author  nipkow 
Wed, 01 May 2013 03:56:57 +0200  
changeset 51848  ed847ce0b70c 
parent 51834  8deb369ee70b 
child 51849  19ee0cebe76d 
permissions  rwrr 
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(* 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 :: (order,order) order 
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begin 
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definition "less_eq_prod p1 p2 = (fst p1 \<le> fst p2 \<and> snd p1 \<le> snd p2)" 
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definition "less_prod p1 p2 = (p1 \<le> p2 \<and> \<not> p2 \<le> (p1::'a*'b))" 
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instance 

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proof 

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case goal1 show ?case by(rule less_prod_def) 
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next 
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case goal2 show ?case by(simp add: less_eq_prod_def) 
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next 
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case goal3 thus ?case unfolding less_eq_prod_def by(metis order_trans) 
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next 
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case goal4 thus ?case by(simp add: less_eq_prod_def)(metis eq_iff surjective_pairing) 
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qed 
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end 

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subsection "Backward Analysis of Expressions" 

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subclass (in bounded_lattice) semilattice_sup_top .. 
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locale Val_abs1_gamma = Gamma where \<gamma> = \<gamma> 
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for \<gamma> :: "'av::bounded_lattice \<Rightarrow> val set" + 

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assumes inter_gamma_subset_gamma_inf: 
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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_inf: "x : \<gamma> a1 \<Longrightarrow> x : \<gamma> a2 \<Longrightarrow> x : \<gamma>(a1 \<sqinter> a2)" 
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by (metis IntI inter_gamma_subset_gamma_inf set_mp) 

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lemma gamma_inf: "\<gamma>(a1 \<sqinter> a2) = \<gamma> a1 \<inter> \<gamma> a2" 
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by(rule equalityI[OF _ inter_gamma_subset_gamma_inf]) 
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(metis inf_le1 inf_le2 le_inf_iff mono_gamma) 
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end 

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locale Val_abs1 = Val_abs1_gamma where \<gamma> = \<gamma> 
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for \<gamma> :: "'av::bounded_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 = Val_abs1 where \<gamma> = \<gamma> 
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for \<gamma> :: "'av::bounded_lattice \<Rightarrow> val set" 

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begin 
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lemma in_gamma_sup_UpI: 
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"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) sup_ge1 sup_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 S) = aval' e S" 
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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)(auto simp add: 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' = \<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) sup 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 (a1,a2) = filter_less' res (aval'' e1 S) (aval'' e2 S) 
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in afilter e1 a1 (afilter e2 a2 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>s 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> (fun S' x)" 

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using V(1,2) 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 
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by(simp add: Let_def \<gamma>_st_def) 
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(metis mono_gamma emptyE in_gamma_inf 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 aval''_sound] 
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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 

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by simp (metis And(1) And(2) in_gamma_sup_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 (lifting) afilter_sound aval''_sound filter_less') 
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qed 
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definition "step' = Step 
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(\<lambda>x e S. case S of None \<Rightarrow> None  Some S \<Rightarrow> Some(update S x (aval' e S))) 
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(\<lambda>b S. bfilter b True S)" 

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definition AI :: "com \<Rightarrow> 'av st option acom option" where 

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"AI c = pfp (step' \<top>) (bot c)" 
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lemma strip_step'[simp]: "strip(step' S c) = strip c" 

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by(simp add: step'_def) 
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lemma top_on_afilter: "\<lbrakk> top_on_opt S X; vars e \<subseteq> X \<rbrakk> \<Longrightarrow> top_on_opt (afilter e a S) X" 
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by(induction e arbitrary: a S) (auto simp: Let_def split: option.splits prod.split) 
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lemma top_on_bfilter: "\<lbrakk>top_on_opt S X; vars b \<subseteq> X\<rbrakk> \<Longrightarrow> top_on_opt (bfilter b r S) X" 
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by(induction b arbitrary: r S) (auto simp: top_on_afilter top_on_sup split: prod.split) 
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lemma top_on_step': "top_on_acom C ( vars C) \<Longrightarrow> top_on_acom (step' \<top> C) ( vars C)" 
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unfolding step'_def 
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by(rule top_on_Step) 
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(auto simp add: top_on_top top_on_bfilter split: option.split) 
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subsubsection "Soundness" 

