(* 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 ‹Delete prexisting code equations› declare [[code drop: "{#}" Multiset.is_empty add_mset "plus :: 'a multiset ⇒ _" "minus :: 'a multiset ⇒ _" inter_mset union_mset image_mset filter_mset count "size :: _ multiset ⇒ nat" sum_mset prod_mset set_mset sorted_list_of_multiset subset_mset subseteq_mset equal_multiset_inst.equal_multiset]] 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 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 ‹<› is \<^prop>‹xs < ys ⟷ 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 ― ‹we cannot use ‹λa n. (+) (a * n)› for folding, since ‹(*)› is not defined in ‹comm_monoid_add›› 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 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