(* Title: Turing machines
   Author: Xu Jian <xujian817@hotmail.com>
   Maintainer: Xu Jian
*)

theory turing_basic
imports Main
begin

section {* Basic definitions of Turing machine *}

datatype action = W0 | W1 | L | R | Nop

datatype cell = Bk | Oc

type_synonym tape = "cell list \<times> cell list"

type_synonym state = nat

type_synonym instr = "action \<times> state"

type_synonym tprog = "instr list \<times> nat"

type_synonym config = "state \<times> tape"

fun nth_of where
  "nth_of xs i = (if i \<ge> length xs then None
                  else Some (xs ! i))"

lemma nth_of_map [simp]:
  shows "nth_of (map f p) n = (case (nth_of p n) of None \<Rightarrow> None | Some x \<Rightarrow> Some (f x))"
apply(induct p arbitrary: n)
apply(auto)
apply(case_tac n)
apply(auto)
done

fun 
  fetch :: "instr list \<Rightarrow> state \<Rightarrow> cell \<Rightarrow> instr"
where
  "fetch p 0 b = (Nop, 0)"
| "fetch p (Suc s) Bk = 
     (case nth_of p (2 * s) of
        Some i \<Rightarrow> i
      | None \<Rightarrow> (Nop, 0))"
|"fetch p (Suc s) Oc = 
     (case nth_of p ((2 * s) + 1) of
         Some i \<Rightarrow> i
       | None \<Rightarrow> (Nop, 0))"

lemma fetch_Nil [simp]:
  shows "fetch [] s b = (Nop, 0)"
apply(case_tac s)
apply(auto)
apply(case_tac b)
apply(auto)
done

fun 
  update :: "action \<Rightarrow> tape \<Rightarrow> tape"
where 
  "update W0 (l, r) = (l, Bk # (tl r))" 
| "update W1 (l, r) = (l, Oc # (tl r))"
| "update L (l, r) = (if l = [] then ([], Bk # r) else (tl l, (hd l) # r))" 
| "update R (l, r) = (if r = [] then (Bk # l, []) else ((hd r) # l, tl r))" 
| "update Nop (l, r) = (l, r)"

abbreviation 
  "read r == if (r = []) then Bk else hd r"

fun step :: "config \<Rightarrow> tprog \<Rightarrow> config"
  where 
  "step (s, l, r) (p, off) = 
  (let (a, s') = fetch p (s - off) (read r) in (s', update a (l, r)))"

fun steps :: "config \<Rightarrow> tprog \<Rightarrow> nat \<Rightarrow> config"
  where
  "steps c p 0 = c" |
  "steps c p (Suc n) = steps (step c p) p n"

lemma step_red [simp]: 
  shows "steps c p (Suc n) = step (steps c p n) p"
by (induct n arbitrary: c) (auto)

lemma steps_add [simp]: 
  shows "steps c p (m + n) = steps (steps c p m) p n"
by (induct m arbitrary: c) (auto)

fun 
  tm_wf :: "tprog \<Rightarrow> bool"
where
  "tm_wf (p, off) = (length p \<ge> 2 \<and> length p mod 2 = 0 \<and> 
                    (\<forall>(a, s) \<in> set p. s \<le> length p div 2
                                             + off \<and> s \<ge> off))"

(* FIXME: needed? *)
lemma halt_lemma: 
  "\<lbrakk>wf LE; \<forall>n. (\<not> P (f n) \<longrightarrow> (f (Suc n), (f n)) \<in> LE)\<rbrakk> \<Longrightarrow> \<exists>n. P (f n)"
by (metis wf_iff_no_infinite_down_chain)

abbreviation exponent :: "'a \<Rightarrow> nat \<Rightarrow> 'a list" ("_ \<up> _" [100, 99] 100)
  where "x \<up> n == replicate n x"

consts tape_of :: "'a \<Rightarrow> cell list" ("<_>" 100)

fun tape_of_nat_list :: "nat list \<Rightarrow> cell list" 
  where 
  "tape_of_nat_list [] = []" |
  "tape_of_nat_list [n] = Oc\<up>(Suc n)" |
  "tape_of_nat_list (n#ns) = Oc\<up>(Suc n) @ 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\<up>(Suc n)"

definition tinres :: "cell list \<Rightarrow> cell list \<Rightarrow> bool"
  where
  "tinres xs ys = (\<exists>n. xs = ys @ Bk \<up> n \<or> ys = xs @ Bk \<up> n)"

