Theory HOL-Library.Disjoint_Sets

```(*  Author:     Johannes Hölzl, TU München
*)

section ‹Partitions and Disjoint Sets›

theory Disjoint_Sets
imports FuncSet
begin

lemma mono_imp_UN_eq_last: "mono A ⟹ (⋃i≤n. A i) = A n"
unfolding mono_def by auto

subsection ‹Set of Disjoint Sets›

abbreviation disjoint :: "'a set set ⇒ bool" where "disjoint ≡ pairwise disjnt"

lemma disjoint_def: "disjoint A ⟷ (∀a∈A. ∀b∈A. a ≠ b ⟶ a ∩ b = {})"
unfolding pairwise_def disjnt_def by auto

lemma disjointI:
"(⋀a b. a ∈ A ⟹ b ∈ A ⟹ a ≠ b ⟹ a ∩ b = {}) ⟹ disjoint A"
unfolding disjoint_def by auto

lemma disjointD:
"disjoint A ⟹ a ∈ A ⟹ b ∈ A ⟹ a ≠ b ⟹ a ∩ b = {}"
unfolding disjoint_def by auto

lemma disjoint_image: "inj_on f (⋃A) ⟹ disjoint A ⟹ disjoint ((`) f ` A)"
unfolding inj_on_def disjoint_def by blast

lemma assumes "disjoint (A ∪ B)"
shows   disjoint_unionD1: "disjoint A" and disjoint_unionD2: "disjoint B"
using assms by (simp_all add: disjoint_def)

lemma disjoint_INT:
assumes *: "⋀i. i ∈ I ⟹ disjoint (F i)"
shows "disjoint {⋂i∈I. X i | X. ∀i∈I. X i ∈ F i}"
proof (safe intro!: disjointI del: equalityI)
fix A B :: "'a ⇒ 'b set" assume "(⋂i∈I. A i) ≠ (⋂i∈I. B i)"
then obtain i where "A i ≠ B i" "i ∈ I"
by auto
moreover assume "∀i∈I. A i ∈ F i" "∀i∈I. B i ∈ F i"
ultimately show "(⋂i∈I. A i) ∩ (⋂i∈I. B i) = {}"
using *[OF ‹i∈I›, THEN disjointD, of "A i" "B i"]
by (auto simp flip: INT_Int_distrib)
qed

lemma diff_Union_pairwise_disjoint:
assumes "pairwise disjnt 𝒜" "ℬ ⊆ 𝒜"
shows "⋃𝒜 - ⋃ℬ = ⋃(𝒜 - ℬ)"
proof -
have "False"
if x: "x ∈ A" "x ∈ B" and AB: "A ∈ 𝒜" "A ∉ ℬ" "B ∈ ℬ" for x A B
proof -
have "A ∩ B = {}"
using assms disjointD AB by blast
with x show ?thesis
by blast
qed
then show ?thesis by auto
qed

lemma Int_Union_pairwise_disjoint:
assumes "pairwise disjnt (𝒜 ∪ ℬ)"
shows "⋃𝒜 ∩ ⋃ℬ = ⋃(𝒜 ∩ ℬ)"
proof -
have "False"
if x: "x ∈ A" "x ∈ B" and AB: "A ∈ 𝒜" "A ∉ ℬ" "B ∈ ℬ" for x A B
proof -
have "A ∩ B = {}"
using assms disjointD AB by blast
with x show ?thesis
by blast
qed
then show ?thesis by auto
qed

lemma psubset_Union_pairwise_disjoint:
assumes ℬ: "pairwise disjnt ℬ" and "𝒜 ⊂ ℬ - {{}}"
shows "⋃𝒜 ⊂ ⋃ℬ"
unfolding psubset_eq
proof
show "⋃𝒜 ⊆ ⋃ℬ"
using assms by blast
have "𝒜 ⊆ ℬ" "⋃(ℬ - 𝒜 ∩ (ℬ - {{}})) ≠ {}"
using assms by blast+
then show "⋃𝒜 ≠ ⋃ℬ"
using diff_Union_pairwise_disjoint [OF ℬ] by blast
qed

subsubsection "Family of Disjoint Sets"

definition disjoint_family_on :: "('i ⇒ 'a set) ⇒ 'i set ⇒ bool" where
"disjoint_family_on A S ⟷ (∀m∈S. ∀n∈S. m ≠ n ⟶ A m ∩ A n = {})"

