author | haftmann |
Wed, 19 Feb 2014 22:05:05 +0100 | |
changeset 55599 | 6535c537b243 |
parent 48480 | cb03acfae211 |
child 55600 | 3c7610b8dcfc |
permissions | -rw-r--r-- |
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(* Author: Tobias Nipkow *) |
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theory Abs_Int2_ivl_ITP |
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imports Abs_Int2_ITP "../Abs_Int_Tests" |
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begin |
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subsection "Interval Analysis" |
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datatype ivl = I "int option" "int option" |
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definition "\<gamma>_ivl i = (case i of |
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I (Some l) (Some h) \<Rightarrow> {l..h} | |
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I (Some l) None \<Rightarrow> {l..} | |
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I None (Some h) \<Rightarrow> {..h} | |
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I None None \<Rightarrow> UNIV)" |
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abbreviation I_Some_Some :: "int \<Rightarrow> int \<Rightarrow> ivl" ("{_\<dots>_}") where |
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"{lo\<dots>hi} == I (Some lo) (Some hi)" |
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abbreviation I_Some_None :: "int \<Rightarrow> ivl" ("{_\<dots>}") where |
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"{lo\<dots>} == I (Some lo) None" |
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abbreviation I_None_Some :: "int \<Rightarrow> ivl" ("{\<dots>_}") where |
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"{\<dots>hi} == I None (Some hi)" |
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abbreviation I_None_None :: "ivl" ("{\<dots>}") where |
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"{\<dots>} == I None None" |
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definition "num_ivl n = {n\<dots>n}" |
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fun in_ivl :: "int \<Rightarrow> ivl \<Rightarrow> bool" where |
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"in_ivl k (I (Some l) (Some h)) \<longleftrightarrow> l \<le> k \<and> k \<le> h" | |
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"in_ivl k (I (Some l) None) \<longleftrightarrow> l \<le> k" | |
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"in_ivl k (I None (Some h)) \<longleftrightarrow> k \<le> h" | |
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"in_ivl k (I None None) \<longleftrightarrow> True" |
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instantiation option :: (plus)plus |
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begin |
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fun plus_option where |
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"Some x + Some y = Some(x+y)" | |
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"_ + _ = None" |
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instance .. |
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end |
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definition empty where "empty = {1\<dots>0}" |
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fun is_empty where |
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"is_empty {l\<dots>h} = (h<l)" | |
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"is_empty _ = False" |
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lemma [simp]: "is_empty(I l h) = |
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(case l of Some l \<Rightarrow> (case h of Some h \<Rightarrow> h<l | None \<Rightarrow> False) | None \<Rightarrow> False)" |
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by(auto split:option.split) |
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lemma [simp]: "is_empty i \<Longrightarrow> \<gamma>_ivl i = {}" |
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by(auto simp add: \<gamma>_ivl_def split: ivl.split option.split) |
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definition "plus_ivl i1 i2 = (if is_empty i1 | is_empty i2 then empty else |
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case (i1,i2) of (I l1 h1, I l2 h2) \<Rightarrow> I (l1+l2) (h1+h2))" |
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instantiation ivl :: SL_top |
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begin |
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definition le_option :: "bool \<Rightarrow> int option \<Rightarrow> int option \<Rightarrow> bool" where |
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"le_option pos x y = |
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(case x of (Some i) \<Rightarrow> (case y of Some j \<Rightarrow> i\<le>j | None \<Rightarrow> pos) |
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| None \<Rightarrow> (case y of Some j \<Rightarrow> \<not>pos | None \<Rightarrow> True))" |
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fun le_aux where |
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"le_aux (I l1 h1) (I l2 h2) = (le_option False l2 l1 & le_option True h1 h2)" |
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definition le_ivl where |
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"i1 \<sqsubseteq> i2 = |
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(if is_empty i1 then True else |
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if is_empty i2 then False else le_aux i1 i2)" |
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definition min_option :: "bool \<Rightarrow> int option \<Rightarrow> int option \<Rightarrow> int option" where |
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"min_option pos o1 o2 = (if le_option pos o1 o2 then o1 else o2)" |
