author | nipkow |
Fri, 21 Sep 2012 13:56:57 +0200 | |
changeset 49497 | 860b7c6bd913 |
parent 49464 | 4e4bdd12280f |
child 50986 | c54ea7f5418f |
permissions | -rw-r--r-- |
47613 | 1 |
(* Author: Tobias Nipkow *) |
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theory Abs_Int2 |
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imports Abs_Int1 |
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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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subsection "Backward Analysis of Expressions" |
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class lattice = semilattice + bot + |
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fixes meet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 65) |
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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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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::lattice \<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::lattice \<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::lattice \<Rightarrow> val set" |
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begin |
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lemma in_gamma_join_UpI: |
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"S1 \<in> L X \<Longrightarrow> S2 \<in> L X \<Longrightarrow> 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 (hide_lams, no_types) semilatticeL_class.join_ge1 semilatticeL_class.join_ge2 mono_gamma_o subsetD) |
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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> S \<in> L X \<Longrightarrow> vars a \<subseteq> X \<Longrightarrow> aval a s : \<gamma>(aval'' a S)" |
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by(simp add: L_option_def L_st_def aval'_sound split: option.splits) |
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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' = fun 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_in_L: "S \<in> L X \<Longrightarrow> vars e \<subseteq> X \<Longrightarrow> afilter e a S \<in> L X" |
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by(induction e arbitrary: a S) |
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(auto simp: Let_def update_def L_st_def |
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split: option.splits prod.split) |
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49396 | 116 |
lemma afilter_sound: "S \<in> L X \<Longrightarrow> vars e \<subseteq> X \<Longrightarrow> |
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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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49497 | 122 |
obtain S' where "S = Some S'" and "s : \<gamma>\<^isub>s S'" using `s : \<gamma>\<^isub>o S` |
47613 | 123 |
by(auto simp: in_gamma_option_iff) |
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moreover hence "s x : \<gamma> (fun S' x)" |
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49396 | 125 |
using V(1,2) by(simp add: \<gamma>_st_def L_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(3) |
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by(simp add: Let_def \<gamma>_st_def) |
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49396 | 129 |
(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.prems(3)] aval''_sound[OF Plus.prems(3)]] |
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49396 | 133 |
by (auto simp: afilter_in_L split: prod.split) |
47613 | 134 |
qed |
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49396 | 136 |
lemma bfilter_in_L: "S \<in> L X \<Longrightarrow> vars b \<subseteq> X \<Longrightarrow> bfilter b bv S \<in> L X" |
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by(induction b arbitrary: bv S)(auto simp: afilter_in_L split: prod.split) |
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47613 | 138 |
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49396 | 139 |
lemma bfilter_sound: "S \<in> L X \<Longrightarrow> vars b \<subseteq> X \<Longrightarrow> |
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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 |
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49396 | 147 |
by simp (metis And(1) And(2) bfilter_in_L 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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49396 | 151 |
(metis (lifting) afilter_in_L afilter_sound aval''_sound filter_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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49344 | 161 |
