)"
+ by simp
+ next
+ show "?Q1 \<turnstile>-> ?P4"
+ apply(simp add: t_imply_def)
+ done
+ qed
+ thus "?thesis"
+ apply(simp add: t_imply_def UTM_pre_def)
+ done
+qed
+
+text {*
+ The correctness of @{text "UTM"}, the unhalt case.
+ *}
+
+lemma UTM_uhalt_lemma:
+ "\<lbrakk>turing_basic.t_correct tp;
+ \<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp));
+ args \<noteq> []\<rbrakk>
+ \<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, [], <code tp # args>) UTM stp)"
+using UTM_uhalt_lemma_pre[of tp l args]
+apply(simp add: UTM_pre_def t_utm_def UTM_def F_aprog_def F_tprog_def)
+apply(case_tac "rec_ci rec_F", simp)
+done
+
end
\ No newline at end of file