--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/Tree23_Map.thy Sun Oct 18 17:25:13 2015 +0200
@@ -0,0 +1,155 @@
+(* Author: Tobias Nipkow *)
+
+section \<open>2-3 Tree Implementation of Maps\<close>
+
+theory Tree23_Map
+imports
+ Tree23_Set
+ Map_by_Ordered
+begin
+
+fun lookup :: "('a::linorder * '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)"
+
+fun upd :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) up\<^sub>i" where
+"upd a b Leaf = Up\<^sub>i Leaf (a,b) Leaf" |
+"upd a b (Node2 l xy r) =
+ (if a < fst xy then
+ (case upd a b l of
+ T\<^sub>i l' => T\<^sub>i (Node2 l' xy r)
+ | Up\<^sub>i l1 q l2 => T\<^sub>i (Node3 l1 q l2 xy r))
+ else if a = fst xy then T\<^sub>i (Node2 l (a,b) r)
+ else
+ (case upd a b r of
+ T\<^sub>i r' => T\<^sub>i (Node2 l xy r')
+ | Up\<^sub>i r1 q r2 => T\<^sub>i (Node3 l xy r1 q r2)))" |
+"upd a b (Node3 l xy1 m xy2 r) =
+ (if a < fst xy1 then
+ (case upd a b l of
+ T\<^sub>i l' => T\<^sub>i (Node3 l' xy1 m xy2 r)
+ | Up\<^sub>i l1 q l2 => Up\<^sub>i (Node2 l1 q l2) xy1 (Node2 m xy2 r))
+ else if a = fst xy1 then T\<^sub>i (Node3 l (a,b) m xy2 r)
+ else if a < fst xy2 then
+ (case upd a b m of
+ T\<^sub>i m' => T\<^sub>i (Node3 l xy1 m' xy2 r)
+ | Up\<^sub>i m1 q m2 => Up\<^sub>i (Node2 l xy1 m1) q (Node2 m2 xy2 r))
+ else if a = fst xy2 then T\<^sub>i (Node3 l xy1 m (a,b) r)
+ else
+ (case upd a b r of
+ T\<^sub>i r' => T\<^sub>i (Node3 l xy1 m xy2 r')
+ | Up\<^sub>i r1 q r2 => Up\<^sub>i (Node2 l xy1 m) xy2 (Node2 r1 q 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)"
+
+fun del :: "'a::linorder \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) up\<^sub>d"
+where
+"del k Leaf = T\<^sub>d Leaf" |
+"del k (Node2 Leaf p Leaf) = (if k=fst p then Up\<^sub>d Leaf else T\<^sub>d(Node2 Leaf p Leaf))" |
+"del k (Node3 Leaf p Leaf q Leaf) = T\<^sub>d(if k=fst p then Node2 Leaf q Leaf
+ else if k=fst q then Node2 Leaf p Leaf else Node3 Leaf p Leaf q Leaf)" |
+"del k (Node2 l a r) = (if k<fst a then node21 (del k l) a r else
+ if k > fst a then node22 l a (del k r) else
+ let (a',t) = del_min r in node22 l a' t)" |
+"del k (Node3 l a m b r) = (if k<fst a then node31 (del k l) a m b r else
+ if k = fst a then let (a',m') = del_min m in node32 l a' m' b r else
+ if k < fst b then node32 l a (del k m) b r else
+ if k = fst b then let (b',r') = del_min r in node33 l a m b' r'
+ else node33 l a m b (del k r))"
+
+definition delete :: "'a::linorder \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) tree23" where
+"delete k t = tree\<^sub>d(del k t)"
+
+
+subsection "Proofs for Lookup"
+
+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)
+
+
+subsection "Proofs for Update"
+
+text {* Balanced trees *}
+
+text{* First a standard proof that @{const upd} preserves @{const bal}. *}
+
+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)
+
+text{* Now an alternative proof (by Brian Huffman) that runs faster because
+two properties (balance and height) are combined in one predicate. *}
+
+lemma full\<^sub>i_ins: "full n t \<Longrightarrow> full\<^sub>i n (upd a b t)"
+by (induct rule: full.induct, auto split: up\<^sub>i.split)
+
+text {* The @{const update} operation preserves balance. *}
+
+lemma bal_update: "bal t \<Longrightarrow> bal (update a b t)"
+unfolding bal_iff_full update_def
+apply (erule exE)
+apply (drule full\<^sub>i_ins [of _ _ a b])
+apply (cases "upd a b t")
+apply (auto intro: full.intros)
+done
+
+text {* Functional correctness of @{const "update"}. *}
+
+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 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)
+
+
+subsection "Proofs for Deletion"
+
+lemma height_del: "bal t \<Longrightarrow> height(del x t) = height t"
+by(induction x t rule: del.induct)
+ (auto simp add: heights max_def height_del_min split: prod.split)
+
+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)
+
+corollary bal_delete: "bal t \<Longrightarrow> bal(delete x t)"
+by(simp add: delete_def bal_tree\<^sub>d_del)
+
+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)
+
+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>Overall Correctness\<close>
+
+interpretation T23_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