(* Author: Stefan Berghofer, Lukas Bulwahn, TU Muenchen *)

header {* A tabled implementation of the reflexive transitive closure *}

theory Transitive_Closure_Table

imports Main

begin

inductive rtrancl_path :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a \<Rightarrow> bool"

  for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

where

  base: "rtrancl_path r x [] x"

| step: "r x y \<Longrightarrow> rtrancl_path r y ys z \<Longrightarrow> rtrancl_path r x (y # ys) z"

lemma rtranclp_eq_rtrancl_path: "r\<^sup>*\<^sup>* x y = (\<exists>xs. rtrancl_path r x xs y)"

proof

  assume "r\<^sup>*\<^sup>* x y"

  then show "\<exists>xs. rtrancl_path r x xs y"

  proof (induct rule: converse_rtranclp_induct)

    case base

    have "rtrancl_path r y [] y" by (rule rtrancl_path.base)

    then show ?case ..

  next

    case (step x z)

    from `\<exists>xs. rtrancl_path r z xs y`

    obtain xs where "rtrancl_path r z xs y" ..

    with `r x z` have "rtrancl_path r x (z # xs) y"

      by (rule rtrancl_path.step)

    then show ?case ..

  qed

next

  assume "\<exists>xs. rtrancl_path r x xs y"

  then obtain xs where "rtrancl_path r x xs y" ..

  then show "r\<^sup>*\<^sup>* x y"

  proof induct

    case (base x)

    show ?case by (rule rtranclp.rtrancl_refl)

  next

    case (step x y ys z)

    from `r x y` `r\<^sup>*\<^sup>* y z` show ?case

      by (rule converse_rtranclp_into_rtranclp)

  qed

qed

lemma rtrancl_path_trans:

  assumes xy: "rtrancl_path r x xs y"

  and yz: "rtrancl_path r y ys z"

  shows "rtrancl_path r x (xs @ ys) z" using xy yz

proof (induct arbitrary: z)

  case (base x)

  then show ?case by simp

next

  case (step x y xs)

  then have "rtrancl_path r y (xs @ ys) z"

    by simp

  with `r x y` have "rtrancl_path r x (y # (xs @ ys)) z"

    by (rule rtrancl_path.step)

  then show ?case by simp

qed

lemma rtrancl_path_appendE:

  assumes xz: "rtrancl_path r x (xs @ y # ys) z"

  obtains "rtrancl_path r x (xs @ [y]) y" and "rtrancl_path r y ys z" using xz

proof (induct xs arbitrary: x)

  case Nil

  then have "rtrancl_path r x (y # ys) z" by simp

  then obtain xy: "r x y" and yz: "rtrancl_path r y ys z"

    by cases auto

  from xy have "rtrancl_path r x [y] y"

    by (rule rtrancl_path.step [OF _ rtrancl_path.base])

  then have "rtrancl_path r x ([] @ [y]) y" by simp

  then show ?thesis using yz by (rule Nil)

next

  case (Cons a as)

  then have "rtrancl_path r x (a # (as @ y # ys)) z" by simp

  then obtain xa: "r x a" and az: "rtrancl_path r a (as @ y # ys) z"

    by cases auto

  show ?thesis

  proof (rule Cons(1) [OF _ az])

    assume "rtrancl_path r y ys z"

    assume "rtrancl_path r a (as @ [y]) y"

    with xa have "rtrancl_path r x (a # (as @ [y])) y"

      by (rule rtrancl_path.step)

    then have "rtrancl_path r x ((a # as) @ [y]) y"

      by simp

    then show ?thesis using `rtrancl_path r y ys z`

      by (rule Cons(2))

  qed

qed

lemma rtrancl_path_distinct:

  assumes xy: "rtrancl_path r x xs y"

  obtains xs' where "rtrancl_path r x xs' y" and "distinct (x # xs')" using xy

proof (induct xs rule: measure_induct_rule [of length])

  case (less xs)

  show ?case

  proof (cases "distinct (x # xs)")

    case True

    with `rtrancl_path r x xs y` show ?thesis by (rule less)

  next

    case False

    then have "\<exists>as bs cs a. x # xs = as @ [a] @ bs @ [a] @ cs"

      by (rule not_distinct_decomp)

    then obtain as bs cs a where xxs: "x # xs = as @ [a] @ bs @ [a] @ cs"

      by iprover

    show ?thesis

    proof (cases as)

      case Nil

      with xxs have x: "x = a" and xs: "xs = bs @ a # cs"

