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(* Title: HOL/Library/While_Combinator.thy
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Author: Tobias Nipkow
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Copyright 2000 TU Muenchen
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*)
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header {* An application of the While combinator *}
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theory While_Combinator_Example
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imports While_Combinator
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begin
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text {* Computation of the @{term lfp} on finite sets via
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iteration. *}
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theorem lfp_conv_while:
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"[| mono f; finite U; f U = U |] ==>
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lfp f = fst (while (\<lambda>(A, fA). A \<noteq> fA) (\<lambda>(A, fA). (fA, f fA)) ({}, f {}))"
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apply (rule_tac P = "\<lambda>(A, B). (A \<subseteq> U \<and> B = f A \<and> A \<subseteq> B \<and> B \<subseteq> lfp f)" and
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r = "((Pow U \<times> UNIV) \<times> (Pow U \<times> UNIV)) \<inter>
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inv_image finite_psubset (op - U o fst)" in while_rule)
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apply (subst lfp_unfold)
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apply assumption
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apply (simp add: monoD)
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apply (subst lfp_unfold)
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apply assumption
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apply clarsimp
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apply (blast dest: monoD)
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apply (fastsimp intro!: lfp_lowerbound)
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apply (blast intro: wf_finite_psubset Int_lower2 [THEN [2] wf_subset])
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apply (clarsimp simp add: finite_psubset_def order_less_le)
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apply (blast intro!: finite_Diff dest: monoD)
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done
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subsection {* Example *}
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text{* Cannot use @{thm[source]set_eq_subset} because it leads to
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looping because the antisymmetry simproc turns the subset relationship
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back into equality. *}
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theorem "P (lfp (\<lambda>N::int set. {0} \<union> {(n + 2) mod 6 | n. n \<in> N})) =
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P {0, 4, 2}"
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proof -
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have seteq: "!!A B. (A = B) = ((!a : A. a:B) & (!b:B. b:A))"
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by blast
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have aux: "!!f A B. {f n | n. A n \<or> B n} = {f n | n. A n} \<union> {f n | n. B n}"
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apply blast
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done
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show ?thesis
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apply (subst lfp_conv_while [where ?U = "{0, 1, 2, 3, 4, 5}"])
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apply (rule monoI)
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apply blast
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apply simp
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apply (simp add: aux set_eq_subset)
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txt {* The fixpoint computation is performed purely by rewriting: *}
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apply (simp add: while_unfold aux seteq del: subset_empty)
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done
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qed
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end |