fun 
  shift :: "instr list \<Rightarrow> nat \<Rightarrow> instr list"
where
  "shift p n = (map (\<lambda> (a, s). (a, (if s = 0 then 0 else s + n))) p)"


lemma length_shift [simp]: 
  "length (shift p n) = length p"
by (simp)

fun 
  adjust :: "instr list \<Rightarrow> instr list"
where
  "adjust p = map (\<lambda> (a, s). (a, if s = 0 then (Suc (length p div 2)) else s)) p"

lemma length_adjust[simp]: 
  shows "length (adjust p) = length p"
by (induct p) (auto)

fun
  tm_comp :: "instr list \<Rightarrow> instr list \<Rightarrow> instr list" ("_ |+| _" [0, 0] 100)
where
  "tm_comp p1 p2 = ((adjust p1) @ (shift p2 ((length p1) div 2)))"

fun
  is_final :: "config \<Rightarrow> bool"
where
  "is_final (s, l, r) = (s = 0)"

lemma is_final_steps:
  assumes "is_final (s, l, r)"
  shows "is_final (steps (s, l, r) (p, off) n)"
using assms by (induct n) (auto)

fun 
  holds_for :: "(tape \<Rightarrow> bool) \<Rightarrow> config \<Rightarrow> bool" ("_ holds'_for _" [100, 99] 100)
where
  "P holds_for (s, l, r) = P (l, r)"  

(*
lemma step_0 [simp]: 
  shows "step (0, (l, r)) p = (0, (l, r))"
by simp

lemma steps_0 [simp]: 
  shows "steps (0, (l, r)) p n = (0, (l, r))"
by (induct n) (simp_all)
*)

lemma is_final_holds[simp]:
  assumes "is_final c"
  shows "Q holds_for (steps c (p, off) n) = Q holds_for c"
using assms 
apply(induct n)
apply(case_tac [!] c)
apply(auto)
done

type_synonym assert = "tape \<Rightarrow> bool"

definition assert_imp :: "assert \<Rightarrow> assert \<Rightarrow> bool" ("_ \<mapsto> _" [0, 0] 100)
  where
  "assert_imp P Q = (\<forall>l r. P (l, r) \<longrightarrow> Q (l, r))"

lemma holds_for_imp:
  assumes "P holds_for c"
  and "P \<mapsto> Q"
  shows "Q holds_for c"
using assms unfolding assert_imp_def by (case_tac c, auto)

lemma test:
  assumes "is_final (steps (1, (l, r)) p n1)"
  and     "is_final (steps (1, (l, r)) p n2)"
  shows "Q holds_for (steps (1, (l, r)) p n1) \<longleftrightarrow> Q holds_for (steps (1, (l, r)) p n2)"
proof -
  obtain n3 where "n1 = n2 + n3 \<or> n2 = n1 + n3"
    by (metis le_iff_add nat_le_linear)
  with assms show "Q holds_for (steps (1, (l, r)) p n1) \<longleftrightarrow> Q holds_for (steps (1, (l, r)) p n2)"  
    by(case_tac p) (auto)
qed

definition
  Hoare :: "assert \<Rightarrow> tprog \<Rightarrow> assert \<Rightarrow> bool" ("({(1_)}/ (_)/ {(1_)})" 50)
where
  "{P} p {Q} \<equiv> 
     (\<forall>l r. P (l, r) \<longrightarrow> (\<exists>n. is_final (steps (1, (l, r)) p n) \<and> Q holds_for (steps (1, (l, r)) p n)))"

lemma HoareI:
  assumes "\<And>l r. P (l, r) \<Longrightarrow> \<exists>n. is_final (steps (1, (l, r)) p n) \<and> Q holds_for (steps (1, (l, r)) p n)"
  shows "{P} p {Q}"
unfolding Hoare_def using assms by auto

text {*
{P1} A {Q1}   {P2} B {Q2}  Q1 \<mapsto> P2
-----------------------------------
    {P1} A |+| B {Q2}
*}

lemma step_0 [simp]: 
  shows "step (0, (l, r)) p = (0, (l, r))"
by (case_tac p, simp)

lemma steps_0 [simp]: 
  shows "steps (0, (l, r)) p n = (0, (l, r))"
by (induct n) (simp_all)

declare steps.simps[simp del]

