Theory HOL-Library.Multiset_Order

(*  Title:      HOL/Library/Multiset_Order.thy
    Author:     Dmitriy Traytel, TU Muenchen
    Author:     Jasmin Blanchette, Inria, LORIA, MPII
    Author:     Martin Desharnais, MPI-INF Saarbruecken
*)

section ‹More Theorems about the Multiset Order›

theory Multiset_Order
imports Multiset
begin

subsection ‹Alternative Characterizations›

subsubsection ‹The Dershowitz--Manna Ordering›

definition multpDM where
  "multpDM r M N ⟷
   (∃X Y. X ≠ {#} ∧ X ⊆# N ∧ M = (N - X) + Y ∧ (∀k. k ∈# Y ⟶ (∃a. a ∈# X ∧ r k a)))"

lemma multpDM_imp_multp:
  "multpDM r M N ⟹ multp r M N"
proof -
  assume "multpDM r M N"
  then obtain X Y where
    "X ≠ {#}" and "X ⊆# N" and "M = N - X + Y" and "∀k. k ∈# Y ⟶ (∃a. a ∈# X ∧ r k a)"
    unfolding multpDM_def by blast
  then have "multp r (N - X + Y) (N - X + X)"
    by (intro one_step_implies_multp) (auto simp: Bex_def trans_def)
  with ‹M = N - X + Y› ‹X ⊆# N› show "multp r M N"
    by (metis subset_mset.diff_add)
qed

subsubsection ‹The Huet--Oppen Ordering›

definition multpHO where
  "multpHO r M N ⟷ M ≠ N ∧ (∀y. count N y < count M y ⟶ (∃x. r y x ∧ count M x < count N x))"

lemma multp_imp_multpHO:
  assumes "asymp r" and "transp r"
  shows "multp r M N ⟹ multpHO r M N"
  unfolding multp_def mult_def
proof (induction rule: trancl_induct)
  case (base P)
  then show ?case
    using ‹asymp r›
    by (auto elim!: mult1_lessE simp: count_eq_zero_iff multpHO_def split: if_splits
        dest!: Suc_lessD)
next
  case (step N P)
  from step(3) have "M ≠ N" and
    **: "⋀y. count N y < count M y ⟹ (∃x. r y x ∧ count M x < count N x)"
    by (simp_all add: multpHO_def)
  from step(2) obtain M0 a K where
    *: "P = add_mset a M0" "N = M0 + K" "a ∉# K" "⋀b. b ∈# K ⟹ r b a"
    using ‹asymp r› by (auto elim: mult1_lessE)
  from ‹M ≠ N› ** *(1,2,3) have "M ≠ P"
    using *(4) ‹asymp r›
    by (metis asympD add_cancel_right_right add_diff_cancel_left' add_mset_add_single count_inI
        count_union diff_diff_add_mset diff_single_trivial in_diff_count multi_member_last)
  moreover
  have count_a: "∃z. r a z ∧ count M z < count P z" if "count P a ≤ count M a"
  proof -
    from ‹a ∉# K› and that have "count N a < count M a"
      unfolding *(1,2) by (auto simp add: not_in_iff)
    with ** obtain z where z: "r a z" "count M z < count N z"
      by blast
    with * have "count N z ≤ count P z"
      using ‹asymp r›
      by (metis add_diff_cancel_left' add_mset_add_single asympD diff_diff_add_mset
          diff_single_trivial in_diff_count not_le_imp_less)
    with z show ?thesis by auto
  qed
  have "∃x. r y x ∧ count M x < count P x" if count_y: "count P y < count M y" for y
  proof (cases "y = a")
    case True
    with count_y count_a show ?thesis by auto
  next
    case False
    show ?thesis
    proof (cases "y ∈# K")
      case True
      with *(4) have "r y a" by simp
      then show ?thesis
        by (cases "count P a ≤ count M a") (auto dest: count_a intro: ‹transp r›[THEN transpD])
    next
      case False
      with ‹y ≠ a› have "count P y = count N y" unfolding *(1,2)
        by (simp add: not_in_iff)
      with count_y ** obtain z where z: "r y z" "count M z < count N z" by auto
      show ?thesis
      proof (cases "z ∈# K")
        case True
        with *(4) have "r z a" by simp
        with z(1) show ?thesis
          by (cases "count P a ≤ count M a") (auto dest!: count_a intro: ‹transp r›[THEN transpD])
      next
        case False
        with ‹a ∉# K› have "count N z ≤ count P z" unfolding *
          by (auto simp add: not_in_iff)
        with z show ?thesis by auto
      qed
    qed
  qed
  ultimately show ?case unfolding multpHO_def by blast
qed

lemma multpHO_imp_multpDM: "multpHO r M N ⟹ multpDM r M N"
unfolding multpDM_def
proof (intro iffI exI conjI)
  assume "multpHO r M N"
  then obtain z where z: "count M z < count N z"
    unfolding multpHO_def by (auto simp: multiset_eq_iff nat_neq_iff)
  define X where "X = N - M"
  define Y where "Y = M - N"
  from z show "X ≠ {#}" unfolding X_def by (auto simp: multiset_eq_iff not_less_eq_eq Suc_le_eq)
  from z show "X ⊆# N" unfolding X_def by auto
  show "M = (N - X) + Y" unfolding X_def Y_def multiset_eq_iff count_union count_diff by force
  show "∀k. k ∈# Y ⟶ (∃a. a ∈# X ∧ r k a)"
  proof (intro allI impI)
    fix k
    assume "k ∈# Y"
    then have "count N k < count M k" unfolding Y_def
      by (auto simp add: in_diff_count)
    with ‹multpHO r M N› obtain a where "r k a" and "count M a < count N a"
      unfolding multpHO_def by blast
    then show "∃a. a ∈# X ∧ r k a" unfolding X_def
      by (auto simp add: in_diff_count)
  qed
qed

