# Theory Finite_Set

theory Finite_Set
imports Product_Type Sum_Type Fields
```(*  Title:      HOL/Finite_Set.thy
Author:     Tobias Nipkow
Author:     Lawrence C Paulson
Author:     Markus Wenzel
Author:     Andrei Popescu
*)

section ‹Finite sets›

theory Finite_Set
imports Product_Type Sum_Type Fields
begin

subsection ‹Predicate for finite sets›

context notes [[inductive_internals]]
begin

inductive finite :: "'a set ⇒ bool"
where
emptyI [simp, intro!]: "finite {}"
| insertI [simp, intro!]: "finite A ⟹ finite (insert a A)"

end

simproc_setup finite_Collect ("finite (Collect P)") = ‹K Set_Comprehension_Pointfree.simproc›

declare [[simproc del: finite_Collect]]

lemma finite_induct [case_names empty insert, induct set: finite]:
― ‹Discharging ‹x ∉ F› entails extra work.›
assumes "finite F"
assumes "P {}"
and insert: "⋀x F. finite F ⟹ x ∉ F ⟹ P F ⟹ P (insert x F)"
shows "P F"
using ‹finite F›
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 ∈ F"
then have "insert x F = F" by (rule insert_absorb)
with P show ?thesis by (simp only:)
next
assume "x ∉ F"
from F this P show ?thesis by (rule insert)
qed
qed

lemma infinite_finite_induct [case_names infinite empty insert]:
assumes infinite: "⋀A. ¬ finite A ⟹ P A"
and empty: "P {}"
and insert: "⋀x F. finite F ⟹ x ∉ F ⟹ P F ⟹ 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 ‹Choice principles›

lemma ex_new_if_finite: ― ‹does not depend on def of finite at all›
assumes "¬ finite (UNIV :: 'a set)" and "finite A"
shows "∃a::'a. a ∉ A"
proof -
from assms have "A ≠ 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 ⟹ ∀x∈A. ∃y. P x y ⟹ ∃f. ∀x∈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: "∀x∈A. P x (f x)" and ab: "P a b"
by auto
show ?case (is "∃f. ?P f")
proof
show "?P (λx. 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 "∃(n::nat) f. A = f ` {i. i < n} ∧ inj_on f {i. i < n}"
using assms
proof induct
case empty
show ?case
proof
show "∃f. {} = f ` {i::nat. i < 0} ∧ inj_on f {i. i < 0}"
by simp
qed
next
case (insert a A)
have notinA: "a ∉ 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} ⟹ 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 "∃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 ⟷ (∃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 "∃f n. f ` A = {i::nat. i < n} ∧ 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])
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 ≤ k}"

subsubsection ‹Finiteness and common set operations›

lemma rev_finite_subset: "finite B ⟹ A ⊆ B ⟹ 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 ⊆ insert x F" and r: "A - {x} ⊆ F ⟹ finite (A - {x})"
by fact+
show "finite A"
proof cases
assume x: "x ∈ A"
with A have "A - {x} ⊆ 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 ⊆ F"
using that by fact
assume "x ∉ A"
with A show "A ⊆ F"
qed
qed

lemma finite_subset: "A ⊆ B ⟹ finite B ⟹ finite A"
by (rule rev_finite_subset)

lemma finite_UnI:
assumes "finite F" and "finite G"
shows "finite (F ∪ G)"
using assms by induct simp_all

lemma finite_Un [iff]: "finite (F ∪ G) ⟷ finite F ∧ finite G"
by (blast intro: finite_UnI finite_subset [of _ "F ∪ G"])

lemma finite_insert [simp]: "finite (insert a A) ⟷ finite A"
proof -
have "finite {a} ∧ finite A ⟷ finite A" by simp
then have "finite ({a} ∪ A) ⟷ finite A" by (simp only: finite_Un)
then show ?thesis by simp
qed

lemma finite_Int [simp, intro]: "finite F ∨ finite G ⟹ finite (F ∩ G)"
by (blast intro: finite_subset)

lemma finite_Collect_conjI [simp, intro]:
"finite {x. P x} ∨ finite {x. Q x} ⟹ finite {x. P x ∧ Q x}"

lemma finite_Collect_disjI [simp]:
"finite {x. P x ∨ Q x} ⟷ finite {x. P x} ∧ finite {x. Q x}"

lemma finite_Diff [simp, intro]: "finite A ⟹ finite (A - B)"
by (rule finite_subset, rule Diff_subset)

lemma finite_Diff2 [simp]:
assumes "finite B"
shows "finite (A - B) ⟷ finite A"
proof -
have "finite A ⟷ finite ((A - B) ∪ (A ∩ B))"
also have "… ⟷ finite (A - B)"
using ‹finite B› by simp
finally show ?thesis ..
