--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/uncomputable.thy Wed Feb 06 02:39:53 2013 +0000
@@ -0,0 +1,1614 @@
+(* Title: Turing machine's definition and its charater
+ Author: XuJian <xujian817@hotmail.com>
+ Maintainer: Xujian
+*)
+
+header {* Undeciablity of the {\em Halting problem} *}
+
+theory uncomputable
+imports Main turing_basic
+begin
+
+text {*
+ The {\em Copying} TM, which duplicates its input.
+*}
+definition tcopy :: "tprog"
+where
+"tcopy \<equiv> [(W0, 0), (R, 2), (R, 3), (R, 2),
+ (W1, 3), (L, 4), (L, 4), (L, 5), (R, 11), (R, 6),
+ (R, 7), (W0, 6), (R, 7), (R, 8), (W1, 9), (R, 8),
+ (L, 10), (L, 9), (L, 10), (L, 5), (R, 12), (R, 12),
+ (W1, 13), (L, 14), (R, 12), (R, 12), (L, 15), (W0, 14),
+ (R, 0), (L, 15)]"
+
+text {*
+ @{text "wipeLastBs tp"} removes all blanks at the end of tape @{text "tp"}.
+*}
+fun wipeLastBs :: "block list \<Rightarrow> block list"
+ where
+ "wipeLastBs bl = rev (dropWhile (\<lambda>a. a = Bk) (rev bl))"
+
+fun isBk :: "block \<Rightarrow> bool"
+ where
+ "isBk b = (b = Bk)"
+
+text {*
+ The following functions are used to expression invariants of {\em Copying} TM.
+*}
+fun tcopy_F0 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F0 x (l, r) = (\<exists> i. l = Bk\<^bsup>i\<^esup> \<and> r = Oc\<^bsup>x\<^esup> @ Bk # Oc\<^bsup>x\<^esup>)"
+
+fun tcopy_F1 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F1 x (l, r) = (l = [] \<and> r = Oc\<^bsup>x\<^esup>)"
+
+fun tcopy_F2 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F2 x (l, r) = (\<exists> i j. i > 0 \<and> i + j = x \<and> l = Oc\<^bsup>i\<^esup> \<and> r = Oc\<^bsup>j\<^esup>)"
+
+fun tcopy_F3 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F3 x (l, r) = (x > 0 \<and> l = Bk # Oc\<^bsup>x\<^esup> \<and> tl r = [])"
+
+fun tcopy_F4 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F4 x (l, r) = (x > 0 \<and> ((l = Oc\<^bsup>x\<^esup> \<and> r = [Bk, Oc]) \<or> (l = Oc\<^bsup>x - 1\<^esup> \<and> r = [Oc, Bk, Oc])))"
+
+fun tcopy_F5_loop :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F5_loop x (l, r) =
+ (\<exists> i j. i + j + 1 = x \<and> l = Oc\<^bsup>i\<^esup> \<and> r = Oc # Oc # Bk\<^bsup>j\<^esup> @ Oc\<^bsup>j\<^esup> \<and> j > 0)"
+
+fun tcopy_F5_exit :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F5_exit x (l, r) =
+ (l = [] \<and> r = Bk # Oc # Bk\<^bsup>x\<^esup> @ Oc\<^bsup>x\<^esup> \<and> x > 0 )"
+
+fun tcopy_F5 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F5 x (l, r) = (tcopy_F5_loop x (l, r) \<or> tcopy_F5_exit x (l, r))"
+
+fun tcopy_F6 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F6 x (l, r) =
+ (\<exists> i j any. i + j = x \<and> x > 0 \<and> i > 0 \<and> j > 0 \<and> l = Oc\<^bsup>i\<^esup> \<and> r = any#Bk\<^bsup>j\<^esup> @ Oc\<^bsup>j\<^esup>)"
+
+fun tcopy_F7 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F7 x (l, r) =
+ (\<exists> i j k t. i + j = x \<and> i > 0 \<and> j > 0 \<and> k + t = Suc j \<and> l = Bk\<^bsup>k\<^esup> @ Oc\<^bsup>i\<^esup> \<and> r = Bk\<^bsup>t\<^esup> @ Oc\<^bsup>j\<^esup>)"
+
+fun tcopy_F8 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F8 x (l, r) =
+ (\<exists> i j k t. i + j = x \<and> i > 0 \<and> j > 0 \<and> k + t = j \<and> l = Oc\<^bsup>k\<^esup> @ Bk\<^bsup>Suc j\<^esup> @ Oc\<^bsup>i\<^esup> \<and> r = Oc\<^bsup>t\<^esup>)"
+
+fun tcopy_F9_loop :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+where
+ "tcopy_F9_loop x (l, r) =
+ (\<exists> i j k t. i + j = Suc x \<and> i > 0 \<and> j > 0 \<and> k + t = j \<and> t > 0\<and> l = Oc\<^bsup>k\<^esup> @ Bk\<^bsup>j\<^esup> @ Oc\<^bsup>i\<^esup> \<and> r = Oc\<^bsup>t\<^esup>)"
+
+fun tcopy_F9_exit :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F9_exit x (l, r) = (\<exists> i j. i + j = Suc x \<and> i > 0 \<and> j > 0 \<and> l = Bk\<^bsup>j - 1\<^esup> @ Oc\<^bsup>i\<^esup> \<and> r = Bk # Oc\<^bsup>j\<^esup>)"
+
+fun tcopy_F9 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F9 x (l, r) = (tcopy_F9_loop x (l, r) \<or>
+ tcopy_F9_exit x (l, r))"
+
+fun tcopy_F10_loop :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F10_loop x (l, r) =
+ (\<exists> i j k t. i + j = Suc x \<and> i > 0 \<and> k + t + 1 = j \<and> l = Bk\<^bsup>k\<^esup> @ Oc\<^bsup>i\<^esup> \<and> r = Bk\<^bsup>Suc t\<^esup> @ Oc\<^bsup>j\<^esup>)"
+
+fun tcopy_F10_exit :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F10_exit x (l, r) =
+ (\<exists> i j. i + j = x \<and> j > 0 \<and> l = Oc\<^bsup>i\<^esup> \<and> r = Oc # Bk\<^bsup>j\<^esup> @ Oc\<^bsup>j\<^esup>)"
+
+fun tcopy_F10 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F10 x (l, r) = (tcopy_F10_loop x (l, r) \<or> tcopy_F10_exit x (l, r))"
+
+fun tcopy_F11 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F11 x (l, r) = (x > 0 \<and> l = [Bk] \<and> r = Oc # Bk\<^bsup>x\<^esup> @ Oc\<^bsup>x\<^esup>)"
+
+fun tcopy_F12 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F12 x (l, r) = (\<exists> i j. i + j = Suc x \<and> x > 0 \<and> l = Oc\<^bsup>i\<^esup> @ [Bk] \<and> r = Bk\<^bsup>j\<^esup> @ Oc\<^bsup>x\<^esup>)"
+
+fun tcopy_F13 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F13 x (l, r) =
+ (\<exists> i j. x > 0 \<and> i + j = x \<and> l = Oc\<^bsup>i\<^esup> @ [Bk] \<and> r = Oc # Bk\<^bsup>j\<^esup> @ Oc\<^bsup>x\<^esup> )"
+
+fun tcopy_F14 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F14 x (l, r) = (\<exists> any. x > 0 \<and> l = Oc\<^bsup>x\<^esup> @ [Bk] \<and> r = any#Oc\<^bsup>x\<^esup>)"
+
+fun tcopy_F15_loop :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F15_loop x (l, r) =
+ (\<exists> i j. i + j = x \<and> x > 0 \<and> j > 0 \<and> l = Oc\<^bsup>i\<^esup> @ [Bk] \<and> r = Oc\<^bsup>j\<^esup> @ Bk # Oc\<^bsup>x\<^esup>)"
+
+fun tcopy_F15_exit :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F15_exit x (l, r) = (x > 0 \<and> l = [] \<and> r = Bk # Oc\<^bsup>x\<^esup> @ Bk # Oc\<^bsup>x\<^esup>)"
+
+fun tcopy_F15 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
+ where
+ "tcopy_F15 x (l, r) = (tcopy_F15_loop x (l, r) \<or> tcopy_F15_exit x (l, r))"
+
+text {*
+ The following @{text "inv_tcopy"} is the invariant of the {\em Copying} TM.