lemma before_final: 
  assumes "steps (1, tp) A n = (0, tp')"
  obtains n' where "\<not> is_final (steps (1, tp) A n')" 
        and "steps (1, tp) A (Suc n') = (0, tp')"
proof -
  from assms have "\<exists> n'. \<not> is_final (steps (1, tp) A n') \<and> 
               steps (1, tp) A (Suc n') = (0, tp')"
  proof(induct n arbitrary: tp', simp add: steps.simps)
    fix n tp'
    assume ind: 
      "\<And>tp'. steps (1, tp) A n = (0, tp') \<Longrightarrow>
      \<exists>n'. \<not> is_final (steps (1, tp) A n') \<and> steps (1, tp) A (Suc n') = (0, tp')"
    and h: " steps (1, tp) A (Suc n) = (0, tp')"
    from h show  "\<exists>n'. \<not> is_final (steps (1, tp) A n') \<and> steps (1, tp) A (Suc n') = (0, tp')"
    proof(simp add: step_red del: steps.simps, 
                     case_tac "(steps (Suc 0, tp) A n)", case_tac "a = 0", simp)
      fix a b c
      assume " steps (Suc 0, tp) A n = (0, tp')"
      hence "\<exists>n'. \<not> is_final (steps (1, tp) A n') \<and> steps (1, tp) A (Suc n') = (0, tp')"
        apply(rule_tac ind, simp)
        done
      thus "\<exists>n'. \<not> is_final (steps (Suc 0, tp) A n') \<and> step (steps (Suc 0, tp) A n') A = (0, tp')"
        apply(simp)
        done
    next
      fix a b c
      assume "steps (Suc 0, tp) A n = (a, b, c)"
             "step (steps (Suc 0, tp) A n) A = (0, tp')"
        "a \<noteq> 0"
      thus "\<exists>n'. \<not> is_final (steps (Suc 0, tp) A n') \<and> 
        step (steps (Suc 0, tp) A n') A = (0, tp')"
        apply(rule_tac x = n in exI, simp)
        done
    qed
  qed
  thus "(\<And>n'. \<lbrakk>\<not> is_final (steps (1, tp) A n'); 
    steps (1, tp) A (Suc n') = (0, tp')\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis"
    by auto
qed

declare tm_comp.simps [simp del] adjust.simps[simp del] shift.simps[simp del]

lemma length_comp:
"length (A |+| B) = length A + length B"
apply(auto simp: tm_comp.simps)
done

lemma tmcomp_fetch_in_first:
  assumes "case (fetch A a x) of (ac, ns) \<Rightarrow> ns \<noteq> 0"
  shows "fetch (A |+| B) a x = fetch A a x"
using assms
apply(case_tac a, case_tac [!] x, 
auto simp: length_comp tm_comp.simps length_adjust nth_append)
apply(simp_all add: adjust.simps)
done


lemma is_final_eq: "is_final (ba, tp) = (ba = 0)"
apply(case_tac tp, simp add: is_final.simps)
done

lemma t_merge_pre_eq_step: 
  assumes step: "step (a, b, c) (A, 0) = cf"
  and     tm_wf: "tm_wf (A, 0)" 
  and     unfinal: "\<not> is_final cf"
  shows "step (a, b, c) (A |+| B, 0) = cf"
proof -
  have "fetch (A |+| B) a (read c) = fetch A a (read c)"
  proof(rule_tac tmcomp_fetch_in_first)
    from step and unfinal show "case fetch A a (read c) of (ac, ns) \<Rightarrow> ns \<noteq> 0"
      apply(auto simp: is_final.simps)
      apply(case_tac "fetch A a (read c)", simp_all add: is_final_eq)
      done
  qed      
  thus "?thesis"
    using step
    apply(auto simp: step.simps is_final.simps)
    done
qed

declare tm_wf.simps[simp del] step.simps[simp del]

