src/HOL/Data_Structures/Tree_Map.thy
changeset 61203 a8a8eca85801
child 61224 759b5299a9f2
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/Data_Structures/Tree_Map.thy	Mon Sep 21 14:44:32 2015 +0200
     1.3 @@ -0,0 +1,72 @@
     1.4 +(* Author: Tobias Nipkow *)
     1.5 +
     1.6 +section {* Unbalanced Tree as Map *}
     1.7 +
     1.8 +theory Tree_Map
     1.9 +imports
    1.10 +  "~~/src/HOL/Library/Tree"
    1.11 +  Map_by_Ordered
    1.12 +begin
    1.13 +
    1.14 +fun lookup :: "('a::linorder*'b) tree \<Rightarrow> 'a \<Rightarrow> 'b option" where
    1.15 +"lookup Leaf x = None" |
    1.16 +"lookup (Node l (a,b) r) x = (if x < a then lookup l x else
    1.17 +  if x > a then lookup r x else Some b)"
    1.18 +
    1.19 +fun update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
    1.20 +"update a b Leaf = Node Leaf (a,b) Leaf" |
    1.21 +"update a b (Node l (x,y) r) =
    1.22 +   (if a < x then Node (update a b l) (x,y) r
    1.23 +    else if a=x then Node l (a,b) r
    1.24 +    else Node l (x,y) (update a b r))"
    1.25 +
    1.26 +fun del_min :: "'a tree \<Rightarrow> 'a * 'a tree" where
    1.27 +"del_min (Node Leaf a r) = (a, r)" |
    1.28 +"del_min (Node l a r) = (let (x,l') = del_min l in (x, Node l' a r))"
    1.29 +
    1.30 +fun delete :: "'a::linorder \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
    1.31 +"delete k Leaf = Leaf" |
    1.32 +"delete k (Node l (a,b) r) = (if k<a then Node (delete k l) (a,b) r else
    1.33 +  if k > a then Node l (a,b) (delete k r) else
    1.34 +  if r = Leaf then l else let (ab',r') = del_min r in Node l ab' r')"
    1.35 +
    1.36 +
    1.37 +subsection "Functional Correctness Proofs"
    1.38 +
    1.39 +lemma lookup_eq: "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
    1.40 +apply (induction t)
    1.41 +apply (auto simp: sorted_lems map_of_append map_of_sorteds split: option.split)
    1.42 +done
    1.43 +
    1.44 +
    1.45 +lemma inorder_update:
    1.46 +  "sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)"
    1.47 +by(induction t) (auto simp: upd_list_sorteds sorted_lems)
    1.48 +
    1.49 +
    1.50 +lemma del_minD:
    1.51 +  "del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> sorted1(inorder t) \<Longrightarrow>
    1.52 +   x # inorder t' = inorder t"
    1.53 +by(induction t arbitrary: t' rule: del_min.induct)
    1.54 +  (auto simp: sorted_lems split: prod.splits)
    1.55 +
    1.56 +lemma inorder_delete:
    1.57 +  "sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
    1.58 +by(induction t)
    1.59 +  (auto simp: del_list_sorted sorted_lems dest!: del_minD split: prod.splits)
    1.60 +
    1.61 +
    1.62 +interpretation Map_by_Ordered
    1.63 +where empty = Leaf and lookup = lookup and update = update and delete = delete
    1.64 +and inorder = inorder and wf = "\<lambda>_. True"
    1.65 +proof (standard, goal_cases)
    1.66 +  case 1 show ?case by simp
    1.67 +next
    1.68 +  case 2 thus ?case by(simp add: lookup_eq)
    1.69 +next
    1.70 +  case 3 thus ?case by(simp add: inorder_update)
    1.71 +next
    1.72 +  case 4 thus ?case by(simp add: inorder_delete)
    1.73 +qed (rule TrueI)+
    1.74 +
    1.75 +end