src/HOL/Library/While_Combinator.thy
author Andreas Lochbihler
Wed Feb 27 10:33:30 2013 +0100 (2013-02-27)
changeset 51288 be7e9a675ec9
parent 50577 cfbad2d08412
child 53217 1a8673a6d669
permissions -rw-r--r--
add wellorder instance for Numeral_Type (suggested by Jesus Aransay)
     1 (*  Title:      HOL/Library/While_Combinator.thy
     2     Author:     Tobias Nipkow
     3     Author:     Alexander Krauss
     4     Copyright   2000 TU Muenchen
     5 *)
     6 
     7 header {* A general ``while'' combinator *}
     8 
     9 theory While_Combinator
    10 imports Main
    11 begin
    12 
    13 subsection {* Partial version *}
    14 
    15 definition while_option :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a option" where
    16 "while_option b c s = (if (\<exists>k. ~ b ((c ^^ k) s))
    17    then Some ((c ^^ (LEAST k. ~ b ((c ^^ k) s))) s)
    18    else None)"
    19 
    20 theorem while_option_unfold[code]:
    21 "while_option b c s = (if b s then while_option b c (c s) else Some s)"
    22 proof cases
    23   assume "b s"
    24   show ?thesis
    25   proof (cases "\<exists>k. ~ b ((c ^^ k) s)")
    26     case True
    27     then obtain k where 1: "~ b ((c ^^ k) s)" ..
    28     with `b s` obtain l where "k = Suc l" by (cases k) auto
    29     with 1 have "~ b ((c ^^ l) (c s))" by (auto simp: funpow_swap1)
    30     then have 2: "\<exists>l. ~ b ((c ^^ l) (c s))" ..
    31     from 1
    32     have "(LEAST k. ~ b ((c ^^ k) s)) = Suc (LEAST l. ~ b ((c ^^ Suc l) s))"
    33       by (rule Least_Suc) (simp add: `b s`)
    34     also have "... = Suc (LEAST l. ~ b ((c ^^ l) (c s)))"
    35       by (simp add: funpow_swap1)
    36     finally
    37     show ?thesis 
    38       using True 2 `b s` by (simp add: funpow_swap1 while_option_def)
    39   next
    40     case False
    41     then have "~ (\<exists>l. ~ b ((c ^^ Suc l) s))" by blast
    42     then have "~ (\<exists>l. ~ b ((c ^^ l) (c s)))"
    43       by (simp add: funpow_swap1)
    44     with False  `b s` show ?thesis by (simp add: while_option_def)
    45   qed
    46 next
    47   assume [simp]: "~ b s"
    48   have least: "(LEAST k. ~ b ((c ^^ k) s)) = 0"
    49     by (rule Least_equality) auto
    50   moreover 
    51   have "\<exists>k. ~ b ((c ^^ k) s)" by (rule exI[of _ "0::nat"]) auto
    52   ultimately show ?thesis unfolding while_option_def by auto 
    53 qed
    54 
    55 lemma while_option_stop2:
    56  "while_option b c s = Some t \<Longrightarrow> EX k. t = (c^^k) s \<and> \<not> b t"
    57 apply(simp add: while_option_def split: if_splits)
    58 by (metis (lifting) LeastI_ex)
    59 
    60 lemma while_option_stop: "while_option b c s = Some t \<Longrightarrow> ~ b t"
    61 by(metis while_option_stop2)
    62 
    63 theorem while_option_rule:
    64 assumes step: "!!s. P s ==> b s ==> P (c s)"
    65 and result: "while_option b c s = Some t"
    66 and init: "P s"
    67 shows "P t"
    68 proof -
    69   def k == "LEAST k. ~ b ((c ^^ k) s)"
    70   from assms have t: "t = (c ^^ k) s"
    71     by (simp add: while_option_def k_def split: if_splits)    
    72   have 1: "ALL i<k. b ((c ^^ i) s)"
    73     by (auto simp: k_def dest: not_less_Least)
    74 
    75   { fix i assume "i <= k" then have "P ((c ^^ i) s)"
    76       by (induct i) (auto simp: init step 1) }
    77   thus "P t" by (auto simp: t)
    78 qed
    79 
    80 lemma funpow_commute: 
    81   "\<lbrakk>\<forall>k' < k. f (c ((c^^k') s)) = c' (f ((c^^k') s))\<rbrakk> \<Longrightarrow> f ((c^^k) s) = (c'^^k) (f s)"
    82 by (induct k arbitrary: s) auto
    83 
    84 lemma while_option_commute:
    85   assumes "\<And>s. b s = b' (f s)" "\<And>s. \<lbrakk>b s\<rbrakk> \<Longrightarrow> f (c s) = c' (f s)" 
