src/HOL/Finite_Set.thy

added lemmas Finite_Set.bex_min_element and Finite_Set.bex_max_element

(*  Title:      HOL/Finite_Set.thy
    Author:     Tobias Nipkow
    Author:     Lawrence C Paulson
    Author:     Markus Wenzel
    Author:     Jeremy Avigad
    Author:     Andrei Popescu
*)

section \<open>Finite sets\<close>

theory Finite_Set
  imports Product_Type Sum_Type Fields Relation
begin

subsection \<open>Predicate for finite sets\<close>

context notes [[inductive_internals]]
begin

inductive finite :: "'a set \<Rightarrow> bool"
  where
    emptyI [simp, intro!]: "finite {}"
  | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"

end

simproc_setup finite_Collect ("finite (Collect P)") = \<open>K Set_Comprehension_Pointfree.simproc\<close>

declare [[simproc del: finite_Collect]]

lemma finite_induct [case_names empty insert, induct set: finite]:
  \<comment> \<open>Discharging \<open>x \<notin> F\<close> entails extra work.\<close>
  assumes "finite F"
  assumes "P {}"
    and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
  shows "P F"
  using \<open>finite F\<close>
proof induct
  show "P {}" by fact
next
  fix x F
  assume F: "finite F" and P: "P F"
  show "P (insert x F)"
  proof cases
    assume "x \<in> F"
    then have "insert x F = F" by (rule insert_absorb)
    with P show ?thesis by (simp only:)
  next
    assume "x \<notin> F"
    from F this P show ?thesis by (rule insert)
  qed
qed

lemma infinite_finite_induct [case_names infinite empty insert]:
  assumes infinite: "\<And>A. \<not> finite A \<Longrightarrow> P A"
    and empty: "P {}"
    and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
  shows "P A"
proof (cases "finite A")
  case False
  with infinite show ?thesis .
next
  case True
  then show ?thesis by (induct A) (fact empty insert)+
qed


subsubsection \<open>Choice principles\<close>

lemma ex_new_if_finite: \<comment> \<open>does not depend on def of finite at all\<close>
  assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
  shows "\<exists>a::'a. a \<notin> A"
proof -
  from assms have "A \<noteq> UNIV" by blast
  then show ?thesis by blast
qed

text \<open>A finite choice principle. Does not need the SOME choice operator.\<close>

lemma finite_set_choice: "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
proof (induct rule: finite_induct)
  case empty
  then show ?case by simp
next
  case (insert a A)
  then obtain f b where f: "\<forall>x\<in>A. P x (f x)" and ab: "P a b"
    by auto
  show ?case (is "\<exists>f. ?P f")
  proof
    show "?P (\<lambda>x. if x = a then b else f x)"
      using f ab by auto
  qed
qed


subsubsection \<open>Finite sets are the images of initial segments of natural numbers\<close>

lemma finite_imp_nat_seg_image_inj_on:
  assumes "finite A"
  shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
  using assms
proof induct
  case empty
  show ?case
  proof
    show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}"
      by simp
  qed
next
  case (insert a A)
  have notinA: "a \<notin> A" by fact
  from insert.hyps obtain n f where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}"
    by blast
  then have "insert a A = f(n:=a) ` {i. i < Suc n}" and "inj_on (f(n:=a)) {i. i < Suc n}"
    using notinA by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
  then show ?case by blast
qed

lemma nat_seg_image_imp_finite: "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
proof (induct n arbitrary: A)
  case 0
  then show ?case by simp
next
  case (Suc n)
  let ?B = "f ` {i. i < n}"
  have finB: "finite ?B" by (rule Suc.hyps[OF refl])
  show ?case
  proof (cases "\<exists>k<n. f n = f k")
    case True
    then have "A = ?B"
      using Suc.prems by (auto simp:less_Suc_eq)
    then show ?thesis
      using finB by simp
  next
    case False
    then have "A = insert (f n) ?B"
      using Suc.prems by (auto simp:less_Suc_eq)
    then show ?thesis using finB by simp
  qed
qed

lemma finite_conv_nat_seg_image: "finite A \<longleftrightarrow> (\<exists>n f. A = f ` {i::nat. i < n})"
  by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)

lemma finite_imp_inj_to_nat_seg:
  assumes "finite A"
  shows "\<exists>f n. f ` A = {i::nat. i < n} \<and> inj_on f A"
proof -
  from finite_imp_nat_seg_image_inj_on [OF \<open>finite A\<close>]
  obtain f and n :: nat where bij: "bij_betw f {i. i<n} A"
    by (auto simp: bij_betw_def)
  let ?f = "the_inv_into {i. i<n} f"
  have "inj_on ?f A \<and> ?f ` A = {i. i<n}"
    by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
  then show ?thesis by blast
qed

lemma finite_Collect_less_nat [iff]: "finite {n::nat. n < k}"
  by (fastforce simp: finite_conv_nat_seg_image)

lemma finite_Collect_le_nat [iff]: "finite {n::nat. n \<le> k}"
  by (simp add: le_eq_less_or_eq Collect_disj_eq)


subsection \<open>Finiteness and common set operations\<close>

lemma rev_finite_subset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
proof (induct arbitrary: A rule: finite_induct)
  case empty
  then show ?case by simp
next
  case (insert x F A)
  have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})"
    by fact+
  show "finite A"
  proof cases
    assume x: "x \<in> A"
    with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
    with r have "finite (A - {x})" .
    then have "finite (insert x (A - {x}))" ..
    also have "insert x (A - {x}) = A"
      using x by (rule insert_Diff)
    finally show ?thesis .
  next
    show ?thesis when "A \<subseteq> F"
      using that by fact
    assume "x \<notin> A"
    with A show "A \<subseteq> F"
      by (simp add: subset_insert_iff)
  qed
qed

lemma finite_subset: "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
  by (rule rev_finite_subset)

simproc_setup finite ("finite A") = \<open>fn _ =>
let
  val finite_subset = @{thm finite_subset}
  val Eq_TrueI = @{thm Eq_TrueI}

  fun is_subset A th = case Thm.prop_of th of
        (_ $ (Const (\<^const_name>\<open>less_eq\<close>, Type (\<^type_name>\<open>fun\<close>, [Type (\<^type_name>\<open>set\<close>, _), _])) $ A' $ B))
        => if A aconv A' then SOME(B,th) else NONE
      | _ => NONE;

  fun is_finite th = case Thm.prop_of th of
        (_ $ (Const (\<^const_name>\<open>finite\<close>, _) $ A)) => SOME(A,th)
      |  _ => NONE;

  fun comb (A,sub_th) (A',fin_th) ths = if A aconv A' then (sub_th,fin_th) :: ths else ths

  fun proc ss ct =
    (let
       val _ $ A = Thm.term_of ct
       val prems = Simplifier.prems_of ss
       val fins = map_filter is_finite prems
       val subsets = map_filter (is_subset A) prems
     in case fold_product comb subsets fins [] of
          (sub_th,fin_th) :: _ => SOME((fin_th RS (sub_th RS finite_subset)) RS Eq_TrueI)
        | _ => NONE
     end)
in proc end
\<close>

(* Needs to be used with care *)
declare [[simproc del: finite]]

lemma finite_UnI:
  assumes "finite F" and "finite G"
  shows "finite (F \<union> G)"
  using assms by induct simp_all

lemma finite_Un [iff]: "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
  by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])

lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
proof -
  have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
  then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
  then show ?thesis by simp
qed

lemma finite_Int [simp, intro]: "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
  by (blast intro: finite_subset)

lemma finite_Collect_conjI [simp, intro]:
  "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
  by (simp add: Collect_conj_eq)

lemma finite_Collect_disjI [simp]:
  "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
  by (simp add: Collect_disj_eq)

lemma finite_Diff [simp, intro]: "finite A \<Longrightarrow> finite (A - B)"
  by (rule finite_subset, rule Diff_subset)

lemma finite_Diff2 [simp]:
  assumes "finite B"
  shows "finite (A - B) \<longleftrightarrow> finite A"
proof -
  have "finite A \<longleftrightarrow> finite ((A - B) \<union> (A \<inter> B))"
    by (simp add: Un_Diff_Int)
  also have "\<dots> \<longleftrightarrow> finite (A - B)"
    using \<open>finite B\<close> by simp
  finally show ?thesis ..
qed

lemma finite_Diff_insert [iff]: "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
proof -
  have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
  moreover have "A - insert a B = A - B - {a}" by auto
  ultimately show ?thesis by simp
qed

lemma finite_compl [simp]:
  "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
  by (simp add: Compl_eq_Diff_UNIV)

lemma finite_Collect_not [simp]:
  "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
  by (simp add: Collect_neg_eq)

lemma finite_Union [simp, intro]:
  "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite (\<Union>A)"
  by (induct rule: finite_induct) simp_all

lemma finite_UN_I [intro]:
  "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
  by (induct rule: finite_induct) simp_all

lemma finite_UN [simp]: "finite A \<Longrightarrow> finite (\<Union>(B ` A)) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
  by (blast intro: finite_subset)

lemma finite_Inter [intro]: "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
  by (blast intro: Inter_lower finite_subset)

lemma finite_INT [intro]: "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
  by (blast intro: INT_lower finite_subset)

lemma finite_imageI [simp, intro]: "finite F \<Longrightarrow> finite (h ` F)"
  by (induct rule: finite_induct) simp_all

lemma finite_image_set [simp]: "finite {x. P x} \<Longrightarrow> finite {f x |x. P x}"
  by (simp add: image_Collect [symmetric])

lemma finite_image_set2:
  "finite {x. P x} \<Longrightarrow> finite {y. Q y} \<Longrightarrow> finite {f x y |x y. P x \<and> Q y}"
  by (rule finite_subset [where B = "\<Union>x \<in> {x. P x}. \<Union>y \<in> {y. Q y}. {f x y}"]) auto

lemma finite_imageD:
  assumes "finite (f ` A)" and "inj_on f A"
  shows "finite A"
  using assms
proof (induct "f ` A" arbitrary: A)
  case empty
  then show ?case by simp
next
  case (insert x B)
  then have B_A: "insert x B = f ` A"
    by simp
  then obtain y where "x = f y" and "y \<in> A"
    by blast
  from B_A \<open>x \<notin> B\<close> have "B = f ` A - {x}"
    by blast
  with B_A \<open>x \<notin> B\<close> \<open>x = f y\<close> \<open>inj_on f A\<close> \<open>y \<in> A\<close> have "B = f ` (A - {y})"
    by (simp add: inj_on_image_set_diff)
  moreover from \<open>inj_on f A\<close> have "inj_on f (A - {y})"
    by (rule inj_on_diff)
  ultimately have "finite (A - {y})"
    by (rule insert.hyps)
  then show "finite A"
    by simp
qed

lemma finite_image_iff: "inj_on f A \<Longrightarrow> finite (f ` A) \<longleftrightarrow> finite A"
  using finite_imageD by blast

lemma finite_surj: "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
  by (erule finite_subset) (rule finite_imageI)

lemma finite_range_imageI: "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
  by (drule finite_imageI) (simp add: range_composition)

lemma finite_subset_image:
  assumes "finite B"
  shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
  using assms
proof induct
  case empty
  then show ?case by simp
next
  case insert
  then show ?case
    by (clarsimp simp del: image_insert simp add: image_insert [symmetric]) blast
qed

lemma all_subset_image: "(\<forall>B. B \<subseteq> f ` A \<longrightarrow> P B) \<longleftrightarrow> (\<forall>B. B \<subseteq> A \<longrightarrow> P(f ` B))"
  by (safe elim!: subset_imageE) (use image_mono in \<open>blast+\<close>) (* slow *)

lemma all_finite_subset_image:
  "(\<forall>B. finite B \<and> B \<subseteq> f ` A \<longrightarrow> P B) \<longleftrightarrow> (\<forall>B. finite B \<and> B \<subseteq> A \<longrightarrow> P (f ` B))"
proof safe
  fix B :: "'a set"
  assume B: "finite B" "B \<subseteq> f ` A" and P: "\<forall>B. finite B \<and> B \<subseteq> A \<longrightarrow> P (f ` B)"
  show "P B"
    using finite_subset_image [OF B] P by blast
qed blast

lemma ex_finite_subset_image:
  "(\<exists>B. finite B \<and> B \<subseteq> f ` A \<and> P B) \<longleftrightarrow> (\<exists>B. finite B \<and> B \<subseteq> A \<and> P (f ` B))"
proof safe
  fix B :: "'a set"
  assume B: "finite B" "B \<subseteq> f ` A" and "P B"
  show "\<exists>B. finite B \<and> B \<subseteq> A \<and> P (f ` B)"
    using finite_subset_image [OF B] \<open>P B\<close> by blast
qed blast

lemma finite_vimage_IntI: "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
proof (induct rule: finite_induct)
  case (insert x F)
  then show ?case
    by (simp add: vimage_insert [of h x F] finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
qed simp

