src/HOL/Library/Multiset_Order.thy

Added the command nxsketch, which is like sketch except it separates these subgoals by "next" fills in the context using "fix"/"assume" rather than "for"/"if". Also the sketch command does the same if there is only one subgoal.

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

section \<open>More Theorems about the Multiset Order\<close>

theory Multiset_Order
imports Multiset
begin

subsection \<open>Alternative Characterizations\<close>

subsubsection \<open>The Dershowitz--Manna Ordering\<close>

definition multp\<^sub>D\<^sub>M where
  "multp\<^sub>D\<^sub>M r M N \<longleftrightarrow>
   (\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = (N - X) + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> r k a)))"

lemma multp\<^sub>D\<^sub>M_imp_multp:
  "multp\<^sub>D\<^sub>M r M N \<Longrightarrow> multp r M N"
proof -
  assume "multp\<^sub>D\<^sub>M r M N"
  then obtain X Y where
    "X \<noteq> {#}" and "X \<subseteq># N" and "M = N - X + Y" and "\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> r k a)"
    unfolding multp\<^sub>D\<^sub>M_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 \<open>M = N - X + Y\<close> \<open>X \<subseteq># N\<close> show "multp r M N"
    by (metis subset_mset.diff_add)
qed

subsubsection \<open>The Huet--Oppen Ordering\<close>

definition multp\<^sub>H\<^sub>O where
  "multp\<^sub>H\<^sub>O r M N \<longleftrightarrow> M \<noteq> N \<and> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x. r y x \<and> count M x < count N x))"

lemma multp_imp_multp\<^sub>H\<^sub>O:
  assumes "asymp r" and "transp r"
  shows "multp r M N \<Longrightarrow> multp\<^sub>H\<^sub>O r M N"
  unfolding multp_def mult_def
proof (induction rule: trancl_induct)
  case (base P)
  then show ?case
    using \<open>asymp r\<close>
    by (auto elim!: mult1_lessE simp: count_eq_zero_iff multp\<^sub>H\<^sub>O_def split: if_splits
        dest!: Suc_lessD)
next
  case (step N P)
  from step(3) have "M \<noteq> N" and
    **: "\<And>y. count N y < count M y \<Longrightarrow> (\<exists>x. r y x \<and> count M x < count N x)"
    by (simp_all add: multp\<^sub>H\<^sub>O_def)
  from step(2) obtain M0 a K where
    *: "P = add_mset a M0" "N = M0 + K" "a \<notin># K" "\<And>b. b \<in># K \<Longrightarrow> r b a"
    using \<open>asymp r\<close> by (auto elim: mult1_lessE)
  from \<open>M \<noteq> N\<close> ** *(1,2,3) have "M \<noteq> P"
    using *(4) \<open>asymp r\<close>
    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
  { assume "count P a \<le> count M a"
    with \<open>a \<notin># K\<close> 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 \<le> count P z"
        using \<open>asymp r\<close>
        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 have "\<exists>z. r a z \<and> count M z < count P z" by auto
  } note count_a = this
  { fix y
    assume count_y: "count P y < count M y"
    have "\<exists>x. r y x \<and> count M x < count P x"
    proof (cases "y = a")
      case True
      with count_y count_a show ?thesis by auto
    next
      case False
      show ?thesis
      proof (cases "y \<in># K")
        case True
        with *(4) have "r y a" by simp
        then show ?thesis
          by (cases "count P a \<le> count M a") (auto dest: count_a intro: \<open>transp r\<close>[THEN transpD])
      next
        case False
        with \<open>y \<noteq> a\<close> 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 \<in># K")
          case True
          with *(4) have "r z a" by simp
          with z(1) show ?thesis
            by (cases "count P a \<le> count M a") (auto dest!: count_a intro: \<open>transp r\<close>[THEN transpD])
        next
          case False
          with \<open>a \<notin># K\<close> have "count N z \<le> count P z" unfolding *
            by (auto simp add: not_in_iff)
          with z show ?thesis by auto
        qed
      qed
    qed
  }
  ultimately show ?case unfolding multp\<^sub>H\<^sub>O_def by blast
qed

lemma multp\<^sub>H\<^sub>O_imp_multp\<^sub>D\<^sub>M: "multp\<^sub>H\<^sub>O r M N \<Longrightarrow> multp\<^sub>D\<^sub>M r M N"
unfolding multp\<^sub>D\<^sub>M_def
proof (intro iffI exI conjI)
  assume "multp\<^sub>H\<^sub>O r M N"
  then obtain z where z: "count M z < count N z"
    unfolding multp\<^sub>H\<^sub>O_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 \<noteq> {#}" unfolding X_def by (auto simp: multiset_eq_iff not_less_eq_eq Suc_le_eq)
  from z show "X \<subseteq># 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 "\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> r k a)"
  proof (intro allI impI)
    fix k
    assume "k \<in># Y"
    then have "count N k < count M k" unfolding Y_def
      by (auto simp add: in_diff_count)
    with \<open>multp\<^sub>H\<^sub>O r M N\<close> obtain a where "r k a" and "count M a < count N a"
      unfolding multp\<^sub>H\<^sub>O_def by blast
    then show "\<exists>a. a \<in># X \<and> r k a" unfolding X_def
      by (auto simp add: in_diff_count)
  qed
qed

