
diff -r 9510e5131e06 -r b51cb9aef3ae thys/Abacus.thy
--- a/thys/Abacus.thy	Fri Feb 15 07:42:47 2013 +0000
+++ b/thys/Abacus.thy	Fri Feb 15 14:05:26 2013 +0000
@@ -1,5 +1,11 @@
+(* Title: thys/Abacus.thy
+   Author: Jian Xu, Xingyuan Zhang, and Christian Urban
+*)
+
+header {* Abacus Machines *}
+
 theory Abacus
-imports Uncomputable
+imports Turing_Hoare Abacus_Mopup
 begin
 
 declare replicate_Suc[simp add]
@@ -603,6 +609,7 @@
     done
 qed
 
+declare fetch.simps[simp]
 lemma append_append_fetch: 
     "\<lbrakk>length tp1 mod 2 = 0; length tp mod 2 = 0;
       length tp1 div 2 < a \<and> a \<le> length tp1 div 2 + length tp div 2\<rbrakk>
@@ -1248,12 +1255,6 @@
 apply(erule disj_forward, auto)
 done
 
-lemma tape_of_nl_cons: "<m # lm> = (if lm = [] then Oc\<up>(Suc m)
-                    else Oc\<up>(Suc m) @ Bk # <lm>)"
-apply(case_tac lm, simp_all add: tape_of_nl_abv  tape_of_nat_abv split: if_splits)
-done
-
-
 lemma locate_a_2_locate_a[simp]: "inv_locate_a (as, am) (q, aaa, Bk # xs) ires
        \<Longrightarrow> inv_locate_a (as, am) (q, aaa, Oc # xs) ires"
 apply(simp only: inv_locate_a.simps at_begin_norm.simps 
@@ -2019,6 +2020,9 @@
 apply(auto simp: inv_check_left_moving.simps inv_check_left_moving_in_middle.simps split: if_splits)
 done
 
+lemma numeral_4_eq_4: "4 =  Suc (Suc (Suc (Suc 0)))"
+by arith
+
 lemma tinc_correct_pre:
   assumes layout: "ly = layout_of ap"
   and inv_start: "inv_locate_a (as, lm) (n, l, r) ires"
@@ -2049,14 +2053,13 @@
       apply(simp add:Q)
       apply(simp add: inc_inv.simps)
       apply(case_tac c, case_tac [2] aa)
-      apply(auto simp: Let_def step.simps tinc_b_def numeral_2_eq_2 numeral_3_eq_3  split: if_splits)
-      apply(simp_all add: inc_inv.simps inc_LE_def lex_triple_def lex_pair_def inc_measure_def numeral_5_eq_5
-                          numeral_6_eq_6 numeral_7_eq_7 numeral_8_eq_8 numeral_9_eq_9)         
+      apply(auto simp: Let_def step.simps tinc_b_def split: if_splits)
+      apply(simp_all add: inc_inv.simps inc_LE_def lex_triple_def lex_pair_def inc_measure_def numeral_5_eq_5 numeral_2_eq_2 numeral_3_eq_3
+                          numeral_4_eq_4 numeral_6_eq_6 numeral_7_eq_7 numeral_8_eq_8 numeral_9_eq_9)         
       done
   qed
 qed
          
-
 lemma tinc_correct: 
   assumes layout: "ly = layout_of ap"
   and inv_start: "inv_locate_a (as, lm) (n, l, r) ires"
@@ -3756,817 +3759,6 @@
 apply(simp_all add: start_of.simps fetch.simps nth_append)
 done
 
-(********for mopup***********)
-fun mopup_a :: "nat \<Rightarrow> instr list"
-  where
-  "mopup_a 0 = []" |
-  "mopup_a (Suc n) = mopup_a n @ 
-       [(R, 2*n + 3), (W0, 2*n + 2), (R, 2*n + 1), (W1, 2*n + 2)]"
-
-definition mopup_b :: "instr list"
-  where
-  "mopup_b \<equiv> [(R, 2), (R, 1), (L, 5), (W0, 3), (R, 4), (W0, 3),
-            (R, 2), (W0, 3), (L, 5), (L, 6), (R, 0), (L, 6)]"
-
-fun mopup :: "nat \<Rightarrow> instr list"
-  where 
-  "mopup n = mopup_a n @ shift mopup_b (2*n)"
-(****)
-
-type_synonym mopup_type = "config \<Rightarrow> nat list \<Rightarrow> nat \<Rightarrow> cell list \<Rightarrow> bool"
-
-fun mopup_stop :: "mopup_type"
-  where
-  "mopup_stop (s, l, r) lm n ires= 
-        (\<exists> ln rn. l = Bk\<up>ln @ Bk # Bk # ires \<and> r = <abc_lm_v lm n> @ Bk\<up>rn)"
-
-fun mopup_bef_erase_a :: "mopup_type"
-  where
-  "mopup_bef_erase_a (s, l, r) lm n ires= 
-         (\<exists> ln m rn. l = Bk\<up>ln @ Bk # Bk # ires \<and> 
-                  r = Oc\<up>m@ Bk # <(drop ((s + 1) div 2) lm)> @ Bk\<up>rn)"
-
-fun mopup_bef_erase_b :: "mopup_type"
-  where
-  "mopup_bef_erase_b (s, l, r) lm n ires = 
-      (\<exists> ln m rn. l = Bk\<up>ln @ Bk # Bk # ires \<and> r = Bk # Oc\<up>m @ Bk # 
-                                      <(drop (s div 2) lm)> @ Bk\<up>rn)"
-
-fun mopup_jump_over1 :: "mopup_type"
-  where
-  "mopup_jump_over1 (s, l, r) lm n ires = 
-      (\<exists> ln m1 m2 rn. m1 + m2 = Suc (abc_lm_v lm n) \<and> 
-        l = Oc\<up>m1 @ Bk\<up>ln @ Bk # Bk # ires \<and> 
-     (r = Oc\<up>m2 @ Bk # <(drop (Suc n) lm)> @ Bk\<up>rn \<or> 
-     (r = Oc\<up>m2 \<and> (drop (Suc n) lm) = [])))"
-
-fun mopup_aft_erase_a :: "mopup_type"
-  where
-  "mopup_aft_erase_a (s, l, r) lm n ires = 
-      (\<exists> lnl lnr rn (ml::nat list) m. 
-          m = Suc (abc_lm_v lm n) \<and> l = Bk\<up>lnr @ Oc\<up>m @ Bk\<up>lnl @ Bk # Bk # ires \<and> 
-                                   (r = <ml> @ Bk\<up>rn))"
-
-fun mopup_aft_erase_b :: "mopup_type"
-  where
-  "mopup_aft_erase_b (s, l, r) lm n ires= 
-   (\<exists> lnl lnr rn (ml::nat list) m. 
