Theory While_Combinator

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theory While_Combinator
imports Main

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


header {* A general ``while'' combinator *}

theory While_Combinator
imports Main
begin


text {*
We define the while combinator as the "mother of all tail recursive functions".
*}


function (tailrec) while :: "('a => bool) => ('a => 'a) => 'a => 'a"
where
while_unfold[simp del]: "while b c s = (if b s then while b c (c s) else s)"

by auto

declare while_unfold[code]

lemma def_while_unfold:
assumes fdef: "f == while test do"
shows "f x = (if test x then f(do x) else x)"

proof -
have "f x = while test do x" using fdef by simp
also have "… = (if test x then while test do (do x) else x)"
by(rule while_unfold)
also have "… = (if test x then f(do x) else x)" by(simp add:fdef[symmetric])
finally show ?thesis .
qed


text {*
The proof rule for @{term while}, where @{term P} is the invariant.
*}


theorem while_rule_lemma:
assumes invariant: "!!s. P s ==> b s ==> P (c s)"
and terminate: "!!s. P s ==> ¬ b s ==> Q s"
and wf: "wf {(t, s). P s ∧ b s ∧ t = c s}"
shows "P s ==> Q (while b c s)"

using wf
apply (induct s)
apply simp
apply (subst while_unfold)
apply (simp add: invariant terminate)
done

theorem while_rule:
"[| P s;
!!s. [| P s; b s |] ==> P (c s);
!!s. [| P s; ¬ b s |] ==> Q s;
wf r;
!!s. [| P s; b s |] ==> (c s, s) ∈ r |] ==>
Q (while b c s)"

apply (rule while_rule_lemma)
prefer 4 apply assumption
apply blast
apply blast
apply (erule wf_subset)
apply blast
done

text {*
\medskip An application: 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 (λ(A, fA). A ≠ fA) (λ(A, fA). (fA, f fA)) ({}, f {}))"

apply (rule_tac P = "λ(A, B). (A ⊆ U ∧ B = f A ∧ A ⊆ B ∧ B ⊆ lfp f)" and
r = "((Pow U × UNIV) × (Pow U × UNIV)) ∩
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


text {*
An example of using the @{term while} combinator.
*}


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 (λN::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