lemma multp_eq_multp\<^sub>D\<^sub>M: "asymp r \<Longrightarrow> transp r \<Longrightarrow> multp r = multp\<^sub>D\<^sub>M r"
  using multp\<^sub>D\<^sub>M_imp_multp multp_imp_multp\<^sub>H\<^sub>O[THEN multp\<^sub>H\<^sub>O_imp_multp\<^sub>D\<^sub>M]
  by blast

lemma multp_eq_multp\<^sub>H\<^sub>O: "asymp r \<Longrightarrow> transp r \<Longrightarrow> multp r = multp\<^sub>H\<^sub>O r"
  using multp\<^sub>H\<^sub>O_imp_multp\<^sub>D\<^sub>M[THEN multp\<^sub>D\<^sub>M_imp_multp] multp_imp_multp\<^sub>H\<^sub>O
  by blast

lemma multp\<^sub>D\<^sub>M_plus_plusI[simp]:
  assumes "multp\<^sub>D\<^sub>M R M1 M2"
  shows "multp\<^sub>D\<^sub>M R (M + M1) (M + M2)"
proof -
  from assms obtain X Y where
    "X \<noteq> {#}" and "X \<subseteq># M2" and "M1 = M2 - X + Y" and "\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> R k a)"
  unfolding multp\<^sub>D\<^sub>M_def by auto

  show "multp\<^sub>D\<^sub>M R (M + M1) (M + M2)"
    unfolding multp\<^sub>D\<^sub>M_def
  proof (intro exI conjI)
    show "X \<noteq> {#}"
      using \<open>X \<noteq> {#}\<close> by simp
  next
    show "X \<subseteq># M + M2"
      using \<open>X \<subseteq># M2\<close>
      by (simp add: subset_mset.add_increasing)
  next
    show "M + M1 = M + M2 - X + Y"
      using \<open>X \<subseteq># M2\<close> \<open>M1 = M2 - X + Y\<close>
      by (metis multiset_diff_union_assoc union_assoc)
  next
    show "\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> R k a)"
      using \<open>\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> R k a)\<close> by simp
  qed
qed

lemma multp\<^sub>H\<^sub>O_plus_plus[simp]: "multp\<^sub>H\<^sub>O R (M + M1) (M + M2) \<longleftrightarrow> multp\<^sub>H\<^sub>O R M1 M2"
  unfolding multp\<^sub>H\<^sub>O_def by simp

lemma strict_subset_implies_multp\<^sub>D\<^sub>M: "A \<subset># B \<Longrightarrow> multp\<^sub>D\<^sub>M r A B"
  unfolding multp\<^sub>D\<^sub>M_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_multp\<^sub>H\<^sub>O: "A \<subset># B \<Longrightarrow> multp\<^sub>H\<^sub>O r A B"
  unfolding multp\<^sub>H\<^sub>O_def
  by (simp add: leD mset_subset_eq_count)

lemma multp\<^sub>H\<^sub>O_implies_one_step_strong:
  assumes "multp\<^sub>H\<^sub>O R A B"
  defines "J \<equiv> B - A" and "K \<equiv> A - B"
  shows "J \<noteq> {#}" and "\<forall>k \<in># K. \<exists>x \<in># J. R k x"
proof -
  show "J \<noteq> {#}"
  using \<open>multp\<^sub>H\<^sub>O R A B\<close>
  by (metis Diff_eq_empty_iff_mset J_def add.right_neutral multp\<^sub>D\<^sub>M_def multp\<^sub>H\<^sub>O_imp_multp\<^sub>D\<^sub>M
      multp\<^sub>H\<^sub>O_plus_plus subset_mset.add_diff_inverse subset_mset.le_zero_eq)

  show "\<forall>k\<in>#K. \<exists>x\<in>#J. R k x"
    using \<open>multp\<^sub>H\<^sub>O R A B\<close>
    by (metis J_def K_def in_diff_count multp\<^sub>H\<^sub>O_def)
qed

lemma multp\<^sub>H\<^sub>O_minus_inter_minus_inter_iff:
  fixes M1 M2 :: "_ multiset"
  shows "multp\<^sub>H\<^sub>O R (M1 - M2) (M2 - M1) \<longleftrightarrow> multp\<^sub>H\<^sub>O R M1 M2"
  by (metis diff_intersect_left_idem multiset_inter_commute multp\<^sub>H\<^sub>O_plus_plus
      subset_mset.add_diff_inverse subset_mset.inf.cobounded1)

lemma multp\<^sub>H\<^sub>O_iff_set_mset_less\<^sub>H\<^sub>O_set_mset:
  "multp\<^sub>H\<^sub>O R M1 M2 \<longleftrightarrow> (set_mset (M1 - M2) \<noteq> set_mset (M2 - M1) \<and>
    (\<forall>y \<in># M1 - M2. (\<exists>x \<in># M2 - M1. R y x)))"
  unfolding multp\<^sub>H\<^sub>O_minus_inter_minus_inter_iff[of R M1 M2, symmetric]
  unfolding multp\<^sub>H\<^sub>O_def
  unfolding count_minus_inter_lt_count_minus_inter_iff
  unfolding minus_inter_eq_minus_inter_iff
  by auto


