diff -r 0b302c0b449a -r 469c26d19f8e Attic/recursive.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/recursive.thy	Wed Feb 06 02:39:53 2013 +0000
@@ -0,0 +1,5024 @@
+theory recursive
+imports Main rec_def abacus
+begin
+
+section {* 
+  Compiling from recursive functions to Abacus machines
+  *}
+
+text {*
+  Some auxilliary Abacus machines used to construct the result Abacus machines.
+*}
+
+text {*
+  @{text "get_paras_num recf"} returns the arity of recursive function @{text "recf"}.
+*}
+fun get_paras_num :: "recf \<Rightarrow> nat"
+  where
+  "get_paras_num z = 1" |
+  "get_paras_num s = 1" |
+  "get_paras_num (id m n) = m" |
+  "get_paras_num (Cn n f gs) = n" |
+  "get_paras_num (Pr n f g) = Suc n"  |
+  "get_paras_num (Mn n f) = n"  
+
+fun addition :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_prog"
+  where
+  "addition m n p = [Dec m 4, Inc n, Inc p, Goto 0, Dec p 7, 
+                       Inc m, Goto 4]"
+
+fun empty :: "nat \<Rightarrow> nat \<Rightarrow> abc_prog"
+  where
+  "empty m n = [Dec m 3, Inc n, Goto 0]"
+
+fun abc_inst_shift :: "abc_inst \<Rightarrow> nat \<Rightarrow> abc_inst"
+  where
+  "abc_inst_shift (Inc m) n = Inc m" |
+  "abc_inst_shift (Dec m e) n = Dec m (e + n)" |
+  "abc_inst_shift (Goto m) n = Goto (m + n)"
+
+fun abc_shift :: "abc_inst list \<Rightarrow> nat \<Rightarrow> abc_inst list" 
+  where
+  "abc_shift xs n = map (\<lambda> x. abc_inst_shift x n) xs" 
+
+fun abc_append :: "abc_inst list \<Rightarrow> abc_inst list \<Rightarrow> 
+                           abc_inst list" (infixl "[+]" 60)
+  where
+  "abc_append al bl = (let al_len = length al in 
+                           al @ abc_shift bl al_len)"
+
+text {*
+  The compilation of @{text "z"}-operator.
+*}
+definition rec_ci_z :: "abc_inst list"
+  where
+  "rec_ci_z \<equiv> [Goto 1]"
+
+text {*
+  The compilation of @{text "s"}-operator.
+*}
+definition rec_ci_s :: "abc_inst list"
+  where
+  "rec_ci_s \<equiv> (addition 0 1 2 [+] [Inc 1])"
+
+
+text {*
+  The compilation of @{text "id i j"}-operator
+*}
+
+fun rec_ci_id :: "nat \<Rightarrow> nat \<Rightarrow> abc_inst list"
+  where
+  "rec_ci_id i j = addition j i (i + 1)"
+
+
+fun mv_boxes :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_inst list"
+  where
+  "mv_boxes ab bb 0 = []" |
+  "mv_boxes ab bb (Suc n) = mv_boxes ab bb n [+] empty (ab + n)
+  (bb + n)"
+
+fun empty_boxes :: "nat \<Rightarrow> abc_inst list"
+  where
+  "empty_boxes 0 = []" |
+  "empty_boxes (Suc n) = empty_boxes n [+] [Dec n 2, Goto 0]"
+
+fun cn_merge_gs ::
+  "(abc_inst list \<times> nat \<times> nat) list \<Rightarrow> nat \<Rightarrow> abc_inst list"
+  where
+  "cn_merge_gs [] p = []" |
+  "cn_merge_gs (g # gs) p = 
+      (let (gprog, gpara, gn) = g in 
+         gprog [+] empty gpara p [+] cn_merge_gs gs (Suc p))"
+
+
+text {*
+  The compiler of recursive functions, where @{text "rec_ci recf"} return 
+  @{text "(ap, arity, fp)"}, where @{text "ap"} is the Abacus program, @{text "arity"} is the 
+  arity of the recursive function @{text "recf"}, 
+@{text "fp"} is the amount of memory which is going to be
+  used by @{text "ap"} for its execution. 
+*}
+
+function rec_ci :: "recf \<Rightarrow> abc_inst list \<times> nat \<times> nat"
+  where
+  "rec_ci z = (rec_ci_z, 1, 2)" |
+  "rec_ci s = (rec_ci_s, 1, 3)" |
+  "rec_ci (id m n) = (rec_ci_id m n, m, m + 2)" |
+  "rec_ci (Cn n f gs) = 
+      (let cied_gs = map (\<lambda> g. rec_ci g) (f # gs) in
+       let (fprog, fpara, fn) = hd cied_gs in 
+       let pstr = 
+        Max (set (Suc n # fn # (map (\<lambda> (aprog, p, n). n) cied_gs))) in
+       let qstr = pstr + Suc (length gs) in 
+       (cn_merge_gs (tl cied_gs) pstr [+] mv_boxes 0 qstr n [+] 
+          mv_boxes pstr 0 (length gs) [+] fprog [+] 
+            empty fpara pstr [+] empty_boxes (length gs) [+] 
+             empty pstr n [+] mv_boxes qstr 0 n, n,  qstr + n))" |
+  "rec_ci (Pr n f g) = 
+         (let (fprog, fpara, fn) = rec_ci f in 
+          let (gprog, gpara, gn) = rec_ci g in 
+          let p = Max (set ([n + 3, fn, gn])) in 
+          let e = length gprog + 7 in 
+           (empty n p [+] fprog [+] empty n (Suc n) [+] 
+               (([Dec p e] [+] gprog [+] 
+                 [Inc n, Dec (Suc n) 3, Goto 1]) @
+                     [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gprog + 4)]),
+             Suc n, p + 1))" |
+  "rec_ci (Mn n f) =
+         (let (fprog, fpara, fn) = rec_ci f in 
+          let len = length (fprog) in 
+            (fprog @ [Dec (Suc n) (len + 5), Dec (Suc n) (len + 3),
+             Goto (len + 1), Inc n, Goto 0], n, max (Suc n) fn) )"
+  by pat_completeness auto
+termination 
+proof
+term size
+  show "wf (measure size)" by auto
+next
+  fix n f gs x
+  assume "(x::recf) \<in> set (f # gs)" 
+  thus "(x, Cn n f gs) \<in> measure size"
+    by(induct gs, auto)
+next
+  fix n f g
+  show "(f, Pr n f g) \<in> measure size" by auto
+next
+  fix n f g x xa y xb ya
+  show "(g, Pr n f g) \<in> measure size" by auto
+next
+  fix n f
+  show "(f, Mn n f) \<in> measure size" by auto
+qed
+
+declare rec_ci.simps [simp del] rec_ci_s_def[simp del] 
+        rec_ci_z_def[simp del] rec_ci_id.simps[simp del]
+        mv_boxes.simps[simp del] abc_append.simps[simp del]
+        empty.simps[simp del] addition.simps[simp del]
+  
+thm rec_calc_rel.induct
+
+declare abc_steps_l.simps[simp del] abc_fetch.simps[simp del] 
+        abc_step_l.simps[simp del] 
+
+lemma abc_steps_add: 
+  "abc_steps_l (as, lm) ap (m + n) = 
+         abc_steps_l (abc_steps_l (as, lm) ap m) ap n"
+apply(induct m arbitrary: n as lm, simp add: abc_steps_l.simps)
+proof -
+  fix m n as lm
+  assume ind: 
+    "\<And>n as lm. abc_steps_l (as, lm) ap (m + n) = 
+                   abc_steps_l (abc_steps_l (as, lm) ap m) ap n"
+  show "abc_steps_l (as, lm) ap (Suc m + n) = 
+             abc_steps_l (abc_steps_l (as, lm) ap (Suc m)) ap n"
+    apply(insert ind[of as lm "Suc n"], simp)
+    apply(insert ind[of as lm "Suc 0"], simp add: abc_steps_l.simps)
+    apply(case_tac "(abc_steps_l (as, lm) ap m)", simp)
+    apply(simp add: abc_steps_l.simps)
+    apply(case_tac "abc_step_l (a, b) (abc_fetch a ap)", 
+          simp add: abc_steps_l.simps)
+    done
+qed
+
+(*lemmas: rec_ci and rec_calc_rel*)
+
+lemma rec_calc_inj_case_z: 
+  "\<lbrakk>rec_calc_rel z l x; rec_calc_rel z l y\<rbrakk> \<Longrightarrow> x = y"
+apply(auto elim: calc_z_reverse)
+done
+
+lemma  rec_calc_inj_case_s: 
+  "\<lbrakk>rec_calc_rel s l x; rec_calc_rel s l y\<rbrakk> \<Longrightarrow> x = y"
+apply(auto elim: calc_s_reverse)
+done
+
+lemma rec_calc_inj_case_id:
+  "\<lbrakk>rec_calc_rel (recf.id nat1 nat2) l x;
+    rec_calc_rel (recf.id nat1 nat2) l y\<rbrakk> \<Longrightarrow> x = y"
+apply(auto elim: calc_id_reverse)
+done
+
+lemma rec_calc_inj_case_mn:
+  assumes ind: "\<And> l x y. \<lbrakk>rec_calc_rel f l x; rec_calc_rel f l y\<rbrakk> 
+           \<Longrightarrow> x = y" 
+  and h: "rec_calc_rel (Mn n f) l x" "rec_calc_rel (Mn n f) l y"
+  shows "x = y"
+  apply(insert h)
+  apply(elim  calc_mn_reverse)
+  apply(case_tac "x > y", simp)
+  apply(erule_tac x = "y" in allE, auto)
+proof -
+  fix v va
+  assume "rec_calc_rel f (l @ [y]) 0" 
+    "rec_calc_rel f (l @ [y]) v"  
+    "0 < v"
+  thus "False"
+    apply(insert ind[of "l @ [y]" 0 v], simp)
+    done
+next
+  fix v va
+  assume 
+    "rec_calc_rel f (l @ [x]) 0" 
+    "\<forall>x<y. \<exists>v. rec_calc_rel f (l @ [x]) v \<and> 0 < v" "\<not> y < x"
+  thus "x = y"
+    apply(erule_tac x = "x" in allE)
+    apply(case_tac "x = y", auto)
+    apply(drule_tac y = v in ind, simp, simp)
+    done
+qed 
+
+lemma rec_calc_inj_case_pr: 
+  assumes f_ind: 
+  "\<And>l x y. \<lbrakk>rec_calc_rel f l x; rec_calc_rel f l y\<rbrakk> \<Longrightarrow> x = y"
+  and g_ind:
+  "\<And>x xa y xb ya l xc yb. 
+  \<lbrakk>x = rec_ci f; (xa, y) = x; (xb, ya) = y; 
+  rec_calc_rel g l xc; rec_calc_rel g l yb\<rbrakk> \<Longrightarrow> xc = yb"
+  and h: "rec_calc_rel (Pr n f g) l x" "rec_calc_rel (Pr n f g) l y"  
+  shows "x = y"
+  apply(case_tac "rec_ci f")
+proof -
+  fix a b c
+  assume "rec_ci f = (a, b, c)"
+  hence ng_ind: 
+    "\<And> l xc yb. \<lbrakk>rec_calc_rel g l xc; rec_calc_rel g l yb\<rbrakk>
+    \<Longrightarrow> xc = yb"
+    apply(insert g_ind[of "(a, b, c)" "a" "(b, c)" b c], simp)
+    done
+  from h show "x = y"
+    apply(erule_tac calc_pr_reverse, erule_tac calc_pr_reverse)
+    apply(erule f_ind, simp, simp)
+    apply(erule_tac calc_pr_reverse, simp, simp)
+  proof -
+    fix la ya ry laa yaa rya
+    assume k1:  "rec_calc_rel g (la @ [ya, ry]) x" 
+      "rec_calc_rel g (la @ [ya, rya]) y"
+      and k2: "rec_calc_rel (Pr (length la) f g) (la @ [ya]) ry"
+              "rec_calc_rel (Pr (length la) f g) (la @ [ya]) rya"
+    from k2 have "ry = rya"
+      apply(induct ya arbitrary: ry rya)
+      apply(erule_tac calc_pr_reverse, 
+        erule_tac calc_pr_reverse, simp)
+      apply(erule f_ind, simp, simp, simp)
+      apply(erule_tac calc_pr_reverse, simp)
+      apply(erule_tac rSucy = rya in calc_pr_reverse, simp, simp)
+    proof -
+      fix ya ry rya l y ryb laa yb ryc
+      assume ind:
+        "\<And>ry rya. \<lbrakk>rec_calc_rel (Pr (length l) f g) (l @ [y]) ry; 
+                   rec_calc_rel (Pr (length l) f g) (l @ [y]) rya\<rbrakk> \<Longrightarrow> ry = rya"
+        and j: "rec_calc_rel (Pr (length l) f g) (l @ [y]) ryb"
+        "rec_calc_rel g (l @ [y, ryb]) ry" 
+        "rec_calc_rel (Pr (length l) f g) (l @ [y]) ryc" 
+        "rec_calc_rel g (l @ [y, ryc]) rya"
+      from j show "ry = rya"
+	apply(insert ind[of ryb ryc], simp)
+	apply(insert ng_ind[of "l @ [y, ryc]" ry rya], simp)
+	done
+    qed 
+    from k1 and this show "x = y"
+      apply(simp)
+      apply(insert ng_ind[of "la @ [ya, rya]" x y], simp)
+      done
+  qed  
+qed
+
+lemma Suc_nth_part_eq:
+  "\<forall>k<Suc (length list). (a # xs) ! k = (aa # list) ! k
+       \<Longrightarrow> \<forall>k<(length list). (xs) ! k = (list) ! k"
+apply(rule allI, rule impI)
+apply(erule_tac x = "Suc k" in allE, simp)
+done
+
+
+lemma list_eq_intro:  
+  "\<lbrakk>length xs = length ys; \<forall> k < length xs. xs ! k = ys ! k\<rbrakk> 
+  \<Longrightarrow> xs = ys"
+apply(induct xs arbitrary: ys, simp)
+apply(case_tac ys, simp, simp)
+proof -
+  fix a xs ys aa list
+  assume ind: 
+    "\<And>ys. \<lbrakk>length list = length ys; \<forall>k<length ys. xs ! k = ys ! k\<rbrakk>
+    \<Longrightarrow> xs = ys"
+    and h: "length xs = length list" 
+    "\<forall>k<Suc (length list). (a # xs) ! k = (aa # list) ! k"
+  from h show "a = aa \<and> xs = list"
+    apply(insert ind[of list], simp)
+    apply(frule Suc_nth_part_eq, simp)
+    apply(erule_tac x = "0" in allE, simp)
+    done
+qed
+
+lemma rec_calc_inj_case_cn: 
+  assumes ind: 
+  "\<And>x l xa y.
+  \<lbrakk>x = f \<or> x \<in> set gs; rec_calc_rel x l xa; rec_calc_rel x l y\<rbrakk>
+  \<Longrightarrow> xa = y"
+  and h: "rec_calc_rel (Cn n f gs) l x" 
+         "rec_calc_rel (Cn n f gs) l y"
+  shows "x = y"
+  apply(insert h, elim  calc_cn_reverse)
+  apply(subgoal_tac "rs = rsa")
+  apply(rule_tac x = f and l = rsa and xa = x and y = y in ind, 
+        simp, simp, simp)
+  apply(intro list_eq_intro, simp, rule allI, rule impI)
+  apply(erule_tac x = k in allE, rule_tac x = k in allE, simp, simp)
+  apply(rule_tac x = "gs ! k" in ind, simp, simp, simp)
+  done
+
+lemma rec_calc_inj:
+  "\<lbrakk>rec_calc_rel f l x; 
+    rec_calc_rel f l y\<rbrakk> \<Longrightarrow> x = y"
+apply(induct f arbitrary: l x y rule: rec_ci.induct)
+apply(simp add: rec_calc_inj_case_z)
+apply(simp add: rec_calc_inj_case_s)
+apply(simp add: rec_calc_inj_case_id, simp)
+apply(erule rec_calc_inj_case_cn,simp, simp)
+apply(erule rec_calc_inj_case_pr, auto)
+apply(erule rec_calc_inj_case_mn, auto)
+done
+
+
+lemma calc_rel_reverse_ind_step_ex: 
+  "\<lbrakk>rec_calc_rel (Pr n f g) (lm @ [Suc x]) rs\<rbrakk> 
+  \<Longrightarrow> \<exists> rs. rec_calc_rel (Pr n f g) (lm @ [x]) rs"
+apply(erule calc_pr_reverse, simp, simp)
+apply(rule_tac x = rk in exI, simp)
+done
+
+lemma [simp]: "Suc x \<le> y \<Longrightarrow> Suc (y - Suc x) = y - x"
+by arith
+
+lemma calc_pr_para_not_null: 
+  "rec_calc_rel (Pr n f g) lm rs \<Longrightarrow> lm \<noteq> []"
+apply(erule calc_pr_reverse, simp, simp)
+done
+
+lemma calc_pr_less_ex: 
+ "\<lbrakk>rec_calc_rel (Pr n f g) lm rs; x \<le> last lm\<rbrakk> \<Longrightarrow> 
+ \<exists>rs. rec_calc_rel (Pr n f g) (butlast lm @ [last lm - x]) rs"
+apply(subgoal_tac "lm \<noteq> []")
+apply(induct x, rule_tac x = rs in exI, simp, simp, erule exE)
+apply(rule_tac rs = xa in calc_rel_reverse_ind_step_ex, simp)
+apply(simp add: calc_pr_para_not_null)
+done
+
+lemma calc_pr_zero_ex:
+  "rec_calc_rel (Pr n f g) lm rs \<Longrightarrow> 
+             \<exists>rs. rec_calc_rel f (butlast lm) rs"
+apply(drule_tac x = "last lm" in calc_pr_less_ex, simp,
+      erule_tac exE, simp)
+apply(erule_tac calc_pr_reverse, simp)
+apply(rule_tac x = rs in exI, simp, simp)
+done
+
+
+lemma abc_steps_ind: 
+  "abc_steps_l (as, am) ap (Suc stp) =
+          abc_steps_l (abc_steps_l (as, am) ap stp) ap (Suc 0)"
+apply(insert abc_steps_add[of as am ap stp "Suc 0"], simp)
+done
+
+lemma abc_steps_zero: "abc_steps_l asm ap 0 = asm"
+apply(case_tac asm, simp add: abc_steps_l.simps)
+done
+
+lemma abc_append_nth: 
+  "n < length ap + length bp \<Longrightarrow> 
+       (ap [+] bp) ! n =
+         (if n < length ap then ap ! n 
+          else abc_inst_shift (bp ! (n - length ap)) (length ap))"
+apply(simp add: abc_append.simps nth_append map_nth split: if_splits)
+done
+
+lemma abc_state_keep:  
+  "as \<ge> length bp \<Longrightarrow> abc_steps_l (as, lm) bp stp = (as, lm)"
+apply(induct stp, simp add: abc_steps_zero)
+apply(simp add: abc_steps_ind)
+apply(simp add: abc_steps_zero)
+apply(simp add: abc_steps_l.simps abc_fetch.simps abc_step_l.simps)
+done
+
+lemma abc_halt_equal: 
+  "\<lbrakk>abc_steps_l (0, lm) bp stpa = (length bp, lm1); 
+    abc_steps_l (0, lm) bp stpb = (length bp, lm2)\<rbrakk> \<Longrightarrow> lm1 = lm2"
+apply(case_tac "stpa - stpb > 0")
+apply(insert abc_steps_add[of 0 lm bp stpb "stpa - stpb"], simp)
+apply(insert abc_state_keep[of bp "length bp" lm2 "stpa - stpb"], 
+      simp, simp add: abc_steps_zero)
+apply(insert abc_steps_add[of 0 lm bp stpa "stpb - stpa"], simp)
+apply(insert abc_state_keep[of bp "length bp" lm1 "stpb - stpa"], 
+      simp)
+done  
+
+lemma abc_halt_point_ex: 
+  "\<lbrakk>\<exists>stp. abc_steps_l (0, lm) bp stp = (bs, lm');
+    bs = length bp; bp \<noteq> []\<rbrakk> 
+  \<Longrightarrow> \<exists> stp. (\<lambda> (s, l). s < bs \<and> 
+              (abc_steps_l (s, l) bp (Suc 0)) = (bs, lm')) 
+      (abc_steps_l (0, lm) bp stp) "
+apply(erule_tac exE)
+proof -
+  fix stp
+  assume "bs = length bp" 
+         "abc_steps_l (0, lm) bp stp = (bs, lm')" 
+         "bp \<noteq> []"
+  thus 
+    "\<exists>stp. (\<lambda>(s, l). s < bs \<and> 
+      abc_steps_l (s, l) bp (Suc 0) = (bs, lm')) 
+                       (abc_steps_l (0, lm) bp stp)"
+    apply(induct stp, simp add: abc_steps_zero, simp)
+  proof -
+    fix stpa
+    assume ind: 
+     "abc_steps_l (0, lm) bp stpa = (length bp, lm')
+       \<Longrightarrow> \<exists>stp. (\<lambda>(s, l). s < length bp  \<and> abc_steps_l (s, l) bp 
+             (Suc 0) = (length bp, lm')) (abc_steps_l (0, lm) bp stp)"
+    and h: "abc_steps_l (0, lm) bp (Suc stpa) = (length bp, lm')" 
+           "abc_steps_l (0, lm) bp stp = (length bp, lm')" 
+           "bp \<noteq> []"
+    from h show 
+      "\<exists>stp. (\<lambda>(s, l). s < length bp \<and> abc_steps_l (s, l) bp (Suc 0)
+                    = (length bp, lm')) (abc_steps_l (0, lm) bp stp)"
+      apply(case_tac "abc_steps_l (0, lm) bp stpa", 
+            case_tac "a = length bp")
+      apply(insert ind, simp)
+      apply(subgoal_tac "b = lm'", simp)
+      apply(rule_tac abc_halt_equal, simp, simp)
+      apply(rule_tac x = stpa in exI, simp add: abc_steps_ind)
+      apply(simp add: abc_steps_zero)
+      apply(rule classical, simp add: abc_steps_l.simps 
+                             abc_fetch.simps abc_step_l.simps)
+      done
+  qed
+qed  
+
+
+lemma abc_append_empty_r[simp]: "[] [+] ab = ab"
+apply(simp add: abc_append.simps abc_inst_shift.simps)
+apply(induct ab, simp, simp)
+apply(case_tac a, simp_all add: abc_inst_shift.simps)
+done
+
+lemma abc_append_empty_l[simp]:  "ab [+] [] = ab"
+apply(simp add: abc_append.simps abc_inst_shift.simps)
+done
+
+
+lemma abc_append_length[simp]:  
+  "length (ap [+] bp) = length ap + length bp"
+apply(simp add: abc_append.simps)
+done
+
+lemma abc_append_commute: "as [+] bs [+] cs = as [+] (bs [+] cs)"
+apply(simp add: abc_append.simps abc_shift.simps abc_inst_shift.simps)
+apply(induct cs, simp, simp)
+apply(case_tac a, auto simp: abc_inst_shift.simps)
+done
+
+lemma abc_halt_point_step[simp]: 
+  "\<lbrakk>a < length bp; abc_steps_l (a, b) bp (Suc 0) = (length bp, lm')\<rbrakk>
+  \<Longrightarrow> abc_steps_l (length ap + a, b) (ap [+] bp [+] cp) (Suc 0) = 
+                                        (length ap + length bp, lm')"
+apply(simp add: abc_steps_l.simps abc_fetch.simps abc_append_nth)
+apply(case_tac "bp ! a", 
+                      auto simp: abc_steps_l.simps abc_step_l.simps)
+done
+
+lemma abc_step_state_in:
+  "\<lbrakk>bs < length bp;  abc_steps_l (a, b) bp (Suc 0) = (bs, l)\<rbrakk>
+  \<Longrightarrow> a < length bp"
+apply(simp add: abc_steps_l.simps abc_fetch.simps)
+apply(rule_tac classical, 
+      simp add: abc_step_l.simps abc_steps_l.simps)
+done
+
+
+lemma abc_append_state_in_exc: 
+  "\<lbrakk>bs < length bp; abc_steps_l (0, lm) bp stpa = (bs, l)\<rbrakk>
+ \<Longrightarrow> abc_steps_l (length ap, lm) (ap [+] bp [+] cp) stpa = 
+                                             (length ap + bs, l)"
+apply(induct stpa arbitrary: bs l, simp add: abc_steps_zero)
+proof -
+  fix stpa bs l
+  assume ind: 
+    "\<And>bs l. \<lbrakk>bs < length bp; abc_steps_l (0, lm) bp stpa = (bs, l)\<rbrakk>
+    \<Longrightarrow> abc_steps_l (length ap, lm) (ap [+] bp [+] cp) stpa = 
+                                                (length ap + bs, l)"
+    and h: "bs < length bp" 
+           "abc_steps_l (0, lm) bp (Suc stpa) = (bs, l)"
+  from h show 
+    "abc_steps_l (length ap, lm) (ap [+] bp [+] cp) (Suc stpa) = 
+                                                (length ap + bs, l)"
+    apply(simp add: abc_steps_ind)
+    apply(case_tac "(abc_steps_l (0, lm) bp stpa)", simp)
+  proof -
+    fix a b
+    assume g: "abc_steps_l (0, lm) bp stpa = (a, b)" 
+              "abc_steps_l (a, b) bp (Suc 0) = (bs, l)"
+    from h and g have k1: "a < length bp"
+      apply(simp add: abc_step_state_in)
+      done
+    from h and g and k1 show 
+   "abc_steps_l (abc_steps_l (length ap, lm) (ap [+] bp [+] cp) stpa) 
+              (ap [+] bp [+] cp) (Suc 0) = (length ap + bs, l)"
+      apply(insert ind[of a b], simp)
+      apply(simp add: abc_steps_l.simps abc_fetch.simps 
+                      abc_append_nth)
+      apply(case_tac "bp ! a", auto simp: 
+                                 abc_steps_l.simps abc_step_l.simps)
+      done
+  qed
+qed
+
+lemma [simp]: "abc_steps_l (0, am) [] stp = (0, am)"
+apply(induct stp, simp add: abc_steps_zero)
+apply(simp add: abc_steps_ind)
+apply(simp add: abc_steps_zero abc_steps_l.simps 
+                abc_fetch.simps abc_step_l.simps)
+done
+
+lemma abc_append_exc1:
+  "\<lbrakk>\<exists> stp. abc_steps_l (0, lm) bp stp = (bs, lm');
+    bs = length bp; 
+    as = length ap\<rbrakk>
+    \<Longrightarrow> \<exists> stp. abc_steps_l (as, lm) (ap [+] bp [+] cp) stp 
+                                                 = (as + bs, lm')"
+apply(case_tac "bp = []", erule_tac exE, simp,
+      rule_tac x = 0 in exI, simp add: abc_steps_zero)
+apply(frule_tac abc_halt_point_ex, simp, simp,
+      erule_tac exE, erule_tac exE) 
+apply(rule_tac x = "stpa + Suc 0" in exI)
+apply(case_tac "(abc_steps_l (0, lm) bp stpa)", 
+      simp add: abc_steps_ind)
+apply(subgoal_tac 
+  "abc_steps_l (length ap, lm) (ap [+] bp [+] cp) stpa 
+                                   = (length ap + a, b)", simp)
+apply(simp add: abc_steps_zero)
+apply(rule_tac abc_append_state_in_exc, simp, simp)
+done
+
+lemma abc_append_exc3: 
+  "\<lbrakk>\<exists> stp. abc_steps_l (0, am) bp stp = (bs, bm); ss = length ap\<rbrakk>
+   \<Longrightarrow>  \<exists> stp. abc_steps_l (ss, am) (ap [+] bp) stp = (bs + ss, bm)"
+apply(erule_tac exE)
+proof -
+  fix stp
+  assume h: "abc_steps_l (0, am) bp stp = (bs, bm)" "ss = length ap"
+  thus " \<exists>stp. abc_steps_l (ss, am) (ap [+] bp) stp = (bs + ss, bm)"
+  proof(induct stp arbitrary: bs bm)
+    fix bs bm
+    assume "abc_steps_l (0, am) bp 0 = (bs, bm)"
+    thus "\<exists>stp. abc_steps_l (ss, am) (ap [+] bp) stp = (bs + ss, bm)"
+      apply(rule_tac x = 0 in exI, simp add: abc_steps_l.simps)
+      done
+  next
+    fix stp bs bm
+    assume ind: 
+      "\<And>bs bm. \<lbrakk>abc_steps_l (0, am) bp stp = (bs, bm);
+                 ss = length ap\<rbrakk> \<Longrightarrow> 
+          \<exists>stp. abc_steps_l (ss, am) (ap [+] bp) stp = (bs + ss, bm)"
+    and g: "abc_steps_l (0, am) bp (Suc stp) = (bs, bm)"
+    from g show 
+      "\<exists>stp. abc_steps_l (ss, am) (ap [+] bp) stp = (bs + ss, bm)"
+      apply(insert abc_steps_add[of 0 am bp stp "Suc 0"], simp)
+      apply(case_tac "(abc_steps_l (0, am) bp stp)", simp)
+    proof -
+      fix a b
+      assume "(bs, bm) = abc_steps_l (a, b) bp (Suc 0)" 
+             "abc_steps_l (0, am) bp (Suc stp) = 
+                       abc_steps_l (a, b) bp (Suc 0)" 
+              "abc_steps_l (0, am) bp stp = (a, b)"
+      thus "?thesis"
+	apply(insert ind[of a b], simp add: h, erule_tac exE)
+	apply(rule_tac x = "Suc stp" in exI)
+	apply(simp only: abc_steps_ind, simp add: abc_steps_zero)
+      proof -
+	fix stp
+	assume "(bs, bm) = abc_steps_l (a, b) bp (Suc 0)"
+	thus "abc_steps_l (a + length ap, b) (ap [+] bp) (Suc 0)
+                                              = (bs + length ap, bm)"
+	  apply(simp add: abc_steps_l.simps abc_steps_zero
+                          abc_fetch.simps split: if_splits)
+	  apply(case_tac "bp ! a", 
+                simp_all add: abc_inst_shift.simps abc_append_nth
+                   abc_steps_l.simps abc_steps_zero abc_step_l.simps)
+	  apply(auto)
+	  done
+      qed
+    qed
+  qed
+qed
+
+lemma abc_add_equal:
+  "\<lbrakk>ap \<noteq> []; 
+    abc_steps_l (0, am) ap astp = (a, b);
+    a < length ap\<rbrakk>
+     \<Longrightarrow> (abc_steps_l (0, am) (ap @ bp) astp) = (a, b)"
+apply(induct astp arbitrary: a b, simp add: abc_steps_l.simps, simp)
+apply(simp add: abc_steps_ind)
+apply(case_tac "(abc_steps_l (0, am) ap astp)")
+proof -
+  fix astp a b aa ba
+  assume ind: 
+    "\<And>a b. \<lbrakk>abc_steps_l (0, am) ap astp = (a, b); 
+             a < length ap\<rbrakk> \<Longrightarrow> 
+                  abc_steps_l (0, am) (ap @ bp) astp = (a, b)"
+  and h: "abc_steps_l (abc_steps_l (0, am) ap astp) ap (Suc 0)
+                                                            = (a, b)"
+        "a < length ap" 
+        "abc_steps_l (0, am) ap astp = (aa, ba)"
+  from h show "abc_steps_l (abc_steps_l (0, am) (ap @ bp) astp)
+                                        (ap @ bp) (Suc 0) = (a, b)"
+    apply(insert ind[of aa ba], simp)
+    apply(subgoal_tac "aa < length ap", simp)
+    apply(simp add: abc_steps_l.simps abc_fetch.simps
+                     nth_append abc_steps_zero)
+    apply(rule abc_step_state_in, auto)
+    done
+qed
+
+
+lemma abc_add_exc1: 
+  "\<lbrakk>\<exists> astp. abc_steps_l (0, am) ap astp = (as, bm); as = length ap\<rbrakk>
+  \<Longrightarrow> \<exists> stp. abc_steps_l (0, am) (ap @ bp) stp = (as, bm)"
+apply(case_tac "ap = []", simp, 
+      rule_tac x = 0 in exI, simp add: abc_steps_zero)
+apply(drule_tac abc_halt_point_ex, simp, simp)
+apply(erule_tac exE, case_tac "(abc_steps_l (0, am) ap astp)", simp)
+apply(rule_tac x = "Suc astp" in exI, simp add: abc_steps_ind, auto)
+apply(frule_tac bp = bp in abc_add_equal, simp, simp, simp)
+apply(simp add: abc_steps_l.simps abc_steps_zero 
+                abc_fetch.simps nth_append)
+done
+
+declare abc_shift.simps[simp del] 
+
+lemma abc_append_exc2: 
+  "\<lbrakk>\<exists> astp. abc_steps_l (0, am) ap astp = (as, bm); as = length ap; 
+    \<exists> bstp. abc_steps_l (0, bm) bp bstp = (bs, bm'); bs = length bp;
+    cs = as + bs; bp \<noteq> []\<rbrakk>
+  \<Longrightarrow> \<exists> stp. abc_steps_l (0, am) (ap [+] bp) stp = (cs, bm')"
+apply(insert abc_append_exc1[of bm bp bs bm' as ap "[]"], simp)
+apply(drule_tac bp = "abc_shift bp (length ap)" in abc_add_exc1, simp)
+apply(subgoal_tac "ap @ abc_shift bp (length ap) = ap [+] bp", 
+      simp, auto)
+apply(rule_tac x = "stpa + stp" in exI, simp add: abc_steps_add)
+apply(simp add: abc_append.simps)
+done
+lemma exp_length[simp]: "length (a\<^bsup>b\<^esup>) = b"
+by(simp add: exponent_def)
+lemma exponent_add_iff: "a\<^bsup>b\<^esup> @ a\<^bsup>c \<^esup>@ xs = a\<^bsup>b + c \<^esup>@ xs"
+apply(auto simp: exponent_def replicate_add)
+done
+lemma exponent_cons_iff: "a # a\<^bsup>c \<^esup>@ xs = a\<^bsup>Suc c \<^esup>@ xs"
+apply(auto simp: exponent_def replicate_add)
+done
+
+
+lemma  [simp]: "length lm = n \<Longrightarrow>  
+  abc_steps_l (Suc 0, lm @ Suc x # 0 # suf_lm) 
+       [Inc n, Dec (Suc n) 3, Goto (Suc 0)] (Suc (Suc 0))
+                                  = (3, lm @ Suc x # 0 # suf_lm)"
+apply(simp add: abc_steps_l.simps abc_fetch.simps 
+                abc_step_l.simps abc_lm_v.simps abc_lm_s.simps 
+                nth_append list_update_append)
+done
+
+lemma [simp]: 
+  "length lm = n \<Longrightarrow> 
+  abc_steps_l (Suc 0, lm @ Suc x # Suc y # suf_lm) 
+     [Inc n, Dec (Suc n) 3, Goto (Suc 0)] (Suc (Suc 0))
+  = (Suc 0, lm @ Suc x # y # suf_lm)"
+apply(simp add: abc_steps_l.simps abc_fetch.simps 
+                abc_step_l.simps abc_lm_v.simps abc_lm_s.simps 
+                nth_append list_update_append)
+done
+
+lemma pr_cycle_part_middle_inv: 
+  "\<lbrakk>length lm = n\<rbrakk> \<Longrightarrow> 
+  \<exists> stp. abc_steps_l (0, lm @ x # y # suf_lm) 
+                         [Inc n, Dec (Suc n) 3, Goto (Suc 0)] stp 
+  = (3, lm @ Suc x # 0 # suf_lm)"
+proof -
+  assume h: "length lm = n"
+  hence k1: "\<exists> stp. abc_steps_l (0, lm @ x # y # suf_lm) 
+                           [Inc n, Dec (Suc n) 3, Goto (Suc 0)] stp 
+    = (Suc 0, lm @ Suc x # y # suf_lm)"
+    apply(rule_tac x = "Suc 0" in exI)
+    apply(simp add: abc_steps_l.simps abc_step_l.simps 
+                    abc_lm_v.simps abc_lm_s.simps nth_append 
+                    list_update_append abc_fetch.simps)
+    done
+  from h have k2: 
+    "\<exists> stp. abc_steps_l (Suc 0, lm @ Suc x # y # suf_lm)
+                      [Inc n, Dec (Suc n) 3, Goto (Suc 0)] stp 
+    = (3, lm @ Suc x # 0 # suf_lm)"
+    apply(induct y)
+    apply(rule_tac x = "Suc (Suc 0)" in exI, simp, simp, 
+          erule_tac exE)
+    apply(rule_tac x = "Suc (Suc 0) + stp" in exI, 
+          simp only: abc_steps_add, simp)
+    done      
+  from k1 and k2 show 
+    "\<exists> stp. abc_steps_l (0, lm @ x # y # suf_lm) 
+                       [Inc n, Dec (Suc n) 3, Goto (Suc 0)] stp 
+    = (3, lm @ Suc x # 0 # suf_lm)"
+    apply(erule_tac exE, erule_tac exE)
+    apply(rule_tac x = "stp + stpa" in exI, simp add: abc_steps_add)
+    done
+qed
+
+lemma [simp]: 
+  "length lm = Suc n \<Longrightarrow> 
+  (abc_steps_l (length ap, lm @ x # Suc y # suf_lm) 
+           (ap @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length ap)]) 