qed

lemma finite_Diff_insert [iff]: "finite (A - insert a B) ⟷ finite (A - B)"
proof -
have "finite (A - B) ⟷ 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) ⟹ finite (- A) ⟷ finite (UNIV :: 'a set)"

lemma finite_Collect_not [simp]:
"finite {x :: 'a. P x} ⟹ finite {x. ¬ P x} ⟷ finite (UNIV :: 'a set)"

lemma finite_Union [simp, intro]:
"finite A ⟹ (⋀M. M ∈ A ⟹ finite M) ⟹ finite (⋃A)"
by (induct rule: finite_induct) simp_all

lemma finite_UN_I [intro]:
"finite A ⟹ (⋀a. a ∈ A ⟹ finite (B a)) ⟹ finite (⋃a∈A. B a)"
by (induct rule: finite_induct) simp_all

lemma finite_UN [simp]: "finite A ⟹ finite (UNION A B) ⟷ (∀x∈A. finite (B x))"
by (blast intro: finite_subset)

lemma finite_Inter [intro]: "∃A∈M. finite A ⟹ finite (⋂M)"
by (blast intro: Inter_lower finite_subset)

lemma finite_INT [intro]: "∃x∈I. finite (A x) ⟹ finite (⋂x∈I. A x)"
by (blast intro: INT_lower finite_subset)

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

lemma finite_image_set [simp]: "finite {x. P x} ⟹ finite {f x |x. P x}"

lemma finite_image_set2:
"finite {x. P x} ⟹ finite {y. Q y} ⟹ finite {f x y |x y. P x ∧ Q y}"
by (rule finite_subset [where B = "⋃x ∈ {x. P x}. ⋃y ∈ {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 ∈ A"
by blast
from B_A ‹x ∉ B› have "B = f ` A - {x}"
by blast
with B_A ‹x ∉ B› ‹x = f y› ‹inj_on f A› ‹y ∈ A› have "B = f ` (A - {y})"
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_image_iff: "inj_on f A ⟹ finite (f ` A) ⟷ finite A"
using finite_imageD by blast

lemma finite_surj: "finite A ⟹ B ⊆ f ` A ⟹ finite B"
by (erule finite_subset) (rule finite_imageI)

lemma finite_range_imageI: "finite (range g) ⟹ finite (range (λx. f (g x)))"
by (drule finite_imageI) (simp add: range_composition)

lemma finite_subset_image:
assumes "finite B"
shows "B ⊆ f ` A ⟹ ∃C⊆A. finite C ∧ 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  (* slow *)
qed

lemma finite_vimage_IntI: "finite F ⟹ inj_on h A ⟹ finite (h -` F ∩ 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_finite_vimage_IntI:
assumes "finite F"
and "⋀y. y ∈ F ⟹ finite ((h -` {y}) ∩ A)"
shows "finite (h -` F ∩ A)"
proof -
have *: "h -` F ∩ A = (⋃ y∈F. (h -` {y}) ∩ A)"
by blast
show ?thesis
by (simp only: * assms finite_UN_I)
qed

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

lemma finite_vimageD': "finite (f -` A) ⟹ A ⊆ range f ⟹ finite A"
by (auto simp add: subset_image_iff intro: finite_subset[rotated])

lemma finite_vimageD: "finite (h -` F) ⟹ surj h ⟹ finite F"
by (auto dest: finite_vimageD')

lemma finite_vimage_iff: "bij h ⟹ finite (h -` F) ⟷ finite F"
unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)

lemma finite_Collect_bex [simp]:
assumes "finite A"
shows "finite {x. ∃y∈A. Q x y} ⟷ (∀y∈A. finite {x. Q x y})"
proof -
have "{x. ∃y∈A. Q x y} = (⋃y∈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. ∃y. P y ∧ Q x y} ⟷ (∀y. P y ⟶ finite {x. Q x y})"
proof -
have "{x. ∃y. P y ∧ Q x y} = (⋃y∈{y. P y}. {x. Q x y})"
by auto
with assms show ?thesis
by simp
qed

lemma finite_Plus: "finite A ⟹ finite B ⟹ 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 ⊆ 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 ⊆ 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) ⟷ finite A ∧ finite B"
by (auto intro: finite_PlusD finite_Plus)

lemma finite_Plus_UNIV_iff [simp]:
"finite (UNIV :: ('a + 'b) set) ⟷ finite (UNIV :: 'a set) ∧ finite (UNIV :: 'b set)"
by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)

lemma finite_SigmaI [simp, intro]:
"finite A ⟹ (⋀a. a∈A ⟹ finite (B a)) ⟹ finite (SIGMA a:A. B a)"
unfolding Sigma_def by blast

lemma finite_SigmaI2:
assumes "finite {x∈A. B x ≠ {}}"
and "⋀a. a ∈ A ⟹ finite (B a)"
shows "finite (Sigma A B)"
proof -
from assms have "finite (Sigma {x∈A. B x ≠ {}} B)"
by auto
also have "Sigma {x:A. B x ≠ {}} B = Sigma A B"
by auto
finally show ?thesis .