-      m = Suc (abc_lm_v lm n) \<and> 
-      l = Bk\<up>lnr @ Oc\<up>m @ Bk\<up>lnl @ Bk # Bk # ires \<and> 
-     (r = Bk # <ml> @ Bk\<up>rn \<or>
-      r = Bk # Bk # <ml> @ Bk\<up>rn))"
-
-fun mopup_aft_erase_c :: "mopup_type"
-  where
-  "mopup_aft_erase_c (s, l, r) lm n ires = 
- (\<exists> lnl lnr rn (ml::nat list) m. 
-     m = Suc (abc_lm_v lm n) \<and> 
-     l = Bk\<up>lnr @ Oc\<up>m @ Bk\<up>lnl @ Bk # Bk # ires \<and> 
-    (r = <ml> @ Bk\<up>rn \<or> r = Bk # <ml> @ Bk\<up>rn))"
-
-fun mopup_left_moving :: "mopup_type"
-  where
-  "mopup_left_moving (s, l, r) lm n ires = 
-  (\<exists> lnl lnr rn m.
-     m = Suc (abc_lm_v lm n) \<and> 
-   ((l = Bk\<up>lnr @ Oc\<up>m @ Bk\<up>lnl @ Bk # Bk # ires \<and> r = Bk\<up>rn) \<or>
-    (l = Oc\<up>(m - 1) @ Bk\<up>lnl @ Bk # Bk # ires \<and> r = Oc # Bk\<up>rn)))"
-
-fun mopup_jump_over2 :: "mopup_type"
-  where
-  "mopup_jump_over2 (s, l, r) lm n ires = 
-     (\<exists> ln rn m1 m2.
-          m1 + m2 = Suc (abc_lm_v lm n) 
-        \<and> r \<noteq> [] 
-        \<and> (hd r = Oc \<longrightarrow> (l = Oc\<up>m1 @ Bk\<up>ln @ Bk # Bk # ires \<and> r = Oc\<up>m2 @ Bk\<up>rn)) 
-        \<and> (hd r = Bk \<longrightarrow> (l = Bk\<up>ln @ Bk # ires \<and> r = Bk # Oc\<up>(m1+m2)@ Bk\<up>rn)))"
-
-
-fun mopup_inv :: "mopup_type"
-  where
-  "mopup_inv (s, l, r) lm n ires = 
-      (if s = 0 then mopup_stop (s, l, r) lm n ires
-       else if s \<le> 2*n then
-               if s mod 2 = 1 then mopup_bef_erase_a (s, l, r) lm n ires
-                   else mopup_bef_erase_b (s, l, r) lm n ires
-            else if s = 2*n + 1 then 
-                mopup_jump_over1 (s, l, r) lm n ires
-            else if s = 2*n + 2 then mopup_aft_erase_a (s, l, r) lm n ires
-            else if s = 2*n + 3 then mopup_aft_erase_b (s, l, r) lm n ires
-            else if s = 2*n + 4 then mopup_aft_erase_c (s, l, r) lm n ires
-            else if s = 2*n + 5 then mopup_left_moving (s, l, r) lm n ires
-            else if s = 2*n + 6 then mopup_jump_over2 (s, l, r) lm n ires
-            else False)"
-
-lemma mopup_fetch_0[simp]: 
-     "(fetch (mopup_a n @ shift mopup_b (2 * n)) 0 b) = (Nop, 0)"
-by(simp add: fetch.simps)
-
-lemma mop_bef_length[simp]: "length (mopup_a n) = 4 * n"
-apply(induct n, simp_all add: mopup_a.simps)
-done
-
-lemma mopup_a_nth: 
-  "\<lbrakk>q < n; x < 4\<rbrakk> \<Longrightarrow> mopup_a n ! (4 * q + x) = 
-                             mopup_a (Suc q) ! ((4 * q) + x)"
-apply(induct n, simp)
-apply(case_tac "q < n", simp add: mopup_a.simps, auto)
-apply(simp add: nth_append)
-apply(subgoal_tac "q = n", simp)
-apply(arith)
-done
-
-lemma fetch_bef_erase_a_o[simp]: 
- "\<lbrakk>0 < s; s \<le> 2 * n; s mod 2 = Suc 0\<rbrakk>
-  \<Longrightarrow> (fetch (mopup_a n @ shift mopup_b (2 * n)) s Oc) = (W0, s + 1)"
-apply(subgoal_tac "\<exists> q. s = 2*q + 1", auto)
-apply(subgoal_tac "length (mopup_a n) = 4*n")
-apply(auto simp: fetch.simps nth_of.simps nth_append)
-apply(subgoal_tac "mopup_a n ! (4 * q + 1) = 
-                      mopup_a (Suc q) ! ((4 * q) + 1)", 
-      simp add: mopup_a.simps nth_append)
-apply(rule mopup_a_nth, auto)
-apply arith
-done
-
-lemma fetch_bef_erase_a_b[simp]:
-  "\<lbrakk>0 < s; s \<le> 2 * n; s mod 2 = Suc 0\<rbrakk>
-   \<Longrightarrow>  (fetch (mopup_a n @ shift mopup_b (2 * n)) s Bk) = (R, s + 2)"
-apply(subgoal_tac "\<exists> q. s = 2*q + 1", auto)
-apply(subgoal_tac "length (mopup_a n) = 4*n")
-apply(auto simp: fetch.simps nth_of.simps nth_append)
-apply(subgoal_tac "mopup_a n ! (4 * q + 0) = 
-                       mopup_a (Suc q) ! ((4 * q + 0))", 
-      simp add: mopup_a.simps nth_append)
-apply(rule mopup_a_nth, auto)
-apply arith
-done
-
-lemma fetch_bef_erase_b_b: 
-  "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n; s mod 2 = 0\<rbrakk> \<Longrightarrow> 
-     (fetch (mopup_a n @ shift mopup_b (2 * n)) s Bk) = (R, s - 1)"
-apply(subgoal_tac "\<exists> q. s = 2 * q", auto)
-apply(case_tac qa, simp, simp)
-apply(auto simp: fetch.simps nth_of.simps nth_append)
-apply(subgoal_tac "mopup_a n ! (4 * nat + 2) = 
-                     mopup_a (Suc nat) ! ((4 * nat) + 2)", 
-      simp add: mopup_a.simps nth_append)
-apply(rule mopup_a_nth, auto)
-done
-
-lemma fetch_jump_over1_o: 
- "fetch (mopup_a n @ shift mopup_b (2 * n)) (Suc (2 * n)) Oc
-  = (R, Suc (2 * n))"
-apply(subgoal_tac "length (mopup_a n) = 4 * n")
-apply(auto simp: fetch.simps nth_of.simps mopup_b_def nth_append 
-                 shift.simps)