+                    (Suc (Suc (Suc 0))))
+  = (length ap, lm @ Suc x # y # suf_lm)"
+apply(simp add: abc_steps_l.simps abc_fetch.simps abc_step_l.simps 
+         abc_lm_v.simps list_update_append nth_append abc_lm_s.simps)
+done
+
+lemma switch_para_inv:
+  assumes bp_def:"bp =  ap @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto ss]"
+  and h: "rec_ci (Pr n f g) = (aprog, rs_pos, a_md)" 
+         "ss = length ap" 
+         "length lm = Suc n"
+  shows " \<exists>stp. abc_steps_l (ss, lm @ x # y # suf_lm) bp stp =
+                               (0, lm @ (x + y) # 0 # suf_lm)"
+apply(induct y arbitrary: x)
+apply(rule_tac x = "Suc 0" in exI,
+  simp add: bp_def empty.simps abc_steps_l.simps 
+            abc_fetch.simps h abc_step_l.simps 
+            abc_lm_v.simps list_update_append nth_append
+            abc_lm_s.simps)
+proof -
+  fix y x
+  assume ind: 
+    "\<And>x. \<exists>stp. abc_steps_l (ss, lm @ x # y # suf_lm) bp stp = 
+                                     (0, lm @ (x + y) # 0 # suf_lm)"
+  show "\<exists>stp. abc_steps_l (ss, lm @ x # Suc y # suf_lm) bp stp = 
+                                  (0, lm @ (x + Suc y) # 0 # suf_lm)"
+    apply(insert ind[of "Suc x"], erule_tac exE)
+    apply(rule_tac x = "Suc (Suc (Suc 0)) + stp" in exI, 
+          simp only: abc_steps_add bp_def h)
+    apply(simp add: h)
+    done
+qed
+
+lemma [simp]:
+  "length lm = rs_pos \<and> Suc (Suc rs_pos) < a_md \<and> 0 < rs_pos \<Longrightarrow> 
+      a_md - Suc 0 < Suc (Suc (Suc (a_md + length suf_lm - 
+                                         Suc (Suc (Suc 0)))))"
+apply(arith)
+done
+
+lemma [simp]: 
+  "Suc (Suc rs_pos) < a_md \<and> 0 < rs_pos \<Longrightarrow> 
+                           \<not> a_md - Suc 0 < rs_pos - Suc 0"
+apply(arith)
+done
+
+lemma [simp]: 
+  "Suc (Suc rs_pos) < a_md \<and> 0 < rs_pos \<Longrightarrow> 
+           \<not> a_md - rs_pos < Suc (Suc (a_md - Suc (Suc rs_pos)))"
+apply(arith)
+done
+
+lemma butlast_append_last: "lm \<noteq> [] \<Longrightarrow> lm = butlast lm @ [last lm]"
+apply(auto)
+done
+
+lemma [simp]: "rec_ci (Pr n f g) = (aprog, rs_pos, a_md)
+           \<Longrightarrow> (Suc (Suc rs_pos)) < a_md"
+apply(simp add: rec_ci.simps)
+apply(case_tac "rec_ci f", simp)
+apply(case_tac "rec_ci g", simp)
+apply(arith)
+done
+
+(*
+lemma pr_para_ge_suc0: "rec_calc_rel (Pr n f g) lm xs \<Longrightarrow> 0 < n"
+apply(erule calc_pr_reverse, simp, simp)
+done
+*)
+
+lemma ci_pr_para_eq: "rec_ci (Pr n f g) = (aprog, rs_pos, a_md)
+                  \<Longrightarrow> rs_pos = Suc n"
+apply(simp add: rec_ci.simps)
+apply(case_tac "rec_ci g",  case_tac "rec_ci f", simp)
+done
+
+lemma [intro]:  
+  "\<lbrakk>rec_ci z = (aprog, rs_pos, a_md); rec_calc_rel z lm xs\<rbrakk>
+  \<Longrightarrow> length lm = rs_pos"
+apply(simp add: rec_ci.simps rec_ci_z_def)
+apply(erule_tac calc_z_reverse, simp)
+done
+
+lemma [intro]: 
+  "\<lbrakk>rec_ci s = (aprog, rs_pos, a_md); rec_calc_rel s lm xs\<rbrakk>
+  \<Longrightarrow> length lm = rs_pos"
+apply(simp add: rec_ci.simps rec_ci_s_def)
+apply(erule_tac calc_s_reverse, simp)
+done
+
+lemma [intro]: 
+  "\<lbrakk>rec_ci (recf.id nat1 nat2) = (aprog, rs_pos, a_md); 
+    rec_calc_rel (recf.id nat1 nat2) lm xs\<rbrakk> \<Longrightarrow> length lm = rs_pos"
+apply(simp add: rec_ci.simps rec_ci_id.simps)
+apply(erule_tac calc_id_reverse, simp)
+done
+
+lemma [intro]: 
+  "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md); 
+    rec_calc_rel (Cn n f gs) lm xs\<rbrakk> \<Longrightarrow> length lm = rs_pos"
+apply(erule_tac calc_cn_reverse, simp)
+apply(simp add: rec_ci.simps)
+apply(case_tac "rec_ci f",  simp)
+done
+
+lemma [intro]:
+  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md); 
+    rec_calc_rel (Pr n f g) lm xs\<rbrakk> \<Longrightarrow> length lm = rs_pos"
+apply(erule_tac  calc_pr_reverse, simp)
+apply(drule_tac ci_pr_para_eq, simp, simp)
+apply(drule_tac ci_pr_para_eq, simp)
+done
+
+lemma [intro]: 
+  "\<lbrakk>rec_ci (Mn n f) = (aprog, rs_pos, a_md);
+    rec_calc_rel (Mn n f) lm xs\<rbrakk> \<Longrightarrow> length lm = rs_pos"
+apply(erule_tac calc_mn_reverse)
+apply(simp add: rec_ci.simps)
+apply(case_tac "rec_ci f",  simp)
+done
+
+lemma para_pattern: 
+  "\<lbrakk>rec_ci f = (aprog, rs_pos, a_md); rec_calc_rel f lm xs\<rbrakk>
+  \<Longrightarrow> length lm = rs_pos"
+apply(case_tac f, auto)
+done
+
+lemma ci_pr_g_paras:
+  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md);
+    rec_ci g = (a, aa, ba);
+    rec_calc_rel (Pr n f g) (lm @ [x]) rs; x > 0\<rbrakk> \<Longrightarrow> 
+    aa = Suc rs_pos "
+apply(erule calc_pr_reverse, simp)
+apply(subgoal_tac "length (args @ [k, rk]) = aa", simp)
+apply(subgoal_tac "rs_pos = Suc n", simp)
+apply(simp add: ci_pr_para_eq)
+apply(erule para_pattern, simp)
+done
+
+lemma ci_pr_g_md_less: 
+  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md); 
+    rec_ci g = (a, aa, ba)\<rbrakk> \<Longrightarrow> ba < a_md"
+apply(simp add: rec_ci.simps)
+apply(case_tac "rec_ci f",  auto)
+done
+
+lemma [intro]: "rec_ci z = (ap, rp, ad) \<Longrightarrow> rp < ad"
+  by(simp add: rec_ci.simps)
+
+lemma [intro]: "rec_ci s = (ap, rp, ad) \<Longrightarrow> rp < ad"
+  by(simp add: rec_ci.simps)
+
+lemma [intro]: "rec_ci (recf.id nat1 nat2) = (ap, rp, ad) \<Longrightarrow> rp < ad"
+  by(simp add: rec_ci.simps)
+
+lemma [intro]: "rec_ci (Cn n f gs) = (ap, rp, ad) \<Longrightarrow> rp < ad"
+apply(simp add: rec_ci.simps)
+apply(case_tac "rec_ci f",  simp)
+done
+
+lemma [intro]: "rec_ci (Pr n f g) = (ap, rp, ad) \<Longrightarrow> rp < ad"
+apply(simp add: rec_ci.simps)
+by(case_tac "rec_ci f", case_tac "rec_ci g",  auto)
+
+lemma [intro]: "rec_ci (Mn n f) = (ap, rp, ad) \<Longrightarrow> rp < ad"
+apply(simp add: rec_ci.simps)
+apply(case_tac "rec_ci f", simp)
+apply(arith)
+done
+
+lemma ci_ad_ge_paras: "rec_ci f = (ap, rp, ad) \<Longrightarrow> ad > rp"
+apply(case_tac f, auto)
+done
+
+lemma [elim]: "\<lbrakk>a [+] b = []; a \<noteq> [] \<or> b \<noteq> []\<rbrakk> \<Longrightarrow> RR"
+apply(auto simp: abc_append.simps abc_shift.simps)
+done
+
+lemma [intro]: "rec_ci z = ([], aa, ba) \<Longrightarrow> False"
+by(simp add: rec_ci.simps rec_ci_z_def)
+
+lemma [intro]: "rec_ci s = ([], aa, ba) \<Longrightarrow> False"
+by(auto simp: rec_ci.simps rec_ci_s_def addition.simps)
+
+lemma [intro]: "rec_ci (id m n) = ([], aa, ba) \<Longrightarrow> False"
+by(auto simp: rec_ci.simps rec_ci_id.simps addition.simps)
+
+lemma [intro]: "rec_ci (Cn n f gs) = ([], aa, ba) \<Longrightarrow> False"
+apply(case_tac "rec_ci f", auto simp: rec_ci.simps abc_append.simps)
+apply(simp add: abc_shift.simps empty.simps)
+done
+
+lemma [intro]: "rec_ci (Pr n f g) = ([], aa, ba) \<Longrightarrow> False"
+apply(simp add: rec_ci.simps)
+apply(case_tac "rec_ci f", case_tac "rec_ci g")
+by(auto)
+
+lemma [intro]: "rec_ci (Mn n f) = ([], aa, ba) \<Longrightarrow> False"
+apply(case_tac "rec_ci f", auto simp: rec_ci.simps)
+done
+
+lemma rec_ci_not_null:  "rec_ci g = (a, aa, ba) \<Longrightarrow> a \<noteq> []"
+by(case_tac g, auto)
+
+lemma calc_pr_g_def:
+ "\<lbrakk>rec_calc_rel (Pr rs_pos f g) (lm @ [Suc x]) rsa;
+   rec_calc_rel (Pr rs_pos f g) (lm @ [x]) rsxa\<rbrakk>
+ \<Longrightarrow> rec_calc_rel g (lm @ [x, rsxa]) rsa"
+apply(erule_tac calc_pr_reverse, simp, simp)
+apply(subgoal_tac "rsxa = rk", simp)
+apply(erule_tac rec_calc_inj, auto)
+done
+
+lemma ci_pr_md_def: 
+  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md);
+    rec_ci g = (a, aa, ba); rec_ci f = (ab, ac, bc)\<rbrakk>
+  \<Longrightarrow> a_md = Suc (max (n + 3) (max bc ba))"
+by(simp add: rec_ci.simps)
+
+lemma  ci_pr_f_paras: 
+  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md);
+    rec_calc_rel (Pr n f g) lm rs;
+    rec_ci f = (ab, ac, bc)\<rbrakk>  \<Longrightarrow> ac = rs_pos - Suc 0"
+apply(subgoal_tac "\<exists>rs. rec_calc_rel f (butlast lm) rs", 
+      erule_tac exE)
+apply(drule_tac f = f and lm = "butlast lm" in para_pattern, 
+      simp, simp)
+apply(drule_tac para_pattern, simp)
+apply(subgoal_tac "lm \<noteq> []", simp)
+apply(erule_tac calc_pr_reverse, simp, simp)
+apply(erule calc_pr_zero_ex)
+done
+
+lemma ci_pr_md_ge_f:  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md);
+        rec_ci f = (ab, ac, bc)\<rbrakk> \<Longrightarrow> Suc bc \<le> a_md"
+apply(case_tac "rec_ci g")
+apply(simp add: rec_ci.simps, auto)
+done
+
+lemma ci_pr_md_ge_g:  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md);
+        rec_ci g = (ab, ac, bc)\<rbrakk> \<Longrightarrow> bc < a_md"
+apply(case_tac "rec_ci f")
+apply(simp add: rec_ci.simps, auto)
+done 
+
+lemma rec_calc_rel_def0: 
+  "\<lbrakk>rec_calc_rel (Pr n f g) lm rs; rec_calc_rel f (butlast lm) rsa\<rbrakk>
+  \<Longrightarrow> rec_calc_rel (Pr n f g) (butlast lm @ [0]) rsa"
+  apply(rule_tac calc_pr_zero, simp)
+apply(erule_tac calc_pr_reverse, simp, simp, simp)
+done
+
+lemma [simp]:  "length (empty m n) = 3"
+by (auto simp: empty.simps)
+(*
+lemma
+  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md);
+  rec_calc_rel (Pr n f g) lm rs;
+  rec_ci g = (a, aa, ba);
+  rec_ci f = (ab, ac, bc)\<rbrakk>
+\<Longrightarrow> \<exists>ap bp cp. aprog = ap [+] bp [+] cp \<and> length ap = 3 + length ab \<and> bp = recursive.empty (n - Suc 0) n 3"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "recursive.empty (n - Suc 0) (max (Suc (Suc n)) (max bc ba)) 3 [+] ab" in exI, simp)
+apply(rule_tac x = "([Dec (max (Suc (Suc n)) (max bc ba)) (length a + 7)] [+] a [+] 
+  [Inc (n - Suc 0), Dec n 3, Goto (Suc 0)]) @ [Dec (Suc n) 0, Inc n, Goto (length a + 4)]" in exI, simp)
+apply(auto simp: abc_append_commute)
+done
+
+lemma  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md);
+        rec_ci g = (a, aa, ba); rec_ci f = (ab, ac, bc)\<rbrakk>
+    \<Longrightarrow> \<exists>ap bp cp. aprog = ap [+] bp [+] cp \<and> length ap = 3 \<and> bp = ab"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "recursive.empty (n - Suc 0) (max (Suc (Suc n)) (max bc ba)) 3" in exI, simp)
+apply(rule_tac x = "recursive.empty (n - Suc 0) n 3 [+]
+     ([Dec (max (Suc (Suc n)) (max bc ba)) (length a + 7)] [+] a 
+  [+] [Inc (n - Suc 0), Dec n 3, Goto (Suc 0)]) @ [Dec (Suc n) 0, Inc n, Goto (length a + 4)]" in exI, auto)
+apply(simp add: abc_append_commute)
+done
+*)
+
+lemma [simp]: "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md); rec_calc_rel (Pr n f g) lm rs\<rbrakk>
+    \<Longrightarrow> rs_pos = Suc n"
+apply(simp add: ci_pr_para_eq)
+done
+
+
+lemma [simp]: "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md); rec_calc_rel (Pr n f g) lm rs\<rbrakk>
+    \<Longrightarrow> length lm = Suc n"
+apply(subgoal_tac "rs_pos = Suc n", rule_tac para_pattern, simp, simp)
+apply(case_tac "rec_ci f", case_tac "rec_ci g", simp add: rec_ci.simps)
+done
+
+lemma [simp]: "rec_ci (Pr n f g) = (a, rs_pos, a_md) \<Longrightarrow> Suc (Suc n) < a_md"
+apply(case_tac "rec_ci f", case_tac "rec_ci g", simp add: rec_ci.simps)
+apply arith
+done
+
+lemma [simp]: "rec_ci (Pr n f g) = (aprog, rs_pos, a_md) \<Longrightarrow> 0 < rs_pos"
+apply(case_tac "rec_ci f", case_tac "rec_ci g")
+apply(simp add: rec_ci.simps)
+done
+
+lemma [simp]: "Suc (Suc rs_pos) < a_md \<Longrightarrow> 
+       butlast lm @ (last lm - xa) # (rsa::nat) # 0 # 0\<^bsup>a_md - Suc (Suc (Suc rs_pos))\<^esup> @ xa # suf_lm =
+       butlast lm @ (last lm - xa) # rsa # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ xa # suf_lm"
+apply(simp add: exp_ind_def[THEN sym])
+done
+
+lemma pr_cycle_part_ind: 
+  assumes g_ind: 
+  "\<And>lm rs suf_lm. rec_calc_rel g lm rs \<Longrightarrow> 
+  \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>ba - aa\<^esup> @ suf_lm) a stp = 
+                    (length a, lm @ rs # 0\<^bsup>ba - Suc aa\<^esup> @ suf_lm)"
+  and ap_def: 
+  "ap = ([Dec (a_md - Suc 0) (length a + 7)] [+]
+        (a [+] [Inc (rs_pos - Suc 0), Dec rs_pos 3, Goto (Suc 0)])) @
+         [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length a + 4)]"
+  and h: "rec_ci (Pr n f g) = (aprog, rs_pos, a_md)" 
+         "rec_calc_rel (Pr n f g) 
+                   (butlast lm @ [last lm - Suc xa]) rsxa" 
+         "Suc xa \<le> last lm" 
+         "rec_ci g = (a, aa, ba)"
+         "rec_calc_rel (Pr n f g) (butlast lm @ [last lm - xa]) rsa"
+         "lm \<noteq> []"
+  shows 
+  "\<exists>stp. abc_steps_l 
+     (0, butlast lm @ (last lm - Suc xa) # rsxa # 
+               0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ Suc xa # suf_lm) ap stp =
+     (0, butlast lm @ (last lm - xa) # rsa
+                 # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ xa # suf_lm)"
+proof -
+  have k1: "\<exists>stp. abc_steps_l (0, butlast lm @ (last lm - Suc xa) #
+    rsxa # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ Suc xa # suf_lm) ap stp =
+         (length a + 4, butlast lm @ (last lm - xa) # 0 # rsa #
+                           0\<^bsup>a_md - Suc (Suc (Suc rs_pos))\<^esup> @ xa # suf_lm)"
+    apply(simp add: ap_def, rule_tac abc_add_exc1)
+    apply(rule_tac as = "Suc 0" and 
+      bm = "butlast lm @ (last lm - Suc xa) # 
+      rsxa # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ xa # suf_lm" in abc_append_exc2,
+      auto)
+  proof -
+    show 
+      "\<exists>astp. abc_steps_l (0, butlast lm @ (last lm - Suc xa) # rsxa 
+                   # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ Suc xa # suf_lm) 
+              [Dec (a_md - Suc 0)(length a + 7)] astp =
+      (Suc 0, butlast lm @ (last lm - Suc xa) # 
+             rsxa # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ xa # suf_lm)"
+      apply(rule_tac x = "Suc 0" in exI, 
+          simp add: abc_steps_l.simps abc_step_l.simps
+                     abc_fetch.simps)
+      apply(subgoal_tac "length lm = Suc n \<and> rs_pos = Suc n \<and>
+                              a_md > Suc (Suc rs_pos)")
+      apply(simp add: abc_lm_v.simps nth_append abc_lm_s.simps)
+      apply(insert nth_append[of 
+                 "(last lm - Suc xa) # rsxa # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup>" 
+                 "Suc xa # suf_lm" "(a_md - rs_pos)"], simp)
+      apply(simp add: list_update_append del: list_update.simps)
+      apply(insert list_update_append[of "(last lm - Suc xa) # rsxa # 
+                                           0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup>" 
+                    "Suc xa # suf_lm" "a_md - rs_pos" "xa"], simp)
+      apply(case_tac a_md, simp, simp)
+      apply(insert h, simp)
+      apply(insert para_pattern[of "Pr n f g" aprog rs_pos a_md 
+                    "(butlast lm @ [last lm - Suc xa])" rsxa], simp)
+      done
+  next
+    show "\<exists>bstp. abc_steps_l (0, butlast lm @ (last lm - Suc xa) # 
+           rsxa # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ xa # suf_lm) (a [+] 
+            [Inc (rs_pos - Suc 0), Dec rs_pos 3, Goto (Suc 0)]) bstp =
+         (3 + length a, butlast lm @ (last lm - xa) # 0 # rsa #
+                          0\<^bsup>a_md - Suc (Suc (Suc rs_pos))\<^esup> @ xa # suf_lm)"
+      apply(rule_tac as = "length a" and
+               bm = "butlast lm @ (last lm - Suc xa) # rsxa # rsa #
+                     0\<^bsup>a_md - Suc (Suc (Suc rs_pos))\<^esup> @ xa # suf_lm" 
+        in abc_append_exc2, simp_all)
+    proof -
+      from h have j1: "aa = Suc rs_pos \<and> a_md > ba \<and> ba > Suc rs_pos"
+	apply(insert h)
+	apply(insert ci_pr_g_paras[of n f g aprog rs_pos
+                 a_md a aa ba "butlast lm" "last lm - xa" rsa], simp)
+	apply(drule_tac ci_pr_md_ge_g, auto)
+	apply(erule_tac ci_ad_ge_paras)
+	done
+      from h have j2: "rec_calc_rel g (butlast lm @ 
+                                  [last lm - Suc xa, rsxa]) rsa"
+	apply(rule_tac  calc_pr_g_def, simp, simp)
+	done
+      from j1 and j2 show 
+        "\<exists>astp. abc_steps_l (0, butlast lm @ (last lm - Suc xa) #
+                rsxa # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ xa # suf_lm) a astp =
+        (length a, butlast lm @ (last lm - Suc xa) # rsxa # rsa 
+                         # 0\<^bsup>a_md - Suc (Suc (Suc rs_pos))\<^esup> @ xa # suf_lm)"
+	apply(insert g_ind[of
+          "butlast lm @ (last lm - Suc xa) # [rsxa]" rsa 
+          "0\<^bsup>a_md - ba - Suc 0 \<^esup> @ xa # suf_lm"], simp, auto)
+	apply(simp add: exponent_add_iff)
+	apply(rule_tac x = stp in exI, simp add: numeral_3_eq_3)
+	done
+    next
+      from h have j3: "length lm = rs_pos \<and> rs_pos > 0"
+	apply(rule_tac conjI)
+	apply(drule_tac lm = "(butlast lm @ [last lm - Suc xa])"
+                          and xs = rsxa in para_pattern, simp, simp, simp)
+        done
+      from h have j4: "Suc (last lm - Suc xa) = last lm - xa"
+	apply(case_tac "last lm", simp, simp)
+	done
+      from j3 and j4 show
+      "\<exists>bstp. abc_steps_l (0, butlast lm @ (last lm - Suc xa) # rsxa #
+                     rsa # 0\<^bsup>a_md - Suc (Suc (Suc rs_pos))\<^esup> @ xa # suf_lm)
+            [Inc (rs_pos - Suc 0), Dec rs_pos 3, Goto (Suc 0)] bstp =
+        (3, butlast lm @ (last lm - xa) # 0 # rsa #
+                       0\<^bsup>a_md - Suc (Suc (Suc rs_pos))\<^esup> @ xa # suf_lm)"
+	apply(insert pr_cycle_part_middle_inv[of "butlast lm" 
+          "rs_pos - Suc 0" "(last lm - Suc xa)" rsxa 
+          "rsa # 0\<^bsup>a_md - Suc (Suc (Suc rs_pos))\<^esup> @ xa # suf_lm"], simp)
+	done
+    qed
+  qed
+  from h have k2: 
+    "\<exists>stp. abc_steps_l (length a + 4, butlast lm @ (last lm - xa) # 0 
+           # rsa # 0\<^bsup>a_md - Suc (Suc (Suc rs_pos))\<^esup> @ xa # suf_lm) ap stp =
+    (0, butlast lm @ (last lm - xa) # rsa # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ xa # suf_lm)"
+    apply(insert switch_para_inv[of ap 
+      "([Dec (a_md - Suc 0) (length a + 7)] [+] 
+      (a [+] [Inc (rs_pos - Suc 0), Dec rs_pos 3, Goto (Suc 0)]))"
+      n "length a + 4" f g aprog rs_pos a_md 
+      "butlast lm @ [last lm - xa]" 0 rsa 
+      "0\<^bsup>a_md - Suc (Suc (Suc rs_pos))\<^esup> @ xa # suf_lm"])
+    apply(simp add: h ap_def)
+    apply(subgoal_tac "length lm = Suc n \<and> Suc (Suc rs_pos) < a_md", 
+          simp)
+    apply(insert h, simp)
+    apply(frule_tac lm = "(butlast lm @ [last lm - Suc xa])" 
+      and xs = rsxa in para_pattern, simp, simp)
+    done   
+  from k1 and k2 show "?thesis"
+    apply(auto)
+    apply(rule_tac x = "stp + stpa" in exI, simp add: abc_steps_add)
+    done
+qed
+
+lemma ci_pr_ex1: 
+  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md);
+    rec_ci g = (a, aa, ba);
+    rec_ci f = (ab, ac, bc)\<rbrakk>
+\<Longrightarrow> \<exists>ap bp. length ap = 6 + length ab \<and>
+    aprog = ap [+] bp \<and>
+    bp = ([Dec (a_md - Suc 0) (length a + 7)] [+] (a [+]
+         [Inc (rs_pos - Suc 0), Dec rs_pos 3, Goto (Suc 0)])) @ 
+         [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length a + 4)]"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "recursive.empty n (max (Suc (Suc (Suc n)))
+    (max bc ba)) [+] ab [+] recursive.empty n (Suc n)" in exI,
+     simp)
+apply(auto simp add: abc_append_commute add3_Suc)
+done
+
+lemma pr_cycle_part:
+  "\<lbrakk>\<And>lm rs suf_lm. rec_calc_rel g lm rs \<Longrightarrow>
+     \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>ba - aa\<^esup> @ suf_lm) a stp = 
+                        (length a, lm @ rs # 0\<^bsup>ba - Suc aa\<^esup> @ suf_lm);
+  rec_ci (Pr n f g) = (aprog, rs_pos, a_md); 
+  rec_calc_rel (Pr n f g) lm rs;
+  rec_ci g = (a, aa, ba);
+  rec_calc_rel (Pr n f g) (butlast lm @ [last lm - x]) rsx;
+  rec_ci f = (ab, ac, bc);
+  lm \<noteq> [];
+  x \<le> last lm\<rbrakk> \<Longrightarrow> 
+  \<exists>stp. abc_steps_l (6 + length ab, butlast lm @ (last lm - x) #
+              rsx # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ x # suf_lm) aprog stp =
+  (6 + length ab, butlast lm @ last lm # rs #
+                                0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ 0 # suf_lm)"
+proof -
+  assume g_ind:
+    "\<And>lm rs suf_lm. rec_calc_rel g lm rs \<Longrightarrow> 
+    \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>ba - aa\<^esup> @ suf_lm) a stp =
+                      (length a, lm @ rs # 0\<^bsup>ba - Suc aa\<^esup> @ suf_lm)"
+    and h: "rec_ci (Pr n f g) = (aprog, rs_pos, a_md)" 
+           "rec_calc_rel (Pr n f g) lm rs" 
+           "rec_ci g = (a, aa, ba)"
+           "rec_calc_rel (Pr n f g) (butlast lm @ [last lm - x]) rsx" 
+           "lm \<noteq> []"
+           "x \<le> last lm" 
+           "rec_ci f = (ab, ac, bc)" 
+  from h show 
+    "\<exists>stp. abc_steps_l (6 + length ab, butlast lm @ (last lm - x) # 
+            rsx # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ x # suf_lm) aprog stp =
+    (6 + length ab, butlast lm @ last lm # rs #
+                               0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ 0 # suf_lm)" 
+  proof(induct x arbitrary: rsx, simp_all)
+    fix rsxa
+    assume "rec_calc_rel (Pr n f g) lm rsxa" 
+           "rec_calc_rel (Pr n f g) lm rs"
+    from h and this have "rs = rsxa"
+      apply(subgoal_tac "lm \<noteq> [] \<and> rs_pos = Suc n", simp)
+      apply(rule_tac rec_calc_inj, simp, simp)
+      apply(simp)
+      done
+    thus "\<exists>stp. abc_steps_l (6 + length ab, butlast lm @  last lm # 
+             rsxa # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ 0 # suf_lm) aprog stp =
+      (6 + length ab, butlast lm @ last lm # rs #
+                               0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ 0 # suf_lm)"
+      by(rule_tac x = 0 in exI, simp add: abc_steps_l.simps)
+  next
+    fix xa rsxa
+    assume ind:
+   "\<And>rsx. rec_calc_rel (Pr n f g) (butlast lm @ [last lm - xa]) rsx 
+  \<Longrightarrow> \<exists>stp. abc_steps_l (6 + length ab, butlast lm @ (last lm - xa) #
+             rsx # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ xa # suf_lm) aprog stp =
+      (6 + length ab, butlast lm @ last lm # rs # 
+                               0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ 0 # suf_lm)"
+      and g: "rec_calc_rel (Pr n f g) 
+                      (butlast lm @ [last lm - Suc xa]) rsxa"
+      "Suc xa \<le> last lm"
+      "rec_ci (Pr n f g) = (aprog, rs_pos, a_md)" 
+      "rec_calc_rel (Pr n f g) lm rs"
+      "rec_ci g = (a, aa, ba)" 
+      "rec_ci f = (ab, ac, bc)" "lm \<noteq> []"
+    from g have k1: 
+      "\<exists> rs. rec_calc_rel (Pr n f g) (butlast lm @ [last lm - xa]) rs"
+      apply(rule_tac rs = rs in  calc_pr_less_ex, simp, simp)
+      done
+    from g and this show 
+      "\<exists>stp. abc_steps_l (6 + length ab, 
+           butlast lm @ (last lm - Suc xa) # rsxa # 
+              0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ Suc xa # suf_lm) aprog stp =
+              (6 + length ab, butlast lm @ last lm # rs # 
+                                0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ 0 # suf_lm)"
+    proof(erule_tac exE)
+      fix rsa
+      assume k2: "rec_calc_rel (Pr n f g) 
+                           (butlast lm @ [last lm - xa]) rsa"
+      from g and k2 have
+      "\<exists>stp. abc_steps_l (6 + length ab, butlast lm @ 
+       (last lm - Suc xa) # rsxa # 
+               0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ Suc xa # suf_lm) aprog stp
+        = (6 + length ab, butlast lm @ (last lm - xa) # rsa # 
+                               0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ xa # suf_lm)"
+	proof -
+	  from g have k2_1: 
+            "\<exists> ap bp. length ap = 6 + length ab \<and>
+                   aprog = ap [+] bp \<and> 
+                   bp = ([Dec (a_md - Suc 0) (length a + 7)] [+]
+                  (a [+] [Inc (rs_pos - Suc 0), Dec rs_pos 3, 
+                  Goto (Suc 0)])) @
+                  [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length a + 4)]"
+            apply(rule_tac ci_pr_ex1, auto)
+	    done
+	  from k2_1 and k2 and g show "?thesis"
+	    proof(erule_tac exE, erule_tac exE)
+	      fix ap bp
+	      assume 
+                "length ap = 6 + length ab \<and> 
+                 aprog = ap [+] bp \<and> bp =
+                ([Dec (a_md - Suc 0) (length a + 7)] [+] 
+                (a [+] [Inc (rs_pos - Suc 0), Dec rs_pos 3,
+                Goto (Suc 0)])) @ 
+                [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length a + 4)]" 
+	      from g and this and k2 and g_ind show "?thesis"
+		apply(insert abc_append_exc3[of 
+                  "butlast lm @ (last lm - Suc xa) # rsxa #
+                  0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ Suc xa # suf_lm" bp 0
+                  "butlast lm @ (last lm - xa) # rsa #
+                0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ xa # suf_lm" "length ap" ap],
+                 simp)
+		apply(subgoal_tac 
+                "\<exists>stp. abc_steps_l (0, butlast lm @ (last lm - Suc xa)
+                           # rsxa # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ Suc xa # 
+                              suf_lm) bp stp =
+	          (0, butlast lm @ (last lm - xa) # rsa #
+                           0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ xa # suf_lm)",
+                      simp, erule_tac conjE, erule conjE)
+		apply(erule pr_cycle_part_ind, auto)
+		done
+	    qed
+	  qed  
+      from g and k2 and this show "?thesis"
+	apply(erule_tac exE)
+	apply(insert ind[of rsa], simp)
+	apply(erule_tac exE)
+	apply(rule_tac x = "stp + stpa" in exI, 
+              simp add: abc_steps_add)
+	done
+    qed
+  qed
+qed
+
+lemma ci_pr_length: 
+  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md); 
+    rec_ci g = (a, aa, ba);  
+    rec_ci f = (ab, ac, bc)\<rbrakk>
+    \<Longrightarrow>  length aprog = 13 + length ab + length a"
+apply(auto simp: rec_ci.simps)
+done
+
+thm empty.simps
+term max
+fun empty_inv :: "nat \<times> nat list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> bool"
+  where
+  "empty_inv (as, lm) m n initlm = 
+         (let plus = initlm ! m + initlm ! n in
+           length initlm > max m n \<and> m \<noteq> n \<and> 
+              (if as = 0 then \<exists> k l. lm = initlm[m := k, n := l] \<and> 
+                    k + l = plus \<and> k \<le> initlm ! m 
+              else if as = 1 then \<exists> k l. lm = initlm[m := k, n := l]
+                             \<and> k + l + 1 = plus \<and> k < initlm ! m 
+              else if as = 2 then \<exists> k l. lm = initlm[m := k, n := l] 
+                              \<and> k + l = plus \<and> k \<le> initlm ! m
+              else if as = 3 then lm = initlm[m := 0, n := plus]
+              else False))"
+
+fun empty_stage1 :: "nat \<times> nat list \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "empty_stage1 (as, lm) m  = 
+            (if as = 3 then 0 
+             else 1)"
+
+fun empty_stage2 :: "nat \<times> nat list \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "empty_stage2 (as, lm) m = (lm ! m)"
+
+fun empty_stage3 :: "nat \<times> nat list \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "empty_stage3 (as, lm) m = (if as = 1 then 3 
+                                else if as = 2 then 2
+                                else if as = 0 then 1 
+                                else 0)"
+
+
+ 
+fun empty_measure :: "((nat \<times> nat list) \<times> nat) \<Rightarrow> (nat \<times> nat \<times> nat)"
+  where
+  "empty_measure ((as, lm), m) = 
+     (empty_stage1 (as, lm) m, empty_stage2 (as, lm) m,
+      empty_stage3 (as, lm) m)"
+
+definition lex_pair :: "((nat \<times> nat) \<times> nat \<times> nat) set"
+  where
+  "lex_pair = 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 empty_LE :: 