lemma finite_finite_vimage_IntI:
  assumes "finite F"
    and "\<And>y. y \<in> F \<Longrightarrow> finite ((h -` {y}) \<inter> A)"
  shows "finite (h -` F \<inter> A)"
proof -
  have *: "h -` F \<inter> A = (\<Union> y\<in>F. (h -` {y}) \<inter> A)"
    by blast
  show ?thesis
    by (simp only: * assms finite_UN_I)
qed

lemma finite_vimageI: "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
  using finite_vimage_IntI[of F h UNIV] by auto

lemma finite_vimageD': "finite (f -` A) \<Longrightarrow> A \<subseteq> range f \<Longrightarrow> finite A"
  by (auto simp add: subset_image_iff intro: finite_subset[rotated])

lemma finite_vimageD: "finite (h -` F) \<Longrightarrow> surj h \<Longrightarrow> finite F"
  by (auto dest: finite_vimageD')

lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
  unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)

lemma finite_inverse_image_gen:
  assumes "finite A" "inj_on f D"
  shows "finite {j\<in>D. f j \<in> A}"
  using finite_vimage_IntI [OF assms]
  by (simp add: Collect_conj_eq inf_commute vimage_def)

lemma finite_inverse_image:
  assumes "finite A" "inj f"
  shows "finite {j. f j \<in> A}"
  using finite_inverse_image_gen [OF assms] by simp

lemma finite_Collect_bex [simp]:
  assumes "finite A"
  shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
proof -
  have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
  with assms show ?thesis by simp
qed

lemma finite_Collect_bounded_ex [simp]:
  assumes "finite {y. P y}"
  shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
proof -
  have "{x. \<exists>y. P y \<and> Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})"
    by auto
  with assms show ?thesis
    by simp
qed

lemma finite_Plus: "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
  by (simp add: Plus_def)

lemma finite_PlusD:
  fixes A :: "'a set" and B :: "'b set"
  assumes fin: "finite (A <+> B)"
  shows "finite A" "finite B"
proof -
  have "Inl ` A \<subseteq> A <+> B"
    by auto
  then have "finite (Inl ` A :: ('a + 'b) set)"
    using fin by (rule finite_subset)
  then show "finite A"
    by (rule finite_imageD) (auto intro: inj_onI)
next
  have "Inr ` B \<subseteq> A <+> B"
    by auto
  then have "finite (Inr ` B :: ('a + 'b) set)"
    using fin by (rule finite_subset)
  then show "finite B"
    by (rule finite_imageD) (auto intro: inj_onI)
qed

lemma finite_Plus_iff [simp]: "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
  by (auto intro: finite_PlusD finite_Plus)

lemma finite_Plus_UNIV_iff [simp]:
  "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
  by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)

lemma finite_SigmaI [simp, intro]:
  "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (SIGMA a:A. B a)"
  unfolding Sigma_def by blast

lemma finite_SigmaI2:
  assumes "finite {x\<in>A. B x \<noteq> {}}"
  and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)"
  shows "finite (Sigma A B)"
proof -
  from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)"
    by auto
  also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B"
    by auto
  finally show ?thesis .
qed

lemma finite_cartesian_product: "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
  by (rule finite_SigmaI)

lemma finite_Prod_UNIV:
  "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
  by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)

lemma finite_cartesian_productD1:
  assumes "finite (A \<times> B)" and "B \<noteq> {}"
  shows "finite A"
proof -
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
    by (auto simp add: finite_conv_nat_seg_image)
  then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}"
    by simp
  with \<open>B \<noteq> {}\<close> have "A = (fst \<circ> f) ` {i::nat. i < n}"
    by (simp add: image_comp)
  then have "\<exists>n f. A = f ` {i::nat. i < n}"
    by blast
  then show ?thesis
    by (auto simp add: finite_conv_nat_seg_image)
qed

lemma finite_cartesian_productD2:
  assumes "finite (A \<times> B)" and "A \<noteq> {}"
  shows "finite B"
proof -
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
    by (auto simp add: finite_conv_nat_seg_image)
  then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}"
    by simp
  with \<open>A \<noteq> {}\<close> have "B = (snd \<circ> f) ` {i::nat. i < n}"
    by (simp add: image_comp)
  then have "\<exists>n f. B = f ` {i::nat. i < n}"
    by blast
  then show ?thesis
    by (auto simp add: finite_conv_nat_seg_image)
qed

lemma finite_cartesian_product_iff:
  "finite (A \<times> B) \<longleftrightarrow> (A = {} \<or> B = {} \<or> (finite A \<and> finite B))"
  by (auto dest: finite_cartesian_productD1 finite_cartesian_productD2 finite_cartesian_product)

lemma finite_prod:
  "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
  using finite_cartesian_product_iff[of UNIV UNIV] by simp

lemma finite_Pow_iff [iff]: "finite (Pow A) \<longleftrightarrow> finite A"
proof
  assume "finite (Pow A)"
  then have "finite ((\<lambda>x. {x}) ` A)"
    by (blast intro: finite_subset)  (* somewhat slow *)
  then show "finite A"
    by (rule finite_imageD [unfolded inj_on_def]) simp
next
  assume "finite A"
  then show "finite (Pow A)"
    by induct (simp_all add: Pow_insert)
qed

corollary finite_Collect_subsets [simp, intro]: "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
  by (simp add: Pow_def [symmetric])

lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
  by (simp only: finite_Pow_iff Pow_UNIV[symmetric])

lemma finite_UnionD: "finite (\<Union>A) \<Longrightarrow> finite A"
  by (blast intro: finite_subset [OF subset_Pow_Union])

lemma finite_bind:
  assumes "finite S"
  assumes "\<forall>x \<in> S. finite (f x)"
  shows "finite (Set.bind S f)"
using assms by (simp add: bind_UNION)

lemma finite_filter [simp]: "finite S \<Longrightarrow> finite (Set.filter P S)"
unfolding Set.filter_def by simp

lemma finite_set_of_finite_funs:
  assumes "finite A" "finite B"
  shows "finite {f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}" (is "finite ?S")
proof -
  let ?F = "\<lambda>f. {(a,b). a \<in> A \<and> b = f a}"
  have "?F ` ?S \<subseteq> Pow(A \<times> B)"
    by auto
  from finite_subset[OF this] assms have 1: "finite (?F ` ?S)"
    by simp
  have 2: "inj_on ?F ?S"
    by (fastforce simp add: inj_on_def set_eq_iff fun_eq_iff)  (* somewhat slow *)
  show ?thesis
    by (rule finite_imageD [OF 1 2])
qed

lemma not_finite_existsD:
  assumes "\<not> finite {a. P a}"
  shows "\<exists>a. P a"
proof (rule classical)
  assume "\<not> ?thesis"
  with assms show ?thesis by auto
qed

lemma finite_converse [iff]: "finite (r\<inverse>) \<longleftrightarrow> finite r"
  unfolding converse_def conversep_iff
  using [[simproc add: finite_Collect]]
  by (auto elim: finite_imageD simp: inj_on_def)

lemma finite_Domain: "finite r \<Longrightarrow> finite (Domain r)"
  by (induct set: finite) auto

lemma finite_Range: "finite r \<Longrightarrow> finite (Range r)"
  by (induct set: finite) auto

lemma finite_Field: "finite r \<Longrightarrow> finite (Field r)"
  by (simp add: Field_def finite_Domain finite_Range)

lemma finite_Image[simp]: "finite R \<Longrightarrow> finite (R `` A)"
  by(rule finite_subset[OF _ finite_Range]) auto


subsection \<open>Further induction rules on finite sets\<close>

lemma finite_ne_induct [case_names singleton insert, consumes 2]:
  assumes "finite F" and "F \<noteq> {}"
  assumes "\<And>x. P {x}"
    and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
  shows "P F"
  using assms
proof induct
  case empty
  then show ?case by simp
next
  case (insert x F)
  then show ?case by cases auto
qed

lemma finite_subset_induct [consumes 2, case_names empty insert]:
  assumes "finite F" and "F \<subseteq> A"
    and empty: "P {}"
    and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
  shows "P F"
  using \<open>finite F\<close> \<open>F \<subseteq> A\<close>
proof induct
  show "P {}" by fact
next
  fix x F
  assume "finite F" and "x \<notin> F" and P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
  show "P (insert x F)"
  proof (rule insert)
    from i show "x \<in> A" by blast
    from i have "F \<subseteq> A" by blast
    with P show "P F" .
    show "finite F" by fact
    show "x \<notin> F" by fact
  qed
qed

lemma finite_empty_induct:
  assumes "finite A"
    and "P A"
    and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
  shows "P {}"
proof -
  have "P (A - B)" if "B \<subseteq> A" for B :: "'a set"
  proof -
    from \<open>finite A\<close> that have "finite B"
      by (rule rev_finite_subset)
    from this \<open>B \<subseteq> A\<close> show "P (A - B)"
    proof induct
      case empty
      from \<open>P A\<close> show ?case by simp
    next
      case (insert b B)
      have "P (A - B - {b})"
      proof (rule remove)
        from \<open>finite A\<close> show "finite (A - B)"
          by induct auto
        from insert show "b \<in> A - B"
          by simp
        from insert show "P (A - B)"
          by simp
      qed
      also have "A - B - {b} = A - insert b B"
        by (rule Diff_insert [symmetric])
      finally show ?case .
    qed
  qed
  then have "P (A - A)" by blast
  then show ?thesis by simp
qed

lemma finite_update_induct [consumes 1, case_names const update]:
  assumes finite: "finite {a. f a \<noteq> c}"
    and const: "P (\<lambda>a. c)"
    and update: "\<And>a b f. finite {a. f a \<noteq> c} \<Longrightarrow> f a = c \<Longrightarrow> b \<noteq> c \<Longrightarrow> P f \<Longrightarrow> P (f(a := b))"
  shows "P f"
  using finite
proof (induct "{a. f a \<noteq> c}" arbitrary: f)
  case empty
  with const show ?case by simp
next
  case (insert a A)
  then have "A = {a'. (f(a := c)) a' \<noteq> c}" and "f a \<noteq> c"
    by auto
  with \<open>finite A\<close> have "finite {a'. (f(a := c)) a' \<noteq> c}"
    by simp
  have "(f(a := c)) a = c"
    by simp
  from insert \<open>A = {a'. (f(a := c)) a' \<noteq> c}\<close> have "P (f(a := c))"
    by simp
  with \<open>finite {a'. (f(a := c)) a' \<noteq> c}\<close> \<open>(f(a := c)) a = c\<close> \<open>f a \<noteq> c\<close>
  have "P ((f(a := c))(a := f a))"
    by (rule update)
  then show ?case by simp
qed

lemma finite_subset_induct' [consumes 2, case_names empty insert]:
  assumes "finite F" and "F \<subseteq> A"
    and empty: "P {}"
    and insert: "\<And>a F. \<lbrakk>finite F; a \<in> A; F \<subseteq> A; a \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert a F)"
  shows "P F"
  using assms(1,2)
proof induct
  show "P {}" by fact
next
  fix x F
  assume "finite F" and "x \<notin> F" and
    P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
  show "P (insert x F)"
  proof (rule insert)
    from i show "x \<in> A" by blast
    from i have "F \<subseteq> A" by blast
    with P show "P F" .
    show "finite F" by fact
    show "x \<notin> F" by fact
    show "F \<subseteq> A" by fact
  qed
qed


subsection \<open>Class \<open>finite\<close>\<close>

class finite =
  assumes finite_UNIV: "finite (UNIV :: 'a set)"
begin

lemma finite [simp]: "finite (A :: 'a set)"
  by (rule subset_UNIV finite_UNIV finite_subset)+

lemma finite_code [code]: "finite (A :: 'a set) \<longleftrightarrow> True"
  by simp

end

instance prod :: (finite, finite) finite
  by standard (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)

lemma inj_graph: "inj (\<lambda>f. {(x, y). y = f x})"
  by (rule inj_onI) (auto simp add: set_eq_iff fun_eq_iff)

instance "fun" :: (finite, finite) finite
proof
  show "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
  proof (rule finite_imageD)
    let ?graph = "\<lambda>f::'a \<Rightarrow> 'b. {(x, y). y = f x}"
    have "range ?graph \<subseteq> Pow UNIV"
      by simp
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
      by (simp only: finite_Pow_iff finite)
    ultimately show "finite (range ?graph)"
      by (rule finite_subset)
    show "inj ?graph"
      by (rule inj_graph)
  qed
qed

instance bool :: finite
  by standard (simp add: UNIV_bool)

instance set :: (finite) finite
  by standard (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)

instance unit :: finite
  by standard (simp add: UNIV_unit)

instance sum :: (finite, finite) finite
  by standard (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)