+*}
+fun inv_tcopy :: "nat \<Rightarrow> t_conf \<Rightarrow> bool"
+ where
+ "inv_tcopy x c = (let (state, tp) = c in
+ if state = 0 then tcopy_F0 x tp
+ else if state = 1 then tcopy_F1 x tp
+ else if state = 2 then tcopy_F2 x tp
+ else if state = 3 then tcopy_F3 x tp
+ else if state = 4 then tcopy_F4 x tp
+ else if state = 5 then tcopy_F5 x tp
+ else if state = 6 then tcopy_F6 x tp
+ else if state = 7 then tcopy_F7 x tp
+ else if state = 8 then tcopy_F8 x tp
+ else if state = 9 then tcopy_F9 x tp
+ else if state = 10 then tcopy_F10 x tp
+ else if state = 11 then tcopy_F11 x tp
+ else if state = 12 then tcopy_F12 x tp
+ else if state = 13 then tcopy_F13 x tp
+ else if state = 14 then tcopy_F14 x tp
+ else if state = 15 then tcopy_F15 x tp
+ else False)"
+declare tcopy_F0.simps [simp del]
+ tcopy_F1.simps [simp del]
+ tcopy_F2.simps [simp del]
+ tcopy_F3.simps [simp del]
+ tcopy_F4.simps [simp del]
+ tcopy_F5.simps [simp del]
+ tcopy_F6.simps [simp del]
+ tcopy_F7.simps [simp del]
+ tcopy_F8.simps [simp del]
+ tcopy_F9.simps [simp del]
+ tcopy_F10.simps [simp del]
+ tcopy_F11.simps [simp del]
+ tcopy_F12.simps [simp del]
+ tcopy_F13.simps [simp del]
+ tcopy_F14.simps [simp del]
+ tcopy_F15.simps [simp del]
+
+lemma exp_zero_simp: "(a\<^bsup>b\<^esup> = []) = (b = 0)"
+apply(auto simp: exponent_def)
+done
+
+lemma exp_zero_simp2: "([] = a\<^bsup>b\<^esup> ) = (b = 0)"
+apply(auto simp: exponent_def)
+done
+
+lemma [elim]: "\<lbrakk>tstep (0, a, b) tcopy = (s, l, r); s \<noteq> 0\<rbrakk> \<Longrightarrow> RR"
+apply(simp add: tstep.simps tcopy_def fetch.simps)
+done
+
+lemma [elim]: "\<lbrakk>tstep (Suc 0, a, b) tcopy = (s, l, r); s \<noteq> 0; s \<noteq> 2\<rbrakk>
+ \<Longrightarrow> RR"
+apply(simp add: tstep.simps tcopy_def fetch.simps)
+apply(simp split: block.splits list.splits)
+done
+
+lemma [elim]: "\<lbrakk>tstep (2, a, b) tcopy = (s, l, r); s \<noteq> 2; s \<noteq> 3\<rbrakk>
+ \<Longrightarrow> RR"
+apply(simp add: tstep.simps tcopy_def fetch.simps)
+apply(simp split: block.splits list.splits)
+done
+
+lemma [elim]: "\<lbrakk>tstep (3, a, b) tcopy = (s, l, r); s \<noteq> 3; s \<noteq> 4\<rbrakk>
+ \<Longrightarrow> RR"
+by(simp add: tstep.simps tcopy_def fetch.simps
+ split: block.splits list.splits)
+
+lemma [elim]: "\<lbrakk>tstep (4, a, b) tcopy = (s, l, r); s \<noteq> 4; s \<noteq> 5\<rbrakk>
+ \<Longrightarrow> RR"
+by(simp add: tstep.simps tcopy_def fetch.simps
+ split: block.splits list.splits)
+
+lemma [elim]: "\<lbrakk>tstep (5, a, b) tcopy = (s, l, r); s \<noteq> 6; s \<noteq> 11\<rbrakk>
+ \<Longrightarrow> RR"
+by(simp add: tstep.simps tcopy_def fetch.simps
+ split: block.splits list.splits)
+
+lemma [elim]: "\<lbrakk>tstep (6, a, b) tcopy = (s, l, r); s \<noteq> 6; s \<noteq> 7\<rbrakk>
+ \<Longrightarrow> RR"
+by(simp add: tstep.simps tcopy_def fetch.simps
+ split: block.splits list.splits)
+
+lemma [elim]: "\<lbrakk>tstep (7, a, b) tcopy = (s, l, r); s \<noteq> 7; s \<noteq> 8\<rbrakk>
+ \<Longrightarrow> RR"
+by(simp add: tstep.simps tcopy_def fetch.simps
+ split: block.splits list.splits)
+
+lemma [elim]: "\<lbrakk>tstep (8, a, b) tcopy = (s, l, r); s \<noteq> 8; s \<noteq> 9\<rbrakk>
+ \<Longrightarrow> RR"
+by(simp add: tstep.simps tcopy_def fetch.simps
+ split: block.splits list.splits)
+
+lemma [elim]: "\<lbrakk>tstep (9, a, b) tcopy = (s, l, r); s \<noteq> 9; s \<noteq> 10\<rbrakk>
+ \<Longrightarrow> RR"
+by(simp add: tstep.simps tcopy_def fetch.simps
+ split: block.splits list.splits)
+
+lemma [elim]: "\<lbrakk>tstep (10, a, b) tcopy = (s, l, r); s \<noteq> 10; s \<noteq> 5\<rbrakk>
+ \<Longrightarrow> RR"
+by(simp add: tstep.simps tcopy_def fetch.simps
+ split: block.splits list.splits)
+
+lemma [elim]: "\<lbrakk>tstep (11, a, b) tcopy = (s, l, r); s \<noteq> 12\<rbrakk> \<Longrightarrow> RR"
+by(simp add: tstep.simps tcopy_def fetch.simps
+ split: block.splits list.splits)
+
+lemma [elim]: "\<lbrakk>tstep (12, a, b) tcopy = (s, l, r); s \<noteq> 13; s \<noteq> 14\<rbrakk>
+ \<Longrightarrow> RR"
+by(simp add: tstep.simps tcopy_def fetch.simps
+ split: block.splits list.splits)
+
+lemma [elim]: "\<lbrakk>tstep (13, a, b) tcopy = (s, l, r); s \<noteq> 12\<rbrakk> \<Longrightarrow> RR"
+by(simp add: tstep.simps tcopy_def fetch.simps
+ split: block.splits list.splits)
+
+lemma [elim]: "\<lbrakk>tstep (14, a, b) tcopy = (s, l, r); s \<noteq> 14; s \<noteq> 15\<rbrakk>
+ \<Longrightarrow> RR"
+by(simp add: tstep.simps tcopy_def fetch.simps
+ split: block.splits list.splits)
+
+lemma [elim]: "\<lbrakk>tstep (15, a, b) tcopy = (s, l, r); s \<noteq> 0; s \<noteq> 15\<rbrakk>
+ \<Longrightarrow> RR"
+by(simp add: tstep.simps tcopy_def fetch.simps
+ split: block.splits list.splits)
+
+(*
+lemma min_Suc4: "min (Suc (Suc x)) x = x"
+by auto
+
+lemma takeWhile2replicate:
+ "\<exists>n. takeWhile (\<lambda>a. a = b) list = replicate n b"
+apply(induct list)
+apply(rule_tac x = 0 in exI, simp)
+apply(auto)
+apply(rule_tac x = "Suc n" in exI, simp)
+done
+
+lemma rev_replicate_same: "rev (replicate x b) = replicate x b"
+by(simp)
+
+lemma rev_equal: "a = b \<Longrightarrow> rev a = rev b"
+by simp
+
+lemma rev_equal_rev: "rev a = rev b \<Longrightarrow> a = b"
+by simp
+
+lemma rep_suc_rev[simp]:"replicate n b @ [b] = replicate (Suc n) b"
+apply(rule rev_equal_rev)
+apply(simp only: rev_append rev_replicate_same)
+apply(auto)
+done
+
+lemma replicate_Cons_simp: "b # replicate n b @ xs =
+ replicate n b @ b # xs"
+apply(simp)
+done
+*)
+
+lemma [elim]: "\<lbrakk>tstep (14, b, c) tcopy = (15, ab, ba);
+ tcopy_F14 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F15 x (ab, ba)"
+apply(auto simp: tstep.simps tcopy_def
+ tcopy_F14.simps tcopy_F15.simps fetch.simps new_tape.simps
+ split: if_splits list.splits block.splits)
+apply(erule_tac [!] x = "x - 1" in allE)
+apply(case_tac [!] x, simp_all add: exp_ind_def exp_zero)
+apply(erule_tac [!] x = "Suc 0" in allE, simp_all)
+done
+
+(*
+lemma dropWhile_drophd: "\<not> p a \<Longrightarrow>
+ (dropWhile p xs @ (a # as)) = (dropWhile p (xs @ [a]) @ as)"
+apply(induct xs)
+apply(auto)
+done
+*)
+(*
+lemma dropWhile_append3: "\<lbrakk>\<not> p a;
+ listall ((dropWhile p xs) @ [a]) isBk\<rbrakk> \<Longrightarrow>
+ listall (dropWhile p (xs @ [a])) isBk"
+apply(drule_tac p = p and xs = xs and a = a in dropWhile_drophd, simp)
+done
+
+lemma takeWhile_append3: "\<lbrakk>\<not>p a; (takeWhile p xs) = b\<rbrakk>
+ \<Longrightarrow> takeWhile p (xs @ (a # as)) = b"
+apply(drule_tac P = p and xs = xs and x = a and l = as in
+ takeWhile_tail)
+apply(simp)
+done
+
+lemma listall_append: "list_all p (xs @ ys) =
+ (list_all p xs \<and> list_all p ys)"
+apply(induct xs)
+apply(simp+)
+done
+*)
+lemma false_case1:
+ "\<lbrakk>Oc\<^bsup>j\<^esup> @ Bk # Oc\<^bsup>i + j\<^esup> = Oc # list;
+ 0 < i;
+ \<forall>ia. tl (Oc\<^bsup>i\<^esup> @ [Bk]) = Oc\<^bsup>ia\<^esup> @ [Bk] \<longrightarrow> (\<forall>ja. ia + ja = i + j
+ \<longrightarrow> hd (Oc\<^bsup>i\<^esup> @ [Bk]) # Oc # list \<noteq> Oc\<^bsup>ja\<^esup> @ Bk # Oc\<^bsup>i + j\<^esup>)\<rbrakk>
+ \<Longrightarrow> RR"
+apply(case_tac i, auto simp: exp_ind_def)
+apply(erule_tac x = nat in allE, simp add:exp_ind_def)
+apply(erule_tac x = "Suc j" in allE, simp)
+done
+
+lemma false_case3:"\<forall>ja. ja \<noteq> i \<Longrightarrow> RR"
+by auto
+
+lemma [elim]: "\<lbrakk>tstep (15, b, c) tcopy = (15, ab, ba);
+ tcopy_F15 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F15 x (ab, ba)"
+apply(auto simp: tstep.simps tcopy_F15.simps
+ tcopy_def fetch.simps new_tape.simps
+ split: if_splits list.splits block.splits elim: false_case1)
+apply(case_tac [!] i, simp_all add: exp_zero exp_ind_def)
+apply(erule_tac [!] x = nat in allE, simp_all add: exp_ind_def)
+apply(auto elim: false_case3)
+done
+
+lemma [elim]: "\<lbrakk>tstep (14, b, c) tcopy = (14, ab, ba);
+ tcopy_F14 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F14 x (ab, ba)"
+apply(auto simp: tcopy_F14.simps tcopy_def tstep.simps
+ tcopy_F14.simps fetch.simps new_tape.simps
+ split: if_splits list.splits block.splits)
+done
+
+
+lemma [elim]: "\<lbrakk>tstep (12, b, c) tcopy = (14, ab, ba);
+ tcopy_F12 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F14 x (ab, ba)"
+apply(auto simp:tcopy_F12.simps tcopy_F14.simps
+ tcopy_def tstep.simps fetch.simps new_tape.simps
+ split: if_splits list.splits block.splits)
+apply(case_tac [!] j, simp_all add: exp_zero exp_ind_def)
+done
+
+lemma [elim]: "\<lbrakk>tstep (12, b, c) tcopy = (13, ab, ba);
+ tcopy_F12 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F13 x (ab, ba)"