subsubsection \<open>Monotonicity\<close>

lemma multp\<^sub>D\<^sub>M_mono_strong:
  "multp\<^sub>D\<^sub>M R M1 M2 \<Longrightarrow> (\<And>x y. x \<in># M1 \<Longrightarrow> y \<in># M2 \<Longrightarrow> R x y \<Longrightarrow> S x y) \<Longrightarrow> multp\<^sub>D\<^sub>M S M1 M2"
  unfolding multp\<^sub>D\<^sub>M_def
  by (metis add_diff_cancel_left' in_diffD subset_mset.diff_add)

lemma multp\<^sub>H\<^sub>O_mono_strong:
  "multp\<^sub>H\<^sub>O R M1 M2 \<Longrightarrow> (\<And>x y. x \<in># M1 \<Longrightarrow> y \<in># M2 \<Longrightarrow> R x y \<Longrightarrow> S x y) \<Longrightarrow> multp\<^sub>H\<^sub>O S M1 M2"
  unfolding multp\<^sub>H\<^sub>O_def
  by (metis count_inI less_zeroE)


subsubsection \<open>Properties of Orders\<close>

paragraph \<open>Asymmetry\<close>

text \<open>The following lemma is a negative result stating that asymmetry of an arbitrary binary
relation cannot be simply lifted to @{const multp\<^sub>H\<^sub>O}. It suffices to have four distinct values to
build a counterexample.\<close>

lemma asymp_not_liftable_to_multp\<^sub>H\<^sub>O:
  fixes a b c d :: 'a
  assumes "distinct [a, b, c, d]"
  shows "\<not> (\<forall>(R :: 'a \<Rightarrow> 'a \<Rightarrow> bool). asymp R \<longrightarrow> asymp (multp\<^sub>H\<^sub>O R))"
proof -
  define R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
    "R = (\<lambda>x y. x = a \<and> y = c \<or> x = b \<and> y = d \<or> x = c \<and> y = b \<or> x = d \<and> y = a)"

  from assms(1) have "{#a, b#} \<noteq> {#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 "\<not> asymp (multp\<^sub>H\<^sub>O R)"
    unfolding asymp_on_def Set.ball_simps not_all not_imp not_not
  proof (intro exI conjI)
    show "multp\<^sub>H\<^sub>O R {#a, b#} {#c, d#}"
      unfolding multp\<^sub>H\<^sub>O_def
      using \<open>{#a, b#} \<noteq> {#c, d#}\<close> R_def assms by auto
  next
    show "multp\<^sub>H\<^sub>O R {#c, d#} {#a, b#}"
      unfolding multp\<^sub>H\<^sub>O_def
      using \<open>{#a, b#} \<noteq> {#c, d#}\<close> R_def assms by auto
  qed
  ultimately show ?thesis
    unfolding not_all not_imp by auto
qed

text \<open>However, if the binary relation is both asymmetric and transitive, then @{const multp\<^sub>H\<^sub>O} is
also asymmetric.\<close>

lemma asymp_on_multp\<^sub>H\<^sub>O:
  assumes "asymp_on A R" and "transp_on A R" and
    B_sub_A: "\<And>M. M \<in> B \<Longrightarrow> set_mset M \<subseteq> A"
  shows "asymp_on B (multp\<^sub>H\<^sub>O R)"
proof (rule asymp_onI)
  fix M1 M2 :: "'a multiset"
  assume "M1 \<in> B" "M2 \<in> B" "multp\<^sub>H\<^sub>O R M1 M2"

  from \<open>transp_on A R\<close> B_sub_A have tran: "transp_on (set_mset (M1 - M2)) R"
    using \<open>M1 \<in> B\<close>
    by (meson in_diffD subset_eq transp_on_subset)

  from \<open>asymp_on A R\<close> B_sub_A have asym: "asymp_on (set_mset (M1 - M2)) R"
    using \<open>M1 \<in> B\<close>
    by (meson in_diffD subset_eq asymp_on_subset)

  show "\<not> multp\<^sub>H\<^sub>O R M2 M1"
  proof (cases "M1 - M2 = {#}")
    case True
    then show ?thesis
      using multp\<^sub>H\<^sub>O_implies_one_step_strong(1) by metis
  next
    case False
    hence "\<exists>m\<in>#M1 - M2. \<forall>x\<in>#M1 - M2. x \<noteq> m \<longrightarrow> \<not> R m x"
      using Finite_Set.bex_max_element[of "set_mset (M1 - M2)" R, OF finite_set_mset asym tran]
      by simp
    with \<open>transp_on A R\<close> B_sub_A have "\<exists>y\<in>#M2 - M1. \<forall>x\<in>#M1 - M2. \<not> R y x"
      using \<open>multp\<^sub>H\<^sub>O R M1 M2\<close>[THEN multp\<^sub>H\<^sub>O_implies_one_step_strong(2)]
      using asym[THEN irreflp_on_if_asymp_on, THEN irreflp_onD]
      by (metis \<open>M1 \<in> B\<close> \<open>M2 \<in> B\<close> in_diffD subsetD transp_onD)
    thus ?thesis
      unfolding multp\<^sub>H\<^sub>O_iff_set_mset_less\<^sub>H\<^sub>O_set_mset by simp
  qed
qed