abbreviation "disjoint_family A ≡ disjoint_family_on A UNIV"

lemma disjoint_family_elem_disjnt:
assumes "infinite A" "finite C"
and df: "disjoint_family_on B A"
obtains x where "x ∈ A" "disjnt C (B x)"
proof -
have "False" if *: "∀x ∈ A. ∃y. y ∈ C ∧ y ∈ B x"
proof -
obtain g where g: "∀x ∈ A. g x ∈ C ∧ g x ∈ B x"
using * by metis
with df have "inj_on g A"
by (fastforce simp add: inj_on_def disjoint_family_on_def)
then have "infinite (g ` A)"
using ‹infinite A› finite_image_iff by blast
then show False
by (meson ‹finite C› finite_subset g image_subset_iff)
qed
then show ?thesis
by (force simp: disjnt_iff intro: that)
qed

lemma disjoint_family_onD:
"disjoint_family_on A I ⟹ i ∈ I ⟹ j ∈ I ⟹ i ≠ j ⟹ A i ∩ A j = {}"
by (auto simp: disjoint_family_on_def)

lemma disjoint_family_subset: "disjoint_family A ⟹ (⋀x. B x ⊆ A x) ⟹ disjoint_family B"

lemma disjoint_family_on_insert:
"i ∉ I ⟹ disjoint_family_on A (insert i I) ⟷ A i ∩ (⋃i∈I. A i) = {} ∧ disjoint_family_on A I"
by (fastforce simp: disjoint_family_on_def)

lemma disjoint_family_on_bisimulation:
assumes "disjoint_family_on f S"
and "⋀n m. n ∈ S ⟹ m ∈ S ⟹ n ≠ m ⟹ f n ∩ f m = {} ⟹ g n ∩ g m = {}"
shows "disjoint_family_on g S"
using assms unfolding disjoint_family_on_def by auto

lemma disjoint_family_on_mono:
"A ⊆ B ⟹ disjoint_family_on f B ⟹ disjoint_family_on f A"
unfolding disjoint_family_on_def by auto

lemma disjoint_family_Suc:
"(⋀n. A n ⊆ A (Suc n)) ⟹ disjoint_family (λi. A (Suc i) - A i)"
using lift_Suc_mono_le[of A]
(metis insert_absorb insert_subset le_SucE le_antisym not_le_imp_less less_imp_le)

lemma disjoint_family_on_disjoint_image:
"disjoint_family_on A I ⟹ disjoint (A ` I)"
unfolding disjoint_family_on_def disjoint_def by force

lemma disjoint_family_on_vimageI: "disjoint_family_on F I ⟹ disjoint_family_on (λi. f -` F i) I"
by (auto simp: disjoint_family_on_def)

lemma disjoint_image_disjoint_family_on:
assumes d: "disjoint (A ` I)" and i: "inj_on A I"
shows "disjoint_family_on A I"
unfolding disjoint_family_on_def
proof (intro ballI impI)
fix n m assume nm: "m ∈ I" "n ∈ I" and "n ≠ m"
with i[THEN inj_onD, of n m] show "A n ∩ A m = {}"
by (intro disjointD[OF d]) auto
qed

lemma disjoint_family_on_iff_disjoint_image:
assumes "⋀i. i ∈ I ⟹ A i ≠ {}"
shows "disjoint_family_on A I ⟷ disjoint (A ` I) ∧ inj_on A I"
proof
assume "disjoint_family_on A I"
then show "disjoint (A ` I) ∧ inj_on A I"
by (metis (mono_tags, lifting) assms disjoint_family_onD disjoint_family_on_disjoint_image inf.idem inj_onI)
