(* Author: Tobias Nipkow *)
theory Abs_Int1
imports Abs_State
begin
lemma le_iff_le_annos_zip: "C1 \<le> C2 \<longleftrightarrow>
(\<forall> (a1,a2) \<in> set(zip (annos C1) (annos C2)). a1 \<le> a2) \<and> strip C1 = strip C2"
by(induct C1 C2 rule: less_eq_acom.induct) (auto simp: size_annos_same2)
lemma le_iff_le_annos: "C1 \<le> C2 \<longleftrightarrow>
strip C1 = strip C2 \<and> (\<forall> i<size(annos C1). annos C1 ! i \<le> annos C2 ! i)"
by(auto simp add: le_iff_le_annos_zip set_zip) (metis size_annos_same2)
lemma mono_fun_L[simp]: "F \<in> L X \<Longrightarrow> F \<le> G \<Longrightarrow> x : X \<Longrightarrow> fun F x \<le> fun G x"
by(simp add: mono_fun L_st_def)
lemma bot_in_L[simp]: "bot c \<in> L(vars c)"
by(simp add: L_acom_def bot_def)
lemma L_acom_simps[simp]: "SKIP {P} \<in> L X \<longleftrightarrow> P \<in> L X"
"(x ::= e {P}) \<in> L X \<longleftrightarrow> x : X \<and> vars e \<subseteq> X \<and> P \<in> L X"
"(C1;C2) \<in> L X \<longleftrightarrow> C1 \<in> L X \<and> C2 \<in> L X"
"(IF b THEN {P1} C1 ELSE {P2} C2 {Q}) \<in> L X \<longleftrightarrow>
vars b \<subseteq> X \<and> C1 \<in> L X \<and> C2 \<in> L X \<and> P1 \<in> L X \<and> P2 \<in> L X \<and> Q \<in> L X"
"({I} WHILE b DO {P} C {Q}) \<in> L X \<longleftrightarrow>
I \<in> L X \<and> vars b \<subseteq> X \<and> P \<in> L X \<and> C \<in> L X \<and> Q \<in> L X"
by(auto simp add: L_acom_def)
lemma post_in_annos: "post C : set(annos C)"
by(induction C) auto
lemma post_in_L[simp]: "C \<in> L X \<Longrightarrow> post C \<in> L X"
by(simp add: L_acom_def post_in_annos)
subsection "Computable Abstract Interpretation"
text{* Abstract interpretation over type @{text st} instead of functions. *}
context Gamma
begin
fun aval' :: "aexp \<Rightarrow> 'av st \<Rightarrow> 'av" where
"aval' (N i) S = num' i" |
"aval' (V x) S = fun S x" |
"aval' (Plus a1 a2) S = plus' (aval' a1 S) (aval' a2 S)"
lemma aval'_sound: "s : \<gamma>\<^isub>s S \<Longrightarrow> vars a \<subseteq> dom S \<Longrightarrow> aval a s : \<gamma>(aval' a S)"
by (induction a) (auto simp: gamma_num' gamma_plus' \<gamma>_st_def)
lemma gamma_Step_subcomm: fixes C1 C2 :: "'a::semilattice_sup acom"
assumes "!!x e S. x : X \<Longrightarrow> vars e \<subseteq> X \<Longrightarrow> S \<in> L X \<Longrightarrow> f1 x e (\<gamma>\<^isub>o S) \<subseteq> \<gamma>\<^isub>o (f2 x e S)"
"!!b S. S \<in> L X \<Longrightarrow> vars b \<subseteq> X \<Longrightarrow> g1 b (\<gamma>\<^isub>o S) \<subseteq> \<gamma>\<^isub>o (g2 b S)"
