diff -r 000000000000 -r aa8656a8dbef abacus.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/abacus.thy	Mon Dec 24 01:26:23 2012 +0000
@@ -0,0 +1,6923 @@
+header {* 
+ {\em abacus} a kind of register machine
+*}
+
+theory abacus
+imports Main turing_basic
+begin
+
+text {*
+  {\em Abacus} instructions:
+*}
+
+datatype abc_inst =
+  -- {* @{text "Inc n"} increments the memory cell (or register) with address @{text "n"} by one.
+     *}
+     Inc nat
+  -- {*
+     @{text "Dec n label"} decrements the memory cell with address @{text "n"} by one. 
+      If cell @{text "n"} is already zero, no decrements happens and the executio jumps to
+      the instruction labeled by @{text "label"}.
+     *}
+   | Dec nat nat
+  -- {*
+  @{text "Goto label"} unconditionally jumps to the instruction labeled by @{text "label"}.
+  *}
+   | Goto nat
+  
+
+text {*
+  Abacus programs are defined as lists of Abacus instructions.
+*}
+type_synonym abc_prog = "abc_inst list"
+
+section {*
+  Sample Abacus programs
+  *}
+
+text {*
+  Abacus for addition and clearance.
+*}
+fun plus_clear :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_prog"  
+  where
+  "plus_clear m n e = [Dec m e, Inc n, Goto 0]"
+
+text {*
+  Abacus for clearing memory untis.
+*}
+fun clear :: "nat \<Rightarrow> nat \<Rightarrow> abc_prog"
+  where
+  "clear n e = [Dec n e, Goto 0]"
+
+fun plus:: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_prog"
+  where
+  "plus m n p e = [Dec m 4, Inc n, Inc p,
+                   Goto 0, Dec p e, Inc m, Goto 4]"
+
+fun mult :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_prog"
+  where
+  "mult m1 m2 n p e = [Dec m1 e]@ plus m1 m2 p 1"
+
+fun expo :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_prog"
+  where
+  "expo n m1 m2 p e = [Inc n, Dec m1 e] @ mult m2 n n p 2"
+
+
+text {*
+  The state of Abacus machine.
+  *}
+type_synonym abc_state = nat
+
+(* text {*
+  The memory of Abacus machine is defined as a function from address to contents.
+*}
+type_synonym abc_mem = "nat \<Rightarrow> nat" *)
+
+text {*
+  The memory of Abacus machine is defined as a list of contents, with 
+  every units addressed by index into the list.
+  *}
+type_synonym abc_lm = "nat list"
+
+text {*
+  Fetching contents out of memory. Units not represented by list elements are considered
+  as having content @{text "0"}.
+*}
+fun abc_lm_v :: "abc_lm \<Rightarrow> nat \<Rightarrow> nat"
+  where 
+    "abc_lm_v lm n = (if (n < length lm) then (lm!n) else 0)"         
+
+(*
+fun abc_l2m :: "abc_lm \<Rightarrow> abc_mem"
+  where 
+    "abc_l2m lm = abc_lm_v lm"
+*)
+
+text {*
+  Set the content of memory unit @{text "n"} to value @{text "v"}.
+  @{text "am"} is the Abacus memory before setting.
+  If address @{text "n"} is outside to scope of @{text "am"}, @{text "am"} 
+  is extended so that @{text "n"} becomes in scope.
+*}
+fun abc_lm_s :: "abc_lm \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_lm"
+  where
+    "abc_lm_s am n v = (if (n < length am) then (am[n:=v]) else 
+                           am@ (replicate (n - length am) 0) @ [v])"
+
+
+text {*
+  The configuration of Abaucs machines consists of its current state and its
+  current memory:
+*}
+type_synonym abc_conf_l = "abc_state \<times> abc_lm"
+
+text {*
+  Fetch instruction out of Abacus program:
+*}
+
+fun abc_fetch :: "nat \<Rightarrow> abc_prog \<Rightarrow> abc_inst option" 
+  where
+  "abc_fetch s p = (if (s < length p) then Some (p ! s)
+                    else None)"
+
+text {*
+  Single step execution of Abacus machine. If no instruction is feteched, 
+  configuration does not change.
+*}
+fun abc_step_l :: "abc_conf_l \<Rightarrow> abc_inst option \<Rightarrow> abc_conf_l"
+  where
+  "abc_step_l (s, lm) a = (case a of 
+               None \<Rightarrow> (s, lm) |
+               Some (Inc n)  \<Rightarrow> (let nv = abc_lm_v lm n in
+                       (s + 1, abc_lm_s lm n (nv + 1))) |
+               Some (Dec n e) \<Rightarrow> (let nv = abc_lm_v lm n in
+                       if (nv = 0) then (e, abc_lm_s lm n 0) 
+                       else (s + 1,  abc_lm_s lm n (nv - 1))) |
+               Some (Goto n) \<Rightarrow> (n, lm) 
+               )"
+
+text {*
+  Multi-step execution of Abacus machine.
+*}
+fun abc_steps_l :: "abc_conf_l \<Rightarrow> abc_prog \<Rightarrow> nat \<Rightarrow> abc_conf_l"
+  where
+  "abc_steps_l (s, lm) p 0 = (s, lm)" |
+  "abc_steps_l (s, lm) p (Suc n) = abc_steps_l (abc_step_l (s, lm) (abc_fetch s p)) p n"
+
+section {*
+  Compiling Abacus machines into Truing machines
+*}
+
+
+subsection {*
+  Compiling functions
+*}
+
+text {*
+  @{text "findnth n"} returns the TM which locates the represention of
+  memory cell @{text "n"} on the tape and changes representation of zero
+  on the way.
+*}
+
+fun findnth :: "nat \<Rightarrow> tprog"
+  where
+  "findnth 0 = []" |
+  "findnth (Suc n) = (findnth n @ [(W1, 2 * n + 1), 
+           (R, 2 * n + 2), (R, 2 * n + 3), (R, 2 * n + 2)])"
+
+text {*
+  @{text "tinc_b"} returns the TM which increments the representation 
+  of the memory cell under rw-head by one and move the representation 
+  of cells afterwards to the right accordingly.
+  *}
+
+definition tinc_b :: "tprog"
+  where
+  "tinc_b \<equiv> [(W1, 1), (R, 2), (W1, 3), (R, 2), (W1, 3), (R, 4), 
+             (L, 7), (W0, 5), (R, 6), (W0, 5), (W1, 3), (R, 6),
+             (L, 8), (L, 7), (R, 9), (L, 7), (R, 10), (W0, 9)]" 
+
+text {*
+  @{text "tshift tm off"} shifts @{text "tm"} by offset @{text "off"}, leaving 
+  instructions concerning state @{text "0"} unchanged, because state @{text "0"} 
+  is the end state, which needs not be changed with shift operation.
+  *}
+
+fun tshift :: "tprog \<Rightarrow> nat \<Rightarrow> tprog"
+  where
+  "tshift tp off = (map (\<lambda> (action, state). 
+       (action, (if state = 0 then 0
+                 else state + off))) tp)"
+
+text {*
+  @{text "tinc ss n"} returns the TM which simulates the execution of 
+  Abacus instruction @{text "Inc n"}, assuming that TM is located at
+  location @{text "ss"} in the final TM complied from the whole
+  Abacus program.
+*}
+
+fun tinc :: "nat \<Rightarrow> nat \<Rightarrow> tprog"
+  where
+  "tinc ss n = tshift (findnth n @ tshift tinc_b (2 * n)) (ss - 1)"
+
+text {*
+  @{text "tinc_b"} returns the TM which decrements the representation 
+  of the memory cell under rw-head by one and move the representation 
+  of cells afterwards to the left accordingly.
+  *}
+
+definition tdec_b :: "tprog"
+  where
+  "tdec_b \<equiv>  [(W1, 1), (R, 2), (L, 14), (R, 3), (L, 4), (R, 3),
+              (R, 5), (W0, 4), (R, 6), (W0, 5), (L, 7), (L, 8),
+              (L, 11), (W0, 7), (W1, 8), (R, 9), (L, 10), (R, 9),
+              (R, 5), (W0, 10), (L, 12), (L, 11), (R, 13), (L, 11),
+              (R, 17), (W0, 13), (L, 15), (L, 14), (R, 16), (L, 14),
+              (R, 0), (W0, 16)]"
+
+text {*
+  @{text "sete tp e"} attaches the termination edges (edges leading to state @{text "0"}) 
+  of TM @{text "tp"} to the intruction labelled by @{text "e"}.
+  *}
+
+fun sete :: "tprog \<Rightarrow> nat \<Rightarrow> tprog"
+  where
+  "sete tp e = map (\<lambda> (action, state). (action, if state = 0 then e else state)) tp"
+
+text {*
+  @{text "tdec ss n label"} returns the TM which simulates the execution of 
+  Abacus instruction @{text "Dec n label"}, assuming that TM is located at
+  location @{text "ss"} in the final TM complied from the whole
+  Abacus program.
+*}
+
+fun tdec :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> tprog"
+  where
+  "tdec ss n e = sete (tshift (findnth n @ tshift tdec_b (2 * n)) 
+                 (ss - 1)) e"
+ 
+text {*
+  @{text "tgoto f(label)"} returns the TM simulating the execution of Abacus instruction
+  @{text "Goto label"}, where @{text "f(label)"} is the corresponding location of
+  @{text "label"} in the final TM compiled from the overall Abacus program.
+*}
+
+fun tgoto :: "nat \<Rightarrow> tprog"
+  where
+  "tgoto n = [(Nop, n), (Nop, n)]"
+
+text {*
+  The layout of the final TM compiled from an Abacus program is represented
+  as a list of natural numbers, where the list element at index @{text "n"} represents the 
+  starting state of the TM simulating the execution of @{text "n"}-th instruction
+  in the Abacus program.
+*}
+
+type_synonym layout = "nat list"
+
+text {*
+  @{text "length_of i"} is the length of the 
+  TM simulating the Abacus instruction @{text "i"}.
+*}
+fun length_of :: "abc_inst \<Rightarrow> nat"
+  where
+  "length_of i = (case i of 
+                    Inc n   \<Rightarrow> 2 * n + 9 |
+                    Dec n e \<Rightarrow> 2 * n + 16 |
+                    Goto n  \<Rightarrow> 1)"
+
+text {*
+  @{text "layout_of ap"} returns the layout of Abacus program @{text "ap"}.
+*}
+fun layout_of :: "abc_prog \<Rightarrow> layout"
+  where "layout_of ap = map length_of ap"
+
+
+text {*
+  @{text "start_of layout n"} looks out the starting state of @{text "n"}-th
+  TM in the finall TM.
+*}
+
+fun start_of :: "nat list \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "start_of ly 0 = Suc 0" |
+  "start_of ly (Suc as) = 
+        (if as < length ly then start_of ly as + (ly ! as)
+         else start_of ly as)"
+
+text {*
+  @{text "ci lo ss i"} complies Abacus instruction @{text "i"}
+  assuming the TM of @{text "i"} starts from state @{text "ss"} 
+  within the overal layout @{text "lo"}.
+*}
+
+fun ci :: "layout \<Rightarrow> nat \<Rightarrow> abc_inst \<Rightarrow> tprog"
+  where
+  "ci ly ss i = (case i of 
+                    Inc n   \<Rightarrow> tinc ss n |
+                    Dec n e \<Rightarrow> tdec ss n (start_of ly e) |
+                    Goto n  \<Rightarrow> tgoto (start_of ly n))"
+
+text {*
+  @{text "tpairs_of ap"} transfroms Abacus program @{text "ap"} pairing
+  every instruction with its starting state.
+*}
+fun tpairs_of :: "abc_prog \<Rightarrow> (nat \<times> abc_inst) list"
+  where "tpairs_of ap = (zip (map (start_of (layout_of ap)) 
+                         [0..<(length ap)]) ap)"
+
+
+text {*
+  @{text "tms_of ap"} returns the list of TMs, where every one of them simulates
+  the corresponding Abacus intruction in @{text "ap"}.
+*}
+
+fun tms_of :: "abc_prog \<Rightarrow> tprog list"
+  where "tms_of ap = map (\<lambda> (n, tm). ci (layout_of ap) n tm) 
+                         (tpairs_of ap)"
+
+text {*
+  @{text "tm_of ap"} returns the final TM machine compiled from Abacus program @{text "ap"}.
+*}
+fun tm_of :: "abc_prog \<Rightarrow> tprog"
+  where "tm_of ap = concat (tms_of ap)"
+
+text {*
+  The following two functions specify the well-formedness of complied TM.
+*}
+fun t_ncorrect :: "tprog \<Rightarrow> bool"
+  where
+  "t_ncorrect tp = (length tp mod 2 = 0)"
+
+fun abc2t_correct :: "abc_prog \<Rightarrow> bool"
+  where 
+  "abc2t_correct ap = list_all (\<lambda> (n, tm). 
+             t_ncorrect (ci (layout_of ap) n tm)) (tpairs_of ap)"
+
+lemma findnth_length: "length (findnth n) div 2 = 2 * n"
+apply(induct n, simp, simp)
+done
+
+lemma ci_length : "length (ci ns n ai) div 2 = length_of ai"
+apply(auto simp: ci.simps tinc_b_def tdec_b_def findnth_length
+                 split: abc_inst.splits)
+done
+
+subsection {*
+  Representation of Abacus memory by TM tape
+*}
+
+consts tape_of :: "'a \<Rightarrow> block list" ("<_>" 100)
+
+text {*
+  @{text "tape_of_nat_list am"} returns the TM tape representing
+  Abacus memory @{text "am"}.
+  *}
+
+fun tape_of_nat_list :: "nat list \<Rightarrow> block list"
+  where 
+  "tape_of_nat_list [] = []" |
+  "tape_of_nat_list [n] = Oc\<^bsup>n+1\<^esup>" |
+  "tape_of_nat_list (n#ns) = (Oc\<^bsup>n+1\<^esup>) @ [Bk] @ (tape_of_nat_list ns)"
+
+defs (overloaded)
+  tape_of_nl_abv: "<am> \<equiv> tape_of_nat_list am"
+  tape_of_nat_abv : "<(n::nat)> \<equiv> Oc\<^bsup>n+1\<^esup>"
+
+text {*
+  @{text "crsp_l acf tcf"} meams the abacus configuration @{text "acf"}
+  is corretly represented by the TM configuration @{text "tcf"}.
+*}
+
+fun crsp_l :: "layout \<Rightarrow> abc_conf_l \<Rightarrow> t_conf \<Rightarrow> block list \<Rightarrow> bool"
+  where 
+  "crsp_l ly (as, lm) (ts, (l, r)) inres = 
+           (ts = start_of ly as \<and> (\<exists> rn. r = <lm> @ Bk\<^bsup>rn\<^esup>)
+            \<and> l = Bk # Bk # inres)"
+
+declare crsp_l.simps[simp del]
+
+subsection {*
+  A more general definition of TM execution. 
+*}
+
+(*
+fun nnth_of :: "(taction \<times> nat) list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> (taction \<times> nat)"
+  where
+  "nnth_of p s b = (if 2*s < length p 
+                    then (p ! (2*s + b)) else (Nop, 0))"
+
+thm nth_of.simps
+
+fun nfetch :: "tprog \<Rightarrow> nat \<Rightarrow> block \<Rightarrow> taction \<times> nat"
+  where
+  "nfetch p 0 b = (Nop, 0)" |
+  "nfetch p (Suc s) b = 
+             (case b of 
+                Bk \<Rightarrow> nnth_of p s 0 |
+                Oc \<Rightarrow> nnth_of p s 1)"
+*)
+
+text {*
+  @{text "t_step tcf (tp, ss)"} returns the result of one step exection of TM @{text "tp"}
+  assuming @{text "tp"} starts from instial state @{text "ss"}.
+*}
+
+fun t_step :: "t_conf \<Rightarrow> (tprog \<times> nat) \<Rightarrow> t_conf"
+  where 
+  "t_step c (p, off) = 
+           (let (state, leftn, rightn) = c in
+            let (action, next_state) = fetch p (state-off)
+                             (case rightn of 
+                                [] \<Rightarrow> Bk | 
+                                Bk # xs \<Rightarrow> Bk |
+                                Oc # xs \<Rightarrow> Oc
+                             ) 
+             in 
+            (next_state, new_tape action (leftn, rightn)))"
+
+
+text {*
+  @{text "t_steps tcf (tp, ss) n"} returns the result of @{text "n"}-step exection 
+  of TM @{text "tp"} assuming @{text "tp"} starts from instial state @{text "ss"}.
+*}
+
+fun t_steps :: "t_conf \<Rightarrow> (tprog \<times> nat) \<Rightarrow> nat \<Rightarrow> t_conf"
+  where
+  "t_steps c (p, off) 0 = c" |
+  "t_steps c (p, off) (Suc n) = t_steps 
+                     (t_step c (p, off)) (p, off) n" 
+
+lemma stepn: "t_steps c (p, off) (Suc n) = 
+              t_step (t_steps c (p, off) n) (p, off)"
+apply(induct n arbitrary: c, simp add: t_steps.simps)
+apply(simp add: t_steps.simps)
+done
+
+text {*
+  The type of invarints expressing correspondence between 
+  Abacus configuration and TM configuration.
+*}
+
+type_synonym inc_inv_t = "abc_conf_l \<Rightarrow> t_conf \<Rightarrow> block list \<Rightarrow> bool"
+
+declare tms_of.simps[simp del] tm_of.simps[simp del]
+        layout_of.simps[simp del] abc_fetch.simps [simp del]  
+        t_step.simps[simp del] t_steps.simps[simp del] 
+        tpairs_of.simps[simp del] start_of.simps[simp del]
+        fetch.simps [simp del] t_ncorrect.simps[simp del]
+        new_tape.simps [simp del] ci.simps [simp del] length_of.simps[simp del] 
+        layout_of.simps[simp del] crsp_l.simps[simp del]
+        abc2t_correct.simps[simp del]
+
+lemma tct_div2: "t_ncorrect tp \<Longrightarrow> (length tp) mod 2 = 0"
+apply(simp add: t_ncorrect.simps)
+done
+
+lemma t_shift_fetch: 
+    "\<lbrakk>t_ncorrect tp1; t_ncorrect tp; 
+      length tp1 div 2 < a \<and> a \<le> length tp1 div 2 + length tp div 2\<rbrakk>
+    \<Longrightarrow> fetch tp (a - length tp1 div 2) b = 
+         fetch (tp1 @ tp @ tp2) a b"
+apply(subgoal_tac "\<exists> x. a = length tp1 div 2 + x", erule exE, simp)
+apply(case_tac x, simp)
+apply(subgoal_tac "length tp1 div 2 + Suc nat = 
+             Suc (length tp1 div 2 + nat)")
+apply(simp only: fetch.simps nth_of.simps, auto)
+apply(case_tac b, simp)
+apply(subgoal_tac "2 * (length tp1 div 2) = length tp1", simp)
+apply(subgoal_tac "2 * nat < length tp", simp add: nth_append, simp)
+apply(simp add: t_ncorrect.simps, auto)
+apply(subgoal_tac "2 * (length tp1 div 2) = length tp1", simp)
+apply(subgoal_tac "2 * nat < length tp", simp add: nth_append, auto)
+apply(simp add: t_ncorrect.simps, auto)
+apply(rule_tac x = "a - length tp1 div 2" in exI, simp)
+done
+
+lemma t_shift_in_step:
+      "\<lbrakk>t_step (a, aa, ba) (tp, length tp1 div 2) = (s, l, r);
+        t_ncorrect tp1; t_ncorrect tp;
+        length tp1 div 2 < a \<and> a \<le> length tp1 div 2 + length tp div 2\<rbrakk>
+       \<Longrightarrow> t_step (a, aa, ba) (tp1 @ tp @ tp2, 0) = (s, l, r)"
+apply(simp add: t_step.simps)
+apply(subgoal_tac "fetch tp (a - length tp1 div 2) (case ba of [] \<Rightarrow> 
+                   Bk | x # xs \<Rightarrow> x)
+             = fetch (tp1 @ tp @ tp2) a (case ba of [] \<Rightarrow> Bk | x # xs
+                   \<Rightarrow> x)")
+apply(case_tac "fetch tp (a - length tp1 div 2) (case ba of [] \<Rightarrow> Bk
+                | x # xs \<Rightarrow> x)")
+apply(auto intro: t_shift_fetch)
+apply(case_tac ba, simp, simp)
+apply(case_tac aaa, simp, simp)
+done
+
+declare add_Suc_right[simp del]
+lemma t_step_add: "t_steps c (p, off) (m + n) = 
+          t_steps (t_steps c (p, off) m) (p, off) n"
+apply(induct m arbitrary: n,  simp add: t_steps.simps, simp)
+apply(subgoal_tac "t_steps c (p, off) (Suc (m + n)) = 
+                         t_steps c (p, off) (m + Suc n)", simp)
+apply(subgoal_tac "t_steps (t_steps c (p, off) m) (p, off) (Suc n) =
+                t_steps (t_step (t_steps c (p, off) m) (p, off)) 
+                         (p, off) n")
+apply(simp, simp add: stepn)
+apply(simp only: t_steps.simps)
+apply(simp only: add_Suc_right)
+done
+declare add_Suc_right[simp]
+
+lemma s_out_fetch: "\<lbrakk>t_ncorrect tp; 
+        \<not> (length tp1 div 2 < a \<and> a \<le> length tp1 div 2 + 
+         length tp div 2)\<rbrakk>
+      \<Longrightarrow> fetch tp (a - length tp1 div 2) b = (Nop, 0)"
+apply(auto)
+apply(simp add: fetch.simps)
+apply(subgoal_tac "\<exists> x. a - length tp1 div 2 = length tp div 2 + x")
+apply(erule exE, simp)
+apply(case_tac x, simp)
+apply(auto simp add: fetch.simps)
+apply(subgoal_tac "2 * (length tp div 2) =  length tp")
+apply(auto simp: t_ncorrect.simps split: block.splits)
+apply(rule_tac x = "a - length tp1 div 2 - length tp div 2" in exI
+     , simp)
+done 
+
+lemma conf_keep_step: 
+      "\<lbrakk>t_ncorrect tp; 
+        \<not> (length tp1 div 2 < a \<and> a \<le> length tp1 div 2 + 
+       length tp div 2)\<rbrakk>
+      \<Longrightarrow> t_step (a, aa, ba) (tp, length tp1 div 2) = (0, aa, ba)"
+apply(simp add: t_step.simps)
+apply(subgoal_tac "fetch tp (a - length tp1 div 2) (case ba of [] \<Rightarrow> 
+  Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc) = (Nop, 0)")
+apply(simp add: new_tape.simps)
+apply(rule s_out_fetch, simp, simp)
+done
+
+lemma conf_keep: 
+      "\<lbrakk>t_ncorrect tp; 
+        \<not> (length tp1 div 2 < a \<and>
+        a \<le> length tp1 div 2 + length tp div 2); n > 0\<rbrakk>
+      \<Longrightarrow> t_steps (a, aa, ba) (tp, length tp1 div 2) n = (0, aa, ba)"
+apply(induct n, simp)
+apply(case_tac n, simp add: t_steps.simps)
+apply(rule_tac conf_keep_step, simp+)
+apply(subgoal_tac " t_steps (a, aa, ba) 
+               (tp, length tp1 div 2) (Suc (Suc nat))
+         = t_step (t_steps (a, aa, ba) 
+            (tp, length tp1 div 2) (Suc nat)) (tp, length tp1 div 2)")
+apply(simp)
+apply(rule_tac conf_keep_step, simp, simp)
+apply(rule stepn)
+done
+
+lemma state_bef_inside: 
+    "\<lbrakk>t_ncorrect tp1; t_ncorrect tp; 
+      t_steps (s0, l0, r0) (tp, length tp1 div 2) stp = (s, l, r);
+      length tp1 div 2 < s0 \<and> 
+         s0 \<le> length tp1 div 2 + length tp div 2;
+      length tp1 div 2 < s \<and> s \<le> length tp1 div 2 + length tp div 2; 
+      n < stp; t_steps (s0, l0, r0) (tp, length tp1 div 2) n = 
+      (a, aa, ba)\<rbrakk>
+      \<Longrightarrow>  length tp1 div 2 < a \<and> 
+         a \<le> length tp1 div 2 + length tp div 2"
+apply(subgoal_tac "\<exists> x. stp = n + x", erule exE)
+apply(simp only: t_step_add)
+apply(rule classical)
+apply(subgoal_tac "t_steps (a, aa, ba) 
+          (tp, length tp1 div 2) x = (0, aa, ba)")
+apply(simp)
+apply(rule conf_keep, simp, simp, simp)
+apply(rule_tac x = "stp - n" in exI, simp)
+done
+
+lemma turing_shift_inside: 
+       "\<lbrakk>t_steps (s0, l0, r0) (tp, length tp1 div 2) stp = (s, l, r);
+         length tp1 div 2 < s0 \<and> 
+         s0 \<le> length tp1 div 2 + length tp div 2; 
+         t_ncorrect tp1; t_ncorrect tp;
+         length tp1 div 2 < s \<and> 
+         s \<le> length tp1 div 2 + length tp div 2\<rbrakk>
+       \<Longrightarrow> t_steps (s0, l0, r0) (tp1 @ tp @ tp2, 0) stp = (s, l, r)"
+apply(induct stp arbitrary: s l r)
+apply(simp add: t_steps.simps)
+apply(subgoal_tac " t_steps (s0, l0, r0) 
+        (tp, length tp1 div 2) (Suc stp)
+                  = t_step (t_steps (s0, l0, r0) 
+        (tp, length tp1 div 2) stp) (tp, length tp1 div 2)")
+apply(case_tac "t_steps (s0, l0, r0) (tp, length tp1 div 2) stp")
+apply(subgoal_tac "length tp1 div 2 < a \<and> 
+            a \<le> length tp1 div 2 + length tp div 2")
+apply(subgoal_tac "t_steps (s0, l0, r0) 
+           (tp1 @ tp @ tp2, 0) stp = (a, b, c)")
+apply(simp only: stepn, simp)
+apply(rule_tac t_shift_in_step, simp+)
+defer
+apply(rule stepn)
+apply(rule_tac n = stp and stp = "Suc stp" and a = a 
+               and aa = b and ba = c in state_bef_inside, simp+)
+done
+
+lemma take_Suc_last[elim]: "Suc as \<le> length xs \<Longrightarrow> 
+            take (Suc as) xs = take as xs @ [xs ! as]"
+apply(induct xs arbitrary: as, simp, simp)
+apply(case_tac as, simp, simp)
+done
+
+lemma concat_suc: "Suc as \<le> length xs \<Longrightarrow> 
+       concat (take (Suc as) xs) = concat (take as xs) @ xs! as"
+apply(subgoal_tac "take (Suc as) xs = take as xs @ [xs ! as]", simp)
+by auto
+
+lemma concat_take_suc_iff: "Suc n \<le> length tps \<Longrightarrow> 
+       concat (take n tps) @ (tps ! n) = concat (take (Suc n) tps)"
+apply(drule_tac concat_suc, simp)
+done
+
+lemma concat_drop_suc_iff: 
+   "Suc n < length tps \<Longrightarrow> concat (drop (Suc n) tps) = 
+           tps ! Suc n @ concat (drop (Suc (Suc n)) tps)"
+apply(induct tps arbitrary: n, simp, simp)
+apply(case_tac tps, simp, simp)
+apply(case_tac n, simp, simp)
+done
+
+declare append_assoc[simp del]
+
+lemma  tm_append: "\<lbrakk>n < length tps; tp = tps ! n\<rbrakk> \<Longrightarrow> 
+           \<exists> tp1 tp2. concat tps = tp1 @ tp @ tp2 \<and> tp1 = 
+              concat (take n tps) \<and> tp2 = concat (drop (Suc n) tps)"
+apply(rule_tac x = "concat (take n tps)" in exI)
+apply(rule_tac x = "concat (drop (Suc n) tps)" in exI)
+apply(auto)
+apply(induct n, simp)
+apply(case_tac tps, simp, simp, simp)
+apply(subgoal_tac "concat (take n tps) @ (tps ! n) = 
+               concat (take (Suc n) tps)")
+apply(simp only: append_assoc[THEN sym], simp only: append_assoc)
+apply(subgoal_tac " concat (drop (Suc n) tps) = tps ! Suc n @ 
+                  concat (drop (Suc (Suc n)) tps)", simp)
+apply(rule_tac concat_drop_suc_iff, simp)
+apply(rule_tac concat_take_suc_iff, simp)
+done
+
+declare append_assoc[simp]
+
+lemma map_of:  "n < length xs \<Longrightarrow> (map f xs) ! n = f (xs ! n)"
+by(auto)
+
+lemma [simp]: "length (tms_of aprog) = length aprog"
+apply(auto simp: tms_of.simps tpairs_of.simps)
+done
+
+lemma ci_nth: "\<lbrakk>ly = layout_of aprog; as < length aprog; 
+                abc_fetch as aprog = Some ins\<rbrakk>
+    \<Longrightarrow> ci ly (start_of ly as) ins = tms_of aprog ! as"
+apply(simp add: tms_of.simps tpairs_of.simps 
+      abc_fetch.simps  map_of del: map_append)
+done
+
+lemma t_split:"\<lbrakk>
+        ly = layout_of aprog;
+        as < length aprog; abc_fetch as aprog = Some ins\<rbrakk>
+      \<Longrightarrow> \<exists> tp1 tp2. concat (tms_of aprog) = 
+            tp1 @ (ci ly (start_of ly as) ins) @ tp2
+            \<and> tp1 = concat (take as (tms_of aprog)) \<and> 
+              tp2 = concat (drop (Suc as) (tms_of aprog))"
+apply(insert tm_append[of "as" "tms_of aprog" 
+                             "ci ly (start_of ly as) ins"], simp)
+apply(subgoal_tac "ci ly (start_of ly as) ins = (tms_of aprog) ! as")
+apply(subgoal_tac "length (tms_of aprog) = length aprog", simp, simp)
+apply(rule_tac ci_nth, auto)
+done
+
+lemma math_sub: "\<lbrakk>x >= Suc 0; x - 1 = z\<rbrakk> \<Longrightarrow> x + y - Suc 0 = z + y"
+by auto
+
+lemma start_more_one: "as \<noteq> 0 \<Longrightarrow> start_of ly as >= Suc 0"
+apply(induct as, simp add: start_of.simps)
+apply(case_tac as, auto simp: start_of.simps)
+done
+
+lemma tm_ct: "\<lbrakk>abc2t_correct aprog; tp \<in> set (tms_of aprog)\<rbrakk> \<Longrightarrow> 
+                           t_ncorrect tp"
+apply(simp add: abc2t_correct.simps tms_of.simps)
+apply(auto)
+apply(simp add:list_all_iff, auto)
+done
+
+lemma div_apart: "\<lbrakk>x mod (2::nat) = 0; y mod 2 = 0\<rbrakk> 
+          \<Longrightarrow> (x + y) div 2 = x div 2 + y div 2"
+apply(drule mod_eqD)+
+apply(auto)
+done
+
+lemma div_apart_iff: "\<lbrakk>x mod (2::nat) = 0; y mod 2 = 0\<rbrakk> \<Longrightarrow> 
+           (x + y) mod 2 = 0"
+apply(auto)
+done
+
+lemma tms_ct: "\<lbrakk>abc2t_correct aprog; n < length aprog\<rbrakk> \<Longrightarrow> 
+         t_ncorrect (concat (take n (tms_of aprog)))"
+apply(induct n, simp add: t_ncorrect.simps, simp)
+apply(subgoal_tac "concat (take (Suc n) (tms_of aprog)) = 
+        concat (take n (tms_of aprog)) @ (tms_of aprog ! n)", simp)
+apply(simp add: t_ncorrect.simps)
+apply(rule_tac div_apart_iff, simp)
+apply(subgoal_tac "t_ncorrect (tms_of aprog ! n)", 
+            simp add: t_ncorrect.simps)
+apply(rule_tac tm_ct, simp)
+apply(rule_tac nth_mem, simp add: tms_of.simps tpairs_of.simps)
+apply(rule_tac concat_suc, simp add: tms_of.simps tpairs_of.simps)
+done
+
+lemma tcorrect_div2: "\<lbrakk>abc2t_correct aprog; Suc as < length aprog\<rbrakk>
+  \<Longrightarrow> (length (concat (take as (tms_of aprog))) + length (tms_of aprog
+ ! as)) div 2 = length (concat (take as (tms_of aprog))) div 2 + 
+                 length (tms_of aprog ! as) div 2"
+apply(subgoal_tac "t_ncorrect (tms_of aprog ! as)")
+apply(subgoal_tac "t_ncorrect (concat (take as (tms_of aprog)))")
+apply(rule_tac div_apart)
+apply(rule tct_div2, simp)+
+apply(erule_tac tms_ct, simp)
+apply(rule_tac tm_ct, simp)
+apply(rule_tac nth_mem)
+apply(simp add: tms_of.simps tpairs_of.simps)
+done
+
+lemma [simp]: "length (layout_of aprog) = length aprog"
+apply(auto simp: layout_of.simps)
+done
+
+lemma start_of_ind: "\<lbrakk>as < length aprog; ly = layout_of aprog\<rbrakk> \<Longrightarrow> 
+       start_of ly (Suc as) = start_of ly as + 
+                          length ((tms_of aprog) ! as) div 2"
+apply(simp only: start_of.simps, simp)
+apply(auto simp: start_of.simps tms_of.simps layout_of.simps 
+                 tpairs_of.simps)
+apply(simp add: ci_length)
+done
+
+lemma concat_take_suc: "Suc n \<le> length xs \<Longrightarrow>
+  concat (take (Suc n) xs) = concat (take n xs) @ (xs ! n)"
+apply(subgoal_tac "take (Suc n) xs =
+                   take n xs @ [xs ! n]")
+apply(auto)
+done
+
+lemma ci_length_not0: "Suc 0 <= length (ci ly as i) div 2"
+apply(subgoal_tac "length (ci ly as i) div 2 = length_of i")
+apply(simp add: length_of.simps split: abc_inst.splits)
+apply(rule ci_length)
+done
+ 
+lemma findnth_length2: "length (findnth n) = 4 * n"
+apply(induct n, simp)
+apply(simp)
+done
+
+lemma ci_length2: "length (ci ly as i) = 2 * (length_of i)"
+apply(simp add: ci.simps length_of.simps tinc_b_def tdec_b_def
+              split: abc_inst.splits, auto)
+apply(simp add: findnth_length2)+
+done
+
+lemma tm_mod2: "as < length aprog \<Longrightarrow> 
+             length (tms_of aprog ! as) mod 2 = 0"
+apply(simp add: tms_of.simps)
+apply(subgoal_tac "map (\<lambda>(x, y). ci (layout_of aprog) x y) 
+              (tpairs_of aprog) ! as
+                = (\<lambda>(x, y). ci (layout_of aprog) x y) 
+              ((tpairs_of aprog) ! as)", simp)
+apply(case_tac "(tpairs_of aprog ! as)", simp)
+apply(subgoal_tac "length (ci (layout_of aprog) a b) =
+                 2 * (length_of b)", simp)
+apply(rule ci_length2)
+apply(rule map_of, simp add: tms_of.simps tpairs_of.simps)
+done
+
+lemma tms_mod2: "as \<le> length aprog \<Longrightarrow> 
+        length (concat (take as (tms_of aprog))) mod 2 = 0"
+apply(induct as, simp, simp)
+apply(subgoal_tac "concat (take (Suc as) (tms_of aprog))
+                  = concat (take as (tms_of aprog)) @ 
+                       (tms_of aprog ! as)", auto)
+apply(rule div_apart_iff, simp, rule tm_mod2, simp)
+apply(rule concat_take_suc, simp add: tms_of.simps tpairs_of.simps)
+done
+
+lemma [simp]: "\<lbrakk>as < length aprog; (abc_fetch as aprog) = Some ins\<rbrakk>
+       \<Longrightarrow> ci (layout_of aprog) 
+          (start_of (layout_of aprog) as) (ins) \<in> set (tms_of aprog)"
+apply(insert ci_nth[of "layout_of aprog" aprog as], simp)
+done
+
+lemma startof_not0: "start_of ly as > 0"
+apply(induct as, simp add: start_of.simps)
+apply(case_tac as, auto simp: start_of.simps)
+done
+
+declare abc_step_l.simps[simp del]
+lemma pre_lheq: "\<lbrakk>tp = concat (take as (tms_of aprog));
+   abc2t_correct aprog; as \<le> length aprog\<rbrakk> \<Longrightarrow> 
+         start_of (layout_of aprog) as - Suc 0 = length tp div 2"
+apply(induct as arbitrary: tp, simp add: start_of.simps, simp)
+proof - 
+  fix as tp
+  assume h1: "\<And>tp. tp = concat (take as (tms_of aprog)) \<Longrightarrow> 
+     start_of (layout_of aprog) as - Suc 0 = 
+            length (concat (take as (tms_of aprog))) div 2"
+  and h2: " abc2t_correct aprog" "Suc as \<le> length aprog" 
+  from h2 show "start_of (layout_of aprog) (Suc as) - Suc 0 = 
+          length (concat (take (Suc as) (tms_of aprog))) div 2"
+    apply(insert h1[of "concat (take as (tms_of aprog))"], simp)
+    apply(insert start_of_ind[of as aprog "layout_of aprog"], simp)
+    apply(subgoal_tac "(take (Suc as) (tms_of aprog)) = 
+            take as (tms_of aprog) @ [(tms_of aprog) ! as]", simp)
+    apply(subgoal_tac "(length (concat (take as (tms_of aprog))) + 
+                       length (tms_of aprog ! as)) div 2
+            = length (concat (take as (tms_of aprog))) div 2 + 
+              length (tms_of aprog ! as) div 2", simp)
+    apply(subgoal_tac "start_of (layout_of aprog) as = 
+       length (concat (take as (tms_of aprog))) div 2 + Suc 0", simp)
+    apply(subgoal_tac "start_of (layout_of aprog) as > 0", simp, 
+           rule_tac startof_not0)
+    apply(insert tm_mod2[of as aprog], simp)
+    apply(insert tms_mod2[of as aprog], simp, arith)
+    apply(rule take_Suc_last, simp)
+    done
+qed
+
+lemma crsp2stateq: 
+  "\<lbrakk>as < length aprog; abc2t_correct aprog;
+       crsp_l (layout_of aprog) (as, am) (a, aa, ba) inres\<rbrakk> \<Longrightarrow> 
+        a = length (concat (take as (tms_of aprog))) div 2 + 1"
+apply(simp add: crsp_l.simps)
+apply(insert pre_lheq[of "(concat (take as (tms_of aprog)))" as aprog]
+, simp)   
+apply(subgoal_tac "start_of (layout_of aprog) as > 0", 
+      auto intro: startof_not0)
+done
+
+lemma turing_shift_outside: 
+     "\<lbrakk>t_steps (s0, l0, r0) (tp, length tp1 div 2) stp = (s, l, r); 
+       s \<noteq> 0; stp > 0;
+       length tp1 div 2 < s0 \<and> 
+       s0 \<le> length tp1 div 2 + length tp div 2; 
+       t_ncorrect tp1; t_ncorrect tp;
+       \<not> (length tp1 div 2 < s \<and> 
+      s \<le> length tp1 div 2 + length tp div 2)\<rbrakk>
+    \<Longrightarrow> \<exists>stp' > 0. t_steps (s0, l0, r0) (tp1 @ tp @ tp2, 0) stp' 
+                = (s, l, r)"
+apply(rule_tac x = stp in exI)
+apply(case_tac stp, simp add: t_steps.simps)
+apply(simp only: stepn)
+apply(case_tac "t_steps (s0, l0, r0) (tp, length tp1 div 2) nat")
+apply(subgoal_tac "length tp1 div 2 < a \<and> 
+                   a \<le> length tp1 div 2 + length tp div 2")
+apply(subgoal_tac "t_steps (s0, l0, r0) (tp1 @ tp @ tp2, 0) nat 
+                   = (a, b, c)", simp)
+apply(rule_tac t_shift_in_step, simp+)
+apply(rule_tac turing_shift_inside, simp+)
+apply(rule classical)
+apply(subgoal_tac "t_step (a,b,c) 
+            (tp, length tp1 div 2) = (0, b, c)", simp)
+apply(rule_tac conf_keep_step, simp+)
+done
+
+lemma turing_shift: 
+  "\<lbrakk>t_steps (s0, (l0, r0)) (tp, (length tp1 div 2)) stp
+   = (s, (l, r)); s \<noteq> 0; stp > 0;
+  (length tp1 div 2 < s0 \<and> s0 <= length tp1 div 2 + length tp div 2);
+  t_ncorrect tp1; t_ncorrect tp\<rbrakk> \<Longrightarrow> 
+         \<exists> stp' > 0. t_steps (s0, (l0, r0)) (tp1 @ tp @ tp2, 0) stp' =
+                    (s, (l, r))"
+apply(case_tac "s > length tp1 div 2 \<and> 
+              s <= length tp1 div 2 + length tp div 2")
+apply(subgoal_tac " t_steps (s0, l0, r0) (tp1 @ tp @ tp2, 0) stp = 
+                   (s, l, r)")
+apply(rule_tac x = stp in exI, simp)
+apply(rule_tac turing_shift_inside, simp+)
+apply(rule_tac turing_shift_outside, simp+)
+done
+
+lemma inc_startof_not0:  "start_of ly as \<ge> Suc 0"
+apply(induct as, simp add: start_of.simps)
+apply(simp add: start_of.simps)
+done
+
+lemma s_crsp:
+  "\<lbrakk>as < length aprog; abc_fetch as aprog = Some ins;
+  abc2t_correct aprog;
+  crsp_l (layout_of aprog) (as, am) (a, aa, ba) inres\<rbrakk> \<Longrightarrow>  
+  length (concat (take as (tms_of aprog))) div 2 < a 
+      \<and> a \<le> length (concat (take as (tms_of aprog))) div 2 + 
+         length (ci (layout_of aprog) (start_of (layout_of aprog) as)
+         ins) div 2"
+apply(subgoal_tac "a = length (concat (take as (tms_of aprog))) div 
+                   2 + 1", simp)
+apply(rule_tac ci_length_not0)
+apply(rule crsp2stateq, simp+)
+done
+
+lemma tms_out_ex:
+  "\<lbrakk>ly = layout_of aprog; tprog = tm_of aprog;
+  abc2t_correct aprog;
+  crsp_l ly (as, am) tc inres; as < length aprog;
+  abc_fetch as aprog = Some ins;
+  t_steps tc (ci ly (start_of ly as) ins, 
+  (start_of ly as) - 1) n = (s, l, r);
+  n > 0; 
+  abc_step_l (as, am) (abc_fetch as aprog) = (as', am');
+  s = start_of ly as'
+  \<rbrakk>
+  \<Longrightarrow> \<exists> stp > 0. (t_steps tc (tprog, 0) stp = (s, (l, r)))"
+apply(simp only: tm_of.simps)
+apply(subgoal_tac "\<exists> tp1 tp2. concat (tms_of aprog) = 
+      tp1 @ (ci ly (start_of ly as) ins) @ tp2
+    \<and> tp1 = concat (take as (tms_of aprog)) \<and> 
+      tp2 = concat (drop (Suc as) (tms_of aprog))")
+apply(erule exE, erule exE, erule conjE, erule conjE,
+      case_tac tc, simp)
+apply(rule turing_shift)
+apply(subgoal_tac "start_of (layout_of aprog) as - Suc 0 
+                = length tp1 div 2", simp)
+apply(rule_tac pre_lheq, simp, simp, simp)
+apply(simp add: startof_not0, simp)
+apply(rule_tac s_crsp, simp, simp, simp, simp)
+apply(rule tms_ct, simp, simp)
+apply(rule tm_ct, simp)
+apply(subgoal_tac "ci (layout_of aprog) 
+                 (start_of (layout_of aprog) as) ins
+                = (tms_of aprog ! as)", simp)
+apply(simp add: tms_of.simps tpairs_of.simps)
+apply(simp add: tms_of.simps tpairs_of.simps abc_fetch.simps)
+apply(erule_tac t_split, auto simp: tm_of.simps)
+done
+
+(*
+subsection {* The compilation of @{text "Inc n"} *}
+*)
+
+text {*
+  The lemmas in this section lead to the correctness of 
+  the compilation of @{text "Inc n"} instruction.
