
diff -r 340738057c8c -r 29800666e526 src/HOL/IMP/Abs_Int_Den/Abs_Int_den1_ivl.thy
--- a/src/HOL/IMP/Abs_Int_Den/Abs_Int_den1_ivl.thy	Mon Feb 22 14:37:56 2016 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,270 +0,0 @@
-(* Author: Tobias Nipkow *)
-
-theory Abs_Int_den1_ivl
-imports Abs_Int_den1
-begin
-
-subsection "Interval Analysis"
-
-datatype ivl = I "int option" "int option"
-
-text{* We assume an important invariant: arithmetic operations are never
-applied to empty intervals @{term"I (Some i) (Some j)"} with @{term"j <
-i"}. This avoids special cases. Why can we assume this? Because an empty
-interval of values for a variable means that the current program point is
-unreachable. But this should actually translate into the bottom state, not a
-state where some variables have empty intervals. *}
-
-definition "rep_ivl i =
- (case i of I (Some l) (Some h) \<Rightarrow> {l..h} | I (Some l) None \<Rightarrow> {l..}
-  | I None (Some h) \<Rightarrow> {..h} | I None None \<Rightarrow> UNIV)"
-
-definition "num_ivl n = I (Some n) (Some n)"
-
-definition
-  [code_abbrev]: "contained_in i k \<longleftrightarrow> k \<in> rep_ivl i"
-
-lemma contained_in_simps [code]:
-  "contained_in (I (Some l) (Some h)) k \<longleftrightarrow> l \<le> k \<and> k \<le> h"
-  "contained_in (I (Some l) None) k \<longleftrightarrow> l \<le> k"
-  "contained_in (I None (Some h)) k \<longleftrightarrow> k \<le> h"
-  "contained_in (I None None) k \<longleftrightarrow> True"
-  by (simp_all add: contained_in_def rep_ivl_def)
-
-instantiation option :: (plus)plus
-begin
-
-fun plus_option where
-"Some x + Some y = Some(x+y)" |
-"_ + _ = None"
-
-instance proof qed
-
-end
-
-definition empty where "empty = I (Some 1) (Some 0)"
-
-fun is_empty where
-"is_empty(I (Some l) (Some h)) = (h<l)" |
-"is_empty _ = False"
-
-lemma [simp]: "is_empty(I l h) =
-  (case l of Some l \<Rightarrow> (case h of Some h \<Rightarrow> h<l | None \<Rightarrow> False) | None \<Rightarrow> False)"
-by(auto split:option.split)
-
-lemma [simp]: "is_empty i \<Longrightarrow> rep_ivl i = {}"
-by(auto simp add: rep_ivl_def split: ivl.split option.split)
-
-definition "plus_ivl i1 i2 = ((*if is_empty i1 | is_empty i2 then empty else*)
-  case (i1,i2) of (I l1 h1, I l2 h2) \<Rightarrow> I (l1+l2) (h1+h2))"
-
-instantiation ivl :: SL_top
-begin
-
-definition le_option :: "bool \<Rightarrow> int option \<Rightarrow> int option \<Rightarrow> bool" where
-"le_option pos x y =
- (case x of (Some i) \<Rightarrow> (case y of Some j \<Rightarrow> i\<le>j | None \<Rightarrow> pos)
-  | None \<Rightarrow> (case y of Some j \<Rightarrow> \<not>pos | None \<Rightarrow> True))"
-
-fun le_aux where
-"le_aux (I l1 h1) (I l2 h2) = (le_option False l2 l1 & le_option True h1 h2)"
-
-definition le_ivl where
-"i1 \<sqsubseteq> i2 =
- (if is_empty i1 then True else
-  if is_empty i2 then False else le_aux i1 i2)"
-
-definition min_option :: "bool \<Rightarrow> int option \<Rightarrow> int option \<Rightarrow> int option" where
-"min_option pos o1 o2 = (if le_option pos o1 o2 then o1 else o2)"
-
-definition max_option :: "bool \<Rightarrow> int option \<Rightarrow> int option \<Rightarrow> int option" where