subsection \<open>A basic fold functional for finite sets\<close>

text \<open>
  The intended behaviour is \<open>fold f z {x\<^sub>1, \<dots>, x\<^sub>n} = f x\<^sub>1 (\<dots> (f x\<^sub>n z)\<dots>)\<close>
  if \<open>f\<close> is ``left-commutative''.
  The commutativity requirement is relativised to the carrier set \<open>S\<close>:
\<close>

locale comp_fun_commute_on =
  fixes S :: "'a set"
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
  assumes comp_fun_commute_on: "x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
begin

lemma fun_left_comm: "x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f y (f x z) = f x (f y z)"
  using comp_fun_commute_on by (simp add: fun_eq_iff)

lemma commute_left_comp: "x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
  by (simp add: o_assoc comp_fun_commute_on)

end

inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
  for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b
  where
    emptyI [intro]: "fold_graph f z {} z"
  | insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y \<Longrightarrow> fold_graph f z (insert x A) (f x y)"

inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"

lemma fold_graph_closed_lemma:
  "fold_graph f z A x \<and> x \<in> B"
  if "fold_graph g z A x"
    "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> f a b = g a b"
    "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> g a b \<in> B"
    "z \<in> B"
  using that(1-3)
proof (induction rule: fold_graph.induct)
  case (insertI x A y)
  have "fold_graph f z A y" "y \<in> B"
    unfolding atomize_conj
    by (rule insertI.IH) (auto intro: insertI.prems)
  then have "g x y \<in> B" and f_eq: "f x y = g x y"
    by (auto simp: insertI.prems)
  moreover have "fold_graph f z (insert x A) (f x y)"
    by (rule fold_graph.insertI; fact)
  ultimately
  show ?case
    by (simp add: f_eq)
qed (auto intro!: that)

lemma fold_graph_closed_eq:
  "fold_graph f z A = fold_graph g z A"
  if "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> f a b = g a b"
     "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> g a b \<in> B"
     "z \<in> B"
  using fold_graph_closed_lemma[of f z A _ B g] fold_graph_closed_lemma[of g z A _ B f] that
  by auto

definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
  where "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"

lemma fold_closed_eq: "fold f z A = fold g z A"
  if "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> f a b = g a b"
     "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> g a b \<in> B"
     "z \<in> B"
  unfolding Finite_Set.fold_def
  by (subst fold_graph_closed_eq[where B=B and g=g]) (auto simp: that)

text \<open>
  A tempting alternative for the definition is
  \<^term>\<open>if finite A then THE y. fold_graph f z A y else e\<close>.
  It allows the removal of finiteness assumptions from the theorems
  \<open>fold_comm\<close>, \<open>fold_reindex\<close> and \<open>fold_distrib\<close>.
  The proofs become ugly. It is not worth the effort. (???)
\<close>

lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
  by (induct rule: finite_induct) auto


subsubsection \<open>From \<^const>\<open>fold_graph\<close> to \<^term>\<open>fold\<close>\<close>

context comp_fun_commute_on
begin

lemma fold_graph_finite:
  assumes "fold_graph f z A y"
  shows "finite A"
  using assms by induct simp_all

lemma fold_graph_insertE_aux:
  assumes "A \<subseteq> S"
  assumes "fold_graph f z A y" "a \<in> A"
  shows "\<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'"
  using assms(2-,1)
proof (induct set: fold_graph)
  case emptyI
  then show ?case by simp
next
  case (insertI x A y)
  show ?case
  proof (cases "x = a")
    case True
    with insertI show ?thesis by auto
  next
    case False
    then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
      using insertI by auto
    from insertI have "x \<in> S" "a \<in> S" by auto
    then have "f x y = f a (f x y')"
      unfolding y by (intro fun_left_comm; simp)
    moreover have "fold_graph f z (insert x A - {a}) (f x y')"
      using y' and \<open>x \<noteq> a\<close> and \<open>x \<notin> A\<close>
      by (simp add: insert_Diff_if fold_graph.insertI)
    ultimately show ?thesis
      by fast
  qed
qed

lemma fold_graph_insertE:
  assumes "insert x A \<subseteq> S"
  assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
  obtains y where "v = f x y" and "fold_graph f z A y"
  using assms by (auto dest: fold_graph_insertE_aux[OF \<open>insert x A \<subseteq> S\<close> _ insertI1])

lemma fold_graph_determ:
  assumes "A \<subseteq> S"
  assumes "fold_graph f z A x" "fold_graph f z A y"
  shows "y = x"
  using assms(2-,1)
proof (induct arbitrary: y set: fold_graph)
  case emptyI
  then show ?case by fast
next
  case (insertI x A y v)
  from \<open>insert x A \<subseteq> S\<close> and \<open>fold_graph f z (insert x A) v\<close> and \<open>x \<notin> A\<close>
  obtain y' where "v = f x y'" and "fold_graph f z A y'"
    by (rule fold_graph_insertE)
  from \<open>fold_graph f z A y'\<close> insertI have "y' = y"
    by simp
  with \<open>v = f x y'\<close> show "v = f x y"
    by simp
qed

lemma fold_equality: "A \<subseteq> S \<Longrightarrow> fold_graph f z A y \<Longrightarrow> fold f z A = y"
  by (cases "finite A") (auto simp add: fold_def intro: fold_graph_determ dest: fold_graph_finite)

lemma fold_graph_fold:
  assumes "A \<subseteq> S"
  assumes "finite A"
  shows "fold_graph f z A (fold f z A)"
proof -
  from \<open>finite A\<close> have "\<exists>x. fold_graph f z A x"
    by (rule finite_imp_fold_graph)
  moreover note fold_graph_determ[OF \<open>A \<subseteq> S\<close>]
  ultimately have "\<exists>!x. fold_graph f z A x"
    by (rule ex_ex1I)
  then have "fold_graph f z A (The (fold_graph f z A))"
    by (rule theI')
  with assms show ?thesis
    by (simp add: fold_def)
qed

text \<open>The base case for \<open>fold\<close>:\<close>

lemma (in -) fold_infinite [simp]: "\<not> finite A \<Longrightarrow> fold f z A = z"
  by (auto simp: fold_def)

lemma (in -) fold_empty [simp]: "fold f z {} = z"
  by (auto simp: fold_def)

text \<open>The various recursion equations for \<^const>\<open>fold\<close>:\<close>

lemma fold_insert [simp]:
  assumes "insert x A \<subseteq> S"
  assumes "finite A" and "x \<notin> A"
  shows "fold f z (insert x A) = f x (fold f z A)"
proof (rule fold_equality[OF \<open>insert x A \<subseteq> S\<close>])
  fix z
  from \<open>insert x A \<subseteq> S\<close> \<open>finite A\<close> have "fold_graph f z A (fold f z A)"
    by (blast intro: fold_graph_fold)
  with \<open>x \<notin> A\<close> have "fold_graph f z (insert x A) (f x (fold f z A))"
    by (rule fold_graph.insertI)
  then show "fold_graph f z (insert x A) (f x (fold f z A))"
    by simp
qed

declare (in -) empty_fold_graphE [rule del] fold_graph.intros [rule del]
  \<comment> \<open>No more proofs involve these.\<close>

lemma fold_fun_left_comm:
  assumes "insert x A \<subseteq> S" "finite A" 
  shows "f x (fold f z A) = fold f (f x z) A"
  using assms(2,1)
proof (induct rule: finite_induct)
  case empty
  then show ?case by simp
next
  case (insert y F)
  then have "fold f (f x z) (insert y F) = f y (fold f (f x z) F)"
    by simp
  also have "\<dots> = f x (f y (fold f z F))"
    using insert by (simp add: fun_left_comm[where ?y=x])
  also have "\<dots> = f x (fold f z (insert y F))"
  proof -
    from insert have "insert y F \<subseteq> S" by simp
    from fold_insert[OF this] insert show ?thesis by simp
  qed
  finally show ?case ..
qed

lemma fold_insert2:
  "insert x A \<subseteq> S \<Longrightarrow> finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A)  = fold f (f x z) A"
  by (simp add: fold_fun_left_comm)

lemma fold_rec:
  assumes "A \<subseteq> S"
  assumes "finite A" and "x \<in> A"
  shows "fold f z A = f x (fold f z (A - {x}))"
proof -
  have A: "A = insert x (A - {x})"
    using \<open>x \<in> A\<close> by blast
  then have "fold f z A = fold f z (insert x (A - {x}))"
    by simp
  also have "\<dots> = f x (fold f z (A - {x}))"
    by (rule fold_insert) (use assms in \<open>auto\<close>)
  finally show ?thesis .
qed

lemma fold_insert_remove:
  assumes "insert x A \<subseteq> S"
  assumes "finite A"
  shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
proof -
  from \<open>finite A\<close> have "finite (insert x A)"
    by auto
  moreover have "x \<in> insert x A"
    by auto
  ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
    using \<open>insert x A \<subseteq> S\<close> by (blast intro: fold_rec)
  then show ?thesis
    by simp
qed

lemma fold_set_union_disj:
  assumes "A \<subseteq> S" "B \<subseteq> S"
  assumes "finite A" "finite B" "A \<inter> B = {}"
  shows "Finite_Set.fold f z (A \<union> B) = Finite_Set.fold f (Finite_Set.fold f z A) B"
  using \<open>finite B\<close> assms(1,2,3,5)
proof induct
  case (insert x F)
  have "fold f z (A \<union> insert x F) = f x (fold f (fold f z A) F)"
    using insert by auto
  also have "\<dots> = fold f (fold f z A) (insert x F)"
    using insert by (blast intro: fold_insert[symmetric])
  finally show ?case .
qed simp


end

text \<open>Other properties of \<^const>\<open>fold\<close>:\<close>

lemma fold_graph_image:
  assumes "inj_on g A"
  shows "fold_graph f z (g ` A) = fold_graph (f \<circ> g) z A"
proof
  fix w
  show "fold_graph f z (g ` A) w = fold_graph (f o g) z A w"
  proof
    assume "fold_graph f z (g ` A) w"
    then show "fold_graph (f \<circ> g) z A w"
      using assms
    proof (induct "g ` A" w arbitrary: A)
      case emptyI
      then show ?case by (auto intro: fold_graph.emptyI)
    next
      case (insertI x A r B)
      from \<open>inj_on g B\<close> \<open>x \<notin> A\<close> \<open>insert x A = image g B\<close> obtain x' A'
        where "x' \<notin> A'" and [simp]: "B = insert x' A'" "x = g x'" "A = g ` A'"
        by (rule inj_img_insertE)
      from insertI.prems have "fold_graph (f \<circ> g) z A' r"
        by (auto intro: insertI.hyps)
      with \<open>x' \<notin> A'\<close> have "fold_graph (f \<circ> g) z (insert x' A') ((f \<circ> g) x' r)"
        by (rule fold_graph.insertI)
      then show ?case
        by simp
    qed
  next
    assume "fold_graph (f \<circ> g) z A w"
    then show "fold_graph f z (g ` A) w"
      using assms
    proof induct
      case emptyI
      then show ?case
        by (auto intro: fold_graph.emptyI)
    next
      case (insertI x A r)
      from \<open>x \<notin> A\<close> insertI.prems have "g x \<notin> g ` A"
        by auto
      moreover from insertI have "fold_graph f z (g ` A) r"
        by simp
      ultimately have "fold_graph f z (insert (g x) (g ` A)) (f (g x) r)"
        by (rule fold_graph.insertI)
      then show ?case
        by simp
    qed
  qed
qed

lemma fold_image:
  assumes "inj_on g A"
  shows "fold f z (g ` A) = fold (f \<circ> g) z A"
proof (cases "finite A")
  case False
  with assms show ?thesis
    by (auto dest: finite_imageD simp add: fold_def)
next
  case True
  then show ?thesis
    by (auto simp add: fold_def fold_graph_image[OF assms])
qed

lemma fold_cong:
  assumes "comp_fun_commute_on S f" "comp_fun_commute_on S g"
    and "A \<subseteq> S" "finite A"
    and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
    and "s = t" and "A = B"
  shows "fold f s A = fold g t B"
proof -
  have "fold f s A = fold g s A"
    using \<open>finite A\<close> \<open>A \<subseteq> S\<close> cong
  proof (induct A)
    case empty
    then show ?case by simp
  next
    case insert
    interpret f: comp_fun_commute_on S f by (fact \<open>comp_fun_commute_on S f\<close>)
    interpret g: comp_fun_commute_on S g by (fact \<open>comp_fun_commute_on S g\<close>)
    from insert show ?case by simp
  qed
  with assms show ?thesis by simp
qed


text \<open>A simplified version for idempotent functions:\<close>

locale comp_fun_idem_on = comp_fun_commute_on +
  assumes comp_fun_idem_on: "x \<in> S \<Longrightarrow> f x \<circ> f x = f x"
begin

lemma fun_left_idem: "x \<in> S \<Longrightarrow> f x (f x z) = f x z"
  using comp_fun_idem_on by (simp add: fun_eq_iff)