qed (use disjoint_image_disjoint_family_on in metis)

lemma card_UN_disjoint':
assumes "disjoint_family_on A I" "⋀i. i ∈ I ⟹ finite (A i)" "finite I"
shows "card (⋃i∈I. A i) = (∑i∈I. card (A i))"
using assms   by (simp add: card_UN_disjoint disjoint_family_on_def)

lemma disjoint_UN:
assumes F: "⋀i. i ∈ I ⟹ disjoint (F i)" and *: "disjoint_family_on (λi. ⋃(F i)) I"
shows "disjoint (⋃i∈I. F i)"
proof (safe intro!: disjointI del: equalityI)
fix A B i j assume "A ≠ B" "A ∈ F i" "i ∈ I" "B ∈ F j" "j ∈ I"
show "A ∩ B = {}"
proof cases
assume "i = j" with F[of i] ‹i ∈ I› ‹A ∈ F i› ‹B ∈ F j› ‹A ≠ B› show "A ∩ B = {}"
by (auto dest: disjointD)
next
assume "i ≠ j"
with * ‹i∈I› ‹j∈I› have "(⋃(F i)) ∩ (⋃(F j)) = {}"
by (rule disjoint_family_onD)
with ‹A∈F i› ‹i∈I› ‹B∈F j› ‹j∈I›
show "A ∩ B = {}"
by auto
qed
qed

lemma distinct_list_bind:
assumes "distinct xs" "⋀x. x ∈ set xs ⟹ distinct (f x)"
"disjoint_family_on (set ∘ f) (set xs)"
shows   "distinct (List.bind xs f)"
using assms
by (induction xs)
(auto simp: disjoint_family_on_def distinct_map inj_on_def set_list_bind)

lemma bij_betw_UNION_disjoint:
assumes disj: "disjoint_family_on A' I"
assumes bij: "⋀i. i ∈ I ⟹ bij_betw f (A i) (A' i)"
shows   "bij_betw f (⋃i∈I. A i) (⋃i∈I. A' i)"
unfolding bij_betw_def
proof
from bij show eq: "f ` ⋃(A ` I) = ⋃(A' ` I)"
by (auto simp: bij_betw_def image_UN)
show "inj_on f (⋃(A ` I))"
proof (rule inj_onI, clarify)
fix i j x y assume A: "i ∈ I" "j ∈ I" "x ∈ A i" "y ∈ A j" and B: "f x = f y"
from A bij[of i] bij[of j] have "f x ∈ A' i" "f y ∈ A' j"
by (auto simp: bij_betw_def)
with B have "A' i ∩ A' j ≠ {}" by auto
with disj A have "i = j" unfolding disjoint_family_on_def by blast
with A B bij[of i] show "x = y" by (auto simp: bij_betw_def dest: inj_onD)
qed
qed

lemma disjoint_union: "disjoint C ⟹ disjoint B ⟹ ⋃C ∩ ⋃B = {} ⟹ disjoint (C ∪ B)"
using disjoint_UN[of "{C, B}" "λx. x"] by (auto simp add: disjoint_family_on_def)

text ‹
Sum/product of the union of a finite disjoint family
›
context comm_monoid_set
begin

lemma UNION_disjoint_family:
assumes "finite I" and "∀i∈I. finite (A i)"
and "disjoint_family_on A I"
shows "F g (⋃(A ` I)) = F (λx. F g (A x)) I"
using assms unfolding disjoint_family_on_def  by (rule UNION_disjoint)

lemma Union_disjoint_sets:
assumes "∀A∈C. finite A" and "disjoint C"
shows "F g (⋃C) = (F ∘ F) g C"
using assms unfolding disjoint_def  by (rule Union_disjoint)

end

text ‹
The union of an infinite disjoint family of non-empty sets is infinite.