shows "C \<in> L X \<Longrightarrow> S \<in> L X \<Longrightarrow> Step f1 g1 (\<gamma>\<^isub>o S) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c (Step f2 g2 S C)"
proof(induction C arbitrary: S)
qed (auto simp: assms intro!: mono_gamma_o post_in_L sup_ge1 sup_ge2)
lemma in_gamma_update: "\<lbrakk> s : \<gamma>\<^isub>s S; i : \<gamma> a \<rbrakk> \<Longrightarrow> s(x := i) : \<gamma>\<^isub>s(update S x a)"
by(simp add: \<gamma>_st_def)
end
lemma Step_in_L: fixes C :: "'a::semilatticeL acom"
assumes "!!x e S. \<lbrakk>S \<in> L X; x \<in> X; vars e \<subseteq> X\<rbrakk> \<Longrightarrow> f x e S \<in> L X"
"!!b S. S \<in> L X \<Longrightarrow> vars b \<subseteq> X \<Longrightarrow> g b S \<in> L X"
shows"\<lbrakk> C \<in> L X; S \<in> L X \<rbrakk> \<Longrightarrow> Step f g S C \<in> L X"
proof(induction C arbitrary: S)
case Assign thus ?case
by(auto simp: L_st_def assms split: option.splits)
qed (auto simp: assms)
text{* The for-clause (here and elsewhere) only serves the purpose of fixing
the name of the type parameter @{typ 'av} which would otherwise be renamed to
@{typ 'a}. *}
locale Abs_Int = Gamma where \<gamma>=\<gamma> for \<gamma> :: "'av::semilattice \<Rightarrow> val set"
begin
definition "step' = Step
(\<lambda>x e S. case S of None \<Rightarrow> None | Some S \<Rightarrow> Some(update S x (aval' e S)))
(\<lambda>b S. S)"
definition AI :: "com \<Rightarrow> 'av st option acom option" where
"AI c = pfp (step' (Top(vars c))) (bot c)"
lemma strip_step'[simp]: "strip(step' S C) = strip C"
by(simp add: step'_def)
text{* Soundness: *}
lemma step_step': "C \<in> L X \<Longrightarrow> S \<in> L X \<Longrightarrow> step (\<gamma>\<^isub>o S) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c (step' S C)"
unfolding step_def step'_def
by(rule gamma_Step_subcomm)
(auto simp: L_st_def intro!: aval'_sound in_gamma_update split: option.splits)
lemma step'_in_L[simp]: "\<lbrakk> C \<in> L X; S \<in> L X \<rbrakk> \<Longrightarrow> step' S C \<in> L X"
unfolding step'_def
by(rule Step_in_L)(auto simp: L_st_def step'_def split: option.splits)
lemma AI_sound: "AI c = Some C \<Longrightarrow> CS c \<le> \<gamma>\<^isub>c C"
proof(simp add: CS_def AI_def)
assume 1: "pfp (step' (Top(vars c))) (bot c) = Some C"
have "C \<in> L(vars c)"
by(rule pfp_inv[where P = "%C. C \<in> L(vars c)", OF 1 _ bot_in_L])
(erule step'_in_L[OF _ Top_in_L])
have pfp': "step' (Top(vars c)) C \<le> C" by(rule pfp_pfp[OF 1])
have 2: "step (\<gamma>\<^isub>o(Top(vars c))) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c C"
proof(rule order_trans)
