author | wenzelm |
Mon, 27 Aug 2012 14:34:54 +0200 | |
changeset 48933 | d25e47e32bc0 |
parent 44890 | 22f665a2e91c |
child 58889 | 5b7a9633cfa8 |
permissions | -rw-r--r-- |
41959 | 1 |
(* Title: HOL/ex/While_Combinator_Example.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 "~~/src/HOL/Library/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 (fastforce 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 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 |