lemma fold_insert_idem:
  assumes "insert x A \<subseteq> S"
  assumes fin: "finite A"
  shows "fold f z (insert x A)  = f x (fold f z A)"
proof cases
  assume "x \<in> A"
  then obtain B where "A = insert x B" and "x \<notin> B"
    by (rule set_insert)
  then show ?thesis
    using assms by (simp add: comp_fun_idem_on fun_left_idem)
next
  assume "x \<notin> A"
  then show ?thesis
    using assms by auto
qed

declare fold_insert [simp del] fold_insert_idem [simp]

lemma fold_insert_idem2: "insert x A \<subseteq> S \<Longrightarrow> finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
  by (simp add: fold_fun_left_comm)

end


subsubsection \<open>Liftings to \<open>comp_fun_commute_on\<close> etc.\<close>
                   
lemma (in comp_fun_commute_on) comp_comp_fun_commute_on:
  "range g \<subseteq> S \<Longrightarrow> comp_fun_commute_on R (f \<circ> g)"
  by standard (force intro: comp_fun_commute_on)

lemma (in comp_fun_idem_on) comp_comp_fun_idem_on:
  assumes "range g \<subseteq> S"
  shows "comp_fun_idem_on R (f \<circ> g)"
proof
  interpret f_g: comp_fun_commute_on R "f o g"
    by (fact comp_comp_fun_commute_on[OF \<open>range g \<subseteq> S\<close>])
  show "x \<in> R \<Longrightarrow> y \<in> R \<Longrightarrow> (f \<circ> g) y \<circ> (f \<circ> g) x = (f \<circ> g) x \<circ> (f \<circ> g) y" for x y
    by (fact f_g.comp_fun_commute_on)
qed (use \<open>range g \<subseteq> S\<close> in \<open>force intro: comp_fun_idem_on\<close>)

lemma (in comp_fun_commute_on) comp_fun_commute_on_funpow:
  "comp_fun_commute_on S (\<lambda>x. f x ^^ g x)"
proof
  fix x y assume "x \<in> S" "y \<in> S"
  show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
  proof (cases "x = y")
    case True
    then show ?thesis by simp
  next
    case False
    show ?thesis
    proof (induct "g x" arbitrary: g)
      case 0
      then show ?case by simp
    next
      case (Suc n g)
      have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
      proof (induct "g y" arbitrary: g)
        case 0
        then show ?case by simp
      next
        case (Suc n g)
        define h where "h z = g z - 1" for z
        with Suc have "n = h y"
          by simp
        with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
          by auto
        from Suc h_def have "g y = Suc (h y)"
          by simp
        with \<open>x \<in> S\<close> \<open>y \<in> S\<close> show ?case
          by (simp add: comp_assoc hyp) (simp add: o_assoc comp_fun_commute_on)
      qed
      define h where "h z = (if z = x then g x - 1 else g z)" for z
      with Suc have "n = h x"
        by simp
      with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
        by auto
      with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y"
        by simp
      from Suc h_def have "g x = Suc (h x)"
        by simp
      then show ?case
        by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2) (simp add: comp_assoc hyp1)
    qed
  qed
qed


subsubsection \<open>\<^term>\<open>UNIV\<close> as carrier set\<close>

locale comp_fun_commute =
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
  assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
begin

lemma (in -) comp_fun_commute_def': "comp_fun_commute f = comp_fun_commute_on UNIV f"
  unfolding comp_fun_commute_def comp_fun_commute_on_def by blast

text \<open>
  We abuse the \<open>rewrites\<close> functionality of locales to remove trivial assumptions that
  result from instantiating the carrier set to \<^term>\<open>UNIV\<close>.
\<close>
sublocale comp_fun_commute_on UNIV f
  rewrites "\<And>X. (X \<subseteq> UNIV) \<equiv> True"
       and "\<And>x. x \<in> UNIV \<equiv> True"
       and "\<And>P. (True \<Longrightarrow> P) \<equiv> Trueprop P"
       and "\<And>P Q. (True \<Longrightarrow> PROP P \<Longrightarrow> PROP Q) \<equiv> (PROP P \<Longrightarrow> True \<Longrightarrow> PROP Q)"
proof -
  show "comp_fun_commute_on UNIV f"
    by standard  (simp add: comp_fun_commute)
qed simp_all

end

lemma (in comp_fun_commute) comp_comp_fun_commute: "comp_fun_commute (f o g)"
  unfolding comp_fun_commute_def' by (fact comp_comp_fun_commute_on)

lemma (in comp_fun_commute) comp_fun_commute_funpow: "comp_fun_commute (\<lambda>x. f x ^^ g x)"
  unfolding comp_fun_commute_def' by (fact comp_fun_commute_on_funpow)

locale comp_fun_idem = comp_fun_commute +
  assumes comp_fun_idem: "f x o f x = f x"
begin

lemma (in -) comp_fun_idem_def': "comp_fun_idem f = comp_fun_idem_on UNIV f"
  unfolding comp_fun_idem_on_def comp_fun_idem_def comp_fun_commute_def'
  unfolding comp_fun_idem_axioms_def comp_fun_idem_on_axioms_def
  by blast

text \<open>
  Again, we abuse the \<open>rewrites\<close> functionality of locales to remove trivial assumptions that
  result from instantiating the carrier set to \<^term>\<open>UNIV\<close>.
\<close>
sublocale comp_fun_idem_on UNIV f
  rewrites "\<And>X. (X \<subseteq> UNIV) \<equiv> True"
       and "\<And>x. x \<in> UNIV \<equiv> True"
       and "\<And>P. (True \<Longrightarrow> P) \<equiv> Trueprop P"
       and "\<And>P Q. (True \<Longrightarrow> PROP P \<Longrightarrow> PROP Q) \<equiv> (PROP P \<Longrightarrow> True \<Longrightarrow> PROP Q)"
proof -
  show "comp_fun_idem_on UNIV f"
    by standard (simp_all add: comp_fun_idem comp_fun_commute)
qed simp_all

end

lemma (in comp_fun_idem) comp_comp_fun_idem: "comp_fun_idem (f o g)"
  unfolding comp_fun_idem_def' by (fact comp_comp_fun_idem_on)


subsubsection \<open>Expressing set operations via \<^const>\<open>fold\<close>\<close>

lemma comp_fun_commute_const: "comp_fun_commute (\<lambda>_. f)"
  by standard (rule refl)

lemma comp_fun_idem_insert: "comp_fun_idem insert"
  by standard auto

lemma comp_fun_idem_remove: "comp_fun_idem Set.remove"
  by standard auto

lemma (in semilattice_inf) comp_fun_idem_inf: "comp_fun_idem inf"
  by standard (auto simp add: inf_left_commute)

lemma (in semilattice_sup) comp_fun_idem_sup: "comp_fun_idem sup"
  by standard (auto simp add: sup_left_commute)

lemma union_fold_insert:
  assumes "finite A"
  shows "A \<union> B = fold insert B A"
proof -
  interpret comp_fun_idem insert
    by (fact comp_fun_idem_insert)
  from \<open>finite A\<close> show ?thesis
    by (induct A arbitrary: B) simp_all
qed

lemma minus_fold_remove:
  assumes "finite A"
  shows "B - A = fold Set.remove B A"
proof -
  interpret comp_fun_idem Set.remove
    by (fact comp_fun_idem_remove)
  from \<open>finite A\<close> have "fold Set.remove B A = B - A"
    by (induct A arbitrary: B) auto  (* slow *)
  then show ?thesis ..
qed

lemma comp_fun_commute_filter_fold:
  "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
proof -
  interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
  show ?thesis by standard (auto simp: fun_eq_iff)
qed

lemma Set_filter_fold:
  assumes "finite A"
  shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
  using assms
proof -
  interpret commute_insert: comp_fun_commute "(\<lambda>x A'. if P x then Set.insert x A' else A')"
    by (fact comp_fun_commute_filter_fold)
  from \<open>finite A\<close> show ?thesis
    by induct (auto simp add: Set.filter_def)
qed

lemma inter_Set_filter:
  assumes "finite B"
  shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
  using assms
  by induct (auto simp: Set.filter_def)

lemma image_fold_insert:
  assumes "finite A"
  shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
proof -
  interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A"
    by standard auto
  show ?thesis
    using assms by (induct A) auto
qed

lemma Ball_fold:
  assumes "finite A"
  shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
proof -
  interpret comp_fun_commute "\<lambda>k s. s \<and> P k"
    by standard auto
  show ?thesis
    using assms by (induct A) auto
qed

lemma Bex_fold:
  assumes "finite A"
  shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
proof -
  interpret comp_fun_commute "\<lambda>k s. s \<or> P k"
    by standard auto
  show ?thesis
    using assms by (induct A) auto
qed

lemma comp_fun_commute_Pow_fold: "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)"
  by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast

lemma Pow_fold:
  assumes "finite A"
  shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
proof -
  interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A"
    by (rule comp_fun_commute_Pow_fold)
  show ?thesis
    using assms by (induct A) (auto simp: Pow_insert)
qed

lemma fold_union_pair:
  assumes "finite B"
  shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
proof -
  interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)"
    by standard auto
  show ?thesis
    using assms by (induct arbitrary: A) simp_all
qed

lemma comp_fun_commute_product_fold:
  "finite B \<Longrightarrow> comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)"
  by standard (auto simp: fold_union_pair [symmetric])

lemma product_fold:
  assumes "finite A" "finite B"
  shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A"
proof -
  interpret commute_product: comp_fun_commute "(\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)"
    by (fact comp_fun_commute_product_fold[OF \<open>finite B\<close>])
  from assms show ?thesis unfolding Sigma_def
    by (induct A) (simp_all add: fold_union_pair)
qed

context complete_lattice
begin

lemma inf_Inf_fold_inf:
  assumes "finite A"
  shows "inf (Inf A) B = fold inf B A"
proof -
  interpret comp_fun_idem inf
    by (fact comp_fun_idem_inf)
  from \<open>finite A\<close> fold_fun_left_comm show ?thesis
    by (induct A arbitrary: B) (simp_all add: inf_commute fun_eq_iff)
qed

lemma sup_Sup_fold_sup:
  assumes "finite A"
  shows "sup (Sup A) B = fold sup B A"
proof -
  interpret comp_fun_idem sup
    by (fact comp_fun_idem_sup)
  from \<open>finite A\<close> fold_fun_left_comm show ?thesis
    by (induct A arbitrary: B) (simp_all add: sup_commute fun_eq_iff)
qed

lemma Inf_fold_inf: "finite A \<Longrightarrow> Inf A = fold inf top A"
  using inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)

lemma Sup_fold_sup: "finite A \<Longrightarrow> Sup A = fold sup bot A"
  using sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)

lemma inf_INF_fold_inf:
  assumes "finite A"
  shows "inf B (\<Sqinter>(f ` A)) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
proof -
  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
  interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
  from \<open>finite A\<close> have "?fold = ?inf"
    by (induct A arbitrary: B) (simp_all add: inf_left_commute)
  then show ?thesis ..
qed

lemma sup_SUP_fold_sup:
  assumes "finite A"
  shows "sup B (\<Squnion>(f ` A)) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
proof -
  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
  interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
  from \<open>finite A\<close> have "?fold = ?sup"
    by (induct A arbitrary: B) (simp_all add: sup_left_commute)
  then show ?thesis ..
qed

lemma INF_fold_inf: "finite A \<Longrightarrow> \<Sqinter>(f ` A) = fold (inf \<circ> f) top A"
  using inf_INF_fold_inf [of A top] by simp

lemma SUP_fold_sup: "finite A \<Longrightarrow> \<Squnion>(f ` A) = fold (sup \<circ> f) bot A"
  using sup_SUP_fold_sup [of A bot] by simp

lemma finite_Inf_in:
  assumes "finite A" "A\<noteq>{}" and inf: "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> inf x y \<in> A"
  shows "Inf A \<in> A"
proof -
  have "Inf B \<in> A" if "B \<le> A" "B\<noteq>{}" for B
    using finite_subset [OF \<open>B \<subseteq> A\<close> \<open>finite A\<close>] that
  by (induction B) (use inf in \<open>force+\<close>)
  then show ?thesis
    by (simp add: assms)
qed

lemma finite_Sup_in:
  assumes "finite A" "A\<noteq>{}" and sup: "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> sup x y \<in> A"
  shows "Sup A \<in> A"
proof -
  have "Sup B \<in> A" if "B \<le> A" "B\<noteq>{}" for B
    using finite_subset [OF \<open>B \<subseteq> A\<close> \<open>finite A\<close>] that
  by (induction B) (use sup in \<open>force+\<close>)
  then show ?thesis
    by (simp add: assms)
qed

end

subsubsection \<open>Expressing relation operations via \<^const>\<open>fold\<close>\<close>

