src/HOL/Finite_Set.thy
 author haftmann Tue, 26 Mar 2013 21:53:56 +0100 changeset 51546 2e26df807dc7 parent 51489 f738e6dbd844 child 51598 5dbe537087aa permissions -rw-r--r--
more uniform style for interpretation and sublocale declarations
```
(*  Title:      HOL/Finite_Set.thy
Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
*)

theory Finite_Set
imports Option Power
begin

subsection {* Predicate for finite sets *}

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

simproc_setup finite_Collect ("finite (Collect P)") = {* K Set_Comprehension_Pointfree.simproc *}

lemma finite_induct [case_names empty insert, induct set: finite]:
-- {* Discharging @{text "x \<notin> F"} entails extra work. *}
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 `finite F`
proof induct
show "P {}" by fact
fix x F assume F: "finite F" and P: "P F"
show "P (insert x F)"
proof cases
assume "x \<in> F"
hence "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

subsubsection {* Choice principles *}

lemma ex_new_if_finite: -- "does not depend on def of finite at all"
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 {* A finite choice principle. Does not need the SOME choice operator. *}

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: "ALL x:A. P x (f x)" and ab: "P a b" by auto
show ?case (is "EX f. ?P f")
proof
show "?P(%x. if x = a then b else f x)" using f ab by auto
qed
qed

subsubsection {* Finite sets are the images of initial segments of natural numbers *}

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
hence "insert a A = f(n:=a) ` {i. i < Suc n}"
"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)
thus ?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 thus ?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
assume "\<exists>k<n. f n = f k"
hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
thus ?thesis using finB by simp
next
assume "\<not>(\<exists> k<n. f n = f k)"
hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
thus ?thesis using finB by simp
qed
qed

lemma finite_conv_nat_seg_image:
"finite A \<longleftrightarrow> (\<exists>(n::nat) 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::nat. f ` A = {i. i < n} \<and> inj_on f A"
proof -
from finite_imp_nat_seg_image_inj_on[OF `finite A`]
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 & ?f ` A = {i. i<n}"
by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
thus ?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}"

subsubsection {* Finiteness and common set operations *}

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})" .
hence "finite (insert x (A - {x}))" ..
also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
finally show ?thesis .
next
show "A \<subseteq> F ==> ?thesis" by fact
assume "x \<notin> A"
with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
qed
qed

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

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))" by (simp add: Un_Diff_Int)
also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` 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 A B) \<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_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 `x \<notin> B` have "B = f ` A - {x}" by blast
with B_A `x \<notin> B` `x = f y` `inj_on f A` `y \<in> A` have "B = f ` (A - {y})" by (simp add: inj_on_image_set_diff)
moreover from `inj_on f A` 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_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 finite_vimage_IntI:
"finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
apply (induct rule: finite_induct)
apply simp_all
apply (subst vimage_insert)
apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
done

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:
assumes fin: "finite (h -` F)" and surj: "surj h"
shows "finite F"
proof -
have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
finally show "finite F" .
qed

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_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. EX y. P y & 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)) ==> finite (SIGMA a:A. B a)"
by (unfold Sigma_def) 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 `B \<noteq> {}` 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 `A \<noteq> {}` 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_prod:
"finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
by(auto simp add: UNIV_Times_UNIV[symmetric] simp del: UNIV_Times_UNIV
dest: finite_cartesian_productD1 finite_cartesian_productD2)

lemma finite_Pow_iff [iff]:
"finite (Pow A) \<longleftrightarrow> finite A"
proof
assume "finite (Pow A)"
then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
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])

subsubsection {* Further induction rules on finite sets *}

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"
assumes 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 `finite F` `F \<subseteq> A`
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"
assumes "P A"
and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
shows "P {}"
proof -
have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
proof -
fix B :: "'a set"
assume "B \<subseteq> A"
with `finite A` have "finite B" by (rule rev_finite_subset)
from this `B \<subseteq> A` show "P (A - B)"
proof induct
case empty
from `P A` show ?case by simp
next
case (insert b B)
have "P (A - B - {b})"
proof (rule remove)
from `finite A` 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

subsection {* Class @{text finite}  *}

class finite =
assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
begin

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

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

end

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

lemma inj_graph: "inj (%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 => 'b) set)"
proof (rule finite_imageD)
let ?graph = "%f::'a => '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 default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)

instance unit :: finite

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

lemma finite_option_UNIV [simp]:
"finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)

instance option :: (finite) finite

subsection {* A basic fold functional for finite sets *}

text {* The intended behaviour is
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
if @{text f} is ``left-commutative'':
*}

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 fun_left_comm: "f y (f x z) = f x (f y z)"
using comp_fun_commute by (simp add: fun_eq_iff)

lemma commute_left_comp:
"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"

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)"

text{*A tempting alternative for the definiens is
@{term "if finite A then THE y. fold_graph f z A y else e"}.