lemma asymp_multp\<^sub>H\<^sub>O:
  assumes "asymp R" and "transp R"
  shows "asymp (multp\<^sub>H\<^sub>O R)"
  using assms asymp_on_multp\<^sub>H\<^sub>O[of UNIV, simplified] by metis


paragraph \<open>Irreflexivity\<close>

lemma irreflp_on_multp\<^sub>H\<^sub>O[simp]: "irreflp_on B (multp\<^sub>H\<^sub>O R)"
    by (simp add: irreflp_onI multp\<^sub>H\<^sub>O_def)


paragraph \<open>Transitivity\<close>

lemma transp_on_multp\<^sub>H\<^sub>O:
  assumes "asymp_on A R" and "transp_on A R" and
    B_sub_A: "\<And>M. M \<in> B \<Longrightarrow> set_mset M \<subseteq> A"
  shows "transp_on B (multp\<^sub>H\<^sub>O R)"
proof (rule transp_onI)
  from assms have "asymp_on B (multp\<^sub>H\<^sub>O R)"
    using asymp_on_multp\<^sub>H\<^sub>O by metis

  fix M1 M2 M3
  assume hyps: "M1 \<in> B" "M2 \<in> B" "M3 \<in> B" "multp\<^sub>H\<^sub>O R M1 M2" "multp\<^sub>H\<^sub>O R M2 M3"

  from assms have
    [intro]: "asymp_on (set_mset M1 \<union> set_mset M2) R" "transp_on (set_mset M1 \<union> set_mset M2) R"
    using \<open>M1 \<in> B\<close> \<open>M2 \<in> B\<close>
    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 \<open>multp\<^sub>H\<^sub>O R M1 M2\<close> have
    "M1 \<noteq> M2" and
    "\<forall>y. count M2 y < count M1 y \<longrightarrow> (\<exists>x. R y x \<and> count M1 x < count M2 x)"
    unfolding multp\<^sub>H\<^sub>O_def by simp_all

  from \<open>multp\<^sub>H\<^sub>O R M2 M3\<close> have
    "M2 \<noteq> M3" and
    "\<forall>y. count M3 y < count M2 y \<longrightarrow> (\<exists>x. R y x \<and> count M2 x < count M3 x)"
    unfolding multp\<^sub>H\<^sub>O_def by simp_all

  show "multp\<^sub>H\<^sub>O R M1 M3"
  proof (rule ccontr)
    let ?P = "\<lambda>x. count M3 x < count M1 x \<and> (\<forall>y. R x y \<longrightarrow> count M1 y \<ge> count M3 y)"

    assume "\<not> multp\<^sub>H\<^sub>O R M1 M3"
    hence "M1 = M3 \<or> (\<exists>x. ?P x)"
      unfolding multp\<^sub>H\<^sub>O_def by force
    thus False
    proof (elim disjE)
      assume "M1 = M3"
      thus False
        using \<open>asymp_on B (multp\<^sub>H\<^sub>O R)\<close>[THEN asymp_onD]
        using \<open>M2 \<in> B\<close> \<open>M3 \<in> B\<close> \<open>multp\<^sub>H\<^sub>O R M1 M2\<close> \<open>multp\<^sub>H\<^sub>O R M2 M3\<close>
        by metis
    next
      assume "\<exists>x. ?P x"
      hence "\<exists>x \<in># M1 + M2. ?P x"
        by (auto simp: count_inI)
      have "\<exists>y \<in># M1 + M2. ?P y \<and> (\<forall>z \<in># M1 + M2. R y z \<longrightarrow> \<not> ?P z)"
      proof (rule Finite_Set.bex_max_element_with_property)
        show "\<exists>x \<in># M1 + M2. ?P x"
          using \<open>\<exists>x. ?P x\<close>
          by (auto simp: count_inI)
      qed auto
      then obtain x where
        "x \<in># M1 + M2" and
        "count M3 x < count M1 x" and
        "\<forall>y. R x y \<longrightarrow> count M1 y \<ge> count M3 y" and
        "\<forall>y \<in># M1 + M2. R x y \<longrightarrow> count M3 y < count M1 y \<longrightarrow> (\<exists>z. R y z \<and> count M1 z < count M3 z)"
        by force

