(*  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 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