-done
-
-lemma fetch_jump_over1_b: 
- "fetch (mopup_a n @ shift mopup_b (2 * n)) (Suc (2 * n)) Bk 
- = (R, Suc (Suc (2 * n)))"
-apply(subgoal_tac "length (mopup_a n) = 4 * n")
-apply(auto simp: fetch.simps nth_of.simps mopup_b_def 
-                 nth_append shift.simps)
-done
-
-lemma fetch_aft_erase_a_o: 
- "fetch (mopup_a n @ shift mopup_b (2 * n)) (Suc (Suc (2 * n))) Oc 
- = (W0, Suc (2 * n + 2))"
-apply(subgoal_tac "length (mopup_a n) = 4 * n")
-apply(auto simp: fetch.simps nth_of.simps mopup_b_def 
-                 nth_append shift.simps)
-done
-
-lemma fetch_aft_erase_a_b: 
- "fetch (mopup_a n @ shift mopup_b (2 * n)) (Suc (Suc (2 * n))) Bk
-  = (L, Suc (2 * n + 4))"
-apply(subgoal_tac "length (mopup_a n) = 4 * n")
-apply(auto simp: fetch.simps nth_of.simps mopup_b_def 
-                 nth_append shift.simps)
-done
-
-lemma fetch_aft_erase_b_b: 
- "fetch (mopup_a n @ shift mopup_b (2 * n)) (2*n + 3) Bk
-  = (R, Suc (2 * n + 3))"
-apply(subgoal_tac "length (mopup_a n) = 4 * n")
-apply(subgoal_tac "2*n + 3 = Suc (2*n + 2)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mopup_b_def nth_append shift.simps)
-done
-
-lemma fetch_aft_erase_c_o: 
- "fetch (mopup_a n @ shift mopup_b (2 * n)) (2 * n + 4) Oc 
- = (W0, Suc (2 * n + 2))"
-apply(subgoal_tac "length (mopup_a n) = 4 * n")
-apply(subgoal_tac "2*n + 4 = Suc (2*n + 3)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mopup_b_def nth_append shift.simps)
-done
-
-lemma fetch_aft_erase_c_b: 
- "fetch (mopup_a n @ shift mopup_b (2 * n)) (2 * n + 4) Bk 
- = (R, Suc (2 * n + 1))"
-apply(subgoal_tac "length (mopup_a n) = 4 * n")
-apply(subgoal_tac "2*n + 4 = Suc (2*n + 3)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mopup_b_def nth_append shift.simps)
-done
-
-lemma fetch_left_moving_o: 
- "(fetch (mopup_a n @ shift mopup_b (2 * n)) (2 * n + 5) Oc) 
- = (L, 2*n + 6)"
-apply(subgoal_tac "length (mopup_a n) = 4 * n")
-apply(subgoal_tac "2*n + 5 = Suc (2*n + 4)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mopup_b_def nth_append shift.simps)
-done
-
-lemma fetch_left_moving_b: 
- "(fetch (mopup_a n @ shift mopup_b (2 * n)) (2 * n + 5) Bk)
-  = (L, 2*n + 5)"
-apply(subgoal_tac "length (mopup_a n) = 4 * n")
-apply(subgoal_tac "2*n + 5 = Suc (2*n + 4)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mopup_b_def nth_append shift.simps)
-done
-
-lemma fetch_jump_over2_b:
-  "(fetch (mopup_a n @ shift mopup_b (2 * n)) (2 * n + 6) Bk) 
- = (R, 0)"
-apply(subgoal_tac "length (mopup_a n) = 4 * n")
-apply(subgoal_tac "2*n + 6 = Suc (2*n + 5)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mopup_b_def nth_append shift.simps)
-done
-
-lemma fetch_jump_over2_o: 
-"(fetch (mopup_a n @ shift mopup_b (2 * n)) (2 * n + 6) Oc) 
- = (L, 2*n + 6)"
-apply(subgoal_tac "length (mopup_a n) = 4 * n")
-apply(subgoal_tac "2*n + 6 = Suc (2*n + 5)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mopup_b_def nth_append shift.simps)
-done
-
-lemmas mopupfetchs = 
-fetch_bef_erase_a_o fetch_bef_erase_a_b fetch_bef_erase_b_b 
-fetch_jump_over1_o fetch_jump_over1_b fetch_aft_erase_a_o 
-fetch_aft_erase_a_b fetch_aft_erase_b_b fetch_aft_erase_c_o 
-fetch_aft_erase_c_b fetch_left_moving_o fetch_left_moving_b 
-fetch_jump_over2_b fetch_jump_over2_o
-
-declare 
-  mopup_jump_over2.simps[simp del] mopup_left_moving.simps[simp del]
-  mopup_aft_erase_c.simps[simp del] mopup_aft_erase_b.simps[simp del] 
-  mopup_aft_erase_a.simps[simp del] mopup_jump_over1.simps[simp del]
-  mopup_bef_erase_a.simps[simp del] mopup_bef_erase_b.simps[simp del]
-  mopup_stop.simps[simp del]
-
-lemma [simp]: 
-  "\<lbrakk>mopup_bef_erase_a (s, l, Oc # xs) lm n ires\<rbrakk> \<Longrightarrow> 
-  mopup_bef_erase_b (Suc s, l, Bk # xs) lm n ires"
-apply(auto simp: mopup_bef_erase_a.simps mopup_bef_erase_b.simps )
-apply(rule_tac x = "m - 1" in exI, rule_tac x = rn in exI)
-apply(case_tac m, simp, simp)
-done
-
-lemma mopup_false1:
-  "\<lbrakk>0 < s; s \<le> 2 * n; s mod 2 = Suc 0;  \<not> Suc s \<le> 2 * n\<rbrakk> 
-  \<Longrightarrow> RR"
-apply(arith)
-done
-
-lemma [simp]: 
- "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n; s mod 2 = Suc 0; 
-   mopup_bef_erase_a (s, l, Oc # xs) lm n ires; r = Oc # xs\<rbrakk>
- \<Longrightarrow> (Suc s \<le> 2 * n \<longrightarrow> mopup_bef_erase_b (Suc s, l, Bk # xs) lm n ires)  \<and>
-     (\<not> Suc s \<le> 2 * n \<longrightarrow> mopup_jump_over1 (Suc s, l, Bk # xs) lm n ires) "
-apply(auto elim: mopup_false1)
-done
-
-lemma drop_tape_of_cons: 
