1(* Author: Tobias Nipkow *) 2 3section \<open>2-3-4 Tree Implementation of Maps\<close> 4 5theory Tree234_Map 6imports 7 Tree234_Set 8 Map_Specs 9begin 10 11subsection \<open>Map operations on 2-3-4 trees\<close> 12 13fun lookup :: "('a::linorder * 'b) tree234 \<Rightarrow> 'a \<Rightarrow> 'b option" where 14"lookup Leaf x = None" | 15"lookup (Node2 l (a,b) r) x = (case cmp x a of 16 LT \<Rightarrow> lookup l x | 17 GT \<Rightarrow> lookup r x | 18 EQ \<Rightarrow> Some b)" | 19"lookup (Node3 l (a1,b1) m (a2,b2) r) x = (case cmp x a1 of 20 LT \<Rightarrow> lookup l x | 21 EQ \<Rightarrow> Some b1 | 22 GT \<Rightarrow> (case cmp x a2 of 23 LT \<Rightarrow> lookup m x | 24 EQ \<Rightarrow> Some b2 | 25 GT \<Rightarrow> lookup r x))" | 26"lookup (Node4 t1 (a1,b1) t2 (a2,b2) t3 (a3,b3) t4) x = (case cmp x a2 of 27 LT \<Rightarrow> (case cmp x a1 of 28 LT \<Rightarrow> lookup t1 x | EQ \<Rightarrow> Some b1 | GT \<Rightarrow> lookup t2 x) | 29 EQ \<Rightarrow> Some b2 | 30 GT \<Rightarrow> (case cmp x a3 of 31 LT \<Rightarrow> lookup t3 x | EQ \<Rightarrow> Some b3 | GT \<Rightarrow> lookup t4 x))" 32 33fun upd :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree234 \<Rightarrow> ('a*'b) up\<^sub>i" where 34"upd x y Leaf = Up\<^sub>i Leaf (x,y) Leaf" | 35"upd x y (Node2 l ab r) = (case cmp x (fst ab) of 36 LT \<Rightarrow> (case upd x y l of 37 T\<^sub>i l' => T\<^sub>i (Node2 l' ab r) 38 | Up\<^sub>i l1 ab' l2 => T\<^sub>i (Node3 l1 ab' l2 ab r)) | 39 EQ \<Rightarrow> T\<^sub>i (Node2 l (x,y) r) | 40 GT \<Rightarrow> (case upd x y r of 41 T\<^sub>i r' => T\<^sub>i (Node2 l ab r') 42 | Up\<^sub>i r1 ab' r2 => T\<^sub>i (Node3 l ab r1 ab' r2)))" | 43"upd x y (Node3 l ab1 m ab2 r) = (case cmp x (fst ab1) of 44 LT \<Rightarrow> (case upd x y l of 45 T\<^sub>i l' => T\<^sub>i (Node3 l' ab1 m ab2 r) 46 | Up\<^sub>i l1 ab' l2 => Up\<^sub>i (Node2 l1 ab' l2) ab1 (Node2 m ab2 r)) | 47 EQ \<Rightarrow> T\<^sub>i (Node3 l (x,y) m ab2 r) | 48 GT \<Rightarrow> (case cmp x (fst ab2) of 49 LT \<Rightarrow> (case upd x y m of 50 T\<^sub>i m' => T\<^sub>i (Node3 l ab1 m' ab2 r) 51 | Up\<^sub>i m1 ab' m2 => Up\<^sub>i (Node2 l ab1 m1) ab' (Node2 m2 ab2 r)) | 52 EQ \<Rightarrow> T\<^sub>i (Node3 l ab1 m (x,y) r) | 53 GT \<Rightarrow> (case upd x y r of 54 T\<^sub>i r' => T\<^sub>i (Node3 l ab1 m ab2 r') 55 | Up\<^sub>i r1 ab' r2 => Up\<^sub>i (Node2 l ab1 m) ab2 (Node2 r1 ab' r2))))" | 56"upd x y (Node4 t1 ab1 t2 ab2 t3 ab3 t4) = (case cmp x (fst ab2) of 57 LT \<Rightarrow> (case cmp x (fst ab1) of 58 LT \<Rightarrow> (case upd x y t1 of 59 T\<^sub>i t1' => T\<^sub>i (Node4 t1' ab1 t2 ab2 t3 ab3 t4) 60 | Up\<^sub>i t11 q t12 => Up\<^sub>i (Node2 t11 q t12) ab1 (Node3 t2 ab2 t3 ab3 t4)) | 61 EQ \<Rightarrow> T\<^sub>i (Node4 t1 (x,y) t2 ab2 t3 ab3 