lemma multp_eq_multpDM: "asymp r ⟹ transp r ⟹ multp r = multpDM r"
  using multpDM_imp_multp multp_imp_multpHO[THEN multpHO_imp_multpDM]
  by blast

lemma multp_eq_multpHO: "asymp r ⟹ transp r ⟹ multp r = multpHO r"
  using multpHO_imp_multpDM[THEN multpDM_imp_multp] multp_imp_multpHO
  by blast

lemma multpDM_plus_plusI[simp]:
  assumes "multpDM R M1 M2"
  shows "multpDM R (M + M1) (M + M2)"
proof -
  from assms obtain X Y where
    "X ≠ {#}" and "X ⊆# M2" and "M1 = M2 - X + Y" and "∀k. k ∈# Y ⟶ (∃a. a ∈# X ∧ R k a)"
  unfolding multpDM_def by auto

  show "multpDM R (M + M1) (M + M2)"
    unfolding multpDM_def
  proof (intro exI conjI)
    show "X ≠ {#}"
      using ‹X ≠ {#}› by simp
  next
    show "X ⊆# M + M2"
      using ‹X ⊆# M2›
      by (simp add: subset_mset.add_increasing)
  next
    show "M + M1 = M + M2 - X + Y"
      using ‹X ⊆# M2› ‹M1 = M2 - X + Y›
      by (metis multiset_diff_union_assoc union_assoc)
  next
    show "∀k. k ∈# Y ⟶ (∃a. a ∈# X ∧ R k a)"
      using ‹∀k. k ∈# Y ⟶ (∃a. a ∈# X ∧ R k a)› by simp
  qed
qed

lemma multpHO_plus_plus[simp]: "multpHO R (M + M1) (M + M2) ⟷ multpHO R M1 M2"
  unfolding multpHO_def by simp

lemma strict_subset_implies_multpDM: "A ⊂# B ⟹ multpDM r A B"
  unfolding multpDM_def
  by (metis add.right_neutral add_diff_cancel_right' empty_iff mset_subset_eq_add_right
      set_mset_empty subset_mset.lessE)

lemma strict_subset_implies_multpHO: "A ⊂# B ⟹ multpHO r A B"
  unfolding multpHO_def
  by (simp add: leD mset_subset_eq_count)

lemma multpHO_implies_one_step_strong:
  assumes "multpHO R A B"
  defines "J ≡ B - A" and "K ≡ A - B"
  shows "J ≠ {#}" and "∀k ∈# K. ∃x ∈# J. R k x"
proof -
  show "J ≠ {#}"
  using ‹multpHO R A B›
  by (metis Diff_eq_empty_iff_mset J_def add.right_neutral multpDM_def multpHO_imp_multpDM
      multpHO_plus_plus subset_mset.add_diff_inverse subset_mset.le_zero_eq)

  show "∀k∈#K. ∃x∈#J. R k x"
    using ‹multpHO R A B›
    by (metis J_def K_def in_diff_count multpHO_def)
qed

lemma multpHO_minus_inter_minus_inter_iff:
  fixes M1 M2 :: "_ multiset"
  shows "multpHO R (M1 - M2) (M2 - M1) ⟷ multpHO R M1 M2"
  by (metis diff_intersect_left_idem multiset_inter_commute multpHO_plus_plus
      subset_mset.add_diff_inverse subset_mset.inf.cobounded1)

lemma multpHO_iff_set_mset_lessHO_set_mset:
  "multpHO R M1 M2 ⟷ (set_mset (M1 - M2) ≠ set_mset (M2 - M1) ∧
    (∀y ∈# M1 - M2. (∃x ∈# M2 - M1. R y x)))"
  unfolding multpHO_minus_inter_minus_inter_iff[of R M1 M2, symmetric]
  unfolding multpHO_def
  unfolding count_minus_inter_lt_count_minus_inter_iff
  unfolding minus_inter_eq_minus_inter_iff
  by auto


subsubsection ‹Monotonicity›

lemma multpDM_mono_strong:
  "multpDM R M1 M2 ⟹ (⋀x y. x ∈# M1 ⟹ y ∈# M2 ⟹ R x y ⟹ S x y) ⟹ multpDM S M1 M2"
  unfolding multpDM_def
  by (metis add_diff_cancel_left' in_diffD subset_mset.diff_add)

lemma multpHO_mono_strong:
  "multpHO R M1 M2 ⟹ (⋀x y. x ∈# M1 ⟹ y ∈# M2 ⟹ R x y ⟹ S x y) ⟹ multpHO S M1 M2"
  unfolding multpHO_def
  by (metis count_inI less_zeroE)


subsubsection ‹Properties of Orders›

paragraph ‹Asymmetry›

text ‹The following lemma is a negative result stating that asymmetry of an arbitrary binary
relation cannot be simply lifted to @{const multpHO}. It suffices to have four distinct values to
build a counterexample.›

