src/HOL/Library/While_Combinator.thy
author nipkow
Thu Oct 03 12:34:32 2013 +0200 (2013-10-03)
changeset 54050 48c800d8ba2d
parent 54047 83fb090dae9e
child 54196 0c188a3c671a
permissions -rw-r--r--
added and generalised lemmas
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(*  Title:      HOL/Library/While_Combinator.thy
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    Author:     Tobias Nipkow
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    Author:     Alexander Krauss
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*)
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header {* A general ``while'' combinator *}
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theory While_Combinator
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imports Main
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begin
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subsection {* Partial version *}
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definition while_option :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a option" where
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"while_option b c s = (if (\<exists>k. ~ b ((c ^^ k) s))
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   then Some ((c ^^ (LEAST k. ~ b ((c ^^ k) s))) s)
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   else None)"
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theorem while_option_unfold[code]:
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"while_option b c s = (if b s then while_option b c (c s) else Some s)"
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proof cases
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  assume "b s"
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  show ?thesis
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  proof (cases "\<exists>k. ~ b ((c ^^ k) s)")
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    case True
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    then obtain k where 1: "~ b ((c ^^ k) s)" ..
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    with `b s` obtain l where "k = Suc l" by (cases k) auto
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    with 1 have "~ b ((c ^^ l) (c s))" by (auto simp: funpow_swap1)
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    then have 2: "\<exists>l. ~ b ((c ^^ l) (c s))" ..
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    from 1
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    have "(LEAST k. ~ b ((c ^^ k) s)) = Suc (LEAST l. ~ b ((c ^^ Suc l) s))"
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      by (rule Least_Suc) (simp add: `b s`)
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    also have "... = Suc (LEAST l. ~ b ((c ^^ l) (c s)))"
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      by (simp add: funpow_swap1)
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    finally
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    show ?thesis 
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      using True 2 `b s` by (simp add: funpow_swap1 while_option_def)
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  next
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    case False
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    then have "~ (\<exists>l. ~ b ((c ^^ Suc l) s))" by blast
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    then have "~ (\<exists>l. ~ b ((c ^^ l) (c s)))"
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      by (simp add: funpow_swap1)
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    with False  `b s` show ?thesis by (simp add: while_option_def)
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  qed
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next
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  assume [simp]: "~ b s"
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  have least: "(LEAST k. ~ b ((c ^^ k) s)) = 0"
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    by (rule Least_equality) auto
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  moreover 
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  have "\<exists>k. ~ b ((c ^^ k) s)" by (rule exI[of _ "0::nat"]) auto
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  ultimately show ?thesis unfolding while_option_def by auto 
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qed
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lemma while_option_stop2:
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 "while_option b c s = Some t \<Longrightarrow> EX k. t = (c^^k) s \<and> \<not> b t"
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apply(simp add: while_option_def split: if_splits)
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by (metis (lifting) LeastI_ex)
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lemma while_option_stop: "while_option b c s = Some t \<Longrightarrow> ~ b t"
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by(metis while_option_stop2)
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theorem while_option_rule:
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assumes step: "!!s. P s ==> b s ==> P (c s)"
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and result: "while_option b c s = Some t"
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and init: "P s"
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shows "P t"
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proof -
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  def k == "LEAST k. ~ b ((c ^^ k) s)"
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  from assms have t: "t = (c ^^ k) s"
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    by (simp add: while_option_def k_def split: if_splits)    
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  have 1: "ALL i<k. b ((c ^^ i) s)"
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    by (auto simp: k_def dest: not_less_Least)
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  { fix i assume "i <= k" then have "P ((c ^^ i) s)"
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      by (induct i) (auto simp: init step 1) }
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  thus "P t" by (auto simp: t)
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qed
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lemma funpow_commute: 
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  "\<lbrakk>\<forall>k' < k. f (c ((c^^k') s)) = c' (f ((c^^k') s))\<rbrakk> \<Longrightarrow> f ((c^^k) s) = (c'^^k) (f s)"
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by (induct k arbitrary: s) auto
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lemma while_option_commute_invariant:
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assumes Invariant: "\<And>s. P s \<Longrightarrow> b s \<Longrightarrow> P (c s)"
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assumes TestCommute: "\<And>s. P s \<Longrightarrow> b s = b' (f s)"
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assumes BodyCommute: "\<And>s. P s \<Longrightarrow> b s \<Longrightarrow> f (c s) = c' (f s)"
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assumes Initial: "P s"
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shows "Option.map f (while_option b c s) = while_option b' c' (f s)"
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unfolding while_option_def