lemma Id_on_fold:
  assumes "finite A"
  shows "Id_on A = Finite_Set.fold (\<lambda>x. Set.insert (Pair x x)) {} A"
proof -
  interpret comp_fun_commute "\<lambda>x. Set.insert (Pair x x)"
    by standard auto
  from assms show ?thesis
    unfolding Id_on_def by (induct A) simp_all
qed

lemma comp_fun_commute_Image_fold:
  "comp_fun_commute (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)"
proof -
  interpret comp_fun_idem Set.insert
    by (fact comp_fun_idem_insert)
  show ?thesis
    by standard (auto simp: fun_eq_iff comp_fun_commute split: prod.split)
qed

lemma Image_fold:
  assumes "finite R"
  shows "R `` S = Finite_Set.fold (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A) {} R"
proof -
  interpret comp_fun_commute "(\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)"
    by (rule comp_fun_commute_Image_fold)
  have *: "\<And>x F. Set.insert x F `` S = (if fst x \<in> S then Set.insert (snd x) (F `` S) else (F `` S))"
    by (force intro: rev_ImageI)
  show ?thesis
    using assms by (induct R) (auto simp: * )
qed

lemma insert_relcomp_union_fold:
  assumes "finite S"
  shows "{x} O S \<union> X = Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') X S"
proof -
  interpret comp_fun_commute "\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A'"
  proof -
    interpret comp_fun_idem Set.insert
      by (fact comp_fun_idem_insert)
    show "comp_fun_commute (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A')"
      by standard (auto simp add: fun_eq_iff split: prod.split)
  qed
  have *: "{x} O S = {(x', z). x' = fst x \<and> (snd x, z) \<in> S}"
    by (auto simp: relcomp_unfold intro!: exI)
  show ?thesis
    unfolding * using \<open>finite S\<close> by (induct S) (auto split: prod.split)
qed

lemma insert_relcomp_fold:
  assumes "finite S"
  shows "Set.insert x R O S =
    Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') (R O S) S"
proof -
  have "Set.insert x R O S = ({x} O S) \<union> (R O S)"
    by auto
  then show ?thesis
    by (auto simp: insert_relcomp_union_fold [OF assms])
qed

lemma comp_fun_commute_relcomp_fold:
  assumes "finite S"
  shows "comp_fun_commute (\<lambda>(x,y) A.
    Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)"
proof -
  have *: "\<And>a b A.
    Finite_Set.fold (\<lambda>(w, z) A'. if b = w then Set.insert (a, z) A' else A') A S = {(a,b)} O S \<union> A"
    by (auto simp: insert_relcomp_union_fold[OF assms] cong: if_cong)
  show ?thesis
    by standard (auto simp: * )
qed

lemma relcomp_fold:
  assumes "finite R" "finite S"
  shows "R O S = Finite_Set.fold
    (\<lambda>(x,y) A. Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S) {} R"
proof -
  interpret commute_relcomp_fold: comp_fun_commute
    "(\<lambda>(x, y) A. Finite_Set.fold (\<lambda>(w, z) A'. if y = w then insert (x, z) A' else A') A S)"
    by (fact comp_fun_commute_relcomp_fold[OF \<open>finite S\<close>])
  from assms show ?thesis
    by (induct R) (auto simp: comp_fun_commute_relcomp_fold insert_relcomp_fold cong: if_cong)
qed


subsection \<open>Locales as mini-packages for fold operations\<close>

subsubsection \<open>The natural case\<close>

locale folding_on =
  fixes S :: "'a set"
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: "'b"
  assumes comp_fun_commute_on: "x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f y o f x = f x o f y"
begin

interpretation fold?: comp_fun_commute_on S f
  by standard (simp add: comp_fun_commute_on)

definition F :: "'a set \<Rightarrow> 'b"
  where eq_fold: "F A = Finite_Set.fold f z A"

lemma empty [simp]: "F {} = z"
  by (simp add: eq_fold)

lemma infinite [simp]: "\<not> finite A \<Longrightarrow> F A = z"
  by (simp add: eq_fold)

lemma insert [simp]:
  assumes "insert x A \<subseteq> S" and "finite A" and "x \<notin> A"
  shows "F (insert x A) = f x (F A)"
proof -
  from fold_insert assms
  have "Finite_Set.fold f z (insert x A) 
      = f x (Finite_Set.fold f z A)"
    by simp
  with \<open>finite A\<close> show ?thesis by (simp add: eq_fold fun_eq_iff)
qed

lemma remove:
  assumes "A \<subseteq> S" and "finite A" and "x \<in> A"
  shows "F A = f x (F (A - {x}))"
proof -
  from \<open>x \<in> A\<close> obtain B where A: "A = insert x B" and "x \<notin> B"
    by (auto dest: mk_disjoint_insert)
  moreover from \<open>finite A\<close> A have "finite B" by simp
  ultimately show ?thesis
    using \<open>A \<subseteq> S\<close> by auto
qed

lemma insert_remove:
  assumes "insert x A \<subseteq> S" and "finite A"
  shows "F (insert x A) = f x (F (A - {x}))"
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)

end


subsubsection \<open>With idempotency\<close>

locale folding_idem_on = folding_on +
  assumes comp_fun_idem_on: "x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f x \<circ> f x = f x"
begin

declare insert [simp del]

interpretation fold?: comp_fun_idem_on S f
  by standard (simp_all add: comp_fun_commute_on comp_fun_idem_on)

lemma insert_idem [simp]:
  assumes "insert x A \<subseteq> S" and "finite A"
  shows "F (insert x A) = f x (F A)"
proof -
  from fold_insert_idem assms
  have "fold f z (insert x A) = f x (fold f z A)" by simp
  with \<open>finite A\<close> show ?thesis by (simp add: eq_fold fun_eq_iff)
qed

end

subsubsection \<open>\<^term>\<open>UNIV\<close> as the carrier set\<close>

locale folding =
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: "'b"
  assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
begin

lemma (in -) folding_def': "folding f = folding_on UNIV f"
  unfolding folding_def folding_on_def by blast

text \<open>
  Again, we abuse the \<open>rewrites\<close> functionality of locales to remove trivial assumptions that
  result from instantiating the carrier set to \<^term>\<open>UNIV\<close>.
\<close>
sublocale folding_on UNIV f
  rewrites "\<And>X. (X \<subseteq> UNIV) \<equiv> True"
       and "\<And>x. x \<in> UNIV \<equiv> True"
       and "\<And>P. (True \<Longrightarrow> P) \<equiv> Trueprop P"
       and "\<And>P Q. (True \<Longrightarrow> PROP P \<Longrightarrow> PROP Q) \<equiv> (PROP P \<Longrightarrow> True \<Longrightarrow> PROP Q)"
proof -
  show "folding_on UNIV f"
    by standard (simp add: comp_fun_commute)
qed simp_all

end

locale folding_idem = folding +
  assumes comp_fun_idem: "f x \<circ> f x = f x"
begin

lemma (in -) folding_idem_def': "folding_idem f = folding_idem_on UNIV f"
  unfolding folding_idem_def folding_def' folding_idem_on_def
  unfolding folding_idem_axioms_def folding_idem_on_axioms_def
  by blast

text \<open>
  Again, we abuse the \<open>rewrites\<close> functionality of locales to remove trivial assumptions that
  result from instantiating the carrier set to \<^term>\<open>UNIV\<close>.
\<close>
sublocale folding_idem_on UNIV f
  rewrites "\<And>X. (X \<subseteq> UNIV) \<equiv> True"
       and "\<And>x. x \<in> UNIV \<equiv> True"
       and "\<And>P. (True \<Longrightarrow> P) \<equiv> Trueprop P"
       and "\<And>P Q. (True \<Longrightarrow> PROP P \<Longrightarrow> PROP Q) \<equiv> (PROP P \<Longrightarrow> True \<Longrightarrow> PROP Q)"
proof -
  show "folding_idem_on UNIV f"
    by standard (simp add: comp_fun_idem)
qed simp_all

end


subsection \<open>Finite cardinality\<close>

text \<open>
  The traditional definition
  \<^prop>\<open>card A \<equiv> LEAST n. \<exists>f. A = {f i |i. i < n}\<close>
  is ugly to work with.
  But now that we have \<^const>\<open>fold\<close> things are easy:
\<close>

global_interpretation card: folding "\<lambda>_. Suc" 0
  defines card = "folding_on.F (\<lambda>_. Suc) 0"
  by standard (rule refl)

lemma card_insert_disjoint: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
  by (fact card.insert)

lemma card_insert_if: "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
  by auto (simp add: card.insert_remove card.remove)

lemma card_ge_0_finite: "card A > 0 \<Longrightarrow> finite A"
  by (rule ccontr) simp

lemma card_0_eq [simp]: "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
  by (auto dest: mk_disjoint_insert)

lemma finite_UNIV_card_ge_0: "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
  by (rule ccontr) simp

lemma card_eq_0_iff: "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
  by auto

lemma card_range_greater_zero: "finite (range f) \<Longrightarrow> card (range f) > 0"
  by (rule ccontr) (simp add: card_eq_0_iff)

lemma card_gt_0_iff: "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
  by (simp add: neq0_conv [symmetric] card_eq_0_iff)

lemma card_Suc_Diff1:
  assumes "finite A" "x \<in> A" shows "Suc (card (A - {x})) = card A"
proof -
  have "Suc (card (A - {x})) = card (insert x (A - {x}))"
    using assms by (simp add: card.insert_remove)
  also have "... = card A"
    using assms by (simp add: card_insert_if)
  finally show ?thesis .
qed

lemma card_insert_le_m1:
  assumes "n > 0" "card y \<le> n - 1" shows  "card (insert x y) \<le> n"
  using assms
  by (cases "finite y") (auto simp: card_insert_if)

lemma card_Diff_singleton:
  assumes "x \<in> A" shows "card (A - {x}) = card A - 1"
proof (cases "finite A")
  case True
  with assms show ?thesis
    by (simp add: card_Suc_Diff1 [symmetric])
qed auto

lemma card_Diff_singleton_if:
  "card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
  by (simp add: card_Diff_singleton)

lemma card_Diff_insert[simp]:
  assumes "a \<in> A" and "a \<notin> B"
  shows "card (A - insert a B) = card (A - B) - 1"
proof -
  have "A - insert a B = (A - B) - {a}"
    using assms by blast
  then show ?thesis
    using assms by (simp add: card_Diff_singleton)
qed

lemma card_insert_le: "card A \<le> card (insert x A)"
proof (cases "finite A")
  case True
  then show ?thesis   by (simp add: card_insert_if)
qed auto

lemma card_Collect_less_nat[simp]: "card {i::nat. i < n} = n"
  by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)

lemma card_Collect_le_nat[simp]: "card {i::nat. i \<le> n} = Suc n"
  using card_Collect_less_nat[of "Suc n"] by (simp add: less_Suc_eq_le)

lemma card_mono:
  assumes "finite B" and "A \<subseteq> B"
  shows "card A \<le> card B"
proof -
  from assms have "finite A"
    by (auto intro: finite_subset)
  then show ?thesis
    using assms
  proof (induct A arbitrary: B)
    case empty
    then show ?case by simp
  next
    case (insert x A)
    then have "x \<in> B"
      by simp
    from insert have "A \<subseteq> B - {x}" and "finite (B - {x})"
      by auto
    with insert.hyps have "card A \<le> card (B - {x})"
      by auto
    with \<open>finite A\<close> \<open>x \<notin> A\<close> \<open>finite B\<close> \<open>x \<in> B\<close> show ?case
      by simp (simp only: card.remove)
  qed
qed

lemma card_seteq: 
  assumes "finite B" and A: "A \<subseteq> B" "card B \<le> card A"
  shows "A = B"
  using assms
proof (induction arbitrary: A rule: finite_induct)
  case (insert b B)
  then have A: "finite A" "A - {b} \<subseteq> B" 
    by force+
  then have "card B \<le> card (A - {b})"
    using insert by (auto simp add: card_Diff_singleton_if)
  then have "A - {b} = B"
    using A insert.IH by auto
  then show ?case 
    using insert.hyps insert.prems by auto
qed auto

lemma psubset_card_mono: "finite B \<Longrightarrow> A < B \<Longrightarrow> card A < card B"
  using card_seteq [of B A] by (auto simp add: psubset_eq)

lemma card_Un_Int:
  assumes "finite A" "finite B"
  shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
  using assms
proof (induct A)
  case empty
  then show ?case by simp
next
  case insert
  then show ?case
    by (auto simp add: insert_absorb Int_insert_left)
qed

lemma card_Un_disjoint: "finite A \<Longrightarrow> finite B \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> card (A \<union> B) = card A + card B"
  using card_Un_Int [of A B] by simp

lemma card_Un_disjnt: "\<lbrakk>finite A; finite B; disjnt A B\<rbrakk> \<Longrightarrow> card (A \<union> B) = card A + card B"
  by (simp add: card_Un_disjoint disjnt_def)