+apply(auto simp:tcopy_F12.simps tcopy_F13.simps
+ tcopy_def tstep.simps fetch.simps new_tape.simps
+ split: if_splits list.splits block.splits)
+apply(case_tac x, simp_all add: exp_ind_def exp_zero)
+apply(rule_tac [!] x = i in exI, simp_all)
+apply(rule_tac [!] x = "j - 1" in exI)
+apply(case_tac [!] j, simp_all add: exp_ind_def exp_zero)
+apply(case_tac x, simp_all add: exp_ind_def exp_zero)
+done
+
+lemma [elim]: "\<lbrakk>tstep (11, b, c) tcopy = (12, ab, ba);
+ tcopy_F11 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F12 x (ab, ba)"
+apply(simp_all add:tcopy_F12.simps tcopy_F11.simps
+ tcopy_def tstep.simps fetch.simps new_tape.simps)
+apply(auto)
+apply(rule_tac x = "Suc 0" in exI,
+ rule_tac x = x in exI, simp add: exp_ind_def exp_zero)
+done
+
+
+lemma [elim]: "\<lbrakk>tstep (13, b, c) tcopy = (12, ab, ba);
+ tcopy_F13 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F12 x (ab, ba)"
+apply(auto simp:tcopy_F12.simps tcopy_F13.simps
+ tcopy_def tstep.simps fetch.simps new_tape.simps
+ split: if_splits list.splits block.splits)
+apply(rule_tac [!] x = "Suc i" in exI, simp_all add: exp_ind_def)
+done
+
+lemma [elim]: "\<lbrakk>tstep (5, b, c) tcopy = (11, ab, ba);
+ tcopy_F5 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F11 x (ab, ba)"
+apply(simp_all add:tcopy_F11.simps tcopy_F5.simps tcopy_def
+ tstep.simps fetch.simps new_tape.simps)
+apply(simp split: if_splits list.splits block.splits)
+done
+
+lemma F10_false: "tcopy_F10 x (b, []) = False"
+apply(auto simp: tcopy_F10.simps exp_ind_def)
+done
+
+lemma F10_false2: "tcopy_F10 x ([], Bk # list) = False"
+apply(auto simp:tcopy_F10.simps)
+apply(case_tac i, simp_all add: exp_ind_def exp_zero)
+done
+
+lemma [simp]: "tcopy_F10_exit x (b, Bk # list) = False"
+apply(auto simp: tcopy_F10.simps)
+done
+
+declare tcopy_F10_loop.simps[simp del] tcopy_F10_exit.simps[simp del]
+
+lemma [simp]: "tcopy_F10_loop x (b, [Bk]) = False"
+apply(auto simp: tcopy_F10_loop.simps)
+apply(simp add: exp_ind_def)
+done
+
+lemma [elim]: "\<lbrakk>tstep (10, b, c) tcopy = (10, ab, ba);
+ tcopy_F10 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F10 x (ab, ba)"
+apply(simp add: tcopy_def tstep.simps fetch.simps
+ new_tape.simps exp_ind_def exp_zero_simp exp_zero_simp2 F10_false F10_false2
+ split: if_splits list.splits block.splits)
+apply(simp add: tcopy_F10.simps del: tcopy_F10_loop.simps tcopy_F10_exit.simps)
+apply(case_tac b, simp, case_tac aa)
+apply(rule_tac disjI1)
+apply(simp only: tcopy_F10_loop.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = i in exI, rule_tac x = j in exI,
+ rule_tac x = "k - 1" in exI, rule_tac x = "Suc t" in exI, simp)
+apply(case_tac k, simp_all add: exp_ind_def exp_zero)
+apply(case_tac i, simp_all add: exp_ind_def exp_zero)
+apply(rule_tac disjI2)
+apply(simp only: tcopy_F10_loop.simps tcopy_F10_exit.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = "i - 1" in exI, rule_tac x = "j" in exI)
+apply(case_tac k, simp_all add: exp_ind_def exp_zero)
+apply(case_tac i, simp_all add: exp_ind_def exp_zero)
+apply(auto)
+apply(simp add: exp_ind_def)
+done
+
+(*
+lemma false_case4: "\<lbrakk>i + (k + t) = Suc x;
+ 0 < i;
+ Bk # list = Oc\<^bsup>t\<^esup>;
+ \<forall>ia j. ia + j = Suc x \<longrightarrow> ia = 0 \<or> (\<forall>ka. tl (Oc\<^bsup>k\<^esup>) @ Bk\<^bsup>k + t\<^esup> @ Oc\<^bsup>i\<^esup> = Bk\<^bsup>ka\<^esup> @ Oc\<^bsup>ia\<^esup> \<longrightarrow> (\<forall>ta. Suc (ka + ta) = j \<longrightarrow> Oc # Oc\<^bsup>t\<^esup> \<noteq> Bk\<^bsup>Suc ta\<^esup> @ Oc\<^bsup>j\<^esup>));
+ 0 < k\<rbrakk>
+ \<Longrightarrow> RR"
+apply(case_tac t, simp_all add: exp_ind_def exp_zero)
+done
+
+lemma false_case5: "
+ \<lbrakk>Suc (i + nata) = x;
+ 0 < i;
+ \<forall>ia j. ia + j = Suc x \<longrightarrow> ia = 0 \<or> (\<forall>k. Bk\<^bsup>nata\<^esup> @ Oc\<^bsup>i\<^esup> = Bk\<^bsup>k\<^esup> @ Oc\<^bsup>ia\<^esup> \<longrightarrow> (\<forall>t. Suc (k + t) = j \<longrightarrow> Bk # Oc # Oc # Oc\<^bsup>nata\<^esup> \<noteq> Bk\<^bsup>t\<^esup> @ Oc\<^bsup>j\<^esup>))\<rbrakk>
+ \<Longrightarrow> False"
+apply(erule_tac x = i in allE, simp)
+apply(erule_tac x = "Suc (Suc nata)" in allE, simp)
+apply(erule_tac x = nata in allE, simp, simp add: exp_ind_def exp_zero)
+done
+
+lemma false_case6: "\<lbrakk>0 < x; \<forall>i. tl (Oc\<^bsup>x\<^esup>) = Oc\<^bsup>i\<^esup> \<longrightarrow> (\<forall>j. i + j = x \<longrightarrow> j = 0 \<or> [Bk, Oc] \<noteq> Bk\<^bsup>j\<^esup> @ Oc\<^bsup>j\<^esup>)\<rbrakk>
+ \<Longrightarrow> False"
+apply(erule_tac x = "x - 1" in allE)
+apply(case_tac x, simp_all add: exp_ind_def exp_zero)
+apply(erule_tac x = "Suc 0" in allE, simp)
+done
+*)
+
+lemma [simp]: "tcopy_F9 x ([], b) = False"
+apply(auto simp: tcopy_F9.simps)
+apply(case_tac [!] i, simp_all add: exp_ind_def exp_zero)
+done
+
+lemma [simp]: "tcopy_F9 x (b, []) = False"
+apply(auto simp: tcopy_F9.simps)
+apply(case_tac [!] t, simp_all add: exp_ind_def exp_zero)
+done
+
+declare tcopy_F9_loop.simps[simp del] tcopy_F9_exit.simps[simp del]
+lemma [simp]: "tcopy_F9_loop x (b, Bk # list) = False"
+apply(auto simp: tcopy_F9_loop.simps)
+apply(case_tac [!] t, simp_all add: exp_ind_def exp_zero)
+done
+
+lemma [elim]: "\<lbrakk>tstep (9, b, c) tcopy = (10, ab, ba);
+ tcopy_F9 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F10 x (ab, ba)"
+apply(auto simp:tcopy_def
+ tstep.simps fetch.simps new_tape.simps exp_zero_simp
+ exp_zero_simp2
+ split: if_splits list.splits block.splits)
+apply(case_tac "hd b", simp add:tcopy_F9.simps tcopy_F10.simps )
+apply(simp only: tcopy_F9_exit.simps tcopy_F10_loop.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = i in exI, rule_tac x = j in exI, simp)
+apply(rule_tac x = "j - 2" in exI, simp add: exp_ind_def)
+apply(case_tac j, simp, simp)
+apply(case_tac nat, simp_all add: exp_zero exp_ind_def)
+apply(case_tac x, simp_all add: exp_ind_def exp_zero)
+apply(simp add: tcopy_F9.simps tcopy_F10.simps)
+apply(rule_tac disjI2)
+apply(simp only: tcopy_F10_exit.simps tcopy_F9_exit.simps)
+apply(erule_tac exE)+
+apply(simp)
+apply(case_tac j, simp_all, case_tac nat, simp_all add: exp_ind_def exp_zero)
+apply(case_tac x, simp_all add: exp_ind_def exp_zero)
+apply(rule_tac x = nata in exI, rule_tac x = 1 in exI, simp add: exp_ind_def exp_zero)
+done
+
+lemma false_case7:
+ "\<lbrakk>i + (n + t) = x; 0 < i; 0 < t; Oc # list = Oc\<^bsup>t\<^esup>; k = Suc n;
+ \<forall>j. i + j = Suc x \<longrightarrow> (\<forall>k. Oc\<^bsup>n\<^esup> @ Bk # Bk\<^bsup>n + t\<^esup> = Oc\<^bsup>k\<^esup> @ Bk\<^bsup>j\<^esup> \<longrightarrow>
+ (\<forall>ta. k + ta = j \<longrightarrow> ta = 0 \<or> Oc # Oc\<^bsup>t\<^esup> \<noteq> Oc\<^bsup>ta\<^esup>))\<rbrakk>
+ \<Longrightarrow> RR"
+apply(erule_tac x = "k + t" in allE, simp)
+apply(erule_tac x = n in allE, simp add: exp_ind_def)
+apply(erule_tac x = "Suc t" in allE, simp)
+done
+
+lemma false_case8:
+ "\<lbrakk>i + t = Suc x;
+ 0 < i;
+ 0 < t;
+ \<forall>ia j. tl (Bk\<^bsup>t\<^esup> @ Oc\<^bsup>i\<^esup>) = Bk\<^bsup>j - Suc 0\<^esup> @ Oc\<^bsup>ia\<^esup> \<longrightarrow>
+ ia + j = Suc x \<longrightarrow> ia = 0 \<or> j = 0 \<or> Oc\<^bsup>t\<^esup> \<noteq> Oc\<^bsup>j\<^esup>\<rbrakk> \<Longrightarrow>
+ RR"
+apply(erule_tac x = i in allE, simp)
+apply(erule_tac x = t in allE, simp)
+apply(case_tac t, simp_all add: exp_ind_def exp_zero)
+done
+
+lemma [elim]: "\<lbrakk>tstep (9, b, c) tcopy = (9, ab, ba);
+ tcopy_F9 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F9 x (ab, ba)"
+apply(auto simp: tcopy_F9.simps tcopy_def
+ tstep.simps fetch.simps new_tape.simps exp_zero_simp exp_zero_simp2
+ tcopy_F9_exit.simps tcopy_F9_loop.simps
+ split: if_splits list.splits block.splits)
+apply(case_tac [!] k, simp_all add: exp_ind_def exp_zero)
+apply(erule_tac [!] x = i in allE, simp)
+apply(erule_tac false_case7, simp_all)+
+apply(case_tac t, simp_all add: exp_zero exp_ind_def)
+apply(erule_tac false_case7, simp_all)+
+apply(erule_tac false_case8, simp_all)
+apply(erule_tac false_case7, simp_all)+
+apply(case_tac t, simp_all add: exp_ind_def exp_zero)
+apply(erule_tac false_case7, simp_all)
+apply(erule_tac false_case8, simp_all)
+apply(erule_tac false_case7, simp_all)
+done
+
+lemma [elim]: "\<lbrakk>tstep (8, b, c) tcopy = (9, ab, ba);
+ tcopy_F8 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F9 x (ab, ba)"
+apply(auto simp:tcopy_F8.simps tcopy_F9.simps tcopy_def
+ tstep.simps fetch.simps new_tape.simps tcopy_F9_loop.simps
+ tcopy_F9_exit.simps
+ split: if_splits list.splits block.splits)
+apply(case_tac [!] t, simp_all add: exp_ind_def exp_zero)
