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Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/Isabelle/HOL/Library/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 26 kB

Quelle Infinite_Set.thy Sprache: Isabelle

(*  Title:      HOL/Library/Infinite_Set.thy
    Author:     Stephan Merz
*)

section \<open>Infinite Sets and Related Concepts\<close>

theory Infinite_Set
  imports Main
begin

subsection \<open>The set of natural numbers is infinite\<close>

lemma infinite_nat_iff_unbounded_le: "infinite S \ (\m. \n\m. n \ S)"
  for S :: "nat set"
  using frequently_cofinite[of "\x. x \ S"]
  by (simp add: cofinite_eq_sequentially frequently_def eventually_sequentially)

lemma infinite_nat_iff_unbounded: "infinite S \ (\m. \n>m. n \ S)"
  for S :: "nat set"
  using frequently_cofinite[of "\x. x \ S"]
  by (simp add: cofinite_eq_sequentially frequently_def eventually_at_top_dense)

lemma finite_nat_iff_bounded: "finite S \ (\k. S \ {..
  for S :: "nat set"
  using infinite_nat_iff_unbounded_le[of S] by (simp add: subset_eq) (metis not_le)

lemma finite_nat_iff_bounded_le: "finite S \ (\k. S \ {.. k})"
  for S :: "nat set"
  using infinite_nat_iff_unbounded[of S] by (simp add: subset_eq) (metis not_le)

lemma finite_nat_bounded: "finite S \ \k. S \ {..
  for S :: "nat set"
  by (simp add: finite_nat_iff_bounded)

text \<open>
  For a set of natural numbers to be infinite, it is enough to know
  that for any number larger than some \<open>k\<close>, there is some larger
  number that is an element of the set.
\<close>

lemma unbounded_k_infinite: "\m>k. \n>m. n \ S \ infinite (S::nat set)"
  by (metis finite_nat_set_iff_bounded gt_ex order_less_not_sym order_less_trans)

lemma nat_not_finite: "finite (UNIV::nat set) \ R"
  by simp

lemma range_inj_infinite:
  fixes f :: "nat \ 'a"
  assumes "inj f"
  shows "infinite (range f)"
proof
  assume "finite (range f)"
  from this assms have "finite (UNIV::nat set)"
    by (rule finite_imageD)
  then show False by simp
qed

subsection \<open>The set of integers is also infinite\<close>

lemma infinite_int_iff_infinite_nat_abs: "infinite S \ infinite ((nat \ abs) ` S)"
  for S :: "int set"
proof (unfold Not_eq_iff, rule iffI)
  assume "finite ((nat \ abs) ` S)"
  then have "finite (nat ` (abs ` S))"
    by (simp add: image_image cong: image_cong)
  moreover have "inj_on nat (abs ` S)"
    by (rule inj_onI) auto
  ultimately have "finite (abs ` S)"
    by (rule finite_imageD)
  then show "finite S"
    by (rule finite_image_absD)
qed simp

proposition infinite_int_iff_unbounded_le: "infinite S \ (\m. \n. \n\ \ m \ n \ S)"
  for S :: "int set"
  by (simp add: infinite_int_iff_infinite_nat_abs infinite_nat_iff_unbounded_le o_def image_def)
    (metis abs_ge_zero nat_le_eq_zle le_nat_iff)

proposition infinite_int_iff_unbounded: "infinite S \ (\m. \n. \n\ > m \ n \ S)"
  for S :: "int set"
  by (simp add: infinite_int_iff_infinite_nat_abs infinite_nat_iff_unbounded o_def image_def)
    (metis (full_types) nat_le_iff nat_mono not_le)

proposition finite_int_iff_bounded: "finite S \ (\k. abs ` S \ {..
  for S :: "int set"
  using infinite_int_iff_unbounded_le[of S] by (simp add: subset_eq) (metis not_le)

proposition finite_int_iff_bounded_le: "finite S \ (\k. abs ` S \ {.. k})"
  for S :: "int set"
  using infinite_int_iff_unbounded[of S] by (simp add: subset_eq) (metis not_le)

