# Theory Abs_Int2

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theory Abs_Int2
imports Abs_Int1
`(* Author: Tobias Nipkow *)theory Abs_Int2imports Abs_Int1begininstantiation prod :: (preord,preord) preordbegindefinition "le_prod p1 p2 = (fst p1 \<sqsubseteq> fst p2 ∧ snd p1 \<sqsubseteq> snd p2)"instanceproof  case goal1 show ?case by(simp add: le_prod_def)next  case goal2 thus ?case unfolding le_prod_def by(metis le_trans)qedendsubsection "Backward Analysis of Expressions"class lattice = semilattice + bot +fixes meet :: "'a => 'a => 'a" (infixl "\<sqinter>" 65)assumes meet_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"and meet_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y"and meet_greatest: "x \<sqsubseteq> y ==> x \<sqsubseteq> z ==> x \<sqsubseteq> y \<sqinter> z"beginlemma mono_meet: "x \<sqsubseteq> x' ==> y \<sqsubseteq> y' ==> x \<sqinter> y \<sqsubseteq> x' \<sqinter> y'"by (metis meet_greatest meet_le1 meet_le2 le_trans)endlocale Val_abs1_gamma =  Gamma where γ = γ for γ :: "'av::lattice => val set" +assumes inter_gamma_subset_gamma_meet:  "γ a1 ∩ γ a2 ⊆ γ(a1 \<sqinter> a2)"and gamma_bot[simp]: "γ ⊥ = {}"beginlemma in_gamma_meet: "x : γ a1 ==> x : γ a2 ==> x : γ(a1 \<sqinter> a2)"by (metis IntI inter_gamma_subset_gamma_meet set_mp)lemma gamma_meet[simp]: "γ(a1 \<sqinter> a2) = γ a1 ∩ γ a2"by (metis equalityI inter_gamma_subset_gamma_meet le_inf_iff mono_gamma meet_le1 meet_le2)endlocale Val_abs1 =  Val_abs1_gamma where γ = γ   for γ :: "'av::lattice => val set" +fixes test_num' :: "val => 'av => bool"and filter_plus' :: "'av => 'av => 'av => 'av * 'av"and filter_less' :: "bool => 'av => 'av => 'av * 'av"assumes test_num': "test_num' n a = (n : γ a)"and filter_plus': "filter_plus' a a1 a2 = (b1,b2) ==>  n1 : γ a1 ==> n2 : γ a2 ==> n1+n2 : γ a ==> n1 : γ b1 ∧ n2 : γ b2"and filter_less': "filter_less' (n1<n2) a1 a2 = (b1,b2) ==>  n1 : γ a1 ==> n2 : γ a2 ==> n1 : γ b1 ∧ n2 : γ b2"locale Abs_Int1 =  Val_abs1 where γ = γ for γ :: "'av::lattice => val set"beginlemma in_gamma_join_UpI:  "S1 ∈ L X ==> S2 ∈ L X ==> s : γ⇣o S1 ∨ s : γ⇣o S2 ==> s : γ⇣o(S1 \<squnion> S2)"by (metis (hide_lams, no_types) semilatticeL_class.join_ge1 semilatticeL_class.join_ge2 mono_gamma_o subsetD)fun aval'' :: "aexp => 'av st option => 'av" where"aval'' e None = ⊥" |"aval'' e (Some sa) = aval' e sa"lemma aval''_sound: "s : γ⇣o S ==> S ∈ L X ==> vars a ⊆ X ==> aval a s : γ(aval'' a S)"by(simp add: L_option_def L_st_def aval'_sound split: option.splits)subsubsection "Backward analysis"fun afilter :: "aexp => 'av => 'av st option => 'av st option" where"afilter (N n) a S = (if test_num' n a then S else None)" |"afilter (V x) a S = (case S of None => None | Some S =>  let a' = fun S x \<sqinter> a in  if a' \<sqsubseteq> ⊥ then None else Some(update S x a'))" |"afilter (Plus e1 e2) a S = (let (a1,a2) = filter_plus' a (aval'' e1 S) (aval'' e2 S)  in afilter e1 a1 (afilter e2 a2 S))"text{* The test for @{const bot} in the @{const V}-case is important: @{constbot} indicates that a variable has no possible values, i.e.