+apply(rule_tac x = i in exI)
+apply(rule_tac x = "Suc k" in exI, simp)
+apply(rule_tac x = "k" in exI, simp add: exp_ind_def exp_zero)
+done
+
+
+lemma [elim]: "\<lbrakk>tstep (8, b, c) tcopy = (8, ab, ba);
+ tcopy_F8 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F8 x (ab, ba)"
+apply(auto simp:tcopy_F8.simps tcopy_def tstep.simps
+ fetch.simps new_tape.simps exp_zero_simp exp_zero split: if_splits list.splits
+
+ block.splits)
+apply(rule_tac x = i in exI, rule_tac x = "k + t" in exI, simp)
+apply(rule_tac x = "Suc k" in exI, simp)
+apply(rule_tac x = "t - 1" in exI, simp)
+apply(case_tac t, simp_all add: exp_zero exp_ind_def)
+done
+
+
+lemma [elim]: "\<lbrakk>tstep (7, b, c) tcopy = (7, ab, ba);
+ tcopy_F7 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F7 x (ab, ba)"
+apply(auto simp:tcopy_F7.simps tcopy_def tstep.simps fetch.simps
+ new_tape.simps exp_ind_def exp_zero_simp
+ split: if_splits list.splits block.splits)
+apply(rule_tac x = i in exI)
+apply(rule_tac x = j in exI, simp)
+apply(rule_tac x = "Suc k" in exI, simp)
+apply(rule_tac x = "t - 1" in exI)
+apply(case_tac t, simp_all add: exp_zero exp_ind_def)
+apply(case_tac j, simp_all add: exp_zero exp_ind_def)
+done
+
+lemma [elim]: "\<lbrakk>tstep (7, b, c) tcopy = (8, ab, ba);
+ tcopy_F7 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F8 x (ab, ba)"
+apply(auto simp:tcopy_F7.simps tcopy_def tstep.simps tcopy_F8.simps
+ fetch.simps new_tape.simps exp_zero_simp
+ split: if_splits list.splits block.splits)
+apply(rule_tac x = i in exI, simp)
+apply(rule_tac x = "Suc 0" in exI, simp add: exp_ind_def exp_zero)
+apply(rule_tac x = "j - 1" in exI, simp)
+apply(case_tac t, simp_all add: exp_ind_def )
+apply(case_tac j, simp_all add: exp_ind_def exp_zero)
+done
+
+lemma [elim]: "\<lbrakk>tstep (6, b, c) tcopy = (7, ab, ba);
+ tcopy_F6 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F7 x (ab, ba)"
+apply(case_tac x)
+apply(auto simp:tcopy_F7.simps tcopy_F6.simps
+ tcopy_def tstep.simps fetch.simps new_tape.simps exp_zero_simp
+ split: if_splits list.splits block.splits)
+apply(case_tac i, simp_all add: exp_ind_def exp_zero)
+apply(rule_tac x = i in exI, simp)
+apply(rule_tac x = j in exI, simp)
+apply(rule_tac x = "Suc 0" in exI, simp add: exp_ind_def exp_zero)
+done
+
+lemma [elim]: "\<lbrakk>tstep (6, b, c) tcopy = (6, ab, ba);
+ tcopy_F6 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F6 x (ab, ba)"
+apply(auto simp:tcopy_F6.simps tcopy_def tstep.simps
+ new_tape.simps fetch.simps
+ split: if_splits list.splits block.splits)
+done
+
+lemma [elim]: "\<lbrakk>tstep (5, b, c) tcopy = (6, ab, ba);
+ tcopy_F5 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F6 x (ab, ba)"
+apply(auto simp:tcopy_F5.simps tcopy_F6.simps tcopy_def
+ tstep.simps fetch.simps new_tape.simps exp_zero_simp2
+ split: if_splits list.splits block.splits)
+apply(rule_tac x = "Suc 0" in exI, simp add: exp_ind_def exp_zero)
+apply(rule_tac x = "Suc i" in exI, simp add: exp_ind_def)
+done
+
+lemma [elim]: "\<lbrakk>tstep (10, b, c) tcopy = (5, ab, ba);
+ tcopy_F10 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F5 x (ab, ba)"
+apply(auto simp:tcopy_F5.simps tcopy_F10.simps tcopy_def
+ tstep.simps fetch.simps new_tape.simps exp_zero_simp exp_zero_simp2
+ exp_ind_def tcopy_F10.simps tcopy_F10_loop.simps tcopy_F10_exit.simps
+ split: if_splits list.splits block.splits )
+apply(erule_tac [!] x = "i - 1" in allE)
+apply(erule_tac [!] x = j in allE, simp_all)
+apply(case_tac [!] i, simp_all add: exp_ind_def)
+done
+
+lemma [elim]: "\<lbrakk>tstep (4, b, c) tcopy = (5, ab, ba);
+ tcopy_F4 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F5 x (ab, ba)"
+apply(auto simp:tcopy_F5.simps tcopy_F4.simps tcopy_def
+ tstep.simps fetch.simps new_tape.simps exp_zero_simp exp_zero_simp2
+ split: if_splits list.splits block.splits)
+apply(case_tac x, simp, simp add: exp_ind_def exp_zero)
+apply(erule_tac [!] x = "x - 2" in allE)
+apply(erule_tac [!] x = "Suc 0" in allE)
+apply(case_tac [!] x, simp_all add: exp_ind_def exp_zero)
+apply(case_tac [!] nat, simp_all add: exp_ind_def exp_zero)
+done
+
+lemma [elim]: "\<lbrakk>tstep (3, b, c) tcopy = (4, ab, ba);
+ tcopy_F3 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F4 x (ab, ba)"
+apply(auto simp:tcopy_F3.simps tcopy_F4.simps
+ tcopy_def tstep.simps fetch.simps new_tape.simps
+ split: if_splits list.splits block.splits)
+done
+
+lemma [elim]: "\<lbrakk>tstep (4, b, c) tcopy = (4, ab, ba);
+ tcopy_F4 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F4 x (ab, ba)"
+apply(case_tac x)
+apply(auto simp:tcopy_F3.simps tcopy_F4.simps
+ tcopy_def tstep.simps fetch.simps new_tape.simps exp_zero_simp exp_zero_simp2 exp_ind_def
+ split: if_splits list.splits block.splits)
+done
+
+lemma [elim]: "\<lbrakk>tstep (3, b, c) tcopy = (3, ab, ba);
+ tcopy_F3 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F3 x (ab, ba)"
+apply(auto simp:tcopy_F3.simps tcopy_F4.simps
+ tcopy_def tstep.simps fetch.simps new_tape.simps
+ split: if_splits list.splits block.splits)
+done
+
+lemma [elim]: "\<lbrakk>tstep (2, b, c) tcopy = (3, ab, ba);
+ tcopy_F2 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F3 x (ab, ba)"
+apply(case_tac x)
+apply(auto simp:tcopy_F3.simps tcopy_F2.simps
+ tcopy_def tstep.simps fetch.simps new_tape.simps
+ exp_zero_simp exp_zero_simp2 exp_zero
+ split: if_splits list.splits block.splits)
+apply(case_tac [!] j, simp_all add: exp_zero exp_ind_def)
+done
+
+lemma [elim]: "\<lbrakk>tstep (2, b, c) tcopy = (2, ab, ba);
+ tcopy_F2 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F2 x (ab, ba)"
+apply(auto simp:tcopy_F3.simps tcopy_F2.simps
+ tcopy_def tstep.simps fetch.simps new_tape.simps
+ exp_zero_simp exp_zero_simp2 exp_zero
+ split: if_splits list.splits block.splits)
+apply(rule_tac x = "Suc i" in exI, simp add: exp_ind_def exp_zero)
+apply(rule_tac x = "j - 1" in exI, simp)
+apply(case_tac j, simp_all add: exp_ind_def exp_zero)
+done
+
+lemma [elim]: "\<lbrakk>tstep (Suc 0, b, c) tcopy = (2, ab, ba);
+ tcopy_F1 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F2 x (ab, ba)"
+apply(auto simp:tcopy_F1.simps tcopy_F2.simps
+ tcopy_def tstep.simps fetch.simps new_tape.simps
+ exp_zero_simp exp_zero_simp2 exp_zero
+ split: if_splits list.splits block.splits)
+apply(rule_tac x = "Suc 0" in exI, simp)
+apply(rule_tac x = "x - 1" in exI, simp)
+apply(case_tac x, simp_all add: exp_ind_def exp_zero)
+done
+
+lemma [elim]: "\<lbrakk>tstep (Suc 0, b, c) tcopy = (0, ab, ba);
+ tcopy_F1 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F0 x (ab, ba)"
+apply(simp_all add:tcopy_F0.simps tcopy_F1.simps
+ tcopy_def tstep.simps fetch.simps new_tape.simps
+ exp_zero_simp exp_zero_simp2 exp_zero
+ split: if_splits list.splits block.splits )
+apply(case_tac x, simp_all add: exp_ind_def exp_zero, auto)
+done
+
+lemma [elim]: "\<lbrakk>tstep (15, b, c) tcopy = (0, ab, ba);
+ tcopy_F15 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F0 x (ab, ba)"
+apply(auto simp: tcopy_F15.simps tcopy_F0.simps
+ tcopy_def tstep.simps new_tape.simps fetch.simps
+ split: if_splits list.splits block.splits)
+apply(rule_tac x = "Suc 0" in exI, simp add: exp_ind_def exp_zero)
+apply(case_tac [!] j, simp_all add: exp_ind_def exp_zero)
+done
+
+
+lemma [elim]: "\<lbrakk>tstep (0, b, c) tcopy = (0, ab, ba);
+ tcopy_F0 x (b, c)\<rbrakk> \<Longrightarrow> tcopy_F0 x (ab, ba)"
+apply(case_tac x)
+apply(simp_all add: tcopy_F0.simps tcopy_def
+ tstep.simps new_tape.simps fetch.simps)
+done
+
+declare tstep.simps[simp del]
+
+text {*
+ Finally establishes the invariant of Copying TM, which is used to dervie
+ the parital correctness of Copying TM.
+*}
+lemma inv_tcopy_step:"inv_tcopy x c \<Longrightarrow> inv_tcopy x (tstep c tcopy)"
+apply(induct c)
+apply(auto split: if_splits block.splits list.splits taction.splits)
+apply(auto simp: tstep.simps tcopy_def fetch.simps new_tape.simps
+ split: if_splits list.splits block.splits taction.splits)
+done
+
+declare inv_tcopy.simps[simp del]
+
+text {*
+ Invariant under mult-step execution.
+ *}
+lemma inv_tcopy_steps:
+ "inv_tcopy x (steps (Suc 0, [], Oc\<^bsup>x\<^esup>) tcopy stp) "
+apply(induct stp)
+apply(simp add: tstep.simps tcopy_def steps.simps
+ tcopy_F1.simps inv_tcopy.simps)
+apply(drule_tac inv_tcopy_step, simp add: tstep_red)
+done
+
+
+
+
+(*----------halt problem of tcopy----------------------------------------*)
+
+section {*
+ The following definitions are used to construct the measure function used to show
+ the termnation of Copying TM.
