--- a/src/HOL/IMP/Abs_Int2.thy Tue May 07 03:24:23 2013 +0200
+++ b/src/HOL/IMP/Abs_Int2.thy Tue May 07 10:34:55 2013 +0200
@@ -48,12 +48,12 @@
locale Val_abs1 = Val_abs1_gamma where \<gamma> = \<gamma>
for \<gamma> :: "'av::bounded_lattice \<Rightarrow> val set" +
fixes test_num' :: "val \<Rightarrow> 'av \<Rightarrow> bool"
-and filter_plus' :: "'av \<Rightarrow> 'av \<Rightarrow> 'av \<Rightarrow> 'av * 'av"
-and filter_less' :: "bool \<Rightarrow> 'av \<Rightarrow> 'av \<Rightarrow> 'av * 'av"
+and constrain_plus' :: "'av \<Rightarrow> 'av \<Rightarrow> 'av \<Rightarrow> 'av * 'av"
+and constrain_less' :: "bool \<Rightarrow> 'av \<Rightarrow> 'av \<Rightarrow> 'av * 'av"
assumes test_num': "test_num' i a = (i : \<gamma> a)"
-and filter_plus': "filter_plus' a a1 a2 = (a\<^isub>1',a\<^isub>2') \<Longrightarrow>
+and constrain_plus': "constrain_plus' a a1 a2 = (a\<^isub>1',a\<^isub>2') \<Longrightarrow>
i1 : \<gamma> a1 \<Longrightarrow> i2 : \<gamma> a2 \<Longrightarrow> i1+i2 : \<gamma> a \<Longrightarrow> i1 : \<gamma> a\<^isub>1' \<and> i2 : \<gamma> a\<^isub>2'"
-and filter_less': "filter_less' (i1<i2) a1 a2 = (a\<^isub>1',a\<^isub>2') \<Longrightarrow>
+and constrain_less': "constrain_less' (i1<i2) a1 a2 = (a\<^isub>1',a\<^isub>2') \<Longrightarrow>
i1 : \<gamma> a1 \<Longrightarrow> i2 : \<gamma> a2 \<Longrightarrow> i1 : \<gamma> a\<^isub>1' \<and> i2 : \<gamma> a\<^isub>2'"
@@ -74,14 +74,14 @@
subsubsection "Backward analysis"
-fun afilter :: "aexp \<Rightarrow> 'av \<Rightarrow> 'av st option \<Rightarrow> '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 \<Rightarrow> None | Some S \<Rightarrow>
+fun aconstrain :: "aexp \<Rightarrow> 'av \<Rightarrow> 'av st option \<Rightarrow> 'av st option" where
+"aconstrain (N n) a S = (if test_num' n a then S else None)" |
+"aconstrain (V x) a S = (case S of None \<Rightarrow> None | Some S \<Rightarrow>
let a' = fun S x \<sqinter> a in
if a' = \<bottom> 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))"
+"aconstrain (Plus e1 e2) a S =
+ (let (a1,a2) = constrain_plus' a (aval'' e1 S) (aval'' e2 S)
+ in aconstrain e1 a1 (aconstrain e2 a2 S))"
text{* The test for @{const bot} in the @{const V}-case is important: @{const
bot} indicates that a variable has no possible values, i.e.\ that the current
@@ -93,17 +93,17 @@
making the analysis less precise. *}
-fun bfilter :: "bexp \<Rightarrow> bool \<Rightarrow> 'av st option \<Rightarrow> 'av st option" where
-"bfilter (Bc v) res S = (if v=res then S else None)" |
-"bfilter (Not b) res S = bfilter b (\<not> 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 (a1,a2) = filter_less' res (aval'' e1 S) (aval'' e2 S)
- in afilter e1 a1 (afilter e2 a2 S))"
+fun bconstrain :: "bexp \<Rightarrow> bool \<Rightarrow> 'av st option \<Rightarrow> 'av st option" where
+"bconstrain (Bc v) res S = (if v=res then S else None)" |
+"bconstrain (Not b) res S = bconstrain b (\<not> res) S" |
+"bconstrain (And b1 b2) res S =
+ (if res then bconstrain b1 True (bconstrain b2 True S)
+ else bconstrain b1 False S \<squnion> bconstrain b2 False S)" |
+"bconstrain (Less e1 e2) res S =
+ (let (a1,a2) = constrain_less' res (aval'' e1 S) (aval'' e2 S)
+ in aconstrain e1 a1 (aconstrain e2 a2 S))"
