(* 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 {* Delete prexisting code equations *}
lemma [code, code del]:
"{#} = {#}"
..
lemma [code, code del]:
"single = single"
..
lemma [code, code del]:
"plus = (plus :: 'a multiset \<Rightarrow> _)"
..
lemma [code, code del]:
"minus = (minus :: 'a multiset \<Rightarrow> _)"
..
lemma [code, code del]:
"inf = (inf :: 'a multiset \<Rightarrow> _)"
..
lemma [code, code del]:
"sup = (sup :: 'a multiset \<Rightarrow> _)"
..
lemma [code, code del]:
"image_mset = image_mset"
..
lemma [code, code del]:
"Multiset.filter = Multiset.filter"
..
lemma [code, code del]:
"count = count"
..
lemma [code, code del]:
"mcard = mcard"
..
lemma [code, code del]:
"msetsum = msetsum"
..
lemma [code, code del]:
"msetprod = msetprod"
..
lemma [code, code del]:
"set_of = set_of"
..
lemma [code, code del]:
"sorted_list_of_multiset = sorted_list_of_multiset"
..
lemma [code, code del]:
"ord_multiset_inst.less_eq_multiset = ord_multiset_inst.less_eq_multiset"
..
lemma [code, code del]:
"ord_multiset_inst.less_multiset = ord_multiset_inst.less_multiset"
..
lemma [code, code del]:
"equal_multiset_inst.equal_multiset = equal_multiset_inst.equal_multiset"
..
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_eq [code]: "HOL.equal (m1::'a::equal multiset) m2 \<longleftrightarrow> m1 \<le> m2 \<and> m2 \<le> m1"
by (metis equal_multiset_def eq_iff)
text{* By default the code for @{text "<"} is @{prop"xs < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> xs = ys"}.
With equality implemented by @{text"\<le>"}, this leads to three calls of @{text"\<le>"}.
Here is a more efficient version: *}
lemma mset_less[code]: "xs < (ys :: 'a multiset) \<longleftrightarrow> xs \<le> ys \<and> \<not> ys \<le> xs"
by (rule less_le_not_le)
lemma mset_less_eq_Bag0:
"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 thus ?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)
thus "count (Bag xs) x \<le> count A x" by (simp add: mset_le_def)
qed
qed
lemma mset_less_eq_Bag [code]:
"Bag xs \<le> (A :: 'a multiset) \<longleftrightarrow> (\<forall>(x, n) \<in> set (DAList.impl_of xs). n \<le> count A x)"
proof -
{
fix x n
assume "(x,n) \<in> set (DAList.impl_of xs)"
hence "count_of (DAList.impl_of xs) x = n"
proof (transfer)
fix x n and xs :: "('a \<times> nat) list"
show "(distinct \<circ> map fst) xs \<Longrightarrow> (x, n) \<in> set xs \<Longrightarrow> count_of xs x = n"
proof (induct xs)
case (Cons ym ys)
obtain y m where ym: "ym = (y,m)" by force
note Cons = Cons[unfolded ym]
show ?case
proof (cases "x = y")
case False
with Cons show ?thesis unfolding ym by auto
next
case True
with Cons(2-3) have "m = n" by force
with True show ?thesis unfolding ym by auto
qed
qed auto
qed
}
thus ?thesis unfolding mset_less_eq_Bag0 by auto
qed
declare multiset_inter_def [code]
declare sup_multiset_def [code]
declare multiset_of.simps [code]
fun fold_impl :: "('a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<times> nat)list \<Rightarrow> 'b" where
"fold_impl fn e ((a,n) # ms) = (fold_impl fn ((fn a n) e) ms)"
| "fold_impl fn e [] = e"
definition fold :: "('a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a, nat)alist \<Rightarrow> 'b" where
"fold f e al = fold_impl f e (DAList.impl_of al)"
hide_const (open) fold
context comp_fun_commute
begin
lemma DAList_Multiset_fold: assumes fn: "\<And> a n x. fn a n x = (f a ^^ n) x"
shows "Multiset.fold f e (Bag al) = DAList_Multiset.fold fn e al"
unfolding DAList_Multiset.fold_def
proof (induct al)
fix ys
let ?inv = "{xs :: ('a \<times> nat)list. (distinct \<circ> map fst) xs}"
note cs[simp del] = count_simps
have count[simp]: "\<And> x. count (Abs_multiset (count_of x)) = count_of x"
by (rule Abs_multiset_inverse[OF count_of_multiset])
assume ys: "ys \<in> ?inv"
thus "Multiset.fold f e (Bag (Alist ys)) = fold_impl fn e (DAList.impl_of (Alist ys))"
unfolding Bag_def unfolding Alist_inverse[OF ys]
proof (induct ys arbitrary: e rule: list.induct)
case Nil
show ?case
by (rule trans[OF arg_cong[of _ "{#}" "Multiset.fold f e", OF multiset_eqI]])
(auto, simp add: cs)
next
case (Cons pair ys e)
obtain a n where pair: "pair = (a,n)" by force
from fn[of a n] have [simp]: "fn a n = (f a ^^ n)" by auto
have inv: "ys \<in> ?inv" using Cons(2) by auto
note IH = Cons(1)[OF inv]
def Ys \<equiv> "Abs_multiset (count_of ys)"
have id: "Abs_multiset (count_of ((a, n) # ys)) = ((op + {# a #}) ^^ n) Ys"
unfolding Ys_def
proof (rule multiset_eqI, unfold count)
fix c
show "count_of ((a, n) # ys) c = count ((op + {#a#} ^^ n) (Abs_multiset (count_of ys))) c" (is "?l = ?r")
proof (cases "c = a")
case False thus ?thesis unfolding cs by (induct n) auto
next
case True
hence "?l = n" by (simp add: cs)
also have "n = ?r" unfolding True
proof (induct n)
case 0
from Cons(2)[unfolded pair] have "a \<notin> fst ` set ys" by auto
thus ?case by (induct ys) (simp, auto simp: cs)
qed auto
finally show ?thesis .