+*}
+
+fun at_begin_fst_bwtn :: "inc_inv_t"
+  where
+  "at_begin_fst_bwtn (as, lm) (s, l, r) ires = 
+      (\<exists> lm1 tn rn. lm1 = (lm @ (0\<^bsup>tn\<^esup>)) \<and> length lm1 = s \<and> 
+          (if lm1 = [] then l = Bk # Bk # ires
+           else l = [Bk]@<rev lm1>@Bk#Bk#ires) \<and> r = (Bk\<^bsup>rn\<^esup>))" 
+
+
+fun at_begin_fst_awtn :: "inc_inv_t"
+  where
+  "at_begin_fst_awtn (as, lm) (s, l, r) ires = 
+      (\<exists> lm1 tn rn. lm1 = (lm @ (0\<^bsup>tn\<^esup>)) \<and> length lm1 = s \<and>
+         (if lm1 = []  then l = Bk # Bk # ires
+          else l = [Bk]@<rev lm1>@Bk#Bk#ires) \<and> r = [Oc]@Bk\<^bsup>rn\<^esup>
+  )"
+
+fun at_begin_norm :: "inc_inv_t"
+  where
+  "at_begin_norm (as, lm) (s, l, r) ires= 
+      (\<exists> lm1 lm2 rn. lm = lm1 @ lm2 \<and> length lm1 = s \<and> 
+        (if lm1 = [] then l = Bk # Bk # ires
+         else l = Bk # <rev lm1> @ Bk# Bk # ires ) \<and> r = <lm2> @ (Bk\<^bsup>rn\<^esup>))"
+
+fun in_middle :: "inc_inv_t"
+  where
+  "in_middle (as, lm) (s, l, r) ires = 
+      (\<exists> lm1 lm2 tn m ml mr rn. lm @ 0\<^bsup>tn\<^esup> = lm1 @ [m] @ lm2
+       \<and> length lm1 = s \<and> m + 1 = ml + mr \<and>  
+         ml \<noteq> 0 \<and> tn = s + 1 - length lm \<and> 
+       (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires 
+        else l = (Oc\<^bsup>ml\<^esup>)@[Bk]@<rev lm1>@
+                 Bk # Bk # ires) \<and> (r = (Oc\<^bsup>mr\<^esup>) @ [Bk] @ <lm2>@ (Bk\<^bsup>rn\<^esup>) \<or> 
+      (lm2 = [] \<and> r = (Oc\<^bsup>mr\<^esup>)))
+      )"
+
+fun inv_locate_a :: "inc_inv_t"
+  where "inv_locate_a (as, lm) (s, l, r) ires = 
+     (at_begin_norm (as, lm) (s, l, r) ires \<or>
+      at_begin_fst_bwtn (as, lm) (s, l, r) ires \<or>
+      at_begin_fst_awtn (as, lm) (s, l, r) ires
+      )"
+
+fun inv_locate_b :: "inc_inv_t"
+  where "inv_locate_b (as, lm) (s, l, r) ires = 
+        (in_middle (as, lm) (s, l, r)) ires "
+
+fun inv_after_write :: "inc_inv_t"
+  where "inv_after_write (as, lm) (s, l, r) ires = 
+           (\<exists> rn m lm1 lm2. lm = lm1 @ m # lm2 \<and>
+             (if lm1 = [] then l = Oc\<^bsup>m\<^esup> @ Bk # Bk # ires
+              else Oc # l = Oc\<^bsup>Suc m \<^esup>@ Bk # <rev lm1> @ 
+                      Bk # Bk # ires) \<and> r = [Oc] @ <lm2> @ (Bk\<^bsup>rn\<^esup>))"
+
+fun inv_after_move :: "inc_inv_t"
+  where "inv_after_move (as, lm) (s, l, r) ires = 
+      (\<exists> rn m lm1 lm2. lm = lm1 @ m # lm2 \<and>
+        (if lm1 = [] then l = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # ires
+         else l = Oc\<^bsup>Suc m\<^esup>@ Bk # <rev lm1> @ Bk # Bk # ires) \<and> 
+        r = <lm2> @ (Bk\<^bsup>rn\<^esup>))"
+
+fun inv_after_clear :: "inc_inv_t"
+  where "inv_after_clear (as, lm) (s, l, r) ires =
+       (\<exists> rn m lm1 lm2 r'. lm = lm1 @ m # lm2 \<and> 
+        (if lm1 = [] then l = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # ires
+         else l = Oc\<^bsup>Suc m\<^esup>@ Bk # <rev lm1> @ Bk # Bk # ires) \<and> 
+          r = Bk # r' \<and> Oc # r' = <lm2> @ (Bk\<^bsup>rn\<^esup>))"
+
+fun inv_on_right_moving :: "inc_inv_t"
+  where "inv_on_right_moving (as, lm) (s, l, r) ires = 
+       (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> 
+            ml + mr = m \<and> 
+          (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
+          else l = (Oc\<^bsup>ml\<^esup>) @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> 
+         ((r = (Oc\<^bsup>mr\<^esup>) @ [Bk] @ <lm2> @ (Bk\<^bsup>rn\<^esup>)) \<or> 
+          (r = (Oc\<^bsup>mr\<^esup>) \<and> lm2 = [])))"
+
+fun inv_on_left_moving_norm :: "inc_inv_t"
+  where "inv_on_left_moving_norm (as, lm) (s, l, r) ires =
+      (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and>  
+             ml + mr = Suc m \<and> mr > 0 \<and> (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
+                                         else l = (Oc\<^bsup>ml\<^esup>) @ Bk # <rev lm1> @ Bk # Bk # ires)
+        \<and> (r = (Oc\<^bsup>mr\<^esup>) @ Bk # <lm2> @ (Bk\<^bsup>rn\<^esup>) \<or> 
+           (lm2 = [] \<and> r = Oc\<^bsup>mr\<^esup>)))"
+
+fun inv_on_left_moving_in_middle_B:: "inc_inv_t"
+  where "inv_on_left_moving_in_middle_B (as, lm) (s, l, r) ires =
+                (\<exists> lm1 lm2 rn. lm = lm1 @ lm2 \<and>  
+                     (if lm1 = [] then l = Bk # ires
+                      else l = <rev lm1> @ Bk # Bk # ires) \<and> 
+                      r = Bk # <lm2> @ (Bk\<^bsup>rn\<^esup>))"
+
+fun inv_on_left_moving :: "inc_inv_t"
+  where "inv_on_left_moving (as, lm) (s, l, r) ires = 
+       (inv_on_left_moving_norm  (as, lm) (s, l, r) ires \<or>
+        inv_on_left_moving_in_middle_B (as, lm) (s, l, r) ires)"
+
+
+fun inv_check_left_moving_on_leftmost :: "inc_inv_t"
+  where "inv_check_left_moving_on_leftmost (as, lm) (s, l, r) ires = 
+                (\<exists> rn. l = ires \<and> r = [Bk, Bk] @ <lm> @ (Bk\<^bsup>rn\<^esup>))"
+
+fun inv_check_left_moving_in_middle :: "inc_inv_t"
+  where "inv_check_left_moving_in_middle (as, lm) (s, l, r) ires = 
+
+              (\<exists> lm1 lm2 r' rn. lm = lm1 @ lm2 \<and>
+                 (Oc # l = <rev lm1> @ Bk # Bk # ires) \<and> r = Oc # Bk # r' \<and> 
+                           r' = <lm2> @ (Bk\<^bsup>rn\<^esup>))"
+
+fun inv_check_left_moving :: "inc_inv_t"
+  where "inv_check_left_moving (as, lm) (s, l, r) ires = 
+             (inv_check_left_moving_on_leftmost (as, lm) (s, l, r) ires \<or>
+             inv_check_left_moving_in_middle (as, lm) (s, l, r) ires)"
+
+fun inv_after_left_moving :: "inc_inv_t"
+  where "inv_after_left_moving (as, lm) (s, l, r) ires= 
+              (\<exists> rn. l = Bk # ires \<and> r = Bk # <lm> @ (Bk\<^bsup>rn\<^esup>))"
+
+fun inv_stop :: "inc_inv_t"
+  where "inv_stop (as, lm) (s, l, r) ires= 
+              (\<exists> rn. l = Bk # Bk # ires \<and> r = <lm> @ (Bk\<^bsup>rn\<^esup>))"
+
+
+fun inc_inv :: "layout \<Rightarrow> nat \<Rightarrow> inc_inv_t"
+  where
+  "inc_inv ly n (as, lm) (s, l, r) ires =
+              (let ss = start_of ly as in
+               let lm' = abc_lm_s lm n ((abc_lm_v lm n)+1) in
+                if s = 0 then False
+                else if s < ss then False
+                else if s < ss + 2 * n then 
+                   if (s - ss) mod 2 = 0 then 
+                       inv_locate_a (as, lm) ((s - ss) div 2, l, r) ires
+                   else inv_locate_b (as, lm) ((s - ss) div 2, l, r) ires
+                else if s = ss + 2 * n then 
+                        inv_locate_a (as, lm) (n, l, r) ires
+                else if s = ss + 2 * n + 1 then 
+                   inv_locate_b (as, lm) (n, l, r) ires
+                else if s = ss + 2 * n + 2 then 
+                   inv_after_write (as, lm') (s - ss, l, r) ires
+                else if s = ss + 2 * n + 3 then 
+                   inv_after_move (as, lm') (s - ss, l, r) ires
+                else if s = ss + 2 * n + 4 then 
+                   inv_after_clear (as, lm') (s - ss, l, r) ires
+                else if s = ss + 2 * n + 5 then 
+                   inv_on_right_moving (as, lm') (s - ss, l, r) ires
+                else if s = ss + 2 * n + 6 then 
+                   inv_on_left_moving (as, lm') (s - ss, l, r) ires
+                else if s = ss + 2 * n + 7 then 
+                   inv_check_left_moving (as, lm') (s - ss, l, r) ires
+                else if s = ss + 2 * n + 8 then 
+                   inv_after_left_moving (as, lm') (s - ss, l, r) ires
+                else if s = ss + 2 * n + 9 then 
+                   inv_stop (as, lm') (s - ss, l, r) ires
+                else False) "
+
+lemma fetch_intro: 
+  "\<lbrakk>\<And>xs.\<lbrakk>ba = Oc # xs\<rbrakk> \<Longrightarrow> P (fetch prog i Oc);
+   \<And>xs.\<lbrakk>ba = Bk # xs\<rbrakk> \<Longrightarrow> P (fetch prog i Bk);
+   ba = [] \<Longrightarrow> P (fetch prog i Bk)
+   \<rbrakk> \<Longrightarrow> P (fetch prog i 
+             (case ba of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))"
+by (auto split:list.splits block.splits)
+
+lemma length_findnth[simp]: "length (findnth n) = 4 * n"
+apply(induct n, simp)
+apply(simp)
+done
+
+declare tshift.simps[simp del]
+declare findnth.simps[simp del]
+
+lemma findnth_nth: 
+ "\<lbrakk>n > q; x < 4\<rbrakk> \<Longrightarrow> 
+        (findnth n) ! (4 * q + x) = (findnth (Suc q) ! (4 * q + x))"
+apply(induct n, simp)
+apply(case_tac "q < n", simp add: findnth.simps, auto)
+apply(simp add: nth_append)
+apply(subgoal_tac "q = n", simp)
+apply(arith)
+done
+
+lemma Suc_pre[simp]: "\<not> a < start_of ly as \<Longrightarrow> 
+          (Suc a - start_of ly as) = Suc (a - start_of ly as)"
+apply(arith)
+done
+
+lemma fetch_locate_a_o: "
+\<And>a  q xs.
+    \<lbrakk>\<not> a < start_of (layout_of aprog) as; 
+      a < start_of (layout_of aprog) as + 2 * n; 
+      a - start_of (layout_of aprog) as = 2 * q; 
+      start_of (layout_of aprog) as > 0\<rbrakk>
+    \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
+         (Inc n)) (Suc (2 * q)) Oc) = (R, a+1)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append Suc_pre)
+apply(subgoal_tac "(findnth n ! Suc (4 * q)) = 
+                 findnth (Suc q) ! (4 * q + 1)")
+apply(simp add: findnth.simps nth_append)
+apply(subgoal_tac " findnth n !(4 * q + 1) = 
+                 findnth (Suc q) ! (4 * q + 1)", simp)
+apply(rule_tac findnth_nth, auto)
+done
+
+lemma fetch_locate_a_b: "
+\<And>a  q xs.
+    \<lbrakk>abc_fetch as aprog = Some (Inc n);  
+     \<not> a < start_of (layout_of aprog) as; 
+     a < start_of (layout_of aprog) as + 2 * n; 
+     a - start_of (layout_of aprog) as = 2 * q; 
+     start_of (layout_of aprog) as > 0\<rbrakk>
+    \<Longrightarrow> (fetch (ci (layout_of aprog) 
+      (start_of (layout_of aprog) as) (Inc n)) (Suc (2 * q)) Bk)
+       = (W1, a)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                 tshift.simps nth_append)
+apply(subgoal_tac "(findnth n ! (4 * q)) = 
+                           findnth (Suc q) ! (4 * q )")
+apply(simp add: findnth.simps nth_append)
+apply(subgoal_tac " findnth n !(4 * q + 0) =
+                            findnth (Suc q) ! (4 * q + 0)", simp)
+apply(rule_tac findnth_nth, auto)
+done
+
+lemma [intro]: "x mod 2 = Suc 0 \<Longrightarrow> \<exists> q. x = Suc (2 * q)"
+apply(drule mod_eqD, auto)
+done
+
+lemma  add3_Suc: "x + 3 = Suc (Suc (Suc x))"
+apply(arith)
+done
+
+declare start_of.simps[simp]
+(*
+lemma layout_not0: "start_of ly as > 0"
+by(induct as, auto)
+*)
+lemma [simp]: 
+ "\<lbrakk>\<not> a < start_of (layout_of aprog) as; 
+   a - start_of (layout_of aprog) as = Suc (2 * q); 
+   abc_fetch as aprog = Some (Inc n); 
+   start_of (layout_of aprog) as > 0\<rbrakk>
+    \<Longrightarrow> Suc (Suc (2 * q + start_of (layout_of aprog) as - Suc 0)) = a"
+apply(subgoal_tac 
+"Suc (Suc (2 * q + start_of (layout_of aprog) as - Suc 0)) 
+              = 2 + 2 * q + start_of (layout_of aprog) as - Suc 0", 
+  simp, simp add: inc_startof_not0)
+done
+
+lemma fetch_locate_b_o: "
+\<And>a  xs.
+    \<lbrakk>0 < a; \<not> a < start_of (layout_of aprog) as; 
+  a < start_of (layout_of aprog) as + 2 * n; 
+ (a - start_of (layout_of aprog) as) mod 2 = Suc 0; 
+ start_of (layout_of aprog) as > 0\<rbrakk>
+    \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+      (Inc n)) (Suc (a - start_of (layout_of aprog) as)) Oc) = (R, a)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                 nth_of.simps tshift.simps nth_append)
+apply(subgoal_tac "\<exists> q. (a - start_of (layout_of aprog) as) = 
+                         2 * q + 1", auto)
+apply(subgoal_tac "(findnth n ! Suc (Suc (Suc (4 * q)))) 
+                  = findnth (Suc q) ! (4 * q + 3)")
+apply(simp add: findnth.simps nth_append)
+apply(subgoal_tac " findnth n ! (4 * q + 3) = 
+                 findnth (Suc q) ! (4 * q + 3)", simp add: add3_Suc)
+apply(rule_tac findnth_nth, auto)
+done
+
+lemma fetch_locate_b_b: "
+\<And>a  xs.
+    \<lbrakk>0 < a;  \<not> a < start_of (layout_of aprog) as; 
+     a < start_of (layout_of aprog) as + 2 * n; 
+     (a - start_of (layout_of aprog) as) mod 2 = Suc 0; 
+     start_of (layout_of aprog) as > 0\<rbrakk>
+    \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
+        (Inc n)) (Suc (a - start_of (layout_of aprog) as)) Bk) 
+        = (R, a + 1)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append)
+apply(subgoal_tac "\<exists> q. (a - start_of (layout_of aprog) as) = 
+                  2 * q + 1", auto)
+apply(subgoal_tac "(findnth n ! Suc ((Suc (4 * q)))) = 
+                    findnth (Suc q) ! (4 * q + 2)")
+apply(simp add: findnth.simps nth_append)
+apply(subgoal_tac " findnth n ! (4 * q + 2) = 
+                    findnth (Suc q) ! (4 * q + 2)", simp)
+apply(rule_tac findnth_nth, auto)
+done
+
+lemma fetch_locate_n_a_o: 
+       "start_of (layout_of aprog) as > 0
+       \<Longrightarrow> (fetch (ci (layout_of aprog) 
+      (start_of (layout_of aprog) as) (Inc n)) (Suc (2 * n)) Oc) = 
+             (R, start_of (layout_of aprog) as + 2 * n + 1)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tinc_b_def)
+done
+
+lemma fetch_locate_n_a_b: "
+       start_of (layout_of aprog) as > 0
+      \<Longrightarrow> (fetch (ci (layout_of aprog) 
+    (start_of (layout_of aprog) as) (Inc n)) (Suc (2 * n)) Bk) 
+   = (W1, start_of (layout_of aprog) as + 2 * n)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tinc_b_def)
+done
+
+lemma fetch_locate_n_b_o: "
+    start_of (layout_of aprog) as > 0
+    \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+     (Inc n)) (Suc (Suc (2 * n))) Oc) = 
+                      (R, start_of (layout_of aprog) as + 2 * n + 1)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tinc_b_def)
+done
+
+lemma fetch_locate_n_b_b: "
+    start_of (layout_of aprog) as > 0
+   \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+   (Inc n)) (Suc (Suc (2 * n))) Bk) = 
+       (W1, start_of (layout_of aprog) as + 2 * n + 2)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tinc_b_def)
+done
+
+lemma fetch_after_write_o: "
+    start_of (layout_of aprog) as > 0
+    \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+            (Inc n)) (Suc (Suc (Suc (2 * n)))) Oc) = 
+        (R, start_of (layout_of aprog) as + 2*n + 3)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tinc_b_def)
+done
+
+lemma fetch_after_move_o: "
+      start_of (layout_of aprog) as > 0
+      \<Longrightarrow> (fetch (ci (layout_of aprog) 
+              (start_of (layout_of aprog) as) (Inc n)) (4 + 2 * n) Oc)
+        = (W0, start_of (layout_of aprog) as + 2 * n + 4)"
+apply(auto simp: ci.simps findnth.simps tshift.simps 
+                 tinc_b_def add3_Suc)
+apply(subgoal_tac "4 + 2*n = Suc (2*n + 3)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma fetch_after_move_b: "
+      start_of (layout_of aprog) as > 0
+      \<Longrightarrow>(fetch (ci (layout_of aprog) 
+            (start_of (layout_of aprog) as) (Inc n)) (4 + 2 * n) Bk)
+       = (L, start_of (layout_of aprog) as + 2 * n + 6)"
+apply(auto simp: ci.simps findnth.simps tshift.simps 
+                 tinc_b_def add3_Suc)
+apply(subgoal_tac "4 + 2*n = Suc (2*n + 3)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma fetch_clear_b: "
+     start_of (layout_of aprog) as > 0
+      \<Longrightarrow> (fetch (ci (layout_of aprog) 
+              (start_of (layout_of aprog) as) (Inc n)) (5 + 2 * n) Bk)
+      = (R, start_of (layout_of aprog) as + 2 * n + 5)"
+apply(auto simp: ci.simps findnth.simps 
+                     tshift.simps tinc_b_def add3_Suc)
+apply(subgoal_tac "5 + 2*n = Suc (2*n + 4)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma fetch_right_move_o: "
+      start_of (layout_of aprog) as > 0
+      \<Longrightarrow> (fetch (ci (layout_of aprog) 
+                (start_of (layout_of aprog) as) (Inc n)) (6 + 2*n) Oc)
+      = (R, start_of (layout_of aprog) as + 2 * n + 5)"
+apply(auto simp: ci.simps findnth.simps tshift.simps 
+                 tinc_b_def add3_Suc)
+apply(subgoal_tac "6 + 2*n = Suc (2*n + 5)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma fetch_right_move_b: "
+      start_of (layout_of aprog) as > 0
+      \<Longrightarrow> (fetch (ci (layout_of aprog) 
+                (start_of (layout_of aprog) as) (Inc n)) (6 + 2*n) Bk)
+      = (W1, start_of (layout_of aprog) as + 2 * n + 2)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tinc_b_def add3_Suc)
+apply(subgoal_tac "6 + 2*n = Suc (2*n + 5)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma fetch_left_move_o: "
+      start_of (layout_of aprog) as > 0
+      \<Longrightarrow> (fetch (ci (layout_of aprog) 
+               (start_of (layout_of aprog) as) (Inc n)) (7 + 2*n) Oc)
+      = (L, start_of (layout_of aprog) as + 2 * n + 6)"
+apply(auto simp: ci.simps findnth.simps tshift.simps 
+                 tinc_b_def add3_Suc)
+apply(subgoal_tac "7 + 2*n = Suc (2*n + 6)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma fetch_left_move_b: "
+      start_of (layout_of aprog) as > 0
+      \<Longrightarrow> (fetch (ci (layout_of aprog) 
+               (start_of (layout_of aprog) as) (Inc n)) (7 + 2*n) Bk)
+      = (L, start_of (layout_of aprog) as + 2 * n + 7)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tinc_b_def add3_Suc)
+apply(subgoal_tac "7 + 2*n = Suc (2*n + 6)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma fetch_check_left_move_o: "
+      start_of (layout_of aprog) as > 0
+      \<Longrightarrow> (fetch (ci (layout_of aprog) 
+               (start_of (layout_of aprog) as) (Inc n)) (8 + 2*n) Oc)
+      = (L, start_of (layout_of aprog) as + 2 * n + 6)"
+apply(auto simp: ci.simps findnth.simps tshift.simps tinc_b_def)
+apply(subgoal_tac "8 + 2 * n = Suc (2 * n + 7)", 
+                                  simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma fetch_check_left_move_b: "
+      start_of (layout_of aprog) as > 0
+      \<Longrightarrow> (fetch (ci (layout_of aprog) 
+              (start_of (layout_of aprog) as) (Inc n)) (8 + 2*n) Bk)
+      = (R, start_of (layout_of aprog) as + 2 * n + 8)  "
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tinc_b_def add3_Suc)
+apply(subgoal_tac "8 + 2*n= Suc (2*n + 7)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma fetch_after_left_move: "
+      start_of (layout_of aprog) as > 0
+      \<Longrightarrow> (fetch (ci (layout_of aprog) 
+              (start_of (layout_of aprog) as) (Inc n)) (9 + 2*n) Bk)
+     = (R, start_of (layout_of aprog) as + 2 * n + 9)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tinc_b_def)
+done
+
+lemma fetch_stop: "
+       start_of (layout_of aprog) as > 0
+      \<Longrightarrow> (fetch (ci (layout_of aprog) 
+             (start_of (layout_of aprog) as) (Inc n)) (10 + 2 *n)  b)
+     = (Nop, 0)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tinc_b_def
+            split: block.splits)
+done
+
+lemma fetch_state0: "
+       (fetch (ci (layout_of aprog) 
+               (start_of (layout_of aprog) as) (Inc n)) 0 b)
+     = (Nop, 0)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tinc_b_def)
+done
+
+lemmas fetch_simps = 
+  fetch_locate_a_o fetch_locate_a_b fetch_locate_b_o fetch_locate_b_b 
+  fetch_locate_n_a_b fetch_locate_n_a_o fetch_locate_n_b_o 
+  fetch_locate_n_b_b fetch_after_write_o fetch_after_move_o 
+  fetch_after_move_b fetch_clear_b fetch_right_move_o 
+  fetch_right_move_b fetch_left_move_o fetch_left_move_b
+  fetch_after_left_move fetch_check_left_move_o fetch_stop 
+  fetch_state0 fetch_check_left_move_b
+
+text {* *}
+declare exponent_def[simp del] tape_of_nat_list.simps[simp del] 
+   at_begin_norm.simps[simp del] at_begin_fst_bwtn.simps[simp del] 
+   at_begin_fst_awtn.simps[simp del] in_middle.simps[simp del] 
+   abc_lm_s.simps[simp del] abc_lm_v.simps[simp del]  
+   ci.simps[simp del] t_step.simps[simp del]
+   inv_after_move.simps[simp del] 
+   inv_on_left_moving_norm.simps[simp del] 
+   inv_on_left_moving_in_middle_B.simps[simp del]
+   inv_after_clear.simps[simp del] 
+   inv_after_write.simps[simp del] inv_on_left_moving.simps[simp del]
+   inv_on_right_moving.simps[simp del] 
+   inv_check_left_moving.simps[simp del] 
+   inv_check_left_moving_in_middle.simps[simp del]
+   inv_check_left_moving_on_leftmost.simps[simp del] 
+   inv_after_left_moving.simps[simp del]
+   inv_stop.simps[simp del] inv_locate_a.simps[simp del] 
+   inv_locate_b.simps[simp del]
+declare tms_of.simps[simp del] tm_of.simps[simp del]
+        layout_of.simps[simp del] abc_fetch.simps [simp del] 
+        t_step.simps[simp del] t_steps.simps[simp del] 
+        tpairs_of.simps[simp del] start_of.simps[simp del]
+        fetch.simps [simp del] new_tape.simps [simp del] 
+        nth_of.simps [simp del] ci.simps [simp del]
+        length_of.simps[simp del]
+
+(*! Start point *)
+lemma [simp]: "Suc (2 * q) mod 2 = Suc 0"
+by arith
+
+lemma [simp]: "Suc (2 * q) div 2 = q"
+by arith
+
+lemma [simp]: "\<lbrakk> \<not> a < start_of ly as; 
+          a < start_of ly as + 2 * n; a - start_of ly as = 2 * q\<rbrakk>
+             \<Longrightarrow> Suc a < start_of ly as + 2 * n"
+apply(arith)
+done
+
+lemma [simp]: "x mod 2 = Suc 0 \<Longrightarrow> (Suc x) mod 2 = 0"
+by arith
+
+lemma [simp]: "x mod 2 = Suc 0 \<Longrightarrow> (Suc x) div 2 = Suc (x div 2)"
+by arith
+lemma exp_def[simp]: "a\<^bsup>Suc n \<^esup>= a # a\<^bsup>n\<^esup>"
+by(simp add: exponent_def)
+lemma [intro]: "Bk # r = Oc\<^bsup>mr\<^esup> @ r' \<Longrightarrow> mr = 0"
+by(case_tac mr, auto simp: exponent_def)
+
+lemma [intro]: "Bk # r = replicate mr Oc \<Longrightarrow> mr = 0"
+by(case_tac mr, auto)
+lemma tape_of_nl_abv_cons[simp]: "xs \<noteq> [] \<Longrightarrow> 
+                   <x # xs> = Oc\<^bsup>Suc x\<^esup>@ Bk # <xs>"
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
+apply(case_tac xs, simp, simp add: tape_of_nat_list.simps)
+done
+
+lemma [simp]: "<[]::nat list> = []"
+by(auto simp: tape_of_nl_abv tape_of_nat_list.simps)
+lemma [simp]: "Oc # r = <(lm::nat list)> @ Bk\<^bsup>rn\<^esup>\<Longrightarrow> lm \<noteq> []"
+apply(auto simp: tape_of_nl_abv tape_of_nat_list.simps)
+apply(case_tac rn, auto simp: exponent_def)
+done
+lemma BkCons_nil: "Bk # xs = <lm::nat list> @ Bk\<^bsup>rn\<^esup>\<Longrightarrow> lm = []"
+apply(case_tac lm, simp)
+apply(case_tac list, auto simp: tape_of_nl_abv tape_of_nat_list.simps)
+done
+lemma BkCons_nil': "Bk # xs = <lm::nat list> @ Bk\<^bsup>ln\<^esup>\<Longrightarrow> lm = []"
+by(auto intro: BkCons_nil)
+
+lemma hd_tl_tape_of_nat_list:  
+   "tl (lm::nat list) \<noteq> [] \<Longrightarrow> <lm> = <hd lm> @ Bk # <tl lm>"
+apply(frule tape_of_nl_abv_cons[of "tl lm" "hd lm"])
+apply(simp add: tape_of_nat_abv Bk_def del: tape_of_nl_abv_cons)
+apply(subgoal_tac "lm = hd lm # tl lm", auto)
+apply(case_tac lm, auto)
+done
+lemma [simp]: "Oc # xs = Oc\<^bsup>mr\<^esup> @ Bk # <lm2> @ Bk\<^bsup>rn\<^esup>\<Longrightarrow> mr > 0"
+apply(case_tac mr, auto simp: exponent_def)
+done
+
+lemma tape_of_nat_list_cons: "xs \<noteq> [] \<Longrightarrow> tape_of_nat_list (x # xs) =
+              replicate (Suc x) Oc @ Bk # tape_of_nat_list xs"
+apply(drule tape_of_nl_abv_cons[of xs x])
+apply(auto simp: tape_of_nl_abv tape_of_nat_abv Oc_def Bk_def exponent_def)
+done
+
+lemma rev_eq: "rev xs = rev ys \<Longrightarrow> xs = ys"
+by simp
+
+lemma tape_of_nat_list_eq: " xs = ys \<Longrightarrow> 
+        tape_of_nat_list xs = tape_of_nat_list ys"
+by simp
+
+lemma tape_of_nl_nil_eq: "<(lm::nat list)> = [] = (lm = [])"
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
+apply(case_tac lm, simp add: tape_of_nat_list.simps)
+apply(case_tac "list")
+apply(auto simp: tape_of_nat_list.simps)
+done
+
+lemma rep_ind: "replicate (Suc n) a = replicate n a @ [a]"
+apply(induct n, simp, simp)
+done
+
+lemma [simp]: "Oc # r = <lm::nat list> @ replicate rn Bk \<Longrightarrow> Suc 0 \<le> length lm"
+apply(rule_tac classical, auto)
+apply(case_tac lm, simp, case_tac rn, auto)
+done
+lemma Oc_Bk_Cons: "Oc # Bk # list = <lm::nat list> @ Bk\<^bsup>ln\<^esup> \<Longrightarrow> 
+                   lm \<noteq> [] \<and> hd lm = 0"
+apply(case_tac lm, simp, case_tac ln, simp add: exponent_def, simp add: exponent_def, simp)
+apply(case_tac lista, auto simp: tape_of_nl_abv tape_of_nat_list.simps)
+done 
+(*lemma Oc_Oc_Cons: "Oc # Oc # list = <lm::nat list> @ Bk\<^bsup>ln\<^esup> \<Longrightarrow> 
+                  lm \<noteq> [] \<and> hd lm > 0"
+apply(case_tac lm, simp add: exponent_def, case_tac ln, simp, simp)
+apply(case_tac lista, 
+        auto simp: tape_of_nl_abv tape_of_nat_list.simps exponent_def)
+apply(case_tac [!] a, auto)
+apply(case_tac ln, auto)
+done
+*)
+lemma Oc_nil_zero[simp]: "[Oc] = <lm::nat list> @ Bk\<^bsup>ln\<^esup> 
+                 \<Longrightarrow> lm = [0] \<and> ln = 0"
+apply(case_tac lm, simp)
+apply(case_tac ln, auto simp: exponent_def)
+apply(case_tac [!] list, 
+        auto simp: tape_of_nl_abv tape_of_nat_list.simps)
+done
+
+lemma  [simp]: "Oc # r = <lm2> @ replicate rn Bk \<Longrightarrow> 
+       (\<exists>rn. r = replicate (hd lm2) Oc @ Bk # <tl lm2> @ 
+                      replicate rn Bk) \<or> 
+          tl lm2 = [] \<and> r = replicate (hd lm2) Oc"
+apply(rule_tac disjCI, simp)
+apply(case_tac "tl lm2 = []", simp)
+apply(case_tac lm2, simp add: tape_of_nl_abv tape_of_nat_list.simps)
+apply(case_tac rn, simp, simp, simp)
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps exponent_def)
+apply(case_tac rn, simp, simp)
+apply(rule_tac x = rn in exI)
+apply(simp add: hd_tl_tape_of_nat_list)
+apply(simp add: tape_of_nat_abv Oc_def exponent_def)
+done
+
+(*inv: from locate_a to locate_b*)
+lemma [simp]: 
+      "inv_locate_a (as, lm) (q, l, Oc # r) ires
+       \<Longrightarrow> inv_locate_b (as, lm) (q, Oc # l, r) ires"
+apply(simp only: inv_locate_a.simps inv_locate_b.simps in_middle.simps
+          at_begin_norm.simps at_begin_fst_bwtn.simps
+          at_begin_fst_awtn.simps)
+apply(erule disjE, erule exE, erule exE, erule exE)
+apply(rule_tac x = lm1 in exI, rule_tac x = "tl lm2" in exI, simp)
+apply(rule_tac x = "0" in exI, rule_tac x = "hd lm2" in exI, 
+                auto simp: exponent_def)
+apply(rule_tac x = "Suc 0" in exI, simp add:exponent_def)
+apply(rule_tac x = "lm @ replicate tn 0" in exI, 
+      rule_tac x = "[]" in exI,    
+      rule_tac x = "Suc tn" in exI, rule_tac x = 0 in exI)
+apply(simp only: rep_ind, simp)
+apply(rule_tac x = "Suc 0" in exI, auto)
+apply(case_tac [1-3] rn, simp_all )
+apply(rule_tac x = "lm @ replicate tn 0" in exI, 
+      rule_tac x = "[]" in exI, 
+      rule_tac x = "Suc tn" in exI, 
+      rule_tac x = 0 in exI, simp add: rep_ind del: replicate_Suc split:if_splits)
+apply(rule_tac x = "Suc 0" in exI, auto)
+apply(case_tac rn, simp, simp)
+apply(rule_tac [!] x = "Suc 0" in exI, auto)
+apply(case_tac [!] rn, simp_all)
+done
+
+(*inv: from locate_a to _locate_a*)
+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 
+                 at_begin_fst_bwtn.simps at_begin_fst_awtn.simps)
+apply(erule_tac disjE, erule exE, erule exE, erule exE, 
+      rule disjI2, rule disjI2)
+defer
+apply(erule_tac disjE, erule exE, erule exE, 
+      erule exE, rule disjI2, rule disjI2)
+prefer 2
+apply(simp)
+proof-
+  fix lm1 tn rn
+  assume k: "lm1 = am @ 0\<^bsup>tn\<^esup> \<and> length lm1 = q \<and> (if lm1 = [] then aaa = Bk # Bk # 
+    ires else aaa = [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> Bk # xs = Bk\<^bsup>rn\<^esup>"
+  thus "\<exists>lm1 tn rn. lm1 = am @ 0\<^bsup>tn\<^esup> \<and> length lm1 = q \<and> (if lm1 = [] then 
+    aaa = Bk # Bk # ires else aaa = [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> Oc # xs = [Oc] @ Bk\<^bsup>rn\<^esup>"
+    (is "\<exists>lm1 tn rn. ?P lm1 tn rn")
+  proof -
+    from k have "?P lm1 tn (rn - 1)"
+      apply(auto simp: Oc_def)
+      by(case_tac [!] "rn::nat", auto simp: exponent_def)
+    thus ?thesis by blast
+  qed
+next
+  fix lm1 lm2 rn
+  assume h1: "am = lm1 @ lm2 \<and> length lm1 = q \<and> (if lm1 = [] 
+    then aaa = Bk # Bk # ires else aaa = Bk # <rev lm1> @ Bk # Bk # ires) \<and>
+    Bk # xs = <lm2> @ Bk\<^bsup>rn\<^esup>"
+  from h1 have h2: "lm2 = []"
+  proof(rule_tac xs = xs and rn = rn in BkCons_nil, simp)
+  qed
+  from h1 and h2 show "\<exists>lm1 tn rn. lm1 = am @ 0\<^bsup>tn\<^esup> \<and> length lm1 = q \<and> 
+    (if lm1 = [] then aaa = Bk # Bk # ires else aaa = [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and>
+    Oc # xs = [Oc] @ Bk\<^bsup>rn\<^esup>" 
+    (is "\<exists>lm1 tn rn. ?P lm1 tn rn")
+  proof -
+    from h1 and h2  have "?P lm1 0 (rn - 1)"
+      apply(auto simp: Oc_def exponent_def 
+                      tape_of_nl_abv tape_of_nat_list.simps)
+      by(case_tac "rn::nat", simp, simp)
+    thus ?thesis by blast
+  qed
+qed
+
+lemma [intro]: "\<exists>rn. [a] = a\<^bsup>rn\<^esup>"
+by(rule_tac x = "Suc 0" in exI, simp add: exponent_def)
+
+lemma [intro]: "\<exists>tn. [] = a\<^bsup>tn\<^esup>"
+apply(rule_tac x = 0 in exI, simp add: exponent_def)
+done
+
+lemma [intro]:  "at_begin_norm (as, am) (q, aaa, []) ires
+             \<Longrightarrow> at_begin_norm (as, am) (q, aaa, [Bk]) ires"
+apply(simp add: at_begin_norm.simps, erule_tac exE, erule_tac exE)
+apply(rule_tac x = lm1 in exI, simp, auto)
+done
+
+lemma [intro]: "at_begin_fst_bwtn (as, am) (q, aaa, []) ires 
+            \<Longrightarrow> at_begin_fst_bwtn (as, am) (q, aaa, [Bk]) ires"
+apply(simp only: at_begin_fst_bwtn.simps, erule_tac exE, erule_tac exE, erule_tac exE)
+apply(rule_tac x = "am @ 0\<^bsup>tn\<^esup>" in exI, auto)
+done
+
+lemma [intro]: "at_begin_fst_awtn (as, am) (q, aaa, []) ires
+           \<Longrightarrow> at_begin_fst_awtn (as, am) (q, aaa, [Bk]) ires"
+apply(auto simp: at_begin_fst_awtn.simps)
+done 
+
+lemma [intro]: "inv_locate_a (as, am) (q, aaa, []) ires
+            \<Longrightarrow> inv_locate_a (as, am) (q, aaa, [Bk]) ires"
+apply(simp only: inv_locate_a.simps)
+apply(erule disj_forward)
+defer
+apply(erule disj_forward, auto)
+done
+
+lemma [simp]: "inv_locate_a (as, am) (q, aaa, []) ires \<Longrightarrow> 
+               inv_locate_a (as, am) (q, aaa, [Oc]) ires"
+apply(insert locate_a_2_locate_a [of as am q aaa "[]"])
+apply(subgoal_tac "inv_locate_a (as, am) (q, aaa, [Bk]) ires", auto)
+done
+
+(*inv: from locate_b to locate_b*)
+lemma [simp]: "inv_locate_b (as, am) (q, aaa, Oc # xs) ires
+         \<Longrightarrow> inv_locate_b (as, am) (q, Oc # aaa, xs) ires"
+apply(simp only: inv_locate_b.simps in_middle.simps)
+apply(erule exE)+
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = tn in exI, rule_tac x = m in exI)
+apply(rule_tac x = "Suc ml" in exI, rule_tac x = "mr - 1" in exI,
+      rule_tac x = rn in exI)
+apply(case_tac mr, simp_all add: exponent_def, auto)
+done
+lemma zero_and_nil[intro]: "(Bk # Bk\<^bsup>n\<^esup> = Oc\<^bsup>mr\<^esup> @ Bk # <lm::nat list> @ 
+                             Bk\<^bsup>rn \<^esup>) \<or> (lm2 = [] \<and> Bk # Bk\<^bsup>n\<^esup> = Oc\<^bsup>mr\<^esup>)
+       \<Longrightarrow> mr = 0 \<and> lm = []"
+apply(rule context_conjI)
+apply(case_tac mr, auto simp:exponent_def)
+apply(insert BkCons_nil[of "replicate (n - 1) Bk" lm rn])
+apply(case_tac n, auto simp: exponent_def Bk_def  tape_of_nl_nil_eq)
+done
+
+lemma tape_of_nat_def: "<[m::nat]> =  Oc # Oc\<^bsup>m\<^esup>"
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
+done
+lemma [simp]: "\<lbrakk>inv_locate_b (as, am) (q, aaa, Bk # xs) ires; \<exists>n. xs = Bk\<^bsup>n\<^esup>\<rbrakk>
+            \<Longrightarrow> inv_locate_a (as, am) (Suc q, Bk # aaa, xs) ires"
+apply(simp add: inv_locate_b.simps inv_locate_a.simps)
+apply(rule_tac disjI2, rule_tac disjI1)
+apply(simp only: in_middle.simps at_begin_fst_bwtn.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = "lm1 @ [m]" in exI, rule_tac x = tn in exI, simp)
+apply(subgoal_tac "mr = 0 \<and> lm2 = []")
+defer
+apply(rule_tac n = n and mr = mr and lm = "lm2" 
+               and rn = rn and n = n in zero_and_nil)
+apply(auto simp: exponent_def)
+apply(case_tac "lm1 = []", auto simp: tape_of_nat_def)
+done
+
+lemma length_equal: "xs = ys \<Longrightarrow> length xs = length ys"
+by auto
+lemma [simp]: "a\<^bsup>0\<^esup> = []" 
+by(simp add: exp_zero)
+(*inv: from locate_b to locate_a*)
+lemma [simp]: "length (a\<^bsup>b\<^esup>) = b"
+apply(simp add: exponent_def)
+done
+
+lemma [simp]: "\<lbrakk>inv_locate_b (as, am) (q, aaa, Bk # xs) ires; 
+                \<not> (\<exists>n. xs = Bk\<^bsup>n\<^esup>)\<rbrakk> 
+       \<Longrightarrow> inv_locate_a (as, am) (Suc q, Bk # aaa, xs) ires"
+apply(simp add: inv_locate_b.simps inv_locate_a.simps)
+apply(rule_tac disjI1)
+apply(simp only: in_middle.simps at_begin_norm.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = "lm1 @ [m]" in exI, rule_tac x = lm2 in exI, simp)
+apply(subgoal_tac "tn = 0", simp add: exponent_def , auto split: if_splits)
+apply(case_tac [!] mr, simp_all add: tape_of_nat_def, auto)
+apply(case_tac lm2, simp, erule_tac x = rn in allE, simp)
+apply(case_tac am, simp, simp)
+apply(case_tac lm2, simp, erule_tac x = rn in allE, simp)
+apply(drule_tac length_equal, simp)
+done
+
+lemma locate_b_2_a[intro]: 
+       "inv_locate_b (as, am) (q, aaa, Bk # xs) ires
