--- a/src/HOL/Data_Structures/Tree234_Map.thy Wed Nov 11 16:42:30 2015 +0100
+++ b/src/HOL/Data_Structures/Tree234_Map.thy Wed Nov 11 18:32:26 2015 +0100
@@ -1,181 +1,181 @@
-(* Author: Tobias Nipkow *)
-
-section \<open>A 2-3-4 Tree Implementation of Maps\<close>
-
-theory Tree234_Map
-imports
- Tree234_Set
- "../Data_Structures/Map_by_Ordered"
-begin
-
-subsection \<open>Map operations on 2-3-4 trees\<close>
-
-fun lookup :: "('a::cmp * 'b) tree234 \<Rightarrow> 'a \<Rightarrow> 'b option" where
-"lookup Leaf x = None" |
-"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))" |
-"lookup (Node4 t1 (a1,b1) t2 (a2,b2) t3 (a3,b3) t4) x = (case cmp x a2 of
- LT \<Rightarrow> (case cmp x a1 of
- LT \<Rightarrow> lookup t1 x | EQ \<Rightarrow> Some b1 | GT \<Rightarrow> lookup t2 x) |
- EQ \<Rightarrow> Some b2 |
- GT \<Rightarrow> (case cmp x a3 of
- LT \<Rightarrow> lookup t3 x | EQ \<Rightarrow> Some b3 | GT \<Rightarrow> lookup t4 x))"
-
-fun upd :: "'a::cmp \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree234 \<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) = (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)) |
- 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) = (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)) |
- 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))))" |
-"upd x y (Node4 t1 ab1 t2 ab2 t3 ab3 t4) = (case cmp x (fst ab2) of
- LT \<Rightarrow> (case cmp x (fst ab1) of
- LT \<Rightarrow> (case upd x y t1 of
- T\<^sub>i t1' => T\<^sub>i (Node4 t1' ab1 t2 ab2 t3 ab3 t4)
- | Up\<^sub>i t11 q t12 => Up\<^sub>i (Node2 t11 q t12) ab1 (Node3 t2 ab2 t3 ab3 t4)) |
- EQ \<Rightarrow> T\<^sub>i (Node4 t1 (x,y) t2 ab2 t3 ab3 t4) |
- GT \<Rightarrow> (case upd x y t2 of
- T\<^sub>i t2' => T\<^sub>i (Node4 t1 ab1 t2' ab2 t3 ab3 t4)
- | Up\<^sub>i t21 q t22 => Up\<^sub>i (Node2 t1 ab1 t21) q (Node3 t22 ab2 t3 ab3 t4))) |
- EQ \<Rightarrow> T\<^sub>i (Node4 t1 ab1 t2 (x,y) t3 ab3 t4) |
- GT \<Rightarrow> (case cmp x (fst ab3) of
- LT \<Rightarrow> (case upd x y t3 of
- T\<^sub>i t3' \<Rightarrow> T\<^sub>i (Node4 t1 ab1 t2 ab2 t3' ab3 t4)
- | Up\<^sub>i t31 q t32 => Up\<^sub>i (Node2 t1 ab1 t2) ab2(*q*) (Node3 t31 q t32 ab3 t4)) |
- EQ \<Rightarrow> T\<^sub>i (Node4 t1 ab1 t2 ab2 t3 (x,y) t4) |
- GT \<Rightarrow> (case upd x y t4 of
- T\<^sub>i t4' => T\<^sub>i (Node4 t1 ab1 t2 ab2 t3 ab3 t4')
- | Up\<^sub>i t41 q t42 => Up\<^sub>i (Node2 t1 ab1 t2) ab2 (Node3 t3 ab3 t41 q t42))))"
-
-definition update :: "'a::cmp \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree234 \<Rightarrow> ('a*'b) tree234" where
-"update x y t = tree\<^sub>i(upd x y t)"
-
-fun del :: "'a::cmp \<Rightarrow> ('a*'b) tree234 \<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 (Node4 Leaf ab1 Leaf ab2 Leaf ab3 Leaf) =
- T\<^sub>d(if x = fst ab1 then Node3 Leaf ab2 Leaf ab3 Leaf else
- if x = fst ab2 then Node3 Leaf ab1 Leaf ab3 Leaf else
- if x = fst ab3 then Node3 Leaf ab1 Leaf ab2 Leaf
- else Node4 Leaf ab1 Leaf ab2 Leaf ab3 Leaf)" |
-"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)))" |