qed

lemma finite_cartesian_product: "finite A ⟹ finite B ⟹ finite (A × B)"
by (rule finite_SigmaI)

lemma finite_Prod_UNIV:
"finite (UNIV :: 'a set) ⟹ finite (UNIV :: 'b set) ⟹ finite (UNIV :: ('a × 'b) set)"
by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)

lemma finite_cartesian_productD1:
assumes "finite (A × B)" and "B ≠ {}"
shows "finite A"
proof -
from assms obtain n f where "A × B = f ` {i::nat. i < n}"
then have "fst ` (A × B) = fst ` f ` {i::nat. i < n}"
by simp
with ‹B ≠ {}› have "A = (fst ∘ f) ` {i::nat. i < n}"
then have "∃n f. A = f ` {i::nat. i < n}"
by blast
then show ?thesis
qed

lemma finite_cartesian_productD2:
assumes "finite (A × B)" and "A ≠ {}"
shows "finite B"
proof -
from assms obtain n f where "A × B = f ` {i::nat. i < n}"
then have "snd ` (A × B) = snd ` f ` {i::nat. i < n}"
by simp
with ‹A ≠ {}› have "B = (snd ∘ f) ` {i::nat. i < n}"
then have "∃n f. B = f ` {i::nat. i < n}"
by blast
then show ?thesis
qed

lemma finite_cartesian_product_iff:
"finite (A × B) ⟷ (A = {} ∨ B = {} ∨ (finite A ∧ finite B))"
by (auto dest: finite_cartesian_productD1 finite_cartesian_productD2 finite_cartesian_product)

lemma finite_prod:
"finite (UNIV :: ('a × 'b) set) ⟷ finite (UNIV :: 'a set) ∧ finite (UNIV :: 'b set)"
using finite_cartesian_product_iff[of UNIV UNIV] by simp

lemma finite_Pow_iff [iff]: "finite (Pow A) ⟷ finite A"
proof
assume "finite (Pow A)"
then have "finite ((λx. {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 ⟹ finite {B. B ⊆ A}"

lemma finite_set: "finite (UNIV :: 'a set set) ⟷ finite (UNIV :: 'a set)"
by (simp only: finite_Pow_iff Pow_UNIV[symmetric])

lemma finite_UnionD: "finite (⋃A) ⟹ finite A"
by (blast intro: finite_subset [OF subset_Pow_Union])

lemma finite_bind:
assumes "finite S"
assumes "∀x ∈ S. finite (f x)"
shows "finite (Set.bind S f)"
using assms by (simp add: bind_UNION)

lemma finite_filter [simp]: "finite S ⟹ 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. ∀x. (x ∈ A ⟶ f x ∈ B) ∧ (x ∉ A ⟶ f x = d)}" (is "finite ?S")
proof -
let ?F = "λf. {(a,b). a ∈ A ∧ b = f a}"
have "?F ` ?S ⊆ Pow(A × 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 "¬ finite {a. P a}"
shows "∃a. P a"
proof (rule classical)
assume "¬ ?thesis"
with assms show ?thesis by auto
qed

subsubsection ‹Further induction rules on finite sets›

lemma finite_ne_induct [case_names singleton insert, consumes 2]:
assumes "finite F" and "F ≠ {}"
assumes "⋀x. P {x}"
and "⋀x F. finite F ⟹ F ≠ {} ⟹ x ∉ F ⟹ P F  ⟹ 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 ⊆ A"
and empty: "P {}"
and insert: "⋀a F. finite F ⟹ a ∈ A ⟹ a ∉ F ⟹ P F ⟹ P (insert a F)"
shows "P F"
using ‹finite F› ‹F ⊆ A›
proof induct
show "P {}" by fact
next
fix x F
assume "finite F" and "x ∉ F" and P: "F ⊆ A ⟹ P F" and i: "insert x F ⊆ A"
show "P (insert x F)"
proof (rule insert)
from i show "x ∈ A" by blast
from i have "F ⊆ A" by blast
with P show "P F" .
show "finite F" by fact
show "x ∉ F" by fact
qed
qed

lemma finite_empty_induct:
assumes "finite A"
and "P A"
and remove: "⋀a A. finite A ⟹ a ∈ A ⟹ P A ⟹ P (A - {a})"
shows "P {}"
proof -
have "P (A - B)" if "B ⊆ A" for B :: "'a set"
proof -
from ‹finite A› that have "finite B"
by (rule rev_finite_subset)
from this ‹B ⊆ 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 ∈ 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 ≠ c}"
and const: "P (λa. c)"
and update: "⋀a b f. finite {a. f a ≠ c} ⟹ f a = c ⟹ b ≠ c ⟹ P f ⟹ P (f(a := b))"
shows "P f"
using finite
proof (induct "{a. f a ≠ c}" arbitrary: f)
case empty
with const show ?case by simp
next
case (insert a A)
then have "A = {a'. (f(a := c)) a' ≠ c}" and "f a ≠ c"
by auto
with ‹finite A› have "finite {a'. (f(a := c)) a' ≠ c}"
by simp
have "(f(a := c)) a = c"
by simp
from insert ‹A = {a'. (f(a := c)) a' ≠ c}› have "P (f(a := c))"
by simp
with ‹finite {a'. (f(a := c)) a' ≠ c}› ‹(f(a := c)) a = c› ‹f a ≠ c›
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 ⊆ A"
and empty: "P {}"
and insert: "⋀a F. ⟦finite F; a ∈ A; F ⊆ A; a ∉ F; P F ⟧ ⟹ 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 ∉ F" and
P: "F ⊆ A ⟹ P F" and i: "insert x F ⊆ A"
show "P (insert x F)"
proof (rule insert)
from i show "x ∈ A" by blast
from i have "F ⊆ A" by blast
with P show "P F" .
