author | nipkow |
Sat, 21 Apr 2018 08:41:42 +0200 | |
changeset 68020 | 6aade817bee5 |
parent 67965 | aaa31cd0caef |
child 68431 | b294e095f64c |
permissions | -rw-r--r-- |
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(* Author: Tobias Nipkow *) |
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section \<open>2-3 Tree Implementation of Maps\<close> |
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theory Tree23_Map |
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imports |
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Tree23_Set |
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Map_Specs |
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begin |
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fun lookup :: "('a::linorder * 'b) tree23 \<Rightarrow> 'a \<Rightarrow> 'b option" where |
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"lookup Leaf x = None" | |
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"lookup (Node2 l (a,b) r) x = (case cmp x a of |
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LT \<Rightarrow> lookup l x | |
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GT \<Rightarrow> lookup r x | |
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EQ \<Rightarrow> Some b)" | |
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"lookup (Node3 l (a1,b1) m (a2,b2) r) x = (case cmp x a1 of |
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LT \<Rightarrow> lookup l x | |
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EQ \<Rightarrow> Some b1 | |
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GT \<Rightarrow> (case cmp x a2 of |
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LT \<Rightarrow> lookup m x | |
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EQ \<Rightarrow> Some b2 | |
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GT \<Rightarrow> lookup r x))" |
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fun upd :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) up\<^sub>i" where |
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"upd x y Leaf = Up\<^sub>i Leaf (x,y) Leaf" | |
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"upd x y (Node2 l ab r) = (case cmp x (fst ab) of |
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LT \<Rightarrow> (case upd x y l of |
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T\<^sub>i l' => T\<^sub>i (Node2 l' ab r) |
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| Up\<^sub>i l1 ab' l2 => T\<^sub>i (Node3 l1 ab' l2 ab r)) | |
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EQ \<Rightarrow> T\<^sub>i (Node2 l (x,y) r) | |
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GT \<Rightarrow> (case upd x y r of |
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T\<^sub>i r' => T\<^sub>i (Node2 l ab r') |
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| Up\<^sub>i r1 ab' r2 => T\<^sub>i (Node3 l ab r1 ab' r2)))" | |
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"upd x y (Node3 l ab1 m ab2 r) = (case cmp x (fst ab1) of |
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LT \<Rightarrow> (case upd x y l of |
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T\<^sub>i l' => T\<^sub>i (Node3 l' ab1 m ab2 r) |
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| Up\<^sub>i l1 ab' l2 => Up\<^sub>i (Node2 l1 ab' l2) ab1 (Node2 m ab2 r)) | |
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EQ \<Rightarrow> T\<^sub>i (Node3 l (x,y) m ab2 r) | |
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GT \<Rightarrow> (case cmp x (fst ab2) of |
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LT \<Rightarrow> (case upd x y m of |
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T\<^sub>i m' => T\<^sub>i (Node3 l ab1 m' ab2 r) |
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| Up\<^sub>i m1 ab' m2 => Up\<^sub>i (Node2 l ab1 m1) ab' (Node2 m2 ab2 r)) | |
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EQ \<Rightarrow> T\<^sub>i (Node3 l ab1 m (x,y) r) | |
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GT \<Rightarrow> (case upd x y r of |
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T\<^sub>i r' => T\<^sub>i (Node3 l ab1 m ab2 r') |
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| Up\<^sub>i r1 ab' r2 => Up\<^sub>i (Node2 l ab1 m) ab2 (Node2 r1 ab' r2))))" |
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definition update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) tree23" where |
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"update a b t = tree\<^sub>i(upd a b t)" |
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fun del :: "'a::linorder \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) up\<^sub>d" where |
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"del x Leaf = T\<^sub>d Leaf" | |
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"del x (Node2 Leaf ab1 Leaf) = (if x=fst ab1 then Up\<^sub>d Leaf else T\<^sub>d(Node2 Leaf ab1 Leaf))" | |
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"del x (Node3 Leaf ab1 Leaf ab2 Leaf) = T\<^sub>d(if x=fst ab1 then Node2 Leaf ab2 Leaf |
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else if x=fst ab2 then Node2 Leaf ab1 Leaf else Node3 Leaf ab1 Leaf ab2 Leaf)" | |
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"del x (Node2 l ab1 r) = (case cmp x (fst ab1) of |
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LT \<Rightarrow> node21 (del x l) ab1 r | |
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GT \<Rightarrow> node22 l ab1 (del x r) | |
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EQ \<Rightarrow> let (ab1',t) = split_min r in node22 l ab1' t)" | |
