src/HOL/Library/While_Combinator.thy
 author Christian Sternagel Wed Aug 29 12:23:14 2012 +0900 (2012-08-29) changeset 49083 01081bca31b6 parent 46365 547d1a1dcaf6 child 50008 eb7f574d0303 permissions -rw-r--r--
dropped ord and bot instance for list prefixes (use locale interpretation instead, which allows users to decide what order to use on lists)
1 (*  Title:      HOL/Library/While_Combinator.thy
2     Author:     Tobias Nipkow
3     Author:     Alexander Krauss
4     Copyright   2000 TU Muenchen
5 *)
7 header {* A general ``while'' combinator *}
9 theory While_Combinator
10 imports Main
11 begin
13 subsection {* Partial version *}
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)"
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
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)
60 lemma while_option_stop: "while_option b c s = Some t \<Longrightarrow> ~ b t"
61 by(metis while_option_stop2)
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)
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
81 subsection {* Total version *}
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)"
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
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)
96 text {*
97  The proof rule for @{term while}, where @{term P} is the invariant.
98 *}
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
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
127 text{* Proving termination: *}
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
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])
145 text{* Kleene iteration starting from the empty set and assuming some finite
146 bounding set: *}
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
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
172 end