+    \<Longrightarrow> inv_locate_a (as, am) (Suc q, Bk # aaa, xs) ires"
+apply(case_tac "\<exists> n. xs = Bk\<^bsup>n\<^esup>", simp, simp)
+done
+
+lemma locate_b_2_locate_a[simp]: 
+    "\<lbrakk>\<not> a < start_of ly as; 
+      a < start_of ly as + 2 * n; 
+      (a - start_of ly as) mod 2 = Suc 0; 
+     inv_locate_b (as, am) ((a - start_of ly as) div 2, aaa, Bk # xs) ires\<rbrakk>
+   \<Longrightarrow> (Suc a < start_of ly as + 2 * n \<longrightarrow> inv_locate_a (as, am)
+       (Suc ((a - start_of ly as) div 2), Bk # aaa, xs) ires) \<and>
+       (\<not> Suc a < start_of ly as + 2 * n \<longrightarrow> 
+                inv_locate_a (as, am) (n, Bk # aaa, xs) ires)"
+apply(auto)
+apply(subgoal_tac "n > 0")
+apply(subgoal_tac "(a - start_of ly as) div 2 = n - 1")
+apply(insert locate_b_2_a [of as am "n - 1" aaa xs], simp)
+apply(arith)
+apply(case_tac n, simp, simp)
+done
+
+lemma [simp]:  "inv_locate_b (as, am) (q, l, []) ires 
+           \<Longrightarrow>  inv_locate_b (as, am) (q, l, [Bk]) ires"
+apply(simp only: inv_locate_b.simps in_middle.simps)
+apply(erule exE)+
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = tn in exI, rule_tac x = m in exI, 
+      rule_tac x = ml in exI, rule_tac x = mr in exI)
+apply(auto)
+done
+
+lemma locate_b_2_locate_a_B[simp]: 
+ "\<lbrakk>\<not> a < start_of ly as; 
+   a < start_of ly as + 2 * n; 
+   (a - start_of ly as) mod 2 = Suc 0; 
+   inv_locate_b (as, am) ((a - start_of ly as) div 2, aaa, []) ires\<rbrakk>
+   \<Longrightarrow> (Suc a < start_of ly as + 2 * n \<longrightarrow> 
+     inv_locate_a (as, am) 
+            (Suc ((a - start_of ly as) div 2), Bk # aaa, []) ires) 
+    \<and> (\<not> Suc a < start_of ly as + 2 * n \<longrightarrow> 
+                  inv_locate_a (as, am) (n, Bk # aaa, []) ires)"
+apply(insert locate_b_2_locate_a [of a ly as n am aaa "[]"], simp)
+done
+
+(*inv: from locate_b to after_write*)
+lemma inv_locate_b_2_after_write[simp]: 
+      "inv_locate_b (as, am) (n, aaa, Bk # xs) ires
+      \<Longrightarrow> inv_after_write (as, abc_lm_s am n (Suc (abc_lm_v am n)))
+          (Suc (Suc (2 * n)), aaa, Oc # xs) ires"
+apply(auto simp: in_middle.simps inv_after_write.simps 
+                 abc_lm_v.simps abc_lm_s.simps  inv_locate_b.simps)
+apply(subgoal_tac [!] "mr = 0", auto simp: exponent_def split: if_splits)
+apply(subgoal_tac "lm2 = []", simp)
+apply(rule_tac x = rn in exI, rule_tac x = "Suc m" in exI,
+      rule_tac x = "lm1" in exI, simp, rule_tac x = "[]" in exI, simp)
+apply(case_tac "Suc (length lm1) - length am", simp, simp only: rep_ind, simp)
+apply(subgoal_tac "length lm1 - length am = nat", simp, arith)
+apply(drule_tac length_equal, simp)
+done
+
+lemma [simp]: "inv_locate_b (as, am) (n, aaa, []) ires \<Longrightarrow> 
+     inv_after_write (as, abc_lm_s am n (Suc (abc_lm_v am n))) 
+                     (Suc (Suc (2 * n)), aaa, [Oc]) ires"
+apply(insert inv_locate_b_2_after_write [of as am n aaa "[]"])
+by(simp)
+
+(*inv: from after_write to after_move*)
+lemma [simp]: "inv_after_write (as, lm) (Suc (Suc (2 * n)), l, Oc # r) ires
+                \<Longrightarrow> inv_after_move (as, lm) (2 * n + 3, Oc # l, r) ires"
+apply(auto simp:inv_after_move.simps inv_after_write.simps split: if_splits)
+done
+
+lemma [simp]: "inv_after_write (as, abc_lm_s am n (Suc (abc_lm_v am n)
+                )) (Suc (Suc (2 * n)), aaa, Bk # xs) ires = False"
+apply(simp add: inv_after_write.simps )
+done
+
+lemma [simp]: 
+ "inv_after_write (as, abc_lm_s am n (Suc (abc_lm_v am n))) 
+                        (Suc (Suc (2 * n)), aaa, []) ires = False"
+apply(simp add: inv_after_write.simps )
+done
+
+(*inv: from after_move to after_clear*)
+lemma [simp]: "inv_after_move (as, lm) (s, l, Oc # r) ires
+                \<Longrightarrow> inv_after_clear (as, lm) (s', l, Bk # r) ires"
+apply(auto simp: inv_after_move.simps inv_after_clear.simps split: if_splits)
+done
+
+(*inv: from after_move to on_leftmoving*)
+lemma inv_after_move_2_inv_on_left_moving[simp]:  
+   "inv_after_move (as, lm) (s, l, Bk # r) ires
+   \<Longrightarrow> (l = [] \<longrightarrow> 
+         inv_on_left_moving (as, lm) (s', [], Bk # Bk # r) ires) \<and>
+      (l \<noteq> [] \<longrightarrow> 
+         inv_on_left_moving (as, lm) (s', tl l, hd l # Bk # r) ires)"
+apply(simp only: inv_after_move.simps inv_on_left_moving.simps)
+apply(subgoal_tac "l \<noteq> []", rule conjI, simp, rule impI, 
+                rule disjI1, simp only: inv_on_left_moving_norm.simps)
+apply(erule exE)+
+apply(subgoal_tac "lm2 = []")
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,  
+    rule_tac x = m in exI, rule_tac x = m in exI, 
+    rule_tac x = 1 in exI,  
+    rule_tac x = "rn - 1" in exI, simp, case_tac rn)
+apply(auto simp: exponent_def  intro: BkCons_nil split: if_splits)
+done
+
+lemma [elim]: "[] = <lm::nat list> \<Longrightarrow> lm = []"
+using tape_of_nl_nil_eq[of lm]
+by simp
+
+lemma inv_after_move_2_inv_on_left_moving_B[simp]: 
+    "inv_after_move (as, lm) (s, l, []) ires
+      \<Longrightarrow> (l = [] \<longrightarrow> inv_on_left_moving (as, lm) (s', [], [Bk]) ires) \<and>
+          (l \<noteq> [] \<longrightarrow> inv_on_left_moving (as, lm) (s', tl l, [hd l]) ires)"
+apply(simp only: inv_after_move.simps inv_on_left_moving.simps)
+apply(subgoal_tac "l \<noteq> []", rule conjI, simp, rule impI, rule disjI1,
+        simp only: inv_on_left_moving_norm.simps)
+apply(erule exE)+
+apply(subgoal_tac "lm2 = []")
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,  
+      rule_tac x = m in exI, rule_tac x = m in exI, 
+      rule_tac x = 1 in exI, rule_tac x = "rn - 1" in exI, simp, case_tac rn)
+apply(auto simp: exponent_def  tape_of_nl_nil_eq  intro: BkCons_nil  split: if_splits)
+done
+
+(*inv: from after_clear to on_right_moving*)
+lemma [simp]: "Oc # r = replicate rn Bk = False"
+apply(case_tac rn, simp, simp)
+done
+
+lemma inv_after_clear_2_inv_on_right_moving[simp]: 
+     "inv_after_clear (as, lm) (2 * n + 4, l, Bk # r) ires
+      \<Longrightarrow> inv_on_right_moving (as, lm) (2 * n + 5, Bk # l, r) ires"
+apply(auto simp: inv_after_clear.simps inv_on_right_moving.simps )
+apply(subgoal_tac "lm2 \<noteq> []")
+apply(rule_tac x = "lm1 @ [m]" in exI, rule_tac x = "tl lm2" in exI, 
+      rule_tac x = "hd lm2" in exI, simp)
+apply(rule_tac x = 0 in exI, rule_tac x = "hd lm2" in exI)
+apply(simp add: exponent_def, rule conjI)
+apply(case_tac [!] "lm2::nat list", auto simp: exponent_def)
+apply(case_tac rn, auto split: if_splits simp: tape_of_nat_def)
+apply(case_tac list, 
+     simp add:  tape_of_nl_abv tape_of_nat_list.simps exponent_def)
+apply(erule_tac x = "rn - 1" in allE, 
+      case_tac rn, auto simp: exponent_def)
+apply(case_tac list, 
+     simp add:  tape_of_nl_abv tape_of_nat_list.simps exponent_def)
+apply(erule_tac x = "rn - 1" in allE, 
+      case_tac rn, auto simp: exponent_def)
+done
+
+
+lemma [simp]: "inv_after_clear (as, lm) (2 * n + 4, l, []) ires\<Longrightarrow> 
+               inv_after_clear (as, lm) (2 * n + 4, l, [Bk]) ires" 
+by(auto simp: inv_after_clear.simps)
+
+lemma [simp]: "inv_after_clear (as, lm) (2 * n + 4, l, []) ires
+             \<Longrightarrow> inv_on_right_moving (as, lm) (2 * n + 5, Bk # l, []) ires"
+by(insert 
+    inv_after_clear_2_inv_on_right_moving[of as lm n l "[]"], simp)
+
+(*inv: from on_right_moving to on_right_movign*)
+lemma [simp]: "inv_on_right_moving (as, lm) (2 * n + 5, l, Oc # r) ires
+      \<Longrightarrow> inv_on_right_moving (as, lm) (2 * n + 5, Oc # l, r) ires"
+apply(auto simp: inv_on_right_moving.simps)
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+           rule_tac x = "ml + mr" in exI, simp)
+apply(rule_tac x = "Suc ml" in exI, 
+           rule_tac x = "mr - 1" in exI, simp)
+apply(case_tac mr, auto simp: exponent_def )
+apply(rule_tac x = lm1 in exI, rule_tac x = "[]" in exI, 
+      rule_tac x = "ml + mr" in exI, simp)
+apply(rule_tac x = "Suc ml" in exI, 
+      rule_tac x = "mr - 1" in exI, simp)
+apply(case_tac mr, auto split: if_splits simp: exponent_def)
+done
+
+lemma inv_on_right_moving_2_inv_on_right_moving[simp]: 
+     "inv_on_right_moving (as, lm) (2 * n + 5, l, Bk # r) ires
+     \<Longrightarrow> inv_after_write (as, lm) (Suc (Suc (2 * n)), l, Oc # r) ires"
+apply(auto simp: inv_on_right_moving.simps inv_after_write.simps )
+apply(case_tac mr, auto simp: exponent_def split: if_splits)
+apply(case_tac [!] mr, simp_all)
+done
+      
+lemma [simp]: "inv_on_right_moving (as, lm) (2 * n + 5, l, []) ires\<Longrightarrow> 
+             inv_on_right_moving (as, lm) (2 * n + 5, l, [Bk]) ires"
+apply(auto simp: inv_on_right_moving.simps exponent_def)
+apply(rule_tac x = lm1 in exI, rule_tac x = "[]" in exI, simp)
+apply (rule_tac x = m in exI, auto split: if_splits simp: exponent_def)
+done
+
+(*inv: from on_right_moving to after_write*)
+lemma [simp]: "inv_on_right_moving (as, lm) (2 * n + 5, l, []) ires
+       \<Longrightarrow> inv_after_write (as, lm) (Suc (Suc (2 * n)), l, [Oc]) ires"
+apply(rule_tac inv_on_right_moving_2_inv_on_right_moving, simp)
+done
+
+(*inv: from on_left_moving to on_left_moving*)
+lemma [simp]: "inv_on_left_moving_in_middle_B (as, lm) 
+               (s, l, Oc # r) ires = False"
+apply(auto simp: inv_on_left_moving_in_middle_B.simps )
+done
+
+lemma [simp]: "inv_on_left_moving_norm (as, lm) (s, l, Bk # r) ires 
+             = False"
+apply(auto simp: inv_on_left_moving_norm.simps)
+apply(case_tac [!] mr, auto simp: )
+done
+
+lemma [intro]: "\<exists>rna. Oc # Oc\<^bsup>m\<^esup> @ Bk # <lm> @ Bk\<^bsup>rn\<^esup> = <m # lm> @ Bk\<^bsup>rna\<^esup>"
+apply(case_tac lm, simp add: tape_of_nl_abv tape_of_nat_list.simps)
+apply(rule_tac x = "Suc rn" in exI, simp)
+apply(case_tac list, simp_all add: tape_of_nl_abv tape_of_nat_list.simps, auto)
+done
+
+
+lemma [simp]: 
+  "\<lbrakk>inv_on_left_moving_norm (as, lm) (s, l, Oc # r) ires;
+    hd l = Bk; l \<noteq> []\<rbrakk> \<Longrightarrow> 
+     inv_on_left_moving_in_middle_B (as, lm) (s, tl l, Bk # Oc # r) ires"
+apply(case_tac l, simp, simp)
+apply(simp only: inv_on_left_moving_norm.simps 
+                 inv_on_left_moving_in_middle_B.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = lm1 in exI, rule_tac x = "m # lm2" in exI, auto)
+apply(case_tac [!] ml, auto)
+apply(rule_tac [!] x = 0 in exI, simp_all add: tape_of_nl_abv tape_of_nat_list.simps)
+done
+
+lemma [simp]: "\<lbrakk>inv_on_left_moving_norm (as, lm) (s, l, Oc # r) ires; 
+                hd l = Oc; l \<noteq> []\<rbrakk>
+            \<Longrightarrow> inv_on_left_moving_norm (as, lm) 
+                                        (s, tl l, Oc # Oc # r) ires"
+apply(simp only: inv_on_left_moving_norm.simps)
+apply(erule exE)+
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = m in exI, rule_tac x = "ml - 1" in exI,
+      rule_tac x = "Suc mr" in exI, rule_tac x = rn in exI, simp)
+apply(case_tac ml, auto simp: exponent_def split: if_splits)
+done
+
+lemma [simp]: "inv_on_left_moving_norm (as, lm) (s, [], Oc # r) ires
+     \<Longrightarrow> inv_on_left_moving_in_middle_B (as, lm) (s, [], Bk # Oc # r) ires"
+apply(auto simp: inv_on_left_moving_norm.simps 
+                 inv_on_left_moving_in_middle_B.simps split: if_splits)
+done
+
+lemma [simp]:"inv_on_left_moving (as, lm) (s, l, Oc # r) ires
+    \<Longrightarrow> (l = [] \<longrightarrow> inv_on_left_moving (as, lm) (s, [], Bk # Oc # r) ires)
+ \<and>  (l \<noteq> [] \<longrightarrow> inv_on_left_moving (as, lm) (s, tl l, hd l # Oc # r) ires)"
+apply(simp add: inv_on_left_moving.simps)
+apply(case_tac "l \<noteq> []", rule conjI, simp, simp)
+apply(case_tac "hd l", simp, simp, simp)
+done
+
+(*inv: from on_left_moving to check_left_moving*)
+lemma [simp]: "inv_on_left_moving_in_middle_B (as, lm) 
+                                      (s, Bk # list, Bk # r) ires
+          \<Longrightarrow> inv_check_left_moving_on_leftmost (as, lm) 
+                                      (s', list, Bk # Bk # r) ires"
+apply(auto simp: inv_on_left_moving_in_middle_B.simps 
+                 inv_check_left_moving_on_leftmost.simps split: if_splits)
+apply(case_tac [!] "rev lm1", simp_all)
+apply(case_tac [!] lista, simp_all add: tape_of_nl_abv tape_of_nat_list.simps)
+done
+
+lemma [simp]:
+    "inv_check_left_moving_in_middle (as, lm) (s, l, Bk # r) ires= False"
+by(auto simp: inv_check_left_moving_in_middle.simps )
+
+lemma [simp]: 
+ "inv_on_left_moving_in_middle_B (as, lm) (s, [], Bk # r) ires\<Longrightarrow> 
+  inv_check_left_moving_on_leftmost (as, lm) (s', [], Bk # Bk # r) ires"
+apply(auto simp: inv_on_left_moving_in_middle_B.simps 
+                 inv_check_left_moving_on_leftmost.simps split: if_splits)
+done
+
+
+lemma [simp]: "inv_check_left_moving_on_leftmost (as, lm) 
+                                       (s, list, Oc # r) ires= False"
+by(auto simp: inv_check_left_moving_on_leftmost.simps split: if_splits)
+
+lemma [simp]: "inv_on_left_moving_in_middle_B (as, lm) 
+                                         (s, Oc # list, Bk # r) ires
+ \<Longrightarrow> inv_check_left_moving_in_middle (as, lm) (s', list, Oc # Bk # r) ires"
+apply(auto simp: inv_on_left_moving_in_middle_B.simps 
+                 inv_check_left_moving_in_middle.simps  split: if_splits)
+done
+
+lemma inv_on_left_moving_2_check_left_moving[simp]:
+ "inv_on_left_moving (as, lm) (s, l, Bk # r) ires
+ \<Longrightarrow> (l = [] \<longrightarrow> inv_check_left_moving (as, lm) (s', [], Bk # Bk # r) ires)
+ \<and> (l \<noteq> [] \<longrightarrow> 
+      inv_check_left_moving (as, lm) (s', tl l, hd l # Bk # r) ires)"
+apply(simp add: inv_on_left_moving.simps inv_check_left_moving.simps)
+apply(case_tac l, simp, simp)
+apply(case_tac a, simp, simp)
+done
+
+lemma [simp]: "inv_on_left_moving_norm (as, lm) (s, l, []) ires = False"
+apply(auto simp: inv_on_left_moving_norm.simps)
+by(case_tac [!] mr, auto)
+
+lemma [simp]: "inv_on_left_moving (as, lm) (s, l, []) ires\<Longrightarrow> 
+     inv_on_left_moving (as, lm) (6 + 2 * n, l, [Bk]) ires"
+apply(simp add: inv_on_left_moving.simps)
+apply(auto simp: inv_on_left_moving_in_middle_B.simps)
+done
+
+lemma [simp]: "inv_on_left_moving (as, lm) (s, l, []) ires = False"
+apply(simp add: inv_on_left_moving.simps)
+apply(simp add: inv_on_left_moving_in_middle_B.simps)
+done
+
+lemma [simp]: "inv_on_left_moving (as, lm) (s, l, []) ires
+ \<Longrightarrow> (l = [] \<longrightarrow> inv_check_left_moving (as, lm) (s', [], [Bk]) ires) \<and>
+    (l \<noteq> [] \<longrightarrow> inv_check_left_moving (as, lm) (s', tl l, [hd l]) ires)"
+by simp
+
+lemma Oc_Bk_Cons_ex[simp]: 
+ "Oc # Bk # list = <lm::nat list> @ Bk\<^bsup>ln\<^esup> \<Longrightarrow> 
+                             \<exists>ln. list = <tl (lm)> @ Bk\<^bsup>ln\<^esup>"
+apply(case_tac "lm", simp)
+apply(case_tac ln, simp_all add: exponent_def)
+apply(case_tac lista, 
+      auto simp: tape_of_nl_abv tape_of_nat_list.simps exponent_def)
+apply(case_tac [!] a, auto simp: )
+apply(case_tac ln, simp, rule_tac x = nat in exI, simp)
+done
+
+lemma [simp]:
+  "Oc # Bk # list = <rev lm1::nat list> @ Bk\<^bsup>ln\<^esup> \<Longrightarrow> 
+      \<exists>rna. Oc # Bk # <lm2> @ Bk\<^bsup>rn\<^esup> = <hd (rev lm1) # lm2> @ Bk\<^bsup>rna\<^esup>"
+apply(frule Oc_Bk_Cons, simp)
+apply(case_tac lm2, 
+     auto simp: tape_of_nl_abv tape_of_nat_list.simps  exponent_def )
+apply(rule_tac x = "Suc rn" in exI, simp)
+done
+
+(*inv: from check_left_moving to on_left_moving*)
+lemma [intro]: "\<exists>rna. a # a\<^bsup>rn\<^esup> = a\<^bsup>rna\<^esup>"
+apply(rule_tac x = "Suc rn" in exI, simp)
+done
+
+lemma 
+inv_check_left_moving_in_middle_2_on_left_moving_in_middle_B[simp]:
+"inv_check_left_moving_in_middle (as, lm) (s, Bk # list, Oc # r) ires
+  \<Longrightarrow> inv_on_left_moving_in_middle_B (as, lm) (s', list, Bk # Oc # r) ires"
+apply(simp only: inv_check_left_moving_in_middle.simps 
+                 inv_on_left_moving_in_middle_B.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = "rev (tl (rev lm1))" in exI, 
+      rule_tac x = "[hd (rev lm1)] @ lm2" in exI, auto)
+apply(case_tac [!] "rev lm1",simp_all add: tape_of_nl_abv tape_of_nat_list.simps)
+apply(case_tac [!] a, simp_all)
+apply(case_tac [1] lm2, simp_all add: tape_of_nat_list.simps, auto)
+apply(case_tac [3] lm2, simp_all add: tape_of_nat_list.simps, auto)
+apply(case_tac [!] lista, simp_all add: tape_of_nat_list.simps)
+done
+
+lemma [simp]: 
+ "inv_check_left_moving_in_middle (as, lm) (s, [], Oc # r) ires\<Longrightarrow>
+     inv_check_left_moving_in_middle (as, lm) (s', [Bk], Oc # r) ires"
+apply(auto simp: inv_check_left_moving_in_middle.simps )
+done
+
+lemma [simp]: 
+ "inv_check_left_moving_in_middle (as, lm) (s, [], Oc # r) ires
+   \<Longrightarrow> inv_on_left_moving_in_middle_B (as, lm) (s', [], Bk # Oc # r) ires"
+apply(insert 
+inv_check_left_moving_in_middle_2_on_left_moving_in_middle_B[of 
+                  as lm n "[]" r], simp)
+done 
+
+lemma [simp]: "a\<^bsup>0\<^esup> = []"
+apply(simp add: exponent_def)
+done
+
+lemma [simp]: "inv_check_left_moving_in_middle (as, lm) 
+                       (s, Oc # list, Oc # r) ires
+   \<Longrightarrow> inv_on_left_moving_norm (as, lm) (s', list, Oc # Oc # r) ires"
+apply(auto simp: inv_check_left_moving_in_middle.simps 
+                 inv_on_left_moving_norm.simps)
+apply(rule_tac x = "rev (tl (rev lm1))" in exI, 
+      rule_tac x = lm2 in exI, rule_tac x = "hd (rev lm1)" in exI)
+apply(rule_tac conjI)
+apply(case_tac "rev lm1", simp, simp)
+apply(rule_tac x = "hd (rev lm1) - 1" in exI, auto)
+apply(rule_tac [!] x = "Suc (Suc 0)" in exI, simp)
+apply(case_tac [!] "rev lm1", simp_all)
+apply(case_tac [!] a, simp_all add: tape_of_nl_abv tape_of_nat_list.simps, auto)
+done 
+
+lemma [simp]: "inv_check_left_moving (as, lm) (s, l, Oc # r) ires
+\<Longrightarrow> (l = [] \<longrightarrow> inv_on_left_moving (as, lm) (s', [], Bk # Oc # r) ires) \<and>
+   (l \<noteq> [] \<longrightarrow> inv_on_left_moving (as, lm) (s', tl l, hd l # Oc # r) ires)"
+apply(case_tac l, 
+      auto simp: inv_check_left_moving.simps inv_on_left_moving.simps)
+apply(case_tac a, simp, simp)
+done
+
+(*inv: check_left_moving to after_left_moving*)
+lemma [simp]: "inv_check_left_moving (as, lm) (s, l, Bk # r) ires
+                \<Longrightarrow> inv_after_left_moving (as, lm) (s', Bk # l, r) ires"
+apply(auto simp: inv_check_left_moving.simps 
+ inv_check_left_moving_on_leftmost.simps inv_after_left_moving.simps)
+done
+
+
+lemma [simp]:"inv_check_left_moving (as, lm) (s, l, []) ires
+      \<Longrightarrow> inv_after_left_moving (as, lm) (s', Bk # l, []) ires"
+by(simp add: inv_check_left_moving.simps  
+inv_check_left_moving_in_middle.simps 
+inv_check_left_moving_on_leftmost.simps)
+
+(*inv: after_left_moving to inv_stop*)
+lemma [simp]: "inv_after_left_moving (as, lm) (s, l, Bk # r) ires
+       \<Longrightarrow> inv_stop (as, lm) (s', Bk # l, r) ires"
+apply(auto simp: inv_after_left_moving.simps inv_stop.simps)
+done
+
+lemma [simp]: "inv_after_left_moving (as, lm) (s, l, []) ires
+             \<Longrightarrow> inv_stop (as, lm) (s', Bk # l, []) ires"
+by(auto simp: inv_after_left_moving.simps)
+
+(*inv: stop to stop*)
+lemma [simp]: "inv_stop (as, lm) (x, l, r) ires \<Longrightarrow> 
+               inv_stop (as, lm) (y, l, r) ires"
+apply(simp add: inv_stop.simps)
+done
+
+lemma [simp]: "inv_after_clear (as, lm) (s, aaa, Oc # xs) ires= False"
+apply(auto simp: inv_after_clear.simps )
+done
+
+lemma [simp]: 
+  "inv_after_left_moving (as, lm) (s, aaa, Oc # xs) ires = False"
+by(auto simp: inv_after_left_moving.simps  )
+
+lemma start_of_not0: "as \<noteq> 0 \<Longrightarrow> start_of ly as > 0"
+apply(rule startof_not0)
+done
+
+text {*
+  The single step currectness of the TM complied from Abacus instruction @{text "Inc n"}.
+  It shows every single step execution of this TM keeps the invariant.
+*}
+
+lemma inc_inv_step: 
+  assumes 
+  -- {* Invariant holds on the start *}
+      h11: "inc_inv ly n (as, am) tc ires" 
+  -- {* The layout of Abacus program @{text "aprog"} is @{text "ly"} *}
+  and h12: "ly = layout_of aprog" 
+  -- {* The instruction at position @{text "as"} is @{text "Inc n"} *}
+  and h21: "abc_fetch as aprog = Some (Inc n)" 
+  -- {* TM not yet reach the final state, where @{text "start_of ly as + 2*n + 9"} is the state
+        where the current TM stops and the next TM starts. *}
+  and h22: "(\<lambda> (s, l, r). s \<noteq> start_of ly as + 2*n + 9) tc"
+  shows 
+  -- {* 
+  Single step execution of the TM keeps the invaraint, where 
+  the TM compiled from @{text "Inc n"} is @{text "(ci ly (start_of ly as) (Inc n))"}
+  @{text "start_of ly as - Suc 0)"} is the offset used to execute this {\em shifted}
+  TM.
+  *}
+  "inc_inv ly n (as, am) (t_step tc (ci ly (start_of ly as) (Inc n), start_of ly as - Suc 0)) ires"
+proof -
+  from h21 h22  have h3 : "start_of (layout_of aprog) as > 0"
+    apply(case_tac as, simp add: start_of.simps abc_fetch.simps)
+    apply(insert start_of_not0[of as "layout_of aprog"], simp)
+    done    
+  from h11 h12 and h21 h22 and this show ?thesis 
+    apply(case_tac tc, simp)
+    apply(case_tac "a = 0", 
+      auto split:if_splits simp add:t_step.simps,
+      tactic {* ALLGOALS (resolve_tac [@{thm fetch_intro}]) *})
+    apply (simp_all add:fetch_simps new_tape.simps)
+    done
+qed
+
+
+lemma t_steps_ind: "t_steps tc (p, off) (Suc n)
+                 = t_step (t_steps tc (p, off) n) (p, off)"
+apply(induct n arbitrary: tc)
+apply(simp add: t_steps.simps)
+apply(simp add: t_steps.simps)
+done
+
+definition lex_pair :: "((nat \<times> nat) \<times> (nat \<times> nat)) set"
+  where 
+  "lex_pair \<equiv> less_than <*lex*> less_than"
+
+definition lex_triple :: 
+   "((nat \<times> (nat \<times> nat)) \<times> (nat \<times> (nat \<times> nat))) set"
+  where "lex_triple \<equiv> less_than <*lex*> lex_pair"
+
+definition lex_square :: 
+    "((nat \<times> nat \<times> nat \<times> nat) \<times> (nat \<times> nat \<times> nat \<times> nat)) set"
+  where "lex_square \<equiv> less_than <*lex*> lex_triple"
+
+fun abc_inc_stage1 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "abc_inc_stage1 (s, l, r) ss n = 
+            (if s = 0 then 0
+             else if s \<le> ss+2*n+1 then 5
+             else if s\<le> ss+2*n+5 then 4
+             else if s \<le> ss+2*n+7 then 3
+             else if s = ss+2*n+8 then 2
+             else 1)"
+
+fun abc_inc_stage2 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "abc_inc_stage2 (s, l, r) ss n =
+                (if s \<le> ss + 2*n + 1 then 0
+                 else if s = ss + 2*n + 2 then length r
+                 else if s = ss + 2*n + 3 then length r
+                 else if s = ss + 2*n + 4 then length r
+                 else if s = ss + 2*n + 5 then 
+                                  if r \<noteq> [] then length r
+                                  else 1
+                 else if s = ss+2*n+6 then length l
+                 else if s = ss+2*n+7 then length l
+                 else 0)"
+
+fun abc_inc_stage3 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> block list \<Rightarrow>  nat"
+  where
+  "abc_inc_stage3 (s, l, r) ss n ires = (
+              if s = ss + 2*n + 3 then 4
+              else if s = ss + 2*n + 4 then 3
+              else if s = ss + 2*n + 5 then 
+                   if r \<noteq> [] \<and> hd r = Oc then 2
+                   else 1
+              else if s = ss + 2*n + 2 then 0
+              else if s = ss + 2*n + 6 then 
+                      if l = Bk # ires \<and> r \<noteq> [] \<and>  hd r = Oc then 2
+                      else 1
+              else if s = ss + 2*n + 7 then 
+                      if r \<noteq> [] \<and> hd r = Oc then 3
+                      else 0
+              else ss+2*n+9 - s)"
+
+fun abc_inc_stage4 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> block list \<Rightarrow> nat"
+  where
+  "abc_inc_stage4 (s, l, r) ss n ires = 
+            (if s \<le> ss+2*n+1 \<and> (s - ss) mod 2 = 0 then 
+                if (r\<noteq>[] \<and> hd r = Oc) then 0
+                else 1
+             else if (s \<le> ss+2*n+1 \<and> (s - ss) mod 2 = Suc 0) 
+                                                 then length r
+             else if s = ss + 2*n + 6 then 
+                  if l = Bk # ires \<and> hd r = Bk then 0
+                  else Suc (length l)
+             else 0)"
+ 
+fun abc_inc_measure :: "(t_conf \<times> nat \<times> nat \<times> block list) \<Rightarrow> 
+                        (nat \<times> nat \<times> nat \<times> nat)"
+  where
+  "abc_inc_measure (c, ss, n, ires) = 
+     (abc_inc_stage1 c ss n, abc_inc_stage2 c ss n, 
+      abc_inc_stage3 c ss n ires, abc_inc_stage4 c ss n ires)"
+
+definition abc_inc_LE :: "(((nat \<times> block list \<times> block list) \<times> nat \<times> 
+       nat \<times> block list) \<times> ((nat \<times> block list \<times> block list) \<times> nat \<times> nat \<times> block list)) set"
+  where "abc_inc_LE \<equiv> (inv_image lex_square abc_inc_measure)"
+
+lemma wf_lex_triple: "wf lex_triple"
+by (auto intro:wf_lex_prod simp:lex_triple_def lex_pair_def)
+
+lemma wf_lex_square: "wf lex_square"
+by (auto intro:wf_lex_triple simp:lex_triple_def lex_square_def lex_pair_def)
+
+lemma wf_abc_inc_le[intro]: "wf abc_inc_LE"
+by(auto intro:wf_inv_image wf_lex_square simp:abc_inc_LE_def)
+
+(********************************************************************)
+declare inc_inv.simps[simp del]
+
+lemma halt_lemma2': 
+  "\<lbrakk>wf LE;  \<forall> n. ((\<not> P (f n) \<and> Q (f n)) \<longrightarrow> 
+    (Q (f (Suc n)) \<and> (f (Suc n), (f n)) \<in> LE)); Q (f 0)\<rbrakk> 
+      \<Longrightarrow> \<exists> n. P (f n)"
+apply(intro exCI, simp)
+apply(subgoal_tac "\<forall> n. Q (f n)", simp)
+apply(drule_tac f = f in wf_inv_image)
+apply(simp add: inv_image_def)
+apply(erule wf_induct, simp)
+apply(erule_tac x = x in allE)
+apply(erule_tac x = n in allE, erule_tac x = n in allE)
+apply(erule_tac x = "Suc x" in allE, simp)
+apply(rule_tac allI)
+apply(induct_tac n, simp)
+apply(erule_tac x = na in allE, simp)
+done
+
+lemma halt_lemma2'': 
+  "\<lbrakk>P (f n); \<not> P (f (0::nat))\<rbrakk> \<Longrightarrow> 
+         \<exists> n. (P (f n) \<and> (\<forall> i < n. \<not> P (f i)))"
+apply(induct n rule: nat_less_induct, auto)
+done
+
+lemma halt_lemma2''':
+ "\<lbrakk>\<forall>n. \<not> P (f n) \<and> Q (f n) \<longrightarrow> Q (f (Suc n)) \<and> (f (Suc n), f n) \<in> LE;
+                 Q (f 0);  \<forall>i<na. \<not> P (f i)\<rbrakk> \<Longrightarrow> Q (f na)"
+apply(induct na, simp, simp)
+done
+
+lemma halt_lemma2: 
+  "\<lbrakk>wf LE;  
+    \<forall> n. ((\<not> P (f n) \<and> Q (f n)) \<longrightarrow> (Q (f (Suc n)) \<and> (f (Suc n), (f n)) \<in> LE)); 
+    Q (f 0); \<not> P (f 0)\<rbrakk> 
+  \<Longrightarrow> \<exists> n. P (f n) \<and> Q (f n)"
+apply(insert halt_lemma2' [of LE P f Q], simp, erule_tac exE)
+apply(subgoal_tac "\<exists> n. (P (f n) \<and> (\<forall> i < n. \<not> P (f i)))")
+apply(erule_tac exE)+
+apply(rule_tac x = na in exI, auto)
+apply(rule halt_lemma2''', simp, simp, simp)
+apply(erule_tac halt_lemma2'', simp)
+done
+
+lemma [simp]: 
+  "\<lbrakk>ly = layout_of aprog; abc_fetch as aprog = Some (Inc n)\<rbrakk>
+    \<Longrightarrow> start_of ly (Suc as) = start_of ly as + 2*n +9"
+apply(case_tac as, auto simp: abc_fetch.simps start_of.simps 
+          layout_of.simps length_of.simps split: if_splits)
+done
+
+lemma inc_inv_init: 
+ "\<lbrakk>abc_fetch as aprog = Some (Inc n); 
+   crsp_l ly (as, am) (start_of ly as, l, r) ires; ly = layout_of aprog\<rbrakk>
+  \<Longrightarrow> inc_inv ly n (as, am) (start_of ly as, l, r) ires"
+apply(auto simp: crsp_l.simps inc_inv.simps 
+      inv_locate_a.simps at_begin_fst_bwtn.simps 
+      at_begin_fst_awtn.simps at_begin_norm.simps )
+apply(auto intro: startof_not0)
+done
+     
+lemma inc_inv_stop_pre[simp]: 
+   "\<lbrakk>ly = layout_of aprog; inc_inv ly n (as, am) (s, l, r) ires; 
+     s = start_of ly as; abc_fetch as aprog = Some (Inc n)\<rbrakk>
+    \<Longrightarrow>  (\<forall>na. \<not> (\<lambda>((s, l, r), ss, n', ires'). s = start_of ly (Suc as)) 
+         (t_steps (s, l, r) (ci ly (start_of ly as) 
+          (Inc n), start_of ly as - Suc 0) na, s, n, ires) \<and>
+       (\<lambda>((s, l, r), ss, n', ires'). inc_inv ly n (as, am) (s, l, r) ires')
+         (t_steps (s, l, r) (ci ly (start_of ly as) 
+             (Inc n), start_of ly as - Suc 0) na, s, n, ires) \<longrightarrow>
+       (\<lambda>((s, l, r), ss, n', ires'). inc_inv ly n (as, am) (s, l, r) ires') 
+      (t_steps (s, l, r) (ci ly (start_of ly as) 
+              (Inc n), start_of ly as - Suc 0) (Suc na), s, n, ires) \<and>
+     ((t_steps (s, l, r) (ci ly (start_of ly as) (Inc n), 
+        start_of ly as - Suc 0) (Suc na), s, n, ires), 
+      t_steps (s, l, r) (ci ly (start_of ly as) 
+        (Inc n), start_of ly as - Suc 0) na, s, n, ires) \<in> abc_inc_LE)"
+apply(rule allI, rule impI, simp add: t_steps_ind,
+       rule conjI, erule_tac conjE)
+apply(rule_tac inc_inv_step, simp, simp, simp)
+apply(case_tac "t_steps (start_of (layout_of aprog) as, l, r) (ci (layout_of aprog) 
+  (start_of (layout_of aprog) as) (Inc n), start_of (layout_of aprog) as - Suc 0) na", simp)
+proof -
+  fix na
+  assume h1: "abc_fetch as aprog = Some (Inc n)"
+    "\<not> (\<lambda>(s, l, r) (ss, n', ires'). s = start_of (layout_of aprog) as + 2 * n + 9)
+    (t_steps (start_of (layout_of aprog) as, l, r) (ci (layout_of aprog) 
+    (start_of (layout_of aprog) as) (Inc n), start_of (layout_of aprog) as - Suc 0) na) 
+    (start_of (layout_of aprog) as, n, ires) \<and>
+    inc_inv (layout_of aprog) n (as, am) (t_steps (start_of (layout_of aprog) as, l, r)
+    (ci (layout_of aprog) (start_of (layout_of aprog) as) (Inc n), start_of (layout_of aprog) as - Suc 0) na) ires"
+  from h1 have h2: "start_of (layout_of aprog) as > 0"
+    apply(rule_tac startof_not0)
+    done
+  from h1 and h2 show "((t_step (t_steps (start_of (layout_of aprog) as, l, r) (ci (layout_of aprog)
+    (start_of (layout_of aprog) as) (Inc n), start_of (layout_of aprog) as - Suc 0) na)
+    (ci (layout_of aprog) (start_of (layout_of aprog) as) (Inc n), start_of (layout_of aprog) as - Suc 0),
+    start_of (layout_of aprog) as, n, ires),
+    t_steps (start_of (layout_of aprog) as, l, r) 
+    (ci (layout_of aprog) (start_of (layout_of aprog) as) (Inc n), start_of (layout_of aprog) as - Suc 0) na, 
+    start_of (layout_of aprog) as, n, ires)
+            \<in> abc_inc_LE"
+    apply(case_tac "(t_steps (start_of (layout_of aprog) as, l, r) 
+               (ci (layout_of aprog)
+         (start_of (layout_of aprog) as) (Inc n), 
+           start_of (layout_of aprog) as - Suc 0) na)", simp)
+    apply(case_tac "a = 0", 
+     auto split:if_splits simp add:t_step.simps inc_inv.simps, 
+       tactic {* ALLGOALS (resolve_tac [@{thm fetch_intro}]) *})
+    apply(simp_all add:fetch_simps new_tape.simps)
+    apply(auto simp add: abc_inc_LE_def  
+    lex_square_def lex_triple_def lex_pair_def
+      inv_after_write.simps inv_after_move.simps inv_after_clear.simps
+      inv_on_left_moving.simps inv_on_left_moving_norm.simps split: if_splits)
+    done
+qed
+
+lemma inc_inv_stop_pre1: 
+  "\<lbrakk>
+  ly = layout_of aprog; 
+  abc_fetch as aprog = Some (Inc n);
+  s = start_of ly as; 
+  inc_inv ly n (as, am) (s, l, r) ires
+  \<rbrakk> \<Longrightarrow> 
+  (\<exists> stp > 0. (\<lambda> (s', l', r').
+           s' = start_of ly (Suc as) \<and> 
+           inc_inv ly n (as, am) (s', l', r') ires) 
+               (t_steps (s, l, r) (ci ly (start_of ly as) (Inc n), 
+                        start_of ly as - Suc 0) stp))"
+apply(insert halt_lemma2[of abc_inc_LE 
+    "\<lambda> ((s, l, r), ss, n', ires'). s = start_of ly (Suc as)" 
+    "(\<lambda> stp. (t_steps (s, l, r) 
+     (ci ly (start_of ly as) (Inc n), 
+     start_of ly as - Suc 0) stp, s, n, ires))" 
+    "\<lambda> ((s, l, r), ss, n'). inc_inv ly n (as, am) (s, l, r) ires"])
+apply(insert  wf_abc_inc_le)
+apply(insert inc_inv_stop_pre[of ly aprog n as am s l r ires], simp)
+apply(simp only: t_steps.simps, auto)
+apply(rule_tac x = na in exI)
+apply(case_tac "(t_steps (start_of (layout_of aprog) as, l, r) 
+   (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+   (Inc n), start_of (layout_of aprog) as - Suc 0) na)", simp)
+apply(case_tac na, simp add: t_steps.simps, simp)
+done
+
+lemma inc_inv_stop: 
+  assumes program_and_layout: 
+  -- {* There is an Abacus program @{text "aprog"} and its layout is @{text "ly"}: *}
+  "ly = layout_of aprog"
+  and an_instruction:
+  -- {* There is an instruction @{text "Inc n"} at postion @{text "as"} of @{text "aprog"} *}
+  "abc_fetch as aprog = Some (Inc n)"
+  and the_start_state:
+  -- {* According to @{text "ly"} and @{text "as"}, 
+        the start state of the TM compiled from this
+        @{text "Inc n"} instruction should be @{text "s"}:
+     *}
+  "s = start_of ly as"
+  and inv:
+  -- {* Invariant holds on configuration @{text "(s, l, r)"} *}
+  "inc_inv ly n (as, am) (s, l, r) ires"
+  shows  -- {* After @{text "stp"} steps of execution, the compiled 
+            TM reaches the start state of next compiled TM and the invariant 
+            still holds.