+ "(((nat \<times> nat list) \<times> nat) \<times> ((nat \<times> nat list) \<times> nat)) set"
+  where 
+  "empty_LE \<equiv> (inv_image lex_triple empty_measure)"
+
+lemma wf_lex_triple: "wf lex_triple"
+  by (auto intro:wf_lex_prod simp:lex_triple_def lex_pair_def)
+
+lemma wf_empty_le[intro]: "wf empty_LE"
+by(auto intro:wf_inv_image wf_lex_triple simp: empty_LE_def)
+
+declare empty_inv.simps[simp del]
+
+lemma empty_inv_init:  
+"\<lbrakk>m < length initlm; n < length initlm; m \<noteq> n\<rbrakk> \<Longrightarrow> 
+  empty_inv (0, initlm) m n initlm"
+apply(simp add: abc_steps_l.simps empty_inv.simps)
+apply(rule_tac x = "initlm ! m" in exI, 
+      rule_tac x = "initlm ! n" in exI, simp)
+done
+
+lemma [simp]: "abc_fetch 0 (recursive.empty m n) = Some (Dec m 3)"
+apply(simp add: empty.simps abc_fetch.simps)
+done
+
+lemma [simp]: "abc_fetch (Suc 0) (recursive.empty m n) =
+               Some (Inc n)"
+apply(simp add: empty.simps abc_fetch.simps)
+done
+
+lemma [simp]: "abc_fetch 2 (recursive.empty m n) = Some (Goto 0)"
+apply(simp add: empty.simps abc_fetch.simps)
+done
+
+lemma [simp]: "abc_fetch 3 (recursive.empty m n) = None"
+apply(simp add: empty.simps abc_fetch.simps)
+done
+
+lemma [simp]: 
+  "\<lbrakk>m \<noteq> n; m < length initlm; n < length initlm;
+    k + l = initlm ! m + initlm ! n; k \<le> initlm ! m; 0 < k\<rbrakk>
+ \<Longrightarrow> \<exists>ka la. initlm[m := k, n := l, m := k - Suc 0] = 
+     initlm[m := ka, n := la] \<and>
+     Suc (ka + la) = initlm ! m + initlm ! n \<and> 
+     ka < initlm ! m"
+apply(rule_tac x = "k - Suc 0" in exI, rule_tac x = l in exI, 
+      simp, auto)
+apply(subgoal_tac 
+      "initlm[m := k, n := l, m := k - Suc 0] = 
+       initlm[n := l, m := k, m := k - Suc 0]")
+apply(simp add: list_update_overwrite )
+apply(simp add: list_update_swap)
+apply(simp add: list_update_swap)
+done
+
+lemma [simp]:
+  "\<lbrakk>m \<noteq> n; m < length initlm; n < length initlm; 
+    Suc (k + l) = initlm ! m + initlm ! n;
+    k < initlm ! m\<rbrakk>
+    \<Longrightarrow> \<exists>ka la. initlm[m := k, n := l, n := Suc l] = 
+                initlm[m := ka, n := la] \<and> 
+                ka + la = initlm ! m + initlm ! n \<and> 
+                ka \<le> initlm ! m"
+apply(rule_tac x = k in exI, rule_tac x = "Suc l" in exI, auto)
+done
+
+lemma [simp]: 
+  "\<lbrakk>length initlm > max m n; m \<noteq> n\<rbrakk> \<Longrightarrow> 
+   \<forall>na. \<not> (\<lambda>(as, lm) m. as = 3) 
+    (abc_steps_l (0, initlm) (recursive.empty m n) na) m \<and> 
+  empty_inv (abc_steps_l (0, initlm) 
+           (recursive.empty m n) na) m n initlm \<longrightarrow>
+  empty_inv (abc_steps_l (0, initlm) 
+           (recursive.empty m n) (Suc na)) m n initlm \<and>
+  ((abc_steps_l (0, initlm) (recursive.empty m n) (Suc na), m),
+   abc_steps_l (0, initlm) (recursive.empty m n) na, m) \<in> empty_LE"
+apply(rule allI, rule impI, simp add: abc_steps_ind)
+apply(case_tac "(abc_steps_l (0, initlm) (recursive.empty m n) na)",
+      simp)
+apply(auto split:if_splits simp add:abc_steps_l.simps empty_inv.simps)
+apply(auto simp add: empty_LE_def lex_triple_def lex_pair_def 
+                     abc_step_l.simps abc_steps_l.simps
+                     empty_inv.simps abc_lm_v.simps abc_lm_s.simps
+                split: if_splits )
+apply(rule_tac x = k in exI, rule_tac x = "Suc l" in exI, simp)
+done
+
+lemma empty_inv_halt: 
+  "\<lbrakk>length initlm > max m n; m \<noteq> n\<rbrakk> \<Longrightarrow> 
+  \<exists> stp. (\<lambda> (as, lm). as = 3 \<and> 
+  empty_inv (as, lm) m n initlm) 
+             (abc_steps_l (0::nat, initlm) (empty m n) stp)"
+apply(insert halt_lemma2[of empty_LE
+  "\<lambda> ((as, lm), m). as = (3::nat)"
+  "\<lambda> stp. (abc_steps_l (0, initlm) (recursive.empty m n) stp, m)" 
+  "\<lambda> ((as, lm), m). empty_inv (as, lm) m n initlm"])
+apply(insert wf_empty_le, simp add: empty_inv_init abc_steps_zero)
+apply(erule_tac exE)
+apply(rule_tac x = na in exI)
+apply(case_tac "(abc_steps_l (0, initlm) (recursive.empty m n) na)",
+      simp, auto)
+done
+
+lemma empty_halt_cond:
+  "\<lbrakk>m \<noteq> n; empty_inv (a, b) m n lm; a = 3\<rbrakk> \<Longrightarrow> 
+  b = lm[n := lm ! m + lm ! n, m := 0]"
+apply(simp add: empty_inv.simps, auto)
+apply(simp add: list_update_swap)
+done
+
+lemma empty_ex:
+  "\<lbrakk>length lm > max m n; m \<noteq> n\<rbrakk> \<Longrightarrow> 
+  \<exists> stp. abc_steps_l (0::nat, lm) (empty m n) stp
+  = (3, (lm[n := (lm ! m + lm ! n)])[m := 0::nat])"
+apply(drule empty_inv_halt, simp, erule_tac exE)
+apply(rule_tac x = stp in exI)
+apply(case_tac "abc_steps_l (0, lm) (recursive.empty m n) stp",
+      simp)
+apply(erule_tac empty_halt_cond, auto)
+done
+
+lemma [simp]: 
+  "\<lbrakk>a_md = Suc (max (Suc (Suc n)) (max bc ba)); 
+   length lm = rs_pos \<and> rs_pos = n \<and> n > 0\<rbrakk>
+  \<Longrightarrow> n - Suc 0 < length lm + 
+  (Suc (max (Suc (Suc n)) (max bc ba)) - rs_pos + length suf_lm) \<and>
+   Suc (Suc n) < length lm + (Suc (max (Suc (Suc n)) (max bc ba)) -
+  rs_pos + length suf_lm) \<and> bc < length lm + (Suc (max (Suc (Suc n)) 
+ (max bc ba)) - rs_pos + length suf_lm) \<and> ba < length lm + 
+  (Suc (max (Suc (Suc n)) (max bc ba)) - rs_pos + length suf_lm)"
+apply(arith)
+done
+
+lemma [simp]:
+  "\<lbrakk>a_md = Suc (max (Suc (Suc n)) (max bc ba)); 
+   length lm = rs_pos \<and> rs_pos = n \<and> n > 0\<rbrakk>
+ \<Longrightarrow> n - Suc 0 < Suc (length suf_lm + max (Suc (Suc n)) (max bc ba)) \<and>
+     Suc n < length suf_lm + max (Suc (Suc n)) (max bc ba) \<and> 
+     bc < Suc (length suf_lm + max (Suc (Suc n)) (max bc ba)) \<and> 
+     ba < Suc (length suf_lm + max (Suc (Suc n)) (max bc ba))"
+apply(arith)
+done
+
+lemma [simp]: "n - Suc 0 \<noteq> max (Suc (Suc n)) (max bc ba)"
+apply(arith)
+done
+
+lemma [simp]: 
+  "a_md \<ge> Suc bc \<and> rs_pos > 0 \<and> bc \<ge> rs_pos \<Longrightarrow> 
+ bc - (rs_pos - Suc 0) + a_md - Suc bc = Suc (a_md - rs_pos - Suc 0)"
+apply(arith)
+done
+
+lemma [simp]: "length lm = n \<and> rs_pos = n \<and> 0 < rs_pos \<and> 
+                                                  Suc rs_pos < a_md 
+       \<Longrightarrow> n - Suc 0 < Suc (Suc (a_md + length suf_lm - Suc (Suc 0))) 
+        \<and> n < Suc (Suc (a_md + length suf_lm - Suc (Suc 0)))"
+apply(arith)
+done
+     
+lemma [simp]: "length lm = n \<and> rs_pos = n \<and> 0 < rs_pos \<and> 
+               Suc rs_pos < a_md \<Longrightarrow> n - Suc 0 \<noteq> n"
+by arith
+
+lemma ci_pr_ex2: 
+  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md);
+    rec_calc_rel (Pr n f g) lm rs; 
+    rec_ci g = (a, aa, ba); 
+    rec_ci f = (ab, ac, bc)\<rbrakk>
+  \<Longrightarrow> \<exists>ap bp. aprog = ap [+] bp \<and> 
+         ap = empty n (max (Suc (Suc (Suc n))) (max bc ba))"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "(ab [+] (recursive.empty n (Suc n) [+]
+              ([Dec (max (n + 3) (max bc ba)) (length a + 7)] 
+      [+] (a [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)])) @ 
+      [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length a + 4)]))" in exI, auto)
+apply(simp add: abc_append_commute add3_Suc)
+done
+
+lemma [simp]: 
+  "max (Suc (Suc (Suc n))) (max bc ba) - n < 
+     Suc (max (Suc (Suc (Suc n))) (max bc ba)) - n"
+apply(arith)
+done
+lemma exp_nth[simp]: "n < m \<Longrightarrow> a\<^bsup>m\<^esup> ! n = a"
+apply(simp add: exponent_def)
+done
+
+lemma [simp]: "length lm = n \<and> rs_pos = n \<and> 0 < n \<Longrightarrow> 
+                      lm[n - Suc 0 := 0::nat] = butlast lm @ [0]"
+apply(auto)
+apply(insert list_update_append[of "butlast lm" "[last lm]" 
+                                   "length lm - Suc 0" "0"], simp)
+done
+
+lemma [simp]: "\<lbrakk>length lm = n; 0 < n\<rbrakk>  \<Longrightarrow> lm ! (n - Suc 0) = last lm"
+apply(insert nth_append[of "butlast lm" "[last lm]" "n - Suc 0"],
+      simp)
+apply(insert butlast_append_last[of lm], auto)
+done
+lemma exp_suc_iff: "a\<^bsup>b\<^esup> @ [a] = a\<^bsup>b + Suc 0\<^esup>"
+apply(simp add: exponent_def rep_ind del: replicate.simps)
+done
+
+lemma less_not_less[simp]: "n > 0 \<Longrightarrow> \<not> n < n - Suc 0"
+by auto
+
+lemma [simp]:
+  "Suc n < length suf_lm + max (Suc (Suc n)) (max bc ba) \<and> 
+  bc < Suc (length suf_lm + max (Suc (Suc n)) 
+  (max bc ba)) \<and> 
+  ba < Suc (length suf_lm + max (Suc (Suc n)) (max bc ba))"
+  by arith
+
+lemma [simp]: "length lm = n \<and> rs_pos = n \<and> n > 0 \<Longrightarrow> 
+(lm @ 0\<^bsup>Suc (max (Suc (Suc n)) (max bc ba)) - n\<^esup> @ suf_lm) 
+  [max (Suc (Suc n)) (max bc ba) :=
+   (lm @ 0\<^bsup>Suc (max (Suc (Suc n)) (max bc ba)) - n\<^esup> @ suf_lm) ! (n - Suc 0) + 
+       (lm @ 0\<^bsup>Suc (max (Suc (Suc n)) (max bc ba)) - n\<^esup> @ suf_lm) ! 
+                   max (Suc (Suc n)) (max bc ba), n - Suc 0 := 0::nat]
+ = butlast lm @ 0 # 0\<^bsup>max (Suc (Suc n)) (max bc ba) - n\<^esup> @ last lm # suf_lm"
+apply(simp add: nth_append exp_nth list_update_append)
+apply(insert list_update_append[of "0\<^bsup>(max (Suc (Suc n)) (max bc ba)) - n\<^esup>"
+         "[0]" "max (Suc (Suc n)) (max bc ba) - n" "last lm"], simp)
+apply(simp add: exp_suc_iff Suc_diff_le del: list_update.simps)
+done
+
+lemma exp_eq: "(a = b) = (c\<^bsup>a\<^esup> = c\<^bsup>b\<^esup>)"
+apply(auto simp: exponent_def)
+done
+
+lemma [simp]:
+  "\<lbrakk>length lm = n; 0 < n;  Suc n < a_md\<rbrakk> \<Longrightarrow> 
+   (butlast lm @ rsa # 0\<^bsup>a_md - Suc n\<^esup> @ last lm # suf_lm)
+    [n := (butlast lm @ rsa # 0\<^bsup>a_md - Suc n\<^esup> @ last lm # suf_lm) ! 
+        (n - Suc 0) + (butlast lm @ rsa # (0::nat)\<^bsup>a_md - Suc n\<^esup> @ 
+                                last lm # suf_lm) ! n, n - Suc 0 := 0]
+ = butlast lm @ 0 # rsa # 0\<^bsup>a_md - Suc (Suc n)\<^esup> @ last lm # suf_lm"
+apply(simp add: nth_append exp_nth list_update_append)
+apply(case_tac "a_md - Suc n", simp, simp add: exponent_def)
+done
+
+lemma [simp]: 
+  "Suc (Suc rs_pos) \<le> a_md \<and> length lm = rs_pos \<and> 0 < rs_pos
+  \<Longrightarrow> a_md - Suc 0 < 
+          Suc (Suc (Suc (a_md + length suf_lm - Suc (Suc (Suc 0)))))"
+by arith
+
+lemma [simp]: 
+  "Suc (Suc rs_pos) \<le> a_md \<and> length lm = rs_pos \<and> 0 < rs_pos \<Longrightarrow> 
+                                   \<not> a_md - Suc 0 < rs_pos - Suc 0"
+by arith
+
+lemma [simp]: "Suc (Suc rs_pos) \<le> a_md \<Longrightarrow> 
+                                \<not> a_md - Suc 0 < rs_pos - Suc 0"
+by arith
+
+lemma [simp]: "\<lbrakk>Suc (Suc rs_pos) \<le> a_md\<rbrakk> \<Longrightarrow> 
+               \<not> a_md - rs_pos < Suc (Suc (a_md - Suc (Suc rs_pos)))"
+by arith 
+
+lemma [simp]: 
+  "Suc (Suc rs_pos) \<le> a_md \<and> length lm = rs_pos \<and> 0 < rs_pos
+ \<Longrightarrow> (abc_lm_v (butlast lm @ last lm # rs # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @
+        0 # suf_lm) (a_md - Suc 0) = 0 \<longrightarrow>
+      abc_lm_s (butlast lm @ last lm # rs # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ 
+        0 # suf_lm) (a_md - Suc 0) 0 = 
+         lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm) \<and>
+     abc_lm_v (butlast lm @ last lm # rs # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ 
+               0 # suf_lm) (a_md - Suc 0) = 0"
+apply(simp add: abc_lm_v.simps nth_append abc_lm_s.simps)
+apply(insert nth_append[of "last lm # rs # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup>" 
+               "0 # suf_lm" "(a_md - rs_pos)"], auto)
+apply(simp only: exp_suc_iff)
+apply(subgoal_tac "a_md - Suc 0 < a_md + length suf_lm", simp)
+apply(case_tac "lm = []", auto)
+done
+
+lemma pr_prog_ex[simp]: "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md); 
+      rec_ci g = (a, aa, ba); rec_ci f = (ab, ac, bc)\<rbrakk>
+    \<Longrightarrow> \<exists>cp. aprog = recursive.empty n (max (n + 3) 
+                    (max bc ba)) [+] cp"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "(ab [+] (recursive.empty n (Suc n) [+]
+              ([Dec (max (n + 3) (max bc ba)) (length a + 7)] 
+             [+] (a [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)]))
+             @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length a + 4)]))" in exI)
+apply(auto simp: abc_append_commute)
+done
+
+lemma [simp]: "empty m n \<noteq> []"
+by (simp add: empty.simps)
+(*
+lemma [simp]: "\<lbrakk>rs_pos = n; 0 < rs_pos ; Suc rs_pos < a_md\<rbrakk> \<Longrightarrow> 
+                        n - Suc 0 < a_md + length suf_lm"
+by arith
+*)
+lemma [intro]: 
+  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md); 
+    rec_ci f = (ab, ac, bc)\<rbrakk> \<Longrightarrow> 
+   \<exists>ap. (\<exists>cp. aprog = ap [+] ab [+] cp) \<and> length ap = 3"
+apply(case_tac "rec_ci g", simp add: rec_ci.simps)
+apply(rule_tac x = "empty n 
+              (max (n + 3) (max bc c))" in exI, simp)
+apply(rule_tac x = "recursive.empty n (Suc n) [+]
+                 ([Dec (max (n + 3) (max bc c)) (length a + 7)]
+                 [+] a [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)])
+               @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length a + 4)]" in exI, 
+      auto)
+apply(simp add: abc_append_commute)
+done
+
+lemma [intro]: 
+  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md);
+    rec_ci g = (a, aa, ba); 
+    rec_ci f = (ab, ac, bc)\<rbrakk> \<Longrightarrow> 
+    \<exists>ap. (\<exists>cp. aprog = ap [+] recursive.empty n (Suc n) [+] cp)
+      \<and> length ap = 3 + length ab"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "recursive.empty n (max (n + 3)
+                                (max bc ba)) [+] ab" in exI, simp)
+apply(rule_tac x = "([Dec (max (n + 3) (max bc ba))
+  (length a + 7)] [+] a [+] 
+  [Inc n, Dec (Suc n) 3, Goto (Suc 0)]) @ 
+  [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length a + 4)]" in exI)
+apply(auto simp: abc_append_commute)
+done
+
+(*
+lemma [simp]:
+  "n - Suc 0 < Suc (max (Suc (Suc n)) (max bc ba) + length suf_lm) \<and>
+  Suc n < max (Suc (Suc n)) (max bc ba) + length suf_lm \<and> 
+  bc < Suc (max (Suc (Suc n)) (max bc ba) + length suf_lm) \<and> 
+  ba < Suc (max (Suc (Suc n)) (max bc ba) + length suf_lm)"
+by arith
+*)
+
+lemma [intro]: 
+  "\<lbrakk>rec_ci (Pr n f g) = (aprog, rs_pos, a_md);
+    rec_ci g = (a, aa, ba); 
+    rec_ci f = (ab, ac, bc)\<rbrakk>
+    \<Longrightarrow> \<exists>ap. (\<exists>cp. aprog = ap [+] ([Dec (a_md - Suc 0) (length a + 7)]
+             [+] (a [+] [Inc (rs_pos - Suc 0), Dec rs_pos 3, 
+             Goto (Suc 0)])) @ [Dec (Suc (Suc n)) 0, Inc (Suc n),
+             Goto (length a + 4)] [+] cp) \<and>
+             length ap = 6 + length ab"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "recursive.empty n
+    (max (n + 3) (max bc ba)) [+] ab [+] 
+     recursive.empty n (Suc n)" in exI, simp)
+apply(rule_tac x = "[]" in exI, auto)
+apply(simp add: abc_append_commute)
+done
+
+(*
+lemma [simp]: "\<lbrakk>rs_pos = n; 0 < rs_pos ; Suc rs_pos < a_md\<rbrakk> \<Longrightarrow> 
+     n - Suc 0 < Suc (Suc (a_md + length suf_lm - 2)) \<and>
+     n < Suc (Suc (a_md + length suf_lm - 2))"
+by arith
+*)
+
+lemma [simp]: 
+  "n < Suc (max (n + 3) (max bc ba) + length suf_lm) \<and> 
+   Suc (Suc n) < max (n + 3) (max bc ba) + length suf_lm \<and> 
+   bc < Suc (max (n + 3) (max bc ba) + length suf_lm) \<and> 
+   ba < Suc (max (n + 3) (max bc ba) + length suf_lm)"
+by arith
+
+lemma [simp]: "n \<noteq> max (n + (3::nat)) (max bc ba)"
+by arith
+
+lemma [simp]:"length lm = Suc n \<Longrightarrow> lm[n := (0::nat)] = butlast lm @ [0]"
+apply(subgoal_tac "\<exists> xs x. lm = xs @ [x]", auto simp: list_update_append)
+apply(rule_tac x = "butlast lm" in exI, rule_tac x = "last lm" in exI)
+apply(case_tac lm, auto)
+done
+
+lemma [simp]:  "length lm = Suc n \<Longrightarrow> lm ! n =last lm"
+apply(subgoal_tac "lm \<noteq> []")
+apply(simp add: last_conv_nth, case_tac lm, simp_all)
+done
+
+lemma [simp]: "length lm = Suc n \<Longrightarrow> 
+      (lm @ (0::nat)\<^bsup>max (n + 3) (max bc ba) - n\<^esup> @ suf_lm)
+           [max (n + 3) (max bc ba) := (lm @ 0\<^bsup>max (n + 3) (max bc ba) - n\<^esup> @ suf_lm) ! n + 
+                  (lm @ 0\<^bsup>max (n + 3) (max bc ba) - n\<^esup> @ suf_lm) ! max (n + 3) (max bc ba), n := 0]
+       = butlast lm @ 0 # 0\<^bsup>max (n + 3) (max bc ba) - Suc n\<^esup> @ last lm # suf_lm"
+apply(auto simp: list_update_append nth_append)
+apply(subgoal_tac "(0\<^bsup>max (n + 3) (max bc ba) - n\<^esup>) = 0\<^bsup>max (n + 3) (max bc ba) - Suc n\<^esup> @ [0::nat]")
+apply(simp add: list_update_append)
+apply(simp add: exp_suc_iff)
+done
+
+lemma [simp]: "Suc (Suc n) < a_md \<Longrightarrow>  
+      n < Suc (Suc (a_md + length suf_lm - 2)) \<and>
+        n < Suc (a_md + length suf_lm - 2)"
+by(arith)
+
+lemma [simp]: "\<lbrakk>length lm = Suc n; Suc (Suc n) < a_md\<rbrakk>
+        \<Longrightarrow>(butlast lm @ (rsa::nat) # 0\<^bsup>a_md - Suc (Suc n)\<^esup> @ last lm # suf_lm)
+          [Suc n := (butlast lm @ rsa # 0\<^bsup>a_md - Suc (Suc n)\<^esup> @ last lm # suf_lm) ! n +
+                  (butlast lm @ rsa # 0\<^bsup>a_md - Suc (Suc n)\<^esup> @ last lm # suf_lm) ! Suc n, n := 0]
+    = butlast lm @ 0 # rsa # 0\<^bsup>a_md - Suc (Suc (Suc n))\<^esup> @ last lm # suf_lm"
+apply(auto simp: list_update_append)
+apply(subgoal_tac "(0\<^bsup>a_md - Suc (Suc n)\<^esup>) = (0::nat) # (0\<^bsup>a_md - Suc (Suc (Suc n))\<^esup>)", simp add: nth_append)
+apply(simp add: exp_ind_def[THEN sym])
+done
+
+lemma pr_case:
+  assumes nf_ind:
+  "\<And> lm rs suf_lm. rec_calc_rel f lm rs \<Longrightarrow> 
+  \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>bc - ac\<^esup> @ suf_lm) ab stp = 
+                (length ab, lm @ rs # 0\<^bsup>bc - Suc ac\<^esup> @ suf_lm)"
+  and ng_ind: "\<And> lm rs suf_lm. rec_calc_rel g lm rs \<Longrightarrow> 
+        \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>ba - aa\<^esup> @ suf_lm) a stp = 
+                       (length a, lm @ rs # 0\<^bsup>ba - Suc aa\<^esup> @ suf_lm)"
+    and h: "rec_ci (Pr n f g) = (aprog, rs_pos, a_md)"  "rec_calc_rel (Pr n f g) lm rs" 
+           "rec_ci g = (a, aa, ba)" "rec_ci f = (ab, ac, bc)" 
+  shows "\<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp = (length aprog, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+proof -
+  from h have k1: "\<exists> stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp
+    = (3, butlast lm @ 0 # 0\<^bsup>a_md - rs_pos - 1\<^esup> @ last lm # suf_lm)"
+  proof -
+    have "\<exists>bp cp. aprog = bp [+] cp \<and> bp = empty n 
+                 (max (n + 3) (max bc ba))"
+      apply(insert h, simp)
+      apply(erule pr_prog_ex, auto)
+      done
+    thus "?thesis"
+      apply(erule_tac exE, erule_tac exE, simp)
+      apply(subgoal_tac 
+           "\<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm)
+              ([] [+] recursive.empty n
+                  (max (n + 3) (max bc ba)) [+] cp) stp =
+             (0 + 3, butlast lm @ 0 # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ 
+                                        last lm # suf_lm)", simp)
+      apply(rule_tac abc_append_exc1, simp_all)
+      apply(insert empty_ex[of "n" "(max (n + 3) 
+                 (max bc ba))" "lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm"], simp)
+      apply(subgoal_tac "a_md = Suc (max (n + 3) (max bc ba))",
+            simp)
+      apply(subgoal_tac "length lm = Suc n \<and> rs_pos = Suc n", simp)
+      apply(insert h)
+      apply(simp add: para_pattern ci_pr_para_eq)
+      apply(rule ci_pr_md_def, auto)
+      done
+  qed
+  from h have k2: 
+  "\<exists> stp. abc_steps_l (3,  butlast lm @ 0 # 0\<^bsup>a_md - rs_pos - 1\<^esup> @ 
+             last lm # suf_lm) aprog stp 
+    = (length aprog, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+  proof -
+    from h have k2_1: "\<exists> rs. rec_calc_rel f (butlast lm) rs"
+      apply(erule_tac calc_pr_zero_ex)
+      done
+    thus "?thesis"
+    proof(erule_tac exE)
+      fix rsa
+      assume k2_2: "rec_calc_rel f (butlast lm) rsa"
+      from h and k2_2 have k2_2_1: 
+       "\<exists> stp. abc_steps_l (3, butlast lm @ 0 # 0\<^bsup>a_md - rs_pos - 1\<^esup> 
+                 @ last lm # suf_lm) aprog stp
+        = (3 + length ab, butlast lm @ rsa # 0\<^bsup>a_md - rs_pos - 1\<^esup> @ 
+                                             last lm # suf_lm)"
+      proof -
+	from h have j1: "
+          \<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and> length ap = 3 \<and> 
+              bp = ab"
+	  apply(auto)
+	  done
+	from h have j2: "ac = rs_pos - 1"
+	  apply(drule_tac ci_pr_f_paras, simp, auto)
+	  done
+	from h and j2 have j3: "a_md \<ge> Suc bc \<and> rs_pos > 0 \<and> bc \<ge> rs_pos"
+	  apply(rule_tac conjI)
+	  apply(erule_tac ab = ab and ac = ac in ci_pr_md_ge_f, simp)
+	  apply(rule_tac context_conjI)
+          apply(simp_all add: rec_ci.simps)
+	  apply(drule_tac ci_ad_ge_paras, drule_tac ci_ad_ge_paras)
+	  apply(arith)
+	  done	  
+	from j1 and j2 show "?thesis"
+	  apply(auto simp del: abc_append_commute)
+	  apply(rule_tac abc_append_exc1, simp_all)
+	  apply(insert nf_ind[of "butlast lm" "rsa" 
+                "0\<^bsup>a_md - bc - Suc 0\<^esup> @ last lm # suf_lm"], 
+               simp add: k2_2 j2, erule_tac exE)
+	  apply(simp add: exponent_add_iff j3)
+	  apply(rule_tac x = "stp" in exI, simp)
+	  done
+      qed
+      from h have k2_2_2: 
+      "\<exists> stp. abc_steps_l (3 + length ab, butlast lm @ rsa # 
+                  0\<^bsup>a_md - rs_pos - 1\<^esup> @ last lm # suf_lm) aprog stp
+        = (6 + length ab, butlast lm @ 0 # rsa # 
+                       0\<^bsup>a_md - rs_pos - 2\<^esup> @ last lm # suf_lm)"
+      proof -	     
+	from h have "\<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and> 
+          length ap = 3 + length ab \<and> bp = recursive.empty n (Suc n)"
+	  by auto
+	thus "?thesis"
+	proof(erule_tac exE, erule_tac exE, erule_tac exE, 
+              erule_tac exE)
+	  fix ap cp bp apa
+	  assume "aprog = ap [+] bp [+] cp \<and> length ap = 3 + 
+                    length ab \<and> bp = recursive.empty n (Suc n)"
+	  thus "?thesis"
+	    apply(simp del: abc_append_commute)
+	    apply(subgoal_tac 
+              "\<exists>stp. abc_steps_l (3 + length ab, 
+               butlast lm @ rsa # 0\<^bsup>a_md - Suc rs_pos\<^esup> @
+                 last lm # suf_lm) (ap [+] 
+                   recursive.empty n (Suc n) [+] cp) stp =
+              ((3 + length ab) + 3, butlast lm @ 0 # rsa # 
+                  0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ last lm # suf_lm)", simp)
+	    apply(rule_tac abc_append_exc1, simp_all)
+	    apply(insert empty_ex[of n "Suc n" 
+                    "butlast lm @ rsa # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ 
+                          last lm # suf_lm"], simp)
+	    apply(subgoal_tac "length lm = Suc n \<and> rs_pos = Suc n \<and> a_md > Suc (Suc n)", simp)
+	    apply(insert h, simp)
+            done
+	qed
+      qed
+      from h have k2_3: "lm \<noteq> []"
+	apply(rule_tac calc_pr_para_not_null, simp)
+	done
+      from h and k2_2 and k2_3 have k2_2_3: 
+      "\<exists> stp. abc_steps_l (6 + length ab, butlast lm @ 
+          (last lm - last lm) # rsa # 
+            0\<^bsup>a_md - (Suc (Suc rs_pos))\<^esup> @ last lm # suf_lm) aprog stp
+        = (6 + length ab, butlast lm @ last lm # rs # 
+                        0\<^bsup>a_md - Suc (Suc (rs_pos))\<^esup> @ 0 # suf_lm)"
+	apply(rule_tac x = "last lm" and g = g in pr_cycle_part, auto)
+	apply(rule_tac ng_ind, simp)
+	apply(rule_tac rec_calc_rel_def0, simp, simp)
+	done
+      from h  have k2_2_4: 
+       "\<exists> stp. abc_steps_l (6 + length ab,
+             butlast lm @ last lm # rs # 0\<^bsup>a_md - rs_pos - 2\<^esup> @
+                  0 # suf_lm) aprog stp
+        = (13 + length ab + length a,
+                   lm @ rs # 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+      proof -
+	from h have 
+        "\<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and>
+                     length ap = 6 + length ab \<and> 
+                    bp = ([Dec (a_md - Suc 0) (length a + 7)] [+] 
+                         (a [+] [Inc (rs_pos - Suc 0), 
+                         Dec rs_pos 3, Goto (Suc 0)])) @ 
+                        [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length a + 4)]"
+	  by auto
+	thus "?thesis"
+	  apply(auto)
+	  apply(subgoal_tac  
+            "\<exists>stp. abc_steps_l (6 + length ab, butlast lm @ 
+                last lm # rs # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ 0 # suf_lm)
+                (ap [+] ([Dec (a_md - Suc 0) (length a + 7)] [+] 
+                (a [+] [Inc (rs_pos - Suc 0), Dec rs_pos 3, 
+                Goto (Suc 0)])) @ [Dec (Suc (Suc n)) 0, Inc (Suc n), 
+                Goto (length a + 4)] [+] cp) stp =
+            (6 + length ab + (length a + 7) , 
+                 lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)", simp)
+	  apply(subgoal_tac "13 + (length ab + length a) = 
+                              13 + length ab + length a", simp)
+	  apply(arith)
+	  apply(rule abc_append_exc1, simp_all)
+	  apply(rule_tac x = "Suc 0" in exI, 
+                simp add: abc_steps_l.simps abc_fetch.simps
+                         nth_append abc_append_nth abc_step_l.simps)
+	  apply(subgoal_tac "a_md > Suc (Suc rs_pos) \<and> 
+                            length lm = rs_pos \<and> rs_pos > 0", simp)
+	  apply(insert h, simp)
+	  apply(subgoal_tac "rs_pos = Suc n", simp, simp)
+          done
+      qed
+      from h have k2_2_5: "length aprog = 13 + length ab + length a"
+	apply(rule_tac ci_pr_length, simp_all)
+	done
+      from k2_2_1 and k2_2_2 and k2_2_3 and k2_2_4 and k2_2_5 
+      show "?thesis"
+	apply(auto)
+	apply(rule_tac x = "stp + stpa + stpb + stpc" in exI, 
+              simp add: abc_steps_add)
+	done
+    qed
+  qed	
+  from k1 and k2 show 
+    "\<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp 
+               = (length aprog, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+    apply(erule_tac exE)
+    apply(erule_tac exE)
+    apply(rule_tac x = "stp + stpa" in exI)
+    apply(simp add: abc_steps_add)
+    done
+qed
+
+thm rec_calc_rel.induct
+
+lemma eq_switch: "x = y \<Longrightarrow> y = x"
+by simp
+
+lemma [simp]: 
+  "\<lbrakk>rec_ci f = (a, aa, ba); 
+    rec_ci (Mn n f) = (aprog, rs_pos, a_md)\<rbrakk> \<Longrightarrow> \<exists>bp. aprog = a @ bp"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "[Dec (Suc n) (length a + 5), 
+      Dec (Suc n) (length a + 3), Goto (Suc (length a)), 
+      Inc n, Goto 0]" in exI, auto)
+done
+
+lemma ci_mn_para_eq[simp]: 
+  "rec_ci (Mn n f) = (aprog, rs_pos, a_md) \<Longrightarrow> rs_pos = n"
+apply(case_tac "rec_ci f", simp add: rec_ci.simps)
+done
+(*
+lemma [simp]: "\<lbrakk>rec_ci f = (a, aa, ba); rec_ci (Mn n f) = (aprog, rs_pos, a_md); rec_calc_rel (Mn n f) lm rs\<rbrakk> \<Longrightarrow> aa = Suc rs_pos"
+apply(rule_tac calc_mn_reverse, simp)
+apply(insert para_pattern [of f a aa ba "lm @ [rs]" 0], simp)
+apply(subgoal_tac "rs_pos = length lm", simp)
+apply(drule_tac ci_mn_para_eq, simp)
+done
+*)
+lemma [simp]: "rec_ci f = (a, aa, ba) \<Longrightarrow> aa < ba"
+apply(simp add: ci_ad_ge_paras)
+done
+
+lemma [simp]: "\<lbrakk>rec_ci f = (a, aa, ba); 
+                rec_ci (Mn n f) = (aprog, rs_pos, a_md)\<rbrakk>
+    \<Longrightarrow> ba \<le> a_md"
+apply(simp add: rec_ci.simps)
+by arith
+
+lemma mn_calc_f: 
+  assumes ind: 
+  "\<And>aprog a_md rs_pos rs suf_lm lm.