-  "\<lbrakk>Suc q < length lm; x = lm ! q\<rbrakk> \<Longrightarrow> <drop q lm> = Oc # Oc \<up> x @ Bk # <drop (Suc q) lm>"
-by (metis Suc_lessD append_Cons list.simps(2) nth_drop' replicate_Suc tape_of_nl_cons)
-
-lemma erase2jumpover1:
-  "\<lbrakk>q < length list; 
-             \<forall>rn. <drop q list> \<noteq> Oc # Oc \<up> abc_lm_v (a # list) (Suc q) @ Bk # <drop (Suc q) list> @ Bk \<up> rn\<rbrakk>
-       \<Longrightarrow> <drop q list> = Oc # Oc \<up> abc_lm_v (a # list) (Suc q)"
-apply(erule_tac x = 0 in allE, simp)
-apply(case_tac "Suc q < length list")
-apply(erule_tac notE)
-apply(rule_tac drop_tape_of_cons, simp_all add: abc_lm_v.simps)
-apply(subgoal_tac "length list = Suc q", auto)
-apply(subgoal_tac "drop q list = [list ! q]")
-apply(simp add: tape_of_nl_abv tape_of_nat_abv)
-by (metis append_Nil2 append_eq_conv_conj drop_Suc_conv_tl lessI)
-
-lemma erase2jumpover2:
-  "\<lbrakk>q < length list; \<forall>rn. <drop q list> @ Bk # Bk \<up> n \<noteq>
-  Oc # Oc \<up> abc_lm_v (a # list) (Suc q) @ Bk # <drop (Suc q) list> @ Bk \<up> rn\<rbrakk>
-  \<Longrightarrow> RR"
-apply(case_tac "Suc q < length list")
-apply(erule_tac x = "Suc n" in allE, simp)
-apply(erule_tac notE)
-apply(rule_tac drop_tape_of_cons, simp_all add: abc_lm_v.simps)
-apply(subgoal_tac "length list = Suc q", auto)
-apply(erule_tac x = "n" in allE, simp add: tape_of_nl_abv)
-by (metis append_Nil2 append_eq_conv_conj drop_Suc_conv_tl lessI replicate_Suc tape_of_nl_abv tape_of_nl_cons)
-
-lemma mopup_bef_erase_a_2_jump_over[simp]: 
- "\<lbrakk>n < length lm; 0 < s; s mod 2 = Suc 0;  s \<le> 2 * n;
-   mopup_bef_erase_a (s, l, Bk # xs) lm n ires; \<not> (Suc (Suc s) \<le> 2 * n)\<rbrakk> 
-\<Longrightarrow> mopup_jump_over1 (s', Bk # l, xs) lm n ires"
-apply(auto simp: mopup_bef_erase_a.simps mopup_jump_over1.simps)
-apply(case_tac m, auto simp: mod_ex1)
-apply(subgoal_tac "n = Suc q", auto)
-apply(rule_tac x = "Suc ln" in exI, rule_tac x = 0 in exI, auto)
-apply(case_tac [!] lm, simp_all)
-apply(case_tac [!] rn, auto elim: erase2jumpover1 erase2jumpover2)
-apply(erule_tac x = 0 in allE, simp)
-apply(rule_tac classical, simp)
-apply(erule_tac notE)
-apply(rule_tac drop_tape_of_cons, simp_all add: abc_lm_v.simps)
-done
-
-lemma Suc_Suc_div:  "\<lbrakk>0 < s; s mod 2 = Suc 0; Suc (Suc s) \<le> 2 * n\<rbrakk>
-           \<Longrightarrow> (Suc (Suc (s div 2))) \<le> n"
-apply(arith)
-done
-
-lemma mopup_bef_erase_a_2_a[simp]: 
- "\<lbrakk>n < length lm; 0 < s; s mod 2 = Suc 0; 
-   mopup_bef_erase_a (s, l, Bk # xs) lm n ires; 
-   Suc (Suc s) \<le> 2 * n\<rbrakk> \<Longrightarrow> 
-   mopup_bef_erase_a (Suc (Suc s), Bk # l, xs) lm n ires"
-apply(auto simp: mopup_bef_erase_a.simps)
-apply(subgoal_tac "drop (Suc (Suc (s div 2))) lm \<noteq> []")
-apply(case_tac m, simp_all)
-apply(rule_tac x = "Suc (abc_lm_v lm (Suc (s div 2)))" in exI, 
-      rule_tac x = rn in exI, auto simp: mod_ex1)
-apply(rule_tac drop_tape_of_cons)
-apply arith
-apply(simp add: abc_lm_v.simps)
-done
-
-lemma mopup_false2: 
- "\<lbrakk>0 < s; s \<le> 2 * n; 
-   s mod 2 = Suc 0; Suc s \<noteq> 2 * n;
-   \<not> Suc (Suc s) \<le> 2 * n\<rbrakk> \<Longrightarrow> RR"
-apply(arith)
-done
-
-lemma [simp]: "mopup_bef_erase_a (s, l, []) lm n ires \<Longrightarrow> 
-                        mopup_bef_erase_a (s, l, [Bk]) lm n ires"
-apply(auto simp: mopup_bef_erase_a.simps)
-done
-
-lemma [simp]:
-   "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n; s mod 2 = Suc 0; \<not> Suc (Suc s) \<le> 2 *n;
-     mopup_bef_erase_a (s, l, []) lm n ires\<rbrakk>
-    \<Longrightarrow>  mopup_jump_over1 (s', Bk # l, []) lm n ires"
-by auto
-
-lemma "mopup_bef_erase_b (s, l, Oc # xs) lm n ires \<Longrightarrow> l \<noteq> []"
-apply(auto simp: mopup_bef_erase_b.simps)
-done
-
-lemma [simp]: "mopup_bef_erase_b (s, l, Oc # xs) lm n ires = False"
-apply(auto simp: mopup_bef_erase_b.simps )
-done
- 
-lemma [simp]: "\<lbrakk>0 < s; s \<le> 2 *n; s mod 2 \<noteq> Suc 0\<rbrakk> \<Longrightarrow> 
-                                      (s - Suc 0) mod 2 = Suc 0"
-apply(arith)
-done
-
-lemma [simp]: "\<lbrakk>0 < s; s \<le> 2 *n; s mod 2 \<noteq> Suc 0\<rbrakk> \<Longrightarrow>
-                                       s - Suc 0 \<le> 2 * n"
-apply(simp)
-done
-
-lemma [simp]: "\<lbrakk>0 < s; s \<le> 2 *n; s mod 2 \<noteq> Suc 0\<rbrakk> \<Longrightarrow> \<not> s \<le> Suc 0"
-apply(arith)
-done
-
-lemma [simp]: "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n; 
