src/HOL/IMP/Abs_Int1.thy
author nipkow
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(* Author: Tobias Nipkow *)
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theory Abs_Int1
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imports Abs_Int0 Vars
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begin
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instantiation prod :: (preord,preord) preord
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begin
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definition "le_prod p1 p2 = (fst p1 \<sqsubseteq> fst p2 \<and> snd p1 \<sqsubseteq> snd p2)"
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instance
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proof
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  case goal1 show ?case by(simp add: le_prod_def)
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next
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  case goal2 thus ?case unfolding le_prod_def by(metis le_trans)
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qed
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end
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instantiation com :: vars
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begin
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fun vars_com :: "com \<Rightarrow> vname set" where
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"vars com.SKIP = {}" |
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"vars (x::=e) = {x} \<union> vars e" |
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"vars (c1;c2) = vars c1 \<union> vars c2" |
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"vars (IF b THEN c1 ELSE c2) = vars b \<union> vars c1 \<union> vars c2" |
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"vars (WHILE b DO c) = vars b \<union> vars c"
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instance ..
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end
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lemma finite_avars: "finite(vars(a::aexp))"
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by(induction a) simp_all
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lemma finite_bvars: "finite(vars(b::bexp))"
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by(induction b) (simp_all add: finite_avars)
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lemma finite_cvars: "finite(vars(c::com))"
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by(induction c) (simp_all add: finite_avars finite_bvars)
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subsection "Backward Analysis of Expressions"
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hide_const bot
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class L_top_bot = SL_top +
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fixes meet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 65)
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and bot :: "'a" ("\<bottom>")
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assumes meet_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"
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and meet_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y"
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and meet_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
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assumes bot[simp]: "\<bottom> \<sqsubseteq> x"
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begin
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lemma mono_meet: "x \<sqsubseteq> x' \<Longrightarrow> y \<sqsubseteq> y' \<Longrightarrow> x \<sqinter> y \<sqsubseteq> x' \<sqinter> y'"
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by (metis meet_greatest meet_le1 meet_le2 le_trans)
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end
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locale Val_abs1_gamma =
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  Gamma where \<gamma> = \<gamma> for \<gamma> :: "'av::L_top_bot \<Rightarrow> val set" +
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assumes inter_gamma_subset_gamma_meet:
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  "\<gamma> a1 \<inter> \<gamma> a2 \<subseteq> \<gamma>(a1 \<sqinter> a2)"
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and gamma_Bot[simp]: "\<gamma> \<bottom> = {}"
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begin
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lemma in_gamma_meet: "x : \<gamma> a1 \<Longrightarrow> x : \<gamma> a2 \<Longrightarrow> x : \<gamma>(a1 \<sqinter> a2)"
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by (metis IntI inter_gamma_subset_gamma_meet set_mp)
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lemma gamma_meet[simp]: "\<gamma>(a1 \<sqinter> a2) = \<gamma> a1 \<inter> \<gamma> a2"
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by (metis equalityI inter_gamma_subset_gamma_meet le_inf_iff mono_gamma meet_le1 meet_le2)
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end
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locale Val_abs1 =
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  Val_abs1_gamma where \<gamma> = \<gamma>
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   for \<gamma> :: "'av::L_top_bot \<Rightarrow> val set" +
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fixes test_num' :: "val \<Rightarrow> 'av \<Rightarrow> bool"
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and filter_plus' :: "'av \<Rightarrow> 'av \<Rightarrow> 'av \<Rightarrow> 'av * 'av"
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and filter_less' :: "bool \<Rightarrow> 'av \<Rightarrow> 'av \<Rightarrow> 'av * 'av"
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assumes test_num': "test_num' n a = (n : \<gamma> a)"
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and filter_plus': "filter_plus' a a1 a2 = (b1,b2) \<Longrightarrow>
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  n1 : \<gamma> a1 \<Longrightarrow> n2 : \<gamma> a2 \<Longrightarrow> n1+n2 : \<gamma> a \<Longrightarrow> n1 : \<gamma> b1 \<and> n2 : \<gamma> b2"