show "finite F" by fact
show "x ∉ F" by fact
show "F ⊆ A" by fact
qed
qed

subsection ‹Class ‹finite››

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) ⟷ 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 (λf. {(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 = "λf::'a ⇒ 'b. {(x, y). y = f x}"
have "range ?graph ⊆ 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 ‹A basic fold functional for finite sets›

text ‹The intended behaviour is
‹fold f z {x⇩1, …, x⇩n} = f x⇩1 (… (f x⇩n z)…)›
if ‹f› is ``left-commutative'':
›

locale comp_fun_commute =
fixes f :: "'a ⇒ 'b ⇒ 'b"
assumes comp_fun_commute: "f y ∘ f x = f x ∘ 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 ∘ (f x ∘ g) = f x ∘ (f y ∘ g)"

end

inductive fold_graph :: "('a ⇒ 'b ⇒ 'b) ⇒ 'b ⇒ 'a set ⇒ 'b ⇒ bool"
for f :: "'a ⇒ 'b ⇒ 'b" and z :: 'b
where
emptyI [intro]: "fold_graph f z {} z"
| insertI [intro]: "x ∉ A ⟹ fold_graph f z A y ⟹ 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 ∧ x ∈ B"
if "fold_graph g z A x"
"⋀a b. a ∈ A ⟹ b ∈ B ⟹ f a b = g a b"
"⋀a b. a ∈ A ⟹ b ∈ B ⟹ g a b ∈ B"
"z ∈ B"
using that(1-3)
proof (induction rule: fold_graph.induct)
case (insertI x A y)
have "fold_graph f z A y" "y ∈ B"
unfolding atomize_conj
by (rule insertI.IH) (auto intro: insertI.prems)
then have "g x y ∈ 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 "⋀a b. a ∈ A ⟹ b ∈ B ⟹ f a b = g a b"
"⋀a b. a ∈ A ⟹ b ∈ B ⟹ g a b ∈ B"
"z ∈ 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 ⇒ 'b ⇒ 'b) ⇒ 'b ⇒ 'a set ⇒ '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 "⋀a b. a ∈ A ⟹ b ∈ B ⟹ f a b = g a b"
"⋀a b. a ∈ A ⟹ b ∈ B ⟹ g a b ∈ B"
"z ∈ B"
unfolding Finite_Set.fold_def
by (subst fold_graph_closed_eq[where B=B and g=g]) (auto simp: that)

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
‹fold_comm›, ‹fold_reindex› and ‹fold_distrib›.
The proofs become ugly. It is not worth the effort. (???)
›

lemma finite_imp_fold_graph: "finite A ⟹ ∃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 ⟹ a ∈ A ⟹ ∃y'. y = f a y' ∧ fold_graph f z (A - {a}) y'"
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
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 ≠ a› and ‹x ∉ A›
ultimately show ?thesis
by fast
qed
qed

lemma fold_graph_insertE:
assumes "fold_graph f z (insert x A) v" and "x ∉ 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 ⟹ fold_graph f z A y ⟹ y = x"
proof (induct arbitrary: y set: fold_graph)
case emptyI
then show ?case by fast
next
case (insertI x A y v)
from ‹fold_graph f z (insert x A) v› and ‹x ∉ 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

lemma fold_equality: "fold_graph f z A y ⟹ 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 "∃x. fold_graph f z A x"
by (rule finite_imp_fold_graph)
moreover note fold_graph_determ
ultimately have "∃!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 ‹The base case for ‹fold›:›

lemma (in -) fold_infinite [simp]: "¬ finite A ⟹ 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 ‹The various recursion equations for @{const fold}:›

lemma fold_insert [simp]:
assumes "finite A" and "x ∉ 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 ∉ 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 ⟹ 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
then show ?case
by (simp add: fun_left_comm [of x])
qed

lemma fold_insert2: "finite A ⟹ x ∉ A ⟹ fold f z (insert x A)  = fold f (f x z) A"

lemma fold_rec:
assumes "finite A" and "x ∈ A"
shows "fold f z A = f x (fold f z (A - {x}))"
proof -
have A: "A = insert x (A - {x})"
using ‹x ∈ A› by blast
then have "fold f z A = fold f z (insert x (A - {x}))"
by simp
also have "… = 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 ∈ 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

lemma fold_set_union_disj:
assumes "finite A" "finite B" "A ∩ B = {}"
shows "Finite_Set.fold f z (A ∪ B) = Finite_Set.fold f (Finite_Set.fold f z A) B"
using assms(2,1,3) by induct simp_all

end

text ‹Other properties of @{const fold}:›

lemma fold_image:
assumes "inj_on g A"
shows "fold f z (g ` A) = fold (f ∘ 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
have "fold_graph f z (g ` A) = fold_graph (f ∘ g) z A"
proof
fix w
show "fold_graph f z (g ` A) w ⟷ fold_graph (f ∘ g) z A w" (is "?P ⟷ ?Q")
proof
assume ?P
then show ?Q
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 ‹inj_on g B› ‹x ∉ A› ‹insert x A = image g B› obtain x' A'
where "x' ∉ 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 ∘ g) z A' r"
by (auto intro: insertI.hyps)
with ‹x' ∉ A'› have "fold_graph (f ∘ g) z (insert x' A') ((f ∘ g) x' r)"
by (rule fold_graph.insertI)
then show ?case
by simp
qed
next
assume ?Q
then show ?P
using assms
proof induct
case emptyI
then show ?case
by (auto intro: fold_graph.emptyI)
next
case (insertI x A r)
from ‹x ∉ A› insertI.prems have "g x ∉ 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
with True assms show ?thesis
qed

lemma fold_cong:
assumes "comp_fun_commute f" "comp_fun_commute g"
and "finite A"
and cong: "⋀x. x ∈ A ⟹ 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
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 ∘ 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 ∈ A"
then obtain B where "A = insert x B" and "x ∉ B"
by (rule set_insert)
then show ?thesis
using assms by (simp add: comp_fun_idem fun_left_idem)
next
assume "x ∉ A"
then show ?thesis
using assms by simp
qed

declare fold_insert [simp del] fold_insert_idem [simp]

lemma fold_insert_idem2: "finite A ⟹ fold f z (insert x A) = fold f (f x z) A"

end

subsubsection ‹Liftings to ‹comp_fun_commute› etc.›

lemma (in comp_fun_commute) comp_comp_fun_commute: "comp_fun_commute (f ∘ g)"

lemma (in comp_fun_idem) comp_comp_fun_idem: "comp_fun_idem (f ∘ 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 (λx. f x ^^ g x)"
proof
show "f y ^^ g y ∘ f x ^^ g x = f x ^^ g x ∘ f y ^^ g y" for x 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 ∘ f x = f x ∘ 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 ∘ f x = f x ∘ f y ^^ h y"
by auto
from Suc h_def have "g y = Suc (h y)"
by simp
then 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 ∘ f x ^^ h x = f x ^^ h x ∘ f y ^^ h y"
by auto
with False h_def have hyp2: "f y ^^ g y ∘ f x ^^ h x = f x ^^ h x ∘ 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 ‹Expressing set operations via @{const fold}›

lemma comp_fun_commute_const: "comp_fun_commute (λ_. f)"
by standard rule

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 ∪ 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  (* slow *)
then show ?thesis ..
