theory turing_hoare
imports turing_basic
begin

type_synonym assert = "tape \<Rightarrow> bool"

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

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 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(auto)
apply(case_tac [!] c)
apply(auto)
done

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)

definition
  Hoare :: "assert \<Rightarrow> tprog0 \<Rightarrow> assert \<Rightarrow> bool" ("({(1_)}/ (_)/ {(1_)})" 50)
where
  "{P} p {Q} \<equiv> 
     (\<forall>l r. P (l, r) \<longrightarrow> (\<exists>n. is_final (steps0 (1, (l, r)) p n) \<and> Q holds_for (steps0 (1, (l, r)) p n)))"

lemma HoareI:
  assumes "\<And>l r. P (l, r) \<Longrightarrow> \<exists>n. is_final (steps0 (1, (l, r)) p n) \<and> Q holds_for (steps0 (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   A, B well-formed
  ------------------------------------------------------
  {P1} A |+| B {Q2}
*}


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 {Q1}"
  and B_halt : "{P2} B {Q2}"
  shows "{P1} A |+| B {Q2}"
proof(rule HoareI)
  fix l r
  assume h: "P1 (l, r)"
  then obtain n1 
    where "is_final (steps0 (1, l, r) A n1)" and "Q1 holds_for (steps0 (1, l, r) A n1)"
    using A_halt unfolding Hoare_def by auto
  then obtain l' r' where "steps0 (1, l, r) A n1 = (0, l', r')" and c: "Q1 holds_for (0, l', r')"
    by(case_tac "steps0 (1, l, r) A n1") (auto)
  then obtain stpa where d: "steps0 (1, l, r) (A |+| B) stpa = (Suc (length A div 2), l', r')"
    using A_wf by(rule_tac t_merge_pre_halt_same) (auto)
  moreover
  from c aimpb have "P2 holds_for (0, l', r')"
    by (rule holds_for_imp)
  then have "P2 (l', r')" by auto
  then obtain n2 
    where "is_final (steps0 (1, l', r') B n2)" and "Q2 holds_for (steps0 (1, l', r') B n2)"
    using B_halt unfolding Hoare_def by auto
  then obtain l'' r'' where "steps0 (1, l', r') B n2 = (0, l'', r'')" and g: "Q2 holds_for (0, l'', r'')"
    by (case_tac "steps0 (1, l', r') B n2") (auto)
  then have "steps0 (Suc (length A div 2), l', r')  (A |+| B) n2 = (0, l'', r'')"
    by (rule_tac t_merge_second_halt_same) (auto simp: A_wf B_wf)
  ultimately show 
    "\<exists>n. is_final (steps0 (1, l, r) (A |+| B) n) \<and> Q2 holds_for (steps0 (1, l, r) (A |+| B) n)"
    using g 
    apply(rule_tac x = "stpa + n2" in exI)
    apply(simp add: steps_add)
    done
qed

definition
  Hoare_unhalt :: "assert \<Rightarrow> tprog0 \<Rightarrow> bool" ("({(1_)}/ (_))" 50)
where
  "{P} p \<equiv> (\<forall>l r. P (l, r) \<longrightarrow> (\<forall> n . \<not> (is_final (steps0 (1, (l, r)) p n))))"

lemma Hoare_unhalt_I:
  assumes "\<And>l r. P (l, r) \<Longrightarrow> \<forall> n. \<not> is_final (steps0 (1, (l, r)) p n)"
  shows "{P} p"
unfolding Hoare_unhalt_def using assms by auto

lemma Hoare_plus_unhalt:
  fixes A B :: tprog0 
  assumes aimpb: "Q1 \<mapsto> P2"
  and A_wf : "tm_wf (A, 0)"
  and B_wf : "tm_wf (B, 0)"
  and A_halt : "{P1} A {Q1}"
  and B_uhalt : "{P2} B"
  shows "{P1} (A |+| B)"
proof(rule_tac Hoare_unhalt_I)
  fix l r
  assume h: "P1 (l, r)"
  then obtain n1 where a: "is_final (steps0 (1, l, r) A n1)" and b: "Q1 holds_for (steps0 (1, l, r) A n1)"
    using A_halt unfolding Hoare_def by auto
  then obtain l' r' where "steps0 (1, l, r) A n1 = (0, l', r')" and c: "Q1 holds_for (0, l', r')"
    by(case_tac "steps0 (1, l, r) A n1", auto)
  then obtain stpa where d: "steps0 (1, l, r) (A |+| B) 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 (steps0 (Suc 0, l', r') B n)  "
    using B_uhalt unfolding Hoare_unhalt_def
    by auto
  from e show "\<forall>n. \<not> is_final (steps0 (1, l, r) (A |+| B) n)"
  proof(rule_tac allI, case_tac "n > stpa")
    fix n
    assume h2: "stpa < n"
    hence "\<not> is_final (steps0 (Suc 0, l', r') B (n - stpa))"
      using e
      apply(erule_tac x = "n - stpa" in allE) by simp
    then obtain s'' l'' r'' where f: "steps0 (Suc 0, l', r') B (n - stpa) = (s'', l'', r'')" and g: "s'' \<noteq> 0"
      apply(case_tac "steps0 (Suc 0, l', r') B (n - stpa)", auto)
      done
    have k: "steps0 (Suc (length A div 2), l', r') (A |+| B) (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 (steps0 (1, l, r) (A |+| B) n)"
    proof -
      have "\<not> is_final (steps0 (1, l, r) (A |+| B) (stpa + (n  - stpa)))"
        using d k A_wf
        apply(simp only: steps_add d, simp add: tm_wf.simps)
        done
      thus "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"
        using h2 by simp
    qed
  next
    fix n
    assume h2: "\<not> stpa < n"
    with d show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"
      apply(auto)
      apply(subgoal_tac "\<exists> d. stpa = n + d", auto)
      apply(case_tac "(steps0 (Suc 0, l, r) (A |+| B) n)", simp)
      apply(rule_tac x = "stpa - n" in exI, simp)
      done
  qed
qed

lemma Hoare_weak:
  fixes p::tprog0
  assumes a: "{P} p {Q}"
  and b: "P' \<mapsto> P" 
  and c: "Q \<mapsto> Q'"
  shows "{P'} p {Q'}"
using assms
unfolding Hoare_def assert_imp_def
by (blast intro: holds_for_imp[simplified assert_imp_def])

end