Quelle  DAList_Multiset.thy

(*  Title:      HOL/Library/DAList_Multiset.thy
  Author: Lukas Bulwahn, TU Muenchen
*)

section ‹Multisets partially implemented by association lists›

theory DAList_Multiset
imports Multiset DAList
begin

text ‹Raw operations on lists›

definition join_raw ::
    "('key ==> 'val × 'val ==> 'val) ==>
      ('key × 'val) list ==> ('key × 'val) list ==> ('key × 'val) list"
  where "join_raw f xs ys = foldr (λ(k, v). map_default k v (λv'. 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 (λv'. 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)
  apply (metis Some_eq_map_of_iff map_of_eq_None_iff option.simps(2))
  done

lemma distinct_join_raw:
  assumes "distinct (map fst xs)"
  shows "distinct (map fst (join_raw f xs ys))"
  using assms
proof (induct ys)
  case Nil
  then show ?case by simp
next
  case (Cons y ys)
  then show ?case by (cases y) (simp add: distinct_map_default)
qed

definition "subtract_entries_raw xs ys = foldr (λ(k, v). AList.map_entry k (λv'. 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))
  apply (metis map_of_eq_None_iff option.simps(4) option.simps(5))
  done

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 ==> 'b × 'b ==> 'b) ==> ('a, 'b) alist ==> ('a, 'b) alist ==> ('a, 'b) alist"
  is join_raw
  by (simp add: distinct_join_raw)

lift_definition subtract_entries :: "('a, ('b :: minus)) alist ==> ('a, 'b) alist ==> ('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 × nat) list ==> 'a ==> nat"
  where "count_of xs x = (case map_of xs x of None ==> 0 | Some n ==> n)"

lemma count_of_multiset: "finite {x. 0 < count_of xs x}"
proof -
  let ?A = "{x::'a. 0 < (case map_of xs x of None ==> 0::nat | Some n ==> n)}"
  have "?A ⊆ dom (map_of xs)"
  proof
    fix x
    assume "x ∈ ?A"
    then have "0 < (case map_of xs x of None ==> 0::nat | Some n ==> n)"
      by simp
    then have "map_of xs x ≠ None"
      by (cases "map_of xs x") auto
    then show "x ∈ 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)
qed

lemma count_simps [simp]:
  "count_of [] = (λ_. 0)"
  "count_of ((x, n) # xs) = (λy. 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 ∉ fst ` set xs ==> count_of xs x = 0"
  by (induct xs) (simp_all add: count_of_def)

lemma count_of_filter: "count_of (List.filter (P ∘ 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 (λx. 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 (λx (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 ==> '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)

lemma Bag_eq:
  ‹Bag ms = (∑(a, n)←alist.impl_of ms. replicate_mset n a)›
  for ms :: ‹('a, nat) alist›
proof -
  have *: ‹count_of xs a = count (∑(a, n)←xs. replicate_mset n a) a›
    if ‹distinct (map fst xs)›
    for xs and a :: 'a
  using that proof (induction xs)
    case Nil
    then show ?case
      by simp
  next
    case (Cons xn xs)
    then show ?case by (cases xn)
      (auto simp add: count_eq_zero_iff if_split_mem2 image_iff)
  qed
  show ?thesis
    by (rule multiset_eqI) (simp add: *)
qed

lemma Mempty_Bag [code]: "{#} = Bag (DAList.empty)"
  by (simp add: multiset_eq_iff alist.Alist_inverse DAList.empty_def)

lift_definition is_empty_Bag_impl :: "('a, nat) alist ==> bool" is
  "λxs. list_all (λx. snd x = 0) xs" .

lemma is_empty_Bag [code]: "Multiset.is_empty (Bag xs) ⟷ is_empty_Bag_impl xs"
proof -
  have "Multiset.is_empty (Bag xs) ⟷ (∀x. count (Bag xs) x = 0)"
    unfolding Multiset.is_empty_def multiset_eq_iff by simp
  also have "… ⟷ (∀x∈fst ` set (alist.impl_of xs). count (Bag xs) x = 0)"
  proof (intro iffI allI ballI)
    fix x assume A: "∀x∈fst ` set (alist.impl_of xs). count (Bag xs) x = 0"
    thus "count (Bag xs) x = 0"
    proof (cases "x ∈ fst ` set (alist.impl_of xs)")
      case False
      thus ?thesis by (force simp: count_of_def split: option.splits)
    qed (insert A, auto)
  qed simp_all
  also have "… ⟷ list_all (λx. snd x = 0) (alist.impl_of xs)" 
    by (auto simp: count_of_def list_all_def)
  finally show ?thesis by (simp add: is_empty_Bag_impl.rep_eq)
qed

lemma union_Bag [code]: "Bag xs + Bag ys = Bag (join (λx (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 add_mset_Bag [code]: "add_mset x (Bag xs) =
    Bag (join (λx (n1, n2). n1 + n2) (DAList.update x 1 DAList.empty) xs)"
  unfolding add_mset_add_single[of x "Bag xs"] union_Bag[symmetric]
  by (simp add: multiset_eq_iff update.rep_eq empty.rep_eq)