+  \<lbrakk>rec_ci f = (aprog, rs_pos, a_md); rec_calc_rel f lm rs\<rbrakk>  
+  \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp    
+           = (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+  and h: "rec_ci f = (a, aa, ba)" 
+         "rec_ci (Mn n f) = (aprog, rs_pos, a_md)"  
+         "rec_calc_rel f (lm @ [x]) rsx" 
+         "aa = Suc n"
+  shows "\<exists>stp. abc_steps_l (0, lm @ x # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm) 
+                  aprog stp = (length a, 
+                   lm @ x # rsx # 0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ suf_lm)"
+proof -
+  from h have k1: "\<exists> ap bp. aprog = ap @ bp \<and> ap = a"
+    by simp
+  from h have k2: "rs_pos = n"
+    apply(erule_tac ci_mn_para_eq)
+    done
+  from h and k1 and k2 show "?thesis"
+  
+  proof(erule_tac exE, erule_tac exE, simp, 
+        rule_tac abc_add_exc1, auto)
+    fix bp
+    show 
+      "\<exists>astp. abc_steps_l (0, lm @ x # 0\<^bsup>a_md - Suc n\<^esup> @ suf_lm) a astp
+      = (length a, lm @ x # rsx # 0\<^bsup>a_md - Suc (Suc n)\<^esup> @ suf_lm)"
+      apply(insert ind[of a "Suc n" ba  "lm @ [x]" rsx 
+             "0\<^bsup>a_md - ba\<^esup> @ suf_lm"], simp add: exponent_add_iff h k2)
+      apply(subgoal_tac "ba > aa \<and> a_md \<ge> ba \<and> aa = Suc n", 
+            insert h, auto)
+      done
+  qed
+qed
+thm rec_ci.simps
+
+fun mn_ind_inv ::
+  "nat \<times> nat list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> nat list \<Rightarrow> bool"
+  where
+  "mn_ind_inv (as, lm') ss x rsx suf_lm lm = 
+           (if as = ss then lm' = lm @ x # rsx # suf_lm
+            else if as = ss + 1 then 
+                 \<exists>y. (lm' = lm @ x # y # suf_lm) \<and> y \<le> rsx
+            else if as = ss + 2 then 
+                 \<exists>y. (lm' = lm @ x # y # suf_lm) \<and> y \<le> rsx
+            else if as = ss + 3 then lm' = lm @ x # 0 # suf_lm
+            else if as = ss + 4 then lm' = lm @ Suc x # 0 # suf_lm
+            else if as = 0 then lm' = lm @ Suc x # 0 # suf_lm
+            else False
+)"
+
+fun mn_stage1 :: "nat \<times> nat list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "mn_stage1 (as, lm) ss n = 
+            (if as = 0 then 0 
+             else if as = ss + 4 then 1
+             else if as = ss + 3 then 2
+             else if as = ss + 2 \<or> as = ss + 1 then 3
+             else if as = ss then 4
+             else 0
+)"
+
+fun mn_stage2 :: "nat \<times> nat list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "mn_stage2 (as, lm) ss n = 
+            (if as = ss + 1 \<or> as = ss + 2 then (lm ! (Suc n))
+             else 0)"
+
+fun mn_stage3 :: "nat \<times> nat list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "mn_stage3 (as, lm) ss n = (if as = ss + 2 then 1 else 0)"
+
+ 
+fun mn_measure :: "((nat \<times> nat list) \<times> nat \<times> nat) \<Rightarrow>
+                                                (nat \<times> nat \<times> nat)"
+  where
+  "mn_measure ((as, lm), ss, n) = 
+     (mn_stage1 (as, lm) ss n, mn_stage2 (as, lm) ss n,
+                                       mn_stage3 (as, lm) ss n)"
+
+definition mn_LE :: "(((nat \<times> nat list) \<times> nat \<times> nat) \<times>
+                     ((nat \<times> nat list) \<times> nat \<times> nat)) set"
+  where "mn_LE \<equiv> (inv_image lex_triple mn_measure)"
+
+thm halt_lemma2
+lemma wf_mn_le[intro]: "wf mn_LE"
+by(auto intro:wf_inv_image wf_lex_triple simp: mn_LE_def)
+
+declare mn_ind_inv.simps[simp del]
+
+lemma mn_inv_init: 
+  "mn_ind_inv (abc_steps_l (length a, lm @ x # rsx # suf_lm) aprog 0)
+                                         (length a) x rsx suf_lm lm"
+apply(simp add: mn_ind_inv.simps abc_steps_zero)
+done
+
+lemma mn_halt_init: 
+  "rec_ci f = (a, aa, ba) \<Longrightarrow> 
+  \<not> (\<lambda>(as, lm') (ss, n). as = 0) 
+    (abc_steps_l (length a, lm @ x # rsx # suf_lm) aprog 0) 
+                                                       (length a, n)"
+apply(simp add: abc_steps_zero)
+apply(erule_tac rec_ci_not_null)
+done
+
+thm rec_ci.simps
+lemma [simp]: 
+  "\<lbrakk>rec_ci f = (a, aa, ba); 
+    rec_ci (Mn n f) = (aprog, rs_pos, a_md)\<rbrakk>
+    \<Longrightarrow> abc_fetch (length a) aprog =
+                      Some (Dec (Suc n) (length a + 5))"
+apply(simp add: rec_ci.simps abc_fetch.simps, 
+                erule_tac conjE, erule_tac conjE, simp)
+apply(drule_tac eq_switch, drule_tac eq_switch, simp)
+done
+
+lemma [simp]: "\<lbrakk>rec_ci f = (a, aa, ba); rec_ci (Mn n f) = (aprog, rs_pos, a_md)\<rbrakk>
+    \<Longrightarrow> abc_fetch (Suc (length a)) aprog = Some (Dec (Suc n) (length a + 3))"
+apply(simp add: rec_ci.simps abc_fetch.simps, erule_tac conjE, erule_tac conjE, simp)
+apply(drule_tac eq_switch, drule_tac eq_switch, simp add: nth_append)
+done
+
+lemma [simp]:
+  "\<lbrakk>rec_ci f = (a, aa, ba);
+    rec_ci (Mn n f) = (aprog, rs_pos, a_md)\<rbrakk>
+    \<Longrightarrow> abc_fetch (Suc (Suc (length a))) aprog = 
+                                     Some (Goto (length a + 1))"
+apply(simp add: rec_ci.simps abc_fetch.simps,
+      erule_tac conjE, erule_tac conjE, simp)
+apply(drule_tac eq_switch, drule_tac eq_switch, simp add: nth_append)
+done
+
+lemma [simp]: 
+  "\<lbrakk>rec_ci f = (a, aa, ba);
+    rec_ci (Mn n f) = (aprog, rs_pos, a_md)\<rbrakk>
+    \<Longrightarrow> abc_fetch (length a + 3) aprog = Some (Inc n)"
+apply(simp add: rec_ci.simps abc_fetch.simps, 
+      erule_tac conjE, erule_tac conjE, simp)
+apply(drule_tac eq_switch, drule_tac eq_switch, simp add: nth_append)
+done
+
+lemma [simp]: "\<lbrakk>rec_ci f = (a, aa, ba); rec_ci (Mn n f) = (aprog, rs_pos, a_md)\<rbrakk>
+    \<Longrightarrow> abc_fetch (length a + 4) aprog = Some (Goto 0)"
+apply(simp add: rec_ci.simps abc_fetch.simps, erule_tac conjE, erule_tac conjE, simp)
+apply(drule_tac eq_switch, drule_tac eq_switch, simp add: nth_append)
+done
+
+lemma [simp]: 
+  "0 < rsx
+   \<Longrightarrow> \<exists>y. (lm @ x # rsx # suf_lm)[Suc (length lm) := rsx - Suc 0]   
+    = lm @ x # y # suf_lm \<and> y \<le> rsx"
+apply(case_tac rsx, simp, simp)
+apply(rule_tac x = nat in exI, simp add: list_update_append)
+done
+
+lemma [simp]: 
+  "\<lbrakk>y \<le> rsx; 0 < y\<rbrakk>
+   \<Longrightarrow> \<exists>ya. (lm @ x # y # suf_lm)[Suc (length lm) := y - Suc 0] 
+          = lm @ x # ya # suf_lm \<and> ya \<le> rsx"
+apply(case_tac y, simp, simp)
+apply(rule_tac x = nat in exI, simp add: list_update_append)
+done
+
+lemma mn_halt_lemma: 
+  "\<lbrakk>rec_ci f = (a, aa, ba);
+    rec_ci (Mn n f) = (aprog, rs_pos, a_md);
+     0 < rsx; length lm = n\<rbrakk>
+    \<Longrightarrow>
+  \<forall>na. \<not> (\<lambda>(as, lm') (ss, n). as = 0)
+  (abc_steps_l (length a, lm @ x # rsx # suf_lm) aprog na) 
+                                                       (length a, n)
+ \<and> mn_ind_inv (abc_steps_l (length a, lm @ x # rsx # suf_lm)
+                       aprog na) (length a) x rsx suf_lm lm 
+\<longrightarrow> mn_ind_inv (abc_steps_l (length a, lm @ x # rsx # suf_lm) aprog 
+                         (Suc na)) (length a) x rsx suf_lm lm
+ \<and> ((abc_steps_l (length a, lm @ x # rsx # suf_lm) aprog (Suc na), 
+                                                    length a, n), 
+    abc_steps_l (length a, lm @ x # rsx # suf_lm) aprog na,
+                              length a, n) \<in> mn_LE"
+apply(rule allI, rule impI, simp add: abc_steps_ind)
+apply(case_tac "(abc_steps_l (length a, lm @ x # rsx # suf_lm) 
+                                                   aprog na)", simp)
+apply(auto split:if_splits simp add:abc_steps_l.simps 
+                           mn_ind_inv.simps abc_steps_zero)
+apply(auto simp add: mn_LE_def lex_triple_def lex_pair_def 
+            abc_step_l.simps abc_steps_l.simps mn_ind_inv.simps
+            abc_lm_v.simps abc_lm_s.simps nth_append
+           split: if_splits)
+apply(drule_tac  rec_ci_not_null, simp)
+done
+
+lemma mn_halt:
+  "\<lbrakk>rec_ci f = (a, aa, ba);
+    rec_ci (Mn n f) = (aprog, rs_pos, a_md);
+    0 < rsx; length lm = n\<rbrakk>
+    \<Longrightarrow> \<exists> stp. (\<lambda> (as, lm'). (as = 0 \<and> 
+           mn_ind_inv (as, lm')  (length a) x rsx suf_lm lm))
+            (abc_steps_l (length a, lm @ x # rsx # suf_lm) aprog stp)"
+apply(insert wf_mn_le)	  
+apply(insert halt_lemma2[of mn_LE
+  "\<lambda> ((as, lm'), ss, n). as = 0"
+  "\<lambda> stp. (abc_steps_l (length a, lm @ x # rsx # suf_lm) aprog stp, 
+   length a, n)"
+   "\<lambda> ((as, lm'), ss, n). mn_ind_inv (as, lm') ss x rsx suf_lm lm"], 
+   simp)
+apply(simp add: mn_halt_init mn_inv_init)
+apply(drule_tac x = x and suf_lm = suf_lm in mn_halt_lemma, auto)
+apply(rule_tac x = n in exI, 
+      case_tac "(abc_steps_l (length a, lm @ x # rsx # suf_lm)
+                              aprog n)", simp)
+done
+
+lemma [simp]: "Suc rs_pos < a_md \<Longrightarrow> 
+                Suc (a_md - Suc (Suc rs_pos)) = a_md - Suc rs_pos"
+by arith
+
+term rec_ci
+(*
+lemma [simp]: "\<lbrakk>rec_ci (Mn n f) = (aprog, rs_pos, a_md); rec_calc_rel (Mn n f) lm rs\<rbrakk>  \<Longrightarrow> Suc rs_pos < a_md"
+apply(case_tac "rec_ci f")
+apply(subgoal_tac "c > b \<and> b = Suc rs_pos \<and> a_md \<ge> c")
+apply(arith, auto)
+done
+*)
+lemma mn_ind_step: 
+  assumes ind:  
+  "\<And>aprog a_md rs_pos rs suf_lm lm.
+  \<lbrakk>rec_ci f = (aprog, rs_pos, a_md);
+   rec_calc_rel f lm rs\<rbrakk> \<Longrightarrow> 
+  \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp
+            = (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+  and h: "rec_ci f = (a, aa, ba)" 
+         "rec_ci (Mn n f) = (aprog, rs_pos, a_md)"  
+         "rec_calc_rel f (lm @ [x]) rsx" 
+         "rsx > 0" 
+         "Suc rs_pos < a_md" 
+         "aa = Suc rs_pos"
+  shows "\<exists>stp. abc_steps_l (0, lm @ x # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm) 
+             aprog stp = (0, lm @ Suc x # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+thm abc_add_exc1
+proof -
+  have k1: 
+    "\<exists> stp. abc_steps_l (0, lm @ x #  0\<^bsup>a_md - Suc (rs_pos)\<^esup> @ suf_lm)
+         aprog stp = 
+       (length a, lm @ x # rsx # 0\<^bsup>a_md  - Suc (Suc rs_pos) \<^esup>@ suf_lm)"
+    apply(insert h)
+    apply(auto intro: mn_calc_f ind)
+    done
+  from h have k2: "length lm = n"
+    apply(subgoal_tac "rs_pos = n")
+    apply(drule_tac  para_pattern, simp, simp, simp)
+    done
+  from h have k3: "a_md > (Suc rs_pos)"
+    apply(simp)
+    done
+  from k2 and h and k3 have k4: 
+    "\<exists> stp. abc_steps_l (length a,
+       lm @ x # rsx # 0\<^bsup>a_md  - Suc (Suc rs_pos)  \<^esup>@ suf_lm) aprog stp = 
+        (0, lm @ Suc x # 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+    apply(frule_tac x = x and 
+       suf_lm = "0\<^bsup>a_md - Suc (Suc rs_pos)\<^esup> @ suf_lm" in mn_halt, auto)
+    apply(rule_tac x = "stp" in exI, 
+          simp add: mn_ind_inv.simps rec_ci_not_null exponent_def)
+    apply(simp only: replicate.simps[THEN sym], simp)
+    done
+  
+  from k1 and k4 show "?thesis"
+    apply(auto)
+    apply(rule_tac x = "stp + stpa" in exI, simp add: abc_steps_add)
+    done
+qed
+
+lemma [simp]: 
+  "\<lbrakk>rec_ci f = (a, aa, ba); rec_ci (Mn n f) = (aprog, rs_pos, a_md);
+    rec_calc_rel (Mn n f) lm rs\<rbrakk> \<Longrightarrow> aa = Suc rs_pos"
+apply(rule_tac calc_mn_reverse, simp)
+apply(insert para_pattern [of f a aa ba "lm @ [rs]" 0], simp)
+apply(subgoal_tac "rs_pos = length lm", simp)
+apply(drule_tac ci_mn_para_eq, simp)
+done
+
+lemma [simp]: "\<lbrakk>rec_ci (Mn n f) = (aprog, rs_pos, a_md);      
+                rec_calc_rel (Mn n f) lm rs\<rbrakk>  \<Longrightarrow> Suc rs_pos < a_md"
+apply(case_tac "rec_ci f")
+apply(subgoal_tac "c > b \<and> b = Suc rs_pos \<and> a_md \<ge> c")
+apply(arith, auto)
+done
+
+lemma mn_ind_steps:  
+  assumes ind:
+  "\<And>aprog a_md rs_pos rs suf_lm lm. 
+  \<lbrakk>rec_ci f = (aprog, rs_pos, a_md); rec_calc_rel f lm rs\<rbrakk> \<Longrightarrow> 
+  \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp = 
+              (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+  and h: "rec_ci f = (a, aa, ba)" 
+  "rec_ci (Mn n f) = (aprog, rs_pos, a_md)" 
+  "rec_calc_rel (Mn n f) lm rs"
+  "rec_calc_rel f (lm @ [rs]) 0" 
+  "\<forall>x<rs. (\<exists> v. rec_calc_rel f (lm @ [x]) v \<and> 0 < v)"
+  "n = length lm" 
+  "x \<le> rs"
+  shows "\<exists>stp. abc_steps_l (0, lm @ 0 # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)
+                 aprog stp = (0, lm @ x # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+apply(insert h, induct x, 
+      rule_tac x = 0 in exI, simp add: abc_steps_zero, simp)
+proof -
+  fix x
+  assume k1: 
+    "\<exists>stp. abc_steps_l (0, lm @ 0 # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)
+                aprog stp = (0, lm @ x # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+  and k2: "rec_ci (Mn (length lm) f) = (aprog, rs_pos, a_md)" 
+          "rec_calc_rel (Mn (length lm) f) lm rs" 
+          "rec_calc_rel f (lm @ [rs]) 0" 
+          "\<forall>x<rs.(\<exists> v. rec_calc_rel f (lm @ [x]) v \<and> v > 0)" 
+          "n = length lm" 
+          "Suc x \<le> rs" 
+          "rec_ci f = (a, aa, ba)"
+  hence k2:
+    "\<exists>stp. abc_steps_l (0, lm @ x # 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm) aprog
+               stp = (0, lm @ Suc x # 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+    apply(erule_tac x = x in allE)
+    apply(auto)
+    apply(rule_tac  x = x in mn_ind_step)
+    apply(rule_tac ind, auto)      
+    done
+  from k1 and k2 show 
+    "\<exists>stp. abc_steps_l (0, lm @ 0 # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)
+          aprog stp = (0, lm @ Suc x # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+    apply(auto)
+    apply(rule_tac x = "stp + stpa" in exI, simp only: abc_steps_add)
+    done
+qed
+    
+lemma [simp]: 
+"\<lbrakk>rec_ci f = (a, aa, ba); 
+  rec_ci (Mn n f) = (aprog, rs_pos, a_md); 
+  rec_calc_rel (Mn n f) lm rs;
+  length lm = n\<rbrakk>
+ \<Longrightarrow> abc_lm_v (lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm) (Suc n) = 0"
+apply(auto simp: abc_lm_v.simps nth_append)
+done
+
+lemma [simp]: 
+  "\<lbrakk>rec_ci f = (a, aa, ba); 
+    rec_ci (Mn n f) = (aprog, rs_pos, a_md); 
+    rec_calc_rel (Mn n f) lm rs;
+     length lm = n\<rbrakk>
+    \<Longrightarrow> abc_lm_s (lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm) (Suc n) 0 =
+                           lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm"
+apply(auto simp: abc_lm_s.simps list_update_append)
+done
+
+lemma mn_length: 
+  "\<lbrakk>rec_ci f = (a, aa, ba);
+    rec_ci (Mn n f) = (aprog, rs_pos, a_md)\<rbrakk>
+  \<Longrightarrow> length aprog = length a + 5"
+apply(simp add: rec_ci.simps, erule_tac conjE)
+apply(drule_tac eq_switch, drule_tac eq_switch, simp)
+done
+
+lemma mn_final_step:
+  assumes ind:
+  "\<And>aprog a_md rs_pos rs suf_lm lm.
+  \<lbrakk>rec_ci f = (aprog, rs_pos, a_md); 
+  rec_calc_rel f lm rs\<rbrakk> \<Longrightarrow> 
+  \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp =
+              (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+  and h: "rec_ci f = (a, aa, ba)" 
+         "rec_ci (Mn n f) = (aprog, rs_pos, a_md)" 
+         "rec_calc_rel (Mn n f) lm rs" 
+         "rec_calc_rel f (lm @ [rs]) 0" 
+  shows "\<exists>stp. abc_steps_l (0, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm) 
+     aprog stp = (length aprog, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+proof -
+  from h and ind have k1:
+    "\<exists>stp.  abc_steps_l (0, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm) 
+        aprog stp = (length a,  lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+    thm mn_calc_f
+    apply(insert mn_calc_f[of f a aa ba n aprog 
+                               rs_pos a_md lm rs 0 suf_lm], simp)
+    apply(subgoal_tac "aa = Suc n", simp add: exponent_cons_iff)
+    apply(subgoal_tac "rs_pos = n", simp, simp)
+    done
+  from h have k2: "length lm = n"
+    apply(subgoal_tac "rs_pos = n")
+    apply(drule_tac f = "Mn n f" in para_pattern, simp, simp, simp)
+    done
+  from h and k2 have k3: 
+  "\<exists>stp. abc_steps_l (length a, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)
+    aprog stp = (length a + 5, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+    apply(rule_tac x = "Suc 0" in exI, 
+          simp add: abc_step_l.simps abc_steps_l.simps)
+    done
+  from h have k4: "length aprog = length a + 5"
+    apply(simp add: mn_length)
+    done
+  from k1 and k3 and k4 show "?thesis"
+    apply(auto)
+    apply(rule_tac x = "stp + stpa" in exI, simp add: abc_steps_add)
+    done
+qed
+
+lemma mn_case: 
+  assumes ind: 
+  "\<And>aprog a_md rs_pos rs suf_lm lm.
