src/HOL/Finite_Set.thy
 author desharna Sat, 27 May 2023 23:34:42 +0200 changeset 78120 a8e5cefeb3ab parent 78099 4d9349989d94 permissions -rw-r--r--
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```
(*  Title:      HOL/Finite_Set.thy
Author:     Tobias Nipkow
Author:     Lawrence C Paulson
Author:     Markus Wenzel
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}"

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"
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>
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 ctxt ct =
(let
val _ \$ A = Thm.term_of ct
val prems = Simplifier.prems_of ctxt
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 K 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}"

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

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))"
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)"

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

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}"

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})"
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)"

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}"
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}"
then have "\<exists>n f. A = f ` {i::nat. i < n}"
by blast
then show ?thesis
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}"
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}"
then have "\<exists>n f. B = f ` {i::nat. i < n}"
by blast
then show ?thesis
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)"
qed

corollary finite_Collect_subsets [simp, intro]: "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"

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
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

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

instance unit :: finite

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)"

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
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>
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
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"

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"

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
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"
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
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
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

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

lemma empty [simp]: "F {} = z"

lemma infinite [simp]: "\<not> finite A \<Longrightarrow> F A = z"

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"
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"
qed simp_all

end

subsection \<open>Finite cardinality\<close>

text \<open>
\<^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
qed auto

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

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"

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
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})"

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"

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)"

text \<open>Relates to equivalence classes.  Based on a theorem of F. KammÃ¼ller.\<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
qed
qed

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

context begin

qualified lemma
assumes "finite A" and "asymp_on A R" and "transp_on A R" and "\<exists>x \<in> A. P x"
shows
bex_min_element_with_property: "\<exists>x \<in> A. P x \<and> (\<forall>y \<in> A. R y x \<longrightarrow> \<not> P y)" and
bex_max_element_with_property: "\<exists>x \<in> A. P x \<and> (\<forall>y \<in> A. R x y \<longrightarrow> \<not> P y)"
unfolding atomize_conj
using assms
proof (induction A rule: finite_induct)
case empty
hence False
by simp_all
thus ?case ..
next
case (insert x F)

from insert.prems have "asymp_on F R"
using asymp_on_subset by blast

from insert.prems have "transp_on F R"
using transp_on_subset by blast

show ?case
proof (cases "P x")
case True
show ?thesis
proof (cases "\<exists>a\<in>F. P a")
case True
with insert.IH obtain min max where
"min \<in> F" and "P min" and "\<forall>z \<in> F. R z min \<longrightarrow> \<not> P z"
"max \<in> F" and "P max" and "\<forall>z \<in> F. R max z \<longrightarrow> \<not> P z"
using \<open>asymp_on F R\<close> \<open>transp_on F R\<close> by auto

