author | haftmann |
Mon, 06 Feb 2017 20:56:34 +0100 | |
changeset 64990 | c6a7de505796 |
parent 63411 | e051eea34990 |
child 66453 | cc19f7ca2ed6 |
permissions | -rw-r--r-- |
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(* Author: Tobias Nipkow *) |
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section \<open>Unbalanced Tree Implementation of Set\<close> |
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theory Tree_Set |
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imports |
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"~~/src/HOL/Library/Tree" |
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Cmp |
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Set_by_Ordered |
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begin |
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fun isin :: "'a::linorder tree \<Rightarrow> 'a \<Rightarrow> bool" where |
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"isin Leaf x = False" | |
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"isin (Node l a r) x = |
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(case cmp x a of |
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LT \<Rightarrow> isin l x | |
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EQ \<Rightarrow> True | |
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GT \<Rightarrow> isin r x)" |
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hide_const (open) insert |
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63411
e051eea34990
got rid of class cmp; added height-size proofs by Daniel Stuewe
nipkow
parents:
61678
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changeset
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fun insert :: "'a::linorder \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where |
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"insert x Leaf = Node Leaf x Leaf" | |
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"insert x (Node l a r) = |
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(case cmp x a of |
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LT \<Rightarrow> Node (insert x l) a r | |
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EQ \<Rightarrow> Node l a r | |
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GT \<Rightarrow> Node l a (insert x r))" |
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fun del_min :: "'a tree \<Rightarrow> 'a * 'a tree" where |
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"del_min (Node l a r) = |
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(if l = Leaf then (a,r) else let (x,l') = del_min l in (x, Node l' a r))" |
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63411
e051eea34990
got rid of class cmp; added height-size proofs by Daniel Stuewe
nipkow
parents:
61678
diff
changeset
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fun delete :: "'a::linorder \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where |
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"delete x Leaf = Leaf" | |
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"delete x (Node l a r) = |
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(case cmp x a of |
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LT \<Rightarrow> Node (delete x l) a r | |
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GT \<Rightarrow> Node l a (delete x r) | |
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EQ \<Rightarrow> if r = Leaf then l else let (a',r') = del_min r in Node l a' r')" |
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subsection "Functional Correctness Proofs" |
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lemma "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))" |
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by (induction t) (auto simp: elems_simps1) |
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lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))" |
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by (induction t) (auto simp: elems_simps2) |
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lemma inorder_insert: |
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"sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)" |
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by(induction t) (auto simp: ins_list_simps) |
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lemma del_minD: |
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"del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> x # inorder t' = inorder t" |
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by(induction t arbitrary: t' rule: del_min.induct) |
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(auto simp: sorted_lems split: prod.splits if_splits) |
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lemma inorder_delete: |
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"sorted(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 Set_by_Ordered |
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where empty = Leaf and isin = isin and insert = insert 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: isin_set) |
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next |
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case 3 thus ?case by(simp add: inorder_insert) |
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next |
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case 4 thus ?case by(simp add: inorder_delete) |
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qed (rule TrueI)+ |
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end |