+            *}
+      "(\<exists> stp > 0. (\<lambda> (s', l', r').
+           s' = start_of ly (Suc as) \<and> 
+           inc_inv ly n (as, am) (s', l', r') ires) 
+               (t_steps (s, l, r) (ci ly (start_of ly as) (Inc n), 
+                        start_of ly as - Suc 0) stp))"
+proof -
+  from inc_inv_stop_pre1 [OF  program_and_layout an_instruction the_start_state inv] 
+  show ?thesis .
+qed
+
+lemma inc_inv_stop_cond: 
+  "\<lbrakk>ly = layout_of aprog; 
+    s' = start_of ly (as + 1); 
+    inc_inv ly n (as, lm) (s', (l', r')) ires; 
+    abc_fetch as aprog = Some (Inc n)\<rbrakk> \<Longrightarrow>
+    crsp_l ly (Suc as, abc_lm_s lm n (Suc (abc_lm_v lm n))) 
+                                                (s', l', r') ires"
+apply(subgoal_tac "s' = start_of ly as + 2*n + 9", simp)
+apply(auto simp: inc_inv.simps inv_stop.simps crsp_l.simps )
+done
+
+lemma inc_crsp_ex_pre:
+  "\<lbrakk>ly = layout_of aprog; 
+    crsp_l ly (as, am) tc ires;  
+    abc_fetch as aprog = Some (Inc n)\<rbrakk>
+ \<Longrightarrow> \<exists>stp > 0. crsp_l ly (abc_step_l (as, am) (Some (Inc n))) 
+                (t_steps tc (ci ly (start_of ly as) (Inc n), 
+                                start_of ly as - Suc 0) stp) ires"
+proof(case_tac tc, simp add: abc_step_l.simps)
+  fix a b c
+  assume h1: "ly = layout_of aprog" 
+         "crsp_l (layout_of aprog) (as, am) (a, b, c) ires"
+         "abc_fetch as aprog = Some (Inc n)"
+  hence h2: "a = start_of ly as"
+    by(auto simp: crsp_l.simps)
+  from h1 and h2 have h3: 
+       "inc_inv ly n (as, am) (start_of ly as, b, c) ires"
+    by(rule_tac inc_inv_init, simp, simp, simp)
+  from h1 and h2 and h3 have h4:
+       "(\<exists> stp > 0. (\<lambda> (s', l', r'). s' = 
+           start_of ly (Suc as) \<and> inc_inv ly n (as, am) (s', l', r') ires)
+         (t_steps (a, b, c) (ci ly (start_of ly as) 
+                 (Inc n), start_of ly as - Suc 0) stp))"
+    apply(rule_tac inc_inv_stop, auto)
+    done
+  from h1 and h2 and h3 and h4 show 
+     "\<exists>stp > 0. crsp_l (layout_of aprog) 
+        (Suc as, abc_lm_s am n (Suc (abc_lm_v am n)))
+       (t_steps (a, b, c) (ci (layout_of aprog) 
+          (start_of (layout_of aprog) as) (Inc n), 
+              start_of (layout_of aprog) as - Suc 0) stp) ires"
+    apply(erule_tac exE)
+    apply(rule_tac x = stp in exI)
+    apply(case_tac "(t_steps (a, b, c) (ci (layout_of aprog) 
+         (start_of (layout_of aprog) as) (Inc n), 
+             start_of (layout_of aprog) as - Suc 0) stp)", simp)
+    apply(rule_tac inc_inv_stop_cond, auto)
+    done
+qed
+
+text {*
+  The total correctness of the compilaton of @{text "Inc n"} instruction.
+*}
+
+lemma inc_crsp_ex:
+  assumes layout: 
+  -- {* For any Abacus program @{text "aprog"}, assuming its layout is @{text "ly"} *}
+  "ly = layout_of aprog"
+  and corresponds: 
+  -- {* Abacus configuration @{text "(as, am)"} is in correspondence with 
+        TM configuration @{text "tc"} *}
+  "crsp_l ly (as, am) tc ires"
+  and inc:
+  -- {* There is an instruction @{text "Inc n"} at postion @{text "as"} of @{text "aprog"} *}
+  "abc_fetch as aprog = Some (Inc n)"
+  shows
+  -- {* 
+  After @{text "stp"} steps of execution, the TM compiled from this @{text "Inc n"}
+  stops with a configuration which corresponds to the Abacus configuration obtained
+  from the execution of this @{text "Inc n"} instruction.
+  *} 
+  "\<exists>stp > 0. crsp_l ly (abc_step_l (as, am) (Some (Inc n))) 
+                       (t_steps tc (ci ly (start_of ly as) (Inc n), 
+                                start_of ly as - Suc 0) stp) ires"
+proof -
+  from inc_crsp_ex_pre [OF layout corresponds inc] show ?thesis .
+qed
+
+(*
+subsection {* The compilation of @{text "Dec n e"} *}
+*)
+
+text {*
+  The lemmas in this section lead to the correctness of the compilation 
+  of @{text "Dec n e"} instruction using the same techniques as 
+  @{text "Inc n"}.
+*}
+
+type_synonym dec_inv_t = "(nat * nat list) \<Rightarrow> t_conf \<Rightarrow> block list \<Rightarrow>  bool"
+
+fun dec_first_on_right_moving :: "nat \<Rightarrow> dec_inv_t"
+  where
+  "dec_first_on_right_moving n (as, lm) (s, l, r) ires = 
+               (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> 
+         ml + mr = Suc m \<and> length lm1 = n \<and> ml > 0 \<and> m > 0 \<and>
+             (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
+                          else  l = (Oc\<^bsup>ml\<^esup>) @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> 
+    ((r = (Oc\<^bsup>mr\<^esup>) @ [Bk] @ <lm2> @ (Bk\<^bsup>rn\<^esup>)) \<or> (r = (Oc\<^bsup>mr\<^esup>) \<and> lm2 = [])))"
+
+fun dec_on_right_moving :: "dec_inv_t"
+  where
+  "dec_on_right_moving (as, lm) (s, l, r) ires =  
+   (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> 
+                             ml + mr = Suc (Suc m) \<and>
+   (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
+                else  l = (Oc\<^bsup>ml\<^esup>) @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> 
+   ((r = (Oc\<^bsup>mr\<^esup>) @ [Bk] @ <lm2> @ (Bk\<^bsup>rn\<^esup>)) \<or> (r = (Oc\<^bsup>mr\<^esup>) \<and> lm2 = [])))"
+
+fun dec_after_clear :: "dec_inv_t"
+  where
+  "dec_after_clear (as, lm) (s, l, r) ires = 
+              (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> 
+                ml + mr = Suc m \<and> ml = Suc m \<and> r \<noteq> [] \<and> r \<noteq> [] \<and>
+               (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
+                            else l = (Oc\<^bsup>ml \<^esup>) @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> 
+               (tl r = Bk # <lm2> @ (Bk\<^bsup>rn\<^esup>) \<or> tl r = [] \<and> lm2 = []))"
+
+fun dec_after_write :: "dec_inv_t"
+  where
+  "dec_after_write (as, lm) (s, l, r) ires = 
+         (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> 
+       ml + mr = Suc m \<and> ml = Suc m \<and> lm2 \<noteq> [] \<and>
+       (if lm1 = [] then l = Bk # Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
+                    else l = Bk # (Oc\<^bsup>ml \<^esup>) @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> 
+       tl r = <lm2> @ (Bk\<^bsup>rn\<^esup>))"
+
+fun dec_right_move :: "dec_inv_t"
+  where
+  "dec_right_move (as, lm) (s, l, r) ires = 
+        (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 
+            \<and> ml = Suc m \<and> mr = (0::nat) \<and> 
+              (if lm1 = [] then l = Bk # Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
+                          else l = Bk # Oc\<^bsup>ml\<^esup>@ [Bk] @ <rev lm1> @ Bk # Bk # ires) 
+           \<and> (r = Bk # <lm2> @ Bk\<^bsup>rn\<^esup>\<or> r = [] \<and> lm2 = []))"
+
+fun dec_check_right_move :: "dec_inv_t"
+  where
+  "dec_check_right_move (as, lm) (s, l, r) ires = 
+        (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> 
+           ml = Suc m \<and> mr = (0::nat) \<and> 
+           (if lm1 = [] then l = Bk # Bk # Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
+                       else l = Bk # Bk # Oc\<^bsup>ml \<^esup>@ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> 
+           r = <lm2> @ Bk\<^bsup>rn\<^esup>)"
+
+fun dec_left_move :: "dec_inv_t"
+  where
+  "dec_left_move (as, lm) (s, l, r) ires = 
+    (\<exists> lm1 m rn. (lm::nat list) = lm1 @ [m::nat] \<and>   
+    rn > 0 \<and> 
+   (if lm1 = [] then l = Bk # Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # ires
+    else l = Bk # Oc\<^bsup>Suc m\<^esup> @ Bk # <rev lm1> @ Bk # Bk # ires) \<and> r = Bk\<^bsup>rn\<^esup>)"
+
+declare
+  dec_on_right_moving.simps[simp del] dec_after_clear.simps[simp del] 
+  dec_after_write.simps[simp del] dec_left_move.simps[simp del] 
+  dec_check_right_move.simps[simp del] dec_right_move.simps[simp del] 
+  dec_first_on_right_moving.simps[simp del]
+
+fun inv_locate_n_b :: "inc_inv_t"
+  where
+  "inv_locate_n_b (as, lm) (s, l, r) ires= 
+    (\<exists> lm1 lm2 tn m ml mr rn. lm @ 0\<^bsup>tn\<^esup> = lm1 @ [m] @ lm2 \<and> 
+     length lm1 = s \<and> m + 1 = ml + mr \<and> 
+     ml = 1 \<and> tn = s + 1 - length lm \<and>
+     (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
+      else l = Oc\<^bsup>ml\<^esup>@Bk#<rev lm1>@Bk#Bk#ires) \<and> 
+     (r = (Oc\<^bsup>mr\<^esup>) @ [Bk] @ <lm2>@ (Bk\<^bsup>rn\<^esup>) \<or> (lm2 = [] \<and> r = (Oc\<^bsup>mr\<^esup>)))
+  )"
+
+fun dec_inv_1 :: "layout \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> dec_inv_t"
+  where
+  "dec_inv_1 ly n e (as, am) (s, l, r) ires = 
+           (let ss = start_of ly as in
+            let am' = abc_lm_s am n (abc_lm_v am n - Suc 0) in
+            let am'' = abc_lm_s am n (abc_lm_v am n) in
+              if s = start_of ly e then  inv_stop (as, am'') (s, l, r) ires
+              else if s = ss then False
+              else if ss \<le> s \<and> s < ss + 2*n then
+                   if (s - ss) mod 2 = 0 then 
+                        inv_locate_a (as, am) ((s - ss) div 2, l, r) ires
+                    \<or> inv_locate_a (as, am'') ((s - ss) div 2, l, r) ires
+                   else 
+                     inv_locate_b (as, am) ((s - ss) div 2, l, r) ires
+                  \<or> inv_locate_b (as, am'') ((s - ss) div 2, l, r) ires
+              else if s = ss + 2 * n then 
+                  inv_locate_a (as, am) (n, l, r) ires
+                \<or> inv_locate_a (as, am'') (n, l, r) ires
+              else if s = ss + 2 * n + 1 then 
+                  inv_locate_b (as, am) (n, l, r) ires
+              else if s = ss + 2 * n + 13 then 
+                  inv_on_left_moving (as, am'') (s, l, r) ires
+              else if s = ss + 2 * n + 14 then 
+                  inv_check_left_moving (as, am'') (s, l, r) ires
+              else if s = ss + 2 * n + 15 then 
+                  inv_after_left_moving (as, am'') (s, l, r) ires
+              else False)"
+
+fun dec_inv_2 :: "layout \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> dec_inv_t"
+  where
+  "dec_inv_2 ly n e (as, am) (s, l, r) ires =
+           (let ss = start_of ly as in
+            let am' = abc_lm_s am n (abc_lm_v am n - Suc 0) in
+            let am'' = abc_lm_s am n (abc_lm_v am n) in
+              if s = 0 then False
+              else if s = ss then False
+              else if ss \<le> s \<and> s < ss + 2*n then
+                   if (s - ss) mod 2 = 0 then 
+                      inv_locate_a (as, am) ((s - ss) div 2, l, r) ires
+                   else inv_locate_b (as, am) ((s - ss) div 2, l, r) ires
+              else if s = ss + 2 * n then 
+                      inv_locate_a (as, am) (n, l, r) ires
+              else if s = ss + 2 * n + 1 then 
+                      inv_locate_n_b (as, am) (n, l, r) ires
+              else if s = ss + 2 * n + 2 then 
+                      dec_first_on_right_moving n (as, am'') (s, l, r) ires
+              else if s = ss + 2 * n + 3 then 
+                      dec_after_clear (as, am') (s, l, r) ires
+              else if s = ss + 2 * n + 4 then 
+                      dec_right_move (as, am') (s, l, r) ires
+              else if s = ss + 2 * n + 5 then 
+                      dec_check_right_move (as, am') (s, l, r) ires
+              else if s = ss + 2 * n + 6 then 
+                      dec_left_move (as, am') (s, l, r) ires
+              else if s = ss + 2 * n + 7 then 
+                      dec_after_write (as, am') (s, l, r) ires
+              else if s = ss + 2 * n + 8 then 
+                      dec_on_right_moving (as, am') (s, l, r) ires
+              else if s = ss + 2 * n + 9 then 
+                      dec_after_clear (as, am') (s, l, r) ires
+              else if s = ss + 2 * n + 10 then 
+                      inv_on_left_moving (as, am') (s, l, r) ires
+              else if s = ss + 2 * n + 11 then 
+                      inv_check_left_moving (as, am') (s, l, r) ires
+              else if s = ss + 2 * n + 12 then 
+                      inv_after_left_moving (as, am') (s, l, r) ires
+              else if s = ss + 2 * n + 16 then 
+                      inv_stop (as, am') (s, l, r) ires
+              else False)"
+
+(*begin: dec_fetch lemmas*)
+
+lemma dec_fetch_locate_a_o: 
+      "\<lbrakk>start_of ly as \<le> a;
+        a < start_of ly as + 2 * n; start_of ly as > 0;
+        a - start_of ly as = 2 * q\<rbrakk>
+       \<Longrightarrow> fetch (ci (layout_of aprog) 
+         (start_of ly as) (Dec n e)) (Suc (2 * q))  Oc = (R, a + 1)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append Suc_pre)
+apply(subgoal_tac "(findnth n ! Suc (4 * q)) = 
+                          findnth (Suc q) ! (4 * q + 1)")
+apply(simp add: findnth.simps nth_append)
+apply(subgoal_tac " findnth n !(4 * q + 1) = 
+                          findnth (Suc q) ! (4 * q + 1)", simp)
+apply(rule_tac findnth_nth, auto)
+done
+
+lemma  dec_fetch_locate_a_b:
+       "\<lbrakk>start_of ly as \<le> a; 
+         a < start_of ly as + 2 * n; 
+         start_of ly as > 0;
+         a - start_of ly as = 2 * q\<rbrakk>
+       \<Longrightarrow> fetch (ci (layout_of aprog) (start_of ly as) (Dec n e)) 
+              (Suc (2 * q))  Bk = (W1, a)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append)
+apply(subgoal_tac "(findnth n ! (4 * q)) = 
+                       findnth (Suc q) ! (4 * q )")
+apply(simp add: findnth.simps nth_append)
+apply(subgoal_tac " findnth n !(4 * q + 0) = 
+                       findnth (Suc q) ! (4 * q + 0)", simp)
+apply(rule_tac findnth_nth, auto)
+done
+
+lemma dec_fetch_locate_b_o:
+      "\<lbrakk>start_of ly as \<le> a; 
+        a < start_of ly as + 2 * n; 
+        (a - start_of ly as) mod 2 = Suc 0; 
+        start_of ly as> 0\<rbrakk>
+       \<Longrightarrow> fetch (ci (layout_of aprog) (start_of ly as) (Dec n e)) 
+                       (Suc (a - start_of ly as)) Oc = (R, a)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append)
+apply(subgoal_tac "\<exists> q. (a - start_of ly as) = 2 * q + 1", auto)
+apply(subgoal_tac "(findnth n ! Suc (Suc (Suc (4 * q)))) = 
+                                findnth (Suc q) ! (4 * q + 3)")
+apply(simp add: findnth.simps nth_append)
+apply(subgoal_tac " findnth n ! (4 * q + 3) = 
+                 findnth (Suc q) ! (4 * q + 3)", simp add: add3_Suc)
+apply(rule_tac findnth_nth, auto)
+done
+
+lemma dec_fetch_locate_b_b: 
+      "\<lbrakk>\<not> a < start_of ly as; 
+        a < start_of ly as + 2 * n; 
+       (a - start_of ly as) mod 2 = Suc 0; 
+        start_of ly as > 0\<rbrakk>
+       \<Longrightarrow> fetch (ci (layout_of aprog) (start_of ly as) (Dec n e)) 
+              (Suc (a - start_of ly as))  Bk = (R, a + 1)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append)
+apply(subgoal_tac "\<exists> q. (a - start_of ly as) = 2 * q + 1", auto)
+apply(subgoal_tac "(findnth n ! Suc ((Suc (4 * q)))) = 
+                          findnth (Suc q) ! (4 * q + 2)")
+apply(simp add: findnth.simps nth_append)
+apply(subgoal_tac " findnth n ! (4 * q + 2) = 
+                          findnth (Suc q) ! (4 * q + 2)", simp)
+apply(rule_tac findnth_nth, auto)
+done
+
+lemma dec_fetch_locate_n_a_o: 
+       "start_of ly as > 0 \<Longrightarrow> fetch (ci (layout_of aprog) 
+                       (start_of ly as) (Dec n e)) (Suc (2 * n))  Oc
+       = (R, start_of ly as + 2*n + 1)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tdec_b_def)
+done
+
+lemma dec_fetch_locate_n_a_b: 
+       "start_of ly as > 0 \<Longrightarrow> fetch (ci (layout_of aprog) 
+                       (start_of ly as) (Dec n e)) (Suc (2 * n))  Bk
+       = (W1, start_of ly as + 2*n)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tdec_b_def)
+done
+
+lemma dec_fetch_locate_n_b_o: 
+       "start_of ly as > 0 \<Longrightarrow> 
+            fetch (ci (layout_of aprog) 
+                (start_of ly as) (Dec n e)) (Suc (Suc (2 * n)))  Oc
+      = (R, start_of ly as + 2*n + 2)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tdec_b_def)
+done
+
+
+lemma dec_fetch_locate_n_b_b: 
+       "start_of ly as > 0 \<Longrightarrow> 
+       fetch (ci (layout_of aprog) 
+                  (start_of ly as) (Dec n e)) (Suc (Suc (2 * n))) Bk
+      = (L, start_of ly as + 2*n + 13)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tdec_b_def)
+done
+
+lemma dec_fetch_first_on_right_move_o: 
+      "start_of ly as > 0 \<Longrightarrow> 
+       fetch (ci (layout_of aprog) 
+             (start_of ly as) (Dec n e)) (Suc (Suc (Suc (2 * n))))  Oc
+     = (R, start_of ly as + 2*n + 2)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tdec_b_def)
+done
+
+lemma dec_fetch_first_on_right_move_b: 
+      "start_of ly as > 0 \<Longrightarrow> 
+      fetch (ci (layout_of aprog) (start_of ly as) (Dec n e)) 
+                             (Suc (Suc (Suc (2 * n))))  Bk
+     = (L, start_of ly as + 2*n + 3)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                  nth_of.simps tshift.simps nth_append tdec_b_def)
+done
+
+lemma [simp]: "fetch x (a + 2 * n) b = fetch x (2 * n + a) b"
+thm arg_cong
+apply(rule_tac x = "a + 2*n" and y = "2*n + a" in arg_cong, simp)
+done
+
+lemma dec_fetch_first_after_clear_o: 
+     "start_of ly as > 0 \<Longrightarrow> fetch (ci (layout_of aprog) 
+                      (start_of ly as) (Dec n e)) (2 * n + 4) Oc
+    = (W0, start_of ly as + 2*n + 3)"
+apply(auto simp: ci.simps findnth.simps tshift.simps 
+                          tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 4 = Suc (2*n + 3)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_first_after_clear_b: 
+     "start_of ly as > 0 \<Longrightarrow> 
+     fetch (ci (layout_of aprog) 
+                   (start_of ly as) (Dec n e)) (2 * n + 4) Bk
+    = (R, start_of ly as + 2*n + 4)"
+apply(auto simp: ci.simps findnth.simps 
+               tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 4= Suc (2*n + 3)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_right_move_b: 
+     "start_of ly as > 0 \<Longrightarrow> fetch (ci (layout_of aprog) 
+                          (start_of ly as) (Dec n e)) (2 * n + 5) Bk
+    = (R, start_of ly as + 2*n + 5)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 5= Suc (2*n + 4)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_check_right_move_b: 
+     "start_of ly as > 0 \<Longrightarrow> 
+      fetch (ci (layout_of aprog)
+                (start_of ly as) (Dec n e)) (2 * n + 6) Bk
+    = (L, start_of ly as + 2*n + 6)"
+apply(auto simp: ci.simps findnth.simps 
+               tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 6 = Suc (2*n + 5)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_check_right_move_o: 
+     "start_of ly as > 0 \<Longrightarrow> 
+    fetch (ci (layout_of aprog) (start_of ly as) 
+                      (Dec n e)) (2 * n + 6) Oc
+    = (L, start_of ly as + 2*n + 7)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 6 = Suc (2*n + 5)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_left_move_b: 
+     "start_of ly as > 0 \<Longrightarrow> 
+     fetch (ci (layout_of aprog) 
+             (start_of ly as) (Dec n e)) (2 * n + 7) Bk
+    = (L, start_of ly as + 2*n + 10)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 7 = Suc (2*n + 6)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_after_write_b: 
+     "start_of ly as > 0 \<Longrightarrow> 
+    fetch (ci (layout_of aprog) 
+                   (start_of ly as) (Dec n e)) (2 * n + 8) Bk
+    = (W1, start_of ly as + 2*n + 7)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 8 = Suc (2*n + 7)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_after_write_o: 
+     "start_of ly as > 0 \<Longrightarrow> 
+     fetch (ci (layout_of aprog) 
+                   (start_of ly as) (Dec n e)) (2 * n + 8) Oc
+    = (R, start_of ly as + 2*n + 8)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 8 = Suc (2*n + 7)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_on_right_move_b: 
+     "start_of ly as > 0 \<Longrightarrow> 
+     fetch (ci (layout_of aprog) 
+                   (start_of ly as) (Dec n e)) (2 * n + 9) Bk
+    = (L, start_of ly as + 2*n + 9)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 9 = Suc (2*n + 8)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_on_right_move_o: 
+     "start_of ly as > 0 \<Longrightarrow> 
+     fetch (ci (layout_of aprog) 
+             (start_of ly as) (Dec n e)) (2 * n + 9) Oc
+    = (R, start_of ly as + 2*n + 8)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 9 = Suc (2*n + 8)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_after_clear_b: 
+     "start_of ly as > 0 \<Longrightarrow> 
+     fetch (ci (layout_of aprog) 
+            (start_of ly as) (Dec n e)) (2 * n + 10) Bk
+    = (R, start_of ly as + 2*n + 4)" 
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 10 = Suc (2*n + 9)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_after_clear_o: 
+     "start_of ly as > 0 \<Longrightarrow> 
+    fetch (ci (layout_of aprog) 
+             (start_of ly as) (Dec n e)) (2 * n + 10) Oc
+    = (W0, start_of ly as + 2*n + 9)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 10= Suc (2*n + 9)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_on_left_move1_o:
+      "start_of ly as > 0 \<Longrightarrow> 
+    fetch (ci (layout_of aprog) 
+           (start_of ly as) (Dec n e)) (2 * n + 11) Oc
+    = (L, start_of ly as + 2*n + 10)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 11= Suc (2*n + 10)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_on_left_move1_b:
+     "start_of ly as > 0 \<Longrightarrow> 
+    fetch (ci (layout_of aprog) 
+             (start_of ly as) (Dec n e)) (2 * n + 11) Bk
+    = (L, start_of ly as + 2*n + 11)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 11 = Suc (2*n + 10)", 
+      simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_check_left_move1_o: 
+    "start_of ly as > 0 \<Longrightarrow> 
+  fetch (ci (layout_of aprog) 
+             (start_of ly as) (Dec n e)) (2 * n + 12) Oc
+    = (L, start_of ly as + 2*n + 10)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 12= Suc (2*n + 11)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_check_left_move1_b: 
+    "start_of ly as > 0 \<Longrightarrow> 
+   fetch (ci (layout_of aprog) 
+                  (start_of ly as) (Dec n e)) (2 * n + 12) Bk
+    = (R, start_of ly as + 2*n + 12)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 12 = Suc (2*n + 11)", 
+      simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_after_left_move1_b: 
+  "start_of ly as > 0 \<Longrightarrow> 
+  fetch (ci (layout_of aprog) 
+                (start_of ly as) (Dec n e)) (2 * n + 13) Bk
+    = (R, start_of ly as + 2*n + 16)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 13 = Suc (2*n + 12)", 
+      simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_on_left_move2_o:
+  "start_of ly as > 0 \<Longrightarrow> 
+  fetch (ci (layout_of aprog) 
+           (start_of ly as) (Dec n e)) (2 * n + 14) Oc
+   = (L, start_of ly as + 2*n + 13)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 14 = Suc (2*n + 13)", 
+      simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_on_left_move2_b:
+  "start_of ly as > 0 \<Longrightarrow> 
+  fetch (ci (layout_of aprog) 
+              (start_of ly as) (Dec n e)) (2 * n + 14) Bk
+ = (L, start_of ly as + 2*n + 14)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 14 = Suc (2*n + 13)", 
+      simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_check_left_move2_o:
+  "start_of ly as > 0 \<Longrightarrow> 
+  fetch (ci (layout_of aprog) 
+                (start_of ly as) (Dec n e)) (2 * n + 15)  Oc
+ = (L, start_of ly as + 2*n + 13)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 15 = Suc (2*n + 14)", 
+      simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_check_left_move2_b:
+  "start_of ly as > 0 \<Longrightarrow> 
+  fetch (ci (layout_of aprog) 
+                (start_of ly as) (Dec n e)) (2 * n + 15)  Bk
+ = (R, start_of ly as + 2*n + 15)"
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 15= Suc (2*n + 14)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_after_left_move2_b: 
+  "\<lbrakk>ly = layout_of aprog; 
+    abc_fetch as aprog = Some (Dec n e); 
+    start_of ly as > 0\<rbrakk> \<Longrightarrow> 
+     fetch (ci (layout_of aprog) (start_of ly as) 
+              (Dec n e)) (2 * n + 16)  Bk
+ = (R, start_of ly e)" 
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac "2*n + 16 = Suc (2*n + 15)",
+      simp only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+lemma dec_fetch_next_state: 
+    "start_of ly as > 0 \<Longrightarrow> 
+    fetch (ci (layout_of aprog) 
+           (start_of ly as) (Dec n e)) (2* n + 17)  b
+    = (Nop, 0)"
+apply(case_tac b)
+apply(auto simp: ci.simps findnth.simps 
+                 tshift.simps tdec_b_def add3_Suc)
+apply(subgoal_tac [!] "2*n + 17 = Suc (2*n + 16)", 
+      simp_all only: fetch.simps)
+apply(auto simp: nth_of.simps nth_append)
+done
+
+(*End: dec_fetch lemmas*)
+lemmas dec_fetch_simps = 
+ dec_fetch_locate_a_o dec_fetch_locate_a_b dec_fetch_locate_b_o 
+ dec_fetch_locate_b_b dec_fetch_locate_n_a_o 
+ dec_fetch_locate_n_a_b dec_fetch_locate_n_b_o 
+ dec_fetch_locate_n_b_b dec_fetch_first_on_right_move_o 
+ dec_fetch_first_on_right_move_b dec_fetch_first_after_clear_b
+ dec_fetch_first_after_clear_o dec_fetch_right_move_b 
+ dec_fetch_on_right_move_b dec_fetch_on_right_move_o 
+ dec_fetch_after_clear_b dec_fetch_after_clear_o
+ dec_fetch_check_right_move_b dec_fetch_check_right_move_o 
+ dec_fetch_left_move_b dec_fetch_on_left_move1_b 
+ dec_fetch_on_left_move1_o dec_fetch_check_left_move1_b 
+ dec_fetch_check_left_move1_o dec_fetch_after_left_move1_b 
+ dec_fetch_on_left_move2_b dec_fetch_on_left_move2_o
+ dec_fetch_check_left_move2_o dec_fetch_check_left_move2_b 
+ dec_fetch_after_left_move2_b dec_fetch_after_write_b 
+ dec_fetch_after_write_o dec_fetch_next_state
+
+lemma [simp]:
+  "\<lbrakk>start_of ly as \<le> a; 
+    a < start_of ly as + 2 * n; 
+    (a - start_of ly as) mod 2 = Suc 0; 
+    inv_locate_b (as, am) ((a - start_of ly as) div 2, aaa, Bk # xs) ires\<rbrakk>
+     \<Longrightarrow> \<not> Suc a < start_of ly as + 2 * n \<longrightarrow> 
+                inv_locate_a (as, am) (n, Bk # aaa, xs) ires"
+apply(insert locate_b_2_locate_a[of a ly as n am aaa xs], simp)
+done
+ 
+lemma [simp]: 
+  "\<lbrakk>start_of ly as \<le> a; 
+    a < start_of ly as + 2 * n; 
+    (a - start_of ly as) mod 2 = Suc 0; 
+    inv_locate_b (as, am) ((a - start_of ly as) div 2, aaa, []) ires\<rbrakk>
+   \<Longrightarrow> \<not> Suc a < start_of ly as + 2 * n \<longrightarrow> 
+                  inv_locate_a (as, am) (n, Bk # aaa, []) ires"
+apply(insert locate_b_2_locate_a_B[of a ly as n am aaa], simp)
+done
+
+(*
+lemma [simp]: "a\<^bsup>0\<^esup>=[]"
+apply(simp add: exponent_def)
+done
+*)
+
+lemma exp_ind: "a\<^bsup>Suc b\<^esup> =  a\<^bsup>b\<^esup> @ [a]"
+apply(simp only: exponent_def rep_ind)
+done
+
+lemma [simp]:
+  "inv_locate_b (as, am) (n, l, Oc # r) ires
+  \<Longrightarrow> dec_first_on_right_moving n (as,  abc_lm_s am n (abc_lm_v am n))
+                      (Suc (Suc (start_of ly as + 2 * n)), Oc # l, r) ires"
+apply(simp only: inv_locate_b.simps 
+     dec_first_on_right_moving.simps in_middle.simps 
+     abc_lm_s.simps abc_lm_v.simps)
+apply(erule_tac exE)+
+apply(erule conjE)+
+apply(case_tac "n < length am", simp)
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = m in exI, simp)
+apply(rule_tac x = "Suc ml" in exI, rule_tac conjI, rule_tac [1-2] impI)
+prefer 3
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = m in exI, simp)
+apply(subgoal_tac "Suc n - length am = Suc (n - length am)",
+      simp only:exponent_def rep_ind, simp)
+apply(rule_tac x = "Suc ml" in exI, simp_all)
+apply(rule_tac [!] x = "mr - 1" in exI, simp_all)
+apply(case_tac [!] mr, auto)
+done
+
+lemma [simp]: 
+  "\<lbrakk>inv_locate_b (as, am) (n, l, r) ires; l \<noteq> []\<rbrakk> \<Longrightarrow> 
+  \<not> inv_on_left_moving_in_middle_B (as, abc_lm_s am n (abc_lm_v am n)) 
+    (s, tl l, hd l # r) ires"
+apply(auto simp: inv_locate_b.simps 
+                 inv_on_left_moving_in_middle_B.simps in_middle.simps)
+apply(case_tac [!] ml, auto split: if_splits)
+done
+
+lemma [simp]: "inv_locate_b (as, am) (n, l, r) ires \<Longrightarrow> l \<noteq> []"
+apply(auto simp: inv_locate_b.simps in_middle.simps split: if_splits)
+done
+
+lemma [simp]: "\<lbrakk>inv_locate_b (as, am) (n, l, Bk # r) ires; n < length am\<rbrakk>
+     \<Longrightarrow> inv_on_left_moving_norm (as, am) (s, tl l, hd l # Bk # r) ires"
+apply(simp only: inv_locate_b.simps inv_on_left_moving_norm.simps 
+                 in_middle.simps)
+apply(erule_tac exE)+
+apply(erule_tac conjE)+
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = m in exI, simp)
+apply(rule_tac x = "ml - 1" in exI, auto)
+apply(rule_tac [!] x = "Suc mr" in exI)
+apply(case_tac [!] mr, auto)
+done
+
+lemma [simp]: "\<lbrakk>inv_locate_b (as, am) (n, l, Bk # r) ires; \<not> n < length am\<rbrakk>
+    \<Longrightarrow> inv_on_left_moving_norm (as, am @ 
+        replicate (n - length am) 0 @ [0]) (s, tl l, hd l # Bk # r) ires"
+apply(simp only: inv_locate_b.simps inv_on_left_moving_norm.simps 
+                 in_middle.simps)
+apply(erule_tac exE)+
+apply(erule_tac conjE)+
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = m in exI, simp)
+apply(subgoal_tac "Suc n - length am = Suc (n - length am)", simp only: rep_ind exponent_def, simp_all)
+apply(rule_tac x = "Suc mr" in exI, auto)
+done
+
+lemma inv_locate_b_2_on_left_moving[simp]: 
+  "\<lbrakk>inv_locate_b (as, am) (n, l, Bk # r) ires\<rbrakk>
+   \<Longrightarrow> (l = [] \<longrightarrow> inv_on_left_moving (as,  
+            abc_lm_s am n (abc_lm_v am n)) (s, [], Bk # Bk # r) ires) \<and>
+       (l \<noteq> [] \<longrightarrow> inv_on_left_moving (as,  
+            abc_lm_s am n (abc_lm_v am n)) (s, tl l, hd l # Bk # r) ires)"
+apply(subgoal_tac "l\<noteq>[]")
+apply(subgoal_tac "\<not> inv_on_left_moving_in_middle_B 
+      (as,  abc_lm_s am n (abc_lm_v am n)) (s, tl l, hd l # Bk # r) ires")
+apply(simp add:inv_on_left_moving.simps 
+          abc_lm_s.simps abc_lm_v.simps split: if_splits, auto)
+done
+
+lemma [simp]: 
+  "inv_locate_b (as, am) (n, l, []) ires \<Longrightarrow> 
+                   inv_locate_b (as, am) (n, l, [Bk]) ires" 
+apply(auto simp: inv_locate_b.simps in_middle.simps)
+apply(rule_tac x = lm1 in exI, rule_tac x = "[]" in exI,
+      rule_tac x = "Suc (length lm1) - length am" in exI, 
+      rule_tac x = m in exI, simp)
+apply(rule_tac x = ml in exI, rule_tac x = mr in exI)
+apply(auto)
+done
+
+lemma nil_2_nil: "<lm::nat list> = [] \<Longrightarrow> lm = []"
+apply(auto simp: tape_of_nl_abv)
+apply(case_tac lm, simp)
+apply(case_tac list, auto simp: tape_of_nat_list.simps)
+done
+
+lemma  inv_locate_b_2_on_left_moving_b[simp]: 
+   "inv_locate_b (as, am) (n, l, []) ires
+     \<Longrightarrow> (l = [] \<longrightarrow> inv_on_left_moving (as, 
+                  abc_lm_s am n (abc_lm_v am n)) (s, [], [Bk]) ires) \<and>
+         (l \<noteq> [] \<longrightarrow> inv_on_left_moving (as, abc_lm_s am n 
+                  (abc_lm_v am n)) (s, tl l, [hd l]) ires)"
+apply(insert inv_locate_b_2_on_left_moving[of as am n l "[]" ires s])
+apply(simp only: inv_on_left_moving.simps, simp)
+apply(subgoal_tac "\<not> inv_on_left_moving_in_middle_B 
+         (as, abc_lm_s am n (abc_lm_v am n)) (s, tl l, [hd l]) ires", simp)
+apply(simp only: inv_on_left_moving_norm.simps)
+apply(erule_tac exE)+
+apply(erule_tac conjE)+
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = m in exI, rule_tac x = ml in exI, 
+      rule_tac x = mr in exI, simp)
+apply(case_tac mr, simp, simp, case_tac nat, auto intro: nil_2_nil)
+done
+
+lemma [simp]: 
+ "\<lbrakk>dec_first_on_right_moving n (as, am) (s, aaa, Oc # xs) ires\<rbrakk>
+   \<Longrightarrow> dec_first_on_right_moving n (as, am) (s', Oc # aaa, xs) ires"
+apply(simp only: dec_first_on_right_moving.simps)
+apply(erule exE)+
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = m in exI, simp)
+apply(rule_tac x = "Suc ml" in exI, 
+      rule_tac x = "mr - 1" in exI, auto)
+apply(case_tac [!] mr, auto)
+done
+
+lemma [simp]: 
+  "dec_first_on_right_moving n (as, am) (s, l, Bk # xs) ires \<Longrightarrow> l \<noteq> []"
+apply(auto simp: dec_first_on_right_moving.simps split: if_splits)
+done
+
+lemma [elim]: 
+  "\<lbrakk>\<not> length lm1 < length am; 
+    am @ replicate (length lm1 - length am) 0 @ [0::nat] = 
+                                                lm1 @ m # lm2;
+    0 < m\<rbrakk>
+   \<Longrightarrow> RR"
+apply(subgoal_tac "lm2 = []", simp)
+apply(drule_tac length_equal, simp)
+done
+
+lemma [simp]: 
+ "\<lbrakk>dec_first_on_right_moving n (as, 
+                   abc_lm_s am n (abc_lm_v am n)) (s, l, Bk # xs) ires\<rbrakk>
+\<Longrightarrow> dec_after_clear (as, abc_lm_s am n 
+                 (abc_lm_v am n - Suc 0)) (s', tl l, hd l # Bk # xs) ires"
+apply(simp only: dec_first_on_right_moving.simps 
+                 dec_after_clear.simps abc_lm_s.simps abc_lm_v.simps)
+apply(erule_tac exE)+
+apply(case_tac "n < length am")
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = "m - 1" in exI, auto simp: )
+apply(case_tac [!] mr, auto)
+done
+
+lemma [simp]: 
+ "\<lbrakk>dec_first_on_right_moving n (as, 
+                   abc_lm_s am n (abc_lm_v am n)) (s, l, []) ires\<rbrakk>
+\<Longrightarrow> (l = [] \<longrightarrow> dec_after_clear (as, 
+             abc_lm_s am n (abc_lm_v am n - Suc 0)) (s', [], [Bk]) ires) \<and>
+    (l \<noteq> [] \<longrightarrow> dec_after_clear (as, abc_lm_s am n 
+                      (abc_lm_v am n - Suc 0)) (s', tl l, [hd l]) ires)"
+apply(subgoal_tac "l \<noteq> []", 
+      simp only: dec_first_on_right_moving.simps 
+                 dec_after_clear.simps abc_lm_s.simps abc_lm_v.simps)
+apply(erule_tac exE)+
+apply(case_tac "n < length am", simp)
+apply(rule_tac x = lm1 in exI, rule_tac x = "m - 1" in exI, auto)
+apply(case_tac [1-2] mr, auto)
+apply(case_tac [1-2] m, auto simp: dec_first_on_right_moving.simps split: if_splits)
+done
+
+lemma [simp]: "\<lbrakk>dec_after_clear (as, am) (s, l, Oc # r) ires\<rbrakk>
+                \<Longrightarrow> dec_after_clear (as, am) (s', l, Bk # r) ires"
+apply(auto simp: dec_after_clear.simps)
+done
+
+lemma [simp]: "\<lbrakk>dec_after_clear (as, am) (s, l, Bk # r) ires\<rbrakk>
+                \<Longrightarrow> dec_right_move (as, am) (s', Bk # l, r) ires"
+apply(auto simp: dec_after_clear.simps dec_right_move.simps split: if_splits)
+done
+
+lemma [simp]: "\<lbrakk>dec_after_clear (as, am) (s, l, []) ires\<rbrakk>
+             \<Longrightarrow> dec_right_move (as, am) (s', Bk # l, []) ires"
+apply(auto simp: dec_after_clear.simps dec_right_move.simps )
+done
+
+lemma [simp]: "\<exists>rn. a::block\<^bsup>rn\<^esup> = []"
+apply(rule_tac x = 0 in exI, simp)
+done
+
+lemma [simp]: "\<lbrakk>dec_after_clear (as, am) (s, l, []) ires\<rbrakk>
+             \<Longrightarrow> dec_right_move (as, am) (s', Bk # l, [Bk]) ires"
+apply(auto simp: dec_after_clear.simps dec_right_move.simps split: if_splits)
+done
+
+lemma [simp]:"dec_right_move (as, am) (s, l, Oc # r) ires = False"
+apply(auto simp: dec_right_move.simps)
+done
+              
+lemma dec_right_move_2_check_right_move[simp]:
+     "\<lbrakk>dec_right_move (as, am) (s, l, Bk # r) ires\<rbrakk>
+      \<Longrightarrow> dec_check_right_move (as, am) (s', Bk # l, r) ires"
+apply(auto simp: dec_right_move.simps dec_check_right_move.simps split: if_splits)
+done
+
+lemma [simp]: 
+ "dec_right_move (as, am) (s, l, []) ires= 
+  dec_right_move (as, am) (s, l, [Bk]) ires"
+apply(simp add: dec_right_move.simps)
+apply(rule_tac iffI)
+apply(erule_tac [!] exE)+
+apply(erule_tac [2] exE)
+apply(rule_tac [!] x = lm1 in exI, rule_tac x = "[]" in exI, 
+      rule_tac [!] x = m in exI, auto)
+apply(auto intro: nil_2_nil)
+done
+
+lemma [simp]: "\<lbrakk>dec_right_move (as, am) (s, l, []) ires\<rbrakk>
+             \<Longrightarrow> dec_check_right_move (as, am) (s, Bk # l, []) ires"
+apply(insert dec_right_move_2_check_right_move[of as am s l "[]" s'], 
+      simp)
+done
+
+lemma [simp]: "dec_check_right_move (as, am) (s, l, r) ires\<Longrightarrow> l \<noteq> []"
+apply(auto simp: dec_check_right_move.simps split: if_splits)
+done
+ 
+lemma [simp]: "\<lbrakk>dec_check_right_move (as, am) (s, l, Oc # r) ires\<rbrakk>
+             \<Longrightarrow> dec_after_write (as, am) (s', tl l, hd l # Oc # r) ires"
+apply(auto simp: dec_check_right_move.simps dec_after_write.simps)
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = m in exI, auto)
+done
+
+lemma [simp]: "\<lbrakk>dec_check_right_move (as, am) (s, l, Bk # r) ires\<rbrakk>
+                \<Longrightarrow> dec_left_move (as, am) (s', tl l, hd l # Bk # r) ires"
+apply(auto simp: dec_check_right_move.simps 
+                 dec_left_move.simps inv_after_move.simps)
+apply(rule_tac x = lm1 in exI, rule_tac x = m in exI, auto)
+apply(auto intro: BkCons_nil nil_2_nil dest: BkCons_nil)
+apply(rule_tac x = "Suc rn" in exI)
+apply(auto intro: BkCons_nil nil_2_nil dest: BkCons_nil)
+done
+
+lemma [simp]: "\<lbrakk>dec_check_right_move (as, am) (s, l, []) ires\<rbrakk>
+             \<Longrightarrow> dec_left_move (as, am) (s', tl l, [hd l]) ires"
+apply(auto simp: dec_check_right_move.simps 
+                 dec_left_move.simps inv_after_move.simps)
+apply(rule_tac x = lm1 in exI, rule_tac x = m in exI, auto)
+apply(auto intro: BkCons_nil nil_2_nil dest: BkCons_nil)
+done
+
+lemma [simp]: "dec_left_move (as, am) (s, aaa, Oc # xs) ires = False"
+apply(auto simp: dec_left_move.simps inv_after_move.simps)
+apply(case_tac [!] rn, auto)
+done
+
+lemma [simp]: "dec_left_move (as, am) (s, l, r) ires
+             \<Longrightarrow> l \<noteq> []"
+apply(auto simp: dec_left_move.simps split: if_splits)
+done
+
+lemma tape_of_nl_abv_cons_ex[simp]: 
+   "\<exists>lna. Oc # Oc\<^bsup>m\<^esup> @ Bk # <rev lm1> @ Bk\<^bsup>ln\<^esup> = <m # rev lm1> @ Bk\<^bsup>lna\<^esup>"
+apply(case_tac "lm1=[]", auto simp: tape_of_nl_abv 
+                                    tape_of_nat_list.simps)
+apply(rule_tac x = "ln" in exI, simp)
+apply(simp add:  tape_of_nat_list_cons exponent_def)
+done
+
+(*
+lemma [simp]: "inv_on_left_moving_in_middle_B (as, lm1 @ [m])
+                 (s', Oc # Oc\<^bsup>m\<^esup> @ Bk # <rev lm1> @ Bk\<^bsup>ln\<^esup>, Bk # Bk\<^bsup>rn\<^esup>) ires"
+apply(simp only: inv_on_left_moving_in_middle_B.simps)
+apply(rule_tac x = "lm1 @ [m]" in exI, rule_tac x = "[]" in exI, auto)
+done    
+*)
+lemma [simp]: "inv_on_left_moving_in_middle_B (as, [m])
+  (s', Oc # Oc\<^bsup>m\<^esup> @ Bk # Bk # ires, Bk # Bk\<^bsup>rn\<^esup>) ires"
+apply(simp add: inv_on_left_moving_in_middle_B.simps)
+apply(rule_tac x = "[m]" in exI, simp, auto simp: tape_of_nat_def)
+done
+
+lemma [simp]: "inv_on_left_moving_in_middle_B (as, [m])
+  (s', Oc # Oc\<^bsup>m\<^esup> @ Bk # Bk # ires, [Bk]) ires"
+apply(simp add: inv_on_left_moving_in_middle_B.simps)
+apply(rule_tac x = "[m]" in exI, simp, auto simp: tape_of_nat_def)
+done
+
+lemma [simp]: "lm1 \<noteq> [] \<Longrightarrow> 
+  inv_on_left_moving_in_middle_B (as, lm1 @ [m]) (s', 
+  Oc # Oc\<^bsup>m\<^esup> @ Bk # <rev lm1> @ Bk # Bk # ires, Bk # Bk\<^bsup>rn\<^esup>) ires"
+apply(simp only: inv_on_left_moving_in_middle_B.simps)
+apply(rule_tac x = "lm1 @ [m ]" in exI, rule_tac x = "[]" in exI, simp, auto)
+done
+
+lemma [simp]: "lm1 \<noteq> [] \<Longrightarrow> 
+  inv_on_left_moving_in_middle_B (as, lm1 @ [m]) (s', 
+  Oc # Oc\<^bsup>m\<^esup> @ Bk # <rev lm1> @ Bk # Bk # ires, [Bk]) ires"
+apply(simp only: inv_on_left_moving_in_middle_B.simps)
+apply(rule_tac x = "lm1 @ [m ]" in exI, rule_tac x = "[]" in exI, simp, auto)
+done
+
+lemma [simp]: "dec_left_move (as, am) (s, l, Bk # r) ires
+       \<Longrightarrow> inv_on_left_moving (as, am) (s', tl l, hd l # Bk # r) ires"
+apply(auto simp: dec_left_move.simps inv_on_left_moving.simps split: if_splits)
+done
+
+(*