t4) | 62 GT \<Rightarrow> (case upd x y t2 of 63 T\<^sub>i t2' => T\<^sub>i (Node4 t1 ab1 t2' ab2 t3 ab3 t4) 64 | Up\<^sub>i t21 q t22 => Up\<^sub>i (Node2 t1 ab1 t21) q (Node3 t22 ab2 t3 ab3 t4))) | 65 EQ \<Rightarrow> T\<^sub>i (Node4 t1 ab1 t2 (x,y) t3 ab3 t4) | 66 GT \<Rightarrow> (case cmp x (fst ab3) of 67 LT \<Rightarrow> (case upd x y t3 of 68 T\<^sub>i t3' \<Rightarrow> T\<^sub>i (Node4 t1 ab1 t2 ab2 t3' ab3 t4) 69 | Up\<^sub>i t31 q t32 => Up\<^sub>i (Node2 t1 ab1 t2) ab2(*q*) (Node3 t31 q t32 ab3 t4)) | 70 EQ \<Rightarrow> T\<^sub>i (Node4 t1 ab1 t2 ab2 t3 (x,y) t4) | 71 GT \<Rightarrow> (case upd x y t4 of 72 T\<^sub>i t4' => T\<^sub>i (Node4 t1 ab1 t2 ab2 t3 ab3 t4') 73 | Up\<^sub>i t41 q t42 => Up\<^sub>i (Node2 t1 ab1 t2) ab2 (Node3 t3 ab3 t41 q t42))))" 74 75definition update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree234 \<Rightarrow> ('a*'b) tree234" where 76"update x y t = tree\<^sub>i(upd x y t)" 77 78fun del :: "'a::linorder \<Rightarrow> ('a*'b) tree234 \<Rightarrow> ('a*'b) up\<^sub>d" where 79"del x Leaf = T\<^sub>d Leaf" | 80"del x (Node2 Leaf ab1 Leaf) = (if x=fst ab1 then Up\<^sub>d Leaf else T\<^sub>d(Node2 Leaf ab1 Leaf))" | 81"del x (Node3 Leaf ab1 Leaf ab2 Leaf) = T\<^sub>d(if x=fst ab1 then Node2 Leaf ab2 Leaf 82 else if x=fst ab2 then Node2 Leaf ab1 Leaf else Node3 Leaf ab1 Leaf ab2 Leaf)" | 83"del x (Node4 Leaf ab1 Leaf ab2 Leaf ab3 Leaf) = 84 T\<^sub>d(if x = fst ab1 then Node3 Leaf ab2 Leaf ab3 Leaf else 85 if x = fst ab2 then Node3 Leaf ab1 Leaf ab3 Leaf else 86 if x = fst ab3 then Node3 Leaf ab1 Leaf ab2 Leaf 87 else Node4 Leaf ab1 Leaf ab2 Leaf ab3 Leaf)" | 88"del x (Node2 l ab1 r) = (case cmp x (fst ab1) of 89 LT \<Rightarrow> node21 (del x l) ab1 r | 90 GT \<Rightarrow> node22 l ab1 (del x r) | 91 EQ \<Rightarrow> let (ab1',t) = split_min r in node22 l ab1' t)" | 92"del x (Node3 l ab1 m ab2 r) = (case cmp x (fst ab1) of 93 LT \<Rightarrow> node31 (del x l) ab1 m ab2 r | 94 EQ \<Rightarrow> let (ab1',m') = split_min m in node32 l ab1' m' ab2 r | 95 GT \<Rightarrow> (case cmp x (fst ab2) of 96 LT \<Rightarrow> node32 l ab1 (del x m) ab2 r | 97 EQ \<Rightarrow> let (ab2',r') = split_min r in node33 l ab1 m ab2' r' | 98 GT \<Rightarrow> node33 l ab1 m ab2 (del x r)))" | 99"del x (Node4 t1 ab1 t2 ab2 t3 ab3 t4) = (case cmp x (fst ab2) of 100 LT \<Rightarrow> (case cmp x (fst ab1) of 101 LT \<Rightarrow> node41 (del x t1) ab1 t2 ab2 t3 ab3 t4 | 102 EQ \<Rightarrow> let (ab',t2') = split_min t2 in node42 t1 ab' t2' ab2 t3 ab3 t4 | 103 GT \<Rightarrow> node42 t1 ab1 (del x t2) ab2 t3 ab3 t4) | 104 EQ \<Rightarrow> let (ab',t3') = split_min t3 in node43 t1 ab1 t2 ab' t3' ab3 t4 | 105 GT \<Rightarrow> (case cmp x (fst ab3) of 106 LT \<Rightarrow> node43 t1 ab1 t2 ab2 (del x t3) ab3 t4 | 107 EQ \<Rightarrow> let (ab',t4') = split_min t4 in node44 t1 ab1 t2 ab2 t3 ab' t4' | 108 GT \<Rightarrow> node44 t1 ab1 t2 ab2 t3 ab3 (del x t4)))" 109 110definition delete :: "'a::linorder \<Rightarrow> ('a*'b) tree234 \<Rightarrow> ('a*'b) tree234" where 111"delete x t = tree\<^sub>d(del x t)" 112 113 114subsection "Functional correctness" 115 116lemma lookup_map_of: 117 "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x" 118by (induction t) (auto simp: map_of_simps split: option.split) 119 120 121lemma inorder_upd: 122 "sorted1(inorder t) \<Longrightarrow> inorder(tree\<^sub>i(upd a b t)) = upd_list a b (inorder t)" 123by(induction t) 124 (auto simp: upd_list_simps, auto simp: upd_list_simps split: up\<^sub>i.splits) 125 126lemma inorder_update: 127 "sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)" 128by(simp add: update_def inorder_upd) 129 130lemma inorder_del: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow> 131 inorder(tree\<^sub>d (del x t)) = del_list x (inorder t)" 132by(induction t rule: del.induct) 133 (auto simp: del_list_simps inorder_nodes split_minD split!: if_splits prod.splits) 134(* 30 secs (2016) *) 135 136lemma inorder_delete: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow> 137 inorder(delete x t) = del_list x (inorder t)" 138by(simp add: delete_def inorder_del) 139 140 141subsection \<open>Balancedness\<close> 142 143lemma bal_upd: "bal t \<Longrightarrow> bal (tree\<^sub>i(upd x y t)) \<and> height(upd x y t) = height t" 144by (induct t) (auto, auto split!: if_split up\<^sub>i.split) (* 20 secs (2015) *) 145 146lemma bal_update: "bal t \<Longrightarrow> bal (update x y t)" 147by (simp add: update_def bal_upd) 148 149lemma height_del: "bal t \<Longrightarrow> height(del x t) = height t" 150by(induction x t rule: del.induct) 151 (auto simp add: heights height_split_min split!: if_split prod.split) 152 153lemma bal_tree\<^sub>d_del: "bal t \<Longrightarrow> bal(tree\<^sub>d(del x t))" 154by(induction x t rule: del.induct) 155 (auto simp: bals bal_split_min height_del height_split_min split!: if_split prod.split) 156 157corollary bal_delete: "bal t \<Longrightarrow> bal(delete x t)" 158by(simp add: delete_def bal_tree\<^sub>d_del) 159 160 161subsection \<open>Overall Correctness\<close> 162 163interpretation M: Map_by_Ordered 164where empty = empty and lookup = lookup and update = update and delete = delete 165and inorder = inorder and inv = bal 166proof (standard, goal_cases) 167 case 2 thus ?case by(simp add: lookup_map_of) 168next 169 case 3 thus ?case by(simp add: inorder_update) 170next 171 case 4 thus ?case by(simp add: inorder_delete) 172next 173 case 6 thus ?case by(simp add: bal_update) 174next 175 case 7 thus ?case by(simp add: bal_delete) 176qed (simp add: empty_def)+ 177 178end 179