(* Author: Tobias Nipkow *)
theory Abs_Int0
imports Abs_State
begin
subsection "Computable Abstract Interpretation"
text{* Abstract interpretation over type @{text st} instead of
functions. *}
context Val_abs
begin
fun aval' :: "aexp \<Rightarrow> 'a st \<Rightarrow> 'a" where
"aval' (N n) _ = num' n" |
"aval' (V x) S = lookup S x" |
"aval' (Plus a1 a2) S = plus' (aval' a1 S) (aval' a2 S)"
lemma aval'_sound: "s : \<gamma>\<^isub>f S \<Longrightarrow> aval a s : \<gamma>(aval' a S)"
by (induct a) (auto simp: rep_num' rep_plus' rep_st_def lookup_def)
end
locale Abs_Int = Val_abs
begin
fun step' :: "'a st option \<Rightarrow> 'a st option acom \<Rightarrow> 'a st option acom" where
"step' S (SKIP {P}) = (SKIP {S})" |
"step' S (x ::= e {P}) =
x ::= e {case S of None \<Rightarrow> None | Some S \<Rightarrow> Some(update S x (aval' e S))}" |
"step' S (c1; c2) = step' S c1; step' (post c1) c2" |
"step' S (IF b THEN c1 ELSE c2 {P}) =
(let c1' = step' S c1; c2' = step' S c2
in IF b THEN c1' ELSE c2' {post c1 \<squnion> post c2})" |
"step' S ({Inv} WHILE b DO c {P}) =
{S \<squnion> post c} WHILE b DO step' Inv c {Inv}"
definition AI :: "com \<Rightarrow> 'a st option acom option" where
"AI = lpfp\<^isub>c (step' \<top>)"
lemma strip_step'[simp]: "strip(step' S c) = strip c"
by(induct c arbitrary: S) (simp_all add: Let_def)
text{* Soundness: *}
lemma in_rep_update:
"\<lbrakk> s : \<gamma>\<^isub>f S; i : \<gamma> a \<rbrakk> \<Longrightarrow> s(x := i) : \<gamma>\<^isub>f(update S x a)"
by(simp add: rep_st_def lookup_update)
text{* The soundness proofs are textually identical to the ones for the step
function operating on states as functions. *}
lemma step_preserves_le2:
"\<lbrakk> S \<subseteq> \<gamma>\<^isub>u sa; cs \<le> \<gamma>\<^isub>c ca; strip cs = c; strip ca = c \<rbrakk>
\<Longrightarrow> step S cs \<le> \<gamma>\<^isub>c (step' sa ca)"
proof(induction c arbitrary: cs ca S sa)
case SKIP thus ?case
by(auto simp:strip_eq_SKIP)
next
case Assign thus ?case
by (fastforce simp: strip_eq_Assign intro: aval'_sound in_rep_update
split: option.splits del:subsetD)
next
case Semi thus ?case apply (auto simp: strip_eq_Semi)
by (metis le_post post_map_acom)
next
case (If b c1 c2)
then obtain cs1 cs2 ca1 ca2 P Pa where
"cs = IF b THEN cs1 ELSE cs2 {P}" "ca = IF b THEN ca1 ELSE ca2 {Pa}"
"P \<subseteq> \<gamma>\<^isub>u Pa" "cs1 \<le> \<gamma>\<^isub>c ca1" "cs2 \<le> \<gamma>\<^isub>c ca2"
"strip cs1 = c1" "strip ca1 = c1" "strip cs2 = c2" "strip ca2 = c2"
by (fastforce simp: strip_eq_If)
moreover have "post cs1 \<subseteq> \<gamma>\<^isub>u(post ca1 \<squnion> post ca2)"
by (metis (no_types) `cs1 \<le> \<gamma>\<^isub>c ca1` join_ge1 le_post mono_rep_u order_trans post_map_acom)