It allows the removal of finiteness assumptions from the theorems
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
The proofs become ugly. It is not worth the effort. (???) *}

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

subsubsection{*From @{const fold_graph} to @{term fold}*}

context comp_fun_commute
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:
"fold_graph f z A y \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'"
proof (induct set: fold_graph)
case (insertI x A y) show ?case
proof (cases "x = a")
assume "x = a" with insertI show ?case by auto
next
assume "x \<noteq> a"
then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
using insertI by auto
have "f x y = f a (f x y')"
unfolding y by (rule fun_left_comm)
moreover have "fold_graph f z (insert x A - {a}) (f x y')"
using y' and `x \<noteq> a` and `x \<notin> A`
ultimately show ?case by fast
qed
qed simp

lemma fold_graph_insertE:
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 _ insertI1])

lemma fold_graph_determ:
"fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
proof (induct arbitrary: y set: fold_graph)
case (insertI x A y v)
from `fold_graph f z (insert x A) v` and `x \<notin> A`
obtain y' where "v = f x y'" and "fold_graph f z A y'"
by (rule fold_graph_insertE)
from `fold_graph f z A y'` have "y' = y" by (rule insertI)
with `v = f x y'` show "v = f x y" by simp
qed fast

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

text {* The base case for @{text fold}: *}

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

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

text{* The various recursion equations for @{const fold}: *}

lemma fold_insert [simp]:
assumes "finite A" and "x \<notin> A"
shows "fold f z (insert x A) = f x (fold f z A)"
proof (rule fold_equality)
fix z
from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
with `x \<notin> A` 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]
-- {* No more proofs involve these. *}

lemma fold_fun_left_comm:
"finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
proof (induct rule: finite_induct)
case empty then show ?case by simp
next
case (insert y A) then show ?case
by (simp add: fun_left_comm [of x])
qed

lemma fold_insert2:
"finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A)  = fold f (f x z) A"

lemma fold_rec:
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 `x \<in> A` 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) (simp add: `finite A`)+
finally show ?thesis .
qed

lemma fold_insert_remove:
assumes "finite A"
shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
proof -
from `finite A` 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}))"
by (rule fold_rec)
then show ?thesis by simp
qed

text{* Other properties of @{const fold}: *}

lemma fold_image:
assumes "finite A" and "inj_on g A"
shows "fold f z (g ` A) = fold (f \<circ> g) z A"
using assms
proof induction
case (insert a F)
interpret comp_fun_commute "\<lambda>x. f (g x)" by default (simp add: comp_fun_commute)
from insert show ?case by auto
qed simp

end

lemma fold_cong:
assumes "comp_fun_commute f" "comp_fun_commute g"
assumes "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 `finite A` cong proof (induct A)
case empty then show ?case by simp
next
case (insert x A)
interpret f: comp_fun_commute f by (fact `comp_fun_commute f`)
interpret g: comp_fun_commute g by (fact `comp_fun_commute g`)
from insert show ?case by simp
qed
with assms show ?thesis by simp
qed

text {* A simplified version for idempotent functions: *}

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

lemma fun_left_idem: "f x (f x z) = f x z"
using comp_fun_idem by (simp add: fun_eq_iff)

lemma fold_insert_idem:
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 fun_left_idem)
next
assume "x \<notin> A" then show ?thesis using assms by simp
qed

declare fold_insert [simp del] fold_insert_idem [simp]

lemma fold_insert_idem2:
"finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"

end

subsubsection {* Liftings to @{text comp_fun_commute} etc. *}

lemma (in comp_fun_commute) comp_comp_fun_commute:
"comp_fun_commute (f \<circ> g)"
proof

lemma (in comp_fun_idem) comp_comp_fun_idem:
"comp_fun_idem (f \<circ> g)"
by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)

lemma (in comp_fun_commute) comp_fun_commute_funpow:
"comp_fun_commute (\<lambda>x. f x ^^ g x)"
proof
fix y x
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)
def h \<equiv> "\<lambda>z. g z - 1"
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
then show ?case by (simp add: comp_assoc hyp)
qed
def h \<equiv> "\<lambda>z. if z = x then g x - 1 else g 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)
qed