-"max_option pos o1 o2 = (if le_option pos o1 o2 then o2 else o1)"
-
-definition "i1 \<squnion> i2 =
- (if is_empty i1 then i2 else if is_empty i2 then i1
-  else case (i1,i2) of (I l1 h1, I l2 h2) \<Rightarrow>
-          I (min_option False l1 l2) (max_option True h1 h2))"
-
-definition "Top = I None None"
-
-instance
-proof
-  case goal1 thus ?case
-    by(cases x, simp add: le_ivl_def le_option_def split: option.split)
-next
-  case goal2 thus ?case
-    by(cases x, cases y, cases z, auto simp: le_ivl_def le_option_def split: option.splits if_splits)
-next
-  case goal3 thus ?case
-    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)
-next
-  case goal4 thus ?case
-    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)
-next
-  case goal5 thus ?case
-    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)
-next
-  case goal6 thus ?case
-    by(cases x, simp add: Top_ivl_def le_ivl_def le_option_def split: option.split)
-qed
-
-end
-
-
-instantiation ivl :: L_top_bot
-begin
-
-definition "i1 \<sqinter> i2 = (if is_empty i1 \<or> is_empty i2 then empty else
-  case (i1,i2) of (I l1 h1, I l2 h2) \<Rightarrow>
-    I (max_option False l1 l2) (min_option True h1 h2))"
-
-definition "Bot = empty"
-
-instance
-proof
-  case goal1 thus ?case
-    by (simp add:meet_ivl_def empty_def meet_ivl_def le_ivl_def le_option_def max_option_def min_option_def split: ivl.splits option.splits)
-next
-  case goal2 thus ?case
-    by (simp add:meet_ivl_def empty_def meet_ivl_def le_ivl_def le_option_def max_option_def min_option_def split: ivl.splits option.splits)
-next
-  case goal3 thus ?case
-    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)
-next
-  case goal4 show ?case by(cases x, simp add: Bot_ivl_def empty_def le_ivl_def)
-qed
-
-end
-
-instantiation option :: (minus)minus
-begin
-
-fun minus_option where
-"Some x - Some y = Some(x-y)" |
-"_ - _ = None"
-
-instance proof qed
-
-end
-
-definition "minus_ivl i1 i2 = ((*if is_empty i1 | is_empty i2 then empty else*)
-  case (i1,i2) of (I l1 h1, I l2 h2) \<Rightarrow> I (l1-h2) (h1-l2))"
-
-lemma rep_minus_ivl:
-  "n1 : rep_ivl i1 \<Longrightarrow> n2 : rep_ivl i2 \<Longrightarrow> n1-n2 : rep_ivl(minus_ivl i1 i2)"
-by(auto simp add: minus_ivl_def rep_ivl_def split: ivl.splits option.splits)
-
-
-definition "filter_plus_ivl i i1 i2 = ((*if is_empty i then empty else*)
-  i1 \<sqinter> minus_ivl i i2, i2 \<sqinter> minus_ivl i i1)"
-
-fun filter_less_ivl :: "bool \<Rightarrow> ivl \<Rightarrow> ivl \<Rightarrow> ivl * ivl" where
-"filter_less_ivl res (I l1 h1) (I l2 h2) =
-  ((*if is_empty(I l1 h1) \<or> is_empty(I l2 h2) then (empty, empty) else*)
-   if res
-   then (I l1 (min_option True h1 (h2 - Some 1)),
-         I (max_option False (l1 + Some 1) l2) h2)
-   else (I (max_option False l1 l2) h1, I l2 (min_option True h1 h2)))"
-
-global_interpretation Rep rep_ivl
-proof
-  case goal1 thus ?case
-    by(auto simp: rep_ivl_def le_ivl_def le_option_def split: ivl.split option.split if_splits)
-qed
-
-global_interpretation Val_abs rep_ivl num_ivl plus_ivl
-proof
-  case goal1 thus ?case by(simp add: rep_ivl_def num_ivl_def)
-next
-  case goal2 thus ?case
-    by(auto simp add: rep_ivl_def plus_ivl_def split: ivl.split option.splits)
