# HG changeset patch # User zhang # Date 1350307432 0 # Node ID 48b231495281a26974fea24d0ca1351ed0528d4e # Parent 1ce04eb1c8adf9f67135bfe74f792a298195e8f7 Some illustration added together with more explanations. diff -r 1ce04eb1c8ad -r 48b231495281 utm/ROOT.ML --- a/utm/ROOT.ML Sat Sep 29 12:38:12 2012 +0000 +++ b/utm/ROOT.ML Mon Oct 15 13:23:52 2012 +0000 @@ -3,10 +3,11 @@ uncomputable.thy : The existence of Turing uncomputable functions. abacus.thy : The basic definitions of Abacus machine (An intermediate langauge underneath recursive functions) and the compilation of Abacus machines into Turing Machines. - recursive.thy : The basic defintions of Recursive Functions and the compilation of Recursive Functions into + rec_def.thy: The basic definitions of Recursive Functions. + recursive.thy : The compilation of Recursive Functions into Abacus machines. UF.thy : The construction of Universal Function, named "rec_F" and the proof of its correctness. UTM.thy: Obtaining Uinversal Turing Machine by scarfolding the Turing Machine compiled from "rec_F" with some initialization and termination processing Turing Machines. *) - no_document use_thys ["turing_basic", "uncomputable", "abacus", "recursive", "UF", "UTM"] + use_thys ["turing_basic", "uncomputable", "abacus", "rec_def", "recursive", "UF", "UTM"] diff -r 1ce04eb1c8ad -r 48b231495281 utm/UF.thy --- a/utm/UF.thy Sat Sep 29 12:38:12 2012 +0000 +++ b/utm/UF.thy Mon Oct 15 13:23:52 2012 +0000 @@ -9,8 +9,9 @@ UTM can easil be obtained by compling @{text "rec_F"} into the corresponding Turing Machine. *} - -section {* The construction of component functions *} +section {* Univeral Function *} + +subsection {* The construction of component functions *} text {* This section constructs a set of component functions used to construct @{text "rec_F"}. @@ -126,7 +127,7 @@ the effect of which is to take out the first @{text "Suc k"} arguments out of the @{text "n"} input arguments. *} -(* get_fstn_args *) + fun get_fstn_args :: "nat \ nat \ recf list" where "get_fstn_args n 0 = []" @@ -341,9 +342,6 @@ arity.simps[simp del] Sigma.simps[simp del] rec_sigma.simps[simp del] - -section {* Properties of @{text rec_sigma} *} - lemma [simp]: "arity z = 1" by(simp add: arity.simps) @@ -366,8 +364,6 @@ rec_exec g ([x] @ [rec_exec (Pr n f g) ([x])])" by(simp add: rec_exec.simps) -thm Sigma.simps - lemma Sigma_0_simp_rewrite_single_param: "Sigma f [0] = f [0]" by(simp add: Sigma.simps) @@ -1106,7 +1102,7 @@ qed text {* - @text "quo"} is the formal specification of division. + @{text "quo"} is the formal specification of division. *} fun quo :: "nat list \ nat" where @@ -1415,7 +1411,6 @@ apply(case_tac [!] "zip rgs list = []", simp) apply(subgoal_tac "rgs = [] \ list = []", simp add: Embranch.simps rec_exec.simps rec_embranch.simps) apply(rule_tac zip_null_iff, simp, simp, simp) -thm Embranch.simps proof - fix aa list assume g: @@ -1565,13 +1560,6 @@ [id 2 0]]) (Cn 3 rec_noteq [Cn 3 rec_mult [id 3 1, id 3 2], id 3 0]))]" -(* -lemma prime_lemma1: - "(rec_exec rec_prime [x] = Suc 0) \ - (rec_exec rec_prime [x] = 0)" -apply(auto simp: rec_exec.simps rec_prime_def) -done -*) declare numeral_2_eq_2[simp del] numeral_3_eq_3[simp del] lemma exec_tmp: @@ -1842,7 +1830,6 @@ by simp qed -text {*lemmas for power*} text {* @{text "rec_power"} is the recursive function used to implement power function. @@ -1888,7 +1875,6 @@ apply(simp add: rec_pi_def rec_exec.simps pi_dummy_lemma) done -text{*follows: lemmas for lo*} fun loR :: "nat list \ bool" where "loR [x, y, u] = (x mod (y^u) = 0)" @@ -2175,7 +2161,8 @@ lemma entry_lemma: "rec_exec rec_entry [str, i] = Entry str i" by(simp add: rec_entry_def rec_exec.simps lo_lemma pi_lemma) -section {* The construction of @{text "F"} *} + +subsection {* The construction of F *} text {* Using the auxilliary functions obtained in last section, @@ -2460,10 +2447,7 @@ apply(case_tac "a > 3", rule_tac x = "3" in exI, auto) apply(auto simp: rec_exec.simps) apply(erule_tac [!] Suc_Suc_Suc_Suc_induct, auto simp: rec_exec.simps) - done(* - have "Embranch (zip (map rec_exec ?rgs) (map (\r args. 0 < rec_exec r args) ?rrs)) [p, r, a] - = Embranch (zip ?gs ?rs) [p, r, a]" - apply(simp add)*) + done have k2: "Embranch (zip (map rec_exec ?rgs) (map (\r args. 0 < rec_exec r args) ?rrs)) [p, r, a] = newleft p r a" apply(simp add: Embranch.simps) apply(simp add: rec_exec.simps) @@ -2613,8 +2597,6 @@ lemma actn_lemma: "rec_exec rec_actn [m, q, r] = actn m q r" by(auto simp: rec_actn_def rec_exec.simps entry_lemma scan_lemma) -(* Stop point *) - fun newstat :: "nat \ nat \ nat \ nat" where "newstat m q r = (if q \ 0 then Entry m (4*(q - 1) + 2*scan r + 1) @@ -2919,8 +2901,6 @@ apply(simp add: NSTD_lemma2, auto) done -text {* GGGGGGGGGGGGGGGGGGGGGGG *} - text{* @{text "nonstep m r t"} means afer @{text "t"} steps of execution, the TM coded by @{text "m"} is not at a stardard final configuration. @@ -2957,8 +2937,6 @@ declare nonstop.simps[simp del] -(* when mn, use rec_calc_rel instead of rec_exec*) - lemma primerec_not0: "primerec f n \ n > 0" by(induct f n rule: primerec.induct, auto) @@ -3494,7 +3472,8 @@ done qed -section {* Coding function of TMs *} + +subsection {* Coding function of TMs *} text {* The purpose of this section is to get the coding function of Turing Machine, which is @@ -3562,7 +3541,7 @@ "code tp = (let nl = modify_tprog tp in godel_code nl)" -section {* Relating interperter functions to the execution of TMs *} +subsection {* Relating interperter functions to the execution of TMs *} lemma [simp]: "bl2wc [] = 0" by(simp add: bl2wc.simps bl2nat.simps) term trpl @@ -3571,7 +3550,6 @@ apply(simp add: fetch.simps) done -thm entry_lemma lemma Pi_gr_1[simp]: "Pi n > Suc 0" proof(induct n, auto simp: Pi.simps Np.simps) fix n @@ -3978,8 +3956,6 @@ Max {u. Pi (Suc i) ^ u dvd godel_code nl} = nl ! i" by(simp add: godel_code.simps godel_code'_get_nth) -thm trpl.simps - lemma "trpl l st r = godel_code' [l, st, r] 0" apply(simp add: trpl.simps godel_code'.simps) done @@ -4218,8 +4194,7 @@ "Entry (godel_code (modify_tprog tp))?i = (modify_tprog tp) ! ?i" by(erule_tac godel_decode) - thm modify_tprog.simps - moreover have + moreover have "modify_tprog tp ! ?i = action_map (fst (tp ! (2 * (st - Suc 0) + r mod 2)))" apply(rule_tac modify_tprog_fetch_action) @@ -4307,8 +4282,7 @@ hence "Entry (godel_code (modify_tprog tp)) (?i) = (modify_tprog tp) ! ?i" by(erule_tac godel_decode) - thm modify_tprog.simps - moreover have + moreover have "modify_tprog tp ! ?i = (snd (tp ! (2 * (st - Suc 0) + r mod 2)))" apply(rule_tac modify_tprog_fetch_state) @@ -4606,7 +4580,6 @@ moreover hence "trpl_code (tstep (a, b, c) tp) = rec_exec rec_newconf [code tp, trpl_code (a, b, c)]" - thm rec_t_eq_step apply(rule_tac rec_t_eq_step) using h g apply(simp add: s_keep) @@ -4783,18 +4756,6 @@ qed qed -(* -lemma halt_steps_ex: - "\steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>); - lm \ []; turing_basic.t_correct tp; 0 \ - \ t. rec_calc_rel (rec_halt (length lm)) (code tp # lm) t" -apply(drule_tac halt_least_step, auto) -apply(rule_tac x = stp in exI) -apply(simp add: halt_lemma nonstop_lemma) -apply(auto) -done*) -thm loR.simps - lemma conf_trpl_ex: "\ p q r. conf m (bl2wc ()) stp = trpl p q r" apply(induct stp, auto simp: conf.simps inpt.simps trpl.simps newconf.simps) @@ -4862,8 +4823,6 @@ apply(simp) done -thm halt_lemma - text {* The correntess of @{text "rec_F"} which relates the interpreter function @{text "rec_F"} with the execution of of TMs. diff -r 1ce04eb1c8ad -r 48b231495281 utm/UTM.thy --- 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 \ - [(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 \ 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 \ nat \ tape \ bool" - -fun wcode_before_double :: "bin_inv_t" - where - "wcode_before_double ires rs (l, r) = - (\ ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ - 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) = - (\ ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ - 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) = - (\ ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Oc # ires \ - r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ - ml + mr > Suc 0 \ 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) = - (\ ln rn. - l = Oc # ires \ - 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) \ 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) = - (\ ln rn. l = ires \ - 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) = - (\ ln rn. l = Oc # ires \ - 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) = - (\ ml mr rn. - l = Bk\<^bsup>ml\<^esup> @ Oc # ires \ - r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ - 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) = - (\ ml mr ln rn. - l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ - r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ - 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) = - (\ ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ - 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) = - (\ ml mr ln rn. - l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ - r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ - ml + mr = Suc (Suc rs) \ 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) \ - 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: " = 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 () = Suc a" -apply(simp add: tape_of_nat_abv tape_of_nat_list.simps) -done - -lemma [simp]: "<[a::nat]> = " -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 () = bl_bin () + 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 () = ()" -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: "() = (Oc\<^bsup>Suc a\<^esup> @ Bk # ())" -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]: "() = Oc\<^bsup>Suc a\<^esup>" -apply(simp add: tape_of_nat_abv) -done -lemma tape_of_nl_cons_app2: "() = ( @ 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: "\xs c. x = length xs \ = - @ Bk # Oc\<^bsup>Suc b\<^esup>" - and h: "Suc x = length (xs::nat list)" - show " = @ 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: " = @ Bk # Oc\<^bsup>Suc b\<^esup>" - apply(rule_tac ind) - using h - apply(simp) - done - from g and k show " = @ Bk # Oc\<^bsup>Suc b\<^esup>" - apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) - done - qed -qed - -lemma [simp]: "length () = Suc (Suc aa) + length ()" -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 () + (4 * rs + 4) * 2 ^ (length () - Suc 0) - = bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # ) + rs * (2 * 2 ^ (aa + length ()))" -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: "(()) = () @ [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 # @ [Oc]) - = bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # ) + - 2^(length (Oc # Oc\<^bsup>aa\<^esup> @ Bk # ))" -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 # "] -apply(simp) -done -lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # ) + (4 * 2 ^ (aa + length ()) + - 4 * (rs * 2 ^ (aa + length ()))) = - bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # ) + - rs * (2 * 2 ^ (aa + length ()))" -apply(simp add: tape_of_nl_app_Suc) -done - -declare tape_of_nat[simp del] - -text{* double case*} -fun wcode_double_case_inv :: "nat \ 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 \ nat" - where - "wcode_double_case_state (st, l, r) = - 13 - st" - -fun wcode_double_case_step :: "t_conf \ 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 \ nat \ 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 \ t_conf) set" - where "wcode_double_case_le \ (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> = [] \ 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) \ b \ []" -apply(simp add: wcode_double_case_inv_simps, auto) -done - - -lemma [elim]: "\wcode_on_left_moving_1 ires rs (b, Bk # list); - tl b = aa \ hd b # Bk # list = ba\ \ - 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]: - "\wcode_on_left_moving_1 ires rs (b, Oc # list); tl b = aa \ hd b # Oc # list = ba\ - \ 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]: "\wcode_on_checking_1 ires rs (b, Oc # ba);Oc # b = aa \ list = ba\ - \ 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]: "\wcode_on_right_moving_1 ires rs (b, Bk # ba); Bk # b = aa \ list = b\ \ - 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]: - "\wcode_on_right_moving_1 ires rs (b, Oc # ba); Oc # b = aa \ list = ba\ - \ 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, []) \ False" -apply(simp add: wcode_double_case_inv_simps exponent_def) -done - -lemma [elim]: "\wcode_erase1 ires rs (b, Bk # ba); Bk # b = aa \ list = ba; c = Bk # ba\ - \ 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]: "\wcode_erase1 ires rs (aa, Oc # list); b = aa \ Bk # list = ba\ \ - 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]: "\wcode_goon_right_moving_1 ires rs (aa, []); b = aa \ [Oc] = ba\ - \ 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]: - "\wcode_goon_right_moving_1 ires rs (aa, Bk # list); b = aa \ Oc # list = ba\ - \ 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]: "\wcode_goon_right_moving_1 ires rs (b, Oc # ba); Oc # b = aa \ list = ba\ - \ 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]: "\wcode_backto_standard_pos ires rs (b, []); Bk # b = aa\ \ 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]: "\wcode_backto_standard_pos ires rs (b, Bk # ba); Bk # b = aa \ list = ba\ - \ 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]: "\wcode_backto_standard_pos ires rs (b, Oc # list); tl b = aa; hd b # Oc # list = ba\ - \ 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 = (\ (st, l, r). st = 13) in - let Q = (\ (st, l, r). wcode_double_case_inv st ires rs (l, r)) in - let f = (\ 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 - \ n .P (f n) \ Q (f (n::nat))" -proof - - let ?P = "(\ (st, l, r). st = 13)" - let ?Q = "(\ (st, l, r). wcode_double_case_inv st ires rs (l, r))" - let ?f = "(\ 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 "\ n. ?P (?f n) \ ?Q (?f (n::nat))" - proof(rule_tac halt_lemma2) - show "wf wcode_double_case_le" - by auto - next - show "\ na. \ ?P (?f na) \ ?Q (?f na) \ - ?Q (?f (Suc na)) \ (?f (Suc na), ?f na) \ wcode_double_case_le" - proof(rule_tac allI, case_tac "?f na", simp add: tstep_red) - fix na a b c - show "a \ 13 \ wcode_double_case_inv a ires rs (b, c) \ - (case tstep (a, b, c) t_wcode_main of (st, x) \ - wcode_double_case_inv st ires rs x) \ - (tstep (a, b, c) t_wcode_main, a, b, c) \ 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 "\ ?P (?f 0)" - apply(simp add: steps.simps) - done - qed - thus "let P = \(st, l, r). st = 13; - Q = \(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 \n. P (f n) \ Q (f n)" - apply(simp add: Let_def) - done -qed - -lemma [elim]: "t_ncorrect tp - \ t_ncorrect (abacus.tshift tp a)" -apply(simp add: t_ncorrect.simps shift_length) -done - -lemma tshift_fetch: "\ fetch tp a b = (aa, st'); 0 < st'\ - \ 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: "\steps (st, l, r) tp stp = (st', l', r'); - 0 < st'; - 0 < st \ st \ length tp div 2; - t_ncorrect tp1; - t_ncorrect tp\ - \ 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: "\st' l' r'. - \a = st' \ b = l' \ c = r'; 0 < st'\ - \ 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 [] \ Bk | x # xs \ x)", simp) - apply(subgoal_tac "fetch (abacus.tshift tp (length tp1 div 2)) a - (case c of [] \ Bk | x # xs \ x) = (aa, st' + length tp1 div 2)", simp) - apply(simp add: tshift_fetch) - done -qed - -lemma t_tshift_lemma: "\ steps (st, l, r) tp stp = (st', l', r'); - st' \ 0; - stp > 0; - 0 < st \ st \ length tp div 2; - t_ncorrect tp1; - t_ncorrect tp; - t_ncorrect tp2 - \ - \ \ 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' \ 0" "stp > 0" - "0 < st \ st \ length tp div 2" - "t_ncorrect tp1" - "t_ncorrect tp" - "t_ncorrect tp2" - from h have - "\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 "\ 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 \ 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: "\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 "\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 "\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: "\fetch ap a b = (aa, st); st > 0\ - \ 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: - "\steps (st, l, r) ap stp = (st', l', r'); st' \ 0\ - \ 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: "\st l r st' l' r'. - \steps (st, l, r) ap stp = (st', l', r'); st' \ 0\ \ - 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' \ 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 [] \ Bk | x # xs \ x)", simp) - apply(subgoal_tac "fetch (change_termi_state ap) a (case c of [] \ Bk | x # xs \ x) - = (aa, st')", simp) - apply(simp add: change_termi_state_fetch) - done - qed -qed - -lemma change_termi_state_fetch0: - "\0 < a; a \ length ap div 2; t_correct ap; fetch ap a b = (aa, 0)\ - \ 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: - "\steps (Suc 0, l, r) ap stp = (0, l', r'); t_correct ap\ - \ \ 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 [] \ Bk | x # xs \ x)", simp) -apply(subgoal_tac "fetch (change_termi_state ap) a - (case c of [] \ Bk | x # xs \ 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: - "\ 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 "\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>) \ - \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>) - \ \ 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: - "\ 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: - "\ 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 "\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 "\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 "\ 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 "\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) = - (\ ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Oc # ires \ - r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ - ml + mr > Suc 0 \ mr > 0)" - -fun wcode_on_left_moving_2_O :: "bin_inv_t" - where - "wcode_on_left_moving_2_O ires rs (l, r) = - (\ ln rn. l = Bk # Oc # ires \ - 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) \ - 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) = - (\ ln rn. l = Oc#ires \ - 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) = - (\ ln rn. l = ires \ - 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) = - (\ ln rn. l = Oc # ires \ - 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) = - (\ ln rn. l = Bk # Oc # ires \ - 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) = - (\ ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # ires \ - r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ ml + mr > Suc 0)" - -fun wcode_goon_right_moving_2 :: "bin_inv_t" - where - "wcode_goon_right_moving_2 ires rs (l, r) = - (\ ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ - r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ 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) = - (\ ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ - 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) = - (\ ml mr ln rn. l = Oc\<^bsup>ml \<^esup>@ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ - r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ - ml + mr = (Suc (Suc rs)) \ 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) \ - 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) = - (\ ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ - 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 \ 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 \ nat" - where - "wcode_fourtimes_case_state (st, l, r) = 13 - st" - -fun wcode_fourtimes_case_step :: "t_conf \ 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 \ nat \ 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 \ t_conf) set" - where "wcode_fourtimes_case_le \ (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) \ b \ []" -apply(simp add: wcode_fourtimes_invs, auto) -done - -lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \ 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) \ b \ []" -apply(auto simp: wcode_fourtimes_invs) -done - -lemma [simp]: "wcode_on_checking_2 ires rs (b, Bk # list) - \ 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) \ b\ []" -apply(simp add: wcode_fourtimes_invs) -done - -lemma [simp]: "wcode_right_move ires rs (b, Bk # list) \ 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) \ b \ []" -apply(auto simp: wcode_fourtimes_invs) -done - -lemma [simp]: "wcode_erase2 ires rs (b, Bk # list) \ 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) \ b \ []" -apply(auto simp:wcode_fourtimes_invs ) -done - -lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list) - \ 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) \ b \ []" -apply(auto simp: wcode_fourtimes_invs) -done - -lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \ - 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) \ b \ []" -apply(simp add: wcode_fourtimes_invs, auto) -done - -lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \ b \ []" -apply(simp add: wcode_fourtimes_invs, auto) -done - -lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \ - 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, []) \ b \ []" -apply(auto simp: wcode_fourtimes_invs) -done - -lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \ - 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) - \ (\ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \ (\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) \ False" -apply(simp add: wcode_fourtimes_invs) -done - -lemma [simp]: "wcode_goon_checking ires rs (b, Oc # list) \ - (b = [] \ wcode_right_move ires rs ([Oc], list)) \ - (b \ [] \ 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) \ b \ []" -apply(simp add: wcode_fourtimes_invs) -done - -lemma [simp]: "wcode_erase2 ires rs (b, Oc # list) - \ 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) \ b \ []" -apply(simp only: wcode_fourtimes_invs) -apply(auto) -done - -lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Oc # list) - \ 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) \ b \ []" -apply(simp only:wcode_fourtimes_invs, auto) -done - -lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list) - \ (\ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \ (\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) \ - 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) \ b \ []" -apply(simp only: wcode_fourtimes_invs, auto) -done - -lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list) - \ 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 = (\ (st, l, r). st = t_twice_len + 14) in - let Q = (\ (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r)) in - let f = (\ 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 - \ n .P (f n) \ Q (f (n::nat))" -proof - - let ?P = "(\ (st, l, r). st = t_twice_len + 14)" - let ?Q = "(\ (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r))" - let ?f = "(\ 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 "\ n . ?P (?f n) \ ?Q (?f (n::nat))" - proof(rule_tac halt_lemma2) - show "wf wcode_fourtimes_case_le" - by auto - next - show "\ na. \ ?P (?f na) \ ?Q (?f na) \ - ?Q (?f (Suc na)) \ (?f (Suc na), ?f na) \ 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 "\ ?