lemma asymp_not_liftable_to_multpHO:
  fixes a b c d :: 'a
  assumes "distinct [a, b, c, d]"
  shows "¬ (∀(R :: 'a ⇒ 'a ⇒ bool). asymp R ⟶ asymp (multpHO R))"
proof -
  define R :: "'a ⇒ 'a ⇒ bool" where
    "R = (λx y. x = a ∧ y = c ∨ x = b ∧ y = d ∨ x = c ∧ y = b ∨ x = d ∧ y = a)"

  from assms(1) have "{#a, b#} ≠ {#c, d#}"
    by (metis add_mset_add_single distinct.simps(2) list.set(1) list.simps(15) multi_member_this
        set_mset_add_mset_insert set_mset_single)

  from assms(1) have "asymp R"
    by (auto simp: R_def intro: asymp_onI)
  moreover have "¬ asymp (multpHO R)"
    unfolding asymp_on_def Set.ball_simps not_all not_imp not_not
  proof (intro exI conjI)
    show "multpHO R {#a, b#} {#c, d#}"
      unfolding multpHO_def
      using ‹{#a, b#} ≠ {#c, d#}› R_def assms by auto
  next
    show "multpHO R {#c, d#} {#a, b#}"
      unfolding multpHO_def
      using ‹{#a, b#} ≠ {#c, d#}› R_def assms by auto
  qed
  ultimately show ?thesis
    unfolding not_all not_imp by auto
qed

text ‹However, if the binary relation is both asymmetric and transitive, then @{const multpHO} is
also asymmetric.›

lemma asymp_on_multpHO:
  assumes "asymp_on A R" and "transp_on A R" and
    B_sub_A: "⋀M. M ∈ B ⟹ set_mset M ⊆ A"
  shows "asymp_on B (multpHO R)"
proof (rule asymp_onI)
  fix M1 M2 :: "'a multiset"
  assume "M1 ∈ B" "M2 ∈ B" "multpHO R M1 M2"

  from ‹transp_on A R› B_sub_A have tran: "transp_on (set_mset (M1 - M2)) R"
    using ‹M1 ∈ B›
    by (meson in_diffD subset_eq transp_on_subset)

  from ‹asymp_on A R› B_sub_A have asym: "asymp_on (set_mset (M1 - M2)) R"
    using ‹M1 ∈ B›
    by (meson in_diffD subset_eq asymp_on_subset)

  show "¬ multpHO R M2 M1"
  proof (cases "M1 - M2 = {#}")
    case True
    then show ?thesis
      using multpHO_implies_one_step_strong(1) by metis
  next
    case False
    hence "∃m∈#M1 - M2. ∀x∈#M1 - M2. x ≠ m ⟶ ¬ R m x"
      using Finite_Set.bex_max_element[of "set_mset (M1 - M2)" R, OF finite_set_mset asym tran]
      by simp
    with ‹transp_on A R› B_sub_A have "∃y∈#M2 - M1. ∀x∈#M1 - M2. ¬ R y x"
      using ‹multpHO R M1 M2›[THEN multpHO_implies_one_step_strong(2)]
      using asym[THEN irreflp_on_if_asymp_on, THEN irreflp_onD]
      by (metis ‹M1 ∈ B› ‹M2 ∈ B› in_diffD subsetD transp_onD)
    thus ?thesis
      unfolding multpHO_iff_set_mset_lessHO_set_mset by simp
  qed
qed

lemma asymp_multpHO:
  assumes "asymp R" and "transp R"
  shows "asymp (multpHO R)"
  using assms asymp_on_multpHO[of UNIV, simplified] by metis


paragraph ‹Irreflexivity›

lemma irreflp_on_multpHO[simp]: "irreflp_on B (multpHO R)"
    by (simp add: irreflp_onI multpHO_def)


paragraph ‹Transitivity›

lemma transp_on_multpHO:
  assumes "asymp_on A R" and "transp_on A R" and B_sub_A: "⋀M. M ∈ B ⟹ set_mset M ⊆ A"
  shows "transp_on B (multpHO R)"
proof (rule transp_onI)
  from assms have "asymp_on B (multpHO R)"
    using asymp_on_multpHO by metis

  fix M1 M2 M3
  assume hyps: "M1 ∈ B" "M2 ∈ B" "M3 ∈ B" "multpHO R M1 M2" "multpHO R M2 M3"

  from assms have
    [intro]: "asymp_on (set_mset M1 ∪ set_mset M2) R" "transp_on (set_mset M1 ∪ set_mset M2) R"
    using ‹M1 ∈ B› ‹M2 ∈ B›
    by (simp_all add: asymp_on_subset transp_on_subset)

  from assms have "transp_on (set_mset M1) R"
    by (meson transp_on_subset hyps(1))

  from ‹multpHO R M1 M2› have
    "M1 ≠ M2" and
    "∀y. count M2 y < count M1 y ⟶ (∃x. R y x ∧ count M1 x < count M2 x)"
    unfolding multpHO_def by simp_all

  from ‹multpHO R M2 M3› have
    "M2 ≠ M3" and
    "∀y. count M3 y < count M2 y ⟶ (∃x. R y x ∧ count M2 x < count M3 x)"
    unfolding multpHO_def by simp_all

  show "multpHO R M1 M3"
  proof (rule ccontr)
    let ?P = "λx. count M3 x < count M1 x ∧ (∀y. R x y ⟶ count M1 y ≥ count M3 y)"