lemma infinite_split: \<comment> \<open>courtesy of Michael Schmidt\<close>
  fixes S :: "'a set"
  assumes "infinite S"
  obtains A B
    where "A \ S" "B \ S" "infinite A" "infinite B" "A \ B = {}"
proof-
  obtain f :: "nat \ 'a"
    where f_inj: "inj f" and f_img: "range f \ S"
    using assms infinite_countable_subset by blast
  let ?A = "range (\n. f (2*n))"
  let ?B = "range (\n. f (2*n+1))"
  have a_inf: "infinite ?A"
    using finite_imageD[of "\n. f (2*n)" UNIV] f_inj infinite_UNIV_nat unfolding inj_def
    by fastforce
  have b_inf: "infinite ?B"
    using finite_imageD[of "\n. f (2*n+1)" UNIV] f_inj infinite_UNIV_nat unfolding inj_def
    by fastforce
  from f_inj have "?A \ ?B = {}" unfolding inj_def disjoint_iff using double_not_eq_Suc_double
    by auto
  from that[OF _ _ a_inf b_inf this] f_img show ?thesis
    by blast
qed

subsection \<open>Infinitely Many and Almost All\<close>

text \<open>
  We often need to reason about the existence of infinitely many
  (resp., all but finitely many) objects satisfying some predicate, so
  we introduce corresponding binders and their proof rules.
\<close>

lemma not_INFM [simp]: "\ (INFM x. P x) \ (MOST x. \ P x)"
  by (rule not_frequently)

lemma not_MOST [simp]: "\ (MOST x. P x) \ (INFM x. \ P x)"
  by (rule not_eventually)

lemma INFM_const [simp]: "(INFM x::'a. P) \ P \ infinite (UNIV::'a set)"
  by (simp add: frequently_const_iff)

lemma MOST_const [simp]: "(MOST x::'a. P) \ P \ finite (UNIV::'a set)"
  by (simp add: eventually_const_iff)

lemma INFM_imp_distrib: "(INFM x. P x \ Q x) \ ((MOST x. P x) \ (INFM x. Q x))"
  by (rule frequently_imp_iff)

lemma MOST_imp_iff: "MOST x. P x \ (MOST x. P x \ Q x) \ (MOST x. Q x)"
  by (auto intro: eventually_rev_mp eventually_mono)

lemma INFM_conjI: "INFM x. P x \ MOST x. Q x \ INFM x. P x \ Q x"
  by (rule frequently_rev_mp[of P]) (auto elim: eventually_mono)

text \<open>Properties of quantifiers with injective functions.\<close>

lemma INFM_inj: "INFM x. P (f x) \ inj f \ INFM x. P x"
  using finite_vimageI[of "{x. P x}" f] by (auto simp: frequently_cofinite)

lemma MOST_inj: "MOST x. P x \ inj f \ MOST x. P (f x)"
  using finite_vimageI[of "{x. \ P x}" f] by (auto simp: eventually_cofinite)

text \<open>Properties of quantifiers with singletons.\<close>

lemma not_INFM_eq [simp]:
  "\ (INFM x. x = a)"
  "\ (INFM x. a = x)"
  unfolding frequently_cofinite by simp_all

lemma MOST_neq [simp]:
  "MOST x. x \ a"
  "MOST x. a \ x"
  unfolding eventually_cofinite by simp_all

lemma INFM_neq [simp]:
  "(INFM x::'a. x \ a) \ infinite (UNIV::'a set)"
  "(INFM x::'a. a \ x) \ infinite (UNIV::'a set)"
  unfolding frequently_cofinite by simp_all

lemma MOST_eq [simp]:
  "(MOST x::'a. x = a) \ finite (UNIV::'a set)"
  "(MOST x::'a. a = x) \ finite (UNIV::'a set)"
  unfolding eventually_cofinite by simp_all

lemma MOST_eq_imp:
  "MOST x. x = a \ P x"
  "MOST x. a = x \ P x"
  unfolding eventually_cofinite by simp_all

text \<open>Properties of quantifiers over the naturals.\<close>

lemma MOST_nat: "(\\<^sub>\n. P n) \ (\m. \n>m. P n)"
  for P :: "nat \ bool"
  by (auto simp add: eventually_cofinite finite_nat_iff_bounded_le subset_eq simp flip: not_le)