-               s mod 2 \<noteq> Suc 0; 
-               mopup_bef_erase_b (s, l, Bk # xs) lm n ires; r = Bk # xs\<rbrakk> 
-           \<Longrightarrow> mopup_bef_erase_a (s - Suc 0, Bk # l, xs) lm n ires"
-apply(auto simp: mopup_bef_erase_b.simps mopup_bef_erase_a.simps)
-done
-
-lemma [simp]: "\<lbrakk>mopup_bef_erase_b (s, l, []) lm n ires\<rbrakk> \<Longrightarrow> 
-                   mopup_bef_erase_a (s - Suc 0, Bk # l, []) lm n ires"
-apply(auto simp: mopup_bef_erase_b.simps mopup_bef_erase_a.simps)
-done
-
-lemma [simp]: 
-   "\<lbrakk>n < length lm;
-    mopup_jump_over1 (Suc (2 * n), l, Oc # xs) lm n ires;
-    r = Oc # xs\<rbrakk>
-  \<Longrightarrow> mopup_jump_over1 (Suc (2 * n), Oc # l, xs) lm n ires"
-apply(auto simp: mopup_jump_over1.simps)
-apply(rule_tac x = ln in exI, rule_tac x = "Suc m1" in exI,
-       rule_tac x = "m2 - 1" in exI, simp)
-apply(case_tac "m2", simp, simp)
-apply(rule_tac x = ln in exI, rule_tac x = "Suc m1" in exI, 
-      rule_tac x = "m2 - 1" in exI)
-apply(case_tac m2, simp, simp)
-done
-
-lemma mopup_jump_over1_2_aft_erase_a[simp]:  
- "\<lbrakk>n < length lm; mopup_jump_over1 (Suc (2 * n), l, Bk # xs) lm n ires\<rbrakk>
-  \<Longrightarrow> mopup_aft_erase_a (Suc (Suc (2 * n)), Bk # l, xs) lm n ires"
-apply(simp only: mopup_jump_over1.simps mopup_aft_erase_a.simps)
-apply(erule_tac exE)+
-apply(rule_tac x = ln in exI, rule_tac x = "Suc 0" in exI)
-apply(case_tac m2, simp)
-apply(rule_tac x = rn in exI, rule_tac x = "drop (Suc n) lm" in exI, 
-      simp)
-apply(simp)
-done
-
-lemma [simp]: 
- "\<lbrakk>n < length lm; mopup_jump_over1 (Suc (2 * n), l, []) lm n ires\<rbrakk> \<Longrightarrow> 
-    mopup_aft_erase_a (Suc (Suc (2 * n)), Bk # l, []) lm n ires"
-apply(rule mopup_jump_over1_2_aft_erase_a, simp)
-apply(auto simp: mopup_jump_over1.simps)
-apply(rule_tac x = ln in exI, rule_tac x = "Suc (abc_lm_v lm n)" in exI, 
-      rule_tac x = 0 in exI, simp add: )
-done
-
-
-lemma [simp]: 
- "\<lbrakk>n < length lm; 
-   mopup_aft_erase_a (Suc (Suc (2 * n)), l, Oc # xs) lm n ires\<rbrakk> 
- \<Longrightarrow> mopup_aft_erase_b (Suc (Suc (Suc (2 * n))), l, Bk # xs) lm n ires"
-apply(auto simp: mopup_aft_erase_a.simps mopup_aft_erase_b.simps )
-apply(case_tac ml)
-apply(simp_all add: tape_of_nl_cons split: if_splits)
-apply(case_tac a, simp_all)
-apply(rule_tac x = rn in exI, rule_tac x = "[]" in exI, simp)
-apply(rule_tac x = rn in exI, rule_tac x = "[nat]" in exI, simp)
-apply(case_tac a, simp_all)
-apply(rule_tac x = rn in exI, rule_tac x = "list" in exI, simp)
-apply(rule_tac x = rn in exI)
-apply(rule_tac x = "nat # list" in exI, simp add: tape_of_nl_cons)
-done
-
-lemma [simp]:
-  "mopup_aft_erase_a (Suc (Suc (2 * n)), l, Bk # xs) lm n ires \<Longrightarrow> l \<noteq> []"
-apply(auto simp: mopup_aft_erase_a.simps)
-done
-
-lemma [simp]:
-  "\<lbrakk>n < length lm;
-    mopup_aft_erase_a (Suc (Suc (2 * n)), l, Bk # xs) lm n ires\<rbrakk>
-  \<Longrightarrow> mopup_left_moving (5 + 2 * n, tl l, hd l # Bk # xs) lm n ires"
-apply(simp only: mopup_aft_erase_a.simps mopup_left_moving.simps)
-apply(erule exE)+
-apply(case_tac lnr, simp)
-apply(case_tac ml, simp, simp add: tape_of_nl_cons split: if_splits)
-apply(auto)
-apply(case_tac ml, simp_all add: tape_of_nl_cons split: if_splits)
-apply(rule_tac x = "Suc rn" in exI, simp)
-done
-
-lemma [simp]:
-  "mopup_aft_erase_a (Suc (Suc (2 * n)), l, []) lm n ires \<Longrightarrow> l \<noteq> []"
-apply(simp only: mopup_aft_erase_a.simps)
-apply(erule exE)+
-apply(auto)
-done
-
-lemma [simp]:
-  "\<lbrakk>n < length lm; mopup_aft_erase_a (Suc (Suc (2 * n)), l, []) lm n ires\<rbrakk>
-  \<Longrightarrow> mopup_left_moving (5 + 2 * n, tl l, [hd l]) lm n ires"
-apply(simp only: mopup_aft_erase_a.simps mopup_left_moving.simps)
-apply(erule exE)+
-apply(subgoal_tac "ml = [] \<and> rn = 0", erule conjE, erule conjE, simp)
-apply(case_tac lnr, simp)
-apply(rule_tac x = lnl in exI, simp)
-apply(rule_tac x = 1 in exI, simp)
-apply(case_tac ml, simp, simp)
-done
-
-
-lemma [simp]: "mopup_aft_erase_b (2 * n + 3, l, Oc # xs) lm n ires = False"
-apply(auto simp: mopup_aft_erase_b.simps )
-done
-
-lemma tape_of_ex1[intro]: 
-  "\<exists>rna ml. Oc \<up> a @ Bk \<up> rn = <ml::nat list> @ Bk \<up> rna \<or> Oc \<up> a @ Bk \<up> rn = Bk # <ml> @ Bk \<up> rna"
-apply(case_tac a, simp_all)
-apply(rule_tac x = rn in exI, rule_tac x = "[]" in exI, simp)
-apply(rule_tac x = rn in exI, rule_tac x = "[nat]" in exI, simp)
-done
-
