diff -r 01f688735b9b -r 2cae8a39803e thys/uncomputable.thy
--- a/thys/uncomputable.thy	Thu Jan 31 11:35:16 2013 +0000
+++ b/thys/uncomputable.thy	Fri Feb 01 01:34:37 2013 +0000
@@ -63,8 +63,6 @@
 | "inv_begin3 n (l, r) = (n > 0 \<and> (l, tl r) = (Bk # Oc \<up> n, []))"
 | "inv_begin4 n (l, r) = (n > 0 \<and> (l, r) = (Oc \<up> n, [Bk, Oc]) \<or> (l, r) = (Oc \<up> (n - 1), [Oc, Bk, Oc]))"
 
-
-
 fun inv_begin :: "nat \<Rightarrow> config \<Rightarrow> bool"
   where
   "inv_begin n (s, tp) = 
@@ -80,9 +78,9 @@
 by (rule_tac x = "Suc i" in exI, simp)
 
 lemma inv_begin_step: 
-  assumes "inv_begin x cf"
-  and "x > 0"
-  shows "inv_begin x (step0 cf tcopy_begin)"
+  assumes "inv_begin n cf"
+  and "n > 0"
+  shows "inv_begin n (step0 cf tcopy_begin)"
 using assms
 unfolding tcopy_begin_def
 apply(case_tac cf)
@@ -94,9 +92,9 @@
 done
 
 lemma inv_begin_steps: 
-  assumes "inv_begin x cf"
-  and "x > 0"
-  shows "inv_begin x (steps0 cf tcopy_begin stp)"
+  assumes "inv_begin n cf"
+  and "n > 0"
+  shows "inv_begin n (steps0 cf tcopy_begin stp)"
 apply(induct stp)
 apply(simp add: assms)
 apply(auto simp del: steps.simps)
@@ -104,23 +102,22 @@
 apply(simp_all add: assms)
 done
 
-(*
 lemma begin_partial_correctness:
-  assumes "\<exists>stp. is_final (steps0 (1, [], Oc \<up> n) tcopy_begin stp)"
+  assumes "is_final (steps0 (1, [], Oc \<up> n) tcopy_begin stp)"
   shows "0 < n \<Longrightarrow> {inv_begin1 n} tcopy_begin {inv_begin0 n}"
-using assms 
-apply(auto simp add: Hoare_halt_def)
-apply(rule_tac x="stp" in exI)
-apply(simp)
-apply(case_tac "steps0 (Suc 0, [], Oc \<up> n) tcopy_begin stp")
-apply(simp)
-apply(case_tac n)
-apply(simp)
-apply(simp)
-apply(case_tac nat)
-apply(simp)
-apply(simp)
-*)
+proof(rule_tac Hoare_haltI)
+  fix l r
+  assume h: "0 < n" "inv_begin1 n (l, r)"
+  have "inv_begin n (steps0 (1, [], Oc \<up> n) tcopy_begin stp)"
+    using h by (rule_tac inv_begin_steps) (simp_all add: inv_begin.simps)
+  then show
+    "\<exists>stp. is_final (steps0 (1, l, r) tcopy_begin stp) \<and> 
+    inv_begin0 n holds_for steps (1, l, r) (tcopy_begin, 0) stp"
+    using h assms
+    apply(rule_tac x = stp in exI)
+    apply(case_tac "(steps0 (1, [], Oc \<up> n) tcopy_begin stp)", simp add: inv_begin.simps)
+    done
+qed
 
 fun measure_begin_state :: "config \<Rightarrow> nat"
   where
@@ -135,69 +132,44 @@
          else 0)"
 
 definition
-  "begin_LE = measures [measure_begin_state, measure_begin_step]"
+  "measure_begin = measures [measure_begin_state, measure_begin_step]"
 
-lemma [simp]: "\<lbrakk>tl r = []; r \<noteq> []; r \<noteq> [Bk]\<rbrakk> \<Longrightarrow> r = [Oc]"
-by (case_tac r, auto, case_tac a, auto)
+lemma wf_measure_begin:
+  shows "wf measure_begin" 
+unfolding measure_begin_def 
+by auto
 