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: - "\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 "\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 "\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: - "\ 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 "\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>) \ - \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>) - \ \ 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 \ 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: - "\ 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: - "\ 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 "\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 "\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 "\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 "\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) = - (\ ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Bk # ires \ - r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ - ml + mr > Suc 0 \ mr > 0 )" - -fun wcode_on_left_moving_3_O :: "bin_inv_t" - where - "wcode_on_left_moving_3_O ires rs (l, r) = - (\ ln rn. l = Bk # Bk # ires \ - 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) \ - 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) = - (\ ln rn. l = Bk # ires \ - 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) = - (\ ln rn. l = ires \ - 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) = - (\ ln rn. l = Bk # ires \ - r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" - -fun wcode_halt_case_inv :: "nat \ 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 \ 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 \ nat" - where - "wcode_halt_case_step (st, l, r) = - (if st = 1 then length l - else 0)" - -fun wcode_halt_case_measure :: "t_conf \ nat \ 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 \ t_conf) set" - where "wcode_halt_case_le \ (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) - \ 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) \ - (b = [] \ wcode_stop ires rs ([Bk], list)) \ - (b \ [] \ 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) \ b \ []" -apply(auto simp: wcode_halt_invs) -done - -lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Oc # list) \ - 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) \ b \ []" -apply(simp add: wcode_halt_invs, auto) -done - - -lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \ b \ []" -apply(auto simp: wcode_halt_invs) -done - -lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \ - 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 = (\ (st, l, r). st = 0) in - let Q = (\ (st, l, r). wcode_halt_case_inv st ires rs (l, r)) in - let f = (\ 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 - \ n .P (f n) \ Q (f (n::nat))" -proof - - let ?P = "(\ (st, l, r). st = 0)" - let ?Q = "(\ (st, l, r). wcode_halt_case_inv st ires rs (l, r))" - let ?f = "(\ 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 "\ n. ?P (?f n) \ ?Q (?f (n::nat))" - proof(rule_tac halt_lemma2) - show "wf wcode_halt_case_le" by auto - next - show "\ na. \ ?P (?f na) \ ?Q (?f na) \ - ?Q (?f (Suc na)) \ (?f (Suc na), ?f na) \ 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 "\ ?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]: "\ xs. ( :: 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: - "\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: - "\args \ []; lm = \ \ - \ 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: - "\args lm rs m n. - \x = length args; (args::nat list) \ []; lm = \ - \ \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) \ []" "lm = " - from h have "\ (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 - "\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 "\stp ln rn. - steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev () @ 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 () + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)" - proof(induct "a" arbitrary: m n rs ires, simp) - fix m n rs ires - show "\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: - "\m n rs ires. - \stp ln rn. - steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev () @ 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 () + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)" - show "\stp ln rn. - steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev () @ 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 () + rs * 2 ^ Suc a\<^esup> @ Bk\<^bsup>rn\<^esup>)" - proof - - have "\stp ln rn. - steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev () @ 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 () @ 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 () @ 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 () @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast - moreover have - "\stp ln rn. - steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev () @ 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 () + (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 () @ 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 () + (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 \ []" "lm = " "args = xs @ [a::nat]" "xs = (aa::nat) # list" - thus "\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 "\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 () @ Bk # Bk # ires, - Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)" - proof(simp add: tape_of_nl_rev) - have "\ xs. () = Oc # xs" by auto - from this obtain xs where "() = Oc # xs" .. - thus "\stp ln rn. - steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # @ Bk # Bk # ires, - Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = - (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ @ 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 () @ 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 () @ Bk # Bk # ires, - Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast - from g have - "\ 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 ()+ (4*rs + 4) * 2^(length () - 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 ()+ (4*rs + 4) * 2^(length () - 1) \<^esup> @ Bk\<^bsup>rnb\<^esup>)" - by blast - from stp1 and stp2 and h - show "\stp ln rn. - steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # @ 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 # ) + rs * (2 * 2 ^ (aa + length ()))\<^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: - "\ m n rs args lm. - \lm = ; args = aa # list @ [ab]\ - \ \stp ln rn. - steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ @ 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 () + rs * 2 ^ (length () - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)" - and k: "args = aa # list @ [Suc ab]" "lm = " - show "\stp ln rn. - steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ @ 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 () + rs * 2 ^ (length () - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)" - proof(simp add: tape_of_nl_cons_app1) - have "\stp ln rn. - steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # @ 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 # @ 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 # @ 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 # @ 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 # @ Bk # Bk # ires, - Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast - from k have - "\ stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ @ 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 ( ) + (2*rs + 2)* 2^(length () - 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> @ @ 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 ( ) + (2*rs + 2)* 2^(length () - Suc 0)\<^esup> @ Bk\<^bsup>rnb\<^esup>)" - by blast - from stp1 and stp2 show - "\stp ln rn. - steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # @ 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 # ) + rs * (2 * 2 ^ (aa + length ()))\<^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 \ - [(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 \ nat list \ tape \ bool" - where - "wprepare_add_one m lm (l, r) = - (\ rn. l = [] \ - (r = @ Bk\<^bsup>rn\<^esup> \ - r = Bk # @ Bk\<^bsup>rn\<^esup>))" - -fun wprepare_goto_first_end :: "nat \ nat list \ tape \ bool" - where - "wprepare_goto_first_end m lm (l, r) = - (\ ml mr rn. l = Oc\<^bsup>ml\<^esup> \ - r = Oc\<^bsup>mr\<^esup> @ Bk # @ Bk\<^bsup>rn\<^esup> \ - ml + mr = Suc (Suc m))" - -fun wprepare_erase :: "nat \ nat list \ tape \ bool" - where - "wprepare_erase m lm (l, r) = - (\ rn. l = Oc\<^bsup>Suc m\<^esup> \ - tl r = Bk # @ Bk\<^bsup>rn\<^esup>)" - -fun wprepare_goto_start_pos_B :: "nat \ nat list \ tape \ bool" - where - "wprepare_goto_start_pos_B m lm (l, r) = - (\ rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \ - r = Bk # @ Bk\<^bsup>rn\<^esup>)" - -fun wprepare_goto_start_pos_O :: "nat \ nat list \ tape \ bool" - where - "wprepare_goto_start_pos_O m lm (l, r) = - (\ rn. l = Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ - r = @ Bk\<^bsup>rn\<^esup>)" - -fun wprepare_goto_start_pos :: "nat \ nat list \ tape \ bool" - where - "wprepare_goto_start_pos m lm (l, r) = - (wprepare_goto_start_pos_B m lm (l, r) \ - wprepare_goto_start_pos_O m lm (l, r))" - -fun wprepare_loop_start_on_rightmost :: "nat \ nat list \ tape \ bool" - where - "wprepare_loop_start_on_rightmost m lm (l, r) = - (\ rn mr. rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # @ Bk\<^bsup>rn\<^esup> \ l \ [] \ - r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>)" - -fun wprepare_loop_start_in_middle :: "nat \ nat list \ tape \ bool" - where - "wprepare_loop_start_in_middle m lm (l, r) = - (\ rn (mr:: nat) (lm1::nat list). - rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # @ Bk\<^bsup>rn\<^esup> \ l \ [] \ - r = Oc\<^bsup>mr\<^esup> @ Bk # @ Bk\<^bsup>rn\<^esup> \ lm1 \ [])" - -fun wprepare_loop_start :: "nat \ nat list \ tape \ bool" - where - "wprepare_loop_start m lm (l, r) = (wprepare_loop_start_on_rightmost m lm (l, r) \ - wprepare_loop_start_in_middle m lm (l, r))" - -fun wprepare_loop_goon_on_rightmost :: "nat \ nat list \ tape \ bool" - where - "wprepare_loop_goon_on_rightmost m lm (l, r) = - (\ rn. l = Bk # @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ - r = Bk\<^bsup>rn\<^esup>)" - -fun wprepare_loop_goon_in_middle :: "nat \ nat list \ tape \ bool" - where - "wprepare_loop_goon_in_middle m lm (l, r) = - (\ rn (mr:: nat) (lm1::nat list). - rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # @ Bk\<^bsup>rn\<^esup> \ l \ [] \ - (if lm1 = [] then r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> - else r = Oc\<^bsup>mr\<^esup> @ Bk # @ Bk\<^bsup>rn\<^esup>) \ mr > 0)" - -fun wprepare_loop_goon :: "nat \ nat list \ tape \ bool" - where - "wprepare_loop_goon m lm (l, r) = - (wprepare_loop_goon_in_middle m lm (l, r) \ - wprepare_loop_goon_on_rightmost m lm (l, r))" - -fun wprepare_add_one2 :: "nat \ nat list \ tape \ bool" - where - "wprepare_add_one2 m lm (l, r) = - (\ rn. l = Bk # Bk # @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ - (r = [] \ tl r = Bk\<^bsup>rn\<^esup>))" - -fun wprepare_stop :: "nat \ nat list \ tape \ bool" - where - "wprepare_stop m lm (l, r) = - (\ rn. l = Bk # @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ - r = Bk # Oc # Bk\<^bsup>rn\<^esup>)" - -fun wprepare_inv :: "nat \ nat \ nat list \ tape \ 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 \ nat" - where - "wprepare_stage (st, l, r) = - (if st \ 1 \ st \ 4 then 3 - else if st = 5 \ st = 6 then 2 - else 1)" - -fun wprepare_state :: "t_conf \ 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 \ 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 \ [] \ hd r = Oc) then 0 - else 1) - else 0)" - -fun wcode_prepare_measure :: "t_conf \ nat \ nat \ 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 \ t_conf) set" - where "wcode_prepare_le \ (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 \ [] \ \ []" -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 \ [] \ wprepare_add_one m lm (b, []) = False" -apply(simp add: wprepare_invs) -apply(simp add: tape_of_nl_not_null) -done - -lemma [simp]: "lm \ [] \ wprepare_goto_first_end m lm (b, []) = False" -apply(simp add: wprepare_invs) -done - -lemma [simp]: "lm \ [] \ wprepare_erase m lm (b, []) = False" -apply(simp add: wprepare_invs) -done - - - -lemma [simp]: "lm \ [] \ wprepare_goto_start_pos m lm (b, []) = False" -apply(simp add: wprepare_invs tape_of_nl_not_null) -done - -lemma [simp]: "\lm \ []; wprepare_loop_start m lm (b, [])\ \ b \ []" -apply(simp add: wprepare_invs tape_of_nl_not_null, auto) -done - -lemma [simp]: "\lm \ []; wprepare_loop_start m lm (b, [])\ \ - 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]: "\lm \ []; wprepare_loop_goon m lm (b, [])\ \ b \ []" -apply(simp only: wprepare_invs tape_of_nl_not_null, auto) -done - -lemma [simp]:"\lm \ []; wprepare_loop_goon m lm (b, [])\ \ - 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, []) \ b \ []" -apply(simp only: wprepare_invs tape_of_nl_not_null, auto) -done - -lemma [simp]: "wprepare_add_one2 m lm (b, []) \ 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]: "\lm \ []; wprepare_add_one m lm (b, Bk # list)\ - \ (b = [] \ wprepare_goto_first_end m lm ([], Oc # list)) \ - (b \ [] \ 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) \ b \ []" -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) \ - 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) \ b \ []" -apply(simp only: wprepare_invs exp_ind_def, auto) -done - -lemma [simp]: "wprepare_erase m lm (b, Bk # list) \ - wprepare_goto_start_pos m lm (Bk # b, list)" -apply(simp only: wprepare_invs, auto) -done - -lemma [simp]: "\wprepare_add_one m lm (b, Bk # list)\ \ list \ []" -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]: "\lm \ []; wprepare_goto_first_end m lm (b, Bk # list)\ \ list \ []" -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]: "\lm \ []; wprepare_goto_first_end m lm (b, Bk # list)\ \ b \ []" -apply(simp only: wprepare_invs, auto) -apply(case_tac mr, simp_all add: exp_ind_def tape_of_nl_not_null) -done - -lemma [simp]: "\lm \ []; wprepare_erase m lm (b, Bk # list)\ \ list \ []" -apply(simp only: wprepare_invs, auto) -done - -lemma [simp]: "\lm \ []; wprepare_erase m lm (b, Bk # list)\ \ b \ []" -apply(simp only: wprepare_invs, auto simp: exp_ind_def) -done - -lemma [simp]: "\lm \ []; wprepare_goto_start_pos m lm (b, Bk # list)\ \ list \ []" -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]: "\lm \ []; wprepare_goto_start_pos m lm (b, Bk # list)\ \ b \ []" -apply(simp only: wprepare_invs) -apply(auto) -done - -lemma [simp]: "\lm \ []; wprepare_loop_goon m lm (b, Bk # list)\ \ b \ []" -apply(simp only: wprepare_invs, auto) -done - -lemma [simp]: "\lm \ []; wprepare_loop_goon m lm (b, Bk # list)\ \ - (list = [] \ wprepare_add_one2 m lm (Bk # b, [])) \ - (list \ [] \ 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) \ b \ []" -apply(simp only: wprepare_invs, simp) -done - -lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \ - (list = [] \ wprepare_add_one2 m lm (b, [Oc])) \ - (list \ [] \ 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) - \ (b = [] \ wprepare_goto_first_end m lm ([Oc], list)) \ - (b \ [] \ 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) \ b \ []" -apply(simp only: wprepare_invs, auto simp: exp_ind_def) -done - -lemma [simp]: "wprepare_erase m lm (b, Oc # list) - \ wprepare_erase m lm (b, Bk # list)" -apply(simp only:wprepare_invs, auto simp: exp_ind_def) -done - -lemma [simp]: "\lm \ []; wprepare_goto_start_pos m lm (b, Bk # list)\ - \ 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) \ b \ []" -apply(simp only:wprepare_invs, auto) -done -lemma [elim]: "Bk # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ \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 \ rev x = rev y" -by simp - -lemma tape_of_nl_false1: - "lm \ [] \ rev b @ [Bk] \ Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>m\<^esup> @ Bk # Bk # " -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]: "\lm \ []; wprepare_loop_start m lm (b, [Bk])\ \ - 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) - \ 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]: "\lm \ []; wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista)\ - \ 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]: "\lm \ []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\ - \ 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]: - "\lm \ []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\ - \ 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]: "\lm \ []; wprepare_loop_start m lm (b, Bk # a # lista)\ \ - 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: - "\lm \ []; wprepare_loop_start m lm (b, Bk # list)\ \ - (list = [] \ wprepare_loop_goon m lm (Bk # b, [])) \ - (list \ [] \ 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) - \ (hd b = Oc \ (b = [] \ wprepare_add_one m lm ([], Bk # Oc # list)) \ - (b \ [] \ wprepare_add_one m lm (tl b, Oc # Oc # list))) \ - (hd b \ Oc \ (b = [] \ wprepare_add_one m lm ([], Bk # Oc # list)) \ - (b \ [] \ 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) \ b \ []" -apply(simp) -done - -lemma [simp]: "wprepare_loop_start_on_rightmost m lm (b, Oc # list) \ - 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) \ - 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) \ - 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) \ b \ []" -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) \ b \ []" -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: "\rev b @ Oc\<^bsup>mr\<^esup> = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # ; - b \ []; 0 < mr; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>\ - \ 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: "\rev b @ Oc\<^bsup>mr\<^esup> @ Bk # = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # ; - b \ []; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk # <(a::nat) # lista> @ Bk\<^bsup>rn\<^esup>\ - \ 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) \ - wprepare_loop_start_on_rightmost m lm (Oc # b, list) \ - 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) - \ 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) - \ b = [] \ 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) - \ 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) - \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]: "\lm \ []; wprepare_goto_start_pos m lm (b, Oc # list)\ - \ 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) \ b \ []" -apply(auto simp: wprepare_add_one2.simps) -done - -lemma add_one_2_stop: - "wprepare_add_one2 m lm (b, Oc # list) - \ 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 \ []" - shows "let P = (\ (st, l, r). st = 0) in - let Q = (\ (st, l, r). wprepare_inv st m lm (l, r)) in - let f = (\ stp. steps (Suc 0, [], ()) t_wcode_prepare stp) in - \ n .P (f n) \ Q (f n)" -proof - - let ?P = "(\ (st, l, r). st = 0)" - let ?Q = "(\ (st, l, r). wprepare_inv st m lm (l, r))" - let ?f = "(\ stp. steps (Suc 0, [], ()) t_wcode_prepare stp)" - have "\ n. ?P (?f n) \ ?Q (?f n)" - proof(rule_tac halt_lemma2) - show "wf wcode_prepare_le" by auto - next - show "\ n. \ ?P (?f n) \ ?Q (?f n) \ - ?Q (?f (Suc n)) \ (?f (Suc n), ?f n) \ 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 "\ ?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 \ - list_all (\(acn, st). (st \ 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 (\(acn, st). (st \ y)) tp \ - list_all (\(acn, st). (st \ 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 (\(acn, s). s \ (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 (\(acn, s). s \ 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 (\(acn, s). s \ 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 (\(acn, s). s \ - (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 (\(acn, s). s \ 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 (\(acn, s). s \ 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 (\(acn, s). s \ (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 \ [] \ - \ stp ln rn. steps (Suc 0, [], ) (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 ()\<^esup> @ Bk\<^bsup>rn\<^esup>)" -proof - - let ?P1 = "\ (l, r). l = [] \ r = " - let ?Q1 = "\ (l, r). wprepare_stop m args (l, r)" - let ?P2 = ?Q1 - let ?Q2 = "\ (l, r). (\ ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \ - r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>)" - let ?P3 = "\ tp. False" - assume h: "args \ []" - have "?P1 \-> \ tp. (\ stp tp'. steps (Suc 0, tp) - (t_wcode_prepare |+| t_wcode_main) stp = (0, tp') \ ?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 "\stp. case steps (Suc 0, [], ) t_wcode_prepare stp of (st, tp') - \ st = 0 \ 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 "\stp. case steps (Suc 0, a, b) t_wcode_main stp of - (st, tp') \ (st = 0) \ (case tp' of (l, r) \ l = Bk # Oc\<^bsup>Suc m\<^esup> \ - (\ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>))" - proof(simp only: wprepare_stop.simps, erule_tac exE) - fix rn - assume "a = Bk # @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ - b = Bk # Oc # Bk\<^bsup>rn\<^esup>" - thus "?thesis" - using t_wcode_main_lemma_pre[of "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 \-> wprepare_stop m args" - by(simp add: t_imply_def) - qed - thus "\ stp ln rn. steps (Suc 0, [], ) (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 ()\<^esup> @ Bk\<^bsup>rn\<^esup>)" - apply(simp add: t_imply_def) - apply(erule_tac exE)+ - apply(auto) - done -qed - - -lemma [simp]: "tinres r r' \ - fetch t ss (case r of [] \ Bk | x # xs \ x) = - fetch t ss (case r' of [] \ Bk | x # xs \ 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]: "\ n. (a::block)\<^bsup>n\<^esup> = []" -by auto - -lemma [simp]: "\tinres r r'; r \ []; r' \ []\ \ 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]: "\tinres r []; r \ []\ \ hd r = Bk" -apply(auto simp: tinres_def) -apply(case_tac n, auto) -done - -lemma [simp]: "\tinres [] r'; r' \ []\ \ hd r' = Bk" -apply(auto simp: tinres_def) -done - -lemma [intro]: "\na. tl r = tl (r @ Bk\<^bsup>n\<^esup>) @ Bk\<^bsup>na\<^esup> \ 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' \ 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]: "\tinres r []; r \ []\ \ 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'\ \ 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' \ tinres (b # r) (b # r')" -apply(auto simp: tinres_def) -done - -lemma tinres_step2: - "\tinres r r'; tstep (ss, l, r) t = (sa, la, ra); tstep (ss, l, r') t = (sb, lb, rb)\ - \ la = lb \ tinres ra rb \ 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 [] \ Bk | x # xs \ x)", simp) -apply(auto simp: new_tape.simps) -apply(simp_all split: taction.splits if_splits) -apply(auto) -done - - -lemma tinres_steps2: - "\tinres r r'; steps (ss, l, r) t stp = (sa, la, ra); steps (ss, l, r') t stp = (sb, lb, rb)\ - \ la = lb \ tinres ra rb \ 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: "\sa la ra sb lb rb. \steps (ss, l, r) t stp = (sa, la, ra); - steps (ss, l, r') t stp = (sb, lb, rb)\ \ la = lb \ tinres ra rb \ 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 \ tinres c ca \ a = aa" - apply(rule_tac ind, simp_all add: h) - done - thus "la = lb \ tinres ra rb \ 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 \ nat \ tape \ bool" - where - "wadjust_start m rs (l, r) = - (\ ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \ - tl r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" - -fun wadjust_loop_start :: "nat \ nat \ tape \ bool" - where - "wadjust_loop_start m rs (l, r) = - (\ ln rn ml mr. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ - r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ - ml + mr = Suc (Suc rs) \ mr > 0)" - -fun wadjust_loop_right_move :: "nat \ nat \ tape \ bool" - where - "wadjust_loop_right_move m rs (l, r) = - (\ ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ - r = Bk\<^bsup>nr\<^esup> @ Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ - ml + mr = Suc (Suc rs) \ mr > 0 \ - nl + nr > 0)" - -fun wadjust_loop_check :: "nat \ nat \ tape \ bool" - where - "wadjust_loop_check m rs (l, r) = - (\ ml mr ln rn. l = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ - r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ ml + mr = (Suc rs))" - -fun wadjust_loop_erase :: "nat \ nat \ tape \ bool" - where - "wadjust_loop_erase m rs (l, r) = - (\ ml mr ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ - tl r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ ml + mr = (Suc rs) \ mr > 0)" - -fun wadjust_loop_on_left_moving_O :: "nat \ nat \ tape \ bool" - where - "wadjust_loop_on_left_moving_O m rs (l, r) = - (\ ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m \<^esup>\ - r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ - ml + mr = Suc rs \ mr > 0)" - -fun wadjust_loop_on_left_moving_B :: "nat \ nat \ tape \ bool" - where - "wadjust_loop_on_left_moving_B m rs (l, r) = - (\ ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ - r = Bk\<^bsup>nr\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ - ml + mr = Suc rs \ mr > 0)" - -fun wadjust_loop_on_left_moving :: "nat \ nat \ tape \ bool" - where - "wadjust_loop_on_left_moving m rs (l, r) = - (wadjust_loop_on_left_moving_O m rs (l, r) \ - wadjust_loop_on_left_moving_B m rs (l, r))" - -fun wadjust_loop_right_move2 :: "nat \ nat \ tape \ bool" - where - "wadjust_loop_right_move2 m rs (l, r) = - (\ ml mr ln rn. l = Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ - r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ - ml + mr = Suc rs \ mr > 0)" - -fun wadjust_erase2 :: "nat \ nat \ tape \ bool" - where - "wadjust_erase2 m rs (l, r) = - (\ ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ - tl r = Bk\<^bsup>rn\<^esup>)" - -fun wadjust_on_left_moving_O :: "nat \ nat \ tape \ bool" - where - "wadjust_on_left_moving_O m rs (l, r) = - (\ rn. l = Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ - r = Oc # Bk\<^bsup>rn\<^esup>)" - -fun wadjust_on_left_moving_B :: "nat \ nat \ tape \ bool" - where - "wadjust_on_left_moving_B m rs (l, r) = - (\ ln rn. l = Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ - r = Bk\<^bsup>rn\<^esup>)" - -fun wadjust_on_left_moving :: "nat \ nat \ tape \ bool" - where - "wadjust_on_left_moving m rs (l, r) = - (wadjust_on_left_moving_O m rs (l, r) \ - wadjust_on_left_moving_B m rs (l, r))" - -fun wadjust_goon_left_moving_B :: "nat \ nat \ tape \ bool" - where - "wadjust_goon_left_moving_B m rs (l, r) = - (\ rn. l = Oc\<^bsup>Suc m\<^esup> \ - r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" - -fun wadjust_goon_left_moving_O :: "nat \ nat \ tape \ bool" - where - "wadjust_goon_left_moving_O m rs (l, r) = - (\ ml mr rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ - r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ - ml + mr = Suc (Suc rs) \ mr > 0)" - -fun wadjust_goon_left_moving :: "nat \ nat \ tape \ bool" - where - "wadjust_goon_left_moving m rs (l, r) = - (wadjust_goon_left_moving_B m rs (l, r) \ - wadjust_goon_left_moving_O m rs (l, r))" - -fun wadjust_backto_standard_pos_B :: "nat \ nat \ tape \ bool" - where - "wadjust_backto_standard_pos_B m rs (l, r) = - (\ rn. l = [] \ - 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 \ nat \ tape \ bool" - where - "wadjust_backto_standard_pos_O m rs (l, r) = - (\ ml mr rn. l = Oc\<^bsup>ml\<^esup> \ - r = Oc\<^bsup>mr\<^esup> @ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup> \ - ml + mr = Suc m \ mr > 0)" - -fun wadjust_backto_standard_pos :: "nat \ nat \ tape \ bool" - where - "wadjust_backto_standard_pos m rs (l, r) = - (wadjust_backto_standard_pos_B m rs (l, r) \ - wadjust_backto_standard_pos_O m rs (l, r))" - -fun wadjust_stop :: "nat \ nat \ tape \ bool" -where - "wadjust_stop m rs (l, r) = - (\ rn. l = [Bk] \ - 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 \ nat \ nat \ tape \ 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 \ t_conf \ nat" - where - "wadjust_phase rs (st, l, r) = - (if st = 1 then 3 - else if st \ 2 \ st \ 7 then 2 - else if st \ 8 \ st \ 11 then 1 - else 0)" - -thm dropWhile.simps - -fun wadjust_stage :: "nat \ t_conf \ nat" - where - "wadjust_stage rs (st, l, r) = - (if st \ 2 \ st \ 7 then - rs - length (takeWhile (\ a. a = Oc) - (tl (dropWhile (\ a. a = Oc) (rev l @ r)))) - else 0)" - -fun wadjust_state :: "nat \ t_conf \ nat" - where - "wadjust_state rs (st, l, r) = - (if st \ 2 \ st \ 7 then 8 - st - else if st \ 8 \ st \ 11 then 12 - st - else 0)" - -fun wadjust_step :: "nat \ t_conf \ 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 \ t_conf) \ nat \ nat \ nat \ 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 \ t_conf) \ nat \ t_conf) set" - where "wadjust_le \ (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, []) \ c \ []" -apply(auto simp: wadjust_loop_right_move.simps) -done - -lemma [simp]: "wadjust_loop_right_move m rs (c, []) - \ 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, []) \ c \ []" -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, []) \ - 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, []) \ 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, []) - \ (c = [] \ wadjust_loop_on_left_moving m rs ([], [Bk])) \ - (c \ [] \ 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]: "\wadjust_erase2 m rs (c, []); c \ []\ \ - 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, []) - \ (c = [] \ wadjust_on_left_moving m rs ([], [Bk])) \ - (c \ [] \ 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]: " \wadjust_on_left_moving_B m rs (c, []); c \ []; hd c = Bk\ \ - 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]: "\wadjust_on_left_moving_B m rs (c, []); c \ []; hd c = Oc\ \ - 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]: "\wadjust_on_left_moving m rs (c, []); c \ []\ \ - 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, []) - \ (c = [] \ wadjust_on_left_moving m rs ([], [Bk])) \ - (c \ [] \ 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) \ - (c = [] \ wadjust_start m rs ([], Oc # list)) \ - (c \ [] \ 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) \ c \ []" -apply(simp only: wadjust_loop_right_move.simps, auto) -done - -lemma [simp]: "wadjust_loop_right_move m rs (c, Bk # list) - \ 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) \ c \ []" -apply(simp only: wadjust_loop_check.simps, auto) -done - -lemma [simp]: "wadjust_loop_check m rs (c, Bk # list) - \ 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) \ c \ []" -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]: "\wadjust_loop_erase m rs (c, Bk # list); hd c = Bk\ - \ 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]: "\wadjust_loop_erase m rs (c, Bk # list); c \ []; hd c = Oc\ \ - 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]: "\wadjust_loop_erase m rs (c, Bk # list); c \ []\ \ - 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) \ c \ []" -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]: "\wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Bk\ - \ 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]: "\wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Oc\ - \ 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) - \ 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) \ c \ []" -apply(simp only: wadjust_loop_right_move2.simps, auto) -done - -lemma [simp]: "wadjust_loop_right_move2 m rs (c, Bk # list) \ 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) \ c \ []" -apply(auto simp:wadjust_erase2.simps ) -done - -lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \ - 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) \ c \ []" -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]: "\wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Bk\ - \ 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]: "\wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Oc\ - \ 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) \ - 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) \ c \ []" -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]: "\wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Bk\ - \ 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]: "\wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Oc\ - \ 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) \ - 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) \ - (c = [] \ wadjust_stop m rs ([Bk], list)) \ (c \ [] \ 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) - \ (c = [] \ wadjust_loop_start m rs ([Oc], list)) \ - (c \ [] \ 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) \ c \ []" -apply(simp add: wadjust_loop_start.simps, auto) -done - -lemma [simp]: "wadjust_loop_start m rs (c, Oc # list) - \ 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) \ - 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) \ - 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) \ - 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) - \ 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) - \ (c = [] \ wadjust_erase2 m rs ([], Bk # list)) - \ (c \ [] \ 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]: "\wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Bk\ \ - 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]: "\wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Oc\ - \ 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) \ - 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) \ - 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]: "\wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Bk\ - \ 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]: "\wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Oc\ \ - 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) \ - 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) \ - 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]: - "\wadjust_backto_standard_pos_O m rs (c, Oc # list); c \ []; hd c = Bk\ - \ 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]: "\wadjust_backto_standard_pos_O m rs (c, Oc # list); c \ []; hd c = Oc\ - \ 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) - \ (c = [] \ wadjust_backto_standard_pos m rs ([], Bk # Oc # list)) \ - (c \ [] \ 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]: "\Suc (Suc rs) = a; wadjust_loop_erase m rs (c, Bk # list)\ - \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) - < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list)))) \ - a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) = - a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\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]: - "\Suc (Suc rs) = a; wadjust_loop_on_left_moving m rs (c, Bk # list)\ - \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) - < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list)))) \ - a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) = - a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list))))" -apply(subgoal_tac "c \ []") -apply(case_tac c, simp_all) -done - -lemma dropWhile_exp1: "dropWhile (\a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = dropWhile (\a. a = Oc) xs" -apply(induct n, simp_all add: exp_ind_def) -done -lemma takeWhile_exp1: "takeWhile (\a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = Oc\<^bsup>n\<^esup> @ takeWhile (\a. a = Oc) xs" -apply(induct n, simp_all add: exp_ind_def) -done - -lemma [simp]: "\Suc (Suc rs) = a; wadjust_loop_right_move2 m rs (c, Bk # list)\ - \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Oc # list)))) - < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\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]: "\Suc (Suc rs) = a; wadjust_loop_check m rs (c, Oc # list)\ - \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Oc # list)))) - < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Oc # list)))) \ - a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Oc # list)))) = - a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Oc # list))))" -apply(case_tac "c", simp_all) -done - -lemma [simp]: - "\Suc (Suc rs) = a; wadjust_loop_erase m rs (c, Oc # list)\ - \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list)))) - < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Oc # list)))) \ - a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list)))) = - a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\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 = (\ (len, st, l, r). st = 0) in - let Q = (\ (len, st, l, r). wadjust_inv st m rs (l, r)) in - let f = (\ 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 - \ n .P (f n) \ Q (f n)" -proof - - let ?P = "(\ (len, st, l, r). st = 0)" - let ?Q = "\ (len, st, l, r). wadjust_inv st m rs (l, r)" - let ?f = "\ 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 "\ n. ?P (?f n) \ ?Q (?f n)" - proof(rule_tac halt_lemma2) - show "wf wadjust_le" by auto - next - show "\ n. \ ?P (?f n) \ ?Q (?f n) \ - ?Q (?f (Suc n)) \ (?f (Suc n), ?f n) \ 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 [] \ Bk | x # xs \ x) - of (ac, ns) \ (ns, new_tape ac (c, d)) of (st, x) \ 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 \ wadjust_inv b m rs (c, d)" - "Suc (Suc rs) = a \ 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 [] \ Bk | x # xs \ x) - of (ac, ns) \ (ns, new_tape ac (c, d))), a, b, c, d) \ 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 "\ ?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 \ [] \ - \ stp rn. steps (Suc 0, [], ) - ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp - = (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>)" -proof - - let ?P1 = "\ (l, r). l = [] \ r = " - let ?Q1 = "\(l, r). l = Bk # Oc\<^bsup>Suc m\<^esup> \ - (\ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>)" - let ?P2 = ?Q1 - let ?Q2 = "\ (l, r). (wadjust_stop m (bl_bin () - 1) (l, r))" - let ?P3 = "\ tp. False" - assume h: "args \ []" - have "?P1 \-> \ tp. (\ stp tp'. steps (Suc 0, tp) - ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp = (0, tp') \ ?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 "\stp. case steps (Suc 0, [], ) (t_wcode_prepare |+| t_wcode_main) stp of - (st, tp') \ st = 0 \ (case tp' of (l, r) \ l = Bk # Oc\<^bsup>Suc m\<^esup> \ - (\ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^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 "\stp. case steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # - Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp of - (st, tp') \ st = 0 \ wadjust_stop m (bl_bin () - Suc 0) tp'" - using wadjust_correctness[of m "bl_bin () - 1" "Suc ln" rn] - apply(subgoal_tac "bl_bin () > 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 \-> ?P2" - by(simp add: t_imply_def) - qed - thus "\stp rn. steps (Suc 0, [], ) ((t_wcode_prepare |+| t_wcode_main) |+| - t_wcode_adjust) stp = (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>)" - apply(simp add: t_imply_def) - apply(erule_tac exE)+ - apply(subgoal_tac "bl_bin () > 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 \ [] \ - \ stp ln rn. steps (Suc 0, [], ) (t_wcode) stp = - (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>)" -apply(simp add: wcode_lemma_pre' t_wcode_def) -done - -lemma wcode_lemma: - "args \ [] \ - \ stp ln rn. steps (Suc 0, [], ) (t_wcode) stp = - (0, [Bk], <[m ,bl_bin ()]> @ 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 \ (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 \ - (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: - "\turing_basic.t_correct tp; - length lm = k; - steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>); - rs > 0\ - \ \ stp m. abc_steps_l (0, [code tp, bl2wc ()]) (F_aprog) stp = - (length (F_aprog), code tp # bl2wc () # (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: - "\rs > 0; - abc_steps_l (0, [code tp, bl2wc ()]) F_aprog stp = - (length F_aprog, code tp # bl2wc () # (rs - 1) # 0\<^bsup>m\<^esup>)\ - \ \stp m n.(steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) 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 ()]" stp "code tp # bl2wc () # (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': - "\turing_basic.t_correct tp; - 0 < rs; - steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\ - \ \stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) 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]: "\rs > 0; Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup> = c @ Bk\<^bsup>n\<^esup>\ - \ \n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" -apply(case_tac "na > n") -apply(subgoal_tac "\ d. na = d + n", auto simp: exp_add) -apply(rule_tac x = "na - n" in exI, simp) -apply(subgoal_tac "\ 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'': - "\turing_basic.t_correct tp; - 0 < rs; - steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\ - \ \stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ 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 ()]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)" - and gr: "rs > 0" - thus "\stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ 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 ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp) - fix a b c - assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)" - "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)" - thus " a = 0 \ b = Bk\<^bsup>ma\<^esup> \ (\n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" - using tinres_steps2[of "<[code tp, bl2wc ()]>" "<[code tp, bl2wc ()]> @ 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> \ \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: - "\turing_basic.t_correct tp; - 0 < rs; - steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\ - \ \stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc ()]> @ 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 ()]> @ 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 "\stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc ()]> @ 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 ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp) - fix a b c - assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ 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 ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)" - thus "a = 0 \ (\m. b = Bk\<^bsup>m\<^esup>) \ (\n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" - using tinres_steps[of "[Bk, Bk]" "[Bk]" "Suc 0" "<[code tp, bl2wc ()]> @ 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: - "\turing_basic.t_correct tp; - 0 < rs; - args \ []; - steps (Suc 0, Bk\<^bsup>i\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\ - \ \stp m n. steps (Suc 0, [], ) UTM_pre stp = - (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" -proof - - let ?Q2 = "\ (l, r). (\ ln rn. l = Bk\<^bsup>ln\<^esup> \ r = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" - term ?Q2 - let ?P1 = "\ (l, r). l = [] \ r = " - let ?Q1 = "\ (l, r). (l = [Bk] \ - (\ rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>))" - let ?P2 = ?Q1 - let ?P3 = "\ (l, r). False" - assume h: "turing_basic.t_correct tp" "0 < rs" - "args \ []" "steps (Suc 0, Bk\<^bsup>i\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)" - have "?P1 \-> \ tp. (\ stp tp'. steps (Suc 0, tp) - (t_wcode |+| t_utm) stp = (0, tp') \ ?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 "\stp. case steps (Suc 0, [], ) t_wcode stp of (st, tp') \ - st = 0 \ (case tp' of (l, r) \ l = [Bk] \ - (\rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^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 "\stp. case steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ - Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp of - (st, tp') \ st = 0 \ (case tp' of (l, r) \ - (\ln. l = Bk\<^bsup>ln\<^esup>) \ (\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 \-> ?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: - "\turing_basic.t_correct tp; - 0 < rs; - args \ []; - steps (Suc 0, Bk\<^bsup>i\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\ - \ \stp m n. steps (Suc 0, [], ) 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 \ bool" - where - "TSTD c = (let (st, l, r) = c in - st = 0 \ (\ m. l = Bk\<^bsup>m\<^esup>) \ (\ rs n. r = Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>))" - -thm abacus_turing_eq_uhalt - -lemma nstd_case1: "0 < a \ NSTD (trpl_code (a, b, c))" -apply(simp add: NSTD.simps trpl_code.simps) -done - -lemma [simp]: "\m. b \ Bk\<^bsup>m\<^esup> \ 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 "\ 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: "\m. b \ Bk\<^bsup>m\<^esup> \ 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 \ RR" -apply(induct x arbitrary: y, simp, simp) -apply(case_tac y, simp, simp) -done - -lemma bl2nat_zero_eq[simp]: "(bl2nat c 0 = 0) = (\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: - "\Suc (bl2wc c) = 2 ^ m\ \ \ 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: - "\m. Suc (bl2nat c 0) = 2 ^ m \ \rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" - and h: - "Suc (Suc (2 * bl2nat c 0)) = 2 * 2 ^ nat" - have "\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 "\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]: - "\\rs n. c \ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>; - bl2wc c = 2 ^ lg (Suc (bl2wc c)) 2 - Suc 0\ \ bl2wc c = 0" -apply(subgoal_tac "\ 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: - "\rs n. c \ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup> \ 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: "\ TSTD (a, b, c) - \ 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: - "\turing_basic.t_correct tp; - steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (a, b, c); - \ TSTD (a, b, c)\ - \ rec_exec rec_nonstop [code tp, bl2wc (), stp] = Suc 0" -apply(simp add: rec_nonstop_def rec_exec.simps) -apply(subgoal_tac - "rec_exec rec_conf [code tp, bl2wc (), 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: - "\turing_basic.t_correct tp; - \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp))\ - \ \y. rec_calc_rel rec_nonstop - ([code tp, bl2wc (), y]) (Suc 0)" -apply(rule_tac allI, erule_tac x = y in allE) -apply(case_tac "steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp y", simp) -apply(rule_tac nonstop_t_uhalt_eq, simp_all) -done - -(* -lemma [simp]: - "\jturing_basic.t_correct tp; - \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)); - rec_ci rec_F = (F_ap, rs_pos, a_md)\ - \ \ stp. case abc_steps_l (0, [code tp, bl2wc ()] @ 0\<^bsup>a_md - rs_pos \<^esup> - @ suflm) (F_ap) stp of (ss, e) \ 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 ()" 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': - "\turing_basic.t_correct tp; - \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)); - rec_ci rec_F = (ap, pos, md)\ - \ \ stp. case abc_steps_l (0, [code tp, bl2wc ()]) ap stp of (ss, e) - \ 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: - "\stp. case abc_steps_l (0, code tp # bl2wc () # 0\<^bsup>md - pos\<^esup>) ap stp of - (ss, e) \ ss < length ap" - "abc_steps_l (0, [code tp, bl2wc ()]) ap stp = (a, b)" - "turing_basic.t_correct tp" - "rec_ci rec_F = (ap, pos, md)" - moreover have "ap \ []" - 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 () # 0\<^bsup>md - pos\<^esup>) ap stp", simp) - fix aa ba - assume g: "aa < length ap" - "abc_steps_l (0, code tp # bl2wc () # 0\<^bsup>md - pos\<^esup>) ap stp = (aa, ba)" - "ap \ []" - thus "?thesis" - using abc_list_crsp_steps[of "[code tp, bl2wc ()]" - "md - pos" ap stp aa ba] h - apply(simp) - done - qed -qed - -lemma uabc_uhalt: - "\turing_basic.t_correct tp; - \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp))\ - \ \ stp. case abc_steps_l (0, [code tp, bl2wc ()]) F_aprog - stp of (ss, e) \ 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 - "\stp. case abc_steps_l (0, [code tp, bl2wc ()]) a stp of (ss, e) - \ ss < length a" - "rec_ci rec_F = (a, b, c)" - thus - "\stp. case abc_steps_l (0, [code tp, bl2wc ()]) - (a [+] dummy_abc (Suc (Suc 0))) stp of (ss, e) \ - ss < Suc (Suc (Suc (length a)))" - using abc_append_uhalt1[of a "[code tp, bl2wc ()]" - "a [+] dummy_abc (Suc (Suc 0))" "[]" "dummy_abc (Suc (Suc 0))"] - apply(simp) - done -qed - -thm abacus_turing_eq_uhalt -lemma tutm_uhalt': - "\turing_basic.t_correct tp; - \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp))\ - \ \ stp. \ isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) t_utm stp)" - using abacus_turing_eq_uhalt[of "layout_of (F_aprog)" - "F_aprog" "F_tprog" "[code tp, bl2wc ()]" - "start_of (layout_of (F_aprog )) (length (F_aprog))" - "Suc (Suc 0)"] -apply(simp add: F_tprog_def) -apply(subgoal_tac "\stp. case abc_steps_l (0, [code tp, bl2wc ()]) - (F_aprog) stp of (as, am) \ 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' \ tinres r' r" -apply(auto simp: tinres_def) -done - -lemma inres_tape: - "\steps (st, l, r) tp stp = (a, b, c); steps (st, l', r') tp stp = (a', b', c'); - tinres l l'; tinres r r'\ - \ a = a' \ tinres b b' \ 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 \ c = ca \ a = aa" - using h - apply(rule_tac tinres_steps, auto) - done - - thm tinres_steps2 - moreover have "b' = ba \ tinres c' ca \ 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: "\ stp. \ isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) t_utm stp) - \ \ stp. \ isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc ()]> @ Bk\<^bsup>n\<^esup>) t_utm stp)" -apply(rule_tac allI, case_tac "(steps (Suc 0, Bk\<^bsup>m\<^esup>, - <[code tp, bl2wc ()]> @ 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 ()]>) 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: - "\turing_basic.t_correct tp; - \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp))\ - \ \ stp. \ isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc ()]> @ 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: - "\turing_basic.t_correct tp; - \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)); - args \ []\ - \ \ stp. \ isS0 (steps (Suc 0, [], ) UTM_pre stp)" -proof - - let ?P1 = "\ (l, r). l = [] \ r = " - let ?Q1 = "\ (l, r). (l = [Bk] \ - (\ rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>))" - let ?P4 = ?Q1 - let ?P3 = "\ (l, r). False" - assume h: "turing_basic.t_correct tp" "\stp. \ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)" - "args \ []" - have "?P1 \-> \ tp. \ (\ 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 "\stp. case steps (Suc 0, [], ) t_wcode stp of (st, tp') \ - st = 0 \ (case tp' of (l, r) \ l = [Bk] \ - (\rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^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 ())\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp) - \ 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 ())\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp) \ - isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp)" - by simp - next - show "?Q1 \-> ?