+  \<lbrakk>rec_ci f = (aprog, rs_pos, a_md); rec_calc_rel f lm rs\<rbrakk> \<Longrightarrow> 
+  \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp = 
+               (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+  and h: "rec_ci (Mn n f) = (aprog, rs_pos, a_md)" 
+         "rec_calc_rel (Mn n f) lm rs"
+  shows "\<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp 
+  = (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+apply(case_tac "rec_ci f", simp)
+apply(insert h, rule_tac calc_mn_reverse, simp)
+proof -
+  fix a b c v
+  assume h: "rec_ci f = (a, b, c)" 
+            "rec_ci (Mn n f) = (aprog, rs_pos, a_md)" 
+            "rec_calc_rel (Mn n f) lm rs" 
+            "rec_calc_rel f (lm @ [rs]) 0" 
+            "\<forall>x<rs. \<exists>v. rec_calc_rel f (lm @ [x]) v \<and> 0 < v"
+            "n = length lm"
+  hence k1:
+    "\<exists>stp. abc_steps_l (0, lm @ 0 # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm) aprog
+                  stp = (0, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+    thm mn_ind_steps
+    apply(auto intro: mn_ind_steps ind)
+    done
+  from h have k2: 
+    "\<exists>stp. abc_steps_l (0, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm) aprog
+         stp = (length aprog, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+    apply(auto intro: mn_final_step ind)
+    done
+  from k1 and k2 show 
+    "\<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp = 
+  (length aprog, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+    apply(auto, insert h)
+    apply(subgoal_tac "Suc rs_pos < a_md")
+    apply(rule_tac x = "stp + stpa" in exI, 
+      simp only: abc_steps_add exponent_cons_iff, simp, simp)
+    done
+qed
+
+lemma z_rs: "rec_calc_rel z lm rs \<Longrightarrow> rs = 0"
+apply(rule_tac calc_z_reverse, auto)
+done
+
+lemma z_case:
+  "\<lbrakk>rec_ci z = (aprog, rs_pos, a_md); rec_calc_rel z lm rs\<rbrakk>
+  \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp =
+           (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+apply(simp add: rec_ci.simps rec_ci_z_def, auto)
+apply(rule_tac x = "Suc 0" in exI, simp add: abc_steps_l.simps 
+                               abc_fetch.simps abc_step_l.simps z_rs)
+done
+thm addition.simps
+
+thm addition.simps
+thm rec_ci_s_def
+fun addition_inv :: "nat \<times> nat list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow>     
+                     nat list \<Rightarrow> bool"
+  where
+  "addition_inv (as, lm') m n p lm = 
+        (let sn = lm ! n in
+         let sm = lm ! m in
+         lm ! p = 0 \<and>
+             (if as = 0 then \<exists> x. x \<le> lm ! m \<and> lm' = lm[m := x,
+                                    n := (sn + sm - x), p := (sm - x)]
+             else if as = 1 then \<exists> x. x < lm ! m \<and> lm' = lm[m := x,
+                            n := (sn + sm - x - 1), p := (sm - x - 1)]
+             else if as = 2 then \<exists> x. x < lm ! m \<and> lm' = lm[m := x, 
+                               n := (sn + sm - x), p := (sm - x - 1)]
+             else if as = 3 then \<exists> x. x < lm ! m \<and> lm' = lm[m := x,
+                                   n := (sn + sm - x), p := (sm - x)]
+             else if as = 4 then \<exists> x. x \<le> lm ! m \<and> lm' = lm[m := x,
+                                       n := (sn + sm), p := (sm - x)] 
+             else if as = 5 then \<exists> x. x < lm ! m \<and> lm' = lm[m := x, 
+                                  n := (sn + sm), p := (sm - x - 1)] 
+             else if as = 6 then \<exists> x. x < lm ! m \<and> lm' =
+                     lm[m := Suc x, n := (sn + sm), p := (sm - x - 1)]
+             else if as = 7 then lm' = lm[m := sm, n := (sn + sm)]
+             else False))"
+
+fun addition_stage1 :: "nat \<times> nat list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "addition_stage1 (as, lm) m p = 
+          (if as = 0 \<or> as = 1 \<or> as = 2 \<or> as = 3 then 2 
+           else if as = 4 \<or> as = 5 \<or> as = 6 then 1
+           else 0)"
+
+fun addition_stage2 :: "nat \<times> nat list \<Rightarrow> nat \<Rightarrow>  nat \<Rightarrow> nat"
+  where
+  "addition_stage2 (as, lm) m p = 
+              (if 0 \<le> as \<and> as \<le> 3 then lm ! m
+               else if 4 \<le> as \<and> as \<le> 6 then lm ! p
+               else 0)"
+
+fun addition_stage3 :: "nat \<times> nat list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "addition_stage3 (as, lm) m p = 
+             (if as = 1 then 4  
+              else if as = 2 then 3 
+              else if as = 3 then 2
+              else if as = 0 then 1 
+              else if as = 5 then 2
+              else if as = 6 then 1 
+              else if as = 4 then 0 
+              else 0)"
+
+fun addition_measure :: "((nat \<times> nat list) \<times> nat \<times> nat) \<Rightarrow> 
+                                                 (nat \<times> nat \<times> nat)"
+  where
+  "addition_measure ((as, lm), m, p) =
+     (addition_stage1 (as, lm) m p, 
+      addition_stage2 (as, lm) m p,
+      addition_stage3 (as, lm) m p)"
+
+definition addition_LE :: "(((nat \<times> nat list) \<times> nat \<times> nat) \<times> 
+                          ((nat \<times> nat list) \<times> nat \<times> nat)) set"
+  where "addition_LE \<equiv> (inv_image lex_triple addition_measure)"
+
+lemma [simp]: "wf addition_LE"
+by(simp add: wf_inv_image wf_lex_triple addition_LE_def)
+
+declare addition_inv.simps[simp del]
+
+lemma addition_inv_init: 
+  "\<lbrakk>m \<noteq> n; max m n < p; length lm > p; lm ! p = 0\<rbrakk> \<Longrightarrow>
+                                   addition_inv (0, lm) m n p lm"
+apply(simp add: addition_inv.simps)
+apply(rule_tac x = "lm ! m" in exI, simp)
+done
+
+thm addition.simps
+
+lemma [simp]: "abc_fetch 0 (addition m n p) = Some (Dec m 4)"
+by(simp add: abc_fetch.simps addition.simps)
+
+lemma [simp]: "abc_fetch (Suc 0) (addition m n p) = Some (Inc n)"
+by(simp add: abc_fetch.simps addition.simps)
+
+lemma [simp]: "abc_fetch 2 (addition m n p) = Some (Inc p)"
+by(simp add: abc_fetch.simps addition.simps)
+
+lemma [simp]: "abc_fetch 3 (addition m n p) = Some (Goto 0)"
+by(simp add: abc_fetch.simps addition.simps)
+
+lemma [simp]: "abc_fetch 4 (addition m n p) = Some (Dec p 7)"
+by(simp add: abc_fetch.simps addition.simps)
+
+lemma [simp]: "abc_fetch 5 (addition m n p) = Some (Inc m)"
+by(simp add: abc_fetch.simps addition.simps)
+
+lemma [simp]: "abc_fetch 6 (addition m n p) = Some (Goto 4)"
+by(simp add: abc_fetch.simps addition.simps)
+
+lemma [simp]:
+  "\<lbrakk>m \<noteq> n; p < length lm; lm ! p = 0; m < p; n < p; x \<le> lm ! m; 0 < x\<rbrakk>
+ \<Longrightarrow> \<exists>xa<lm ! m. lm[m := x, n := lm ! n + lm ! m - x, 
+                    p := lm ! m - x, m := x - Suc 0] =
+                 lm[m := xa, n := lm ! n + lm ! m - Suc xa,
+                    p := lm ! m - Suc xa]"
+apply(case_tac x, simp, simp)
+apply(rule_tac x = nat in exI, simp add: list_update_swap 
+                                         list_update_overwrite)
+done
+
+lemma [simp]:
+  "\<lbrakk>m \<noteq> n; p < length lm; lm ! p = 0; m < p; n < p; x < lm ! m\<rbrakk>
+   \<Longrightarrow> \<exists>xa<lm ! m. lm[m := x, n := lm ! n + lm ! m - Suc x,
+                      p := lm ! m - Suc x, n := lm ! n + lm ! m - x]
+                 = lm[m := xa, n := lm ! n + lm ! m - xa, 
+                      p := lm ! m - Suc xa]"
+apply(rule_tac x = x in exI, 
+      simp add: list_update_swap list_update_overwrite)
+done
+
+lemma [simp]: 
+  "\<lbrakk>m \<noteq> n; p < length lm; lm ! p = 0; m < p; n < p; x < lm ! m\<rbrakk>
+   \<Longrightarrow> \<exists>xa<lm ! m. lm[m := x, n := lm ! n + lm ! m - x, 
+                          p := lm ! m - Suc x, p := lm ! m - x]
+                 = lm[m := xa, n := lm ! n + lm ! m - xa, 
+                          p := lm ! m - xa]"
+apply(rule_tac x = x in exI, simp add: list_update_overwrite)
+done
+
+lemma [simp]: 
+  "\<lbrakk>m \<noteq> n; p < length lm; lm ! p = (0::nat); m < p; n < p; x < lm ! m\<rbrakk>
+  \<Longrightarrow> \<exists>xa\<le>lm ! m. lm[m := x, n := lm ! n + lm ! m - x,
+                                   p := lm ! m - x] = 
+                  lm[m := xa, n := lm ! n + lm ! m - xa, 
+                                   p := lm ! m - xa]"
+apply(rule_tac x = x in exI, simp)
+done
+
+lemma [simp]: 
+  "\<lbrakk>m \<noteq> n; p < length lm; lm ! p = 0; m < p; n < p;
+    x \<le> lm ! m; lm ! m \<noteq> x\<rbrakk>
+  \<Longrightarrow> \<exists>xa<lm ! m. lm[m := x, n := lm ! n + lm ! m, 
+                       p := lm ! m - x, p := lm ! m - Suc x] 
+               = lm[m := xa, n := lm ! n + lm ! m, 
+                       p := lm ! m - Suc xa]"
+apply(rule_tac x = x in exI, simp add: list_update_overwrite)
+done
+
+lemma [simp]:
+  "\<lbrakk>m \<noteq> n; p < length lm; lm ! p = 0; m < p; n < p; x < lm ! m\<rbrakk>
+  \<Longrightarrow> \<exists>xa<lm ! m. lm[m := x, n := lm ! n + lm ! m,
+                             p := lm ! m - Suc x, m := Suc x]
+                = lm[m := Suc xa, n := lm ! n + lm ! m, 
+                             p := lm ! m - Suc xa]"
+apply(rule_tac x = x in exI, 
+     simp add: list_update_swap list_update_overwrite)
+done
+
+lemma [simp]: 
+  "\<lbrakk>m \<noteq> n; p < length lm; lm ! p = 0; m < p; n < p; x < lm ! m\<rbrakk>
+  \<Longrightarrow> \<exists>xa\<le>lm ! m. lm[m := Suc x, n := lm ! n + lm ! m, 
+                             p := lm ! m - Suc x] 
+               = lm[m := xa, n := lm ! n + lm ! m, p := lm ! m - xa]"
+apply(rule_tac x = "Suc x" in exI, simp)
+done
+
+lemma addition_halt_lemma: 
+  "\<lbrakk>m \<noteq> n; max m n < p; length lm > p; lm ! p = 0\<rbrakk> \<Longrightarrow>
+  \<forall>na. \<not> (\<lambda>(as, lm') (m, p). as = 7) 
+        (abc_steps_l (0, lm) (addition m n p) na) (m, p) \<and> 
+  addition_inv (abc_steps_l (0, lm) (addition m n p) na) m n p lm 
+\<longrightarrow> addition_inv (abc_steps_l (0, lm) (addition m n p) 
+                                 (Suc na)) m n p lm 
+  \<and> ((abc_steps_l (0, lm) (addition m n p) (Suc na), m, p), 
+     abc_steps_l (0, lm) (addition m n p) na, m, p) \<in> addition_LE"
+apply(rule allI, rule impI, simp add: abc_steps_ind)
+apply(case_tac "(abc_steps_l (0, lm) (addition m n p) na)", simp)
+apply(auto split:if_splits simp add: addition_inv.simps
+                                 abc_steps_zero)
+apply(simp_all add: abc_steps_l.simps abc_steps_zero)
+apply(auto simp add: addition_LE_def lex_triple_def lex_pair_def 
+                     abc_step_l.simps addition_inv.simps 
+                     abc_lm_v.simps abc_lm_s.simps nth_append
+                split: if_splits)
+apply(rule_tac x = x in exI, simp)
+done
+
+lemma  addition_ex: 
+  "\<lbrakk>m \<noteq> n; max m n < p; length lm > p; lm ! p = 0\<rbrakk> \<Longrightarrow> 
+  \<exists> stp. (\<lambda> (as, lm'). as = 7 \<and> addition_inv (as, lm') m n p lm) 
+                        (abc_steps_l (0, lm) (addition m n p) stp)"
+apply(insert halt_lemma2[of addition_LE 
+  "\<lambda> ((as, lm'), m, p). as = 7"
+  "\<lambda> stp. (abc_steps_l (0, lm) (addition m n p) stp, m, p)"
+  "\<lambda> ((as, lm'), m, p). addition_inv (as, lm') m n p lm"], 
+  simp add: abc_steps_zero addition_inv_init)
+apply(drule_tac addition_halt_lemma, simp, simp, simp,
+      simp, erule_tac exE)
+apply(rule_tac x = na in exI, 
+      case_tac "(abc_steps_l (0, lm) (addition m n p) na)", auto)
+done
+
+lemma [simp]: "length (addition m n p) = 7"
+by (simp add: addition.simps)
+
+lemma [elim]: "addition 0 (Suc 0) 2 = [] \<Longrightarrow> RR"
+by(simp add: addition.simps)
+
+lemma [simp]: "(0\<^bsup>2\<^esup>)[0 := n] = [n, 0::nat]"
+apply(subgoal_tac "2 = Suc 1", 
+      simp only: replicate.simps exponent_def)
+apply(auto)
+done
+
+lemma [simp]: 
+  "\<exists>stp. abc_steps_l (0, n # 0\<^bsup>2\<^esup> @ suf_lm) 
+     (addition 0 (Suc 0) 2 [+] [Inc (Suc 0)]) stp = 
+                                      (8, n # Suc n # 0 # suf_lm)"
+apply(rule_tac bm = "n # n # 0 # suf_lm" in abc_append_exc2, auto)
+apply(insert addition_ex[of 0 "Suc 0" 2 "n # 0\<^bsup>2\<^esup> @ suf_lm"], 
+      simp add: nth_append numeral_2_eq_2, erule_tac exE)
+apply(rule_tac x = stp in exI,
+      case_tac "(abc_steps_l (0, n # 0\<^bsup>2\<^esup> @ suf_lm)
+                      (addition 0 (Suc 0) 2) stp)", 
+      simp add: addition_inv.simps nth_append list_update_append numeral_2_eq_2)
+apply(simp add: nth_append numeral_2_eq_2, erule_tac exE)
+apply(rule_tac x = "Suc 0" in exI,
+      simp add: abc_steps_l.simps abc_fetch.simps 
+      abc_steps_zero abc_step_l.simps abc_lm_s.simps abc_lm_v.simps)
+done
+
+lemma s_case:
+  "\<lbrakk>rec_ci s = (aprog, rs_pos, a_md); rec_calc_rel s lm rs\<rbrakk>
+  \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp =
+               (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+apply(simp add: rec_ci.simps rec_ci_s_def, auto)
+apply(rule_tac calc_s_reverse, auto)
+done
+
+lemma [simp]: 
+  "\<lbrakk>n < length lm; lm ! n = rs\<rbrakk>
+    \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm @ 0 # 0 #suf_lm)
+                     (addition n (length lm) (Suc (length lm))) stp 
+             = (7, lm @ rs # 0 # suf_lm)"
+apply(insert addition_ex[of n "length lm"
+                           "Suc (length lm)" "lm @ 0 # 0 # suf_lm"])
+apply(simp add: nth_append, erule_tac exE)
+apply(rule_tac x = stp in exI)
+apply(case_tac "abc_steps_l (0, lm @ 0 # 0 # suf_lm) (addition n (length lm)
+                 (Suc (length lm))) stp", simp)
+apply(simp add: addition_inv.simps)
+apply(insert nth_append[of lm "0 # 0 # suf_lm" "n"], simp)
+done
+
+lemma [simp]: "0\<^bsup>2\<^esup> = [0, 0::nat]"
+apply(auto simp: exponent_def numeral_2_eq_2)
+done
+
+lemma id_case: 
+  "\<lbrakk>rec_ci (id m n) = (aprog, rs_pos, a_md); 
+    rec_calc_rel (id m n) lm rs\<rbrakk>
+  \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp = 
+               (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+apply(simp add: rec_ci.simps rec_ci_id.simps, auto)
+apply(rule_tac calc_id_reverse, simp, simp)
+done   
+
+lemma list_tl_induct:
+  "\<lbrakk>P []; \<And>a list. P list \<Longrightarrow> P (list @ [a::'a])\<rbrakk> \<Longrightarrow> 
+                                            P ((list::'a list))"
+apply(case_tac "length list", simp)
+proof -
+  fix nat
+  assume ind: "\<And>a list. P list \<Longrightarrow> P (list @ [a])"
+  and h: "length list = Suc nat" "P []"
+  from h show "P list"
+  proof(induct nat arbitrary: list, case_tac lista, simp, simp)
+    fix lista a listaa
+    from h show "P [a]"
+      by(insert ind[of "[]"], simp add: h)
+  next
+    fix nat list
+    assume nind: "\<And>list. \<lbrakk>length list = Suc nat; P []\<rbrakk> \<Longrightarrow> P list" 
+    and g: "length (list:: 'a list) = Suc (Suc nat)"
+    from g show "P (list::'a list)"
+      apply(insert nind[of "butlast list"], simp add: h)
+      apply(insert ind[of "butlast list" "last list"], simp)
+      apply(subgoal_tac "butlast list @ [last list] = list", simp)
+      apply(case_tac "list::'a list", simp, simp)
+      done
+  qed
+qed      
+  
+thm list.induct
+
+lemma nth_eq_butlast_nth: "\<lbrakk>length ys > Suc k\<rbrakk> \<Longrightarrow> 
+                                        ys ! k = butlast ys ! k"
+apply(subgoal_tac "\<exists> xs y. ys = xs @ [y]", auto simp: nth_append)
+apply(rule_tac x = "butlast ys" in exI, rule_tac x = "last ys" in exI)
+apply(case_tac "ys = []", simp, simp)
+done
+
+lemma [simp]: 
+"\<lbrakk>\<forall>k<Suc (length list). rec_calc_rel ((list @ [a]) ! k) lm (ys ! k);
+  length ys = Suc (length list)\<rbrakk>
+   \<Longrightarrow> \<forall>k<length list. rec_calc_rel (list ! k) lm (butlast ys ! k)"
+apply(rule allI, rule impI)
+apply(erule_tac  x = k in allE, simp add: nth_append)
+apply(subgoal_tac "ys ! k = butlast ys ! k", simp)
+apply(rule_tac nth_eq_butlast_nth, arith)
+done
+
+
+thm cn_merge_gs.simps
+lemma cn_merge_gs_tl_app: 
+  "cn_merge_gs (gs @ [g]) pstr = 
+        cn_merge_gs gs pstr [+] cn_merge_gs [g] (pstr + length gs)"
+apply(induct gs arbitrary: pstr, simp add: cn_merge_gs.simps, simp)
+apply(case_tac a, simp add: abc_append_commute)
+done
+
+lemma cn_merge_gs_length: 
+  "length (cn_merge_gs (map rec_ci list) pstr) = 
+      (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci list. length ap) + 3 * length list "
+apply(induct list arbitrary: pstr, simp, simp)
+apply(case_tac "rec_ci a", simp)
+done
+
+lemma [simp]: "Suc n \<le> pstr \<Longrightarrow> pstr + x - n > 0"
+by arith
+
+lemma [simp]:
+  "\<lbrakk>Suc (pstr + length list) \<le> a_md; 
+    length ys = Suc (length list);
+    length lm = n;
+     Suc n \<le> pstr\<rbrakk>
+   \<Longrightarrow>  (ys ! length list # 0\<^bsup>pstr - Suc n\<^esup> @ butlast ys @
+             0\<^bsup>a_md - (pstr + length list)\<^esup> @ suf_lm) ! 
+                      (pstr + length list - n) = (0 :: nat)"
+apply(insert nth_append[of "ys ! length list # 0\<^bsup>pstr - Suc n\<^esup> @
+     butlast ys" "0\<^bsup>a_md - (pstr + length list)\<^esup> @ suf_lm"
+      "(pstr + length list - n)"], simp add: nth_append)
+done
+
+lemma [simp]:
+  "\<lbrakk>Suc (pstr + length list) \<le> a_md; 
+    length ys = Suc (length list);
+    length lm = n;
+     Suc n \<le> pstr\<rbrakk>
+    \<Longrightarrow> (lm @ last ys # 0\<^bsup>pstr - Suc n\<^esup> @ butlast ys @
+         0\<^bsup>a_md - (pstr + length list)\<^esup> @ suf_lm)[pstr + length list := 
+                                        last ys, n := 0] =
+        lm @ 0::nat\<^bsup>pstr - n\<^esup> @ ys @ 0\<^bsup>a_md - Suc (pstr + length list)\<^esup> @ suf_lm"
+apply(insert list_update_length[of 
+   "lm @ last ys # 0\<^bsup>pstr - Suc n\<^esup> @ butlast ys" 0 
+   "0\<^bsup>a_md - Suc (pstr + length list)\<^esup> @ suf_lm" "last ys"], simp)
+apply(simp add: exponent_cons_iff)
+apply(insert list_update_length[of "lm" 
+        "last ys" "0\<^bsup>pstr - Suc n\<^esup> @ butlast ys @ 
+      last ys # 0\<^bsup>a_md - Suc (pstr + length list)\<^esup> @ suf_lm" 0], simp)
+apply(simp add: exponent_cons_iff)
+apply(case_tac "ys = []", simp_all add: append_butlast_last_id)
+done
+
+
+lemma cn_merge_gs_ex: 
+  "\<lbrakk>\<And>x aprog a_md rs_pos rs suf_lm lm.
+    \<lbrakk>x \<in> set gs; rec_ci x = (aprog, rs_pos, a_md);
+     rec_calc_rel x lm rs\<rbrakk>
+     \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp 
+           = (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm); 
+   pstr + length gs\<le> a_md;
+   \<forall>k<length gs. rec_calc_rel (gs ! k) lm (ys ! k);
+   length ys = length gs; length lm = n;
+   pstr \<ge> Max (set (Suc n # map (\<lambda>(aprog, p, n). n) (map rec_ci gs)))\<rbrakk>
+  \<Longrightarrow> \<exists> stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - n\<^esup> @ suf_lm)
+                   (cn_merge_gs (map rec_ci gs) pstr) stp 
+   = (listsum (map ((\<lambda>(ap, pos, n). length ap) \<circ> rec_ci) gs) +
+  3 * length gs, lm @ 0\<^bsup>pstr - n\<^esup> @ ys @ 0\<^bsup>a_md - (pstr + length gs)\<^esup> @ suf_lm)"
+apply(induct gs arbitrary: ys rule: list_tl_induct)
+apply(simp add: exponent_add_iff, simp)
+proof -
+  fix a list ys
+  assume ind: "\<And>x aprog a_md rs_pos rs suf_lm lm.
+    \<lbrakk>x = a \<or> x \<in> set list; rec_ci x = (aprog, rs_pos, a_md); 
+     rec_calc_rel x lm rs\<rbrakk>
+    \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp =
+                (length aprog, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+  and ind2: 
+    "\<And>ys. \<lbrakk>\<And>x aprog a_md rs_pos rs suf_lm lm.
+    \<lbrakk>x \<in> set list; rec_ci x = (aprog, rs_pos, a_md);
+     rec_calc_rel x lm rs\<rbrakk>
+    \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp
+        = (length aprog, lm @ rs # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm);
+    \<forall>k<length list. rec_calc_rel (list ! k) lm (ys ! k); 
+    length ys = length list\<rbrakk>
+    \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - n\<^esup> @ suf_lm) 
+                   (cn_merge_gs (map rec_ci list) pstr) stp =
+    (listsum (map ((\<lambda>(ap, pos, n). length ap) \<circ> rec_ci) list) +
+     3 * length list,
+                lm @ 0\<^bsup>pstr - n\<^esup> @ ys @ 0\<^bsup>a_md - (pstr + length list)\<^esup> @ suf_lm)"
+    and h: "Suc (pstr + length list) \<le> a_md" 
+            "\<forall>k<Suc (length list). 
+                   rec_calc_rel ((list @ [a]) ! k) lm (ys ! k)" 
+            "length ys = Suc (length list)" 
+            "length lm = n"
+            "Suc n \<le> pstr \<and> (\<lambda>(aprog, p, n). n) (rec_ci a) \<le> pstr \<and> 
+            (\<forall>a\<in>set list. (\<lambda>(aprog, p, n). n) (rec_ci a) \<le> pstr)"
+  from h have k1: 
+    "\<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - n\<^esup> @ suf_lm)
+                     (cn_merge_gs (map rec_ci list) pstr) stp =
+    (listsum (map ((\<lambda>(ap, pos, n). length ap) \<circ> rec_ci) list) +
+     3 * length list, lm @ 0\<^bsup>pstr - n\<^esup> @ butlast ys @
+                               0\<^bsup>a_md - (pstr + length list)\<^esup> @ suf_lm) "
+    apply(rule_tac ind2)
+    apply(rule_tac ind, auto)
+    done
+  from k1 and h show 
+    "\<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - n\<^esup> @ suf_lm) 
+          (cn_merge_gs (map rec_ci list @ [rec_ci a]) pstr) stp =
+        (listsum (map ((\<lambda>(ap, pos, n). length ap) \<circ> rec_ci) list) + 
+        (\<lambda>(ap, pos, n). length ap) (rec_ci a) + (3 + 3 * length list),
+             lm @ 0\<^bsup>pstr - n\<^esup> @ ys @ 0\<^bsup>a_md - Suc (pstr + length list)\<^esup> @ suf_lm)"
+    apply(simp add: cn_merge_gs_tl_app)
+    thm abc_append_exc2
+    apply(rule_tac as = 
+  "(\<Sum>(ap, pos, n)\<leftarrow>map rec_ci list. length ap) + 3 * length list"    
+      and bm = "lm @ 0\<^bsup>pstr - n\<^esup> @ butlast ys @ 
+                              0\<^bsup>a_md - (pstr + length list)\<^esup> @ suf_lm" 
+      and bs = "(\<lambda>(ap, pos, n). length ap) (rec_ci a) + 3" 
+      and bm' = "lm @ 0\<^bsup>pstr - n\<^esup> @ ys @ 0\<^bsup>a_md - Suc (pstr + length list)\<^esup> @ 
+                                  suf_lm" in abc_append_exc2, simp)
+    apply(simp add: cn_merge_gs_length)
+  proof -
+    from h show 
+      "\<exists>bstp. abc_steps_l (0, lm @ 0\<^bsup>pstr - n\<^esup> @ butlast ys @ 
+                                  0\<^bsup>a_md - (pstr + length list)\<^esup> @ suf_lm) 
+              ((\<lambda>(gprog, gpara, gn). gprog [+] recursive.empty gpara 
+              (pstr + length list)) (rec_ci a)) bstp =
+              ((\<lambda>(ap, pos, n). length ap) (rec_ci a) + 3, 
+             lm @ 0\<^bsup>pstr - n\<^esup> @ ys @ 0\<^bsup>a_md - Suc (pstr + length list)\<^esup> @ suf_lm)"
+      apply(case_tac "rec_ci a", simp)
+      apply(rule_tac as = "length aa" and 
+                     bm = "lm @ (ys ! (length list)) # 
+          0\<^bsup>pstr - Suc n\<^esup> @ butlast ys @ 0\<^bsup>a_md - (pstr + length list)\<^esup> @ suf_lm" 
+        and bs = "3" and bm' = "lm @ 0\<^bsup>pstr - n\<^esup> @ ys @
+             0\<^bsup>a_md - Suc (pstr + length list)\<^esup> @ suf_lm" in abc_append_exc2)
+    proof -
+      fix aa b c
+      assume g: "rec_ci a = (aa, b, c)"
+      from h and g have k2: "b = n"
+	apply(erule_tac x = "length list" in allE, simp)
+	apply(subgoal_tac "length lm = b", simp)
+	apply(rule para_pattern, simp, simp)
+	done
+      from h and g and this show 
+        "\<exists>astp. abc_steps_l (0, lm @ 0\<^bsup>pstr - n\<^esup> @ butlast ys @ 
+                         0\<^bsup>a_md - (pstr + length list)\<^esup> @ suf_lm) aa astp =
+        (length aa, lm @ ys ! length list # 0\<^bsup>pstr - Suc n\<^esup> @ 
+                       butlast ys @ 0\<^bsup>a_md - (pstr + length list)\<^esup> @ suf_lm)"
+	apply(subgoal_tac "c \<ge> Suc n")
+	apply(insert ind[of a aa b c lm "ys ! length list" 
+     "0\<^bsup>pstr - c\<^esup> @ butlast ys @ 0\<^bsup>a_md - (pstr + length list)\<^esup> @ suf_lm"], simp)
+	apply(erule_tac x = "length list" in allE, 
+              simp add: exponent_add_iff)
+	apply(rule_tac Suc_leI, rule_tac ci_ad_ge_paras, simp)
+	done
+    next
+      fix aa b c
+      show "length aa = length aa" by simp 
+    next
+      fix aa b c
+      assume "rec_ci a = (aa, b, c)"
+      from h and this show     
+      "\<exists>bstp. abc_steps_l (0, lm @ ys ! length list #
+          0\<^bsup>pstr - Suc n\<^esup> @ butlast ys @ 0\<^bsup>a_md - (pstr + length list)\<^esup> @ suf_lm)
+                 (recursive.empty b (pstr + length list)) bstp =
+       (3, lm @ 0\<^bsup>pstr - n\<^esup> @ ys @ 0\<^bsup>a_md - Suc (pstr + length list)\<^esup> @ suf_lm)"
+	apply(insert empty_ex [of b "pstr + length list" 
+         "lm @ ys ! length list # 0\<^bsup>pstr - Suc n\<^esup> @ butlast ys @ 
+         0\<^bsup>a_md - (pstr + length list)\<^esup> @ suf_lm"], simp)
+        apply(subgoal_tac "b = n")
+	apply(simp add: nth_append split: if_splits)
+	apply(erule_tac x = "length list" in allE, simp)
+        apply(drule para_pattern, simp, simp)
+	done
+    next
+      fix aa b c
+      show "3 = length (recursive.empty b (pstr + length list))" 
+        by simp
+    next
+      fix aa b aaa ba
+      show "length aa + 3 = length aa + 3" by simp
+    next
+      fix aa b c
+      show "empty b (pstr + length list) \<noteq> []" 
+        by(simp add: empty.simps)
+    qed
+  next
+    show "(\<lambda>(ap, pos, n). length ap) (rec_ci a) + 3 = 
+        length ((\<lambda>(gprog, gpara, gn). gprog [+]
+           recursive.empty gpara (pstr + length list)) (rec_ci a))"
+      by(case_tac "rec_ci a", simp)
+  next
+    show "listsum (map ((\<lambda>(ap, pos, n). length ap) \<circ> rec_ci) list) +
+      (\<lambda>(ap, pos, n). length ap) (rec_ci a) + (3 + 3 * length list)=
+      (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci list. length ap) + 3 * length list + 
+                ((\<lambda>(ap, pos, n). length ap) (rec_ci a) + 3)" by simp
+  next
+    show "(\<lambda>(gprog, gpara, gn). gprog [+] 
+      recursive.empty gpara (pstr + length list)) (rec_ci a) \<noteq> []"
+      by(case_tac "rec_ci a", 
+         simp add: abc_append.simps abc_shift.simps)
+  qed
+qed
+   
+declare drop_abc_lm_v_simp[simp del]
+
+lemma [simp]: "length (mv_boxes aa ba n) = 3*n"
+by(induct n, auto simp: mv_boxes.simps)
+
+lemma exp_suc: "a\<^bsup>Suc b\<^esup> = a\<^bsup>b\<^esup> @ [a]"
+by(simp add: exponent_def rep_ind del: replicate.simps)
+
+lemma [simp]: 
+  "\<lbrakk>Suc n \<le> ba - aa;  length lm2 = Suc n;
+    length lm3 = ba - Suc (aa + n)\<rbrakk>
+  \<Longrightarrow> (last lm2 # lm3 @ butlast lm2 @ 0 # lm4) ! (ba - aa) = (0::nat)"
+proof -
+  assume h: "Suc n \<le> ba - aa"
+  and g: "length lm2 = Suc n" "length lm3 = ba - Suc (aa + n)"
+  from h and g have k: "ba - aa = Suc (length lm3 + n)"
+    by arith
+  from  k show 
+    "(last lm2 # lm3 @ butlast lm2 @ 0 # lm4) ! (ba - aa) = 0"
+    apply(simp, insert g)
+    apply(simp add: nth_append)
+    done
+qed
+
+lemma [simp]: "length lm1 = aa \<Longrightarrow>
+      (lm1 @ 0\<^bsup>n\<^esup> @ last lm2 # lm3 @ butlast lm2 @ 0 # lm4) ! (aa + n) = last lm2"
+apply(simp add: nth_append)
+done
+
+lemma [simp]: "\<lbrakk>Suc n \<le> ba - aa; aa < ba\<rbrakk> \<Longrightarrow> 
+                    (ba < Suc (aa + (ba - Suc (aa + n) + n))) = False"
+apply arith
+done
+
+lemma [simp]: "\<lbrakk>Suc n \<le> ba - aa; aa < ba; length lm1 = aa; 
+       length lm2 = Suc n; length lm3 = ba - Suc (aa + n)\<rbrakk>
+     \<Longrightarrow> (lm1 @ 0\<^bsup>n\<^esup> @ last lm2 # lm3 @ butlast lm2 @ 0 # lm4) ! (ba + n) = 0"
+using nth_append[of "lm1 @ 0\<Colon>'a\<^bsup>n\<^esup> @ last lm2 # lm3 @ butlast lm2" 
+                     "(0\<Colon>'a) # lm4" "ba + n"]
+apply(simp)
+done
+
+lemma [simp]: 
+ "\<lbrakk>Suc n \<le> ba - aa; aa < ba; length lm1 = aa; length lm2 = Suc n;
+                 length lm3 = ba - Suc (aa + n)\<rbrakk>
+  \<Longrightarrow> (lm1 @ 0\<^bsup>n\<^esup> @ last lm2 # lm3 @ butlast lm2 @ (0::nat) # lm4)
+  [ba + n := last lm2, aa + n := 0] = 
+  lm1 @ 0 # 0\<^bsup>n\<^esup> @ lm3 @ lm2 @ lm4"
+using list_update_append[of "lm1 @ 0\<^bsup>n\<^esup> @ last lm2 # lm3 @ butlast lm2" "0 # lm4" 
+                            "ba + n" "last lm2"]
+apply(simp)
+apply(simp add: list_update_append)
+apply(simp only: exponent_cons_iff exp_suc, simp)
+apply(case_tac lm2, simp, simp)
+done
+
+
+lemma mv_boxes_ex:
+  "\<lbrakk>n \<le> ba - aa; ba > aa; length lm1 = aa; 
+    length (lm2::nat list) = n; length lm3 = ba - aa - n\<rbrakk>
+     \<Longrightarrow> \<exists> stp. abc_steps_l (0, lm1 @ lm2 @ lm3 @ 0\<^bsup>n\<^esup> @ lm4)
+       (mv_boxes aa ba n) stp = (3 * n, lm1 @ 0\<^bsup>n\<^esup> @ lm3 @ lm2 @ lm4)"
+apply(induct n arbitrary: lm2 lm3 lm4, simp)
+apply(rule_tac x = 0 in exI, simp add: abc_steps_zero, 
+              simp add: mv_boxes.simps del: exp_suc_iff)
+apply(rule_tac as = "3 *n" and bm = "lm1 @ 0\<^bsup>n\<^esup> @ last lm2 # lm3 @
+               butlast lm2 @ 0 # lm4" in abc_append_exc2, simp_all)
+apply(simp only: exponent_cons_iff, simp only: exp_suc, simp)
+proof -
+  fix n lm2 lm3 lm4
+  assume ind:
+    "\<And>lm2 lm3 lm4. \<lbrakk>length lm2 = n; length lm3 = ba - (aa + n)\<rbrakk> \<Longrightarrow>
+    \<exists>stp. abc_steps_l (0, lm1 @ lm2 @ lm3 @ 0\<^bsup>n\<^esup> @ lm4) 
+       (mv_boxes aa ba n) stp = (3 * n, lm1 @ 0\<^bsup>n\<^esup> @ lm3 @ lm2 @ lm4)"
+  and h: "Suc n \<le> ba - aa" "aa < ba" "length (lm1::nat list) = aa" 
+         "length (lm2::nat list) = Suc n" 
+         "length (lm3::nat list) = ba - Suc (aa + n)"
+  from h show 
+    "\<exists>astp. abc_steps_l (0, lm1 @ lm2 @ lm3 @ 0\<^bsup>n\<^esup> @ 0 # lm4) 
+                       (mv_boxes aa ba n) astp = 
+        (3 * n, lm1 @ 0\<^bsup>n\<^esup> @ last lm2 # lm3 @ butlast lm2 @ 0 # lm4)"
+    apply(insert ind[of "butlast lm2" "last lm2 # lm3" "0 # lm4"], 
+          simp)
+    apply(subgoal_tac "lm1 @ butlast lm2 @ last lm2 # lm3 @ 0\<^bsup>n\<^esup> @ 
+              0 # lm4 = lm1 @ lm2 @ lm3 @ 0\<^bsup>n\<^esup> @ 0 # lm4", simp, simp)
+    apply(case_tac "lm2 = []", simp, simp)
+    done
+next
+  fix n lm2 lm3 lm4
+  assume h: "Suc n \<le> ba - aa"
+            "aa < ba" 
+            "length (lm1::nat list) = aa" 
+            "length (lm2::nat list) = Suc n" 
+            "length (lm3::nat list) = ba - Suc (aa + n)"
+  thus " \<exists>bstp. abc_steps_l (0, lm1 @ 0\<^bsup>n\<^esup> @ last lm2 # lm3 @
+                       butlast lm2 @ 0 # lm4) 
+                         (recursive.empty (aa + n) (ba + n)) bstp
+               = (3, lm1 @ 0 # 0\<^bsup>n\<^esup> @ lm3 @ lm2 @ lm4)"
+    apply(insert empty_ex[of "aa + n" "ba + n" 
+       "lm1 @ 0\<^bsup>n\<^esup> @ last lm2 # lm3 @ butlast lm2 @ 0 # lm4"], simp)
+    done
+qed
+(*    
+lemma [simp]: "\<lbrakk>Suc n \<le> aa - ba; 
+                ba < aa; 
+               length lm2 = aa - Suc (ba + n)\<rbrakk>
+      \<Longrightarrow> ((0::nat) # lm2 @ 0\<^bsup>n\<^esup> @ last lm3 # lm4) ! (aa - ba)
+         = last lm3"
+proof -
+  assume h: "Suc n \<le> aa - ba"
+    and g: " ba < aa" "length lm2 = aa - Suc (ba + n)"
+  from h and g have k: "aa - ba = Suc (length lm2 + n)"
+    by arith
+  thus "((0::nat) # lm2 @ 0\<^bsup>n\<^esup> @ last lm3 # lm4) ! (aa - ba) = last lm3"
+    apply(simp,  simp add: nth_append)
+    done
+qed
+*)
+
+lemma [simp]: "\<lbrakk>Suc n \<le> aa - ba; ba < aa; length lm1 = ba; 
+        length lm2 = aa - Suc (ba + n); length lm3 = Suc n\<rbrakk>
+   \<Longrightarrow> (lm1 @ butlast lm3 @ 0 # lm2 @ 0\<^bsup>n\<^esup> @ last lm3 # lm4) ! (aa + n) = last lm3"
+using nth_append[of "lm1 @ butlast lm3 @ 0 # lm2 @ 0\<^bsup>n\<^esup>" "last lm3 # lm4" "aa + n"]
+apply(simp)
+done
+
+lemma [simp]: "\<lbrakk>Suc n \<le> aa - ba; ba < aa; length lm1 = ba; 
+        length lm2 = aa - Suc (ba + n); length lm3 = Suc n\<rbrakk>
+     \<Longrightarrow> (lm1 @ butlast lm3 @ 0 # lm2 @ 0\<^bsup>n\<^esup> @ last lm3 # lm4) ! (ba + n) = 0"
+apply(simp add: nth_append)
+done
+
+
+lemma [simp]: "\<lbrakk>Suc n \<le> aa - ba; ba < aa; length lm1 = ba; 
+        length lm2 = aa - Suc (ba + n); length lm3 = Suc n\<rbrakk> 
+     \<Longrightarrow> (lm1 @ butlast lm3 @ 0 # lm2 @ 0\<^bsup>n\<^esup> @ last lm3 # lm4)[ba + n := last lm3, aa + n := 0]
+      = lm1 @ lm3 @ lm2 @ 0 # 0\<^bsup>n\<^esup> @ lm4"
+using list_update_append[of "lm1 @ butlast lm3" "(0\<Colon>'a) # lm2 @ 0\<Colon>'a\<^bsup>n\<^esup> @ last lm3 # lm4"]
+apply(simp)
+using list_update_append[of "lm1 @ butlast lm3 @ last lm3 # lm2 @ 0\<Colon>'a\<^bsup>n\<^esup>"
+                            "last lm3 # lm4" "aa + n" "0"]
+apply(simp)
+apply(simp only: exp_ind_def[THEN sym] exp_suc, simp)
+apply(case_tac lm3, simp, simp)
+done
+
+
+lemma mv_boxes_ex2:
+  "\<lbrakk>n \<le> aa - ba; 
+    ba < aa; 
+    length (lm1::nat list) = ba;
+    length (lm2::nat list) = aa - ba - n; 
+    length (lm3::nat list) = n\<rbrakk>
+     \<Longrightarrow> \<exists> stp. abc_steps_l (0, lm1 @ 0\<^bsup>n\<^esup> @ lm2 @ lm3 @ lm4) 
+                (mv_boxes aa ba n) stp =
+                    (3 * n, lm1 @ lm3 @ lm2 @ 0\<^bsup>n\<^esup> @ lm4)"
+apply(induct n arbitrary: lm2 lm3 lm4, simp)
+apply(rule_tac x = 0 in exI, simp add: abc_steps_zero, 
+                   simp add: mv_boxes.simps del: exp_suc_iff)
+apply(rule_tac as = "3 *n" and bm = "lm1 @ butlast lm3 @ 0 # lm2 @
+                  0\<^bsup>n\<^esup> @ last lm3 # lm4" in abc_append_exc2, simp_all)
+apply(simp only: exponent_cons_iff, simp only: exp_suc, simp)
+proof -
+  fix n lm2 lm3 lm4
+  assume ind: 
+"\<And>lm2 lm3 lm4. \<lbrakk>length lm2 = aa - (ba + n); length lm3 = n\<rbrakk> \<Longrightarrow> 
+  \<exists>stp. abc_steps_l (0, lm1 @ 0\<^bsup>n\<^esup> @ lm2 @ lm3 @ lm4) 
+                 (mv_boxes aa ba n) stp = 
+                            (3 * n, lm1 @ lm3 @ lm2 @ 0\<^bsup>n\<^esup> @ lm4)"
+  and h: "Suc n \<le> aa - ba" 
+         "ba < aa"  
+         "length (lm1::nat list) = ba" 
+         "length (lm2::nat list) = aa - Suc (ba + n)" 
+         "length (lm3::nat list) = Suc n"
+  from h show
+    "\<exists>astp. abc_steps_l (0, lm1 @ 0\<^bsup>n\<^esup> @ 0 # lm2 @ lm3 @ lm4)
+        (mv_boxes aa ba n) astp = 
+          (3 * n, lm1 @ butlast lm3 @ 0 # lm2 @ 0\<^bsup>n\<^esup> @ last lm3 # lm4)"
+    apply(insert ind[of "0 # lm2" "butlast lm3" "last lm3 # lm4"],
+          simp)
+    apply(subgoal_tac
+      "lm1 @ 0\<^bsup>n\<^esup> @ 0 # lm2 @ butlast lm3 @ last lm3 # lm4 =
+           lm1 @ 0\<^bsup>n\<^esup> @ 0 # lm2 @ lm3 @ lm4", simp, simp)
+    apply(case_tac "lm3 = []", simp, simp)
+    done
+next
+  fix n lm2 lm3 lm4
+  assume h:
+    "Suc n \<le> aa - ba" 
+    "ba < aa"
+    "length lm1 = ba"
+    "length (lm2::nat list) = aa - Suc (ba + n)" 
+    "length (lm3::nat list) = Suc n"
+  thus
+    "\<exists>bstp. abc_steps_l (0, lm1 @ butlast lm3 @ 0 # lm2 @ 0\<^bsup>n\<^esup> @ 
+                               last lm3 # lm4) 
+           (recursive.empty (aa + n) (ba + n)) bstp =
+                 (3, lm1 @ lm3 @ lm2 @ 0 # 0\<^bsup>n\<^esup> @ lm4)"
+    apply(insert empty_ex[of "aa + n" "ba + n" "lm1 @ butlast lm3 @ 
+                          0 # lm2 @ 0\<^bsup>n\<^esup> @ last lm3 # lm4"], simp)
+    done
+qed
+
+lemma cn_merge_gs_len: 
+  "length (cn_merge_gs (map rec_ci gs) pstr) = 
+      (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 3 * length gs"
+apply(induct gs arbitrary: pstr, simp, simp)
+apply(case_tac "rec_ci a", simp)
+done
+
+lemma [simp]: "n < pstr \<Longrightarrow>
+     Suc (pstr + length ys - n) = Suc (pstr + length ys) - n"
+by arith
+
+lemma save_paras':  
+  "\<lbrakk>length lm = n; pstr > n; a_md > pstr + length ys + n\<rbrakk>
+  \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>pstr - n\<^esup> @ ys @
+               0\<^bsup>a_md - pstr - length ys\<^esup> @ suf_lm) 
+                 (mv_boxes 0 (pstr + Suc (length ys)) n) stp
+        = (3 * n, 0\<^bsup>pstr\<^esup> @ ys @ 0 # lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)"
+thm mv_boxes_ex
+apply(insert mv_boxes_ex[of n "pstr + Suc (length ys)" 0 "[]" "lm" 
+         "0\<^bsup>pstr - n\<^esup> @ ys @ [0]" "0\<^bsup>a_md - pstr - length ys - n - Suc 0\<^esup> @ suf_lm"], simp)
+apply(erule_tac exE, rule_tac x = stp in exI,
+                            simp add: exponent_add_iff)
+apply(simp only: exponent_cons_iff, simp)
+done
+
+lemma [simp]:
+ "(max ba (Max (insert ba (((\<lambda>(aprog, p, n). n) o rec_ci) ` set gs))))
+ = (Max (insert ba (((\<lambda>(aprog, p, n). n) o rec_ci) ` set gs)))"
+apply(rule min_max.sup_absorb2, auto)
+done
+
+lemma [simp]:
+  "((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs) = 
+                  (((\<lambda>(aprog, p, n). n) o rec_ci) ` set gs)"
+apply(induct gs)
+apply(simp, simp)
+done
+
+lemma ci_cn_md_def:  
+  "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md); 
+  rec_ci f = (a, aa, ba)\<rbrakk>
+    \<Longrightarrow> a_md = max (Suc n) (Max (insert ba (((\<lambda>(aprog, p, n). n) o 
+  rec_ci) ` set gs))) + Suc (length gs) + n"
+apply(simp add: rec_ci.simps, auto)
+done
+
+lemma save_paras_prog_ex:
+  "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md); 
+    rec_ci f = (a, aa, ba); 
+    pstr = Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n)
+                                    (map rec_ci (f # gs))))\<rbrakk>
+    \<Longrightarrow> \<exists>ap bp cp. 