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]: "filter_mset P (Bag xs) = Bag (DAList.filter (P ∘ 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 ⟷ m1 ⊆# m2 ∧ m2 ⊆# m1"
  by (metis equal_multiset_def subset_mset.order_eq_iff)

text ‹By default the code for ‹🚫close> is 🍋‹xs 🚫 ⟷ xs ≤ ys ∧ ¬ xs = ys›.
 With equality implemented by ‹≤›, this leads to three calls of ‹≤›.
 Here is a more efficient version:›
 lemma mset_less[code]: "xs ⊂# (ys :: 'a multiset) ⟷ xs ⊆# ys ∧ ¬ ys ⊆# xs"
  by (rule subset_mset.less_le_not_le)
 
 lemma mset_less_eq_Bag0:
  "Bag xs ⊆# A ⟷ (∀(x, n) ∈ set (DAList.impl_of xs). count_of (DAList.impl_of xs) x ≤ count A x)"
  (is "?lhs ⟷ ?rhs")
 proof
  assume ?lhs
  then show ?rhs by (auto simp add: subseteq_mset_def)
 next
  assume ?rhs
  show ?lhs
  proof (rule mset_subset_eqI)
  fix x
  from ‹?rhs› have "count_of (DAList.impl_of xs) x ≤ count A x"
  by (cases "x ∈ fst ` set (DAList.impl_of xs)") (auto simp add: count_of_empty)
  then show "count (Bag xs) x ≤ count A x" by (simp add: subset_mset_def)
  qed
 qed
 
 lemma mset_less_eq_Bag [code]:
  "Bag xs ⊆# (A :: 'a multiset) ⟷ (∀(x, n) ∈ set (DAList.impl_of xs). n ≤ count A x)"
 proof -
  {
  fix x n
  assume "(x,n) ∈ set (DAList.impl_of xs)"
  then have "count_of (DAList.impl_of xs) x = n"
  proof transfer
  fix x n
  fix xs :: "('a × nat) list"
  show "(distinct ∘ map fst) xs ==> (x, n) ∈ set xs ==> count_of xs x = n"
  proof (induct xs)
  case Nil
  then show ?case by simp
  next
  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
  qed
  }
  then show ?thesis
  unfolding mset_less_eq_Bag0 by auto
 qed
 
 declare inter_mset_def [code]
 declare union_mset_def [code]
 declare mset.simps [code]
 
 
 fun fold_impl :: "('a ==> nat ==> 'b ==> 'b) ==> 'b ==> ('a × nat) list ==> 'b"
 where
  "fold_impl fn e ((a,n) # ms) = (fold_impl fn ((fn a n) e) ms)"
 | "fold_impl fn e [] = e"
 
 context
 begin
 
 qualified definition fold :: "('a ==> nat ==> 'b ==> 'b) ==> 'b ==> ('a, nat) alist ==> 'b"
  where "fold f e al = fold_impl f e (DAList.impl_of al)"
 