\ that the currentprogram point is unreachable. But then the abstract state should collapse to@{const None}. Put differently, we maintain the invariant that in an abstractstate of the form @{term"Some s"}, all variables are mapped to non-@{constbot} values. Otherwise the (pointwise) join of two abstract states, one ofwhich contains @{const bot} values, may produce too large a result, thusmaking the analysis less precise. *}fun bfilter :: "bexp => bool => 'av st option => 'av st option" where"bfilter (Bc v) res S = (if v=res then S else None)" |"bfilter (Not b) res S = bfilter b (¬ res) S" |"bfilter (And b1 b2) res S =  (if res then bfilter b1 True (bfilter b2 True S)   else bfilter b1 False S \<squnion> bfilter b2 False S)" |"bfilter (Less e1 e2) res S =  (let (res1,res2) = filter_less' res (aval'' e1 S) (aval'' e2 S)   in afilter e1 res1 (afilter e2 res2 S))"lemma afilter_in_L: "S ∈ L X ==> vars e ⊆ X ==> afilter e a S ∈ L X"by(induction e arbitrary: a S)  (auto simp: Let_def update_def L_st_def           split: option.splits prod.split)lemma afilter_sound: "S ∈ L X ==> vars e ⊆ X ==>  s : γ⇣o S ==> aval e s : γ a ==> s : γ⇣o (afilter e a S)"proof(induction e arbitrary: a S)  case N thus ?case by simp (metis test_num')next  case (V x)  obtain S' where "S = Some S'" and "s : γ⇣s S'" using `s : γ⇣o S`    by(auto simp: in_gamma_option_iff)  moreover hence "s x : γ (fun S' x)"    using V(1,2) by(simp add: γ_st_def L_st_def)  moreover have "s x : γ a" using V by simp  ultimately show ?case using V(3)    by(simp add: Let_def γ_st_def)      (metis mono_gamma emptyE in_gamma_meet gamma_bot subset_empty)next  case (Plus e1 e2) thus ?case    using filter_plus'[OF _ aval''_sound[OF Plus.prems(3)] aval''_sound[OF Plus.prems(3)]]    by (auto simp: afilter_in_L split: prod.split)qedlemma bfilter_in_L: "S ∈ L X ==> vars b ⊆ X ==> bfilter b bv S ∈ L X"by(induction b arbitrary: bv S)(auto simp: afilter_in_L split: prod.split)lemma bfilter_sound: "S ∈ L X ==> vars b ⊆ X ==>  s : γ⇣o S ==> bv = bval b s ==> s : γ⇣o(bfilter b bv S)"proof(induction b arbitrary: S bv)  case Bc thus ?case by simpnext  case (Not b) thus ?case by simpnext  case (And b1 b2) thus ?case    by simp (metis And(1) And(2) bfilter_in_L in_gamma_join_UpI)next  case (Less e1 e2) thus ?case    by(auto split: prod.split)      (metis (lifting) afilter_in_L afilter_sound aval''_sound filter_less')qedfun step' :: "'av st option => 'av st option acom => 'av st option acom" where"step' S (SKIP {P}) = (SKIP {S})" |"step' S (x ::= e {P}) =  x ::= e {case S of None => None | Some S => Some(update S x (aval' e S))}" |"step' S (C1; C2) = step' S C1; step' (post C1) C2" |"step' S (IF b THEN {P1} C1 ELSE {P2} C2 {Q}) =  (let P1' = bfilter b True S; C1' = step' P1 C1; P2' = bfilter b False S; C2' = step' P2 C2   in IF b THEN {P1'} C1' ELSE {P2'} C2' {post C1 \<squnion> post C2})" |"step' S ({I} WHILE b DO {p} C {Q}) =   {S \<squnion> post C}   WHILE b DO {bfilter b True I} step' p C   {bfilter b False I}"definition AI :: "com => 'av st option acom option" where"AI c = pfp (step' \<top>⇘c⇙) (bot c)"lemma strip_step'[simp]: "strip(step' S c) = strip c"by(induct c arbitrary: S) (simp_all add: Let_def)subsubsection "Soundness"lemma in_gamma_update:  "[| s : γ⇣s S; i : γ a |] ==> s(x := i) : γ⇣s(update S x a)"by(simp add: γ_st_def)lemma step_step': "C ∈ L X ==> S ∈ L X ==> step (γ⇣o S) (γ⇣c C) ≤ γ⇣c (step' S C)"proof(induction C arbitrary: S)  case SKIP thus ?case by autonext  case Assign thus ?case    by (fastforce simp: L_st_def intro: aval'_sound in_gamma_update split: option.splits)next  case Seq thus ?case by autonext  case (If b _ C1 _ C2)  hence 0: "post C1 \<sqsubseteq> post C1 \<squnion> post C2 ∧ post C2 \<sqsubseteq> post C1 \<squnion> post C2"    by(simp, metis post_in_L join_ge1 join_ge2)  have "vars b ⊆ X" using If.prems by simp  note vars = `S ∈ L X` `vars b ⊆ X`  show ?case using If 0    by (auto simp: mono_gamma_o bfilter_sound[OF vars] bfilter_in_L[OF