src/HOL/Data_Structures/Tree_Map.thy
 author nipkow Mon Sep 21 14:44:32 2015 +0200 (2015-09-21) changeset 61203 a8a8eca85801 child 61224 759b5299a9f2 permissions -rw-r--r--
New subdirectory for functional data structures
```     1 (* Author: Tobias Nipkow *)
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```     2
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```     3 section {* Unbalanced Tree as Map *}
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```     4
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```     5 theory Tree_Map
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```     6 imports
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```     7   "~~/src/HOL/Library/Tree"
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```     8   Map_by_Ordered
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```     9 begin
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```    10
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```    11 fun lookup :: "('a::linorder*'b) tree \<Rightarrow> 'a \<Rightarrow> 'b option" where
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```    12 "lookup Leaf x = None" |
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```    13 "lookup (Node l (a,b) r) x = (if x < a then lookup l x else
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```    14   if x > a then lookup r x else Some b)"
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```    15
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```    16 fun update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
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```    17 "update a b Leaf = Node Leaf (a,b) Leaf" |
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```    18 "update a b (Node l (x,y) r) =
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```    19    (if a < x then Node (update a b l) (x,y) r
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```    20     else if a=x then Node l (a,b) r
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```    21     else Node l (x,y) (update a b r))"
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```    22
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```    23 fun del_min :: "'a tree \<Rightarrow> 'a * 'a tree" where
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```    24 "del_min (Node Leaf a r) = (a, r)" |
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```    25 "del_min (Node l a r) = (let (x,l') = del_min l in (x, Node l' a r))"
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```    26
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```    27 fun delete :: "'a::linorder \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
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```    28 "delete k Leaf = Leaf" |
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```    29 "delete k (Node l (a,b) r) = (if k<a then Node (delete k l) (a,b) r else
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```    30   if k > a then Node l (a,b) (delete k r) else
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```    31   if r = Leaf then l else let (ab',r') = del_min r in Node l ab' r')"
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```    32
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```    33
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```    34 subsection "Functional Correctness Proofs"
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```    35
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```    36 lemma lookup_eq: "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
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```    37 apply (induction t)
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```    38 apply (auto simp: sorted_lems map_of_append map_of_sorteds split: option.split)
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```    39 done
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```    40
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```    41
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```    42 lemma inorder_update:
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```    43   "sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)"
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```    44 by(induction t) (auto simp: upd_list_sorteds sorted_lems)
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```    45
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```    46
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```    47 lemma del_minD:
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```    48   "del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> sorted1(inorder t) \<Longrightarrow>
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```    49    x # inorder t' = inorder t"
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```    50 by(induction t arbitrary: t' rule: del_min.induct)
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```    51   (auto simp: sorted_lems split: prod.splits)
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```    52
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```    53 lemma inorder_delete:
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```    54   "sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
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```    55 by(induction t)
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```    56   (auto simp: del_list_sorted sorted_lems dest!: del_minD split: prod.splits)
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```    57
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```    58
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```    59 interpretation Map_by_Ordered
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```    60 where empty = Leaf and lookup = lookup and update = update and delete = delete
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```    61 and inorder = inorder and wf = "\<lambda>_. True"
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```    62 proof (standard, goal_cases)
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```    63   case 1 show ?case by simp
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```    64 next
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```    65   case 2 thus ?case by(simp add: lookup_eq)
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```    66 next
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```    67   case 3 thus ?case by(simp add: inorder_update)
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```    68 next
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```    69   case 4 thus ?case by(simp add: inorder_delete)
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```    70 qed (rule TrueI)+
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```    71
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```    72 end
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