(* Title: HOL/Library/While_Combinator.thy ID: $Id: While_Combinator.thy,v 1.28 2007/07/16 19:39:56 krauss Exp $ 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
lemma def_while_unfold:
f == while test do ==> f x = (if test x then f (do x) else x)
theorem while_rule_lemma:
[| !!s. [| P s; b s |] ==> P (c s); !!s. [| P s; ¬ b s |] ==> Q s;
wf {(t, s). P s ∧ b s ∧ t = c s}; P s |]
==> Q (while b c s)
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)
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 {}))
theorem
P (lfp (λN. {0} ∪ {(n + 2) mod 6 |n. n ∈ N})) = P {0, 4, 2}