-"del x (Node4 t1 ab1 t2 ab2 t3 ab3 t4) = (case cmp x (fst ab2) of
- LT \<Rightarrow> (case cmp x (fst ab1) of
- LT \<Rightarrow> node41 (del x t1) ab1 t2 ab2 t3 ab3 t4 |
- EQ \<Rightarrow> let (ab',t2') = del_min t2 in node42 t1 ab' t2' ab2 t3 ab3 t4 |
- GT \<Rightarrow> node42 t1 ab1 (del x t2) ab2 t3 ab3 t4) |
- EQ \<Rightarrow> let (ab',t3') = del_min t3 in node43 t1 ab1 t2 ab' t3' ab3 t4 |
- GT \<Rightarrow> (case cmp x (fst ab3) of
- LT \<Rightarrow> node43 t1 ab1 t2 ab2 (del x t3) ab3 t4 |
- EQ \<Rightarrow> let (ab',t4') = del_min t4 in node44 t1 ab1 t2 ab2 t3 ab' t4' |
- GT \<Rightarrow> node44 t1 ab1 t2 ab2 t3 ab3 (del x t4)))"
-
-definition delete :: "'a::cmp \<Rightarrow> ('a*'b) tree234 \<Rightarrow> ('a*'b) tree234" where
-"delete x t = tree\<^sub>d(del x t)"
-
-
-subsection "Functional correctness"
-
-lemma lookup: "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
-by (induction t) (auto simp: map_of_simps split: option.split)
-
-
-lemma inorder_upd:
- "sorted1(inorder t) \<Longrightarrow> inorder(tree\<^sub>i(upd a b t)) = upd_list a b (inorder t)"
-by(induction t)
- (auto simp: upd_list_simps, auto simp: upd_list_simps split: up\<^sub>i.splits)
-
-lemma inorder_update:
- "sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)"
-by(simp add: update_def inorder_upd)
-
-
-lemma inorder_del: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
- inorder(tree\<^sub>d (del x t)) = del_list x (inorder t)"
-by(induction t rule: del.induct)
- ((auto simp: del_list_simps inorder_nodes del_minD split: prod.splits)[1])+
-(* 200 secs (2015) *)
-
-lemma inorder_delete: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
- inorder(delete x t) = del_list x (inorder t)"
-by(simp add: delete_def inorder_del)
-
-
-subsection \<open>Balancedness\<close>
-
-lemma bal_upd: "bal t \<Longrightarrow> bal (tree\<^sub>i(upd x y t)) \<and> height(upd x y t) = height t"
-by (induct t) (auto, auto split: up\<^sub>i.split) (* 20 secs (2015) *)
-
-lemma bal_update: "bal t \<Longrightarrow> bal (update x y t)"
-by (simp add: update_def bal_upd)
-
-
-lemma height_del: "bal t \<Longrightarrow> height(del x t) = height t"
-by(induction x t rule: del.induct)
- (auto simp add: heights height_del_min split: prod.split)
-(* 20 secs (2015) *)
-
-lemma bal_tree\<^sub>d_del: "bal t \<Longrightarrow> bal(tree\<^sub>d(del x t))"
-by(induction x t rule: del.induct)
- (auto simp: bals bal_del_min height_del height_del_min split: prod.split)
-(* 100 secs (2015) *)
-
-corollary bal_delete: "bal t \<Longrightarrow> bal(delete x t)"
-by(simp add: delete_def bal_tree\<^sub>d_del)
-
-
-subsection \<open>Overall Correctness\<close>
-
-interpretation T234_Map: Map_by_Ordered
-where empty = Leaf and lookup = lookup and update = update and delete = delete
-and inorder = inorder and wf = bal
-proof (standard, goal_cases)
- case 2 thus ?case by(simp add: lookup)
-next
- case 3 thus ?case by(simp add: inorder_update)
-next
- case 4 thus ?case by(simp add: inorder_delete)
-next
- case 6 thus ?case by(simp add: bal_update)
