--- a/src/HOL/Data_Structures/Tree23_Map.thy Wed Nov 04 15:07:23 2015 +0100
+++ b/src/HOL/Data_Structures/Tree23_Map.thy Thu Nov 05 08:27:14 2015 +0100
@@ -8,65 +8,65 @@
Map_by_Ordered
begin
-fun lookup :: "('a::linorder * 'b) tree23 \<Rightarrow> 'a \<Rightarrow> 'b option" where
+fun lookup :: "('a::cmp * 'b) tree23 \<Rightarrow> 'a \<Rightarrow> 'b option" where
"lookup Leaf x = None" |
-"lookup (Node2 l (a,b) r) x =
- (if x < a then lookup l x else
- if a < x then lookup r x else Some b)" |
-"lookup (Node3 l (a1,b1) m (a2,b2) r) x =
- (if x < a1 then lookup l x else
- if x = a1 then Some b1 else
- if x < a2 then lookup m x else
- if x = a2 then Some b2
- else lookup r x)"
+"lookup (Node2 l (a,b) r) x = (case cmp x a of
+ LT \<Rightarrow> lookup l x |
+ GT \<Rightarrow> lookup r x |
+ EQ \<Rightarrow> Some b)" |
+"lookup (Node3 l (a1,b1) m (a2,b2) r) x = (case cmp x a1 of
+ LT \<Rightarrow> lookup l x |
+ EQ \<Rightarrow> Some b1 |
+ GT \<Rightarrow> (case cmp x a2 of
+ LT \<Rightarrow> lookup m x |
+ EQ \<Rightarrow> Some b2 |
+ GT \<Rightarrow> lookup r x))"
-fun upd :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) up\<^sub>i" where
+fun upd :: "'a::cmp \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) up\<^sub>i" where
"upd x y Leaf = Up\<^sub>i Leaf (x,y) Leaf" |
-"upd x y (Node2 l ab r) =
- (if x < fst ab then
- (case upd x y l of
+"upd x y (Node2 l ab r) = (case cmp x (fst ab) of
+ LT \<Rightarrow> (case upd x y l of
T\<^sub>i l' => T\<^sub>i (Node2 l' ab r)
- | Up\<^sub>i l1 ab' l2 => T\<^sub>i (Node3 l1 ab' l2 ab r))
- else if x = fst ab then T\<^sub>i (Node2 l (x,y) r)
- else
- (case upd x y r of
+ | Up\<^sub>i l1 ab' l2 => T\<^sub>i (Node3 l1 ab' l2 ab r)) |
+ EQ \<Rightarrow> T\<^sub>i (Node2 l (x,y) r) |
+ GT \<Rightarrow> (case upd x y r of
T\<^sub>i r' => T\<^sub>i (Node2 l ab r')
| Up\<^sub>i r1 ab' r2 => T\<^sub>i (Node3 l ab r1 ab' r2)))" |
-"upd x y (Node3 l ab1 m ab2 r) =
- (if x < fst ab1 then
- (case upd x y l of
+"upd x y (Node3 l ab1 m ab2 r) = (case cmp x (fst ab1) of
+ LT \<Rightarrow> (case upd x y l of
T\<^sub>i l' => T\<^sub>i (Node3 l' ab1 m ab2 r)
- | Up\<^sub>i l1 ab' l2 => Up\<^sub>i (Node2 l1 ab' l2) ab1 (Node2 m ab2 r))
- else if x = fst ab1 then T\<^sub>i (Node3 l (x,y) m ab2 r)
- else if x < fst ab2 then
- (case upd x y m of
- T\<^sub>i m' => T\<^sub>i (Node3 l ab1 m' ab2 r)
- | Up\<^sub>i m1 ab' m2 => Up\<^sub>i (Node2 l ab1 m1) ab' (Node2 m2 ab2 r))
- else if x = fst ab2 then T\<^sub>i (Node3 l ab1 m (x,y) r)
- else
- (case upd x y r of
- T\<^sub>i r' => T\<^sub>i (Node3 l ab1 m ab2 r')
- | Up\<^sub>i r1 ab' r2 => Up\<^sub>i (Node2 l ab1 m) ab2 (Node2 r1 ab' r2)))"
+ | Up\<^sub>i l1 ab' l2 => Up\<^sub>i (Node2 l1 ab' l2) ab1 (Node2 m ab2 r)) |
+ EQ \<Rightarrow> T\<^sub>i (Node3 l (x,y) m ab2 r) |
+ GT \<Rightarrow> (case cmp x (fst ab2) of
