section \<open>An application of the While combinator\<close>
theory While_Combinator_Example imports"HOL-Library.While_Combinator" begin
text\<open>Computation of the \<^term>\<open>lfp\<close> on finite sets via
iteration.\<close>
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 = "\(A, B). (A \ U \ B = f A \ A \ B \ B \ lfp f)" and
r = "((Pow U \ UNIV) \ (Pow U \ UNIV)) \
inv_image finite_psubset ((-) 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 (fastforce 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 dest: monoD) done
subsection \<open>Example\<close>
text\<open>Cannot use @{thm[source]set_eq_subset} because it leads to
looping because the antisymmetry simproc turns the subset relationship back into equality.\<close>
theorem"P (lfp (\N::int set. {0} \ {(n + 2) mod 6 | n. n \ 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 \ B n} = {f n | n. A n} \ {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\<open>The fixpoint computation is performed purely by rewriting:\<close> apply (simp add: while_unfold aux seteq del: subset_empty) done qed
end
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