    assume "¬ multpHO R M1 M3"
    hence "M1 = M3 ∨ (∃x. ?P x)"
      unfolding multpHO_def by force
    thus False
    proof (elim disjE)
      assume "M1 = M3"
      thus False
        using ‹asymp_on B (multpHO R)›[THEN asymp_onD]
        using ‹M2 ∈ B› ‹M3 ∈ B› ‹multpHO R M1 M2› ‹multpHO R M2 M3›
        by metis
    next
      assume "∃x. ?P x"
      hence "∃x ∈# M1 + M2. ?P x"
        by (auto simp: count_inI)
      have "∃y ∈# M1 + M2. ?P y ∧ (∀z ∈# M1 + M2. R y z ⟶ ¬ ?P z)"
      proof (rule Finite_Set.bex_max_element_with_property)
        show "∃x ∈# M1 + M2. ?P x"
          using ‹∃x. ?P x›
          by (auto simp: count_inI)
      qed auto
      then obtain x where
        "x ∈# M1 + M2" and
        "count M3 x < count M1 x" and
        "∀y. R x y ⟶ count M1 y ≥ count M3 y" and
        "∀y ∈# M1 + M2. R x y ⟶ count M3 y < count M1 y ⟶ (∃z. R y z ∧ count M1 z < count M3 z)"
        by force

      let ?Q = "λx'. R== x x' ∧ count M3 x' < count M2 x'"
      show False
      proof (cases "∃x'. ?Q x'")
        case True
        have "∃y ∈# M1 + M2. ?Q y ∧ (∀z ∈# M1 + M2. R y z ⟶ ¬ ?Q z)"
        proof (rule Finite_Set.bex_max_element_with_property)
          show "∃x ∈# M1 + M2. ?Q x"
            using ‹∃x. ?Q x›
            by (auto simp: count_inI)
        qed auto
        then obtain x' where
          "x' ∈# M1 + M2" and
          "R== x x'" and
          "count M3 x' < count M2 x'" and
          maximality_x': "∀z ∈# M1 + M2. R x' z ⟶ ¬ (R== x z) ∨ count M3 z ≥ count M2 z"
          by (auto simp: linorder_not_less)
        with ‹multpHO R M2 M3› obtain y' where
          "R x' y'" and "count M2 y' < count M3 y'"
          unfolding multpHO_def by auto
        hence "count M2 y' < count M1 y'"
          by (smt (verit) ‹R== x x'› ‹∀y. R x y ⟶ count M3 y ≤ count M1 y›
              ‹count M3 x < count M1 x› ‹count M3 x' < count M2 x'› assms(2) count_inI
              dual_order.strict_trans1 hyps(1) hyps(2) hyps(3) less_nat_zero_code B_sub_A subsetD
              sup2E transp_onD)
        with ‹multpHO R M1 M2› obtain y'' where
          "R y' y''" and "count M1 y'' < count M2 y''"
          unfolding multpHO_def by auto
        hence "count M3 y'' < count M2 y''"
          by (smt (verit, del_insts) ‹R x' y'› ‹R== x x'› ‹∀y. R x y ⟶ count M3 y ≤ count M1 y›
              ‹count M2 y' < count M3 y'› ‹count M3 x < count M1 x› ‹count M3 x' < count M2 x'›
              assms(2) count_greater_zero_iff dual_order.strict_trans1 hyps(1) hyps(2) hyps(3)
              less_nat_zero_code linorder_not_less B_sub_A subset_iff sup2E transp_onD)

        moreover have "count M2 y'' ≤ count M3 y''"
        proof -
          have "y'' ∈# M1 + M2"
            by (metis ‹count M1 y'' < count M2 y''› count_inI not_less_iff_gr_or_eq union_iff)

          moreover have "R x' y''"
            by (metis ‹R x' y'› ‹R y' y''› ‹count M2 y' < count M1 y'›
                ‹transp_on (set_mset M1 ∪ set_mset M2) R› ‹x' ∈# M1 + M2› calculation count_inI
                nat_neq_iff set_mset_union transp_onD union_iff)

          moreover have "R== x y''"
            using ‹R== x x'›
            by (metis (mono_tags, opaque_lifting) ‹transp_on (set_mset M1 ∪ set_mset M2) R›
                ‹x ∈# M1 + M2› ‹x' ∈# M1 + M2› calculation(1) calculation(2) set_mset_union sup2I1
                transp_onD transp_on_reflclp)

          ultimately show ?thesis
            using maximality_x'[rule_format, of y''] by metis
        qed

        ultimately show ?thesis
          by linarith
      next
        case False
        hence "⋀x'. R== x x' ⟹ count M2 x' ≤ count M3 x'"
          by auto
        hence "count M2 x ≤ count M3 x"
          by simp
        hence "count M2 x < count M1 x"
          using ‹count M3 x < count M1 x› by linarith
        with ‹multpHO R M1 M2› obtain y where
          "R x y" and "count M1 y < count M2 y"
          unfolding multpHO_def by auto
        hence "count M3 y < count M2 y"
          using ‹∀y. R x y ⟶ count M3 y ≤ count M1 y› dual_order.strict_trans2 by metis
        then show ?thesis
          using False ‹R x y› by auto
      qed
    qed
  qed
qed

lemma transp_multpHO:
  assumes "asymp R" and "transp R"
  shows "transp (multpHO R)"
  using assms transp_on_multpHO[of UNIV, simplified] by metis