vars])next  case (While I b)  hence vars: "I ∈ L X" "vars b ⊆ X" by simp_all  thus ?case using While    by (auto simp: mono_gamma_o bfilter_sound[OF vars] bfilter_in_L[OF vars])qedlemma step'_in_L[simp]: "[| C ∈ L X; S ∈ L X |] ==> step' S C ∈ L X"proof(induction C arbitrary: S)  case Assign thus ?case by(simp add: L_option_def L_st_def update_def split: option.splits)qed (auto simp add: bfilter_in_L)lemma AI_sound: "AI c = Some C ==> CS c ≤ γ⇣c C"proof(simp add: CS_def AI_def)  assume 1: "pfp (step' (top c)) (bot c) = Some C"  have "C ∈ L(vars c)"    by(rule pfp_inv[where P = "%C. C ∈ L(vars c)", OF 1 _ bot_in_L])      (erule step'_in_L[OF _ top_in_L])  have pfp': "step' (top c) C \<sqsubseteq> C" by(rule pfp_pfp[OF 1])  have 2: "step (γ⇣o(top c)) (γ⇣c C) ≤ γ⇣c C"  proof(rule order_trans)    show "step (γ⇣o (top c)) (γ⇣c C) ≤  γ⇣c (step' (top c) C)"      by(rule step_step'[OF `C ∈ L(vars c)` top_in_L])    show "γ⇣c (step' (top c) C) ≤ γ⇣c C"      by(rule mono_gamma_c[OF pfp'])  qed  have 3: "strip (γ⇣c C) = c" by(simp add: strip_pfp[OF _ 1])  have "lfp c (step (γ⇣o(top c))) ≤ γ⇣c C"    by(rule lfp_lowerbound[simplified,where f="step (γ⇣o(top c))", OF 3 2])  thus "lfp c (step UNIV) ≤ γ⇣c C" by simpqedendsubsubsection "Monotonicity"locale Abs_Int1_mono = Abs_Int1 +assumes mono_plus': "a1 \<sqsubseteq> b1 ==> a2 \<sqsubseteq> b2 ==> plus' a1 a2 \<sqsubseteq> plus' b1 b2"and mono_filter_plus': "a1 \<sqsubseteq> b1 ==> a2 \<sqsubseteq> b2 ==> r \<sqsubseteq> r' ==>  filter_plus' r a1 a2 \<sqsubseteq> filter_plus' r' b1 b2"and mono_filter_less': "a1 \<sqsubseteq> b1 ==> a2 \<sqsubseteq> b2 ==>  filter_less' bv a1 a2 \<sqsubseteq> filter_less' bv b1 b2"beginlemma mono_aval':  "S1 \<sqsubseteq> S2 ==> S1 ∈ L X ==> vars e ⊆ X ==> aval' e S1 \<sqsubseteq> aval' e S2"by(induction e) (auto simp: le_st_def mono_plus' L_st_def)lemma mono_aval'':  "S1 \<sqsubseteq> S2 ==> S1 ∈ L X ==> vars e ⊆ X ==> aval'' e S1 \<sqsubseteq> aval'' e S2"apply(cases S1) apply simpapply(cases S2) apply simpby (simp add: mono_aval')lemma mono_afilter: "S1 ∈ L X ==> S2 ∈ L X ==> vars e ⊆ X ==>  r1 \<sqsubseteq> r2 ==> S1 \<sqsubseteq> S2 ==> afilter e r1 S1 \<sqsubseteq> afilter e r2 S2"apply(induction e arbitrary: r1 r2 S1 S2)apply(auto simp: test_num' Let_def mono_meet split: option.splits prod.splits)apply (metis mono_gamma subsetD)apply(drule (2) mono_fun_L)apply (metis mono_meet le_trans)apply(metis mono_aval'' mono_filter_plus'[simplified le_prod_def] fst_conv snd_conv afilter_in_L)donelemma mono_bfilter: "S1 ∈ L X ==> S2 ∈ L X ==> vars b ⊆ X ==>  S1 \<sqsubseteq> S2 ==> bfilter b bv S1 \<sqsubseteq> bfilter b bv S2"apply(induction b arbitrary: bv S1 S2)apply(simp)apply(simp)apply simpapply(metis join_least le_trans[OF _ join_ge1] le_trans[OF _ join_ge2] bfilter_in_L)apply (simp split: prod.splits)apply(metis mono_aval'' mono_afilter mono_filter_less'[simplified le_prod_def] fst_conv snd_conv afilter_in_L)donetheorem mono_step': "S1 ∈ L X ==> S2 ∈ L X ==> C1 ∈ L X ==> C2 ∈ L X ==>  S1 \<sqsubseteq> S2 ==> C1 \<sqsubseteq> C2 ==> step' S1 C1 \<sqsubseteq> step' S2 C2"apply(induction C1 C2 arbitrary: S1 S2 rule: le_acom.induct)apply (auto simp: Let_def mono_bfilter mono_aval' mono_post  le_join_disj le_join_disj[OF  post_in_L post_in_L] bfilter_in_L            split: option.split)donelemma mono_step'_top: "C1 ∈ L(vars c) ==> C2 ∈ L(vars c) ==>  C1 \<sqsubseteq> C2 ==> step' (top c) C1 \<sqsubseteq> step' (top c) C2"by (metis top_in_L mono_step' preord_class.le_refl)endend`