qed

lemma comp_fun_commute_filter_fold:
"comp_fun_commute (λx 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 (λx A'. if P x then Set.insert x A' else A') {} A"
using assms
by induct
(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 ∩ B = Set.filter (λx. x ∈ A) B"
using assms
by induct (auto simp: Set.filter_def)

lemma image_fold_insert:
assumes "finite A"
shows "image f A = fold (λk A. Set.insert (f k) A) {} A"
proof -
interpret comp_fun_commute "λk 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 (λk s. s ∧ P k) True A"
proof -
interpret comp_fun_commute "λk s. s ∧ 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 (λk s. s ∨ P k) False A"
proof -
interpret comp_fun_commute "λk s. s ∨ P k"
by standard auto
show ?thesis
using assms by (induct A) auto
qed

lemma comp_fun_commute_Pow_fold: "comp_fun_commute (λx A. A ∪ Set.insert x ` A)"
by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast  (* somewhat slow *)

lemma Pow_fold:
assumes "finite A"
shows "Pow A = fold (λx A. A ∪ Set.insert x ` A) {{}} A"
proof -
interpret comp_fun_commute "λx A. A ∪ 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 "(⋃y∈B. {(x, y)}) ∪ A = fold (λy. Set.insert (x, y)) A B"
proof -
interpret comp_fun_commute "λy. 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 ⟹ comp_fun_commute (λx z. fold (λy. Set.insert (x, y)) z B)"
by standard (auto simp: fold_union_pair [symmetric])

lemma product_fold:
assumes "finite A" "finite B"
shows "A × B = fold (λx z. fold (λy. 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) (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 ‹finite A› 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 ⟹ 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 ⟹ 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 (INFIMUM A f) = fold (inf ∘ f) B A" (is "?inf = ?fold")
proof -
interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
interpret comp_fun_idem "inf ∘ f" by (fact comp_comp_fun_idem)
from ‹finite A› 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 (SUPREMUM A f) = fold (sup ∘ f) B A" (is "?sup = ?fold")
proof -
interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
interpret comp_fun_idem "sup ∘ f" by (fact comp_comp_fun_idem)
from ‹finite A› have "?fold = ?sup"
by (induct A arbitrary: B) (simp_all add: sup_left_commute)
then show ?thesis ..
qed

lemma INF_fold_inf: "finite A ⟹ INFIMUM A f = fold (inf ∘ f) top A"
using inf_INF_fold_inf [of A top] by simp

lemma SUP_fold_sup: "finite A ⟹ SUPREMUM A f = fold (sup ∘ f) bot A"
using 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 ⇒ 'b ⇒ 'b" and z :: "'b"
assumes comp_fun_commute: "f y ∘ f x = f x ∘ f y"
begin

interpretation fold?: comp_fun_commute f
by standard (use comp_fun_commute in ‹simp add: fun_eq_iff›)

definition F :: "'a set ⇒ 'b"
where eq_fold: "F A = fold f z A"

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

lemma infinite [simp]: "¬ finite A ⟹ F A = z"

lemma insert [simp]:
assumes "finite A" and "x ∉ A"
shows "F (insert x A) = f x (F A)"
proof -
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 ∈ A"
shows "F A = f x (F (A - {x}))"
proof -
from ‹x ∈ A› obtain B where A: "A = insert x B" and "x ∉ B"
by (auto dest: mk_disjoint_insert)
moreover from ‹finite A› A have "finite B" by simp
ultimately show ?thesis by simp
qed

lemma insert_remove: "finite A ⟹ F (insert x A) = f x (F (A - {x}))"
by (cases "x ∈ A") (simp_all add: remove insert_absorb)

end

subsubsection ‹With idempotency›

locale folding_idem = folding +
assumes comp_fun_idem: "f x ∘ f x = f x"
begin

declare insert [simp del]

interpretation fold?: comp_fun_idem f
by standard (insert comp_fun_commute comp_fun_idem, simp add: fun_eq_iff)

lemma insert_idem [simp]:
assumes "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 ‹finite A› show ?thesis by (simp add: eq_fold fun_eq_iff)
qed

end

subsection ‹Finite cardinality›

text ‹
@{prop "card A ≡ LEAST n. ∃f. A = {f i |i. i < n}"}
is ugly to work with.