›
lemma infinite_disjoint_family_imp_infinite_UNION:
assumes "¬finite A" "⋀x. x ∈ A ⟹ f x ≠ {}" "disjoint_family_on f A"
shows   "¬finite (⋃(f ` A))"
proof -
define g where "g x = (SOME y. y ∈ f x)" for x
have g: "g x ∈ f x" if "x ∈ A" for x
unfolding g_def by (rule someI_ex, insert assms(2) that) blast
have inj_on_g: "inj_on g A"
proof (rule inj_onI, rule ccontr)
fix x y assume A: "x ∈ A" "y ∈ A" "g x = g y" "x ≠ y"
with g[of x] g[of y] have "g x ∈ f x" "g x ∈ f y" by auto
with A ‹x ≠ y› assms show False
by (auto simp: disjoint_family_on_def inj_on_def)
qed
from g have "g ` A ⊆ ⋃(f ` A)" by blast
moreover from inj_on_g ‹¬finite A› have "¬finite (g ` A)"
using finite_imageD by blast
ultimately show ?thesis using finite_subset by blast
qed

subsection ‹Construct Disjoint Sequences›

definition disjointed :: "(nat ⇒ 'a set) ⇒ nat ⇒ 'a set" where
"disjointed A n = A n - (⋃i∈{0..<n}. A i)"

lemma finite_UN_disjointed_eq: "(⋃i∈{0..<n}. disjointed A i) = (⋃i∈{0..<n}. A i)"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n)
thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
qed

lemma UN_disjointed_eq: "(⋃i. disjointed A i) = (⋃i. A i)"
by (rule UN_finite2_eq [where k=0])

lemma less_disjoint_disjointed: "m < n ⟹ disjointed A m ∩ disjointed A n = {}"

lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
(metis neq_iff Int_commute less_disjoint_disjointed)

lemma disjointed_subset: "disjointed A n ⊆ A n"

lemma disjointed_0[simp]: "disjointed A 0 = A 0"

lemma disjointed_mono: "mono A ⟹ disjointed A (Suc n) = A (Suc n) - A n"
using mono_imp_UN_eq_last[of A] by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)

subsection ‹Partitions›

text ‹
Partitions \<^term>‹P› of a set \<^term>‹A›. We explicitly disallow empty sets.
›

definition partition_on :: "'a set ⇒ 'a set set ⇒ bool"
where
"partition_on A P ⟷ ⋃P = A ∧ disjoint P ∧ {} ∉ P"

lemma partition_onI:
"⋃P = A ⟹ (⋀p q. p ∈ P ⟹ q ∈ P ⟹ p ≠ q ⟹ disjnt p q) ⟹ {} ∉ P ⟹ partition_on A P"
by (auto simp: partition_on_def pairwise_def)

lemma partition_onD1: "partition_on A P ⟹ A = ⋃P"
by (auto simp: partition_on_def)

lemma partition_onD2: "partition_on A P ⟹ disjoint P"
by (auto simp: partition_on_def)

lemma partition_onD3: "partition_on A P ⟹ {} ∉ P"
by (auto simp: partition_on_def)

subsection ‹Constructions of partitions›

lemma partition_on_empty: "partition_on {} P ⟷ P = {}"
unfolding partition_on_def by fastforce

lemma partition_on_space: "A ≠ {} ⟹ partition_on A {A}"
by (auto simp: partition_on_def disjoint_def)

lemma partition_on_singletons: "partition_on A ((λx. {x}) ` A)"
by (auto simp: partition_on_def disjoint_def)

lemma partition_on_transform:
assumes P: "partition_on A P"
assumes F_UN: "⋃(F ` P) = F (⋃P)" and F_disjnt: "⋀p q. p ∈ P ⟹ q ∈ P ⟹ disjnt p q ⟹ disjnt (F p) (F q)"
shows "partition_on (F A) (F ` P - {{}})"
proof -
have "⋃(F ` P - {{}}) = F A"
unfolding P[THEN partition_onD1] F_UN[symmetric] by auto