    86   shows "Option.map f (while_option b c s) = while_option b' c' (f s)"
    87 unfolding while_option_def
    88 proof (rule trans[OF if_distrib if_cong], safe, unfold option.inject)
    89   fix k assume "\<not> b ((c ^^ k) s)"
    90   thus "\<exists>k. \<not> b' ((c' ^^ k) (f s))"
    91   proof (induction k arbitrary: s)
    92     case 0 thus ?case by (auto simp: assms(1) intro: exI[of _ 0])
    93   next
    94     case (Suc k)
    95     hence "\<not> b ((c^^k) (c s))" by (auto simp: funpow_swap1)
    96     then guess k by (rule exE[OF Suc.IH[of "c s"]])
    97     with assms show ?case by (cases "b s") (auto simp: funpow_swap1 intro: exI[of _ "Suc k"] exI[of _ "0"])
    98   qed
    99 next
   100   fix k assume "\<not> b' ((c' ^^ k) (f s))"
   101   thus "\<exists>k. \<not> b ((c ^^ k) s)"
   102   proof (induction k arbitrary: s)
   103     case 0 thus ?case by (auto simp: assms(1) intro: exI[of _ 0])
   104   next
   105     case (Suc k)
   106     hence *: "\<not> b' ((c'^^k) (c' (f s)))" by (auto simp: funpow_swap1)
   107     show ?case
   108     proof (cases "b s")
   109       case True
   110       with assms(2) * have "\<not> b' ((c'^^k) (f (c s)))" by simp 
   111       then guess k by (rule exE[OF Suc.IH[of "c s"]])
   112       thus ?thesis by (auto simp: funpow_swap1 intro: exI[of _ "Suc k"])
   113     qed (auto intro: exI[of _ "0"])
   114   qed
   115 next
   116   fix k assume k: "\<not> b' ((c' ^^ k) (f s))"
   117   have *: "(LEAST k. \<not> b' ((c' ^^ k) (f s))) = (LEAST k. \<not> b ((c ^^ k) s))" (is "?k' = ?k")
   118   proof (cases ?k')
   119     case 0
   120     have "\<not> b' ((c'^^0) (f s))" unfolding 0[symmetric] by (rule LeastI[of _ k]) (rule k)
   121     hence "\<not> b s" unfolding assms(1) by simp
   122     hence "?k = 0" by (intro Least_equality) auto
   123     with 0 show ?thesis by auto
   124   next
   125     case (Suc k')
   126     have "\<not> b' ((c'^^Suc k') (f s))" unfolding Suc[symmetric] by (rule LeastI) (rule k)
   127     moreover
   128     { fix k assume "k \<le> k'"
   129       hence "k < ?k'" unfolding Suc by simp
   130       hence "b' ((c' ^^ k) (f s))" by (rule iffD1[OF not_not, OF not_less_Least])
   131     } note b' = this
   132     { fix k assume "k \<le> k'"
   133       hence "f ((c ^^ k) s) = (c'^^k) (f s)" by (induct k) (auto simp: b' assms)
   134       with `k \<le> k'` have "b ((c^^k) s)"
   135       proof (induct k)
   136         case (Suc k) thus ?case unfolding assms(1) by (simp only: b')
   137       qed (simp add: b'[of 0, simplified] assms(1))
   138     } note b = this
   139     hence k': "f ((c^^k') s) = (c'^^k') (f s)" by (induct k') (auto simp: assms(2))
   140     ultimately show ?thesis unfolding Suc using b
   141     by (intro sym[OF Least_equality])
   142        (auto simp add: assms(1) assms(2)[OF b] k' not_less_eq_eq[symmetric])
   143   qed
   144   have "f ((c ^^ ?k) s) = (c' ^^ ?k') (f s)" unfolding *
   145     by (auto intro: funpow_commute assms(2) dest: not_less_Least)
   146   thus "\<exists>z. (c ^^ ?k) s = z \<and> f z = (c' ^^ ?k') (f s)" by blast
   147 qed
   148 
   149 subsection {* Total version *}
   150 
   151 definition while :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
   152 where "while b c s = the (while_option b c s)"
   153 
   154 lemma while_unfold [code]:
   155   "while b c s = (if b s then while b c (c s) else s)"
   156 unfolding while_def by (subst while_option_unfold) simp
   157 
   158 lemma def_while_unfold:
   159   assumes fdef: "f == while test do"
   160   shows "f x = (if test x then f(do x) else x)"
   161 unfolding fdef by (fact while_unfold)