lemma MOST_nat_le: "(\\<^sub>\n. P n) \ (\m. \n\m. P n)"
  for P :: "nat \ bool"
  by (auto simp add: eventually_cofinite finite_nat_iff_bounded subset_eq simp flip: not_le)

lemma INFM_nat: "(\\<^sub>\n. P n) \ (\m. \n>m. P n)"
  for P :: "nat \ bool"
  by (simp add: frequently_cofinite infinite_nat_iff_unbounded)

lemma INFM_nat_le: "(\\<^sub>\n. P n) \ (\m. \n\m. P n)"
  for P :: "nat \ bool"
  by (simp add: frequently_cofinite infinite_nat_iff_unbounded_le)

lemma MOST_INFM: "infinite (UNIV::'a set) \ MOST x::'a. P x \ INFM x::'a. P x"
  by (simp add: eventually_frequently)

lemma MOST_Suc_iff: "(MOST n. P (Suc n)) \ (MOST n. P n)"
  by (simp add: cofinite_eq_sequentially)

lemma MOST_SucI: "MOST n. P n \ MOST n. P (Suc n)"
  and MOST_SucD: "MOST n. P (Suc n) \ MOST n. P n"
  by (simp_all add: MOST_Suc_iff)

lemma MOST_ge_nat: "MOST n::nat. m \ n"
  by (simp add: cofinite_eq_sequentially)

\<comment> \<open>legacy names\<close>
lemma Inf_many_def: "Inf_many P \ infinite {x. P x}" by (fact frequently_cofinite)
lemma Alm_all_def: "Alm_all P \ \ (INFM x. \ P x)" by simp
lemma INFM_iff_infinite: "(INFM x. P x) \ infinite {x. P x}" by (fact frequently_cofinite)
lemma MOST_iff_cofinite: "(MOST x. P x) \ finite {x. \ P x}" by (fact eventually_cofinite)
lemma INFM_EX: "(\\<^sub>\x. P x) \ (\x. P x)" by (fact frequently_ex)
lemma ALL_MOST: "\x. P x \ \\<^sub>\x. P x" by (fact always_eventually)
lemma INFM_mono: "\\<^sub>\x. P x \ (\x. P x \ Q x) \ \\<^sub>\x. Q x" by (fact frequently_elim1)
lemma MOST_mono: "\\<^sub>\x. P x \ (\x. P x \ Q x) \ \\<^sub>\x. Q x" by (fact eventually_mono)
lemma INFM_disj_distrib: "(\\<^sub>\x. P x \ Q x) \ (\\<^sub>\x. P x) \ (\\<^sub>\x. Q x)" by (fact frequently_disj_iff)
lemma MOST_rev_mp: "\\<^sub>\x. P x \ \\<^sub>\x. P x \ Q x \ \\<^sub>\x. Q x" by (fact eventually_rev_mp)
lemma MOST_conj_distrib: "(\\<^sub>\x. P x \ Q x) \ (\\<^sub>\x. P x) \ (\\<^sub>\x. Q x)" by (fact eventually_conj_iff)
lemma MOST_conjI: "MOST x. P x \ MOST x. Q x \ MOST x. P x \ Q x" by (fact eventually_conj)
lemma INFM_finite_Bex_distrib: "finite A \ (INFM y. \x\A. P x y) \ (\x\A. INFM y. P x y)" by (fact frequently_bex_finite_distrib)
lemma MOST_finite_Ball_distrib: "finite A \ (MOST y. \x\A. P x y) \ (\x\A. MOST y. P x y)" by (fact eventually_ball_finite_distrib)
lemma INFM_E: "INFM x. P x \ (\x. P x \ thesis) \ thesis" by (fact frequentlyE)
lemma MOST_I: "(\x. P x) \ MOST x. P x" by (rule eventuallyI)
lemmas MOST_iff_finiteNeg = MOST_iff_cofinite

subsection \<open>Enumeration of an Infinite Set\<close>

text \<open>The set's element type must be wellordered (e.g. the natural numbers).\<close>

text \<open>
  Could be generalized to
    \<^prop>\<open>enumerate' S n = (SOME t. t \<in> s \<and> finite {s\<in>S. s < t} \<and> card {s\<in>S. s < t} = n)\<close>.
\<close>

