(* Title: thys/Turing_Hoare.thy

   Author: Jian Xu, Xingyuan Zhang, and Christian Urban

*)

chapter {* Hoare Rules for TMs *}

theory Turing_Hoare

imports Turing

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)"

lemma [intro, simp]:

  "P \<mapsto> P"

unfolding assert_imp_def by simp

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 n) = Q holds_for c"

using assms 

apply(induct n)

apply(auto)

apply(case_tac [!] c)

apply(auto)

done

(* Hoare Rules *)

(* halting case *)

definition

  Hoare_halt :: "assert \<Rightarrow> tprog0 \<Rightarrow> assert \<Rightarrow> bool" ("({(1_)}/ (_)/ {(1_)})" 50)

where

  "{P} p {Q} \<equiv> \<forall>tp. P tp \<longrightarrow> (\<exists>n. is_final (steps0 (1, tp) p n) \<and> Q holds_for (steps0 (1, tp) p n))"

(* not halting case *)

definition

  Hoare_unhalt :: "assert \<Rightarrow> tprog0 \<Rightarrow> bool" ("({(1_)}/ (_)) \<up>" 50)

where

  "{P} p \<up> \<equiv> \<forall>tp. P tp \<longrightarrow> (\<forall> n . \<not> (is_final (steps0 (1, tp) p n)))"

lemma Hoare_haltI:

  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_halt_def 

using assms by auto

lemma Hoare_unhaltI:

  assumes "\<And>l r n. P (l, r) \<Longrightarrow> \<not> is_final (steps0 (1, (l, r)) p n)"

  shows "{P} p \<up>"

unfolding Hoare_unhalt_def 

using assms by auto

text {*

  {P} A {Q}   {Q} B {S}  A well-formed

  -----------------------------------------

  {P} A |+| B {S}

*}

lemma Hoare_plus_halt [case_names A_halt B_halt A_wf]: 

  assumes A_halt : "{P} A {Q}"

  and B_halt : "{Q} B {S}"

  and A_wf : "tm_wf (A, 0)"

  shows "{P} A |+| B {S}"

proof(rule Hoare_haltI)

  fix l r

  assume h: "P (l, r)"

  then obtain n1 l' r' 

    where "is_final (steps0 (1, l, r) A n1)"  

      and a1: "Q holds_for (steps0 (1, l, r) A n1)"

      and a2: "steps0 (1, l, r) A n1 = (0, l', r')"

    using A_halt unfolding Hoare_halt_def

    by (metis is_final_eq surj_pair) 

  then obtain n2 

    where "steps0 (1, l, r) (A |+| B) n2 = (Suc (length A div 2), l', r')"

    using A_wf by (rule_tac tm_comp_next) 

  moreover

  from a1 a2 have "Q (l', r')" by (simp)

  then obtain n3 l'' r''

    where "is_final (steps0 (1, l', r') B n3)" 

    and b1: "S holds_for (steps0 (1, l', r') B n3)"

    and b2: "steps0 (1, l', r') B n3 = (0, l'', r'')"

    using B_halt unfolding Hoare_halt_def 

    by (metis is_final_eq surj_pair) 

  then have "steps0 (Suc (length A div 2), l', r')  (A |+| B) n3 = (0, l'', r'')"

    using A_wf by (rule_tac tm_comp_final) 

  ultimately show 

    "\<exists>n. is_final (steps0 (1, l, r) (A |+| B) n) \<and> S holds_for (steps0 (1, l, r) (A |+| B) n)"

    using b1 b2 by (rule_tac x = "n2 + n3" in exI) (simp)

qed

text {*

  {P} A {Q}   {Q} B loops   A well-formed

  ------------------------------------------

          {P} A |+| B  loops

*}

lemma Hoare_plus_unhalt [case_names A_halt B_unhalt A_wf]:

  assumes A_halt: "{P} A {Q}"

  and B_uhalt: "{Q} B \<up>"

  and A_wf : "tm_wf (A, 0)"

  shows "{P} (A |+| B) \<up>"

proof(rule_tac Hoare_unhaltI)

  fix n l r 

  assume h: "P (l, r)"

  then obtain n1 l' r'

    where a: "is_final (steps0 (1, l, r) A n1)" 

    and b: "Q holds_for (steps0 (1, l, r) A n1)"

    and c: "steps0 (1, l, r) A n1 = (0, l', r')"

    using A_halt unfolding Hoare_halt_def 

    by (metis is_final_eq surj_pair) 

  then obtain n2 where eq: "steps0 (1, l, r) (A |+| B) n2 = (Suc (length A div 2), l', r')"

    using A_wf by (rule_tac tm_comp_next)

  then show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"

  proof(cases "n2 \<le> n")

    case True

    from b c have "Q (l', r')" by simp

    then have "\<forall> n. \<not> is_final (steps0 (1, l', r') B n)  "

      using B_uhalt unfolding Hoare_unhalt_def by simp

    then have "\<not> is_final (steps0 (1, l', r') B (n - n2))" by auto

    then obtain s'' l'' r'' 

      where "steps0 (1, l', r') B (n - n2) = (s'', l'', r'')" 

      and "\<not> is_final (s'', l'', r'')" by (metis surj_pair)

    then have "steps0 (Suc (length A div 2), l', r') (A |+| B) (n - n2) = (s''+ length A div 2, l'', r'')"

      using A_wf by (auto dest: tm_comp_second simp del: tm_wf.simps)

    then have "\<not> is_final (steps0 (1, l, r) (A |+| B) (n2 + (n  - n2)))"

      using A_wf by (simp only: steps_add eq) (simp add: tm_wf.simps)

    then show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)" 

      using `n2 \<le> n` by simp

  next 

    case False

    then obtain n3 where "n = n2 - n3"

      using diff_le_self le_imp_diff_is_add nat_le_linear

      add.commute by metis

    moreover

    with eq show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"

      by (simp add: not_is_final[where ?n1.0="n2"])

  qed

qed

lemma Hoare_consequence:

  assumes "P' \<mapsto> P" "{P} p {Q}" "Q \<mapsto> Q'"

  shows "{P'} p {Q'}"

using assms

unfolding Hoare_halt_def assert_imp_def

by (metis holds_for.simps surj_pair)

end