src/HOL/Library/RBT.thy
author kuncar
Tue Jul 31 13:55:39 2012 +0200 (2012-07-31)
changeset 48622 caaa1a02c650
parent 47450 2ada2be850cb
child 49834 b27bbb021df1
permissions -rw-r--r--
use lifting/transfer formalization of RBT from Lift_RBT
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(*  Title:      HOL/Library/RBT.thy
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    Author:     Lukas Bulwahn and Ondrej Kuncar
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*)
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header {* Abstract type of RBT trees *}
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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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setup_lifting type_definition_rbt
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lift_definition lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b" is "rbt_lookup" 
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by simp
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lift_definition empty :: "('a\<Colon>linorder, 'b) rbt" is RBT_Impl.Empty 
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by (simp add: empty_def)
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lift_definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_insert" 
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by simp
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lift_definition delete :: "'a\<Colon>linorder \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_delete" 
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by simp
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lift_definition entries :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list" is RBT_Impl.entries
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by simp
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lift_definition keys :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a list" is RBT_Impl.keys 
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by simp
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lift_definition bulkload :: "('a\<Colon>linorder \<times> 'b) list \<Rightarrow> ('a, 'b) rbt" is "rbt_bulkload" 
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by simp
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lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is rbt_map_entry 
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by simp
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lift_definition map :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is RBT_Impl.map
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by simp
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lift_definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c"  is RBT_Impl.fold 
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by simp
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lift_definition union :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_union"
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by (simp add: rbt_union_is_rbt)
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lift_definition foldi :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a :: linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c"
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  is RBT_Impl.foldi by simp
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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_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_lookup (impl_of t) = lookup t"
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  by transfer (rule refl)
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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 transfer (rule refl)
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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 transfer (rule refl)
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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 transfer (rule rbt_lookup_rbt_insert)
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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 transfer (simp add: rbt_lookup_rbt_delete 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 transfer (simp add: map_of_entries)
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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 transfer (simp add: entries_rbt_lookup)
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lemma lookup_bulkload [simp]:
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  "lookup (bulkload xs) = map_of xs"
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  by transfer (rule rbt_lookup_rbt_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 transfer (rule rbt_lookup_rbt_map_entry)
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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 transfer (rule rbt_lookup_map)
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lemma fold_fold:
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  "fold f t = List.fold (prod_case f) (entries t)"
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  by transfer (rule RBT_Impl.fold_def)
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lemma impl_of_empty:
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  "impl_of empty = RBT_Impl.Empty"
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  by transfer (rule refl)
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lemma is_empty_empty [simp]:
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  "is_empty t \<longleftrightarrow> t = empty"
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  unfolding is_empty_def by transfer (simp split: rbt.split)
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lemma RBT_lookup_empty [simp]: (*FIXME*)
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  "rbt_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 transfer (rule RBT_lookup_empty)
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lemma sorted_keys [iff]:
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  "sorted (keys t)"
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  by transfer (simp add: RBT_Impl.keys_def rbt_sorted_entries)
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lemma distinct_keys [iff]:
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  "distinct (keys t)"
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  by transfer (simp add: RBT_Impl.keys_def distinct_entries)
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lemma finite_dom_lookup [simp, intro!]: "finite (dom (lookup t))"
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  by transfer simp
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lemma lookup_union: "lookup (union s t) = lookup s ++ lookup t"
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  by transfer (simp add: rbt_lookup_rbt_union)
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lemma lookup_in_tree: "(lookup t k = Some v) = ((k, v) \<in> set (entries t))"
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  by transfer (simp add: rbt_lookup_in_tree)
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lemma keys_entries: "(k \<in> set (keys t)) = (\<exists>v. (k, v) \<in> set (entries t))"
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  by transfer (simp add: keys_entries)
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lemma fold_def_alt:
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  "fold f t = List.fold (prod_case f) (entries t)"
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  by transfer (auto simp: RBT_Impl.fold_def)
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lemma distinct_entries: "distinct (List.map fst (entries t))"
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  by transfer (simp add: distinct_entries)
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lemma non_empty_keys: "t \<noteq> empty \<Longrightarrow> keys t \<noteq> []"
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  by transfer (simp add: non_empty_rbt_keys)
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lemma keys_def_alt:
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  "keys t = List.map fst (entries t)"
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  by transfer (simp add: RBT_Impl.keys_def)
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subsection {* Quickcheck generators *}
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quickcheck_generator rbt predicate: is_rbt constructors: empty, insert
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end