show ?thesis
proof (rule conjI)
show "\<exists>y \<in> insert x F. P y \<and> (\<forall>z \<in> insert x F. R y z \<longrightarrow> \<not> P z)"
proof (cases "R max x")
case True
show ?thesis
proof (intro bexI conjI ballI impI)
show "x \<in> insert x F"
by simp
next
show "P x"
using \<open>P x\<close> by simp
next
fix z assume "z \<in> insert x F" and "R x z"
hence "z = x \<or> z \<in> F"
by simp
thus "\<not> P z"
proof (rule disjE)
assume "z = x"
hence "R x x"
using \<open>R x z\<close> by simp
moreover have "\<not> R x x"
using \<open>asymp_on (insert x F) R\<close>[THEN irreflp_on_if_asymp_on, THEN irreflp_onD]
by simp
ultimately have False
by simp
thus ?thesis ..
next
assume "z \<in> F"
moreover have "R max z"
using \<open>R max x\<close> \<open>R x z\<close>
using \<open>transp_on (insert x F) R\<close>[THEN transp_onD, of max x z]
using \<open>max \<in> F\<close> \<open>z \<in> F\<close> by simp
ultimately show ?thesis
using \<open>\<forall>z \<in> F. R max z \<longrightarrow> \<not> P z\<close> by simp
qed
qed
next
case False
show ?thesis
proof (intro bexI conjI ballI impI)
show "max \<in> insert x F"
using \<open>max \<in> F\<close> by simp
next
show "P max"
using \<open>P max\<close> by simp
next
fix z assume "z \<in> insert x F" and "R max z"
hence "z = x \<or> z \<in> F"
by simp
thus "\<not> P z"
proof (rule disjE)
assume "z = x"
hence False
using \<open>\<not> R max x\<close> \<open>R max z\<close> by simp
thus ?thesis ..
next
assume "z \<in> F"
thus ?thesis
using \<open>R max z\<close> \<open>\<forall>z\<in>F. R max z \<longrightarrow> \<not> P z\<close> by simp
qed
qed
qed
next
show "\<exists>y \<in> insert x F. P y \<and> (\<forall>z \<in> insert x F. R z y \<longrightarrow> \<not> P z)"
proof (cases "R x min")
case True
show ?thesis
proof (intro bexI conjI ballI impI)
show "x \<in> insert x F"
by simp
next
show "P x"
using \<open>P x\<close> by simp
next
fix z assume "z \<in> insert x F" and "R z x"
hence "z = x \<or> z \<in> F"
by simp
thus "\<not> P z"
proof (rule disjE)
assume "z = x"
hence "R x x"
using \<open>R z x\<close> by simp
moreover have "\<not> R x x"
using \<open>asymp_on (insert x F) R\<close>[THEN irreflp_on_if_asymp_on, THEN irreflp_onD]
by simp
ultimately have False
by simp
thus ?thesis ..
next
assume "z \<in> F"
moreover have "R z min"
using \<open>R z x\<close> \<open>R x min\<close>
using \<open>transp_on (insert x F) R\<close>[THEN transp_onD, of z x min]
using \<open>min \<in> F\<close> \<open>z \<in> F\<close> by simp
ultimately show ?thesis
using \<open>\<forall>z \<in> F. R z min \<longrightarrow> \<not> P z\<close> by simp
qed
qed
next
case False
show ?thesis
proof (intro bexI conjI ballI impI)
show "min \<in> insert x F"
using \<open>min \<in> F\<close> by simp
next
show "P min"
using \<open>P min\<close> by simp
next
fix z assume "z \<in> insert x F" and "R z min"
hence "z = x \<or> z \<in> F"
by simp
thus "\<not> P z"
proof (rule disjE)
assume "z = x"
hence False
using \<open>\<not> R x min\<close> \<open>R z min\<close> by simp
thus ?thesis ..
next
assume "z \<in> F"
thus ?thesis
using \<open>R z min\<close> \<open>\<forall>z\<in>F. R z min \<longrightarrow> \<not> P z\<close> by simp
qed
qed
qed
qed
next
case False
then show ?thesis
using \<open>\<exists>a\<in>insert x F. P a\<close>
using \<open>asymp_on (insert x F) R\<close>[THEN asymp_onD, of x] insert_iff[of _ x F]
by blast
qed
next
case False
with insert.prems have "\<exists>x \<in> F. P x"
by simp
with insert.IH have
"\<exists>y \<in> F. P y \<and> (\<forall>z\<in>F. R z y \<longrightarrow> \<not> P z)"
"\<exists>y \<in> F. P y \<and> (\<forall>z\<in>F. R y z \<longrightarrow> \<not> P z)"
using \<open>asymp_on F R\<close> \<open>transp_on F R\<close> by auto
thus ?thesis
using False by auto
qed
qed

qualified lemma
assumes "finite A" and "asymp_on A R" and "transp_on A R" and "A \<noteq> {}"
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"
using \<open>A \<noteq> {}\<close>
bex_min_element_with_property[OF assms(1,2,3), of "\<lambda>_. True", simplified]
bex_max_element_with_property[OF assms(1,2,3), of "\<lambda>_. True", simplified]
by blast+

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)

context begin

qualified lemma
assumes "finite A" and "A \<noteq> {}" and "transp_on A R" and "totalp_on A R"
shows
bex_least_element: "\<exists>l \<in> A. \<forall>x \<in> A. x \<noteq> l \<longrightarrow> R l x" and
bex_greatest_element: "\<exists>g \<in> A. \<forall>x \<in> A. x \<noteq> g \<longrightarrow> R x g"
unfolding atomize_conj
using assms
proof (induction A rule: finite_induct)
case empty
hence False by simp
thus ?case ..
next
case (insert a A')

from insert.prems(2) have transp_on_A': "transp_on A' R"
by (auto intro: transp_onI dest: transp_onD)

