--- a/src/HOL/Library/While_Combinator.thy Fri Jul 09 17:00:42 2010 +0200
+++ b/src/HOL/Library/While_Combinator.thy Fri Jul 09 17:15:03 2010 +0200
@@ -10,7 +10,7 @@
imports Main
begin
-subsection {* Option result *}
+subsection {* Partial version *}
definition while_option :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a option" where
"while_option b c s = (if (\<exists>k. ~ b ((c ^^ k) s))
@@ -81,7 +81,7 @@
qed
-subsection {* Totalized version *}
+subsection {* Total version *}
definition while :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
where "while b c s = the (while_option b c s)"
@@ -127,54 +127,5 @@
apply blast
done
-text {*
- \medskip An application: computation of the @{term lfp} on finite
- sets via iteration.
-*}
-
-theorem lfp_conv_while:
- "[| mono f; finite U; f U = U |] ==>
- lfp f = fst (while (\<lambda>(A, fA). A \<noteq> fA) (\<lambda>(A, fA). (fA, f fA)) ({}, f {}))"
-apply (rule_tac P = "\<lambda>(A, B). (A \<subseteq> U \<and> B = f A \<and> A \<subseteq> B \<and> B \<subseteq> lfp f)" and
- r = "((Pow U \<times> UNIV) \<times> (Pow U \<times> UNIV)) \<inter>
- inv_image finite_psubset (op - U o fst)" in while_rule)
- apply (subst lfp_unfold)
- apply assumption
- apply (simp add: monoD)
- apply (subst lfp_unfold)
- apply assumption
- apply clarsimp
- apply (blast dest: monoD)
- apply (fastsimp intro!: lfp_lowerbound)
- apply (blast intro: wf_finite_psubset Int_lower2 [THEN [2] wf_subset])
-apply (clarsimp simp add: finite_psubset_def order_less_le)
-apply (blast intro!: finite_Diff dest: monoD)
-done
-
-
-subsection {* Example *}
-
-text{* Cannot use @{thm[source]set_eq_subset} because it leads to
-looping because the antisymmetry simproc turns the subset relationship
-back into equality. *}
-
-theorem "P (lfp (\<lambda>N::int set. {0} \<union> {(n + 2) mod 6 | n. n \<in> N})) =
- P {0, 4, 2}"
-proof -
- have seteq: "!!A B. (A = B) = ((!a : A. a:B) & (!b:B. b:A))"
- by blast
- have aux: "!!f A B. {f n | n. A n \<or> B n} = {f n | n. A n} \<union> {f n | n. B n}"
- apply blast
- done
- show ?thesis
- apply (subst lfp_conv_while [where ?U = "{0, 1, 2, 3, 4, 5}"])
- apply (rule monoI)
- apply blast
- apply simp
- apply (simp add: aux set_eq_subset)
- txt {* The fixpoint computation is performed purely by rewriting: *}
- apply (simp add: while_unfold aux seteq del: subset_empty)
- done
-qed
end