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definition max_option :: "bool \<Rightarrow> int option \<Rightarrow> int option \<Rightarrow> int option" where |
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"max_option pos o1 o2 = (if le_option pos o1 o2 then o2 else o1)" |
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definition "i1 \<squnion> i2 = |
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(if is_empty i1 then i2 else if is_empty i2 then i1 |
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else case (i1,i2) of (I l1 h1, I l2 h2) \<Rightarrow> |
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I (min_option False l1 l2) (max_option True h1 h2))" |
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definition "\<top> = {\<dots>}" |
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instance |
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proof |
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case goal1 thus ?case |
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by(cases x, simp add: le_ivl_def le_option_def split: option.split) |
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next |
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case goal2 thus ?case |
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by(cases x, cases y, cases z, auto simp: le_ivl_def le_option_def split: option.splits if_splits) |
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next |
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case goal3 thus ?case |
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by(cases x, cases y, simp add: le_ivl_def join_ivl_def le_option_def min_option_def max_option_def split: option.splits) |
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next |
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case goal4 thus ?case |
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by(cases x, cases y, simp add: le_ivl_def join_ivl_def le_option_def min_option_def max_option_def split: option.splits) |
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next |
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case goal5 thus ?case |
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by(cases x, cases y, cases z, auto simp add: le_ivl_def join_ivl_def le_option_def min_option_def max_option_def split: option.splits if_splits) |
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next |
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case goal6 thus ?case |
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by(cases x, simp add: Top_ivl_def le_ivl_def le_option_def split: option.split) |
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qed |
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end |
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instantiation ivl :: L_top_bot |
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begin |
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definition "i1 \<sqinter> i2 = (if is_empty i1 \<or> is_empty i2 then empty else |
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case (i1,i2) of (I l1 h1, I l2 h2) \<Rightarrow> |
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I (max_option False l1 l2) (min_option True h1 h2))" |
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definition "\<bottom> = empty" |
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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:meet_ivl_def empty_def le_ivl_def le_option_def max_option_def min_option_def split: ivl.splits option.splits) |
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next |
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case goal2 thus ?case |
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by (simp add: empty_def meet_ivl_def le_ivl_def le_option_def max_option_def min_option_def split: ivl.splits option.splits) |
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next |
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case goal3 thus ?case |
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by (cases x, cases y, cases z, auto simp add: le_ivl_def meet_ivl_def empty_def le_option_def max_option_def min_option_def split: option.splits if_splits) |
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next |
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case goal4 show ?case by(cases x, simp add: bot_ivl_def empty_def le_ivl_def) |
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qed |
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end |
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instantiation option :: (minus)minus |
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begin |
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fun minus_option where |
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"Some x - Some y = Some(x-y)" | |
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"_ - _ = None" |
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instance .. |
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end |
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definition "minus_ivl i1 i2 = (if is_empty i1 | is_empty i2 then empty else |
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case (i1,i2) of (I l1 h1, I l2 h2) \<Rightarrow> I (l1-h2) (h1-l2))" |
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lemma gamma_minus_ivl: |
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"n1 : \<gamma>_ivl i1 \<Longrightarrow> n2 : \<gamma>_ivl i2 \<Longrightarrow> n1-n2 : \<gamma>_ivl(minus_ivl i1 i2)" |
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by(auto simp add: minus_ivl_def \<gamma>_ivl_def split: ivl.splits option.splits) |
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definition "filter_plus_ivl i i1 i2 = ((*if is_empty i then empty else*) |
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i1 \<sqinter> minus_ivl i i2, i2 \<sqinter> minus_ivl i i1)" |
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fun filter_less_ivl :: "bool \<Rightarrow> ivl \<Rightarrow> ivl \<Rightarrow> ivl * ivl" where |