+lemma [simp]: "inv_on_left_moving_in_middle_B (as, lm1 @ [m]) 
+                        (s', Oc # Oc\<^bsup>m\<^esup> @ Bk # <rev lm1> @ Bk\<^bsup>ln\<^esup>, [Bk])  ires"
+apply(auto simp: inv_on_left_moving_in_middle_B.simps)
+apply(rule_tac x = "lm1 @ [m]" in exI, rule_tac x = "[]" in exI, auto)
+done
+*)
+
+lemma [simp]: "dec_left_move (as, am) (s, l, []) ires
+             \<Longrightarrow> inv_on_left_moving (as, am) (s', tl l, [hd l]) ires"
+apply(auto simp: dec_left_move.simps inv_on_left_moving.simps split: if_splits)
+done
+
+lemma [simp]: "dec_after_write (as, am) (s, l, Oc # r) ires
+       \<Longrightarrow> dec_on_right_moving (as, am) (s', Oc # l, r) ires"
+apply(auto simp: dec_after_write.simps dec_on_right_moving.simps)
+apply(rule_tac x = "lm1 @ [m]" in exI, rule_tac x = "tl lm2" in exI, 
+      rule_tac x = "hd lm2" in exI, simp)
+apply(rule_tac x = "Suc 0" in exI,rule_tac x =  "Suc (hd lm2)" in exI)
+apply(case_tac lm2, simp, simp)
+apply(case_tac "list = []", 
+      auto simp: tape_of_nl_abv tape_of_nat_list.simps split: if_splits )
+apply(case_tac rn, auto)
+apply(case_tac "rev lm1", simp, simp add: tape_of_nat_list.simps)
+apply(case_tac rn, auto)
+apply(case_tac list, simp_all add: tape_of_nat_list.simps, auto)
+apply(case_tac "rev lm1", simp, simp add: tape_of_nat_list.simps)
+apply(case_tac list, simp_all add: tape_of_nat_list.simps, auto)
+done
+
+lemma [simp]: "dec_after_write (as, am) (s, l, Bk # r) ires
+       \<Longrightarrow> dec_after_write (as, am) (s', l, Oc # r) ires"
+apply(auto simp: dec_after_write.simps)
+done
+
+lemma [simp]: "dec_after_write (as, am) (s, aaa, []) ires
+             \<Longrightarrow> dec_after_write (as, am) (s', aaa, [Oc]) ires"
+apply(auto simp: dec_after_write.simps)
+done
+
+lemma [simp]: "dec_on_right_moving (as, am) (s, l, Oc # r) ires
+       \<Longrightarrow> dec_on_right_moving (as, am) (s', Oc # l, r) ires"
+apply(simp only: dec_on_right_moving.simps)
+apply(erule_tac exE)+
+apply(erule conjE)+
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
+      rule_tac x = "m" in exI, rule_tac x = "Suc ml" in exI, 
+      rule_tac x = "mr - 1" in exI, simp)
+apply(case_tac mr, auto)
+done
+
+lemma [simp]: "dec_on_right_moving (as, am) (s, l, r) ires\<Longrightarrow>  l \<noteq> []"
+apply(auto simp: dec_on_right_moving.simps split: if_splits)
+done
+
+lemma [simp]: "dec_on_right_moving (as, am) (s, l, Bk # r) ires
+      \<Longrightarrow>  dec_after_clear (as, am) (s', tl l, hd l # Bk # r) ires"
+apply(auto simp: dec_on_right_moving.simps dec_after_clear.simps)
+apply(case_tac [!] mr, auto split: if_splits)
+done
+
+lemma [simp]: "dec_on_right_moving (as, am) (s, l, []) ires
+             \<Longrightarrow> dec_after_clear (as, am) (s', tl l, [hd l]) ires"
+apply(auto simp: dec_on_right_moving.simps dec_after_clear.simps)
+apply(case_tac mr, simp_all split: if_splits)
+apply(rule_tac x = lm1 in exI, simp)
+done
+
+lemma start_of_le: "a < b \<Longrightarrow> start_of ly a \<le> start_of ly b"
+proof(induct b arbitrary: a, simp, case_tac "a = b", simp)
+  fix b a
+  show "start_of ly b \<le> start_of ly (Suc b)"
+    apply(case_tac "b::nat", 
+          simp add: start_of.simps, simp add: start_of.simps)
+    done
+next
+  fix b a
+  assume h1: "\<And>a. a < b \<Longrightarrow> start_of ly a \<le> start_of ly b" 
+             "a < Suc b" "a \<noteq> b"
+  hence "a < b"
+    by(simp)
+  from h1 and this have h2: "start_of ly a \<le> start_of ly b"
+    by(drule_tac h1, simp)
+  from h2 show "start_of ly a \<le> start_of ly (Suc b)"
+  proof -
+    have "start_of ly b \<le> start_of ly (Suc b)"
+      apply(case_tac "b::nat", 
+            simp add: start_of.simps, simp add: start_of.simps)
+      done
+    from h2 and this show "start_of ly a \<le> start_of ly (Suc b)"
+      by simp
+  qed
+qed
+
+lemma start_of_dec_length[simp]: 
+  "\<lbrakk>abc_fetch a aprog = Some (Dec n e)\<rbrakk> \<Longrightarrow> 
+    start_of (layout_of aprog) (Suc a)
+          = start_of (layout_of aprog) a + 2*n + 16"
+apply(case_tac a, auto simp: abc_fetch.simps start_of.simps 
+                             layout_of.simps length_of.simps 
+                       split: if_splits)
+done
+
+lemma start_of_ge: 
+ "\<lbrakk>abc_fetch a aprog = Some (Dec n e); a < e\<rbrakk> \<Longrightarrow>
+  start_of (layout_of aprog) e > 
+              start_of (layout_of aprog) a + 2*n + 15"
+apply(case_tac "e = Suc a", 
+      simp add: start_of.simps abc_fetch.simps layout_of.simps 
+                length_of.simps split: if_splits)
+apply(subgoal_tac "Suc a < e", drule_tac a = "Suc a" 
+             and ly = "layout_of aprog" in start_of_le)
+apply(subgoal_tac "start_of (layout_of aprog) (Suc a)
+         = start_of (layout_of aprog) a + 2*n + 16", simp)
+apply(rule_tac start_of_dec_length, simp)
+apply(arith)
+done
+
+lemma starte_not_equal[simp]: 
+ "\<lbrakk>abc_fetch as aprog = Some (Dec n e); ly = layout_of aprog\<rbrakk>
+   \<Longrightarrow> (start_of ly e \<noteq> Suc (Suc (start_of ly as + 2 * n)) \<and>  
+        start_of ly e \<noteq> start_of ly as + 2 * n + 3 \<and> 
+        start_of ly e \<noteq> start_of ly as + 2 * n + 4 \<and>
+        start_of ly e \<noteq> start_of ly as + 2 * n + 5 \<and> 
+        start_of ly e \<noteq> start_of ly as + 2 * n + 6 \<and> 
+        start_of ly e \<noteq> start_of ly as + 2 * n + 7 \<and>
+        start_of ly e \<noteq> start_of ly as + 2 * n + 8 \<and> 
+        start_of ly e \<noteq> start_of ly as + 2 * n + 9 \<and> 
+        start_of ly e \<noteq> start_of ly as + 2 * n + 10 \<and>
+        start_of ly e \<noteq> start_of ly as + 2 * n + 11 \<and> 
+        start_of ly e \<noteq> start_of ly as + 2 * n + 12 \<and> 
+        start_of ly e \<noteq> start_of ly as + 2 * n + 13 \<and>
+        start_of ly e \<noteq> start_of ly as + 2 * n + 14 \<and> 
+        start_of ly e \<noteq> start_of ly as + 2 * n + 15)" 
+apply(case_tac "e = as", simp) 
+apply(case_tac "e < as")
+apply(drule_tac a = e and b = as and ly = ly in start_of_le, simp)
+apply(drule_tac a = as and e = e in start_of_ge, simp, simp)
+done
+
+lemma [simp]: "\<lbrakk>abc_fetch as aprog = Some (Dec n e); ly = layout_of aprog\<rbrakk>
+      \<Longrightarrow> (Suc (Suc (start_of ly as + 2 * n)) \<noteq> start_of ly e \<and> 
+          start_of ly as + 2 * n + 3 \<noteq> start_of ly e \<and> 
+          start_of ly as + 2 * n + 4 \<noteq> start_of ly e \<and>
+          start_of ly as + 2 * n + 5 \<noteq>start_of ly e \<and> 
+          start_of ly as + 2 * n + 6 \<noteq> start_of ly e \<and>
+          start_of ly as + 2 * n + 7 \<noteq> start_of ly e \<and> 
+          start_of ly as + 2 * n + 8 \<noteq> start_of ly e \<and> 
+          start_of ly as + 2 * n + 9 \<noteq> start_of ly e \<and> 
+          start_of ly as + 2 * n + 10 \<noteq> start_of ly e \<and>
+          start_of ly as + 2 * n + 11 \<noteq> start_of ly e \<and> 
+          start_of ly as + 2 * n + 12 \<noteq> start_of ly e \<and> 
+          start_of ly as + 2 * n + 13 \<noteq> start_of ly e \<and> 
+          start_of ly as + 2 * n + 14 \<noteq> start_of ly e \<and> 
+          start_of ly as + 2 * n + 15 \<noteq> start_of ly e)"
+apply(insert starte_not_equal[of as aprog n e ly], 
+                            simp del: starte_not_equal)
+apply(erule_tac conjE)+
+apply(rule_tac conjI, simp del: starte_not_equal)+
+apply(rule not_sym, simp)
+done
+
+lemma [simp]: "start_of (layout_of aprog) as > 0 \<Longrightarrow> 
+  fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
+                       (Dec n as)) (Suc 0) Oc =
+ (R, Suc (start_of (layout_of aprog) as))"
+
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                 nth_of.simps tshift.simps nth_append 
+                 Suc_pre tdec_b_def)
+apply(insert findnth_nth[of 0 n "Suc 0"], simp)
+apply(simp add: findnth.simps)
+done
+
+lemma start_of_inj[simp]: 
+  "\<lbrakk>abc_fetch as aprog = Some (Dec n e); e \<noteq> as; ly = layout_of aprog\<rbrakk>
+   \<Longrightarrow> start_of ly as \<noteq> start_of ly e"
+apply(case_tac "e < as")
+apply(case_tac "as", simp, simp)
+apply(case_tac "e = nat", simp add: start_of.simps 
+                                    layout_of.simps length_of.simps)
+apply(subgoal_tac "e < length aprog", simp add: length_of.simps 
+                                         split: abc_inst.splits)
+apply(simp add: abc_fetch.simps split: if_splits)
+apply(subgoal_tac "e < nat", drule_tac a = e and b = nat 
+                                   and ly =ly in start_of_le, simp)
+apply(subgoal_tac "start_of ly nat < start_of ly (Suc nat)", 
+          simp, simp add: start_of.simps layout_of.simps)
+apply(subgoal_tac "nat < length aprog", simp)
+apply(case_tac "aprog ! nat", auto simp: length_of.simps)
+apply(simp add: abc_fetch.simps split: if_splits)
+apply(subgoal_tac "e > as", drule_tac start_of_ge, auto)
+done
+
+lemma [simp]: "\<lbrakk>abc_fetch as aprog = Some (Dec n e); e < as\<rbrakk>
+    \<Longrightarrow> Suc (start_of (layout_of aprog) e) - 
+                               start_of (layout_of aprog) as = 0"
+apply(frule_tac ly = "layout_of aprog" in start_of_le, simp)
+apply(subgoal_tac "start_of (layout_of aprog) as \<noteq> 
+                            start_of (layout_of aprog) e", arith)
+apply(rule start_of_inj, auto)
+done
+
+lemma [simp]:
+   "\<lbrakk>abc_fetch as aprog = Some (Dec n e); 
+     0 < start_of (layout_of aprog) as\<rbrakk>
+ \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+     (Dec n e)) (Suc (start_of (layout_of aprog) e) - 
+                 start_of (layout_of aprog) as) Oc)
+    = (if e = as then (R, start_of (layout_of aprog) as + 1)
+                 else (Nop, 0))"
+apply(auto split: if_splits)
+apply(case_tac "e < as", simp add: fetch.simps)
+apply(subgoal_tac " e > as")
+apply(drule start_of_ge, simp,
+      auto simp: fetch.simps ci_length nth_of.simps)
+apply(subgoal_tac 
+ "length (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+                        (Dec n e)) div 2= length_of (Dec n e)")
+defer
+apply(simp add: ci_length)
+apply(subgoal_tac 
+ "length (ci (layout_of aprog) (start_of (layout_of aprog) as)
+                  (Dec n e)) mod 2 = 0", auto simp: length_of.simps)
+done
+
+lemma [simp]:
+    "start_of (layout_of aprog) as > 0 \<Longrightarrow> 
+ fetch (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+                                          (Dec n as)) (Suc 0)  Bk 
+      = (W1, start_of (layout_of aprog) as)"
+apply(auto simp: ci.simps findnth.simps fetch.simps nth_of.simps 
+                 tshift.simps nth_append Suc_pre tdec_b_def)
+apply(insert findnth_nth[of 0 n "0"], simp)
+apply(simp add: findnth.simps)
+done
+
+lemma [simp]: 
+ "\<lbrakk>abc_fetch as aprog = Some (Dec n e); 
+   0 < start_of (layout_of aprog) as\<rbrakk>
+\<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
+         (Dec n e)) (Suc (start_of (layout_of aprog) e) - 
+              start_of (layout_of aprog) as)  Bk)
+   = (if e = as then (W1, start_of (layout_of aprog) as)
+                  else (Nop, 0))"
+apply(auto split: if_splits)
+apply(case_tac "e < as", simp add: fetch.simps)
+apply(subgoal_tac " e > as")
+apply(drule start_of_ge, simp, auto simp: fetch.simps 
+                                          ci_length nth_of.simps)
+apply(subgoal_tac 
+ "length (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+                            (Dec n e)) div 2= length_of (Dec n e)")
+defer
+apply(simp add: ci_length)
+apply(subgoal_tac 
+ "length (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+                   (Dec n e)) mod 2 = 0", auto simp: length_of.simps)
+apply(simp add: ci.simps tshift.simps tdec_b_def)
+done
+
+lemma [simp]: 
+ "inv_stop (as, abc_lm_s am n (abc_lm_v am n)) (s, l, r) ires \<Longrightarrow> l \<noteq> []"
+apply(auto simp: inv_stop.simps)
+done
+
+lemma [simp]: 
+ "\<lbrakk>abc_fetch as aprog = Some (Dec n e); e \<noteq> as; ly = layout_of aprog\<rbrakk>
+  \<Longrightarrow> (\<not> (start_of ly as \<le> start_of ly e \<and> 
+      start_of ly e < start_of ly as + 2 * n))
+    \<and> start_of ly e \<noteq> start_of ly as + 2*n \<and> 
+      start_of ly e \<noteq> Suc (start_of ly as + 2*n) "
+apply(case_tac "e < as")
+apply(drule_tac ly = ly in start_of_le, simp)
+apply(case_tac n, simp, drule start_of_inj, simp, simp, simp, simp)
+apply(drule_tac start_of_ge, simp, simp)
+done
+
+lemma [simp]: 
+   "\<lbrakk>abc_fetch as aprog = Some (Dec n e); start_of ly as \<le> s; 
+     s < start_of ly as + 2 * n; ly = layout_of aprog\<rbrakk>
+     \<Longrightarrow> Suc s \<noteq> start_of ly e "
+apply(case_tac "e = as", simp)
+apply(case_tac "e < as")
+apply(drule_tac a = e and b = as and ly = ly in start_of_le, simp)
+apply(drule_tac start_of_ge, auto)
+done
+
+lemma [simp]: "\<lbrakk>abc_fetch as aprog = Some (Dec n e); 
+                ly = layout_of aprog\<rbrakk>
+         \<Longrightarrow> Suc (start_of ly as + 2 * n) \<noteq> start_of ly e"
+apply(case_tac "e = as", simp)
+apply(case_tac "e < as")
+apply(drule_tac a = e and b = as and ly = ly in start_of_le, simp)
+apply(drule_tac start_of_ge, auto)
+done
+
+lemma dec_false_1[simp]:
+ "\<lbrakk>abc_lm_v am n = 0; inv_locate_b (as, am) (n, aaa, Oc # xs) ires\<rbrakk>
+  \<Longrightarrow> False"
+apply(auto simp: inv_locate_b.simps in_middle.simps exponent_def)
+apply(case_tac "length lm1 \<ge> length am", auto)
+apply(subgoal_tac "lm2 = []", simp, subgoal_tac "m = 0", simp)
+apply(case_tac mr, auto simp: )
+apply(subgoal_tac "Suc (length lm1) - length am = 
+                   Suc (length lm1 - length am)", 
+      simp add: rep_ind del: replicate.simps, simp)
+apply(drule_tac xs = "am @ replicate (Suc (length lm1) - length am) 0"
+                and ys = "lm1 @ m # lm2" in length_equal, simp)
+apply(case_tac mr, auto simp: abc_lm_v.simps)
+apply(case_tac "mr = 0", simp_all add:  exponent_def split: if_splits)
+apply(subgoal_tac "Suc (length lm1) - length am = 
+                       Suc (length lm1 - length am)", 
+      simp add: rep_ind del: replicate.simps, simp)
+done
+
+lemma [simp]: 
+ "\<lbrakk>inv_locate_b (as, am) (n, aaa, Bk # xs) ires; 
+   abc_lm_v am n = 0\<rbrakk>
+   \<Longrightarrow> inv_on_left_moving (as, abc_lm_s am n 0) 
+                         (s, tl aaa, hd aaa # Bk # xs) ires" 
+apply(insert inv_locate_b_2_on_left_moving[of as am n aaa xs ires s], simp)
+done
+
+lemma [simp]:
+ "\<lbrakk>abc_lm_v am n = 0; inv_locate_b (as, am) (n, aaa, []) ires\<rbrakk>
+   \<Longrightarrow> inv_on_left_moving (as, abc_lm_s am n 0) (s, tl aaa, [hd aaa]) ires"
+apply(insert inv_locate_b_2_on_left_moving_b[of as am n aaa ires s], simp)
+done
+
+lemma [simp]: "\<lbrakk>am ! n = (0::nat); n < length am\<rbrakk> \<Longrightarrow> am[n := 0] = am"
+apply(simp add: list_update_same_conv)
+done
+
+lemma [simp]: "\<lbrakk>abc_lm_v am n = 0; 
+                inv_locate_b (as, abc_lm_s am n 0) (n, Oc # aaa, xs) ires\<rbrakk>
+     \<Longrightarrow> inv_locate_b (as, am) (n, Oc # aaa, xs) ires"
+apply(simp only: inv_locate_b.simps in_middle.simps abc_lm_s.simps 
+                 abc_lm_v.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, simp)
+apply(case_tac "n < length am", simp_all)
+apply(erule_tac conjE)+
+apply(rule_tac x = tn in exI, rule_tac x = m in exI, simp)
+apply(rule_tac x = ml in exI, rule_tac x = mr in exI, simp)
+defer
+apply(rule_tac x = "Suc n - length am" in exI, rule_tac x = m in exI)
+apply(subgoal_tac "Suc n - length am = Suc (n - length am)")
+apply(simp add: exponent_def rep_ind del: replicate.simps, auto)
+done
+
+lemma  [intro]: "\<lbrakk>abc_lm_v (a # list) 0 = 0\<rbrakk> \<Longrightarrow> a = 0"
+apply(simp add: abc_lm_v.simps split: if_splits)
+done
+
+lemma [simp]: 
+ "inv_stop (as, abc_lm_s am n 0) 
+          (start_of (layout_of aprog) e, aaa, Oc # xs) ires
+  \<Longrightarrow> inv_locate_a (as, abc_lm_s am n 0) (0, aaa, Oc # xs) ires"
+apply(simp add: inv_locate_a.simps)
+apply(rule disjI1)
+apply(auto simp: inv_stop.simps at_begin_norm.simps)
+done
+
+lemma [simp]: 
+ "\<lbrakk>abc_lm_v am 0 = 0; 
+  inv_stop (as, abc_lm_s am 0 0) 
+      (start_of (layout_of aprog) e, aaa, Oc # xs) ires\<rbrakk> \<Longrightarrow> 
+  inv_locate_b (as, am) (0, Oc # aaa, xs) ires"
+apply(auto simp: inv_stop.simps inv_locate_b.simps 
+                 in_middle.simps abc_lm_s.simps)
+apply(case_tac "am = []", simp)
+apply(rule_tac x = "[]" in exI, rule_tac x = "Suc 0" in exI, 
+      rule_tac x = 0 in exI, simp)
+apply(rule_tac x = "Suc 0" in exI, rule_tac x = 0 in exI, 
+  simp add: tape_of_nl_abv tape_of_nat_list.simps, auto)
+apply(case_tac rn, auto)
+apply(rule_tac x = "tl am" in exI, rule_tac x = 0 in exI, 
+      rule_tac x = "hd am" in exI, simp)
+apply(rule_tac x = "Suc 0" in exI, rule_tac x = "hd am" in exI, simp)
+apply(case_tac am, simp, simp)
+apply(subgoal_tac "a = 0", case_tac list, 
+      auto simp: tape_of_nat_list.simps tape_of_nl_abv)
+apply(case_tac rn, auto)
+done
+
+lemma [simp]: 
+ "\<lbrakk>inv_stop (as, abc_lm_s am n 0) 
+          (start_of (layout_of aprog) e, aaa, Oc # xs) ires\<rbrakk>
+  \<Longrightarrow> inv_locate_b (as, am) (0, Oc # aaa, xs) ires \<or> 
+      inv_locate_b (as, abc_lm_s am n 0) (0, Oc # aaa, xs) ires"
+apply(simp)
+done
+
+lemma [simp]: 
+"\<lbrakk>abc_lm_v am n = 0; 
+  inv_stop (as, abc_lm_s am n 0) 
+          (start_of (layout_of aprog) e, aaa, Oc # xs) ires\<rbrakk>
+ \<Longrightarrow> \<not> Suc 0 < 2 * n \<longrightarrow> e = as \<longrightarrow> 
+            inv_locate_b (as, am) (n, Oc # aaa, xs) ires"
+apply(case_tac n, simp, simp)
+done
+
+lemma dec_false2: 
+ "inv_stop (as, abc_lm_s am n 0) 
+  (start_of (layout_of aprog) e, aaa, Bk # xs) ires = False"
+apply(auto simp: inv_stop.simps abc_lm_s.simps)
+apply(case_tac "am", simp, case_tac n, simp add: tape_of_nl_abv)
+apply(case_tac list, simp add: tape_of_nat_list.simps )
+apply(simp add: tape_of_nat_list.simps , simp)
+apply(case_tac "list[nat := 0]", 
+      simp add: tape_of_nat_list.simps  tape_of_nl_abv)
+apply(simp add: tape_of_nat_list.simps )
+apply(case_tac "am @ replicate (n - length am) 0 @ [0]", simp)
+apply(case_tac list, auto simp: tape_of_nl_abv 
+                                tape_of_nat_list.simps )
+done	
+
+lemma dec_false3:
+   "inv_stop (as, abc_lm_s am n 0) 
+              (start_of (layout_of aprog) e, aaa, []) ires = False"
+apply(auto simp: inv_stop.simps abc_lm_s.simps)
+apply(case_tac "am", case_tac n, auto)
+apply(case_tac n, auto simp: tape_of_nl_abv)
+apply(case_tac "list::nat list",
+            simp add: tape_of_nat_list.simps tape_of_nat_list.simps)
+apply(simp add: tape_of_nat_list.simps)
+apply(case_tac "list[nat := 0]", 
+            simp add: tape_of_nat_list.simps tape_of_nat_list.simps)
+apply(simp add: tape_of_nat_list.simps)
+apply(case_tac "(am @ replicate (n - length am) 0 @ [0])", simp)
+apply(case_tac list, auto simp: tape_of_nat_list.simps)
+done
+
+lemma [simp]:
+  "fetch (ci (layout_of aprog) 
+       (start_of (layout_of aprog) as) (Dec n e)) 0 b = (Nop, 0)"
+by(simp add: fetch.simps)
+
+declare dec_inv_1.simps[simp del]
+
+declare inv_locate_n_b.simps [simp del]
+
+lemma [simp]:
+"\<lbrakk>0 < abc_lm_v am n; 0 < n; 
+  at_begin_norm (as, am) (n, aaa, Oc # xs) ires\<rbrakk>
+  \<Longrightarrow> inv_locate_n_b (as, am) (n, Oc # aaa, xs) ires"
+apply(simp only: at_begin_norm.simps inv_locate_n_b.simps)
+apply(erule_tac exE)+
+apply(rule_tac x = lm1 in exI, simp)
+apply(case_tac "length lm2", simp)
+apply(case_tac rn, simp, simp)
+apply(rule_tac x = "tl lm2" in exI, rule_tac x = "hd lm2" in exI, simp)
+apply(rule conjI)
+apply(case_tac "lm2", simp, simp)
+apply(case_tac "lm2", auto simp: tape_of_nl_abv tape_of_nat_list.simps)
+apply(case_tac [!] "list", auto simp: tape_of_nl_abv tape_of_nat_list.simps)
+apply(case_tac rn, auto)
+done 
+lemma [simp]: "(\<exists>rn. Oc # xs = Bk\<^bsup>rn\<^esup>) = False"
+apply(auto)
+apply(case_tac rn, auto simp: )
+done
+
+lemma [simp]:
+  "\<lbrakk>0 < abc_lm_v am n; 0 < n; 
+    at_begin_fst_bwtn (as, am) (n, aaa, Oc # xs) ires\<rbrakk>
+ \<Longrightarrow> inv_locate_n_b (as, am) (n, Oc # aaa, xs) ires"
+apply(simp add: at_begin_fst_bwtn.simps inv_locate_n_b.simps )
+done
+ 
+lemma Suc_minus:"length am + tn = n
+       \<Longrightarrow> Suc tn = Suc n - length am "
+apply(arith)
+done
+
+lemma [simp]: 
+ "\<lbrakk>0 < abc_lm_v am n; 0 < n; 
+   at_begin_fst_awtn (as, am) (n, aaa, Oc # xs) ires\<rbrakk>
+ \<Longrightarrow> inv_locate_n_b (as, am) (n, Oc # aaa, xs) ires"
+apply(simp only: at_begin_fst_awtn.simps inv_locate_n_b.simps )
+apply(erule exE)+
+apply(erule conjE)+
+apply(rule_tac x = lm1 in exI, rule_tac x = "[]" in exI, 
+      rule_tac x = "Suc tn" in exI, rule_tac x = 0 in exI)
+apply(simp add: exponent_def rep_ind del: replicate.simps)
+apply(rule conjI)+
+apply(auto)
+apply(case_tac [!] rn, auto)
+done
+
+lemma [simp]: 
+ "\<lbrakk>0 < abc_lm_v am n; 0 < n; inv_locate_a (as, am) (n, aaa, Oc # xs) ires\<rbrakk>
+ \<Longrightarrow> inv_locate_n_b (as, am) (n, Oc#aaa, xs) ires"
+apply(auto simp: inv_locate_a.simps)
+done
+
+lemma [simp]:
+ "\<lbrakk>inv_locate_n_b (as, am) (n, aaa, Oc # xs) ires\<rbrakk>
+ \<Longrightarrow> dec_first_on_right_moving n (as, abc_lm_s am n (abc_lm_v am n))  
+                                      (s, Oc # aaa, xs) ires"
+apply(auto simp: inv_locate_n_b.simps dec_first_on_right_moving.simps 
+                 abc_lm_s.simps abc_lm_v.simps)
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = m in exI, simp)
+apply(rule_tac x = "Suc (Suc 0)" in exI, 
+      rule_tac x = "m - 1" in exI, simp)
+apply(case_tac m, auto simp:  exponent_def)
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = m in exI, 
+      simp add: Suc_diff_le rep_ind del: replicate.simps)
+apply(rule_tac x = "Suc (Suc 0)" in exI, 
+      rule_tac x = "m - 1" in exI, simp)
+apply(case_tac m, auto simp:  exponent_def)
+apply(rule_tac x = lm1 in exI, rule_tac x = "[]" in exI, 
+      rule_tac x = m in exI, simp)
+apply(rule_tac x = "Suc (Suc 0)" in exI, 
+      rule_tac x = "m - 1" in exI, simp)
+apply(case_tac m, auto)
+apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, 
+      rule_tac x = m in exI, 
+      simp add: Suc_diff_le rep_ind del: replicate.simps, simp)
+done
+
+lemma dec_false_2: 
+ "\<lbrakk>0 < abc_lm_v am n; inv_locate_n_b (as, am) (n, aaa, Bk # xs) ires\<rbrakk> 
+ \<Longrightarrow> False"
+apply(auto simp: inv_locate_n_b.simps abc_lm_v.simps split: if_splits)
+apply(case_tac [!] m, auto)
+done
+ 
+lemma dec_false_2_b:
+ "\<lbrakk>0 < abc_lm_v am n; inv_locate_n_b (as, am) 
+                                (n, aaa, []) ires\<rbrakk> \<Longrightarrow> False"
+apply(auto simp: inv_locate_n_b.simps abc_lm_v.simps split: if_splits)
+apply(case_tac [!] m, auto simp: )
+done
+
+
+(*begin: dec halt1 lemmas*)
+thm abc_inc_stage1.simps
+fun abc_dec_1_stage1:: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "abc_dec_1_stage1 (s, l, r) ss n = 
+       (if s > ss \<and> s \<le> ss + 2*n + 1 then 4
+        else if s = ss + 2 * n + 13 \<or> s = ss + 2*n + 14 then 3
+        else if s = ss + 2*n + 15 then 2
+        else 0)"
+
+fun abc_dec_1_stage2:: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "abc_dec_1_stage2 (s, l, r) ss n = 
+       (if s \<le> ss + 2 * n + 1 then (ss + 2 * n + 16 - s)
+        else if s = ss + 2*n + 13 then length l
+        else if s = ss + 2*n + 14 then length l
+        else 0)"
+
+fun abc_dec_1_stage3 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> block list \<Rightarrow> nat"
+  where
+  "abc_dec_1_stage3 (s, l, r) ss n ires = 
+        (if s \<le> ss + 2*n + 1 then 
+             if (s - ss) mod 2 = 0 then 
+                         if r \<noteq> [] \<and> hd r = Oc then 0 else 1  
+                         else length r
+         else if s = ss + 2 * n + 13 then 
+             if l = Bk # ires \<and> r \<noteq> [] \<and> hd r = Oc then 2 
+             else 1
+         else if s = ss + 2 * n + 14 then 
+             if r \<noteq> [] \<and> hd r = Oc then 3 else 0 
+         else 0)"
+
+fun abc_dec_1_measure :: "(t_conf \<times> nat \<times> nat \<times> block list) \<Rightarrow> (nat \<times> nat \<times> nat)"
+  where
+  "abc_dec_1_measure (c, ss, n, ires) = (abc_dec_1_stage1 c ss n, 
+                   abc_dec_1_stage2 c ss n, abc_dec_1_stage3 c ss n ires)"
+
+definition abc_dec_1_LE ::
+  "(((nat \<times> block list \<times> block list) \<times> nat \<times>
+  nat \<times> block list) \<times> ((nat \<times> block list \<times> block list) \<times> nat \<times> nat \<times> block list)) set"
+  where "abc_dec_1_LE \<equiv> (inv_image lex_triple abc_dec_1_measure)"
+
+lemma wf_dec_le: "wf abc_dec_1_LE"
+by(auto intro:wf_inv_image wf_lex_triple simp:abc_dec_1_LE_def)
+
+declare dec_inv_1.simps[simp del] dec_inv_2.simps[simp del]
+
+lemma [elim]: 
+ "\<lbrakk>abc_fetch as aprog = Some (Dec n e); 
+   start_of (layout_of aprog) as < start_of (layout_of aprog) e;
+   start_of (layout_of aprog) e \<le> 
+         Suc (start_of (layout_of aprog) as + 2 * n)\<rbrakk> \<Longrightarrow> False"
+apply(case_tac "e = as", simp)
+apply(case_tac "e < as")
+apply(drule_tac a = e and b = as and ly = "layout_of aprog" in 
+                                                 start_of_le, simp)
+apply(drule_tac start_of_ge, auto)
+done
+
+lemma [elim]: "\<lbrakk>abc_fetch as aprog = Some (Dec n e); 
+                                start_of (layout_of aprog) e 
+    = start_of (layout_of aprog) as + 2 * n + 13\<rbrakk>
+         \<Longrightarrow> False"
+apply(insert starte_not_equal[of as aprog n e "layout_of aprog"], 
+      simp)
+done
+
+lemma [elim]: "\<lbrakk>abc_fetch as aprog = Some (Dec n e);
+                 start_of (layout_of aprog) e = 
+               start_of (layout_of aprog) as + 2 * n + 14\<rbrakk>
+        \<Longrightarrow> False"
+apply(insert starte_not_equal[of as aprog n e "layout_of aprog"],
+      simp)
+done
+
+lemma [elim]: 
+ "\<lbrakk>abc_fetch as aprog = Some (Dec n e);
+   start_of (layout_of aprog) as < start_of (layout_of aprog) e;
+   start_of (layout_of aprog) e \<le> 
+              Suc (start_of (layout_of aprog) as + 2 * n)\<rbrakk>
+   \<Longrightarrow> False"
+apply(case_tac "e = as", simp)
+apply(case_tac "e < as")
+apply(drule_tac a = e and b = as and ly = "layout_of aprog" in 
+                                                    start_of_le, simp)
+apply(drule_tac start_of_ge, auto)
+done
+
+lemma [elim]: 
+ "\<lbrakk>abc_fetch as aprog = Some (Dec n e); 
+   start_of (layout_of aprog) e = 
+               start_of (layout_of aprog) as + 2 * n + 13\<rbrakk>
+    \<Longrightarrow> False"
+apply(insert starte_not_equal[of as aprog n e "layout_of aprog"], 
+      simp)
+done
+
+lemma [simp]: 
+ "abc_fetch as aprog = Some (Dec n e) \<Longrightarrow> 
+   Suc (start_of (layout_of aprog) as) \<noteq> start_of (layout_of aprog) e"
+apply(case_tac "e = as", simp) 
+apply(case_tac "e < as")
+apply(drule_tac a = e and b = as and ly = "(layout_of aprog)" in 
+                                                 start_of_le, simp)
+apply(drule_tac a = as and e = e in start_of_ge, simp, simp)
+done
+
+lemma [simp]: "inv_on_left_moving (as, am) (s, [], r) ires 
+  = False"
+apply(simp add: inv_on_left_moving.simps inv_on_left_moving_norm.simps
+                inv_on_left_moving_in_middle_B.simps)
+done
+
+lemma [simp]: 
+  "inv_check_left_moving (as, abc_lm_s am n 0)
+  (start_of (layout_of aprog) as + 2 * n + 14, [], Oc # xs) ires
+ = False"
+apply(simp add: inv_check_left_moving.simps inv_check_left_moving_in_middle.simps)
+done
+
+lemma dec_inv_stop1_pre: 
+    "\<lbrakk>abc_fetch as aprog = Some (Dec n e); abc_lm_v am n = 0;
+      start_of (layout_of aprog) as > 0\<rbrakk>
+ \<Longrightarrow> \<forall>na. \<not> (\<lambda>(s, l, r) (ss, n', ires'). s = start_of (layout_of aprog) e)
+            (t_steps (Suc (start_of (layout_of aprog) as), l, r) 
+              (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+                 (Dec n e), start_of (layout_of aprog) as - Suc 0) na)
+                      (start_of (layout_of aprog) as, n, ires) \<and>
+           dec_inv_1 (layout_of aprog) n e (as, am)
+            (t_steps (Suc (start_of (layout_of aprog) as), l, r) 
+              (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+                (Dec n e), start_of (layout_of aprog) as - Suc 0) na) ires
+       \<longrightarrow> dec_inv_1 (layout_of aprog) n e (as, am)
+            (t_steps (Suc (start_of (layout_of aprog) as), l, r) 
+              (ci (layout_of aprog) (start_of (layout_of aprog) as)
+                 (Dec n e), start_of (layout_of aprog) as - Suc 0) 
+                    (Suc na)) ires \<and>
+            ((t_steps (Suc (start_of (layout_of aprog) as), l, r) 
+            (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+           (Dec n e), start_of (layout_of aprog) as - Suc 0) (Suc na),
+             start_of (layout_of aprog) as, n, ires),
+         t_steps (Suc (start_of (layout_of aprog) as), l, r) 
+            (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+               (Dec n e), start_of (layout_of aprog) as - Suc 0) na,
+            start_of (layout_of aprog) as, n, ires)
+           \<in> abc_dec_1_LE"
+apply(rule allI, rule impI, simp add: t_steps_ind)
+apply(case_tac "(t_steps (Suc (start_of (layout_of aprog) as), l, r) 
+(ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), 
+start_of (layout_of aprog) as - Suc 0) na)", simp)
+apply(auto split:if_splits simp add:t_step.simps dec_inv_1.simps, 
+          tactic {* ALLGOALS (resolve_tac [@{thm fetch_intro}]) *})
+apply(simp_all add:dec_fetch_simps new_tape.simps dec_inv_1.simps)
+apply(auto simp add: abc_dec_1_LE_def lex_square_def 
+                     lex_triple_def lex_pair_def  
+                split: if_splits)
+apply(rule dec_false_1, simp, simp)
+done
+
+lemma dec_inv_stop1: 
+  "\<lbrakk>ly = layout_of aprog; 
+    dec_inv_1 ly n e (as, am) (start_of ly as + 1, l, r) ires; 
+    abc_fetch as aprog = Some (Dec n e); abc_lm_v am n = 0\<rbrakk> \<Longrightarrow> 
+  (\<exists> stp. (\<lambda> (s', l', r'). s' = start_of ly e \<and> 
+           dec_inv_1 ly n e (as, am) (s', l' , r') ires) 
+  (t_steps (start_of ly as + 1, l, r)
+     (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp))"
+apply(insert halt_lemma2[of abc_dec_1_LE 
+    "\<lambda> ((s, l, r), ss, n', ires'). s = start_of ly e" 
+     "(\<lambda> stp. (t_steps (start_of ly as + 1, l, r) 
+          (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) 
+               stp, start_of ly as, n, ires))"
+     "\<lambda> ((s, l, r), ss, n, ires'). dec_inv_1 ly n e (as, am) (s, l, r) ires'"],
+     simp)
+apply(insert wf_dec_le, simp)
+apply(insert dec_inv_stop1_pre[of as aprog n e am l r], simp)
+apply(subgoal_tac "start_of (layout_of aprog) as > 0", 
+                                      simp add: t_steps.simps)
+apply(erule_tac exE, rule_tac x = na in exI)
+apply(case_tac
+     "(t_steps (Suc (start_of (layout_of aprog) as), l, r) 
+         (ci (layout_of aprog) (start_of (layout_of aprog) as)
+           (Dec n e), start_of (layout_of aprog) as - Suc 0) na)",
+      case_tac b, auto)
+apply(rule startof_not0)
+done
+
+(*begin: dec halt2 lemmas*)
+
+lemma [simp]:
+  "\<lbrakk>abc_fetch as aprog = Some (Dec n e); 
+    ly = layout_of aprog\<rbrakk> \<Longrightarrow> 
+              start_of ly (Suc as) = start_of ly as + 2*n + 16"
+by simp
+
+fun abc_dec_2_stage1 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "abc_dec_2_stage1 (s, l, r) ss n = 
+              (if s \<le> ss + 2*n + 1 then 7
+               else if s = ss + 2*n + 2 then 6 
+               else if s = ss + 2*n + 3 then 5
+               else if s \<ge> ss + 2*n + 4 \<and> s \<le> ss + 2*n + 9 then 4
+               else if s = ss + 2*n + 6 then 3
+               else if s = ss + 2*n + 10 \<or> s = ss + 2*n + 11 then 2
+               else if s = ss + 2*n + 12 then 1
+               else 0)"
+
+thm new_tape.simps
+
+fun abc_dec_2_stage2 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "abc_dec_2_stage2 (s, l, r) ss n = 
+       (if s \<le> ss + 2 * n + 1 then (ss + 2 * n + 16 - s)
+        else if s = ss + 2*n + 10 then length l
+        else if s = ss + 2*n + 11 then length l
+        else if s = ss + 2*n + 4 then length r - 1
+        else if s = ss + 2*n + 5 then length r 
+        else if s = ss + 2*n + 7 then length r - 1
+        else if s = ss + 2*n + 8 then  
+              length r + length (takeWhile (\<lambda> a. a = Oc) l) - 1
+        else if s = ss + 2*n + 9 then 
+              length r + length (takeWhile (\<lambda> a. a = Oc) l) - 1
+        else 0)"
+
+fun abc_dec_2_stage3 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> block list \<Rightarrow> nat"
+  where
+  "abc_dec_2_stage3 (s, l, r) ss n ires =
+        (if s \<le> ss + 2*n + 1 then 
+            if (s - ss) mod 2 = 0 then if r \<noteq> [] \<and> 
+                                          hd r = Oc then 0 else 1  
+            else length r
+         else if s = ss + 2 * n + 10 then 
+             if l = Bk # ires \<and> r \<noteq> [] \<and> hd r = Oc then 2
+             else 1
+         else if s = ss + 2 * n + 11 then 
+             if r \<noteq> [] \<and> hd r = Oc then 3 
+             else 0 
+         else (ss + 2 * n + 16 - s))"
+
+fun abc_dec_2_stage4 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+  where
+  "abc_dec_2_stage4 (s, l, r) ss n = 
+          (if s = ss + 2*n + 2 then length r
+           else if s = ss + 2*n + 8 then length r
+           else if s = ss + 2*n + 3 then 
+               if r \<noteq> [] \<and> hd r = Oc then 1
+               else 0
+           else if s = ss + 2*n + 7 then 
+               if r \<noteq> [] \<and> hd r = Oc then 0 
+               else 1
+           else if s = ss + 2*n + 9 then 
+               if r \<noteq> [] \<and> hd r = Oc then 1
+               else 0 
+           else 0)"
+
+fun abc_dec_2_measure :: "(t_conf \<times> nat \<times> nat \<times> block list) \<Rightarrow> 
+                                    (nat \<times> nat \<times> nat \<times> nat)"
+  where
+  "abc_dec_2_measure (c, ss, n, ires) = 
+       (abc_dec_2_stage1 c ss n, abc_dec_2_stage2 c ss n,
+        abc_dec_2_stage3 c ss n ires, abc_dec_2_stage4 c ss n)"
+
+definition abc_dec_2_LE :: 
+       "(((nat \<times> block list \<times> block list) \<times> nat \<times> nat \<times> block list) \<times> 
+        ((nat \<times> block list \<times> block list) \<times> nat \<times> nat \<times> block list)) set"
+  where "abc_dec_2_LE \<equiv> (inv_image lex_square abc_dec_2_measure)"
+
+lemma wf_dec_2_le: "wf abc_dec_2_LE"
+by(auto intro:wf_inv_image wf_lex_triple wf_lex_square 
+   simp:abc_dec_2_LE_def)
+
+lemma [simp]: "dec_after_write (as, am) (s, aa, r) ires
+           \<Longrightarrow> takeWhile (\<lambda>a. a = Oc) aa = []"
+apply(simp only : dec_after_write.simps)
+apply(erule exE)+
+apply(erule_tac conjE)+
+apply(case_tac aa, simp)
+apply(case_tac a, simp only: takeWhile.simps , simp, simp split: if_splits)
+done
+
+lemma [simp]: 
+     "\<lbrakk>dec_on_right_moving (as, lm) (s, aa, []) ires; 
+       length (takeWhile (\<lambda>a. a = Oc) (tl aa)) 
+           \<noteq> length (takeWhile (\<lambda>a. a = Oc) aa) - Suc 0\<rbrakk>
+    \<Longrightarrow> length (takeWhile (\<lambda>a. a = Oc) (tl aa)) < 
+                       length (takeWhile (\<lambda>a. a = Oc) aa) - Suc 0"
+apply(simp only: dec_on_right_moving.simps)
+apply(erule_tac exE)+
+apply(erule_tac conjE)+
+apply(case_tac mr, auto split: if_splits)
+done
+
+lemma [simp]: 
+  "dec_after_clear (as, abc_lm_s am n (abc_lm_v am n - Suc 0)) 
+             (start_of (layout_of aprog) as + 2 * n + 9, aa, Bk # xs) ires
+ \<Longrightarrow> length xs - Suc 0 < length xs + 
+                             length (takeWhile (\<lambda>a. a = Oc) aa)"
+apply(simp only: dec_after_clear.simps)
+apply(erule_tac exE)+
+apply(erule conjE)+
+apply(simp split: if_splits )
+done
+
+lemma [simp]: 
+ "\<lbrakk>dec_after_clear (as, abc_lm_s am n (abc_lm_v am n - Suc 0))
+       (start_of (layout_of aprog) as + 2 * n + 9, aa, []) ires\<rbrakk>
+    \<Longrightarrow> Suc 0 < length (takeWhile (\<lambda>a. a = Oc) aa)"
+apply(simp add: dec_after_clear.simps split: if_splits)
+done
+
+lemma [simp]: 
+ "\<lbrakk>dec_on_right_moving (as, am) (s, aa, Bk # xs) ires; 
+   Suc (length (takeWhile (\<lambda>a. a = Oc) (tl aa)))
+   \<noteq> length (takeWhile (\<lambda>a. a = Oc) aa)\<rbrakk>
+  \<Longrightarrow> Suc (length (takeWhile (\<lambda>a. a = Oc) (tl aa))) 
+    < length (takeWhile (\<lambda>a. a = Oc) aa)"
+apply(simp only: dec_on_right_moving.simps)
+apply(erule exE)+
+apply(erule conjE)+
+apply(case_tac ml, auto split: if_splits )
+done
+
+(*
+lemma abc_dec_2_wf: 
+     "\<lbrakk>ly = layout_of aprog; dec_inv_2 ly n e (as, am) (start_of ly as + 1, l, r);  abc_fetch as aprog = Dec n e; abc_lm_v am n > 0\<rbrakk>
+       \<Longrightarrow> \<forall>na. \<not> (\<lambda>(s, l, r) (ss, n'). s = start_of (layout_of aprog) as + 2*n + 16)
+        (t_steps (Suc (start_of (layout_of aprog) as), l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) na)
+           (start_of (layout_of aprog) as, n) \<longrightarrow>
+        ((t_steps (Suc (start_of (layout_of aprog) as), l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) (Suc na),
+            start_of (layout_of aprog) as, n),
+          t_steps (Suc (start_of (layout_of aprog) as), l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) na,
+           start_of (layout_of aprog) as, n)
+        \<in> abc_dec_2_LE"
+proof(rule allI, rule impI, simp add: t_steps_ind)
+  fix na
+  assume h1 :"ly = layout_of aprog" "dec_inv_2 (layout_of aprog) n e (as, am) (Suc (start_of (layout_of aprog) as), l, r)" 
+          "abc_fetch as aprog = Dec n e" "abc_lm_v am n > 0"
+         "\<not> (\<lambda>(s, l, r) (ss, n'). s = start_of (layout_of aprog) as + 2*n + 16)
+             (t_steps (Suc (start_of (layout_of aprog) as), l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) na)
+             (start_of (layout_of aprog) as, n)"
+  thus "((t_step (t_steps (Suc (start_of (layout_of aprog) as), l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) na)
+               (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0),
+              start_of (layout_of aprog) as, n),
+             t_steps (Suc (start_of (layout_of aprog) as), l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) na,
+             start_of (layout_of aprog) as, n)
+            \<in> abc_dec_2_LE"
+  proof(insert dec_inv_2_steps[of "layout_of aprog" n e as am "(start_of (layout_of aprog) as + 1, l, r)" aprog na], 
+        case_tac "(t_steps (start_of (layout_of aprog) as + 1, l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) na)", case_tac b, simp)
+    fix a b aa ba
+    assume "dec_inv_2 (layout_of aprog) n e (as, am) (a, aa, ba)" " a \<noteq> start_of (layout_of aprog) as + 2*n + 16" "abc_lm_v am n > 0" "abc_fetch as aprog = Dec n e "
+    thus "((t_step (a, aa, ba) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0), start_of (layout_of aprog) as, n), (a, aa, ba),
+                    start_of (layout_of aprog) as, n)
+                   \<in> abc_dec_2_LE"
+      apply(case_tac "a = 0", auto split:if_splits simp add:t_step.simps dec_inv_2.simps, 
+                tactic {* ALLGOALS (resolve_tac (thms "fetch_intro")) *})
+      apply(simp_all add:dec_fetch_simps new_tape.simps)
+      apply(auto simp add: abc_dec_2_LE_def  lex_square_def lex_triple_def lex_pair_def  
+                           split: if_splits)
+      
+      done
+  qed
+qed
+*)
+
+lemma [simp]: "inv_check_left_moving (as, abc_lm_s am n (abc_lm_v am n - Suc 0)) 
+  (start_of (layout_of aprog) as + 2 * n + 11, [], Oc # xs) ires = False"
+apply(simp add: inv_check_left_moving.simps inv_check_left_moving_in_middle.simps)
+done
+
+lemma dec_inv_stop2_pre: 
+  "\<lbrakk>abc_fetch as aprog = Some (Dec n e); abc_lm_v am n > 0\<rbrakk> \<Longrightarrow> 
+    \<forall>na. \<not> (\<lambda>(s, l, r) (ss, n', ires'). 