"step' S (IF b THEN {P1} C1 ELSE {P2} C2 {Q}) = |
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(let P1' = bfilter b True S; C1' = step' P1 C1; P2' = bfilter b False S; C2' = step' P2 C2 |
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in IF b THEN {P1'} C1' ELSE {P2'} C2' {post C1 \<squnion> post C2})" | |
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"step' S ({I} WHILE b DO {p} C {Q}) = |
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47613 | 165 |
{S \<squnion> post C} |
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WHILE b DO {bfilter b True I} step' p C |
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{bfilter b False I}" |
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47613 | 168 |
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definition AI :: "com \<Rightarrow> 'av st option acom option" where |
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49464 | 170 |
"AI c = pfp (step' \<top>\<^bsub>c\<^esub>) (bot c)" |
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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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49497 | 179 |
"\<lbrakk> s : \<gamma>\<^isub>s S; i : \<gamma> a \<rbrakk> \<Longrightarrow> s(x := i) : \<gamma>\<^isub>s(update S x a)" |
47613 | 180 |
by(simp add: \<gamma>_st_def) |
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theorem step_preserves_le: |
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49396 | 183 |
"\<lbrakk> S \<subseteq> \<gamma>\<^isub>o S'; C \<le> \<gamma>\<^isub>c C'; C' \<in> L X; S' \<in> L X \<rbrakk> \<Longrightarrow> step S C \<le> \<gamma>\<^isub>c (step' S' C')" |
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proof(induction C arbitrary: C' 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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49396 | 188 |
by (fastforce simp: Assign_le map_acom_Assign L_option_def L_st_def |
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intro: aval'_sound in_gamma_update split: option.splits del:subsetD) |
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next |
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47818 | 191 |
case Seq thus ?case apply (auto simp: Seq_le map_acom_Seq) |
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by (metis le_post post_map_acom post_in_L) |
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next |
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case (If b P1 C1 P2 C2 Q) |
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then obtain P1' C1' P2' C2' Q' where |
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"C' = IF b THEN {P1'} C1' ELSE {P2'} C2' {Q'}" |
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"P1 \<subseteq> \<gamma>\<^isub>o P1'" "P2 \<subseteq> \<gamma>\<^isub>o P2'" "C1 \<le> \<gamma>\<^isub>c C1'" "C2 \<le> \<gamma>\<^isub>c C2'" "Q \<subseteq> \<gamma>\<^isub>o Q'" |
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by (fastforce simp: If_le map_acom_If) |
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moreover from this(1) `C' \<in> L X` |
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have L: "C1' \<in> L X" "C2' \<in> L X" "P1' \<in> L X" "P2' \<in> L X" and "vars b \<subseteq> X" |
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by simp_all |
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47613 | 202 |
moreover have "post C1 \<subseteq> \<gamma>\<^isub>o(post C1' \<squnion> post C2')" |
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by (metis (no_types) `C1 \<le> \<gamma>\<^isub>c C1'` join_ge1 le_post mono_gamma_o order_trans post_map_acom post_in_L L) |
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moreover have "post C2 \<subseteq> \<gamma>\<^isub>o(post C1' \<squnion> post C2')" |
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by (metis (no_types) `C2 \<le> \<gamma>\<^isub>c C2'` join_ge2 le_post mono_gamma_o order_trans post_map_acom post_in_L L) |
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moreover note vars = `S' \<in> L X` `vars b \<subseteq> X` |
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47613 | 207 |
ultimately show ?case using `S \<subseteq> \<gamma>\<^isub>o S'` |
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by (simp add: If.IH subset_iff bfilter_sound[OF vars] bfilter_in_L[OF vars]) |
47613 | 209 |
next |
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case (While I b P C1 Q) |