-
-lemma halt_lemma: 
-  "\<lbrakk>wf LE; \<forall>n. (\<not> P (f n) \<longrightarrow> (f (Suc n), (f n)) \<in> LE)\<rbrakk> \<Longrightarrow> \<exists>n. P (f n)"
+lemma measure_begin_induct [case_names Step]: 
+  "\<lbrakk>\<And>n. \<not> P (f n) \<Longrightarrow> (f (Suc n), (f n)) \<in> measure_begin\<rbrakk> \<Longrightarrow> \<exists>n. P (f n)"
+using wf_measure_begin
 by (metis wf_iff_no_infinite_down_chain)
 
-lemma begin_halt: 
-  "x > 0 \<Longrightarrow> \<exists> stp. is_final (steps0 (1, [], Oc \<up> x) tcopy_begin stp)"
-proof(rule_tac LE = begin_LE in halt_lemma) 
-  show "wf begin_LE" unfolding begin_LE_def by (auto) 
-next
-  assume h: "0 < x"
-  show "\<forall>n. \<not> is_final (steps (1, [], Oc \<up> x) (tcopy_begin, 0) n) \<longrightarrow>
-        (steps (1, [], Oc \<up> x) (tcopy_begin, 0) (Suc n), 
-            steps (1, [], Oc \<up> x) (tcopy_begin, 0) n) \<in> begin_LE"
-  proof(rule_tac allI, rule_tac impI)
-    fix n
-    assume a: "\<not> is_final (steps (1, [], Oc \<up> x) (tcopy_begin, 0) n)"
-    have b: "inv_begin x (steps (1, [], Oc \<up> x) (tcopy_begin, 0) n)"
-      apply(rule_tac inv_begin_steps)
-      apply(simp_all add: inv_begin.simps h)
-      done 
-    obtain s l r where c: "(steps (1, [], Oc \<up> x) (tcopy_begin, 0) n) = (s, l, r)"
-      apply(case_tac "steps (1, [], Oc \<up> x) (tcopy_begin, 0) n", auto)
-      done
-    moreover hence "inv_begin x (s, l, r)" 
-      using c b by simp
-    ultimately show "(steps (1, [], Oc \<up> x) (tcopy_begin, 0) (Suc n), 
-      steps (1, [], Oc \<up> x) (tcopy_begin, 0) n) \<in> begin_LE"
-      using a 
-    proof(simp del: inv_begin.simps step.simps steps.simps)
-      assume "inv_begin x (s, l, r)" "0 < s"
-      thus "(step (s, l, r) (tcopy_begin, 0), s, l, r) \<in> begin_LE"
-        apply(auto simp: begin_LE_def step.simps tcopy_begin_def numeral split: if_splits)
-        apply(case_tac r, auto, case_tac a, auto)
-        done
-    qed
-  qed
+lemma begin_halts: 
+  assumes h: "x > 0"
+  shows "\<exists> stp. is_final (steps0 (1, [], Oc \<up> x) tcopy_begin stp)"
+proof (induct rule: measure_begin_induct) 
+  case (Step n)
+  have "\<not> is_final (steps0 (1, [], Oc \<up> x) tcopy_begin n)" by fact
+  moreover
+  have "inv_begin x (steps0 (1, [], Oc \<up> x) tcopy_begin n)"
+    by (rule_tac inv_begin_steps) (simp_all add: inv_begin.simps h)
+  moreover
+  obtain s l r where eq: "(steps0 (1, [], Oc \<up> x) tcopy_begin n) = (s, l, r)"
+    by (metis measure_begin_state.cases)
+  ultimately 
+  have "(step0 (s, l, r) tcopy_begin, s, l, r) \<in> measure_begin"
+    apply(auto simp: measure_begin_def tcopy_begin_def numeral split: if_splits)
+    apply(subgoal_tac "r = [Oc]")
+    apply(auto)
+    by (metis cell.exhaust list.exhaust tl.simps(2))
+  then 
+  show "(steps0 (1, [], Oc \<up> x) tcopy_begin (Suc n), steps0 (1, [], Oc \<up> x) tcopy_begin n) \<in> measure_begin"
+    using eq by (simp only: step_red)
 qed
-    
+
 lemma begin_correct: 
-  "0 < n \<Longrightarrow> {inv_begin1 n} tcopy_begin {inv_begin0 n}"
-proof(rule_tac Hoare_haltI)
-  fix l r
-  assume h: "0 < n" "inv_begin1 n (l, r)"
-  then have "\<exists> stp. is_final (steps0 (1, [], Oc \<up> n) tcopy_begin stp)"
-    by (rule_tac begin_halt)    
-  then obtain stp where "is_final (steps0 (1, [], Oc \<up> n) tcopy_begin stp)" ..
-  moreover have "inv_begin n (steps0 (1, [], Oc \<up> n) tcopy_begin stp)"
-    apply(rule_tac inv_begin_steps)
-    using h by(simp_all add: inv_begin.simps)
-  ultimately show
-    "\<exists>stp. is_final (steps0 (1, l, r) tcopy_begin stp) \<and> 
-    inv_begin0 n holds_for steps (1, l, r) (tcopy_begin, 0) stp"
-    using h
-    apply(rule_tac x = stp in exI)
-    apply(case_tac "(steps0 (1, [], Oc \<up> n) tcopy_begin stp)", simp add: inv_begin.simps)
-    done
-qed
+  shows "0 < n \<Longrightarrow> {inv_begin1 n} tcopy_begin {inv_begin0 n}"
+using begin_partial_correctness begin_halts by blast
 
