author | nipkow |
Mon, 29 Apr 2013 11:31:40 +0200 | |
changeset 51807 | d694233adeae |
parent 51791 | c4db685eaed0 |
child 51826 | 054a40461449 |
permissions | -rw-r--r-- |
47613 | 1 |
(* Author: Tobias Nipkow *) |
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theory Abs_Int0 |
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imports Abs_Int_init |
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begin |
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subsection "Orderings" |
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text{* The basic type classes @{class order}, @{class semilattice_sup} and @{class top} are |
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defined in @{theory Main}, more precisely in theories @{theory Orderings} and @{theory Lattices}. |
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If you view this theory with jedit, just click on the names to get there. *} |
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class semilattice = semilattice_sup + top |
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instance "fun" :: (type, semilattice) semilattice .. |
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instantiation option :: (order)order |
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begin |
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fun less_eq_option where |
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"Some x \<le> Some y = (x \<le> y)" | |
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"None \<le> y = True" | |
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"Some _ \<le> None = False" |
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definition less_option where "x < (y::'a option) = (x \<le> y \<and> \<not> y \<le> x)" |
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lemma le_None[simp]: "(x \<le> None) = (x = None)" |
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by (cases x) simp_all |
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lemma Some_le[simp]: "(Some x \<le> u) = (\<exists>y. u = Some y \<and> x \<le> y)" |
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by (cases u) auto |
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instance proof |
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case goal1 show ?case by(rule less_option_def) |
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next |
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case goal2 show ?case by(cases x, simp_all) |
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next |
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case goal3 thus ?case by(cases z, simp, cases y, simp, cases x, auto) |
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next |
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case goal4 thus ?case by(cases y, simp, cases x, auto) |
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qed |
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end |
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instantiation option :: (sup)sup |
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begin |
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fun sup_option where |
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"Some x \<squnion> Some y = Some(x \<squnion> y)" | |
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"None \<squnion> y = y" | |
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"x \<squnion> None = x" |
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lemma sup_None2[simp]: "x \<squnion> None = x" |
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by (cases x) simp_all |
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instance .. |
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end |
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instantiation option :: (semilattice)semilattice |
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begin |
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definition top_option where "\<top> = Some \<top>" |
