author | nipkow |
Fri, 25 Jan 2013 16:45:09 +0100 | |
changeset 50995 | 3371f5ee4ace |
parent 50986 | c54ea7f5418f |
child 51036 | e7b54119c436 |
permissions | -rw-r--r-- |
47613 | 1 |
(* Author: Tobias Nipkow *) |
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theory Abs_Int3 |
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imports Abs_Int2_ivl |
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begin |
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subsubsection "Welltypedness" |
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class Lc = |
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fixes Lc :: "com \<Rightarrow> 'a set" |
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instantiation st :: (type)Lc |
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begin |
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definition Lc_st :: "com \<Rightarrow> 'a st set" where |
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"Lc_st c = L (vars c)" |
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47613 | 17 |
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instance .. |
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end |
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instantiation acom :: (Lc)Lc |
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begin |
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definition Lc_acom :: "com \<Rightarrow> 'a acom set" where |
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"Lc c = {C. strip C = c \<and> (\<forall>a\<in>set(annos C). a \<in> Lc c)}" |
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47613 | 27 |
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instance .. |
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end |
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instantiation option :: (Lc)Lc |
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begin |
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definition Lc_option :: "com \<Rightarrow> 'a option set" where |
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"Lc c = {None} \<union> Some ` Lc c" |
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lemma Lc_option_simps[simp]: "None \<in> Lc c" "(Some x \<in> Lc c) = (x \<in> Lc c)" |
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by(auto simp: Lc_option_def) |
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47613 | 40 |
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instance .. |
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end |
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lemma Lc_option_iff_wt[simp]: fixes a :: "_ st option" |
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shows "(a \<in> Lc c) = (a \<in> L (vars c))" |
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by(auto simp add: L_option_def Lc_st_def split: option.splits) |
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47613 | 48 |
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context Abs_Int1 |
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begin |
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lemma step'_in_Lc: "C \<in> Lc c \<Longrightarrow> S \<in> Lc c \<Longrightarrow> step' S C \<in> Lc c" |
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apply(auto simp add: Lc_acom_def) |
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by(metis step'_in_L[simplified L_acom_def mem_Collect_eq] order_refl) |
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47613 | 56 |
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end |
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subsection "Widening and Narrowing" |
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class widen = |
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fixes widen :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infix "\<nabla>" 65) |
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class narrow = |
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fixes narrow :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infix "\<triangle>" 65) |
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class WN = widen + narrow + preord + |
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assumes widen1: "x \<sqsubseteq> x \<nabla> y" |
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assumes widen2: "y \<sqsubseteq> x \<nabla> y" |
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assumes narrow1: "y \<sqsubseteq> x \<Longrightarrow> y \<sqsubseteq> x \<triangle> y" |
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assumes narrow2: "y \<sqsubseteq> x \<Longrightarrow> x \<triangle> y \<sqsubseteq> x" |
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class WN_Lc = widen + narrow + preord + Lc + |
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assumes widen1: "x \<in> Lc c \<Longrightarrow> y \<in> Lc c \<Longrightarrow> x \<sqsubseteq> x \<nabla> y" |
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assumes widen2: "x \<in> Lc c \<Longrightarrow> y \<in> Lc c \<Longrightarrow> y \<sqsubseteq> x \<nabla> y" |
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assumes narrow1: "y \<sqsubseteq> x \<Longrightarrow> y \<sqsubseteq> x \<triangle> y" |
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assumes narrow2: "y \<sqsubseteq> x \<Longrightarrow> x \<triangle> y \<sqsubseteq> x" |
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assumes Lc_widen[simp]: "x \<in> Lc c \<Longrightarrow> y \<in> Lc c \<Longrightarrow> x \<nabla> y \<in> Lc c" |
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assumes Lc_narrow[simp]: "x \<in> Lc c \<Longrightarrow> y \<in> Lc c \<Longrightarrow> x \<triangle> y \<in> Lc c" |