-lemma afilter_sound: "s : \<gamma>\<^isub>o S \<Longrightarrow> aval e s : \<gamma> a \<Longrightarrow> s : \<gamma>\<^isub>o (afilter e a S)"
+lemma aconstrain_sound: "s : \<gamma>\<^isub>o S \<Longrightarrow> aval e s : \<gamma> a \<Longrightarrow> s : \<gamma>\<^isub>o (aconstrain e a S)"
proof(induction e arbitrary: a S)
case N thus ?case by simp (metis test_num')
next
@@ -118,11 +118,11 @@
(metis mono_gamma emptyE in_gamma_inf gamma_bot subset_empty)
next
case (Plus e1 e2) thus ?case
- using filter_plus'[OF _ aval''_sound aval''_sound]
+ using constrain_plus'[OF _ aval''_sound aval''_sound]
by (auto split: prod.split)
qed
-lemma bfilter_sound: "s : \<gamma>\<^isub>o S \<Longrightarrow> bv = bval b s \<Longrightarrow> s : \<gamma>\<^isub>o(bfilter b bv S)"
+lemma bconstrain_sound: "s : \<gamma>\<^isub>o S \<Longrightarrow> bv = bval b s \<Longrightarrow> s : \<gamma>\<^isub>o(bconstrain b bv S)"
proof(induction b arbitrary: S bv)
case Bc thus ?case by simp
next
@@ -133,12 +133,12 @@
next
case (Less e1 e2) thus ?case
by(auto split: prod.split)
- (metis (lifting) afilter_sound aval''_sound filter_less')
+ (metis (lifting) aconstrain_sound aval''_sound constrain_less')
qed
definition "step' = Step
(\<lambda>x e S. case S of None \<Rightarrow> None | Some S \<Rightarrow> Some(update S x (aval' e S)))
- (\<lambda>b S. bfilter b True S)"
+ (\<lambda>b S. bconstrain b True S)"
definition AI :: "com \<Rightarrow> 'av st option acom option" where
"AI c = pfp (step' \<top>) (bot c)"
@@ -146,23 +146,23 @@
lemma strip_step'[simp]: "strip(step' S c) = strip c"
by(simp add: step'_def)
-lemma top_on_afilter: "\<lbrakk> top_on_opt S X; vars e \<subseteq> -X \<rbrakk> \<Longrightarrow> top_on_opt (afilter e a S) X"
+lemma top_on_aconstrain: "\<lbrakk> top_on_opt S X; vars e \<subseteq> -X \<rbrakk> \<Longrightarrow> top_on_opt (aconstrain e a S) X"
by(induction e arbitrary: a S) (auto simp: Let_def split: option.splits prod.split)
-lemma top_on_bfilter: "\<lbrakk>top_on_opt S X; vars b \<subseteq> -X\<rbrakk> \<Longrightarrow> top_on_opt (bfilter b r S) X"
-by(induction b arbitrary: r S) (auto simp: top_on_afilter top_on_sup split: prod.split)
+lemma top_on_bconstrain: "\<lbrakk>top_on_opt S X; vars b \<subseteq> -X\<rbrakk> \<Longrightarrow> top_on_opt (bconstrain b r S) X"
+by(induction b arbitrary: r S) (auto simp: top_on_aconstrain top_on_sup split: prod.split)
lemma top_on_step': "top_on_acom C (- vars C) \<Longrightarrow> top_on_acom (step' \<top> C) (- vars C)"
unfolding step'_def
by(rule top_on_Step)
- (auto simp add: top_on_top top_on_bfilter split: option.split)
+ (auto simp add: top_on_top top_on_bconstrain split: option.split)
subsubsection "Soundness"
lemma step_step': "step (\<gamma>\<^isub>o S) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c (step' S C)"
unfolding step_def step'_def
by(rule gamma_Step_subcomm)
- (auto simp: intro!: aval'_sound bfilter_sound in_gamma_update split: option.splits)
+ (auto simp: intro!: aval'_sound bconstrain_sound in_gamma_update split: option.splits)
lemma AI_sound: "AI c = Some C \<Longrightarrow> CS c \<le> \<gamma>\<^isub>c C"
proof(simp add: CS_def AI_def)
@@ -186,10 +186,10 @@
locale Abs_Int1_mono = Abs_Int1 +
assumes mono_plus': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow> plus' a1 a2 \<le> plus' b1 b2"