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instance proof |
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case goal4 show ?case by(cases a, simp_all add: top_option_def) |
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next |
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case goal1 thus ?case by(cases x, simp, cases y, simp_all) |
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next |
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case goal2 thus ?case by(cases y, simp, cases x, simp_all) |
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next |
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case goal3 thus ?case by(cases z, simp, cases y, simp, cases x, simp_all) |
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qed |
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end |
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lemma [simp]: "(Some x < Some y) = (x < y)" |
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by(auto simp: less_le) |
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instantiation option :: (order)bot |
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begin |
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definition bot_option :: "'a option" where |
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"\<bottom> = None" |
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instance |
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proof |
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case goal1 thus ?case by(auto simp: bot_option_def) |
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qed |
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end |
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definition bot :: "com \<Rightarrow> 'a option acom" where |
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"bot c = anno None c" |
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lemma bot_least: "strip C = c \<Longrightarrow> bot c \<le> C" |
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by(induct C arbitrary: c)(auto simp: bot_def) |
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lemma strip_bot[simp]: "strip(bot c) = c" |
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by(simp add: bot_def) |
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subsubsection "Pre-fixpoint iteration" |
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definition pfp :: "(('a::order) \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a option" where |
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"pfp f = while_option (\<lambda>x. \<not> f x \<le> x) f" |
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lemma pfp_pfp: assumes "pfp f x0 = Some x" shows "f x \<le> x" |
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using while_option_stop[OF assms[simplified pfp_def]] by simp |
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lemma while_least: |
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fixes q :: "'a::order" |
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assumes "\<forall>x\<in>L.\<forall>y\<in>L. x \<le> y \<longrightarrow> f x \<le> f y" and "\<forall>x. x \<in> L \<longrightarrow> f x \<in> L" |
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and "\<forall>x \<in> L. b \<le> x" and "b \<in> L" and "f q \<le> q" and "q \<in> L" |
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and "while_option P f b = Some p" |
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shows "p \<le> q" |
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using while_option_rule[OF _ assms(7)[unfolded pfp_def], |
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where P = "%x. x \<in> L \<and> x \<le> q"] |
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by (metis assms(1-6) order_trans) |
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lemma pfp_bot_least: |
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assumes "\<forall>x\<in>{C. strip C = c}.\<forall>y\<in>{C. strip C = c}. x \<le> y \<longrightarrow> f x \<le> f y" |