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"filter_less_ivl res (I l1 h1) (I l2 h2) = |
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(if is_empty(I l1 h1) \<or> is_empty(I l2 h2) then (empty, empty) else |
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if res |
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then (I l1 (min_option True h1 (h2 - Some 1)), |
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I (max_option False (l1 + Some 1) l2) h2) |
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else (I (max_option False l1 l2) h1, I l2 (min_option True h1 h2)))" |
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permanent_interpretation Val_abs |
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where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = plus_ivl |
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proof |
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case goal1 thus ?case |
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by(auto simp: \<gamma>_ivl_def le_ivl_def le_option_def split: ivl.split option.split if_splits) |
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next |
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case goal2 show ?case by(simp add: \<gamma>_ivl_def Top_ivl_def) |
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next |
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case goal3 thus ?case by(simp add: \<gamma>_ivl_def num_ivl_def) |
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next |
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case goal4 thus ?case |
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by(auto simp add: \<gamma>_ivl_def plus_ivl_def split: ivl.split option.splits) |
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qed |
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permanent_interpretation Val_abs1_gamma |
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where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = plus_ivl |
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defines aval_ivl is aval' |
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proof |
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case goal1 thus ?case |
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by(auto simp add: \<gamma>_ivl_def meet_ivl_def empty_def min_option_def max_option_def split: ivl.split option.split) |
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next |
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case goal2 show ?case by(auto simp add: bot_ivl_def \<gamma>_ivl_def empty_def) |
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qed |
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lemma mono_minus_ivl: |
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"i1 \<sqsubseteq> i1' \<Longrightarrow> i2 \<sqsubseteq> i2' \<Longrightarrow> minus_ivl i1 i2 \<sqsubseteq> minus_ivl i1' i2'" |
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apply(auto simp add: minus_ivl_def empty_def le_ivl_def le_option_def split: ivl.splits) |
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apply(simp split: option.splits) |
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apply(simp split: option.splits) |
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apply(simp split: option.splits) |
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done |
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permanent_interpretation Val_abs1 |
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where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = plus_ivl |
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and test_num' = in_ivl |
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and filter_plus' = filter_plus_ivl and filter_less' = filter_less_ivl |
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proof |
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case goal1 thus ?case |
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by (simp add: \<gamma>_ivl_def split: ivl.split option.split) |
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next |
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case goal2 thus ?case |
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by(auto simp add: filter_plus_ivl_def) |
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(metis gamma_minus_ivl add_diff_cancel add_commute)+ |
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next |
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case goal3 thus ?case |
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by(cases a1, cases a2, |
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auto simp: \<gamma>_ivl_def min_option_def max_option_def le_option_def split: if_splits option.splits) |
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qed |
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permanent_interpretation Abs_Int1 |
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where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = plus_ivl |
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and test_num' = in_ivl |
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and filter_plus' = filter_plus_ivl and filter_less' = filter_less_ivl |
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defines afilter_ivl is afilter |
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and bfilter_ivl is bfilter |
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and step_ivl is step' |
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and AI_ivl is AI |
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and aval_ivl' is aval'' |
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.. |
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text{* Monotonicity: *} |
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permanent_interpretation Abs_Int1_mono |
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where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = plus_ivl |