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: - "\turing_basic.t_correct tp; - \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)); - args \ []\ - \ \ stp. \ isS0 (steps (Suc 0, [], ) 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 \ + [(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 \ 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 \ nat \ tape \ bool" + +fun wcode_before_double :: "bin_inv_t" + where + "wcode_before_double ires rs (l, r) = + (\ ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + 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) = + (\ ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + 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) = + (\ ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Oc # ires \ + r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr > Suc 0 \ 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) = + (\ ln rn. + l = Oc # ires \ + 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) \ 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) = + (\ ln rn. l = ires \ + 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) = + (\ ln rn. l = Oc # ires \ + 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) = + (\ ml mr rn. + l = Bk\<^bsup>ml\<^esup> @ Oc # ires \ + r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ + 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) = + (\ ml mr ln rn. + l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + 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) = + (\ ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + 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) = + (\ ml mr ln rn. + l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc (Suc rs) \ 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) \ + 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: " = 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 () = Suc a" +apply(simp add: tape_of_nat_abv tape_of_nat_list.simps) +done + +lemma [simp]: "<[a::nat]> = " +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 () = bl_bin () + 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 () = ()" +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: "() = (Oc\<^bsup>Suc a\<^esup> @ Bk # ())" +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]: "() = Oc\<^bsup>Suc a\<^esup>" +apply(simp add: tape_of_nat_abv) +done +lemma tape_of_nl_cons_app2: "() = ( @ 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: "\xs c. x = length xs \ = + @ Bk # Oc\<^bsup>Suc b\<^esup>" + and h: "Suc x = length (xs::nat list)" + show " = @ 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: " = @ Bk # Oc\<^bsup>Suc b\<^esup>" + apply(rule_tac ind) + using h + apply(simp) + done + from g and k show " = @ Bk # Oc\<^bsup>Suc b\<^esup>" + apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) + done + qed +qed + +lemma [simp]: "length () = Suc (Suc aa) + length ()" +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 () + (4 * rs + 4) * 2 ^ (length () - Suc 0) + = bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # ) + rs * (2 * 2 ^ (aa + length ()))" +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: "(()) = () @ [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 # @ [Oc]) + = bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # ) + + 2^(length (Oc # Oc\<^bsup>aa\<^esup> @ Bk # ))" +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 # "] +apply(simp) +done +lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # ) + (4 * 2 ^ (aa + length ()) + + 4 * (rs * 2 ^ (aa + length ()))) = + bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # ) + + rs * (2 * 2 ^ (aa + length ()))" +apply(simp add: tape_of_nl_app_Suc) +done + +declare tape_of_nat[simp del] + +fun wcode_double_case_inv :: "nat \ 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 \ nat" + where + "wcode_double_case_state (st, l, r) = + 13 - st" + +fun wcode_double_case_step :: "t_conf \ 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 \ nat \ 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 \ t_conf) set" + where "wcode_double_case_le \ (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> = [] \ 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) \ b \ []" +apply(simp add: wcode_double_case_inv_simps, auto) +done + + +lemma [elim]: "\wcode_on_left_moving_1 ires rs (b, Bk # list); + tl b = aa \ hd b # Bk # list = ba\ \ + 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]: + "\wcode_on_left_moving_1 ires rs (b, Oc # list); tl b = aa \ hd b # Oc # list = ba\ + \ 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]: "\wcode_on_checking_1 ires rs (b, Oc # ba);Oc # b = aa \ list = ba\ + \ 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]: "\wcode_on_right_moving_1 ires rs (b, Bk # ba); Bk # b = aa \ list = b\ \ + 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]: + "\wcode_on_right_moving_1 ires rs (b, Oc # ba); Oc # b = aa \ list = ba\ + \ 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, []) \ False" +apply(simp add: wcode_double_case_inv_simps exponent_def) +done + +lemma [elim]: "\wcode_erase1 ires rs (b, Bk # ba); Bk # b = aa \ list = ba; c = Bk # ba\ + \ 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]: "\wcode_erase1 ires rs (aa, Oc # list); b = aa \ Bk # list = ba\ \ + 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]: "\wcode_goon_right_moving_1 ires rs (aa, []); b = aa \ [Oc] = ba\ + \ 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]: + "\wcode_goon_right_moving_1 ires rs (aa, Bk # list); b = aa \ Oc # list = ba\ + \ 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]: "\wcode_goon_right_moving_1 ires rs (b, Oc # ba); Oc # b = aa \ list = ba\ + \ 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]: "\wcode_backto_standard_pos ires rs (b, []); Bk # b = aa\ \ 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]: "\wcode_backto_standard_pos ires rs (b, Bk # ba); Bk # b = aa \ list = ba\ + \ 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]: "\wcode_backto_standard_pos ires rs (b, Oc # list); tl b = aa; hd b # Oc # list = ba\ + \ 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 = (\ (st, l, r). st = 13) in + let Q = (\ (st, l, r). wcode_double_case_inv st ires rs (l, r)) in + let f = (\ 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 + \ n .P (f n) \ Q (f (n::nat))" +proof - + let ?P = "(\ (st, l, r). st = 13)" + let ?Q = "(\ (st, l, r). wcode_double_case_inv st ires rs (l, r))" + let ?f = "(\ 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 "\ n. ?P (?f n) \ ?Q (?f (n::nat))" + proof(rule_tac halt_lemma2) + show "wf wcode_double_case_le" + by auto + next + show "\ na. \ ?P (?f na) \ ?Q (?f na) \ + ?Q (?f (Suc na)) \ (?f (Suc na), ?f na) \ wcode_double_case_le" + proof(rule_tac allI, case_tac "?f na", simp add: tstep_red) + fix na a b c + show "a \ 13 \ wcode_double_case_inv a ires rs (b, c) \ + (case tstep (a, b, c) t_wcode_main of (st, x) \ + wcode_double_case_inv st ires rs x) \ + (tstep (a, b, c) t_wcode_main, a, b, c) \ 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 "\ ?P (?f 0)" + apply(simp add: steps.simps) + done + qed + thus "let P = \(st, l, r). st = 13; + Q = \(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 \n. P (f n) \ Q (f n)" + apply(simp add: Let_def) + done +qed + +lemma [elim]: "t_ncorrect tp + \ t_ncorrect (abacus.tshift tp a)" +apply(simp add: t_ncorrect.simps shift_length) +done + +lemma tshift_fetch: "\ fetch tp a b = (aa, st'); 0 < st'\ + \ 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: "\steps (st, l, r) tp stp = (st', l', r'); + 0 < st'; + 0 < st \ st \ length tp div 2; + t_ncorrect tp1; + t_ncorrect tp\ + \ 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: "\st' l' r'. + \a = st' \ b = l' \ c = r'; 0 < st'\ + \ 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 [] \ Bk | x # xs \ x)", simp) + apply(subgoal_tac "fetch (abacus.tshift tp (length tp1 div 2)) a + (case c of [] \ Bk | x # xs \ x) = (aa, st' + length tp1 div 2)", simp) + apply(simp add: tshift_fetch) + done +qed + +lemma t_tshift_lemma: "\ steps (st, l, r) tp stp = (st', l', r'); + st' \ 0; + stp > 0; + 0 < st \ st \ length tp div 2; + t_ncorrect tp1; + t_ncorrect tp; + t_ncorrect tp2 + \ + \ \ 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' \ 0" "stp > 0" + "0 < st \ st \ length tp div 2" + "t_ncorrect tp1" + "t_ncorrect tp" + "t_ncorrect tp2" + from h have + "\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 "\ 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 \ 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: "\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 "\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 "\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: "\fetch ap a b = (aa, st); st > 0\ + \ 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: + "\steps (st, l, r) ap stp = (st', l', r'); st' \ 0\ + \ 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: "\st l r st' l' r'. + \steps (st, l, r) ap stp = (st', l', r'); st' \ 0\ \ + 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' \ 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 [] \ Bk | x # xs \ x)", simp) + apply(subgoal_tac "fetch (change_termi_state ap) a (case c of [] \ Bk | x # xs \ x) + = (aa, st')", simp) + apply(simp add: change_termi_state_fetch) + done + qed +qed + +lemma change_termi_state_fetch0: + "\0 < a; a \ length ap div 2; t_correct ap; fetch ap a b = (aa, 0)\ + \ 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: + "\steps (Suc 0, l, r) ap stp = (0, l', r'); t_correct ap\ + \ \ 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 [] \ Bk | x # xs \ x)", simp) +apply(subgoal_tac "fetch (change_termi_state ap) a + (case c of [] \ Bk | x # xs \ 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: + "\ 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 "\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>) \ + \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>) + \ \ 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: + "\ 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: + "\ 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 "\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 "\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 "\ 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 "\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) = + (\ ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Oc # ires \ + r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr > Suc 0 \ mr > 0)" + +fun wcode_on_left_moving_2_O :: "bin_inv_t" + where + "wcode_on_left_moving_2_O ires rs (l, r) = + (\ ln rn. l = Bk # Oc # ires \ + 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) \ + 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) = + (\ ln rn. l = Oc#ires \ + 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) = + (\ ln rn. l = ires \ + 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) = + (\ ln rn. l = Oc # ires \ + 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) = + (\ ln rn. l = Bk # Oc # ires \ + 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) = + (\ ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # ires \ + r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ ml + mr > Suc 0)" + +fun wcode_goon_right_moving_2 :: "bin_inv_t" + where + "wcode_goon_right_moving_2 ires rs (l, r) = + (\ ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ 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) = + (\ ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + 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) = + (\ ml mr ln rn. l = Oc\<^bsup>ml \<^esup>@ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = (Suc (Suc rs)) \ 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) \ + 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) = + (\ ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + 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 \ 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 \ nat" + where + "wcode_fourtimes_case_state (st, l, r) = 13 - st" + +fun wcode_fourtimes_case_step :: "t_conf \ 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 \ nat \ 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 \ t_conf) set" + where "wcode_fourtimes_case_le \ (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) \ b \ []" +apply(simp add: wcode_fourtimes_invs, auto) +done + +lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \ 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) \ b \ []" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_on_checking_2 ires rs (b, Bk # list) + \ 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) \ b\ []" +apply(simp add: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_right_move ires rs (b, Bk # list) \ 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) \ b \ []" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_erase2 ires rs (b, Bk # list) \ 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) \ b \ []" +apply(auto simp:wcode_fourtimes_invs ) +done + +lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list) + \ 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) \ b \ []" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \ + 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) \ b \ []" +apply(simp add: wcode_fourtimes_invs, auto) +done + +lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \ b \ []" +apply(simp add: wcode_fourtimes_invs, auto) +done + +lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \ + 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, []) \ b \ []" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \ + 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) + \ (\ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \ (\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) \ False" +apply(simp add: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_goon_checking ires rs (b, Oc # list) \ + (b = [] \ wcode_right_move ires rs ([Oc], list)) \ + (b \ [] \ 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) \ b \ []" +apply(simp add: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_erase2 ires rs (b, Oc # list) + \ 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) \ b \ []" +apply(simp only: wcode_fourtimes_invs) +apply(auto) +done + +lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Oc # list) + \ 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) \ b \ []" +apply(simp only:wcode_fourtimes_invs, auto) +done + +lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list) + \ (\ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \ (\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) \ + 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) \ b \ []" +apply(simp only: wcode_fourtimes_invs, auto) +done + +lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list) + \ 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 = (\ (st, l, r). st = t_twice_len + 14) in + let Q = (\ (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r)) in + let f = (\ 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 + \ n .P (f n) \ Q (f (n::nat))" +proof - + let ?P = "(\ (st, l, r). st = t_twice_len + 14)" + let ?Q = "(\ (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r))" + let ?f = "(\ 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 "\ n . ?P (?f n) \ ?Q (?f (n::nat))" + proof(rule_tac halt_lemma2) + show "wf wcode_fourtimes_case_le" + by auto + next + show "\ na. \ ?P (?f na) \ ?Q (?f na) \ + ?Q (?f (Suc na)) \ (?f (Suc na), ?f na) \ 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 "\ ?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: + "\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 "\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 "\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: + "\ 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 "\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>) \ + \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>) + \ \ 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 \ 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: + "\ 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: + "\ 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 "\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 "\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 "\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 "\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) = + (\ ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Bk # ires \ + r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr > Suc 0 \ mr > 0 )" + +fun wcode_on_left_moving_3_O :: "bin_inv_t" + where + "wcode_on_left_moving_3_O ires rs (l, r) = + (\ ln rn. l = Bk # Bk # ires \ + 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) \ + 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) = + (\ ln rn. l = Bk # ires \ + 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) = + (\ ln rn. l = ires \ + 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) = + (\ ln rn. l = Bk # ires \ + r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wcode_halt_case_inv :: "nat \ 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 \ 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 \ nat" + where + "wcode_halt_case_step (st, l, r) = + (if st = 1 then length l + else 0)" + +fun wcode_halt_case_measure :: "t_conf \ nat \ 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 \ t_conf) set" + where "wcode_halt_case_le \ (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) + \ 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) \ + (b = [] \ wcode_stop ires rs ([Bk], list)) \ + (b \ [] \ 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) \ b \ []" +apply(auto simp: wcode_halt_invs) +done + +lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Oc # list) \ + 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) \ b \ []" +apply(simp add: wcode_halt_invs, auto) +done + + +lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \ b \ []" +apply(auto simp: wcode_halt_invs) +done + +lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \ + 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 = (\ (st, l, r). st = 0) in + let Q = (\ (st, l, r). wcode_halt_case_inv st ires rs (l, r)) in + let f = (\ 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 + \ n .P (f n) \ Q (f (n::nat))" +proof - + let ?P = "(\ (st, l, r). st = 0)" + let ?Q = "(\ (st, l, r). wcode_halt_case_inv st ires rs (l, r))" + let ?f = "(\ 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 "\ n. ?P (?f n) \ ?Q (?f (n::nat))" + proof(rule_tac halt_lemma2) + show "wf wcode_halt_case_le" by auto + next + show "\ na. \ ?P (?f na) \ ?Q (?f na) \ + ?Q (?f (Suc na)) \ (?f (Suc na), ?f na) \ 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 "\ ?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]: "\ xs. ( :: 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: + "\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: + "\args \ []; lm = \ \ + \ 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: + "\args lm rs m n. + \x = length args; (args::nat list) \ []; lm = \ + \ \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) \ []" "lm = " + from h have "\ (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 + "\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 "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev () @ 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 () + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)" + proof(induct "a" arbitrary: m n rs ires, simp) + fix m n rs ires + show "\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: + "\m n rs ires. + \stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev () @ 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 () + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)" + show "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev () @ 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 () + rs * 2 ^ Suc a\<^esup> @ Bk\<^bsup>rn\<^esup>)" + proof - + have "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev () @ 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 () @ 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 () @ 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 () @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast + moreover have + "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev () @ 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 () + (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 () @ 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 () + (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 \ []" "lm = " "args = xs @ [a::nat]" "xs = (aa::nat) # list" + thus "\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 "\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 () @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + proof(simp add: tape_of_nl_rev) + have "\ xs. () = Oc # xs" by auto + from this obtain xs where "() = Oc # xs" .. + thus "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ @ 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 () @ 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 () @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast + from g have + "\ 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 ()+ (4*rs + 4) * 2^(length () - 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 ()+ (4*rs + 4) * 2^(length () - 1) \<^esup> @ Bk\<^bsup>rnb\<^esup>)" + by blast + from stp1 and stp2 and h + show "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # @ 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 # ) + rs * (2 * 2 ^ (aa + length ()))\<^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: + "\ m n rs args lm. + \lm = ; args = aa # list @ [ab]\ + \ \stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ @ 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 () + rs * 2 ^ (length () - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + and k: "args = aa # list @ [Suc ab]" "lm = " + show "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ @ 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 () + rs * 2 ^ (length () - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + proof(simp add: tape_of_nl_cons_app1) + have "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # @ 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 # @ 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 # @ 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 # @ 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 # @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast + from k have + "\ stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ @ 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 ( ) + (2*rs + 2)* 2^(length () - 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> @ @ 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 ( ) + (2*rs + 2)* 2^(length () - Suc 0)\<^esup> @ Bk\<^bsup>rnb\<^esup>)" + by blast + from stp1 and stp2 show + "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # @ 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 # ) + rs * (2 * 2 ^ (aa + length ()))\<^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 \ + [(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 \ nat list \ tape \ bool" + where + "wprepare_add_one m lm (l, r) = + (\ rn. l = [] \ + (r = @ Bk\<^bsup>rn\<^esup> \ + r = Bk # @ Bk\<^bsup>rn\<^esup>))" + +fun wprepare_goto_first_end :: "nat \ nat list \ tape \ bool" + where + "wprepare_goto_first_end m lm (l, r) = + (\ ml mr rn. l = Oc\<^bsup>ml\<^esup> \ + r = Oc\<^bsup>mr\<^esup> @ Bk # @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc (Suc m))" + +fun wprepare_erase :: "nat \ nat list \ tape \ bool" + where + "wprepare_erase m lm (l, r) = + (\ rn. l = Oc\<^bsup>Suc m\<^esup> \ + tl r = Bk # @ Bk\<^bsup>rn\<^esup>)" + +fun wprepare_goto_start_pos_B :: "nat \ nat list \ tape \ bool" + where + "wprepare_goto_start_pos_B m lm (l, r) = + (\ rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk # @ Bk\<^bsup>rn\<^esup>)" + +fun wprepare_goto_start_pos_O :: "nat \ nat list \ tape \ bool" + where + "wprepare_goto_start_pos_O m lm (l, r) = + (\ rn. l = Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ + r = @ Bk\<^bsup>rn\<^esup>)" + +fun wprepare_goto_start_pos :: "nat \ nat list \ tape \ bool" + where + "wprepare_goto_start_pos m lm (l, r) = + (wprepare_goto_start_pos_B m lm (l, r) \ + wprepare_goto_start_pos_O m lm (l, r))" + +fun wprepare_loop_start_on_rightmost :: "nat \ nat list \ tape \ bool" + where + "wprepare_loop_start_on_rightmost m lm (l, r) = + (\ rn mr. rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # @ Bk\<^bsup>rn\<^esup> \ l \ [] \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wprepare_loop_start_in_middle :: "nat \ nat list \ tape \ bool" + where + "wprepare_loop_start_in_middle m lm (l, r) = + (\ rn (mr:: nat) (lm1::nat list). + rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # @ Bk\<^bsup>rn\<^esup> \ l \ [] \ + r = Oc\<^bsup>mr\<^esup> @ Bk # @ Bk\<^bsup>rn\<^esup> \ lm1 \ [])" + +fun wprepare_loop_start :: "nat \ nat list \ tape \ bool" + where + "wprepare_loop_start m lm (l, r) = (wprepare_loop_start_on_rightmost m lm (l, r) \ + wprepare_loop_start_in_middle m lm (l, r))" + +fun wprepare_loop_goon_on_rightmost :: "nat \ nat list \ tape \ bool" + where + "wprepare_loop_goon_on_rightmost m lm (l, r) = + (\ rn. l = Bk # @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk\<^bsup>rn\<^esup>)" + +fun wprepare_loop_goon_in_middle :: "nat \ nat list \ tape \ bool" + where + "wprepare_loop_goon_in_middle m lm (l, r) = + (\ rn (mr:: nat) (lm1::nat list). + rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # @ Bk\<^bsup>rn\<^esup> \ l \ [] \ + (if lm1 = [] then r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> + else r = Oc\<^bsup>mr\<^esup> @ Bk # @ Bk\<^bsup>rn\<^esup>) \ mr > 0)" + +fun wprepare_loop_goon :: "nat \ nat list \ tape \ bool" + where + "wprepare_loop_goon m lm (l, r) = + (wprepare_loop_goon_in_middle m lm (l, r) \ + wprepare_loop_goon_on_rightmost m lm (l, r))" + +fun wprepare_add_one2 :: "nat \ nat list \ tape \ bool" + where + "wprepare_add_one2 m lm (l, r) = + (\ rn. l = Bk # Bk # @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ + (r = [] \ tl r = Bk\<^bsup>rn\<^esup>))" + +fun wprepare_stop :: "nat \ nat list \ tape \ bool" + where + "wprepare_stop m lm (l, r) = + (\ rn. l = Bk # @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk # Oc # Bk\<^bsup>rn\<^esup>)" + +fun wprepare_inv :: "nat \ nat \ nat list \ tape \ 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 \ nat" + where + "wprepare_stage (st, l, r) = + (if st \ 1 \ st \ 4 then 3 + else if st = 5 \ st = 6 then 2 + else 1)" + +fun wprepare_state :: "t_conf \ 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 \ 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 \ [] \ hd r = Oc) then 0 + else 1) + else 0)" + +fun wcode_prepare_measure :: "t_conf \ nat \ nat \ 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 \ t_conf) set" + where "wcode_prepare_le \ (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 \ [] \ \ []" +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 \ [] \ wprepare_add_one m lm (b, []) = False" +apply(simp add: wprepare_invs) +apply(simp add: tape_of_nl_not_null) +done + +lemma [simp]: "lm \ [] \ wprepare_goto_first_end m lm (b, []) = False" +apply(simp add: wprepare_invs) +done + +lemma [simp]: "lm \ [] \ wprepare_erase m lm (b, []) = False" +apply(simp add: wprepare_invs) +done + + + +lemma [simp]: "lm \ [] \ wprepare_goto_start_pos m lm (b, []) = False" +apply(simp add: wprepare_invs tape_of_nl_not_null) +done + +lemma [simp]: "\lm \ []; wprepare_loop_start m lm (b, [])\ \ b \ []" +apply(simp add: wprepare_invs tape_of_nl_not_null, auto) +done + +lemma [simp]: "\lm \ []; wprepare_loop_start m lm (b, [])\ \ + 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]: "\lm \ []; wprepare_loop_goon m lm (b, [])\ \ b \ []" +apply(simp only: wprepare_invs tape_of_nl_not_null, auto) +done + +lemma [simp]:"\lm \ []; wprepare_loop_goon m lm (b, [])\ \ + 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, []) \ b \ []" +apply(simp only: wprepare_invs tape_of_nl_not_null, auto) +done + +lemma [simp]: "wprepare_add_one2 m lm (b, []) \ 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]: "\lm \ []; wprepare_add_one m lm (b, Bk # list)\ + \ (b = [] \ wprepare_goto_first_end m lm ([], Oc # list)) \ + (b \ [] \ 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) \ b \ []" +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) \ + 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) \ b \ []" +apply(simp only: wprepare_invs exp_ind_def, auto) +done + +lemma [simp]: "wprepare_erase m lm (b, Bk # list) \ + wprepare_goto_start_pos m lm (Bk # b, list)" +apply(simp only: wprepare_invs, auto) +done + +lemma [simp]: "\wprepare_add_one m lm (b, Bk # list)\ \ list \ []" +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]: "\lm \ []; wprepare_goto_first_end m lm (b, Bk # list)\ \ list \ []" +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]: "\lm \ []; wprepare_goto_first_end m lm (b, Bk # list)\ \ b \ []" +apply(simp only: wprepare_invs, auto) +apply(case_tac mr, simp_all add: exp_ind_def tape_of_nl_not_null) +done + +lemma [simp]: "\lm \ []; wprepare_erase m lm (b, Bk # list)\ \ list \ []" +apply(simp only: wprepare_invs, auto) +done + +lemma [simp]: "\lm \ []; wprepare_erase m lm (b, Bk # list)\ \ b \ []" +apply(simp only: wprepare_invs, auto simp: exp_ind_def) +done + +lemma [simp]: "\lm \ []; wprepare_goto_start_pos m lm (b, Bk # list)\ \ list \ []" +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]: "\lm \ []; wprepare_goto_start_pos m lm (b, Bk # list)\ \ b \ []" +apply(simp only: wprepare_invs) +apply(auto) +done + +lemma [simp]: "\lm \ []; wprepare_loop_goon m lm (b, Bk # list)\ \ b \ []" +apply(simp only: wprepare_invs, auto) +done + +lemma [simp]: "\lm \ []; wprepare_loop_goon m lm (b, Bk # list)\ \ + (list = [] \ wprepare_add_one2 m lm (Bk # b, [])) \ + (list \ [] \ 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) \ b \ []" +apply(simp only: wprepare_invs, simp) +done + +lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \ + (list = [] \ wprepare_add_one2 m lm (b, [Oc])) \ + (list \ [] \ 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) + \ (b = [] \ wprepare_goto_first_end m lm ([Oc], list)) \ + (b \ [] \ 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) \ b \ []" +apply(simp only: wprepare_invs, auto simp: exp_ind_def) +done + +lemma [simp]: "wprepare_erase m lm (b, Oc # list) + \ wprepare_erase m lm (b, Bk # list)" +apply(simp only:wprepare_invs, auto simp: exp_ind_def) +done + +lemma [simp]: "\lm \ []; wprepare_goto_start_pos m lm (b, Bk # list)\ + \ 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) \ b \ []" +apply(simp only:wprepare_invs, auto) +done +lemma [elim]: "Bk # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ \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 \ rev x = rev y" +by simp + +lemma tape_of_nl_false1: + "lm \ [] \ rev b @ [Bk] \ Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>m\<^esup> @ Bk # Bk # " +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]: "\lm \ []; wprepare_loop_start m lm (b, [Bk])\ \ + 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) + \ 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]: "\lm \ []; wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista)\ + \ 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]: "\lm \ []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\ + \ 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]: + "\lm \ []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\ + \ 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]: "\lm \ []; wprepare_loop_start m lm (b, Bk # a # lista)\ \ + 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: + "\lm \ []; wprepare_loop_start m lm (b, Bk # list)\ \ + (list = [] \ wprepare_loop_goon m lm (Bk # b, [])) \ + (list \ [] \ 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) + \ (hd b = Oc \ (b = [] \ wprepare_add_one m lm ([], Bk # Oc # list)) \ + (b \ [] \ wprepare_add_one m lm (tl b, Oc # Oc # list))) \ + (hd b \ Oc \ (b = [] \ wprepare_add_one m lm ([], Bk # Oc # list)) \ + (b \ [] \ 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) \ b \ []" +apply(simp) +done + +lemma [simp]: "wprepare_loop_start_on_rightmost m lm (b, Oc # list) \ + 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) \ + 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) \ + 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) \ b \ []" +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) \ b \ []" +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: "\rev b @ Oc\<^bsup>mr\<^esup> = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # ; + b \ []; 0 < mr; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>\ + \ 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: "\rev b @ Oc\<^bsup>mr\<^esup> @ Bk # = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # ; + b \ []; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk # <(a::nat) # lista> @ Bk\<^bsup>rn\<^esup>\ + \ 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) \ + wprepare_loop_start_on_rightmost m lm (Oc # b, list) \ + 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) + \ 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) + \ b = [] \ 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) + \ 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) + \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]: "\lm \ []; wprepare_goto_start_pos m lm (b, Oc # list)\ + \ 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) \ b \ []" +apply(auto simp: wprepare_add_one2.simps) +done + +lemma add_one_2_stop: + "wprepare_add_one2 m lm (b, Oc # list) + \ 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 \ []" + shows "let P = (\ (st, l, r). st = 0) in + let Q = (\ (st, l, r). wprepare_inv st m lm (l, r)) in + let f = (\ stp. steps (Suc 0, [], ()) t_wcode_prepare stp) in + \ n .P (f n) \ Q (f n)" +proof - + let ?P = "(\ (st, l, r). st = 0)" + let ?Q = "(\ (st, l, r). wprepare_inv st m lm (l, r))" + let ?f = "(\ stp. steps (Suc 0, [], ()) t_wcode_prepare stp)" + have "\ n. ?P (?f n) \ ?Q (?f n)" + proof(rule_tac halt_lemma2) + show "wf wcode_prepare_le" by auto + next + show "\ n. \ ?P (?f n) \ ?Q (?f n) \ + ?Q (?f (Suc n)) \ (?f (Suc n), ?f n) \ 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 "\ ?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 \ + list_all (\(acn, st). (st \ 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 (\(acn, st). (st \ y)) tp \ + list_all (\(acn, st). (st \ 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 (\(acn, s). s \ (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 (\(acn, s). s \ 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 (\(acn, s). s \ 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 (\(acn, s). s \ + (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 (\(acn, s). s \ 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 (\(acn, s). s \ 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 (\(acn, s). s \ (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 \ [] \ + \ stp ln rn. steps (Suc 0, [], ) (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 ()\<^esup> @ Bk\<^bsup>rn\<^esup>)" +proof - + let ?P1 = "\ (l, r). l = [] \ r = " + let ?Q1 = "\ (l, r). wprepare_stop m args (l, r)" + let ?P2 = ?Q1 + let ?Q2 = "\ (l, r). (\ ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>)" + let ?P3 = "\ tp. False" + assume h: "args \ []" + have "?P1 \-> \ tp. (\ stp tp'. steps (Suc 0, tp) + (t_wcode_prepare |+| t_wcode_main) stp = (0, tp') \ ?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 "\stp. case steps (Suc 0, [], ) t_wcode_prepare stp of (st, tp') + \ st = 0 \ 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 "\stp. case steps (Suc 0, a, b) t_wcode_main stp of + (st, tp') \ (st = 0) \ (case tp' of (l, r) \ l = Bk # Oc\<^bsup>Suc m\<^esup> \ + (\ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>))" + proof(simp only: wprepare_stop.simps, erule_tac exE) + fix rn + assume "a = Bk # @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ + b = Bk # Oc # Bk\<^bsup>rn\<^esup>" + thus "?thesis" + using t_wcode_main_lemma_pre[of "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 \-> wprepare_stop m args" + by(simp add: t_imply_def) + qed + thus "\ stp ln rn. steps (Suc 0, [], ) (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 ()\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(simp add: t_imply_def) + apply(erule_tac exE)+ + apply(auto) + done +qed + + +lemma [simp]: "tinres r r' \ + fetch t ss (case r of [] \ Bk | x # xs \ x) = + fetch t ss (case r' of [] \ Bk | x # xs \ 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]: "\ n. (a::block)\<^bsup>n\<^esup> = []" +by auto + +lemma [simp]: "\tinres r r'; r \ []; r' \ []\ \ 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]: "\tinres r []; r \ []\ \ hd r = Bk" +apply(auto simp: tinres_def) +apply(case_tac n, auto) +done + +lemma [simp]: "\tinres [] r'; r' \ []\ \ hd r' = Bk" +apply(auto simp: tinres_def) +done + +lemma [intro]: "\na. tl r = tl (r @ Bk\<^bsup>n\<^esup>) @ Bk\<^bsup>na\<^esup> \ 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' \ 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]: "\tinres r []; r \ []\ \ 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'\ \ 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' \ tinres (b # r) (b # r')" +apply(auto simp: tinres_def) +done + +lemma tinres_step2: + "\tinres r r'; tstep (ss, l, r) t = (sa, la, ra); tstep (ss, l, r') t = (sb, lb, rb)\ + \ la = lb \ tinres ra rb \ 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 [] \ Bk | x # xs \ x)", simp) +apply(auto simp: new_tape.simps) +apply(simp_all split: taction.splits if_splits) +apply(auto) +done + + +lemma tinres_steps2: + "\tinres r r'; steps (ss, l, r) t stp = (sa, la, ra); steps (ss, l, r') t stp = (sb, lb, rb)\ + \ la = lb \ tinres ra rb \ 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: "\sa la ra sb lb rb. \steps (ss, l, r) t stp = (sa, la, ra); + steps (ss, l, r') t stp = (sb, lb, rb)\ \ la = lb \ tinres ra rb \ 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 \ tinres c ca \ a = aa" + apply(rule_tac ind, simp_all add: h) + done + thus "la = lb \ tinres ra rb \ 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 \ nat \ tape \ bool" + where + "wadjust_start m rs (l, r) = + (\ ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \ + tl r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wadjust_loop_start :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_start m rs (l, r) = + (\ ln rn ml mr. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc (Suc rs) \ mr > 0)" + +fun wadjust_loop_right_move :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_right_move m rs (l, r) = + (\ ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk\<^bsup>nr\<^esup> @ Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc (Suc rs) \ mr > 0 \ + nl + nr > 0)" + +fun wadjust_loop_check :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_check m rs (l, r) = + (\ ml mr ln rn. l = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ ml + mr = (Suc rs))" + +fun wadjust_loop_erase :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_erase m rs (l, r) = + (\ ml mr ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + tl r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ ml + mr = (Suc rs) \ mr > 0)" + +fun wadjust_loop_on_left_moving_O :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_on_left_moving_O m rs (l, r) = + (\ ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m \<^esup>\ + r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc rs \ mr > 0)" + +fun wadjust_loop_on_left_moving_B :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_on_left_moving_B m rs (l, r) = + (\ ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk\<^bsup>nr\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc rs \ mr > 0)" + +fun wadjust_loop_on_left_moving :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_on_left_moving m rs (l, r) = + (wadjust_loop_on_left_moving_O m rs (l, r) \ + wadjust_loop_on_left_moving_B m rs (l, r))" + +fun wadjust_loop_right_move2 :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_right_move2 m rs (l, r) = + (\ ml mr ln rn. l = Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc rs \ mr > 0)" + +fun wadjust_erase2 :: "nat \ nat \ tape \ bool" + where + "wadjust_erase2 m rs (l, r) = + (\ ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + tl r = Bk\<^bsup>rn\<^esup>)" + +fun wadjust_on_left_moving_O :: "nat \ nat \ tape \ bool" + where + "wadjust_on_left_moving_O m rs (l, r) = + (\ rn. l = Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Oc # Bk\<^bsup>rn\<^esup>)" + +fun wadjust_on_left_moving_B :: "nat \ nat \ tape \ bool" + where + "wadjust_on_left_moving_B m rs (l, r) = + (\ ln rn. l = Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk\<^bsup>rn\<^esup>)" + +fun wadjust_on_left_moving :: "nat \ nat \ tape \ bool" + where + "wadjust_on_left_moving m rs (l, r) = + (wadjust_on_left_moving_O m rs (l, r) \ + wadjust_on_left_moving_B m rs (l, r))" + +fun wadjust_goon_left_moving_B :: "nat \ nat \ tape \ bool" + where + "wadjust_goon_left_moving_B m rs (l, r) = + (\ rn. l = Oc\<^bsup>Suc m\<^esup> \ + r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wadjust_goon_left_moving_O :: "nat \ nat \ tape \ bool" + where + "wadjust_goon_left_moving_O m rs (l, r) = + (\ ml mr rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc (Suc rs) \ mr > 0)" + +fun wadjust_goon_left_moving :: "nat \ nat \ tape \ bool" + where + "wadjust_goon_left_moving m rs (l, r) = + (wadjust_goon_left_moving_B m rs (l, r) \ + wadjust_goon_left_moving_O m rs (l, r))" + +fun wadjust_backto_standard_pos_B :: "nat \ nat \ tape \ bool" + where + "wadjust_backto_standard_pos_B m rs (l, r) = + (\ rn. l = [] \ + 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 \ nat \ tape \ bool" + where + "wadjust_backto_standard_pos_O m rs (l, r) = + (\ ml mr rn. l = Oc\<^bsup>ml\<^esup> \ + r = Oc\<^bsup>mr\<^esup> @ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc m \ mr > 0)" + +fun wadjust_backto_standard_pos :: "nat \ nat \ tape \ bool" + where + "wadjust_backto_standard_pos m rs (l, r) = + (wadjust_backto_standard_pos_B m rs (l, r) \ + wadjust_backto_standard_pos_O m rs (l, r))" + +fun wadjust_stop :: "nat \ nat \ tape \ bool" +where + "wadjust_stop m rs (l, r) = + (\ rn. l = [Bk] \ + 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 \ nat \ nat \ tape \ 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 \ t_conf \ nat" + where + "wadjust_phase rs (st, l, r) = + (if st = 1 then 3 + else if st \ 2 \ st \ 7 then 2 + else if st \ 8 \ st \ 11 then 1 + else 0)" + +thm dropWhile.simps + +fun wadjust_stage :: "nat \ t_conf \ nat" + where + "wadjust_stage rs (st, l, r) = + (if st \ 2 \ st \ 7 then + rs - length (takeWhile (\ a. a = Oc) + (tl (dropWhile (\ a. a = Oc) (rev l @ r)))) + else 0)" + +fun wadjust_state :: "nat \ t_conf \ nat" + where + "wadjust_state rs (st, l, r) = + (if st \ 2 \ st \ 7 then 8 - st + else if st \ 8 \ st \ 11 then 12 - st + else 0)" + +fun wadjust_step :: "nat \ t_conf \ 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 \ t_conf) \ nat \ nat \ nat \ 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 \ t_conf) \ nat \ t_conf) set" + where "wadjust_le \ (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, []) \ c \ []" +apply(auto simp: wadjust_loop_right_move.simps) +done + +lemma [simp]: "wadjust_loop_right_move m rs (c, []) + \ 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, []) \ c \ []" +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, []) \ + 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, []) \ 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, []) + \ (c = [] \ wadjust_loop_on_left_moving m rs ([], [Bk])) \ + (c \ [] \ 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]: "\wadjust_erase2 m rs (c, []); c \ []\ \ + 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, []) + \ (c = [] \ wadjust_on_left_moving m rs ([], [Bk])) \ + (c \ [] \ 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]: " \wadjust_on_left_moving_B m rs (c, []); c \ []; hd c = Bk\ \ + 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]: "\wadjust_on_left_moving_B m rs (c, []); c \ []; hd c = Oc\ \ + 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]: "\wadjust_on_left_moving m rs (c, []); c \ []\ \ + 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, []) + \ (c = [] \ wadjust_on_left_moving m rs ([], [Bk])) \ + (c \ [] \ 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) \ + (c = [] \ wadjust_start m rs ([], Oc # list)) \ + (c \ [] \ 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) \ c \ []" +apply(simp only: wadjust_loop_right_move.simps, auto) +done + +lemma [simp]: "wadjust_loop_right_move m rs (c, Bk # list) + \ 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) \ c \ []" +apply(simp only: wadjust_loop_check.simps, auto) +done + +lemma [simp]: "wadjust_loop_check m rs (c, Bk # list) + \ 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) \ c \ []" +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]: "\wadjust_loop_erase m rs (c, Bk # list); hd c = Bk\ + \ 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]: "\wadjust_loop_erase m rs (c, Bk # list); c \ []; hd c = Oc\ \ + 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]: "\wadjust_loop_erase m rs (c, Bk # list); c \ []\ \ + 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) \ c \ []" +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]: "\wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Bk\ + \ 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]: "\wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Oc\ + \ 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) + \ 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) \ c \ []" +apply(simp only: wadjust_loop_right_move2.simps, auto) +done + +lemma [simp]: "wadjust_loop_right_move2 m rs (c, Bk # list) \ 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) \ c \ []" +apply(auto simp:wadjust_erase2.simps ) +done + +lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \ + 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) \ c \ []" +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]: "\wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Bk\ + \ 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]: "\wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Oc\ + \ 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) \ + 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) \ c \ []" +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]: "\wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Bk\ + \ 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]: "\wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Oc\ + \ 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) \ + 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) \ + (c = [] \ wadjust_stop m rs ([Bk], list)) \ (c \ [] \ 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) + \ (c = [] \ wadjust_loop_start m rs ([Oc], list)) \ + (c \ [] \ 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) \ c \ []" +apply(simp add: wadjust_loop_start.simps, auto) +done + +lemma [simp]: "wadjust_loop_start m rs (c, Oc # list) + \ 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) \ + 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) \ + 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) \ + 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) + \ 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) + \ (c = [] \ wadjust_erase2 m rs ([], Bk # list)) + \ (c \ [] \ 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]: "\wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Bk\ \ + 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]: "\wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Oc\ + \ 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) \ + 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) \ + 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]: "\wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Bk\ + \ 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]: "\wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Oc\ \ + 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) \ + 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) \ + 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]: + "\wadjust_backto_standard_pos_O m rs (c, Oc # list); c \ []; hd c = Bk\ + \ 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]: "\wadjust_backto_standard_pos_O m rs (c, Oc # list); c \ []; hd c = Oc\ + \ 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) + \ (c = [] \ wadjust_backto_standard_pos m rs ([], Bk # Oc # list)) \ + (c \ [] \ 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]: "\Suc (Suc rs) = a; wadjust_loop_erase m rs (c, Bk # list)\ + \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) + < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list)))) \ + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) = + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\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]: + "\Suc (Suc rs) = a; wadjust_loop_on_left_moving m rs (c, Bk # list)\ + \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) + < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list)))) \ + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) = + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list))))" +apply(subgoal_tac "c \ []") +apply(case_tac c, simp_all) +done + +lemma dropWhile_exp1: "dropWhile (\a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = dropWhile (\a. a = Oc) xs" +apply(induct n, simp_all add: exp_ind_def) +done +lemma takeWhile_exp1: "takeWhile (\a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = Oc\<^bsup>n\<^esup> @ takeWhile (\a. a = Oc) xs" +apply(induct n, simp_all add: exp_ind_def) +done + +lemma [simp]: "\Suc (Suc rs) = a; wadjust_loop_right_move2 m rs (c, Bk # list)\ + \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Oc # list)))) + < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\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]: "\Suc (Suc rs) = a; wadjust_loop_check m rs (c, Oc # list)\ + \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Oc # list)))) + < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Oc # list)))) \ + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Oc # list)))) = + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Oc # list))))" +apply(case_tac "c", simp_all) +done + +lemma [simp]: + "\Suc (Suc rs) = a; wadjust_loop_erase m rs (c, Oc # list)\ + \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list)))) + < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Oc # list)))) \ + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list)))) = + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\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 = (\ (len, st, l, r). st = 0) in + let Q = (\ (len, st, l, r). wadjust_inv st m rs (l, r)) in + let f = (\ 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 + \ n .P (f n) \ Q (f n)" +proof - + let ?P = "(\ (len, st, l, r). st = 0)" + let ?Q = "\ (len, st, l, r). wadjust_inv st m rs (l, r)" + let ?f = "\ 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 "\ n. ?P (?f n) \ ?Q (?f n)" + proof(rule_tac halt_lemma2) + show "wf wadjust_le" by auto + next + show "\ n. \ ?P (?f n) \ ?Q (?f n) \ + ?Q (?f (Suc n)) \ (?f (Suc n), ?f n) \ 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 [] \ Bk | x # xs \ x) + of (ac, ns) \ (ns, new_tape ac (c, d)) of (st, x) \ 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 \ wadjust_inv b m rs (c, d)" + "Suc (Suc rs) = a \ 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 [] \ Bk | x # xs \ x) + of (ac, ns) \ (ns, new_tape ac (c, d))), a, b, c, d) \ 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 "\ ?