But now that we have @{const fold} things are easy:
›

global_interpretation card: folding "λ_. Suc" 0
defines card = "folding.F (λ_. Suc) 0"
by standard rule

lemma card_infinite: "¬ finite A ⟹ card A = 0"
by (fact card.infinite)

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

lemma card_insert_disjoint: "finite A ⟹ x ∉ A ⟹ card (insert x A) = Suc (card A)"
by (fact card.insert)

lemma card_insert_if: "finite A ⟹ card (insert x A) = (if x ∈ 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 ⟹ finite A"
by (rule ccontr) simp

lemma card_0_eq [simp]: "finite A ⟹ card A = 0 ⟷ A = {}"
by (auto dest: mk_disjoint_insert)

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

lemma card_eq_0_iff: "card A = 0 ⟷ A = {} ∨ ¬ finite A"
by auto

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

lemma card_gt_0_iff: "0 < card A ⟷ A ≠ {} ∧ finite A"
by (simp add: neq0_conv [symmetric] card_eq_0_iff)

lemma card_Suc_Diff1: "finite A ⟹ x ∈ A ⟹ Suc (card (A - {x})) = card A"
apply (rule insert_Diff [THEN subst, where t = A])
apply assumption
apply (simp del: insert_Diff_single)
done

lemma card_insert_le_m1: "n > 0 ⟹ card y ≤ n - 1 ⟹ card (insert x y) ≤ n"
apply (cases "finite y")
apply (cases "x ∈ y")
apply (auto simp: insert_absorb)
done

lemma card_Diff_singleton: "finite A ⟹ x ∈ A ⟹ card (A - {x}) = card A - 1"

lemma card_Diff_singleton_if:
"finite A ⟹ card (A - {x}) = (if x ∈ A then card A - 1 else card A)"

lemma card_Diff_insert[simp]:
assumes "finite A" and "a ∈ A" and "a ∉ 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 ⊆ B"
shows "card A ≤ 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 ∈ B"
by simp
from insert have "A ⊆ B - {x}" and "finite (B - {x})"
by auto
with insert.hyps have "card A ≤ card (B - {x})"
by auto
with ‹finite A› ‹x ∉ A› ‹finite B› ‹x ∈ 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 (simp add: card_Diff_singleton_if split: if_split_asm)
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" "finite B"
shows "card A + card B = card (A ∪ B) + card (A ∩ 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 ⟹ finite B ⟹ A ∩ B = {} ⟹ card (A ∪ B) = card A + card B"
using card_Un_Int [of A B] by simp

lemma card_Un_le: "card (A ∪ B) ≤ card A + card B"
apply (cases "finite A")
apply (cases "finite B")
apply (use le_iff_add card_Un_Int in blast)
apply simp
apply simp
done

lemma card_Diff_subset:
assumes "finite B"
and "B ⊆ 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 ∩ B)"
shows "card (A - B) = card A - card (A ∩ B)"
proof -
have "A - B = A - A ∩ B" by auto
with assms show ?thesis
qed

lemma diff_card_le_card_Diff:
assumes "finite B"
shows "card A - card B ≤ card (A - B)"
proof -
have "card A - card B ≤ card A - card (A ∩ B)"
using card_mono[OF assms Int_lower2, of A] by arith
also have "… = 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"
by (rule Suc_less_SucD) (simp add: card_Suc_Diff1 del: card_Diff_insert)

lemma card_Diff2_less: "finite A ⟹ x ∈ A ⟹ y ∈ A ⟹ card (A - {x} - {y}) < card A"
apply (cases "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"
by (cases "x ∈ A") (simp_all add: card_Diff1_less less_imp_le)

lemma card_psubset: "finite B ⟹ A ⊆ B ⟹ card A < card B ⟹ A < B"
by (erule psubsetI) blast

lemma card_le_inj:
assumes fA: "finite A"
and fB: "finite B"
and c: "card A ≤ card B"
shows "∃f. f ` A ⊆ B ∧ 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 ≤ card t"
by simp
from "2.prems"(3) [OF "2.hyps"(1) cst]
obtain f where "f ` s ⊆ t" "inj_on f s"
by blast
with "2.prems"(2) "2.hyps"(2) show ?case
apply -
apply (rule exI[where x = "λz. if z = x then y else f z"])
done
qed
qed

lemma card_subset_eq:
assumes fB: "finite B"
and AB: "A ⊆ 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 ∩ (B - A) = {}"
by blast
have eq: "A ∪ (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 ∉ F ⟹ ∀c1 ∈ insert x F. ∀c2 ∈ insert x F. c1 ≠ c2 ⟶ c1 ∩ c2 = {} ⟹ x ∩ ⋃F = {}"
by auto  (* somewhat slow *)

lemma finite_psubset_induct [consumes 1, case_names psubset]:
assumes finite: "finite A"
and major: "⋀A. finite A ⟹ (⋀B. B ⊂ A ⟹ P B) ⟹ 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 ⟹ finite B ⟹ P B" for B by fact
have "P B" if "B ⊂ A" for B
proof -
from that have "card B < card A"
using psubset_card_mono fin by blast
moreover
from that have "B ⊆ 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: "⋀T. T ⊂ S ⟹ P T ⟹ ∃s∈S - T. P (insert s T)"
shows "P S"
proof -
have "0 ≤ card S" by simp
then have "∃T ⊆ S. card T = card S ∧ P T"
proof (induct rule: dec_induct)
case base with ‹P {}›
show ?case
by (intro exI[of _ "{}"]) auto
next
case (step n)
then obtain T where T: "T ⊆ S" "card T = n" "P T"
by auto
with ‹n < card S› have "T ⊂ S" "P T"
by auto
with select[of T] obtain s where "s ∈ S" "s ∉ T" "P (insert s T)"
by auto
with step(2) T ‹finite S› show ?case
by (intro exI[of _ "insert s T"]) (auto dest: finite_subset)
qed
with ‹finite S› 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: "¬ finite B ⟹ P B"