with P show ?thesis
by (auto simp add: partition_on_def pairwise_def intro!: F_disjnt)
qed

lemma partition_on_restrict: "partition_on A P ⟹ partition_on (B ∩ A) ((∩) B ` P - {{}})"
by (intro partition_on_transform) (auto simp: disjnt_def)

lemma partition_on_vimage: "partition_on A P ⟹ partition_on (f -` A) ((-`) f ` P - {{}})"
by (intro partition_on_transform) (auto simp: disjnt_def)

lemma partition_on_inj_image:
assumes P: "partition_on A P" and f: "inj_on f A"
shows "partition_on (f ` A) ((`) f ` P - {{}})"
proof (rule partition_on_transform[OF P])
show "p ∈ P ⟹ q ∈ P ⟹ disjnt p q ⟹ disjnt (f ` p) (f ` q)" for p q
using f[THEN inj_onD] P[THEN partition_onD1] by (auto simp: disjnt_def)
qed auto

lemma partition_on_insert:
assumes "disjnt p (⋃P)"
shows "partition_on A (insert p P) ⟷ partition_on (A-p) P ∧ p ⊆ A ∧ p ≠ {}"
using assms
by (auto simp: partition_on_def disjnt_iff pairwise_insert)

subsection ‹Finiteness of partitions›

lemma finitely_many_partition_on:
assumes "finite A"
shows "finite {P. partition_on A P}"
proof (rule finite_subset)
show "{P. partition_on A P} ⊆ Pow (Pow A)"
unfolding partition_on_def by auto
show "finite (Pow (Pow A))"
using assms by simp
qed

lemma finite_elements: "finite A ⟹ partition_on A P ⟹ finite P"
using partition_onD1[of A P] by (simp add: finite_UnionD)

lemma product_partition:
assumes "partition_on A P" and "⋀p. p ∈ P ⟹ finite p"
shows "card A = (∑p∈P. card p)"
using assms unfolding partition_on_def  by (meson card_Union_disjoint)

subsection ‹Equivalence of partitions and equivalence classes›

lemma partition_on_quotient:
assumes r: "equiv A r"
shows "partition_on A (A // r)"
proof (rule partition_onI)
from r have "refl_on A r"
by (auto elim: equivE)
then show "⋃(A // r) = A" "{} ∉ A // r"
by (auto simp: refl_on_def quotient_def)

fix p q assume "p ∈ A // r" "q ∈ A // r" "p ≠ q"
then obtain x y where "x ∈ A" "y ∈ A" "p = r `` {x}" "q = r `` {y}"
by (auto simp: quotient_def)
with r equiv_class_eq_iff[OF r, of x y] ‹p ≠ q› show "disjnt p q"
by (auto simp: disjnt_equiv_class)
qed

lemma equiv_partition_on:
assumes P: "partition_on A P"
shows "equiv A {(x, y). ∃p ∈ P. x ∈ p ∧ y ∈ p}"
proof (rule equivI)
have "A = ⋃P"
using P by (auto simp: partition_on_def)

have "{(x, y). ∃p ∈ P. x ∈ p ∧ y ∈ p} ⊆ A × A"
unfolding ‹A = ⋃P› by blast
then show "refl_on A {(x, y). ∃p∈P. x ∈ p ∧ y ∈ p}"
unfolding refl_on_def ‹A = ⋃P› by auto
next
show "trans {(x, y). ∃p∈P. x ∈ p ∧ y ∈ p}"
using P by (auto simp only: trans_def disjoint_def partition_on_def)
next
show "sym {(x, y). ∃p∈P. x ∈ p ∧ y ∈ p}"
by (auto simp only: sym_def)
qed

lemma partition_on_eq_quotient:
assumes P: "partition_on A P"
shows "A // {(x, y). ∃p ∈ P. x ∈ p ∧ y ∈ p} = P"
unfolding quotient_def
proof safe
fix x assume "x ∈ A"
then obtain p where "p ∈ P" "x ∈ p" "⋀q. q ∈ P ⟹ x ∈ q ⟹ p = q"
using P by (auto simp: partition_on_def disjoint_def)
then have "{y. ∃p∈P. x ∈ p ∧ y ∈ p} = p"