+      aprog = ap [+] bp [+] cp \<and>
+      length ap = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 
+              3 * length gs \<and> bp = mv_boxes 0 (pstr + Suc (length gs)) n"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = 
+  "cn_merge_gs (map rec_ci gs) (max (Suc n) (Max (insert ba 
+      (((\<lambda>(aprog, p, n). n) o rec_ci) ` set gs))))" in exI,
+      simp add: cn_merge_gs_len)
+apply(rule_tac x = 
+  "mv_boxes (max (Suc n) (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))))
+   0 (length gs) [+] a [+]recursive.empty aa (max (Suc n) 
+   (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) [+]
+   empty_boxes (length gs) [+] recursive.empty (max (Suc n) 
+  (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) n [+]
+   mv_boxes (Suc (max (Suc n) (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) 
+   ` set gs))) + length gs)) 0 n" in exI, auto)
+apply(simp add: abc_append_commute)
+done
+
+lemma save_paras: 
+  "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md);
+    rs_pos = n;
+    \<forall>k<length gs. rec_calc_rel (gs ! k) lm (ys ! k);
+    length ys = length gs;
+    length lm = n;
+    rec_ci f = (a, aa, ba);
+    pstr = Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n)
+                                          (map rec_ci (f # gs))))\<rbrakk>
+  \<Longrightarrow> \<exists>stp. abc_steps_l ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) +
+          3 * length gs, lm @ 0\<^bsup>pstr - n\<^esup> @ ys @
+                 0\<^bsup>a_md - pstr - length ys\<^esup> @ suf_lm) aprog stp = 
+           ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) +
+                      3 * length gs + 3 * n, 
+             0\<^bsup>pstr\<^esup> @ ys @ 0 # lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup>  @ suf_lm)"
+proof -
+  assume h:
+    "rec_ci (Cn n f gs) = (aprog, rs_pos, a_md)"
+    "rs_pos = n" 
+    "\<forall>k<length gs. rec_calc_rel (gs ! k) lm (ys ! k)" 
+    "length ys = length gs"  
+    "length lm = n"    
+    "rec_ci f = (a, aa, ba)"
+    and g: "pstr = Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n) 
+                                        (map rec_ci (f # gs))))"
+  from h and g have k1: 
+    "\<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and> 
+    length ap = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 
+                3 *length gs \<and> bp = mv_boxes 0 (pstr + Suc (length ys)) n"
+    apply(drule_tac save_paras_prog_ex, auto)
+    done
+  from h have k2: 
+    "\<exists> stp. abc_steps_l (0, lm @ 0\<^bsup>pstr - n\<^esup> @ ys @ 
+                         0\<^bsup>a_md - pstr - length ys\<^esup> @ suf_lm)
+         (mv_boxes 0 (pstr + Suc (length ys)) n) stp = 
+        (3 * n, 0\<^bsup>pstr\<^esup> @ ys @ 0 # lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup>  @ suf_lm)"
+    apply(rule_tac save_paras', simp, simp_all add: g)
+    apply(drule_tac a = a and aa = aa and ba = ba in 
+                                        ci_cn_md_def, simp, simp)
+    done
+  from k1 show 
+    "\<exists>stp. abc_steps_l ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 
+         3 * length gs, lm @ 0\<^bsup>pstr - n\<^esup> @ ys @ 
+                 0\<^bsup>a_md - pstr - length ys\<^esup> @ suf_lm) aprog stp =
+             ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 
+               3 * length gs + 3 * n, 
+                0\<^bsup>pstr\<^esup> @ ys @ 0 # lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)"
+  proof(erule_tac exE, erule_tac exE, erule_tac exE, erule_tac exE)
+    fix ap bp apa cp
+    assume "aprog = ap [+] bp [+] cp \<and> length ap = 
+            (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 3 * length gs
+            \<and> bp = mv_boxes 0 (pstr + Suc (length ys)) n"
+    from this and k2 show "?thesis"
+      apply(simp)
+      apply(rule_tac abc_append_exc1, simp, simp, simp)
+      done
+  qed
+qed
+ 
+lemma ci_cn_para_eq:
+  "rec_ci (Cn n f gs) = (aprog, rs_pos, a_md) \<Longrightarrow> rs_pos = n"
+apply(simp add: rec_ci.simps, case_tac "rec_ci f", simp)
+done
+
+lemma calc_gs_prog_ex: 
+  "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md); 
+    rec_ci f = (a, aa, ba);
+    Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n) 
+                         (map rec_ci (f # gs)))) = pstr\<rbrakk>
+   \<Longrightarrow> \<exists>ap bp. aprog = ap [+] bp \<and> 
+                 ap = cn_merge_gs (map rec_ci gs) pstr"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "mv_boxes 0 (Suc (max (Suc n)  
+   (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))) + length gs)) n [+]
+   mv_boxes (max (Suc n) (Max (insert ba 
+  (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) 0 (length gs) [+]
+   a [+] recursive.empty aa (max (Suc n)
+    (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) [+]
+   empty_boxes (length gs) [+] recursive.empty (max (Suc n)
+    (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) n [+]
+    mv_boxes (Suc (max (Suc n) (Max 
+    (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))) + length gs)) 0 n"
+   in exI)
+apply(auto simp: abc_append_commute)
+done
+
+lemma cn_calc_gs: 
+  assumes ind: 
+  "\<And>x aprog a_md rs_pos rs suf_lm lm.
+  \<lbrakk>x \<in> set gs; 
+   rec_ci x = (aprog, rs_pos, a_md); 
+   rec_calc_rel x lm rs\<rbrakk>
+  \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp = 
+     (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+  and h:  "rec_ci (Cn n f gs) = (aprog, rs_pos, a_md)"  
+          "\<forall>k<length gs. rec_calc_rel (gs ! k) lm (ys ! k)"
+          "length ys = length gs" 
+          "length lm = n" 
+          "rec_ci f = (a, aa, ba)" 
+          "Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n) 
+                               (map rec_ci (f # gs)))) = pstr"
+  shows  
+  "\<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp =
+  ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 3 * length gs, 
+   lm @ 0\<^bsup>pstr - n\<^esup> @ ys @ 0\<^bsup>a_md -pstr - length ys\<^esup> @ suf_lm) "
+proof -
+  from h have k1:
+    "\<exists> ap bp. aprog = ap [+] bp \<and> ap = 
+                        cn_merge_gs (map rec_ci gs) pstr"
+    by(erule_tac calc_gs_prog_ex, auto)
+  from h have j1: "rs_pos = n"
+    by(simp add: ci_cn_para_eq)
+  from h have j2: "a_md \<ge> pstr"
+    by(drule_tac a = a and aa = aa and ba = ba in 
+                                ci_cn_md_def, simp, simp)
+  from h have j3: "pstr > n"
+    by(auto)    
+  from j1 and j2 and j3 and h have k2:
+    "\<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) 
+                         (cn_merge_gs (map rec_ci gs) pstr) stp 
+    = ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 3 * length gs, 
+                  lm @ 0\<^bsup>pstr - n\<^esup> @ ys @ 0\<^bsup>a_md - pstr - length ys\<^esup> @ suf_lm)"
+    apply(simp)
+    apply(rule_tac cn_merge_gs_ex, rule_tac ind, simp, simp, auto)
+    apply(drule_tac a = a and aa = aa and ba = ba in 
+                                 ci_cn_md_def, simp, simp)
+    apply(rule min_max.le_supI2, auto)
+    done
+  from k1 show "?thesis"
+  proof(erule_tac exE, erule_tac exE, simp)
+    fix ap bp
+    from k2 show 
+      "\<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm)
+           (cn_merge_gs (map rec_ci gs) pstr [+] bp) stp =
+      (listsum (map ((\<lambda>(ap, pos, n). length ap) \<circ> rec_ci) gs) +
+         3 * length gs, 
+         lm @ 0\<^bsup>pstr - n\<^esup> @ ys @ 0\<^bsup>a_md - (pstr + length ys)\<^esup> @ suf_lm)"
+      apply(insert abc_append_exc1[of 
+        "lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm" 
+        "(cn_merge_gs (map rec_ci gs) pstr)" 
+        "length (cn_merge_gs (map rec_ci gs) pstr)" 
+        "lm @ 0\<^bsup>pstr - n\<^esup> @ ys @ 0\<^bsup>a_md - pstr - length ys\<^esup> @ suf_lm" 0 
+        "[]" bp], simp add: cn_merge_gs_len)
+      done      
+  qed
+qed
+
+lemma reset_new_paras': 
+  "\<lbrakk>length lm = n; 
+    pstr > 0; 
+    a_md \<ge> pstr + length ys + n;
+     pstr > length ys\<rbrakk> \<Longrightarrow>
+   \<exists>stp. abc_steps_l (0, 0\<^bsup>pstr\<^esup> @ ys @ 0 # lm @  0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @
+          suf_lm) (mv_boxes pstr 0 (length ys)) stp =
+  (3 * length ys, ys @ 0\<^bsup>pstr\<^esup> @ 0 # lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)"
+thm mv_boxes_ex2
+apply(insert mv_boxes_ex2[of "length ys" "pstr" 0 "[]"
+     "0\<^bsup>pstr - length ys\<^esup>" "ys" 
+     "0 # lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm"], 
+     simp add: exponent_add_iff)
+done
+
+lemma [simp]:  
+  "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md);
+  rec_calc_rel f ys rs; rec_ci f = (a, aa, ba);
+  pstr = Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n) 
+               (map rec_ci (f # gs))))\<rbrakk>
+  \<Longrightarrow> length ys < pstr"
+apply(subgoal_tac "length ys = aa", simp)
+apply(subgoal_tac "aa < ba \<and> ba \<le> pstr", 
+      rule basic_trans_rules(22), auto)
+apply(rule min_max.le_supI2)
+apply(auto)
+apply(erule_tac para_pattern, simp)
+done
+
+lemma reset_new_paras_prog_ex: 
+  "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md); 
+   rec_ci f = (a, aa, ba);
+   Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n) 
+  (map rec_ci (f # gs)))) = pstr\<rbrakk>
+  \<Longrightarrow> \<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and> 
+  length ap = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) +
+           3 *length gs + 3 * n \<and> bp = mv_boxes pstr 0 (length gs)"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "cn_merge_gs (map rec_ci gs) (max (Suc n) 
+          (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) [+] 
+          mv_boxes 0 (Suc (max (Suc n) (Max (insert ba 
+           (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))) + length gs)) n" in exI, 
+       simp add: cn_merge_gs_len)
+apply(rule_tac x = "a [+]
+     recursive.empty aa (max (Suc n) (Max (insert ba 
+     (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) [+]
+     empty_boxes (length gs) [+] recursive.empty 
+     (max (Suc n) (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) n
+      [+] mv_boxes (Suc (max (Suc n) (Max (insert ba 
+     (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))) + length gs)) 0 n" in exI,
+       auto simp: abc_append_commute)
+done
+
+
+lemma reset_new_paras:
+       "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md);
+        rs_pos = n;
+        \<forall>k<length gs. rec_calc_rel (gs ! k) lm (ys ! k);
+        length ys = length gs;
+        length lm = n;
+        length ys = aa;
+        rec_ci f = (a, aa, ba);
+        pstr = Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n)
+                                    (map rec_ci (f # gs))))\<rbrakk>
+\<Longrightarrow> \<exists>stp. abc_steps_l ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 
+                                               3 * length gs + 3 * n,
+        0\<^bsup>pstr \<^esup>@ ys @ 0 # lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm) aprog stp =
+  ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 6 * length gs + 3 * n,
+           ys @ 0\<^bsup>pstr\<^esup> @ 0 # lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)"
+proof -
+  assume h:
+    "rec_ci (Cn n f gs) = (aprog, rs_pos, a_md)"
+    "rs_pos = n" 
+    "length ys = aa"
+    "\<forall>k<length gs. rec_calc_rel (gs ! k) lm (ys ! k)"
+    "length ys = length gs"  "length lm = n"    
+    "rec_ci f = (a, aa, ba)"
+    and g: "pstr = Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n)
+                                         (map rec_ci (f # gs))))"
+  thm rec_ci.simps
+  from h and g have k1:
+    "\<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and> length ap = 
+    (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) +
+          3 *length gs + 3 * n \<and> bp = mv_boxes pstr 0 (length ys)"
+    by(drule_tac reset_new_paras_prog_ex, auto)
+  from h have k2:
+    "\<exists> stp. abc_steps_l (0, 0\<^bsup>pstr \<^esup>@ ys @ 0 # lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @
+              suf_lm) (mv_boxes pstr 0 (length ys)) stp = 
+    (3 * (length ys), 
+     ys @ 0\<^bsup>pstr\<^esup> @ 0 # lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)"
+    apply(rule_tac reset_new_paras', simp)
+    apply(simp add: g)
+    apply(drule_tac a = a and aa = aa and ba = ba in ci_cn_md_def,
+      simp, simp add: g, simp)
+    apply(subgoal_tac "length gs = aa \<and> aa < ba \<and> ba \<le> pstr", arith,
+          simp add: para_pattern)
+    apply(insert g, auto intro: min_max.le_supI2)
+    done
+  from k1 show 
+    "\<exists>stp. abc_steps_l ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 3
+    * length gs + 3 * n, 0\<^bsup>pstr\<^esup> @ ys @ 0 # lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ 
+     suf_lm) aprog stp =
+    ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 6 * length gs +
+      3 * n, ys @ 0\<^bsup>pstr\<^esup> @ 0 # lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)"
+  proof(erule_tac exE, erule_tac exE, erule_tac exE, erule_tac exE)
+    fix ap bp apa cp
+    assume "aprog = ap [+] bp [+] cp \<and> length ap = 
+      (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 3 * length gs +
+                  3 * n \<and> bp = mv_boxes pstr 0 (length ys)"
+    from this and k2 show "?thesis"
+      apply(simp)
+      apply(drule_tac as = 
+        "(\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 3 * length gs +
+        3 * n" and ap = ap and cp = cp in abc_append_exc1, auto)
+      apply(rule_tac x = stp in exI, simp add: h)
+      using h
+      apply(simp)
+      done
+  qed
+qed
+
+thm rec_ci.simps 
+
+lemma calc_f_prog_ex: 
+  "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md);
+    rec_ci f = (a, aa, ba);
+    Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n) 
+                   (map rec_ci (f # gs)))) = pstr\<rbrakk>
+   \<Longrightarrow> \<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and>
+  length ap = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) +
+                                6 *length gs + 3 * n \<and> bp = a"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "cn_merge_gs (map rec_ci gs) (max (Suc n) (Max (insert ba
+     (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) [+] 
+     mv_boxes 0 (Suc (max (Suc n) (Max (insert ba 
+            (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))) + length gs)) n [+]
+     mv_boxes (max (Suc n) (Max (insert ba 
+      (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) 0 (length gs)" in exI,
+     simp add: cn_merge_gs_len)
+apply(rule_tac x = "recursive.empty aa (max (Suc n) (Max (insert ba 
+     (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) [+]
+     empty_boxes (length gs) [+] recursive.empty (max (Suc n) (
+     Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) n [+]
+     mv_boxes (Suc (max (Suc n) (Max (insert ba 
+     (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))) + length gs)) 0 n" in exI,
+  auto simp: abc_append_commute)
+done
+
+lemma calc_cn_f:
+  assumes ind:
+  "\<And>x aprog a_md rs_pos rs suf_lm lm.
+  \<lbrakk>x \<in> set (f # gs);
+  rec_ci x = (aprog, rs_pos, a_md); 
+  rec_calc_rel x lm rs\<rbrakk>
+  \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp =
+  (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+  and h: "rec_ci (Cn n f gs) = (aprog, rs_pos, a_md)"
+  "rec_calc_rel (Cn n f gs) lm rs"
+  "length ys = length gs"
+  "rec_calc_rel f ys rs"
+  "length lm = n"
+  "rec_ci f = (a, aa, ba)" 
+  and p: "pstr = Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n)
+                                (map rec_ci (f # gs))))"
+  shows "\<exists>stp. abc_steps_l   
+  ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 6 * length gs + 3 * n,
+  ys @ 0\<^bsup>pstr\<^esup> @ 0 # lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm) aprog stp =
+  ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 6 * length gs + 
+                3 * n + length a,
+  ys @ rs # 0\<^bsup>pstr\<^esup> @ lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)"
+proof -
+  from h have k1: 
+    "\<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and> 
+    length ap = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 
+    6 *length gs + 3 * n \<and> bp = a"
+    by(drule_tac calc_f_prog_ex, auto)
+  from h and k1 show "?thesis"
+  proof (erule_tac exE, erule_tac exE, erule_tac exE, erule_tac exE)
+    fix ap bp apa cp
+    assume
+      "aprog = ap [+] bp [+] cp \<and> 
+      length ap = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 
+      6 * length gs + 3 * n \<and> bp = a"
+    from h and this show "?thesis"
+      apply(simp, rule_tac abc_append_exc1, simp_all)
+      apply(insert ind[of f "a" aa ba ys rs 
+        "0\<^bsup>pstr - ba + length gs \<^esup> @ 0 # lm @ 
+        0\<^bsup>a_md - Suc (pstr + length gs + n)\<^esup> @ suf_lm"], simp)
+      apply(subgoal_tac "ba > aa \<and> aa = length gs\<and> pstr \<ge> ba", simp)
+      apply(simp add: exponent_add_iff)
+      apply(case_tac pstr, simp add: p)
+      apply(simp only: exp_suc, simp)
+      apply(rule conjI, rule ci_ad_ge_paras, simp, rule conjI)
+      apply(subgoal_tac "length ys = aa", simp,
+        rule para_pattern, simp, simp)
+      apply(insert p, simp)
+      apply(auto intro: min_max.le_supI2)
+      done
+  qed
+qed
+(*
+lemma [simp]: 
+  "\<lbrakk>pstr + length ys + n \<le> a_md; ys \<noteq> []\<rbrakk> \<Longrightarrow> 
+                          pstr < a_md + length suf_lm"
+apply(case_tac "length ys", simp)
+apply(arith)
+done
+*)
+lemma [simp]: 
+  "pstr > length ys 
+  \<Longrightarrow> (ys @ rs # 0\<^bsup>pstr\<^esup> @ lm @
+  0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm) ! pstr = (0::nat)"
+apply(simp add: nth_append)
+done
+
+(*
+lemma [simp]: "\<lbrakk>length ys < pstr; pstr - length ys = Suc x\<rbrakk>
+  \<Longrightarrow> pstr - Suc (length ys) = x"
+by arith
+*)
+lemma [simp]: "pstr > length ys \<Longrightarrow> 
+      (ys @ rs # 0\<^bsup>pstr\<^esup> @ lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)
+                                         [pstr := rs, length ys := 0] =
+       ys @ 0\<^bsup>pstr - length ys\<^esup> @ (rs::nat) # 0\<^bsup>length ys\<^esup> @ lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm"
+apply(auto simp: list_update_append)
+apply(case_tac "pstr - length ys",simp_all)
+using list_update_length[of 
+  "0\<^bsup>pstr - Suc (length ys)\<^esup>" "0" "0\<^bsup>length ys\<^esup> @ lm @ 
+  0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm" rs]
+apply(simp only: exponent_cons_iff exponent_add_iff, simp)
+apply(subgoal_tac "pstr - Suc (length ys) = nat", simp, simp)
+done
+
+lemma save_rs': 
+  "\<lbrakk>pstr > length ys\<rbrakk>
+  \<Longrightarrow> \<exists>stp. abc_steps_l (0, ys @ rs # 0\<^bsup>pstr\<^esup> @ lm @ 
+  0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm) 
+  (recursive.empty (length ys) pstr) stp =
+  (3, ys @ 0\<^bsup>pstr - (length ys)\<^esup> @ rs # 
+  0\<^bsup>length ys \<^esup> @ lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)"
+apply(insert empty_ex[of "length ys" pstr 
+  "ys @ rs # 0\<^bsup>pstr\<^esup> @ lm @ 0\<^bsup>a_md - Suc(pstr + length ys + n)\<^esup> @ suf_lm"], 
+  simp)
+done
+
+
+lemma save_rs_prog_ex:
+  "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md);
+  rec_ci f = (a, aa, ba);
+  Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n) 
+                        (map rec_ci (f # gs)))) = pstr\<rbrakk>
+  \<Longrightarrow> \<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and>
+  length ap = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) +
+              6 *length gs + 3 * n + length a
+  \<and> bp = empty aa pstr"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x =
+  "cn_merge_gs (map rec_ci gs) (max (Suc n) (Max (insert ba 
+   (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))))
+   [+] mv_boxes 0 (Suc (max (Suc n) (Max (insert ba 
+   (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))) + length gs)) n [+]
+   mv_boxes (max (Suc n) (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))))
+    0 (length gs) [+] a" 
+  in exI, simp add: cn_merge_gs_len)
+apply(rule_tac x = 
+  "empty_boxes (length gs) [+]
+   recursive.empty (max (Suc n) (Max (insert ba 
+    (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) n [+]
+   mv_boxes (Suc (max (Suc n) (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))
+    + length gs)) 0 n" in exI, 
+  auto simp: abc_append_commute)
+done
+
+lemma save_rs:  
+  assumes h: 
+  "rec_ci (Cn n f gs) = (aprog, rs_pos, a_md)" 
+  "rec_calc_rel (Cn n f gs) lm rs"
+  "\<forall>k<length gs. rec_calc_rel (gs ! k) lm (ys ! k)"
+  "length ys = length gs" 
+  "rec_calc_rel f ys rs" 
+  "rec_ci f = (a, aa, ba)"  
+  "length lm = n"
+  and pdef: "pstr = Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n) 
+                                            (map rec_ci (f # gs))))"
+  shows "\<exists>stp. abc_steps_l
+           ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 6 * length gs
+          + 3 * n + length a, ys @ rs # 0\<^bsup>pstr\<^esup> @ lm @
+             0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm) aprog stp =
+  ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 6 * length gs 
+  + 3 * n + length a + 3,
+  ys @ 0\<^bsup>pstr - length ys \<^esup> @ rs # 0\<^bsup>length ys\<^esup> @ lm @ 
+                               0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)"
+proof -
+  thm rec_ci.simps
+  from h and pdef have k1: 
+    "\<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and>
+    length ap = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) +
+    6 *length gs + 3 * n + length a \<and> bp = empty (length ys) pstr "
+    apply(subgoal_tac "length ys = aa")
+    apply(drule_tac a = a and aa = aa and ba = ba in save_rs_prog_ex, 
+      simp, simp, simp)
+    by(rule_tac para_pattern, simp, simp)
+  from k1 show 
+    "\<exists>stp. abc_steps_l
+    ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 6 * length gs + 3 * n
+    + length a, ys @ rs # 0\<^bsup>pstr \<^esup>@ lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> 
+    @ suf_lm) aprog stp =
+    ((\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 6 * length gs + 3 * n
+    + length a + 3, ys @ 0\<^bsup>pstr - length ys\<^esup> @ rs # 
+    0\<^bsup>length ys\<^esup> @ lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)"
+  proof (erule_tac exE, erule_tac exE, erule_tac exE, erule_tac exE)
+    fix ap bp apa cp
+    assume "aprog = ap [+] bp [+] cp \<and> length ap = 
+      (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 6 * length gs + 
+      3 * n + length a \<and> bp = recursive.empty (length ys) pstr"
+    thus"?thesis"
+      apply(simp, rule_tac abc_append_exc1, simp_all)
+      apply(rule_tac save_rs', insert h)
+      apply(subgoal_tac "length gs = aa \<and> pstr \<ge> ba \<and> ba > aa",
+            arith)
+      apply(simp add: para_pattern, insert pdef, auto)
+      apply(rule_tac min_max.le_supI2, simp)
+      done
+  qed
+qed
+
+lemma [simp]: "length (empty_boxes n) = 2*n"
+apply(induct n, simp, simp)
+done
+
+lemma empty_step_ex: "length lm = n \<Longrightarrow> 
+      \<exists>stp. abc_steps_l (0, lm @ Suc x # suf_lm) [Dec n 2, Goto 0] stp
+  = (0, lm @ x # suf_lm)"
+apply(rule_tac x = "Suc (Suc 0)" in exI, 
+  simp add: abc_steps_l.simps abc_step_l.simps abc_fetch.simps 
+         abc_lm_v.simps abc_lm_s.simps nth_append list_update_append)
+done
+
+lemma empty_box_ex: 
+  "\<lbrakk>length lm = n\<rbrakk> \<Longrightarrow> 
+  \<exists> stp. abc_steps_l (0, lm @ x # suf_lm) [Dec n 2, Goto 0] stp =
+  (Suc (Suc 0), lm @ 0 # suf_lm)"
+apply(induct x)
+apply(rule_tac x = "Suc 0" in exI, 
+  simp add: abc_steps_l.simps abc_fetch.simps abc_step_l.simps
+            abc_lm_v.simps nth_append abc_lm_s.simps, simp)
+apply(drule_tac x = x and suf_lm = suf_lm in empty_step_ex, 
+      erule_tac exE, erule_tac exE)
+apply(rule_tac x = "stpa + stp" in exI, simp add: abc_steps_add)
+done
+
+lemma [simp]: "drop n lm = a # list \<Longrightarrow> list = drop (Suc n) lm"
+apply(induct n arbitrary: lm a list, simp)
+apply(case_tac "lm", simp, simp)
+done
+
+lemma empty_boxes_ex: "\<lbrakk>length lm \<ge> n\<rbrakk>
+     \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm) (empty_boxes n) stp = 
+                                          (2*n, 0\<^bsup>n\<^esup> @ drop n lm)"
+apply(induct n, simp, simp)
+apply(rule_tac abc_append_exc2, auto)
+apply(case_tac "drop n lm", simp, simp)
+proof -
+  fix n stp a list
+  assume h: "Suc n \<le> length lm"  "drop n lm = a # list"
+  thus "\<exists>bstp. abc_steps_l (0, 0\<^bsup>n\<^esup> @ a # list) [Dec n 2, Goto 0] bstp =
+                       (Suc (Suc 0), 0 # 0\<^bsup>n\<^esup> @ drop (Suc n) lm)"
+    apply(insert empty_box_ex[of "0\<^bsup>n\<^esup>" n a list], simp, erule_tac exE)
+    apply(rule_tac x = stp in exI, simp, simp only: exponent_cons_iff)
+    apply(simp add: exponent_def rep_ind del: replicate.simps)
+    done
+qed
+
+
+lemma empty_paras_prog_ex:
+  "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md);
+  rec_ci f = (a, aa, ba); 
+  Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n) 
+                    (map rec_ci (f # gs)))) = pstr\<rbrakk>
+  \<Longrightarrow> \<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and> 
+  length ap = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 
+  6 *length gs + 3 * n + length a + 3 \<and> bp = empty_boxes (length gs)"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "cn_merge_gs (map rec_ci gs) (max (Suc n) 
+    (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) [+] 
+    mv_boxes 0 (Suc (max (Suc n) (Max 
+     (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))) + length gs)) n
+    [+] mv_boxes (max (Suc n) (Max (insert ba 
+    (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) 0 (length gs) [+]
+     a [+] recursive.empty aa (max (Suc n) 
+   (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))))" 
+    in exI, simp add: cn_merge_gs_len)
+apply(rule_tac x = " recursive.empty (max (Suc n) (Max (insert ba
+     (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) n [+]
+     mv_boxes (Suc (max (Suc n) (Max (insert ba 
+     (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))) + length gs)) 0 n" in exI, 
+  auto simp: abc_append_commute)
+done
+
+lemma empty_paras: 
+ assumes h: 
+  "rec_ci (Cn n f gs) = (aprog, rs_pos, a_md)" 
+  "rec_calc_rel (Cn n f gs) lm rs" 
+  "\<forall>k<length gs. rec_calc_rel (gs ! k) lm (ys ! k)"
+  "length ys = length gs" 
+  "rec_calc_rel f ys rs" 
+  "rec_ci f = (a, aa, ba)" 
+  and pdef: "pstr = Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n) 
+                                             (map rec_ci (f # gs))))"
+  and starts: "ss = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) +
+                              6 * length gs + 3 * n + length a + 3"
+  shows "\<exists>stp. abc_steps_l
+           (ss, ys @ 0\<^bsup>pstr - length ys\<^esup> @ rs # 0\<^bsup>length ys\<^esup> 
+               @ lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm) aprog stp =
+   (ss + 2 * length gs, 0\<^bsup>pstr\<^esup> @ rs # 0\<^bsup>length ys \<^esup> @ lm @ 
