author | haftmann |
Wed, 19 Feb 2014 22:05:15 +0100 | |
changeset 55600 | 3c7610b8dcfc |
parent 55599 | 6535c537b243 |
child 58249 | 180f1b3508ed |
permissions | -rw-r--r-- |
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(* Author: Tobias Nipkow *) |
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theory Abs_Int1_const_ITP |
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imports Abs_Int1_ITP "../Abs_Int_Tests" |
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begin |
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subsection "Constant Propagation" |
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datatype const = Const val | Any |
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fun \<gamma>_const where |
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"\<gamma>_const (Const n) = {n}" | |
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"\<gamma>_const (Any) = UNIV" |
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fun plus_const where |
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"plus_const (Const m) (Const n) = Const(m+n)" | |
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"plus_const _ _ = Any" |
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lemma plus_const_cases: "plus_const a1 a2 = |
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(case (a1,a2) of (Const m, Const n) \<Rightarrow> Const(m+n) | _ \<Rightarrow> Any)" |
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by(auto split: prod.split const.split) |
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instantiation const :: SL_top |
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begin |
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fun le_const where |
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"_ \<sqsubseteq> Any = True" | |
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"Const n \<sqsubseteq> Const m = (n=m)" | |
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"Any \<sqsubseteq> Const _ = False" |
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fun join_const where |
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"Const m \<squnion> Const n = (if n=m then Const m else Any)" | |
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"_ \<squnion> _ = Any" |
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definition "\<top> = Any" |
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instance |
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proof |
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case goal1 thus ?case by (cases x) simp_all |
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next |
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case goal2 thus ?case by(cases z, cases y, cases x, simp_all) |
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next |
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case goal3 thus ?case by(cases x, cases y, simp_all) |
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next |
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case goal4 thus ?case by(cases y, cases x, simp_all) |
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next |
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case goal5 thus ?case by(cases z, cases y, cases x, simp_all) |
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next |
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case goal6 thus ?case by(simp add: Top_const_def) |
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qed |
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end |
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45127
d2eb07a1e01b
separated monotonicity reasoning and defined narrowing with while_option
nipkow
parents:
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diff
changeset
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55599
6535c537b243
aggiornamento for "interpretation with definitions": operate uniformly on theory and locale level under the brand of "permanent interpretation"
haftmann
parents:
48480
diff
changeset
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permanent_interpretation Val_abs |
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where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const |
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proof |
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case goal1 thus ?case |
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by(cases a, cases b, simp, simp, cases b, simp, simp) |
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next |
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case goal2 show ?case by(simp add: Top_const_def) |
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next |
45127
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separated monotonicity reasoning and defined narrowing with while_option
nipkow
parents:
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diff
changeset
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case goal3 show ?case by simp |
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next |
45127
d2eb07a1e01b
separated monotonicity reasoning and defined narrowing with while_option
nipkow
parents:
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changeset
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case goal4 thus ?case |
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by(auto simp: plus_const_cases split: const.split) |
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qed |
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55599
6535c537b243
aggiornamento for "interpretation with definitions": operate uniformly on theory and locale level under the brand of "permanent interpretation"
haftmann
parents:
48480
diff
changeset
|
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permanent_interpretation Abs_Int |
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where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const |
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more convenient syntax for permanent interpretation
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parents:
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defining AI_const = AI and step_const = step' and aval'_const = aval' |
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removed accidental dependance of abstract interpreter on gamma
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.. |
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subsubsection "Tests" |
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value "show_acom (((step_const \<top>)^^0) (\<bottom>\<^sub>c test1_const))" |
