author | wenzelm |
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parent 43124 | fdb7e1d5f762 |
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permissions | -rw-r--r-- |
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(* Author: Florian Haftmann, TU Muenchen *) |
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header {* Abstract type of Red-Black Trees *} |
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(*<*) |
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theory RBT |
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imports Main RBT_Impl |
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begin |
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subsection {* Type definition *} |
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typedef (open) ('a, 'b) rbt = "{t :: ('a\<Colon>linorder, 'b) RBT_Impl.rbt. is_rbt t}" |
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morphisms impl_of RBT |
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proof - |
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have "RBT_Impl.Empty \<in> ?rbt" by simp |
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then show ?thesis .. |
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qed |
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lemma rbt_eq_iff: |
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"t1 = t2 \<longleftrightarrow> impl_of t1 = impl_of t2" |
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by (simp add: impl_of_inject) |
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lemma rbt_eqI: |
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"impl_of t1 = impl_of t2 \<Longrightarrow> t1 = t2" |
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by (simp add: rbt_eq_iff) |
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lemma is_rbt_impl_of [simp, intro]: |
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"is_rbt (impl_of t)" |
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using impl_of [of t] by simp |
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lemma RBT_impl_of [simp, code abstype]: |
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"RBT (impl_of t) = t" |
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by (simp add: impl_of_inverse) |
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subsection {* Primitive operations *} |
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definition lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b" where |
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[code]: "lookup t = RBT_Impl.lookup (impl_of t)" |
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definition empty :: "('a\<Colon>linorder, 'b) rbt" where |
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"empty = RBT RBT_Impl.Empty" |
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lemma impl_of_empty [code abstract]: |
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"impl_of empty = RBT_Impl.Empty" |
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by (simp add: empty_def RBT_inverse) |
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definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where |
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"insert k v t = RBT (RBT_Impl.insert k v (impl_of t))" |
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lemma impl_of_insert [code abstract]: |
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"impl_of (insert k v t) = RBT_Impl.insert k v (impl_of t)" |
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by (simp add: insert_def RBT_inverse) |
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definition delete :: "'a\<Colon>linorder \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where |
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"delete k t = RBT (RBT_Impl.delete k (impl_of t))" |
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lemma impl_of_delete [code abstract]: |
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"impl_of (delete k t) = RBT_Impl.delete k (impl_of t)" |
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by (simp add: delete_def RBT_inverse) |
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definition entries :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list" where |
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[code]: "entries t = RBT_Impl.entries (impl_of t)" |
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definition keys :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a list" where |
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[code]: "keys t = RBT_Impl.keys (impl_of t)" |
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definition bulkload :: "('a\<Colon>linorder \<times> 'b) list \<Rightarrow> ('a, 'b) rbt" where |
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"bulkload xs = RBT (RBT_Impl.bulkload xs)" |
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lemma impl_of_bulkload [code abstract]: |
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"impl_of (bulkload xs) = RBT_Impl.bulkload xs" |
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by (simp add: bulkload_def RBT_inverse) |
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definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where |
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"map_entry k f t = RBT (RBT_Impl.map_entry k f (impl_of t))" |
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lemma impl_of_map_entry [code abstract]: |
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"impl_of (map_entry k f t) = RBT_Impl.map_entry k f (impl_of t)" |
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by (simp add: map_entry_def RBT_inverse) |
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definition map :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where |
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"map f t = RBT (RBT_Impl.map f (impl_of t))" |
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lemma impl_of_map [code abstract]: |
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"impl_of (map f t) = RBT_Impl.map f (impl_of t)" |
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by (simp add: map_def RBT_inverse) |
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definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where |
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[code]: "fold f t = RBT_Impl.fold f (impl_of t)" |
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subsection {* Derived operations *} |
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definition is_empty :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> bool" where |
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[code]: "is_empty t = (case impl_of t of RBT_Impl.Empty \<Rightarrow> True | _ \<Rightarrow> False)" |
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subsection {* Abstract lookup properties *} |
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lemma lookup_RBT: |
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"is_rbt t \<Longrightarrow> lookup (RBT t) = RBT_Impl.lookup t" |
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by (simp add: lookup_def RBT_inverse) |
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lemma lookup_impl_of: |
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"RBT_Impl.lookup (impl_of t) = lookup t" |
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by (simp add: lookup_def) |
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lemma entries_impl_of: |
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"RBT_Impl.entries (impl_of t) = entries t" |
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by (simp add: entries_def) |
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lemma keys_impl_of: |
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"RBT_Impl.keys (impl_of t) = keys t" |
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by (simp add: keys_def) |
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lemma lookup_empty [simp]: |
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"lookup empty = Map.empty" |
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by (simp add: empty_def lookup_RBT fun_eq_iff) |
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lemma lookup_insert [simp]: |
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"lookup (insert k v t) = (lookup t)(k \<mapsto> v)" |
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by (simp add: insert_def lookup_RBT lookup_insert lookup_impl_of) |
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lemma lookup_delete [simp]: |
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"lookup (delete k t) = (lookup t)(k := None)" |
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by (simp add: delete_def lookup_RBT RBT_Impl.lookup_delete lookup_impl_of restrict_complement_singleton_eq) |
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lemma map_of_entries [simp]: |
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"map_of (entries t) = lookup t" |
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by (simp add: entries_def map_of_entries lookup_impl_of) |
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lemma entries_lookup: |
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"entries t1 = entries t2 \<longleftrightarrow> lookup t1 = lookup t2" |
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by (simp add: entries_def lookup_def entries_lookup) |
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lemma lookup_bulkload [simp]: |
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"lookup (bulkload xs) = map_of xs" |
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by (simp add: bulkload_def lookup_RBT RBT_Impl.lookup_bulkload) |
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lemma lookup_map_entry [simp]: |
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"lookup (map_entry k f t) = (lookup t)(k := Option.map f (lookup t k))" |
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by (simp add: map_entry_def lookup_RBT RBT_Impl.lookup_map_entry lookup_impl_of) |
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lemma lookup_map [simp]: |
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"lookup (map f t) k = Option.map (f k) (lookup t k)" |
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by (simp add: map_def lookup_RBT RBT_Impl.lookup_map lookup_impl_of) |
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lemma fold_fold: |
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"fold f t = More_List.fold (prod_case f) (entries t)" |
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by (simp add: fold_def fun_eq_iff RBT_Impl.fold_def entries_impl_of) |
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lemma is_empty_empty [simp]: |
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"is_empty t \<longleftrightarrow> t = empty" |
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by (simp add: rbt_eq_iff is_empty_def impl_of_empty split: rbt.split) |
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lemma RBT_lookup_empty [simp]: (*FIXME*) |
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"RBT_Impl.lookup t = Map.empty \<longleftrightarrow> t = RBT_Impl.Empty" |
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by (cases t) (auto simp add: fun_eq_iff) |
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lemma lookup_empty_empty [simp]: |
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"lookup t = Map.empty \<longleftrightarrow> t = empty" |
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by (cases t) (simp add: empty_def lookup_def RBT_inject RBT_inverse) |
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lemma sorted_keys [iff]: |
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"sorted (keys t)" |
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by (simp add: keys_def RBT_Impl.keys_def sorted_entries) |
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lemma distinct_keys [iff]: |
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"distinct (keys t)" |
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by (simp add: keys_def RBT_Impl.keys_def distinct_entries) |
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end |