-and mono_filter_plus': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow> r \<le> r' \<Longrightarrow>
- filter_plus' r a1 a2 \<le> filter_plus' r' b1 b2"
-and mono_filter_less': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow>
- filter_less' bv a1 a2 \<le> filter_less' bv b1 b2"
+and mono_constrain_plus': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow> r \<le> r' \<Longrightarrow>
+ constrain_plus' r a1 a2 \<le> constrain_plus' r' b1 b2"
+and mono_constrain_less': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow>
+ constrain_less' bv a1 a2 \<le> constrain_less' bv b1 b2"
begin
lemma mono_aval':
@@ -204,28 +204,28 @@
apply simp
by (simp add: mono_aval')
-lemma mono_afilter: "r1 \<le> r2 \<Longrightarrow> S1 \<le> S2 \<Longrightarrow> afilter e r1 S1 \<le> afilter e r2 S2"
+lemma mono_aconstrain: "r1 \<le> r2 \<Longrightarrow> S1 \<le> S2 \<Longrightarrow> aconstrain e r1 S1 \<le> aconstrain e r2 S2"
apply(induction e arbitrary: r1 r2 S1 S2)
apply(auto simp: test_num' Let_def inf_mono split: option.splits prod.splits)
apply (metis mono_gamma subsetD)
apply (metis le_bot inf_mono le_st_iff)
apply (metis inf_mono mono_update le_st_iff)
-apply(metis mono_aval'' mono_filter_plus'[simplified less_eq_prod_def] fst_conv snd_conv)
+apply(metis mono_aval'' mono_constrain_plus'[simplified less_eq_prod_def] fst_conv snd_conv)
done
-lemma mono_bfilter: "S1 \<le> S2 \<Longrightarrow> bfilter b bv S1 \<le> bfilter b bv S2"
+lemma mono_bconstrain: "S1 \<le> S2 \<Longrightarrow> bconstrain b bv S1 \<le> bconstrain b bv S2"
apply(induction b arbitrary: bv S1 S2)
apply(simp)
apply(simp)
apply simp
apply(metis order_trans[OF _ sup_ge1] order_trans[OF _ sup_ge2])
apply (simp split: prod.splits)
-apply(metis mono_aval'' mono_afilter mono_filter_less'[simplified less_eq_prod_def] fst_conv snd_conv)
+apply(metis mono_aval'' mono_aconstrain mono_constrain_less'[simplified less_eq_prod_def] fst_conv snd_conv)
done
theorem mono_step': "S1 \<le> S2 \<Longrightarrow> C1 \<le> C2 \<Longrightarrow> step' S1 C1 \<le> step' S2 C2"
unfolding step'_def
-by(rule mono2_Step) (auto simp: mono_aval' mono_bfilter split: option.split)
+by(rule mono2_Step) (auto simp: mono_aval' mono_bconstrain split: option.split)
lemma mono_step'_top: "C1 \<le> C2 \<Longrightarrow> step' \<top> C1 \<le> step' \<top> C2"
by (metis mono_step' order_refl)
--- a/src/HOL/IMP/Abs_Int2_ivl.thy Tue May 07 03:24:23 2013 +0200
+++ b/src/HOL/IMP/Abs_Int2_ivl.thy Tue May 07 10:34:55 2013 +0200
@@ -259,8 +259,8 @@
end
-definition filter_plus_ivl :: "ivl \<Rightarrow> ivl \<Rightarrow> ivl \<Rightarrow> ivl*ivl" where
-"filter_plus_ivl iv iv1 iv2 = (iv1 \<sqinter> (iv - iv2), iv2 \<sqinter> (iv - iv1))"
+definition constrain_plus_ivl :: "ivl \<Rightarrow> ivl \<Rightarrow> ivl \<Rightarrow> ivl*ivl" where
+"constrain_plus_ivl iv iv1 iv2 = (iv1 \<sqinter> (iv - iv2), iv2 \<sqinter> (iv - iv1))"
definition above_rep :: "eint2 \<Rightarrow> eint2" where
"above_rep p = (if is_empty_rep p then empty_rep else let (l,h) = p in (l,\<infinity>))"
@@ -284,8 +284,8 @@
(auto simp add: below_rep_def \<gamma>_rep_cases is_empty_rep_def
split: extended.splits)
-definition filter_less_ivl :: "bool \<Rightarrow> ivl \<Rightarrow> ivl \<Rightarrow> ivl * ivl" where