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proof (rule trans[OF if_distrib if_cong], safe, unfold option.inject)
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  fix k
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  assume "\<not> b ((c ^^ k) s)"
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  with Initial show "\<exists>k. \<not> b' ((c' ^^ k) (f s))"
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  proof (induction k arbitrary: s)
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    case 0 thus ?case by (auto simp: TestCommute intro: exI[of _ 0])
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  next
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    case (Suc k) thus ?case
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    proof (cases "b s")
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      assume "b s"
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      with Suc.IH[of "c s"] Suc.prems show ?thesis
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        by (metis BodyCommute Invariant comp_apply funpow.simps(2) funpow_swap1)
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    next
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      assume "\<not> b s"
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      with Suc show ?thesis by (auto simp: TestCommute intro: exI [of _ 0])
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    qed
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  qed
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next
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  fix k
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  assume "\<not> b' ((c' ^^ k) (f s))"
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  with Initial show "\<exists>k. \<not> b ((c ^^ k) s)"
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  proof (induction k arbitrary: s)
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    case 0 thus ?case by (auto simp: TestCommute intro: exI[of _ 0])
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  next
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    case (Suc k) thus ?case
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    proof (cases "b s")
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       assume "b s"
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      with Suc.IH[of "c s"] Suc.prems show ?thesis
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        by (metis BodyCommute Invariant comp_apply funpow.simps(2) funpow_swap1)
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    next
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      assume "\<not> b s"
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      with Suc show ?thesis by (auto simp: TestCommute intro: exI [of _ 0])
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    qed
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  qed
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next
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  fix k
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  assume k: "\<not> b' ((c' ^^ k) (f s))"
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  have *: "(LEAST k. \<not> b' ((c' ^^ k) (f s))) = (LEAST k. \<not> b ((c ^^ k) s))"
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          (is "?k' = ?k")
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  proof (cases ?k')
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    case 0
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    have "\<not> b' ((c' ^^ 0) (f s))"
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      unfolding 0[symmetric] by (rule LeastI[of _ k]) (rule k)
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    hence "\<not> b s" by (auto simp: TestCommute Initial)
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    hence "?k = 0" by (intro Least_equality) auto
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    with 0 show ?thesis by auto
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  next
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    case (Suc k')
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    have "\<not> b' ((c' ^^ Suc k') (f s))"
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      unfolding Suc[symmetric] by (rule LeastI) (rule k)
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    moreover
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    { fix k assume "k \<le> k'"
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      hence "k < ?k'" unfolding Suc by simp
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      hence "b' ((c' ^^ k) (f s))" by (rule iffD1[OF not_not, OF not_less_Least])
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    }
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    note b' = this
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    { fix k assume "k \<le> k'"
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      hence "f ((c ^^ k) s) = (c' ^^ k) (f s)"
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      and "b ((c ^^ k) s) = b' ((c' ^^ k) (f s))"
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      and "P ((c ^^ k) s)"
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        by (induct k) (auto simp: b' assms)
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      with `k \<le> k'`
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      have "b ((c ^^ k) s)"
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      and "f ((c ^^ k) s) = (c' ^^ k) (f s)"
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      and "P ((c ^^ k) s)"
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        by (auto simp: b')
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    }
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    note b = this(1) and body = this(2) and inv = this(3)
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    hence k': "f ((c ^^ k') s) = (c' ^^ k') (f s)" by auto
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    ultimately show ?thesis unfolding Suc using b
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    proof (intro Least_equality[symmetric])
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      case goal1
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      hence Test: "\<not> b' (f ((c ^^ Suc k') s))"
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        by (auto simp: BodyCommute inv b)
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      have "P ((c ^^ Suc k') s)" by (auto simp: Invariant inv b)
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      with Test show ?case by (auto simp: TestCommute)
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    next
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      case goal2 thus ?case by (metis not_less_eq_eq)
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    qed
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  qed
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  have "f ((c ^^ ?k) s) = (c' ^^ ?k') (f s)" unfolding *
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  proof (rule funpow_commute, clarify)
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    fix k assume "k < ?k"
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    hence TestTrue: "b ((c ^^ k) s)" by (auto dest: not_less_Least)
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    from `k < ?k` have "P ((c ^^ k) s)"
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    proof (induct k)
