association lists with distinct keys uses the quotient infrastructure to obtain code certificates;
added remarks about further improvements
(* 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 (open) ('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 *}
(* FIXME: improve quotient_definition so that type annotations on the right hand sides can be removed *)
quotient_definition lookup :: "('key, 'value) alist \<Rightarrow> 'key \<Rightarrow> 'value option"
where "lookup" is "map_of :: ('key * 'value) list \<Rightarrow> 'key \<Rightarrow> 'value option" ..
quotient_definition empty :: "('key, 'value) alist"
where "empty" is "[] :: ('key * 'value) list" by simp
quotient_definition update :: "'key \<Rightarrow> 'value \<Rightarrow> ('key, 'value) alist \<Rightarrow> ('key, 'value) alist"
where "update" is "AList.update :: 'key \<Rightarrow> 'value \<Rightarrow> ('key * 'value) list \<Rightarrow> ('key * 'value) list"
by (simp add: distinct_update)
(* FIXME: we use an unoptimised delete operation. *)
quotient_definition delete :: "'key \<Rightarrow> ('key, 'value) alist \<Rightarrow> ('key, 'value) alist"
where "delete" is "AList.delete :: 'key \<Rightarrow> ('key * 'value) list \<Rightarrow> ('key * 'value) list"
by (simp add: distinct_delete)
quotient_definition map_entry :: "'key \<Rightarrow> ('value \<Rightarrow> 'value) \<Rightarrow> ('key, 'value) alist \<Rightarrow> ('key, 'value) alist"
where "map_entry" is "AList.map_entry :: 'key \<Rightarrow> ('value \<Rightarrow> 'value) \<Rightarrow> ('key * 'value) list \<Rightarrow> ('key * 'value) list"
by (simp add: distinct_map_entry)
quotient_definition filter :: "('key \<times> 'value \<Rightarrow> bool) \<Rightarrow> ('key, 'value) alist \<Rightarrow> ('key, 'value) alist"
where "filter" is "List.filter :: ('key \<times> 'value \<Rightarrow> bool) \<Rightarrow> ('key * 'value) list \<Rightarrow> ('key * 'value) list"
by (simp add: distinct_map_fst_filter)
quotient_definition map_default :: "'key => 'value => ('value => 'value) => ('key, 'value) alist => ('key, 'value) alist"
where "map_default" is "AList.map_default :: 'key => 'value => ('value => 'value) => ('key * 'value) list => ('key * 'value) list"
by (simp add: distinct_map_default)
(* FIXME: theorems are still used in Multiset; make code certificates available to the user *)
lemma impl_of_empty: "impl_of empty = []"
by (simp add: empty_def Alist_inverse)
lemma impl_of_update: "impl_of (update k v xs) = AList.update k v (impl_of xs)"
by (simp add: update_def Alist_inverse distinct_update)
lemma impl_of_filter:
"impl_of (filter P xs) = List.filter P (impl_of xs)"
unfolding filter_def by (simp add: Alist_inverse distinct_map_fst_filter)
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.collapse (Random.select_weight [(i, Quickcheck.random j \<circ>\<rightarrow> (%k. Quickcheck.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) => code_numeral => (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) => code_numeral => (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