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and filter_less': "filter_less' (n1<n2) a1 a2 = (b1,b2) \<Longrightarrow>
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  n1 : \<gamma> a1 \<Longrightarrow> n2 : \<gamma> a2 \<Longrightarrow> n1 : \<gamma> b1 \<and> n2 : \<gamma> b2"
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locale Abs_Int1 =
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  Val_abs1 where \<gamma> = \<gamma> for \<gamma> :: "'av::L_top_bot \<Rightarrow> val set"
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begin
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lemma in_gamma_join_UpI: "s : \<gamma>\<^isub>o S1 \<or> s : \<gamma>\<^isub>o S2 \<Longrightarrow> s : \<gamma>\<^isub>o(S1 \<squnion> S2)"
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by (metis (no_types) join_ge1 join_ge2 mono_gamma_o set_rev_mp)
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fun aval'' :: "aexp \<Rightarrow> 'av st option \<Rightarrow> 'av" where
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"aval'' e None = \<bottom>" |
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"aval'' e (Some sa) = aval' e sa"
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lemma aval''_sound: "s : \<gamma>\<^isub>o S \<Longrightarrow> aval a s : \<gamma>(aval'' a S)"
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by(cases S)(simp add: aval'_sound)+
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subsubsection "Backward analysis"
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fun afilter :: "aexp \<Rightarrow> 'av \<Rightarrow> 'av st option \<Rightarrow> 'av st option" where
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"afilter (N n) a S = (if test_num' n a then S else None)" |
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"afilter (V x) a S = (case S of None \<Rightarrow> None | Some S \<Rightarrow>
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  let a' = lookup S x \<sqinter> a in
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  if a' \<sqsubseteq> \<bottom> then None else Some(update S x a'))" |
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"afilter (Plus e1 e2) a S =
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 (let (a1,a2) = filter_plus' a (aval'' e1 S) (aval'' e2 S)
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  in afilter e1 a1 (afilter e2 a2 S))"
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text{* The test for @{const bot} in the @{const V}-case is important: @{const
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bot} indicates that a variable has no possible values, i.e.\ that the current
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program point is unreachable. But then the abstract state should collapse to
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@{const None}. Put differently, we maintain the invariant that in an abstract
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state of the form @{term"Some s"}, all variables are mapped to non-@{const
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bot} values. Otherwise the (pointwise) join of two abstract states, one of
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which contains @{const bot} values, may produce too large a result, thus
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making the analysis less precise. *}
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fun bfilter :: "bexp \<Rightarrow> bool \<Rightarrow> 'av st option \<Rightarrow> 'av st option" where
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"bfilter (Bc v) res S = (if v=res then S else None)" |
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"bfilter (Not b) res S = bfilter b (\<not> res) S" |
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"bfilter (And b1 b2) res S =
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  (if res then bfilter b1 True (bfilter b2 True S)
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   else bfilter b1 False S \<squnion> bfilter b2 False S)" |
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"bfilter (Less e1 e2) res S =
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  (let (res1,res2) = filter_less' res (aval'' e1 S) (aval'' e2 S)
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   in afilter e1 res1 (afilter e2 res2 S))"
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lemma afilter_sound: "s : \<gamma>\<^isub>o S \<Longrightarrow> aval e s : \<gamma> a \<Longrightarrow> s : \<gamma>\<^isub>o (afilter e a S)"
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proof(induction e arbitrary: a S)
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  case N thus ?case by simp (metis test_num')
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next
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  case (V x)
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  obtain S' where "S = Some S'" and "s : \<gamma>\<^isub>f S'" using `s : \<gamma>\<^isub>o S`
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    by(auto simp: in_gamma_option_iff)
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  moreover hence "s x : \<gamma> (lookup S' x)" by(simp add: \<gamma>_st_def)
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  moreover have "s x : \<gamma> a" using V by simp
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  ultimately show ?case using V(1)
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    by(simp add: lookup_update Let_def \<gamma>_st_def)
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      (metis mono_gamma emptyE in_gamma_meet gamma_Bot subset_empty)
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next
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  case (Plus e1 e2) thus ?case
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    using filter_plus'[OF _ aval''_sound[OF Plus(3)] aval''_sound[OF Plus(3)]]
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    by (auto split: prod.split)
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qed
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lemma bfilter_sound: "s : \<gamma>\<^isub>o S \<Longrightarrow> bv = bval b s \<Longrightarrow> s : \<gamma>\<^isub>o(bfilter b bv S)"