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47613 | 81 |
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instantiation ivl :: WN |
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begin |
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definition "widen_ivl ivl1 ivl2 = |
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((*if is_empty ivl1 then ivl2 else |
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if is_empty ivl2 then ivl1 else*) |
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49579 | 89 |
case (ivl1,ivl2) of (Ivl l1 h1, Ivl l2 h2) \<Rightarrow> |
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Ivl (if le_option False l2 l1 \<and> l2 \<noteq> l1 then None else l1) |
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(if le_option True h1 h2 \<and> h1 \<noteq> h2 then None else h1))" |
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47613 | 92 |
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definition "narrow_ivl ivl1 ivl2 = |
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((*if is_empty ivl1 \<or> is_empty ivl2 then empty else*) |
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49579 | 95 |
case (ivl1,ivl2) of (Ivl l1 h1, Ivl l2 h2) \<Rightarrow> |
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Ivl (if l1 = None then l2 else l1) |
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(if h1 = None then h2 else h1))" |
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47613 | 98 |
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instance |
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proof qed |
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(auto simp add: widen_ivl_def narrow_ivl_def le_option_def le_ivl_def empty_def split: ivl.split option.split if_splits) |
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end |
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instantiation st :: (WN)WN_Lc |
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begin |
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definition "widen_st F1 F2 = FunDom (\<lambda>x. fun F1 x \<nabla> fun F2 x) (dom F1)" |
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definition "narrow_st F1 F2 = FunDom (\<lambda>x. fun F1 x \<triangle> fun F2 x) (dom F1)" |
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instance |
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proof |
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case goal1 thus ?case |
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by(simp add: widen_st_def le_st_def WN_class.widen1) |
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next |
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case goal2 thus ?case |
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49396 | 119 |
by(simp add: widen_st_def le_st_def WN_class.widen2 Lc_st_def L_st_def) |
47613 | 120 |
next |
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case goal3 thus ?case |
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by(auto simp: narrow_st_def le_st_def WN_class.narrow1) |
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next |
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case goal4 thus ?case |
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by(auto simp: narrow_st_def le_st_def WN_class.narrow2) |
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next |
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49396 | 127 |
case goal5 thus ?case by(auto simp: widen_st_def Lc_st_def L_st_def) |
47613 | 128 |
next |
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case goal6 thus ?case by(auto simp: narrow_st_def Lc_st_def L_st_def) |
47613 | 130 |
qed |
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end |
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49396 | 135 |
instantiation option :: (WN_Lc)WN_Lc |
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begin |
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fun widen_option where |
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"None \<nabla> x = x" | |
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"x \<nabla> None = x" | |
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"(Some x) \<nabla> (Some y) = Some(x \<nabla> y)" |
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fun narrow_option where |
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"None \<triangle> x = None" | |
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"x \<triangle> None = None" | |
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"(Some x) \<triangle> (Some y) = Some(x \<triangle> y)" |
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instance |
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proof |
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case goal1 thus ?case |
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by(induct x y rule: widen_option.induct)(simp_all add: widen1) |
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next |
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case goal2 thus ?case |
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by(induct x y rule: widen_option.induct)(simp_all add: widen2) |
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next |
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case goal3 thus ?case |
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by(induct x y rule: narrow_option.induct) (simp_all add: narrow1) |
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next |
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case goal4 thus ?case |
