(*  Title:      HOL/Library/DAList.thy

    Author:     Lukas Bulwahn, TU Muenchen *)

header {* Abstract type of association lists with unique keys *}

theory DAList

imports AList

begin

text {* This was based on some existing fragments in the AFP-Collection framework. *}

subsection {* Preliminaries *}

lemma distinct_map_fst_filter:

   "distinct (map fst xs) ==> distinct (map fst (List.filter P xs))"

by (induct xs) auto

subsection {* Type @{text "('key, 'value) alist" } *}

typedef ('key, 'value) alist = "{xs :: ('key \<times> 'value) list. (distinct o map fst) xs}"

  morphisms impl_of Alist

proof

  show "[] \<in> {xs. (distinct o map fst) xs}" by simp

qed

setup_lifting type_definition_alist

lemma alist_ext: "impl_of xs = impl_of ys \<Longrightarrow> xs = ys"

by(simp add: impl_of_inject)

lemma alist_eq_iff: "xs = ys \<longleftrightarrow> impl_of xs = impl_of ys"

by(simp add: impl_of_inject)

lemma impl_of_distinct [simp, intro]: "distinct (map fst (impl_of xs))"

using impl_of[of xs] by simp

lemma Alist_impl_of [code abstype]: "Alist (impl_of xs) = xs"

by(rule impl_of_inverse)

subsection {* Primitive operations *}

lift_definition lookup :: "('key, 'value) alist \<Rightarrow> 'key \<Rightarrow> 'value option" is map_of  ..

lift_definition empty :: "('key, 'value) alist" is "[]" by simp

lift_definition update :: "'key \<Rightarrow> 'value \<Rightarrow> ('key, 'value) alist \<Rightarrow> ('key, 'value) alist"

  is AList.update

by (simp add: distinct_update)

(* FIXME: we use an unoptimised delete operation. *)

lift_definition delete :: "'key \<Rightarrow> ('key, 'value) alist \<Rightarrow> ('key, 'value) alist"

  is AList.delete

by (simp add: distinct_delete)

lift_definition map_entry :: "'key \<Rightarrow> ('value \<Rightarrow> 'value) \<Rightarrow> ('key, 'value) alist \<Rightarrow> ('key, 'value) alist"

  is AList.map_entry

by (simp add: distinct_map_entry)

lift_definition filter :: "('key \<times> 'value \<Rightarrow> bool) \<Rightarrow> ('key, 'value) alist \<Rightarrow> ('key, 'value) alist"

  is List.filter

by (simp add: distinct_map_fst_filter)

lift_definition map_default :: "'key => 'value => ('value => 'value) => ('key, 'value) alist => ('key, 'value) alist"

  is AList.map_default

by (simp add: distinct_map_default)

subsection {* Abstract operation properties *}

(* FIXME: to be completed *)

lemma lookup_empty [simp]: "lookup empty k = None"

by(simp add: empty_def lookup_def Alist_inverse)

lemma lookup_delete [simp]: "lookup (delete k al) = (lookup al)(k := None)"

by (simp add: lookup_def delete_def Alist_inverse distinct_delete delete_conv')

subsection {* Further operations *}

subsubsection {* Equality *}

instantiation alist :: (equal, equal) equal begin

definition "HOL.equal (xs :: ('a, 'b) alist) ys == impl_of xs = impl_of ys"

instance

proof

qed (simp add: equal_alist_def impl_of_inject)

end

subsubsection {* Size *}

instantiation alist :: (type, type) size begin

definition "size (al :: ('a, 'b) alist) = length (impl_of al)"

instance ..

end

subsection {* Quickcheck generators *}

notation fcomp (infixl "\<circ>>" 60)

notation scomp (infixl "\<circ>\<rightarrow>" 60)

definition (in term_syntax)

  valterm_empty :: "('key :: typerep, 'value :: typerep) alist \<times> (unit \<Rightarrow> Code_Evaluation.term)"

where

  "valterm_empty = Code_Evaluation.valtermify empty"

definition (in term_syntax)

  valterm_update :: "'key :: typerep \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow>

  'value :: typerep \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow>

  ('key, 'value) alist \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow>

  ('key, 'value) alist \<times> (unit \<Rightarrow> Code_Evaluation.term)" where

  [code_unfold]: "valterm_update k v a = Code_Evaluation.valtermify update {\<cdot>} k {\<cdot>} v {\<cdot>}a"

fun (in term_syntax) random_aux_alist 

where

  "random_aux_alist i j = (if i = 0 then Pair valterm_empty else Quickcheck_Random.collapse (Random.select_weight [(i, Quickcheck_Random.random j \<circ>\<rightarrow> (%k. Quickcheck_Random.random j \<circ>\<rightarrow> (%v. random_aux_alist (i - 1) j \<circ>\<rightarrow> (%a. Pair (valterm_update k v a))))), (1, Pair valterm_empty)]))"

instantiation alist :: (random, random) random

begin

definition random_alist

where

  "random_alist i = random_aux_alist i i"

instance ..

end

no_notation fcomp (infixl "\<circ>>" 60)

no_notation scomp (infixl "\<circ>\<rightarrow>" 60)

instantiation alist :: (exhaustive, exhaustive) exhaustive

begin

fun exhaustive_alist :: "(('a, 'b) alist => (bool * term list) option) => natural => (bool * term list) option"

where

  "exhaustive_alist f i = (if i = 0 then None else case f empty of Some ts => Some ts | None =>

     exhaustive_alist (%a. Quickcheck_Exhaustive.exhaustive (%k. Quickcheck_Exhaustive.exhaustive (%v. f (update k v a)) (i - 1)) (i - 1)) (i - 1))"

instance ..

end

instantiation alist :: (full_exhaustive, full_exhaustive) full_exhaustive

begin

fun full_exhaustive_alist :: "(('a, 'b) alist * (unit => term) => (bool * term list) option) => natural => (bool * term list) option"

where

  "full_exhaustive_alist f i = (if i = 0 then None else case f valterm_empty of Some ts => Some ts | None =>

     full_exhaustive_alist (%a. Quickcheck_Exhaustive.full_exhaustive (%k. Quickcheck_Exhaustive.full_exhaustive (%v. f (valterm_update k v a)) (i - 1)) (i - 1)) (i - 1))"

instance ..

end

hide_const valterm_empty valterm_update random_aux_alist

hide_fact (open) lookup_def empty_def update_def delete_def map_entry_def filter_def map_default_def

hide_const (open) impl_of lookup empty update delete map_entry filter map_default 

end