 declare tm_comp.simps [simp del] 
 declare adjust.simps[simp del] 
@@ -206,7 +178,6 @@
 declare step.simps[simp del]
 declare steps.simps[simp del]
 
-
 (* tcopy_loop *)
 
 fun 
@@ -217,8 +188,8 @@
   inv_loop6_loop :: "nat \<Rightarrow> tape \<Rightarrow> bool" and
   inv_loop6_exit :: "nat \<Rightarrow> tape \<Rightarrow> bool"
 where
-  "inv_loop1_loop x (l, r) = (\<exists> i j. i + j + 1 = x \<and> (l, r) = (Oc\<up>i, Oc#Oc#Bk\<up>j @ Oc\<up>j) \<and> j > 0)"
-| "inv_loop1_exit x (l, r) = ((l, r) = ([], Bk#Oc#Bk\<up>x @ Oc\<up>x) \<and> x > 0)"
+  "inv_loop1_loop n (l, r) = (\<exists> i j. i + j + 1 = n \<and> (l, r) = (Oc\<up>i, Oc#Oc#Bk\<up>j @ Oc\<up>j) \<and> j > 0)"
+| "inv_loop1_exit n (l, r) = (0 < n \<and> (l, r) = ([], Bk#Oc#Bk\<up>n @ Oc\<up>n))"
 | "inv_loop5_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, r) = (Oc\<up>k@Bk\<up>j@Oc\<up>i, Oc\<up>t))"
 | "inv_loop5_exit x (l, r) = 
@@ -237,15 +208,15 @@
   inv_loop5 :: "nat \<Rightarrow> tape \<Rightarrow> bool" and
   inv_loop6 :: "nat \<Rightarrow> tape \<Rightarrow> bool"
 where
-  "inv_loop0 x (l, r) =  ((l, r) = ([Bk], Oc # Bk\<up>x @ Oc\<up>x) \<and> x > 0)"
-| "inv_loop1 x (l, r) = (inv_loop1_loop x (l, r) \<or> inv_loop1_exit x (l, r))"
-| "inv_loop2 x (l, r) = (\<exists> i j any. i + j = x \<and> x > 0 \<and> i > 0 \<and> j > 0 \<and> (l, r) = (Oc\<up>i, any#Bk\<up>j@Oc\<up>j))"
-| "inv_loop3 x (l, r) = 
-     (\<exists> i j k t. i + j = x \<and> i > 0 \<and> j > 0 \<and>  k + t = Suc j \<and> (l, r) = (Bk\<up>k@Oc\<up>i, Bk\<up>t@Oc\<up>j))"
-| "inv_loop4 x (l, r) = 
-     (\<exists> i j k t. i + j = x \<and> i > 0 \<and> j > 0 \<and>  k + t = j \<and> (l, r) = (Oc\<up>k @ Bk\<up>(Suc j)@Oc\<up>i, Oc\<up>t))"
-| "inv_loop5 x (l, r) = (inv_loop5_loop x (l, r) \<or> inv_loop5_exit x (l, r))"
-| "inv_loop6 x (l, r) = (inv_loop6_loop x (l, r) \<or> inv_loop6_exit x (l, r))"
+  "inv_loop0 n (l, r) =  (0 < n \<and> (l, r) = ([Bk], Oc # Bk\<up>n @ Oc\<up>n))"
+| "inv_loop1 n (l, r) = (inv_loop1_loop n (l, r) \<or> inv_loop1_exit n (l, r))"
+| "inv_loop2 n (l, r) = (\<exists> i j any. i + j = n \<and> n > 0 \<and> i > 0 \<and> j > 0 \<and> (l, r) = (Oc\<up>i, any#Bk\<up>j@Oc\<up>j))"
+| "inv_loop3 n (l, r) = 
+     (\<exists> i j k t. i + j = n \<and> i > 0 \<and> j > 0 \<and>  k + t = Suc j \<and> (l, r) = (Bk\<up>k@Oc\<up>i, Bk\<up>t@Oc\<up>j))"
+| "inv_loop4 n (l, r) = 
+     (\<exists> i j k t. i + j = n \<and> i > 0 \<and> j > 0 \<and>  k + t = j \<and> (l, r) = (Oc\<up>k @ Bk\<up>(Suc j)@Oc\<up>i, Oc\<up>t))"
+| "inv_loop5 n (l, r) = (inv_loop5_loop n (l, r) \<or> inv_loop5_exit n (l, r))"
+| "inv_loop6 n (l, r) = (inv_loop6_loop n (l, r) \<or> inv_loop6_exit n (l, r))"
 