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by(induct x y rule: narrow_option.induct) (simp_all add: narrow2) |
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next |
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case goal5 thus ?case |
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49396 | 163 |
by(induction x y rule: widen_option.induct)(auto simp: Lc_st_def) |
47613 | 164 |
next |
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case goal6 thus ?case |
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by(induction x y rule: narrow_option.induct)(auto simp: Lc_st_def) |
47613 | 167 |
qed |
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end |
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fun map2_acom :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a acom \<Rightarrow> 'a acom \<Rightarrow> 'a acom" where |
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"map2_acom f (SKIP {a1}) (SKIP {a2}) = (SKIP {f a1 a2})" | |
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"map2_acom f (x ::= e {a1}) (x' ::= e' {a2}) = (x ::= e {f a1 a2})" | |
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"map2_acom f (C1;C2) (D1;D2) = (map2_acom f C1 D1; map2_acom f C2 D2)" | |
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49095 | 176 |
"map2_acom f (IF b THEN {p1} C1 ELSE {p2} C2 {a1}) (IF b' THEN {q1} D1 ELSE {q2} D2 {a2}) = |
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(IF b THEN {f p1 q1} map2_acom f C1 D1 ELSE {f p2 q2} map2_acom f C2 D2 {f a1 a2})" | |
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"map2_acom f ({a1} WHILE b DO {p} C {a2}) ({a3} WHILE b' DO {p'} C' {a4}) = |
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({f a1 a3} WHILE b DO {f p p'} map2_acom f C C' {f a2 a4})" |
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47613 | 180 |
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49548 | 181 |
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instantiation acom :: (widen)widen |
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begin |
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definition "widen_acom = map2_acom (op \<nabla>)" |
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instance .. |
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end |
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instantiation acom :: (narrow)narrow |
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begin |
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definition "narrow_acom = map2_acom (op \<triangle>)" |
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instance .. |
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end |
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instantiation acom :: (WN_Lc)WN_Lc |
47613 | 195 |
begin |
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49548 | 197 |
lemma widen_acom1: fixes C1 :: "'a acom" shows |
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"\<lbrakk>\<forall>a\<in>set(annos C1). a \<in> Lc c; \<forall>a\<in>set (annos C2). a \<in> Lc c; strip C1 = strip C2\<rbrakk> |
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\<Longrightarrow> C1 \<sqsubseteq> C1 \<nabla> C2" |
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by(induct C1 C2 rule: le_acom.induct) |
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(auto simp: widen_acom_def widen1 Lc_acom_def) |
47613 | 202 |
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49548 | 203 |
lemma widen_acom2: fixes C1 :: "'a acom" shows |
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"\<lbrakk>\<forall>a\<in>set(annos C1). a \<in> Lc c; \<forall>a\<in>set (annos C2). a \<in> Lc c; strip C1 = strip C2\<rbrakk> |
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\<Longrightarrow> C2 \<sqsubseteq> C1 \<nabla> C2" |
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by(induct C1 C2 rule: le_acom.induct) |
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(auto simp: widen_acom_def widen2 Lc_acom_def) |
47613 | 208 |
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lemma strip_map2_acom[simp]: |
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"strip C1 = strip C2 \<Longrightarrow> strip(map2_acom f C1 C2) = strip C1" |
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by(induct f C1 C2 rule: map2_acom.induct) simp_all |
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lemma strip_widen_acom[simp]: |
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"strip C1 = strip C2 \<Longrightarrow> strip(C1 \<nabla> C2) = strip C1" |
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49548 | 215 |
by(simp add: widen_acom_def) |
47613 | 216 |
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lemma strip_narrow_acom[simp]: |
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"strip C1 = strip C2 \<Longrightarrow> strip(C1 \<triangle> C2) = strip C1" |
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49548 | 219 |
by(simp add: narrow_acom_def) |
47613 | 220 |
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lemma annos_map2_acom[simp]: "strip C2 = strip C1 \<Longrightarrow> |
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annos(map2_acom f C1 C2) = map (%(x,y).f x y) (zip (annos C1) (annos C2))" |
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by(induction f C1 C2 rule: map2_acom.induct)(simp_all add: size_annos_same2) |
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instance |
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226 |
proof |
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49396 | 227 |
case goal1 thus ?case by(auto simp: Lc_acom_def widen_acom1) |
47613 | 228 |
next |
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case goal2 thus ?case by(auto simp: Lc_acom_def widen_acom2) |