and remove: "⋀A. finite A ⟹ A ≠ {} ⟹ A ⊆ B ⟹ (⋀x. x ∈ A ⟹ P (A - {x})) ⟹ 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 ⊆ 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 ‹A ⊆ B› and ‹finite B› have "finite A"
by (rule finite_subset)
moreover from Suc.hyps have "A ≠ {}" by auto
moreover note ‹A ⊆ B›
moreover have "P (A - {x})" if x: "x ∈ A" for x
using x Suc.prems ‹Suc n = card A› 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 ⇒ bool"
assumes "finite B"
and "P {}"
and "⋀A. finite A ⟹ A ≠ {} ⟹ A ⊆ B ⟹ (⋀x. x ∈ A ⟹ P (A - {x})) ⟹ P A"
defines "B' ≡ B"
shows "P B'"
by (induct B' rule: remove_induct) (simp_all add: assms)

text ‹Main cardinality theorem.›
lemma card_partition [rule_format]:
"finite C ⟹ finite (⋃C) ⟹ (∀c∈C. card c = k) ⟹
(∀c1 ∈ C. ∀c2 ∈ C. c1 ≠ c2 ⟶ c1 ∩ c2 = {}) ⟹
k * card C = card (⋃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 _ "⋃(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 ⊆ UNIV" by simp
show "UNIV ⊆ A"
proof
show "x ∈ A" for x
proof (rule ccontr)
assume "x ∉ A"
then have "A ⊂ 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 "∃b B. A = insert b B ∧ b ∉ B ∧ card B = k ∧ (k = 0 ⟶ B = {})"
proof -
have fin: "finite A"
using assms by (auto intro: ccontr)
moreover have "card A ≠ 0"
using assms by auto
ultimately obtain b where b: "b ∈ A"
by auto
show ?thesis
proof (intro exI conjI)
show "A = insert b (A - {b})"
using b by blast
show "b ∉ A - {b}"
by blast
show "card (A - {b}) = k" and "k = 0 ⟶ A - {b} = {}"
using assms b fin by (fastforce dest: mk_disjoint_insert)+
qed
qed

lemma card_Suc_eq:
"card A = Suc k ⟷
(∃b B. A = insert b B ∧ b ∉ B ∧ card B = k ∧ (k = 0 ⟶ B = {}))"
apply (auto elim!: card_eq_SucD)
apply (subst card.insert)
done

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 ⟷ card A = 1"
unfolding is_singleton_def
by (auto elim!: card_1_singletonE is_singletonE simp del: One_nat_def)

lemma card_le_Suc_iff:
"finite A ⟹ Suc n ≤ card A = (∃a B. A = insert a B ∧ a ∉ B ∧ n ≤ card B ∧ 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 ⇒ 'b) set)"
shows "finite (UNIV :: 'b set)"
proof -
from fin have "finite (range (λf :: 'a ⇒ 'b. f arbitrary))" for arbitrary
by (rule finite_imageI)
moreover have "UNIV = range (λf :: 'a ⇒ '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 "¬ finite A"
shows "∃B. finite B ∧ card B = n ∧ B ⊆ A"
proof (induction n)
case 0
show ?case by (intro exI[of _ "{}"]) auto
next
case (Suc n)
then obtain B where B: "finite B ∧ card B = n ∧ B ⊆ A" ..
with ‹¬ finite A› have "A ≠ B" by auto
with B have "B ⊂ A" by auto
then have "∃x. x ∈ A - B"
by (elim psubset_imp_ex_mem)
then obtain x where x: "x ∈ A - B" ..
with B have "finite (insert x B) ∧ card (insert x B) = Suc n ∧ insert x B ⊆ A"
by auto
then show "∃B. finite B ∧ card B = Suc n ∧ B ⊆ A" ..
qed

text ‹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.›

lemma finite_if_finite_subsets_card_bdd:
assumes "⋀G. G ⊆ F ⟹ finite G ⟹ card G ≤ C"
shows "finite F ∧ card F ≤ C"
proof (cases "finite F")
case False
obtain n::nat where n: "n > max C 0" by auto
obtain G where G: "G ⊆ F" "card G = n" using infinite_arbitrarily_large[OF False] by auto
hence "finite G" using ‹n > max C 0› 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

subsubsection ‹Cardinality of image›

lemma card_image_le: "finite A ⟹ card (f ` A) ≤ card A"
by (induct rule: finite_induct) (simp_all add: le_SucI card_insert_if)

lemma card_image: "inj_on f A ⟹ card (f ` A) = card A"
proof (induct A rule: infinite_finite_induct)
case (infinite A)
then have "¬ 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 ⟹ card A = card B"
by (auto simp: card_image bij_betw_def)

lemma endo_inj_surj: "finite A ⟹ f ` A ⊆ A ⟹ inj_on f A ⟹ 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 ⟹ inj_on f A ⟷ 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 ⊆ B" "finite B"
shows "card A ≤ 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 surj_card_le: "finite A ⟹ B ⊆ f ` A ⟹ card B ≤ card A"
by (blast intro: card_image_le card_mono le_trans)

lemma card_bij_eq:
"inj_on f A ⟹ f ` A ⊆ B ⟹ inj_on g B ⟹ g ` B ⊆ A ⟹ finite A ⟹ finite B
⟹ card A = card B"
by (auto intro: le_antisym card_inj_on_le)

lemma bij_betw_finite: "bij_betw f A B ⟹ finite A ⟷ finite B"
unfolding bij_betw_def using finite_imageD [of f A] by auto

lemma inj_on_finite: "inj_on f A ⟹ f ` A ≤ B ⟹ finite B ⟹ finite A"
using finite_imageD finite_subset by blast

lemma card_vimage_inj: "inj f ⟹ A ⊆ range f ⟹ 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])

subsubsection ‹Pigeonhole Principles›

lemma pigeonhole: "card A > card (f ` A) ⟹ ¬ inj_on f A "
by (auto dest: card_image less_irrefl_nat)

lemma pigeonhole_infinite:
assumes "¬ finite A" and "finite (f`A)"
shows "∃a0∈A. ¬ finite {a∈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∈A. f a = b}")
case True
with ‹¬ finite A› have "¬ finite (A - {a∈A. f a = b})"
by simp
also have "A - {a∈A. f a = b} = {a∈A. f a ≠ b}"
by blast
finally have "¬ finite {a∈A. f a ≠ b}" .