primrec (in wellorder) enumerate :: "'a set \ nat \ 'a"
  where
    enumerate_0: "enumerate S 0 = (LEAST n. n \ S)"
  | enumerate_Suc: "enumerate S (Suc n) = enumerate (S - {LEAST n. n \ S}) n"

lemma enumerate_Suc': "enumerate S (Suc n) = enumerate (S - {enumerate S 0}) n"
  by simp

lemma enumerate_in_set: "infinite S \ enumerate S n \ S"
proof (induct n arbitrary: S)
  case 0
  then show ?case
    by (fastforce intro: LeastI dest!: infinite_imp_nonempty)
next
  case (Suc n)
  then show ?case
    by simp (metis DiffE infinite_remove)
qed

declare enumerate_0 [simp del] enumerate_Suc [simp del]

lemma enumerate_step: "infinite S \ enumerate S n < enumerate S (Suc n)"
proof (induction n arbitrary: S)
  case 0
  then have "enumerate S 0 \ enumerate S (Suc 0)"
    by (simp add: enumerate_0 Least_le enumerate_in_set)
  moreover have "enumerate (S - {enumerate S 0}) 0 \ S - {enumerate S 0}"
    by (meson "0.prems" enumerate_in_set infinite_remove)
  then have "enumerate S 0 \ enumerate (S - {enumerate S 0}) 0"
    by auto
  ultimately show ?case
    by (simp add: enumerate_Suc')
next
  case (Suc n)
  then show ?case
    by (simp add: enumerate_Suc')
qed

lemma enumerate_mono: "m < n \ infinite S \ enumerate S m < enumerate S n"
  by (induct m n rule: less_Suc_induct) (auto intro: enumerate_step)

lemma enumerate_mono_iff [simp]:
  "infinite S \ enumerate S m < enumerate S n \ m < n"
  by (metis enumerate_mono less_asym less_linear)

lemma enumerate_mono_le_iff [simp]:
  "infinite S \ enumerate S m \ enumerate S n \ m \ n"
  by (meson enumerate_mono_iff not_le)

lemma le_enumerate:
  assumes S: "infinite S"
  shows "n \ enumerate S n"
  using S
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  then have "n \ enumerate S n" by simp
  also note enumerate_mono[of n "Suc n", OF _ \<open>infinite S\<close>]
  finally show ?case by simp
qed

lemma infinite_enumerate:
  assumes fS: "infinite S"
  shows "\r::nat\nat. strict_mono r \ (\n. r n \ S)"
  unfolding strict_mono_def
  using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by blast

lemma enumerate_Suc'':
  fixes S :: "'a::wellorder set"
  assumes "infinite S"
  shows "enumerate S (Suc n) = (LEAST s. s \ S \ enumerate S n < s)"
  using assms
proof (induct n arbitrary: S)
  case 0
  then have "\s \ S. enumerate S 0 \ s"
    by (auto simp: enumerate.simps intro: Least_le)
  then show ?case
    unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"]
    by (intro arg_cong[where f = Least] ext) auto
next
  case (Suc n S)
  show ?case
  proof (rule order.antisym)
    have S: "infinite (S - {wellorder_class.enumerate S 0})"
      using Suc by auto
    show "wellorder_class.enumerate S (Suc (Suc n)) \ (LEAST s. s \ S \ wellorder_class.enumerate S (Suc n) < s)"
      using enumerate_mono[OF zero_less_Suc] \<open>infinite S\<close> S
      by (smt (verit, best) LeastI_ex Suc.hyps enumerate_0 enumerate_Suc enumerate_in_set
          enumerate_step insertE insert_Diff linorder_not_less not_less_Least)
  qed (simp add: Least_le Suc.prems enumerate_in_set)
qed

lemma enumerate_Ex:
  fixes S :: "nat set"
  assumes S: "infinite S"
    and s: "s \ S"
  shows "\n. enumerate S n = s"
  using s
proof (induct s rule: less_induct)
  case (less s)
  show ?case
  proof (cases "\y\S. y < s")
    case True
    let ?y = "Max {s'\S. s' < s}"
    from True have y: "\x. ?y < x \ (\s'\S. s' < s \ s' < x)"
      by (subst Max_less_iff) auto
    then have y_in: "?y \ {s'\S. s' < s}"
      by (intro Max_in) auto
    with less.hyps[of ?y] obtain n where "enumerate S n = ?y"
      by auto
    with S have "enumerate S (Suc n) = s"
      by (auto simp: y less enumerate_Suc'' intro!: Least_equality)
    then show ?thesis by auto
  next
    case False
    then have "\t\S. s \ t" by auto
    with \<open>s \<in> S\<close> show ?thesis
      by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0)
  qed
qed