 fun inv_loop :: "nat \<Rightarrow> config \<Rightarrow> bool"
   where
@@ -464,8 +435,7 @@
 done
 
 lemma inv_loop_steps:
-  "\<lbrakk>inv_loop x cf; x > 0\<rbrakk>
-  \<Longrightarrow> inv_loop x (steps cf (tcopy_loop, 0) stp)"
+  "\<lbrakk>inv_loop x cf; x > 0\<rbrakk> \<Longrightarrow> inv_loop x (steps cf (tcopy_loop, 0) stp)"
 apply(induct stp, simp add: steps.simps, simp)
 apply(erule_tac inv_loop_step, simp)
 done
@@ -489,14 +459,21 @@
                           else if s = 6 then length l
                           else 0)"
 
-definition loop_LE :: "(config \<times> config) set"
+definition measure_loop :: "(config \<times> config) set"
   where
-   "loop_LE = measures [loop_stage, loop_state, loop_step]"
+   "measure_loop = measures [loop_stage, loop_state, loop_step]"
 
-lemma wf_loop_le: "wf loop_LE"
-unfolding loop_LE_def
+lemma wf_measure_loop: "wf measure_loop"
+unfolding measure_loop_def
 by (auto)
 
+lemma measure_loop_induct [case_names Step]: 
+  "\<lbrakk>\<And>n. \<not> P (f n) \<Longrightarrow> (f (Suc n), (f n)) \<in> measure_loop\<rbrakk> \<Longrightarrow> \<exists>n. P (f n)"
+using wf_measure_loop
+by (metis wf_iff_no_infinite_down_chain)
+
+
+
 lemma [simp]: "inv_loop2 x ([], b) = False"
 by (auto simp: inv_loop2.simps)
 
@@ -509,6 +486,7 @@
 lemma [simp]: "inv_loop4 x ([], b) = False"
 by (auto simp: inv_loop4.simps)
 
+
 lemma [elim]: 
   "\<lbrakk>inv_loop4 x (l', []); l' \<noteq> []; x > 0;
   length (takeWhile (\<lambda>a. a = Oc) (rev l' @ [Oc])) \<noteq> 
@@ -581,7 +559,6 @@
 apply(case_tac l', auto)
 done
 
-
 lemma[elim]: 
   "\<lbrakk>inv_loop6 x (l', Oc # list); l' \<noteq> []; 0 < x;
   length (takeWhile (\<lambda>a. a = Oc) (rev (tl l') @ hd l' # Oc # list))
@@ -592,63 +569,58 @@
 apply(auto simp: inv_loop6.simps)
 done
 
