src/HOL/Library/While_Combinator.thy
 author wenzelm Thu Feb 16 22:53:24 2012 +0100 (2012-02-16) changeset 46507 1b24c24017dd parent 46365 547d1a1dcaf6 child 50008 eb7f574d0303 permissions -rw-r--r--
tuned proofs;
```     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
```
```    81 subsection {* Total version *}
```
```    82
```
```    83 definition while :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
```
```    84 where "while b c s = the (while_option b c s)"
```
```    85
```
```    86 lemma while_unfold:
```
```    87   "while b c s = (if b s then while b c (c s) else s)"
```
```    88 unfolding while_def by (subst while_option_unfold) simp
```
```    89
```
```    90 lemma def_while_unfold:
```
```    91   assumes fdef: "f == while test do"
```
```    92   shows "f x = (if test x then f(do x) else x)"
```
```    93 unfolding fdef by (fact while_unfold)
```
```    94
```
```    95
```
```    96 text {*
```
```    97  The proof rule for @{term while}, where @{term P} is the invariant.
```
```    98 *}
```
```    99
```
```   100 theorem while_rule_lemma:
```
```   101   assumes invariant: "!!s. P s ==> b s ==> P (c s)"
```
```   102     and terminate: "!!s. P s ==> \<not> b s ==> Q s"
```
```   103     and wf: "wf {(t, s). P s \<and> b s \<and> t = c s}"
```
```   104   shows "P s \<Longrightarrow> Q (while b c s)"
```
```   105   using wf
```
```   106   apply (induct s)
```
```   107   apply simp
```
```   108   apply (subst while_unfold)
```
```   109   apply (simp add: invariant terminate)
```
```   110   done
```
```   111
```
```   112 theorem while_rule:
```
```   113   "[| P s;
```
```   114       !!s. [| P s; b s  |] ==> P (c s);
```
```   115       !!s. [| P s; \<not> b s  |] ==> Q s;
```
```   116       wf r;
```
```   117       !!s. [| P s; b s  |] ==> (c s, s) \<in> r |] ==>
```
```   118    Q (while b c s)"
```
```   119   apply (rule while_rule_lemma)
```
```   120      prefer 4 apply assumption
```
```   121     apply blast
```
```   122    apply blast
```
```   123   apply (erule wf_subset)
```
```   124   apply blast
```
```   125   done
```
```   126
```
```   127 text{* Proving termination: *}
```
```   128
```
```   129 theorem wf_while_option_Some:
```
```   130   assumes "wf {(t, s). (P s \<and> b s) \<and> t = c s}"
```
```   131   and "!!s. P s \<Longrightarrow> b s \<Longrightarrow> P(c s)" and "P s"
```
```   132   shows "EX t. while_option b c s = Some t"
```
```   133 using assms(1,3)
```
```   134 apply (induct s)
```
```   135 using assms(2)
```
```   136 apply (subst while_option_unfold)
```
```   137 apply simp
```
```   138 done
```
```   139
```
```   140 theorem measure_while_option_Some: fixes f :: "'s \<Rightarrow> nat"
```
```   141 shows "(!!s. P s \<Longrightarrow> b s \<Longrightarrow> P(c s) \<and> f(c s) < f s)
```
```   142   \<Longrightarrow> P s \<Longrightarrow> EX t. while_option b c s = Some t"
```
```   143 by(blast intro: wf_while_option_Some[OF wf_if_measure, of P b f])
```
```   144
```
```   145 text{* Kleene iteration starting from the empty set and assuming some finite
```
```   146 bounding set: *}
```
```   147
```
```   148 lemma while_option_finite_subset_Some: fixes C :: "'a set"
```
```   149   assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
```
```   150   shows "\<exists>P. while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
```
```   151 proof(rule measure_while_option_Some[where
```
```   152     f= "%A::'a set. card C - card A" and P= "%A. A \<subseteq> C \<and> A \<subseteq> f A" and s= "{}"])
```
```   153   fix A assume A: "A \<subseteq> C \<and> A \<subseteq> f A" "f A \<noteq> A"
```
```   154   show "(f A \<subseteq> C \<and> f A \<subseteq> f (f A)) \<and> card C - card (f A) < card C - card A"
```
```   155     (is "?L \<and> ?R")
```
```   156   proof
```
```   157     show ?L by(metis A(1) assms(2) monoD[OF `mono f`])
```
```   158     show ?R by (metis A assms(2,3) card_seteq diff_less_mono2 equalityI linorder_le_less_linear rev_finite_subset)
```
```   159   qed
```
```   160 qed simp
```
```   161
```
```   162 lemma lfp_the_while_option:
```
```   163   assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
```
```   164   shows "lfp f = the(while_option (\<lambda>A. f A \<noteq> A) f {})"
```
```   165 proof-
```
```   166   obtain P where "while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
```
```   167     using while_option_finite_subset_Some[OF assms] by blast
```
```   168   with while_option_stop2[OF this] lfp_Kleene_iter[OF assms(1)]
```
```   169   show ?thesis by auto
```
```   170 qed
```
```   171
```
```   172 end
```