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proof(induction b arbitrary: S bv)
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  case Bc thus ?case by simp
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next
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  case (Not b) thus ?case by simp
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next
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  case (And b1 b2) thus ?case by(fastforce simp: in_gamma_join_UpI)
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next
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  case (Less e1 e2) thus ?case
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    by (auto split: prod.split)
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       (metis afilter_sound filter_less' aval''_sound Less)
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qed
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fun step' :: "'av st option \<Rightarrow> 'av st option acom \<Rightarrow> 'av st option acom"
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 where
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"step' S (SKIP {P}) = (SKIP {S})" |
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"step' S (x ::= e {P}) =
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  x ::= e {case S of None \<Rightarrow> None | Some S \<Rightarrow> Some(update S x (aval' e S))}" |
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"step' S (c1; c2) = step' S c1; step' (post c1) c2" |
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"step' S (IF b THEN c1 ELSE c2 {P}) =
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  (let c1' = step' (bfilter b True S) c1; c2' = step' (bfilter b False S) c2
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   in IF b THEN c1' ELSE c2' {post c1 \<squnion> post c2})" |
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"step' S ({Inv} WHILE b DO c {P}) =
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   {S \<squnion> post c}
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   WHILE b DO step' (bfilter b True Inv) c
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   {bfilter b False Inv}"
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definition AI :: "com \<Rightarrow> 'av st option acom option" where
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"AI = lpfp\<^isub>c (step' \<top>)"
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lemma strip_step'[simp]: "strip(step' S c) = strip c"
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by(induct c arbitrary: S) (simp_all add: Let_def)
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subsubsection "Soundness"
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lemma in_gamma_update:
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  "\<lbrakk> s : \<gamma>\<^isub>f S; i : \<gamma> a \<rbrakk> \<Longrightarrow> s(x := i) : \<gamma>\<^isub>f(update S x a)"
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by(simp add: \<gamma>_st_def lookup_update)
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lemma step_preserves_le:
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  "\<lbrakk> S \<subseteq> \<gamma>\<^isub>o S'; cs \<le> \<gamma>\<^isub>c ca \<rbrakk> \<Longrightarrow> step S cs \<le> \<gamma>\<^isub>c (step' S' ca)"
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proof(induction cs arbitrary: ca S S')
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  case SKIP thus ?case by(auto simp:SKIP_le map_acom_SKIP)
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next
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  case Assign thus ?case
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    by (fastforce simp: Assign_le  map_acom_Assign intro: aval'_sound in_gamma_update
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      split: option.splits del:subsetD)
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next
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  case Semi thus ?case apply (auto simp: Semi_le map_acom_Semi)
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    by (metis le_post post_map_acom)
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next
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  case (If b cs1 cs2 P)
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  then obtain ca1 ca2 Pa where
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      "ca= IF b THEN ca1 ELSE ca2 {Pa}"
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   211
      "P \<subseteq> \<gamma>\<^isub>o Pa" "cs1 \<le> \<gamma>\<^isub>c ca1" "cs2 \<le> \<gamma>\<^isub>c ca2"
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   212
    by (fastforce simp: If_le map_acom_If)
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   213
  moreover have "post cs1 \<subseteq> \<gamma>\<^isub>o(post ca1 \<squnion> post ca2)"
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    by (metis (no_types) `cs1 \<le> \<gamma>\<^isub>c ca1` join_ge1 le_post mono_gamma_o order_trans post_map_acom)
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  moreover have "post cs2 \<subseteq> \<gamma>\<^isub>o(post ca1 \<squnion> post ca2)"
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   216
    by (metis (no_types) `cs2 \<le> \<gamma>\<^isub>c ca2` join_ge2 le_post mono_gamma_o order_trans post_map_acom)
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   217
  ultimately show ?case using `S \<subseteq> \<gamma>\<^isub>o S'`
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    by (simp add: If.IH subset_iff bfilter_sound)
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next
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   220
  case (While I b cs1 P)
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  then obtain ca1 Ia Pa where
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   222
    "ca = {Ia} WHILE b DO ca1 {Pa}"
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   223
    "I \<subseteq> \<gamma>\<^isub>o Ia" "P \<subseteq> \<gamma>\<^isub>o Pa" "cs1 \<le> \<gamma>\<^isub>c ca1"