      let ?Q = "\<lambda>x'. R\<^sup>=\<^sup>= x x' \<and> count M3 x' < count M2 x'"
      show False
      proof (cases "\<exists>x'. ?Q x'")
        case True
        have "\<exists>y \<in># M1 + M2. ?Q y \<and> (\<forall>z \<in># M1 + M2. R y z \<longrightarrow> \<not> ?Q z)"
        proof (rule Finite_Set.bex_max_element_with_property)
          show "\<exists>x \<in># M1 + M2. ?Q x"
            using \<open>\<exists>x. ?Q x\<close>
            by (auto simp: count_inI)
        qed auto
        then obtain x' where
          "x' \<in># M1 + M2" and
          "R\<^sup>=\<^sup>= x x'" and
          "count M3 x' < count M2 x'" and
          maximality_x': "\<forall>z \<in># M1 + M2. R x' z \<longrightarrow> \<not> (R\<^sup>=\<^sup>= x z) \<or> count M3 z \<ge> count M2 z"
          by (auto simp: linorder_not_less)
        with \<open>multp\<^sub>H\<^sub>O R M2 M3\<close> obtain y' where
          "R x' y'" and "count M2 y' < count M3 y'"
          unfolding multp\<^sub>H\<^sub>O_def by auto
        hence "count M2 y' < count M1 y'"
          by (smt (verit) \<open>R\<^sup>=\<^sup>= x x'\<close> \<open>\<forall>y. R x y \<longrightarrow> count M3 y \<le> count M1 y\<close>
              \<open>count M3 x < count M1 x\<close> \<open>count M3 x' < count M2 x'\<close> 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 \<open>multp\<^sub>H\<^sub>O R M1 M2\<close> obtain y'' where
          "R y' y''" and "count M1 y'' < count M2 y''"
          unfolding multp\<^sub>H\<^sub>O_def by auto
        hence "count M3 y'' < count M2 y''"
          by (smt (verit, del_insts) \<open>R x' y'\<close> \<open>R\<^sup>=\<^sup>= x x'\<close> \<open>\<forall>y. R x y \<longrightarrow> count M3 y \<le> count M1 y\<close>
              \<open>count M2 y' < count M3 y'\<close> \<open>count M3 x < count M1 x\<close> \<open>count M3 x' < count M2 x'\<close>
              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'' \<le> count M3 y''"
        proof -
          have "y'' \<in># M1 + M2"
            by (metis \<open>count M1 y'' < count M2 y''\<close> count_inI not_less_iff_gr_or_eq union_iff)

          moreover have "R x' y''"
            by (metis \<open>R x' y'\<close> \<open>R y' y''\<close> \<open>count M2 y' < count M1 y'\<close>
                \<open>transp_on (set_mset M1 \<union> set_mset M2) R\<close> \<open>x' \<in># M1 + M2\<close> calculation count_inI
                nat_neq_iff set_mset_union transp_onD union_iff)

          moreover have "R\<^sup>=\<^sup>= x y''"
            using \<open>R\<^sup>=\<^sup>= x x'\<close>
            by (metis (mono_tags, opaque_lifting) \<open>transp_on (set_mset M1 \<union> set_mset M2) R\<close>
                \<open>x \<in># M1 + M2\<close> \<open>x' \<in># M1 + M2\<close> 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 "\<And>x'. R\<^sup>=\<^sup>= x x' \<Longrightarrow> count M2 x' \<le> count M3 x'"
          by auto
        hence "count M2 x \<le> count M3 x"
          by simp
        hence "count M2 x < count M1 x"
          using \<open>count M3 x < count M1 x\<close> by linarith
        with \<open>multp\<^sub>H\<^sub>O R M1 M2\<close> obtain y where
          "R x y" and "count M1 y < count M2 y"
          unfolding multp\<^sub>H\<^sub>O_def by auto
        hence "count M3 y < count M2 y"
          using \<open>\<forall>y. R x y \<longrightarrow> count M3 y \<le> count M1 y\<close> dual_order.strict_trans2 by metis
        then show ?thesis
          using False \<open>R x y\<close> by auto
      qed
    qed
  qed
qed

lemma transp_multp\<^sub>H\<^sub>O:
  assumes "asymp R" and "transp R"
  shows "transp (multp\<^sub>H\<^sub>O R)"
  using assms transp_on_multp\<^sub>H\<^sub>O[of UNIV, simplified] by metis


paragraph \<open>Totality\<close>

lemma totalp_on_multp\<^sub>D\<^sub>M:
  "totalp_on A R \<Longrightarrow> (\<And>M. M \<in> B \<Longrightarrow> set_mset M \<subseteq> A) \<Longrightarrow> totalp_on B (multp\<^sub>D\<^sub>M R)"
  by (smt (verit, ccfv_SIG) count_inI in_mono multp\<^sub>H\<^sub>O_def multp\<^sub>H\<^sub>O_imp_multp\<^sub>D\<^sub>M not_less_iff_gr_or_eq
      totalp_onD totalp_onI)

lemma totalp_multp\<^sub>D\<^sub>M: "totalp R \<Longrightarrow> totalp (multp\<^sub>D\<^sub>M R)"
  by (rule totalp_on_multp\<^sub>D\<^sub>M[of UNIV R UNIV, simplified])

lemma totalp_on_multp\<^sub>H\<^sub>O:
  "totalp_on A R \<Longrightarrow> (\<And>M. M \<in> B \<Longrightarrow> set_mset M \<subseteq> A) \<Longrightarrow> totalp_on B (multp\<^sub>H\<^sub>O R)"
  by (smt (verit, ccfv_SIG) count_inI in_mono multp\<^sub>H\<^sub>O_def not_less_iff_gr_or_eq totalp_onD
      totalp_onI)