lemma card_Un_le: "card (A \<union> B) \<le> card A + card B"
proof (cases "finite A \<and> finite B")
  case True
  then show ?thesis
    using le_iff_add card_Un_Int [of A B] by auto
qed auto

lemma card_Diff_subset:
  assumes "finite B"
    and "B \<subseteq> A"
  shows "card (A - B) = card A - card B"
  using assms
proof (cases "finite A")
  case False
  with assms show ?thesis
    by simp
next
  case True
  with assms show ?thesis
    by (induct B arbitrary: A) simp_all
qed

lemma card_Diff_subset_Int:
  assumes "finite (A \<inter> B)"
  shows "card (A - B) = card A - card (A \<inter> B)"
proof -
  have "A - B = A - A \<inter> B" by auto
  with assms show ?thesis
    by (simp add: card_Diff_subset)
qed

lemma diff_card_le_card_Diff:
  assumes "finite B"
  shows "card A - card B \<le> card (A - B)"
proof -
  have "card A - card B \<le> card A - card (A \<inter> B)"
    using card_mono[OF assms Int_lower2, of A] by arith
  also have "\<dots> = card (A - B)"
    using assms by (simp add: card_Diff_subset_Int)
  finally show ?thesis .
qed

lemma card_le_sym_Diff:
  assumes "finite A" "finite B" "card A \<le> card B"
  shows "card(A - B) \<le> card(B - A)"
proof -
  have "card(A - B) = card A - card (A \<inter> B)" using assms(1,2) by(simp add: card_Diff_subset_Int)
  also have "\<dots> \<le> card B - card (A \<inter> B)" using assms(3) by linarith
  also have "\<dots> = card(B - A)" using assms(1,2) by(simp add: card_Diff_subset_Int Int_commute)
  finally show ?thesis .
qed

lemma card_less_sym_Diff:
  assumes "finite A" "finite B" "card A < card B"
  shows "card(A - B) < card(B - A)"
proof -
  have "card(A - B) = card A - card (A \<inter> B)" using assms(1,2) by(simp add: card_Diff_subset_Int)
  also have "\<dots> < card B - card (A \<inter> B)" using assms(1,3) by (simp add: card_mono diff_less_mono)
  also have "\<dots> = card(B - A)" using assms(1,2) by(simp add: card_Diff_subset_Int Int_commute)
  finally show ?thesis .
qed

lemma card_Diff1_less_iff: "card (A - {x}) < card A \<longleftrightarrow> finite A \<and> x \<in> A"
proof (cases "finite A \<and> x \<in> A")
  case True
  then show ?thesis
    by (auto simp: card_gt_0_iff intro: diff_less)
qed auto

lemma card_Diff1_less: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) < card A"
  unfolding card_Diff1_less_iff by auto

lemma card_Diff2_less:
  assumes "finite A" "x \<in> A" "y \<in> A" shows "card (A - {x} - {y}) < card A"
proof (cases "x = y")
  case True
  with assms show ?thesis
    by (simp add: card_Diff1_less del: card_Diff_insert)
next
  case False
  then have "card (A - {x} - {y}) < card (A - {x})" "card (A - {x}) < card A"
    using assms by (intro card_Diff1_less; simp)+
  then show ?thesis
    by (blast intro: less_trans)
qed

lemma card_Diff1_le: "card (A - {x}) \<le> card A"
proof (cases "finite A")
  case True
  then show ?thesis  
    by (cases "x \<in> A") (simp_all add: card_Diff1_less less_imp_le)
qed auto

lemma card_psubset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> card A < card B \<Longrightarrow> A < B"
  by (erule psubsetI) blast

lemma card_le_inj:
  assumes fA: "finite A"
    and fB: "finite B"
    and c: "card A \<le> card B"
  shows "\<exists>f. f ` A \<subseteq> B \<and> inj_on f A"
  using fA fB c
proof (induct arbitrary: B rule: finite_induct)
  case empty
  then show ?case by simp
next
  case (insert x s t)
  then show ?case
  proof (induct rule: finite_induct [OF insert.prems(1)])
    case 1
    then show ?case by simp
  next
    case (2 y t)
    from "2.prems"(1,2,5) "2.hyps"(1,2) have cst: "card s \<le> card t"
      by simp
    from "2.prems"(3) [OF "2.hyps"(1) cst]
    obtain f where *: "f ` s \<subseteq> t" "inj_on f s"
      by blast
    let ?g = "(\<lambda>a. if a = x then y else f a)"
    have "?g ` insert x s \<subseteq> insert y t \<and> inj_on ?g (insert x s)"
      using * "2.prems"(2) "2.hyps"(2) unfolding inj_on_def by auto
    then show ?case by (rule exI[where ?x="?g"])
  qed
qed

lemma card_subset_eq:
  assumes fB: "finite B"
    and AB: "A \<subseteq> B"
    and c: "card A = card B"
  shows "A = B"
proof -
  from fB AB have fA: "finite A"
    by (auto intro: finite_subset)
  from fA fB have fBA: "finite (B - A)"
    by auto
  have e: "A \<inter> (B - A) = {}"
    by blast
  have eq: "A \<union> (B - A) = B"
    using AB by blast
  from card_Un_disjoint[OF fA fBA e, unfolded eq c] have "card (B - A) = 0"
    by arith
  then have "B - A = {}"
    unfolding card_eq_0_iff using fA fB by simp
  with AB show "A = B"
    by blast
qed

lemma insert_partition:
  "x \<notin> F \<Longrightarrow> \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<Longrightarrow> x \<inter> \<Union>F = {}"
  by auto

lemma finite_psubset_induct [consumes 1, case_names psubset]:
  assumes finite: "finite A"
    and major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
  shows "P A"
  using finite
proof (induct A taking: card rule: measure_induct_rule)
  case (less A)
  have fin: "finite A" by fact
  have ih: "card B < card A \<Longrightarrow> finite B \<Longrightarrow> P B" for B by fact
  have "P B" if "B \<subset> A" for B
  proof -
    from that have "card B < card A"
      using psubset_card_mono fin by blast
    moreover
    from that have "B \<subseteq> A"
      by auto
    then have "finite B"
      using fin finite_subset by blast
    ultimately show ?thesis using ih by simp
  qed
  with fin show "P A" using major by blast
qed

lemma finite_induct_select [consumes 1, case_names empty select]:
  assumes "finite S"
    and "P {}"
    and select: "\<And>T. T \<subset> S \<Longrightarrow> P T \<Longrightarrow> \<exists>s\<in>S - T. P (insert s T)"
  shows "P S"
proof -
  have "0 \<le> card S" by simp
  then have "\<exists>T \<subseteq> S. card T = card S \<and> P T"
  proof (induct rule: dec_induct)
    case base with \<open>P {}\<close>
    show ?case
      by (intro exI[of _ "{}"]) auto
  next
    case (step n)
    then obtain T where T: "T \<subseteq> S" "card T = n" "P T"
      by auto
    with \<open>n < card S\<close> have "T \<subset> S" "P T"
      by auto
    with select[of T] obtain s where "s \<in> S" "s \<notin> T" "P (insert s T)"
      by auto
    with step(2) T \<open>finite S\<close> show ?case
      by (intro exI[of _ "insert s T"]) (auto dest: finite_subset)
  qed
  with \<open>finite S\<close> show "P S"
    by (auto dest: card_subset_eq)
qed

lemma remove_induct [case_names empty infinite remove]:
  assumes empty: "P ({} :: 'a set)"
    and infinite: "\<not> finite B \<Longrightarrow> P B"
    and remove: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
  shows "P B"
proof (cases "finite B")
  case False
  then show ?thesis by (rule infinite)
next
  case True
  define A where "A = B"
  with True have "finite A" "A \<subseteq> B"
    by simp_all
  then show "P A"
  proof (induct "card A" arbitrary: A)
    case 0
    then have "A = {}" by auto
    with empty show ?case by simp
  next
    case (Suc n A)
    from \<open>A \<subseteq> B\<close> and \<open>finite B\<close> have "finite A"
      by (rule finite_subset)
    moreover from Suc.hyps have "A \<noteq> {}" by auto
    moreover note \<open>A \<subseteq> B\<close>
    moreover have "P (A - {x})" if x: "x \<in> A" for x
      using x Suc.prems \<open>Suc n = card A\<close> by (intro Suc) auto
    ultimately show ?case by (rule remove)
  qed
qed

lemma finite_remove_induct [consumes 1, case_names empty remove]:
  fixes P :: "'a set \<Rightarrow> bool"
  assumes "finite B"
    and "P {}"
    and "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
  defines "B' \<equiv> B"
  shows "P B'"
  by (induct B' rule: remove_induct) (simp_all add: assms)


text \<open>Main cardinality theorem.\<close>
lemma card_partition [rule_format]:
  "finite C \<Longrightarrow> finite (\<Union>C) \<Longrightarrow> (\<forall>c\<in>C. card c = k) \<Longrightarrow>
    (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}) \<Longrightarrow>
    k * card C = card (\<Union>C)"
proof (induct rule: finite_induct)
  case empty
  then show ?case by simp
next
  case (insert x F)
  then show ?case
    by (simp add: card_Un_disjoint insert_partition finite_subset [of _ "\<Union>(insert _ _)"])
qed

lemma card_eq_UNIV_imp_eq_UNIV:
  assumes fin: "finite (UNIV :: 'a set)"
    and card: "card A = card (UNIV :: 'a set)"
  shows "A = (UNIV :: 'a set)"
proof
  show "A \<subseteq> UNIV" by simp
  show "UNIV \<subseteq> A"
  proof
    show "x \<in> A" for x
    proof (rule ccontr)
      assume "x \<notin> A"
      then have "A \<subset> UNIV" by auto
      with fin have "card A < card (UNIV :: 'a set)"
        by (fact psubset_card_mono)
      with card show False by simp
    qed
  qed
qed

text \<open>The form of a finite set of given cardinality\<close>

lemma card_eq_SucD:
  assumes "card A = Suc k"
  shows "\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {})"
proof -
  have fin: "finite A"
    using assms by (auto intro: ccontr)
  moreover have "card A \<noteq> 0"
    using assms by auto
  ultimately obtain b where b: "b \<in> A"
    by auto
  show ?thesis
  proof (intro exI conjI)
    show "A = insert b (A - {b})"
      using b by blast
    show "b \<notin> A - {b}"
      by blast
    show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
      using assms b fin by (fastforce dest: mk_disjoint_insert)+
  qed
qed

lemma card_Suc_eq:
  "card A = Suc k \<longleftrightarrow>
    (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))"
  by (auto simp: card_insert_if card_gt_0_iff elim!: card_eq_SucD)

lemma card_Suc_eq_finite:
  "card A = Suc k \<longleftrightarrow> (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> finite B)"
  unfolding card_Suc_eq using card_gt_0_iff by fastforce

lemma card_1_singletonE:
  assumes "card A = 1"
  obtains x where "A = {x}"
  using assms by (auto simp: card_Suc_eq)

lemma is_singleton_altdef: "is_singleton A \<longleftrightarrow> card A = 1"
  unfolding is_singleton_def
  by (auto elim!: card_1_singletonE is_singletonE simp del: One_nat_def)

lemma card_1_singleton_iff: "card A = Suc 0 \<longleftrightarrow> (\<exists>x. A = {x})"
  by (simp add: card_Suc_eq)

lemma card_le_Suc0_iff_eq:
  assumes "finite A"
  shows "card A \<le> Suc 0 \<longleftrightarrow> (\<forall>a1 \<in> A. \<forall>a2 \<in> A. a1 = a2)" (is "?C = ?A")
proof
  assume ?C thus ?A using assms by (auto simp: le_Suc_eq dest: card_eq_SucD)
next
  assume ?A
  show ?C
  proof cases
    assume "A = {}" thus ?C using \<open>?A\<close> by simp
  next
    assume "A \<noteq> {}"
    then obtain a where "A = {a}" using \<open>?A\<close> by blast
    thus ?C by simp
  qed
qed

lemma card_le_Suc_iff:
  "Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
proof (cases "finite A")
  case True
  then show ?thesis
    by (fastforce simp: card_Suc_eq less_eq_nat.simps split: nat.splits)
qed auto

lemma finite_fun_UNIVD2:
  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
  shows "finite (UNIV :: 'b set)"
proof -
  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))" for arbitrary
    by (rule finite_imageI)
  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)" for arbitrary
    by (rule UNIV_eq_I) auto
  ultimately show "finite (UNIV :: 'b set)"
    by simp
qed

lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
  unfolding UNIV_unit by simp