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and "\<forall>C. C \<in> {C. strip C = c} \<longrightarrow> f C \<in> {C. strip C = c}" |
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and "f C' \<le> C'" "strip C' = c" "pfp f (bot c) = Some C" |
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shows "C \<le> C'" |
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by(rule while_least[OF assms(1,2) _ _ assms(3) _ assms(5)[unfolded pfp_def]]) |
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(simp_all add: assms(4) bot_least) |
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lemma pfp_inv: |
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"pfp f x = Some y \<Longrightarrow> (\<And>x. P x \<Longrightarrow> P(f x)) \<Longrightarrow> P x \<Longrightarrow> P y" |
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unfolding pfp_def by (metis (lifting) while_option_rule) |
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lemma strip_pfp: |
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assumes "\<And>x. g(f x) = g x" and "pfp f x0 = Some x" shows "g x = g x0" |
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using pfp_inv[OF assms(2), where P = "%x. g x = g x0"] assms(1) by simp |
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subsection "Abstract Interpretation" |
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definition \<gamma>_fun :: "('a \<Rightarrow> 'b set) \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> ('c \<Rightarrow> 'b)set" where |
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"\<gamma>_fun \<gamma> F = {f. \<forall>x. f x \<in> \<gamma>(F x)}" |
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fun \<gamma>_option :: "('a \<Rightarrow> 'b set) \<Rightarrow> 'a option \<Rightarrow> 'b set" where |
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"\<gamma>_option \<gamma> None = {}" | |
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"\<gamma>_option \<gamma> (Some a) = \<gamma> a" |
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text{* The interface for abstract values: *} |
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locale Val_abs = |
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fixes \<gamma> :: "'av::semilattice \<Rightarrow> val set" |
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assumes mono_gamma: "a \<le> b \<Longrightarrow> \<gamma> a \<le> \<gamma> b" |
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and gamma_Top[simp]: "\<gamma> \<top> = UNIV" |
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fixes num' :: "val \<Rightarrow> 'av" |
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and plus' :: "'av \<Rightarrow> 'av \<Rightarrow> 'av" |
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assumes gamma_num': "i \<in> \<gamma>(num' i)" |
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and gamma_plus': "i1 \<in> \<gamma> a1 \<Longrightarrow> i2 \<in> \<gamma> a2 \<Longrightarrow> i1+i2 \<in> \<gamma>(plus' a1 a2)" |
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type_synonym 'av st = "(vname \<Rightarrow> 'av)" |
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locale Abs_Int_fun = Val_abs \<gamma> for \<gamma> :: "'av::semilattice \<Rightarrow> val set" |
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begin |
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fun aval' :: "aexp \<Rightarrow> 'av st \<Rightarrow> 'av" where |
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"aval' (N i) S = num' i" | |
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"aval' (V x) S = S x" | |
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"aval' (Plus a1 a2) S = plus' (aval' a1 S) (aval' a2 S)" |
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definition "asem x e S = (case S of None \<Rightarrow> None | Some S \<Rightarrow> Some(S(x := aval' e S)))" |
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definition "step' = Step asem (\<lambda>b S. S)" |
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lemma strip_step'[simp]: "strip(step' S C) = strip C" |