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   224
    by (fastforce simp: map_acom_While While_le)
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  moreover have "S \<union> post cs1 \<subseteq> \<gamma>\<^isub>o (S' \<squnion> post ca1)"
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    using `S \<subseteq> \<gamma>\<^isub>o S'` le_post[OF `cs1 \<le> \<gamma>\<^isub>c ca1`, simplified]
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    by (metis (no_types) join_ge1 join_ge2 le_sup_iff mono_gamma_o order_trans)
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  ultimately show ?case by (simp add: While.IH subset_iff bfilter_sound)
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qed
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   231
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: "lpfp\<^isub>c (step' \<top>) c = Some c'"
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   234
  have 2: "step' \<top> c' \<sqsubseteq> c'" by(rule lpfpc_pfp[OF 1])
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   235
  have 3: "strip (\<gamma>\<^isub>c (step' \<top> c')) = c"
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    by(simp add: strip_lpfpc[OF _ 1])
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   237
  have "lfp (step UNIV) c \<le> \<gamma>\<^isub>c (step' \<top> c')"
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   238
  proof(rule lfp_lowerbound[simplified,OF 3])
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   239
    show "step UNIV (\<gamma>\<^isub>c (step' \<top> c')) \<le> \<gamma>\<^isub>c (step' \<top> c')"
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   240
    proof(rule step_preserves_le[OF _ _])
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   241
      show "UNIV \<subseteq> \<gamma>\<^isub>o \<top>" by simp
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   242
      show "\<gamma>\<^isub>c (step' \<top> c') \<le> \<gamma>\<^isub>c c'" by(rule mono_gamma_c[OF 2])
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   243
    qed
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   244
  qed
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   245
  from this 2 show "lfp (step UNIV) c \<le> \<gamma>\<^isub>c c'"
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   246
    by (blast intro: mono_gamma_c order_trans)
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   247
qed
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   248
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   249
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   250
subsubsection "Commands over a set of variables"
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text{* Key invariant: the domains of all abstract states are subsets of the
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set of variables of the program. *}
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   254
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   255
definition "domo S = (case S of None \<Rightarrow> {} | Some S' \<Rightarrow> set(dom S'))"
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   257
definition Com :: "vname set \<Rightarrow> 'a st option acom set" where
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   258
"Com X = {c. \<forall>S \<in> set(annos c). domo S \<subseteq> X}"
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   259
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   260
lemma domo_Top[simp]: "domo \<top> = {}"
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   261
by(simp add: domo_def Top_st_def Top_option_def)
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   262
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   263
lemma bot_acom_Com[simp]: "\<bottom>\<^sub>c c \<in> Com X"
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   264
by(simp add: bot_acom_def Com_def domo_def set_annos_anno)
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   265
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   266
lemma post_in_annos: "post c : set(annos c)"
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   267
by(induction c) simp_all
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   268
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   269
lemma domo_join: "domo (S \<squnion> T) \<subseteq> domo S \<union> domo T"
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   270
by(auto simp: domo_def join_st_def split: option.split)
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   271
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   272
lemma domo_afilter: "vars a \<subseteq> X \<Longrightarrow> domo S \<subseteq> X \<Longrightarrow> domo(afilter a i S) \<subseteq> X"
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   273
apply(induction a arbitrary: i S)
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   274
apply(simp add: domo_def)
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   275
apply(simp add: domo_def Let_def update_def lookup_def split: option.splits)
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   276
apply blast
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   277
apply(simp split: prod.split)
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   278
done
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   279
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   280
lemma domo_bfilter: "vars b \<subseteq> X \<Longrightarrow> domo S \<subseteq> X \<Longrightarrow> domo(bfilter b bv S) \<subseteq> X"
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   281
apply(induction b arbitrary: bv S)
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   282
apply(simp add: domo_def)
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   283
apply(simp)
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   284
apply(simp)
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   285
apply rule
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   286
apply (metis le_sup_iff subset_trans[OF domo_join])
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   287
apply(simp split: prod.split)
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   288
