(* Title: HOL/Library/While_Combinator.thy
Author: Tobias Nipkow
Copyright 2000 TU Muenchen
*)
header {* An application of the While combinator *}
theory While_Combinator_Example
imports While_Combinator
begin
text {* 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