lemma infinite_arbitrarily_large:
  assumes "\<not> finite A"
  shows "\<exists>B. finite B \<and> card B = n \<and> B \<subseteq> A"
proof (induction n)
  case 0
  show ?case by (intro exI[of _ "{}"]) auto
next
  case (Suc n)
  then obtain B where B: "finite B \<and> card B = n \<and> B \<subseteq> A" ..
  with \<open>\<not> finite A\<close> have "A \<noteq> B" by auto
  with B have "B \<subset> A" by auto
  then have "\<exists>x. x \<in> A - B"
    by (elim psubset_imp_ex_mem)
  then obtain x where x: "x \<in> A - B" ..
  with B have "finite (insert x B) \<and> card (insert x B) = Suc n \<and> insert x B \<subseteq> A"
    by auto
  then show "\<exists>B. finite B \<and> card B = Suc n \<and> B \<subseteq> A" ..
qed

text \<open>Sometimes, to prove that a set is finite, it is convenient to work with finite subsets
and to show that their cardinalities are uniformly bounded. This possibility is formalized in
the next criterion.\<close>

lemma finite_if_finite_subsets_card_bdd:
  assumes "\<And>G. G \<subseteq> F \<Longrightarrow> finite G \<Longrightarrow> card G \<le> C"
  shows "finite F \<and> card F \<le> C"
proof (cases "finite F")
  case False
  obtain n::nat where n: "n > max C 0" by auto
  obtain G where G: "G \<subseteq> F" "card G = n" using infinite_arbitrarily_large[OF False] by auto
  hence "finite G" using \<open>n > max C 0\<close> using card.infinite gr_implies_not0 by blast
  hence False using assms G n not_less by auto
  thus ?thesis ..
next
  case True thus ?thesis using assms[of F] by auto
qed

lemma obtain_subset_with_card_n:
  assumes "n \<le> card S"
  obtains T where "T \<subseteq> S" "card T = n" "finite T"
proof -
  obtain n' where "card S = n + n'"
    using le_Suc_ex[OF assms] by blast
  with that show thesis
  proof (induct n' arbitrary: S)
    case 0 
    thus ?case by (cases "finite S") auto
  next
    case Suc 
    thus ?case by (auto simp add: card_Suc_eq)
  qed
qed

lemma exists_subset_between: 
  assumes 
    "card A \<le> n" 
    "n \<le> card C"
    "A \<subseteq> C"
    "finite C"
  shows "\<exists>B. A \<subseteq> B \<and> B \<subseteq> C \<and> card B = n" 
  using assms 
proof (induct n arbitrary: A C)
  case 0
  thus ?case using finite_subset[of A C] by (intro exI[of _ "{}"], auto)
next
  case (Suc n A C)
  show ?case
  proof (cases "A = {}")
    case True
    from obtain_subset_with_card_n[OF Suc(3)]
    obtain B where "B \<subseteq> C" "card B = Suc n" by blast
    thus ?thesis unfolding True by blast
  next
    case False
    then obtain a where a: "a \<in> A" by auto
    let ?A = "A - {a}" 
    let ?C = "C - {a}" 
    have 1: "card ?A \<le> n" using Suc(2-) a 
      using finite_subset by fastforce 
    have 2: "card ?C \<ge> n" using Suc(2-) a by auto
    from Suc(1)[OF 1 2 _ finite_subset[OF _ Suc(5)]] Suc(2-)
    obtain B where "?A \<subseteq> B" "B \<subseteq> ?C" "card B = n" by blast
    thus ?thesis using a Suc(2-) 
      by (intro exI[of _ "insert a B"], auto intro!: card_insert_disjoint finite_subset[of B C])
  qed
qed


subsubsection \<open>Cardinality of image\<close>

lemma card_image_le: "finite A \<Longrightarrow> card (f ` A) \<le> card A"
  by (induct rule: finite_induct) (simp_all add: le_SucI card_insert_if)

lemma card_image: "inj_on f A \<Longrightarrow> card (f ` A) = card A"
proof (induct A rule: infinite_finite_induct)
  case (infinite A)
  then have "\<not> finite (f ` A)" by (auto dest: finite_imageD)
  with infinite show ?case by simp
qed simp_all

lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
  by (auto simp: card_image bij_betw_def)

lemma endo_inj_surj: "finite A \<Longrightarrow> f ` A \<subseteq> A \<Longrightarrow> inj_on f A \<Longrightarrow> f ` A = A"
  by (simp add: card_seteq card_image)

lemma eq_card_imp_inj_on:
  assumes "finite A" "card(f ` A) = card A"
  shows "inj_on f A"
  using assms
proof (induct rule:finite_induct)
  case empty
  show ?case by simp
next
  case (insert x A)
  then show ?case
    using card_image_le [of A f] by (simp add: card_insert_if split: if_splits)
qed

lemma inj_on_iff_eq_card: "finite A \<Longrightarrow> inj_on f A \<longleftrightarrow> card (f ` A) = card A"
  by (blast intro: card_image eq_card_imp_inj_on)

lemma card_inj_on_le:
  assumes "inj_on f A" "f ` A \<subseteq> B" "finite B"
  shows "card A \<le> card B"
proof -
  have "finite A"
    using assms by (blast intro: finite_imageD dest: finite_subset)
  then show ?thesis
    using assms by (force intro: card_mono simp: card_image [symmetric])
qed

lemma inj_on_iff_card_le:
  "\<lbrakk> finite A; finite B \<rbrakk> \<Longrightarrow> (\<exists>f. inj_on f A \<and> f ` A \<le> B) = (card A \<le> card B)"
using card_inj_on_le[of _ A B] card_le_inj[of A B] by blast

lemma surj_card_le: "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> card B \<le> card A"
  by (blast intro: card_image_le card_mono le_trans)

lemma card_bij_eq:
  "inj_on f A \<Longrightarrow> f ` A \<subseteq> B \<Longrightarrow> inj_on g B \<Longrightarrow> g ` B \<subseteq> A \<Longrightarrow> finite A \<Longrightarrow> finite B
    \<Longrightarrow> card A = card B"
  by (auto intro: le_antisym card_inj_on_le)

lemma bij_betw_finite: "bij_betw f A B \<Longrightarrow> finite A \<longleftrightarrow> finite B"
  unfolding bij_betw_def using finite_imageD [of f A] by auto

lemma inj_on_finite: "inj_on f A \<Longrightarrow> f ` A \<le> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
  using finite_imageD finite_subset by blast

lemma card_vimage_inj_on_le:
  assumes "inj_on f D" "finite A"
  shows "card (f-`A \<inter> D) \<le> card A"
proof (rule card_inj_on_le)
  show "inj_on f (f -` A \<inter> D)"
    by (blast intro: assms inj_on_subset)
qed (use assms in auto)

lemma card_vimage_inj: "inj f \<Longrightarrow> A \<subseteq> range f \<Longrightarrow> card (f -` A) = card A"
  by (auto 4 3 simp: subset_image_iff inj_vimage_image_eq
      intro: card_image[symmetric, OF subset_inj_on])

lemma card_inverse[simp]: "card (R\<inverse>) = card R"
proof -
  have *: "\<And>R. prod.swap ` R = R\<inverse>" by auto
  {
    assume "\<not>finite R"
    hence ?thesis
      by auto
  } moreover {
    assume "finite R"
    with card_image_le[of R prod.swap] card_image_le[of "R\<inverse>" prod.swap]
    have ?thesis by (auto simp: * )
  } ultimately show ?thesis by blast
qed

subsubsection \<open>Pigeonhole Principles\<close>

lemma pigeonhole: "card A > card (f ` A) \<Longrightarrow> \<not> inj_on f A "
  by (auto dest: card_image less_irrefl_nat)

lemma pigeonhole_infinite:
  assumes "\<not> finite A" and "finite (f`A)"
  shows "\<exists>a0\<in>A. \<not> finite {a\<in>A. f a = f a0}"
  using assms(2,1)
proof (induct "f`A" arbitrary: A rule: finite_induct)
  case empty
  then show ?case by simp
next
  case (insert b F)
  show ?case
  proof (cases "finite {a\<in>A. f a = b}")
    case True
    with \<open>\<not> finite A\<close> have "\<not> finite (A - {a\<in>A. f a = b})"
      by simp
    also have "A - {a\<in>A. f a = b} = {a\<in>A. f a \<noteq> b}"
      by blast
    finally have "\<not> finite {a\<in>A. f a \<noteq> b}" .
    from insert(3)[OF _ this] insert(2,4) show ?thesis
      by simp (blast intro: rev_finite_subset)
  next
    case False
    then have "{a \<in> A. f a = b} \<noteq> {}" by force
    with False show ?thesis by blast
  qed
qed

lemma pigeonhole_infinite_rel:
  assumes "\<not> finite A"
    and "finite B"
    and "\<forall>a\<in>A. \<exists>b\<in>B. R a b"
  shows "\<exists>b\<in>B. \<not> finite {a:A. R a b}"
proof -
  let ?F = "\<lambda>a. {b\<in>B. R a b}"
  from finite_Pow_iff[THEN iffD2, OF \<open>finite B\<close>] have "finite (?F ` A)"
    by (blast intro: rev_finite_subset)
  from pigeonhole_infinite [where f = ?F, OF assms(1) this]
  obtain a0 where "a0 \<in> A" and infinite: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
  obtain b0 where "b0 \<in> B" and "R a0 b0"
    using \<open>a0 \<in> A\<close> assms(3) by blast
  have "finite {a\<in>A. ?F a = ?F a0}" if "finite {a\<in>A. R a b0}"
    using \<open>b0 \<in> B\<close> \<open>R a0 b0\<close> that by (blast intro: rev_finite_subset)
  with infinite \<open>b0 \<in> B\<close> show ?thesis
    by blast
qed


subsubsection \<open>Cardinality of sums\<close>

lemma card_Plus:
  assumes "finite A" "finite B"
  shows "card (A <+> B) = card A + card B"
proof -
  have "Inl`A \<inter> Inr`B = {}" by fast
  with assms show ?thesis
    by (simp add: Plus_def card_Un_disjoint card_image)
qed

lemma card_Plus_conv_if:
  "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
  by (auto simp add: card_Plus)

text \<open>Relates to equivalence classes.  Based on a theorem of F. Kammüller.\<close>

lemma dvd_partition:
  assumes f: "finite (\<Union>C)"
    and "\<forall>c\<in>C. k dvd card c" "\<forall>c1\<in>C. \<forall>c2\<in>C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}"
  shows "k dvd card (\<Union>C)"
proof -
  have "finite C"
    by (rule finite_UnionD [OF f])
  then show ?thesis
    using assms
  proof (induct rule: finite_induct)
    case empty
    show ?case by simp
  next
    case (insert c C)
    then have "c \<inter> \<Union>C = {}"
      by auto
    with insert show ?case
      by (simp add: card_Un_disjoint)
  qed
qed


subsection \<open>Minimal and maximal elements of finite sets\<close>

context begin

qualified lemma
  assumes "finite A" and "A \<noteq> {}" and "transp_on A R" and "asymp_on A R"
  shows
    bex_min_element: "\<exists>m \<in> A. \<forall>x \<in> A. x \<noteq> m \<longrightarrow> \<not> R x m" and
    bex_max_element: "\<exists>m \<in> A. \<forall>x \<in> A. x \<noteq> m \<longrightarrow> \<not> R m x"
  unfolding atomize_conj
  using assms
proof (induction A rule: finite_induct)
  case empty
  hence False
    by simp
  thus ?case ..
next
  case (insert a A')
  show ?case
  proof (cases "A' = {}")
    case True
    show ?thesis
    proof (intro conjI bexI)
      show "\<forall>x\<in>insert a A'. x \<noteq> a \<longrightarrow> \<not> R x a" and "\<forall>x\<in>insert a A'. x \<noteq> a \<longrightarrow> \<not> R a x"
        using True by blast+
    qed simp_all
  next
    case False
    moreover have "transp_on A' R"
      using insert.prems transp_on_subset by blast
    moreover have "asymp_on A' R"
      using insert.prems asymp_on_subset by blast
    ultimately obtain min max where
      "min \<in> A'" and "max \<in> A'" and
      min_is_min: "\<forall>x\<in>A'. x \<noteq> min \<longrightarrow> \<not> R x min" and
      max_is_max: "\<forall>x\<in>A'. x \<noteq> max \<longrightarrow> \<not> R max x"
      using insert.IH by auto

