(* Title: HOL/ex/While_Combinator_Example.thy
Author: Tobias Nipkow
Copyright 2000 TU Muenchen
*)
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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