paragraph ‹Totality›

lemma totalp_on_multpDM:
  "totalp_on A R ⟹ (⋀M. M ∈ B ⟹ set_mset M ⊆ A) ⟹ totalp_on B (multpDM R)"
  by (smt (verit, ccfv_SIG) count_inI in_mono multpHO_def multpHO_imp_multpDM not_less_iff_gr_or_eq
      totalp_onD totalp_onI)

lemma totalp_multpDM: "totalp R ⟹ totalp (multpDM R)"
  by (rule totalp_on_multpDM[of UNIV R UNIV, simplified])

lemma totalp_on_multpHO:
  "totalp_on A R ⟹ (⋀M. M ∈ B ⟹ set_mset M ⊆ A) ⟹ totalp_on B (multpHO R)"
  by (smt (verit, ccfv_SIG) count_inI in_mono multpHO_def not_less_iff_gr_or_eq totalp_onD
      totalp_onI)

lemma totalp_multpHO: "totalp R ⟹ totalp (multpHO R)"
  by (rule totalp_on_multpHO[of UNIV R UNIV, simplified])


paragraph ‹Type Classes›

context preorder
begin

lemma order_mult: "class.order
  (λM N. (M, N) ∈ mult {(x, y). x < y} ∨ M = N)
  (λM N. (M, N) ∈ mult {(x, y). x < y})"
  (is "class.order ?le ?less")
proof -
  have irrefl: "⋀M :: 'a multiset. ¬ ?less M M"
  proof
    fix M :: "'a multiset"
    have "trans {(x'::'a, x). x' < x}"
      by (rule transI) (blast intro: less_trans)
    moreover
    assume "(M, M) ∈ mult {(x, y). x < y}"
    ultimately have "∃I J K. M = I + J ∧ M = I + K
      ∧ J ≠ {#} ∧ (∀k∈set_mset K. ∃j∈set_mset J. (k, j) ∈ {(x, y). x < y})"
      by (rule mult_implies_one_step)
    then obtain I J K where "M = I + J" and "M = I + K"
      and "J ≠ {#}" and "(∀k∈set_mset K. ∃j∈set_mset J. (k, j) ∈ {(x, y). x < y})" by blast
    then have aux1: "K ≠ {#}" and aux2: "∀k∈set_mset K. ∃j∈set_mset K. k < j" by auto
    have "finite (set_mset K)" by simp
    moreover note aux2
    ultimately have "set_mset K = {}"
      by (induct rule: finite_induct)
       (simp, metis (mono_tags) insert_absorb insert_iff insert_not_empty less_irrefl less_trans)
    with aux1 show False by simp
  qed
  have trans: "⋀K M N :: 'a multiset. ?less K M ⟹ ?less M N ⟹ ?less K N"
    unfolding mult_def by (blast intro: trancl_trans)
  show "class.order ?le ?less"
    by standard (auto simp add: less_eq_multiset_def irrefl dest: trans)
qed

text ‹The Dershowitz--Manna ordering:›

definition less_multisetDM where
  "less_multisetDM M N ⟷
   (∃X Y. X ≠ {#} ∧ X ⊆# N ∧ M = (N - X) + Y ∧ (∀k. k ∈# Y ⟶ (∃a. a ∈# X ∧ k < a)))"


text ‹The Huet--Oppen ordering:›

definition less_multisetHO where
  "less_multisetHO M N ⟷ M ≠ N ∧ (∀y. count N y < count M y ⟶ (∃x. y < x ∧ count M x < count N x))"

lemma mult_imp_less_multisetHO:
  "(M, N) ∈ mult {(x, y). x < y} ⟹ less_multisetHO M N"
  unfolding multp_def[of "(<)", symmetric]
  using multp_imp_multpHO[of "(<)"]
  by (simp add: less_multisetHO_def multpHO_def)

lemma less_multisetDM_imp_mult:
  "less_multisetDM M N ⟹ (M, N) ∈ mult {(x, y). x < y}"
  unfolding multp_def[of "(<)", symmetric]
  by (rule multpDM_imp_multp[of "(<)" M N]) (simp add: less_multisetDM_def multpDM_def)

lemma less_multisetHO_imp_less_multisetDM: "less_multisetHO M N ⟹ less_multisetDM M N"
  unfolding less_multisetDM_def less_multisetHO_def
  unfolding multpDM_def[symmetric] multpHO_def[symmetric]
  by (rule multpHO_imp_multpDM)

lemma mult_less_multisetDM: "(M, N) ∈ mult {(x, y). x < y} ⟷ less_multisetDM M N"
  unfolding multp_def[of "(<)", symmetric]
  using multp_eq_multpDM[of "(<)", simplified]
  by (simp add: multpDM_def less_multisetDM_def)

lemma mult_less_multisetHO: "(M, N) ∈ mult {(x, y). x < y} ⟷ less_multisetHO M N"
  unfolding multp_def[of "(<)", symmetric]
  using multp_eq_multpHO[of "(<)", simplified]
  by (simp add: multpHO_def less_multisetHO_def)

lemmas multDM = mult_less_multisetDM[unfolded less_multisetDM_def]
lemmas multHO = mult_less_multisetHO[unfolded less_multisetHO_def]

end

lemma less_multiset_less_multisetHO: "M < N ⟷ less_multisetHO M N"
  unfolding less_multiset_def multp_def multHO less_multisetHO_def ..