+                     s = start_of (layout_of aprog) as + 2 * n + 16)
+   (t_steps (Suc (start_of (layout_of aprog) as), l, r) 
+      (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
+            start_of (layout_of aprog) as - Suc 0) na)
+    (start_of (layout_of aprog) as, n, ires) \<and>
+ dec_inv_2 (layout_of aprog) n e (as, am)
+     (t_steps (Suc (start_of (layout_of aprog) as), l, r) 
+      (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
+          start_of (layout_of aprog) as - Suc 0) na) ires
+ \<longrightarrow>
+ dec_inv_2 (layout_of aprog) n e (as, am)
+     (t_steps (Suc (start_of (layout_of aprog) as), l, r) 
+      (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
+              start_of (layout_of aprog) as - Suc 0) (Suc na)) ires \<and>
+ ((t_steps (Suc (start_of (layout_of aprog) as), l, r) 
+      (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
+            start_of (layout_of aprog) as - Suc 0) (Suc na),  
+              start_of (layout_of aprog) as, n, ires),
+  t_steps (Suc (start_of (layout_of aprog) as), l, r) 
+      (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
+             start_of (layout_of aprog) as - Suc 0) na,
+                          start_of (layout_of aprog) as, n, ires)
+   \<in> abc_dec_2_LE"
+apply(subgoal_tac "start_of (layout_of aprog) as > 0")
+apply(rule allI, rule impI, simp add: t_steps_ind)
+apply(case_tac "(t_steps (Suc (start_of (layout_of aprog) as), l, r)
+     (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), 
+             start_of (layout_of aprog) as - Suc 0) na)", simp)
+apply(auto split:if_splits simp add:t_step.simps dec_inv_2.simps, 
+           tactic {* ALLGOALS (resolve_tac [@{thm fetch_intro}]) *})
+apply(simp_all add:dec_fetch_simps new_tape.simps dec_inv_2.simps)
+apply(auto simp add: abc_dec_2_LE_def lex_square_def lex_triple_def 
+                     lex_pair_def split: if_splits)
+apply(auto intro: dec_false_2_b dec_false_2)
+apply(rule startof_not0)
+done
+
+lemma dec_stop2: 
+ "\<lbrakk>ly = layout_of aprog; 
+   dec_inv_2 ly n e (as, am) (start_of ly as + 1, l, r) ires; 
+   abc_fetch as aprog = Some (Dec n e); 
+   abc_lm_v am n > 0\<rbrakk> \<Longrightarrow> 
+  (\<exists> stp. (\<lambda> (s', l', r'). s' = start_of ly (Suc as) \<and> 
+   dec_inv_2 ly n e (as, am) (s', l', r') ires)
+       (t_steps (start_of ly as+1, l, r) (ci ly (start_of ly as)
+                           (Dec n e), start_of ly as - Suc 0) stp))"
+apply(insert halt_lemma2[of abc_dec_2_LE 
+      "\<lambda> ((s, l, r), ss, n', ires'). s = start_of ly (Suc as)"
+      "(\<lambda> stp. (t_steps (start_of ly as + 1, l, r) 
+       (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp,
+                 start_of ly as, n, ires))"
+      "(\<lambda> ((s, l, r), ss, n, ires'). dec_inv_2 ly n e (as, am) (s, l, r) ires')"])
+apply(insert wf_dec_2_le, simp)
+apply(insert dec_inv_stop2_pre[of as aprog n e am l r], 
+      simp add: t_steps.simps)
+apply(erule_tac exE)
+apply(rule_tac x = na in exI)
+apply(case_tac "(t_steps (Suc (start_of (layout_of aprog) as), l, r) 
+(ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), 
+            start_of (layout_of aprog) as - Suc 0) na)",
+      case_tac b, auto)
+done
+
+lemma dec_inv_stop_cond1: 
+  "\<lbrakk>ly = layout_of aprog; 
+    dec_inv_1 ly n e (as, lm) (s, (l, r)) ires; s = start_of ly e;
+    abc_fetch as aprog = Some (Dec n e); abc_lm_v lm n = 0\<rbrakk> 
+   \<Longrightarrow> crsp_l ly (e, abc_lm_s lm n 0) (s, l, r) ires"
+apply(simp add: dec_inv_1.simps split: if_splits)
+apply(auto simp: crsp_l.simps inv_stop.simps )
+done
+
+lemma dec_inv_stop_cond2: 
+   "\<lbrakk>ly = layout_of aprog; s = start_of ly (Suc as); 
+     dec_inv_2 ly n e (as, lm) (s, (l, r)) ires;
+     abc_fetch as aprog = Some (Dec n e); 
+     abc_lm_v lm n > 0\<rbrakk>
+   \<Longrightarrow> crsp_l ly (Suc as,
+                  abc_lm_s lm n (abc_lm_v lm n - Suc 0)) (s, l, r) ires"
+apply(simp add: dec_inv_2.simps split: if_splits)
+apply(auto simp: crsp_l.simps inv_stop.simps )
+done
+
+lemma [simp]: "(case Bk\<^bsup>rn\<^esup> of [] \<Rightarrow> Bk |
+                 Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc) = Bk"
+apply(case_tac rn, auto)
+done
+
+lemma [simp]: "t_steps tc (p,off) (m + n) = 
+                   t_steps (t_steps tc (p, off) m) (p, off) n"
+apply(induct m arbitrary: n)
+apply(simp add: t_steps.simps)
+proof -
+  fix m n
+  assume h1: "\<And>n. t_steps tc (p, off) (m + n) =
+                     t_steps (t_steps tc (p, off) m) (p, off) n"
+  hence h2: "t_steps tc (p, off) (Suc m + n) = 
+                     t_steps tc (p, off) (m + Suc n)"
+    by simp
+  from h1 and this show 
+    "t_steps tc (p, off) (Suc m + n) = 
+         t_steps (t_steps tc (p, off) (Suc m)) (p, off) n"
+  proof(simp only: h2, simp add: t_steps.simps)
+    have h3: "(t_step (t_steps tc (p, off) m) (p, off)) = 
+                      (t_steps (t_step tc (p, off)) (p, off) m)"
+      apply(simp add: t_steps.simps[THEN sym] t_steps_ind[THEN sym])
+      done
+    from h3 show 
+      "t_steps (t_step (t_steps tc (p, off) m) (p, off)) (p, off) n =          t_steps (t_steps (t_step tc (p, off)) (p, off) m) (p, off) n"
+      by simp
+  qed
+qed
+
+lemma [simp]: " abc_fetch as aprog = Some (Dec n e) \<Longrightarrow> 
+          Suc (start_of (layout_of aprog) as) \<noteq> 
+                           start_of (layout_of aprog) e"
+apply(case_tac "e = as", simp)
+apply(case_tac "e < as")
+apply(drule_tac a = e and b = as and ly = "layout_of aprog" 
+                                           in start_of_le, simp)
+apply(drule_tac start_of_ge, auto)
+done
+
+lemma [simp]: "inv_locate_b (as, []) (0, Oc # Bk # Bk # ires, Bk\<^bsup>rn - Suc 0\<^esup>) ires"
+apply(auto simp: inv_locate_b.simps in_middle.simps)
+apply(rule_tac x = "[]" in exI, rule_tac x = "Suc 0" in exI, 
+      rule_tac x = 0 in exI, simp)
+apply(rule_tac x = "Suc 0" in exI, rule_tac x = 0 in exI, auto)
+apply(case_tac rn, simp, case_tac nat, auto)
+done
+
+lemma [simp]: 
+       "inv_locate_n_b (as, []) (0, Oc # Bk # Bk # ires, Bk\<^bsup>rn - Suc 0\<^esup>) ires"
+apply(auto simp: inv_locate_n_b.simps in_middle.simps)
+apply(case_tac rn, simp, case_tac nat, auto)
+done 
+
+lemma [simp]:
+"abc_fetch as aprog = Some (Dec n e) \<Longrightarrow>
+   dec_inv_1 (layout_of aprog) n e (as, []) 
+    (Suc (start_of (layout_of aprog) as), Oc # Bk # Bk # ires, Bk\<^bsup>rn - Suc 0\<^esup>) ires
+\<and>
+   dec_inv_2 (layout_of aprog) n e (as, []) 
+    (Suc (start_of (layout_of aprog) as), Oc # Bk # Bk # ires, Bk\<^bsup>rn - Suc 0\<^esup>) ires"
+apply(simp add: dec_inv_1.simps dec_inv_2.simps)
+apply(case_tac n, auto)
+done
+
+lemma [simp]: 
+ "\<lbrakk>am \<noteq> []; <am> = Oc # r'; 
+   abc_fetch as aprog = Some (Dec n e)\<rbrakk> 
+ \<Longrightarrow> inv_locate_b (as, am) (0, Oc # Bk # Bk # ires, r' @ Bk\<^bsup>rn\<^esup>) ires"
+apply(auto simp: inv_locate_b.simps in_middle.simps)
+apply(rule_tac x = "tl am" in exI, rule_tac x = 0 in exI,
+      rule_tac x = "hd am" in exI, simp)
+apply(rule_tac x = "Suc 0" in exI)
+apply(rule_tac x = "hd am" in exI, simp)
+apply(case_tac am, simp, case_tac list, auto simp: tape_of_nl_abv tape_of_nat_list.simps)
+apply(case_tac rn, auto)
+done
+
+lemma [simp]: 
+  "\<lbrakk><am> = Oc # r'; abc_fetch as aprog = Some (Dec n e)\<rbrakk> \<Longrightarrow> 
+  inv_locate_n_b (as, am) (0, Oc # Bk # Bk # ires, r' @ Bk\<^bsup>rn\<^esup>) ires"
+apply(auto simp: inv_locate_n_b.simps)
+apply(rule_tac x = "tl am" in exI, rule_tac x = "hd am" in exI, auto)
+apply(case_tac [!] am, auto simp: tape_of_nl_abv tape_of_nat_list.simps )
+apply(case_tac [!]list, auto simp: tape_of_nl_abv tape_of_nat_list.simps)
+apply(case_tac rn, simp, simp)
+apply(erule_tac x = nat in allE, simp)
+done
+
+lemma [simp]:
+   "\<lbrakk>am \<noteq> [];  
+     <am> = Oc # r'; 
+     abc_fetch as aprog = Some (Dec n e)\<rbrakk> \<Longrightarrow>
+    dec_inv_1 (layout_of aprog) n e (as, am) 
+      (Suc (start_of (layout_of aprog) as), 
+           Oc # Bk # Bk # ires, r' @ Bk\<^bsup>rn\<^esup>) ires \<and>
+    dec_inv_2 (layout_of aprog) n e (as, am) 
+      (Suc (start_of (layout_of aprog) as), 
+           Oc # Bk # Bk # ires, r' @ Bk\<^bsup>rn\<^esup>) ires"
+apply(simp add: dec_inv_1.simps dec_inv_2.simps)
+apply(case_tac n, auto)
+done
+
+lemma [simp]: "am \<noteq> [] \<Longrightarrow>  \<exists>r'. <am::nat list> = Oc # r'"
+apply(case_tac am, simp, case_tac list)
+apply(auto simp: tape_of_nl_abv tape_of_nat_list.simps )
+done
+
+lemma [simp]: "start_of (layout_of aprog) as > 0 \<Longrightarrow> 
+      (fetch (ci (layout_of aprog) 
+           (start_of (layout_of aprog) as) (Dec n e)) (Suc 0)  Bk)
+    = (W1, start_of (layout_of aprog) as)"
+apply(auto simp: ci.simps findnth.simps fetch.simps
+             nth_of.simps tshift.simps nth_append Suc_pre tdec_b_def)
+thm findnth_nth
+apply(insert findnth_nth[of 0 n 0], simp)
+apply(simp add: findnth.simps)
+done
+
+lemma [simp]:
+    "start_of (layout_of aprog) as > 0
+   \<Longrightarrow> (t_step (start_of (layout_of aprog) as, Bk # Bk # ires, Bk\<^bsup>rn\<^esup>)
+    (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), 
+                             start_of (layout_of aprog) as - Suc 0))
+   = (start_of (layout_of aprog) as, Bk # Bk # ires, Oc # Bk\<^bsup>rn- Suc 0\<^esup>)"
+apply(simp add: t_step.simps)
+apply(case_tac "start_of (layout_of aprog) as",
+      auto simp: new_tape.simps)
+apply(case_tac rn, auto)
+done
+
+lemma [simp]: "start_of (layout_of aprog) as > 0 \<Longrightarrow> 
+ (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+         (Dec n e)) (Suc 0)  Oc)
+  = (R, Suc (start_of (layout_of aprog) as))"
+
+apply(auto simp: ci.simps findnth.simps fetch.simps
+                 nth_of.simps tshift.simps nth_append 
+                 Suc_pre tdec_b_def)
+apply(insert findnth_nth[of 0 n "Suc 0"], simp)
+apply(simp add: findnth.simps)
+done
+
+lemma [simp]: "start_of (layout_of aprog) as > 0 \<Longrightarrow> 
+ (t_step (start_of (layout_of aprog) as, Bk # Bk # ires, Oc # Bk\<^bsup>rn - Suc 0\<^esup>)
+     (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), 
+        start_of (layout_of aprog) as - Suc 0)) =
+  (Suc (start_of (layout_of aprog) as), Oc # Bk # Bk # ires, Bk\<^bsup>rn-Suc 0\<^esup>)"
+apply(simp add: t_step.simps)
+apply(case_tac "start_of (layout_of aprog) as", 
+      auto simp: new_tape.simps)
+done
+
+lemma [simp]: "start_of (layout_of aprog) as > 0 \<Longrightarrow> 
+ t_step (start_of (layout_of aprog) as, Bk # Bk # ires, Oc # r' @ Bk\<^bsup>rn\<^esup>) 
+      (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
+                 start_of (layout_of aprog) as - Suc 0) =
+      (Suc (start_of (layout_of aprog) as), Oc # Bk # Bk # ires, r' @ Bk\<^bsup>rn\<^esup>)"
+apply(simp add: t_step.simps)
+apply(case_tac "start_of (layout_of aprog) as", 
+      auto simp: new_tape.simps)
+done
+
+lemma crsp_next_state:
+  "\<lbrakk>crsp_l (layout_of aprog) (as, am) tc ires; 
+    abc_fetch as aprog = Some (Dec n e)\<rbrakk>
+  \<Longrightarrow> \<exists> stp' > 0. (\<lambda> (s, l, r). 
+           (s = Suc (start_of (layout_of aprog) as) 
+ \<and> (dec_inv_1 (layout_of aprog) n e (as, am) (s, l, r) ires) 
+ \<and> (dec_inv_2 (layout_of aprog) n e (as, am) (s, l, r)) ires)) 
+     (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
+             (Dec n e), start_of (layout_of aprog) as - Suc 0) stp')"
+apply(subgoal_tac "start_of (layout_of aprog) as > 0")
+apply(case_tac tc, case_tac b, auto simp: crsp_l.simps)
+apply(case_tac "am = []", simp)
+apply(rule_tac x = "Suc (Suc 0)" in exI, simp add: t_steps.simps)
+proof-
+  fix  rn
+  assume h1: "am \<noteq> []" "abc_fetch as aprog = Some (Dec n e)" 
+             "start_of (layout_of aprog) as > 0"
+  hence h2: "\<exists> r'. <am> = Oc # r'"
+    by simp
+  from h1 and h2 show 
+   "\<exists>stp'>0. case t_steps (start_of (layout_of aprog) as, Bk # Bk # ires, <am> @ Bk\<^bsup>rn\<^esup>)
+    (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
+    start_of (layout_of aprog) as - Suc 0) stp' of
+    (s, ab) \<Rightarrow> s = Suc (start_of (layout_of aprog) as) \<and>
+    dec_inv_1 (layout_of aprog) n e (as, am) (s, ab) ires \<and> 
+    dec_inv_2 (layout_of aprog) n e (as, am) (s, ab) ires"
+  proof(erule_tac exE, simp, rule_tac x = "Suc 0" in exI, 
+        simp add: t_steps.simps)
+  qed
+next
+  assume "abc_fetch as aprog = Some (Dec n e)"
+  thus "0 < start_of (layout_of aprog) as"
+   apply(insert startof_not0[of "layout_of aprog" as], simp)
+   done
+qed
+
+lemma dec_crsp_ex1: 
+  "\<lbrakk>crsp_l (layout_of aprog) (as, am) tc ires;
+  abc_fetch as aprog = Some (Dec n e); 
+  abc_lm_v am n = 0\<rbrakk>
+  \<Longrightarrow> \<exists>stp > 0. crsp_l (layout_of aprog) (e, abc_lm_s am n 0) 
+    (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
+ (Dec n e), start_of (layout_of aprog) as - Suc 0) stp) ires"
+proof -
+  assume h1: "crsp_l (layout_of aprog) (as, am) tc ires" 
+       "abc_fetch as aprog = Some (Dec n e)" "abc_lm_v am n = 0"
+  hence h2: "\<exists> stp' > 0. (\<lambda> (s, l, r). 
+    (s = Suc (start_of (layout_of aprog) as) \<and> 
+ (dec_inv_1 (layout_of aprog) n e (as, am) (s, l, r)) ires)) 
+   (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+      (Dec n e), start_of (layout_of aprog) as - Suc 0) stp')"
+    apply(insert crsp_next_state[of aprog as am tc ires n e], auto)
+    done
+  from h1 and h2 show 
+ "\<exists>stp > 0. crsp_l (layout_of aprog) (e, abc_lm_s am n 0) 
+           (t_steps tc (ci (layout_of aprog) (start_of 
+                (layout_of aprog) as) (Dec n e), 
+                    start_of (layout_of aprog) as - Suc 0) stp) ires" 
+  proof(erule_tac exE, case_tac "(t_steps tc (ci (layout_of aprog)
+       (start_of (layout_of aprog) as) (Dec n e), start_of 
+          (layout_of aprog) as - Suc 0) stp')",  simp)
+    fix stp' a b c
+    assume h3: "stp' > 0 \<and> a = Suc (start_of (layout_of aprog) as) \<and> 
+               dec_inv_1 (layout_of aprog) n e (as, am) (a, b, c) ires" 
+             "abc_fetch as aprog = Some (Dec n e)" "abc_lm_v am n = 0"
+     "t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
+          (Dec n e), start_of (layout_of aprog) as - Suc 0) stp' 
+        = (Suc (start_of (layout_of aprog) as), b, c)" 
+    thus "\<exists>stp > 0. crsp_l (layout_of aprog) (e, abc_lm_s am n 0) 
+     (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
+           (Dec n e), start_of (layout_of aprog) as - Suc 0) stp) ires"
+    proof(erule_tac conjE, simp)
+      assume "dec_inv_1 (layout_of aprog) n e (as, am) 
+                    (Suc (start_of (layout_of aprog) as), b, c) ires"     
+             "abc_fetch as aprog = Some (Dec n e)" 
+             "abc_lm_v am n = 0"
+             " t_steps tc (ci (layout_of aprog) 
+              (start_of (layout_of aprog) as) (Dec n e), 
+               start_of (layout_of aprog) as - Suc 0) stp' 
+             = (Suc (start_of (layout_of aprog) as), b, c)"
+      hence h4: "\<exists>stp. (\<lambda>(s', l', r'). s' = 
+                     start_of (layout_of aprog) e \<and> 
+                dec_inv_1 (layout_of aprog) n e (as, am) (s', l', r') ires)
+                 (t_steps (start_of (layout_of aprog) as + 1, b, c) 
+                  (ci (layout_of aprog) 
+                      (start_of (layout_of aprog) as) (Dec n e), 
+                         start_of (layout_of aprog) as - Suc 0) stp)"
+	apply(rule_tac dec_inv_stop1, auto)
+	done
+      from  h3 and h4 show ?thesis
+	apply(erule_tac exE)
+	apply(rule_tac x = "stp' + stp" in exI, simp)
+	apply(case_tac "(t_steps (Suc (start_of (layout_of aprog) as),
+                     b, c) (ci (layout_of aprog) 
+                     (start_of (layout_of aprog) as) (Dec n e), 
+                      start_of (layout_of aprog) as - Suc 0) stp)", 
+              simp)
+	apply(rule_tac dec_inv_stop_cond1, auto)
+	done
+    qed
+  qed
+qed
+	  
+lemma dec_crsp_ex2:
+  "\<lbrakk>crsp_l (layout_of aprog) (as, am) tc ires; 
+    abc_fetch as aprog = Some (Dec n e);
+    0 < abc_lm_v am n\<rbrakk>
+ \<Longrightarrow> \<exists>stp > 0. crsp_l (layout_of aprog) 
+               (Suc as, abc_lm_s am n (abc_lm_v am n - Suc 0))
+   (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+              (Dec n e), start_of (layout_of aprog) as - Suc 0) stp) ires"
+proof -
+  assume h1: 
+ "crsp_l (layout_of aprog) (as, am) tc ires" 
+ "abc_fetch as aprog = Some (Dec n e)"
+  "abc_lm_v am n > 0"
+  hence h2: 
+ "\<exists> stp' > 0. (\<lambda> (s, l, r). (s = Suc (start_of (layout_of aprog) as)
+ \<and> (dec_inv_2 (layout_of aprog) n e (as, am) (s, l, r)) ires)) 
+(t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+              (Dec n e), start_of (layout_of aprog) as - Suc 0) stp')"
+    apply(insert crsp_next_state[of aprog as am tc ires n e], auto)
+    done
+  from h1 and h2 show 
+ "\<exists>stp >0. crsp_l (layout_of aprog) 
+   (Suc as, abc_lm_s am n (abc_lm_v am n - Suc 0))
+   (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+               (Dec n e), start_of (layout_of aprog) as - Suc 0) stp) ires"
+  proof(erule_tac exE, 
+        case_tac 
+ "(t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+      (Dec n e), start_of (layout_of aprog) as - Suc 0) stp')",  simp)
+    fix stp' a b c
+    assume h3: "0 < stp' \<and> a = Suc (start_of (layout_of aprog) as) \<and>
+               dec_inv_2 (layout_of aprog) n e (as, am) (a, b, c) ires" 
+               "abc_fetch as aprog = Some (Dec n e)" 
+               "abc_lm_v am n > 0"
+               "t_steps tc (ci (layout_of aprog) 
+                   (start_of (layout_of aprog) as) (Dec n e), 
+                     start_of (layout_of aprog) as - Suc 0) stp' 
+                  = (Suc (start_of (layout_of aprog) as), b, c)"
+    thus "?thesis"
+    proof(erule_tac conjE, simp)
+      assume 
+    "dec_inv_2 (layout_of aprog) n e (as, am) 
+      (Suc (start_of (layout_of aprog) as), b, c) ires" 
+    "abc_fetch as aprog = Some (Dec n e)" "abc_lm_v am n > 0"
+    "t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
+         (Dec n e), start_of (layout_of aprog) as - Suc 0) stp'
+             = (Suc (start_of (layout_of aprog) as), b, c)"
+      hence h4: 
+   "\<exists>stp. (\<lambda>(s', l', r'). s' = start_of (layout_of aprog) (Suc as) \<and>
+           dec_inv_2 (layout_of aprog) n e (as, am) (s', l', r') ires)
+             (t_steps (start_of (layout_of aprog) as + 1, b, c) 
+              (ci (layout_of aprog) (start_of (layout_of aprog) as) 
+               (Dec n e), start_of (layout_of aprog) as - Suc 0) stp)"
+	apply(rule_tac dec_stop2, auto)
+	done
+      from  h3 and h4 show ?thesis
+	apply(erule_tac exE)
+	apply(rule_tac x = "stp' + stp" in exI, simp)
+	apply(case_tac 
+         "(t_steps (Suc (start_of (layout_of aprog) as), b, c) 
+           (ci (layout_of aprog) (start_of (layout_of aprog) as)
+             (Dec n e), start_of (layout_of aprog) as - Suc 0) stp)"
+              ,simp)
+	apply(rule_tac dec_inv_stop_cond2, auto)
+	done
+    qed
+  qed
+qed
+
+lemma dec_crsp_ex_pre:
+  "\<lbrakk>ly = layout_of aprog; crsp_l ly (as, am) tc ires; 
+     abc_fetch as aprog = Some (Dec n e)\<rbrakk>
+ \<Longrightarrow> \<exists>stp > 0. crsp_l ly (abc_step_l (as, am) (Some (Dec n e))) 
+      (t_steps tc (ci (layout_of aprog) (start_of ly as) (Dec n e),
+                                       start_of ly as - Suc 0) stp) ires"
+apply(auto simp: abc_step_l.simps intro: dec_crsp_ex2 dec_crsp_ex1)
+done
+
+lemma dec_crsp_ex:
+  assumes layout: -- {* There is an Abacus program @{text "aprog"} with layout @{text "ly"} *}
+  "ly = layout_of aprog"
+  and dec: -- {* There is an @{text "Dec n e"} instruction at postion @{text "as"} of @{text "aprog"} *}
+      "abc_fetch as aprog = Some (Dec n e)"
+  and correspond: 
+  -- {* Abacus configuration @{text "(as, am)"} is in correspondence with TM 
+         configuration @{text "tc"}
+      *}
+  "crsp_l ly (as, am) tc ires"
+shows 
+   "\<exists>stp > 0. crsp_l ly (abc_step_l (as, am) (Some (Dec n e))) 
+      (t_steps tc (ci (layout_of aprog) (start_of ly as) (Dec n e),
+                                       start_of ly as - Suc 0) stp) ires"
+proof -
+  from dec_crsp_ex_pre layout dec correspond  show ?thesis by blast
+qed
+
+(*
+subsection {* Compilation of @{text "Goto n"}*}
+*)
+
+lemma goto_fetch: 
+     "fetch (ci (layout_of aprog) 
+         (start_of (layout_of aprog) as) (Goto n)) (Suc 0)  b
+     = (Nop, start_of (layout_of aprog) n)"
+apply(auto simp: ci.simps fetch.simps nth_of.simps 
+           split: block.splits)
+done
+
+text {*
+  Correctness of complied @{text "Goto n"}
+  *}
+
+lemma goto_crsp_ex_pre: 
+  "\<lbrakk>ly = layout_of aprog; 
+    crsp_l ly (as, am) tc ires;
+    abc_fetch as aprog = Some (Goto n)\<rbrakk>
+ \<Longrightarrow> \<exists>stp > 0. crsp_l ly (abc_step_l (as, am) (Some (Goto n))) 
+      (t_steps tc (ci (layout_of aprog) (start_of ly as) (Goto n), 
+                                        start_of ly as - Suc 0) stp) ires"
+apply(rule_tac x = 1 in exI)
+apply(simp add: abc_step_l.simps t_steps.simps t_step.simps)
+apply(case_tac tc, simp)
+apply(subgoal_tac "a = start_of (layout_of aprog) as", auto)
+apply(subgoal_tac "start_of (layout_of aprog) as > 0", simp)
+apply(auto simp: goto_fetch new_tape.simps crsp_l.simps)
+apply(rule startof_not0)
+done
+
+lemma goto_crsp_ex:
+  assumes layout: "ly = layout_of aprog"
+  and goto: "abc_fetch as aprog = Some (Goto n)"
+  and correspondence: "crsp_l ly (as, am) tc ires"
+  shows "\<exists>stp>0. crsp_l ly (abc_step_l (as, am) (Some (Goto n))) 
+              (t_steps tc (ci (layout_of aprog) (start_of ly as) (Goto n),
+                                           start_of ly as - Suc 0) stp) ires"
+proof -
+  from goto_crsp_ex_pre and layout goto correspondence show "?thesis" by blast
+qed
+
+subsection {*
+  The correctness of the compiler
+  *}
+
+declare abc_step_l.simps[simp del]
+
+lemma tm_crsp_ex: 
+         "\<lbrakk>ly = layout_of aprog;
+           crsp_l ly (as, am) tc ires; 
+           as < length aprog;
+           abc_fetch as aprog = Some ins\<rbrakk>
+      \<Longrightarrow> \<exists> n > 0. crsp_l ly (abc_step_l (as,am) (Some ins))
+               (t_steps tc (ci (layout_of aprog) (start_of ly as) 
+                  (ins), (start_of ly as) - (Suc 0)) n) ires"
+apply(case_tac "ins", simp)
+apply(auto intro: inc_crsp_ex_pre dec_crsp_ex goto_crsp_ex)
+done
+
+lemma start_of_pre: 
+  "n < length aprog \<Longrightarrow> start_of (layout_of aprog) n
+                     = start_of (layout_of (butlast aprog)) n"
+apply(induct n, simp add: start_of.simps, simp)
+apply(simp add: layout_of.simps start_of.simps)
+apply(subgoal_tac "n < length aprog - Suc 0", simp)
+apply(subgoal_tac "(aprog ! n) = (butlast aprog ! n)", simp)
+proof -
+  fix n
+  assume h1: "Suc n < length aprog"
+  thus "aprog ! n = butlast aprog ! n"
+    apply(case_tac "length aprog", simp, simp)
+    apply(insert nth_append[of "butlast aprog" "[last aprog]" n])
+    apply(subgoal_tac "(butlast aprog @ [last aprog]) = aprog")
+    apply(simp split: if_splits)
+    apply(rule append_butlast_last_id, case_tac aprog, simp, simp)
+    done
+next
+  fix n
+  assume "Suc n < length aprog"
+  thus "n < length aprog - Suc 0"
+    apply(case_tac aprog, simp, simp)
+    done
+qed
+    
+lemma zip_eq: "xs = ys \<Longrightarrow> zip xs zs = zip ys zs"
+by simp
+
+lemma tpairs_of_append_iff: "length aprog = Suc n \<Longrightarrow> 
+         tpairs_of aprog = tpairs_of (butlast aprog) @ 
+                     [(start_of (layout_of aprog) n, aprog ! n)]"
+apply(simp add: tpairs_of.simps)
+apply(insert zip_append[of "map (start_of (layout_of aprog)) [0..<n]"
+     "butlast aprog" "[start_of (layout_of aprog) n]" "[last aprog]"])
+apply(simp del: zip_append)
+apply(subgoal_tac "(butlast aprog @ [last aprog]) = aprog", auto)
+apply(rule_tac zip_eq, auto)
+apply(rule_tac start_of_pre, simp)
+apply(insert last_conv_nth[of aprog], case_tac aprog, simp, simp)
+apply(rule append_butlast_last_id, case_tac aprog, simp, simp)
+done
+
+lemma [simp]: "list_all (\<lambda>(n, tm). abacus.t_ncorrect (ci layout n tm))
+         (zip (map (start_of layout) [0..<length aprog]) aprog)"
+proof(induct "length aprog" arbitrary: aprog, simp)
+  fix x aprog
+  assume ind: "\<And>aprog. x = length aprog \<Longrightarrow> 
+        list_all (\<lambda>(n, tm). abacus.t_ncorrect (ci layout n tm))
+           (zip (map (start_of layout) [0..<length aprog]) aprog)"
+  and h: "Suc x = length (aprog::abc_inst list)"
+  have g1: "list_all (\<lambda>(n, tm). abacus.t_ncorrect (ci layout n tm)) 
+    (zip (map (start_of layout) [0..<length (butlast aprog)]) 
+                                                 (butlast aprog))"
+    using h
+    apply(rule_tac ind, auto)
+    done
+  have g2: "(map (start_of layout) [0..<length aprog]) = 
+                     map (start_of layout) ([0..<length aprog - 1] 
+         @ [length aprog - 1])"
+    using h
+    apply(case_tac aprog, simp, simp)
+    done
+  have "\<exists> xs a. aprog = xs @ [a]"
+    using h
+    apply(rule_tac x = "butlast aprog" in exI, 
+          rule_tac x = "last aprog" in exI)
+    apply(case_tac "aprog = []", simp, simp)
+    done
+  from this obtain xs where "\<exists> a. aprog = xs @ [a]" ..
