src/HOL/Library/While_Combinator.thy

factored syntactic type classes for bot and top (by Alessandro Coglio)

(*  Title:      HOL/Library/While_Combinator.thy
    Author:     Tobias Nipkow
    Author:     Alexander Krauss
    Copyright   2000 TU Muenchen
*)

header {* A general ``while'' combinator *}

theory While_Combinator
imports Main
begin

subsection {* Partial version *}

definition while_option :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a option" where
"while_option b c s = (if (\<exists>k. ~ b ((c ^^ k) s))
   then Some ((c ^^ (LEAST k. ~ b ((c ^^ k) s))) s)
   else None)"

theorem while_option_unfold[code]:
"while_option b c s = (if b s then while_option b c (c s) else Some s)"
proof cases
  assume "b s"
  show ?thesis
  proof (cases "\<exists>k. ~ b ((c ^^ k) s)")
    case True
    then obtain k where 1: "~ b ((c ^^ k) s)" ..
    with `b s` obtain l where "k = Suc l" by (cases k) auto
    with 1 have "~ b ((c ^^ l) (c s))" by (auto simp: funpow_swap1)
    then have 2: "\<exists>l. ~ b ((c ^^ l) (c s))" ..
    from 1
    have "(LEAST k. ~ b ((c ^^ k) s)) = Suc (LEAST l. ~ b ((c ^^ Suc l) s))"
      by (rule Least_Suc) (simp add: `b s`)
    also have "... = Suc (LEAST l. ~ b ((c ^^ l) (c s)))"
      by (simp add: funpow_swap1)
    finally
    show ?thesis 
      using True 2 `b s` by (simp add: funpow_swap1 while_option_def)
  next
    case False
    then have "~ (\<exists>l. ~ b ((c ^^ Suc l) s))" by blast
    then have "~ (\<exists>l. ~ b ((c ^^ l) (c s)))"
      by (simp add: funpow_swap1)
    with False  `b s` show ?thesis by (simp add: while_option_def)
  qed
next
  assume [simp]: "~ b s"
  have least: "(LEAST k. ~ b ((c ^^ k) s)) = 0"
    by (rule Least_equality) auto
  moreover 
  have "\<exists>k. ~ b ((c ^^ k) s)" by (rule exI[of _ "0::nat"]) auto
  ultimately show ?thesis unfolding while_option_def by auto 
qed

lemma while_option_stop2:
 "while_option b c s = Some t \<Longrightarrow> EX k. t = (c^^k) s \<and> \<not> b t"
apply(simp add: while_option_def split: if_splits)
by (metis (lifting) LeastI_ex)

lemma while_option_stop: "while_option b c s = Some t \<Longrightarrow> ~ b t"
by(metis while_option_stop2)

theorem while_option_rule:
assumes step: "!!s. P s ==> b s ==> P (c s)"
and result: "while_option b c s = Some t"
and init: "P s"
shows "P t"
proof -
  def k == "LEAST k. ~ b ((c ^^ k) s)"
  from assms have t: "t = (c ^^ k) s"
    by (simp add: while_option_def k_def split: if_splits)    
  have 1: "ALL i<k. b ((c ^^ i) s)"
    by (auto simp: k_def dest: not_less_Least)

  { fix i assume "i <= k" then have "P ((c ^^ i) s)"
      by (induct i) (auto simp: init step 1) }
  thus "P t" by (auto simp: t)
qed

lemma funpow_commute: 
  "\<lbrakk>\<forall>k' < k. f (c ((c^^k') s)) = c' (f ((c^^k') s))\<rbrakk> \<Longrightarrow> f ((c^^k) s) = (c'^^k) (f s)"
by (induct k arbitrary: s) auto

lemma while_option_commute:
  assumes "\<And>s. b s = b' (f s)" "\<And>s. \<lbrakk>b s\<rbrakk> \<Longrightarrow> f (c s) = c' (f s)" 
  shows "Option.map f (while_option b c s) = while_option b' c' (f s)"
unfolding while_option_def
proof (rule trans[OF if_distrib if_cong], safe, unfold option.inject)
  fix k assume "\<not> b ((c ^^ k) s)"
  thus "\<exists>k. \<not> b' ((c' ^^ k) (f s))"
  proof (induction k arbitrary: s)
    case 0 thus ?case by (auto simp: assms(1) intro: exI[of _ 0])
  next
    case (Suc k)
    hence "\<not> b ((c^^k) (c s))" by (auto simp: funpow_swap1)
    then guess k by (rule exE[OF Suc.IH[of "c s"]])
    with assms show ?case by (cases "b s") (auto simp: funpow_swap1 intro: exI[of _ "Suc k"] exI[of _ "0"])
  qed
next
  fix k assume "\<not> b' ((c' ^^ k) (f s))"
  thus "\<exists>k. \<not> b ((c ^^ k) s)"
  proof (induction k arbitrary: s)
    case 0 thus ?case by (auto simp: assms(1) intro: exI[of _ 0])
  next
    case (Suc k)
    hence *: "\<not> b' ((c'^^k) (c' (f s)))" by (auto simp: funpow_swap1)
    show ?case
    proof (cases "b s")
      case True
      with assms(2) * have "\<not> b' ((c'^^k) (f (c s)))" by simp 
      then guess k by (rule exE[OF Suc.IH[of "c s"]])
      thus ?thesis by (auto simp: funpow_swap1 intro: exI[of _ "Suc k"])
    qed (auto intro: exI[of _ "0"])
  qed
next
  fix k assume k: "\<not> b' ((c' ^^ k) (f s))"
  have *: "(LEAST k. \<not> b' ((c' ^^ k) (f s))) = (LEAST k. \<not> b ((c ^^ k) s))" (is "?k' = ?k")
  proof (cases ?k')
    case 0
    have "\<not> b' ((c'^^0) (f s))" unfolding 0[symmetric] by (rule LeastI[of _ k]) (rule k)
    hence "\<not> b s" unfolding assms(1) by simp
    hence "?k = 0" by (intro Least_equality) auto
    with 0 show ?thesis by auto
  next
    case (Suc k')
    have "\<not> b' ((c'^^Suc k') (f s))" unfolding Suc[symmetric] by (rule LeastI) (rule k)
    moreover
    { fix k assume "k \<le> k'"
      hence "k < ?k'" unfolding Suc by simp
      hence "b' ((c' ^^ k) (f s))" by (rule iffD1[OF not_not, OF not_less_Least])
    } note b' = this
    { fix k assume "k \<le> k'"
      hence "f ((c ^^ k) s) = (c'^^k) (f s)" by (induct k) (auto simp: b' assms)
      with `k \<le> k'` have "b ((c^^k) s)"
      proof (induct k)
        case (Suc k) thus ?case unfolding assms(1) by (simp only: b')
      qed (simp add: b'[of 0, simplified] assms(1))
    } note b = this
    hence k': "f ((c^^k') s) = (c'^^k') (f s)" by (induct k') (auto simp: assms(2))
    ultimately show ?thesis unfolding Suc using b
    by (intro sym[OF Least_equality])
       (auto simp add: assms(1) assms(2)[OF b] k' not_less_eq_eq[symmetric])
  qed
  have "f ((c ^^ ?k) s) = (c' ^^ ?k') (f s)" unfolding *
    by (auto intro: funpow_commute assms(2) dest: not_less_Least)
  thus "\<exists>z. (c ^^ ?k) s = z \<and> f z = (c' ^^ ?k') (f s)" by blast
qed