qed
qed

subsubsection {* Expressing set operations via @{const fold} *}

lemma comp_fun_commute_const:
"comp_fun_commute (\<lambda>_. f)"
proof
qed rule

lemma comp_fun_idem_insert:
"comp_fun_idem insert"
proof
qed auto

lemma comp_fun_idem_remove:
"comp_fun_idem Set.remove"
proof
qed auto

lemma (in semilattice_inf) comp_fun_idem_inf:
"comp_fun_idem inf"
proof

lemma (in semilattice_sup) comp_fun_idem_sup:
"comp_fun_idem sup"
proof

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 `finite A` 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 `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
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 default (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
by (induct A)
(auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])

lemma inter_Set_filter:
assumes "finite B"
shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
using assms
by (induct B) (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"
using assms
proof -
interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by default 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"
using assms
proof -
interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by default 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"
using assms
proof -
interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by default 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"
using assms
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 default auto
show ?thesis using assms  by (induct B arbitrary: A) simp_all
qed

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

lemma product_fold:
assumes "finite A"
assumes "finite B"
shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A"
using assms unfolding Sigma_def
by (induct A)

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 `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
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 `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
qed

lemma Inf_fold_inf:
assumes "finite A"
shows "Inf A = fold inf top A"
using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)

lemma Sup_fold_sup:
assumes "finite A"
shows "Sup A = fold sup bot A"
using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)

lemma inf_INF_fold_inf:
assumes "finite A"
shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
proof (rule sym)
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 `finite A` show "?fold = ?inf"
by (induct A arbitrary: B)
qed

lemma sup_SUP_fold_sup:
assumes "finite A"
shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
proof (rule sym)
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 `finite A` show "?fold = ?sup"
by (induct A arbitrary: B)
qed

lemma INF_fold_inf:
assumes "finite A"
shows "INFI A f = fold (inf \<circ> f) top A"
using assms inf_INF_fold_inf [of A top] by simp

lemma SUP_fold_sup:
assumes "finite A"
shows "SUPR A f = fold (sup \<circ> f) bot A"
using assms sup_SUP_fold_sup [of A bot] by simp

end

subsection {* Locales as mini-packages for fold operations *}

subsubsection {* The natural case *}

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

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

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

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

lemma insert [simp]:
assumes "finite A" and "x \<notin> A"
shows "F (insert x A) = f x (F A)"
proof -
interpret comp_fun_commute f
by default (insert comp_fun_commute, simp add: fun_eq_iff)
from fold_insert assms
have "fold f z (insert x A) = f x (fold f z A)" by simp
with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
qed

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

lemma insert_remove:
assumes "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 {* With idempotency *}

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

declare insert [simp del]

lemma insert_idem [simp]:
assumes "finite A"
shows "F (insert x A) = f x (F A)"
proof -
interpret comp_fun_idem f
by default (insert comp_fun_commute comp_fun_idem, simp add: fun_eq_iff)
from fold_insert_idem assms
have "fold f z (insert x A) = f x (fold f z A)" by simp
with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
qed

end

subsection {* Finite cardinality *}

text {*
@{prop "card A \<equiv> LEAST n. EX f. A = {f i | i. i < n}"}
is ugly to work with.
But now that we have @{const fold} things are easy:
*}

definition card :: "'a set \<Rightarrow> nat" where
"card = folding.F (\<lambda>_. Suc) 0"

interpretation card!: folding "\<lambda>_. Suc" 0
where
"folding.F (\<lambda>_. Suc) 0 = card"
proof -
show "folding (\<lambda>_. Suc)" by default rule
then interpret card!: folding "\<lambda>_. Suc" 0 .