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by(simp add: step'_def) |
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definition AI :: "com \<Rightarrow> 'av st option acom option" where |
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"AI c = pfp (step' \<top>) (bot c)" |
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abbreviation \<gamma>\<^isub>s :: "'av st \<Rightarrow> state set" |
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where "\<gamma>\<^isub>s == \<gamma>_fun \<gamma>" |
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abbreviation \<gamma>\<^isub>o :: "'av st option \<Rightarrow> state set" |
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where "\<gamma>\<^isub>o == \<gamma>_option \<gamma>\<^isub>s" |
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abbreviation \<gamma>\<^isub>c :: "'av st option acom \<Rightarrow> state set acom" |
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where "\<gamma>\<^isub>c == map_acom \<gamma>\<^isub>o" |
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lemma gamma_s_Top[simp]: "\<gamma>\<^isub>s \<top> = UNIV" |
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by(simp add: top_fun_def \<gamma>_fun_def) |
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lemma gamma_o_Top[simp]: "\<gamma>\<^isub>o \<top> = UNIV" |
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by (simp add: top_option_def) |
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lemma mono_gamma_s: "f1 \<le> f2 \<Longrightarrow> \<gamma>\<^isub>s f1 \<subseteq> \<gamma>\<^isub>s f2" |
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by(auto simp: le_fun_def \<gamma>_fun_def dest: mono_gamma) |
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lemma mono_gamma_o: |
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"S1 \<le> S2 \<Longrightarrow> \<gamma>\<^isub>o S1 \<subseteq> \<gamma>\<^isub>o S2" |
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by(induction S1 S2 rule: less_eq_option.induct)(simp_all add: mono_gamma_s) |
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lemma mono_gamma_c: "C1 \<le> C2 \<Longrightarrow> \<gamma>\<^isub>c C1 \<le> \<gamma>\<^isub>c C2" |
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by (induction C1 C2 rule: less_eq_acom.induct) (simp_all add:mono_gamma_o) |
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text{* Soundness: *} |
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lemma aval'_sound: "s : \<gamma>\<^isub>s S \<Longrightarrow> aval a s : \<gamma>(aval' a S)" |
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by (induct a) (auto simp: gamma_num' gamma_plus' \<gamma>_fun_def) |
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lemma in_gamma_update: "\<lbrakk> s : \<gamma>\<^isub>s S; i : \<gamma> a \<rbrakk> \<Longrightarrow> s(x := i) : \<gamma>\<^isub>s(S(x := a))" |
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by(simp add: \<gamma>_fun_def) |
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lemma gamma_Step_subcomm: |
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assumes "!!x e S. f1 x e (\<gamma>\<^isub>o S) \<subseteq> \<gamma>\<^isub>o (f2 x e S)" "!!b S. g1 b (\<gamma>\<^isub>o S) \<subseteq> \<gamma>\<^isub>o (g2 b S)" |
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shows "Step f1 g1 (\<gamma>\<^isub>o S) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c (Step f2 g2 S C)" |
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proof(induction C arbitrary: S) |
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qed (auto simp: mono_gamma_o assms) |
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lemma step_step': "step (\<gamma>\<^isub>o S) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c (step' S C)" |
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unfolding step_def step'_def |
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by(rule gamma_Step_subcomm) |
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(auto simp: aval'_sound in_gamma_update asem_def split: option.splits) |