lemma t_merge_pre_eq:  
  "\<lbrakk>steps (Suc 0, tp) (A, 0) stp = cf; \<not> is_final cf; tm_wf (A, 0)\<rbrakk>
  \<Longrightarrow> steps (Suc 0, tp) (A |+| B, 0) stp = cf"
proof(induct stp arbitrary: cf, simp add: steps.simps)
  fix stp cf
  assume ind: "\<And>cf. \<lbrakk>steps (Suc 0, tp) (A, 0) stp = cf; \<not> is_final cf; tm_wf (A, 0)\<rbrakk> 
    \<Longrightarrow> steps (Suc 0, tp) (A |+| B, 0) stp = cf"
  and h: "steps (Suc 0, tp) (A, 0) (Suc stp) = cf"
      "\<not> is_final cf" "tm_wf (A, 0)"
  from h show "steps (Suc 0, tp) (A |+| B, 0) (Suc stp) = cf"
  proof(simp add: step_red, case_tac "(steps (Suc 0, tp) (A, 0) stp)", simp)
    fix a b c
    assume g: "steps (Suc 0, tp) (A, 0) stp = (a, b, c)"
      "step (a, b, c) (A, 0) = cf"
    have "(steps (Suc 0, tp) (A |+| B, 0) stp) = (a, b, c)"
    proof(rule ind, simp_all add: h g)
      show "0 < a"
        using g h
        apply(simp add: step_red)
        apply(case_tac a, auto simp: step_0)
        apply(case_tac "steps (Suc 0, tp) (A, 0) stp", simp)
        done
    qed
    thus "step (steps (Suc 0, tp) (A |+| B, 0) stp) (A |+| B, 0) = cf"
      apply(simp)
      apply(rule_tac t_merge_pre_eq_step, simp_all add: g h)
      done
  qed
qed

lemma tmcomp_fetch_in_first2:
  assumes "fetch A a x = (ac, 0)"
          "tm_wf (A, 0)"
          "a \<le> length A div 2" "a > 0"
  shows "fetch (A |+| B) a x = (ac, Suc (length A div 2))"
using assms
apply(case_tac a, case_tac [!] x, 
auto simp: length_comp tm_comp.simps length_adjust nth_append)
apply(simp_all add: adjust.simps)
done

lemma tmcomp_exec_after_first:
  "\<lbrakk>0 < a; step (a, b, c) (A, 0) = (0, tp'); tm_wf (A, 0); 
       a \<le> length A div 2\<rbrakk>
       \<Longrightarrow> step (a, b, c) (A |+| B, 0) = (Suc (length A div 2), tp')"
apply(simp add: step.simps, auto)
apply(case_tac "fetch A a Bk", simp add: tmcomp_fetch_in_first2)
apply(case_tac "fetch A a (hd c)", simp add: tmcomp_fetch_in_first2)
done

lemma step_nothalt_pre: "\<lbrakk>step (aa, ba, ca) (A, 0) = (a, b, c);  0 < a\<rbrakk> \<Longrightarrow> 0 < aa"
apply(case_tac "aa = 0", simp add: step_0, simp)
done

lemma nth_in_set: 
  "\<lbrakk> A ! i = x; i <  length A\<rbrakk> \<Longrightarrow> x \<in> set A"
by auto

lemma step_nothalt: 
  "\<lbrakk>step (aa, ba, ca) (A, 0) = (a, b, c); 0 < a; tm_wf (A, 0)\<rbrakk> \<Longrightarrow> 
  a \<le> length A div 2"
apply(simp add: step.simps)
apply(case_tac aa, case_tac [!] aa, auto split: if_splits simp: tm_wf.simps)
apply(case_tac "A ! (2 * nat)", simp)
apply(erule_tac x = "(aa, a)" in ballE, simp_all add: nth_in_set)
apply(case_tac "hd ca", auto split: if_splits simp: tm_wf.simps)
apply(case_tac "A ! (2 * nat)", simp)
apply(erule_tac x = "(aa, a)" in ballE, simp_all add: nth_in_set)
apply(case_tac "A ! (Suc (2 * nat))")
apply(erule_tac x = "(aa,bb)" in ballE, simp_all add: nth_in_set)
done