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lemma step_step': "step (\<gamma>\<^isub>o S) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c (step' S C)" 
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unfolding step_def step'_def 
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by(rule gamma_Step_subcomm) 

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(auto simp: intro!: aval'_sound bfilter_sound in_gamma_update split: option.splits) 
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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: "pfp (step' \<top>) (bot c) = Some C" 
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have pfp': "step' \<top> C \<le> C" by(rule pfp_pfp[OF 1]) 
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have 2: "step (\<gamma>\<^isub>o \<top>) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c C" "transfer the pfp'" 
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proof(rule order_trans) 
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show "step (\<gamma>\<^isub>o \<top>) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c (step' \<top> C)" by(rule step_step') 
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show "... \<le> \<gamma>\<^isub>c C" by (metis mono_gamma_c[OF pfp']) 
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qed 
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have 3: "strip (\<gamma>\<^isub>c C) = c" by(simp add: strip_pfp[OF _ 1] step'_def) 
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have "lfp c (step (\<gamma>\<^isub>o \<top>)) \<le> \<gamma>\<^isub>c C" 
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by(rule lfp_lowerbound[simplified,where f="step (\<gamma>\<^isub>o \<top>)", OF 3 2]) 
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thus "lfp c (step UNIV) \<le> \<gamma>\<^isub>c C" by simp 
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qed 
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end 

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subsubsection "Monotonicity" 

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locale Abs_Int1_mono = Abs_Int1 + 

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assumes mono_plus': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow> plus' a1 a2 \<le> plus' b1 b2" 
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and mono_filter_plus': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow> r \<le> r' \<Longrightarrow> 
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filter_plus' r a1 a2 \<le> filter_plus' r' b1 b2" 
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and mono_filter_less': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow> 
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filter_less' bv a1 a2 \<le> filter_less' bv b1 b2" 
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begin 
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lemma mono_aval': 

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"S1 \<le> S2 \<Longrightarrow> aval' e S1 \<le> aval' e S2" 
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by(induction e) (auto simp: mono_plus' mono_fun) 
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lemma mono_aval'': 

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"S1 \<le> S2 \<Longrightarrow> aval'' e S1 \<le> aval'' e S2" 
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apply(cases S1) 
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apply simp 

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apply(cases S2) 

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apply simp 

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by (simp add: mono_aval') 

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lemma mono_afilter: "r1 \<le> r2 \<Longrightarrow> S1 \<le> S2 \<Longrightarrow> afilter e r1 S1 \<le> afilter e r2 S2" 
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apply(induction e arbitrary: r1 r2 S1 S2) 
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apply(auto simp: test_num' Let_def inf_mono split: option.splits prod.splits) 
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apply (metis mono_gamma subsetD) 

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apply (metis le_bot inf_mono le_st_iff) 
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apply (metis inf_mono mono_update le_st_iff) 
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apply(metis mono_aval'' mono_filter_plus'[simplified less_eq_prod_def] fst_conv snd_conv) 
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done 
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lemma mono_bfilter: "S1 \<le> S2 \<Longrightarrow> bfilter b bv S1 \<le> bfilter b bv S2" 
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apply(induction b arbitrary: bv S1 S2) 
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apply(simp) 
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apply(simp) 

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apply simp 

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apply(metis order_trans[OF _ sup_ge1] order_trans[OF _ sup_ge2]) 
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apply (simp split: prod.splits) 
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apply(metis mono_aval'' mono_afilter mono_filter_less'[simplified less_eq_prod_def] fst_conv snd_conv) 
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done 
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theorem mono_step': "S1 \<le> S2 \<Longrightarrow> C1 \<le> C2 \<Longrightarrow> step' S1 C1 \<le> step' S2 C2" 
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unfolding step'_def 
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by(rule mono2_Step) (auto simp: mono_aval' mono_bfilter split: option.split) 

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lemma mono_step'_top: "C1 \<le> C2 \<Longrightarrow> step' \<top> C1 \<le> step' \<top> C2" 
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by (metis mono_step' order_refl) 
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end 

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end 