author | nipkow |
Thu, 07 Jul 2016 18:08:02 +0200 | |
changeset 63411 | e051eea34990 |
parent 61790 | 0494964bb226 |
child 67965 | aaa31cd0caef |
permissions | -rw-r--r-- |
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(* Author: Tobias Nipkow *) |
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e051eea34990
got rid of class cmp; added height-size proofs by Daniel Stuewe
nipkow
parents:
61790
diff
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section \<open>Unbalanced Tree Implementation of Map\<close> |
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theory Tree_Map |
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imports |
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Tree_Set |
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Map_by_Ordered |
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begin |
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63411
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got rid of class cmp; added height-size proofs by Daniel Stuewe
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parents:
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fun lookup :: "('a::linorder*'b) tree \<Rightarrow> 'a \<Rightarrow> 'b option" where |
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"lookup Leaf x = None" | |
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"lookup (Node l (a,b) r) x = |
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(case cmp x a of LT \<Rightarrow> lookup l x | GT \<Rightarrow> lookup r x | EQ \<Rightarrow> Some b)" |
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63411
e051eea34990
got rid of class cmp; added height-size proofs by Daniel Stuewe
nipkow
parents:
61790
diff
changeset
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fun update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where |
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"update x y Leaf = Node Leaf (x,y) Leaf" | |
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"update x y (Node l (a,b) r) = (case cmp x a of |
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LT \<Rightarrow> Node (update x y l) (a,b) r | |
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EQ \<Rightarrow> Node l (x,y) r | |
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GT \<Rightarrow> Node l (a,b) (update x y r))" |
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63411
e051eea34990
got rid of class cmp; added height-size proofs by Daniel Stuewe
nipkow
parents:
61790
diff
changeset
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fun delete :: "'a::linorder \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where |
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"delete x Leaf = Leaf" | |
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"delete x (Node l (a,b) r) = (case cmp x a of |
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LT \<Rightarrow> Node (delete x l) (a,b) r | |
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GT \<Rightarrow> Node l (a,b) (delete x r) | |
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EQ \<Rightarrow> if r = Leaf then l else let (ab',r') = del_min r in Node l ab' r')" |
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subsection "Functional Correctness Proofs" |
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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_update: |
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"sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)" |
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by(induction t) (auto simp: upd_list_simps) |
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lemma inorder_delete: |
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"sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)" |
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by(induction t) (auto simp: del_list_simps del_minD split: prod.splits) |
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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 = "\<lambda>_. True" |
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proof (standard, goal_cases) |
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case 1 show ?case by simp |
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next |
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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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qed auto |
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end |