# Theory While_Combinator

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theory While_Combinator
imports Main
`(*  Title:      HOL/Library/While_Combinator.thy    Author:     Tobias Nipkow    Author:     Alexander Krauss    Copyright   2000 TU Muenchen*)header {* A general ``while'' combinator *}theory While_Combinatorimports Mainbeginsubsection {* Partial version *}definition while_option :: "('a => bool) => ('a => 'a) => 'a => 'a option" where"while_option b c s = (if (∃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 "∃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: "∃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 "~ (∃l. ~ b ((c ^^ Suc l) s))" by blast    then have "~ (∃l. ~ b ((c ^^ l) (c s)))"      by (simp add: funpow_swap1)    with False  `b s` show ?thesis by (simp add: while_option_def)  qednext  assume [simp]: "~ b s"  have least: "(LEAST k. ~ b ((c ^^ k) s)) = 0"    by (rule Least_equality) auto  moreover   have "∃k. ~ b ((c ^^ k) s)" by (rule exI[of _ "0::nat"]) auto  ultimately show ?thesis unfolding while_option_def by auto qedlemma while_option_stop2: "while_option b c s = Some t ==> EX k. t = (c^^k) s ∧ ¬ 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 ==> ~ 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)qedlemma funpow_commute:   "[|∀k' < k. f (c ((c^^k') s)) = c' (f ((c^^k') s))|] ==> f ((c^^k) s) = (c'^^k) (f s)"by (induct k arbitrary: s) autolemma while_option_commute:  assumes "!!s. b s = b' (f s)" "!!s. [|b s|] ==> f (c s) = c' (f s)"   shows "Option.map f (while_option b c s) = while_option b' c' (f s)"unfolding while_option_defproof (rule trans[OF if_distrib if_cong], safe, unfold option.inject)  fix k assume "¬ b ((c ^^ k) s)"  thus "∃k. ¬ 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 "¬ 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"])  qednext  fix k assume "¬ b' ((c' ^^ k) (f s))"  thus "∃k. ¬ 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 *: "¬ b' ((c'^^k) (c' (f s)))" by (auto simp: funpow_swap1)    show ?case    proof (cases "b s")      case True      with assms(2) * have "¬ 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"])  qednext  fix k assume k: "¬ b' ((c' ^^ k) (f s))"  have *: "(LEAST k. ¬ b' ((c' ^^ k) (f s))) = (LEAST k. ¬ b ((c ^^ k) s))" (is "?k' = ?k")  proof (cases ?k')    case 0    have "¬ b' ((c'^^0) (f s))" unfolding 0[symmetric] by (rule LeastI[of _ k]) (rule k)    hence "¬ 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 "¬ b' ((c'^^Suc k') (f s))" unfolding Suc[symmetric] by (rule LeastI) (rule k)    moreover    { fix k assume "k ≤ 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 ≤ k'"      hence "f ((c ^^ k) s) = (c'^^k) (f s)" by (induct k) (auto simp: b' assms)      with `k ≤ 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 "∃z. (c ^^ ?k) s = z ∧ f z = (c' ^^ ?k') (f s)" by blastqedsubsection {* Total version *}definition while :: "('a => bool) => ('a => 'a) => 'a => '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) simplemma 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 ==> ¬ b s ==> Q s"    and wf: "wf {(t, s). P s ∧ b s ∧ t = c s}"  shows "P s ==> Q (while b c s)"  using wf  apply (induct s)  apply simp  apply (subst while_unfold)  apply (simp add: invariant terminate)  donetheorem while_rule:  "[| P s;      !!s. [| P s; b s  |] ==> P (c s);      !!s. [| P s; ¬ b s  |] ==> Q s;      wf r;      !!s. [| P s; b s  |] ==> (c s, s) ∈ 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  donetext{* Proving termination: *}theorem wf_while_option_Some:  assumes "wf {(t, s). (P s ∧ b s) ∧ t = c s}"  and "!!s. P s ==> b s ==> 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 simpdonetheorem measure_while_option_Some: fixes f :: "'s => nat"shows "(!!s. P s ==> b s ==> P(c s) ∧ f(c s) < f s)  ==> P s ==> 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 finitebounding set: *}lemma while_option_finite_subset_Some: fixes C :: "'a set"  assumes "mono f" and "!!X. X ⊆ C ==> f X ⊆ C" and "finite C"  shows "∃P. while_option (λA. f A ≠ A) f {} = Some P"proof(rule measure_while_option_Some[where    f= "%A::'a set. card C - card A" and P= "%A. A ⊆ C ∧ A ⊆ f A" and s= "{}"])  fix A assume A: "A ⊆ C ∧ A ⊆ f A" "f A ≠ A"  show "(f A ⊆ C ∧ f A ⊆ f (f A)) ∧ card C - card (f A) < card C - card A"    (is "?L ∧ ?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)  qedqed simplemma lfp_the_while_option:  assumes "mono f" and "!!X. X ⊆ C ==> f X ⊆ C" and "finite C"  shows "lfp f = the(while_option (λA. f A ≠ A) f {})"proof-  obtain P where "while_option (λA. f A ≠ 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 autoqedlemma lfp_while:  assumes "mono f" and "!!X. X ⊆ C ==> f X ⊆ C" and "finite C"  shows "lfp f = while (λA. f A ≠ A) f {}"unfolding while_def using assms by (rule lfp_the_while_option) blastend`