+                                0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)"
+proof -
+  from h and pdef and starts have k1: 
+    "\<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and>
+    length ap = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 
+                               6 *length gs + 3 * n + length a + 3
+    \<and> bp = empty_boxes (length ys)"
+    by(drule_tac empty_paras_prog_ex, auto)
+  from h have j1: "aa < ba"
+    by(simp add: ci_ad_ge_paras)
+  from h have j2: "length gs = aa"
+    by(drule_tac f = f in para_pattern, simp, simp)
+  from h and pdef have j3: "ba \<le> pstr"
+    apply simp 
+    apply(rule_tac min_max.le_supI2, simp)
+    done
+  from k1 show "?thesis"
+  proof (erule_tac exE, erule_tac exE, erule_tac exE, erule_tac exE)
+    fix ap bp apa cp
+    assume "aprog = ap [+] bp [+] cp \<and> 
+      length ap = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) +
+      6 * length gs + 3 * n + length a + 3 \<and> 
+      bp = empty_boxes (length ys)"
+    thus"?thesis"
+      apply(simp, rule_tac abc_append_exc1, simp_all add: starts h)
+      apply(insert empty_boxes_ex[of 
+        "length gs" "ys @ 0\<^bsup>pstr - (length gs)\<^esup> @ rs #
+        0\<^bsup>length gs\<^esup> @ lm @ 0\<^bsup>a_md - Suc (pstr + length gs + n)\<^esup> @ suf_lm"], 
+        simp add: h)
+      apply(erule_tac exE, rule_tac x = stp in exI, 
+        simp add: exponent_def replicate.simps[THEN sym]
+        replicate_add[THEN sym] del: replicate.simps)
+      apply(subgoal_tac "pstr >(length gs)", simp)
+      apply(subgoal_tac "ba > aa \<and> length gs = aa \<and> pstr \<ge> ba", simp)
+      apply(simp add: j1 j2 j3)
+      done     
+  qed
+qed
+
+(*
+lemma [simp]: " n < pstr \<Longrightarrow> 
+  (0\<^bsup>pstr\<^esup>)[n := rs] @ [0::nat] = 0\<^bsup>n\<^esup> @ rs # 0\<^bsup>pstr - n\<^esup>"
+apply(insert list_update_length[of "0\<^bsup>n\<^esup>" 0 "0\<^bsup>pstr - Suc n\<^esup>" rs])
+apply(insert exponent_cons_iff[of "0::nat" "pstr - Suc n" "[]"], simp)
+apply(insert exponent_add_iff[of "0::nat" n "pstr - n" "[]"], simp)
+apply(case_tac "pstr - n", simp, simp only: exp_suc, simp)
+apply(subgoal_tac "pstr - Suc n = nat", simp)
+by arith
+*)
+
+lemma restore_rs_prog_ex:
+  "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md);
+  rec_ci f = (a, aa, ba);
+  Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n)
+  (map rec_ci (f # gs)))) = pstr;
+  ss = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) +
+  8 * length gs + 3 * n + length a + 3\<rbrakk>
+  \<Longrightarrow> \<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and> length ap = ss \<and> 
+                                           bp = empty pstr n"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "cn_merge_gs (map rec_ci gs) (max (Suc n) 
+      (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) [+] 
+      mv_boxes 0 (Suc (max (Suc n) (Max (insert ba (((\<lambda>(aprog, p, n). n)
+        \<circ> rec_ci) ` set gs))) + length gs)) n [+]
+     mv_boxes (max (Suc n) (Max (insert ba 
+      (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) 0 (length gs) [+]
+     a [+] recursive.empty aa (max (Suc n)
+       (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) [+]
+     empty_boxes (length gs)" in exI, simp add: cn_merge_gs_len)
+apply(rule_tac x = "mv_boxes (Suc (max (Suc n) 
+       (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))) 
+        + length gs)) 0 n" 
+  in exI, auto simp: abc_append_commute)
+done
+
+lemma exp_add: "a\<^bsup>b+c\<^esup> = a\<^bsup>b\<^esup> @ a\<^bsup>c\<^esup>"
+apply(simp add: exponent_def replicate_add)
+done
+
+lemma [simp]: "n < pstr \<Longrightarrow> (0\<^bsup>pstr\<^esup>)[n := rs] @ [0::nat] = 0\<^bsup>n\<^esup> @ rs # 0\<^bsup>pstr - n\<^esup>"
+using list_update_length[of "0\<^bsup>n\<^esup>" "0::nat" "0\<^bsup>pstr - Suc n\<^esup>" rs]
+apply(simp add: exp_ind_def[THEN sym] exp_add[THEN sym] exp_suc[THEN sym])
+done
+
+lemma restore_rs:
+  assumes h: "rec_ci (Cn n f gs) = (aprog, rs_pos, a_md)" 
+  "rec_calc_rel (Cn n f gs) lm rs" 
+  "\<forall>k<length gs. rec_calc_rel (gs ! k) lm (ys ! k)"
+  "length ys = length gs"
+  "rec_calc_rel f ys rs" 
+  "rec_ci f = (a, aa, ba)" 
+  and pdef: "pstr = Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n)
+                                        (map rec_ci (f # gs))))"
+  and starts: "ss = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 
+                              8 * length gs + 3 * n + length a + 3" 
+  shows "\<exists>stp. abc_steps_l
+           (ss, 0\<^bsup>pstr\<^esup> @ rs # 0\<^bsup>length ys \<^esup> @ lm @
+                    0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm) aprog stp =
+  (ss + 3, 0\<^bsup>n\<^esup> @ rs # 0\<^bsup>pstr - n\<^esup> @ 0\<^bsup>length ys \<^esup> @ lm @ 
+                                   0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)"
+proof -
+ from h and pdef and starts have k1:
+   "\<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and> length ap = ss \<and> 
+                                            bp = empty pstr n"
+   by(drule_tac restore_rs_prog_ex, auto)
+ from k1 show "?thesis"
+ proof (erule_tac exE, erule_tac exE, erule_tac exE, erule_tac exE)
+   fix ap bp apa cp
+   assume "aprog = ap [+] bp [+] cp \<and> length ap = ss \<and> 
+                                 bp = recursive.empty pstr n"
+   thus"?thesis"
+     apply(simp, rule_tac abc_append_exc1, simp_all add: starts h)
+     apply(insert empty_ex[of pstr n "0\<^bsup>pstr\<^esup> @ rs # 0\<^bsup>length gs\<^esup> @
+                     lm @ 0\<^bsup>a_md - Suc (pstr + length gs + n)\<^esup> @ suf_lm"], simp)
+     apply(subgoal_tac "pstr > n", simp)
+     apply(erule_tac exE, rule_tac x = stp in exI, 
+                         simp add: nth_append list_update_append)
+     apply(simp add: pdef)
+     done
+  qed
+qed
+
+lemma [simp]:"xs \<noteq> [] \<Longrightarrow> length xs \<ge> Suc 0"
+by(case_tac xs, auto)
+
+lemma  [simp]: "n < max (Suc n) (max ba (Max (((\<lambda>(aprog, p, n). n) o 
+                                                  rec_ci) ` set gs)))"
+by(simp)
+
+lemma restore_paras_prog_ex: 
+  "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md); 
+  rec_ci f = (a, aa, ba);
+  Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n)
+                          (map rec_ci (f # gs)))) = pstr;
+  ss = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) +
+                         8 * length gs + 3 * n + length a + 6\<rbrakk>
+  \<Longrightarrow> \<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and> length ap = ss \<and> 
+                      bp = mv_boxes (pstr + Suc (length gs)) (0::nat) n"
+apply(simp add: rec_ci.simps)
+apply(rule_tac x = "cn_merge_gs (map rec_ci gs) (max (Suc n) 
+      (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))))
+      [+] mv_boxes 0 (Suc (max (Suc n) 
+       (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))) 
+     + length gs)) n [+] mv_boxes (max (Suc n) 
+    (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) 0 (length gs) [+]
+     a [+] recursive.empty aa (max (Suc n) 
+      (Max (insert ba (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) [+]
+     empty_boxes (length gs) [+]
+     recursive.empty (max (Suc n) (Max (insert ba 
+     (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) n" in exI, simp add: cn_merge_gs_len)
+apply(rule_tac x = "[]" in exI, auto simp: abc_append_commute)
+done
+
+lemma restore_paras: 
+  assumes h: "rec_ci (Cn n f gs) = (aprog, rs_pos, a_md)" 
+  "rec_calc_rel (Cn n f gs) lm rs" 
+  "\<forall>k<length gs. rec_calc_rel (gs ! k) lm (ys ! k)"
+  "length ys = length gs"
+  "rec_calc_rel f ys rs" 
+  "rec_ci f = (a, aa, ba)"
+  and pdef: 
+  "pstr = Max (set (Suc n # ba # map (\<lambda>(aprog, p, n). n) 
+                         (map rec_ci (f # gs))))"
+  and starts: "ss = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) + 
+                              8 * length gs + 3 * n + length a + 6" 
+  shows "\<exists>stp. abc_steps_l (ss, 0\<^bsup>n\<^esup> @ rs # 0\<^bsup>pstr - n+ length ys\<^esup> @
+                         lm @ 0\<^bsup>a_md - Suc (pstr + length ys + n)\<^esup> @ suf_lm)
+  aprog stp = (ss + 3 * n, lm @ rs # 0\<^bsup>a_md - Suc n\<^esup> @ suf_lm)"
+proof -
+  thm rec_ci.simps
+  from h and pdef and starts have k1:
+    "\<exists> ap bp cp. aprog = ap [+] bp [+] cp \<and> length ap = ss \<and>
+                     bp = mv_boxes (pstr + Suc (length gs)) (0::nat) n"
+    by(drule_tac restore_paras_prog_ex, auto)
+  from k1 show "?thesis"
+  proof (erule_tac exE, erule_tac exE, erule_tac exE, erule_tac exE)
+    fix ap bp apa cp
+    assume "aprog = ap [+] bp [+] cp \<and> length ap = ss \<and> 
+                              bp = mv_boxes (pstr + Suc (length gs)) 0 n"
+    thus"?thesis"
+      apply(simp, rule_tac abc_append_exc1, simp_all add: starts h)
+      apply(insert mv_boxes_ex2[of n "pstr + Suc (length gs)" 0 "[]" 
+        "rs # 0\<^bsup>pstr - n + length gs\<^esup>" "lm" 
+        "0\<^bsup>a_md - Suc (pstr + length gs + n)\<^esup> @ suf_lm"], simp)
+      apply(subgoal_tac "pstr > n \<and> 
+        a_md > pstr + length gs + n \<and> length lm = n" , simp add: exponent_add_iff h)
+      using h pdef
+      apply(simp)     
+      apply(frule_tac a = a and 
+        aa = aa and ba = ba in ci_cn_md_def, simp, simp)
+      apply(subgoal_tac "length lm = rs_pos",
+        simp add: ci_cn_para_eq, erule_tac para_pattern, simp)
+      done
+  qed
+qed
+
+lemma ci_cn_length:
+  "\<lbrakk>rec_ci (Cn n f gs) = (aprog, rs_pos, a_md); 
+  rec_calc_rel (Cn n f gs) lm rs;
+  rec_ci f = (a, aa, ba)\<rbrakk>
+  \<Longrightarrow> length aprog = (\<Sum>(ap, pos, n)\<leftarrow>map rec_ci gs. length ap) +
+                             8 * length gs + 6 * n + length a + 6"
+apply(simp add: rec_ci.simps, auto simp: cn_merge_gs_len)
+done
+
+
+lemma  cn_case: 
+  assumes ind:
+  "\<And>x aprog a_md rs_pos rs suf_lm lm.
+  \<lbrakk>x \<in> set (f # gs);
+  rec_ci x = (aprog, rs_pos, a_md);
+  rec_calc_rel x lm rs\<rbrakk>
+  \<Longrightarrow> \<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp = 
+               (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+  and h: "rec_ci (Cn n f gs) = (aprog, rs_pos, a_md)"
+         "rec_calc_rel (Cn n f gs) lm rs"
+         
+  shows "\<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp 
+  = (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+apply(insert h, case_tac "rec_ci f",  rule_tac calc_cn_reverse, simp)
+proof -
+  fix a b c ys
+  let ?pstr = "Max (set (Suc n # c # (map (\<lambda>(aprog, p, n). n) 
+                                         (map rec_ci (f # gs)))))"  
+  let ?gs_len = "listsum (map (\<lambda> (ap, pos, n). length ap) 
+                                                (map rec_ci (gs)))"
+  assume g: "rec_ci (Cn n f gs) = (aprog, rs_pos, a_md)"
+    "rec_calc_rel (Cn n f gs) lm rs"
+    "\<forall>k<length gs. rec_calc_rel (gs ! k) lm (ys ! k)" 
+    "length ys = length gs" 
+    "rec_calc_rel f ys rs"
+    "n = length lm"
+    "rec_ci f = (a, b, c)"  
+  hence k1:
+    "\<exists> stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp = 
+    (?gs_len + 3 * length gs, lm @ 0\<^bsup>?pstr - n\<^esup> @ ys @
+                               0\<^bsup>a_md - ?pstr - length ys\<^esup> @ suf_lm)"	
+    apply(rule_tac a = a and aa = b and ba = c in cn_calc_gs)
+    apply(rule_tac ind, auto)
+    done  
+  thm rec_ci.simps
+  from g have k2: 
+    "\<exists> stp. abc_steps_l (?gs_len + 3 * length gs,  lm @ 
+        0\<^bsup>?pstr - n\<^esup> @ ys @ 0\<^bsup>a_md - ?pstr - length ys\<^esup> @ suf_lm) aprog stp = 
+    (?gs_len + 3 * length gs + 3 * n, 0\<^bsup>?pstr\<^esup> @ ys @ 0 # lm @ 
+                              0\<^bsup>a_md - Suc (?pstr + length ys + n )\<^esup>  @ suf_lm)"
+    thm save_paras
+    apply(erule_tac ba = c in save_paras, auto intro: ci_cn_para_eq)
+    done
+  from g have k3: 
+    "\<exists> stp. abc_steps_l (?gs_len + 3 * length gs + 3 * n,
+    0\<^bsup>?pstr\<^esup> @ ys @ 0 # lm @ 0\<^bsup>a_md - Suc (?pstr + length ys + n)\<^esup>  @ suf_lm) aprog stp =
+    (?gs_len + 6 * length gs + 3 * n,  
+           ys @ 0\<^bsup>?pstr\<^esup> @ 0 # lm @ 0\<^bsup>a_md - Suc (?pstr + length ys + n)\<^esup>  @ suf_lm)"
+    apply(erule_tac ba = c in reset_new_paras, 
+          auto intro: ci_cn_para_eq)
+    using para_pattern[of f a b c ys rs]
+    apply(simp)
+    done
+  from g have k4: 
+    "\<exists>stp. abc_steps_l  (?gs_len + 6 * length gs + 3 * n,  
+    ys @ 0\<^bsup>?pstr\<^esup> @ 0 # lm @ 0\<^bsup>a_md - Suc (?pstr + length ys + n)\<^esup>  @ suf_lm) aprog stp =
+    (?gs_len + 6 * length gs + 3 * n + length a, 
+   ys @ rs # 0\<^bsup>?pstr \<^esup> @ lm @ 0\<^bsup>a_md - Suc (?pstr + length ys + n)\<^esup>  @ suf_lm)"
+    apply(rule_tac ba = c in calc_cn_f, rule_tac ind, auto)
+    done
+thm rec_ci.simps
+  from g h have k5:
+    "\<exists> stp. abc_steps_l (?gs_len + 6 * length gs + 3 * n + length a,
+    ys @ rs # 0\<^bsup>?pstr \<^esup>@ lm @ 0\<^bsup>a_md - Suc (?pstr + length ys + n)\<^esup>  @ suf_lm)
+    aprog stp =
+    (?gs_len + 6 * length gs + 3 * n + length a + 3,
+    ys @ 0\<^bsup>?pstr - length ys\<^esup> @ rs # 0\<^bsup>length ys\<^esup> @ lm @ 
+    0\<^bsup>a_md  - Suc (?pstr + length ys + n)\<^esup> @ suf_lm)"
+    apply(rule_tac save_rs, auto simp: h)
+    done
+  thm rec_ci.simps
+  thm empty_boxes.simps
+  from g have k6: 
+    "\<exists> stp. abc_steps_l (?gs_len + 6 * length gs + 3 * n + 
+    length a + 3, ys @ 0\<^bsup>?pstr - length ys\<^esup> @ rs # 0\<^bsup>length ys\<^esup> @ lm @ 
+    0\<^bsup>a_md  - Suc (?pstr + length ys + n)\<^esup> @ suf_lm) 
+    aprog stp =
+    (?gs_len + 8 * length gs + 3 *n + length a + 3,
+    0\<^bsup>?pstr \<^esup> @ rs # 0\<^bsup>length ys\<^esup> @ lm @ 
+                        0\<^bsup>a_md -Suc (?pstr + length ys + n)\<^esup> @ suf_lm)"
+    apply(drule_tac suf_lm = suf_lm in empty_paras, auto)
+    apply(rule_tac x = stp in exI, simp)
+    done
+  from g have k7: 
+    "\<exists> stp. abc_steps_l (?gs_len + 8 * length gs + 3 *n + 
+    length a + 3, 0\<^bsup>?pstr \<^esup> @ rs # 0\<^bsup>length ys\<^esup> @ lm @ 
+    0\<^bsup>a_md -Suc (?pstr + length ys + n)\<^esup> @ suf_lm) aprog stp =
+    (?gs_len + 8 * length gs + 3 * n + length a + 6, 
+    0\<^bsup>n\<^esup> @ rs # 0\<^bsup>?pstr  - n\<^esup> @ 0\<^bsup>length ys\<^esup> @ lm @
+                        0\<^bsup>a_md -Suc (?pstr + length ys + n) \<^esup> @ suf_lm)"
+    apply(drule_tac suf_lm = suf_lm in restore_rs, auto)
+    apply(rule_tac x = stp in exI, simp)
+    done
+  from g have k8: "\<exists> stp. abc_steps_l (?gs_len + 8 * length gs + 
+    3 * n + length a + 6,
+    0\<^bsup>n\<^esup> @ rs # 0\<^bsup>?pstr  - n\<^esup> @ 0\<^bsup>length ys\<^esup> @ lm @
+                      0\<^bsup>a_md -Suc (?pstr + length ys + n) \<^esup> @ suf_lm) aprog stp =
+    (?gs_len + 8 * length gs + 6 * n + length a + 6,
+                           lm @ rs # 0\<^bsup>a_md - Suc n \<^esup>@ suf_lm)"
+    apply(drule_tac suf_lm = suf_lm in restore_paras, auto)
+    apply(simp add: exponent_add_iff)
+    apply(rule_tac x = stp in exI, simp)
+    done
+  from g have j1: 
+    "length aprog = ?gs_len + 8 * length gs + 6 * n + length a + 6"
+    by(drule_tac a = a and aa = b and ba = c in ci_cn_length,
+      simp, simp, simp)
+  from g have j2: "rs_pos = n"
+    by(simp add: ci_cn_para_eq)
+  from k1 and k2 and k3 and k4 and k5 and k6 and k7 and k8
+    and j1 and j2 show 
+    "\<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp = 
+    (length aprog, lm @ [rs] @ 0\<^bsup>a_md - rs_pos - 1\<^esup> @ suf_lm)"
+    apply(auto)
+    apply(rule_tac x = "stp + stpa + stpb + stpc +
+      stpd + stpe + stpf + stpg" in exI, simp add: abc_steps_add)
+    done
+qed
+
+text {*
+  Correctness of the complier (terminate case), which says if the execution of 
+  a recursive function @{text "recf"} terminates and gives result, then 
+  the Abacus program compiled from @{text "recf"} termintes and gives the same result.
+  Additionally, to facilitate induction proof, we append @{text "anything"} to the
+  end of Abacus memory.
+*}
+
+lemma aba_rec_equality:
+  "\<lbrakk>rec_ci recf = (ap, arity, fp);
+    rec_calc_rel recf args r\<rbrakk>
+  \<Longrightarrow> (\<exists> stp. (abc_steps_l (0, args @ 0\<^bsup>fp - arity\<^esup> @ anything) ap stp) = 
+              (length ap, args@[r]@0\<^bsup>fp - arity - 1\<^esup> @ anything))"
+apply(induct arbitrary: ap fp arity r anything args
+  rule: rec_ci.induct)
+prefer 5
+proof(case_tac "rec_ci g", case_tac "rec_ci f", simp)
+  fix n f g ap fp arity r anything args  a b c aa ba ca
+  assume f_ind:
+    "\<And>ap fp arity r anything args.
+    \<lbrakk>aa = ap \<and> ba = arity \<and> ca = fp; rec_calc_rel f args r\<rbrakk> \<Longrightarrow> 
+    \<exists>stp. abc_steps_l (0, args @ 0\<^bsup>fp - arity\<^esup> @ anything) ap stp =
+    (length ap, args @ r # 0\<^bsup>fp - Suc arity\<^esup> @ anything)"
+    and g_ind:
+    "\<And>x xa y xb ya ap fp arity r anything args.
+    \<lbrakk>x = (aa, ba, ca); xa = aa \<and> y = (ba, ca); xb = ba \<and> ya = ca; 
+    a = ap \<and> b = arity \<and> c = fp; rec_calc_rel g args r\<rbrakk>
+    \<Longrightarrow> \<exists>stp. abc_steps_l (0, args @ 0\<^bsup>fp - arity\<^esup> @ anything) ap stp =
+    (length ap, args @ r # 0\<^bsup>fp - Suc arity\<^esup> @ anything)"
+    and h: "rec_ci (Pr n f g) = (ap, arity, fp)" 
+    "rec_calc_rel (Pr n f g) args r" 
+    "rec_ci g = (a, b, c)" 
+    "rec_ci f = (aa, ba, ca)"
+  from h have nf_ind: 
+    "\<And> args r anything. rec_calc_rel f args r \<Longrightarrow> 
+    \<exists>stp. abc_steps_l (0, args @ 0\<^bsup>ca - ba\<^esup> @ anything) aa stp = 
+    (length aa, args @ r # 0\<^bsup>ca - Suc ba\<^esup> @ anything)"
+    and ng_ind: 
+    "\<And> args r anything. rec_calc_rel g args r \<Longrightarrow> 
+    \<exists>stp. abc_steps_l (0, args @ 0\<^bsup>c - b\<^esup> @ anything) a stp = 
+         (length a, args @ r # 0\<^bsup>c - Suc b \<^esup> @ anything)"
+    apply(insert f_ind[of aa ba ca], simp)
+    apply(insert g_ind[of "(aa, ba, ca)" aa "(ba, ca)" ba ca a b c],
+      simp)
+    done
+  from nf_ind and ng_ind and h show 
+    "\<exists>stp. abc_steps_l (0, args @ 0\<^bsup>fp - arity\<^esup> @ anything) ap stp = 
+    (length ap, args @ r # 0\<^bsup>fp - Suc arity\<^esup> @ anything)"
+    apply(auto intro: nf_ind ng_ind pr_case)
+    done
+next
+  fix ap fp arity r anything args
+  assume h:
+    "rec_ci z = (ap, arity, fp)" "rec_calc_rel z args r"
+  thus "\<exists>stp. abc_steps_l (0, args @ 0\<^bsup>fp - arity\<^esup> @ anything) ap stp =
+    (length ap, args @ [r] @ 0\<^bsup>fp - arity - 1\<^esup> @ anything)"
+    by (rule_tac z_case)    
+next
+  fix ap fp arity r anything args
+  assume h: 
+    "rec_ci s = (ap, arity, fp)" 
+    "rec_calc_rel s args r"
+  thus 
+    "\<exists>stp. abc_steps_l (0, args @ 0\<^bsup>fp - arity\<^esup> @ anything) ap stp =
+    (length ap, args @ [r] @ 0\<^bsup>fp - arity - 1\<^esup> @ anything)"
+    by(erule_tac s_case, simp)
+next
+  fix m n ap fp arity r anything args
+  assume h: "rec_ci (id m n) = (ap, arity, fp)" 
+    "rec_calc_rel (id m n) args r"
+  thus "\<exists>stp. abc_steps_l (0, args @ 0\<^bsup>fp - arity\<^esup> @ anything) ap stp 
+    = (length ap, args @ [r] @ 0\<^bsup>fp - arity - 1\<^esup> @ anything)"
+    by(erule_tac id_case)
+next
+  fix n f gs ap fp arity r anything args
+  assume ind: "\<And>x ap fp arity r anything args.
+    \<lbrakk>x \<in> set (f # gs); 
+    rec_ci x = (ap, arity, fp); 
+    rec_calc_rel x args r\<rbrakk>
+    \<Longrightarrow> \<exists>stp. abc_steps_l (0, args @ 0\<^bsup>fp - arity\<^esup> @ anything) ap stp =
+    (length ap, args @ [r] @ 0\<^bsup>fp - arity - 1\<^esup> @ anything)"
+  and h: "rec_ci (Cn n f gs) = (ap, arity, fp)" 
+    "rec_calc_rel (Cn n f gs) args r"
+  from h show
+    "\<exists>stp. abc_steps_l (0, args @ 0\<^bsup>fp - arity\<^esup> @ anything) 
+       ap stp = (length ap, args @ [r] @ 0\<^bsup>fp - arity - 1\<^esup> @ anything)"
+    apply(rule_tac cn_case, rule_tac ind, auto)
+    done
+next
+  fix n f ap fp arity r anything args
+  assume ind:
+    "\<And>ap fp arity r anything args.
+    \<lbrakk>rec_ci f = (ap, arity, fp); rec_calc_rel f args r\<rbrakk> \<Longrightarrow> 
+    \<exists>stp. abc_steps_l (0, args @ 0\<^bsup>fp - arity\<^esup> @ anything) ap stp =
+    (length ap, args @ [r] @ 0\<^bsup>fp - arity - 1\<^esup> @ anything)"
+  and h: "rec_ci (Mn n f) = (ap, arity, fp)" 
+    "rec_calc_rel (Mn n f) args r"
+  from h show 
+    "\<exists>stp. abc_steps_l (0, args @ 0\<^bsup>fp - arity\<^esup> @ anything) ap stp = 
+              (length ap, args @ [r] @ 0\<^bsup>fp - arity - 1\<^esup> @ anything)"
+    apply(rule_tac mn_case, rule_tac ind, auto)
+    done    
+qed
+
+
+thm abc_append_state_in_exc
+lemma abc_append_uhalt1:
+  "\<lbrakk>\<forall> stp. (\<lambda> (ss, e). ss < length bp) (abc_steps_l (0, lm) bp stp);
+    p = ap [+] bp [+] cp\<rbrakk>
+  \<Longrightarrow> \<forall> stp. (\<lambda> (ss, e). ss < length p) 
+                     (abc_steps_l (length ap, lm) p stp)"
+apply(auto)
+apply(erule_tac x = stp in allE, auto)
+apply(frule_tac ap = ap and cp = cp in abc_append_state_in_exc, auto)
+done
+
+
+lemma abc_append_unhalt2:
+  "\<lbrakk>abc_steps_l (0, am) ap stp = (length ap, lm); bp \<noteq> [];
+  \<forall> stp. (\<lambda> (ss, e). ss < length bp) (abc_steps_l (0, lm) bp stp);
+  p = ap [+] bp [+] cp\<rbrakk>
+  \<Longrightarrow> \<forall> stp. (\<lambda> (ss, e). ss < length p) (abc_steps_l (0, am) p stp)"
+proof -
+  assume h: 
+    "abc_steps_l (0, am) ap stp = (length ap, lm)" 
+    "bp \<noteq> []"
+    "\<forall> stp. (\<lambda> (ss, e). ss < length bp) (abc_steps_l (0, lm) bp stp)"
+    "p = ap [+] bp [+] cp"
+  have "\<exists> stp. (abc_steps_l (0, am) p stp) = (length ap, lm)"
+    using h
+    thm abc_add_exc1
+    apply(simp add: abc_append.simps)
+    apply(rule_tac abc_add_exc1, auto)
+    done
+  from this obtain stpa where g1: 
+    "(abc_steps_l (0, am) p stpa) = (length ap, lm)" ..
+  moreover have g2: "\<forall> stp. (\<lambda> (ss, e). ss < length p) 
+                          (abc_steps_l (length ap, lm) p stp)"
+    using h
+    apply(erule_tac abc_append_uhalt1, simp)
+    done
+  moreover from g1 and g2 have
+    "\<forall> stp. (\<lambda> (ss, e). ss < length p) 
+                    (abc_steps_l (0, am) p (stpa + stp))"
+    apply(simp add: abc_steps_add)
+    done
+  thus "\<forall> stp. (\<lambda> (ss, e). ss < length p) 
+                           (abc_steps_l (0, am) p stp)"
+    apply(rule_tac allI, auto)
+    apply(case_tac "stp \<ge>  stpa")
+    apply(erule_tac x = "stp - stpa" in allE, simp)
+  proof - 	
+    fix stp a b
+    assume g3:  "abc_steps_l (0, am) p stp = (a, b)" 
+                "\<not> stpa \<le> stp"
+    thus "a < length p"
+      using g1 h
+      apply(case_tac "a < length p", simp, simp)
+      apply(subgoal_tac "\<exists> d. stpa = stp + d")
+      using  abc_state_keep[of p a b "stpa - stp"]
+      apply(erule_tac exE, simp add: abc_steps_add)
+      apply(rule_tac x = "stpa - stp" in exI, simp)
+      done
+  qed
+qed
+
+text {*
+  Correctness of the complier (non-terminating case for Mn). There are many cases when a 
+  recursive function does not terminate. For the purpose of Uiversal Turing Machine, we only 
+  need to prove the case for @{text "Mn"} and @{text "Cn"}.
+  This lemma is for @{text "Mn"}. For @{text "Mn n f"}, this lemma describes what 
+  happens when @{text "f"} always terminates but always does not return zero, so that
+  @{text "Mn"} has to loop forever.
+  *}
+
+lemma Mn_unhalt:
+  assumes mn_rf: "rf = Mn n f"
+  and compiled_mnrf: "rec_ci rf = (aprog, rs_pos, a_md)"
+  and compiled_f: "rec_ci f = (aprog', rs_pos', a_md')"
+  and args: "length lm = n"
+  and unhalt_condition: "\<forall> y. (\<exists> rs. rec_calc_rel f (lm @ [y]) rs \<and> rs \<noteq> 0)"
+  shows "\<forall> stp. case abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm)
+               aprog stp of (ss, e) \<Rightarrow> ss < length aprog"
+  using mn_rf compiled_mnrf compiled_f args unhalt_condition
+proof(rule_tac allI)
+  fix stp
+  assume h: "rf = Mn n f" 
+            "rec_ci rf = (aprog, rs_pos, a_md)"
+            "rec_ci f = (aprog', rs_pos', a_md')" 
+            "\<forall>y. \<exists>rs. rec_calc_rel f (lm @ [y]) rs \<and> rs \<noteq> 0" "length lm = n"
+  thm mn_ind_step
+  have "\<exists>stpa \<ge> stp. abc_steps_l (0, lm @ 0 # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm) aprog stpa 
+         = (0, lm @ stp # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+  proof(induct stp, auto)
+    show "\<exists>stpa. abc_steps_l (0, lm @ 0 # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm) 
+          aprog stpa = (0, lm @ 0 # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+      apply(rule_tac x = 0 in exI, simp add: abc_steps_l.simps)
+      done
+  next
+    fix stp stpa
+    assume g1: "stp \<le> stpa"
+      and g2: "abc_steps_l (0, lm @ 0 # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)
+                            aprog stpa
+               = (0, lm @ stp # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+    have "\<exists>rs. rec_calc_rel f (lm @ [stp]) rs \<and> rs \<noteq> 0"
+      using h
+      apply(erule_tac x = stp in allE, simp)
+      done
+    from this obtain rs where g3:
+      "rec_calc_rel f (lm @ [stp]) rs \<and> rs \<noteq> 0" ..
+    hence "\<exists> stpb. abc_steps_l (0, lm @ stp # 0\<^bsup>a_md - Suc rs_pos\<^esup> @
+                     suf_lm) aprog stpb 
+      = (0, lm @ Suc stp # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+      using h
+      apply(rule_tac mn_ind_step)
+      apply(rule_tac aba_rec_equality, simp, simp)
+    proof -
+      show "rec_ci f = ((aprog', rs_pos', a_md'))" using h by simp
+    next
+      show "rec_ci (Mn n f) = (aprog, rs_pos, a_md)" using h by simp
+    next
+      show "rec_calc_rel f (lm @ [stp]) rs" using g3 by simp
+    next
+      show "0 < rs" using g3 by simp
+    next
+      show "Suc rs_pos < a_md"
+        using g3 h
+        apply(auto)
+        apply(frule_tac f = f in para_pattern, simp, simp)
+        apply(simp add: rec_ci.simps, auto)
+        apply(subgoal_tac "Suc (length lm) < a_md'")
+        apply(arith)
+        apply(simp add: ci_ad_ge_paras)
+        done
+    next
+      show "rs_pos' = Suc rs_pos"
+        using g3 h
+        apply(auto)
+        apply(frule_tac f = f in para_pattern, simp, simp)
+        apply(simp add: rec_ci.simps)
+        done
+    qed
+    thus "\<exists>stpa\<ge>Suc stp. abc_steps_l (0, lm @ 0 # 0\<^bsup>a_md - Suc rs_pos\<^esup> @
+                 suf_lm) aprog stpa 
+      = (0, lm @ Suc stp # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)"
+      using g2
+      apply(erule_tac exE)
+      apply(case_tac "stpb = 0", simp add: abc_steps_l.simps)
+      apply(rule_tac x = "stpa + stpb" in exI, simp add:
+        abc_steps_add)
+      using g1
+      apply(arith)
+      done
+  qed
+  from this obtain stpa where 
+    "stp \<le> stpa \<and> abc_steps_l (0, lm @ 0 # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)
+         aprog stpa = (0, lm @ stp # 0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm)" ..