-"filter_less_ivl res iv1 iv2 =
+definition constrain_less_ivl :: "bool \<Rightarrow> ivl \<Rightarrow> ivl \<Rightarrow> ivl * ivl" where
+"constrain_less_ivl res iv1 iv2 =
(if res
then (iv1 \<sqinter> (below iv2 - [Fin 1\<dots>Fin 1]),
iv2 \<sqinter> (above iv1 + [Fin 1\<dots>Fin 1]))
@@ -333,18 +333,18 @@
interpretation Val_abs1
where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
and test_num' = in_ivl
-and filter_plus' = filter_plus_ivl and filter_less' = filter_less_ivl
+and constrain_plus' = constrain_plus_ivl and constrain_less' = constrain_less_ivl
proof
case goal1 thus ?case by transfer (auto simp: \<gamma>_rep_def)
next
case goal2 thus ?case
- unfolding filter_plus_ivl_def minus_ivl_def
+ unfolding constrain_plus_ivl_def minus_ivl_def
apply(clarsimp simp add: \<gamma>_inf)
using gamma_plus'[of "i1+i2" _ "-i1"] gamma_plus'[of "i1+i2" _ "-i2"]
by(simp add: \<gamma>_uminus)
next
case goal3 thus ?case
- unfolding filter_less_ivl_def minus_ivl_def
+ unfolding constrain_less_ivl_def minus_ivl_def
apply(clarsimp simp add: \<gamma>_inf split: if_splits)
using gamma_plus'[of "i1+1" _ "-1"] gamma_plus'[of "i2 - 1" _ "1"]
apply(simp add: \<gamma>_belowI[of i2] \<gamma>_aboveI[of i1]
@@ -356,9 +356,9 @@
interpretation Abs_Int1
where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
and test_num' = in_ivl
-and filter_plus' = filter_plus_ivl and filter_less' = filter_less_ivl
-defines afilter_ivl is afilter
-and bfilter_ivl is bfilter
+and constrain_plus' = constrain_plus_ivl and constrain_less' = constrain_less_ivl
+defines aconstrain_ivl is aconstrain
+and bconstrain_ivl is bconstrain
and step_ivl is step'
and AI_ivl is AI
and aval_ivl' is aval''
@@ -390,16 +390,16 @@
interpretation Abs_Int1_mono
where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
and test_num' = in_ivl
-and filter_plus' = filter_plus_ivl and filter_less' = filter_less_ivl
+and constrain_plus' = constrain_plus_ivl and constrain_less' = constrain_less_ivl
proof
case goal1 thus ?case by (rule mono_plus_ivl)
next
case goal2 thus ?case
- unfolding filter_plus_ivl_def minus_ivl_def less_eq_prod_def
+ unfolding constrain_plus_ivl_def minus_ivl_def less_eq_prod_def
by (auto simp: le_infI1 le_infI2 mono_plus_ivl mono_minus_ivl)
next
case goal3 thus ?case
- unfolding less_eq_prod_def filter_less_ivl_def minus_ivl_def
+ unfolding less_eq_prod_def constrain_less_ivl_def minus_ivl_def
by (auto simp: le_infI1 le_infI2 mono_plus_ivl mono_above mono_below)
qed
--- a/src/HOL/IMP/Abs_Int3.thy Tue May 07 03:24:23 2013 +0200
+++ b/src/HOL/IMP/Abs_Int3.thy Tue May 07 10:34:55 2013 +0200
@@ -271,7 +271,7 @@
interpretation Abs_Int2
where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
and test_num' = in_ivl
-and filter_plus' = filter_plus_ivl and filter_less' = filter_less_ivl
+and constrain_plus' = constrain_plus_ivl and constrain_less' = constrain_less_ivl
defines AI_ivl' is AI_wn
..
@@ -551,7 +551,7 @@
interpretation Abs_Int2_measure
where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
and test_num' = in_ivl
-and filter_plus' = filter_plus_ivl and filter_less' = filter_less_ivl
+and constrain_plus' = constrain_plus_ivl and constrain_less' = constrain_less_ivl
and m = m_ivl and n = n_ivl and h = 3
proof
case goal2 thus ?case by(rule m_ivl_anti_mono)