lemma steps_in_range: 
  " \<lbrakk>0 < a; steps (Suc 0, tp) (A, 0) stp = (a, b, c); tm_wf (A, 0)\<rbrakk>
  \<Longrightarrow> a \<le> length A div 2"
proof(induct stp arbitrary: a b c)
  fix a b c
  assume h: "0 < a" "steps (Suc 0, tp) (A, 0) 0 = (a, b, c)" 
            "tm_wf (A, 0)"
  thus "a \<le> length A div 2"
    apply(simp add: steps.simps tm_wf.simps, auto)
    done
next
  fix stp a b c
  assume ind: "\<And>a b c. \<lbrakk>0 < a; steps (Suc 0, tp) (A, 0) stp = (a, b, c); 
    tm_wf (A, 0)\<rbrakk> \<Longrightarrow> a \<le> length A div 2"
  and h: "0 < a" "steps (Suc 0, tp) (A, 0) (Suc stp) = (a, b, c)" "tm_wf (A, 0)"
  from h show "a \<le> length A div 2"
  proof(simp add: step_red, case_tac "(steps (Suc 0, tp) (A, 0) stp)", simp)
    fix aa ba ca
    assume g: "step (aa, ba, ca) (A, 0) = (a, b, c)" 
           "steps (Suc 0, tp) (A, 0) stp = (aa, ba, ca)"
    hence "aa \<le> length A div 2"
      apply(rule_tac ind, auto simp: h step_nothalt_pre)
      done
    thus "?thesis"
      using g h
      apply(rule_tac step_nothalt, auto)
      done
  qed
qed

lemma t_merge_pre_halt_same: 
  assumes a_ht: "steps (1, tp) (A, 0) n = (0, tp')"
  and a_wf: "tm_wf (A, 0)"
  obtains n' where "steps (1, tp) (A |+| B, 0) n' = (Suc (length A div 2), tp')"
proof -
  assume a: "\<And>n. steps (1, tp) (A |+| B, 0) n = (Suc (length A div 2), tp') \<Longrightarrow> thesis"
  obtain stp' where "\<not> is_final (steps (1, tp) (A, 0) stp')" and 
                          "steps (1, tp) (A, 0) (Suc stp') = (0, tp')"
  using a_ht before_final by blast
  then have "steps (1, tp) (A |+| B, 0) (Suc stp') = (Suc (length A div 2), tp')"
  proof(simp add: step_red)
    assume "\<not> is_final (steps (Suc 0, tp) (A, 0) stp')"
           " step (steps (Suc 0, tp) (A, 0) stp') (A, 0) = (0, tp')"
    moreover hence "(steps (Suc 0, tp) (A |+| B, 0) stp') = (steps (Suc 0, tp) (A, 0) stp')"
      apply(rule_tac t_merge_pre_eq)
      apply(simp_all add: a_wf a_ht)
      done
    ultimately show "step (steps (Suc 0, tp) (A |+| B, 0) stp') (A |+| B, 0) = (Suc (length A div 2), tp')"
      apply(case_tac " steps (Suc 0, tp) (A, 0) stp'", simp)
      apply(rule tmcomp_exec_after_first, simp_all add: a_wf)
      apply(erule_tac steps_in_range, auto simp: a_wf)
      done
  qed
  with a show thesis by blast
qed

lemma tm_comp_fetch_second_zero:
  "\<lbrakk>fetch B sa' x = (a, 0); tm_wf (A, 0); tm_wf (B, 0); sa' > 0\<rbrakk>
     \<Longrightarrow> fetch (A |+| B) (sa' + (length A div 2)) x = (a, 0)"
apply(case_tac x)
apply(case_tac [!] sa',
  auto simp: fetch.simps length_comp length_adjust nth_append tm_comp.simps
             tm_wf.simps shift.simps split: if_splits)
done 

lemma tm_comp_fetch_second_inst:
  "\<lbrakk>sa > 0; s > 0;  tm_wf (A, 0); tm_wf (B, 0); fetch B sa x = (a, s)\<rbrakk>
     \<Longrightarrow> fetch (A |+| B) (sa + length A div 2) x = (a, s + length A div 2)"
apply(case_tac x)
apply(case_tac [!] sa,
  auto simp: fetch.simps length_comp length_adjust nth_append tm_comp.simps
             tm_wf.simps shift.simps split: if_splits)
done 

