(* Title: HOL/ex/While_Combinator_Example.thy
Author: Tobias Nipkow
Copyright 2000 TU Muenchen
*)
section ‹An application of the While combinator
›
theory While_Combinator_Example
imports "HOL-Library.While_Combinator"
begin
text ‹Computation of the
🍋‹lfp
› on finite sets via
iteration.
›
theorem lfp_conv_while:
"[| mono f; finite U; f U = U |] ==>
lfp f = fst (while (λ(A, fA). A
≠ fA) (λ(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 ‹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 (\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 ‹The fixpoint computation
is performed purely
by rewriting:
›
apply (simp add: while_unfold aux seteq del: subset_empty)
done
qed
end