lemma inj_enumerate:
  fixes S :: "'a::wellorder set"
  assumes S: "infinite S"
  shows "inj (enumerate S)"
  unfolding inj_on_def
proof clarsimp
  show "\x y. enumerate S x = enumerate S y \ x = y"
    by (metis neq_iff enumerate_mono[OF _ \<open>infinite S\<close>])
qed

text \<open>To generalise this, we'd need a condition that all initial segments were finite\<close>
lemma bij_enumerate:
  fixes S :: "nat set"
  assumes S: "infinite S"
  shows "bij_betw (enumerate S) UNIV S"
proof -
  have "\s \ S. \i. enumerate S i = s"
    using enumerate_Ex[OF S] by auto
  moreover note \<open>infinite S\<close> inj_enumerate
  ultimately show ?thesis
    unfolding bij_betw_def by (auto intro: enumerate_in_set)
qed

lemma
  fixes S :: "nat set"
  assumes S: "infinite S"
  shows range_enumerate: "range (enumerate S) = S"
    and strict_mono_enumerate: "strict_mono (enumerate S)"
  by (auto simp add: bij_betw_imp_surj_on bij_enumerate assms strict_mono_def)

text \<open>A pair of weird and wonderful lemmas from HOL Light.\<close>
lemma finite_transitivity_chain:
  assumes "finite A"
    and R: "\x. \ R x x" "\x y z. \R x y; R y z\ \ R x z"
    and A: "\x. x \ A \ \y. y \ A \ R x y"
  shows "A = {}"
  using \<open>finite A\<close> A
proof (induct A)
  case empty
  then show ?case by simp
next
  case (insert a A)
  have False
    using R(1)[of a] R(2)[of _ a] insert(3,4) by blast
  thus ?case ..
qed

corollary Union_maximal_sets:
  assumes "finite \"
  shows "\{T \ \. \U\\. \ T \ U} = \\"
    (is "?lhs = ?rhs")
proof
  show "?lhs \ ?rhs" by force
  show "?rhs \ ?lhs"
  proof (rule Union_subsetI)
    fix S
    assume "S \ \"
    have "{T \ \. S \ T} = {}"
      if "\ (\y. y \ {T \ \. \U\\. \ T \ U} \ S \ y)"
    proof -
      have \<section>: "\<And>x. x \<in> \<F> \<and> S \<subseteq> x \<Longrightarrow> \<exists>y. y \<in> \<F> \<and> S \<subseteq> y \<and> x \<subset> y"
        using that by (blast intro: dual_order.trans psubset_imp_subset)
      show ?thesis
      proof (rule finite_transitivity_chain [of _ "\T U. S \ T \ T \ U"])
      qed (use assms in \<open>auto intro: \<section>\<close>)
    qed
    with \<open>S \<in> \<F>\<close> show "\<exists>y. y \<in> {T \<in> \<F>. \<forall>U\<in>\<F>. \<not> T \<subset> U} \<and> S \<subseteq> y"
      by blast
  qed
qed

subsection \<open>Properties of @{term enumerate} on finite sets\<close>

lemma finite_enumerate_in_set: "\finite S; n < card S\ \ enumerate S n \ S"
proof (induction n arbitrary: S)
  case 0
  then show ?case
    by (metis all_not_in_conv card.empty enumerate.simps(1) not_less0 wellorder_Least_lemma(1))
next
  case (Suc n)
  then have "wellorder_class.enumerate (S - {LEAST n. n \ S}) n \ S"
    by (metis Diff_empty Diff_insert0 Suc_lessD Suc_less_eq card.insert_remove
      finite_Diff insert_Diff insert_Diff_single insert_iff)
  then
  show ?case
    using Suc.prems Suc.IH [of "S - {LEAST n. n \ S}"]
    by (simp add: enumerate.simps)
qed