    show ?thesis
    proof (rule conjI)
      show "\<exists>min\<in>insert a A'. \<forall>x\<in>insert a A'. x \<noteq> min \<longrightarrow> \<not> R x min"
      proof (cases "R a min")
        case True
        show ?thesis
        proof (intro bexI ballI impI)
          show "a \<in> insert a A'"
            by simp
        next
          fix x
          show "x \<in> insert a A' \<Longrightarrow> x \<noteq> a \<Longrightarrow> \<not> R x a"
            using True \<open>min \<in> A'\<close> min_is_min[rule_format, of x] insert.prems(2,3)
            by (auto dest: asymp_onD transp_onD)
        qed
      next
        case False
        show ?thesis
        proof (rule bexI)
          show "min \<in> insert a A'"
            using \<open>min \<in> A'\<close> by auto
        next
          show "\<forall>x\<in>insert a A'. x \<noteq> min \<longrightarrow> \<not> R x min"
            using False min_is_min by blast
        qed
      qed
    next
      show "\<exists>max\<in>insert a A'. \<forall>x\<in>insert a A'. x \<noteq> max \<longrightarrow> \<not> R max x"
      proof (cases "R max a")
        case True
        show ?thesis
        proof (intro bexI ballI impI)
          show "a \<in> insert a A'"
            by simp
        next
          fix x
          show "x \<in> insert a A' \<Longrightarrow> x \<noteq> a \<Longrightarrow> \<not> R a x"
            using True \<open>max \<in> A'\<close> max_is_max[rule_format, of x] insert.prems(2,3)
            by (auto dest: asymp_onD transp_onD)
        qed
      next
        case False
        show ?thesis
        proof (rule bexI)
          show "max \<in> insert a A'"
            using \<open>max \<in> A'\<close> by auto
        next
          show "\<forall>x\<in>insert a A'. x \<noteq> max \<longrightarrow> \<not> R max x"
            using False max_is_max by blast
        qed
      qed
    qed
  qed
qed

end

text \<open>The following alternative form might sometimes be easier to work with.\<close>

lemma is_min_element_in_set_iff:
  "asymp_on A R \<Longrightarrow> (\<forall>y \<in> A. y \<noteq> x \<longrightarrow> \<not> R y x) \<longleftrightarrow> (\<forall>y. R y x \<longrightarrow> y \<notin> A)"
  by (auto dest: asymp_onD)

lemma is_max_element_in_set_iff:
  "asymp_on A R \<Longrightarrow> (\<forall>y \<in> A. y \<noteq> x \<longrightarrow> \<not> R x y) \<longleftrightarrow> (\<forall>y. R x y \<longrightarrow> y \<notin> A)"
  by (auto dest: asymp_onD)


subsubsection \<open>Finite orders\<close>

context order
begin

lemma finite_has_maximal:
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<exists> m \<in> A. \<forall> b \<in> A. m \<le> b \<longrightarrow> m = b"
proof (induction rule: finite_psubset_induct)
  case (psubset A)
  from \<open>A \<noteq> {}\<close> obtain a where "a \<in> A" by auto
  let ?B = "{b \<in> A. a < b}"
  show ?case
  proof cases
    assume "?B = {}"
    hence "\<forall> b \<in> A. a \<le> b \<longrightarrow> a = b" using le_neq_trans by blast
    thus ?thesis using \<open>a \<in> A\<close> by blast
  next
    assume "?B \<noteq> {}"
    have "a \<notin> ?B" by auto
    hence "?B \<subset> A" using \<open>a \<in> A\<close> by blast
    from psubset.IH[OF this \<open>?B \<noteq> {}\<close>] show ?thesis using order.strict_trans2 by blast
  qed
qed

lemma finite_has_maximal2:
  "\<lbrakk> finite A; a \<in> A \<rbrakk> \<Longrightarrow> \<exists> m \<in> A. a \<le> m \<and> (\<forall> b \<in> A. m \<le> b \<longrightarrow> m = b)"
using finite_has_maximal[of "{b \<in> A. a \<le> b}"] by fastforce

lemma finite_has_minimal:
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<exists> m \<in> A. \<forall> b \<in> A. b \<le> m \<longrightarrow> m = b"
proof (induction rule: finite_psubset_induct)
  case (psubset A)
  from \<open>A \<noteq> {}\<close> obtain a where "a \<in> A" by auto
  let ?B = "{b \<in> A. b < a}"
  show ?case
  proof cases
    assume "?B = {}"
    hence "\<forall> b \<in> A. b \<le> a \<longrightarrow> a = b" using le_neq_trans by blast
    thus ?thesis using \<open>a \<in> A\<close> by blast
  next
    assume "?B \<noteq> {}"
    have "a \<notin> ?B" by auto
    hence "?B \<subset> A" using \<open>a \<in> A\<close> by blast
    from psubset.IH[OF this \<open>?B \<noteq> {}\<close>] show ?thesis using order.strict_trans1 by blast
  qed
qed

lemma finite_has_minimal2:
  "\<lbrakk> finite A; a \<in> A \<rbrakk> \<Longrightarrow> \<exists> m \<in> A. m \<le> a \<and> (\<forall> b \<in> A. b \<le> m \<longrightarrow> m = b)"
using finite_has_minimal[of "{b \<in> A. b \<le> a}"] by fastforce

end

subsubsection \<open>Relating injectivity and surjectivity\<close>

lemma finite_surj_inj:
  assumes "finite A" "A \<subseteq> f ` A"
  shows "inj_on f A"
proof -
  have "f ` A = A"
    by (rule card_seteq [THEN sym]) (auto simp add: assms card_image_le)
  then show ?thesis using assms
    by (simp add: eq_card_imp_inj_on)
qed

lemma finite_UNIV_surj_inj: "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
  for f :: "'a \<Rightarrow> 'a"
  by (blast intro: finite_surj_inj subset_UNIV)

lemma finite_UNIV_inj_surj: "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
  for f :: "'a \<Rightarrow> 'a"
  by (fastforce simp:surj_def dest!: endo_inj_surj)

lemma surjective_iff_injective_gen:
  assumes fS: "finite S"
    and fT: "finite T"
    and c: "card S = card T"
    and ST: "f ` S \<subseteq> T"
  shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S"
  (is "?lhs \<longleftrightarrow> ?rhs")
proof
  assume h: "?lhs"
  {
    fix x y
    assume x: "x \<in> S"
    assume y: "y \<in> S"
    assume f: "f x = f y"
    from x fS have S0: "card S \<noteq> 0"
      by auto
    have "x = y"
    proof (rule ccontr)
      assume xy: "\<not> ?thesis"
      have th: "card S \<le> card (f ` (S - {y}))"
        unfolding c
      proof (rule card_mono)
        show "finite (f ` (S - {y}))"
          by (simp add: fS)
        have "\<lbrakk>x \<noteq> y; x \<in> S; z \<in> S; f x = f y\<rbrakk>
         \<Longrightarrow> \<exists>x \<in> S. x \<noteq> y \<and> f z = f x" for z
          by (cases "z = y \<longrightarrow> z = x") auto
        then show "T \<subseteq> f ` (S - {y})"
          using h xy x y f by fastforce
      qed
      also have " \<dots> \<le> card (S - {y})"
        by (simp add: card_image_le fS)
      also have "\<dots> \<le> card S - 1" using y fS by simp
      finally show False using S0 by arith
    qed
  }
  then show ?rhs
    unfolding inj_on_def by blast
next
  assume h: ?rhs
  have "f ` S = T"
    by (simp add: ST c card_image card_subset_eq fT h)
  then show ?lhs by blast
qed

hide_const (open) Finite_Set.fold


subsection \<open>Infinite Sets\<close>

text \<open>
  Some elementary facts about infinite sets, mostly by Stephan Merz.
  Beware! Because "infinite" merely abbreviates a negation, these
  lemmas may not work well with \<open>blast\<close>.
\<close>

abbreviation infinite :: "'a set \<Rightarrow> bool"
  where "infinite S \<equiv> \<not> finite S"

text \<open>
  Infinite sets are non-empty, and if we remove some elements from an
  infinite set, the result is still infinite.
\<close>

lemma infinite_UNIV_nat [iff]: "infinite (UNIV :: nat set)"
proof
  assume "finite (UNIV :: nat set)"
  with finite_UNIV_inj_surj [of Suc] show False
    by simp (blast dest: Suc_neq_Zero surjD)
qed

lemma infinite_UNIV_char_0: "infinite (UNIV :: 'a::semiring_char_0 set)"
proof
  assume "finite (UNIV :: 'a set)"
  with subset_UNIV have "finite (range of_nat :: 'a set)"
    by (rule finite_subset)
  moreover have "inj (of_nat :: nat \<Rightarrow> 'a)"
    by (simp add: inj_on_def)
  ultimately have "finite (UNIV :: nat set)"
    by (rule finite_imageD)
  then show False
    by simp
qed

lemma infinite_imp_nonempty: "infinite S \<Longrightarrow> S \<noteq> {}"
  by auto

lemma infinite_remove: "infinite S \<Longrightarrow> infinite (S - {a})"
  by simp

lemma Diff_infinite_finite:
  assumes "finite T" "infinite S"
  shows "infinite (S - T)"
  using \<open>finite T\<close>
proof induct
  from \<open>infinite S\<close> show "infinite (S - {})"
    by auto
next
  fix T x
  assume ih: "infinite (S - T)"
  have "S - (insert x T) = (S - T) - {x}"
    by (rule Diff_insert)
  with ih show "infinite (S - (insert x T))"
    by (simp add: infinite_remove)
qed

lemma Un_infinite: "infinite S \<Longrightarrow> infinite (S \<union> T)"
  by simp

lemma infinite_Un: "infinite (S \<union> T) \<longleftrightarrow> infinite S \<or> infinite T"
  by simp

lemma infinite_super:
  assumes "S \<subseteq> T"
    and "infinite S"
  shows "infinite T"
proof
  assume "finite T"
  with \<open>S \<subseteq> T\<close> have "finite S" by (simp add: finite_subset)
  with \<open>infinite S\<close> show False by simp
qed

proposition infinite_coinduct [consumes 1, case_names infinite]:
  assumes "X A"
    and step: "\<And>A. X A \<Longrightarrow> \<exists>x\<in>A. X (A - {x}) \<or> infinite (A - {x})"
  shows "infinite A"
proof
  assume "finite A"
  then show False
    using \<open>X A\<close>
  proof (induction rule: finite_psubset_induct)
    case (psubset A)
    then obtain x where "x \<in> A" "X (A - {x}) \<or> infinite (A - {x})"
      using local.step psubset.prems by blast
    then have "X (A - {x})"
      using psubset.hyps by blast
    show False
    proof (rule psubset.IH [where B = "A - {x}"])
      show "A - {x} \<subset> A"
        using \<open>x \<in> A\<close> by blast
    qed fact
  qed
qed

text \<open>
  For any function with infinite domain and finite range there is some
  element that is the image of infinitely many domain elements.  In
  particular, any infinite sequence of elements from a finite set
  contains some element that occurs infinitely often.
\<close>

lemma inf_img_fin_dom':
  assumes img: "finite (f ` A)"
    and dom: "infinite A"
  shows "\<exists>y \<in> f ` A. infinite (f -` {y} \<inter> A)"
proof (rule ccontr)
  have "A \<subseteq> (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by auto
  moreover assume "\<not> ?thesis"
  with img have "finite (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by blast
  ultimately have "finite A" by (rule finite_subset)
  with dom show False by contradiction
qed

lemma inf_img_fin_domE':
  assumes "finite (f ` A)" and "infinite A"
  obtains y where "y \<in> f`A" and "infinite (f -` {y} \<inter> A)"
  using assms by (blast dest: inf_img_fin_dom')

lemma inf_img_fin_dom:
  assumes img: "finite (f`A)" and dom: "infinite A"
  shows "\<exists>y \<in> f`A. infinite (f -` {y})"
  using inf_img_fin_dom'[OF assms] by auto

lemma inf_img_fin_domE:
  assumes "finite (f`A)" and "infinite A"
  obtains y where "y \<in> f`A" and "infinite (f -` {y})"
  using assms by (blast dest: inf_img_fin_dom)

proposition finite_image_absD: "finite (abs ` S) \<Longrightarrow> finite S"
  for S :: "'a::linordered_ring set"
  by (rule ccontr) (auto simp: abs_eq_iff vimage_def dest: inf_img_fin_dom)


subsection \<open>The finite powerset operator\<close>

definition Fpow :: "'a set \<Rightarrow> 'a set set"
where "Fpow A \<equiv> {X. X \<subseteq> A \<and> finite X}"

lemma Fpow_mono: "A \<subseteq> B \<Longrightarrow> Fpow A \<subseteq> Fpow B"
unfolding Fpow_def by auto

lemma empty_in_Fpow: "{} \<in> Fpow A"
unfolding Fpow_def by auto

lemma Fpow_not_empty: "Fpow A \<noteq> {}"
using empty_in_Fpow by blast

lemma Fpow_subset_Pow: "Fpow A \<subseteq> Pow A"
unfolding Fpow_def by auto

lemma Fpow_Pow_finite: "Fpow A = Pow A Int {A. finite A}"
unfolding Fpow_def Pow_def by blast

lemma inj_on_image_Fpow:
  assumes "inj_on f A"
  shows "inj_on (image f) (Fpow A)"
  using assms Fpow_subset_Pow[of A] subset_inj_on[of "image f" "Pow A"]
    inj_on_image_Pow by blast

lemma image_Fpow_mono:
  assumes "f ` A \<subseteq> B"
  shows "(image f) ` (Fpow A) \<subseteq> Fpow B"
  using assms by(unfold Fpow_def, auto)

end