lemma totalp_multp\<^sub>H\<^sub>O: "totalp R \<Longrightarrow> totalp (multp\<^sub>H\<^sub>O R)"
  by (rule totalp_on_multp\<^sub>H\<^sub>O[of UNIV R UNIV, simplified])


paragraph \<open>Type Classes\<close>

context preorder
begin

lemma order_mult: "class.order
  (\<lambda>M N. (M, N) \<in> mult {(x, y). x < y} \<or> M = N)
  (\<lambda>M N. (M, N) \<in> mult {(x, y). x < y})"
  (is "class.order ?le ?less")
proof -
  have irrefl: "\<And>M :: 'a multiset. \<not> ?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) \<in> mult {(x, y). x < y}"
    ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
      \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset J. (k, j) \<in> {(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 \<noteq> {#}" and "(\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset J. (k, j) \<in> {(x, y). x < y})" by blast
    then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_mset K. \<exists>j\<in>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: "\<And>K M N :: 'a multiset. ?less K M \<Longrightarrow> ?less M N \<Longrightarrow> ?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 \<open>The Dershowitz--Manna ordering:\<close>

definition less_multiset\<^sub>D\<^sub>M where
  "less_multiset\<^sub>D\<^sub>M M N \<longleftrightarrow>
   (\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = (N - X) + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> k < a)))"


text \<open>The Huet--Oppen ordering:\<close>

definition less_multiset\<^sub>H\<^sub>O where
  "less_multiset\<^sub>H\<^sub>O M N \<longleftrightarrow> M \<noteq> N \<and> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x. y < x \<and> count M x < count N x))"

lemma mult_imp_less_multiset\<^sub>H\<^sub>O:
  "(M, N) \<in> mult {(x, y). x < y} \<Longrightarrow> less_multiset\<^sub>H\<^sub>O M N"
  unfolding multp_def[of "(<)", symmetric]
  using multp_imp_multp\<^sub>H\<^sub>O[of "(<)"]
  by (simp add: less_multiset\<^sub>H\<^sub>O_def multp\<^sub>H\<^sub>O_def)

lemma less_multiset\<^sub>D\<^sub>M_imp_mult:
  "less_multiset\<^sub>D\<^sub>M M N \<Longrightarrow> (M, N) \<in> mult {(x, y). x < y}"
  unfolding multp_def[of "(<)", symmetric]
  by (rule multp\<^sub>D\<^sub>M_imp_multp[of "(<)" M N]) (simp add: less_multiset\<^sub>D\<^sub>M_def multp\<^sub>D\<^sub>M_def)

lemma less_multiset\<^sub>H\<^sub>O_imp_less_multiset\<^sub>D\<^sub>M: "less_multiset\<^sub>H\<^sub>O M N \<Longrightarrow> less_multiset\<^sub>D\<^sub>M M N"
  unfolding less_multiset\<^sub>D\<^sub>M_def less_multiset\<^sub>H\<^sub>O_def
  unfolding multp\<^sub>D\<^sub>M_def[symmetric] multp\<^sub>H\<^sub>O_def[symmetric]
  by (rule multp\<^sub>H\<^sub>O_imp_multp\<^sub>D\<^sub>M)

lemma mult_less_multiset\<^sub>D\<^sub>M: "(M, N) \<in> mult {(x, y). x < y} \<longleftrightarrow> less_multiset\<^sub>D\<^sub>M M N"
  unfolding multp_def[of "(<)", symmetric]
  using multp_eq_multp\<^sub>D\<^sub>M[of "(<)", simplified]
  by (simp add: multp\<^sub>D\<^sub>M_def less_multiset\<^sub>D\<^sub>M_def)

lemma mult_less_multiset\<^sub>H\<^sub>O: "(M, N) \<in> mult {(x, y). x < y} \<longleftrightarrow> less_multiset\<^sub>H\<^sub>O M N"
  unfolding multp_def[of "(<)", symmetric]
  using multp_eq_multp\<^sub>H\<^sub>O[of "(<)", simplified]
  by (simp add: multp\<^sub>H\<^sub>O_def less_multiset\<^sub>H\<^sub>O_def)

lemmas mult\<^sub>D\<^sub>M = mult_less_multiset\<^sub>D\<^sub>M[unfolded less_multiset\<^sub>D\<^sub>M_def]
lemmas mult\<^sub>H\<^sub>O = mult_less_multiset\<^sub>H\<^sub>O[unfolded less_multiset\<^sub>H\<^sub>O_def]

end

lemma less_multiset_less_multiset\<^sub>H\<^sub>O: "M < N \<longleftrightarrow> less_multiset\<^sub>H\<^sub>O M N"
  unfolding less_multiset_def multp_def mult\<^sub>H\<^sub>O less_multiset\<^sub>H\<^sub>O_def ..