+ LT \<Rightarrow> (case upd x y m of
+ T\<^sub>i m' => T\<^sub>i (Node3 l ab1 m' ab2 r)
+ | Up\<^sub>i m1 ab' m2 => Up\<^sub>i (Node2 l ab1 m1) ab' (Node2 m2 ab2 r)) |
+ EQ \<Rightarrow> T\<^sub>i (Node3 l ab1 m (x,y) r) |
+ GT \<Rightarrow> (case upd x y r of
+ T\<^sub>i r' => T\<^sub>i (Node3 l ab1 m ab2 r')
+ | Up\<^sub>i r1 ab' r2 => Up\<^sub>i (Node2 l ab1 m) ab2 (Node2 r1 ab' r2))))"
-definition update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) tree23" where
+definition update :: "'a::cmp \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) tree23" where
"update a b t = tree\<^sub>i(upd a b t)"
-fun del :: "'a::linorder \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) up\<^sub>d"
-where
+fun del :: "'a::cmp \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) up\<^sub>d" where
"del x Leaf = T\<^sub>d Leaf" |
"del x (Node2 Leaf ab1 Leaf) = (if x=fst ab1 then Up\<^sub>d Leaf else T\<^sub>d(Node2 Leaf ab1 Leaf))" |
"del x (Node3 Leaf ab1 Leaf ab2 Leaf) = T\<^sub>d(if x=fst ab1 then Node2 Leaf ab2 Leaf
else if x=fst ab2 then Node2 Leaf ab1 Leaf else Node3 Leaf ab1 Leaf ab2 Leaf)" |
-"del x (Node2 l ab1 r) = (if x<fst ab1 then node21 (del x l) ab1 r else
- if x > fst ab1 then node22 l ab1 (del x r) else
- let (ab1',t) = del_min r in node22 l ab1' t)" |
-"del x (Node3 l ab1 m ab2 r) = (if x<fst ab1 then node31 (del x l) ab1 m ab2 r else
- if x = fst ab1 then let (ab1',m') = del_min m in node32 l ab1' m' ab2 r else
- if x < fst ab2 then node32 l ab1 (del x m) ab2 r else
- if x = fst ab2 then let (ab2',r') = del_min r in node33 l ab1 m ab2' r'
- else node33 l ab1 m ab2 (del x r))"
+"del x (Node2 l ab1 r) = (case cmp x (fst ab1) of
+ LT \<Rightarrow> node21 (del x l) ab1 r |
+ GT \<Rightarrow> node22 l ab1 (del x r) |
+ EQ \<Rightarrow> let (ab1',t) = del_min r in node22 l ab1' t)" |
+"del x (Node3 l ab1 m ab2 r) = (case cmp x (fst ab1) of
+ LT \<Rightarrow> node31 (del x l) ab1 m ab2 r |
+ EQ \<Rightarrow> let (ab1',m') = del_min m in node32 l ab1' m' ab2 r |
+ GT \<Rightarrow> (case cmp x (fst ab2) of
+ LT \<Rightarrow> node32 l ab1 (del x m) ab2 r |
+ EQ \<Rightarrow> let (ab2',r') = del_min r in node33 l ab1 m ab2' r' |
+ GT \<Rightarrow> node33 l ab1 m ab2 (del x r)))"
-definition delete :: "'a::linorder \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) tree23" where
+definition delete :: "'a::cmp \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) tree23" where
"delete x t = tree\<^sub>d(del x t)"
@@ -98,7 +98,7 @@
subsection \<open>Balancedness\<close>
lemma bal_upd: "bal t \<Longrightarrow> bal (tree\<^sub>i(upd a b t)) \<and> height(upd a b t) = height t"
-by (induct t) (auto split: up\<^sub>i.split)(* 30 secs in 2015 *)
+by (induct t) (auto split: up\<^sub>i.split)(* 16 secs in 2015 *)
corollary bal_update: "bal t \<Longrightarrow> bal (update a b t)"
by (simp add: update_def bal_upd)