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value "show_acom (((step_const \<top>)^^1) (\<bottom>\<^sub>c test1_const))" |
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value "show_acom (((step_const \<top>)^^2) (\<bottom>\<^sub>c test1_const))" |
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value "show_acom (((step_const \<top>)^^3) (\<bottom>\<^sub>c test1_const))" |
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value "show_acom_opt (AI_const test1_const)" |
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value "show_acom_opt (AI_const test2_const)" |
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value "show_acom_opt (AI_const test3_const)" |
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value "show_acom (((step_const \<top>)^^0) (\<bottom>\<^sub>c test4_const))" |
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value "show_acom (((step_const \<top>)^^1) (\<bottom>\<^sub>c test4_const))" |
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value "show_acom (((step_const \<top>)^^2) (\<bottom>\<^sub>c test4_const))" |
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value "show_acom (((step_const \<top>)^^3) (\<bottom>\<^sub>c test4_const))" |
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value "show_acom_opt (AI_const test4_const)" |
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value "show_acom (((step_const \<top>)^^0) (\<bottom>\<^sub>c test5_const))" |
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value "show_acom (((step_const \<top>)^^1) (\<bottom>\<^sub>c test5_const))" |
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value "show_acom (((step_const \<top>)^^2) (\<bottom>\<^sub>c test5_const))" |
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value "show_acom (((step_const \<top>)^^3) (\<bottom>\<^sub>c test5_const))" |
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value "show_acom (((step_const \<top>)^^4) (\<bottom>\<^sub>c test5_const))" |
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value "show_acom (((step_const \<top>)^^5) (\<bottom>\<^sub>c test5_const))" |
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value "show_acom_opt (AI_const test5_const)" |
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value "show_acom (((step_const \<top>)^^0) (\<bottom>\<^sub>c test6_const))" |
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value "show_acom (((step_const \<top>)^^1) (\<bottom>\<^sub>c test6_const))" |
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value "show_acom (((step_const \<top>)^^2) (\<bottom>\<^sub>c test6_const))" |
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value "show_acom (((step_const \<top>)^^3) (\<bottom>\<^sub>c test6_const))" |
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value "show_acom (((step_const \<top>)^^4) (\<bottom>\<^sub>c test6_const))" |
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value "show_acom (((step_const \<top>)^^5) (\<bottom>\<^sub>c test6_const))" |
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value "show_acom (((step_const \<top>)^^6) (\<bottom>\<^sub>c test6_const))" |
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value "show_acom (((step_const \<top>)^^7) (\<bottom>\<^sub>c test6_const))" |
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value "show_acom (((step_const \<top>)^^8) (\<bottom>\<^sub>c test6_const))" |
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value "show_acom (((step_const \<top>)^^9) (\<bottom>\<^sub>c test6_const))" |
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value "show_acom (((step_const \<top>)^^10) (\<bottom>\<^sub>c test6_const))" |
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value "show_acom (((step_const \<top>)^^11) (\<bottom>\<^sub>c test6_const))" |
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value "show_acom_opt (AI_const test6_const)" |
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45623
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Abstract interpretation is now based uniformly on annotated programs,
nipkow
parents:
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text{* Monotonicity: *} |
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55599
6535c537b243
aggiornamento for "interpretation with definitions": operate uniformly on theory and locale level under the brand of "permanent interpretation"
haftmann
parents:
48480
diff
changeset
|
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permanent_interpretation Abs_Int_mono |
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where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const |
45623
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Abstract interpretation is now based uniformly on annotated programs,
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parents:
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proof |
f682f3f7b726
Abstract interpretation is now based uniformly on annotated programs,
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parents:
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case goal1 thus ?case |
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by(auto simp: plus_const_cases split: const.split) |
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Abstract interpretation is now based uniformly on annotated programs,
nipkow
parents:
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qed |
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text{* Termination: *} |
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definition "m_const x = (case x of Const _ \<Rightarrow> 1 | Any \<Rightarrow> 0)" |
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lemma measure_const: |
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"(strict{(x::const,y). x \<sqsubseteq> y})^-1 \<subseteq> measure m_const" |
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by(auto simp: m_const_def split: const.splits) |
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lemma measure_const_eq: |
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"\<forall> x y::const. x \<sqsubseteq> y \<and> y \<sqsubseteq> x \<longrightarrow> m_const x = m_const y" |
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by(auto simp: m_const_def split: const.splits) |
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lemma "EX c'. AI_const c = Some c'" |
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by(rule AI_Some_measure[OF measure_const measure_const_eq]) |
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end |