lemma less_multisetDM:
  "M < N ⟷ (∃X Y. X ≠ {#} ∧ X ⊆# N ∧ M = N - X + Y ∧ (∀k. k ∈# Y ⟶ (∃a. a ∈# X ∧ k < a)))"
  by (rule multDM[folded multp_def less_multiset_def])

lemma less_multisetHO:
  "M < N ⟷ M ≠ N ∧ (∀y. count N y < count M y ⟶ (∃x>y. count M x < count N x))"
  by (rule multHO[folded multp_def less_multiset_def])

lemma subset_eq_imp_le_multiset:
  shows "M ⊆# N ⟹ M ≤ N"
  unfolding less_eq_multiset_def less_multisetHO
  by (simp add: less_le_not_le subseteq_mset_def)

(* FIXME: "le" should be "less" in this and other names *)
lemma le_multiset_right_total: "M < add_mset x M"
  unfolding less_eq_multiset_def less_multisetHO by simp

lemma less_eq_multiset_empty_left[simp]: "{#} ≤ M"
  by (simp add: subset_eq_imp_le_multiset)

lemma ex_gt_imp_less_multiset: "(∃y. y ∈# N ∧ (∀x. x ∈# M ⟶ x < y)) ⟹ M < N"
  unfolding less_multisetHO
  by (metis count_eq_zero_iff count_greater_zero_iff less_le_not_le)

lemma less_eq_multiset_empty_right[simp]: "M ≠ {#} ⟹ ¬ M ≤ {#}"
  by (metis less_eq_multiset_empty_left antisym)

(* FIXME: "le" should be "less" in this and other names *)
lemma le_multiset_empty_left[simp]: "M ≠ {#} ⟹ {#} < M"
  by (simp add: less_multisetHO)

(* FIXME: "le" should be "less" in this and other names *)
lemma le_multiset_empty_right[simp]: "¬ M < {#}"
  using subset_mset.le_zero_eq less_multiset_def multp_def less_multisetDM by blast

(* FIXME: "le" should be "less" in this and other names *)
lemma union_le_diff_plus: "P ⊆# M ⟹ N < P ⟹ M - P + N < M"
  by (drule subset_mset.diff_add[symmetric]) (metis union_le_mono2)

instantiation multiset :: (preorder) ordered_ab_semigroup_monoid_add_imp_le
begin

lemma less_eq_multisetHO:
  "M ≤ N ⟷ (∀y. count N y < count M y ⟶ (∃x. y < x ∧ count M x < count N x))"
  by (auto simp: less_eq_multiset_def less_multisetHO)

instance by standard (auto simp: less_eq_multisetHO)

lemma
  fixes M N :: "'a multiset"
  shows less_eq_multiset_plus_left: "N ≤ (M + N)"
    and less_eq_multiset_plus_right: "M ≤ (M + N)"
  by simp_all

lemma
  fixes M N :: "'a multiset"
  shows le_multiset_plus_left_nonempty: "M ≠ {#} ⟹ N < M + N"
    and le_multiset_plus_right_nonempty: "N ≠ {#} ⟹ M < M + N"
    by simp_all

end

lemma all_lt_Max_imp_lt_mset: "N ≠ {#} ⟹ (∀a ∈# M. a < Max (set_mset N)) ⟹ M < N"
  by (meson Max_in[OF finite_set_mset] ex_gt_imp_less_multiset set_mset_eq_empty_iff)

lemma lt_imp_ex_count_lt: "M < N ⟹ ∃y. count M y < count N y"
  by (meson less_eq_multisetHO less_le_not_le)

lemma subset_imp_less_mset: "A ⊂# B ⟹ A < B"
  by (simp add: order.not_eq_order_implies_strict subset_eq_imp_le_multiset)

lemma image_mset_strict_mono:
  assumes mono_f: "∀x ∈ set_mset M. ∀y ∈ set_mset N. x < y ⟶ f x < f y"
    and less: "M < N"
  shows "image_mset f M < image_mset f N"
proof -
  obtain Y X where
    y_nemp: "Y ≠ {#}" and y_sub_N: "Y ⊆# N" and M_eq: "M = N - Y + X" and
    ex_y: "∀x. x ∈# X ⟶ (∃y. y ∈# Y ∧ x < y)"
    using less[unfolded less_multisetDM] by blast

  have x_sub_M: "X ⊆# M"
    using M_eq by simp

  let ?fY = "image_mset f Y"
  let ?fX = "image_mset f X"

  show ?thesis
    unfolding less_multisetDM
  proof (intro exI conjI)
    show "image_mset f M = image_mset f N - ?fY + ?fX"
      using M_eq[THEN arg_cong, of "image_mset f"] y_sub_N
      by (metis image_mset_Diff image_mset_union)
  next
    obtain y where y: "∀x. x ∈# X ⟶ y x ∈# Y ∧ x < y x"
      using ex_y by metis

    show "∀fx. fx ∈# ?fX ⟶ (∃fy. fy ∈# ?fY ∧ fx < fy)"
    proof (intro allI impI)
      fix fx
      assume "fx ∈# ?fX"
      then obtain x where fx: "fx = f x" and x_in: "x ∈# X"
        by auto
      hence y_in: "y x ∈# Y" and y_gt: "x < y x"
        using y[rule_format, OF x_in] by blast+
      hence "f (y x) ∈# ?fY ∧ f x < f (y x)"
        using mono_f y_sub_N x_sub_M x_in
        by (metis image_eqI in_image_mset mset_subset_eqD)
      thus "∃fy. fy ∈# ?fY ∧ fx < fy"
        unfolding fx by auto
    qed
  qed (auto simp: y_nemp y_sub_N image_mset_subseteq_mono)
qed