-lemma loop_halt: 
-  "\<lbrakk>x > 0; inv_loop x (Suc 0, l, r)\<rbrakk> \<Longrightarrow> 
-      \<exists> stp. is_final (steps (Suc 0, l, r) (tcopy_loop, 0) stp)"
-proof(rule_tac LE = loop_LE in halt_lemma)
-  show "wf loop_LE" by(intro wf_loop_le)
-next
-  assume h: "0 < x"  and g: "inv_loop x (Suc 0, l, r)"
-  show "\<forall>n. \<not> is_final (steps (Suc 0, l, r) (tcopy_loop, 0) n) \<longrightarrow> 
-        (steps (Suc 0, l, r) (tcopy_loop, 0) (Suc n), steps (Suc 0, l, r) (tcopy_loop, 0) n) \<in> loop_LE"
-  proof(rule_tac allI, rule_tac impI)
-    fix n
-    assume a: "\<not> is_final (steps (Suc 0, l, r) (tcopy_loop, 0) n)"
-    obtain s' l' r' where d: "(steps (Suc 0, l, r) (tcopy_loop, 0) n) = (s', l', r')"
-      by(case_tac "(steps (Suc 0, l, r) (tcopy_loop, 0) n)", auto)
-    hence "inv_loop x (s', l', r') \<and> s' \<noteq> 0"
-      using h g
-      apply(drule_tac stp = n in inv_loop_steps, auto)
-      using a
-      apply(simp)
-      done
-    hence "(step(s', l', r') (tcopy_loop, 0), s', l', r') \<in> loop_LE"
-      using h 
-      apply(case_tac r', case_tac [2] a)
-      apply(auto simp: inv_loop.simps step.simps tcopy_loop_def numeral loop_LE_def split: if_splits)
-      done
-    thus "(steps (Suc 0, l, r) (tcopy_loop, 0) (Suc n), 
-          steps (Suc 0, l, r) (tcopy_loop, 0) n) \<in> loop_LE"
-      using d
-      apply(simp add: step_red)
-      done
-  qed
+lemma loop_halts: 
+  assumes h: "n > 0" "inv_loop n (1, l, r)"
+  shows "\<exists> stp. is_final (steps0 (1, l, r) tcopy_loop stp)"
+proof (induct rule: measure_loop_induct) 
+  case (Step stp)
+  have "\<not> is_final (steps0 (1, l, r) tcopy_loop stp)" by fact
+  moreover
+  have "inv_loop n (steps0 (1, l, r) tcopy_loop stp)"
+    by (rule_tac inv_loop_steps) (simp_all only: h)
+  moreover
+  obtain s l' r' where eq: "(steps0 (1, l, r) tcopy_loop stp) = (s, l', r')"
+    by (metis measure_begin_state.cases)
+  ultimately 
+  have "(step0 (s, l', r') tcopy_loop, s, l', r') \<in> measure_loop"
+    using h(1)
+    apply(case_tac r')
+    apply(case_tac [2] a)
+    apply(auto simp: inv_loop.simps step.simps tcopy_loop_def numeral measure_loop_def split: if_splits)
+    done
+  then 
+  show "(steps0 (1, l, r) tcopy_loop (Suc stp), steps0 (1, l, r) tcopy_loop stp) \<in> measure_loop"
+    using eq by (simp only: step_red)
 qed
 
 lemma loop_correct:
-  "x > 0 \<Longrightarrow> {inv_loop1 x} tcopy_loop {inv_loop0 x}"
+  shows "0 < n \<Longrightarrow> {inv_loop1 n} tcopy_loop {inv_loop0 n}"
+  using assms
 proof(rule_tac Hoare_haltI)
   fix l r
-  assume h: "0 < x"
-    "inv_loop1 x (l, r)"
-  hence "\<exists> stp. is_final (steps (Suc 0, l, r) (tcopy_loop, 0) stp)"
-    apply(rule_tac loop_halt, simp_all add: inv_loop.simps)
+  assume h: "0 < n" "inv_loop1 n (l, r)"
+  then obtain stp where k: "is_final (steps0 (1, l, r) tcopy_loop stp)" 
+    using loop_halts
+    apply(simp add: inv_loop.simps)
+    apply(blast)
     done
-  then obtain stp where "is_final (steps (Suc 0, l, r) (tcopy_loop, 0) stp)" ..
-  moreover have "inv_loop x (steps (Suc 0, l, r) (tcopy_loop, 0) stp)"
-    apply(rule_tac inv_loop_steps)
-    using h by(simp_all add: inv_loop.simps)
+  moreover
+  have "inv_loop n (steps0 (1, l, r) tcopy_loop stp)"
+    using h 
+    by (rule_tac inv_loop_steps) (simp_all add: inv_loop.simps)
   ultimately show
-    "\<exists>n. is_final (steps (1, l, r) (tcopy_loop, 0) n) \<and> 
-    inv_loop0 x holds_for steps (1, l, r) (tcopy_loop, 0) n"    
-    using h
+    "\<exists>stp. is_final (steps0 (1, l, r) tcopy_loop stp) \<and> 
+    inv_loop0 n holds_for steps0 (1, l, r) tcopy_loop stp"
+    using h(1) 
     apply(rule_tac x = stp in exI)
-    apply(case_tac "(steps (Suc 0, l, r) (tcopy_loop, 0) stp)", 
-      simp add: inv_begin.simps inv_loop.simps)
+    apply(case_tac "(steps0 (1, l, r) tcopy_loop stp)")
+    apply(simp add: inv_loop.simps)
     done
 qed
 