lemma finite_enumerate_step: "\finite S; Suc n < card S\ \ enumerate S n < enumerate S (Suc n)"
proof (induction n arbitrary: S)
  case 0
  then have "enumerate S 0 \ enumerate S (Suc 0)"
    by (simp add: Least_le enumerate.simps(1) finite_enumerate_in_set)
  moreover have "enumerate (S - {enumerate S 0}) 0 \ S - {enumerate S 0}"
    by (metis 0 Suc_lessD Suc_less_eq card_Suc_Diff1 enumerate_in_set finite_enumerate_in_set)
  then have "enumerate S 0 \ enumerate (S - {enumerate S 0}) 0"
    by auto
  ultimately show ?case
    by (simp add: enumerate_Suc')
next
  case (Suc n)
  then show ?case
    by (simp add: enumerate_Suc' finite_enumerate_in_set)
qed

lemma finite_enumerate_mono: "\m < n; finite S; n < card S\ \ enumerate S m < enumerate S n"
  by (induct m n rule: less_Suc_induct) (auto intro: finite_enumerate_step)

lemma finite_enumerate_mono_iff [simp]:
  "\finite S; m < card S; n < card S\ \ enumerate S m < enumerate S n \ m < n"
  by (metis finite_enumerate_mono less_asym less_linear)

lemma finite_le_enumerate:
  assumes "finite S" "n < card S"
  shows "n \ enumerate S n"
  using assms
proof (induction n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  then have "n \ enumerate S n" by simp
  also note finite_enumerate_mono[of n "Suc n", OF _ \<open>finite S\<close>]
  finally show ?case
    using Suc.prems(2) Suc_leI by blast
qed

lemma finite_enumerate:
  assumes fS: "finite S"
  shows "\r::nat\nat. strict_mono_on {.. (\n S)"
  unfolding strict_mono_def
  using finite_enumerate_in_set[OF fS] finite_enumerate_mono[of _ _ S] fS
  by (metis lessThan_iff strict_mono_on_def)

lemma finite_enumerate_Suc'':
  fixes S :: "'a::wellorder set"
  assumes "finite S" "Suc n < card S"
  shows "enumerate S (Suc n) = (LEAST s. s \ S \ enumerate S n < s)"
  using assms
proof (induction n arbitrary: S)
  case 0
  then have "\s \ S. enumerate S 0 \ s"
    by (auto simp: enumerate.simps intro: Least_le)
  then show ?case
    unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"]
    by (metis Diff_iff dual_order.strict_iff_order singletonD singletonI)
next
  case (Suc n S)
  then have "Suc n < card (S - {enumerate S 0})"
    using Suc.prems(2) finite_enumerate_in_set by force
  then show ?case
    apply (subst (1 2) enumerate_Suc')
    apply (simp add: Suc)
    apply (intro arg_cong[where f = Least] HOL.ext)
    using finite_enumerate_mono[OF zero_less_Suc \<open>finite S\<close>, of n] Suc.prems
    by (auto simp flip: enumerate_Suc')
qed

lemma finite_enumerate_initial_segment:
  fixes S :: "'a::wellorder set"
  assumes "finite S" and n: "n < card (S \ {..
  shows "enumerate (S \ {..
  using n
proof (induction n)
  case 0
  have "(LEAST n. n \ S \ n < s) = (LEAST n. n \ S)"
  proof (rule Least_equality)
    have "\t. t \ S \ t < s"
      by (metis "0" card_gt_0_iff disjoint_iff_not_equal lessThan_iff)
    then show "(LEAST n. n \ S) \ S \ (LEAST n. n \ S) < s"
      by (meson LeastI Least_le le_less_trans)
  qed (simp add: Least_le)
  then show ?case
    by (auto simp: enumerate_0)
next
  case (Suc n)
  then have less_card: "Suc n < card S"
    by (meson assms(1) card_mono inf_sup_ord(1) leD le_less_linear order.trans)
  obtain T where T: "T \ {s \ S. enumerate S n < s}"
    by (metis Infinite_Set.enumerate_step enumerate_in_set finite_enumerate_in_set finite_enumerate_step less_card mem_Collect_eq)
  have "(LEAST x. x \ S \ x < s \ enumerate S n < x) = (LEAST x. x \ S \ enumerate S n < x)"
       (is "_ = ?r")
  proof (intro Least_equality conjI)
    show "?r \ S"
      by (metis (mono_tags, lifting) LeastI mem_Collect_eq T)
    have "\ s \ ?r"
      using not_less_Least [of _ "\x. x \ S \ enumerate S n < x"] Suc assms
      by (metis (mono_tags, lifting) Int_Collect Suc_lessD finite_Int finite_enumerate_in_set finite_enumerate_step lessThan_def less_le_trans)
    then show "?r < s"
      by auto
    show "enumerate S n < ?r"
      by (metis (no_types, lifting) LeastI mem_Collect_eq T)
  qed (auto simp: Least_le)
  then show ?case
    using Suc assms by (simp add: finite_enumerate_Suc'' less_card)
qed

