(* Author: Lukas Bulwahn, TU Muenchen *)
header {* Lifting operations of RBT trees *}
theory Lift_RBT
imports Main "~~/src/HOL/Library/RBT_Impl"
begin
subsection {* Type definition *}
typedef (open) ('a, 'b) rbt = "{t :: ('a\<Colon>linorder, 'b) RBT_Impl.rbt. is_rbt t}"
morphisms impl_of RBT
proof -
have "RBT_Impl.Empty \<in> ?rbt" by simp
then show ?thesis ..
qed
local_setup {* fn lthy =>
let
val quotients = {qtyp = @{typ "('a, 'b) rbt"}, rtyp = @{typ "('a, 'b) RBT_Impl.rbt"},
equiv_rel = @{term "dummy"}, equiv_thm = @{thm refl}}
val qty_full_name = @{type_name "rbt"}
fun qinfo phi = Quotient_Info.transform_quotients phi quotients
in lthy
|> Local_Theory.declaration {syntax = false, pervasive = true}
(fn phi => Quotient_Info.update_quotients qty_full_name (qinfo phi)
#> Quotient_Info.update_abs_rep qty_full_name (Quotient_Info.transform_abs_rep phi
{abs = @{term "RBT"}, rep = @{term "impl_of"}}))
end
*}
lemma rbt_eq_iff:
"t1 = t2 \<longleftrightarrow> impl_of t1 = impl_of t2"
by (simp add: impl_of_inject)
lemma rbt_eqI:
"impl_of t1 = impl_of t2 \<Longrightarrow> t1 = t2"
by (simp add: rbt_eq_iff)
lemma is_rbt_impl_of [simp, intro]:
"is_rbt (impl_of t)"
using impl_of [of t] by simp
lemma RBT_impl_of [simp, code abstype]:
"RBT (impl_of t) = t"
by (simp add: impl_of_inverse)
subsection {* Primitive operations *}
quotient_definition lookup where "lookup :: ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b" is "RBT_Impl.lookup"
declare lookup_def[unfolded map_fun_def comp_def id_def, code]
(* FIXME: some bug in quotient?*)
(*
quotient_definition empty where "empty :: ('a\<Colon>linorder, 'b) rbt"
is "RBT_Impl.Empty"
*)
definition empty :: "('a\<Colon>linorder, 'b) rbt" where
"empty = RBT RBT_Impl.Empty"
lemma impl_of_empty [code abstract]:
"impl_of empty = RBT_Impl.Empty"
by (simp add: empty_def RBT_inverse)
quotient_definition insert where "insert :: 'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
is "RBT_Impl.insert"
lemma impl_of_insert [code abstract]:
"impl_of (insert k v t) = RBT_Impl.insert k v (impl_of t)"
by (simp add: insert_def RBT_inverse)
quotient_definition delete where "delete :: 'a\<Colon>linorder \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
is "RBT_Impl.delete"
lemma impl_of_delete [code abstract]:
"impl_of (delete k t) = RBT_Impl.delete k (impl_of t)"
by (simp add: delete_def RBT_inverse)
(* FIXME: bug in quotient? *)
(*
quotient_definition entries where "entries :: ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list"
is "RBT_Impl.entries"
*)
definition entries :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list" where
[code]: "entries t = RBT_Impl.entries (impl_of t)"
(* FIXME: bug in quotient? *)
(*
quotient_definition keys where "keys :: ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a list"
is "RBT_Impl.keys"
*)
quotient_definition "bulkload :: ('a\<Colon>linorder \<times> 'b) list \<Rightarrow> ('a, 'b) rbt"
is "RBT_Impl.bulkload"
lemma impl_of_bulkload [code abstract]:
"impl_of (bulkload xs) = RBT_Impl.bulkload xs"
by (simp add: bulkload_def RBT_inverse)
quotient_definition "map_entry :: 'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
is "RBT_Impl.map_entry"
lemma impl_of_map_entry [code abstract]:
"impl_of (map_entry k f t) = RBT_Impl.map_entry k f (impl_of t)"
by (simp add: map_entry_def RBT_inverse)
(* Another bug: map is supposed to be a new definition, not using the old one.
quotient_definition "map :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
is "RBT_Impl.map"
*)
(* But this works, and then shows the other bug... *)
(*
quotient_definition map where "map :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
is "RBT_Impl.map"
*)
definition map :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
"map f t = RBT (RBT_Impl.map f (impl_of t))"
lemma impl_of_map [code abstract]:
"impl_of (map f t) = RBT_Impl.map f (impl_of t)"
by (simp add: map_def RBT_inverse)
(* FIXME: bug?
quotient_definition fold where "fold :: ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" is "RBT_Impl.fold"
*)
(*FIXME: definition of fold should have the code attribute. *)
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where
[code]: "fold f t = RBT_Impl.fold f (impl_of t)"
end