diff -r 0b302c0b449a -r 469c26d19f8e recursive.thy
--- a/recursive.thy	Wed Feb 06 02:25:00 2013 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,5024 +0,0 @@
-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
-
-    
-  
-
-
-  
-