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      case 0 thus ?case by (auto simp: assms)
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    next
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      case (Suc h)
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      hence "P ((c ^^ h) s)" by auto
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      with Suc show ?case
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        by (auto, metis (lifting, no_types) Invariant Suc_lessD not_less_Least)
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    qed
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    with TestTrue show "f (c ((c ^^ k) s)) = c' (f ((c ^^ k) s))"
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      by (metis BodyCommute)
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  qed
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  thus "\<exists>z. (c ^^ ?k) s = z \<and> f z = (c' ^^ ?k') (f s)" by blast
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qed
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lemma while_option_commute:
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  assumes "\<And>s. b s = b' (f s)" "\<And>s. \<lbrakk>b s\<rbrakk> \<Longrightarrow> f (c s) = c' (f s)" 
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  shows "Option.map f (while_option b c s) = while_option b' c' (f s)"
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by(rule while_option_commute_invariant[where P = "\<lambda>_. True"])
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  (auto simp add: assms)
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subsection {* Total version *}
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definition while :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
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where "while b c s = the (while_option b c s)"
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lemma while_unfold [code]:
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  "while b c s = (if b s then while b c (c s) else s)"
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unfolding while_def by (subst while_option_unfold) simp
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lemma def_while_unfold:
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  assumes fdef: "f == while test do"
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  shows "f x = (if test x then f(do x) else x)"
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unfolding fdef by (fact while_unfold)
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text {*
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 The proof rule for @{term while}, where @{term P} is the invariant.
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*}
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theorem while_rule_lemma:
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  assumes invariant: "!!s. P s ==> b s ==> P (c s)"
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    and terminate: "!!s. P s ==> \<not> b s ==> Q s"
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    and wf: "wf {(t, s). P s \<and> b s \<and> t = c s}"
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  shows "P s \<Longrightarrow> Q (while b c s)"
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  using wf
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  apply (induct s)
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  apply simp
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  apply (subst while_unfold)
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  apply (simp add: invariant terminate)
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  done
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theorem while_rule:
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  "[| P s;
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      !!s. [| P s; b s  |] ==> P (c s);
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      !!s. [| P s; \<not> b s  |] ==> Q s;
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      wf r;
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      !!s. [| P s; b s  |] ==> (c s, s) \<in> r |] ==>
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   Q (while b c s)"
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  apply (rule while_rule_lemma)
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     prefer 4 apply assumption
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    apply blast
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   apply blast
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  apply (erule wf_subset)
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  apply blast
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  done
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text{* Proving termination: *}
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theorem wf_while_option_Some:
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  assumes "wf {(t, s). (P s \<and> b s) \<and> t = c s}"
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  and "!!s. P s \<Longrightarrow> b s \<Longrightarrow> P(c s)" and "P s"
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  shows "EX t. while_option b c s = Some t"
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using assms(1,3)
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proof (induction s)
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  case less thus ?case using assms(2)
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    by (subst while_option_unfold) simp
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qed
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lemma wf_rel_while_option_Some:
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assumes wf: "wf R"
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assumes smaller: "\<And>s. P s \<and> b s \<Longrightarrow> (c s, s) \<in> R"
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assumes inv: "\<And>s. P s \<and> b s \<Longrightarrow> P(c s)"
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assumes init: "P s"
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shows "\<exists>t. while_option b c s = Some t"
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proof -
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 from smaller have "{(t,s). P s \<and> b s \<and> t = c s} \<subseteq> R" by auto
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 with wf have "wf {(t,s). P s \<and> b s \<and> t = c s}" by (auto simp: wf_subset)
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 with inv init show ?thesis by (auto simp: wf_while_option_Some)
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qed
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theorem measure_while_option_Some: fixes f :: "'s \<Rightarrow> nat"
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shows "(!!s. P s \<Longrightarrow> b s \<Longrightarrow> P(c s) \<and> f(c s) < f s)
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  \<Longrightarrow> P s \<Longrightarrow> EX t. while_option b c s = Some t"
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by(blast intro: wf_while_option_Some[OF wf_if_measure, of P b f])
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text{* Kleene iteration starting from the empty set and assuming some finite
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bounding set: *}