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lemma AI_sound: "AI c = Some C \<Longrightarrow> CS c \<le> \<gamma>\<^isub>c C" |
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proof(simp add: CS_def AI_def) |
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assume 1: "pfp (step' \<top>) (bot c) = Some C" |
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have pfp': "step' \<top> C \<le> C" by(rule pfp_pfp[OF 1]) |
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have 2: "step (\<gamma>\<^isub>o \<top>) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c C" --"transfer the pfp'" |
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proof(rule order_trans) |
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show "step (\<gamma>\<^isub>o \<top>) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c (step' \<top> C)" by(rule step_step') |
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show "... \<le> \<gamma>\<^isub>c C" by (metis mono_gamma_c[OF pfp']) |
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47613 | 233 |
qed |
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have 3: "strip (\<gamma>\<^isub>c C) = c" by(simp add: strip_pfp[OF _ 1] step'_def) |
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have "lfp c (step (\<gamma>\<^isub>o \<top>)) \<le> \<gamma>\<^isub>c C" |
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by(rule lfp_lowerbound[simplified,where f="step (\<gamma>\<^isub>o \<top>)", OF 3 2]) |
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thus "lfp c (step UNIV) \<le> \<gamma>\<^isub>c C" by simp |
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47613 | 238 |
qed |
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end |
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subsubsection "Monotonicity" |
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lemma mono_post: "C1 \<le> C2 \<Longrightarrow> post C1 \<le> post C2" |
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by(induction C1 C2 rule: less_eq_acom.induct) (auto) |
47613 | 247 |
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locale Abs_Int_fun_mono = Abs_Int_fun + |
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assumes mono_plus': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow> plus' a1 a2 \<le> plus' b1 b2" |
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begin |
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lemma mono_aval': "S \<le> S' \<Longrightarrow> aval' e S \<le> aval' e S'" |
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by(induction e)(auto simp: le_fun_def mono_plus') |
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lemma mono_update: "a \<le> a' \<Longrightarrow> S \<le> S' \<Longrightarrow> S(x := a) \<le> S'(x := a')" |
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by(simp add: le_fun_def) |
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lemma mono_step': "S1 \<le> S2 \<Longrightarrow> C1 \<le> C2 \<Longrightarrow> step' S1 C1 \<le> step' S2 C2" |
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unfolding step'_def |
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by(rule mono2_Step) |
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(auto simp: mono_update mono_aval' asem_def split: option.split) |
47613 | 262 |
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51722 | 263 |
lemma mono_step'_top: "C \<le> C' \<Longrightarrow> step' \<top> C \<le> step' \<top> C'" |
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by (metis mono_step' order_refl) |
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lemma AI_least_pfp: assumes "AI c = Some C" "step' \<top> C' \<le> C'" "strip C' = c" |
|
267 |
shows "C \<le> C'" |
|
268 |
by(rule pfp_bot_least[OF _ _ assms(2,3) assms(1)[unfolded AI_def]]) |
|
269 |
(simp_all add: mono_step'_top) |
|
270 |
||
271 |
end |
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272 |
||
273 |
||
274 |
instantiation acom :: (type) vars |
|
275 |
begin |
|
276 |
||
277 |
