author | nipkow |
Tue, 12 May 2020 10:24:53 +0200 | |
changeset 71829 | 6f2663df8374 |
parent 71463 | a31a9da43694 |
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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session-qualified theory imports: isabelle imports -U -i -d '~~/src/Benchmarks' -a;
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"HOL-Library.Tree" |
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Cmp |
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Set_Specs |
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begin |
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definition empty :: "'a tree" where |
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"empty = Leaf" |
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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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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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text \<open>Deletion by replacing:\<close> |
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fun split_min :: "'a tree \<Rightarrow> 'a * 'a tree" where |
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"split_min (Node l a r) = |
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(if l = Leaf then (a,r) else let (x,l') = split_min l in (x, Node l' a r))" |
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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') = split_min r in Node l a' r')" |
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text \<open>Deletion by joining:\<close> |
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fun join :: "('a::linorder)tree \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where |
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"join t Leaf = t" | |
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"join Leaf t = t" | |
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"join (Node t1 a t2) (Node t3 b t4) = |
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(case join t2 t3 of |
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Leaf \<Rightarrow> Node t1 a (Node Leaf b t4) | |
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Node u2 x u3 \<Rightarrow> Node (Node t1 a u2) x (Node u3 b t4))" |
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fun delete2 :: "'a::linorder \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where |
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"delete2 x Leaf = Leaf" | |
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"delete2 x (Node l a r) = |
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(case cmp x a of |
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LT \<Rightarrow> Node (delete2 x l) a r | |
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GT \<Rightarrow> Node l a (delete2 x r) | |
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EQ \<Rightarrow> join l r)" |
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subsection "Functional Correctness Proofs" |
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lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> set (inorder t))" |
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by (induction t) (auto simp: isin_simps) |
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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 split_minD: |
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"split_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> x # inorder t' = inorder t" |
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by(induction t arbitrary: t' rule: split_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 split_minD split: prod.splits) |
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interpretation S: Set_by_Ordered |
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where empty = empty 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 add: empty_def) |
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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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lemma inorder_join: |
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"inorder(join l r) = inorder l @ inorder r" |
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by(induction l r rule: join.induct) (auto split: tree.split) |
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lemma inorder_delete2: |
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"sorted(inorder t) \<Longrightarrow> inorder(delete2 x t) = del_list x (inorder t)" |
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by(induction t) (auto simp: inorder_join del_list_simps) |
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interpretation S2: Set_by_Ordered |
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where empty = empty and isin = isin and insert = insert and delete = delete2 |
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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 add: empty_def) |
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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_delete2) |
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qed (rule TrueI)+ |
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end |