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then obtain C1' I' P' Q' where |
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"C' = {I'} WHILE b DO {P'} C1' {Q'}" |
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"I \<subseteq> \<gamma>\<^isub>o I'" "P \<subseteq> \<gamma>\<^isub>o P'" "Q \<subseteq> \<gamma>\<^isub>o Q'" "C1 \<le> \<gamma>\<^isub>c C1'" |
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47613 | 214 |
by (fastforce simp: map_acom_While While_le) |
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moreover from this(1) `C' \<in> L X` |
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have L: "C1' \<in> L X" "I' \<in> L X" "P' \<in> L X" and "vars b \<subseteq> X" by simp_all |
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moreover note compat = `S' \<in> L X` post_in_L[OF L(1)] |
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moreover note vars = `I' \<in> L X` `vars b \<subseteq> X` |
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47613 | 219 |
moreover have "S \<union> post C1 \<subseteq> \<gamma>\<^isub>o (S' \<squnion> post C1')" |
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using `S \<subseteq> \<gamma>\<^isub>o S'` le_post[OF `C1 \<le> \<gamma>\<^isub>c C1'`, simplified] |
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by (metis (no_types) join_ge1[OF compat] join_ge2[OF compat] le_sup_iff mono_gamma_o order_trans) |
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ultimately show ?case |
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49396 | 223 |
by (simp add: While.IH subset_iff bfilter_sound[OF vars] bfilter_in_L[OF vars]) |
47613 | 224 |
qed |
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49396 | 226 |
lemma step'_in_L[simp]: |
227 |
"\<lbrakk> C \<in> L X; S \<in> L X \<rbrakk> \<Longrightarrow> step' S C \<in> L X" |
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47613 | 228 |
proof(induction C arbitrary: S) |
49396 | 229 |
case Assign thus ?case by(simp add: L_option_def L_st_def update_def split: option.splits) |
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qed (auto simp add: bfilter_in_L) |
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47613 | 231 |
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theorem 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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49464 | 234 |
assume 1: "pfp (step' (top c)) (bot c) = Some C" |
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have "C \<in> L(vars c)" |
49464 | 236 |
by(rule pfp_inv[where P = "%C. C \<in> L(vars c)", OF 1 _ bot_in_L]) |
49396 | 237 |
(erule step'_in_L[OF _ top_in_L]) |
49464 | 238 |
have 2: "step' (top c) C \<sqsubseteq> C" by(rule pfp_pfp[OF 1]) |
47613 | 239 |
have 3: "strip (\<gamma>\<^isub>c (step' (top c) C)) = c" |
49464 | 240 |
by(simp add: strip_pfp[OF _ 1]) |
48759
ff570720ba1c
Improved complete lattice formalisation - no more index set.
nipkow
parents:
47818
diff
changeset
|
241 |
have "lfp c (step UNIV) \<le> \<gamma>\<^isub>c (step' (top c) C)" |
47613 | 242 |
proof(rule lfp_lowerbound[simplified,OF 3]) |
243 |
show "step UNIV (\<gamma>\<^isub>c (step' (top c) C)) \<le> \<gamma>\<^isub>c (step' (top c) C)" |
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49396 | 244 |
proof(rule step_preserves_le[OF _ _ `C \<in> L(vars c)` top_in_L]) |
47613 | 245 |
show "UNIV \<subseteq> \<gamma>\<^isub>o (top c)" by simp |
246 |
show "\<gamma>\<^isub>c (step' (top c) C) \<le> \<gamma>\<^isub>c C" by(rule mono_gamma_c[OF 2]) |
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qed |
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248 |
qed |
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48759
ff570720ba1c
Improved complete lattice formalisation - no more index set.
nipkow
parents:
47818
diff
changeset
|
249 |
from this 2 show "lfp c (step UNIV) \<le> \<gamma>\<^isub>c C" |
47613 | 250 |
by (blast intro: mono_gamma_c order_trans) |
251 |
qed |
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252 |
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253 |
end |
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254 |
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255 |
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256 |
subsubsection "Monotonicity" |
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257 |
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258 |
locale Abs_Int1_mono = Abs_Int1 + |
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259 |
assumes mono_plus': "a1 \<sqsubseteq> b1 \<Longrightarrow> a2 \<sqsubseteq> b2 \<Longrightarrow> plus' a1 a2 \<sqsubseteq> plus' b1 b2" |