47613 | 230 |
next |
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case goal3 thus ?case |
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by(induct x y rule: le_acom.induct)(simp_all add: narrow_acom_def narrow1) |
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next |
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case goal4 thus ?case |
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by(induct x y rule: le_acom.induct)(simp_all add: narrow_acom_def narrow2) |
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next |
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case goal5 thus ?case |
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49396 | 238 |
by(auto simp: Lc_acom_def widen_acom_def split_conv elim!: in_set_zipE) |
47613 | 239 |
next |
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case goal6 thus ?case |
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49396 | 241 |
by(auto simp: Lc_acom_def narrow_acom_def split_conv elim!: in_set_zipE) |
47613 | 242 |
qed |
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244 |
end |
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49396 | 246 |
lemma widen_o_in_L[simp]: fixes x1 x2 :: "_ st option" |
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shows "x1 \<in> L X \<Longrightarrow> x2 \<in> L X \<Longrightarrow> x1 \<nabla> x2 \<in> L X" |
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47613 | 248 |
by(induction x1 x2 rule: widen_option.induct) |
49396 | 249 |
(simp_all add: widen_st_def L_st_def) |
47613 | 250 |
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49396 | 251 |
lemma narrow_o_in_L[simp]: fixes x1 x2 :: "_ st option" |
252 |
shows "x1 \<in> L X \<Longrightarrow> x2 \<in> L X \<Longrightarrow> x1 \<triangle> x2 \<in> L X" |
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47613 | 253 |
by(induction x1 x2 rule: narrow_option.induct) |
49396 | 254 |
(simp_all add: narrow_st_def L_st_def) |
47613 | 255 |
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49396 | 256 |
lemma widen_c_in_L: fixes C1 C2 :: "_ st option acom" |
257 |
shows "strip C1 = strip C2 \<Longrightarrow> C1 \<in> L X \<Longrightarrow> C2 \<in> L X \<Longrightarrow> C1 \<nabla> C2 \<in> L X" |
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47613 | 258 |
by(induction C1 C2 rule: le_acom.induct) |
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(auto simp: widen_acom_def) |
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49396 | 261 |
lemma narrow_c_in_L: fixes C1 C2 :: "_ st option acom" |
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shows "strip C1 = strip C2 \<Longrightarrow> C1 \<in> L X \<Longrightarrow> C2 \<in> L X \<Longrightarrow> C1 \<triangle> C2 \<in> L X" |
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47613 | 263 |
by(induction C1 C2 rule: le_acom.induct) |
264 |
(auto simp: narrow_acom_def) |
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49396 | 266 |
lemma bot_in_Lc[simp]: "bot c \<in> Lc c" |
267 |
by(simp add: Lc_acom_def bot_def) |
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47613 | 268 |
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subsubsection "Post-fixed point computation" |
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271 |
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272 |
definition iter_widen :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> ('a::{preord,widen})option" |
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49548 | 273 |
where "iter_widen f = while_option (\<lambda>x. \<not> f x \<sqsubseteq> x) (\<lambda>x. x \<nabla> f x)" |
47613 | 274 |
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275 |
definition iter_narrow :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> ('a::{preord,narrow})option" |
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49548 | 276 |
where "iter_narrow f = while_option (\<lambda>x. \<not> x \<sqsubseteq> x \<triangle> f x) (\<lambda>x. x \<triangle> f x)" |
47613 | 277 |
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49548 | 278 |
definition pfp_wn :: "('a::{preord,widen,narrow} \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a option" |
279 |
where "pfp_wn f x = |
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49576 | 280 |
(case iter_widen f x of None \<Rightarrow> None | Some p \<Rightarrow> iter_narrow f p)" |
47613 | 281 |
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282 |
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49548 | 283 |
lemma iter_widen_pfp: "iter_widen f x = Some p \<Longrightarrow> f p \<sqsubseteq> p" |
47613 | 284 |
by(auto simp add: iter_widen_def dest: while_option_stop) |
285 |
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286 |
lemma iter_widen_inv: |
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287 |
assumes "!!x. P x \<Longrightarrow> P(f x)" "!!x1 x2. P x1 \<Longrightarrow> P x2 \<Longrightarrow> P(x1 \<nabla> x2)" and "P x" |
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288 |
and "iter_widen f x = Some y" shows "P y" |
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using while_option_rule[where P = "P", OF _ assms(4)[unfolded iter_widen_def]] |
|
290 |
by (blast intro: assms(1-3)) |
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291 |
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292 |
lemma strip_while: fixes f :: "'a acom \<Rightarrow> 'a acom" |
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293 |
assumes "\<forall>C. strip (f C) = strip C" and "while_option P f C = Some C'" |
|
294 |
shows "strip C' = strip C" |