 
+
+
 (* tcopy_end *)
 
 fun
@@ -664,27 +636,26 @@
   inv_end1 :: "nat \<Rightarrow> tape \<Rightarrow> bool" and
   inv_end2 :: "nat \<Rightarrow> tape \<Rightarrow> bool" and
   inv_end3 :: "nat \<Rightarrow> tape \<Rightarrow> bool" and
-  inv_end4 :: "nat \<Rightarrow> tape \<Rightarrow> bool" and
- 
+  inv_end4 :: "nat \<Rightarrow> tape \<Rightarrow> bool" and 
   inv_end5 :: "nat \<Rightarrow> tape \<Rightarrow> bool" 
 where
-  "inv_end0 x (l, r) = (x > 0 \<and> l = [Bk] \<and> r = Oc\<up>x @ Bk # Oc\<up>x)"
-| "inv_end1 x (l, r) = (x > 0 \<and> l = [Bk] \<and> r = Oc # Bk\<up>x @ Oc\<up>x)"
-| "inv_end2 x (l, r) = (\<exists> i j. i + j = Suc x \<and> x > 0 \<and> l = Oc\<up>i @ [Bk] \<and> r = Bk\<up>j @ Oc\<up>x)"
-| "inv_end3 x (l, r) =
-     (\<exists> i j. x > 0 \<and> i + j = x \<and> l = Oc\<up>i @ [Bk] \<and> r = Oc # Bk\<up>j@ Oc\<up>x)"
-| "inv_end4 x (l, r) = (\<exists> any. x > 0 \<and> l = Oc\<up>x @ [Bk] \<and> r = any#Oc\<up>x)"
-| "inv_end5 x (l, r) = (inv_end5_loop x (l, r) \<or> inv_end5_exit x (l, r))"
+  "inv_end0 n (l, r) = (n > 0 \<and> (l, r) = ([Bk], Oc\<up>n @ Bk # Oc\<up>n))"
+| "inv_end1 n (l, r) = (n > 0 \<and> (l, r) = ([Bk], Oc # Bk\<up>n @ Oc\<up>n))"
+| "inv_end2 n (l, r) = (\<exists> i j. i + j = Suc n \<and> n > 0 \<and> l = Oc\<up>i @ [Bk] \<and> r = Bk\<up>j @ Oc\<up>n)"
+| "inv_end3 n (l, r) =
+     (\<exists> i j. n > 0 \<and> i + j = n \<and> l = Oc\<up>i @ [Bk] \<and> r = Oc # Bk\<up>j@ Oc\<up>n)"
+| "inv_end4 n (l, r) = (\<exists> any. n > 0 \<and> l = Oc\<up>n @ [Bk] \<and> r = any#Oc\<up>n)"
+| "inv_end5 n (l, r) = (inv_end5_loop n (l, r) \<or> inv_end5_exit n (l, r))"
 
 fun 
   inv_end :: "nat \<Rightarrow> config \<Rightarrow> bool"
 where
-  "inv_end x (s, l, r) = (if s = 0 then inv_end0 x (l, r)
-                          else if s = 1 then inv_end1 x (l, r)
-                          else if s = 2 then inv_end2 x (l, r)
-                          else if s = 3 then inv_end3 x (l, r)
-                          else if s = 4 then inv_end4 x (l, r)
-                          else if s = 5 then inv_end5 x (l, r)
+  "inv_end n (s, l, r) = (if s = 0 then inv_end0 n (l, r)
+                          else if s = 1 then inv_end1 n (l, r)
+                          else if s = 2 then inv_end2 n (l, r)
+                          else if s = 3 then inv_end3 n (l, r)
+                          else if s = 4 then inv_end4 n (l, r)
+                          else if s = 5 then inv_end5 n (l, r)
                           else False)"
 
 declare inv_end.simps[simp del] inv_end1.simps[simp del]
@@ -843,6 +814,10 @@
 lemma [simp]: "inv_end5 x ([], Oc # list) = False"
 by (auto simp: inv_end5.simps inv_end5_loop.simps)
 