-lemma [intro]: "\<exists>rna ml. Oc \<up> a @ Bk # <list::nat list> @ Bk \<up> rn = 
-  <ml> @ Bk \<up> rna \<or> Oc \<up> a @ Bk # <list> @ Bk \<up> rn = Bk # <ml::nat list> @ Bk \<up> rna"
-apply(case_tac "list = []", simp add: replicate_Suc[THEN sym] del: replicate_Suc)
-apply(rule_tac rn = "Suc rn" in tape_of_ex1)
-apply(case_tac a, simp)
-apply(rule_tac x = rn in exI, rule_tac x = list in exI, simp)
-apply(rule_tac x = rn in exI, rule_tac x = "nat # list" in exI)
-apply(simp add: tape_of_nl_cons)
-done
-
-lemma [simp]: 
- "\<lbrakk>n < length lm; 
-   mopup_aft_erase_c (2 * n + 4, l, Oc # xs) lm n ires\<rbrakk>
-  \<Longrightarrow> mopup_aft_erase_b (Suc (Suc (Suc (2 * n))), l, Bk # xs) lm n ires"
-apply(auto simp: mopup_aft_erase_c.simps mopup_aft_erase_b.simps )
-apply(case_tac ml, simp_all add: tape_of_nl_cons split: if_splits, auto)
-done
-
-lemma mopup_aft_erase_c_aft_erase_a[simp]: 
- "\<lbrakk>n < length lm; mopup_aft_erase_c (2 * n + 4, l, Bk # xs) lm n ires\<rbrakk> 
- \<Longrightarrow> mopup_aft_erase_a (Suc (Suc (2 * n)), Bk # l, xs) lm n ires"
-apply(simp only: mopup_aft_erase_c.simps mopup_aft_erase_a.simps )
-apply(erule_tac exE)+
-apply(erule conjE, erule conjE, erule disjE)
-apply(subgoal_tac "ml = []", simp, case_tac rn, 
-      simp, simp, rule conjI)
-apply(rule_tac x = lnl in exI, rule_tac x = "Suc lnr" in exI, simp)
-apply(rule_tac x = nat in exI, rule_tac x = "[]" in exI, simp)
-apply(case_tac ml, simp, simp add: tape_of_nl_cons split: if_splits)
-apply(rule_tac x = lnl in exI, rule_tac x = "Suc lnr" in exI, simp)
-apply(rule_tac x = rn in exI, rule_tac x = "ml" in exI, simp)
-done
-
-lemma [simp]: 
- "\<lbrakk>n < length lm; mopup_aft_erase_c (2 * n + 4, l, []) lm n ires\<rbrakk> 
- \<Longrightarrow> mopup_aft_erase_a (Suc (Suc (2 * n)), Bk # l, []) lm n ires"
-apply(rule mopup_aft_erase_c_aft_erase_a, simp)
-apply(simp only: mopup_aft_erase_c.simps)
-apply(erule exE)+
-apply(rule_tac x = lnl in exI, rule_tac x = lnr in exI, simp add: )
-apply(rule_tac x = 0 in exI, rule_tac x = "[]" in exI, simp)
-done
-
-lemma mopup_aft_erase_b_2_aft_erase_c[simp]:
-  "\<lbrakk>n < length lm; mopup_aft_erase_b (2 * n + 3, l, Bk # xs) lm n ires\<rbrakk>  
- \<Longrightarrow> mopup_aft_erase_c (4 + 2 * n, Bk # l, xs) lm n ires"
-apply(auto simp: mopup_aft_erase_b.simps mopup_aft_erase_c.simps)
-apply(rule_tac x = "lnl" in exI, rule_tac x = "Suc lnr" in exI, simp)
-apply(rule_tac x = "lnl" in exI, rule_tac x = "Suc lnr" in exI, simp)
-done
-
-lemma [simp]: 
- "\<lbrakk>n < length lm; mopup_aft_erase_b (2 * n + 3, l, []) lm n ires\<rbrakk> 
- \<Longrightarrow> mopup_aft_erase_c (4 + 2 * n, Bk # l, []) lm n ires"
-apply(rule_tac mopup_aft_erase_b_2_aft_erase_c, simp)
-apply(simp add: mopup_aft_erase_b.simps)
-done
-
-lemma [simp]: 
-    "mopup_left_moving (2 * n + 5, l, Oc # xs) lm n ires \<Longrightarrow> l \<noteq> []"
-apply(auto simp: mopup_left_moving.simps)
-done
-
-lemma [simp]:  
- "\<lbrakk>n < length lm; mopup_left_moving (2 * n + 5, l, Oc # xs) lm n ires\<rbrakk>
-  \<Longrightarrow> mopup_jump_over2 (2 * n + 6, tl l, hd l # Oc # xs) lm n ires"
-apply(simp only: mopup_left_moving.simps mopup_jump_over2.simps)
-apply(erule_tac exE)+
-apply(erule conjE, erule disjE, erule conjE)
-apply(case_tac rn, simp, simp add: )
-apply(case_tac "hd l", simp add:  )
-apply(case_tac "abc_lm_v lm n", simp)
-apply(rule_tac x = "lnl" in exI, rule_tac x = rn in exI, 
-      rule_tac x = "Suc 0" in exI, rule_tac x = 0 in exI)
-apply(case_tac lnl, simp, simp, simp add: exp_ind[THEN sym], simp)
-apply(case_tac "abc_lm_v lm n", simp)
-apply(case_tac lnl, simp, simp)
-apply(rule_tac x = lnl in exI, rule_tac x = rn in exI)
-apply(rule_tac x = nat in exI, rule_tac x = "Suc (Suc 0)" in exI, simp)
-done
-
-lemma [simp]: "mopup_left_moving (2 * n + 5, l, xs) lm n ires \<Longrightarrow> l \<noteq> []"
-apply(auto simp: mopup_left_moving.simps)
-done
-
-lemma [simp]:
-  "\<lbrakk>n < length lm; mopup_left_moving (2 * n + 5, l, Bk # xs) lm n ires\<rbrakk> 
- \<Longrightarrow> mopup_left_moving (2 * n + 5, tl l, hd l # Bk # xs) lm n ires"
-apply(simp only: mopup_left_moving.simps)
-apply(erule exE)+
-apply(case_tac lnr, simp)
-apply(rule_tac x = lnl in exI, rule_tac x = nat in exI, simp)
-apply(rule_tac x = "Suc rn" in exI, simp)
-done
-
-lemma [simp]: 
-"\<lbrakk>n < length lm; mopup_left_moving (2 * n + 5, l, []) lm n ires\<rbrakk>
-    \<Longrightarrow> mopup_left_moving (2 * n + 5, tl l, [hd l]) lm n ires"
-apply(simp only: mopup_left_moving.simps)