+*}
+
+definition lex_pair :: "((nat \<times> nat) \<times> nat \<times> nat) set"
+ where
+ "lex_pair \<equiv> less_than <*lex*> less_than"
+
+definition lex_triple ::
+ "((nat \<times> (nat \<times> nat)) \<times> (nat \<times> (nat \<times> nat))) set"
+ where
+"lex_triple \<equiv> less_than <*lex*> lex_pair"
+
+definition lex_square ::
+ "((nat \<times> nat \<times> nat \<times> nat) \<times> (nat \<times> nat \<times> nat \<times> nat)) set"
+ where
+"lex_square \<equiv> less_than <*lex*> lex_triple"
+
+lemma wf_lex_triple: "wf lex_triple"
+ by (auto intro:wf_lex_prod simp:lex_triple_def lex_pair_def)
+
+lemma wf_lex_square: "wf lex_square"
+ by (auto intro:wf_lex_prod
+ simp:lex_triple_def lex_square_def lex_pair_def)
+
+text {*
+ A measurement functions used to show the termination of copying machine:
+*}
+fun tcopy_phase :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_phase c = (let (state, tp) = c in
+ if state > 0 & state <= 4 then 5
+ else if state >=5 & state <= 10 then 4
+ else if state = 11 then 3
+ else if state = 12 | state = 13 then 2
+ else if state = 14 | state = 15 then 1
+ else 0)"
+
+fun tcopy_phase4_stage :: "tape \<Rightarrow> nat"
+ where
+ "tcopy_phase4_stage (ln, rn) =
+ (let lrn = (rev ln) @ rn
+ in length (takeWhile (\<lambda>a. a = Oc) lrn))"
+
+fun tcopy_stage :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_stage c = (let (state, ln, rn) = c in
+ if tcopy_phase c = 5 then 0
+ else if tcopy_phase c = 4 then
+ tcopy_phase4_stage (ln, rn)
+ else if tcopy_phase c = 3 then 0
+ else if tcopy_phase c = 2 then length rn
+ else if tcopy_phase c = 1 then 0
+ else 0)"
+
+fun tcopy_phase4_state :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_phase4_state c = (let (state, ln, rn) = c in
+ if state = 6 & hd rn = Oc then 0
+ else if state = 5 then 1
+ else 12 - state)"
+
+fun tcopy_state :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_state c = (let (state, ln, rn) = c in
+ if tcopy_phase c = 5 then 4 - state
+ else if tcopy_phase c = 4 then
+ tcopy_phase4_state c
+ else if tcopy_phase c = 3 then 0
+ else if tcopy_phase c = 2 then 13 - state
+ else if tcopy_phase c = 1 then 15 - state
+ else 0)"
+
+fun tcopy_step2 :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_step2 (s, l, r) = length r"
+
+fun tcopy_step3 :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_step3 (s, l, r) = (if r = [] | r = [Bk] then Suc 0 else 0)"
+
+fun tcopy_step4 :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_step4 (s, l, r) = length l"
+
+fun tcopy_step7 :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_step7 (s, l, r) = length r"
+
+fun tcopy_step8 :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_step8 (s, l, r) = length r"
+
+fun tcopy_step9 :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_step9 (s, l, r) = length l"
+
+fun tcopy_step10 :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_step10 (s, l, r) = length l"
+
+fun tcopy_step14 :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_step14 (s, l, r) = (case hd r of
+ Oc \<Rightarrow> 1 |
+ Bk \<Rightarrow> 0)"
+
+fun tcopy_step15 :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_step15 (s, l, r) = length l"
+
+fun tcopy_step :: "t_conf \<Rightarrow> nat"
+ where
+ "tcopy_step c = (let (state, ln, rn) = c in
+ if state = 0 | state = 1 | state = 11 |
+ state = 5 | state = 6 | state = 12 | state = 13 then 0
+ else if state = 2 then tcopy_step2 c
+ else if state = 3 then tcopy_step3 c
+ else if state = 4 then tcopy_step4 c
+ else if state = 7 then tcopy_step7 c
+ else if state = 8 then tcopy_step8 c
+ else if state = 9 then tcopy_step9 c
+ else if state = 10 then tcopy_step10 c
+ else if state = 14 then tcopy_step14 c
+ else if state = 15 then tcopy_step15 c
+ else 0)"
+
+text {*
+ The measure function used to show the termination of Copying TM.
+*}
+fun tcopy_measure :: "t_conf \<Rightarrow> (nat * nat * nat * nat)"
+ where
+ "tcopy_measure c =
+ (tcopy_phase c, tcopy_stage c, tcopy_state c, tcopy_step c)"
+
+definition tcopy_LE :: "((nat \<times> block list \<times> block list) \<times>
+ (nat \<times> block list \<times> block list)) set"
+ where
+ "tcopy_LE \<equiv> (inv_image lex_square tcopy_measure)"
+
+lemma wf_tcopy_le: "wf tcopy_LE"
+by(auto intro:wf_inv_image wf_lex_square simp:tcopy_LE_def)
+
+
+declare steps.simps[simp del]
+
+declare tcopy_phase.simps[simp del] tcopy_stage.simps[simp del]
+ tcopy_state.simps[simp del] tcopy_step.simps[simp del]
+ inv_tcopy.simps[simp del]
+declare tcopy_F0.simps [simp]
+ tcopy_F1.simps [simp]
+ tcopy_F2.simps [simp]
+ tcopy_F3.simps [simp]
+ tcopy_F4.simps [simp]
+ tcopy_F5.simps [simp]
+ tcopy_F6.simps [simp]
+ tcopy_F7.simps [simp]
+ tcopy_F8.simps [simp]
+ tcopy_F9.simps [simp]
+ tcopy_F10.simps [simp]
+ tcopy_F11.simps [simp]
+ tcopy_F12.simps [simp]
+ tcopy_F13.simps [simp]
+ tcopy_F14.simps [simp]
+ tcopy_F15.simps [simp]
+ fetch.simps[simp]
+ new_tape.simps[simp]
+lemma [elim]: "tcopy_F1 x (b, c) \<Longrightarrow>
+ (tstep (Suc 0, b, c) tcopy, Suc 0, b, c) \<in> tcopy_LE"
+apply(simp add: tcopy_F1.simps tstep.simps tcopy_def tcopy_LE_def
+ lex_square_def lex_triple_def lex_pair_def tcopy_phase.simps
+ tcopy_stage.simps tcopy_state.simps tcopy_step.simps)
+apply(simp split: if_splits list.splits block.splits taction.splits)
+done
+
+lemma [elim]: "tcopy_F2 x (b, c) \<Longrightarrow>
+ (tstep (2, b, c) tcopy, 2, b, c) \<in> tcopy_LE"
+apply(simp add:tstep.simps tcopy_def tcopy_LE_def lex_square_def
+ lex_triple_def lex_pair_def tcopy_phase.simps tcopy_stage.simps
+ tcopy_state.simps tcopy_step.simps)
+apply(simp split: if_splits list.splits block.splits taction.splits)
+done
+
+lemma [elim]: "tcopy_F3 x (b, c) \<Longrightarrow>
+ (tstep (3, b, c) tcopy, 3, b, c) \<in> tcopy_LE"
+apply(simp add: tstep.simps tcopy_def tcopy_LE_def lex_square_def
+ lex_triple_def lex_pair_def tcopy_phase.simps tcopy_stage.simps
+ tcopy_state.simps tcopy_step.simps)
+apply(simp split: if_splits list.splits block.splits taction.splits)
+apply(auto)
+done
+
+lemma [elim]: "tcopy_F4 x (b, c) \<Longrightarrow>
+ (tstep (4, b, c) tcopy, 4, b, c) \<in> tcopy_LE"
+apply(case_tac x, simp)
+apply(simp add: tcopy_F4.simps tstep.simps tcopy_def tcopy_LE_def
+ lex_square_def lex_triple_def lex_pair_def tcopy_phase.simps
+ tcopy_stage.simps tcopy_state.simps tcopy_step.simps)
+apply(simp split: if_splits list.splits block.splits taction.splits)
+apply(auto simp: exp_ind_def)
+done
+
+lemma[simp]: "takeWhile (\<lambda>a. a = b) (replicate x b @ ys) =
+ replicate x b @ (takeWhile (\<lambda>a. a = b) ys)"
+apply(induct x)
+apply(simp+)
+done
+
+lemma [elim]: "tcopy_F5 x (b, c) \<Longrightarrow>
+ (tstep (5, b, c) tcopy, 5, b, c) \<in> tcopy_LE"
+apply(case_tac x, simp)
+apply(simp add: tstep.simps tcopy_def tcopy_LE_def
+ lex_square_def lex_triple_def lex_pair_def tcopy_phase.simps)
+apply(simp split: if_splits list.splits block.splits taction.splits)
+apply(auto)
+apply(simp_all add: tcopy_phase.simps
+ tcopy_stage.simps tcopy_state.simps)
+done
+
+lemma [elim]: "\<lbrakk>replicate n x = []; n > 0\<rbrakk> \<Longrightarrow> RR"
+apply(case_tac n, simp+)
+done
+
+lemma [elim]: "tcopy_F6 x (b, c) \<Longrightarrow>
+ (tstep (6, b, c) tcopy, 6, b, c) \<in> tcopy_LE"
+apply(case_tac x, simp)
+apply(simp add: tstep.simps tcopy_def tcopy_LE_def
+ lex_square_def lex_triple_def lex_pair_def
+ tcopy_phase.simps)
+apply(simp split: if_splits list.splits block.splits taction.splits)
+apply(auto)
+apply(simp_all add: tcopy_phase.simps tcopy_stage.simps
+ tcopy_state.simps tcopy_step.simps exponent_def)
+done
+
+lemma [elim]: "tcopy_F7 x (b, c) \<Longrightarrow>
+ (tstep (7, b, c) tcopy, 7, b, c) \<in> tcopy_LE"
+apply(case_tac x, simp)
+apply(simp add: tstep.simps tcopy_def tcopy_LE_def lex_square_def
+ lex_triple_def lex_pair_def tcopy_phase.simps)
+apply(simp split: if_splits list.splits block.splits taction.splits)
+apply(auto simp: exp_zero_simp)
+apply(simp_all add: tcopy_phase.simps tcopy_stage.simps
+ tcopy_state.simps tcopy_step.simps)
+done
+
+lemma [elim]: "tcopy_F8 x (b, c) \<Longrightarrow>
+ (tstep (8, b, c) tcopy, 8, b, c) \<in> tcopy_LE"
+apply(case_tac x, simp)