lemma image_mset_mono:
  assumes mono_f: "∀x ∈ set_mset M. ∀y ∈ set_mset N. x < y ⟶ f x < f y"
    and less: "M ≤ N"
  shows "image_mset f M ≤ image_mset f N"
  by (metis eq_iff image_mset_strict_mono less less_imp_le mono_f order.not_eq_order_implies_strict)

lemma mset_lt_single_right_iff[simp]: "M < {#y#} ⟷ (∀x ∈# M. x < y)" for y :: "'a::linorder"
proof (rule iffI)
  assume M_lt_y: "M < {#y#}"
  show "∀x ∈# M. x < y"
  proof
    fix x
    assume x_in: "x ∈# M"
    hence M: "M - {#x#} + {#x#} = M"
      by (meson insert_DiffM2)
    hence "¬ {#x#} < {#y#} ⟹ x < y"
      using x_in M_lt_y
      by (metis diff_single_eq_union le_multiset_empty_left less_add_same_cancel2 mset_le_trans)
    also have "¬ {#y#} < M"
      using M_lt_y mset_le_not_sym by blast
    ultimately show "x < y"
      by (metis (no_types) Max_ge all_lt_Max_imp_lt_mset empty_iff finite_set_mset insertE
        less_le_trans linorder_less_linear mset_le_not_sym set_mset_add_mset_insert
        set_mset_eq_empty_iff x_in)
  qed
next
  assume y_max: "∀x ∈# M. x < y"
  show "M < {#y#}"
    by (rule all_lt_Max_imp_lt_mset) (auto intro!: y_max)
qed

lemma mset_le_single_right_iff[simp]:
  "M ≤ {#y#} ⟷ M = {#y#} ∨ (∀x ∈# M. x < y)" for y :: "'a::linorder"
  by (meson less_eq_multiset_def mset_lt_single_right_iff)


subsubsection ‹Simplifications›

lemma multpHO_repeat_mset_repeat_mset[simp]:
  assumes "n ≠ 0"
  shows "multpHO R (repeat_mset n A) (repeat_mset n B) ⟷ multpHO R A B"
proof (rule iffI)
  assume hyp: "multpHO R (repeat_mset n A) (repeat_mset n B)"
  hence
    1: "repeat_mset n A ≠ repeat_mset n B" and
    2: "∀y. n * count B y < n * count A y ⟶ (∃x. R y x ∧ n * count A x < n * count B x)"
    by (simp_all add: multpHO_def)

  from 1 ‹n ≠ 0› have "A ≠ B"
    by auto

  moreover from 2 ‹n ≠ 0› have "∀y. count B y < count A y ⟶ (∃x. R y x ∧ count A x < count B x)"
    by auto

  ultimately show "multpHO R A B"
    by (simp add: multpHO_def)
next
  assume "multpHO R A B"
  hence 1: "A ≠ B" and 2: "∀y. count B y < count A y ⟶ (∃x. R y x ∧ count A x < count B x)"
    by (simp_all add: multpHO_def)

  from 1 have "repeat_mset n A ≠ repeat_mset n B"
    by (simp add: assms repeat_mset_cancel1)

  moreover from 2 have "∀y. n * count B y < n * count A y ⟶
    (∃x. R y x ∧ n * count A x < n * count B x)"
    by auto

  ultimately show "multpHO R (repeat_mset n A) (repeat_mset n B)"
    by (simp add: multpHO_def)
qed

lemma multpHO_double_double[simp]: "multpHO R (A + A) (B + B) ⟷ multpHO R A B"
  using multpHO_repeat_mset_repeat_mset[of 2]
  by (simp add: numeral_Bit0)


subsection ‹Simprocs›

lemma mset_le_add_iff1:
  "j ≤ (i::nat) ⟹ (repeat_mset i u + m ≤ repeat_mset j u + n) = (repeat_mset (i-j) u + m ≤ n)"
proof -
  assume "j ≤ i"
  then have "j + (i - j) = i"
    using le_add_diff_inverse by blast
  then show ?thesis
    by (metis (no_types) add_le_cancel_left left_add_mult_distrib_mset)
qed

lemma mset_le_add_iff2:
  "i ≤ (j::nat) ⟹ (repeat_mset i u + m ≤ repeat_mset j u + n) = (m ≤ repeat_mset (j-i) u + n)"
proof -
  assume "i ≤ j"
  then have "i + (j - i) = j"
    using le_add_diff_inverse by blast
  then show ?thesis
    by (metis (no_types) add_le_cancel_left left_add_mult_distrib_mset)
qed

simproc_setup msetless_cancel
  ("(l::'a::preorder multiset) + m < n" | "(l::'a multiset) < m + n" |
   "add_mset a m < n" | "m < add_mset a n" |
   "replicate_mset p a < n" | "m < replicate_mset p a" |
   "repeat_mset p m < n" | "m < repeat_mset p n") =
  ‹K Cancel_Simprocs.less_cancel›

simproc_setup msetle_cancel
  ("(l::'a::preorder multiset) + m ≤ n" | "(l::'a multiset) ≤ m + n" |
   "add_mset a m ≤ n" | "m ≤ add_mset a n" |
   "replicate_mset p a ≤ n" | "m ≤ replicate_mset p a" |
   "repeat_mset p m ≤ n" | "m ≤ repeat_mset p n") =
  ‹K Cancel_Simprocs.less_eq_cancel›


