--- a/src/HOL/Data_Structures/Tree23_Map.thy Thu May 16 19:43:21 2019 +0200
+++ b/src/HOL/Data_Structures/Tree23_Map.thy Mon May 20 17:33:13 2019 +0200
@@ -22,37 +22,37 @@
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
-"upd x y Leaf = Up\<^sub>i Leaf (x,y) Leaf" |
+fun upd :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) upI" where
+"upd x y Leaf = OF 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) |
+ TI l' => TI (Node2 l' ab r)
+ | OF l1 ab' l2 => TI (Node3 l1 ab' l2 ab r)) |
+ EQ \<Rightarrow> TI (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)))" |
+ TI r' => TI (Node2 l ab r')
+ | OF r1 ab' r2 => TI (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) |
+ TI l' => TI (Node3 l' ab1 m ab2 r)
+ | OF l1 ab' l2 => OF (Node2 l1 ab' l2) ab1 (Node2 m ab2 r)) |
+ EQ \<Rightarrow> TI (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) |
+ TI m' => TI (Node3 l ab1 m' ab2 r)
+ | OF m1 ab' m2 => OF (Node2 l ab1 m1) ab' (Node2 m2 ab2 r)) |
+ EQ \<Rightarrow> TI (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))))"
+ TI r' => TI (Node3 l ab1 m ab2 r')
+ | OF r1 ab' r2 => OF (Node2 l ab1 m) ab2 (Node2 r1 ab' r2))))"
definition update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) tree23" where
-"update a b t = tree\<^sub>i(upd a b t)"
+"update a b t = treeI(upd a b t)"
-fun del :: "'a::linorder \<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
+fun del :: "'a::linorder \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) upD" where
+"del x Leaf = TD Leaf" |
+"del x (Node2 Leaf ab1 Leaf) = (if x=fst ab1 then UF Leaf else TD(Node2 Leaf ab1 Leaf))" |
+"del x (Node3 Leaf ab1 Leaf ab2 Leaf) = TD(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) = (case cmp x (fst ab1) of
LT \<Rightarrow> node21 (del x l) ab1 r |
@@ -67,7 +67,7 @@
GT \<Rightarrow> node33 l ab1 m ab2 (del x r)))"
definition delete :: "'a::linorder \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) tree23" where
-"delete x t = tree\<^sub>d(del x t)"
+"delete x t = treeD(del x t)"
subsection \<open>Functional Correctness\<close>
@@ -78,8 +78,8 @@
lemma inorder_upd:
- "sorted1(inorder t) \<Longrightarrow> inorder(tree\<^sub>i(upd x y t)) = upd_list x y (inorder t)"
-by(induction t) (auto simp: upd_list_simps split: up\<^sub>i.splits)
+ "sorted1(inorder t) \<Longrightarrow> inorder(treeI(upd x y t)) = upd_list x y (inorder t)"
+by(induction t) (auto simp: upd_list_simps split: upI.splits)
corollary inorder_update:
"sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
@@ -87,7 +87,7 @@
lemma inorder_del: "\<lbrakk> complete t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
- inorder(tree\<^sub>d (del x t)) = del_list x (inorder t)"
+ inorder(treeD (del x t)) = del_list x (inorder t)"
by(induction t rule: del.induct)
(auto simp: del_list_simps inorder_nodes split_minD split!: if_split prod.splits)
@@ -98,8 +98,8 @@
subsection \<open>Balancedness\<close>
-lemma complete_upd: "complete t \<Longrightarrow> complete (tree\<^sub>i(upd x y t)) \<and> height(upd x y t) = height t"
-by (induct t) (auto split!: if_split up\<^sub>i.split)(* 16 secs in 2015 *)
+lemma complete_upd: "complete t \<Longrightarrow> complete (treeI(upd x y t)) \<and> height(upd x y t) = height t"
+by (induct t) (auto split!: if_split upI.split)(* 16 secs in 2015 *)
corollary complete_update: "complete t \<Longrightarrow> complete (update x y t)"
by (simp add: update_def complete_upd)
@@ -109,12 +109,12 @@
by(induction x t rule: del.induct)
(auto simp add: heights max_def height_split_min split: prod.split)
-lemma complete_tree\<^sub>d_del: "complete t \<Longrightarrow> complete(tree\<^sub>d(del x t))"
+lemma complete_treeD_del: "complete t \<Longrightarrow> complete(treeD(del x t))"
by(induction x t rule: del.induct)
(auto simp: completes complete_split_min height_del height_split_min split: prod.split)
corollary complete_delete: "complete t \<Longrightarrow> complete(delete x t)"
-by(simp add: delete_def complete_tree\<^sub>d_del)
+by(simp add: delete_def complete_treeD_del)
subsection \<open>Overall Correctness\<close>