+apply(simp add: tstep.simps tcopy_def tcopy_LE_def lex_square_def
+ lex_triple_def lex_pair_def tcopy_phase.simps)
+apply(simp split: if_splits list.splits block.splits taction.splits)
+apply(auto simp: exp_zero_simp)
+apply(simp_all add: tcopy_phase.simps tcopy_stage.simps
+ tcopy_state.simps tcopy_step.simps exponent_def)
+done
+
+lemma rev_equal_rev: "rev a = rev b \<Longrightarrow> a = b"
+by simp
+
+lemma app_app_app_equal: "xs @ ys @ zs = (xs @ ys) @ zs"
+by simp
+
+lemma append_cons_assoc: "as @ b # bs = (as @ [b]) @ bs"
+apply(rule rev_equal_rev)
+apply(simp)
+done
+
+lemma rev_tl_hd_merge: "bs \<noteq> [] \<Longrightarrow>
+ rev (tl bs) @ hd bs # as = rev bs @ as"
+apply(rule rev_equal_rev)
+apply(simp)
+done
+
+lemma[simp]: "takeWhile (\<lambda>a. a = b) (replicate x b @ ys) =
+ replicate x b @ (takeWhile (\<lambda>a. a = b) ys)"
+apply(induct x)
+apply(simp+)
+done
+
+lemma [elim]: "tcopy_F9 x (b, c) \<Longrightarrow>
+ (tstep (9, b, c) tcopy, 9, b, c) \<in> tcopy_LE"
+apply(simp add: tstep.simps tcopy_def tcopy_LE_def lex_square_def
+ lex_triple_def lex_pair_def tcopy_phase.simps tcopy_F9.simps
+ tcopy_F9_loop.simps tcopy_F9_exit.simps)
+apply(simp split: if_splits list.splits block.splits taction.splits)
+apply(auto)
+apply(simp_all add: tcopy_phase.simps tcopy_stage.simps tcopy_F9_loop.simps
+ tcopy_state.simps tcopy_step.simps tstep.simps exp_zero_simp
+ exponent_def)
+apply(case_tac [1-2] t, simp_all add: rev_tl_hd_merge)
+apply(case_tac j, simp, simp)
+apply(case_tac nat, simp_all)
+apply(case_tac nata, simp_all)
+done
+
+lemma [elim]: "tcopy_F10 x (b, c) \<Longrightarrow>
+ (tstep (10, b, c) tcopy, 10, b, c) \<in> tcopy_LE"
+apply(simp add: tstep.simps tcopy_def tcopy_LE_def lex_square_def
+ lex_triple_def lex_pair_def tcopy_phase.simps tcopy_F10_loop.simps
+ tcopy_F10_exit.simps exp_zero_simp)
+apply(simp split: if_splits list.splits block.splits taction.splits)
+apply(auto simp: exp_zero_simp)
+apply(simp_all add: tcopy_phase.simps tcopy_stage.simps
+ tcopy_state.simps tcopy_step.simps exponent_def
+ rev_tl_hd_merge)
+apply(case_tac k, simp_all)
+apply(case_tac nat, simp_all)
+done
+
+lemma [elim]: "tcopy_F11 x (b, c) \<Longrightarrow>
+ (tstep (11, b, c) tcopy, 11, b, c) \<in> tcopy_LE"
+apply(case_tac x, simp)
+apply(simp add: tstep.simps tcopy_def tcopy_LE_def
+ lex_square_def lex_triple_def lex_pair_def
+ tcopy_phase.simps)
+done
+
+lemma [elim]: "tcopy_F12 x (b, c) \<Longrightarrow>
+ (tstep (12, b, c) tcopy, 12, b, c) \<in> tcopy_LE"
+apply(case_tac x, simp)
+apply(simp add: tstep.simps tcopy_def tcopy_LE_def lex_square_def
+ lex_triple_def lex_pair_def tcopy_phase.simps)
+apply(simp split: if_splits list.splits block.splits taction.splits)
+apply(auto)
+apply(simp_all add: tcopy_phase.simps tcopy_stage.simps
+ tcopy_state.simps tcopy_step.simps)
+apply(simp_all add: exp_ind_def)
+done
+
+lemma [elim]: "tcopy_F13 x (b, c) \<Longrightarrow>
+ (tstep (13, b, c) tcopy, 13, b, c) \<in> tcopy_LE"
+apply(case_tac x, simp)
+apply(simp add: tstep.simps tcopy_def tcopy_LE_def lex_square_def
+ lex_triple_def lex_pair_def tcopy_phase.simps)
+apply(simp split: if_splits list.splits block.splits taction.splits)
+apply(auto)
+apply(simp_all add: tcopy_phase.simps tcopy_stage.simps
+ tcopy_state.simps tcopy_step.simps)
+done
+
+lemma [elim]: "tcopy_F14 x (b, c) \<Longrightarrow>
+ (tstep (14, b, c) tcopy, 14, b, c) \<in> tcopy_LE"
+apply(case_tac x, simp)
+apply(simp add: tstep.simps tcopy_def tcopy_LE_def lex_square_def
+ lex_triple_def lex_pair_def tcopy_phase.simps)
+apply(simp split: if_splits list.splits block.splits taction.splits)
+apply(auto)
+apply(simp_all add: tcopy_phase.simps tcopy_stage.simps
+ tcopy_state.simps tcopy_step.simps)
+done
+
+lemma [elim]: "tcopy_F15 x (b, c) \<Longrightarrow>
+ (tstep (15, b, c) tcopy, 15, b, c) \<in> tcopy_LE"
+apply(case_tac x, simp)
+apply(simp add: tstep.simps tcopy_def tcopy_LE_def lex_square_def
+ lex_triple_def lex_pair_def tcopy_phase.simps )
+apply(simp split: if_splits list.splits block.splits taction.splits)
+apply(auto)
+apply(simp_all add: tcopy_phase.simps tcopy_stage.simps
+ tcopy_state.simps tcopy_step.simps)
+done
+
+lemma exp_length: "length (a\<^bsup>b\<^esup>) = b"
+apply(induct b, simp_all add: exp_zero exp_ind_def)
+done
+
+declare tcopy_F9.simps[simp del] tcopy_F10.simps[simp del]
+
+lemma length_eq: "xs = ys \<Longrightarrow> length xs = length ys"
+by simp
+
+lemma tcopy_wf_step:"\<lbrakk>a > 0; inv_tcopy x (a, b, c)\<rbrakk> \<Longrightarrow>
+ (tstep (a, b, c) tcopy, (a, b, c)) \<in> tcopy_LE"
+apply(simp add:inv_tcopy.simps split: if_splits, auto)
+apply(auto simp: tstep.simps tcopy_def tcopy_LE_def lex_square_def
+ lex_triple_def lex_pair_def tcopy_phase.simps
+ tcopy_stage.simps tcopy_state.simps tcopy_step.simps
+ exp_length exp_zero_simp exponent_def
+ split: if_splits list.splits block.splits taction.splits)
+apply(case_tac [!] t, simp_all)
+apply(case_tac j, simp_all)
+apply(drule_tac length_eq, simp)
+done
+
+lemma tcopy_wf:
+"\<forall>n. let nc = steps (Suc 0, [], Oc\<^bsup>x\<^esup>) tcopy n in
+ let Sucnc = steps (Suc 0, [], Oc\<^bsup>x\<^esup>) tcopy (Suc n) in
+ \<not> isS0 nc \<longrightarrow> ((Sucnc, nc) \<in> tcopy_LE)"
+proof(rule allI, case_tac
+ "steps (Suc 0, [], Oc\<^bsup>x\<^esup>) tcopy n", auto simp: tstep_red)
+ fix n a b c
+ assume h: "\<not> isS0 (a, b, c)"
+ "steps (Suc 0, [], Oc\<^bsup>x\<^esup>) tcopy n = (a, b, c)"
+ hence "inv_tcopy x (a, b, c)"
+ using inv_tcopy_steps[of x n] by(simp)
+ thus "(tstep (a, b, c) tcopy, a, b, c) \<in> tcopy_LE"
+ using h
+ by(rule_tac tcopy_wf_step, auto simp: isS0_def)
+qed
+
+text {*
+ The termination of Copying TM:
+*}
+lemma tcopy_halt:
+ "\<exists>n. isS0 (steps (Suc 0, [], Oc\<^bsup>x\<^esup>) tcopy n)"
+apply(insert halt_lemma
+ [of tcopy_LE isS0 "steps (Suc 0, [], Oc\<^bsup>x\<^esup>) tcopy"])
+apply(insert tcopy_wf [of x])
+apply(simp only: Let_def)
+apply(insert wf_tcopy_le)
+apply(simp)
+done
+
+text {*
+ The total correntess of Copying TM:
+*}
+theorem tcopy_halt_rs:
+ "\<exists>stp m.
+ steps (Suc 0, [], Oc\<^bsup>x\<^esup>) tcopy stp =
+ (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>x\<^esup> @ Bk # Oc\<^bsup>x\<^esup>)"
+using tcopy_halt[of x]
+proof(erule_tac exE)
+ fix n
+ assume h: "isS0 (steps (Suc 0, [], Oc\<^bsup>x\<^esup>) tcopy n)"
+ have "inv_tcopy x (steps (Suc 0, [], Oc\<^bsup>x\<^esup>) tcopy n)"
+ using inv_tcopy_steps[of x n] by simp
+ thus "?thesis"
+ using h
+ apply(cases "(steps (Suc 0, [], Oc\<^bsup>x\<^esup>) tcopy n)",
+ auto simp: isS0_def inv_tcopy.simps)
+ done
+qed
+
+section {*
+ The {\em Dithering} Turing Machine
+*}
+
+text {*
+ The {\em Dithering} TM, when the input is @{text "1"}, it will loop forever, otherwise, it will
+ terminate.
+*}
+definition dither :: "tprog"
+ where
+ "dither \<equiv> [(W0, 1), (R, 2), (L, 1), (L, 0)] "
+
+lemma dither_halt_rs:
+ "\<exists> stp. steps (Suc 0, Bk\<^bsup>m\<^esup>, [Oc, Oc]) dither stp =
+ (0, Bk\<^bsup>m\<^esup>, [Oc, Oc])"
+apply(rule_tac x = "Suc (Suc (Suc 0))" in exI)
+apply(simp add: dither_def steps.simps
+ tstep.simps fetch.simps new_tape.simps)
+done
+
+lemma dither_unhalt_state:
+ "(steps (Suc 0, Bk\<^bsup>m\<^esup>, [Oc]) dither stp =
+ (Suc 0, Bk\<^bsup>m\<^esup>, [Oc])) \<or>
+ (steps (Suc 0, Bk\<^bsup>m\<^esup>, [Oc]) dither stp = (2, Oc # Bk\<^bsup>m\<^esup>, []))"
+ apply(induct stp, simp add: steps.simps)
+ apply(simp add: tstep_red, auto)
+ apply(auto simp: tstep.simps fetch.simps dither_def new_tape.simps)
+ done
+
+lemma dither_unhalt_rs:
+ "\<not> (\<exists> stp. isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, [Oc]) dither stp))"
+proof(auto)
+ fix stp
+ assume h1: "isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, [Oc]) dither stp)"
+ have "\<not> isS0 ((steps (Suc 0, Bk\<^bsup>m\<^esup>, [Oc]) dither stp))"
+ using dither_unhalt_state[of m stp]
+ by(auto simp: isS0_def)
+ from h1 and this show False by (auto)
+qed
+
+section {*
+ The final diagnal arguments to show the undecidability of Halting problem.
+*}
+
+text {*
+ @{text "haltP tp x"} means TM @{text "tp"} terminates on input @{text "x"}
+ and the final configuration is standard.