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using while_option_rule[where P = "\<lambda>C'. strip C' = strip C", OF _ assms(2)] |
|
296 |
by (metis assms(1)) |
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297 |
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49548 | 298 |
lemma strip_iter_widen: fixes f :: "'a::{preord,widen} acom \<Rightarrow> 'a acom" |
47613 | 299 |
assumes "\<forall>C. strip (f C) = strip C" and "iter_widen f C = Some C'" |
300 |
shows "strip C' = strip C" |
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301 |
proof- |
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302 |
have "\<forall>C. strip(C \<nabla> f C) = strip C" |
|
303 |
by (metis assms(1) strip_map2_acom widen_acom_def) |
|
304 |
from strip_while[OF this] assms(2) show ?thesis by(simp add: iter_widen_def) |
|
305 |
qed |
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306 |
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307 |
lemma iter_narrow_pfp: |
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49548 | 308 |
assumes mono: "!!x1 x2::_::WN_Lc. P x1 \<Longrightarrow> P x2 \<Longrightarrow> x1 \<sqsubseteq> x2 \<Longrightarrow> f x1 \<sqsubseteq> f x2" |
49576 | 309 |
and Pinv: "!!x. P x \<Longrightarrow> P(f x)" "!!x1 x2. P x1 \<Longrightarrow> P x2 \<Longrightarrow> P(x1 \<triangle> x2)" |
310 |
and "P p0" and "f p0 \<sqsubseteq> p0" and "iter_narrow f p0 = Some p" |
|
311 |
shows "P p \<and> f p \<sqsubseteq> p" |
|
47613 | 312 |
proof- |
49576 | 313 |
let ?Q = "%p. P p \<and> f p \<sqsubseteq> p \<and> p \<sqsubseteq> p0" |
314 |
{ fix p assume "?Q p" |
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47613 | 315 |
note P = conjunct1[OF this] and 12 = conjunct2[OF this] |
316 |
note 1 = conjunct1[OF 12] and 2 = conjunct2[OF 12] |
|
49576 | 317 |
let ?p' = "p \<triangle> f p" |
318 |
have "?Q ?p'" |
|
47613 | 319 |
proof auto |
49576 | 320 |
show "P ?p'" by (blast intro: P Pinv) |
321 |
have "f ?p' \<sqsubseteq> f p" by(rule mono[OF `P (p \<triangle> f p)` P narrow2[OF 1]]) |
|
322 |
also have "\<dots> \<sqsubseteq> ?p'" by(rule narrow1[OF 1]) |
|
323 |
finally show "f ?p' \<sqsubseteq> ?p'" . |
|
324 |
have "?p' \<sqsubseteq> p" by (rule narrow2[OF 1]) |
|
325 |
also have "p \<sqsubseteq> p0" by(rule 2) |
|
326 |
finally show "?p' \<sqsubseteq> p0" . |
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47613 | 327 |
qed |
328 |
} |
|
329 |
thus ?thesis |
|
330 |
using while_option_rule[where P = ?Q, OF _ assms(6)[simplified iter_narrow_def]] |
|
331 |
by (blast intro: assms(4,5) le_refl) |
|
332 |
qed |
|
333 |
||
334 |
lemma pfp_wn_pfp: |
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49576 | 335 |
assumes mono: "!!x1 x2::_::WN_Lc. P x1 \<Longrightarrow> P x2 \<Longrightarrow> x1 \<sqsubseteq> x2 \<Longrightarrow> f x1 \<sqsubseteq> f x2" |
49548 | 336 |
and Pinv: "P x" "!!x. P x \<Longrightarrow> P(f x)" |
337 |
"!!x1 x2. P x1 \<Longrightarrow> P x2 \<Longrightarrow> P(x1 \<nabla> x2)" |
|
338 |
"!!x1 x2. P x1 \<Longrightarrow> P x2 \<Longrightarrow> P(x1 \<triangle> x2)" |
|
49576 | 339 |
and pfp_wn: "pfp_wn f x = Some p" shows "P p \<and> f p \<sqsubseteq> p" |
47613 | 340 |
proof- |
49576 | 341 |
from pfp_wn obtain p0 |
342 |
where its: "iter_widen f x = Some p0" "iter_narrow f p0 = Some p" |
|
47613 | 343 |
by(auto simp: pfp_wn_def split: option.splits) |
49576 | 344 |
have "P p0" by (blast intro: iter_widen_inv[where P="P"] its(1) Pinv(1-3)) |
47613 | 345 |
thus ?thesis |
346 |
by - (assumption | |
|
347 |
rule iter_narrow_pfp[where P=P] mono Pinv(2,4) iter_widen_pfp its)+ |
|
348 |
qed |
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349 |
||
350 |
lemma strip_pfp_wn: |
|
49548 | 351 |
"\<lbrakk> \<forall>C. strip(f C) = strip C; pfp_wn f C = Some C' \<rbrakk> \<Longrightarrow> strip C' = strip C" |
47613 | 352 |
by(auto simp add: pfp_wn_def iter_narrow_def split: option.splits) |
49548 | 353 |
(metis (no_types) narrow_acom_def strip_iter_widen strip_map2_acom strip_while) |
47613 | 354 |
|
355 |
||
356 |
locale Abs_Int2 = Abs_Int1_mono |
|
49396 | 357 |
where \<gamma>=\<gamma> for \<gamma> :: "'av::{WN,lattice} \<Rightarrow> val set" |
47613 | 358 |
begin |
359 |
||
360 |
definition AI_wn :: "com \<Rightarrow> 'av st option acom option" where |
|
49548 | 361 |
"AI_wn c = pfp_wn (step' (top c)) (bot c)" |
47613 | 362 |
|
363 |
lemma AI_wn_sound: "AI_wn c = Some C \<Longrightarrow> CS c \<le> \<gamma>\<^isub>c C" |
|
364 |
proof(simp add: CS_def AI_wn_def) |
|
49548 | 365 |
assume 1: "pfp_wn (step' (top c)) (bot c) = Some C" |
49396 | 366 |
have 2: "(strip C = c & C \<in> L(vars c)) \<and> step' \<top>\<^bsub>c\<^esub> C \<sqsubseteq> C" |
49548 | 367 |
by(rule pfp_wn_pfp[where x="bot c"]) |
49396 | 368 |
(simp_all add: 1 mono_step'_top widen_c_in_L narrow_c_in_L) |
50986 | 369 |
have pfp: "step (\<gamma>\<^isub>o(top c)) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c C" |
370 |
proof(rule order_trans) |
|
371 |
show "step (\<gamma>\<^isub>o (top c)) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c (step' (top c) C)" |
|
372 |
by(rule step_step'[OF conjunct2[OF conjunct1[OF 2]] top_in_L]) |
|
373 |
show "... \<le> \<gamma>\<^isub>c C" |
|
374 |
by(rule mono_gamma_c[OF conjunct2[OF 2]]) |
|
47613 | 375 |
qed |
50986 | 376 |
have 3: "strip (\<gamma>\<^isub>c C) = c" by(simp add: strip_pfp_wn[OF _ 1]) |
377 |
have "lfp c (step (\<gamma>\<^isub>o (top c))) \<le> \<gamma>\<^isub>c C" |
|
378 |
by(rule lfp_lowerbound[simplified,where f="step (\<gamma>\<^isub>o(top c))", OF 3 pfp]) |
|
379 |
thus "lfp c (step UNIV) \<le> \<gamma>\<^isub>c C" by simp |
|
47613 | 380 |
qed |
381 |
||
382 |
end |
|
383 |
||
384 |
interpretation Abs_Int2 |
|
385 |