definition "vars_acom = vars o strip" |
|
278 |
||
279 |
instance .. |
|
280 |
||
281 |
end |
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282 |
||
283 |
lemma finite_Cvars: "finite(vars(C::'a acom))" |
|
284 |
by(simp add: vars_acom_def) |
|
285 |
||
286 |
||
287 |
subsubsection "Termination" |
|
288 |
||
289 |
lemma pfp_termination: |
|
290 |
fixes x0 :: "'a::order" and m :: "'a \<Rightarrow> nat" |
|
291 |
assumes mono: "\<And>x y. I x \<Longrightarrow> I y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
|
292 |
and m: "\<And>x y. I x \<Longrightarrow> I y \<Longrightarrow> x < y \<Longrightarrow> m x > m y" |
|
293 |
and I: "\<And>x y. I x \<Longrightarrow> I(f x)" and "I x0" and "x0 \<le> f x0" |
|
294 |
shows "\<exists>x. pfp f x0 = Some x" |
|
295 |
proof(simp add: pfp_def, rule wf_while_option_Some[where P = "%x. I x & x \<le> f x"]) |
|
296 |
show "wf {(y,x). ((I x \<and> x \<le> f x) \<and> \<not> f x \<le> x) \<and> y = f x}" |
|
297 |
by(rule wf_subset[OF wf_measure[of m]]) (auto simp: m I) |
|
298 |
next |
|
299 |
show "I x0 \<and> x0 \<le> f x0" using `I x0` `x0 \<le> f x0` by blast |
|
300 |
next |
|
301 |
fix x assume "I x \<and> x \<le> f x" thus "I(f x) \<and> f x \<le> f(f x)" |
|
302 |
by (blast intro: I mono) |
|
303 |
qed |
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304 |
||
305 |
||
306 |
locale Measure1_fun = |
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fixes m :: "'av::top \<Rightarrow> nat" |
51722 | 308 |
fixes h :: "nat" |
309 |
assumes h: "m x \<le> h" |
|
310 |
begin |
|
311 |
||
51791 | 312 |
definition m_s :: "'av st \<Rightarrow> vname set \<Rightarrow> nat" ("m\<^isub>s") where |
313 |
"m_s S X = (\<Sum> x \<in> X. m(S x))" |
|
51722 | 314 |
|
51791 | 315 |
lemma m_s_h: "finite X \<Longrightarrow> m_s S X \<le> h * card X" |
51722 | 316 |
by(simp add: m_s_def) (metis nat_mult_commute of_nat_id setsum_bounded[OF h]) |
317 |
||
51791 | 318 |
fun m_o :: "'av st option \<Rightarrow> vname set \<Rightarrow> nat" ("m\<^isub>o") where |
319 |
"m_o (Some S) X = m_s S X" | |
|
320 |
"m_o None X = h * card X + 1" |
|
51722 | 321 |
|
51791 | 322 |
lemma m_o_h: "finite X \<Longrightarrow> m_o opt X \<le> (h*card X + 1)" |
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by(cases opt)(auto simp add: m_s_h le_SucI dest: m_s_h) |
51722 | 324 |
|
325 |
definition m_c :: "'av st option acom \<Rightarrow> nat" ("m\<^isub>c") where |
|
51791 | 326 |
"m_c C = listsum (map (\<lambda>a. m_o a (vars C)) (annos C))" |
51722 | 327 |
|
328 |
text{* Upper complexity bound: *} |
|
329 |
lemma m_c_h: "m_c C \<le> size(annos C) * (h * card(vars C) + 1)" |
|
330 |
proof- |
|
331 |
let ?X = "vars C" let ?n = "card ?X" let ?a = "size(annos C)" |
|
51791 | 332 |
have "m_c C = (\<Sum>i<?a. m_o (annos C ! i) ?X)" |
51783 | 333 |
by(simp add: m_c_def listsum_setsum_nth atLeast0LessThan) |
51722 | 334 |
also have "\<dots> \<le> (\<Sum>i<?a. h * ?n + 1)" |
335 |
apply(rule setsum_mono) using m_o_h[OF finite_Cvars] by simp |
|
336 |
also have "\<dots> = ?a * (h * ?n + 1)" by simp |
|
337 |
finally show ?thesis . |
|
338 |
qed |
|
339 |
||
340 |
end |
|
341 |
||
342 |
||
343 |
lemma le_iff_le_annos_zip: "C1 \<le> C2 \<longleftrightarrow> |
|
344 |
(\<forall> (a1,a2) \<in> set(zip (annos C1) (annos C2)). a1 \<le> a2) \<and> strip C1 = strip C2" |
|
345 |
by(induct C1 C2 rule: less_eq_acom.induct) (auto simp: size_annos_same2) |
|
346 |
||
347 |
lemma le_iff_le_annos: "C1 \<le> C2 \<longleftrightarrow> |
|
348 |
strip C1 = strip C2 \<and> (\<forall> i<size(annos C1). annos C1 ! i \<le> annos C2 ! i)" |
|
349 |
by(auto simp add: le_iff_le_annos_zip set_zip) (metis size_annos_same2) |
|
350 |
||
351 |
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locale Measure_fun = Measure1_fun where m=m for m :: "'av::semilattice \<Rightarrow> nat" + |
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assumes m2: "x < y \<Longrightarrow> m x > m y" |
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begin |
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355 |