from card_def show "folding.F (\<lambda>_. Suc) 0 = card" by rule
qed

lemma card_infinite:
"\<not> finite A \<Longrightarrow> card A = 0"
by (fact card.infinite)

lemma card_empty:
"card {} = 0"
by (fact card.empty)

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, no_atp]:
"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_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:
"finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
apply(simp del:insert_Diff_single)
done

lemma card_Diff_singleton:
"finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"

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

lemma card_Diff_insert[simp]:
assumes "finite A" and "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: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
by (fact card.insert_remove)

lemma card_insert_le: "finite A ==> card A <= card (insert x A)"

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 <= 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 `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
qed
qed

lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
apply (induct rule: finite_induct)
apply simp
apply clarify
apply (subgoal_tac "finite A & A - {x} <= F")
prefer 2 apply (blast intro: finite_subset, atomize)
apply (drule_tac x = "A - {x}" in spec)
apply (case_tac "card A", auto)
done

lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
apply (simp add: psubset_eq linorder_not_le [symmetric])
apply (blast dest: card_seteq)
done

lemma card_Un_Int:
assumes "finite A" and "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 x A) then show ?case
by (auto simp add: insert_absorb Int_insert_left)
qed

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

lemma card_Diff_subset:
assumes "finite B" and "B \<subseteq> A"
shows "card (A - B) = card A - card B"
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 AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
proof -
have "A - B = A - A \<inter> B" by auto
thus ?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_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
apply (rule Suc_less_SucD)
done

lemma card_Diff2_less:
"finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
apply (case_tac "x = y")
apply (rule less_trans)
prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
done

lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
apply (case_tac "x : A")
done

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

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

lemma finite_psubset_induct[consumes 1, case_names psubset]:
assumes fin: "finite A"
and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
shows "P A"
using fin
proof (induct A taking: card rule: measure_induct_rule)
case (less A)
have fin: "finite A" by fact
have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
{ fix B
assume asm: "B \<subset> A"
from asm have "card B < card A" using psubset_card_mono fin by blast
moreover
from asm have "B \<subseteq> A" by auto
then have "finite B" using fin finite_subset by blast
ultimately
have "P B" using ih by simp
}
with fin show "P A" using major by blast
qed

text{* main cardinality theorem *}
lemma card_partition [rule_format]:
"finite C ==>
finite (\<Union> C) -->
(\<forall>c\<in>C. card c = k) -->
(\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
k * card(C) = card (\<Union> C)"
apply (erule finite_induct, simp)
finite_subset [of _ "\<Union> (insert x F)"])
done

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
fix x
show "x \<in> A"
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{*The form of a finite set of given cardinality*}

lemma card_eq_SucD:
assumes "card A = Suc k"
shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (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) =
(\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
apply(rule iffI)
apply(erule card_eq_SucD)
apply(auto)
apply(subst card.insert)
apply(auto intro:ccontr)
done

lemma card_le_Suc_iff: "finite A \<Longrightarrow>
Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
dest: subset_singletonD split: nat.splits if_splits)

lemma finite_fun_UNIVD2:
assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
shows "finite (UNIV :: 'b set)"
proof -
from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
by (rule finite_imageI)
moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f 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 card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
unfolding UNIV_bool by simp

subsubsection {* Cardinality of image *}

lemma card_image_le: "finite A ==> card (f ` A) <= card A"
apply (induct rule: finite_induct)
apply simp
done

lemma card_image:
assumes "inj_on f A"
shows "card (f ` A) = card A"
proof (cases "finite A")
case True then show ?thesis using assms by (induct A) simp_all
next
case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
with False show ?thesis by simp
qed

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 ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"

lemma eq_card_imp_inj_on:
"[| finite A; card(f ` A) = card A |] ==> inj_on f A"
apply (induct rule:finite_induct)
apply simp
apply(frule card_image_le[where f = f])
done

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

lemma card_inj_on_le:
"[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
apply (subgoal_tac "finite A")
apply (force intro: card_mono simp add: card_image [symmetric])
apply (blast intro: finite_imageD dest: finite_subset)
done

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

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

subsubsection {* Pigeonhole Principles *}

lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
by (auto dest: card_image less_irrefl_nat)

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

lemma pigeonhole_infinite_rel:
assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
shows "EX b:B. ~finite{a:A. R a b}"
proof -
let ?F = "%a. {b:B. R a b}"
from finite_Pow_iff[THEN iffD2, OF `finite B`]
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 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
{ assume "finite{a:A. R a b0}"
then have "finite {a\<in>A. ?F a = ?F a0}"
using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
}
with 1 `b0 : B` show ?thesis by blast
qed

subsubsection {* Cardinality of sums *}

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

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

subsubsection {* Cardinality of the Powerset *}

lemma card_Pow: "finite A ==> card (Pow A) = 2 ^ card A"
apply (induct rule: finite_induct)
apply (subst card_Un_disjoint, blast)
apply (blast, blast)
apply (subgoal_tac "inj_on (insert x) (Pow F)")
apply (subst mult_2)
apply (unfold inj_on_def)
apply (blast elim!: equalityE)
done

text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}

lemma dvd_partition:
"finite (Union C) ==>
ALL c : C. k dvd card c ==>
(ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
k dvd card (Union C)"
apply (frule finite_UnionD)
apply (rotate_tac -1)
apply (induct rule: finite_induct)
apply simp_all
apply clarify
apply (subst card_Un_disjoint)
done

subsubsection {* Relating injectivity and surjectivity *}

lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
apply(rule eq_card_imp_inj_on, assumption)
apply(frule finite_imageI)
apply(drule (1) card_seteq)
apply(erule card_image_le)
apply simp
done

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

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

corollary infinite_UNIV_nat [iff]:
"\<not> finite (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

(* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
lemma infinite_UNIV_char_0 [no_atp]:
"\<not> finite (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)"