subsection ‹Additional facts and instantiations›

lemma ex_gt_count_imp_le_multiset:
  "(∀y :: 'a :: order. y ∈# M + N ⟶ y ≤ x) ⟹ count M x < count N x ⟹ M < N"
  unfolding less_multisetHO
  by (metis count_greater_zero_iff le_imp_less_or_eq less_imp_not_less not_gr_zero union_iff)

lemma mset_lt_single_iff[iff]: "{#x#} < {#y#} ⟷ x < y"
  unfolding less_multisetHO by simp

lemma mset_le_single_iff[iff]: "{#x#} ≤ {#y#} ⟷ x ≤ y" for x y :: "'a::order"
  unfolding less_eq_multisetHO by force

instance multiset :: (linorder) linordered_cancel_ab_semigroup_add
  by standard (metis less_eq_multisetHO not_less_iff_gr_or_eq)

lemma less_eq_multiset_total: "¬ M ≤ N ⟹ N ≤ M" for M N :: "'a :: linorder multiset"
  by simp

instantiation multiset :: (wellorder) wellorder
begin

lemma wf_less_multiset: "wf {(M :: 'a multiset, N). M < N}"
  unfolding less_multiset_def multp_def by (auto intro: wf_mult wf)

instance
proof intro_classes
  fix P :: "'a multiset ⇒ bool" and a :: "'a multiset"
  have "wfp ((<) :: 'a ⇒ 'a ⇒ bool)"
    using wfp_on_less .
  hence "wfp ((<) :: 'a multiset ⇒ 'a multiset ⇒ bool)"
    unfolding less_multiset_def by (rule wfp_multp)
  thus "(⋀x. (⋀y. y < x ⟹ P y) ⟹ P x) ⟹ P a"
    unfolding wfp_on_def[of UNIV, simplified] by metis
qed

end

instantiation multiset :: (preorder) order_bot
begin

definition bot_multiset :: "'a multiset" where "bot_multiset = {#}"

instance by standard (simp add: bot_multiset_def)

end

instance multiset :: (preorder) no_top
proof standard
  fix x :: "'a multiset"
  obtain a :: 'a where True by simp
  have "x < x + (x + {#a#})"
    by simp
  then show "∃y. x < y"
    by blast
qed

instance multiset :: (preorder) ordered_cancel_comm_monoid_add
  by standard

instantiation multiset :: (linorder) distrib_lattice
begin

definition inf_multiset :: "'a multiset ⇒ 'a multiset ⇒ 'a multiset" where
  "inf_multiset A B = (if A < B then A else B)"

definition sup_multiset :: "'a multiset ⇒ 'a multiset ⇒ 'a multiset" where
  "sup_multiset A B = (if B > A then B else A)"

instance
  by intro_classes (auto simp: inf_multiset_def sup_multiset_def)

end

lemma add_mset_lt_left_lt: "a < b ⟹ add_mset a A < add_mset b A"
  by fastforce

lemma add_mset_le_left_le: "a ≤ b ⟹ add_mset a A ≤ add_mset b A" for a :: "'a :: linorder"
  by fastforce

lemma add_mset_lt_right_lt: "A < B ⟹ add_mset a A < add_mset a B"
  by fastforce

lemma add_mset_le_right_le: "A ≤ B ⟹ add_mset a A ≤ add_mset a B"
  by fastforce

lemma add_mset_lt_lt_lt:
  assumes a_lt_b: "a < b" and A_le_B: "A < B"
  shows "add_mset a A < add_mset b B"
  by (rule less_trans[OF add_mset_lt_left_lt[OF a_lt_b] add_mset_lt_right_lt[OF A_le_B]])

lemma add_mset_lt_lt_le: "a < b ⟹ A ≤ B ⟹ add_mset a A < add_mset b B"
  using add_mset_lt_lt_lt le_neq_trans by fastforce

lemma add_mset_lt_le_lt: "a ≤ b ⟹ A < B ⟹ add_mset a A < add_mset b B" for a :: "'a :: linorder"
  using add_mset_lt_lt_lt by (metis add_mset_lt_right_lt le_less)

lemma add_mset_le_le_le:
  fixes a :: "'a :: linorder"
  assumes a_le_b: "a ≤ b" and A_le_B: "A ≤ B"
  shows "add_mset a A ≤ add_mset b B"
  by (rule order.trans[OF add_mset_le_left_le[OF a_le_b] add_mset_le_right_le[OF A_le_B]])

lemma Max_lt_imp_lt_mset:
  assumes n_nemp: "N ≠ {#}" and max: "Max_mset M < Max_mset N" (is "?max_M < ?max_N")
  shows "M < N"
proof (cases "M = {#}")
  case m_nemp: False

  have max_n_in_n: "?max_N ∈# N"
    using n_nemp by simp
  have max_n_nin_m: "?max_N ∉# M"
    using max Max_ge leD by auto

  have "M ≠ N"
    using max by auto
  moreover
  have "∃x > y. count M x < count N x" if "count N y < count M y" for y
  proof -
    from that have "y ∈# M"
      by (simp add: count_inI)
    then have "?max_M ≥ y"
      by simp
    then have "?max_N > y"
      using max by auto
    then show ?thesis
      using max_n_nin_m max_n_in_n count_inI by force
  qed
  ultimately show ?thesis
    unfolding less_multisetHO by blast
qed (auto simp: n_nemp)

end