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 \ [] \ + \ stp rn. steps (Suc 0, [], ) + ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp + = (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>)" +proof - + let ?P1 = "\ (l, r). l = [] \ r = " + let ?Q1 = "\(l, r). l = Bk # Oc\<^bsup>Suc m\<^esup> \ + (\ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>)" + let ?P2 = ?Q1 + let ?Q2 = "\ (l, r). (wadjust_stop m (bl_bin () - 1) (l, r))" + let ?P3 = "\ tp. False" + assume h: "args \ []" + have "?P1 \-> \ tp. (\ stp tp'. steps (Suc 0, tp) + ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp = (0, tp') \ ?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 "\stp. case steps (Suc 0, [], ) (t_wcode_prepare |+| t_wcode_main) stp of + (st, tp') \ st = 0 \ (case tp' of (l, r) \ l = Bk # Oc\<^bsup>Suc m\<^esup> \ + (\ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^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 "\stp. case steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # + Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp of + (st, tp') \ st = 0 \ wadjust_stop m (bl_bin () - Suc 0) tp'" + using wadjust_correctness[of m "bl_bin () - 1" "Suc ln" rn] + apply(subgoal_tac "bl_bin () > 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 \-> ?P2" + by(simp add: t_imply_def) + qed + thus "\stp rn. steps (Suc 0, [], ) ((t_wcode_prepare |+| t_wcode_main) |+| + t_wcode_adjust) stp = (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(simp add: t_imply_def) + apply(erule_tac exE)+ + apply(subgoal_tac "bl_bin () > 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 \ [] \ + \ stp ln rn. steps (Suc 0, [], ) (t_wcode) stp = + (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>)" +apply(simp add: wcode_lemma_pre' t_wcode_def) +done + +lemma wcode_lemma: + "args \ [] \ + \ stp ln rn. steps (Suc 0, [], ) (t_wcode) stp = + (0, [Bk], <[m ,bl_bin ()]> @ 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 \ (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 \ + (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: + "\turing_basic.t_correct tp; + length lm = k; + steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>); + rs > 0\ + \ \ stp m. abc_steps_l (0, [code tp, bl2wc ()]) (F_aprog) stp = + (length (F_aprog), code tp # bl2wc () # (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: + "\rs > 0; + abc_steps_l (0, [code tp, bl2wc ()]) F_aprog stp = + (length F_aprog, code tp # bl2wc () # (rs - 1) # 0\<^bsup>m\<^esup>)\ + \ \stp m n.(steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) 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 ()]" stp "code tp # bl2wc () # (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': + "\turing_basic.t_correct tp; + 0 < rs; + steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\ + \ \stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) 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]: "\rs > 0; Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup> = c @ Bk\<^bsup>n\<^esup>\ + \ \n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" +apply(case_tac "na > n") +apply(subgoal_tac "\ d. na = d + n", auto simp: exp_add) +apply(rule_tac x = "na - n" in exI, simp) +apply(subgoal_tac "\ 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'': + "\turing_basic.t_correct tp; + 0 < rs; + steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\ + \ \stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ 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 ()]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)" + and gr: "rs > 0" + thus "\stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ 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 ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp) + fix a b c + assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)" + "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)" + thus " a = 0 \ b = Bk\<^bsup>ma\<^esup> \ (\n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" + using tinres_steps2[of "<[code tp, bl2wc ()]>" "<[code tp, bl2wc ()]> @ 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> \ \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: + "\turing_basic.t_correct tp; + 0 < rs; + steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\ + \ \stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc ()]> @ 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 ()]> @ 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 "\stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc ()]> @ 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 ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp) + fix a b c + assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ 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 ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)" + thus "a = 0 \ (\m. b = Bk\<^bsup>m\<^esup>) \ (\n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" + using tinres_steps[of "[Bk, Bk]" "[Bk]" "Suc 0" "<[code tp, bl2wc ()]> @ 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: + "\turing_basic.t_correct tp; + 0 < rs; + args \ []; + steps (Suc 0, Bk\<^bsup>i\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\ + \ \stp m n. steps (Suc 0, [], ) UTM_pre stp = + (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" +proof - + let ?Q2 = "\ (l, r). (\ ln rn. l = Bk\<^bsup>ln\<^esup> \ r = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + term ?Q2 + let ?P1 = "\ (l, r). l = [] \ r = " + let ?Q1 = "\ (l, r). (l = [Bk] \ + (\ rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>))" + let ?P2 = ?Q1 + let ?P3 = "\ (l, r). False" + assume h: "turing_basic.t_correct tp" "0 < rs" + "args \ []" "steps (Suc 0, Bk\<^bsup>i\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)" + have "?P1 \-> \ tp. (\ stp tp'. steps (Suc 0, tp) + (t_wcode |+| t_utm) stp = (0, tp') \ ?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 "\stp. case steps (Suc 0, [], ) t_wcode stp of (st, tp') \ + st = 0 \ (case tp' of (l, r) \ l = [Bk] \ + (\rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^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 "\stp. case steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ + Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp of + (st, tp') \ st = 0 \ (case tp' of (l, r) \ + (\ln. l = Bk\<^bsup>ln\<^esup>) \ (\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 \-> ?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: + "\turing_basic.t_correct tp; + 0 < rs; + args \ []; + steps (Suc 0, Bk\<^bsup>i\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\ + \ \stp m n. steps (Suc 0, [], ) 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 \ bool" + where + "TSTD c = (let (st, l, r) = c in + st = 0 \ (\ m. l = Bk\<^bsup>m\<^esup>) \ (\ rs n. r = Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>))" + +thm abacus_turing_eq_uhalt + +lemma nstd_case1: "0 < a \ NSTD (trpl_code (a, b, c))" +apply(simp add: NSTD.simps trpl_code.simps) +done + +lemma [simp]: "\m. b \ Bk\<^bsup>m\<^esup> \ 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 "\ 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: "\m. b \ Bk\<^bsup>m\<^esup> \ 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 \ RR" +apply(induct x arbitrary: y, simp, simp) +apply(case_tac y, simp, simp) +done + +lemma bl2nat_zero_eq[simp]: "(bl2nat c 0 = 0) = (\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: + "\Suc (bl2wc c) = 2 ^ m\ \ \ 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: + "\m. Suc (bl2nat c 0) = 2 ^ m \ \rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" + and h: + "Suc (Suc (2 * bl2nat c 0)) = 2 * 2 ^ nat" + have "\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 "\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]: + "\\rs n. c \ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>; + bl2wc c = 2 ^ lg (Suc (bl2wc c)) 2 - Suc 0\ \ bl2wc c = 0" +apply(subgoal_tac "\ 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: + "\rs n. c \ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup> \ 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: "\ TSTD (a, b, c) + \ 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: + "\turing_basic.t_correct tp; + steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (a, b, c); + \ TSTD (a, b, c)\ + \ rec_exec rec_nonstop [code tp, bl2wc (), stp] = Suc 0" +apply(simp add: rec_nonstop_def rec_exec.simps) +apply(subgoal_tac + "rec_exec rec_conf [code tp, bl2wc (), 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: + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp))\ + \ \y. rec_calc_rel rec_nonstop + ([code tp, bl2wc (), y]) (Suc 0)" +apply(rule_tac allI, erule_tac x = y in allE) +apply(case_tac "steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp y", simp) +apply(rule_tac nonstop_t_uhalt_eq, simp_all) +done + +(* +lemma [simp]: + "\jturing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)); + rec_ci rec_F = (F_ap, rs_pos, a_md)\ + \ \ stp. case abc_steps_l (0, [code tp, bl2wc ()] @ 0\<^bsup>a_md - rs_pos \<^esup> + @ suflm) (F_ap) stp of (ss, e) \ 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 ()" 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': + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)); + rec_ci rec_F = (ap, pos, md)\ + \ \ stp. case abc_steps_l (0, [code tp, bl2wc ()]) ap stp of (ss, e) + \ 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: + "\stp. case abc_steps_l (0, code tp # bl2wc () # 0\<^bsup>md - pos\<^esup>) ap stp of + (ss, e) \ ss < length ap" + "abc_steps_l (0, [code tp, bl2wc ()]) ap stp = (a, b)" + "turing_basic.t_correct tp" + "rec_ci rec_F = (ap, pos, md)" + moreover have "ap \ []" + 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 () # 0\<^bsup>md - pos\<^esup>) ap stp", simp) + fix aa ba + assume g: "aa < length ap" + "abc_steps_l (0, code tp # bl2wc () # 0\<^bsup>md - pos\<^esup>) ap stp = (aa, ba)" + "ap \ []" + thus "?thesis" + using abc_list_crsp_steps[of "[code tp, bl2wc ()]" + "md - pos" ap stp aa ba] h + apply(simp) + done + qed +qed + +lemma uabc_uhalt: + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp))\ + \ \ stp. case abc_steps_l (0, [code tp, bl2wc ()]) F_aprog + stp of (ss, e) \ 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 + "\stp. case abc_steps_l (0, [code tp, bl2wc ()]) a stp of (ss, e) + \ ss < length a" + "rec_ci rec_F = (a, b, c)" + thus + "\stp. case abc_steps_l (0, [code tp, bl2wc ()]) + (a [+] dummy_abc (Suc (Suc 0))) stp of (ss, e) \ + ss < Suc (Suc (Suc (length a)))" + using abc_append_uhalt1[of a "[code tp, bl2wc ()]" + "a [+] dummy_abc (Suc (Suc 0))" "[]" "dummy_abc (Suc (Suc 0))"] + apply(simp) + done +qed + +lemma tutm_uhalt': + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp))\ + \ \ stp. \ isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) t_utm stp)" + using abacus_turing_eq_uhalt[of "layout_of (F_aprog)" + "F_aprog" "F_tprog" "[code tp, bl2wc ()]" + "start_of (layout_of (F_aprog )) (length (F_aprog))" + "Suc (Suc 0)"] +apply(simp add: F_tprog_def) +apply(subgoal_tac "\stp. case abc_steps_l (0, [code tp, bl2wc ()]) + (F_aprog) stp of (as, am) \ 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' \ tinres r' r" +apply(auto simp: tinres_def) +done + +lemma inres_tape: + "\steps (st, l, r) tp stp = (a, b, c); steps (st, l', r') tp stp = (a', b', c'); + tinres l l'; tinres r r'\ + \ a = a' \ tinres b b' \ 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 \ c = ca \ a = aa" + using h + apply(rule_tac tinres_steps, auto) + done + + thm tinres_steps2 + moreover have "b' = ba \ tinres c' ca \ 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: "\ stp. \ isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) t_utm stp) + \ \ stp. \ isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc ()]> @ Bk\<^bsup>n\<^esup>) t_utm stp)" +apply(rule_tac allI, case_tac "(steps (Suc 0, Bk\<^bsup>m\<^esup>, + <[code tp, bl2wc ()]> @ 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 ()]>) 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: + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp))\ + \ \ stp. \ isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc ()]> @ 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: + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)); + args \ []\ + \ \ stp. \ isS0 (steps (Suc 0, [], ) UTM_pre stp)" +proof - + let ?P1 = "\ (l, r). l = [] \ r = " + let ?Q1 = "\ (l, r). (l = [Bk] \ + (\ rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>))" + let ?P4 = ?Q1 + let ?P3 = "\ (l, r). False" + assume h: "turing_basic.t_correct tp" "\stp. \ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)" + "args \ []" + have "?P1 \-> \ tp. \ (\ 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 "\stp. case steps (Suc 0, [], ) t_wcode stp of (st, tp') \ + st = 0 \ (case tp' of (l, r) \ l = [Bk] \ + (\rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^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 ())\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp) + \ 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 ())\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp) \ + isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp)" + by simp + next + show "?Q1 \-> ?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: + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)); + args \ []\ + \ \ stp. \ isS0 (steps (Suc 0, [], ) 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 diff -r 1ce04eb1c8ad -r 48b231495281 utm/abacus.thy --- a/utm/abacus.thy Sat Sep 29 12:38:12 2012 +0000 +++ b/utm/abacus.thy Mon Oct 15 13:23:52 2012 +0000 @@ -1,5 +1,5 @@ header {* - {\em Abacus} (a kind of register machine) + {\em abacus} a kind of register machine *} theory abacus @@ -935,14 +935,15 @@ apply(erule_tac t_split, auto simp: tm_of.simps) done -subsubsection {* The compilation of @{text "Inc n"} *} +(* +subsection {* The compilation of @{text "Inc n"} *} +*) text {* The lemmas in this section lead to the correctness of the compilation of @{text "Inc n"} instruction. *} -(*****Begin: inc crsp*******) fun at_begin_fst_bwtn :: "inc_inv_t" where "at_begin_fst_bwtn (as, lm) (s, l, r) ires = @@ -2568,12 +2569,9 @@ from inc_crsp_ex_pre [OF layout corresponds inc] show ?thesis . qed -(*******End: inc crsp********) - -(*******Begin: dec crsp******) - -subsubsection {* The compilation of @{text "Dec n e"} *} - +(* +subsection {* The compilation of @{text "Dec n e"} *} +*) text {* The lemmas in this section lead to the correctness of the compilation @@ -4834,14 +4832,10 @@ from dec_crsp_ex_pre layout dec correspond show ?thesis by blast qed - -(*******End: dec crsp********) - - -subsubsection {* Compilation of @{text "Goto n"}*} - - -(*******Begin: goto crsp********) +(* +subsection {* Compilation of @{text "Goto n"}*} +*) + lemma goto_fetch: "fetch (ci (layout_of aprog) (start_of (layout_of aprog) as) (Goto n)) (Suc 0) b @@ -4880,9 +4874,8 @@ proof - from goto_crsp_ex_pre and layout goto correspondence show "?thesis" by blast qed -(*******End : goto crsp*********) - -subsubsection {* + +subsection {* The correctness of the compiler *} @@ -5158,8 +5151,7 @@ from steps_crsp_pre [OF layout compiled correspond execution] show ?thesis . qed - -subsubsection {* The Mop-up machine *} +subsection {* The Mop-up machine *} fun mop_bef :: "nat \ tprog" where @@ -6001,7 +5993,6 @@ apply(erule_tac x = rn in allE, simp_all) done -(***Begin: mopup stop***) fun abc_mopup_stage1 :: "t_conf \ nat \ nat" where "abc_mopup_stage1 (s, l, r) n = @@ -6107,26 +6098,6 @@ apply(rule_tac mopup_init, auto) done (***End: mopup stop****) -(* -lemma mopup_stop_cond: "mopup_inv (0, l, r) lm n ires \ - (\ln rn. ?l = Bk\<^bsup>ln\<^esup> @ Bk # Bk # ?ires \ ?r = @ Bk\<^bsup>rn\<^esup>) " - t_halt_conf (0, l, r) \ t_result r = Suc (abc_lm_v lm n)" -apply(simp add: mopup_inv.simps mopup_stop.simps t_halt_conf.simps - t_result.simps, auto simp: tape_of_nat_abv) -apply(rule_tac x = rn in exI, - rule_tac x = "Suc (abc_lm_v lm n)" in exI, - simp add: tape_of_nat_abv) -apply(simp add: tape_of_nat_abv exponent_def) -apply(subgoal_tac "takeWhile (\a. a = Oc) - (replicate (abc_lm_v lm n) Oc @ replicate rn Bk) - = replicate (abc_lm_v lm n) Oc @ takeWhile (\a. a = Oc) - (replicate rn Bk)", simp) -apply(case_tac rn, simp, simp) -apply(rule takeWhile_append2) -apply(case_tac x, auto) -done -*) - lemma mopup_halt_conf_pre: "\n < length lm; crsp_l ly (as, lm) (s, l, r) ires\ @@ -6148,24 +6119,20 @@ apply(rule_tac mopup_halt, simp, simp) done -thm mopup_stop.simps - lemma mopup_halt_conf: assumes len: "n < length lm" and correspond: "crsp_l ly (as, lm) (s, l, r) ires" shows - "\ na. (\ (s', l', r'). ((\ln rn. s' = 0 \ l' = Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires \ r' = Oc\<^bsup>Suc (abc_lm_v lm n)\<^esup> @ Bk\<^bsup>rn\<^esup>))) + "\ na. (\ (s', l', r'). ((\ln rn. s' = 0 \ l' = Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires + \ r' = Oc\<^bsup>Suc (abc_lm_v lm n)\<^esup> @ Bk\<^bsup>rn\<^esup>))) (t_steps (Suc 0, l, r) ((mop_bef n @ tshift mp_up (2 * n)), 0) na)" using len correspond mopup_halt_conf_pre[of n lm ly as s l r ires] apply(simp add: mopup_stop.simps tape_of_nat_abv tape_of_nat_list.simps) done -(*********End: mop_up****************************) - - -subsubsection {* Final results about Abacus machine *} - -thm mopup_halt + +subsection {* Final results about Abacus machine *} + lemma mopup_halt_bef: "\n < length lm; crsp_l ly (as, lm) (s, l, r) ires\ \ \stp. (\(s, l, r). s \ 0 \ ((\ (s', l', r'). s' = 0) (t_step (s, l, r) (mop_bef n @ tshift mp_up (2 * n), 0)))) @@ -6293,29 +6260,6 @@ apply(rule startof_not0, auto) done -(* -lemma stop_conf: "mopup_inv (0, aca, bc) am n - \ t_halt_conf (0, aca, bc) \ t_result bc = Suc (abc_lm_v am n)" -apply(case_tac n, - auto simp: mopup_inv.simps mopup_stop.simps t_halt_conf.simps - t_result.simps tape_of_nl_abv tape_of_nat_abv ) -apply(rule_tac x = "rn" in exI, - rule_tac x = "Suc (abc_lm_v am 0)" in exI, simp) -apply(subgoal_tac "takeWhile (\a. a = Oc) (Oc\<^bsup>abc_lm_v am 0\<^esup> @ Bk\<^bsup>rn\<^esup>) - = Oc\<^bsup>abc_lm_v am 0\<^esup> @ takeWhile (\a. a = Oc) (Bk\<^bsup>rn\<^esup>)", simp) -apply(simp add: exponent_def, case_tac rn, simp, simp) -apply(rule_tac takeWhile_append2, simp add: exponent_def) -apply(rule_tac x = rn in exI, - rule_tac x = "Suc (abc_lm_v am (Suc nat))" in exI, simp) -apply(subgoal_tac - "takeWhile (\a. a = Oc) (Oc\<^bsup>abc_lm_v am (Suc nat)\<^esup> @ Bk\<^bsup>rn\<^esup>) = - Oc\<^bsup>abc_lm_v am (Suc nat)\<^esup> @ takeWhile (\a. a = Oc) (Bk\<^bsup>rn\<^esup>)", simp) -apply(simp add: exponent_def, case_tac rn, simp, simp) -apply(rule_tac takeWhile_append2, simp add: exponent_def) -done -*) - - lemma start_of_out_range: "as \ length aprog \ start_of (layout_of aprog) as = @@ -6448,7 +6392,8 @@ TM @{text "(tMp n (mop_ss - 1))"} will halt and gives rise to a configuration which only hold the content of memory cell @{text "n"}: *} - "\ stp. (\ (s, l, r). \ ln rn. s = 0 \ l = Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires \ r = Oc\<^bsup>Suc (abc_lm_v am n)\<^esup> @ Bk\<^bsup>rn\<^esup>) + "\ stp. (\ (s, l, r). \ ln rn. s = 0 \ l = Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires + \ r = Oc\<^bsup>Suc (abc_lm_v am n)\<^esup> @ Bk\<^bsup>rn\<^esup>) (t_steps tc (tprog @ (tMp n (mop_ss - 1)), 0) stp)" proof - from layout complied correspond halt_state abc_exec rs_len mopup_start @@ -6656,6 +6601,7 @@ apply(rule_tac abacus_turing_eq_unhalt_case_pre, auto) done + definition abc_list_crsp:: "nat list \ nat list \ bool" where "abc_list_crsp xs ys = (\ n. xs = ys @ 0\<^bsup>n\<^esup> \ ys = xs @ 0\<^bsup>n\<^esup>)" @@ -6663,7 +6609,6 @@ apply(auto simp: abc_list_crsp_def) done -thm abc_lm_v.simps lemma abc_list_crsp_lm_v: "abc_list_crsp lma lmb \ abc_lm_v lma n = abc_lm_v lmb n" apply(auto simp: abc_list_crsp_def abc_lm_v.simps @@ -6748,8 +6693,6 @@ split: abc_inst.splits if_splits) done -thm abc_step_l.simps - lemma abc_steps_red: "abc_steps_l ac aprog stp = (as, am) \ abc_steps_l ac aprog (Suc stp) = @@ -6799,13 +6742,10 @@ done qed -text {* Begin: equvilence between steps and t_steps*} lemma [simp]: "(case ca of [] \ Bk | Bk # xs \ Bk | Oc # xs \ Oc) = (case ca of [] \ Bk | x # xs \ x)" by(case_tac ca, simp_all, case_tac a, simp, simp) -text {* needed to interpret*} - lemma steps_eq: "length t mod 2 = 0 \ t_steps c (t, 0) stp = steps c t stp" apply(induct stp) @@ -6815,8 +6755,6 @@ apply(auto simp: t_step.simps tstep.simps) done -text{* end: equvilence between steps and t_steps*} - lemma crsp_l_start: "crsp_l ly (0, lm) (Suc 0, Bk # Bk # ires, @ Bk\<^bsup>rn\<^esup>) ires" apply(simp add: crsp_l.simps, auto simp: start_of.simps) done @@ -6867,7 +6805,6 @@ done -thm tinres_steps lemma list_length: "xs = ys \ length xs = length ys" by simp lemma [elim]: "tinres (Bk\<^bsup>m\<^esup>) b \ \m. b = Bk\<^bsup>m\<^esup>" @@ -6903,6 +6840,7 @@ text {* Main theorem for the case when the original Abacus program does halt. *} + lemma abacus_turing_eq_halt: assumes layout: "ly = layout_of aprog" @@ -6952,26 +6890,7 @@ (start_of ly (length aprog) - Suc 0)) mod 2 = 0") apply(simp add: steps_eq, auto simp: isS0_def) done -(* -lemma abacus_turing_eq_uhalt_pre: - "\ly = layout_of aprog; - tprog = tm_of aprog; - \ stp. ((\ (as, am). as < length aprog) - (abc_steps_l (0, lm) aprog stp)); - mop_ss = start_of ly (length aprog)\ - \ (\ (\ stp. isS0 (steps (Suc 0, [Bk, Bk], ) - (tprog @ (tMp n (mop_ss - 1))) stp)))" -apply(drule_tac k = 0 and n = n in abacus_turing_eq_uhalt', auto) -apply(erule_tac x = stp in allE, erule_tac x = stp in allE) -apply(subgoal_tac "tinres ([Bk]) (Bk\<^bsup>k\<^esup>)") -apply(case_tac "steps (Suc 0, Bk\<^bsup>k\<^esup>, ) - (tm_of aprog @ tMp n (start_of ly (length aprog) - Suc 0)) stp") -apply(case_tac - "steps (Suc 0, [Bk], ) - (tm_of aprog @ tMp n (start_of ly (length aprog) - Suc 0)) stp") -apply(drule_tac tinres_steps, auto simp: isS0_def) -done -*) + text {* Main theorem for the case when the original Abacus program does not halt. *} @@ -7000,6 +6919,5 @@ layout compiled abc_unhalt mop_start by(auto) - end diff -r 1ce04eb1c8ad -r 48b231495281 utm/document/root.tex --- a/utm/document/root.tex Sat Sep 29 12:38:12 2012 +0000 +++ b/utm/document/root.tex Mon Oct 15 13:23:52 2012 +0000 @@ -1,6 +1,29 @@ \documentclass[11pt,a4paper]{article} \usepackage{isabelle,isabellesym} - +%begin adding +%\usepackage{pdfsetup} +\usepackage{fancyhdr} +\usepackage{beamerarticle} +\usepackage[english]{babel} +%\usepackage{enumitem} +\usepackage{enumerate} +\usepackage{cases} +%\usepackage{CJK,cjknumb} +%\usepackage{pgf,pgfarrows,pgfnodes,pgfautomata,pgfheaps,pgfshade} +\usepackage{amsmath,amssymb} +%\usepackage[latin1]{inputenc} +%\usepackage{colortbl} +\usepackage{tikz} +\usetikzlibrary{arrows,automata,decorations,fit,calc} +\usetikzlibrary{shapes,shapes.arrows,snakes,positioning} +\usepgflibrary{shapes.misc} % LATEX and plain TEX and pure pgf +\usetikzlibrary{matrix} +\usepackage[latin1]{inputenc} +\usepackage{verbatim} +\usepackage{romannum} +\usepackage{makeidx} +\usepackage{listings} +%end adding % further packages required for unusual symbols (see also % isabellesym.sty), use only when needed @@ -32,7 +55,7 @@ % for uniform font size %\renewcommand{\isastyle}{\isastyleminor} - +\newcommand{\wuhao}{\fontsize{6pt}{10pt}\selectfont} % ÎåºÅ, µ¥±¶Ðоà \begin{document} diff -r 1ce04eb1c8ad -r 48b231495281 utm/turing_basic.thy --- a/utm/turing_basic.thy Sat Sep 29 12:38:12 2012 +0000 +++ b/utm/turing_basic.thy Mon Oct 15 13:23:52 2012 +0000 @@ -1,747 +1,736 @@ -theory turing_basic -imports Main -begin - -section {* Basic definitions of Turing machine *} - -(* Title: Turing machine's definition and its charater - Author: Xu Jian - Maintainer: Xu Jian -*) - -text {* -\label{description of turing machine} -*} - -section {* Basic definitions of Turing machine *} - -(* Title: Turing machine's definition and its charater - Author: Xu Jian - Maintainer: Xu Jian -*) - -text {* - Actions of Turing machine (Abbreviated TM in the following* ). -*} - -datatype taction = - -- {* Write zero *} - W0 | - -- {* Write one *} - W1 | - -- {* Move left *} - L | - -- {* Move right *} - R | - -- {* Do nothing *} - Nop - -text {* - Tape contents in every block. -*} - -datatype block = - -- {* Blank *} - Bk | - -- {* Occupied *} - Oc - -text {* - Tape is represented as a pair of lists $(L_{left}, L_{right})$, - where $L_left$, named {\em left list}, is used to represent - the tape to the left of RW-head and - $L_{right}$, named {\em right list}, is used to represent the tape - under and to the right of RW-head. -*} - -type_synonym tape = "block list \ block list" - -text {* The state of turing machine.