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')"

  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 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

  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