 end
 
 context comp_fun_commute
 begin
 
 lemma DAList_Multiset_fold:
  assumes fn: "∧a n x. fn a n x = (f a ^^ n) x"
  shows "fold_mset f e (Bag al) = DAList_Multiset.fold fn e al"
  unfolding DAList_Multiset.fold_def
 proof (induct al)
  fix ys
  let ?inv = "{xs :: ('a × nat) list. (distinct ∘ map fst) xs}"
  note cs[simp del] = count_simps
  have count[simp]: "∧x. count (Abs_multiset (count_of x)) = count_of x"
  by (rule Abs_multiset_inverse) (simp add: count_of_multiset)
  assume ys: "ys ∈ ?inv"
  then show "fold_mset 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 _ "{#}" "fold_mset 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 ∈ ?inv"
  using Cons(2) by auto
  note IH = Cons(1)[OF inv]
  define Ys where "Ys = Abs_multiset (count_of ys)"
  have id: "Abs_multiset (count_of ((a, n) # ys)) = (((+) {# a #}) ^^ n) Ys"
  unfolding Ys_def
  proof (rule multiset_eqI, unfold count)
  fix c
  show "count_of ((a, n) # ys) c =
  count (((+) {#a#} ^^ n) (Abs_multiset (count_of ys))) c" (is "?l = ?r")
  proof (cases "c = a")
  case False
  then show ?thesis
  unfolding cs by (induct n) auto
  next
  case True
  then have "?l = n" by (simp add: cs)
  also have "n = ?r" unfolding True
  proof (induct n)
  case 0
  from Cons(2)[unfolded pair] have "a ∉ fst ` set ys" by auto
  then show ?case by (induct ys) (simp, auto simp: cs)
  next
  case Suc
  then show ?case by simp
  qed
  finally show ?thesis .
  qed
  qed
  show ?case
  unfolding pair
  apply (simp add: IH[symmetric])
  unfolding id Ys_def[symmetric]
  apply (induct n)
  apply (auto simp: fold_mset_fun_left_comm[symmetric])
  done
  qed
 qed
 
 end
 
 context
 begin
 
 private lift_definition single_alist_entry :: "'a ==> 'b ==> ('a, 'b) alist" is "λa b. [(a, b)]"
  by auto
 
 lemma image_mset_Bag [code]:
  "image_mset f (Bag ms) =
  DAList_Multiset.fold (λa 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 = ((add_mset ∘ 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 "… = count ?l x"
  by (simp add: single_alist_entry.rep_eq)
  finally show "count ?l x = count ?r x" ..
  qed
 qed
 
 end
 
\<comment> \<open>we cannot use \<open>\<lambda>a n. (+) (a * n)\<close> for folding, since \<open>(*)
lemma sum_mset_Bag[code]: "sum_mset (Bag ms) = DAList_Multiset.fold (λa n. (((+) a) ^^ n)) 0 ms"
  unfolding sum_mset.eq_fold
  apply (rule comp_fun_commute.DAList_Multiset_fold)
  apply unfold_locales
  apply (auto simp: ac_simps)
  done

🍋 ‹we cannot use ‹λa n. (*) (a ^ n)› for folding, since ‹(^)› is not defined in ‹comm_monoid_mult›\›
lemma prod_mset_Bag[code]: "prod_mset (Bag ms) = DAList_Multiset.fold (λa n. (((*) a) ^^ n)) 1 ms"
  unfolding prod_mset.eq_fold
  apply (rule comp_fun_commute.DAList_Multiset_fold)
  apply unfold_locales
  apply (auto simp: ac_simps)
  done

lemma size_fold: "size A = fold_mset (λ_. Suc) 0 A" (is "_ = fold_mset ?f _ _")
proof -
  interpret comp_fun_commute ?f by standard auto
  show ?thesis by (induct A) auto
qed

lemma size_Bag[code]: "size (Bag ms) = DAList_Multiset.fold (λa n. (+) n) 0 ms"
  unfolding size_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_mset_fold: "set_mset A = fold_mset insert {} A" (is "_ = fold_mset ?f _ _")
proof -
  interpret comp_fun_commute ?f by standard auto
  show ?thesis by (induct A) auto
qed

lemma set_mset_Bag[code]:
  "set_mset (Bag ms) = DAList_Multiset.fold (λa n. (if n = 0 then (λm. m) else insert a)) {} ms"
  unfolding set_mset_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 λm. m else insert a) x = (insert a ^^ n) x" (is "?l n = ?r n")
  proof (cases n)
    case 0
    then show ?thesis by simp
  next
    case (Suc m)
    then have "?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
qed

lemma sorted_list_of_multiset_Bag [code]:
  ‹sorted_list_of_multiset (Bag ms) = concat (map (λ(a, n). replicate n a)
  (sort_key fst (DAList.impl_of ms)))›
proof -
  have *: ‹sorted (concat (map (λ(a, n). replicate n a) ans))›
    if ‹sorted (map fst ans)›
    for ans :: ‹('a × nat) list›
    using that by (induction ans) (auto simp add: sorted_append)
  have ‹mset ?rhs = mset ?lhs›
    by (simp add: Bag_eq mset_concat comp_def split_def flip: sum_mset_sum_list)
  moreover have ‹sorted ?rhs›
    by (rule *) simp
  ultimately have ‹sort ?lhs = ?rhs›
    by (rule properties_for_sort)
  then show ?thesis
    by simp
qed

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 (λxs. f (Bag xs)) i"

instance ..

end

end