*} -type_synonym tstate = nat - -text {* - Turing machine instruction is represented as a - pair @{text "(action, next_state)"}, - where @{text "action"} is the action to take at the current state - and @{text "next_state"} is the next state the machine is getting into - after the action. -*} -type_synonym tinst = "taction \ tstate" - -text {* - Program of Turing machine is represented as a list of Turing instructions - and the execution of the program starts from the head of the list. - *} -type_synonym tprog = "tinst list" - - -text {* - Turing machine configuration, which consists of the current state - and the tape. -*} -type_synonym t_conf = "tstate \ tape" - -fun nth_of :: "'a list \ nat \ 'a option" - where - "nth_of xs n = (if n < length xs then Some (xs!n) - else None)" - -text {* - The function used to fetech instruction out of Turing program. - *} - -fun fetch :: "tprog \ tstate \ block \ tinst" - where - "fetch p s b = (if s = 0 then (Nop, 0) else - case b of - Bk \ case nth_of p (2 * (s - 1)) of - Some i \ i - | None \ (Nop, 0) - | Oc \ case nth_of p (2 * (s - 1) +1) of - Some i \ i - | None \ (Nop, 0))" - - -fun new_tape :: "taction \ tape \ tape" -where - "new_tape action (leftn, rightn) = (case action of - W0 \ (leftn, Bk#(tl rightn)) | - W1 \ (leftn, Oc#(tl rightn)) | - L \ (if leftn = [] then (tl leftn, Bk#rightn) - else (tl leftn, (hd leftn) # rightn)) | - R \ if rightn = [] then (Bk#leftn,tl rightn) - else ((hd rightn)#leftn, tl rightn) | - Nop \ (leftn, rightn) - )" - -text {* - The one step function used to transfer Turing machine configuration. -*} -fun tstep :: "t_conf \ tprog \ t_conf" - where - "tstep c p = (let (s, l, r) = c in - let (ac, ns) = (fetch p s (case r of [] \ Bk | - x # xs \ x)) in - (ns, new_tape ac (l, r)))" - -text {* - The many-step function. -*} -fun steps :: "t_conf \ tprog \ nat \ t_conf" - where - "steps c p 0 = c" | - "steps c p (Suc n) = steps (tstep c p) p n" - -lemma tstep_red: "steps c p (Suc n) = tstep (steps c p n) p" -proof(induct n arbitrary: c) - fix c - show "steps c p (Suc 0) = tstep (steps c p 0) p" by(simp add: steps.simps) -next - fix n c - assume ind: "\ c. steps c p (Suc n) = tstep (steps c p n) p" - have "steps (tstep c p) p (Suc n) = tstep (steps (tstep c p) p n) p" - by(rule ind) - thus "steps c p (Suc (Suc n)) = tstep (steps c p (Suc n)) p" by(simp add: steps.simps) -qed - -declare Let_def[simp] option.split[split] - -definition - "iseven n \ \ x. n = 2 * x" - - -text {* - The following @{text "t_correct"} function is used to specify the wellformedness of Turing - machine. -*} -fun t_correct :: "tprog \ bool" - where - "t_correct p = (length p \ 2 \ iseven (length p) \ - list_all (\ (acn, s). s \ length p div 2) p)" - -declare t_correct.simps[simp del] - -lemma allimp: "\\x. P x \ Q x; \x. P x\ \ \x. Q x" -by(auto elim: allE) - -lemma halt_lemma: "\wf LE; \ n. (\ P (f n) \ (f (Suc n), (f n)) \ LE)\ \ \ n. P (f n)" -apply(rule exCI, drule allimp, auto) -apply(drule_tac f = f in wf_inv_image, simp add: inv_image_def) -apply(erule wf_induct, auto) -done - -lemma steps_add: "steps c t (x + y) = steps (steps c t x) t y" -by(induct x arbitrary: c, auto simp: steps.simps tstep_red) - -lemma listall_set: "list_all p t \ \ a \ set t. p a" -by(induct t, auto) - -lemma fetch_ex: "\b a. fetch T aa ab = (b, a)" -by(simp add: fetch.simps) -definition exponent :: "'a \ nat \ 'a list" ("_\<^bsup>_\<^esup>" [0, 0]100) - where "exponent x n = replicate n x" - -text {* - @{text "tinres l1 l2"} means left list @{text "l1"} is congruent with - @{text "l2"} with respect to the execution of Turing machine. - Appending Blank to the right of eigther one does not affect the - outcome of excution. -*} - -definition tinres :: "block list \ block list \ bool" - where - "tinres bx by = (\ n. bx = by@Bk\<^bsup>n\<^esup> \ by = bx @ Bk\<^bsup>n\<^esup>)" - -lemma exp_zero: "a\<^bsup>0\<^esup> = []" -by(simp add: exponent_def) -lemma exp_ind_def: "a\<^bsup>Suc x \<^esup> = a # a\<^bsup>x\<^esup>" -by(simp add: exponent_def) - -text {* - The following lemma shows the meaning of @{text "tinres"} with respect to - one step execution. - *} -lemma tinres_step: - "\tinres l l'; tstep (ss, l, r) t = (sa, la, ra); tstep (ss, l', r) t = (sb, lb, rb)\ - \ tinres la lb \ ra = rb \ sa = sb" -apply(auto simp: tstep.simps fetch.simps new_tape.simps - split: if_splits taction.splits list.splits - block.splits) -apply(case_tac [!] "t ! (2 * (ss - Suc 0))", - auto simp: exponent_def tinres_def split: if_splits taction.splits list.splits - block.splits) -apply(case_tac [!] "t ! (2 * (ss - Suc 0) + Suc 0)", - auto simp: exponent_def tinres_def split: if_splits taction.splits list.splits - block.splits) -done - -declare tstep.simps[simp del] steps.simps[simp del] - -text {* - The following lemma shows the meaning of @{text "tinres"} with respect to - many step execution. - *} -lemma tinres_steps: - "\tinres l l'; steps (ss, l, r) t stp = (sa, la, ra); steps (ss, l', r) t stp = (sb, lb, rb)\ - \ tinres la lb \ ra = rb \ 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: "\sa la ra sb lb rb. \steps (ss, l, r) t stp = (sa, la, ra); - steps (ss, l', r) t stp = (sb, lb, rb)\ \ tinres la lb \ ra = rb \ sa = sb" - and h: " tinres l l'" "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 "tinres b ba \ c = ca \ a = aa" - apply(rule_tac ind, simp_all add: h) - done - thus "tinres la lb \ ra = rb \ sa = sb" - apply(rule_tac l = b and l' = ba and r = c and ss = a - and t = t in tinres_step) - using h - apply(simp, simp, simp) - done -qed - -text {* - The following function @{text "tshift tp n"} is used to shift Turing programs - @{text "tp"} by @{text "n"} when it is going to be combined with others. - *} - -fun tshift :: "tprog \ nat \ tprog" - where - "tshift tp off = (map (\ (action, state). (action, (if state = 0 then 0 - else state + off))) tp)" - -text {* - When two Turing programs are combined, the end state (state @{text "0"}) of the one - at the prefix position needs to be connected to the start state - of the one at postfix position. If @{text "tp"} is the Turing program - to be at the prefix, @{text "change_termi_state tp"} is the transformed Turing program. - *} -fun change_termi_state :: "tprog \ tprog" - where - "change_termi_state t = - (map (\ (acn, ns). if ns = 0 then (acn, Suc ((length t) div 2)) else (acn, ns)) t)" - -text {* - @{text "t_add tp1 tp2"} is the combined Truing program. -*} - -fun t_add :: "tprog \ tprog \ tprog" ("_ |+| _" [0, 0] 100) - where - "t_add t1 t2 = ((change_termi_state t1) @ (tshift t2 ((length t1) div 2)))" - -text {* - Tests whether the current configuration is at state @{text "0"}. -*} -definition isS0 :: "t_conf \ bool" - where - "isS0 c = (let (s, l, r) = c in s = 0)" - -declare tstep.simps[simp del] steps.simps[simp del] - t_add.simps[simp del] fetch.simps[simp del] - new_tape.simps[simp del] - - -text {* - Single step execution starting from state @{text "0"} will not make any progress. -*} -lemma tstep_0: "tstep (0, tp) p = (0, tp)" -apply(simp add: tstep.simps fetch.simps new_tape.simps) -done - - -text {* - Many step executions starting from state @{text "0"} will not make any progress. -*} - -lemma steps_0: "steps (0, tp) p stp = (0, tp)" -apply(induct stp) -apply(simp add: steps.simps) -apply(simp add: tstep_red tstep_0) -done - -lemma s_keep_step: "\a \ length A div 2; tstep (a, b, c) A = (s, l, r); t_correct A\ - \ s \ length A div 2" -apply(simp add: tstep.simps fetch.simps t_correct.simps iseven_def - split: if_splits block.splits list.splits) -apply(case_tac [!] a, auto simp: list_all_length) -apply(erule_tac x = "2 * nat" in allE, auto) -apply(erule_tac x = "2 * nat" in allE, auto) -apply(erule_tac x = "Suc (2 * nat)" in allE, auto) -done - -lemma s_keep: "\steps (Suc 0, tp) A stp = (s, l, r); t_correct A\ \ s \ length A div 2" -proof(induct stp arbitrary: s l r) - case 0 thus "?case" by(auto simp: t_correct.simps steps.simps) -next - fix stp s l r - assume ind: "\s l r. \steps (Suc 0, tp) A stp = (s, l, r); t_correct A\ \ s \ length A div 2" - and h1: "steps (Suc 0, tp) A (Suc stp) = (s, l, r)" - and h2: "t_correct A" - from h1 h2 show "s \ length A div 2" - proof(simp add: tstep_red, cases "(steps (Suc 0, tp) A stp)", simp) - fix a b c - assume h3: "tstep (a, b, c) A = (s, l, r)" - and h4: "steps (Suc 0, tp) A stp = (a, b, c)" - have "a \ length A div 2" - using h2 h4 - by(rule_tac l = b and r = c in ind, auto) - thus "?thesis" - using h3 h2 - by(simp add: s_keep_step) - qed -qed - -lemma t_merge_fetch_pre: - "\fetch A s b = (ac, ns); s \ length A div 2; t_correct A; s \ 0\ \ - fetch (A |+| B) s b = (ac, if ns = 0 then Suc (length A div 2) else ns)" -apply(subgoal_tac "2 * (s - Suc 0) < length A \ Suc (2 * (s - Suc 0)) < length A") -apply(auto simp: fetch.simps t_add.simps split: if_splits block.splits) -apply(simp_all add: nth_append change_termi_state.simps) -done - -lemma [simp]: "\\ a \ length A div 2; t_correct A\ \ fetch A a b = (Nop, 0)" -apply(auto simp: fetch.simps del: nth_of.simps split: block.splits) -apply(case_tac [!] a, auto simp: t_correct.simps iseven_def) -done - -lemma [elim]: "\t_correct A; \ isS0 (tstep (a, b, c) A)\ \ a \ length A div 2" -apply(rule_tac classical, auto simp: tstep.simps new_tape.simps isS0_def) -done - -lemma [elim]: "\t_correct A; \ isS0 (tstep (a, b, c) A)\ \ 0 < a" -apply(rule_tac classical, simp add: tstep_0 isS0_def) -done - - -lemma t_merge_pre_eq_step: "\tstep (a, b, c) A = cf; t_correct A; \ isS0 cf\ - \ tstep (a, b, c) (A |+| B) = cf" -apply(subgoal_tac "a \ length A div 2 \ a \ 0") -apply(simp add: tstep.simps) -apply(case_tac "fetch A a (case c of [] \ Bk | x # xs \ x)", simp) -apply(drule_tac B = B in t_merge_fetch_pre, simp, simp, simp, simp add: isS0_def, auto) -done - -lemma t_merge_pre_eq: "\steps (Suc 0, tp) A stp = cf; \ isS0 cf; t_correct A\ - \ steps (Suc 0, tp) (A |+| B) stp = cf" -proof(induct stp arbitrary: cf) - case 0 thus "?case" by(simp add: steps.simps) -next - fix stp cf - assume ind: "\cf. \steps (Suc 0, tp) A stp = cf; \ isS0 cf; t_correct A\ - \ steps (Suc 0, tp) (A |+| B) stp = cf" - and h1: "steps (Suc 0, tp) A (Suc stp) = cf" - and h2: "\ isS0 cf" - and h3: "t_correct A" - from h1 h2 h3 show "steps (Suc 0, tp) (A |+| B) (Suc stp) = cf" - proof(simp add: tstep_red, cases "steps (Suc 0, tp) (A) stp", simp) - fix a b c - assume h4: "tstep (a, b, c) A = cf" - and h5: "steps (Suc 0, tp) A stp = (a, b, c)" - have "steps (Suc 0, tp) (A |+| B) stp = (a, b, c)" - proof(cases a) - case 0 thus "?thesis" - using h4 h2 - apply(simp add: tstep_0, cases cf, simp add: isS0_def) - done - next - case (Suc n) thus "?thesis" - using h5 h3 - apply(rule_tac ind, auto simp: isS0_def) - done - qed - thus "tstep (steps (Suc 0, tp) (A |+| B) stp) (A |+| B) = cf" - using h4 h5 h3 h2 - apply(simp) - apply(rule t_merge_pre_eq_step, auto) - done - qed -qed - -declare nth.simps[simp del] tshift.simps[simp del] change_termi_state.simps[simp del] - -lemma [simp]: "length (change_termi_state A) = length A" -by(simp add: change_termi_state.simps) - -lemma first_halt_point: "steps (Suc 0, tp) A stp = (0, tp') - \ \stp. \ isS0 (steps (Suc 0, tp) A stp) \ steps (Suc 0, tp) A (Suc stp) = (0, tp')" -proof(induct stp) - case 0 thus "?case" by(simp add: steps.simps) -next - case (Suc n) - fix stp - assume ind: "steps (Suc 0, tp) A stp = (0, tp') \ - \stp. \ isS0 (steps (Suc 0, tp) A stp) \ steps (Suc 0, tp) A (Suc stp) = (0, tp')" - and h: "steps (Suc 0, tp) A (Suc stp) = (0, tp')" - from h show "\stp. \ isS0 (steps (Suc 0, tp) A stp) \ steps (Suc 0, tp) A (Suc stp) = (0, tp')" - proof(simp add: tstep_red, cases "steps (Suc 0, tp) A stp", simp, case_tac a) - fix a b c - assume g1: "a = (0::nat)" - and g2: "tstep (a, b, c) A = (0, tp')" - and g3: "steps (Suc 0, tp) A stp = (a, b, c)" - have "steps (Suc 0, tp) A stp = (0, tp')" - using g2 g1 g3 - by(simp add: tstep_0) - hence "\ stp. \ isS0 (steps (Suc 0, tp) A stp) \ steps (Suc 0, tp) A (Suc stp) = (0, tp')" - by(rule ind) - thus "\stp. \ isS0 (steps (Suc 0, tp) A stp) \ tstep (steps (Suc 0, tp) A stp) A = (0, tp')" - apply(simp add: tstep_red) - done - next - fix a b c nat - assume g1: "steps (Suc 0, tp) A stp = (a, b, c)" - and g2: "steps (Suc 0, tp) A (Suc stp) = (0, tp')" "a= Suc nat" - thus "\stp. \ isS0 (steps (Suc 0, tp) A stp) \ tstep (steps (Suc 0, tp) A stp) A = (0, tp')" - apply(rule_tac x = stp in exI) - apply(simp add: isS0_def tstep_red) - done - qed -qed - -lemma t_merge_pre_halt_same': - "\\ isS0 (steps (Suc 0, tp) A stp) ; steps (Suc 0, tp) A (Suc stp) = (0, tp'); t_correct A\ - \ steps (Suc 0, tp) (A |+| B) (Suc stp) = (Suc (length A div 2), tp')" -proof(simp add: tstep_red, cases "steps (Suc 0, tp) A stp", simp) - fix a b c - assume h1: "\ isS0 (a, b, c)" - and h2: "tstep (a, b, c) A = (0, tp')" - and h3: "t_correct A" - and h4: "steps (Suc 0, tp) A stp = (a, b, c)" - have "steps (Suc 0, tp) (A |+| B) stp = (a, b, c)" - using h1 h4 h3 - apply(rule_tac t_merge_pre_eq, auto) - done - moreover have "tstep (a, b, c) (A |+| B) = (Suc (length A div 2), tp')" - using h2 h3 h1 h4 - apply(simp add: tstep.simps) - apply(case_tac " fetch A a (case c of [] \ Bk | x # xs \ x)", simp) - apply(drule_tac B = B in t_merge_fetch_pre, auto simp: isS0_def intro: s_keep) - done - ultimately show "tstep (steps (Suc 0, tp) (A |+| B) stp) (A |+| B) = (Suc (length A div 2), tp')" - by(simp) -qed - -text {* - When Turing machine @{text "A"} and @{text "B"} are combined and the execution - of @{text "A"} can termination within @{text "stp"} steps, - the combined machine @{text "A |+| B"} will eventually get into the starting - state of machine @{text "B"}. -*} -lemma t_merge_pre_halt_same: " - \steps (Suc 0, tp) A stp = (0, tp'); t_correct A; t_correct B\ - \ \ stp. steps (Suc 0, tp) (A |+| B) stp = (Suc (length A div 2), tp')" -proof - - assume a_wf: "t_correct A" - and b_wf: "t_correct B" - and a_ht: "steps (Suc 0, tp) A stp = (0, tp')" - have halt_point: "\ stp. \ isS0 (steps (Suc 0, tp) A stp) \ steps (Suc 0, tp) A (Suc stp) = (0, tp')" - using a_ht - by(erule_tac first_halt_point) - then obtain stp' where "\ isS0 (steps (Suc 0, tp) A stp') \ steps (Suc 0, tp) A (Suc stp') = (0, tp')".. - hence "steps (Suc 0, tp) (A |+| B) (Suc stp') = (Suc (length A div 2), tp')" - using a_wf - apply(rule_tac t_merge_pre_halt_same', auto) - done - thus "?thesis" .. -qed - -lemma fetch_0: "fetch p 0 b = (Nop, 0)" -by(simp add: fetch.simps) - -lemma [simp]: "length (tshift B x) = length B" -by(simp add: tshift.simps) - -lemma [simp]: "t_correct A \ 2 * (length A div 2) = length A" -apply(simp add: t_correct.simps iseven_def, auto) -done - -lemma t_merge_fetch_snd: - "\fetch B a b = (ac, ns); t_correct A; t_correct B; a > 0 \ - \ fetch (A |+| B) (a + length A div 2) b - = (ac, if ns = 0 then 0 else ns + length A div 2)" -apply(auto simp: fetch.simps t_add.simps split: if_splits block.splits) -apply(case_tac [!] a, simp_all) -apply(simp_all add: nth_append change_termi_state.simps tshift.simps) -done - -lemma t_merge_snd_eq_step: - "\tstep (s, l, r) B = (s', l', r'); t_correct A; t_correct B; s > 0\ - \ tstep (s + length A div 2, l, r) (A |+| B) = - (if s' = 0 then 0 else s' + length A div 2, l' ,r') " -apply(simp add: tstep.simps) -apply(cases "fetch B s (case r of [] \ Bk | x # xs \ x)") -apply(auto simp: t_merge_fetch_snd) -apply(frule_tac [!] t_merge_fetch_snd, auto) -done - -text {* - Relates the executions of TM @{text "B"}, one is when @{text "B"} is executed alone, - the other is the execution when @{text "B"} is in the combined TM. -*} -lemma t_merge_snd_eq_steps: - "\t_correct A; t_correct B; steps (s, l, r) B stp = (s', l', r'); s > 0\ - \ steps (s + length A div 2, l, r) (A |+| B) stp = - (if s' = 0 then 0 else s' + length A div 2, l', r')" -proof(induct stp arbitrary: s' l' r') - case 0 thus "?case" - by(simp add: steps.simps) -next - fix stp s' l' r' - assume ind: "\s' l' r'. \t_correct A; t_correct B; steps (s, l, r) B stp = (s', l', r'); 0 < s\ - \ steps (s + length A div 2, l, r) (A |+| B) stp = - (if s' = 0 then 0 else s' + length A div 2, l', r')" - and h1: "steps (s, l, r) B (Suc stp) = (s', l', r')" - and h2: "t_correct A" - and h3: "t_correct B" - and h4: "0 < s" - from h1 show "steps (s + length A div 2, l, r) (A |+| B) (Suc stp) - = (if s' = 0 then 0 else s' + length A div 2, l', r')" - proof(simp only: tstep_red, cases "steps (s, l, r) B stp") - fix a b c - assume h5: "steps (s, l, r) B stp = (a, b, c)" "tstep (steps (s, l, r) B stp) B = (s', l', r')" - hence h6: "(steps (s + length A div 2, l, r) (A |+| B) stp) = - ((if a = 0 then 0 else a + length A div 2, b, c))" - using h2 h3 h4 - by(rule_tac ind, auto) - thus "tstep (steps (s + length A div 2, l, r) (A |+| B) stp) (A |+| B) = - (if s' = 0 then 0 else s'+ length A div 2, l', r')" - using h5 - proof(auto) - assume "tstep (0, b, c) B = (0, l', r')" thus "tstep (0, b, c) (A |+| B) = (0, l', r')" - by(simp add: tstep_0) - next - assume "tstep (0, b, c) B = (s', l', r')" "0 < s'" - thus "tstep (0, b, c) (A |+| B) = (s' + length A div 2, l', r')" - by(simp add: tstep_0) - next - assume "tstep (a, b, c) B = (0, l', r')" "0 < a" - thus "tstep (a + length A div 2, b, c) (A |+| B) = (0, l', r')" - using h2 h3 - by(drule_tac t_merge_snd_eq_step, auto) - next - assume "tstep (a, b, c) B = (s', l', r')" "0 < a" "0 < s'" - thus "tstep (a + length A div 2, b, c) (A |+| B) = (s' + length A div 2, l', r')" - using h2 h3 - by(drule_tac t_merge_snd_eq_step, auto) - qed - qed -qed - -lemma t_merge_snd_halt_eq: - "\steps (Suc 0, tp) B stp = (0, tp'); t_correct A; t_correct B\ - \ \stp. steps (Suc (length A div 2), tp) (A |+| B) stp = (0, tp')" -apply(case_tac tp, cases tp', simp) -apply(drule_tac s = "Suc 0" in t_merge_snd_eq_steps, auto) -done - -lemma t_inj: "\steps (Suc 0, tp) A stpa = (0, tp1); steps (Suc 0, tp) A stpb = (0, tp2)\ - \ tp1 = tp2" -proof - - assume h1: "steps (Suc 0, tp) A stpa = (0, tp1)" - and h2: "steps (Suc 0, tp) A stpb = (0, tp2)" - thus "?thesis" - proof(cases "stpa < stpb") - case True thus "?thesis" - using h1 h2 - apply(drule_tac less_imp_Suc_add, auto) - apply(simp del: add_Suc_right add_Suc add: add_Suc_right[THEN sym] steps_add steps_0) - done - next - case False thus "?thesis" - using h1 h2 - apply(drule_tac leI) - apply(case_tac "stpb = stpa", auto) - apply(subgoal_tac "stpb < stpa") - apply(drule_tac less_imp_Suc_add, auto) - apply(simp del: add_Suc_right add_Suc add: add_Suc_right[THEN sym] steps_add steps_0) - done - qed -qed - -type_synonym t_assert = "tape \ bool" - -definition t_imply :: "t_assert \ t_assert \ bool" ("_ \-> _" [0, 0] 100) - where - "t_imply a1 a2 = (\ tp. a1 tp \ a2 tp)" - - -locale turing_merge = - fixes A :: "tprog" and B :: "tprog" and P1 :: "t_assert" - and P2 :: "t_assert" - and P3 :: "t_assert" - and P4 :: "t_assert" - and Q1:: "t_assert" - and Q2 :: "t_assert" - assumes - A_wf : "t_correct A" - and B_wf : "t_correct B" - and A_halt : "P1 tp \ \ stp. let (s, tp') = steps (Suc 0, tp) A stp in s = 0 \ Q1 tp'" - and B_halt : "P2 tp \ \ stp. let (s, tp') = steps (Suc 0, tp) B stp in s = 0 \ Q2 tp'" - and A_uhalt : "P3 tp \ \ (\ stp. isS0 (steps (Suc 0, tp) A stp))" - and B_uhalt: "P4 tp \ \ (\ stp. isS0 (steps (Suc 0, tp) B stp))" -begin - - -text {* - The following lemma tries to derive the Hoare logic rule for sequentially combined TMs. - It deals with the situtation when both @{text "A"} and @{text "B"} are terminated. -*} - -lemma t_merge_halt: - assumes aimpb: "Q1 \-> P2" - shows "P1 \-> \ tp. (\ stp tp'. steps (Suc 0, tp) (A |+| B) stp = (0, tp') \ Q2 tp')" -proof(simp add: t_imply_def, auto) - fix a b - assume h: "P1 (a, b)" - hence "\ stp. let (s, tp') = steps (Suc 0, a, b) A stp in s = 0 \ Q1 tp'" - using A_halt by simp - from this obtain stp1 where "let (s, tp') = steps (Suc 0, a, b) A stp1 in s = 0 \ Q1 tp'" .. - thus "\stp aa ba. steps (Suc 0, a, b) (A |+| B) stp = (0, aa, ba) \ Q2 (aa, ba)" - proof(case_tac "steps (Suc 0, a, b) A stp1", simp, erule_tac conjE) - fix aa ba c - assume g1: "Q1 (ba, c)" - and g2: "steps (Suc 0, a, b) A stp1 = (0, ba, c)" - hence "P2 (ba, c)" - using aimpb apply(simp add: t_imply_def) - done - hence "\ stp. let (s, tp') = steps (Suc 0, ba, c) B stp in s = 0 \ Q2 tp'" - using B_halt by simp - from this obtain stp2 where "let (s, tp') = steps (Suc 0, ba, c) B stp2 in s = 0 \ Q2 tp'" .. - thus "?thesis" - proof(case_tac "steps (Suc 0, ba, c) B stp2", simp, erule_tac conjE) - fix aa bb ca - assume g3: " Q2 (bb, ca)" "steps (Suc 0, ba, c) B stp2 = (0, bb, ca)" - have "\ stp. steps (Suc 0, a, b) (A |+| B) stp = (Suc (length A div 2), ba , c)" - using g2 A_wf B_wf - by(rule_tac t_merge_pre_halt_same, auto) - moreover have "\ stp. steps (Suc (length A div 2), ba, c) (A |+| B) stp = (0, bb, ca)" - using g3 A_wf B_wf - apply(rule_tac t_merge_snd_halt_eq, auto) - done - ultimately show "\stp aa ba. steps (Suc 0, a, b) (A |+| B) stp = (0, aa, ba) \ Q2 (aa, ba)" - apply(erule_tac exE, erule_tac exE) - apply(rule_tac x = "stp + stpa" in exI, simp add: steps_add) - using g3 by simp - qed - qed -qed - -lemma t_merge_uhalt_tmp: - assumes B_uh: "\stp. \ isS0 (steps (Suc 0, b, c) B stp)" - and merge_ah: "steps (Suc 0, tp) (A |+| B) stpa = (Suc (length A div 2), b, c)" - shows "\ stp. \ isS0 (steps (Suc 0, tp) (A |+| B) stp)" - using B_uh merge_ah -apply(rule_tac allI) -apply(case_tac "stp > stpa") -apply(erule_tac x = "stp - stpa" in allE) -apply(case_tac "(steps (Suc 0, b, c) B (stp - stpa))", simp) -proof - - fix stp a ba ca - assume h1: "\ isS0 (a, ba, ca)" "stpa < stp" - and h2: "steps (Suc 0, b, c) B (stp - stpa) = (a, ba, ca)" - have "steps (Suc 0 + length A div 2, b, c) (A |+| B) (stp - stpa) = - (if a = 0 then 0 else a + length A div 2, ba, ca)" - using A_wf B_wf h2 - by(rule_tac t_merge_snd_eq_steps, auto) - moreover have "a > 0" using h1 by(simp add: isS0_def) - moreover have "\ stpb. stp = stpa + stpb" - using h1 by(rule_tac x = "stp - stpa" in exI, simp) - ultimately show "\ isS0 (steps (Suc 0, tp) (A |+| B) stp)" - using merge_ah - by(auto simp: steps_add isS0_def) -next - fix stp - assume h: "steps (Suc 0, tp) (A |+| B) stpa = (Suc (length A div 2), b, c)" "\ stpa < stp" - hence "\ stpb. stpa = stp + stpb" apply(rule_tac x = "stpa - stp" in exI, auto) done - thus "\ isS0 (steps (Suc 0, tp) (A |+| B) stp)" - using h - apply(auto) - apply(cases "steps (Suc 0, tp) (A |+| B) stp", simp add: steps_add isS0_def steps_0) - done -qed - -text {* - The following lemma deals with the situation when TM @{text "B"} can not terminate. - *} - -lemma t_merge_uhalt: - assumes aimpb: "Q1 \-> P4" - shows "P1 \-> \ tp. \ (\ stp. isS0 (steps (Suc 0, tp) (A |+| B) stp))" -proof(simp only: t_imply_def, rule_tac allI, rule_tac impI) - fix tp - assume init_asst: "P1 tp" - show "\ (\stp. isS0 (steps (Suc 0, tp) (A |+| B) stp))" - proof - - have "\ stp. let (s, tp') = steps (Suc 0, tp) A stp in s = 0 \ Q1 tp'" - using A_halt[of tp] init_asst - by(simp) - from this obtain stpx where "let (s, tp') = steps (Suc 0, tp) A stpx in s = 0 \ Q1 tp'" .. - thus "?thesis" - proof(cases "steps (Suc 0, tp) A stpx", simp, erule_tac conjE) - fix a b c - assume "Q1 (b, c)" - and h3: "steps (Suc 0, tp) A stpx = (0, b, c)" - hence h2: "P4 (b, c)" using aimpb - by(simp add: t_imply_def) - have "\ stp. steps (Suc 0, tp) (A |+| B) stp = (Suc (length A div 2), b, c)" - using h3 A_wf B_wf - apply(rule_tac stp = stpx in t_merge_pre_halt_same, auto) - done - from this obtain stpa where h4:"steps (Suc 0, tp) (A |+| B) stpa = (Suc (length A div 2), b, c)" .. - have " \ (\ stp. isS0 (steps (Suc 0, b, c) B stp))" - using B_uhalt [of "(b, c)"] h2 apply simp - done - from this and h4 show "\stp. \ isS0 (steps (Suc 0, tp) (A |+| B) stp)" - by(rule_tac t_merge_uhalt_tmp, auto) - qed - qed -qed -end - -end - +theory turing_basic +imports Main +begin + +section {* Basic definitions of Turing machine *} + +(* Title: Turing machine's definition and its charater + Author: Xu Jian + Maintainer: Xu Jian +*) + +text {* + Actions of Turing machine (Abbreviated TM in the following* ). +*} + +datatype taction = + -- {* Write zero *} + W0 | + -- {* Write one *} + W1 | + -- {* Move left *} + L | + -- {* Move right *} + R | + -- {* Do nothing *} + Nop + +text {* + Tape contents in every block. +*} + +datatype block = + -- {* Blank *} + Bk | + -- {* Occupied *} + Oc + +text {* + Tape is represented as a pair of lists $(L_{left}, L_{right})$, + where $L_left$, named {\em left list}, is used to represent + the tape to the left of RW-head and + $L_{right}$, named {\em right list}, is used to represent the tape + under and to the right of RW-head. +*} + +type_synonym tape = "block list \ block list" + +text {* The state of turing machine.