show "step (\<gamma>\<^isub>o (Top(vars c))) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c (step' (Top(vars c)) C)"
by(rule step_step'[OF `C \<in> L(vars c)` Top_in_L])
show "\<gamma>\<^isub>c (step' (Top(vars c)) C) \<le> \<gamma>\<^isub>c C"
by(rule mono_gamma_c[OF pfp'])
qed
have 3: "strip (\<gamma>\<^isub>c C) = c" by(simp add: strip_pfp[OF _ 1] step'_def)
have "lfp c (step (\<gamma>\<^isub>o(Top(vars c)))) \<le> \<gamma>\<^isub>c C"
by(rule lfp_lowerbound[simplified,where f="step (\<gamma>\<^isub>o(Top(vars c)))", OF 3 2])
thus "lfp c (step UNIV) \<le> \<gamma>\<^isub>c C" by simp
qed
end
subsubsection "Monotonicity"
lemma le_sup_disj: "y \<in> L X \<Longrightarrow> (z::_::semilatticeL) \<in> L X \<Longrightarrow>
x \<le> y \<or> x \<le> z \<Longrightarrow> x \<le> y \<squnion> z"
by (metis sup_ge1 sup_ge2 order_trans)
theorem mono2_Step: fixes C1 :: "'a::semilatticeL acom"
assumes "!!x e S1 S2. \<lbrakk>S1 \<in> L X; S2 \<in> L X; x \<in> X; vars e \<subseteq> X; S1 \<le> S2\<rbrakk> \<Longrightarrow> f x e S1 \<le> f x e S2"
"!!b S1 S2. S1 \<in> L X \<Longrightarrow> S2 \<in> L X \<Longrightarrow> vars b \<subseteq> X \<Longrightarrow> S1 \<le> S2 \<Longrightarrow> g b S1 \<le> g b S2"
shows "S1 \<in> L X \<Longrightarrow> S2 \<in> L X \<Longrightarrow> C1 \<in> L X \<Longrightarrow> C2 \<in> L X \<Longrightarrow>
S1 \<le> S2 \<Longrightarrow> C1 \<le> C2 \<Longrightarrow> Step f g S1 C1 \<le> Step f g S2 C2"
by(induction C1 C2 arbitrary: S1 S2 rule: less_eq_acom.induct)
(auto simp: mono_post assms le_sup_disj le_sup_disj[OF post_in_L post_in_L])
locale Abs_Int_mono = Abs_Int +
assumes mono_plus': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow> plus' a1 a2 \<le> plus' b1 b2"
begin
lemma mono_aval':
"S1 \<le> S2 \<Longrightarrow> S1 \<in> L X \<Longrightarrow> S2 \<in> L X \<Longrightarrow> vars e \<subseteq> X \<Longrightarrow> aval' e S1 \<le> aval' e S2"
by(induction e) (auto simp: less_eq_st_def mono_plus' L_st_def)
theorem mono_step': "S1 \<in> L X \<Longrightarrow> S2 \<in> L X \<Longrightarrow> C1 \<in> L X \<Longrightarrow> C2 \<in> L X \<Longrightarrow>
S1 \<le> S2 \<Longrightarrow> C1 \<le> C2 \<Longrightarrow> step' S1 C1 \<le> step' S2 C2"
unfolding step'_def
by(rule mono2_Step) (auto simp: mono_aval' split: option.split)
lemma mono_step'_top: "C \<in> L X \<Longrightarrow> C' \<in> L X \<Longrightarrow>
C \<le> C' \<Longrightarrow> step' (Top X) C \<le> step' (Top X) C'"
by (metis Top_in_L mono_step' order_refl)
lemma pfp_bot_least:
assumes "\<forall>x\<in>L(vars c)\<inter>{C. strip C = c}.\<forall>y\<in>L(vars c)\<inter>{C. strip C = c}.