+lemma halt_lemma: 
+  "\<lbrakk>wf LE; \<forall>n. (\<not> P (f n) \<longrightarrow> (f (Suc n), (f n)) \<in> LE)\<rbrakk> \<Longrightarrow> \<exists>n. P (f n)"
+by (metis wf_iff_no_infinite_down_chain)
+
 lemma end_halt: 
   "\<lbrakk>x > 0; inv_end x (Suc 0, l, r)\<rbrakk> \<Longrightarrow> 
       \<exists> stp. is_final (steps (Suc 0, l, r) (tcopy_end, 0) stp)"
@@ -875,20 +850,20 @@
 qed
 
 lemma end_correct:
-  "x > 0 \<Longrightarrow> {inv_end1 x} tcopy_end {inv_end0 x}"
+  "n > 0 \<Longrightarrow> {inv_end1 n} tcopy_end {inv_end0 n}"
 proof(rule_tac Hoare_haltI)
   fix l r
-  assume h: "0 < x"
-    "inv_end1 x (l, r)"
+  assume h: "0 < n"
+    "inv_end1 n (l, r)"
   then have "\<exists> stp. is_final (steps0 (1, l, r) tcopy_end stp)"
     by (simp add: end_halt inv_end.simps)
   then obtain stp where "is_final (steps0 (1, l, r) tcopy_end stp)" ..
-  moreover have "inv_end x (steps0 (1, l, r) tcopy_end stp)"
+  moreover have "inv_end n (steps0 (1, l, r) tcopy_end stp)"
     apply(rule_tac inv_end_steps)
     using h by(simp_all add: inv_end.simps)
   ultimately show
-    "\<exists>n. is_final (steps (1, l, r) (tcopy_end, 0) n) \<and> 
-    inv_end0 x holds_for steps (1, l, r) (tcopy_end, 0) n"        
+    "\<exists>stp. is_final (steps (1, l, r) (tcopy_end, 0) stp) \<and> 
+    inv_end0 n holds_for steps (1, l, r) (tcopy_end, 0) stp"        
     using h
     apply(rule_tac x = stp in exI)
     apply(case_tac "(steps0 (1, l, r) tcopy_end stp)") 
@@ -940,10 +915,10 @@
     by (rule_tac Hoare_plus_halt) (auto)
 qed
 
-abbreviation
-  "pre_tcopy n \<equiv> \<lambda>tp. tp = ([]::cell list, <[n::nat]>)"
-abbreviation
-  "post_tcopy n \<equiv> \<lambda>tp. tp= ([Bk], <[n, n::nat]>)"
+abbreviation (input)
+  "pre_tcopy n \<equiv> \<lambda>tp. tp = ([]::cell list, <(n::nat)>)"
+abbreviation (input)
+  "post_tcopy n \<equiv> \<lambda>tp. tp= ([Bk], <(n, n::nat)>)"
 
 lemma tcopy_correct:
   shows "{pre_tcopy n} tcopy {post_tcopy n}"
@@ -955,7 +930,7 @@
     by (auto simp add: tape_of_nl_abv tape_of_nat_abv)
   moreover
   have "inv_end0 (Suc n) = post_tcopy n" 
-    by (auto simp add: inv_end0.simps tape_of_nat_abv tape_of_nl_abv)
+    by (auto simp add: inv_end0.simps tape_of_nat_abv tape_of_nat_pair)
   ultimately
   show "{pre_tcopy n} tcopy {post_tcopy n}" 
     by simp
@@ -1044,30 +1019,30 @@
   The following two assumptions specifies that @{text "H"} does solve the Halting problem.
   *}
   and h_case: 
-  "\<And> M ns. haltP M ns \<Longrightarrow> {(\<lambda>tp. tp = ([Bk], <code M#ns>))} H {(\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <0::nat>))}"
+  "\<And> M ns. haltP M ns \<Longrightarrow> {(\<lambda>tp. tp = ([Bk], <(code M, ns)>))} H {(\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <0::nat>))}"
   and nh_case: 
-  "\<And> M ns. \<not> haltP M ns \<Longrightarrow> {(\<lambda>tp. tp = ([Bk], <code M#ns>))} H {(\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1::nat>))}"
+  "\<And> M ns. \<not> haltP M ns \<Longrightarrow> {(\<lambda>tp. tp = ([Bk], <(code M, ns)>))} H {(\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1::nat>))}"
 begin
 