|
51785 | 356 |
text{* The predicates @{text "top_on_ty a X"} that follow describe that any abstract |
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state in @{text a} maps all variables in @{text X} to @{term \<top>}. |
51722 | 358 |
This is an important invariant for the termination proof where we argue that only |
359 |
the finitely many variables in the program change. That the others do not change |
|
360 |
follows because they remain @{term \<top>}. *} |
|
361 |
||
51785 | 362 |
fun top_on_st :: "'av st \<Rightarrow> vname set \<Rightarrow> bool" ("top'_on\<^isub>s") where |
363 |
"top_on_st S X = (\<forall>x\<in>X. S x = \<top>)" |
|
51722 | 364 |
|
51785 | 365 |
fun top_on_opt :: "'av st option \<Rightarrow> vname set \<Rightarrow> bool" ("top'_on\<^isub>o") where |
366 |
"top_on_opt (Some S) X = top_on_st S X" | |
|
367 |
"top_on_opt None X = True" |
|
51722 | 368 |
|
51785 | 369 |
definition top_on_acom :: "'av st option acom \<Rightarrow> vname set \<Rightarrow> bool" ("top'_on\<^isub>c") where |
370 |
"top_on_acom C X = (\<forall>a \<in> set(annos C). top_on_opt a X)" |
|
51722 | 371 |
|
51785 | 372 |
lemma top_on_top: "top_on_opt \<top> X" |
51722 | 373 |
by(auto simp: top_option_def) |
374 |
||
51785 | 375 |
lemma top_on_bot: "top_on_acom (bot c) X" |
51722 | 376 |
by(auto simp add: top_on_acom_def bot_def) |
377 |
||
51785 | 378 |
lemma top_on_post: "top_on_acom C X \<Longrightarrow> top_on_opt (post C) X" |
51722 | 379 |
by(simp add: top_on_acom_def post_in_annos) |
380 |
||
381 |
lemma top_on_acom_simps: |
|
51785 | 382 |
"top_on_acom (SKIP {Q}) X = top_on_opt Q X" |
383 |
"top_on_acom (x ::= e {Q}) X = top_on_opt Q X" |
|
384 |
"top_on_acom (C1;C2) X = (top_on_acom C1 X \<and> top_on_acom C2 X)" |
|
385 |
"top_on_acom (IF b THEN {P1} C1 ELSE {P2} C2 {Q}) X = |
|
386 |
(top_on_opt P1 X \<and> top_on_acom C1 X \<and> top_on_opt P2 X \<and> top_on_acom C2 X \<and> top_on_opt Q X)" |
|
387 |
"top_on_acom ({I} WHILE b DO {P} C {Q}) X = |
|
388 |
(top_on_opt I X \<and> top_on_acom C X \<and> top_on_opt P X \<and> top_on_opt Q X)" |
|
51722 | 389 |
by(auto simp add: top_on_acom_def) |
390 |
||
391 |
lemma top_on_sup: |
|
51785 | 392 |
"top_on_opt o1 X \<Longrightarrow> top_on_opt o2 X \<Longrightarrow> top_on_opt (o1 \<squnion> o2) X" |
51722 | 393 |
apply(induction o1 o2 rule: sup_option.induct) |
394 |
apply(auto) |
|
395 |
done |
|
396 |
||
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397 |
lemma top_on_Step: fixes C :: "'av st option acom" |
51785 | 398 |
assumes "!!x e S. \<lbrakk>top_on_opt S X; x \<notin> X; vars e \<subseteq> -X\<rbrakk> \<Longrightarrow> top_on_opt (f x e S) X" |
399 |
"!!b S. top_on_opt S X \<Longrightarrow> vars b \<subseteq> -X \<Longrightarrow> top_on_opt (g b S) X" |
|
400 |
shows "\<lbrakk> vars C \<subseteq> -X; top_on_opt S X; top_on_acom C X \<rbrakk> \<Longrightarrow> top_on_acom (Step f g S C) X" |
|
51722 | 401 |
proof(induction C arbitrary: S) |
402 |
qed (auto simp: top_on_acom_simps vars_acom_def top_on_post top_on_sup assms) |
|
403 |
||
404 |
lemma m1: "x \<le> y \<Longrightarrow> m x \<ge> m y" |
|
405 |
by(auto simp: le_less m2) |
|
406 |
||
407 |
lemma m_s2_rep: assumes "finite(X)" and "S1 = S2 on -X" and "\<forall>x. S1 x \<le> S2 x" and "S1 \<noteq> S2" |
|
408 |
shows "(\<Sum>x\<in>X. m (S2 x)) < (\<Sum>x\<in>X. m (S1 x))" |
|
409 |
proof- |
|
410 |
from assms(3) have 1: "\<forall>x\<in>X. m(S1 x) \<ge> m(S2 x)" by (simp add: m1) |
|
411 |
from assms(2,3,4) have "EX x:X. S1 x < S2 x" |
|
412 |
by(simp add: fun_eq_iff) (metis Compl_iff le_neq_trans) |
|
413 |
hence 2: "\<exists>x\<in>X. m(S1 x) > m(S2 x)" by (metis m2) |
|
414 |
from setsum_strict_mono_ex1[OF `finite X` 1 2] |
|
415 |
show "(\<Sum>x\<in>X. m (S2 x)) < (\<Sum>x\<in>X. m (S1 x))" . |
|
416 |
qed |
|
417 |
||
51791 | 418 |
lemma m_s2: "finite(X) \<Longrightarrow> S1 = S2 on -X \<Longrightarrow> S1 < S2 \<Longrightarrow> m_s S1 X > m_s S2 X" |
51722 | 419 |
apply(auto simp add: less_fun_def m_s_def) |
420 |
apply(simp add: m_s2_rep le_fun_def) |