from insert(3)[OF _ this] insert(2,4) show ?thesis
by simp (blast intro: rev_finite_subset)
next
case False
then have "{a ∈ A. f a = b} ≠ {}" by force
with False show ?thesis by blast
qed
qed

lemma pigeonhole_infinite_rel:
assumes "¬ finite A"
and "finite B"
and "∀a∈A. ∃b∈B. R a b"
shows "∃b∈B. ¬ finite {a:A. R a b}"
proof -
let ?F = "λa. {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 ∈ A" and infinite: "¬ finite {a∈A. ?F a = ?F a0}" ..
obtain b0 where "b0 ∈ B" and "R a0 b0"
using ‹a0 ∈ A› assms(3) by blast
have "finite {a∈A. ?F a = ?F a0}" if "finite {a∈A. R a b0}"
using ‹b0 ∈ B› ‹R a0 b0› that by (blast intro: rev_finite_subset)
with infinite ‹b0 ∈ B› show ?thesis
by blast
qed

subsubsection ‹Cardinality of sums›

lemma card_Plus:
assumes "finite A" "finite B"
shows "card (A <+> B) = card A + card B"
proof -
have "Inl`A ∩ 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 ∧ finite B then card A + card B else 0)"

text ‹Relates to equivalence classes.  Based on a theorem of F. KammÃ¼ller.›

lemma dvd_partition:
assumes f: "finite (⋃C)"
and "∀c∈C. k dvd card c" "∀c1∈C. ∀c2∈C. c1 ≠ c2 ⟶ c1 ∩ c2 = {}"
shows "k dvd card (⋃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
then show ?case
apply simp
apply (subst card_Un_disjoint)
done
qed
qed

subsubsection ‹Relating injectivity and surjectivity›

lemma finite_surj_inj:
assumes "finite A" "A ⊆ 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) ⟹ surj f ⟹ inj f"
for f :: "'a ⇒ 'a"
by (blast intro: finite_surj_inj subset_UNIV)

lemma finite_UNIV_inj_surj: "finite(UNIV:: 'a set) ⟹ inj f ⟹ surj f"
for f :: "'a ⇒ 'a"
by (fastforce simp:surj_def dest!: endo_inj_surj)

corollary infinite_UNIV_nat [iff]: "¬ 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

lemma infinite_UNIV_char_0: "¬ 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 ⇒ 'a)"
ultimately have "finite (UNIV :: nat set)"
by (rule finite_imageD)
then show False
by simp
qed

hide_const (open) Finite_Set.fold

subsection ‹Infinite Sets›

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

abbreviation infinite :: "'a set ⇒ bool"
where "infinite S ≡ ¬ finite S"

text ‹
Infinite sets are non-empty, and if we remove some elements from an
infinite set, the result is still infinite.
›

lemma infinite_imp_nonempty: "infinite S ⟹ S ≠ {}"
by auto

lemma infinite_remove: "infinite S ⟹ infinite (S - {a})"
by simp

lemma Diff_infinite_finite:
assumes "finite T" "infinite S"
shows "infinite (S - T)"
using ‹finite T›
proof induct
from ‹infinite S› 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 ⟹ infinite (S ∪ T)"
by simp

lemma infinite_Un: "infinite (S ∪ T) ⟷ infinite S ∨ infinite T"
by simp

lemma infinite_super:
assumes "S ⊆ T"
and "infinite S"
shows "infinite T"
proof
assume "finite T"
with ‹S ⊆ T› have "finite S" by (simp add: finite_subset)
with ‹infinite S› show False by simp
qed

proposition infinite_coinduct [consumes 1, case_names infinite]:
assumes "X A"
and step: "⋀A. X A ⟹ ∃x∈A. X (A - {x}) ∨ infinite (A - {x})"
shows "infinite A"
proof
assume "finite A"
then show False
using ‹X A›
proof (induction rule: finite_psubset_induct)
case (psubset A)
then obtain x where "x ∈ A" "X (A - {x}) ∨ infinite (A - {x})"
using local.step psubset.prems by blast
then have "X (A - {x})"
using psubset.hyps by blast
show False
apply (rule psubset.IH [where B = "A - {x}"])
apply (use ‹x ∈ A› in blast)
apply (simp add: ‹X (A - {x})›)
done
qed
qed

text ‹
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.
›

lemma inf_img_fin_dom':
assumes img: "finite (f ` A)"
and dom: "infinite A"
shows "∃y ∈ f ` A. infinite (f -` {y} ∩ A)"
proof (rule ccontr)
have "A ⊆ (⋃y∈f ` A. f -` {y} ∩ A)" by auto
moreover assume "¬ ?thesis"
with img have "finite (⋃y∈f ` A. f -` {y} ∩ 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 ∈ f`A" and "infinite (f -` {y} ∩ 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 "∃y ∈ 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 ∈ f`A" and "infinite (f -` {y})"
using assms by (blast dest: inf_img_fin_dom)

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

end
```