   162 
   163 
   164 text {*
   165  The proof rule for @{term while}, where @{term P} is the invariant.
   166 *}
   167 
   168 theorem while_rule_lemma:
   169   assumes invariant: "!!s. P s ==> b s ==> P (c s)"
   170     and terminate: "!!s. P s ==> \<not> b s ==> Q s"
   171     and wf: "wf {(t, s). P s \<and> b s \<and> t = c s}"
   172   shows "P s \<Longrightarrow> Q (while b c s)"
   173   using wf
   174   apply (induct s)
   175   apply simp
   176   apply (subst while_unfold)
   177   apply (simp add: invariant terminate)
   178   done
   179 
   180 theorem while_rule:
   181   "[| P s;
   182       !!s. [| P s; b s  |] ==> P (c s);
   183       !!s. [| P s; \<not> b s  |] ==> Q s;
   184       wf r;
   185       !!s. [| P s; b s  |] ==> (c s, s) \<in> r |] ==>
   186    Q (while b c s)"
   187   apply (rule while_rule_lemma)
   188      prefer 4 apply assumption
   189     apply blast
   190    apply blast
   191   apply (erule wf_subset)
   192   apply blast
   193   done
   194 
   195 text{* Proving termination: *}
   196 
   197 theorem wf_while_option_Some:
   198   assumes "wf {(t, s). (P s \<and> b s) \<and> t = c s}"
   199   and "!!s. P s \<Longrightarrow> b s \<Longrightarrow> P(c s)" and "P s"
   200   shows "EX t. while_option b c s = Some t"
   201 using assms(1,3)
   202 apply (induct s)
   203 using assms(2)
   204 apply (subst while_option_unfold)
   205 apply simp
   206 done
   207 
   208 theorem measure_while_option_Some: fixes f :: "'s \<Rightarrow> nat"
   209 shows "(!!s. P s \<Longrightarrow> b s \<Longrightarrow> P(c s) \<and> f(c s) < f s)
   210   \<Longrightarrow> P s \<Longrightarrow> EX t. while_option b c s = Some t"
   211 by(blast intro: wf_while_option_Some[OF wf_if_measure, of P b f])
   212 
   213 text{* Kleene iteration starting from the empty set and assuming some finite
   214 bounding set: *}
   215 
   216 lemma while_option_finite_subset_Some: fixes C :: "'a set"
   217   assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
   218   shows "\<exists>P. while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
   219 proof(rule measure_while_option_Some[where
   220     f= "%A::'a set. card C - card A" and P= "%A. A \<subseteq> C \<and> A \<subseteq> f A" and s= "{}"])
   221   fix A assume A: "A \<subseteq> C \<and> A \<subseteq> f A" "f A \<noteq> A"
   222   show "(f A \<subseteq> C \<and> f A \<subseteq> f (f A)) \<and> card C - card (f A) < card C - card A"
   223     (is "?L \<and> ?R")
   224   proof
   225     show ?L by(metis A(1) assms(2) monoD[OF `mono f`])
   226     show ?R by (metis A assms(2,3) card_seteq diff_less_mono2 equalityI linorder_le_less_linear rev_finite_subset)
   227   qed
   228 qed simp
   229 
   230 lemma lfp_the_while_option:
   231   assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
   232   shows "lfp f = the(while_option (\<lambda>A. f A \<noteq> A) f {})"
   233 proof-
   234   obtain P where "while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
   235     using while_option_finite_subset_Some[OF assms] by blast
   236   with while_option_stop2[OF this] lfp_Kleene_iter[OF assms(1)]
   237   show ?thesis by auto
   238 qed
   239 
   240 lemma lfp_while:
   241   assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
   242   shows "lfp f = while (\<lambda>A. f A \<noteq> A) f {}"
   243 unfolding while_def using assms by (rule lfp_the_while_option) blast
   244 
   245 end