from insert.prems(3) have
totalp_on_a_A'_raw: "\<forall>y \<in> A'. a \<noteq> y \<longrightarrow> R a y \<or> R y a" and
totalp_on_A': "totalp_on A' R"

show ?case
proof (cases "A' = {}")
case True
thus ?thesis by simp
next
case False
then obtain least greatest where
"least \<in> A'" and least_of_A': "\<forall>x\<in>A'. x \<noteq> least \<longrightarrow> R least x" and
"greatest \<in> A'" and greatest_of_A': "\<forall>x\<in>A'. x \<noteq> greatest \<longrightarrow> R x greatest"
using insert.IH[OF _ transp_on_A' totalp_on_A'] by auto

show ?thesis
proof (rule conjI)
show "\<exists>l\<in>insert a A'. \<forall>x\<in>insert a A'. x \<noteq> l \<longrightarrow> R l x"
proof (cases "R a least")
case True
show ?thesis
proof (intro bexI ballI impI)
show "a \<in> insert a A'"
by simp
next
fix x
show "\<And>x. x \<in> insert a A' \<Longrightarrow> x \<noteq> a \<Longrightarrow> R a x"
using True \<open>least \<in> A'\<close> least_of_A'
using insert.prems(2)[THEN transp_onD, of a least]
by auto
qed
next
case False
show ?thesis
proof (intro bexI ballI impI)
show "least \<in> insert a A'"
using \<open>least \<in> A'\<close> by simp
next
fix x
show "x \<in> insert a A' \<Longrightarrow> x \<noteq> least \<Longrightarrow> R least x"
using False \<open>least \<in> A'\<close> least_of_A' totalp_on_a_A'_raw
by (cases "x = a") auto
qed
qed
next
show "\<exists>g \<in> insert a A'. \<forall>x \<in> insert a A'. x \<noteq> g \<longrightarrow> R x g"
proof (cases "R greatest a")
case True
show ?thesis
proof (intro bexI ballI impI)
show "a \<in> insert a A'"
by simp
next
fix x
show "\<And>x. x \<in> insert a A' \<Longrightarrow> x \<noteq> a \<Longrightarrow> R x a"
using True \<open>greatest \<in> A'\<close> greatest_of_A'
using insert.prems(2)[THEN transp_onD, of _ greatest a]
by auto
qed
next
case False
show ?thesis
proof (intro bexI ballI impI)
show "greatest \<in> insert a A'"
using \<open>greatest \<in> A'\<close> by simp
next
fix x
show "x \<in> insert a A' \<Longrightarrow> x \<noteq> greatest \<Longrightarrow> R x greatest"
using False \<open>greatest \<in> A'\<close> greatest_of_A' totalp_on_a_A'_raw
by (cases "x = a") auto
qed
qed
qed
qed
qed

end

subsubsection \<open>Finite orders\<close>

context order
begin

lemma finite_has_maximal:
assumes "finite A" and "A \<noteq> {}"
shows "\<exists> m \<in> A. \<forall> b \<in> A. m \<le> b \<longrightarrow> m = b"
proof -
obtain m where "m \<in> A" and m_is_max: "\<forall>x\<in>A. x \<noteq> m \<longrightarrow> \<not> m < x"
using Finite_Set.bex_max_element[OF \<open>finite A\<close> _ _ \<open>A \<noteq> {}\<close>, of "(<)"] by auto
moreover have "\<forall>b \<in> A. m \<le> b \<longrightarrow> m = b"
using m_is_max by (auto simp: le_less)
ultimately show ?thesis
by auto
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:
assumes "finite A" and "A \<noteq> {}"
shows "\<exists> m \<in> A. \<forall> b \<in> A. b \<le> m \<longrightarrow> m = b"
proof -
obtain m where "m \<in> A" and m_is_min: "\<forall>x\<in>A. x \<noteq> m \<longrightarrow> \<not> x < m"
using Finite_Set.bex_min_element[OF \<open>finite A\<close> _ _ \<open>A \<noteq> {}\<close>, of "(<)"] by auto
moreover have "\<forall>b \<in> A. b \<le> m \<longrightarrow> m = b"
using m_is_min by (auto simp: le_less)
ultimately show ?thesis
by auto
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
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}))"
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})"
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)"
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))"
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
```