moreover have "post cs2 \<subseteq> \<gamma>\<^isub>u(post ca1 \<squnion> post ca2)"
by (metis (no_types) `cs2 \<le> \<gamma>\<^isub>c ca2` join_ge2 le_post mono_rep_u order_trans post_map_acom)
ultimately show ?case using If.prems(1) by (simp add: If.IH subset_iff)
next
case (While b c1)
then obtain cs1 ca1 I P Ia Pa where
"cs = {I} WHILE b DO cs1 {P}" "ca = {Ia} WHILE b DO ca1 {Pa}"
"I \<subseteq> \<gamma>\<^isub>u Ia" "P \<subseteq> \<gamma>\<^isub>u Pa" "cs1 \<le> \<gamma>\<^isub>c ca1"
"strip cs1 = c1" "strip ca1 = c1"
by (fastforce simp: strip_eq_While)
moreover have "S \<union> post cs1 \<subseteq> \<gamma>\<^isub>u (sa \<squnion> post ca1)"
using `S \<subseteq> \<gamma>\<^isub>u sa` le_post[OF `cs1 \<le> \<gamma>\<^isub>c ca1`, simplified]
by (metis (no_types) join_ge1 join_ge2 le_sup_iff mono_rep_u order_trans)
ultimately show ?case by (simp add: While.IH subset_iff)
qed
lemma step_preserves_le:
"\<lbrakk> S \<subseteq> \<gamma>\<^isub>u sa; cs \<le> \<gamma>\<^isub>c ca; strip cs = c \<rbrakk>
\<Longrightarrow> step S cs \<le> \<gamma>\<^isub>c(step' sa ca)"
by (metis le_strip step_preserves_le2 strip_acom)
lemma AI_sound: "AI c = Some c' \<Longrightarrow> CS UNIV c \<le> \<gamma>\<^isub>c c'"
proof(simp add: CS_def AI_def)
assume 1: "lpfp\<^isub>c (step' \<top>) c = Some c'"
have 2: "step' \<top> c' \<sqsubseteq> c'" by(rule lpfpc_pfp[OF 1])
have 3: "strip (\<gamma>\<^isub>c (step' \<top> c')) = c"
by(simp add: strip_lpfpc[OF _ 1])
have "lfp c (step UNIV) \<le> \<gamma>\<^isub>c (step' \<top> c')"
proof(rule lfp_lowerbound[simplified,OF 3])
show "step UNIV (\<gamma>\<^isub>c (step' \<top> c')) \<le> \<gamma>\<^isub>c (step' \<top> c')"
proof(rule step_preserves_le[OF _ _ 3])
show "UNIV \<subseteq> \<gamma>\<^isub>u \<top>" by simp
show "\<gamma>\<^isub>c (step' \<top> c') \<le> \<gamma>\<^isub>c c'" by(rule mono_rep_c[OF 2])
qed
qed
from this 2 show "lfp c (step UNIV) \<le> \<gamma>\<^isub>c c'"
by (blast intro: mono_rep_c order_trans)
qed
end
subsubsection "Monotonicity"
locale Abs_Int_mono = Abs_Int +
assumes mono_plus': "a1 \<sqsubseteq> b1 \<Longrightarrow> a2 \<sqsubseteq> b2 \<Longrightarrow> plus' a1 a2 \<sqsubseteq> plus' b1 b2"
begin
lemma mono_aval': "S \<sqsubseteq> S' \<Longrightarrow> aval' e S \<sqsubseteq> aval' e S'"
by(induction e) (auto simp: le_st_def lookup_def mono_plus')
lemma mono_update: "a \<sqsubseteq> a' \<Longrightarrow> S \<sqsubseteq> S' \<Longrightarrow> update S x a \<sqsubseteq> update S' x a'"
by(auto simp add: le_st_def lookup_def update_def)
lemma step'_mono: "S \<sqsubseteq> S' \<Longrightarrow> step' S c \<sqsubseteq> step' S' c"
apply(induction c arbitrary: S S')
apply (auto simp: Let_def mono_update mono_aval' le_join_disj split: option.split)
done
end
end