qed
qed
show ?case unfolding pair
by (simp add: IH[symmetric], unfold id Ys_def[symmetric],
induct n, auto simp: fold_mset_fun_left_comm[symmetric])
qed
qed
end
lift_definition single_alist_entry :: "'a \<Rightarrow> 'b \<Rightarrow> ('a,'b)alist" is "\<lambda> a b. [(a,b)]" by auto
lemma image_mset_Bag[code]:
"image_mset f (Bag ms) =
DAList_Multiset.fold (\<lambda> a n m. Bag (single_alist_entry (f a) n) + m) {#} ms"
unfolding image_mset_def
proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, (auto simp: ac_simps)[1])
fix a n m
show "Bag (single_alist_entry (f a) n) + m = ((op + \<circ> single \<circ> f) a ^^ n) m" (is "?l = ?r")
proof (rule multiset_eqI)
fix x
have "count ?r x = (if x = f a then n + count m x else count m x)"
by (induct n, auto)
also have "\<dots> = count ?l x" by (simp add: single_alist_entry.rep_eq)
finally show "count ?l x = count ?r x" ..
qed
qed
hide_const single_alist_entry
(* we cannot use (\<lambda> a n. op + (a * n)) for folding, since * is not defined
in comm_monoid_add *)
lemma msetsum_Bag[code]:
"msetsum (Bag ms) = DAList_Multiset.fold (\<lambda> a n. ((op + a) ^^ n)) 0 ms"
unfolding msetsum.eq_fold
by (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, auto simp: ac_simps)
(* we cannot use (\<lambda> a n. op * (a ^ n)) for folding, since ^ is not defined
in comm_monoid_mult *)
lemma msetprod_Bag[code]:
"msetprod (Bag ms) = DAList_Multiset.fold (\<lambda> a n. ((op * a) ^^ n)) 1 ms"
unfolding msetprod.eq_fold
by (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, auto simp: ac_simps)
lemma mcard_fold: "mcard A = Multiset.fold (\<lambda> _. Suc) 0 A" (is "_ = Multiset.fold ?f _ _")
proof -
interpret comp_fun_commute ?f by (default, auto)
show ?thesis by (induct A) auto
qed
lemma mcard_Bag[code]:
"mcard (Bag ms) = DAList_Multiset.fold (\<lambda> a n. op + n) 0 ms"
unfolding mcard_fold
proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, simp)
fix a n x
show "n + x = (Suc ^^ n) x" by (induct n) auto
qed
lemma set_of_fold: "set_of A = Multiset.fold insert {} A" (is "_ = Multiset.fold ?f _ _")
proof -
interpret comp_fun_commute ?f by (default, auto)
show ?thesis by (induct A, auto)
qed
lemma set_of_Bag[code]:
"set_of (Bag ms) = DAList_Multiset.fold (\<lambda> a n. (if n = 0 then (\<lambda> m. m) else insert a)) {} ms"
unfolding set_of_fold
proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, (auto simp: ac_simps)[1])
fix a n x
show "(if n = 0 then \<lambda>m. m else insert a) x = (insert a ^^ n) x" (is "?l n = ?r n")
proof (cases n)
case (Suc m)
hence "?l n = insert a x" by simp
moreover have "?r n = insert a x" unfolding Suc by (induct m) auto
ultimately show ?thesis by auto
qed auto
qed
instantiation multiset :: (exhaustive) exhaustive
begin
definition exhaustive_multiset :: "('a multiset \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool * term list) option"
where
"exhaustive_multiset f i = Quickcheck_Exhaustive.exhaustive (\<lambda>xs. f (Bag xs)) i"
instance ..
end
end