where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = plus_ivl |
|
386 |
and test_num' = in_ivl |
|
387 |
and filter_plus' = filter_plus_ivl and filter_less' = filter_less_ivl |
|
388 |
defines AI_ivl' is AI_wn |
|
389 |
.. |
|
390 |
||
391 |
||
392 |
subsubsection "Tests" |
|
393 |
||
49399 | 394 |
(* Trick to make the code generator happy. *) |
49396 | 395 |
lemma [code]: "equal_class.equal (x::'a st) y = equal_class.equal x y" |
396 |
by(rule refl) |
|
397 |
||
47613 | 398 |
definition "step_up_ivl n = |
399 |
((\<lambda>C. C \<nabla> step_ivl (top(strip C)) C)^^n)" |
|
400 |
definition "step_down_ivl n = |
|
401 |
((\<lambda>C. C \<triangle> step_ivl (top (strip C)) C)^^n)" |
|
402 |
||
403 |
text{* For @{const test3_ivl}, @{const AI_ivl} needed as many iterations as |
|
404 |
the loop took to execute. In contrast, @{const AI_ivl'} converges in a |
|
405 |
constant number of steps: *} |
|
406 |
||
407 |
value "show_acom (step_up_ivl 1 (bot test3_ivl))" |
|
408 |
value "show_acom (step_up_ivl 2 (bot test3_ivl))" |
|
409 |
value "show_acom (step_up_ivl 3 (bot test3_ivl))" |
|
410 |
value "show_acom (step_up_ivl 4 (bot test3_ivl))" |
|
411 |
value "show_acom (step_up_ivl 5 (bot test3_ivl))" |
|
49188 | 412 |
value "show_acom (step_up_ivl 6 (bot test3_ivl))" |
413 |
value "show_acom (step_up_ivl 7 (bot test3_ivl))" |
|
414 |
value "show_acom (step_up_ivl 8 (bot test3_ivl))" |
|
415 |
value "show_acom (step_down_ivl 1 (step_up_ivl 8 (bot test3_ivl)))" |
|
416 |
value "show_acom (step_down_ivl 2 (step_up_ivl 8 (bot test3_ivl)))" |
|
417 |
value "show_acom (step_down_ivl 3 (step_up_ivl 8 (bot test3_ivl)))" |
|
418 |
value "show_acom (step_down_ivl 4 (step_up_ivl 8 (bot test3_ivl)))" |
|
50995 | 419 |
value "show_acom (the(AI_ivl' test3_ivl))" |
47613 | 420 |
|
421 |
||
422 |
text{* Now all the analyses terminate: *} |
|
423 |
||
50995 | 424 |
value "show_acom (the(AI_ivl' test4_ivl))" |
425 |
value "show_acom (the(AI_ivl' test5_ivl))" |
|
426 |
value "show_acom (the(AI_ivl' test6_ivl))" |
|
47613 | 427 |
|
428 |
||
429 |
subsubsection "Generic Termination Proof" |
|
430 |
||
49547 | 431 |
locale Measure_WN = Measure1 where m=m for m :: "'av::WN \<Rightarrow> nat" + |
47613 | 432 |
fixes n :: "'av \<Rightarrow> nat" |
433 |
assumes m_widen: "~ y \<sqsubseteq> x \<Longrightarrow> m(x \<nabla> y) < m x" |
|
434 |
assumes n_mono: "x \<sqsubseteq> y \<Longrightarrow> n x \<le> n y" |
|
49576 | 435 |
assumes n_narrow: "y \<sqsubseteq> x \<Longrightarrow> ~ x \<sqsubseteq> x \<triangle> y \<Longrightarrow> n(x \<triangle> y) < n x" |
47613 | 436 |
|
437 |
begin |
|
438 |
||
49547 | 439 |
lemma m_s_widen: "S1 \<in> L X \<Longrightarrow> S2 \<in> L X \<Longrightarrow> finite X \<Longrightarrow> |
440 |
~ S2 \<sqsubseteq> S1 \<Longrightarrow> m_s(S1 \<nabla> S2) < m_s S1" |
|
441 |
proof(auto simp add: le_st_def m_s_def L_st_def widen_st_def) |
|
47613 | 442 |
assume "finite(dom S1)" |
443 |
have 1: "\<forall>x\<in>dom S1. m(fun S1 x) \<ge> m(fun S1 x \<nabla> fun S2 x)" |
|
49547 | 444 |
by (metis m1 WN_class.widen1) |
47613 | 445 |
fix x assume "x \<in> dom S1" "\<not> fun S2 x \<sqsubseteq> fun S1 x" |
446 |
hence 2: "EX x : dom S1. m(fun S1 x) > m(fun S1 x \<nabla> fun S2 x)" |
|
447 |
using m_widen by blast |
|
448 |
from setsum_strict_mono_ex1[OF `finite(dom S1)` 1 2] |
|
449 |
show "(\<Sum>x\<in>dom S1. m (fun S1 x \<nabla> fun S2 x)) < (\<Sum>x\<in>dom S1. m (fun S1 x))" . |
|
450 |
qed |
|
451 |
||
49547 | 452 |
lemma m_o_widen: "\<lbrakk> S1 \<in> L X; S2 \<in> L X; finite X; \<not> S2 \<sqsubseteq> S1 \<rbrakk> \<Longrightarrow> |
453 |
m_o (card X) (S1 \<nabla> S2) < m_o (card X) S1" |
|
454 |
by(auto simp: m_o_def L_st_def m_s_h less_Suc_eq_le m_s_widen |
|
455 |
split: option.split) |
|
47613 | 456 |
|
49547 | 457 |
lemma m_c_widen: |
458 |
"C1 \<in> Lc c \<Longrightarrow> C2 \<in> Lc c \<Longrightarrow> \<not> C2 \<sqsubseteq> C1 \<Longrightarrow> m_c (C1 \<nabla> C2) < m_c C1" |
|
459 |
apply(auto simp: Lc_acom_def m_c_def Let_def widen_acom_def) |
|
460 |
apply(subgoal_tac "length(annos C2) = length(annos C1)") |
|
461 |
prefer 2 apply (simp add: size_annos_same2) |
|
462 |
apply (auto) |
|
463 |
apply(rule setsum_strict_mono_ex1) |
|
464 |
apply simp |
|
465 |
apply (clarsimp) |
|
466 |
apply(simp add: m_o1 finite_cvars widen1[where c = "strip C2"]) |
|
467 |
apply(auto simp: le_iff_le_annos listrel_iff_nth) |
|
468 |
apply(rule_tac x=i in bexI) |
|
469 |
prefer 2 apply simp |
|
470 |
apply(rule m_o_widen) |
|
471 |
apply (simp add: finite_cvars)+ |
|
472 |
done |
|
473 |
||
474 |
||
49576 | 475 |
definition n_s :: "'av st \<Rightarrow> nat" ("n\<^isub>s") where |
476 |
"n\<^isub>s S = (\<Sum>x\<in>dom S. n(fun S x))" |
|
49547 | 477 |
|
49576 | 478 |
lemma n_s_mono: assumes "S1 \<sqsubseteq> S2" shows "n\<^isub>s S1 \<le> n\<^isub>s S2" |
47613 | 479 |
proof- |
480 |
from assms have [simp]: "dom S1 = dom S2" "\<forall>x\<in>dom S1. fun S1 x \<sqsubseteq> fun S2 x" |
|
481 |
by(simp_all add: le_st_def) |
|
482 |
have "(\<Sum>x\<in>dom S1. n(fun S1 x)) \<le> (\<Sum>x\<in>dom S1. n(fun S2 x))" |
|
483 |
by(rule setsum_mono)(simp add: le_st_def n_mono) |
|
49547 | 484 |
thus ?thesis by(simp add: n_s_def) |
47613 | 485 |
qed |
486 |
||
49547 | 487 |
lemma n_s_narrow: |
47613 | 488 |
assumes "finite(dom S1)" and "S2 \<sqsubseteq> S1" "\<not> S1 \<sqsubseteq> S1 \<triangle> S2" |
49576 | 489 |
shows "n\<^isub>s (S1 \<triangle> S2) < n\<^isub>s S1" |
47613 | 490 |
proof- |
491 |
from `S2\<sqsubseteq>S1` have [simp]: "dom S1 = dom S2" "\<forall>x\<in>dom S1. fun S2 x \<sqsubseteq> fun S1 x" |
|
492 |
by(simp_all add: le_st_def) |
|
493 |
have 1: "\<forall>x\<in>dom S1. n(fun (S1 \<triangle> S2) x) \<le> n(fun S1 x)" |