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260 |
and mono_filter_plus': "a1 \<sqsubseteq> b1 \<Longrightarrow> a2 \<sqsubseteq> b2 \<Longrightarrow> r \<sqsubseteq> r' \<Longrightarrow> |
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261 |
filter_plus' r a1 a2 \<sqsubseteq> filter_plus' r' b1 b2" |
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262 |
and mono_filter_less': "a1 \<sqsubseteq> b1 \<Longrightarrow> a2 \<sqsubseteq> b2 \<Longrightarrow> |
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263 |
filter_less' bv a1 a2 \<sqsubseteq> filter_less' bv b1 b2" |
|
264 |
begin |
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265 |
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266 |
lemma mono_aval': |
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49396 | 267 |
"S1 \<sqsubseteq> S2 \<Longrightarrow> S1 \<in> L X \<Longrightarrow> vars e \<subseteq> X \<Longrightarrow> aval' e S1 \<sqsubseteq> aval' e S2" |
268 |
by(induction e) (auto simp: le_st_def mono_plus' L_st_def) |
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47613 | 269 |
|
270 |
lemma mono_aval'': |
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49396 | 271 |
"S1 \<sqsubseteq> S2 \<Longrightarrow> S1 \<in> L X \<Longrightarrow> vars e \<subseteq> X \<Longrightarrow> aval'' e S1 \<sqsubseteq> aval'' e S2" |
47613 | 272 |
apply(cases S1) |
273 |
apply simp |
|
274 |
apply(cases S2) |
|
275 |
apply simp |
|
276 |
by (simp add: mono_aval') |
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277 |
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49396 | 278 |
lemma mono_afilter: "S1 \<in> L X \<Longrightarrow> S2 \<in> L X \<Longrightarrow> vars e \<subseteq> X \<Longrightarrow> |
47613 | 279 |
r1 \<sqsubseteq> r2 \<Longrightarrow> S1 \<sqsubseteq> S2 \<Longrightarrow> afilter e r1 S1 \<sqsubseteq> afilter e r2 S2" |
280 |
apply(induction e arbitrary: r1 r2 S1 S2) |
|
281 |
apply(auto simp: test_num' Let_def mono_meet split: option.splits prod.splits) |
|
282 |
apply (metis mono_gamma subsetD) |
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49396 | 283 |
apply(drule (2) mono_fun_L) |
47613 | 284 |
apply (metis mono_meet le_trans) |
49396 | 285 |
apply(metis mono_aval'' mono_filter_plus'[simplified le_prod_def] fst_conv snd_conv afilter_in_L) |
47613 | 286 |
done |
287 |
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49396 | 288 |
lemma mono_bfilter: "S1 \<in> L X \<Longrightarrow> S2 \<in> L X \<Longrightarrow> vars b \<subseteq> X \<Longrightarrow> |
47613 | 289 |
S1 \<sqsubseteq> S2 \<Longrightarrow> bfilter b bv S1 \<sqsubseteq> bfilter b bv S2" |
290 |
apply(induction b arbitrary: bv S1 S2) |
|
291 |
apply(simp) |
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292 |
apply(simp) |
|
293 |
apply simp |
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49396 | 294 |
apply(metis join_least le_trans[OF _ join_ge1] le_trans[OF _ join_ge2] bfilter_in_L) |
47613 | 295 |
apply (simp split: prod.splits) |
49396 | 296 |
apply(metis mono_aval'' mono_afilter mono_filter_less'[simplified le_prod_def] fst_conv snd_conv afilter_in_L) |
47613 | 297 |
done |
298 |
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theorem mono_step': "S1 \<in> L X \<Longrightarrow> S2 \<in> L X \<Longrightarrow> C1 \<in> L X \<Longrightarrow> C2 \<in> L X \<Longrightarrow> |
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S1 \<sqsubseteq> S2 \<Longrightarrow> C1 \<sqsubseteq> C2 \<Longrightarrow> step' S1 C1 \<sqsubseteq> step' S2 C2" |
301 |
apply(induction C1 C2 arbitrary: S1 S2 rule: le_acom.induct) |
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apply (auto simp: Let_def mono_bfilter mono_aval' mono_post |
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le_join_disj le_join_disj[OF post_in_L post_in_L] bfilter_in_L |
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split: option.split) |
305 |
done |
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306 |
||
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lemma mono_step'_top: "C1 \<in> L(vars c) \<Longrightarrow> C2 \<in> L(vars c) \<Longrightarrow> |
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C1 \<sqsubseteq> C2 \<Longrightarrow> step' (top c) C1 \<sqsubseteq> step' (top c) C2" |
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309 |
by (metis top_in_L mono_step' preord_class.le_refl) |
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|
311 |
end |
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312 |
||
313 |
end |