+*}
+definition haltP :: "tprog \<Rightarrow> nat \<Rightarrow> bool"
+ where
+ "haltP t x = (\<exists>n a b c. steps (Suc 0, [], Oc\<^bsup>x\<^esup>) t n = (0, Bk\<^bsup>a\<^esup>, Oc\<^bsup>b\<^esup> @ Bk\<^bsup>c\<^esup>))"
+
+lemma [simp]: "length (A |+| B) = length A + length B"
+by(auto simp: t_add.simps tshift.simps)
+
+lemma [intro]: "\<lbrakk>iseven (x::nat); iseven y\<rbrakk> \<Longrightarrow> iseven (x + y)"
+apply(auto simp: iseven_def)
+apply(rule_tac x = "x + xa" in exI, simp)
+done
+
+lemma t_correct_add[intro]:
+ "\<lbrakk>t_correct A; t_correct B\<rbrakk> \<Longrightarrow> t_correct (A |+| B)"
+apply(auto simp: t_correct.simps tshift.simps t_add.simps
+ change_termi_state.simps list_all_iff)
+apply(erule_tac x = "(a, b)" in ballE, auto)
+apply(case_tac "ba = 0", auto)
+done
+
+lemma [intro]: "t_correct tcopy"
+apply(simp add: t_correct.simps tcopy_def iseven_def)
+apply(rule_tac x = 15 in exI, simp)
+done
+
+lemma [intro]: "t_correct dither"
+apply(simp add: t_correct.simps dither_def iseven_def)
+apply(rule_tac x = 2 in exI, simp)
+done
+
+text {*
+ The following locale specifies that TM @{text "H"} can be used to solve
+ the {\em Halting Problem} and @{text "False"} is going to be derived
+ under this locale. Therefore, the undecidability of {\em Halting Problem}
+ is established.
+*}
+locale uncomputable =
+ -- {* The coding function of TM, interestingly, the detailed definition of this
+ funciton @{text "code"} does not affect the final result. *}
+ fixes code :: "tprog \<Rightarrow> nat"
+ -- {*
+ The TM @{text "H"} is the one which is assummed being able to solve the Halting problem.
+ *}
+ and H :: "tprog"
+ assumes h_wf[intro]: "t_correct H"
+ -- {*
+ The following two assumptions specifies that @{text "H"} does solve the Halting problem.
+ *}
+ and h_case:
+ "\<And> M n. \<lbrakk>(haltP M n)\<rbrakk> \<Longrightarrow>
+ \<exists> na nb. (steps (Suc 0, [], Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na = (0, Bk\<^bsup>nb\<^esup>, [Oc]))"
+ and nh_case:
+ "\<And> M n. \<lbrakk>(\<not> haltP M n)\<rbrakk> \<Longrightarrow>
+ \<exists> na nb. (steps (Suc 0, [], Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na = (0, Bk\<^bsup>nb\<^esup>, [Oc, Oc]))"
+begin
+
+term t_correct
+declare haltP_def[simp del]
+definition tcontra :: "tprog \<Rightarrow> tprog"
+ where
+ "tcontra h \<equiv> ((tcopy |+| h) |+| dither)"
+
+lemma [simp]: "a\<^bsup>0\<^esup> = []"
+ by(simp add: exponent_def)
+
+lemma tinres_ex1:
+ "tinres (Bk\<^bsup>nb\<^esup>) b \<Longrightarrow> \<exists>nb. b = Bk\<^bsup>nb\<^esup>"
+apply(auto simp: tinres_def)
+proof -
+ fix n
+ assume "Bk\<^bsup>nb\<^esup> = b @ Bk\<^bsup>n\<^esup>"
+ thus "\<exists>nb. b = Bk\<^bsup>nb\<^esup>"
+ proof(induct b arbitrary: nb)
+ show "\<exists>nb. [] = Bk\<^bsup>nb\<^esup>"
+ by(rule_tac x = 0 in exI, simp add: exp_zero)
+ next
+ fix a b nb
+ assume ind: "\<And>nb. Bk\<^bsup>nb\<^esup> = b @ Bk\<^bsup>n\<^esup> \<Longrightarrow> \<exists>nb. b = Bk\<^bsup>nb\<^esup>"
+ and h: "Bk\<^bsup>nb\<^esup> = (a # b) @ Bk\<^bsup>n\<^esup>"
+ from h show "\<exists>nb. a # b = Bk\<^bsup>nb\<^esup>"
+ proof(case_tac a, case_tac nb, simp_all add: exp_ind_def)
+ fix nat
+ assume "Bk\<^bsup>nat\<^esup> = b @ Bk\<^bsup>n\<^esup>"
+ thus "\<exists>nb. Bk # b = Bk\<^bsup>nb\<^esup>"
+ using ind[of nat]
+ apply(auto)
+ apply(rule_tac x = "Suc nb" in exI, simp add: exp_ind_def)
+ done
+ next
+ assume "Bk\<^bsup>nb\<^esup> = Oc # b @ Bk\<^bsup>n\<^esup>"
+ thus "\<exists>nb. Oc # b = Bk\<^bsup>nb\<^esup>"
+ apply(case_tac nb, simp_all add: exp_ind_def)
+ done
+ qed
+ qed
+next
+ fix n
+ show "\<exists>nba. Bk\<^bsup>nb\<^esup> @ Bk\<^bsup>n\<^esup> = Bk\<^bsup>nba\<^esup>"
+ apply(rule_tac x = "nb + n" in exI)
+ apply(simp add: exponent_def replicate_add)
+ done
+qed
+
+lemma h_newcase: "\<And> M n. \<lbrakk>(haltP M n)\<rbrakk> \<Longrightarrow>
+ \<exists> na nb. (steps (Suc 0, Bk\<^bsup>x\<^esup>, Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na = (0, Bk\<^bsup>nb\<^esup>, [Oc]))"
+proof -
+ fix M n x
+ assume "haltP M n"
+ hence " \<exists> na nb. (steps (Suc 0, [], Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na
+ = (0, Bk\<^bsup>nb\<^esup>, [Oc]))"
+ apply(erule_tac h_case)
+ done
+ from this obtain na nb where
+ cond1:"(steps (Suc 0, [], Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na
+ = (0, Bk\<^bsup>nb\<^esup>, [Oc]))" by blast
+ thus "\<exists> na nb. (steps (Suc 0, Bk\<^bsup>x\<^esup>, Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na = (0, Bk\<^bsup>nb\<^esup>, [Oc]))"
+ proof(rule_tac x = na in exI, case_tac "steps (Suc 0, Bk\<^bsup>x\<^esup>, Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na", simp)
+ fix a b c
+ assume cond2: "steps (Suc 0, Bk\<^bsup>x\<^esup>, Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na = (a, b, c)"
+ have "tinres (Bk\<^bsup>nb\<^esup>) b \<and> [Oc] = c \<and> 0 = a"
+ proof(rule_tac tinres_steps)
+ show "tinres [] (Bk\<^bsup>x\<^esup>)"
+ apply(simp add: tinres_def)
+ apply(auto)
+ done
+ next
+ show "(steps (Suc 0, [], Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na
+ = (0, Bk\<^bsup>nb\<^esup>, [Oc]))"
+ by(simp add: cond1)
+ next
+ show "steps (Suc 0, Bk\<^bsup>x\<^esup>, Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na = (a, b, c)"
+ by(simp add: cond2)
+ qed
+ thus "a = 0 \<and> (\<exists>nb. b = Bk\<^bsup>nb\<^esup>) \<and> c = [Oc]"
+ apply(auto simp: tinres_ex1)
+ done
+ qed
+qed
+
+lemma nh_newcase: "\<And> M n. \<lbrakk>\<not> (haltP M n)\<rbrakk> \<Longrightarrow>
+ \<exists> na nb. (steps (Suc 0, Bk\<^bsup>x\<^esup>, Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na = (0, Bk\<^bsup>nb\<^esup>, [Oc, Oc]))"
+proof -
+ fix M n
+ assume "\<not> haltP M n"
+ hence "\<exists> na nb. (steps (Suc 0, [], Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na
+ = (0, Bk\<^bsup>nb\<^esup>, [Oc, Oc]))"
+ apply(erule_tac nh_case)
+ done
+ from this obtain na nb where
+ cond1: "(steps (Suc 0, [], Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na
+ = (0, Bk\<^bsup>nb\<^esup>, [Oc, Oc]))" by blast
+ thus "\<exists> na nb. (steps (Suc 0, Bk\<^bsup>x\<^esup>, Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na = (0, Bk\<^bsup>nb\<^esup>, [Oc, Oc]))"
+ proof(rule_tac x = na in exI, case_tac "steps (Suc 0, Bk\<^bsup>x\<^esup>, Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na", simp)
+ fix a b c
+ assume cond2: "steps (Suc 0, Bk\<^bsup>x\<^esup>, Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na = (a, b, c)"
+ have "tinres (Bk\<^bsup>nb\<^esup>) b \<and> [Oc, Oc] = c \<and> 0 = a"
+ proof(rule_tac tinres_steps)
+ show "tinres [] (Bk\<^bsup>x\<^esup>)"
+ apply(simp add: tinres_def)
+ apply(auto)
+ done
+ next
+ show "(steps (Suc 0, [], Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na
+ = (0, Bk\<^bsup>nb\<^esup>, [Oc, Oc]))"
+ by(simp add: cond1)
+ next
+ show "steps (Suc 0, Bk\<^bsup>x\<^esup>, Oc\<^bsup>code M\<^esup> @ Bk # Oc\<^bsup>n\<^esup>) H na = (a, b, c)"
+ by(simp add: cond2)
+ qed
+ thus "a = 0 \<and> (\<exists>nb. b = Bk\<^bsup>nb\<^esup>) \<and> c = [Oc, Oc]"
+ apply(auto simp: tinres_ex1)
+ done
+ qed
+qed
+
+lemma haltP_weaking:
+ "haltP (tcontra H) (code (tcontra H)) \<Longrightarrow>
+ \<exists>stp. isS0 (steps (Suc 0, [], Oc\<^bsup>code (tcontra H)\<^esup>)
+ ((tcopy |+| H) |+| dither) stp)"
+ apply(simp add: haltP_def, auto)
+ apply(rule_tac x = n in exI, simp add: isS0_def tcontra_def)
+ done
+
+lemma h_uh: "haltP (tcontra H) (code (tcontra H))
+ \<Longrightarrow> \<not> haltP (tcontra H) (code (tcontra H))"
+proof -
+ let ?cn = "code (tcontra H)"