-next
- case 7 thus ?case by(simp add: bal_delete)
-qed simp+
-
-end
+(* Author: Tobias Nipkow *)
+
+section \<open>A 2-3-4 Tree Implementation of Maps\<close>
+
+theory Tree234_Map
+imports
+ Tree234_Set
+ "../Data_Structures/Map_by_Ordered"
+begin
+
+subsection \<open>Map operations on 2-3-4 trees\<close>
+
+fun lookup :: "('a::cmp * 'b) tree234 \<Rightarrow> 'a \<Rightarrow> 'b option" where
+"lookup Leaf x = None" |
+"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))" |
+"lookup (Node4 t1 (a1,b1) t2 (a2,b2) t3 (a3,b3) t4) x = (case cmp x a2 of
+ LT \<Rightarrow> (case cmp x a1 of
+ LT \<Rightarrow> lookup t1 x | EQ \<Rightarrow> Some b1 | GT \<Rightarrow> lookup t2 x) |
+ EQ \<Rightarrow> Some b2 |
+ GT \<Rightarrow> (case cmp x a3 of
+ LT \<Rightarrow> lookup t3 x | EQ \<Rightarrow> Some b3 | GT \<Rightarrow> lookup t4 x))"
+
+fun upd :: "'a::cmp \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree234 \<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) = (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)) |
+ 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) = (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)) |
+ 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))))" |
+"upd x y (Node4 t1 ab1 t2 ab2 t3 ab3 t4) = (case cmp x (fst ab2) of
+ LT \<Rightarrow> (case cmp x (fst ab1) of
+ LT \<Rightarrow> (case upd x y t1 of
+ T\<^sub>i t1' => T\<^sub>i (Node4 t1' ab1 t2 ab2 t3 ab3 t4)
+ | Up\<^sub>i t11 q t12 => Up\<^sub>i (Node2 t11 q t12) ab1 (Node3 t2 ab2 t3 ab3 t4)) |
+ EQ \<Rightarrow> T\<^sub>i (Node4 t1 (x,y) t2 ab2 t3 ab3 t4) |
+ GT \<Rightarrow> (case upd x y t2 of
+ T\<^sub>i t2' => T\<^sub>i (Node4 t1 ab1 t2' ab2 t3 ab3 t4)
+ | Up\<^sub>i t21 q t22 => Up\<^sub>i (Node2 t1 ab1 t21) q (Node3 t22 ab2 t3 ab3 t4))) |
+ EQ \<Rightarrow> T\<^sub>i (Node4 t1 ab1 t2 (x,y) t3 ab3 t4) |
+ GT \<Rightarrow> (case cmp x (fst ab3) of
+ LT \<Rightarrow> (case upd x y t3 of
+ T\<^sub>i t3' \<Rightarrow> T\<^sub>i (Node4 t1 ab1 t2 ab2 t3' ab3 t4)
+ | Up\<^sub>i t31 q t32 => Up\<^sub>i (Node2 t1 ab1 t2) ab2(*q*) (Node3 t31 q t32 ab3 t4)) |
+ EQ \<Rightarrow> T\<^sub>i (Node4 t1 ab1 t2 ab2 t3 (x,y) t4) |
+ GT \<Rightarrow> (case upd x y t4 of
+ T\<^sub>i t4' => T\<^sub>i (Node4 t1 ab1 t2 ab2 t3 ab3 t4')
+ | Up\<^sub>i t41 q t42 => Up\<^sub>i (Node2 t1 ab1 t2) ab2 (Node3 t3 ab3 t41 q t42))))"
+
+definition update :: "'a::cmp \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree234 \<Rightarrow> ('a*'b) tree234" where
+"update x y t = tree\<^sub>i(upd x y t)"
+
+fun del :: "'a::cmp \<Rightarrow> ('a*'b) tree234 \<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 (Node4 Leaf ab1 Leaf ab2 Leaf ab3 Leaf) =
+ T\<^sub>d(if x = fst ab1 then Node3 Leaf ab2 Leaf ab3 Leaf else
+ if x = fst ab2 then Node3 Leaf ab1 Leaf ab3 Leaf else
+ if x = fst ab3 then Node3 Leaf ab1 Leaf ab2 Leaf
+ else Node4 Leaf ab1 Leaf ab2 Leaf ab3 Leaf)" |
+"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)))" |
+"del x (Node4 t1 ab1 t2 ab2 t3 ab3 t4) = (case cmp x (fst ab2) of