x \<le> y \<longrightarrow> f x \<le> f y"
and "\<forall>C. C \<in> L(vars c)\<inter>{C. strip C = c} \<longrightarrow> f C \<in> L(vars c)\<inter>{C. strip C = c}"
and "f C' \<le> C'" "strip C' = c" "C' \<in> L(vars c)" "pfp f (bot c) = Some C"
shows "C \<le> C'"
by(rule while_least[OF assms(1,2) _ _ assms(3) _ assms(6)[unfolded pfp_def]])
(simp_all add: assms(4,5) bot_least)
lemma AI_least_pfp: assumes "AI c = Some C"
and "step' (Top(vars c)) C' \<le> C'" "strip C' = c" "C' \<in> L(vars c)"
shows "C \<le> C'"
by(rule pfp_bot_least[OF _ _ assms(2-4) assms(1)[unfolded AI_def]])
(simp_all add: mono_step'_top)
end
subsubsection "Termination"
lemma pfp_termination:
fixes x0 :: "'a::order" and m :: "'a \<Rightarrow> nat"
assumes mono: "\<And>x y. I x \<Longrightarrow> I y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
and m: "\<And>x y. I x \<Longrightarrow> I y \<Longrightarrow> x < y \<Longrightarrow> m x > m y"
and I: "\<And>x y. I x \<Longrightarrow> I(f x)" and "I x0" and "x0 \<le> f x0"
shows "\<exists>x. pfp f x0 = Some x"
proof(simp add: pfp_def, rule wf_while_option_Some[where P = "%x. I x & x \<le> f x"])
show "wf {(y,x). ((I x \<and> x \<le> f x) \<and> \<not> f x \<le> x) \<and> y = f x}"
by(rule wf_subset[OF wf_measure[of m]]) (auto simp: m I)
next
show "I x0 \<and> x0 \<le> f x0" using `I x0` `x0 \<le> f x0` by blast
next
fix x assume "I x \<and> x \<le> f x" thus "I(f x) \<and> f x \<le> f(f x)"
by (blast intro: I mono)
qed
locale Measure1 =
fixes m :: "'av::order \<Rightarrow> nat"
fixes h :: "nat"
assumes h: "m x \<le> h"
begin
definition m_s :: "'av st \<Rightarrow> nat" ("m\<^isub>s") where
"m_s S = (\<Sum> x \<in> dom S. m(fun S x))"
lemma m_s_h: "x \<in> L X \<Longrightarrow> finite X \<Longrightarrow> m_s x \<le> h * card X"
by(simp add: L_st_def m_s_def)
(metis nat_mult_commute of_nat_id setsum_bounded[OF h])
definition m_o :: "nat \<Rightarrow> 'av st option \<Rightarrow> nat" ("m\<^isub>o") where
"m_o d opt = (case opt of None \<Rightarrow> h*d+1 | Some S \<Rightarrow> m_s S)"
lemma m_o_h: "ost \<in> L X \<Longrightarrow> finite X \<Longrightarrow> m_o (card X) ost \<le> (h*card X + 1)"
by(auto simp add: m_o_def m_s_h split: option.split dest!:m_s_h)
definition m_c :: "'av st option acom \<Rightarrow> nat" ("m\<^isub>c") where
"m_c C = (\<Sum>i<size(annos C). m_o (card(vars(strip C))) (annos C ! i))"
lemma m_c_h: assumes "C \<in> L(vars(strip C))"
shows "m_c C \<le> size(annos C) * (h * card(vars(strip C)) + 1)"
proof-
let ?X = "vars(strip C)" let ?n = "card ?X" let ?a = "size(annos C)"
{ fix i assume "i < ?a"
hence "annos C ! i \<in> L ?X" using assms by(simp add: L_acom_def)
note m_o_h[OF this finite_cvars]
} note 1 = this
have "m_c C = (\<Sum>i<?a. m_o ?n (annos C ! i))" by(simp add: m_c_def)
also have "\<dots> \<le> (\<Sum>i<?a. h * ?n + 1)"
apply(rule setsum_mono) using 1 by simp
also have "\<dots> = ?a * (h * ?n + 1)" by simp
finally show ?thesis .