+  from this obtain a where g3: "aprog = xs @ [a]" ..
+  from g1 and g2 and g3 show "list_all (\<lambda>(n, tm). 
+                              abacus.t_ncorrect (ci layout n tm)) 
+              (zip (map (start_of layout) [0..<length aprog]) aprog)"
+    apply(simp)
+    apply(auto simp: t_ncorrect.simps ci.simps  tshift.simps 
+          tinc_b_def tdec_b_def split: abc_inst.splits)
+    apply arith+
+    done
+qed
+
+lemma [intro]: "abc2t_correct aprog"
+apply(simp add: abc2t_correct.simps tpairs_of.simps 
+          split: abc_inst.splits)
+done
+
+lemma as_out: "\<lbrakk>ly = layout_of aprog; tprog = tm_of aprog; 
+                crsp_l ly (as, am) tc ires; length aprog \<le> as\<rbrakk> 
+            \<Longrightarrow> abc_step_l (as, am) (abc_fetch as aprog) = (as, am)"
+apply(simp add: abc_fetch.simps abc_step_l.simps)
+done
+
+lemma tm_merge_ex: 
+  "\<lbrakk>crsp_l (layout_of aprog) (as, am) tc ires; 
+    as < length aprog; 
+    abc_fetch as aprog = Some a; 
+    abc2t_correct aprog;
+    crsp_l (layout_of aprog) (abc_step_l (as, am) (Some a))
+     (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
+         a, start_of (layout_of aprog) as - Suc 0) n) ires; 
+    n > 0\<rbrakk>
+   \<Longrightarrow> \<exists>stp > 0. crsp_l (layout_of aprog) (abc_step_l (as, am) 
+                       (Some a)) (t_steps tc (tm_of aprog, 0) stp) ires"
+apply(case_tac "(t_steps tc (ci (layout_of aprog) 
+           (start_of (layout_of aprog) as) a, 
+            start_of (layout_of aprog) as - Suc 0) n)",  simp)
+apply(case_tac "(abc_step_l (as, am) (Some a))", simp)
+proof -
+  fix aa b c aaa ba 
+  assume h: 
+  "crsp_l (layout_of aprog) (as, am) tc ires" 
+  "as < length aprog" 
+  "abc_fetch as aprog = Some a" 
+  "crsp_l (layout_of aprog) (aaa, ba) (aa, b, c) ires" 
+  "abc2t_correct aprog" 
+  "n>0"
+  "t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as) a,
+      start_of (layout_of aprog) as - Suc 0) n = (aa, b, c)" 
+   "abc_step_l (as, am) (Some a) = (aaa, ba)"
+  hence "aa = start_of (layout_of aprog) aaa"
+    apply(simp add: crsp_l.simps)
+    done
+  from this and h show 
+  "\<exists>stp > 0. crsp_l (layout_of aprog) (aaa, ba) 
+                          (t_steps tc (tm_of aprog, 0) stp) ires"
+    apply(insert tms_out_ex[of "layout_of aprog" aprog 
+                "tm_of aprog" as am tc ires a n aa b c aaa ba], auto)
+    done
+qed
+ 
+lemma crsp_inside: 
+  "\<lbrakk>ly = layout_of aprog; 
+    tprog = tm_of aprog;
+    crsp_l ly (as, am) tc ires;
+    as < length aprog\<rbrakk> \<Longrightarrow> 
+    (\<exists> stp > 0. crsp_l ly (abc_step_l (as,am) (abc_fetch as aprog)) 
+                                         (t_steps tc (tprog, 0) stp) ires)"
+apply(case_tac "abc_fetch as aprog", simp add: abc_fetch.simps)
+proof -
+  fix a
+  assume "ly = layout_of aprog" 
+     "tprog = tm_of aprog" 
+     "crsp_l ly (as, am) tc ires" 
+     "as < length aprog" 
+     "abc_fetch as aprog = Some a"
+  thus "\<exists>stp > 0. crsp_l ly (abc_step_l (as, am) 
+                 (abc_fetch as aprog)) (t_steps tc (tprog, 0) stp) ires"
+    proof(insert tm_crsp_ex[of ly aprog as am tc ires a], 
+          auto intro: tm_merge_ex)
+  qed
+qed
+
+lemma crsp_outside: 
+  "\<lbrakk>ly = layout_of aprog; tprog = tm_of aprog;
+    crsp_l ly (as, am) tc ires; as \<ge> length aprog\<rbrakk>
+    \<Longrightarrow> (\<exists> stp. crsp_l ly (abc_step_l (as,am) (abc_fetch as aprog)) 
+                                         (t_steps tc (tprog, 0) stp) ires)"
+apply(subgoal_tac "abc_step_l (as, am) (abc_fetch as aprog)
+                = (as, am)", simp)
+apply(rule_tac x = 0 in exI, simp add: t_steps.simps)
+apply(rule as_out, simp+)
+done
+
+text {*
+  Single-step correntess of the compiler.
+*}
+lemma astep_crsp_pre: 
+      "\<lbrakk>ly = layout_of aprog; 
+        tprog = tm_of aprog;
+        crsp_l ly (as, am) tc ires\<rbrakk>
+       \<Longrightarrow> (\<exists> stp. crsp_l ly (abc_step_l (as,am) 
+                  (abc_fetch as aprog)) (t_steps tc (tprog, 0) stp) ires)"
+apply(case_tac "as < length aprog")
+apply(drule_tac crsp_inside, auto)
+apply(rule_tac crsp_outside, simp+)
+done
+
+text {*
+  Single-step correntess of the compiler.
+*}
+lemma astep_crsp_pre1: 
+      "\<lbrakk>ly = layout_of aprog;
+        tprog = tm_of aprog;
+        crsp_l ly (as, am) tc ires\<rbrakk>
+       \<Longrightarrow> (\<exists> stp. crsp_l ly (abc_step_l (as,am) 
+                  (abc_fetch as aprog)) (t_steps tc (tprog, 0) stp) ires)"
+apply(case_tac "as < length aprog")
+apply(drule_tac crsp_inside, auto)
+apply(rule_tac crsp_outside, simp+)
+done
+
+lemma astep_crsp:
+  assumes layout: 
+  -- {* There is a Abacus program @{text "aprog"} with layout @{text "ly"} *}
+  "ly = layout_of aprog"
+  and compiled: 
+  -- {* @{text "tprog"} is the TM compiled from @{text "aprog"} *}
+  "tprog = tm_of aprog"
+  and corresponds: 
+  -- {* Abacus configuration @{text "(as, am)"} is in correspondence with TM configuration
+   @{text "tc"} *}
+  "crsp_l ly (as, am) tc ires"
+  -- {* One step execution of @{text "aprog"} can be simulated by multi-step execution 
+  of @{text "tprog"} *}
+  shows "(\<exists> stp. crsp_l ly (abc_step_l (as,am) 
+                  (abc_fetch as aprog)) (t_steps tc (tprog, 0) stp) ires)"
+proof -
+  from astep_crsp_pre1 [OF layout compiled corresponds] show ?thesis .
+qed
+
+lemma steps_crsp_pre: 
+    "\<lbrakk>ly = layout_of aprog; tprog = tm_of aprog; 
+      crsp_l ly ac tc ires; ac' = abc_steps_l ac aprog n\<rbrakk> \<Longrightarrow> 
+        (\<exists> n'. crsp_l ly ac' (t_steps tc (tprog, 0) n') ires)"
+apply(induct n arbitrary: ac' ac tc, simp add: abc_steps_l.simps)
+apply(rule_tac x = 0 in exI)
+apply(case_tac ac, simp add: abc_steps_l.simps t_steps.simps)
+apply(case_tac ac, simp add: abc_steps_l.simps)
+apply(subgoal_tac 
+   "(\<exists> stp. crsp_l ly (abc_step_l (a, b)
+            (abc_fetch a aprog)) (t_steps tc (tprog, 0) stp) ires)")
+apply(erule exE)
+apply(subgoal_tac 
+   "\<exists>n'. crsp_l (layout_of aprog) 
+    (abc_steps_l (abc_step_l (a, b) (abc_fetch a aprog)) aprog n)
+         (t_steps ((t_steps tc (tprog, 0) stp)) (tm_of aprog, 0) n') ires")
+apply(erule exE)
+apply(subgoal_tac 
+    "t_steps (t_steps tc (tprog, 0) stp) (tm_of aprog, 0) n' =
+     t_steps tc (tprog, 0) (stp + n')")
+apply(rule_tac x = "stp + n'" in exI, simp)
+apply(auto intro: astep_crsp simp: t_step_add)
+done
+
+text {*
+  Multi-step correctess of the compiler.
+*}
+
+lemma steps_crsp: 
+  assumes layout: 
+  -- {* There is an Abacus program @{text "aprog"} with layout @{text "ly"} *}
+    "ly = layout_of aprog"
+  and compiled: 
+  -- {* @{text "tprog"} is the TM compiled from @{text "aprog"} *}
+  "tprog = tm_of aprog"
+  and correspond: 
+  -- {* Abacus configuration @{text "ac"} is in correspondence with TM configuration @{text "tc"} *}
+      "crsp_l ly ac tc ires"
+  and execution: 
+  -- {* @{text "ac'"} is the configuration obtained from @{text "n"}-step execution 
+      of @{text "aprog"} starting from configuration @{text "ac"} *}
+  "ac' = abc_steps_l ac aprog n" 
+  -- {* There exists steps @{text "n'"} steps, after these steps of execution, 
+  the Turing configuration such obtained is in correspondence with @{text "ac'"} *}
+  shows "(\<exists> n'. crsp_l ly ac' (t_steps tc (tprog, 0) n') ires)"
+proof -
+  from steps_crsp_pre [OF layout compiled correspond execution] show ?thesis .
+qed
+
+subsection {* The Mop-up machine *}
+
+fun mop_bef :: "nat \<Rightarrow> tprog"
+  where
+  "mop_bef 0 = []" |
+  "mop_bef (Suc n) = mop_bef n @ 
+       [(R, 2*n + 3), (W0, 2*n + 2), (R, 2*n + 1), (W1, 2*n + 2)]"
+
+definition mp_up :: "tprog"
+  where
+  "mp_up \<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 tMp :: "nat \<Rightarrow> nat \<Rightarrow> tprog"
+  where 
+  "tMp n off = tshift (mop_bef n @ tshift mp_up (2*n)) off"
+
+declare  mp_up_def[simp del]  tMp.simps[simp del] mop_bef.simps[simp del]
+(**********Begin: equiv among aba and turing***********)
+
+lemma tm_append_step: 
+ "\<lbrakk>t_ncorrect tp1; t_step tc (tp1, 0) = (s, l, r); s \<noteq> 0\<rbrakk> 
+ \<Longrightarrow> t_step tc (tp1 @ tp2, 0) = (s, l, r)"
+apply(simp add: t_step.simps)
+apply(case_tac tc, simp)
+apply(case_tac 
+       "(fetch tp1 a (case c of [] \<Rightarrow> Bk |
+               Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))", simp)
+apply(case_tac a, simp add: fetch.simps)
+apply(simp add: fetch.simps)
+apply(case_tac c, simp)
+apply(case_tac [!] "ab::block")
+apply(auto simp: nth_of.simps nth_append t_ncorrect.simps 
+           split: if_splits)
+done
+
+lemma state0_ind: "t_steps (0, l, r) (tp, 0) stp = (0, l, r)"
+apply(induct stp, simp add: t_steps.simps)
+apply(simp add: t_steps.simps t_step.simps fetch.simps new_tape.simps)
+done
+
+lemma tm_append_steps:  
+ "\<lbrakk>t_ncorrect tp1; t_steps tc (tp1, 0) stp = (s, l ,r); s \<noteq> 0\<rbrakk>
+  \<Longrightarrow> t_steps tc (tp1 @ tp2, 0) stp = (s, l, r)"
+apply(induct stp arbitrary: tc s l r)
+apply(case_tac tc,  simp)
+apply(simp add: t_steps.simps)
+proof -
+  fix stp tc s l r
+  assume h1: "\<And>tc s l r. \<lbrakk>t_ncorrect tp1; t_steps tc (tp1, 0) stp = 
+   (s, l, r); s \<noteq> 0\<rbrakk> \<Longrightarrow> t_steps tc (tp1 @ tp2, 0) stp = (s, l, r)"
+    and h2: "t_steps tc (tp1, 0) (Suc stp) = (s, l, r)" "s \<noteq> 0" 
+            "t_ncorrect tp1"
+  thus "t_steps tc (tp1 @ tp2, 0) (Suc stp) = (s, l, r)"
+    apply(simp add: t_steps.simps)
+    apply(case_tac "(t_step tc (tp1, 0))", simp)
+    proof-
+      fix a b c 
+      assume g1: "\<And>tc s l r. \<lbrakk>t_steps tc (tp1, 0) stp = (s, l, r); 
+                0 < s\<rbrakk> \<Longrightarrow> t_steps tc (tp1 @ tp2, 0) stp = (s, l, r)"
+	and g2: "t_step tc (tp1, 0) = (a, b, c)" 
+                "t_steps (a, b, c) (tp1, 0) stp = (s, l, r)" 
+                "0 < s" 
+                "t_ncorrect tp1"
+      hence g3: "a > 0"
+	apply(case_tac "a::nat", auto simp: t_steps.simps)
+	apply(simp add: state0_ind)
+	done
+      from g1 and g2 and this have g4: 
+                    "(t_step tc (tp1 @ tp2, 0)) = (a, b, c)"
+	apply(rule_tac tm_append_step, simp, simp, simp)
+	done
+      from g1 and g2 and g3 and g4 show 
+          "t_steps (t_step tc (tp1 @ tp2, 0)) (tp1 @ tp2, 0) stp
+                                                         = (s, l, r)"
+	apply(simp)
+	done
+    qed
+qed
+
+lemma shift_fetch: 
+ "\<lbrakk>n < length tp; 
+  (tp:: (taction \<times> nat) list) ! n = (aa, ba);
+   ba \<noteq> 0\<rbrakk> 
+   \<Longrightarrow> (tshift tp (length tp div 2)) ! n = 
+                     (aa , ba + length tp div 2)"
+apply(simp add: tshift.simps)
+done
+
+lemma tshift_length_equal: "length (tshift tp q) = length tp"
+apply(auto simp: tshift.simps)
+done
+
+thm nth_of.simps
+
+lemma [simp]: "t_ncorrect tp \<Longrightarrow> 2 * (length tp div 2) = length tp"
+apply(auto simp: t_ncorrect.simps)
+done
+
+lemma  tm_append_step_equal': 
+   "\<lbrakk>t_ncorrect tp; t_ncorrect tp'; off = length tp div 2\<rbrakk> \<Longrightarrow> 
+    (\<lambda> (s, l, r). ((\<lambda> (s', l', r'). 
+      (s'\<noteq> 0 \<longrightarrow> (s = s' + off \<and> l = l' \<and> r = r'))) 
+         (t_step (a, b, c) (tp', 0))))
+               (t_step (a + off, b, c) (tp @ tshift tp' off, 0))"
+apply(simp add: t_step.simps)
+apply(case_tac a, simp add: fetch.simps)
+apply(case_tac 
+"(fetch tp' a (case c of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))",
+ simp)
+apply(case_tac 
+"(fetch (tp @ tshift tp' (length tp div 2))
+        (Suc (nat + length tp div 2)) 
+           (case c of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))", 
+ simp)
+apply(case_tac "(new_tape aa (b, c))",
+      case_tac "(new_tape aaa (b, c))", simp, 
+      rule impI, simp add: fetch.simps split: block.splits option.splits)
+apply (auto simp: nth_of.simps t_ncorrect.simps 
+                      nth_append tshift_length_equal tshift.simps split: if_splits)
+done
+
+
+lemma  tm_append_step_equal: 
+ "\<lbrakk>t_ncorrect tp; t_ncorrect tp'; off = length tp div 2; 
+   t_step (a, b, c) (tp', 0) = (aa, ab, bb);  aa \<noteq> 0\<rbrakk>
+ \<Longrightarrow> t_step (a + length tp div 2, b, c) 
+        (tp @ tshift tp' (length tp div 2), 0)
+                          = (aa + length tp div 2, ab, bb)"
+apply(insert tm_append_step_equal'[of tp tp' off a b c], simp)
+apply(case_tac "(t_step (a + length tp div 2, b, c) 
+                   (tp @ tshift tp' (length tp div 2), 0))", simp)
+done
+
+lemma tm_append_steps_equal: 
+ "\<lbrakk>t_ncorrect tp; t_ncorrect tp'; off = length tp div 2\<rbrakk> \<Longrightarrow> 
+   (\<lambda> (s, l, r). ((\<lambda> (s', l', r'). ((s'\<noteq> 0 \<longrightarrow> s = s' + off \<and> l = l'
+                     \<and> r = r'))) (t_steps (a, b, c) (tp', 0) stp)))
+   (t_steps (a + off, b, c) (tp @ tshift tp' off, 0) stp)"
+apply(induct stp arbitrary : a b c, simp add: t_steps.simps)
+apply(simp add: t_steps.simps)
+apply(case_tac "(t_step (a, b, c) (tp', 0))", simp)
+apply(case_tac "aa = 0", simp add: state0_ind)
+apply(subgoal_tac "(t_step (a + length tp div 2, b, c) 
+                      (tp @ tshift tp' (length tp div 2), 0)) 
+  = (aa + length tp div 2, ba, ca)", simp)
+apply(rule tm_append_step_equal, auto)
+done
+
+(*********Begin: mop_up***************)
+type_synonym mopup_type = "t_conf \<Rightarrow> nat list \<Rightarrow> nat \<Rightarrow> block list \<Rightarrow> bool"
+
+fun mopup_stop :: "mopup_type"
+  where
+  "mopup_stop (s, l, r) lm n ires= 
+        (\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires \<and> r = <abc_lm_v lm n> @ Bk\<^bsup>rn\<^esup>)"
+
+fun mopup_bef_erase_a :: "mopup_type"
+  where
+  "mopup_bef_erase_a (s, l, r) lm n ires= 
+         (\<exists> ln m rn. l = Bk \<^bsup>ln\<^esup> @ Bk # Bk # ires \<and> 
+                  r = Oc\<^bsup>m \<^esup>@ Bk # <(drop ((s + 1) div 2) lm)> @ Bk\<^bsup>rn\<^esup>)"
+
+fun mopup_bef_erase_b :: "mopup_type"
+  where
+  "mopup_bef_erase_b (s, l, r) lm n ires = 
+      (\<exists> ln m rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires \<and> r = Bk # Oc\<^bsup>m\<^esup> @ Bk # 
+                                      <(drop (s div 2) lm)> @ Bk\<^bsup>rn\<^esup>)"
+
+
+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\<^bsup>m1\<^esup> @ Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires \<and> 
+     (r = Oc\<^bsup>m2\<^esup> @ Bk # <(drop (Suc n) lm)> @ Bk\<^bsup>rn \<^esup>\<or> 
+     (r = Oc\<^bsup>m2\<^esup> \<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\<^bsup>lnr \<^esup>@ Oc\<^bsup>m \<^esup>@ Bk\<^bsup>lnl\<^esup> @ Bk # Bk # ires \<and> 
+                                   (r = <ml> @ Bk\<^bsup>rn\<^esup>))"
+
+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\<^bsup>lnr \<^esup>@ Oc\<^bsup>m \<^esup>@ Bk\<^bsup>lnl\<^esup> @ Bk # Bk # ires \<and> 
+     (r = Bk # <ml> @ Bk\<^bsup>rn \<^esup>\<or>
+      r = Bk # Bk # <ml> @ Bk\<^bsup>rn\<^esup>))"
+
+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\<^bsup>lnr \<^esup>@ Oc\<^bsup>m \<^esup>@ Bk\<^bsup>lnl\<^esup> @ Bk # Bk # ires \<and> 
+    (r = <ml> @ Bk\<^bsup>rn \<^esup>\<or> r = Bk # <ml> @ Bk\<^bsup>rn\<^esup>))"
+
+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\<^bsup>lnr \<^esup>@ Oc\<^bsup>m \<^esup>@ Bk\<^bsup>lnl\<^esup> @ Bk # Bk # ires \<and> r = Bk\<^bsup>rn\<^esup>) \<or>
+    (l = Oc\<^bsup>m - 1\<^esup> @ Bk\<^bsup>lnl\<^esup> @ Bk # Bk # ires \<and> r = Oc # Bk\<^bsup>rn\<^esup>)))"
+
+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\<^bsup>m1\<^esup> @ Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires \<and> r = Oc\<^bsup>m2\<^esup> @ Bk\<^bsup>rn\<^esup>)) 
+        \<and> (hd r = Bk \<longrightarrow> (l = Bk\<^bsup>ln\<^esup> @ Bk # ires \<and> r = Bk # Oc\<^bsup>m1 + m2\<^esup> @ Bk\<^bsup>rn\<^esup>)))"
+
+
+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)"
+
+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 mopup_fetch_0[simp]: 
+     "(fetch (mop_bef n @ tshift mp_up (2 * n)) 0 b) = (Nop, 0)"
+by(simp add: fetch.simps)
+
+lemma mop_bef_length[simp]: "length (mop_bef n) = 4 * n"
+apply(induct n, simp add: mop_bef.simps, simp add: mop_bef.simps)
+done
+
+thm findnth_nth
+lemma mop_bef_nth: 
+  "\<lbrakk>q < n; x < 4\<rbrakk> \<Longrightarrow> mop_bef n ! (4 * q + x) = 
+                             mop_bef (Suc q) ! ((4 * q) + x)"
+apply(induct n, simp)
+apply(case_tac "q < n", simp add: mop_bef.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 (mop_bef n @ tshift mp_up (2 * n)) s Oc) = (W0, s + 1)"
+apply(subgoal_tac "\<exists> q. s = 2*q + 1", auto)
+apply(subgoal_tac "length (mop_bef n) = 4*n")
+apply(auto simp: fetch.simps nth_of.simps nth_append)
+apply(subgoal_tac "mop_bef n ! (4 * q + 1) = 
+                      mop_bef (Suc q) ! ((4 * q) + 1)", 
+      simp add: mop_bef.simps nth_append)
+apply(rule mop_bef_nth, auto)
+done
+
+lemma fetch_bef_erase_a_b[simp]:
+  "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n; s mod 2 = Suc 0\<rbrakk>
+   \<Longrightarrow>  (fetch (mop_bef n @ tshift mp_up (2 * n)) s Bk) = (R, s + 2)"
+apply(subgoal_tac "\<exists> q. s = 2*q + 1", auto)
+apply(subgoal_tac "length (mop_bef n) = 4*n")
+apply(auto simp: fetch.simps nth_of.simps nth_append)
+apply(subgoal_tac "mop_bef n ! (4 * q + 0) = 
+                       mop_bef (Suc q) ! ((4 * q + 0))", 
+      simp add: mop_bef.simps nth_append)
+apply(rule mop_bef_nth, auto)
+done
+
+lemma fetch_bef_erase_b_b: 
+  "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n; s mod 2 = 0\<rbrakk> \<Longrightarrow> 
+     (fetch (mop_bef n @ tshift mp_up (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 "mop_bef n ! (4 * nat + 2) = 
+                     mop_bef (Suc nat) ! ((4 * nat) + 2)", 
+      simp add: mop_bef.simps nth_append)
+apply(rule mop_bef_nth, auto)
+done
+
+lemma fetch_jump_over1_o: 
+ "fetch (mop_bef n @ tshift mp_up (2 * n)) (Suc (2 * n)) Oc
+  = (R, Suc (2 * n))"
+apply(subgoal_tac "length (mop_bef n) = 4 * n")
+apply(auto simp: fetch.simps nth_of.simps mp_up_def nth_append 
+                 tshift.simps)
+done
+
+lemma fetch_jump_over1_b: 
+ "fetch (mop_bef n @ tshift mp_up (2 * n)) (Suc (2 * n)) Bk 
+ = (R, Suc (Suc (2 * n)))"
+apply(subgoal_tac "length (mop_bef n) = 4 * n")
+apply(auto simp: fetch.simps nth_of.simps mp_up_def 
+                 nth_append tshift.simps)
+done
+
+lemma fetch_aft_erase_a_o: 
+ "fetch (mop_bef n @ tshift mp_up (2 * n)) (Suc (Suc (2 * n))) Oc 
+ = (W0, Suc (2 * n + 2))"
+apply(subgoal_tac "length (mop_bef n) = 4 * n")
+apply(auto simp: fetch.simps nth_of.simps mp_up_def 
+                 nth_append tshift.simps)
+done
+
+lemma fetch_aft_erase_a_b: 
+ "fetch (mop_bef n @ tshift mp_up (2 * n)) (Suc (Suc (2 * n))) Bk
+  = (L, Suc (2 * n + 4))"
+apply(subgoal_tac "length (mop_bef n) = 4 * n")
+apply(auto simp: fetch.simps nth_of.simps mp_up_def 
+                 nth_append tshift.simps)
+done
+
+lemma fetch_aft_erase_b_b: 
+ "fetch (mop_bef n @ tshift mp_up (2 * n)) (2*n + 3) Bk
+  = (R, Suc (2 * n + 3))"
+apply(subgoal_tac "length (mop_bef n) = 4 * n")
+apply(subgoal_tac "2*n + 3 = Suc (2*n + 2)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps mp_up_def nth_append tshift.simps)
+done
+
+lemma fetch_aft_erase_c_o: 
+ "fetch (mop_bef n @ tshift mp_up (2 * n)) (2 * n + 4) Oc 
+ = (W0, Suc (2 * n + 2))"
+apply(subgoal_tac "length (mop_bef n) = 4 * n")
+apply(subgoal_tac "2*n + 4 = Suc (2*n + 3)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps mp_up_def nth_append tshift.simps)
+done
+
+lemma fetch_aft_erase_c_b: 
+ "fetch (mop_bef n @ tshift mp_up (2 * n)) (2 * n + 4) Bk 
+ = (R, Suc (2 * n + 1))"
+apply(subgoal_tac "length (mop_bef n) = 4 * n")
+apply(subgoal_tac "2*n + 4 = Suc (2*n + 3)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps mp_up_def nth_append tshift.simps)
+done
+
+lemma fetch_left_moving_o: 
+ "(fetch (mop_bef n @ tshift mp_up (2 * n)) (2 * n + 5) Oc) 
+ = (L, 2*n + 6)"
+apply(subgoal_tac "length (mop_bef n) = 4 * n")
+apply(subgoal_tac "2*n + 5 = Suc (2*n + 4)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps mp_up_def nth_append tshift.simps)
+done
+
+lemma fetch_left_moving_b: 
+ "(fetch (mop_bef n @ tshift mp_up (2 * n)) (2 * n + 5) Bk)
+  = (L, 2*n + 5)"
+apply(subgoal_tac "length (mop_bef n) = 4 * n")
+apply(subgoal_tac "2*n + 5 = Suc (2*n + 4)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps mp_up_def nth_append tshift.simps)
+done
+
+lemma fetch_jump_over2_b:
+  "(fetch (mop_bef n @ tshift mp_up (2 * n)) (2 * n + 6) Bk) 
+ = (R, 0)"
+apply(subgoal_tac "length (mop_bef n) = 4 * n")
+apply(subgoal_tac "2*n + 6 = Suc (2*n + 5)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps mp_up_def nth_append tshift.simps)
+done
+
+lemma fetch_jump_over2_o: 
+"(fetch (mop_bef n @ tshift mp_up (2 * n)) (2 * n + 6) Oc) 
+ = (L, 2*n + 6)"
+apply(subgoal_tac "length (mop_bef n) = 4 * n")
+apply(subgoal_tac "2*n + 6 = Suc (2*n + 5)", simp only: fetch.simps)
+apply(auto simp: nth_of.simps mp_up_def nth_append tshift.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
+
+lemma [simp]: 
+"\<lbrakk>n < length lm; 0 < s; s mod 2 = Suc 0; 
+  mopup_bef_erase_a (s, l, Oc # xs) lm n ires; 
+  Suc s \<le> 2 * n\<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_abc_lm_v_simp[simp]: 
+   "n < length lm \<Longrightarrow> drop n lm = abc_lm_v lm n # drop (Suc n) lm"
+apply(auto simp: abc_lm_v.simps)
+apply(drule drop_Suc_conv_tl, simp)
+done
+lemma [simp]: "(\<exists>rna. Bk\<^bsup>rn\<^esup> = Bk # Bk\<^bsup>rna\<^esup>) \<or> Bk\<^bsup>rn\<^esup> = []"
+apply(case_tac rn, simp, auto)
+done
+
+lemma [simp]: "\<exists>lna. Bk # Bk\<^bsup>ln\<^esup> = Bk\<^bsup>lna\<^esup>"
+apply(rule_tac x = "Suc ln" in exI, auto)
+done
+
+lemma mopup_bef_erase_a_2_jump_over[simp]: 
+ "\<lbrakk>n < length lm; 0 < s; s mod 2 = Suc 0; 
+   mopup_bef_erase_a (s, l, Bk # xs) lm n ires; Suc s = 2 * n\<rbrakk> 
+\<Longrightarrow> mopup_jump_over1 (Suc (2 * n), Bk # l, xs) lm n ires"
+apply(auto simp: mopup_bef_erase_a.simps mopup_jump_over1.simps)
+apply(case_tac m, simp)
+apply(rule_tac x = "Suc ln" in exI, rule_tac x = 0 in exI, 
+      simp add: tape_of_nl_abv)
+apply(case_tac "drop (Suc n) lm", auto simp: tape_of_nat_list.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)
+apply(rule_tac x = "Suc (abc_lm_v lm (Suc (s div 2)))" in exI, 
+      rule_tac x = rn in exI, simp, simp)
+apply(subgoal_tac "(Suc (Suc (s div 2))) \<le> n", simp, 
+      rule_tac Suc_Suc_div, auto)
+done
+
+lemma mopup_false2: 
+ "\<lbrakk>n < length lm; 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]:
+  "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n; 
+   s mod 2 = Suc 0; 
+   mopup_bef_erase_a (s, l, Bk # xs) lm n ires; 
+   r = Bk # xs\<rbrakk>
+ \<Longrightarrow> (Suc s = 2 * n \<longrightarrow> 
+             mopup_jump_over1 (Suc (2 * n), Bk # l, xs) lm n ires) \<and>
+     (Suc s \<noteq> 2 * n \<longrightarrow> 
+       (Suc (Suc s) \<le> 2 * n \<longrightarrow> 
+          mopup_bef_erase_a (Suc (Suc s), Bk # l, xs) lm n ires) \<and> 
+       (\<not> Suc (Suc s) \<le> 2 * n \<longrightarrow> 
+          mopup_aft_erase_a (Suc (Suc s), Bk # l, xs) lm n ires))"
+apply(auto elim: mopup_false2)
+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;
+     mopup_bef_erase_a (s, l, []) lm n ires; r = []\<rbrakk>
+    \<Longrightarrow> (Suc s = 2 * n \<longrightarrow> 
+              mopup_jump_over1 (Suc (2 * n), Bk # l, []) lm n ires) \<and>
+        (Suc s \<noteq> 2 * n \<longrightarrow> 
+             (Suc (Suc s) \<le> 2 * n \<longrightarrow> 
+                 mopup_bef_erase_a (Suc (Suc s), Bk # l, []) lm n ires) \<and>
+             (\<not> Suc (Suc s) \<le> 2 * n \<longrightarrow> 
+                 mopup_aft_erase_a (Suc (Suc s), Bk # l, []) lm n ires))"
+apply(auto)
+done
+
+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)
+apply(case_tac "m2", simp, simp, rule_tac x = rn in exI, 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 = m1 in exI, 
+      rule_tac x = m2 in exI, simp add: )
+apply(rule_tac x = 0 in exI, auto)
+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, simp, case_tac rn, simp, simp)
+apply(case_tac list, auto simp: tape_of_nl_abv 
+                                tape_of_nat_list.simps )
+apply(case_tac a, simp, rule_tac x = rn in exI, 
+      rule_tac x = "[]" in exI,
+       simp add: tape_of_nat_list.simps, simp)
+apply(rule_tac x = rn in exI, rule_tac x = "[nat]" in exI, 
+      simp add: tape_of_nat_list.simps )
+apply(case_tac a, simp, rule_tac x = rn in exI, 
+       rule_tac x = "aa # lista" in exI, simp, simp)
+apply(rule_tac x = rn in exI, rule_tac x = "nat # aa # lista" in exI, 
+       simp add: tape_of_nat_list.simps )
+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(rule_tac x = lnl in exI, simp, rule_tac x = rn in exI, simp)
+apply(subgoal_tac "ml = []", simp)
+apply(rule_tac xs = xs and rn = rn in BkCons_nil, simp, auto)
+apply(subgoal_tac "ml = []", auto)
+apply(rule_tac xs = xs and rn = rn in BkCons_nil, 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, rule_tac x = lnl in exI, simp, 
+      rule_tac x = 0 in exI, simp)
+apply(rule_tac x = lnl in exI, rule_tac x = nat in exI, 
+      rule_tac x = "Suc 0" in exI, simp)
+apply(case_tac ml, simp, case_tac rn, simp, simp)
+apply(case_tac list, auto simp: tape_of_nl_abv tape_of_nat_list.simps)
+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 [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, case_tac rn, simp, simp, simp)
+apply(case_tac list, auto simp: tape_of_nl_abv 
+                        tape_of_nat_list.simps tape_of_nat_abv )
+apply(case_tac a, rule_tac x = rn in exI, 
+      rule_tac x = "[]" in exI, simp add: tape_of_nat_list.simps)
+apply(rule_tac x = rn in exI, rule_tac x = "[nat]" in exI, 
+      simp add: tape_of_nat_list.simps )
+apply(case_tac a, simp, rule_tac x = rn in exI, 
+      rule_tac x = "aa # lista" in exI, simp)
+apply(rule_tac x = rn in exI, rule_tac x = "nat # aa # lista" in exI, 
+      simp add: tape_of_nat_list.simps )
+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(rule_tac xs = xs and rn = rn in BkCons_nil, simp, simp, 
+      rule conjI)
+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 = 0 in exI, 
+      rule_tac x = rn in exI, simp, simp)
+apply(rule_tac x = lnl in exI, rule_tac x = nat 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, simp)
+apply(rule_tac x = lnl in exI, rule_tac x = 0 in exI, 
+      rule_tac x = 0 in exI, simp, 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 [intro]: "\<exists>lna. Bk # Bk\<^bsup>ln\<^esup> = Bk\<^bsup>lna\<^esup> @ [Bk]"
+apply(simp only: exp_ind[THEN sym], auto)
+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]: "\<exists>rna. Oc # Oc\<^bsup>a\<^esup> @ Bk\<^bsup>rn\<^esup> = <a> @ Bk\<^bsup>rna\<^esup>"
+apply(case_tac a, auto simp: tape_of_nat_abv )
+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)
+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 (t_step (s, l, r) 
+       ((mop_bef n @ tshift mp_up (2 * n)), 0)) lm n ires"
+apply(auto split:if_splits simp add:t_step.simps,
+      tactic {* ALLGOALS (resolve_tac [@{thm "fetch_intro"}]) *})
+apply(simp_all add: mopupfetchs new_tape.simps)
+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 (t_steps (s, l, r) 
+                   ((mop_bef n @ tshift mp_up (2 * n)), 0) stp) lm n ires"
+apply(induct stp, simp add: t_steps.simps)
+apply(simp add: t_steps_ind)
+apply(case_tac "(t_steps (s, l, r) 
+                (mop_bef n @ tshift mp_up (2 * n), 0) stp)", simp)
+apply(rule_tac mopup_inv_step, simp, simp)
+done
+
+lemma [simp]: 
+ "\<lbrakk>n < length lm; Suc 0 \<le> n\<rbrakk> \<Longrightarrow> 
+            mopup_bef_erase_a (Suc 0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>) lm n ires"
+apply(auto simp: mopup_bef_erase_a.simps  abc_lm_v.simps)
+apply(case_tac lm, simp, case_tac list, simp, simp)
+apply(rule_tac x = "Suc a" in exI, rule_tac x = rn in exI, simp)
+done
+  
+lemma [simp]:
+  "lm \<noteq> [] \<Longrightarrow> mopup_jump_over1 (Suc 0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>) lm 0  ires"
+apply(auto simp: mopup_jump_over1.simps)
+apply(rule_tac x = ln in exI, rule_tac x = 0 in exI, simp add: )
+apply(case_tac lm, simp, simp add: abc_lm_v.simps)
+apply(case_tac rn, simp)
+apply(case_tac list, rule_tac disjI2, 
+      simp add: tape_of_nl_abv tape_of_nat_list.simps)
+apply(rule_tac disjI1,
+      simp add: tape_of_nl_abv tape_of_nat_list.simps )
+apply(rule_tac disjI1, case_tac list, 
+      simp add: tape_of_nl_abv tape_of_nat_list.simps, 
+      rule_tac x = nat in exI, simp)
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps )
+done
+
+lemma mopup_init: 
+ "\<lbrakk>n < length lm; crsp_l ly (as, lm) (ac, l, r) ires\<rbrakk> \<Longrightarrow> 
+                               mopup_inv (Suc 0, l, r) lm n ires"
+apply(auto simp: crsp_l.simps mopup_inv.simps)
+apply(case_tac n, simp, auto simp: mopup_bef_erase_a.simps )
+apply(rule_tac x = "Suc (hd lm)" in exI, rule_tac x = rn in exI, simp)
+apply(case_tac lm, simp, case_tac list, simp, case_tac lista, simp add: abc_lm_v.simps)
+apply(simp add: tape_of_nl_abv tape_of_nat_list.simps abc_lm_v.simps)
+apply(simp add: mopup_jump_over1.simps)
+apply(rule_tac x = 0 in exI, rule_tac x = 0 in exI, auto)
+apply(case_tac [!] n, simp_all)
+apply(case_tac [!] lm, simp, case_tac list, simp)
+apply(case_tac rn, simp add: tape_of_nl_abv tape_of_nat_list.simps abc_lm_v.simps)
+apply(erule_tac x = nat in allE, simp add: tape_of_nl_abv tape_of_nat_list.simps abc_lm_v.simps)
+apply(simp add: abc_lm_v.simps, auto)
+apply(case_tac list, simp_all add: tape_of_nl_abv tape_of_nat_list.simps abc_lm_v.simps) 
+apply(erule_tac x = rn in allE, simp_all)
+done
+
+fun abc_mopup_stage1 :: "t_conf \<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 :: "t_conf \<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 :: "t_conf \<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)"
+
+fun abc_mopup_measure :: "(t_conf \<times> nat) \<Rightarrow> (nat \<times> nat \<times> nat)"
+  where
+  "abc_mopup_measure (c, n) = 
+    (abc_mopup_stage1 c n, abc_mopup_stage2 c n, 
+                                       abc_mopup_stage3 c n)"
+
+definition abc_mopup_LE ::
+   "(((nat \<times> block list \<times> block list) \<times> nat) \<times> 
+    ((nat \<times> block list \<times> block list) \<times> nat)) set"
+  where
+  "abc_mopup_LE \<equiv> (inv_image lex_triple abc_mopup_measure)"
+
+lemma wf_abc_mopup_le[intro]: "wf abc_mopup_LE"
+by(auto intro:wf_inv_image wf_lex_triple simp:abc_mopup_LE_def)
+
+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
+
+lemma mopup_halt_pre: 
+ "\<lbrakk>n < length lm; mopup_inv (Suc 0, l, r) lm n ires; wf abc_mopup_LE\<rbrakk>
+ \<Longrightarrow>  \<forall>na. \<not> (\<lambda>(s, l, r) n. s = 0) (t_steps (Suc 0, l, r)
+      (mop_bef n @ tshift mp_up (2 * n), 0) na) n \<longrightarrow>
+       ((t_steps (Suc 0, l, r) (mop_bef n @ tshift mp_up (2 * n), 0) 
+        (Suc na), n),
+       t_steps (Suc 0, l, r) (mop_bef n @ tshift mp_up (2 * n), 0)
+         na, n) \<in> abc_mopup_LE"
+apply(rule allI, rule impI, simp add: t_steps_ind)
+apply(subgoal_tac "mopup_inv (t_steps (Suc 0, l, r) 
+                     (mop_bef n @ tshift mp_up (2 * n), 0) na) lm n ires")
+apply(case_tac "(t_steps (Suc 0, l, r) 
+               (mop_bef n @ tshift mp_up (2 * n), 0) na)",  simp)
+proof -
+  fix na a b c
+  assume  "n < length lm" "mopup_inv (a, b, c) lm n ires" "0 < a"
+  thus "((t_step (a, b, c) (mop_bef n @ tshift mp_up (2 * n), 0), n),
+         (a, b, c), n) \<in> abc_mopup_LE"
+    apply(auto split:if_splits simp add:t_step.simps mopup_inv.simps,
+      tactic {* ALLGOALS (resolve_tac [@{thm "fetch_intro"}]) *})
+    apply(simp_all add: mopupfetchs new_tape.simps abc_mopup_LE_def 
+                   lex_triple_def lex_pair_def )
+    done
+next
+  fix na
+  assume "n < length lm" "mopup_inv (Suc 0, l, r) lm n ires"
+  thus "mopup_inv (t_steps (Suc 0, l, r) 
+       (mop_bef n @ tshift mp_up (2 * n), 0) na) lm n ires"
+    apply(rule mopup_inv_steps)
+    done
+qed
+
+lemma mopup_halt: "\<lbrakk>n < length lm; crsp_l ly (as, lm) (s, l, r) ires\<rbrakk> \<Longrightarrow> 
+  \<exists> stp. (\<lambda> (s, l, r). s = 0) (t_steps (Suc 0, l, r) 
+        ((mop_bef n @ tshift mp_up (2 * n)), 0) stp)"
+apply(subgoal_tac "mopup_inv (Suc 0, l, r) lm n ires")
+apply(insert wf_abc_mopup_le)
+apply(insert halt_lemma[of abc_mopup_LE 
+    "\<lambda> ((s, l, r), n). s = 0" 
+    "\<lambda> stp. (t_steps (Suc 0, l, r) ((mop_bef n @ tshift mp_up (2 * n))
+           , 0) stp, n)"], auto)
+apply(insert mopup_halt_pre[of n lm l r], simp, erule exE)
+apply(rule_tac x = na in exI, case_tac "(t_steps (Suc 0, l, r) 
+          (mop_bef n @ tshift mp_up (2 * n), 0) na)", simp)
+apply(rule_tac mopup_init, auto)
+done
+(***End: mopup stop****)
+
+lemma mopup_halt_conf_pre: 
+ "\<lbrakk>n < length lm; crsp_l ly (as, lm) (s, l, r) ires\<rbrakk> 
+  \<Longrightarrow> \<exists> na. (\<lambda> (s', l', r').  s' = 0 \<and> mopup_stop (s', l', r') lm n ires)
+      (t_steps (Suc 0, l, r) 
+            ((mop_bef n @ tshift mp_up (2 * n)), 0) na)"
+apply(subgoal_tac "\<exists> stp. (\<lambda> (s, l, r). s = 0) 
+ (t_steps (Suc 0, l, r) ((mop_bef n @ tshift mp_up (2 * n)), 0) stp)",
+       erule exE)
+apply(rule_tac x = stp in exI, 
+      case_tac "(t_steps (Suc 0, l, r) 
+          (mop_bef n @ tshift mp_up (2 * n), 0) stp)", simp)
+apply(subgoal_tac " mopup_inv (Suc 0, l, r) lm n ires")
+apply(subgoal_tac "mopup_inv (t_steps (Suc 0, l, r) 