subsection {* Total version *}

definition while :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
where "while b c s = the (while_option b c s)"

lemma while_unfold [code]:
  "while b c s = (if b s then while b c (c s) else s)"
unfolding while_def by (subst while_option_unfold) simp

lemma def_while_unfold:
  assumes fdef: "f == while test do"
  shows "f x = (if test x then f(do x) else x)"
unfolding fdef by (fact while_unfold)


text {*
 The proof rule for @{term while}, where @{term P} is the invariant.
*}

theorem while_rule_lemma:
  assumes invariant: "!!s. P s ==> b s ==> P (c s)"
    and terminate: "!!s. P s ==> \<not> b s ==> Q s"
    and wf: "wf {(t, s). P s \<and> b s \<and> t = c s}"
  shows "P s \<Longrightarrow> Q (while b c s)"
  using wf
  apply (induct s)
  apply simp
  apply (subst while_unfold)
  apply (simp add: invariant terminate)
  done

theorem while_rule:
  "[| P s;
      !!s. [| P s; b s  |] ==> P (c s);
      !!s. [| P s; \<not> b s  |] ==> Q s;
      wf r;
      !!s. [| P s; b s  |] ==> (c s, s) \<in> r |] ==>
   Q (while b c s)"
  apply (rule while_rule_lemma)
     prefer 4 apply assumption
    apply blast
   apply blast
  apply (erule wf_subset)
  apply blast
  done

text{* Proving termination: *}

theorem wf_while_option_Some:
  assumes "wf {(t, s). (P s \<and> b s) \<and> t = c s}"
  and "!!s. P s \<Longrightarrow> b s \<Longrightarrow> P(c s)" and "P s"
  shows "EX t. while_option b c s = Some t"
using assms(1,3)
apply (induct s)
using assms(2)
apply (subst while_option_unfold)
apply simp
done

theorem measure_while_option_Some: fixes f :: "'s \<Rightarrow> nat"
shows "(!!s. P s \<Longrightarrow> b s \<Longrightarrow> P(c s) \<and> f(c s) < f s)
  \<Longrightarrow> P s \<Longrightarrow> EX t. while_option b c s = Some t"
by(blast intro: wf_while_option_Some[OF wf_if_measure, of P b f])

text{* Kleene iteration starting from the empty set and assuming some finite
bounding set: *}

lemma while_option_finite_subset_Some: fixes C :: "'a set"
  assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
  shows "\<exists>P. while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
proof(rule measure_while_option_Some[where
    f= "%A::'a set. card C - card A" and P= "%A. A \<subseteq> C \<and> A \<subseteq> f A" and s= "{}"])
  fix A assume A: "A \<subseteq> C \<and> A \<subseteq> f A" "f A \<noteq> A"
  show "(f A \<subseteq> C \<and> f A \<subseteq> f (f A)) \<and> card C - card (f A) < card C - card A"
    (is "?L \<and> ?R")
  proof
    show ?L by(metis A(1) assms(2) monoD[OF `mono f`])
    show ?R by (metis A assms(2,3) card_seteq diff_less_mono2 equalityI linorder_le_less_linear rev_finite_subset)
  qed
qed simp

lemma lfp_the_while_option:
  assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
  shows "lfp f = the(while_option (\<lambda>A. f A \<noteq> A) f {})"
proof-
  obtain P where "while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
    using while_option_finite_subset_Some[OF assms] by blast
  with while_option_stop2[OF this] lfp_Kleene_iter[OF assms(1)]
  show ?thesis by auto
qed

lemma lfp_while:
  assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
  shows "lfp f = while (\<lambda>A. f A \<noteq> A) f {}"
unfolding while_def using assms by (rule lfp_the_while_option) blast

end