+ LT \<Rightarrow> (case cmp x (fst ab1) of
+ LT \<Rightarrow> node41 (del x t1) ab1 t2 ab2 t3 ab3 t4 |
+ EQ \<Rightarrow> let (ab',t2') = del_min t2 in node42 t1 ab' t2' ab2 t3 ab3 t4 |
+ GT \<Rightarrow> node42 t1 ab1 (del x t2) ab2 t3 ab3 t4) |
+ EQ \<Rightarrow> let (ab',t3') = del_min t3 in node43 t1 ab1 t2 ab' t3' ab3 t4 |
+ GT \<Rightarrow> (case cmp x (fst ab3) of
+ LT \<Rightarrow> node43 t1 ab1 t2 ab2 (del x t3) ab3 t4 |
+ EQ \<Rightarrow> let (ab',t4') = del_min t4 in node44 t1 ab1 t2 ab2 t3 ab' t4' |
+ GT \<Rightarrow> node44 t1 ab1 t2 ab2 t3 ab3 (del x t4)))"
+
+definition delete :: "'a::cmp \<Rightarrow> ('a*'b) tree234 \<Rightarrow> ('a*'b) tree234" where
+"delete x t = tree\<^sub>d(del x t)"
+
+
+subsection "Functional correctness"
+
+lemma lookup: "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
+by (induction t) (auto simp: map_of_simps split: option.split)
+
+
+lemma inorder_upd:
+ "sorted1(inorder t) \<Longrightarrow> inorder(tree\<^sub>i(upd a b t)) = upd_list a b (inorder t)"
+by(induction t)
+ (auto simp: upd_list_simps, auto simp: upd_list_simps split: up\<^sub>i.splits)
+
+lemma inorder_update:
+ "sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)"
+by(simp add: update_def inorder_upd)
+
+
+lemma inorder_del: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
+ inorder(tree\<^sub>d (del x t)) = del_list x (inorder t)"
+by(induction t rule: del.induct)
+ ((auto simp: del_list_simps inorder_nodes del_minD split: prod.splits)[1])+
+(* 200 secs (2015) *)
+
+lemma inorder_delete: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
+ inorder(delete x t) = del_list x (inorder t)"
+by(simp add: delete_def inorder_del)
+
+
+subsection \<open>Balancedness\<close>
+
+lemma bal_upd: "bal t \<Longrightarrow> bal (tree\<^sub>i(upd x y t)) \<and> height(upd x y t) = height t"
+by (induct t) (auto, auto split: up\<^sub>i.split) (* 20 secs (2015) *)
+
+lemma bal_update: "bal t \<Longrightarrow> bal (update x y t)"
+by (simp add: update_def bal_upd)
+
+
+lemma height_del: "bal t \<Longrightarrow> height(del x t) = height t"
+by(induction x t rule: del.induct)
+ (auto simp add: heights height_del_min split: prod.split)
+(* 20 secs (2015) *)
+
+lemma bal_tree\<^sub>d_del: "bal t \<Longrightarrow> bal(tree\<^sub>d(del x t))"
+by(induction x t rule: del.induct)
+ (auto simp: bals bal_del_min height_del height_del_min split: prod.split)
+(* 100 secs (2015) *)
+
+corollary bal_delete: "bal t \<Longrightarrow> bal(delete x t)"
+by(simp add: delete_def bal_tree\<^sub>d_del)
+
+
+subsection \<open>Overall Correctness\<close>
+
+interpretation T234_Map: Map_by_Ordered
+where empty = Leaf and lookup = lookup and update = update and delete = delete
+and inorder = inorder and wf = bal
+proof (standard, goal_cases)
+ case 2 thus ?case by(simp add: lookup)
+next
+ case 3 thus ?case by(simp add: inorder_update)
+next
+ case 4 thus ?case by(simp add: inorder_delete)
+next
+ case 6 thus ?case by(simp add: bal_update)
+next
+ case 7 thus ?case by(simp add: bal_delete)
+qed simp+
+
+end