by (safe intro!: bexI[of _ p]) simp
then show "{(x, y). ∃p∈P. x ∈ p ∧ y ∈ p} `` {x} ∈ P"
by (simp add: ‹p ∈ P›)
next
fix p assume "p ∈ P"
then have "p ≠ {}"
using P by (auto simp: partition_on_def)
then obtain x where "x ∈ p"
by auto
then have "x ∈ A" "⋀q. q ∈ P ⟹ x ∈ q ⟹ p = q"
using P ‹p ∈ P› by (auto simp: partition_on_def disjoint_def)
with ‹p∈P› ‹x ∈ p› have "{y. ∃p∈P. x ∈ p ∧ y ∈ p} = p"
by (safe intro!: bexI[of _ p]) simp
then show "p ∈ (⋃x∈A. {{(x, y). ∃p∈P. x ∈ p ∧ y ∈ p} `` {x}})"
by (auto intro: ‹x ∈ A›)
qed

lemma partition_on_alt: "partition_on A P ⟷ (∃r. equiv A r ∧ P = A // r)"
by (auto simp: partition_on_eq_quotient intro!: partition_on_quotient intro: equiv_partition_on)

subsection ‹Refinement of partitions›

definition refines :: "'a set ⇒ 'a set set ⇒ 'a set set ⇒ bool"
where "refines A P Q ≡
partition_on A P ∧ partition_on A Q ∧ (∀X∈P. ∃Y∈Q. X ⊆ Y)"

lemma refines_refl: "partition_on A P ⟹ refines A P P"
using refines_def by blast

lemma refines_asym1:
assumes "refines A P Q" "refines A Q P"
shows "P ⊆ Q"
proof
fix X
assume "X ∈ P"
then obtain Y X' where "Y ∈ Q" "X ⊆ Y" "X' ∈ P" "Y ⊆ X'"
by (meson assms refines_def)
then have "X' = X"
using assms(2) unfolding partition_on_def refines_def
by (metis ‹X ∈ P› ‹X ⊆ Y› disjnt_self_iff_empty disjnt_subset1 pairwiseD)
then show "X ∈ Q"
using ‹X ⊆ Y› ‹Y ∈ Q› ‹Y ⊆ X'› by force
qed

lemma refines_asym: "⟦refines A P Q; refines A Q P⟧ ⟹ P=Q"
by (meson antisym_conv refines_asym1)

lemma refines_trans: "⟦refines A P Q; refines A Q R⟧ ⟹ refines A P R"
by (meson order.trans refines_def)

lemma refines_obtains_subset:
assumes "refines A P Q" "q ∈ Q"
shows "partition_on q {p ∈ P. p ⊆ q}"
proof -
have "p ⊆ q ∨ disjnt p q" if "p ∈ P" for p
using that assms unfolding refines_def partition_on_def disjoint_def
by (metis disjnt_def disjnt_subset1)
with assms have "q ⊆ Union {p ∈ P. p ⊆ q}"
using assms
by (clarsimp simp: refines_def disjnt_iff partition_on_def) (metis Union_iff)
with assms have "q = Union {p ∈ P. p ⊆ q}"
by auto
then show ?thesis
using assms by (auto simp: refines_def disjoint_def partition_on_def)
qed

subsection ‹The coarsest common refinement of a set of partitions›

definition common_refinement :: "'a set set set ⇒ 'a set set"
where "common_refinement 𝒫 ≡ (⋃f ∈ (Π⇩E P∈𝒫. P). {⋂ (f ` 𝒫)}) - {{}}"

text ‹With non-extensional function space›
lemma common_refinement: "common_refinement 𝒫 = (⋃f ∈ (Π P∈𝒫. P). {⋂ (f ` 𝒫)}) - {{}}"
(is "?lhs = ?rhs")
proof
show "?rhs ⊆ ?lhs"
apply (clarsimp simp add: common_refinement_def PiE_def Ball_def)
by (metis restrict_Pi_cancel image_restrict_eq restrict_extensional)
qed (auto simp add: common_refinement_def PiE_def)

lemma common_refinement_exists: "⟦X ∈ common_refinement 𝒫; P ∈ 𝒫⟧ ⟹ ∃R∈P. X ⊆ R"

lemma Union_common_refinement: "⋃ (common_refinement 𝒫) = (⋂ P∈𝒫. ⋃P)"
proof
show "(⋂ P∈𝒫. ⋃P) ⊆ ⋃ (common_refinement 𝒫)"
proof (clarsimp simp: common_refinement)
fix x
assume "∀P∈𝒫. ∃X∈P. x ∈ X"