lemma t_merge_second_same:
  assumes a_wf: "tm_wf (A, 0)"
  and b_wf: "tm_wf (B, 0)"
  and steps: "steps (Suc 0, l, r) (B, 0) stp = (s, l', r')"
  shows "steps (Suc (length A div 2), l, r)  (A |+| B, 0) stp
       = (if s = 0 then 0
          else s + length A div 2, l', r')"
using a_wf b_wf steps
proof(induct stp arbitrary: s l' r', simp add: steps.simps, simp)
  fix stpa sa l'a r'a
  assume ind: "\<And>s l' r'. steps (Suc 0, l, r) (B, 0) stpa = (s, l', r') \<Longrightarrow>
    steps (Suc (length A div 2), l, r) (A |+| B, 0) stpa = 
                (if s = 0 then 0 else s + length A div 2, l', r')"
  and h: "step (steps (Suc 0, l, r) (B, 0) stpa) (B, 0) = (sa, l'a, r'a)"
  obtain sa' l'' r'' where a: "(steps (Suc 0, l, r) (B, 0) stpa) = (sa', l'', r'')"
    apply(case_tac "steps (Suc 0, l, r) (B, 0) stpa", auto)
    done
  from this have b: "steps (Suc (length A div 2), l, r) (A |+| B, 0) stpa = 
                (if sa' = 0 then 0 else sa' + length A div 2, l'', r'')"
    apply(erule_tac ind)
    done
  from a b h show 
    "(sa = 0 \<longrightarrow> step (steps (Suc (length A div 2), l, r) (A |+| B, 0) stpa) (A |+| B, 0) = (0, l'a, r'a)) \<and>
    (0 < sa \<longrightarrow> step (steps (Suc (length A div 2), l, r) (A |+| B, 0) stpa) (A |+| B, 0) = (sa + length A div 2, l'a, r'a))"
  proof(case_tac "sa' = 0", auto)
    assume "step (sa', l'', r'') (B, 0) = (0, l'a, r'a)" "0 < sa'"
    thus "step (sa' + length A div 2, l'', r'') (A |+| B, 0) = (0, l'a, r'a)"
      using a_wf b_wf
      apply(simp add:  step.simps)
      apply(case_tac "fetch B sa' (read r'')", auto)
      apply(simp_all add: step.simps tm_comp_fetch_second_zero)
      done
  next
    assume "step (sa', l'', r'') (B, 0) = (sa, l'a, r'a)" "0 < sa'" "0 < sa"
    thus "step (sa' + length A div 2, l'', r'') (A |+| B, 0) = (sa + length A div 2, l'a, r'a)"
      using a_wf b_wf
      apply(simp add: step.simps)
      apply(case_tac "fetch B sa' (read r'')", auto)
      apply(simp_all add: step.simps tm_comp_fetch_second_inst)
      done
  qed
qed

lemma t_merge_second_halt_same:
  "\<lbrakk>tm_wf (A, 0); tm_wf (B, 0); 
   steps (1, l, r) (B, 0) stp = (0, l', r')\<rbrakk>
     \<Longrightarrow> steps (Suc (length A div 2), l, r)  (A |+| B, 0) stp
       = (0, l', r')"
using t_merge_second_same[where s = "0"]
apply(auto)
done

lemma Hoare_plus_halt: 
  assumes aimpb: "Q1 \<mapsto> P2"
  and A_wf : "tm_wf (A, 0)"
  and B_wf : "tm_wf (B, 0)"
  and A_halt : "{P1} (A, 0) {Q1}"
  and B_halt : "{P2} (B, 0) {Q2}"
  shows "{P1} (A |+| B, 0) {Q2}"
proof(rule HoareI)
  fix l r
  assume h: "P1 (l, r)"
  then obtain n1 where a: "is_final (steps (1, l, r) (A, 0) n1)" and b: "Q1 holds_for (steps (1, l, r) (A, 0) n1)"
    using A_halt unfolding Hoare_def by auto
  then obtain l' r' where "steps (1, l, r) (A, 0) n1 = (0, l', r')" and c: "Q1 holds_for (0, l', r')"
    by(case_tac "steps (1, l, r) (A, 0) n1", auto)
  then obtain stpa where d: "steps (1, l, r) (A |+| B, 0) stpa = (Suc (length A div 2), l', r')"
    using A_wf
    by(rule_tac t_merge_pre_halt_same, auto)
  from c aimpb have "P2 holds_for (0, l', r')"
    by(rule holds_for_imp)
  from this have "P2 (l', r')" by auto
  from this obtain n2 where e: "is_final (steps (1, l', r') (B, 0) n2)" and f: "Q2 holds_for (steps (1, l', r') (B, 0) n2)"
    using B_halt unfolding Hoare_def
    by auto
  then obtain l'' r'' where "steps (1, l', r') (B, 0) n2 = (0, l'', r'')" and g: "Q2 holds_for (0, l'', r'')"
    by(case_tac "steps (1, l', r') (B, 0) n2", auto)
  from this have "steps (Suc (length A div 2), l', r')  (A |+| B, 0) n2
    = (0, l'', r'')"
    apply(rule_tac t_merge_second_halt_same, auto simp: A_wf B_wf)
    done
  thus "\<exists>n. is_final (steps (1, l, r) (A |+| B, 0) n) \<and> Q2 holds_for (steps (1, l, r) (A |+| B, 0) n)"
    using d g
    apply(rule_tac x = "stpa + n2" in exI)
    apply(simp add: steps_add)
    done
qed