|
494 |
by(auto simp: le_st_def narrow_st_def n_mono WN_class.narrow2) |
|
495 |
have 2: "\<exists>x\<in>dom S1. n(fun (S1 \<triangle> S2) x) < n(fun S1 x)" using `\<not> S1 \<sqsubseteq> S1 \<triangle> S2` |
|
49576 | 496 |
by(force simp: le_st_def narrow_st_def intro: n_narrow) |
47613 | 497 |
have "(\<Sum>x\<in>dom S1. n(fun (S1 \<triangle> S2) x)) < (\<Sum>x\<in>dom S1. n(fun S1 x))" |
498 |
apply(rule setsum_strict_mono_ex1[OF `finite(dom S1)`]) using 1 2 by blast+ |
|
499 |
moreover have "dom (S1 \<triangle> S2) = dom S1" by(simp add: narrow_st_def) |
|
49547 | 500 |
ultimately show ?thesis by(simp add: n_s_def) |
47613 | 501 |
qed |
502 |
||
503 |
||
49576 | 504 |
definition n_o :: "'av st option \<Rightarrow> nat" ("n\<^isub>o") where |
505 |
"n\<^isub>o opt = (case opt of None \<Rightarrow> 0 | Some S \<Rightarrow> n\<^isub>s S + 1)" |
|
47613 | 506 |
|
49576 | 507 |
lemma n_o_mono: "S1 \<sqsubseteq> S2 \<Longrightarrow> n\<^isub>o S1 \<le> n\<^isub>o S2" |
49547 | 508 |
by(induction S1 S2 rule: le_option.induct)(auto simp: n_o_def n_s_mono) |
47613 | 509 |
|
510 |
lemma n_o_narrow: |
|
49396 | 511 |
"S1 \<in> L X \<Longrightarrow> S2 \<in> L X \<Longrightarrow> finite X |
49576 | 512 |
\<Longrightarrow> S2 \<sqsubseteq> S1 \<Longrightarrow> \<not> S1 \<sqsubseteq> S1 \<triangle> S2 \<Longrightarrow> n\<^isub>o (S1 \<triangle> S2) < n\<^isub>o S1" |
47613 | 513 |
apply(induction S1 S2 rule: narrow_option.induct) |
49547 | 514 |
apply(auto simp: n_o_def L_st_def n_s_narrow) |
47613 | 515 |
done |
516 |
||
49576 | 517 |
|
518 |
definition n_c :: "'av st option acom \<Rightarrow> nat" ("n\<^isub>c") where |
|
519 |
"n\<^isub>c C = (let as = annos C in \<Sum>i<size as. n\<^isub>o (as!i))" |
|
47613 | 520 |
|
49396 | 521 |
lemma n_c_narrow: "C1 \<in> Lc c \<Longrightarrow> C2 \<in> Lc c \<Longrightarrow> |
49576 | 522 |
C2 \<sqsubseteq> C1 \<Longrightarrow> \<not> C1 \<sqsubseteq> C1 \<triangle> C2 \<Longrightarrow> n\<^isub>c (C1 \<triangle> C2) < n\<^isub>c C1" |
49396 | 523 |
apply(auto simp: n_c_def Let_def Lc_acom_def narrow_acom_def) |
47613 | 524 |
apply(subgoal_tac "length(annos C2) = length(annos C1)") |
525 |
prefer 2 apply (simp add: size_annos_same2) |
|
526 |
apply (auto) |
|
527 |
apply(rule setsum_strict_mono_ex1) |
|
528 |
apply simp |
|
529 |
apply (clarsimp) |
|
530 |
apply(rule n_o_mono) |
|
531 |
apply(rule narrow2) |
|
532 |
apply(fastforce simp: le_iff_le_annos listrel_iff_nth) |
|
49496 | 533 |
apply(auto simp: le_iff_le_annos listrel_iff_nth) |
47613 | 534 |
apply(rule_tac x=i in bexI) |
535 |
prefer 2 apply simp |
|
536 |
apply(rule n_o_narrow[where X = "vars(strip C1)"]) |
|
537 |
apply (simp add: finite_cvars)+ |
|
538 |
done |
|
539 |
||
540 |
end |
|
541 |
||
542 |
||
543 |
lemma iter_widen_termination: |
|
49396 | 544 |
fixes m :: "'a::WN_Lc \<Rightarrow> nat" |
47613 | 545 |
assumes P_f: "\<And>C. P C \<Longrightarrow> P(f C)" |
546 |
and P_widen: "\<And>C1 C2. P C1 \<Longrightarrow> P C2 \<Longrightarrow> P(C1 \<nabla> C2)" |
|
547 |
and m_widen: "\<And>C1 C2. P C1 \<Longrightarrow> P C2 \<Longrightarrow> ~ C2 \<sqsubseteq> C1 \<Longrightarrow> m(C1 \<nabla> C2) < m C1" |
|
548 |
and "P C" shows "EX C'. iter_widen f C = Some C'" |
|
49547 | 549 |
proof(simp add: iter_widen_def, |
550 |
rule measure_while_option_Some[where P = P and f=m]) |
|
47613 | 551 |
show "P C" by(rule `P C`) |
552 |
next |
|
49547 | 553 |
fix C assume "P C" "\<not> f C \<sqsubseteq> C" thus "P (C \<nabla> f C) \<and> m (C \<nabla> f C) < m C" |
554 |
by(simp add: P_f P_widen m_widen) |
|
47613 | 555 |
qed |
49496 | 556 |
|
47613 | 557 |
lemma iter_narrow_termination: |
49396 | 558 |
fixes n :: "'a::WN_Lc \<Rightarrow> nat" |
47613 | 559 |
assumes P_f: "\<And>C. P C \<Longrightarrow> P(f C)" |
560 |
and P_narrow: "\<And>C1 C2. P C1 \<Longrightarrow> P C2 \<Longrightarrow> P(C1 \<triangle> C2)" |
|
561 |
and mono: "\<And>C1 C2. P C1 \<Longrightarrow> P C2 \<Longrightarrow> C1 \<sqsubseteq> C2 \<Longrightarrow> f C1 \<sqsubseteq> f C2" |
|
562 |
and n_narrow: "\<And>C1 C2. P C1 \<Longrightarrow> P C2 \<Longrightarrow> C2 \<sqsubseteq> C1 \<Longrightarrow> ~ C1 \<sqsubseteq> C1 \<triangle> C2 \<Longrightarrow> n(C1 \<triangle> C2) < n C1" |
|
563 |
and init: "P C" "f C \<sqsubseteq> C" shows "EX C'. iter_narrow f C = Some C'" |
|
49547 | 564 |
proof(simp add: iter_narrow_def, |
565 |
rule measure_while_option_Some[where f=n and P = "%C. P C \<and> f C \<sqsubseteq> C"]) |
|
47613 | 566 |
show "P C \<and> f C \<sqsubseteq> C" using init by blast |
567 |
next |
|
49547 | 568 |
fix C assume 1: "P C \<and> f C \<sqsubseteq> C" and 2: "\<not> C \<sqsubseteq> C \<triangle> f C" |
47613 | 569 |
hence "P (C \<triangle> f C)" by(simp add: P_f P_narrow) |
570 |
moreover then have "f (C \<triangle> f C) \<sqsubseteq> C \<triangle> f C" |
|
571 |
by (metis narrow1 narrow2 1 mono preord_class.le_trans) |
|
49547 | 572 |
moreover have "n (C \<triangle> f C) < n C" using 1 2 by(simp add: n_narrow P_f) |
573 |
ultimately show "(P (C \<triangle> f C) \<and> f (C \<triangle> f C) \<sqsubseteq> C \<triangle> f C) \<and> n(C \<triangle> f C) < n C" |
|
574 |
by blast |
|
47613 | 575 |
qed |
576 |
||
49547 | 577 |
locale Abs_Int2_measure = |
578 |
Abs_Int2 where \<gamma>=\<gamma> + Measure_WN where m=m |
|
579 |
for \<gamma> :: "'av::{WN,lattice} \<Rightarrow> val set" and m :: "'av \<Rightarrow> nat" |
|
580 |
||
47613 | 581 |
|
582 |
subsubsection "Termination: Intervals" |
|
583 |
||
584 |
definition m_ivl :: "ivl \<Rightarrow> nat" where |
|
49579 | 585 |
"m_ivl ivl = (case ivl of Ivl l h \<Rightarrow> |
47613 | 586 |