+            (mop_bef n @ tshift mp_up (2 * n), 0) stp) lm n ires", simp)
+apply(simp only: mopup_inv.simps)
+apply(rule_tac mopup_inv_steps, simp, simp)
+apply(rule_tac mopup_init, simp, simp)
+apply(rule_tac mopup_halt, simp, simp)
+done
+
+lemma  mopup_halt_conf:
+  assumes len: "n < length lm"
+  and correspond: "crsp_l ly (as, lm) (s, l, r) ires"
+  shows 
+  "\<exists> na. (\<lambda> (s', l', r'). ((\<exists>ln rn. s' = 0 \<and> l' = Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires
+                           \<and> r' = Oc\<^bsup>Suc (abc_lm_v lm n)\<^esup> @ Bk\<^bsup>rn\<^esup>)))
+             (t_steps (Suc 0, l, r) 
+                  ((mop_bef n @ tshift mp_up (2 * n)), 0) na)"
+using len correspond mopup_halt_conf_pre[of n lm ly as s l r ires]
+apply(simp add: mopup_stop.simps tape_of_nat_abv tape_of_nat_list.simps)
+done
+
+subsection {* Final results about Abacus machine *}
+
+lemma mopup_halt_bef: "\<lbrakk>n < length lm; crsp_l ly (as, lm) (s, l, r) ires\<rbrakk> 
+    \<Longrightarrow> \<exists>stp. (\<lambda>(s, l, r). s \<noteq> 0 \<and> ((\<lambda> (s', l', r'). s' = 0)
+   (t_step (s, l, r) (mop_bef n @ tshift mp_up (2 * n), 0))))
+    (t_steps (Suc 0, l, r) (mop_bef n @ tshift mp_up (2 * n), 0) stp)"
+apply(insert mopup_halt[of n lm ly as s l r ires], simp, erule_tac exE)
+proof -
+  fix stp
+  assume "n < length lm" 
+         "crsp_l ly (as, lm) (s, l, r) ires" 
+         "(\<lambda>(s, l, r). s = 0) 
+            (t_steps (Suc 0, l, r) 
+              (mop_bef n @ tshift mp_up (2 * n), 0) stp)"
+  thus "\<exists>stp. (\<lambda>(s, ab). 0 < s \<and> (\<lambda>(s', l', r'). s' = 0) 
+   (t_step (s, ab) (mop_bef n @ tshift mp_up (2 * n), 0))) 
+    (t_steps (Suc 0, l, r) (mop_bef n @ tshift mp_up (2 * n), 0) stp)"
+  proof(induct stp, simp add: t_steps.simps, simp)
+    fix stpa
+    assume h1: 
+      "(\<lambda>(s, l, r). s = 0) (t_steps (Suc 0, l, r) 
+           (mop_bef n @ tshift mp_up (2 * n), 0) stpa) \<Longrightarrow>
+       \<exists>stp. (\<lambda>(s, ab). 0 < s \<and> (\<lambda>(s', l', r'). s' = 0) 
+         (t_step (s, ab) (mop_bef n @ tshift mp_up (2 * n), 0))) 
+            (t_steps (Suc 0, l, r) 
+              (mop_bef n @ tshift mp_up (2 * n), 0) stp)"
+      and h2: 
+        "(\<lambda>(s, l, r). s = 0) (t_steps (Suc 0, l, r) 
+                    (mop_bef n @ tshift mp_up (2 * n), 0) (Suc stpa))"
+         "n < length lm" 
+         "crsp_l ly (as, lm) (s, l, r) ires"
+    thus "\<exists>stp. (\<lambda>(s, ab). 0 < s \<and> (\<lambda>(s', l', r'). s' = 0) 
+             (t_step (s, ab) (mop_bef n @ tshift mp_up (2 * n), 0))) (
+                t_steps (Suc 0, l, r) 
+                  (mop_bef n @ tshift mp_up (2 * n), 0) stp)"
+      apply(case_tac "(\<lambda>(s, l, r). s = 0) (t_steps (Suc 0, l, r) 
+                     (mop_bef n @ tshift mp_up (2 * n), 0) stpa)", 
+            simp)
+      apply(rule_tac x = "stpa" in exI)
+      apply(case_tac "(t_steps (Suc 0, l, r) 
+                         (mop_bef n @ tshift mp_up (2 * n), 0) stpa)",
+            simp add: t_steps_ind)
+      done
+  qed
+qed
+
+lemma tshift_nth_state0: "\<lbrakk>n < length tp; tp ! n = (a, 0)\<rbrakk>
+    \<Longrightarrow> tshift tp off ! n = (a, 0)"
+apply(induct n, case_tac tp, simp)
+apply(auto simp: tshift.simps)
+done
+
+lemma shift_length: "length (tshift tp n) = length tp"
+apply(auto simp: tshift.simps)
+done
+
+lemma even_Suc_le: "\<lbrakk>y mod 2 = 0; 2 * x < y\<rbrakk> \<Longrightarrow> Suc (2 * x) < y"
+by arith
+
+lemma [simp]: "(4::nat) * n mod 2 = 0"
+by arith
+
+lemma tm_append_fetch_equal: 
+  "\<lbrakk>t_ncorrect (tm_of aprog); s'> 0;
+    fetch (mop_bef n @ tshift mp_up (2 * n)) s' b = (a, 0)\<rbrakk>
+\<Longrightarrow> fetch (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n)) 
+    (length (tm_of aprog) div 2)) (s' + length (tm_of aprog) div 2) b 
+   = (a, 0)"
+apply(case_tac s', simp)
+apply(auto simp: fetch.simps nth_of.simps t_ncorrect.simps shift_length nth_append
+                 tshift.simps split: list.splits block.splits split: if_splits)
+done
+
+lemma [simp]:
+  "\<lbrakk>t_ncorrect (tm_of aprog);
+    t_step (s', l', r') (mop_bef n @ tshift mp_up (2 * n), 0) = 
+                                               (0, l'', r''); s' > 0\<rbrakk>
+  \<Longrightarrow> t_step (s' + length (tm_of aprog) div 2, l', r') 
+        (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n))
+           (length (tm_of aprog) div 2), 0) = (0, l'', r'')"
+apply(simp add: t_step.simps)
+apply(subgoal_tac 
+   "(fetch (mop_bef n @ tshift mp_up (2 * n)) s' 
+              (case r' of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))
+  = (fetch (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n)) 
+       (length (tm_of aprog) div 2)) (s' + length (tm_of aprog) div 2)
+    (case r' of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))", simp)
+apply(case_tac "(fetch (mop_bef n @ tshift mp_up (2 * n)) s' 
+       (case r' of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))", simp)
+apply(drule_tac tm_append_fetch_equal, auto)
+done
+
+lemma [intro]: 
+  "start_of (layout_of aprog) (length aprog) - Suc 0 = 
+                                      length (tm_of aprog) div 2"
+apply(subgoal_tac  "abc2t_correct aprog")
+apply(insert pre_lheq[of "concat (take (length aprog) 
+       (tms_of aprog))" "length aprog" aprog], simp add: tm_of.simps)
+by auto
+
+lemma tm_append_stop_step: 
+  "\<lbrakk>t_ncorrect (tm_of aprog); 
+    t_ncorrect (mop_bef n @ tshift mp_up (2 * n)); n < length lm; 
+   (t_steps (Suc 0, l, r) (mop_bef n @ tshift mp_up (2 * n), 0) stp) =
+                         (s', l', r');
+    s' \<noteq> 0;
+    t_step (s', l', r') (mop_bef n @ tshift mp_up (2 * n), 0) 
+                                                     = (0, l'', r'')\<rbrakk>
+     \<Longrightarrow>
+(t_steps ((start_of (layout_of aprog) (length aprog), l, r)) 
+  (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n))
+   (start_of (layout_of aprog) (length aprog) - Suc 0), 0) (Suc stp))
+  = (0, l'', r'')"
+apply(insert tm_append_steps_equal[of "tm_of aprog" 
+      "(mop_bef n @ tshift mp_up (2 * n))"
+      "(start_of (layout_of aprog) (length aprog) - Suc 0)" 
+      "Suc 0" l r stp], simp)
+apply(subgoal_tac "(start_of (layout_of aprog) (length aprog) - Suc 0)
+              = (length (tm_of aprog) div 2)", simp add: t_steps_ind)
+apply(case_tac 
+ "(t_steps (start_of (layout_of aprog) (length aprog), l, r) 
+      (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n))
+           (length (tm_of aprog) div 2), 0) stp)", simp)
+apply(subgoal_tac "start_of (layout_of aprog) (length aprog) > 0", 
+      case_tac "start_of (layout_of aprog) (length aprog)", 
+      simp, simp)
+apply(rule startof_not0, auto)
+done
+
+lemma start_of_out_range: 
+"as \<ge> length aprog \<Longrightarrow> 
+   start_of (layout_of aprog) as = 
+             start_of (layout_of aprog) (length aprog)"
+apply(induct as, simp)
+apply(case_tac "length aprog = Suc as", simp)
+apply(simp add: start_of.simps)
+done
+
+lemma [intro]: "t_ncorrect (tm_of aprog)"
+apply(simp add: tm_of.simps)
+apply(insert tms_mod2[of "length aprog" aprog], 
+                                simp add: t_ncorrect.simps)
+done
+
+lemma abacus_turing_eq_halt_case_pre: 
+   "\<lbrakk>ly = layout_of aprog; 
+     tprog = tm_of aprog; 
+     crsp_l ly ac tc ires;
+     n < length am;
+     abc_steps_l ac aprog stp = (as, am); 
+     mop_ss = start_of ly (length aprog);
+     as \<ge> length aprog\<rbrakk>
+     \<Longrightarrow> \<exists> stp. (\<lambda> (s, l, r). s = 0 \<and> mopup_inv (0, l, r) am n ires)
+                (t_steps tc (tprog @ (tMp n (mop_ss - 1)), 0) stp)"
+apply(insert steps_crsp[of ly aprog tprog ac tc ires "(as, am)" stp], auto)
+apply(case_tac "(t_steps tc (tm_of aprog, 0) n')",  
+      simp add: tMp.simps)
+apply(subgoal_tac "t_ncorrect (mop_bef n @ tshift mp_up (2 * n))")
+proof -
+  fix n' a b c
+  assume h1: 
+    "crsp_l (layout_of aprog) ac tc ires" 
+    "abc_steps_l ac aprog stp = (as, am)" 
+    "length aprog \<le> as"
+    "crsp_l (layout_of aprog) (as, am) (a, b, c) ires"
+    "t_steps tc (tm_of aprog, 0) n' = (a, b, c)"
+    "n < length am"
+    "t_ncorrect (mop_bef n @ tshift mp_up (2 * n))"
+  hence h2:
+  "t_steps tc (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n))
+    (start_of (layout_of aprog) (length aprog) - Suc 0), 0) n' 
+                                    = (a, b, c)" 
+    apply(rule_tac tm_append_steps, simp)
+    apply(simp add: crsp_l.simps, auto)
+    apply(simp add: crsp_l.simps)
+    apply(rule startof_not0)
+    done
+  from h1 have h3: 
+  "\<exists>stp. (\<lambda>(s, l, r). s \<noteq> 0 \<and> ((\<lambda> (s', l', r'). s' = 0) 
+           (t_step (s, l, r) (mop_bef n @ tshift mp_up (2 * n), 0))))
+         (t_steps (Suc 0, b, c) 
+               (mop_bef n @ tshift mp_up (2 * n), 0) stp)"
+    apply(rule_tac mopup_halt_bef, auto)
+    done
+  from h1 and h2 and h3 show 
+    "\<exists>stp. case t_steps tc (tm_of aprog @ abacus.tshift (mop_bef n @ abacus.tshift mp_up (2 * n))
+    (start_of (layout_of aprog) (length aprog) - Suc 0), 0) stp of (s, ab)
+    \<Rightarrow> s = 0 \<and> mopup_inv (0, ab) am n ires"
+  proof(erule_tac exE, 
+    case_tac "(t_steps (Suc 0, b, c) 
+              (mop_bef n @ tshift mp_up (2 * n), 0) stpa)", simp,
+    case_tac "(t_step (aa, ba, ca) 
+              (mop_bef n @ tshift mp_up (2 * n), 0))", simp)
+    fix stpa aa ba ca aaa baa caa
+    assume g1: "0 < aa \<and> aaa = 0" 
+      "t_steps (Suc 0, b, c) 
+      (mop_bef n @ tshift mp_up (2 * n), 0) stpa = (aa, ba,ca)" 
+      "t_step (aa, ba, ca) (mop_bef n @ tshift mp_up (2 * n), 0)
+      = (0, baa, caa)"
+    from h1 and this have g2: 
+      "t_steps (start_of (layout_of aprog) (length aprog), b, c) 
+         (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n)) 
+           (start_of (layout_of aprog) (length aprog) - Suc 0), 0) 
+                (Suc stpa) = (0, baa, caa)"
+      apply(rule_tac tm_append_stop_step, auto)
+      done
+    from h1 and h2 and g1 and this show "?thesis"
+      apply(rule_tac x = "n' + Suc stpa" in exI)
+      apply(simp add: t_steps_ind del: t_steps.simps)
+      apply(subgoal_tac "a = start_of (layout_of aprog) 
+                                          (length aprog)", simp)
+      apply(insert mopup_inv_steps[of n am "Suc 0" b c ires "Suc stpa"],
+            simp add: t_steps_ind)
+      apply(subgoal_tac "mopup_inv (Suc 0, b, c) am n ires", simp)
+      apply(rule_tac mopup_init, simp, simp)
+      apply(simp add: crsp_l.simps)
+      apply(erule_tac start_of_out_range)
+      done
+  qed
+next
+  show " t_ncorrect (mop_bef n @ tshift mp_up (2 * n))"
+    apply(auto simp: t_ncorrect.simps tshift.simps mp_up_def)
+    done   
+qed
+
+text {*
+  One of the main theorems about Abacus compilation.
+*}
+
+lemma abacus_turing_eq_halt_case: 
+  assumes layout: 
+  -- {* There is an Abacus program @{text "aprog"} with layout @{text "ly"}: *}
+  "ly = layout_of aprog"
+  and complied: 
+  -- {* The TM compiled from @{text "aprog"} is @{text "tprog"}: *}
+  "tprog = tm_of aprog"
+  and correspond: 
+  -- {* TM configuration @{text "tc"} and Abacus configuration @{text "ac"}
+  are in correspondence: *}
+  "crsp_l ly ac tc ires"
+  and halt_state: 
+  -- {* @{text "as"} is a program label outside the range of @{text "aprog"}. So 
+  if Abacus is in such a state, it is in halt state: *}
+  "as \<ge> length aprog"
+  and abc_exec: 
+  -- {* Supposing after @{text "stp"} step of execution, Abacus program @{text "aprog"}
+  reaches such a halt state: *}
+  "abc_steps_l ac aprog stp = (as, am)"
+  and rs_len: 
+  -- {* @{text "n"} is a memory address in the range of Abacus memory @{text "am"}: *}
+  "n < length am"
+  and mopup_start:
+  -- {* The startling label for mopup mahines, according to the layout and Abacus program 
+   should be @{text "mop_ss"}: *}
+  "mop_ss = start_of ly (length aprog)"
+  shows 
+  -- {* 
+  After @{text "stp"} steps of execution of the TM composed of @{text "tprog"} and the mopup 
+  TM @{text "(tMp n (mop_ss - 1))"} will halt and gives rise to a configuration which 
+  only hold the content of memory cell @{text "n"}:
+  *}
+  "\<exists> stp. (\<lambda> (s, l, r). \<exists> ln rn. s = 0 \<and>  l = Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires
+     \<and> r = Oc\<^bsup>Suc (abc_lm_v am n)\<^esup> @ Bk\<^bsup>rn\<^esup>)
+           (t_steps tc (tprog @ (tMp n (mop_ss - 1)), 0) stp)"
+proof -
+  from layout complied correspond halt_state abc_exec rs_len mopup_start
+       and abacus_turing_eq_halt_case_pre [of ly aprog tprog ac tc ires n am stp as mop_ss]
+  show "?thesis" 
+    apply(simp add: mopup_inv.simps mopup_stop.simps tape_of_nat_abv)
+    done
+qed
+
+lemma abc_unhalt_case_zero: 
+"\<lbrakk>crsp_l (layout_of aprog) ac tc ires;
+  n < length am; 
+  \<forall>stp. (\<lambda>(as, am). as < length aprog) (abc_steps_l ac aprog stp)\<rbrakk>
+ \<Longrightarrow> \<exists>astp bstp. 0 \<le> bstp \<and> 
+          crsp_l (layout_of aprog) (abc_steps_l ac aprog astp) 
+                (t_steps tc (tm_of aprog, 0) bstp) ires"
+apply(rule_tac x = "Suc 0" in exI)
+apply(case_tac " abc_steps_l ac aprog (Suc 0)", simp)
+proof -
+  fix a b
+  assume "crsp_l (layout_of aprog) ac tc ires" 
+         "abc_steps_l ac aprog (Suc 0) = (a, b)"
+  thus "\<exists>bstp. crsp_l (layout_of aprog) (a, b) 
+               (t_steps tc (tm_of aprog, 0) bstp) ires"
+    apply(insert steps_crsp[of "layout_of aprog" aprog 
+                  "tm_of aprog" ac tc ires "(a, b)" "Suc 0"], auto)
+    done
+qed
+
+declare abc_steps_l.simps[simp del]
+
+lemma abc_steps_ind: 
+ "let (as, am) = abc_steps_l ac aprog stp in 
+   abc_steps_l ac aprog (Suc stp) =
+              abc_step_l (as, am) (abc_fetch as aprog) "
+proof(simp)
+  show "(\<lambda>(as, am). abc_steps_l ac aprog (Suc stp) = 
+        abc_step_l (as, am) (abc_fetch as aprog)) 
+              (abc_steps_l ac aprog stp)"
+  proof(induct stp arbitrary: ac)
+    fix ac
+    show "(\<lambda>(as, am). abc_steps_l ac aprog (Suc 0) = 
+            abc_step_l (as, am) (abc_fetch as aprog))  
+                    (abc_steps_l ac aprog 0)"
+      apply(case_tac "ac:: nat \<times> nat list", 
+            simp add: abc_steps_l.simps)
+      apply(case_tac "(abc_step_l (a, b) (abc_fetch a aprog))",
+            simp add: abc_steps_l.simps)
+      done
+  next
+    fix stp ac
+    assume h1:
+      "(\<And>ac. (\<lambda>(as, am). abc_steps_l ac aprog (Suc stp) =
+                            abc_step_l (as, am) (abc_fetch as aprog)) 
+             (abc_steps_l ac aprog stp))"
+    thus 
+      "(\<lambda>(as, am). abc_steps_l ac aprog (Suc (Suc stp)) =
+              abc_step_l (as, am) (abc_fetch as aprog)) 
+                             (abc_steps_l ac aprog (Suc stp))"
+      apply(case_tac "ac::nat \<times> nat list", simp)
+      apply(subgoal_tac 
+           "abc_steps_l (a, b) aprog (Suc (Suc stp)) =
+            abc_steps_l (abc_step_l (a, b) (abc_fetch a aprog)) 
+                                              aprog (Suc stp)", simp)
+      apply(case_tac "(abc_step_l (a, b) (abc_fetch a aprog))", simp)
+    proof -
+      fix a b aa ba
+      assume h2: "abc_step_l (a, b) (abc_fetch a aprog) = (aa, ba)"
+      from h1 and h2  show 
+      "(\<lambda>(as, am). abc_steps_l (aa, ba) aprog (Suc stp) = 
+          abc_step_l (as, am) (abc_fetch as aprog)) 
+                    (abc_steps_l (a, b) aprog (Suc stp))"
+	apply(insert h1[of "(aa, ba)"])
+	apply(simp add: abc_steps_l.simps)
+	apply(insert h2, simp)
+	done
+    next
+      fix a b
+      show 
+        "abc_steps_l (a, b) aprog (Suc (Suc stp)) = 
+         abc_steps_l (abc_step_l (a, b) (abc_fetch a aprog)) 
+                                                   aprog (Suc stp)"
+	apply(simp only: abc_steps_l.simps)
+	done
+    qed
+  qed
+qed
+
+lemma abc_unhalt_case_induct: 
+  "\<lbrakk>crsp_l (layout_of aprog) ac tc ires;
+    n < length am; 
+    \<forall>stp. (\<lambda>(as, am). as < length aprog) (abc_steps_l ac aprog stp); 
+    stp \<le> bstp;
+    crsp_l (layout_of aprog) (abc_steps_l ac aprog astp) 
+                           (t_steps tc (tm_of aprog, 0) bstp) ires\<rbrakk>
+ \<Longrightarrow> \<exists>astp bstp. Suc stp \<le> bstp \<and> crsp_l (layout_of aprog) 
+       (abc_steps_l ac aprog astp) (t_steps tc (tm_of aprog, 0) bstp) ires"
+apply(rule_tac x = "Suc astp" in exI)
+apply(case_tac "abc_steps_l ac aprog astp")
+proof -
+  fix a b
+  assume 
+    "\<forall>stp. (\<lambda>(as, am). as < length aprog)  
+                 (abc_steps_l ac aprog stp)" 
+    "stp \<le> bstp"
+    "crsp_l (layout_of aprog) (abc_steps_l ac aprog astp) 
+      (t_steps tc (tm_of aprog, 0) bstp) ires" 
+    "abc_steps_l ac aprog astp = (a, b)"
+  thus 
+ "\<exists>bstp\<ge>Suc stp. crsp_l (layout_of aprog)
+       (abc_steps_l ac aprog (Suc astp)) 
+   (t_steps tc (tm_of aprog, 0) bstp) ires"
+    apply(insert crsp_inside[of "layout_of aprog" aprog 
+      "tm_of aprog" a b "(t_steps tc (tm_of aprog, 0) bstp)" "ires"], auto)
+    apply(erule_tac x = astp in allE, auto)
+    apply(rule_tac x = "bstp + stpa" in exI, simp)
+    apply(insert abc_steps_ind[of ac aprog "astp"], simp)
+    done
+qed   
+
+lemma abc_unhalt_case: 
+  "\<lbrakk>crsp_l (layout_of aprog) ac tc ires;  
+    \<forall>stp. (\<lambda>(as, am). as < length aprog) (abc_steps_l ac aprog stp)\<rbrakk>
+ \<Longrightarrow> (\<exists> astp bstp. bstp \<ge> stp \<and> 
+         crsp_l (layout_of aprog) (abc_steps_l ac aprog astp) 
+                                (t_steps tc (tm_of aprog, 0) bstp) ires)"
+apply(induct stp)
+apply(rule_tac abc_unhalt_case_zero, auto)
+apply(rule_tac abc_unhalt_case_induct, auto)
+done
+  
+lemma abacus_turing_eq_unhalt_case_pre: 
+  "\<lbrakk>ly = layout_of aprog; 
+    tprog = tm_of aprog;
+    crsp_l ly ac tc ires;
+    \<forall> stp. ((\<lambda> (as, am). as < length aprog)
+                       (abc_steps_l ac aprog stp));
+    mop_ss = start_of ly (length aprog)\<rbrakk>
+  \<Longrightarrow> (\<not> (\<exists> stp. (\<lambda> (s, l, r). s = 0)
+              (t_steps tc (tprog @ (tMp n (mop_ss - 1)), 0) stp)))"
+  apply(auto)
+proof -
+  fix stp a b
+  assume h1: 
+    "crsp_l (layout_of aprog) ac tc ires" 
+    "\<forall>stp. (\<lambda>(as, am). as < length aprog) (abc_steps_l ac aprog stp)"
+    "t_steps tc (tm_of aprog @ tMp n (start_of (layout_of aprog) 
+    (length aprog) - Suc 0), 0) stp = (0, a, b)"
+  thus "False"
+  proof(insert abc_unhalt_case[of aprog ac tc ires stp], auto, 
+        case_tac "(abc_steps_l ac aprog astp)", 
+        case_tac "(t_steps tc (tm_of aprog, 0) bstp)", simp)
+    fix astp bstp aa ba aaa baa c
+    assume h2: 
+      "abc_steps_l ac aprog astp = (aa, ba)" "stp \<le> bstp"
+      "t_steps tc (tm_of aprog, 0) bstp = (aaa, baa, c)" 
+      "crsp_l (layout_of aprog) (aa, ba) (aaa, baa, c) ires"
+    hence h3: 
+      "t_steps tc (tm_of aprog @ tMp n 
+       (start_of (layout_of aprog) (length aprog) - Suc 0), 0) bstp 
+                    = (aaa, baa, c)"
+      apply(intro tm_append_steps, auto)
+      apply(simp add: crsp_l.simps, rule startof_not0)
+      done
+    from h2 have h4: "\<exists> diff. bstp = stp + diff"
+      apply(rule_tac x = "bstp - stp" in exI, simp)
+      done
+    from h4 and h3 and h2  and h1 show "?thesis"
+      apply(auto)
+      apply(simp add: state0_ind crsp_l.simps)
+      apply(subgoal_tac "start_of (layout_of aprog) aa > 0", simp)
+      apply(rule startof_not0)
+      done
+  qed
+qed
+
+lemma abacus_turing_eq_unhalt_case:
+  assumes layout: 
+  -- {* There is an Abacus program @{text "aprog"} with layout @{text "ly"}: *}
+  "ly = layout_of aprog"
+  and compiled: 
+  -- {* The TM compiled from @{text "aprog"} is @{text "tprog"}: *}
+  "tprog = tm_of aprog"
+  and correspond: 
+  -- {* 
+  TM configuration @{text "tc"} and Abacus configuration @{text "ac"}
+  are in correspondence: 
+  *}
+  "crsp_l ly ac tc ires"
+  and abc_unhalt: 
+  -- {*
+  If, no matter how many steps the Abacus program @{text "aprog"} executes, it
+  may never reach a halt state. 
+  *}
+  "\<forall> stp. ((\<lambda> (as, am). as < length aprog)
+                       (abc_steps_l ac aprog stp))"
+  and mopup_start: "mop_ss = start_of ly (length aprog)"
+  shows
+  -- {*
+  The the TM composed of TM @{text "tprog"} and the moupup TM may never reach a halt state as well.
+  *}
+  "\<not> (\<exists> stp. (\<lambda> (s, l, r). s = 0)
+              (t_steps tc (tprog @ (tMp n (mop_ss - 1)), 0) stp))"
+  using layout compiled correspond abc_unhalt mopup_start
+  apply(rule_tac abacus_turing_eq_unhalt_case_pre, auto)
+  done
+
+
+definition abc_list_crsp:: "nat list \<Rightarrow> nat list \<Rightarrow> bool"
+  where
+  "abc_list_crsp xs ys = (\<exists> n. xs = ys @ 0\<^bsup>n\<^esup> \<or> ys = xs @ 0\<^bsup>n\<^esup>)"
+lemma [intro]: "abc_list_crsp (lm @ 0\<^bsup>m\<^esup>) lm"
+apply(auto simp: abc_list_crsp_def)
+done
+
+lemma abc_list_crsp_lm_v: 
+  "abc_list_crsp lma lmb \<Longrightarrow> abc_lm_v lma n = abc_lm_v lmb n"
+apply(auto simp: abc_list_crsp_def abc_lm_v.simps 
+                 nth_append exponent_def)
+done
+
+lemma  rep_app_cons_iff: 
+  "k < n \<Longrightarrow> replicate n a[k:=b] = 
+          replicate k a @ b # replicate (n - k - 1) a"
+apply(induct n arbitrary: k, simp)
+apply(simp split:nat.splits)
+done
+
+lemma abc_list_crsp_lm_s: 
+  "abc_list_crsp lma lmb \<Longrightarrow> 
+      abc_list_crsp (abc_lm_s lma m n) (abc_lm_s lmb m n)"
+apply(auto simp: abc_list_crsp_def abc_lm_v.simps abc_lm_s.simps)
+apply(simp_all add: list_update_append, auto simp: exponent_def)
+proof -
+  fix na
+  assume h: "m < length lmb + na" " \<not> m < length lmb"
+  hence "m - length lmb < na" by simp
+  hence "replicate na 0[(m- length lmb):= n] = 
+           replicate (m - length lmb) 0 @ n # 
+              replicate (na - (m - length lmb) - 1) 0"
+    apply(erule_tac rep_app_cons_iff)
+    done
+  thus "\<exists>nb. replicate na 0[m - length lmb := n] =
+                 replicate (m - length lmb) 0 @ n # replicate nb 0 \<or>
+                 replicate (m - length lmb) 0 @ [n] =
+                 replicate na 0[m - length lmb := n] @ replicate nb 0"
+    apply(auto)
+    done
+next
+  fix na
+  assume h: "\<not> m < length lmb + na"
+  show 
+    "\<exists>nb. replicate na 0 @ replicate (m - (length lmb + na)) 0 @ [n] =
+           replicate (m - length lmb) 0 @ n # replicate nb 0 \<or>
+          replicate (m - length lmb) 0 @ [n] =
+            replicate na 0 @
+            replicate (m - (length lmb + na)) 0 @ n # replicate nb 0"
+    apply(rule_tac x = 0 in exI, simp, auto)
+    using h
+    apply(simp add: replicate_add[THEN sym])
+    done
+next
+  fix na
+  assume h: "\<not> m < length lma" "m < length lma + na"
+  hence "m - length lma < na" by simp
+  hence 
+    "replicate na 0[(m- length lma):= n] = replicate (m - length lma) 
+                  0 @ n # replicate (na - (m - length lma) - 1) 0"
+    apply(erule_tac rep_app_cons_iff)
+    done
+  thus "\<exists>nb. replicate (m - length lma) 0 @ [n] =
+                 replicate na 0[m - length lma := n] @ replicate nb 0 
+           \<or> replicate na 0[m - length lma := n] =
+                 replicate (m - length lma) 0 @ n # replicate nb 0"
+    apply(auto)
+    done
+next
+  fix na
+  assume "\<not> m < length lma + na"
+  thus " \<exists>nb. replicate (m - length lma) 0 @ [n] =
+            replicate na 0 @
+            replicate (m - (length lma + na)) 0 @ n # replicate nb 0 
+        \<or>   replicate na 0 @ 
+               replicate (m - (length lma + na)) 0 @ [n] =
+            replicate (m - length lma) 0 @ n # replicate nb 0"
+    apply(rule_tac x = 0 in exI, simp, auto)
+    apply(simp add: replicate_add[THEN sym])
+    done
+qed
+
+lemma abc_list_crsp_step: 
+  "\<lbrakk>abc_list_crsp lma lmb; abc_step_l (aa, lma) i = (a, lma'); 
+    abc_step_l (aa, lmb) i = (a', lmb')\<rbrakk>
+    \<Longrightarrow> a' = a \<and> abc_list_crsp lma' lmb'"
+apply(case_tac i, auto simp: abc_step_l.simps 
+       abc_list_crsp_lm_s abc_list_crsp_lm_v Let_def 
+                       split: abc_inst.splits if_splits)
+done
+
+lemma abc_steps_red: 
+  "abc_steps_l ac aprog stp = (as, am) \<Longrightarrow>
+     abc_steps_l ac aprog (Suc stp) = 
+           abc_step_l (as, am) (abc_fetch as aprog)"
+using abc_steps_ind[of ac aprog stp]
+apply(simp)
+done
+
+lemma abc_list_crsp_steps: 
+  "\<lbrakk>abc_steps_l (0, lm @ 0\<^bsup>m\<^esup>) aprog stp = (a, lm'); aprog \<noteq> []\<rbrakk> 
+      \<Longrightarrow> \<exists> lma. abc_steps_l (0, lm) aprog stp = (a, lma) \<and> 
+                                          abc_list_crsp lm' lma"
+apply(induct stp arbitrary: a lm', simp add: abc_steps_l.simps, auto)
+apply(case_tac "abc_steps_l (0, lm @ 0\<^bsup>m\<^esup>) aprog stp", 
+      simp add: abc_steps_ind)
+proof -
+  fix stp a lm' aa b
+  assume ind:
+    "\<And>a lm'. aa = a \<and> b = lm' \<Longrightarrow> 
+     \<exists>lma. abc_steps_l (0, lm) aprog stp = (a, lma) \<and>
+                                          abc_list_crsp lm' lma"
+    and h: "abc_steps_l (0, lm @ 0\<^bsup>m\<^esup>) aprog (Suc stp) = (a, lm')" 
+           "abc_steps_l (0, lm @ 0\<^bsup>m\<^esup>) aprog stp = (aa, b)" 
+           "aprog \<noteq> []"
+  hence g1: "abc_steps_l (0, lm @ 0\<^bsup>m\<^esup>) aprog (Suc stp)
+          = abc_step_l (aa, b) (abc_fetch aa aprog)"
+    apply(rule_tac abc_steps_red, simp)
+    done
+  have "\<exists>lma. abc_steps_l (0, lm) aprog stp = (aa, lma) \<and> 
+              abc_list_crsp b lma"
+    apply(rule_tac ind, simp)
+    done
+  from this obtain lma where g2: 
+    "abc_steps_l (0, lm) aprog stp = (aa, lma) \<and> 
+     abc_list_crsp b lma"   ..
+  hence g3: "abc_steps_l (0, lm) aprog (Suc stp)
+          = abc_step_l (aa, lma) (abc_fetch aa aprog)"
+    apply(rule_tac abc_steps_red, simp)
+    done
+  show "\<exists>lma. abc_steps_l (0, lm) aprog (Suc stp) = (a, lma) \<and>  
+              abc_list_crsp lm' lma"
+    using g1 g2 g3 h
+    apply(auto)
+    apply(case_tac "abc_step_l (aa, b) (abc_fetch aa aprog)",
+          case_tac "abc_step_l (aa, lma) (abc_fetch aa aprog)", simp)
+    apply(rule_tac abc_list_crsp_step, auto)
+    done
+qed
+
+lemma [simp]: "(case ca of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc) =
+                (case ca of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)"
+by(case_tac ca, simp_all, case_tac a, simp, simp)
+
+lemma steps_eq: "length t mod 2 = 0 \<Longrightarrow> 
+                    t_steps c (t, 0) stp = steps c t stp"
+apply(induct stp)
+apply(simp add: steps.simps t_steps.simps)
+apply(simp add:tstep_red t_steps_ind)
+apply(case_tac "steps c t stp", simp)
+apply(auto simp: t_step.simps tstep.simps)
+done
+
+lemma crsp_l_start: "crsp_l ly (0, lm) (Suc 0, Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>) ires"
+apply(simp add: crsp_l.simps, auto simp: start_of.simps)
+done
+
+lemma t_ncorrect_app: "\<lbrakk>t_ncorrect t1; t_ncorrect t2\<rbrakk> \<Longrightarrow> 
+                                          t_ncorrect (t1 @ t2)"
+apply(simp add: t_ncorrect.simps, auto)
+done
+
+lemma [simp]: 
+  "(length (tm_of aprog) + 
+    length (tMp n (start_of ly (length aprog) - Suc 0))) mod 2 = 0"
+apply(subgoal_tac 
+ "t_ncorrect (tm_of aprog @ tMp n 
+             (start_of ly (length aprog) - Suc 0))")
+apply(simp add: t_ncorrect.simps)
+apply(rule_tac t_ncorrect_app, 
+      auto simp: tMp.simps t_ncorrect.simps tshift.simps mp_up_def)
+apply(subgoal_tac
+       "t_ncorrect (tm_of aprog)", simp add: t_ncorrect.simps)
+apply(auto)
+done
+
+lemma [simp]: "takeWhile (\<lambda>a. a = Oc) 
+              (replicate rs Oc @ replicate rn Bk) = replicate rs Oc"
+apply(induct rs, auto)
+apply(induct rn, auto)
+done
+
+lemma abacus_turing_eq_halt': 
+  "\<lbrakk>ly = layout_of aprog; 
+    tprog = tm_of aprog; 
+    n < length am;
+    abc_steps_l (0, lm) aprog stp = (as, am); 
+    mop_ss = start_of ly (length aprog);
+    as \<ge> length aprog\<rbrakk>
+    \<Longrightarrow> \<exists> stp m l. steps (Suc 0, Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>) 
+                (tprog @ (tMp n (mop_ss - 1))) stp
+                  = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (abc_lm_v am n)\<^esup> @ Bk\<^bsup>l\<^esup>)"
+apply(drule_tac tc = "(Suc 0, Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>)" in 
+               abacus_turing_eq_halt_case, auto intro: crsp_l_start)
+apply(subgoal_tac 
+         "length (tm_of aprog @ tMp n 
+                  (start_of ly (length aprog) - Suc 0)) mod 2 = 0")
+apply(simp add: steps_eq)
+apply(rule_tac x = stpa in exI, 
+       simp add:  exponent_def, auto)
+done
+
+
+lemma list_length: "xs = ys \<Longrightarrow> length xs = length ys"
+by simp
+lemma [elim]: "tinres (Bk\<^bsup>m\<^esup>) b \<Longrightarrow> \<exists>m. b = Bk\<^bsup>m\<^esup>"
+apply(auto simp: tinres_def)
+apply(rule_tac x = "m-n" in exI, 
+             auto simp: exponent_def replicate_add[THEN sym]) 
+apply(case_tac "m < n", auto)
+apply(drule_tac list_length, auto)
+apply(subgoal_tac "\<exists> d. m = d + n", auto simp: replicate_add)
+apply(rule_tac x = "m - n" in exI, simp)
+done
+lemma [intro]: "tinres [Bk] (Bk\<^bsup>k\<^esup>) "
+apply(auto simp: tinres_def exponent_def)
+apply(case_tac k, auto)
+apply(rule_tac x = "Suc 0" in exI, simp)
+done
+
+lemma abacus_turing_eq_halt_pre: 
+ "\<lbrakk>ly = layout_of aprog; 
+   tprog = tm_of aprog; 
+   n < length am;     
+   abc_steps_l (0, lm) aprog stp = (as, am);  
+   mop_ss = start_of ly (length aprog);
+   as \<ge> length aprog\<rbrakk>
+  \<Longrightarrow> \<exists> stp m l. steps  (Suc 0, Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>)
+               (tprog @ (tMp n (mop_ss - 1))) stp
+                 = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (abc_lm_v am n)\<^esup> @ Bk\<^bsup>l\<^esup>)"
+using abacus_turing_eq_halt'
+apply(simp)
+done
+
+
+text {*
+  Main theorem for the case when the original Abacus program does halt.
+*}
+
+lemma abacus_turing_eq_halt: 
+  assumes layout:
+  "ly = layout_of aprog"
+  -- {* There is an Abacus program @{text "aprog"} with layout @{text "ly"}: *}
+  and compiled: "tprog = tm_of aprog"
+  -- {* The TM compiled from @{text "aprog"} is @{text "tprog"}: *}
+  and halt_state: 
+   -- {* @{text "as"} is a program label outside the range of @{text "aprog"}. So 
+  if Abacus is in such a state, it is in halt state: *}
+  "as \<ge> length aprog"
+  and abc_exec: 
+  -- {* Supposing after @{text "stp"} step of execution, Abacus program @{text "aprog"}
+  reaches such a halt state: *}
+  "abc_steps_l (0, lm) aprog stp = (as, am)"
+  and rs_locate: 
+   -- {* @{text "n"} is a memory address in the range of Abacus memory @{text "am"}: *}
+  "n < length am"  
+  and mopup_start: 
+   -- {* The startling label for mopup mahines, according to the layout and Abacus program 
+   should be @{text "mop_ss"}: *}
+  "mop_ss = start_of ly (length aprog)"
+  shows 
+  -- {* 
+  After @{text "stp"} steps of execution of the TM composed of @{text "tprog"} and the mopup 
+  TM @{text "(tMp n (mop_ss - 1))"} will halt and gives rise to a configuration which 
+  only hold the content of memory cell @{text "n"}:
+  *}
+  "\<exists> stp m l. steps (Suc 0, Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>) (tprog @ (tMp n (mop_ss - 1))) stp
+                      = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (abc_lm_v am n)\<^esup> @ Bk\<^bsup>l\<^esup>)"
+  using layout compiled halt_state abc_exec rs_locate mopup_start
+  by(rule_tac abacus_turing_eq_halt_pre, auto)
+
+lemma abacus_turing_eq_uhalt': 
+ "\<lbrakk>ly = layout_of aprog; 
+   tprog = tm_of aprog; 
+   \<forall> stp. ((\<lambda> (as, am). as < length aprog) 
+                   (abc_steps_l (0, lm) aprog stp));
+   mop_ss = start_of ly (length aprog)\<rbrakk>
+  \<Longrightarrow> (\<not> (\<exists> stp. isS0 (steps (Suc 0, [Bk, Bk], <lm>) 
+                      (tprog @ (tMp n (mop_ss - 1))) stp)))"
+apply(drule_tac tc = "(Suc 0, [Bk, Bk], <lm>)" and n = n and ires = "[]" in 
+         abacus_turing_eq_unhalt_case, auto intro: crsp_l_start)
+apply(simp add: crsp_l.simps start_of.simps)
+apply(erule_tac x = stp in allE, erule_tac x = stp in allE)
+apply(subgoal_tac 
+   "length (tm_of aprog @ tMp n 
+         (start_of ly (length aprog) - Suc 0)) mod 2 = 0")
+apply(simp add: steps_eq, auto simp: isS0_def)
+done
+
+text {*
+  Main theorem for the case when the original Abacus program does not halt.
+  *}
+lemma abacus_turing_eq_uhalt:
+  assumes layout: 
+  -- {* There is an Abacus program @{text "aprog"} with layout @{text "ly"}: *}
+  "ly = layout_of aprog"
+  and compiled:
+   -- {* The TM compiled from @{text "aprog"} is @{text "tprog"}: *}
+  "tprog = tm_of aprog"
+  and abc_unhalt:
+  -- {*
+  If, no matter how many steps the Abacus program @{text "aprog"} executes, it
+  may never reach a halt state. 
+  *}
+  "\<forall> stp. ((\<lambda> (as, am). as < length aprog) 
+                      (abc_steps_l (0, lm) aprog stp))"
+  and mop_start: "mop_ss = start_of ly (length aprog)"
+  shows 
+   -- {*
+  The the TM composed of TM @{text "tprog"} and the moupup TM may never reach a halt state as well.
+  *}
+  "\<not> (\<exists> stp. isS0 (steps (Suc 0, [Bk, Bk], <lm>) 
+                    (tprog @ (tMp n (mop_ss - 1))) stp))"
+  using abacus_turing_eq_uhalt'
+        layout compiled abc_unhalt mop_start
+  by(auto)
+
+end
+