+ let ?P1 = "\<lambda> tp. let (l, r) = tp in (l = [] \<and>
+ (r::block list) = Oc\<^bsup>(?cn)\<^esup>)"
+ let ?Q1 = "\<lambda> (l, r).(\<exists> nb. l = Bk\<^bsup>nb\<^esup> \<and>
+ r = Oc\<^bsup>(?cn)\<^esup> @ Bk # Oc\<^bsup>(?cn)\<^esup>)"
+ let ?P2 = ?Q1
+ let ?Q2 = "\<lambda> (l, r). (\<exists> nd. l = Bk\<^bsup>nd \<^esup>\<and> r = [Oc])"
+ let ?P3 = "\<lambda> tp. False"
+ assume h: "haltP (tcontra H) (code (tcontra H))"
+ hence h1: "\<And> x. \<exists> na nb. steps (Suc 0, Bk\<^bsup>x\<^esup>, Oc\<^bsup>code (tcontra H)\<^esup> @ Bk #
+ Oc\<^bsup>code (tcontra H)\<^esup>) H na = (0, Bk\<^bsup>nb\<^esup>, [Oc])"
+ by(drule_tac x = x in h_newcase, simp)
+ have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp) (tcopy |+| H) stp = (0, tp') \<and> ?Q2 tp')"
+ proof(rule_tac turing_merge.t_merge_halt[of tcopy H "?P1" "?P2" "?P3"
+ "?P3" "?Q1" "?Q2"], auto simp: turing_merge_def)
+ show "\<exists>stp. case steps (Suc 0, [], Oc\<^bsup>?cn\<^esup>) tcopy stp of (s, tp') \<Rightarrow>
+ s = 0 \<and> (case tp' of (l, r) \<Rightarrow> (\<exists>nb. l = Bk\<^bsup>nb\<^esup>) \<and> r = Oc\<^bsup>?cn\<^esup> @ Bk # Oc\<^bsup>?cn\<^esup>)"
+ using tcopy_halt_rs[of "?cn"]
+ apply(auto)
+ apply(rule_tac x = stp in exI, auto simp: exponent_def)
+ done
+ next
+ fix nb
+ show "\<exists>stp. case steps (Suc 0, Bk\<^bsup>nb\<^esup>, Oc\<^bsup>code (tcontra H)\<^esup> @ Bk # Oc\<^bsup>code (tcontra H)\<^esup>) H stp of
+ (s, tp') \<Rightarrow> s = 0 \<and> (case tp' of (l, r) \<Rightarrow> (\<exists>nd. l = Bk\<^bsup>nd\<^esup>) \<and> r = [Oc])"
+ using h1[of nb]
+ apply(auto)
+ apply(rule_tac x = na in exI, auto)
+ done
+ next
+ show "\<lambda>(l, r). ((\<exists>nb. l = Bk\<^bsup>nb\<^esup>) \<and> r = Oc\<^bsup>code (tcontra H)\<^esup> @ Bk # Oc\<^bsup>code (tcontra H)\<^esup>) \<turnstile>->
+ \<lambda>(l, r). ((\<exists>nb. l = Bk\<^bsup>nb\<^esup>) \<and> r = Oc\<^bsup>code (tcontra H)\<^esup> @ Bk # Oc\<^bsup>code (tcontra H)\<^esup>)"
+ apply(simp add: t_imply_def)
+ done
+ qed
+ hence "\<exists>stp tp'. steps (Suc 0, [], Oc\<^bsup>?cn\<^esup>) (tcopy |+| H) stp = (0, tp') \<and>
+ (case tp' of (l, r) \<Rightarrow> \<exists>nd. l = Bk\<^bsup>nd\<^esup> \<and> r = [Oc])"
+ apply(simp add: t_imply_def)
+ done
+ hence "?P1 \<turnstile>-> \<lambda> tp. \<not> (\<exists> stp. isS0 (steps (Suc 0, tp) ((tcopy |+| H) |+| dither) stp))"
+ proof(rule_tac turing_merge.t_merge_uhalt[of "tcopy |+| H" dither "?P1" "?P3" "?P3"
+ "?Q2" "?Q2" "?Q2"], simp add: turing_merge_def, auto)
+ fix stp nd
+ assume "steps (Suc 0, [], Oc\<^bsup>code (tcontra H)\<^esup>) (tcopy |+| H) stp = (0, Bk\<^bsup>nd\<^esup>, [Oc])"
+ thus "\<exists>stp. case steps (Suc 0, [], Oc\<^bsup>code (tcontra H)\<^esup>) (tcopy |+| H) stp of (s, tp')
+ \<Rightarrow> s = 0 \<and> (case tp' of (l, r) \<Rightarrow> (\<exists>nd. l = Bk\<^bsup>nd\<^esup>) \<and> r = [Oc])"
+ apply(rule_tac x = stp in exI, auto)
+ done
+ next
+ fix stp nd nda stpa
+ assume "isS0 (steps (Suc 0, Bk\<^bsup>nda\<^esup>, [Oc]) dither stpa)"
+ thus "False"
+ using dither_unhalt_rs[of nda]
+ apply auto
+ done
+ next
+ fix stp nd
+ show "\<lambda>(l, r). ((\<exists>nd. l = Bk\<^bsup>nd\<^esup>) \<and> r = [Oc]) \<turnstile>->
+ \<lambda>(l, r). ((\<exists>nd. l = Bk\<^bsup>nd\<^esup>) \<and> r = [Oc])"
+ by (simp add: t_imply_def)
+ qed
+ thus "\<not> haltP (tcontra H) (code (tcontra H))"
+ apply(simp add: t_imply_def haltP_def tcontra_def, auto)
+ apply(erule_tac x = n in allE, simp add: isS0_def)
+ done
+qed
+
+lemma uh_h:
+ assumes uh: "\<not> haltP (tcontra H) (code (tcontra H))"
+ shows "haltP (tcontra H) (code (tcontra H))"
+proof -
+ let ?cn = "code (tcontra H)"
+ have h1: "\<And> x. \<exists> na nb. steps (Suc 0, Bk\<^bsup>x\<^esup>, Oc\<^bsup>?cn\<^esup> @ Bk # Oc\<^bsup>?cn\<^esup>)
+ H na = (0, Bk\<^bsup>nb\<^esup>, [Oc, Oc])"
+ using uh
+ by(drule_tac x = x in nh_newcase, simp)
+ let ?P1 = "\<lambda> tp. let (l, r) = tp in (l = [] \<and>
+ (r::block list) = Oc\<^bsup>(?cn)\<^esup>)"
+ let ?Q1 = "\<lambda> (l, r).(\<exists> na. l = Bk\<^bsup>na\<^esup> \<and> r = [Oc, Oc])"
+ let ?P2 = ?Q1
+ let ?Q2 = ?Q1
+ let ?P3 = "\<lambda> tp. False"
+ have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp) ((tcopy |+| H ) |+| dither)
+ stp = (0, tp') \<and> ?Q2 tp')"
+ proof(rule_tac turing_merge.t_merge_halt[of "tcopy |+| H" dither ?P1 ?P2 ?P3 ?P3
+ ?Q1 ?Q2], auto simp: turing_merge_def)
+ show "\<exists>stp. case steps (Suc 0, [], Oc\<^bsup>code (tcontra H)\<^esup>) (tcopy |+| H) stp of (s, tp') \<Rightarrow>
+
+ s = 0 \<and> (case tp' of (l, r) \<Rightarrow> (\<exists>na. l = Bk\<^bsup>na\<^esup>) \<and> r = [Oc, Oc])"
+ proof -
+ let ?Q1 = "\<lambda> (l, r).(\<exists> nb. l = Bk\<^bsup>nb\<^esup> \<and> r = Oc\<^bsup>(?cn)\<^esup> @ Bk # Oc\<^bsup>(?cn)\<^esup>)"
+ let ?P2 = "?Q1"
+ let ?Q2 = "\<lambda> (l, r).(\<exists> na. l = Bk\<^bsup>na\<^esup> \<and> r = [Oc, Oc])"
+ have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp) (tcopy |+| H )
+ stp = (0, tp') \<and> ?Q2 tp')"
+ proof(rule_tac turing_merge.t_merge_halt[of tcopy H ?P1 ?P2 ?P3 ?P3
+ ?Q1 ?Q2], auto simp: turing_merge_def)
+ show "\<exists>stp. case steps (Suc 0, [], Oc\<^bsup>code (tcontra H)\<^esup>) tcopy stp of (s, tp') \<Rightarrow> s = 0
+ \<and> (case tp' of (l, r) \<Rightarrow> (\<exists>nb. l = Bk\<^bsup>nb\<^esup>) \<and> r = Oc\<^bsup>code (tcontra H)\<^esup> @ Bk # Oc\<^bsup>code (tcontra H)\<^esup>)"
+ using tcopy_halt_rs[of "?cn"]
+ apply(auto)
+ apply(rule_tac x = stp in exI, simp add: exponent_def)
+ done
+ next
+ fix nb
+ show "\<exists>stp. case steps (Suc 0, Bk\<^bsup>nb\<^esup>, Oc\<^bsup>code (tcontra H)\<^esup> @ Bk # Oc\<^bsup>code (tcontra H)\<^esup>) H stp of
+ (s, tp') \<Rightarrow> s = 0 \<and> (case tp' of (l, r) \<Rightarrow> (\<exists>na. l = Bk\<^bsup>na\<^esup>) \<and> r = [Oc, Oc])"
+ using h1[of nb]
+ apply(auto)
+ apply(rule_tac x = na in exI, auto)
+ done
+ next
+ show "\<lambda>(l, r). ((\<exists>nb. l = Bk\<^bsup>nb\<^esup>) \<and> r = Oc\<^bsup>code (tcontra H)\<^esup> @ Bk # Oc\<^bsup>code (tcontra H)\<^esup>) \<turnstile>->
+ \<lambda>(l, r). ((\<exists>nb. l = Bk\<^bsup>nb\<^esup>) \<and> r = Oc\<^bsup>code (tcontra H)\<^esup> @ Bk # Oc\<^bsup>code (tcontra H)\<^esup>)"
+ by(simp add: t_imply_def)
+ qed
+ hence "(\<exists> stp tp'. steps (Suc 0, [], Oc\<^bsup>?cn\<^esup>) (tcopy |+| H ) stp = (0, tp') \<and> ?Q2 tp')"
+ apply(simp add: t_imply_def)
+ done
+ thus "?thesis"
+ apply(auto)
+ apply(rule_tac x = stp in exI, auto)
+ done
+ qed
+ next
+ fix na
+ show "\<exists>stp. case steps (Suc 0, Bk\<^bsup>na\<^esup>, [Oc, Oc]) dither stp of (s, tp')
+ \<Rightarrow> s = 0 \<and> (case tp' of (l, r) \<Rightarrow> (\<exists>na. l = Bk\<^bsup>na\<^esup>) \<and> r = [Oc, Oc])"
+ using dither_halt_rs[of na]
+ apply(auto)
+ apply(rule_tac x = stp in exI, auto)
+ done
+ next
+ show "\<lambda>(l, r). ((\<exists>na. l = Bk\<^bsup>na\<^esup>) \<and> r = [Oc, Oc]) \<turnstile>->
+ (\<lambda>(l, r). (\<exists>na. l = Bk\<^bsup>na\<^esup>) \<and> r = [Oc, Oc])"
+ by (simp add: t_imply_def)
+ qed
+ hence "\<exists> stp tp'. steps (Suc 0, [], Oc\<^bsup>?cn\<^esup>) ((tcopy |+| H ) |+| dither)
+ stp = (0, tp') \<and> ?Q2 tp'"
+ apply(simp add: t_imply_def)
+ done
+ thus "haltP (tcontra H) (code (tcontra H))"
+ apply(auto simp: haltP_def tcontra_def)
+ apply(rule_tac x = stp in exI,
+ rule_tac x = na in exI,
+ rule_tac x = "Suc (Suc 0)" in exI,
+ rule_tac x = "0" in exI, simp add: exp_ind_def)
+ done
+qed
+
+text {*
+ @{text "False"} is finally derived.
+*}
+
+lemma "False"
+using uh_h h_uh
+by auto
+end
+
+end
+