 (* invariants for H *)
-abbreviation
-  "pre_H_inv M n \<equiv> \<lambda>tp. tp = ([Bk], <[code M, n]>)"
+abbreviation (input)
+  "pre_H_inv M ns \<equiv> \<lambda>tp. tp = ([Bk], <(code M, ns::nat list)>)"
 
-abbreviation
+abbreviation (input)
   "post_H_halt_inv \<equiv> \<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1::nat>)"
 
-abbreviation
+abbreviation (input)
   "post_H_unhalt_inv \<equiv> \<lambda>tp. \<exists>k. tp = (Bk \<up> k, <0::nat>)"
 
 
 lemma H_halt_inv:
-  assumes "\<not> haltP M [n]" 
-  shows "{pre_H_inv M n} H {post_H_halt_inv}"
+  assumes "\<not> haltP M ns" 
+  shows "{pre_H_inv M ns} H {post_H_halt_inv}"
 using assms nh_case by auto
 
 lemma H_unhalt_inv:
-  assumes "haltP M [n]" 
-  shows "{pre_H_inv M n} H {post_H_unhalt_inv}"
+  assumes "haltP M ns" 
+  shows "{pre_H_inv M ns} H {post_H_unhalt_inv}"
 using assms h_case by auto
    
 (* TM that produces the contradiction and its code *)
@@ -1083,8 +1058,8 @@
   shows "False"
 proof -
   (* invariants *)
-  def P1 \<equiv> "\<lambda>tp. tp = ([]::cell list, <[code_tcontra]>)"
-  def P2 \<equiv> "\<lambda>tp. tp = ([Bk], <[code_tcontra, code_tcontra]>)"
+  def P1 \<equiv> "\<lambda>tp. tp = ([]::cell list, <code_tcontra>)"
+  def P2 \<equiv> "\<lambda>tp. tp = ([Bk], <(code_tcontra, code_tcontra)>)"
   def P3 \<equiv> "\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1::nat>)"
 
   (*
@@ -1106,8 +1081,9 @@
   next
     case B_halt (* of H *)
     show "{P2} H {P3}"
-      unfolding P2_def P3_def
-      by (rule H_halt_inv[OF assms])
+      unfolding P2_def P3_def 
+      using H_halt_inv[OF assms]
+      by (simp add: tape_of_nat_pair tape_of_nl_abv)
   qed (simp)
 
   (* {P3} dither {P3} *)
@@ -1123,10 +1099,10 @@
     unfolding P1_def P3_def
     unfolding haltP_def
     unfolding Hoare_halt_def 
-    apply(auto)
+    apply(auto)    
     apply(drule_tac x = n in spec)
-    apply(case_tac "steps0 (Suc 0, [], <[code tcontra]>) tcontra n")
-    apply(auto)
+    apply(case_tac "steps0 (Suc 0, [], <code tcontra>) tcontra n")
+    apply(auto simp add: tape_of_nl_abv)
     done
 qed
 
@@ -1136,8 +1112,8 @@
   shows "False"
 proof - 
   (* invariants *)
-  def P1 \<equiv> "\<lambda>tp. tp = ([]::cell list, <[code_tcontra]>)"
-  def P2 \<equiv> "\<lambda>tp. tp = ([Bk], <[code_tcontra, code_tcontra]>)"
+  def P1 \<equiv> "\<lambda>tp. tp = ([]::cell list, <code_tcontra>)"
+  def P2 \<equiv> "\<lambda>tp. tp = ([Bk], <(code_tcontra, code_tcontra)>)"
   def Q3 \<equiv> "\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <0::nat>)"
 
   (*
@@ -1159,8 +1135,8 @@
   next
     case B_halt (* of H *)
     then show "{P2} H {Q3}"
-      unfolding P2_def Q3_def
-      by (rule H_unhalt_inv[OF assms])
+      unfolding P2_def Q3_def using H_unhalt_inv[OF assms]
+      by(simp add: tape_of_nat_pair tape_of_nl_abv)
   qed (simp)
 
   (* {P3} dither loops *)
@@ -1176,7 +1152,7 @@
     unfolding P1_def
     unfolding haltP_def
     unfolding Hoare_halt_def Hoare_unhalt_def
-    by auto
+    by (auto simp add: tape_of_nl_abv)
 qed
       
 text {*