(case l of None \<Rightarrow> 0 | Some _ \<Rightarrow> 1) + (case h of None \<Rightarrow> 0 | Some _ \<Rightarrow> 1))" |
587 |
||
588 |
lemma m_ivl_height: "m_ivl ivl \<le> 2" |
|
589 |
by(simp add: m_ivl_def split: ivl.split option.split) |
|
590 |
||
591 |
lemma m_ivl_anti_mono: "(y::ivl) \<sqsubseteq> x \<Longrightarrow> m_ivl x \<le> m_ivl y" |
|
592 |
by(auto simp: m_ivl_def le_option_def le_ivl_def |
|
593 |
split: ivl.split option.split if_splits) |
|
594 |
||
595 |
lemma m_ivl_widen: |
|
596 |
"~ y \<sqsubseteq> x \<Longrightarrow> m_ivl(x \<nabla> y) < m_ivl x" |
|
597 |
by(auto simp: m_ivl_def widen_ivl_def le_option_def le_ivl_def |
|
598 |
split: ivl.splits option.splits if_splits) |
|
599 |
||
600 |
definition n_ivl :: "ivl \<Rightarrow> nat" where |
|
601 |
"n_ivl ivl = 2 - m_ivl ivl" |
|
602 |
||
603 |
lemma n_ivl_mono: "(x::ivl) \<sqsubseteq> y \<Longrightarrow> n_ivl x \<le> n_ivl y" |
|
604 |
unfolding n_ivl_def by (metis diff_le_mono2 m_ivl_anti_mono) |
|
605 |
||
606 |
lemma n_ivl_narrow: |
|
607 |
"~ x \<sqsubseteq> x \<triangle> y \<Longrightarrow> n_ivl(x \<triangle> y) < n_ivl x" |
|
608 |
by(auto simp: n_ivl_def m_ivl_def narrow_ivl_def le_option_def le_ivl_def |
|
609 |
split: ivl.splits option.splits if_splits) |
|
610 |
||
611 |
||
612 |
interpretation Abs_Int2_measure |
|
613 |
where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = plus_ivl |
|
614 |
and test_num' = in_ivl |
|
615 |
and filter_plus' = filter_plus_ivl and filter_less' = filter_less_ivl |
|
616 |
and m = m_ivl and n = n_ivl and h = 2 |
|
617 |
proof |
|
618 |
case goal1 thus ?case by(rule m_ivl_anti_mono) |
|
619 |
next |
|
49547 | 620 |
case goal2 thus ?case by(rule m_ivl_height) |
47613 | 621 |
next |
49547 | 622 |
case goal3 thus ?case by(rule m_ivl_widen) |
47613 | 623 |
next |
624 |
case goal4 thus ?case by(rule n_ivl_mono) |
|
625 |
next |
|
49576 | 626 |
case goal5 from goal5(2) show ?case by(rule n_ivl_narrow) |
627 |
-- "note that the first assms is unnecessary for intervals" |
|
47613 | 628 |
qed |
629 |
||
630 |
||
631 |
lemma iter_winden_step_ivl_termination: |
|
632 |
"\<exists>C. iter_widen (step_ivl (top c)) (bot c) = Some C" |
|
49396 | 633 |
apply(rule iter_widen_termination[where m = "m_c" and P = "%C. C \<in> Lc c"]) |
634 |
apply (simp_all add: step'_in_Lc m_c_widen) |
|
47613 | 635 |
done |
636 |
||
637 |
lemma iter_narrow_step_ivl_termination: |
|
49396 | 638 |
"C0 \<in> Lc c \<Longrightarrow> step_ivl (top c) C0 \<sqsubseteq> C0 \<Longrightarrow> |
47613 | 639 |
\<exists>C. iter_narrow (step_ivl (top c)) C0 = Some C" |
49396 | 640 |
apply(rule iter_narrow_termination[where n = "n_c" and P = "%C. C \<in> Lc c"]) |
641 |
apply (simp add: step'_in_Lc) |
|
47613 | 642 |
apply (simp) |
643 |
apply(rule mono_step'_top) |
|
49396 | 644 |
apply(simp add: Lc_acom_def L_acom_def) |
645 |
apply(simp add: Lc_acom_def L_acom_def) |
|
47613 | 646 |
apply assumption |
647 |
apply(erule (3) n_c_narrow) |
|
648 |
apply assumption |
|
649 |
apply assumption |
|
650 |
done |
|
651 |
||
652 |
theorem AI_ivl'_termination: |
|
653 |
"\<exists>C. AI_ivl' c = Some C" |
|
654 |
apply(auto simp: AI_wn_def pfp_wn_def iter_winden_step_ivl_termination |
|
655 |
split: option.split) |
|
656 |
apply(rule iter_narrow_step_ivl_termination) |
|
49396 | 657 |
apply(blast intro: iter_widen_inv[where f = "step' \<top>\<^bsub>c\<^esub>" and P = "%C. C \<in> Lc c"] bot_in_Lc Lc_widen step'_in_Lc[where S = "\<top>\<^bsub>c\<^esub>" and c=c, simplified]) |
47613 | 658 |
apply(erule iter_widen_pfp) |
659 |
done |
|
660 |
||
661 |
(*unused_thms Abs_Int_init -*) |
|
662 |
||
49578 | 663 |
subsubsection "Counterexamples" |
664 |
||
665 |
text{* Widening is increasing by assumption, |
|
666 |
but @{prop"x \<sqsubseteq> f x"} is not an invariant of widening. It can already |
|
667 |
be lost after the first step: *} |
|
668 |
||
669 |
lemma assumes "!!x y::'a::WN. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y" |
|
670 |
and "x \<sqsubseteq> f x" and "\<not> f x \<sqsubseteq> x" shows "x \<nabla> f x \<sqsubseteq> f(x \<nabla> f x)" |
|
671 |
nitpick[card = 3, expect = genuine, show_consts] |
|
672 |
(* |
|
673 |
1 < 2 < 3, |
|
674 |
f x = 2, |
|
675 |
x widen y = 3 -- guarantees termination with top=3 |
|
676 |
x = 1 |
|
677 |
Now f is mono, x <= f x, not f x <= x |
|
678 |
but x widen f x = 3, f 3 = 2, but not 3 <= 2 |
|
679 |
*) |
|
680 |
oops |
|
681 |
||
682 |
text{* Widening terminates but may converge more slowly than Kleene iteration. |
|
683 |
In the following model, Kleene iteration goes from 0 to the least pfp |
|
684 |
in one step but widening takes 2 steps to reach a strictly larger pfp: *} |
|
685 |
lemma assumes "!!x y::'a::WN. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y" |
|
686 |
and "x \<sqsubseteq> f x" and "\<not> f x \<sqsubseteq> x" and "f(f x) \<sqsubseteq> f x" |
|
687 |
shows "f(x \<nabla> f x) \<sqsubseteq> x \<nabla> f x" |
|
688 |
nitpick[card = 4, expect = genuine, show_consts] |
|
689 |
(* |
|
690 |
||
691 |
0 < 1 < 2 < 3 |
|
692 |
f: 1 1 3 3 |
|
693 |
||
694 |
0 widen 1 = 2 |
|
695 |
2 widen 3 = 3 |
|
696 |
and x widen y arbitrary, eg 3, which guarantees termination |
|
697 |
||
698 |
Kleene: f(f 0) = f 1 = 1 <= 1 = f 1 |
|
699 |
||
700 |
but |
|
701 |
||
702 |
because not f 0 <= 0, we obtain 0 widen f 0 = 0 wide 1 = 2, |
|
703 |
which is again not a pfp: not f 2 = 3 <= 2 |
|
704 |
Another widening step yields 2 widen f 2 = 2 widen 3 = 3 |
|
705 |
*) |
|
49892
09956f7a00af
proper 'oops' to force sequential checking here, and avoid spurious *** Interrupt stemming from crash of forked outer syntax element;
wenzelm
parents:
49579
diff
changeset
|
706 |
oops |
49578 | 707 |
|
47613 | 708 |
end |