+  thus "case abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog stp
+    of (ss, e) \<Rightarrow> ss < length aprog"
+    apply(case_tac "abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm) aprog
+      stp", simp, case_tac "a \<ge> length aprog", 
+        simp, simp)
+    apply(subgoal_tac "\<exists> d. stpa = stp + d", erule_tac exE)
+    apply(subgoal_tac "lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suf_lm = lm @ 0 # 
+             0\<^bsup>a_md - Suc rs_pos\<^esup> @ suf_lm", simp add: abc_steps_add)
+    apply(frule_tac as = a and lm = b and stp = d in abc_state_keep, 
+          simp)
+    using h  
+    apply(simp add: rec_ci.simps, simp, 
+              simp only: exp_ind_def[THEN sym])
+    apply(case_tac rs_pos, simp, simp)
+    apply(rule_tac x = "stpa - stp" in exI, simp, simp)
+    done
+qed   
+
+
+lemma abc_append_cons_eq[intro!]: 
+  "\<lbrakk>ap = bp; cp = dp\<rbrakk> \<Longrightarrow> ap [+] cp = bp [+] dp"
+by simp 
+
+lemma cn_merge_gs_split: 
+  "\<lbrakk>i < length gs; rec_ci (gs!i) = (ga, gb, gc)\<rbrakk> \<Longrightarrow> 
+     cn_merge_gs (map rec_ci gs) p = 
+        cn_merge_gs (map rec_ci (take i gs)) p [+] ga [+] 
+       empty gb (p + i) [+] 
+      cn_merge_gs (map rec_ci (drop (Suc i) gs)) (p + Suc i)"
+apply(induct i arbitrary: gs p, case_tac gs, simp, simp)
+apply(case_tac gs, simp, case_tac "rec_ci a", 
+       simp add: abc_append_commute[THEN sym])
+done
+
+text {*
+  Correctness of the complier (non-terminating case for Mn). There are many cases when a 
+  recursive function does not terminate. For the purpose of Uiversal Turing Machine, we only 
+  need to prove the case for @{text "Mn"} and @{text "Cn"}.
+  This lemma is for @{text "Cn"}. For @{text "Cn f g1 g2 \<dots>gi, gi+1, \<dots> gn"}, this lemma describes what 
+  happens when every one of @{text "g1, g2, \<dots> gi"} terminates, but 
+  @{text "gi+1"} does not terminate, so that whole function @{text "Cn f g1 g2 \<dots>gi, gi+1, \<dots> gn"}
+  does not terminate.
+  *}
+
+lemma cn_gi_uhalt: 
+  assumes cn_recf: "rf = Cn n f gs"
+  and compiled_cn_recf: "rec_ci rf = (aprog, rs_pos, a_md)"
+  and args_length: "length lm = n"
+  and exist_unhalt_recf: "i < length gs" "gi = gs ! i"
+  and complied_unhalt_recf: "rec_ci gi = (ga, gb, gc)"  "gb = n"
+  and all_halt_before_gi: "\<forall> j < i. (\<exists> rs. rec_calc_rel (gs!j) lm rs)" 
+  and unhalt_condition: "\<And> slm. \<forall> stp. case abc_steps_l (0, lm @ 0\<^bsup>gc - gb\<^esup> @ slm) 
+     ga stp of (se, e) \<Rightarrow> se < length ga"
+  shows " \<forall> stp. case abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suflm) aprog
+  stp of (ss, e) \<Rightarrow> ss < length aprog"
+  using cn_recf compiled_cn_recf args_length exist_unhalt_recf complied_unhalt_recf
+        all_halt_before_gi unhalt_condition
+proof(case_tac "rec_ci f", simp)
+  fix a b c
+  assume h1: "rf = Cn n f gs" 
+    "rec_ci (Cn n f gs) = (aprog, rs_pos, a_md)" 
+    "length lm = n" 
+    "gi = gs ! i" 
+    "rec_ci (gs!i) = (ga, n, gc)" 
+    "gb = n" "rec_ci f = (a, b, c)"
+    and h2: "\<forall>j<i. \<exists>rs. rec_calc_rel (gs ! j) lm rs"
+    "i < length gs"
+  and ind:
+    "\<And> slm. \<forall> stp. case abc_steps_l (0, lm @ 0\<^bsup>gc - n\<^esup> @ slm) ga stp of (se, e) \<Rightarrow> se < length ga"
+  have h3: "rs_pos = n"
+    using h1
+    by(rule_tac ci_cn_para_eq, simp)
+  let ?ggs = "take i gs"
+  have "\<exists> ys. (length ys = i \<and> 
+    (\<forall> k < i. rec_calc_rel (?ggs ! k) lm (ys ! k)))"
+    using h2
+    apply(induct i, simp, simp)
+    apply(erule_tac exE)
+    apply(erule_tac x = ia in allE, simp)
+    apply(erule_tac exE)
+    apply(rule_tac x = "ys @ [x]" in exI, simp add: nth_append, auto)
+    apply(subgoal_tac "k = length ys", simp, simp)
+    done
+  from this obtain ys where g1:
+    "(length ys = i \<and> (\<forall> k < i. rec_calc_rel (?ggs ! k)
+                        lm (ys ! k)))" ..
+  let ?pstr = "Max (set (Suc n # c # map (\<lambda>(aprog, p, n). n)
+    (map rec_ci (f # gs))))"
+  have "\<exists>stp. abc_steps_l (0, lm @ 0\<^bsup>a_md - n\<^esup> @ suflm) 
+    (cn_merge_gs (map rec_ci ?ggs) ?pstr) stp =
+    (listsum (map ((\<lambda>(ap, pos, n). length ap) \<circ> rec_ci) ?ggs) +
+    3 * length ?ggs, lm @ 0\<^bsup>?pstr - n\<^esup> @ ys @ 0\<^bsup>a_md -(?pstr + length ?ggs)\<^esup> @
+    suflm) "
+    apply(rule_tac  cn_merge_gs_ex)
+    apply(rule_tac  aba_rec_equality, simp, simp)
+    using h1
+    apply(simp add: rec_ci.simps, auto)
+    using g1
+    apply(simp)
+    using h2 g1
+    apply(simp)
+    apply(rule_tac min_max.le_supI2)
+    apply(rule_tac Max_ge, simp, simp, rule_tac disjI2)
+    apply(subgoal_tac "aa \<in> set gs", simp)
+    using h2
+    apply(rule_tac A = "set (take i gs)" in subsetD, 
+      simp add: set_take_subset, simp)
+    done
+  thm cn_merge_gs.simps
+  from this obtain stpa where g2: 
+    "abc_steps_l (0, lm @ 0\<^bsup>a_md - n\<^esup> @ suflm) 
+    (cn_merge_gs (map rec_ci ?ggs) ?pstr) stpa =
+    (listsum (map ((\<lambda>(ap, pos, n). length ap) \<circ> rec_ci) ?ggs) +
+    3 * length ?ggs, lm @ 0\<^bsup>?pstr - n\<^esup> @ ys @ 0\<^bsup>a_md -(?pstr + length ?ggs)\<^esup> @
+    suflm)" ..
+  moreover have 
+    "\<exists> cp. aprog = (cn_merge_gs
+    (map rec_ci ?ggs) ?pstr) [+] ga [+] cp"
+    using h1
+    apply(simp add: rec_ci.simps)
+    apply(rule_tac x = "empty n (?pstr + i) [+] 
+      (cn_merge_gs (map rec_ci (drop (Suc i) gs)) (?pstr + Suc i))
+      [+]mv_boxes 0 (Suc (max (Suc n) (Max (insert c 
+     (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))) +
+      length gs)) n [+] mv_boxes (max (Suc n) (Max (insert c 
+      (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) 0 (length gs) [+]
+      a [+] recursive.empty b (max (Suc n) 
+      (Max (insert c (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) [+]
+     empty_boxes (length gs) [+] recursive.empty (max (Suc n) 
+      (Max (insert c (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs)))) n [+]
+      mv_boxes (Suc (max (Suc n) (Max (insert c 
+    (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))) + length gs)) 0 n" in exI)
+    apply(simp add: abc_append_commute [THEN sym])
+    apply(auto)
+    using cn_merge_gs_split[of i gs ga "length lm" gc 
+      "(max (Suc (length lm))
+       (Max (insert c (((\<lambda>(aprog, p, n). n) \<circ> rec_ci) ` set gs))))"] 
+      h2
+    apply(simp)
+    done
+  from this obtain cp where g3: 
+    "aprog = (cn_merge_gs (map rec_ci ?ggs) ?pstr) [+] ga [+] cp" ..
+  show "\<forall> stp. case abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suflm) 
+    aprog stp of (ss, e) \<Rightarrow> ss < length aprog"
+  proof(rule_tac abc_append_unhalt2)
+    show "abc_steps_l (0, lm @ 0\<^bsup>a_md - rs_pos\<^esup> @ suflm) (
+      cn_merge_gs (map rec_ci ?ggs) ?pstr) stpa =
+         (length ((cn_merge_gs (map rec_ci ?ggs) ?pstr)),  
+          lm @ 0\<^bsup>?pstr - n\<^esup> @ ys @ 0\<^bsup>a_md -(?pstr + length ?ggs)\<^esup> @ suflm)"
+      using h3 g2
+      apply(simp add: cn_merge_gs_length)
+      done
+  next
+    show "ga \<noteq> []"
+      using h1
+      apply(simp add: rec_ci_not_null)
+      done
+  next
+    show "\<forall>stp. case abc_steps_l (0, lm @ 0\<^bsup>?pstr - n\<^esup> @ ys
+      @ 0\<^bsup>a_md - (?pstr + length (take i gs))\<^esup> @ suflm) ga  stp of
+          (ss, e) \<Rightarrow> ss < length ga"
+      using ind[of "0\<^bsup>?pstr -gc\<^esup> @ ys @ 0\<^bsup>a_md - (?pstr + length (take i gs))\<^esup>
+        @ suflm"]
+      apply(subgoal_tac "lm @ 0\<^bsup>?pstr - n\<^esup> @ ys
+        @ 0\<^bsup>a_md - (?pstr + length (take i gs))\<^esup> @ suflm
+                       = lm @ 0\<^bsup>gc - n \<^esup>@ 
+        0\<^bsup>?pstr -gc\<^esup> @ ys @ 0\<^bsup>a_md - (?pstr + length (take i gs))\<^esup> @ suflm", simp)
+      apply(simp add: exponent_def replicate_add[THEN sym])
+      apply(subgoal_tac "gc > n \<and> ?pstr \<ge> gc")
+      apply(erule_tac conjE)
+      apply(simp add: h1)
+      using h1
+      apply(auto)
+      apply(rule_tac min_max.le_supI2)
+      apply(rule_tac Max_ge, simp, simp)
+      apply(rule_tac disjI2)
+      using h2
+      thm rev_image_eqI
+      apply(rule_tac x = "gs!i" in rev_image_eqI, simp, simp)
+      done
+  next
+    show "aprog = cn_merge_gs (map rec_ci (take i gs)) 
+              ?pstr [+] ga [+] cp"
+      using g3 by simp
+  qed
+qed
+
+
+lemma abc_rec_halt_eq': 
+  "\<lbrakk>rec_ci re = (ap, ary, fp); 
+    rec_calc_rel re args r\<rbrakk>
+  \<Longrightarrow> (\<exists> stp. (abc_steps_l (0, args @ 0\<^bsup>fp - ary\<^esup>) ap stp) = 
+                     (length ap, args@[r]@0\<^bsup>fp - ary - 1\<^esup>))"
+using aba_rec_equality[of re ap ary fp args r "[]"]
+by simp
+
+thm abc_step_l.simps
+definition dummy_abc :: "nat \<Rightarrow> abc_inst list"
+where
+"dummy_abc k = [Inc k, Dec k 0, Goto 3]"
+
+lemma abc_rec_halt_eq'': 
+  "\<lbrakk>rec_ci re = (aprog, rs_pos, a_md);  
+  rec_calc_rel re lm rs\<rbrakk>
+  \<Longrightarrow> (\<exists> stp lm' m. (abc_steps_l (0, lm) aprog stp) = 
+  (length aprog, lm') \<and> abc_list_crsp lm' (lm @ rs # 0\<^bsup>m\<^esup>))"
+apply(frule_tac abc_rec_halt_eq', auto)
+apply(drule_tac abc_list_crsp_steps)
+apply(rule_tac rec_ci_not_null, simp)
+apply(erule_tac exE, rule_tac x = stp in exI, 
+  auto simp: abc_list_crsp_def)
+done
+
+lemma [simp]: "length (dummy_abc (length lm)) = 3"
+apply(simp add: dummy_abc_def)
+done
+
+lemma [simp]: "dummy_abc (length lm) \<noteq> []"
+apply(simp add: dummy_abc_def)
+done
+
+lemma dummy_abc_steps_ex: 
+  "\<exists>bstp. abc_steps_l (0, lm') (dummy_abc (length lm)) bstp = 
+  ((Suc (Suc (Suc 0))), abc_lm_s lm' (length lm) (abc_lm_v lm' (length lm)))"
+apply(rule_tac x = "Suc (Suc (Suc 0))" in exI)
+apply(auto simp: abc_steps_l.simps abc_step_l.simps 
+  dummy_abc_def abc_fetch.simps)
+apply(auto simp: abc_lm_s.simps abc_lm_v.simps nth_append)
+apply(simp add: butlast_append)
+done
+
+lemma [elim]: 
+  "lm @ rs # 0\<^bsup>m\<^esup> = lm' @ 0\<^bsup>n\<^esup> \<Longrightarrow> 
+  \<exists>m. abc_lm_s lm' (length lm) (abc_lm_v lm' (length lm)) = 
+                            lm @ rs # 0\<^bsup>m\<^esup>"
+proof(cases "length lm' > length lm")
+  case True 
+  assume h: "lm @ rs # 0\<^bsup>m\<^esup> = lm' @ 0\<^bsup>n\<^esup>" "length lm < length lm'"
+  hence "m \<ge> n"
+    apply(drule_tac list_length)
+    apply(simp)
+    done
+  hence "\<exists> d. m = d + n"
+    apply(rule_tac x = "m - n" in exI, simp)
+    done
+  from this obtain d where "m = d + n" ..
+  from h and this show "?thesis"
+    apply(auto simp: abc_lm_s.simps abc_lm_v.simps 
+                     exponent_def replicate_add)
+    done
+next
+  case False
+  assume h:"lm @ rs # 0\<^bsup>m\<^esup> = lm' @ 0\<^bsup>n\<^esup>" 
+    and    g: "\<not> length lm < length lm'"
+  have "take (Suc (length lm)) (lm @ rs # 0\<^bsup>m\<^esup>) = 
+                        take (Suc (length lm)) (lm' @ 0\<^bsup>n\<^esup>)"
+    using h by simp
+  moreover have "n \<ge> (Suc (length lm) - length lm')"
+    using h g
+    apply(drule_tac list_length)
+    apply(simp)
+    done
+  ultimately show 
+    "\<exists>m. abc_lm_s lm' (length lm) (abc_lm_v lm' (length lm)) =
+                                                       lm @ rs # 0\<^bsup>m\<^esup>"
+    using g h
+    apply(simp add: abc_lm_s.simps abc_lm_v.simps 
+                                        exponent_def min_def)
+    apply(rule_tac x = 0 in exI, 
+      simp add:replicate_append_same replicate_Suc[THEN sym]
+                                      del:replicate_Suc)
+    done
+qed
+
+lemma [elim]: 
+  "abc_list_crsp lm' (lm @ rs # 0\<^bsup>m\<^esup>)
+  \<Longrightarrow> \<exists>m. abc_lm_s lm' (length lm) (abc_lm_v lm' (length lm)) 
+             = lm @ rs # 0\<^bsup>m\<^esup>"
+apply(auto simp: abc_list_crsp_def)
+apply(simp add: abc_lm_v.simps abc_lm_s.simps)
+apply(rule_tac x =  "m + n" in exI, 
+      simp add: exponent_def replicate_add)
+done
+
+
+lemma abc_append_dummy_complie:
+  "\<lbrakk>rec_ci recf = (ap, ary, fp);  
+    rec_calc_rel recf args r; 
+    length args = k\<rbrakk>
+  \<Longrightarrow> (\<exists> stp m. (abc_steps_l (0, args) (ap [+] dummy_abc k) stp) = 
+                  (length ap + 3, args @ r # 0\<^bsup>m\<^esup>))"
+apply(drule_tac abc_rec_halt_eq'', auto simp: numeral_3_eq_3)
+proof -
+  fix stp lm' m
+  assume h: "rec_calc_rel recf args r"  
+    "abc_steps_l (0, args) ap stp = (length ap, lm')" 
+    "abc_list_crsp lm' (args @ r # 0\<^bsup>m\<^esup>)"
+  thm abc_append_exc2
+  thm abc_lm_s.simps
+  have "\<exists>stp. abc_steps_l (0, args) (ap [+] 
+    (dummy_abc (length args))) stp = (length ap + 3, 
+    abc_lm_s lm' (length args) (abc_lm_v lm' (length args)))"
+    using h
+    apply(rule_tac bm = lm' in abc_append_exc2,
+          auto intro: dummy_abc_steps_ex simp: numeral_3_eq_3)
+    done
+  thus "\<exists>stp m. abc_steps_l (0, args) (ap [+] 
+    dummy_abc (length args)) stp = (Suc (Suc (Suc (length ap))), args @ r # 0\<^bsup>m\<^esup>)"
+    using h
+    apply(erule_tac exE)
+    apply(rule_tac x = stpa in exI, auto)
+    done
+qed
+
+lemma [simp]: "length (dummy_abc k) = 3"
+apply(simp add: dummy_abc_def)
+done
+
+lemma [simp]: "length args = k \<Longrightarrow> abc_lm_v (args @ r # 0\<^bsup>m\<^esup>) k = r "
+apply(simp add: abc_lm_v.simps nth_append)
+done
+
+lemma t_compiled_by_rec: 
+  "\<lbrakk>rec_ci recf = (ap, ary, fp); 
+    rec_calc_rel recf args r;
+    length args = k;
+    ly = layout_of (ap [+] dummy_abc k);
+    mop_ss = start_of ly (length (ap [+] dummy_abc k));
+    tp = tm_of (ap [+] dummy_abc k)\<rbrakk>
+  \<Longrightarrow> \<exists> stp m l. steps (Suc 0, Bk # Bk # ires, <args> @ Bk\<^bsup>rn\<^esup>) (tp @ (tMp k (mop_ss - 1))) stp
+                      = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc r\<^esup> @ Bk\<^bsup>l\<^esup>)"
+  using abc_append_dummy_complie[of recf ap ary fp args r k]
+apply(simp)
+apply(erule_tac exE)+
+apply(frule_tac tprog = tp and as = "length ap + 3" and n = k 
+               and ires = ires and rn = rn in abacus_turing_eq_halt, simp_all, simp)
+apply(erule_tac exE)+
+apply(simp)
+apply(rule_tac x = stpa in exI, rule_tac x = ma in exI, rule_tac x = l in exI, simp)
+done
+
+thm tms_of.simps
+
+lemma [simp]:
+  "list_all (\<lambda>(acn, s). s \<le> Suc (Suc (Suc (Suc (Suc (Suc (2 * n))))))) xs \<Longrightarrow>
+  list_all (\<lambda>(acn, s). s \<le> Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (2 * n))))))))) xs"
+apply(induct xs, simp, simp)
+apply(case_tac a, simp)
+done
+
+(*
+lemma [simp]: "t_correct (tMp n 0)"
+apply(simp add: t_correct.simps tMp.simps shift_length mp_up_def iseven_def, auto)
+apply(rule_tac x = "2*n + 6" in exI, simp)
+apply(induct n, auto simp: mop_bef.simps)
+apply(simp add: tshift.simps)
+done
+*)
+
+lemma tshift_append: "tshift (xs @ ys) n = tshift xs n @ tshift ys n"
+apply(simp add: tshift.simps)
+done
+
+lemma [simp]: "length (tMp n ss) = 4 * n + 12"
+apply(auto simp: tMp.simps tshift_append shift_length mp_up_def)
+done
+
+lemma length_tm_even[intro]: "\<exists>x. length (tm_of ap) = 2*x"
+apply(subgoal_tac "t_ncorrect (tm_of ap)")
+apply(simp add: t_ncorrect.simps, auto)
+done
+
+lemma [simp]: "k < length ap \<Longrightarrow> tms_of ap ! k  = 
+ ci (layout_of ap) (start_of (layout_of ap) k) (ap ! k)"
+apply(simp add: tms_of.simps tpairs_of.simps)
+done
+
+lemma [elim]: "\<lbrakk>k < length ap; ap ! k = Inc n; 
+       (a, b) \<in> set (tshift (tshift tinc_b (2 * n)) 
+                            (start_of (layout_of ap) k - Suc 0))\<rbrakk>
+       \<Longrightarrow> b \<le> start_of (layout_of ap) (length ap)"
+apply(subgoal_tac "b \<le> start_of (layout_of ap) (Suc k)")
+apply(subgoal_tac "start_of (layout_of ap) (Suc k) \<le> start_of (layout_of ap) (length ap) ")
+apply(arith)
+apply(case_tac "Suc k = length ap", simp)
+apply(rule_tac start_of_le, simp)
+apply(auto simp: tinc_b_def tshift.simps start_of.simps 
+  layout_of.simps length_of.simps startof_not0)
+done
+
+lemma findnth_le[elim]: "(a, b) \<in> set (tshift (findnth n) (start_of (layout_of ap) k - Suc 0))
+        \<Longrightarrow> b \<le> Suc (start_of (layout_of ap) k + 2 * n)"
+apply(induct n, simp add: findnth.simps tshift.simps)
+apply(simp add: findnth.simps tshift_append, auto)
+apply(auto simp: tshift.simps)
+done
+
+
+lemma  [elim]: "\<lbrakk>k < length ap; ap ! k = Inc n; (a, b) \<in> 
+  set (tshift (findnth n) (start_of (layout_of ap) k - Suc 0))\<rbrakk> 
+  \<Longrightarrow> b \<le> start_of (layout_of ap) (length ap)"
+apply(subgoal_tac "b \<le> start_of (layout_of ap) (Suc k)")
+apply(subgoal_tac "start_of (layout_of ap) (Suc k) \<le> start_of (layout_of ap) (length ap) ")
+apply(arith)
+apply(case_tac "Suc k = length ap", simp)
+apply(rule_tac start_of_le, simp)
+apply(subgoal_tac "b \<le> start_of (layout_of ap) k + 2*n + 1 \<and> 
+     start_of (layout_of ap) k + 2*n + 1 \<le>  start_of (layout_of ap) (Suc k)", auto)
+apply(auto simp: tinc_b_def tshift.simps start_of.simps 
+  layout_of.simps length_of.simps startof_not0)
+done
+
+lemma start_of_eq: "length ap < as \<Longrightarrow> start_of (layout_of ap) as = start_of (layout_of ap) (length ap)"
+apply(induct as, simp)
+apply(case_tac "length ap < as", simp add: start_of.simps)
+apply(subgoal_tac "as = length ap")
+apply(simp add: start_of.simps)
+apply arith
+done
+
+lemma start_of_all_le: "start_of (layout_of ap) as \<le> start_of (layout_of ap) (length ap)"
+apply(subgoal_tac "as > length ap \<or> as = length ap \<or> as < length ap", 
+      auto simp: start_of_eq start_of_le)
+done
+
+lemma [elim]: "\<lbrakk>k < length ap; 
+        ap ! k = Dec n e;
+         (a, b) \<in> set (tshift (findnth n) (start_of (layout_of ap) k - Suc 0))\<rbrakk>
+       \<Longrightarrow> b \<le> start_of (layout_of ap) (length ap)"
+apply(subgoal_tac "b \<le> start_of (layout_of ap) k + 2*n + 1 \<and> 
+     start_of (layout_of ap) k + 2*n + 1 \<le>  start_of (layout_of ap) (Suc k) \<and>
+      start_of (layout_of ap) (Suc k) \<le> start_of (layout_of ap) (length ap)", auto)
+apply(simp add:  tshift.simps start_of.simps 
+  layout_of.simps length_of.simps startof_not0)
+apply(rule_tac start_of_all_le)
+done
+
+thm length_of.simps
+lemma [elim]: "\<lbrakk>k < length ap; ap ! k = Dec n e; (a, b) \<in> set (tshift (tshift tdec_b (2 * n))
+                  (start_of (layout_of ap) k - Suc 0))\<rbrakk>
+       \<Longrightarrow> b \<le> start_of (layout_of ap) (length ap)"
+apply(subgoal_tac "2*n + start_of (layout_of ap) k + 16 \<le> start_of (layout_of ap) (length ap) \<and> start_of (layout_of ap) k > 0")
+prefer 2
+apply(subgoal_tac "2 * n + start_of (layout_of ap) k + 16 = start_of (layout_of ap) (Suc k)
+                 \<and> start_of (layout_of ap) (Suc k) \<le> start_of (layout_of ap) (length ap)")
+apply(simp add: startof_not0, rule_tac conjI)
+apply(simp add: start_of.simps layout_of.simps length_of.simps)
+apply(rule_tac start_of_all_le)
+apply(auto simp: tdec_b_def tshift.simps)
+done
+
+lemma tms_any_less: "\<lbrakk>k < length ap; (a, b) \<in> set (tms_of ap ! k)\<rbrakk> \<Longrightarrow> b \<le> start_of (layout_of ap) (length ap)"
+apply(simp)
+apply(case_tac "ap!k", simp_all add: ci.simps tshift_append, auto intro: start_of_all_le)
+done
+lemma concat_in: "i < length (concat xs) \<Longrightarrow> \<exists>k < length xs. concat xs ! i \<in> set (xs ! k)"
+apply(induct xs rule: list_tl_induct, simp, simp)
+apply(case_tac "i < length (concat list)", simp)
+apply(erule_tac exE, rule_tac x = k in exI)
+apply(simp add: nth_append)
+apply(rule_tac x = "length list" in exI, simp)
+apply(simp add: nth_append)
+done 
+
+lemma [simp]: "length (tms_of ap) = length ap"
+apply(simp add: tms_of.simps tpairs_of.simps)
+done
+
+lemma in_tms: "i < length (tm_of ap) \<Longrightarrow> \<exists> k < length ap. (tm_of ap ! i) \<in> set (tms_of ap ! k)"
+apply(simp add: tm_of.simps)
+using concat_in[of i "tms_of ap"]
+by simp
+
+lemma all_le_start_of: "list_all (\<lambda>(acn, s). s \<le> start_of (layout_of ap) (length ap)) (tm_of ap)"
+apply(simp add: list_all_length)
+apply(rule_tac allI, rule_tac impI)
+apply(drule_tac in_tms, auto elim: tms_any_less)
+done
+
+lemma length_ci: "\<lbrakk>k < length ap; length (ci ly y (ap ! k)) = 2 * qa\<rbrakk>
+      \<Longrightarrow> layout_of ap ! k = qa"
+apply(case_tac "ap ! k")
+apply(auto simp: layout_of.simps ci.simps 
+  length_of.simps shift_length tinc_b_def tdec_b_def)
+done
+
+lemma [intro]: "length (ci ly y i) mod 2 = 0"
+apply(auto simp: ci.simps shift_length tinc_b_def tdec_b_def
+      split: abc_inst.splits)
+done
+
+lemma [intro]: "listsum (map (length \<circ> (\<lambda>(x, y). ci ly x y)) zs) mod 2 = 0"
+apply(induct zs rule: list_tl_induct, simp)
+apply(case_tac a, simp)
+apply(subgoal_tac "length (ci ly aa b) mod 2 = 0")
+apply(auto)
+done
+
+lemma zip_pre:
+  "(length ys) \<le> length ap \<Longrightarrow>
+  zip ys ap = zip ys (take (length ys) (ap::'a list))"
+proof(induct ys arbitrary: ap, simp, case_tac ap, simp)
+  fix a ys ap aa list
+  assume ind: "\<And>(ap::'a list). length ys \<le> length ap \<Longrightarrow> 
+    zip ys ap = zip ys (take (length ys) ap)"
+  and h: "length (a # ys) \<le> length ap" "(ap::'a list) = aa # (list::'a list)"
+  from h show "zip (a # ys) ap = zip (a # ys) (take (length (a # ys)) ap)"
+    using ind[of list]
+    apply(simp)
+    done
+qed
+ 
+lemma start_of_listsum: 
+  "\<lbrakk>k \<le> length ap; length ss = k\<rbrakk> \<Longrightarrow> start_of (layout_of ap) k = 
+        Suc (listsum (map (length \<circ> (\<lambda>(x, y). ci ly x y)) (zip ss ap)) div 2)"
+proof(induct k arbitrary: ss, simp add: start_of.simps, simp)
+  fix k ss
+  assume ind: "\<And>ss. length ss = k \<Longrightarrow> start_of (layout_of ap) k = 
+            Suc (listsum (map (length \<circ> (\<lambda>(x, y). ci ly x y)) (zip ss ap)) div 2)"
+  and h: "Suc k \<le>  length ap" "length (ss::nat list) = Suc k"
+  have "\<exists> ys y. ss = ys @ [y]"
+    using h
+    apply(rule_tac x = "butlast ss" in exI,
+          rule_tac x = "last ss" in exI)
+    apply(case_tac "ss = []", auto)
+    done
+  from this obtain ys y where k1: "ss = (ys::nat list) @ [y]"
+    by blast
+  from h and this have k2: 
+    "start_of (layout_of ap) k = 
+    Suc (listsum (map (length \<circ> (\<lambda>(x, y). ci ly x y)) (zip ys ap)) div 2)"
+    apply(rule_tac ind, simp)
+    done
+  have k3: "zip ys ap = zip ys (take k ap)"
+    using zip_pre[of ys ap] k1 h
+    apply(simp)
+    done
+  have k4: "(zip [y] (drop (length ys) ap)) = [(y, ap ! length ys)]"
+    using k1 h
+    apply(case_tac "drop (length ys) ap", simp)
+    apply(subgoal_tac "hd (drop (length ys) ap) = ap ! length ys")
+    apply(simp)
+    apply(rule_tac hd_drop_conv_nth, simp)
+    done
+  from k1 and h k2 k3 k4 show "start_of (layout_of ap) (Suc k) = 
+    Suc (listsum (map (length \<circ> (\<lambda>(x, y). ci ly x y)) (zip ss ap)) div 2)"
+    apply(simp add: zip_append1 start_of.simps)
+    apply(subgoal_tac 
+      "listsum (map (length \<circ> (\<lambda>(x, y). ci ly x y)) (zip ys (take k ap))) mod 2 = 0 \<and> 
+      length (ci ly y (ap!k)) mod 2 = 0")
+    apply(auto)
+    apply(rule_tac length_ci, simp, simp)
+    done
+qed
+
+lemma length_start_of_tm: "start_of (layout_of ap) (length ap) = Suc (length (tm_of ap)  div 2)"
+apply(simp add: tm_of.simps length_concat tms_of.simps tpairs_of.simps)
+apply(rule_tac start_of_listsum, simp, simp)
+done
+
+lemma tm_even: "length (tm_of ap) mod 2 = 0" 
+apply(subgoal_tac "t_ncorrect (tm_of ap)", auto)
+apply(simp add: t_ncorrect.simps)
+done
+
+lemma [elim]: "list_all (\<lambda>(acn, s). s \<le> Suc q) xs
+        \<Longrightarrow> list_all (\<lambda>(acn, s). s \<le> q + (2 * n + 6)) xs"
+apply(simp add: list_all_length)
+apply(auto)
+done
+
+lemma [simp]: "length mp_up = 12"
+apply(simp add: mp_up_def)
+done
+
+lemma [elim]: "\<lbrakk>na < 4 * n; tshift (mop_bef n) q ! na = (a, b)\<rbrakk> \<Longrightarrow> b \<le> q + (2 * n + 6)"
+apply(induct n, simp, simp add: mop_bef.simps nth_append tshift_append shift_length)
+apply(case_tac "na < 4*n", simp, simp)
+apply(subgoal_tac "na = 4*n \<or> na = 1 + 4*n \<or> na = 2 + 4*n \<or> na = 3 + 4*n",
+  auto simp: shift_length)
+apply(simp_all add: tshift.simps)
+done
+
+lemma mp_up_all_le: "list_all  (\<lambda>(acn, s). s \<le> q + (2 * n + 6)) 
+  [(R, Suc (Suc (2 * n + q))), (R, Suc (2 * n + q)), 
+  (L, 5 + 2 * n + q), (W0, Suc (Suc (Suc (2 * n + q)))), (R, 4 + 2 * n + q),
+  (W0, Suc (Suc (Suc (2 * n + q)))), (R, Suc (Suc (2 * n + q))),
+  (W0, Suc (Suc (Suc (2 * n + q)))), (L, 5 + 2 * n + q),
+  (L, 6 + 2 * n + q), (R, 0),  (L, 6 + 2 * n + q)]"
+apply(auto)
+done
+
+
+lemma [intro]: "list_all (\<lambda>(acn, s). s \<le> q + (2 * n + 6)) (tMp n q)"
+apply(auto simp: list_all_length tMp.simps tshift_append nth_append shift_length)
+apply(auto simp: tshift.simps mp_up_def)
+apply(subgoal_tac "na - 4*n \<ge> 0 \<and> na - 4 *n < 12", auto split: nat.splits)
+apply(insert mp_up_all_le[of q n])
+apply(simp add: list_all_length)
+apply(erule_tac x = "na - 4 * n" in allE, simp add: numeral_3_eq_3)
+done
+
+lemma t_compiled_correct: 
+  "\<lbrakk>tp = tm_of ap; ly = layout_of ap; mop_ss = start_of ly (length ap)\<rbrakk> \<Longrightarrow> 
+       t_correct (tp @ tMp n (mop_ss - Suc 0))"
+  using tm_even[of ap] length_start_of_tm[of ap] all_le_start_of[of ap]
+apply(auto simp: t_correct.simps iseven_def)
+apply(rule_tac x = "q + 2*n + 6" in exI, simp)
+done
+
+end
+
+    
+  
+
+
+  
+