lemma less_multiset\<^sub>D\<^sub>M:
  "M < N \<longleftrightarrow> (\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = N - X + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> k < a)))"
  by (rule mult\<^sub>D\<^sub>M[folded multp_def less_multiset_def])

lemma less_multiset\<^sub>H\<^sub>O:
  "M < N \<longleftrightarrow> M \<noteq> N \<and> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x>y. count M x < count N x))"
  by (rule mult\<^sub>H\<^sub>O[folded multp_def less_multiset_def])

lemma subset_eq_imp_le_multiset:
  shows "M \<subseteq># N \<Longrightarrow> M \<le> N"
  unfolding less_eq_multiset_def less_multiset\<^sub>H\<^sub>O
  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_multiset\<^sub>H\<^sub>O by simp

lemma less_eq_multiset_empty_left[simp]:
  shows "{#} \<le> M"
  by (simp add: subset_eq_imp_le_multiset)

lemma ex_gt_imp_less_multiset: "(\<exists>y. y \<in># N \<and> (\<forall>x. x \<in># M \<longrightarrow> x < y)) \<Longrightarrow> M < N"
  unfolding less_multiset\<^sub>H\<^sub>O
  by (metis count_eq_zero_iff count_greater_zero_iff less_le_not_le)

lemma less_eq_multiset_empty_right[simp]: "M \<noteq> {#} \<Longrightarrow> \<not> M \<le> {#}"
  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 \<noteq> {#} \<Longrightarrow> {#} < M"
  by (simp add: less_multiset\<^sub>H\<^sub>O)

(* FIXME: "le" should be "less" in this and other names *)
lemma le_multiset_empty_right[simp]: "\<not> M < {#}"
  using subset_mset.le_zero_eq less_multiset_def multp_def less_multiset\<^sub>D\<^sub>M by blast

(* FIXME: "le" should be "less" in this and other names *)
lemma union_le_diff_plus: "P \<subseteq># M \<Longrightarrow> N < P \<Longrightarrow> 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_multiset\<^sub>H\<^sub>O:
  "M \<le> N \<longleftrightarrow> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x. y < x \<and> count M x < count N x))"
  by (auto simp: less_eq_multiset_def less_multiset\<^sub>H\<^sub>O)

instance by standard (auto simp: less_eq_multiset\<^sub>H\<^sub>O)

lemma
  fixes M N :: "'a multiset"
  shows
    less_eq_multiset_plus_left: "N \<le> (M + N)" and
    less_eq_multiset_plus_right: "M \<le> (M + N)"
  by simp_all

lemma
  fixes M N :: "'a multiset"
  shows
    le_multiset_plus_left_nonempty: "M \<noteq> {#} \<Longrightarrow> N < M + N" and
    le_multiset_plus_right_nonempty: "N \<noteq> {#} \<Longrightarrow> M < M + N"
    by simp_all

end

lemma all_lt_Max_imp_lt_mset: "N \<noteq> {#} \<Longrightarrow> (\<forall>a \<in># M. a < Max (set_mset N)) \<Longrightarrow> 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 \<Longrightarrow> \<exists>y. count M y < count N y"
  by (meson less_eq_multiset\<^sub>H\<^sub>O less_le_not_le)

lemma subset_imp_less_mset: "A \<subset># B \<Longrightarrow> A < B"
  by (simp add: order.not_eq_order_implies_strict subset_eq_imp_le_multiset)

lemma image_mset_strict_mono:
  assumes
    mono_f: "\<forall>x \<in> set_mset M. \<forall>y \<in> set_mset N. x < y \<longrightarrow> 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 \<noteq> {#}" and y_sub_N: "Y \<subseteq># N" and M_eq: "M = N - Y + X" and
    ex_y: "\<forall>x. x \<in># X \<longrightarrow> (\<exists>y. y \<in># Y \<and> x < y)"
    using less[unfolded less_multiset\<^sub>D\<^sub>M] by blast

  have x_sub_M: "X \<subseteq># M"
    using M_eq by simp

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

  show ?thesis
    unfolding less_multiset\<^sub>D\<^sub>M
  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: "\<forall>x. x \<in># X \<longrightarrow> y x \<in># Y \<and> x < y x"
      using ex_y by moura

    show "\<forall>fx. fx \<in># ?fX \<longrightarrow> (\<exists>fy. fy \<in># ?fY \<and> fx < fy)"
    proof (intro allI impI)
      fix fx
      assume "fx \<in># ?fX"
      then obtain x where fx: "fx = f x" and x_in: "x \<in># X"
        by auto
      hence y_in: "y x \<in># Y" and y_gt: "x < y x"
        using y[rule_format, OF x_in] by blast+
      hence "f (y x) \<in># ?fY \<and> 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 "\<exists>fy. fy \<in># ?fY \<and> 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: "\<forall>x \<in> set_mset M. \<forall>y \<in> set_mset N. x < y \<longrightarrow> f x < f y" and
    less: "M \<le> N"
  shows "image_mset f M \<le> 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#} \<longleftrightarrow> (\<forall>x \<in># M. x < y)" for y :: "'a::linorder"
proof (rule iffI)
  assume M_lt_y: "M < {#y#}"
  show "\<forall>x \<in># M. x < y"
  proof
    fix x
    assume x_in: "x \<in># M"
    hence M: "M - {#x#} + {#x#} = M"
      by (meson insert_DiffM2)
    hence "\<not> {#x#} < {#y#} \<Longrightarrow> 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 "\<not> {#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: "\<forall>x \<in># 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 \<le> {#y#} \<longleftrightarrow> M = {#y#} \<or> (\<forall>x \<in># M. x < y)" for y :: "'a::linorder"
  by (meson less_eq_multiset_def mset_lt_single_right_iff)