then obtain F where F: "⋀P. P∈𝒫 ⟹ F P ∈ P ∧ x ∈ F P"
by metis
then have "x ∈ ⋂ (F ` 𝒫)"
by force
with F show "∃X∈(⋃x∈Π P∈𝒫. P. {⋂ (x ` 𝒫)}) - {{}}. x ∈ X"
by (auto simp add: Pi_iff Bex_def)
qed
qed (auto simp: common_refinement_def)

lemma partition_on_common_refinement:
assumes A: "⋀P. P ∈ 𝒫 ⟹ partition_on A P" and "𝒫 ≠ {}"
shows "partition_on A (common_refinement 𝒫)"
proof (rule partition_onI)
show "⋃ (common_refinement 𝒫) = A"
using assms by (simp add: partition_on_def Union_common_refinement)
fix P Q
assume "P ∈ common_refinement 𝒫" and "Q ∈ common_refinement 𝒫" and "P ≠ Q"
then obtain f g where f: "f ∈ (Π⇩E P∈𝒫. P)" and P: "P = ⋂ (f ` 𝒫)" and "P ≠ {}"
and   g: "g ∈ (Π⇩E P∈𝒫. P)" and Q: "Q = ⋂ (g ` 𝒫)" and "Q ≠ {}"
have "f=g" if "x ∈ P" "x ∈ Q" for x
proof (rule extensionalityI [of _ 𝒫])
fix R
assume "R ∈ 𝒫"
with that P Q f g A [unfolded partition_on_def, OF ‹R ∈ 𝒫›]
show "f R = g R"
by (metis INT_E Int_iff PiE_iff disjointD emptyE)
qed (use PiE_iff f g in auto)
then show "disjnt P Q"
by (metis P Q ‹P ≠ Q› disjnt_iff)

lemma refines_common_refinement:
assumes "⋀P. P ∈ 𝒫 ⟹ partition_on A P" "P ∈ 𝒫"
shows "refines A (common_refinement 𝒫) P"
unfolding refines_def
proof (intro conjI strip)
fix X
assume "X ∈ common_refinement 𝒫"
with assms show "∃Y∈P. X ⊆ Y"
by (auto simp: common_refinement_def)
qed (use assms partition_on_common_refinement in auto)

text ‹The common refinement is itself refined by any other›
lemma common_refinement_coarsest:
assumes "⋀P. P ∈ 𝒫 ⟹ partition_on A P" "partition_on A R" "⋀P. P ∈ 𝒫 ⟹ refines A R P" "𝒫 ≠ {}"
shows "refines A R (common_refinement 𝒫)"
unfolding refines_def
proof (intro conjI ballI partition_on_common_refinement)
fix X
assume "X ∈ R"
have "∃p ∈ P. X ⊆ p" if "P ∈ 𝒫" for P
by (meson ‹X ∈ R› assms(3) refines_def that)
then obtain F where f: "⋀P. P ∈ 𝒫 ⟹ F P ∈ P ∧ X ⊆ F P"
by metis
with ‹partition_on A R› ‹X ∈ R› ‹𝒫 ≠ {}›
have "⋂ (F ` 𝒫) ∈ common_refinement 𝒫"
apply (simp add: partition_on_def common_refinement Pi_iff Bex_def)
by (metis (no_types, lifting) cINF_greatest subset_empty)
with f show "∃Y∈common_refinement 𝒫. X ⊆ Y"
by (metis ‹𝒫 ≠ {}› cINF_greatest)
qed (use assms in auto)

lemma finite_common_refinement:
assumes "finite 𝒫" "⋀P. P ∈ 𝒫 ⟹ finite P"
shows "finite (common_refinement 𝒫)"
proof -
have "finite (Π⇩E P∈𝒫. P)"
then show ?thesis
by (auto simp: common_refinement_def)
qed

lemma card_common_refinement:
assumes "finite 𝒫" "⋀P. P ∈ 𝒫 ⟹ finite P"
shows "card (common_refinement 𝒫) ≤ (∏P ∈ 𝒫. card P)"
proof -
have "card (common_refinement 𝒫) ≤ card (⋃f ∈ (Π⇩E P∈𝒫. P). {⋂ (f ` 𝒫)})"
unfolding common_refinement_def by (meson card_Diff1_le)
also have "… ≤ (∑f∈(Π⇩E P∈𝒫. P). card{⋂ (f ` 𝒫)})"
by (metis assms finite_PiE card_UN_le)
also have "… = card(Π⇩E P∈𝒫. P)"
by simp
also have "… = (∏P ∈ 𝒫. card P)"
by (simp add: assms(1) card_PiE dual_order.eq_iff)
finally show ?thesis .
qed

end
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