definition
  Hoare_unhalt :: "assert \<Rightarrow> tprog \<Rightarrow> bool" ("({(1_)}/ (_))" 50)
where
  "{P} p \<equiv> 
     (\<forall>l r. P (l, r) \<longrightarrow> (\<forall> n . \<not> (is_final (steps (1, (l, r)) p n))))"

lemma Hoare_unhalt_I:
  assumes "\<And>l r. P (l, r) \<Longrightarrow> \<forall> n. \<not> is_final (steps (1, (l, r)) p n)"
  shows "{P} p"
unfolding Hoare_unhalt_def using assms by auto

lemma Hoare_plus_unhalt: 
  assumes aimpb: "Q1 \<mapsto> P2"
  and A_wf : "tm_wf (A, 0)"
  and B_wf : "tm_wf (B, 0)"
  and A_halt : "{P1} (A, 0) {Q1}"
  and B_uhalt : "{P2} (B, 0)"
  shows "{P1} (A |+| B, 0)"
proof(rule_tac Hoare_unhalt_I)
  fix l r
  assume h: "P1 (l, r)"
  then obtain n1 where a: "is_final (steps (1, l, r) (A, 0) n1)" and b: "Q1 holds_for (steps (1, l, r) (A, 0) n1)"
    using A_halt unfolding Hoare_def by auto
  then obtain l' r' where "steps (1, l, r) (A, 0) n1 = (0, l', r')" and c: "Q1 holds_for (0, l', r')"
    by(case_tac "steps (1, l, r) (A, 0) n1", auto)
  then obtain stpa where d: "steps (1, l, r) (A |+| B, 0) stpa = (Suc (length A div 2), l', r')"
    using A_wf
    by(rule_tac t_merge_pre_halt_same, auto)
  from c aimpb have "P2 holds_for (0, l', r')"
    by(rule holds_for_imp)
  from this have "P2 (l', r')" by auto
  from this have e: "\<forall> n. \<not> is_final (steps (Suc 0, l', r') (B, 0) n)  "
    using B_uhalt unfolding Hoare_unhalt_def
    by auto
  from e show "\<forall>n. \<not> is_final (steps (1, l, r) (A |+| B, 0) n)"
  proof(rule_tac allI, case_tac "n > stpa")
    fix n
    assume h2: "stpa < n"
    hence "\<not> is_final (steps (Suc 0, l', r') (B, 0) (n - stpa))"
      using e
      apply(erule_tac x = "n - stpa" in allE) by simp
    then obtain s'' l'' r'' where f: "steps (Suc 0, l', r') (B, 0) (n - stpa) = (s'', l'', r'')" and g: "s'' \<noteq> 0"
      apply(case_tac "steps (Suc 0, l', r') (B, 0) (n - stpa)", auto)
      done
    have k: "steps (Suc (length A div 2), l', r') (A |+| B, 0) (n - stpa) = (s''+ length A div 2, l'', r'') "
      using A_wf B_wf f g
      apply(drule_tac t_merge_second_same, auto)
      done
    show "\<not> is_final (steps (1, l, r) (A |+| B, 0) n)"
    proof -
      have "\<not> is_final (steps (1, l, r) (A |+| B, 0) (stpa + (n  - stpa)))"
        using d k A_wf
        apply(simp only: steps_add d, simp add: tm_wf.simps)
        done
      thus "\<not> is_final (steps (1, l, r) (A |+| B, 0) n)"
        using h2 by simp
    qed
  next
    fix n
    assume h2: "\<not> stpa < n"
    with d show "\<not> is_final (steps (1, l, r) (A |+| B, 0) n)"
      apply(auto)
      apply(subgoal_tac "\<exists> d. stpa = n + d", auto)
      apply(case_tac "(steps (Suc 0, l, r) (A |+| B, 0) n)", simp)
      apply(rule_tac x = "stpa - n" in exI, simp)
      done
  qed
qed
        
end