by (metis domo_afilter)
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   289
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   290
lemma step'_Com:
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   291
  "domo S \<subseteq> X \<Longrightarrow> vars(strip c) \<subseteq> X \<Longrightarrow> c : Com X \<Longrightarrow> step' S c : Com X"
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   292
apply(induction c arbitrary: S)
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   293
apply(simp add: Com_def)
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   294
apply(simp add: Com_def domo_def update_def split: option.splits)
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   295
apply(simp (no_asm_use) add: Com_def ball_Un)
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   296
apply (metis post_in_annos)
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   297
apply(simp (no_asm_use) add: Com_def ball_Un)
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   298
apply rule
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   299
apply (metis Un_assoc domo_join order_trans post_in_annos subset_Un_eq)
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   300
apply (metis domo_bfilter)
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apply(simp (no_asm_use) add: Com_def)
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   302
apply rule
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   303
apply (metis domo_join le_sup_iff post_in_annos subset_trans)
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   304
apply rule
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   305
apply (metis domo_bfilter)
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   306
by (metis domo_bfilter)
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   307
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   308
end
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   309
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   310
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   311
subsubsection "Monotonicity"
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locale Abs_Int1_mono = Abs_Int1 +
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assumes mono_plus': "a1 \<sqsubseteq> b1 \<Longrightarrow> a2 \<sqsubseteq> b2 \<Longrightarrow> plus' a1 a2 \<sqsubseteq> plus' b1 b2"
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and mono_filter_plus': "a1 \<sqsubseteq> b1 \<Longrightarrow> a2 \<sqsubseteq> b2 \<Longrightarrow> r \<sqsubseteq> r' \<Longrightarrow>
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  filter_plus' r a1 a2 \<sqsubseteq> filter_plus' r' b1 b2"
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and mono_filter_less': "a1 \<sqsubseteq> b1 \<Longrightarrow> a2 \<sqsubseteq> b2 \<Longrightarrow>
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  filter_less' bv a1 a2 \<sqsubseteq> filter_less' bv b1 b2"
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begin
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lemma mono_aval': "S \<sqsubseteq> S' \<Longrightarrow> aval' e S \<sqsubseteq> aval' e S'"
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by(induction e) (auto simp: le_st_def lookup_def mono_plus')
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lemma mono_aval'': "S \<sqsubseteq> S' \<Longrightarrow> aval'' e S \<sqsubseteq> aval'' e S'"
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apply(cases S)
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 apply simp
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apply(cases S')
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 apply simp
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by (simp add: mono_aval')
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lemma mono_afilter: "r \<sqsubseteq> r' \<Longrightarrow> S \<sqsubseteq> S' \<Longrightarrow> afilter e r S \<sqsubseteq> afilter e r' S'"
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apply(induction e arbitrary: r r' S S')
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apply(auto simp: test_num' Let_def split: option.splits prod.splits)
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apply (metis mono_gamma subsetD)
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apply(drule_tac x = "list" in mono_lookup)
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apply (metis mono_meet le_trans)
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apply (metis mono_meet mono_lookup mono_update)
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apply(metis mono_aval'' mono_filter_plus'[simplified le_prod_def] fst_conv snd_conv)
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done
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lemma mono_bfilter: "S \<sqsubseteq> S' \<Longrightarrow> bfilter b r S \<sqsubseteq> bfilter b r S'"
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apply(induction b arbitrary: r S S')
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apply(auto simp: le_trans[OF _ join_ge1] le_trans[OF _ join_ge2] split: prod.splits)
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apply(metis mono_aval'' mono_afilter mono_filter_less'[simplified le_prod_def] fst_conv snd_conv)
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done
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lemma mono_step': "S \<sqsubseteq> S' \<Longrightarrow> c \<sqsubseteq> c' \<Longrightarrow> step' S c \<sqsubseteq> step' S' c'"
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apply(induction c c' arbitrary: S S' rule: le_acom.induct)
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apply (auto simp: mono_post mono_bfilter mono_update mono_aval' Let_def le_join_disj
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  split: option.split)
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done
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lemma mono_step'2: "mono (step' S)"
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by(simp add: mono_def mono_step'[OF le_refl])
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end
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end