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lemma while_option_finite_subset_Some: fixes C :: "'a set"
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  assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
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  shows "\<exists>P. while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
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proof(rule measure_while_option_Some[where
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    f= "%A::'a set. card C - card A" and P= "%A. A \<subseteq> C \<and> A \<subseteq> f A" and s= "{}"])
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  fix A assume A: "A \<subseteq> C \<and> A \<subseteq> f A" "f A \<noteq> A"
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  show "(f A \<subseteq> C \<and> f A \<subseteq> f (f A)) \<and> card C - card (f A) < card C - card A"
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    (is "?L \<and> ?R")
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  proof
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    show ?L by(metis A(1) assms(2) monoD[OF `mono f`])
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    show ?R by (metis A assms(2,3) card_seteq diff_less_mono2 equalityI linorder_le_less_linear rev_finite_subset)
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  qed
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qed simp
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lemma lfp_the_while_option:
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  assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
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  shows "lfp f = the(while_option (\<lambda>A. f A \<noteq> A) f {})"
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proof-
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  obtain P where "while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
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    using while_option_finite_subset_Some[OF assms] by blast
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  with while_option_stop2[OF this] lfp_Kleene_iter[OF assms(1)]
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  show ?thesis by auto
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qed
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lemma lfp_while:
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  assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
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  shows "lfp f = while (\<lambda>A. f A \<noteq> A) f {}"
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unfolding while_def using assms by (rule lfp_the_while_option) blast
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text{* Computing the reflexive, transitive closure by iterating a successor
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function. Stops when an element is found that dos not satisfy the test.
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More refined (and hence more efficient) versions can be found in ITP 2011 paper
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by Nipkow (the theories are in the AFP entry Flyspeck by Nipkow)
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and the AFP article Executable Transitive Closures by René Thiemann. *}
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definition rtrancl_while :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a list) \<Rightarrow> 'a
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  \<Rightarrow> ('a list * 'a set) option"
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where "rtrancl_while p f x =
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  while_option (%(ws,_). ws \<noteq> [] \<and> p(hd ws))
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    ((%(ws,Z).
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     let x = hd ws; new = filter (\<lambda>y. y \<notin> Z) (f x)
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     in (new @ tl ws, set new \<union> Z)))
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    ([x],{x})"
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lemma rtrancl_while_Some: assumes "rtrancl_while p f x = Some(ws,Z)"
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shows "if ws = []
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       then Z = {(x,y). y \<in> set(f x)}^* `` {x} \<and> (\<forall>z\<in>Z. p z)
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       else \<not>p(hd ws) \<and> hd ws \<in> {(x,y). y \<in> set(f x)}^* `` {x}"
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proof-
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  let ?test = "(%(ws,_). ws \<noteq> [] \<and> p(hd ws))"
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  let ?step = "(%(ws,Z).
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     let x = hd ws; new = filter (\<lambda>y. y \<notin> Z) (f x)
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     in (new @ tl ws, set new \<union> Z))"
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  let ?R = "{(x,y). y \<in> set(f x)}"
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  let ?Inv = "%(ws,Z). x \<in> Z \<and> set ws \<subseteq> Z \<and> ?R `` (Z - set ws) \<subseteq> Z \<and>
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                       Z \<subseteq> ?R^* `` {x} \<and> (\<forall>z\<in>Z - set ws. p z)"
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  { fix ws Z assume 1: "?Inv(ws,Z)" and 2: "?test(ws,Z)"
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    from 2 obtain v vs where [simp]: "ws = v#vs" by (auto simp: neq_Nil_conv)
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    have "?Inv(?step (ws,Z))" using 1 2
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      by (auto intro: rtrancl.rtrancl_into_rtrancl)
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  } note inv = this
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  hence "!!p. ?Inv p \<Longrightarrow> ?test p \<Longrightarrow> ?Inv(?step p)"
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    apply(tactic {* split_all_tac @{context} 1 *})
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    using inv by iprover
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  moreover have "?Inv ([x],{x})" by (simp)
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  ultimately have I: "?Inv (ws,Z)"
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    by (rule while_option_rule[OF _ assms[unfolded rtrancl_while_def]])
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  { assume "ws = []"
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    hence ?thesis using I
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      by (auto simp del:Image_Collect_split dest: Image_closed_trancl)
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  } moreover
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  { assume "ws \<noteq> []"
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    hence ?thesis using I while_option_stop[OF assms[unfolded rtrancl_while_def]]
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      by (simp add: subset_iff)
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  }
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  ultimately show ?thesis by simp
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qed
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end