-qed
-
-global_interpretation Rep1 rep_ivl
-proof
-  case goal1 thus ?case
-    by(auto simp add: rep_ivl_def meet_ivl_def empty_def min_option_def max_option_def split: ivl.split option.split)
-next
-  case goal2 show ?case by(auto simp add: Bot_ivl_def rep_ivl_def empty_def)
-qed
-
-global_interpretation
-  Val_abs1 rep_ivl num_ivl plus_ivl filter_plus_ivl filter_less_ivl
-proof
-  case goal1 thus ?case
-    by(auto simp add: filter_plus_ivl_def)
-      (metis rep_minus_ivl add_diff_cancel add.commute)+
-next
-  case goal2 thus ?case
-    by(cases a1, cases a2,
-      auto simp: rep_ivl_def min_option_def max_option_def le_option_def split: if_splits option.splits)
-qed
-
-global_interpretation
-  Abs_Int1 rep_ivl num_ivl plus_ivl filter_plus_ivl filter_less_ivl "(iter' 3)"
-defines afilter_ivl = afilter
-and bfilter_ivl = bfilter
-and AI_ivl = AI
-and aval_ivl = aval'
-proof qed (auto simp: iter'_pfp_above)
-
-
-fun list_up where
-"list_up bot = bot" |
-"list_up (Up x) = Up(list x)"
-
-value "list_up(afilter_ivl (N 5) (I (Some 4) (Some 5)) Top)"
-value "list_up(afilter_ivl (N 5) (I (Some 4) (Some 4)) Top)"
-value "list_up(afilter_ivl (V ''x'') (I (Some 4) (Some 4))
- (Up(FunDom(Top(''x'':=I (Some 5) (Some 6))) [''x''])))"
-value "list_up(afilter_ivl (V ''x'') (I (Some 4) (Some 5))
- (Up(FunDom(Top(''x'':=I (Some 5) (Some 6))) [''x''])))"
-value "list_up(afilter_ivl (Plus (V ''x'') (V ''x'')) (I (Some 0) (Some 10))
-  (Up(FunDom(Top(''x'':= I (Some 0) (Some 100)))[''x''])))"
-value "list_up(afilter_ivl (Plus (V ''x'') (N 7)) (I (Some 0) (Some 10))
-  (Up(FunDom(Top(''x'':= I (Some 0) (Some 100)))[''x''])))"
-
-value "list_up(bfilter_ivl (Less (V ''x'') (V ''x'')) True
-  (Up(FunDom(Top(''x'':= I (Some 0) (Some 0)))[''x''])))"
-value "list_up(bfilter_ivl (Less (V ''x'') (V ''x'')) True
-  (Up(FunDom(Top(''x'':= I (Some 0) (Some 2)))[''x''])))"
-value "list_up(bfilter_ivl (Less (V ''x'') (Plus (N 10) (V ''y''))) True
-  (Up(FunDom(Top(''x'':= I (Some 15) (Some 20),''y'':= I (Some 5) (Some 7)))[''x'', ''y''])))"
-
-definition "test_ivl1 =
- ''y'' ::= N 7;;
- IF Less (V ''x'') (V ''y'')
- THEN ''y'' ::= Plus (V ''y'') (V ''x'')
- ELSE ''x'' ::= Plus (V ''x'') (V ''y'')"
-value "list_up(AI_ivl test_ivl1 Top)"
-
-value "list_up (AI_ivl test3_const Top)"
-
-value "list_up (AI_ivl test5_const Top)"
-
-value "list_up (AI_ivl test6_const Top)"
-
-definition "test2_ivl =
- ''y'' ::= N 7;;
- WHILE Less (V ''x'') (N 100) DO ''y'' ::= Plus (V ''y'') (N 1)"
-value "list_up(AI_ivl test2_ivl Top)"
-
-definition "test3_ivl =
- ''x'' ::= N 0;; ''y'' ::= N 100;; ''z'' ::= Plus (V ''x'') (V ''y'');;
- WHILE Less (V ''x'') (N 11)
- DO (''x'' ::= Plus (V ''x'') (N 1);; ''y'' ::= Plus (V ''y'') (N (- 1)))"
-value "list_up(AI_ivl test3_ivl Top)"
-
-definition "test4_ivl =
- ''x'' ::= N 0;; ''y'' ::= N 0;;
- WHILE Less (V ''x'') (N 1001)
- DO (''y'' ::= V ''x'';; ''x'' ::= Plus (V ''x'') (N 1))"
-value "list_up(AI_ivl test4_ivl Top)"
-
-text{* Nontermination not detected: *}
-definition "test5_ivl =
- ''x'' ::= N 0;;
- WHILE Less (V ''x'') (N 1) DO ''x'' ::= Plus (V ''x'') (N (- 1))"
-value "list_up(AI_ivl test5_ivl Top)"
-
-end