qed
end
locale Measure = Measure1 +
assumes m2: "x < y \<Longrightarrow> m x > m y"
begin
lemma m1: "x \<le> y \<Longrightarrow> m x \<ge> m y"
by(auto simp: le_less m2)
lemma m_s2: "finite(dom S1) \<Longrightarrow> S1 < S2 \<Longrightarrow> m_s S1 > m_s S2"
proof(auto simp add: less_st_def less_eq_st_def m_s_def)
assume "finite(dom S2)" and 0: "\<forall>x\<in>dom S2. fun S1 x \<le> fun S2 x"
hence 1: "\<forall>x\<in>dom S2. m(fun S1 x) \<ge> m(fun S2 x)" by (metis m1)
fix x assume "x \<in> dom S2" "\<not> fun S2 x \<le> fun S1 x"
hence 2: "\<exists>x\<in>dom S2. m(fun S1 x) > m(fun S2 x)" by (metis 0 m2 less_le_not_le)
from setsum_strict_mono_ex1[OF `finite(dom S2)` 1 2]
show "(\<Sum>x\<in>dom S2. m (fun S2 x)) < (\<Sum>x\<in>dom S2. m (fun S1 x))" .
qed
lemma m_o2: "finite X \<Longrightarrow> o1 \<in> L X \<Longrightarrow> o2 \<in> L X \<Longrightarrow>
o1 < o2 \<Longrightarrow> m_o (card X) o1 > m_o (card X) o2"
proof(induction o1 o2 rule: less_eq_option.induct)
case 1 thus ?case by (auto simp: m_o_def L_st_def m_s2 less_option_def)
next
case 2 thus ?case by(auto simp: m_o_def less_option_def le_imp_less_Suc m_s_h)
next
case 3 thus ?case by (auto simp: less_option_def)
qed
lemma m_o1: "finite X \<Longrightarrow> o1 \<in> L X \<Longrightarrow> o2 \<in> L X \<Longrightarrow>
o1 \<le> o2 \<Longrightarrow> m_o (card X) o1 \<ge> m_o (card X) o2"
by(auto simp: le_less m_o2)
lemma m_c2: "C1 \<in> L(vars(strip C1)) \<Longrightarrow> C2 \<in> L(vars(strip C2)) \<Longrightarrow>
C1 < C2 \<Longrightarrow> m_c C1 > m_c C2"
proof(auto simp add: le_iff_le_annos m_c_def size_annos_same[of C1 C2] less_acom_def L_acom_def)
let ?X = "vars(strip C2)"
let ?n = "card ?X"
assume V1: "\<forall>a\<in>set(annos C1). a \<in> L ?X"
and V2: "\<forall>a\<in>set(annos C2). a \<in> L ?X"
and strip_eq: "strip C1 = strip C2"
and 0: "\<forall>i<size(annos C2). annos C1 ! i \<le> annos C2 ! i"
hence 1: "\<forall>i<size(annos C2). m_o ?n (annos C1 ! i) \<ge> m_o ?n (annos C2 ! i)"
by (auto simp: all_set_conv_all_nth)
(metis finite_cvars m_o1 size_annos_same2)
fix i assume "i < size(annos C2)" "\<not> annos C2 ! i \<le> annos C1 ! i"
hence "m_o ?n (annos C1 ! i) > m_o ?n (annos C2 ! i)" (is "?P i")
by(metis m_o2[OF finite_cvars] V1 V2 nth_mem size_annos_same[OF strip_eq] 0 less_option_def)
hence 2: "\<exists>i < size(annos C2). ?P i" using `i < size(annos C2)` by blast
show "(\<Sum>i<size(annos C2). m_o ?n (annos C2 ! i))
< (\<Sum>i<size(annos C2). m_o ?n (annos C1 ! i))"
apply(rule setsum_strict_mono_ex1) using 1 2 by (auto)
qed
end
locale Abs_Int_measure =
Abs_Int_mono where \<gamma>=\<gamma> + Measure where m=m
for \<gamma> :: "'av::semilattice \<Rightarrow> val set" and m :: "'av \<Rightarrow> nat"
begin
lemma AI_Some_measure: "\<exists>C. AI c = Some C"
unfolding AI_def
apply(rule pfp_termination[where I = "\<lambda>C. strip C = c \<and> C \<in> L(vars c)" and m="m_c"])
apply(simp_all add: m_c2 mono_step'_top bot_least)
done
end
end