(* Title: HOL/Library/While.thy ID: $Id: While_Combinator.thy,v 1.22 2005/12/08 19:15:50 wenzelm Exp $ Author: Tobias Nipkow Copyright 2000 TU Muenchen *) header {* A general ``while'' combinator *} theory While_Combinator imports Main begin text {* We define a while-combinator @{term while} and prove: (a) an unrestricted unfolding law (even if while diverges!) (I got this idea from Wolfgang Goerigk), and (b) the invariant rule for reasoning about @{term while}. *} consts while_aux :: "('a => bool) × ('a => 'a) × 'a => 'a" recdef (permissive) while_aux "same_fst (λb. True) (λb. same_fst (λc. True) (λc. {(t, s). b s ∧ c s = t ∧ ¬ (∃f. f (0::nat) = s ∧ (∀i. b (f i) ∧ c (f i) = f (i + 1)))}))" "while_aux (b, c, s) = (if (∃f. f (0::nat) = s ∧ (∀i. b (f i) ∧ c (f i) = f (i + 1))) then arbitrary else if b s then while_aux (b, c, c s) else s)" recdef_tc while_aux_tc: while_aux apply (rule wf_same_fst) apply (rule wf_same_fst) apply (simp add: wf_iff_no_infinite_down_chain) apply blast done constdefs while :: "('a => bool) => ('a => 'a) => 'a => 'a" "while b c s == while_aux (b, c, s)" lemma while_aux_unfold: "while_aux (b, c, s) = (if ∃f. f (0::nat) = s ∧ (∀i. b (f i) ∧ c (f i) = f (i + 1)) then arbitrary else if b s then while_aux (b, c, c s) else s)" apply (rule while_aux_tc [THEN while_aux.simps [THEN trans]]) apply (rule refl) done text {* The recursion equation for @{term while}: directly executable! *} theorem while_unfold [code]: "while b c s = (if b s then while b c (c s) else s)" apply (unfold while_def) apply (rule while_aux_unfold [THEN trans]) apply auto apply (subst while_aux_unfold) apply simp apply clarify apply (erule_tac x = "λi. f (Suc i)" in allE) apply blast done hide const while_aux 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)" apply (induct s rule: wf [THEN wf_induct]) 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: inv_image_def 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. *} lemma seteq: "(A = B) = ((!a : A. a:B) & (!b:B. b:A))" by blast theorem "P (lfp (λN::int set. {0} ∪ {(n + 2) mod 6 | n. n ∈ N})) = P {0, 4, 2}" proof - 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 while_aux_unfold:
while_aux (b, c, s) = (if ∃f. f 0 = s ∧ (∀i. b (f i) ∧ c (f i) = f (i + 1)) then arbitrary else if b s then while_aux (b, c, c s) else s)
theorem while_unfold:
while b c s = (if b s then while b c (c s) else s)
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 {}))
lemma seteq:
(A = B) = ((∀a∈A. a ∈ B) ∧ (∀b∈B. b ∈ A))
theorem
P (lfp (λN. {0} ∪ {(n + 2) mod 6 |n. n ∈ N})) = P {0, 4, 2}