(* Title: HOL/Library/DAList_Multiset.thy
Author: Lukas Bulwahn, TU Muenchen
*)
header {* Multisets partially implemented by association lists *}
theory DAList_Multiset
imports Multiset DAList
begin
text {* Raw operations on lists *}
definition join_raw :: "('key \<Rightarrow> 'val \<times> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
where
"join_raw f xs ys = foldr (\<lambda>(k, v). map_default k v (%v'. f k (v', v))) ys xs"
lemma join_raw_Nil [simp]:
"join_raw f xs [] = xs"
by (simp add: join_raw_def)
lemma join_raw_Cons [simp]:
"join_raw f xs ((k, v) # ys) = map_default k v (%v'. f k (v', v)) (join_raw f xs ys)"
by (simp add: join_raw_def)
lemma map_of_join_raw:
assumes "distinct (map fst ys)"
shows "map_of (join_raw f xs ys) x = (case map_of xs x of None => map_of ys x | Some v =>
(case map_of ys x of None => Some v | Some v' => Some (f x (v, v'))))"
using assms
apply (induct ys)
apply (auto simp add: map_of_map_default split: option.split)
apply (metis map_of_eq_None_iff option.simps(2) weak_map_of_SomeI)
by (metis Some_eq_map_of_iff map_of_eq_None_iff option.simps(2))
lemma distinct_join_raw:
assumes "distinct (map fst xs)"
shows "distinct (map fst (join_raw f xs ys))"
using assms
proof (induct ys)
case (Cons y ys)
thus ?case by (cases y) (simp add: distinct_map_default)
qed auto
definition
"subtract_entries_raw xs ys = foldr (%(k, v). AList.map_entry k (%v'. v' - v)) ys xs"
lemma map_of_subtract_entries_raw:
assumes "distinct (map fst ys)"
shows "map_of (subtract_entries_raw xs ys) x = (case map_of xs x of None => None | Some v =>
(case map_of ys x of None => Some v | Some v' => Some (v - v')))"
using assms unfolding subtract_entries_raw_def
apply (induct ys)
apply auto
apply (simp split: option.split)
apply (simp add: map_of_map_entry)
apply (auto split: option.split)
apply (metis map_of_eq_None_iff option.simps(3) option.simps(4))
by (metis map_of_eq_None_iff option.simps(4) option.simps(5))
lemma distinct_subtract_entries_raw:
assumes "distinct (map fst xs)"
shows "distinct (map fst (subtract_entries_raw xs ys))"
using assms
unfolding subtract_entries_raw_def by (induct ys) (auto simp add: distinct_map_entry)
text {* Operations on alists with distinct keys *}
lift_definition join :: "('a \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist"
is join_raw
by (simp add: distinct_join_raw)
lift_definition subtract_entries :: "('a, ('b :: minus)) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist"
is subtract_entries_raw
by (simp add: distinct_subtract_entries_raw)
text {* Implementing multisets by means of association lists *}
definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat" where
"count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> n)"
lemma count_of_multiset:
"count_of xs \<in> multiset"
proof -
let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)}"
have "?A \<subseteq> dom (map_of xs)"
proof
fix x
assume "x \<in> ?A"
then have "0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)" by simp
then have "map_of xs x \<noteq> None" by (cases "map_of xs x") auto
then show "x \<in> dom (map_of xs)" by auto
qed
with finite_dom_map_of [of xs] have "finite ?A"
by (auto intro: finite_subset)
then show ?thesis
by (simp add: count_of_def fun_eq_iff multiset_def)
qed
lemma count_simps [simp]:
"count_of [] = (\<lambda>_. 0)"
"count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)"
by (simp_all add: count_of_def fun_eq_iff)
lemma count_of_empty:
"x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0"
by (induct xs) (simp_all add: count_of_def)
lemma count_of_filter:
"count_of (List.filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)"
by (induct xs) auto
lemma count_of_map_default [simp]:
"count_of (map_default x b (%x. x + b) xs) y = (if x = y then count_of xs x + b else count_of xs y)"
unfolding count_of_def by (simp add: map_of_map_default split: option.split)
lemma count_of_join_raw:
"distinct (map fst ys) ==> count_of xs x + count_of ys x = count_of (join_raw (%x (x, y). x + y) xs ys) x"
unfolding count_of_def by (simp add: map_of_join_raw split: option.split)
lemma count_of_subtract_entries_raw:
"distinct (map fst ys) ==> count_of xs x - count_of ys x = count_of (subtract_entries_raw xs ys) x"
unfolding count_of_def by (simp add: map_of_subtract_entries_raw split: option.split)
text {* Code equations for multiset operations *}
definition Bag :: "('a, nat) alist \<Rightarrow> 'a multiset" where
"Bag xs = Abs_multiset (count_of (DAList.impl_of xs))"
code_datatype Bag
lemma count_Bag [simp, code]:
"count (Bag xs) = count_of (DAList.impl_of xs)"
by (simp add: Bag_def count_of_multiset Abs_multiset_inverse)
lemma Mempty_Bag [code]:
"{#} = Bag (DAList.empty)"
by (simp add: multiset_eq_iff alist.Alist_inverse DAList.empty_def)
lemma single_Bag [code]:
"{#x#} = Bag (DAList.update x 1 DAList.empty)"
by (simp add: multiset_eq_iff alist.Alist_inverse update.rep_eq empty.rep_eq)
lemma union_Bag [code]:
"Bag xs + Bag ys = Bag (join (\<lambda>x (n1, n2). n1 + n2) xs ys)"
by (rule multiset_eqI) (simp add: count_of_join_raw alist.Alist_inverse distinct_join_raw join_def)
lemma minus_Bag [code]:
"Bag xs - Bag ys = Bag (subtract_entries xs ys)"
by (rule multiset_eqI)
(simp add: count_of_subtract_entries_raw alist.Alist_inverse distinct_subtract_entries_raw subtract_entries_def)
lemma filter_Bag [code]:
"Multiset.filter P (Bag xs) = Bag (DAList.filter (P \<circ> fst) xs)"
by (rule multiset_eqI) (simp add: count_of_filter DAList.filter.rep_eq)
lemma mset_less_eq_Bag [code]:
"Bag xs \<le> A \<longleftrightarrow> (\<forall>(x, n) \<in> set (DAList.impl_of xs). count_of (DAList.impl_of xs) x \<le> count A x)"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?lhs then show ?rhs
by (auto simp add: mset_le_def)
next
assume ?rhs
show ?lhs
proof (rule mset_less_eqI)
fix x
from `?rhs` have "count_of (DAList.impl_of xs) x \<le> count A x"
by (cases "x \<in> fst ` set (DAList.impl_of xs)") (auto simp add: count_of_empty)
then show "count (Bag xs) x \<le> count A x"
by (simp add: mset_le_def)
qed
qed
instantiation multiset :: (equal) equal
begin
definition
[code]: "HOL.equal A B \<longleftrightarrow> (A::'a multiset) \<le> B \<and> B \<le> A"
instance
by default (simp add: equal_multiset_def eq_iff)
end
text {* Quickcheck generators *}
definition (in term_syntax)
bagify :: "('a\<Colon>typerep, nat) alist \<times> (unit \<Rightarrow> Code_Evaluation.term)
\<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
[code_unfold]: "bagify xs = Code_Evaluation.valtermify Bag {\<cdot>} xs"
notation fcomp (infixl "\<circ>>" 60)
notation scomp (infixl "\<circ>\<rightarrow>" 60)
instantiation multiset :: (random) random
begin
definition
"Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (bagify xs))"
instance ..
end
no_notation fcomp (infixl "\<circ>>" 60)
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
instantiation multiset :: (exhaustive) exhaustive
begin
definition exhaustive_multiset :: "('a multiset => (bool * term list) option) => natural => (bool * term list) option"
where
"exhaustive_multiset f i = Quickcheck_Exhaustive.exhaustive (%xs. f (Bag xs)) i"
instance ..
end
instantiation multiset :: (full_exhaustive) full_exhaustive
begin
definition full_exhaustive_multiset :: "('a multiset * (unit => term) => (bool * term list) option) => natural => (bool * term list) option"
where
"full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (%xs. f (bagify xs)) i"
instance ..
end
hide_const (open) bagify
end