lemma finite_enumerate_Ex:
  fixes S :: "'a::wellorder set"
  assumes S: "finite S"
    and s: "s \ S"
  shows "\n
  using s S
proof (induction s arbitrary: S rule: less_induct)
  case (less s)
  show ?case
  proof (cases "\y\S. y < s")
    case True
    let ?T = "S \ {..
    have "finite ?T"
      using less.prems(2) by blast
    have TS: "card ?T < card S"
      using less.prems by (blast intro: psubset_card_mono [OF \<open>finite S\<close>])
    from True have y: "\x. Max ?T < x \ (\s'\S. s' < s \ s' < x)"
      by (subst Max_less_iff) (auto simp: \<open>finite ?T\<close>)
    then have y_in: "Max ?T \ {s'\S. s' < s}"
      using Max_in \<open>finite ?T\<close> by fastforce
    with less.IH[of "Max ?T" ?T] obtain n where n: "enumerate ?T n = Max ?T" "n < card ?T"
      using \<open>finite ?T\<close> by blast
    then have "Suc n < card S"
      using TS less_trans_Suc by blast
    with S n have "enumerate S (Suc n) = s"
      by (subst finite_enumerate_Suc'') (auto simp: y finite_enumerate_initial_segment less finite_enumerate_Suc'' intro!: Least_equality)
    then show ?thesis
      using \<open>Suc n < card S\<close> by blast
  next
    case False
    then have "\t\S. s \ t" by auto
    moreover have "0 < card S"
      using card_0_eq less.prems by blast
    ultimately show ?thesis
      using \<open>s \<in> S\<close>
      by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0)
  qed
qed

lemma finite_enum_subset:
  assumes "\i. i < card X \ enumerate X i = enumerate Y i" and "finite X" "finite Y" "card X \ card Y"
  shows "X \ Y"
  by (metis assms finite_enumerate_Ex finite_enumerate_in_set less_le_trans subsetI)

lemma finite_enum_ext:
  assumes "\i. i < card X \ enumerate X i = enumerate Y i" and "finite X" "finite Y" "card X = card Y"
  shows "X = Y"
  by (intro antisym finite_enum_subset) (auto simp: assms)

lemma finite_bij_enumerate:
  fixes S :: "'a::wellorder set"
  assumes S: "finite S"
  shows "bij_betw (enumerate S) {..
proof -
  have "\n m. \n \ m; n < card S; m < card S\ \ enumerate S n \ enumerate S m"
    using finite_enumerate_mono[OF _ \<open>finite S\<close>] by (auto simp: neq_iff)
  then have "inj_on (enumerate S) {..
    by (auto simp: inj_on_def)
  moreover have "\s \ S. \i
    using finite_enumerate_Ex[OF S] by auto
  moreover note \<open>finite S\<close>
  ultimately show ?thesis
    unfolding bij_betw_def by (auto intro: finite_enumerate_in_set)
qed

lemma ex_bij_betw_strict_mono_card:
  fixes M :: "'a::wellorder set"
  assumes "finite M"
  obtains h where "bij_betw h {.. and "strict_mono_on {..
proof
  show "bij_betw (enumerate M) {..
    by (simp add: assms finite_bij_enumerate)
  show "strict_mono_on {..
    by (simp add: assms finite_enumerate_mono strict_mono_on_def)
qed

end

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