*} +type_synonym tstate = nat + +text {* + Turing machine instruction is represented as a + pair @{text "(action, next_state)"}, + where @{text "action"} is the action to take at the current state + and @{text "next_state"} is the next state the machine is getting into + after the action. +*} +type_synonym tinst = "taction \ tstate" + +text {* + Program of Turing machine is represented as a list of Turing instructions + and the execution of the program starts from the head of the list. + *} +type_synonym tprog = "tinst list" + + +text {* + Turing machine configuration, which consists of the current state + and the tape. +*} +type_synonym t_conf = "tstate \ tape" + +fun nth_of :: "'a list \ nat \ 'a option" + where + "nth_of xs n = (if n < length xs then Some (xs!n) + else None)" + +text {* + The function used to fetech instruction out of Turing program. + *} + +fun fetch :: "tprog \ tstate \ block \ tinst" + where + "fetch p s b = (if s = 0 then (Nop, 0) else + case b of + Bk \ case nth_of p (2 * (s - 1)) of + Some i \ i + | None \ (Nop, 0) + | Oc \ case nth_of p (2 * (s - 1) +1) of + Some i \ i + | None \ (Nop, 0))" + + +fun new_tape :: "taction \ tape \ tape" +where + "new_tape action (leftn, rightn) = (case action of + W0 \ (leftn, Bk#(tl rightn)) | + W1 \ (leftn, Oc#(tl rightn)) | + L \ (if leftn = [] then (tl leftn, Bk#rightn) + else (tl leftn, (hd leftn) # rightn)) | + R \ if rightn = [] then (Bk#leftn,tl rightn) + else ((hd rightn)#leftn, tl rightn) | + Nop \ (leftn, rightn) + )" + +text {* + The one step function used to transfer Turing machine configuration. +*} +fun tstep :: "t_conf \ tprog \ t_conf" + where + "tstep c p = (let (s, l, r) = c in + let (ac, ns) = (fetch p s (case r of [] \ Bk | + x # xs \ x)) in + (ns, new_tape ac (l, r)))" + +text {* + The many-step function. +*} +fun steps :: "t_conf \ tprog \ nat \ t_conf" + where + "steps c p 0 = c" | + "steps c p (Suc n) = steps (tstep c p) p n" + +lemma tstep_red: "steps c p (Suc n) = tstep (steps c p n) p" +proof(induct n arbitrary: c) + fix c + show "steps c p (Suc 0) = tstep (steps c p 0) p" by(simp add: steps.simps) +next + fix n c + assume ind: "\ c. steps c p (Suc n) = tstep (steps c p n) p" + have "steps (tstep c p) p (Suc n) = tstep (steps (tstep c p) p n) p" + by(rule ind) + thus "steps c p (Suc (Suc n)) = tstep (steps c p (Suc n)) p" by(simp add: steps.simps) +qed + +declare Let_def[simp] option.split[split] + +definition + "iseven n \ \ x. n = 2 * x" + + +text {* + The following @{text "t_correct"} function is used to specify the wellformedness of Turing + machine. +*} +fun t_correct :: "tprog \ bool" + where + "t_correct p = (length p \ 2 \ iseven (length p) \ + list_all (\ (acn, s). s \ length p div 2) p)" + +declare t_correct.simps[simp del] + +lemma allimp: "\\x. P x \ Q x; \x. P x\ \ \x. Q x" +by(auto elim: allE) + +lemma halt_lemma: "\wf LE; \ n. (\ P (f n) \ (f (Suc n), (f n)) \ LE)\ \ \ n. P (f n)" +apply(rule exCI, drule allimp, auto) +apply(drule_tac f = f in wf_inv_image, simp add: inv_image_def) +apply(erule wf_induct, auto) +done + +lemma steps_add: "steps c t (x + y) = steps (steps c t x) t y" +by(induct x arbitrary: c, auto simp: steps.simps tstep_red) + +lemma listall_set: "list_all p t \ \ a \ set t. p a" +by(induct t, auto) + +lemma fetch_ex: "\b a. fetch T aa ab = (b, a)" +by(simp add: fetch.simps) +definition exponent :: "'a \ nat \ 'a list" ("_\<^bsup>_\<^esup>" [0, 0]100) + where "exponent x n = replicate n x" + +text {* + @{text "tinres l1 l2"} means left list @{text "l1"} is congruent with + @{text "l2"} with respect to the execution of Turing machine. + Appending Blank to the right of eigther one does not affect the + outcome of excution. +*} + +definition tinres :: "block list \ block list \ bool" + where + "tinres bx by = (\ n. bx = by@Bk\<^bsup>n\<^esup> \ by = bx @ Bk\<^bsup>n\<^esup>)" + +lemma exp_zero: "a\<^bsup>0\<^esup> = []" +by(simp add: exponent_def) +lemma exp_ind_def: "a\<^bsup>Suc x \<^esup> = a # a\<^bsup>x\<^esup>" +by(simp add: exponent_def) + +text {* + The following lemma shows the meaning of @{text "tinres"} with respect to + one step execution. + *} +lemma tinres_step: + "\tinres l l'; tstep (ss, l, r) t = (sa, la, ra); tstep (ss, l', r) t = (sb, lb, rb)\ + \ tinres la lb \ ra = rb \ sa = sb" +apply(auto simp: tstep.simps fetch.simps new_tape.simps + split: if_splits taction.splits list.splits + block.splits) +apply(case_tac [!] "t ! (2 * (ss - Suc 0))", + auto simp: exponent_def tinres_def split: if_splits taction.splits list.splits + block.splits) +apply(case_tac [!] "t ! (2 * (ss - Suc 0) + Suc 0)", + auto simp: exponent_def tinres_def split: if_splits taction.splits list.splits + block.splits) +done + +declare tstep.simps[simp del] steps.simps[simp del] + +text {* + The following lemma shows the meaning of @{text "tinres"} with respect to + many step execution. + *} +lemma tinres_steps: + "\tinres l l'; steps (ss, l, r) t stp = (sa, la, ra); steps (ss, l', r) t stp = (sb, lb, rb)\ + \ tinres la lb \ ra = rb \ 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: "\sa la ra sb lb rb. \steps (ss, l, r) t stp = (sa, la, ra); + steps (ss, l', r) t stp = (sb, lb, rb)\ \ tinres la lb \ ra = rb \ sa = sb" + and h: " tinres l l'" "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 "tinres b ba \ c = ca \ a = aa" + apply(rule_tac ind, simp_all add: h) + done + thus "tinres la lb \ ra = rb \ sa = sb" + apply(rule_tac l = b and l' = ba and r = c and ss = a + and t = t in tinres_step) + using h + apply(simp, simp, simp) + done +qed + +text {* + The following function @{text "tshift tp n"} is used to shift Turing programs + @{text "tp"} by @{text "n"} when it is going to be combined with others. + *} + +fun tshift :: "tprog \ nat \ tprog" + where + "tshift tp off = (map (\ (action, state). (action, (if state = 0 then 0 + else state + off))) tp)" + +text {* + When two Turing programs are combined, the end state (state @{text "0"}) of the one + at the prefix position needs to be connected to the start state + of the one at postfix position. If @{text "tp"} is the Turing program + to be at the prefix, @{text "change_termi_state tp"} is the transformed Turing program. + *} +fun change_termi_state :: "tprog \ tprog" + where + "change_termi_state t = + (map (\ (acn, ns). if ns = 0 then (acn, Suc ((length t) div 2)) else (acn, ns)) t)" + +text {* + @{text "t_add tp1 tp2"} is the combined Truing program. +*} + +fun t_add :: "tprog \ tprog \ tprog" ("_ |+| _" [0, 0] 100) + where + "t_add t1 t2 = ((change_termi_state t1) @ (tshift t2 ((length t1) div 2)))" + +text {* + Tests whether the current configuration is at state @{text "0"}. +*} +definition isS0 :: "t_conf \ bool" + where + "isS0 c = (let (s, l, r) = c in s = 0)" + +declare tstep.simps[simp del] steps.simps[simp del] + t_add.simps[simp del] fetch.simps[simp del] + new_tape.simps[simp del] + + +text {* + Single step execution starting from state @{text "0"} will not make any progress. +*} +lemma tstep_0: "tstep (0, tp) p = (0, tp)" +apply(simp add: tstep.simps fetch.simps new_tape.simps) +done + + +text {* + Many step executions starting from state @{text "0"} will not make any progress. +*} + +lemma steps_0: "steps (0, tp) p stp = (0, tp)" +apply(induct stp) +apply(simp add: steps.simps) +apply(simp add: tstep_red tstep_0) +done + +lemma s_keep_step: "\a \ length A div 2; tstep (a, b, c) A = (s, l, r); t_correct A\ + \ s \ length A div 2" +apply(simp add: tstep.simps fetch.simps t_correct.simps iseven_def + split: if_splits block.splits list.splits) +apply(case_tac [!] a, auto simp: list_all_length) +apply(erule_tac x = "2 * nat" in allE, auto) +apply(erule_tac x = "2 * nat" in allE, auto) +apply(erule_tac x = "Suc (2 * nat)" in allE, auto) +done + +lemma s_keep: "\steps (Suc 0, tp) A stp = (s, l, r); t_correct A\ \ s \ length A div 2" +proof(induct stp arbitrary: s l r) + case 0 thus "?case" by(auto simp: t_correct.simps steps.simps) +next + fix stp s l r + assume ind: "\s l r. \steps (Suc 0, tp) A stp = (s, l, r); t_correct A\ \ s \ length A div 2" + and h1: "steps (Suc 0, tp) A (Suc stp) = (s, l, r)" + and h2: "t_correct A" + from h1 h2 show "s \ length A div 2" + proof(simp add: tstep_red, cases "(steps (Suc 0, tp) A stp)", simp) + fix a b c + assume h3: "tstep (a, b, c) A = (s, l, r)" + and h4: "steps (Suc 0, tp) A stp = (a, b, c)" + have "a \ length A div 2" + using h2 h4 + by(rule_tac l = b and r = c in ind, auto) + thus "?thesis" + using h3 h2 + by(simp add: s_keep_step) + qed +qed + +lemma t_merge_fetch_pre: + "\fetch A s b = (ac, ns); s \ length A div 2; t_correct A; s \ 0\ \ + fetch (A |+| B) s b = (ac, if ns = 0 then Suc (length A div 2) else ns)" +apply(subgoal_tac "2 * (s - Suc 0) < length A \ Suc (2 * (s - Suc 0)) < length A") +apply(auto simp: fetch.simps t_add.simps split: if_splits block.splits) +apply(simp_all add: nth_append change_termi_state.simps) +done + +lemma [simp]: "\\ a \ length A div 2; t_correct A\ \ fetch A a b = (Nop, 0)" +apply(auto simp: fetch.simps del: nth_of.simps split: block.splits) +apply(case_tac [!] a, auto simp: t_correct.simps iseven_def) +done + +lemma [elim]: "\t_correct A; \ isS0 (tstep (a, b, c) A)\ \ a \ length A div 2" +apply(rule_tac classical, auto simp: tstep.simps new_tape.simps isS0_def) +done + +lemma [elim]: "\t_correct A; \ isS0 (tstep (a, b, c) A)\ \ 0 < a" +apply(rule_tac classical, simp add: tstep_0 isS0_def) +done + + +lemma t_merge_pre_eq_step: "\tstep (a, b, c) A = cf; t_correct A; \ isS0 cf\ + \ tstep (a, b, c) (A |+| B) = cf" +apply(subgoal_tac "a \ length A div 2 \ a \ 0") +apply(simp add: tstep.simps) +apply(case_tac "fetch A a (case c of [] \ Bk | x # xs \ x)", simp) +apply(drule_tac B = B in t_merge_fetch_pre, simp, simp, simp, simp add: isS0_def, auto) +done + +lemma t_merge_pre_eq: "\steps (Suc 0, tp) A stp = cf; \ isS0 cf; t_correct A\ + \ steps (Suc 0, tp) (A |+| B) stp = cf" +proof(induct stp arbitrary: cf) + case 0 thus "?case" by(simp add: steps.simps) +next + fix stp cf + assume ind: "\cf. \steps (Suc 0, tp) A stp = cf; \ isS0 cf; t_correct A\ + \ steps (Suc 0, tp) (A |+| B) stp = cf" + and h1: "steps (Suc 0, tp) A (Suc stp) = cf" + and h2: "\ isS0 cf" + and h3: "t_correct A" + from h1 h2 h3 show "steps (Suc 0, tp) (A |+| B) (Suc stp) = cf" + proof(simp add: tstep_red, cases "steps (Suc 0, tp) (A) stp", simp) + fix a b c + assume h4: "tstep (a, b, c) A = cf" + and h5: "steps (Suc 0, tp) A stp = (a, b, c)" + have "steps (Suc 0, tp) (A |+| B) stp = (a, b, c)" + proof(cases a) + case 0 thus "?thesis" + using h4 h2 + apply(simp add: tstep_0, cases cf, simp add: isS0_def) + done + next + case (Suc n) thus "?thesis" + using h5 h3 + apply(rule_tac ind, auto simp: isS0_def) + done + qed + thus "tstep (steps (Suc 0, tp) (A |+| B) stp) (A |+| B) = cf" + using h4 h5 h3 h2 + apply(simp) + apply(rule t_merge_pre_eq_step, auto) + done + qed +qed + +declare nth.simps[simp del] tshift.simps[simp del] change_termi_state.simps[simp del] + +lemma [simp]: "length (change_termi_state A) = length A" +by(simp add: change_termi_state.simps) + +lemma first_halt_point: "steps (Suc 0, tp) A stp = (0, tp') + \ \stp. \ isS0 (steps (Suc 0, tp) A stp) \ steps (Suc 0, tp) A (Suc stp) = (0, tp')" +proof(induct stp) + case 0 thus "?case" by(simp add: steps.simps) +next + case (Suc n) + fix stp + assume ind: "steps (Suc 0, tp) A stp = (0, tp') \ + \stp. \ isS0 (steps (Suc 0, tp) A stp) \ steps (Suc 0, tp) A (Suc stp) = (0, tp')" + and h: "steps (Suc 0, tp) A (Suc stp) = (0, tp')" + from h show "\stp. \ isS0 (steps (Suc 0, tp) A stp) \ steps (Suc 0, tp) A (Suc stp) = (0, tp')" + proof(simp add: tstep_red, cases "steps (Suc 0, tp) A stp", simp, case_tac a) + fix a b c + assume g1: "a = (0::nat)" + and g2: "tstep (a, b, c) A = (0, tp')" + and g3: "steps (Suc 0, tp) A stp = (a, b, c)" + have "steps (Suc 0, tp) A stp = (0, tp')" + using g2 g1 g3 + by(simp add: tstep_0) + hence "\ stp. \ isS0 (steps (Suc 0, tp) A stp) \ steps (Suc 0, tp) A (Suc stp) = (0, tp')" + by(rule ind) + thus "\stp. \ isS0 (steps (Suc 0, tp) A stp) \ tstep (steps (Suc 0, tp) A stp) A = (0, tp')" + apply(simp add: tstep_red) + done + next + fix a b c nat + assume g1: "steps (Suc 0, tp) A stp = (a, b, c)" + and g2: "steps (Suc 0, tp) A (Suc stp) = (0, tp')" "a= Suc nat" + thus "\stp. \ isS0 (steps (Suc 0, tp) A stp) \ tstep (steps (Suc 0, tp) A stp) A = (0, tp')" + apply(rule_tac x = stp in exI) + apply(simp add: isS0_def tstep_red) + done + qed +qed + +lemma t_merge_pre_halt_same': + "\\ isS0 (steps (Suc 0, tp) A stp) ; steps (Suc 0, tp) A (Suc stp) = (0, tp'); t_correct A\ + \ steps (Suc 0, tp) (A |+| B) (Suc stp) = (Suc (length A div 2), tp')" +proof(simp add: tstep_red, cases "steps (Suc 0, tp) A stp", simp) + fix a b c + assume h1: "\ isS0 (a, b, c)" + and h2: "tstep (a, b, c) A = (0, tp')" + and h3: "t_correct A" + and h4: "steps (Suc 0, tp) A stp = (a, b, c)" + have "steps (Suc 0, tp) (A |+| B) stp = (a, b, c)" + using h1 h4 h3 + apply(rule_tac t_merge_pre_eq, auto) + done + moreover have "tstep (a, b, c) (A |+| B) = (Suc (length A div 2), tp')" + using h2 h3 h1 h4 + apply(simp add: tstep.simps) + apply(case_tac " fetch A a (case c of [] \ Bk | x # xs \ x)", simp) + apply(drule_tac B = B in t_merge_fetch_pre, auto simp: isS0_def intro: s_keep) + done + ultimately show "tstep (steps (Suc 0, tp) (A |+| B) stp) (A |+| B) = (Suc (length A div 2), tp')" + by(simp) +qed + +text {* + When Turing machine @{text "A"} and @{text "B"} are combined and the execution + of @{text "A"} can termination within @{text "stp"} steps, + the combined machine @{text "A |+| B"} will eventually get into the starting + state of machine @{text "B"}. +*} +lemma t_merge_pre_halt_same: " + \steps (Suc 0, tp) A stp = (0, tp'); t_correct A; t_correct B\ + \ \ stp. steps (Suc 0, tp) (A |+| B) stp = (Suc (length A div 2), tp')" +proof - + assume a_wf: "t_correct A" + and b_wf: "t_correct B" + and a_ht: "steps (Suc 0, tp) A stp = (0, tp')" + have halt_point: "\ stp. \ isS0 (steps (Suc 0, tp) A stp) \ steps (Suc 0, tp) A (Suc stp) = (0, tp')" + using a_ht + by(erule_tac first_halt_point) + then obtain stp' where "\ isS0 (steps (Suc 0, tp) A stp') \ steps (Suc 0, tp) A (Suc stp') = (0, tp')".. + hence "steps (Suc 0, tp) (A |+| B) (Suc stp') = (Suc (length A div 2), tp')" + using a_wf + apply(rule_tac t_merge_pre_halt_same', auto) + done + thus "?thesis" .. +qed + +lemma fetch_0: "fetch p 0 b = (Nop, 0)" +by(simp add: fetch.simps) + +lemma [simp]: "length (tshift B x) = length B" +by(simp add: tshift.simps) + +lemma [simp]: "t_correct A \ 2 * (length A div 2) = length A" +apply(simp add: t_correct.simps iseven_def, auto) +done + +lemma t_merge_fetch_snd: + "\fetch B a b = (ac, ns); t_correct A; t_correct B; a > 0 \ + \ fetch (A |+| B) (a + length A div 2) b + = (ac, if ns = 0 then 0 else ns + length A div 2)" +apply(auto simp: fetch.simps t_add.simps split: if_splits block.splits) +apply(case_tac [!] a, simp_all) +apply(simp_all add: nth_append change_termi_state.simps tshift.simps) +done + +lemma t_merge_snd_eq_step: + "\tstep (s, l, r) B = (s', l', r'); t_correct A; t_correct B; s > 0\ + \ tstep (s + length A div 2, l, r) (A |+| B) = + (if s' = 0 then 0 else s' + length A div 2, l' ,r') " +apply(simp add: tstep.simps) +apply(cases "fetch B s (case r of [] \ Bk | x # xs \ x)") +apply(auto simp: t_merge_fetch_snd) +apply(frule_tac [!] t_merge_fetch_snd, auto) +done + +text {* + Relates the executions of TM @{text "B"}, one is when @{text "B"} is executed alone, + the other is the execution when @{text "B"} is in the combined TM. +*} +lemma t_merge_snd_eq_steps: + "\t_correct A; t_correct B; steps (s, l, r) B stp = (s', l', r'); s > 0\ + \ steps (s + length A div 2, l, r) (A |+| B) stp = + (if s' = 0 then 0 else s' + length A div 2, l', r')" +proof(induct stp arbitrary: s' l' r') + case 0 thus "?case" + by(simp add: steps.simps) +next + fix stp s' l' r' + assume ind: "\s' l' r'. \t_correct A; t_correct B; steps (s, l, r) B stp = (s', l', r'); 0 < s\ + \ steps (s + length A div 2, l, r) (A |+| B) stp = + (if s' = 0 then 0 else s' + length A div 2, l', r')" + and h1: "steps (s, l, r) B (Suc stp) = (s', l', r')" + and h2: "t_correct A" + and h3: "t_correct B" + and h4: "0 < s" + from h1 show "steps (s + length A div 2, l, r) (A |+| B) (Suc stp) + = (if s' = 0 then 0 else s' + length A div 2, l', r')" + proof(simp only: tstep_red, cases "steps (s, l, r) B stp") + fix a b c + assume h5: "steps (s, l, r) B stp = (a, b, c)" "tstep (steps (s, l, r) B stp) B = (s', l', r')" + hence h6: "(steps (s + length A div 2, l, r) (A |+| B) stp) = + ((if a = 0 then 0 else a + length A div 2, b, c))" + using h2 h3 h4 + by(rule_tac ind, auto) + thus "tstep (steps (s + length A div 2, l, r) (A |+| B) stp) (A |+| B) = + (if s' = 0 then 0 else s'+ length A div 2, l', r')" + using h5 + proof(auto) + assume "tstep (0, b, c) B = (0, l', r')" thus "tstep (0, b, c) (A |+| B) = (0, l', r')" + by(simp add: tstep_0) + next + assume "tstep (0, b, c) B = (s', l', r')" "0 < s'" + thus "tstep (0, b, c) (A |+| B) = (s' + length A div 2, l', r')" + by(simp add: tstep_0) + next + assume "tstep (a, b, c) B = (0, l', r')" "0 < a" + thus "tstep (a + length A div 2, b, c) (A |+| B) = (0, l', r')" + using h2 h3 + by(drule_tac t_merge_snd_eq_step, auto) + next + assume "tstep (a, b, c) B = (s', l', r')" "0 < a" "0 < s'" + thus "tstep (a + length A div 2, b, c) (A |+| B) = (s' + length A div 2, l', r')" + using h2 h3 + by(drule_tac t_merge_snd_eq_step, auto) + qed + qed +qed + +lemma t_merge_snd_halt_eq: + "\steps (Suc 0, tp) B stp = (0, tp'); t_correct A; t_correct B\ + \ \stp. steps (Suc (length A div 2), tp) (A |+| B) stp = (0, tp')" +apply(case_tac tp, cases tp', simp) +apply(drule_tac s = "Suc 0" in t_merge_snd_eq_steps, auto) +done + +lemma t_inj: "\steps (Suc 0, tp) A stpa = (0, tp1); steps (Suc 0, tp) A stpb = (0, tp2)\ + \ tp1 = tp2" +proof - + assume h1: "steps (Suc 0, tp) A stpa = (0, tp1)" + and h2: "steps (Suc 0, tp) A stpb = (0, tp2)" + thus "?thesis" + proof(cases "stpa < stpb") + case True thus "?thesis" + using h1 h2 + apply(drule_tac less_imp_Suc_add, auto) + apply(simp del: add_Suc_right add_Suc add: add_Suc_right[THEN sym] steps_add steps_0) + done + next + case False thus "?thesis" + using h1 h2 + apply(drule_tac leI) + apply(case_tac "stpb = stpa", auto) + apply(subgoal_tac "stpb < stpa") + apply(drule_tac less_imp_Suc_add, auto) + apply(simp del: add_Suc_right add_Suc add: add_Suc_right[THEN sym] steps_add steps_0) + done + qed +qed + +type_synonym t_assert = "tape \ bool" + +definition t_imply :: "t_assert \ t_assert \ bool" ("_ \-> _" [0, 0] 100) + where + "t_imply a1 a2 = (\ tp. a1 tp \ a2 tp)" + + +locale turing_merge = + fixes A :: "tprog" and B :: "tprog" and P1 :: "t_assert" + and P2 :: "t_assert" + and P3 :: "t_assert" + and P4 :: "t_assert" + and Q1:: "t_assert" + and Q2 :: "t_assert" + assumes + A_wf : "t_correct A" + and B_wf : "t_correct B" + and A_halt : "P1 tp \ \ stp. let (s, tp') = steps (Suc 0, tp) A stp in s = 0 \ Q1 tp'" + and B_halt : "P2 tp \ \ stp. let (s, tp') = steps (Suc 0, tp) B stp in s = 0 \ Q2 tp'" + and A_uhalt : "P3 tp \ \ (\ stp. isS0 (steps (Suc 0, tp) A stp))" + and B_uhalt: "P4 tp \ \ (\ stp. isS0 (steps (Suc 0, tp) B stp))" +begin + + +text {* + The following lemma tries to derive the Hoare logic rule for sequentially combined TMs. + It deals with the situtation when both @{text "A"} and @{text "B"} are terminated. +*} + +lemma t_merge_halt: + assumes aimpb: "Q1 \-> P2" + shows "P1 \-> \ tp. (\ stp tp'. steps (Suc 0, tp) (A |+| B) stp = (0, tp') \ Q2 tp')" +proof(simp add: t_imply_def, auto) + fix a b + assume h: "P1 (a, b)" + hence "\ stp. let (s, tp') = steps (Suc 0, a, b) A stp in s = 0 \ Q1 tp'" + using A_halt by simp + from this obtain stp1 where "let (s, tp') = steps (Suc 0, a, b) A stp1 in s = 0 \ Q1 tp'" .. + thus "\stp aa ba. steps (Suc 0, a, b) (A |+| B) stp = (0, aa, ba) \ Q2 (aa, ba)" + proof(case_tac "steps (Suc 0, a, b) A stp1", simp, erule_tac conjE) + fix aa ba c + assume g1: "Q1 (ba, c)" + and g2: "steps (Suc 0, a, b) A stp1 = (0, ba, c)" + hence "P2 (ba, c)" + using aimpb apply(simp add: t_imply_def) + done + hence "\ stp. let (s, tp') = steps (Suc 0, ba, c) B stp in s = 0 \ Q2 tp'" + using B_halt by simp + from this obtain stp2 where "let (s, tp') = steps (Suc 0, ba, c) B stp2 in s = 0 \ Q2 tp'" .. + thus "?thesis" + proof(case_tac "steps (Suc 0, ba, c) B stp2", simp, erule_tac conjE) + fix aa bb ca + assume g3: " Q2 (bb, ca)" "steps (Suc 0, ba, c) B stp2 = (0, bb, ca)" + have "\ stp. steps (Suc 0, a, b) (A |+| B) stp = (Suc (length A div 2), ba , c)" + using g2 A_wf B_wf + by(rule_tac t_merge_pre_halt_same, auto) + moreover have "\ stp. steps (Suc (length A div 2), ba, c) (A |+| B) stp = (0, bb, ca)" + using g3 A_wf B_wf + apply(rule_tac t_merge_snd_halt_eq, auto) + done + ultimately show "\stp aa ba. steps (Suc 0, a, b) (A |+| B) stp = (0, aa, ba) \ Q2 (aa, ba)" + apply(erule_tac exE, erule_tac exE) + apply(rule_tac x = "stp + stpa" in exI, simp add: steps_add) + using g3 by simp + qed + qed +qed + +lemma t_merge_uhalt_tmp: + assumes B_uh: "\stp. \ isS0 (steps (Suc 0, b, c) B stp)" + and merge_ah: "steps (Suc 0, tp) (A |+| B) stpa = (Suc (length A div 2), b, c)" + shows "\ stp. \ isS0 (steps (Suc 0, tp) (A |+| B) stp)" + using B_uh merge_ah +apply(rule_tac allI) +apply(case_tac "stp > stpa") +apply(erule_tac x = "stp - stpa" in allE) +apply(case_tac "(steps (Suc 0, b, c) B (stp - stpa))", simp) +proof - + fix stp a ba ca + assume h1: "\ isS0 (a, ba, ca)" "stpa < stp" + and h2: "steps (Suc 0, b, c) B (stp - stpa) = (a, ba, ca)" + have "steps (Suc 0 + length A div 2, b, c) (A |+| B) (stp - stpa) = + (if a = 0 then 0 else a + length A div 2, ba, ca)" + using A_wf B_wf h2 + by(rule_tac t_merge_snd_eq_steps, auto) + moreover have "a > 0" using h1 by(simp add: isS0_def) + moreover have "\ stpb. stp = stpa + stpb" + using h1 by(rule_tac x = "stp - stpa" in exI, simp) + ultimately show "\ isS0 (steps (Suc 0, tp) (A |+| B) stp)" + using merge_ah + by(auto simp: steps_add isS0_def) +next + fix stp + assume h: "steps (Suc 0, tp) (A |+| B) stpa = (Suc (length A div 2), b, c)" "\ stpa < stp" + hence "\ stpb. stpa = stp + stpb" apply(rule_tac x = "stpa - stp" in exI, auto) done + thus "\ isS0 (steps (Suc 0, tp) (A |+| B) stp)" + using h + apply(auto) + apply(cases "steps (Suc 0, tp) (A |+| B) stp", simp add: steps_add isS0_def steps_0) + done +qed + +text {* + The following lemma deals with the situation when TM @{text "B"} can not terminate. + *} + +lemma t_merge_uhalt: + assumes aimpb: "Q1 \-> P4" + shows "P1 \-> \ tp. \ (\ stp. isS0 (steps (Suc 0, tp) (A |+| B) stp))" +proof(simp only: t_imply_def, rule_tac allI, rule_tac impI) + fix tp + assume init_asst: "P1 tp" + show "\ (\stp. isS0 (steps (Suc 0, tp) (A |+| B) stp))" + proof - + have "\ stp. let (s, tp') = steps (Suc 0, tp) A stp in s = 0 \ Q1 tp'" + using A_halt[of tp] init_asst + by(simp) + from this obtain stpx where "let (s, tp') = steps (Suc 0, tp) A stpx in s = 0 \ Q1 tp'" .. + thus "?thesis" + proof(cases "steps (Suc 0, tp) A stpx", simp, erule_tac conjE) + fix a b c + assume "Q1 (b, c)" + and h3: "steps (Suc 0, tp) A stpx = (0, b, c)" + hence h2: "P4 (b, c)" using aimpb + by(simp add: t_imply_def) + have "\ stp. steps (Suc 0, tp) (A |+| B) stp = (Suc (length A div 2), b, c)" + using h3 A_wf B_wf + apply(rule_tac stp = stpx in t_merge_pre_halt_same, auto) + done + from this obtain stpa where h4:"steps (Suc 0, tp) (A |+| B) stpa = (Suc (length A div 2), b, c)" .. + have " \ (\ stp. isS0 (steps (Suc 0, b, c) B stp))" + using B_uhalt [of "(b, c)"] h2 apply simp + done + from this and h4 show "\stp. \ isS0 (steps (Suc 0, tp) (A |+| B) stp)" + by(rule_tac t_merge_uhalt_tmp, auto) + qed + qed +qed +end + +end +