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and test_num' = in_ivl |
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and filter_plus' = filter_plus_ivl and filter_less' = filter_less_ivl |
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proof |
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case goal1 thus ?case |
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by(auto simp: plus_ivl_def le_ivl_def le_option_def empty_def split: if_splits ivl.splits option.splits) |
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next |
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case goal2 thus ?case |
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by(auto simp: filter_plus_ivl_def le_prod_def mono_meet mono_minus_ivl) |
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next |
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case goal3 thus ?case |
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apply(cases a1, cases b1, cases a2, cases b2, auto simp: le_prod_def) |
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by(auto simp add: empty_def le_ivl_def le_option_def min_option_def max_option_def split: option.splits) |
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qed |
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subsubsection "Tests" |
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value "show_acom_opt (AI_ivl test1_ivl)" |
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text{* Better than @{text AI_const}: *} |
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value "show_acom_opt (AI_ivl test3_const)" |
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value "show_acom_opt (AI_ivl test4_const)" |
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value "show_acom_opt (AI_ivl test6_const)" |
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value "show_acom_opt (AI_ivl test2_ivl)" |
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value "show_acom (((step_ivl \<top>)^^0) (\<bottom>\<^sub>c test2_ivl))" |
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value "show_acom (((step_ivl \<top>)^^1) (\<bottom>\<^sub>c test2_ivl))" |
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value "show_acom (((step_ivl \<top>)^^2) (\<bottom>\<^sub>c test2_ivl))" |
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text{* Fixed point reached in 2 steps. Not so if the start value of x is known: *} |
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|
265 |
value "show_acom_opt (AI_ivl test3_ivl)" |
42a01315d998
removed accidental dependance of abstract interpreter on gamma
nipkow
parents:
46346
diff
changeset
|
266 |
value "show_acom (((step_ivl \<top>)^^0) (\<bottom>\<^sub>c test3_ivl))" |
42a01315d998
removed accidental dependance of abstract interpreter on gamma
nipkow
parents:
46346
diff
changeset
|
267 |
value "show_acom (((step_ivl \<top>)^^1) (\<bottom>\<^sub>c test3_ivl))" |
42a01315d998
removed accidental dependance of abstract interpreter on gamma
nipkow
parents:
46346
diff
changeset
|
268 |
value "show_acom (((step_ivl \<top>)^^2) (\<bottom>\<^sub>c test3_ivl))" |
42a01315d998
removed accidental dependance of abstract interpreter on gamma
nipkow
parents:
46346
diff
changeset
|
269 |
value "show_acom (((step_ivl \<top>)^^3) (\<bottom>\<^sub>c test3_ivl))" |
42a01315d998
removed accidental dependance of abstract interpreter on gamma
nipkow
parents:
46346
diff
changeset
|
270 |
value "show_acom (((step_ivl \<top>)^^4) (\<bottom>\<^sub>c test3_ivl))" |
45623
f682f3f7b726
Abstract interpretation is now based uniformly on annotated programs,
nipkow
parents:
45127
diff
changeset
|
271 |
|
f682f3f7b726
Abstract interpretation is now based uniformly on annotated programs,
nipkow
parents:
45127
diff
changeset
|
272 |
text{* Takes as many iterations as the actual execution. Would diverge if |
f682f3f7b726
Abstract interpretation is now based uniformly on annotated programs,
nipkow
parents:
45127
diff
changeset
|
273 |
loop did not terminate. Worse still, as the following example shows: even if |
f682f3f7b726
Abstract interpretation is now based uniformly on annotated programs,
nipkow
parents:
45127
diff
changeset
|
274 |
the actual execution terminates, the analysis may not. The value of y keeps |
f682f3f7b726
Abstract interpretation is now based uniformly on annotated programs,
nipkow
parents:
45127
diff
changeset
|
275 |
decreasing as the analysis is iterated, no matter how long: *} |
f682f3f7b726
Abstract interpretation is now based uniformly on annotated programs,
nipkow
parents:
45127
diff
changeset
|
276 |
|
46355
42a01315d998
removed accidental dependance of abstract interpreter on gamma
nipkow
parents:
46346
diff
changeset
|
277 |
value "show_acom (((step_ivl \<top>)^^50) (\<bottom>\<^sub>c test4_ivl))" |
45623
f682f3f7b726
Abstract interpretation is now based uniformly on annotated programs,
nipkow
parents:
45127
diff
changeset
|
278 |
|
f682f3f7b726
Abstract interpretation is now based uniformly on annotated programs,
nipkow
parents:
45127
diff
changeset
|
279 |
text{* Relationships between variables are NOT captured: *} |
46355
42a01315d998
removed accidental dependance of abstract interpreter on gamma
nipkow
parents:
46346
diff
changeset
|
280 |
value "show_acom_opt (AI_ivl test5_ivl)" |
45623
f682f3f7b726
Abstract interpretation is now based uniformly on annotated programs,
nipkow
parents:
45127
diff
changeset
|
281 |
|
f682f3f7b726
Abstract interpretation is now based uniformly on annotated programs,
nipkow
parents:
45127
diff
changeset
|
282 |
text{* Again, the analysis would not terminate: *} |
46355
42a01315d998
removed accidental dependance of abstract interpreter on gamma
nipkow
parents:
46346
diff
changeset
|
283 |
value "show_acom (((step_ivl \<top>)^^50) (\<bottom>\<^sub>c test6_ivl))" |
45623
f682f3f7b726
Abstract interpretation is now based uniformly on annotated programs,
nipkow
parents:
45127
diff
changeset
|
284 |
|
45111 | 285 |
end |