-apply(erule exE)+
-apply(case_tac lnr, auto)
-done
-
-
-lemma [simp]: 
- "mopup_jump_over2 (2 * n + 6, l, Oc # xs) lm n ires \<Longrightarrow> l \<noteq> []"
-apply(auto simp: mopup_jump_over2.simps )
-done
-
-lemma [simp]: 
-"\<lbrakk>n < length lm; mopup_jump_over2 (2 * n + 6, l, Oc # xs) lm n ires\<rbrakk>
- \<Longrightarrow>  mopup_jump_over2 (2 * n + 6, tl l, hd l # Oc # xs) lm n ires"
-apply(simp only: mopup_jump_over2.simps)
-apply(erule_tac exE)+
-apply(simp add:  , erule conjE, erule_tac conjE)
-apply(case_tac m1, simp)
-apply(rule_tac x = ln in exI, rule_tac x = rn in exI, 
-      rule_tac x = 0 in exI, simp)
-apply(case_tac ln, simp, simp, simp only: exp_ind[THEN sym], simp)
-apply(rule_tac x = ln in exI, rule_tac x = rn in exI, 
-      rule_tac x = nat in exI, rule_tac x = "Suc m2" in exI, simp)
-done
-
-lemma [simp]: 
- "\<lbrakk>n < length lm; mopup_jump_over2 (2 * n + 6, l, Bk # xs) lm n ires\<rbrakk> 
-  \<Longrightarrow> mopup_stop (0, Bk # l, xs) lm n ires"
-apply(auto simp: mopup_jump_over2.simps mopup_stop.simps)
-apply(simp_all add: tape_of_nat_abv exp_ind[THEN sym])
-done
-
-lemma [simp]: "mopup_jump_over2 (2 * n + 6, l, []) lm n ires = False"
-apply(simp only: mopup_jump_over2.simps, simp)
-done
-
-lemma mopup_inv_step:
-  "\<lbrakk>n < length lm; mopup_inv (s, l, r) lm n ires\<rbrakk>
-  \<Longrightarrow> mopup_inv (step (s, l, r) (mopup_a n @ shift mopup_b (2 * n), 0)) lm n ires"
-apply(case_tac r, case_tac [2] a)
-apply(auto split:if_splits simp add:step.simps)
-apply(simp_all add: mopupfetchs)
-done
-
-declare mopup_inv.simps[simp del]
-lemma mopup_inv_steps: 
-"\<lbrakk>n < length lm; mopup_inv (s, l, r) lm n ires\<rbrakk> \<Longrightarrow> 
-     mopup_inv (steps (s, l, r) (mopup_a n @ shift mopup_b (2 * n), 0)  stp) lm n ires"
-apply(induct_tac stp, simp add: steps.simps)
-apply(simp add: step_red)
-apply(case_tac "steps (s, l, r) 
-                (mopup_a n @ shift mopup_b (2 * n), 0) na", simp)
-apply(rule_tac mopup_inv_step, simp, simp)
-done
-
-fun abc_mopup_stage1 :: "config \<Rightarrow> nat \<Rightarrow> nat"
-  where
-  "abc_mopup_stage1 (s, l, r) n = 
-           (if s > 0 \<and> s \<le> 2*n then 6
-            else if s = 2*n + 1 then 4
-            else if s \<ge> 2*n + 2 \<and> s \<le> 2*n + 4 then 3
-            else if s = 2*n + 5 then 2
-            else if s = 2*n + 6 then 1
-            else 0)"
-
-fun abc_mopup_stage2 :: "config \<Rightarrow> nat \<Rightarrow> nat"
-  where
-  "abc_mopup_stage2 (s, l, r) n = 
-           (if s > 0 \<and> s \<le> 2*n then length r
-            else if s = 2*n + 1 then length r
-            else if s = 2*n + 5 then length l
-            else if s = 2*n + 6 then length l
-            else if s \<ge> 2*n + 2 \<and> s \<le> 2*n + 4 then length r
-            else 0)"
-
-fun abc_mopup_stage3 :: "config \<Rightarrow> nat \<Rightarrow> nat"
-  where
-  "abc_mopup_stage3 (s, l, r) n = 
-          (if s > 0 \<and> s \<le> 2*n then 
-              if hd r = Bk then 0
-              else 1
-           else if s = 2*n + 2 then 1 
-           else if s = 2*n + 3 then 0
-           else if s = 2*n + 4 then 2
-           else 0)"
-
-definition
-  "abc_mopup_measure = measures [\<lambda>(c, n). abc_mopup_stage1 c n, 
-                                 \<lambda>(c, n). abc_mopup_stage2 c n, 
-                                 \<lambda>(c, n). abc_mopup_stage3 c n]"
-
-lemma wf_abc_mopup_measure:
-  shows "wf abc_mopup_measure" 
-unfolding abc_mopup_measure_def 
-by auto
-
-lemma abc_mopup_measure_induct [case_names Step]: 
-  "\<lbrakk>\<And>n. \<not> P (f n) \<Longrightarrow> (f (Suc n), (f n)) \<in> abc_mopup_measure\<rbrakk> \<Longrightarrow> \<exists>n. P (f n)"
-using wf_abc_mopup_measure
-by (metis wf_iff_no_infinite_down_chain)
-
-lemma [simp]: "mopup_bef_erase_a (a, aa, []) lm n ires = False"
-apply(auto simp: mopup_bef_erase_a.simps)
-done
-
-lemma [simp]: "mopup_bef_erase_b (a, aa, []) lm n ires = False"
-apply(auto simp: mopup_bef_erase_b.simps) 
-done
-
-lemma [simp]: "mopup_aft_erase_b (2 * n + 3, aa, []) lm n ires = False"
-apply(auto simp: mopup_aft_erase_b.simps)
-done
-
-declare mopup_inv.simps[simp del]
-
-lemma [simp]: 
-  "\<lbrakk>0 < q; q \<le> n\<rbrakk> \<Longrightarrow> 
-     (fetch (mopup_a n @ shift mopup_b (2 * n)) (2*q) Bk) = (R, 2*q - 1)"
-apply(case_tac q, simp, simp)
-apply(auto simp: fetch.simps nth_of.simps nth_append)
-apply(subgoal_tac "mopup_a n ! (4 * nat + 2) = 
-                     mopup_a (Suc nat) ! ((4 * nat) + 2)", 