subsubsection \<open>Simplifications\<close>

lemma multp\<^sub>H\<^sub>O_repeat_mset_repeat_mset[simp]:
  assumes "n \<noteq> 0"
  shows "multp\<^sub>H\<^sub>O R (repeat_mset n A) (repeat_mset n B) \<longleftrightarrow> multp\<^sub>H\<^sub>O R A B"
proof (rule iffI)
  assume hyp: "multp\<^sub>H\<^sub>O R (repeat_mset n A) (repeat_mset n B)"
  hence
    1: "repeat_mset n A \<noteq> repeat_mset n B" and
    2: "\<forall>y. n * count B y < n * count A y \<longrightarrow> (\<exists>x. R y x \<and> n * count A x < n * count B x)"
    by (simp_all add: multp\<^sub>H\<^sub>O_def)

  from 1 \<open>n \<noteq> 0\<close> have "A \<noteq> B"
    by auto

  moreover from 2 \<open>n \<noteq> 0\<close> have "\<forall>y. count B y < count A y \<longrightarrow> (\<exists>x. R y x \<and> count A x < count B x)"
    by auto

  ultimately show "multp\<^sub>H\<^sub>O R A B"
    by (simp add: multp\<^sub>H\<^sub>O_def)
next
  assume "multp\<^sub>H\<^sub>O R A B"
  hence 1: "A \<noteq> B" and 2: "\<forall>y. count B y < count A y \<longrightarrow> (\<exists>x. R y x \<and> count A x < count B x)"
    by (simp_all add: multp\<^sub>H\<^sub>O_def)

  from 1 have "repeat_mset n A \<noteq> repeat_mset n B"
    by (simp add: assms repeat_mset_cancel1)

  moreover from 2 have "\<forall>y. n * count B y < n * count A y \<longrightarrow>
    (\<exists>x. R y x \<and> n * count A x < n * count B x)"
    by auto

  ultimately show "multp\<^sub>H\<^sub>O R (repeat_mset n A) (repeat_mset n B)"
    by (simp add: multp\<^sub>H\<^sub>O_def)
qed

lemma multp\<^sub>H\<^sub>O_double_double[simp]: "multp\<^sub>H\<^sub>O R (A + A) (B + B) \<longleftrightarrow> multp\<^sub>H\<^sub>O R A B"
  using multp\<^sub>H\<^sub>O_repeat_mset_repeat_mset[of 2]
  by (simp add: numeral_Bit0)


subsection \<open>Simprocs\<close>

lemma mset_le_add_iff1:
  "j \<le> (i::nat) \<Longrightarrow> (repeat_mset i u + m \<le> repeat_mset j u + n) = (repeat_mset (i-j) u + m \<le> n)"
proof -
  assume "j \<le> 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 \<le> (j::nat) \<Longrightarrow> (repeat_mset i u + m \<le> repeat_mset j u + n) = (m \<le> repeat_mset (j-i) u + n)"
proof -
  assume "i \<le> 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") =
  \<open>K Cancel_Simprocs.less_cancel\<close>

simproc_setup msetle_cancel
  ("(l::'a::preorder multiset) + m \<le> n" | "(l::'a multiset) \<le> m + n" |
   "add_mset a m \<le> n" | "m \<le> add_mset a n" |
   "replicate_mset p a \<le> n" | "m \<le> replicate_mset p a" |
   "repeat_mset p m \<le> n" | "m \<le> repeat_mset p n") =
  \<open>K Cancel_Simprocs.less_eq_cancel\<close>


subsection \<open>Additional facts and instantiations\<close>

lemma ex_gt_count_imp_le_multiset:
  "(\<forall>y :: 'a :: order. y \<in># M + N \<longrightarrow> y \<le> x) \<Longrightarrow> count M x < count N x \<Longrightarrow> M < N"
  unfolding less_multiset\<^sub>H\<^sub>O
  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#} \<longleftrightarrow> x < y"
  unfolding less_multiset\<^sub>H\<^sub>O by simp

lemma mset_le_single_iff[iff]: "{#x#} \<le> {#y#} \<longleftrightarrow> x \<le> y" for x y :: "'a::order"
  unfolding less_eq_multiset\<^sub>H\<^sub>O by force

instance multiset :: (linorder) linordered_cancel_ab_semigroup_add
  by standard (metis less_eq_multiset\<^sub>H\<^sub>O not_less_iff_gr_or_eq)

lemma less_eq_multiset_total:
  fixes M N :: "'a :: linorder multiset"
  shows "\<not> M \<le> N \<Longrightarrow> N \<le> M"
  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 by standard (metis less_multiset_def multp_def wf wf_def wf_mult)

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 "\<exists>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 \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
  "inf_multiset A B = (if A < B then A else B)"

definition sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> '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

end