
diff -r 0b302c0b449a -r 469c26d19f8e Attic/uncomputable.thy
--- /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
+