-      simp add: mopup_a.simps nth_append)
-apply(rule mopup_a_nth, auto)
-done
-
-lemma mopup_halt:
-  assumes 
-  less: "n < length lm"
-  and inv: "mopup_inv (Suc 0, l, r) lm n ires"
-  and f: "f = (\<lambda> stp. (steps (Suc 0, l, r) (mopup_a n @ shift mopup_b (2 * n), 0) stp, n))"
-  and P: "P = (\<lambda> (c, n). is_final c)"
-  shows "\<exists> stp. P (f stp)"
-proof (induct rule: abc_mopup_measure_induct) 
-  case (Step na)
-  have h: "\<not> P (f na)" by fact
-  show "(f (Suc na), f na) \<in> abc_mopup_measure"
-  proof(simp add: f)
-    obtain a b c where g:"steps (Suc 0, l, r) (mopup_a n @ shift mopup_b (2 * n), 0) na = (a, b, c)"
-      apply(case_tac "steps (Suc 0, l, r) (mopup_a n @ shift mopup_b (2 * n), 0) na", auto)
-      done
-    then have "mopup_inv (a, b, c) lm n ires"
-      using inv less mopup_inv_steps[of n lm "Suc 0" l r ires na]
-      apply(simp)
-      done
-    moreover have "a > 0"
-      using h g
-      apply(simp add: f P)
-      done
-    ultimately 
-    have "((step (a, b, c) (mopup_a n @ shift mopup_b (2 * n), 0), n), (a, b, c), n) \<in> abc_mopup_measure"
-      apply(case_tac c, case_tac [2] aa)
-      apply(auto split:if_splits simp add:step.simps mopup_inv.simps)
-      apply(simp_all add: mopupfetchs abc_mopup_measure_def lex_triple_def lex_pair_def )
-      done
-    thus "((step (steps (Suc 0, l, r) (mopup_a n @ shift mopup_b (2 * n), 0) na) 
-      (mopup_a n @ shift mopup_b (2 * n), 0), n),
-      steps (Suc 0, l, r) (mopup_a n @ shift mopup_b (2 * n), 0) na, n)
-      \<in> abc_mopup_measure"
-      using g by simp
-  qed
-qed
-
-lemma mopup_inv_start: 
-  "n < length am \<Longrightarrow> mopup_inv (Suc 0, Bk # Bk # ires, <am> @ Bk \<up> k) am n ires"
-apply(auto simp: mopup_inv.simps mopup_bef_erase_a.simps mopup_jump_over1.simps)
-apply(case_tac [!] am, auto split: if_splits simp: tape_of_nl_cons)
-apply(rule_tac x = "Suc a" in exI, rule_tac x = k in exI, simp)
-apply(case_tac [!] n, simp_all add: abc_lm_v.simps)
-apply(case_tac k, simp, simp_all)
-done
-      
-lemma mopup_correct:
-  assumes less: "n < length (am::nat list)"
-  and rs: "abc_lm_v am n = rs"
-  shows "\<exists> stp i j. (steps (Suc 0, Bk # Bk # ires, <am> @ Bk \<up> k) (mopup_a n @ shift mopup_b (2 * n), 0) stp)
-    = (0, Bk\<up>i @ Bk # Bk # ires, Oc # Oc\<up> rs @ Bk\<up>j)"
-using less
-proof -
-  have a: "mopup_inv (Suc 0, Bk # Bk # ires, <am> @ Bk \<up> k) am n ires"
-    using less
-    apply(simp add: mopup_inv_start)
-    done    
-  then have "\<exists> stp. is_final (steps (Suc 0, Bk # Bk # ires, <am> @ Bk \<up> k) (mopup_a n @ shift mopup_b (2 * n), 0) stp)"
-    using less mopup_halt[of n am  "Bk # Bk # ires" "<am> @ Bk \<up> k" ires
-      "(\<lambda>stp. (steps (Suc 0, Bk # Bk # ires, <am> @ Bk \<up> k) (mopup_a n @ shift mopup_b (2 * n), 0) stp, n))"
-      "(\<lambda>(c, n). is_final c)"]
-    apply(simp)
-    done
-  from this obtain stp where b:
-    "is_final (steps (Suc 0, Bk # Bk # ires, <am> @ Bk \<up> k) (mopup_a n @ shift mopup_b (2 * n), 0) stp)" ..
-  from a b have
-    "mopup_inv (steps (Suc 0, Bk # Bk # ires, <am> @ Bk \<up> k) (mopup_a n @ shift mopup_b (2 * n), 0) stp)
-    am n ires"
-    apply(rule_tac mopup_inv_steps, simp_all add: less)
-    done    
-  from b and this show "?thesis"
-    apply(rule_tac x = stp in exI, simp)
-    apply(case_tac "steps (Suc 0, Bk # Bk # ires, <am> @ Bk \<up> k) 
-      (mopup_a n @ shift mopup_b (2 * n), 0) stp")
-    apply(simp add: mopup_inv.simps mopup_stop.simps rs)
-    using rs
-    apply(simp add: tape_of_nat_abv)
-    done
-qed
-
-(*we can use Hoare_plus here*)
-
-lemma wf_mopup[intro]: "tm_wf (mopup n, 0)"
-apply(induct n, simp add: mopup.simps shift.simps mopup_b_def tm_wf.simps)
-apply(auto simp: mopup.simps shift.simps mopup_b_def tm_wf.simps)
-done
-
 lemma length_tp:
   "\<lbrakk>ly = layout_of ap; tp = tm_of ap\<rbrakk> \<Longrightarrow> 
   start_of ly (length ap) = Suc (length tp div 2)"
@@ -4595,7 +3787,7 @@
   have "\<exists> stp i j. (steps (Suc 0, Bk # Bk # ires, <am> @ Bk \<up> k) (mopup_a n @ shift mopup_b (2 * n), 0) stp)
     = (0, Bk\<up>i @ Bk # Bk # ires, Oc # Oc\<up> rs @ Bk\<up>j)"
     using assms
-    by(auto intro: mopup_correct)
+    by(rule_tac mopup_correct, auto simp: abc_lm_v.simps)
   then obtain stpb i j where 
     "steps (Suc 0, Bk # Bk # ires, <am> @ Bk \<up> k) (mopup_a n @ shift mopup_b (2 * n), 0) stpb
     = (0, Bk\<up>i @ Bk # Bk # ires, Oc # Oc\<up> rs @ Bk\<up>j)" by blast