|
421 |
done |
|
422 |
||
51785 | 423 |
lemma m_o2: "finite X \<Longrightarrow> top_on_opt o1 (-X) \<Longrightarrow> top_on_opt o2 (-X) \<Longrightarrow> |
51791 | 424 |
o1 < o2 \<Longrightarrow> m_o o1 X > m_o o2 X" |
51722 | 425 |
proof(induction o1 o2 rule: less_eq_option.induct) |
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51722
diff
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|
426 |
case 1 thus ?case by (auto simp: m_s2 less_option_def) |
51722 | 427 |
next |
51749
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nipkow
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51722
diff
changeset
|
428 |
case 2 thus ?case by(auto simp: less_option_def le_imp_less_Suc m_s_h) |
51722 | 429 |
next |
430 |
case 3 thus ?case by (auto simp: less_option_def) |
|
431 |
qed |
|
432 |
||
51785 | 433 |
lemma m_o1: "finite X \<Longrightarrow> top_on_opt o1 (-X) \<Longrightarrow> top_on_opt o2 (-X) \<Longrightarrow> |
51791 | 434 |
o1 \<le> o2 \<Longrightarrow> m_o o1 X \<ge> m_o o2 X" |
51722 | 435 |
by(auto simp: le_less m_o2) |
436 |
||
437 |
||
51785 | 438 |
lemma m_c2: "top_on_acom C1 (-vars C1) \<Longrightarrow> top_on_acom C2 (-vars C2) \<Longrightarrow> |
51722 | 439 |
C1 < C2 \<Longrightarrow> m_c C1 > m_c C2" |
51783 | 440 |
proof(auto simp add: le_iff_le_annos size_annos_same[of C1 C2] vars_acom_def less_acom_def) |
51722 | 441 |
let ?X = "vars(strip C2)" |
51785 | 442 |
assume top: "top_on_acom C1 (- vars(strip C2))" "top_on_acom C2 (- vars(strip C2))" |
51722 | 443 |
and strip_eq: "strip C1 = strip C2" |
444 |
and 0: "\<forall>i<size(annos C2). annos C1 ! i \<le> annos C2 ! i" |
|
51791 | 445 |
hence 1: "\<forall>i<size(annos C2). m_o (annos C1 ! i) ?X \<ge> m_o (annos C2 ! i) ?X" |
51722 | 446 |
apply (auto simp: all_set_conv_all_nth vars_acom_def top_on_acom_def) |
447 |
by (metis (lifting, no_types) finite_cvars m_o1 size_annos_same2) |
|
448 |
fix i assume i: "i < size(annos C2)" "\<not> annos C2 ! i \<le> annos C1 ! i" |
|
51785 | 449 |
have topo1: "top_on_opt (annos C1 ! i) (- ?X)" |
51722 | 450 |
using i(1) top(1) by(simp add: top_on_acom_def size_annos_same[OF strip_eq]) |
51785 | 451 |
have topo2: "top_on_opt (annos C2 ! i) (- ?X)" |
51722 | 452 |
using i(1) top(2) by(simp add: top_on_acom_def size_annos_same[OF strip_eq]) |
51791 | 453 |
from i have "m_o (annos C1 ! i) ?X > m_o (annos C2 ! i) ?X" (is "?P i") |
51722 | 454 |
by (metis 0 less_option_def m_o2[OF finite_cvars topo1] topo2) |
455 |
hence 2: "\<exists>i < size(annos C2). ?P i" using `i < size(annos C2)` by blast |
|
51791 | 456 |
have "(\<Sum>i<size(annos C2). m_o (annos C2 ! i) ?X) |
457 |
< (\<Sum>i<size(annos C2). m_o (annos C1 ! i) ?X)" |
|
51722 | 458 |
apply(rule setsum_strict_mono_ex1) using 1 2 by (auto) |
51783 | 459 |
thus ?thesis |
460 |
by(simp add: m_c_def vars_acom_def strip_eq listsum_setsum_nth atLeast0LessThan size_annos_same[OF strip_eq]) |
|
51722 | 461 |
qed |
462 |
||
463 |
end |
|
464 |
||
465 |
||
466 |
locale Abs_Int_fun_measure = |
|
467 |
Abs_Int_fun_mono where \<gamma>=\<gamma> + Measure_fun where m=m |
|
468 |
for \<gamma> :: "'av::semilattice \<Rightarrow> val set" and m :: "'av \<Rightarrow> nat" |
|
469 |
begin |
|
470 |
||
51785 | 471 |
lemma top_on_step': "top_on_acom C (-vars C) \<Longrightarrow> top_on_acom (step' \<top> C) (-vars C)" |
51722 | 472 |
unfolding step'_def |
473 |
by(rule top_on_Step) |
|
51807 | 474 |
(auto simp add: top_option_def asem_def split: option.splits) |
51722 | 475 |
|
476 |
lemma AI_Some_measure: "\<exists>C. AI c = Some C" |
|
477 |
unfolding AI_def |
|
51785 | 478 |
apply(rule pfp_termination[where I = "\<lambda>C. top_on_acom C (- vars C)" and m="m_c"]) |
51722 | 479 |
apply(simp_all add: m_c2 mono_step'_top bot_least top_on_bot) |
51754 | 480 |
using top_on_step' apply(auto simp add: vars_acom_def) |
51722 | 481 |
done |
482 |
||
47613 | 483 |
end |
484 |
||
485 |
text{* Problem: not executable because of the comparison of abstract states, |
|
486 |
i.e. functions, in the post-fixedpoint computation. *} |
|
487 |
||
488 |
end |