src/HOL/Library/DAList_Multiset.thy

'oops' requires proper goal statement -- exclude 'notepad' to avoid disrupting begin/end structure;

(*  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