        by auto

      from x xs `rtrancl_path r x xs y` have cs: "rtrancl_path r x cs y"

        by (auto elim: rtrancl_path_appendE)

      from xs have "length cs < length xs" by simp

      then show ?thesis

        by (rule less(1)) (iprover intro: cs less(2))+

    next

      case (Cons d ds)

      with xxs have xs: "xs = ds @ a # (bs @ [a] @ cs)"

        by auto

      with `rtrancl_path r x xs y` obtain xa: "rtrancl_path r x (ds @ [a]) a"

        and ay: "rtrancl_path r a (bs @ a # cs) y"

        by (auto elim: rtrancl_path_appendE)

      from ay have "rtrancl_path r a cs y" by (auto elim: rtrancl_path_appendE)

      with xa have xy: "rtrancl_path r x ((ds @ [a]) @ cs) y"

        by (rule rtrancl_path_trans)

      from xs have "length ((ds @ [a]) @ cs) < length xs" by simp

      then show ?thesis

        by (rule less(1)) (iprover intro: xy less(2))+

    qed

  qed

qed

inductive rtrancl_tab :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"

  for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

where

  base: "rtrancl_tab r xs x x"

| step: "x \<notin> set xs \<Longrightarrow> r x y \<Longrightarrow> rtrancl_tab r (x # xs) y z \<Longrightarrow> rtrancl_tab r xs x z"

lemma rtrancl_path_imp_rtrancl_tab:

  assumes path: "rtrancl_path r x xs y"

  and x: "distinct (x # xs)"

  and ys: "({x} \<union> set xs) \<inter> set ys = {}"

  shows "rtrancl_tab r ys x y" using path x ys

proof (induct arbitrary: ys)

  case base

  show ?case by (rule rtrancl_tab.base)

next

  case (step x y zs z)

  then have "x \<notin> set ys" by auto

  from step have "distinct (y # zs)" by simp

  moreover from step have "({y} \<union> set zs) \<inter> set (x # ys) = {}"

    by auto

  ultimately have "rtrancl_tab r (x # ys) y z"

    by (rule step)

  with `x \<notin> set ys` `r x y`

  show ?case by (rule rtrancl_tab.step)

qed

lemma rtrancl_tab_imp_rtrancl_path:

  assumes tab: "rtrancl_tab r ys x y"

  obtains xs where "rtrancl_path r x xs y" using tab

proof induct

  case base

  from rtrancl_path.base show ?case by (rule base)

next

  case step show ?case by (iprover intro: step rtrancl_path.step)

qed

lemma rtranclp_eq_rtrancl_tab_nil: "r\<^sup>*\<^sup>* x y = rtrancl_tab r [] x y"

proof

  assume "r\<^sup>*\<^sup>* x y"

  then obtain xs where "rtrancl_path r x xs y"

    by (auto simp add: rtranclp_eq_rtrancl_path)

  then obtain xs' where xs': "rtrancl_path r x xs' y"

    and distinct: "distinct (x # xs')"

    by (rule rtrancl_path_distinct)

  have "({x} \<union> set xs') \<inter> set [] = {}" by simp

  with xs' distinct show "rtrancl_tab r [] x y"

    by (rule rtrancl_path_imp_rtrancl_tab)

next

  assume "rtrancl_tab r [] x y"

  then obtain xs where "rtrancl_path r x xs y"

    by (rule rtrancl_tab_imp_rtrancl_path)

  then show "r\<^sup>*\<^sup>* x y"

    by (auto simp add: rtranclp_eq_rtrancl_path)

qed

declare rtranclp_rtrancl_eq[code del]

declare rtranclp_eq_rtrancl_tab_nil[THEN iffD2, code_pred_intro]

code_pred rtranclp using rtranclp_eq_rtrancl_tab_nil [THEN iffD1] by fastforce

subsection {* A simple example *}

datatype ty = A | B | C

inductive test :: "ty \<Rightarrow> ty \<Rightarrow> bool"

where

  "test A B"

| "test B A"

| "test B C"

subsubsection {* Invoking with the predicate compiler and the generic code generator *}

code_pred test .

values "{x. test\<^sup>*\<^sup>* A C}"

values "{x. test\<^sup>*\<^sup>* C A}"

values "{x. test\<^sup>*\<^sup>* A x}"

values "{x. test\<^sup>*\<^sup>* x C}"

value "test\<^sup>*\<^sup>* A C"

value "test\<^sup>*\<^sup>* C A"

hide_type ty

hide_const test A B C

end