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"del x (Node3 l ab1 m ab2 r) = (case cmp x (fst ab1) of |
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LT \<Rightarrow> node31 (del x l) ab1 m ab2 r | |
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EQ \<Rightarrow> let (ab1',m') = split_min m in node32 l ab1' m' ab2 r | |
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GT \<Rightarrow> (case cmp x (fst ab2) of |
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LT \<Rightarrow> node32 l ab1 (del x m) ab2 r | |
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EQ \<Rightarrow> let (ab2',r') = split_min r in node33 l ab1 m ab2' r' | |
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GT \<Rightarrow> node33 l ab1 m ab2 (del x r)))" |
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definition delete :: "'a::linorder \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) tree23" where |
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"delete x t = tree\<^sub>d(del x t)" |
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subsection \<open>Functional Correctness\<close> |
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lemma lookup_map_of: |
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"sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x" |
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by (induction t) (auto simp: map_of_simps split: option.split) |
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lemma inorder_upd: |
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"sorted1(inorder t) \<Longrightarrow> inorder(tree\<^sub>i(upd x y t)) = upd_list x y (inorder t)" |
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by(induction t) (auto simp: upd_list_simps split: up\<^sub>i.splits) |
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corollary inorder_update: |
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"sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)" |
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by(simp add: update_def inorder_upd) |
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lemma inorder_del: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow> |
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inorder(tree\<^sub>d (del x t)) = del_list x (inorder t)" |
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by(induction t rule: del.induct) |
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(auto simp: del_list_simps inorder_nodes split_minD split!: if_split prod.splits) |
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corollary inorder_delete: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow> |
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inorder(delete x t) = del_list x (inorder t)" |
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by(simp add: delete_def inorder_del) |
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subsection \<open>Balancedness\<close> |
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lemma bal_upd: "bal t \<Longrightarrow> bal (tree\<^sub>i(upd x y t)) \<and> height(upd x y t) = height t" |
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by (induct t) (auto split!: if_split up\<^sub>i.split)(* 16 secs in 2015 *) |
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corollary bal_update: "bal t \<Longrightarrow> bal (update x y t)" |
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by (simp add: update_def bal_upd) |
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lemma height_del: "bal t \<Longrightarrow> height(del x t) = height t" |
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by(induction x t rule: del.induct) |
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(auto simp add: heights max_def height_split_min split: prod.split) |
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lemma bal_tree\<^sub>d_del: "bal t \<Longrightarrow> bal(tree\<^sub>d(del x t))" |
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by(induction x t rule: del.induct) |
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(auto simp: bals bal_split_min height_del height_split_min split: prod.split) |
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corollary bal_delete: "bal t \<Longrightarrow> bal(delete x t)" |
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by(simp add: delete_def bal_tree\<^sub>d_del) |
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subsection \<open>Overall Correctness\<close> |
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interpretation Map_by_Ordered |
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where empty = Leaf and lookup = lookup and update = update and delete = delete |
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and inorder = inorder and inv = bal |
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proof (standard, goal_cases) |
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case 2 thus ?case by(simp add: lookup_map_of) |
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next |
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case 3 thus ?case by(simp add: inorder_update) |
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next |
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case 4 thus ?case by(simp add: inorder_delete) |
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next |
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case 6 thus ?case by(simp add: bal_update) |
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next |
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case 7 thus ?case by(simp add: bal_delete) |
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qed simp+ |
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end |