Finite sets #

This file defines predicates for finite and infinite sets and provides Fintype instances for many set constructions. It also proves basic facts about finite sets and gives ways to manipulate Set.Finite expressions.

Main definitions #

• Set.Finite : Set α → Prop
• Set.Infinite : Set α → Prop
• Set.toFinite to prove Set.Finite for a Set from a Finite instance.
• Set.Finite.toFinset to noncomputably produce a Finset from a Set.Finite proof. (See Set.toFinset for a computable version.)

Implementation #

A finite set is defined to be a set whose coercion to a type has a Finite instance.

There are two components to finiteness constructions. The first is Fintype instances for each construction. This gives a way to actually compute a Finset that represents the set, and these may be accessed using set.toFinset. This gets the Finset in the correct form, since otherwise Finset.univ : Finset s is a Finset for the subtype for s. The second component is "constructors" for Set.Finite that give proofs that Fintype instances exist classically given other Set.Finite proofs. Unlike the Fintype instances, these do not use any decidability instances since they do not compute anything.

Tags #

finite sets

def Set.Finite {α : Type u} (s : Set α) :

A set is finite if the corresponding Subtype is finite, i.e., if there exists a natural n : ℕ and an equivalence s ≃ Fin n.

Equations
Instances For
theorem Set.finite_def {α : Type u} {s : Set α} :
s.Finite Nonempty (Fintype s)
theorem Set.Finite.nonempty_fintype {α : Type u} {s : Set α} :
s.FiniteNonempty (Fintype s)

Alias of the forward direction of Set.finite_def.

theorem Set.finite_coe_iff {α : Type u} {s : Set α} :
Finite s s.Finite
theorem Set.toFinite {α : Type u} (s : Set α) [Finite s] :
s.Finite

Constructor for Set.Finite using a Finite instance.

theorem Set.Finite.ofFinset {α : Type u} {p : Set α} (s : ) (H : ∀ (x : α), x s x p) :
p.Finite

Construct a Finite instance for a Set from a Finset with the same elements.

theorem Set.Finite.to_subtype {α : Type u} {s : Set α} (h : s.Finite) :
Finite s

Projection of Set.Finite to its Finite instance. This is intended to be used with dot notation. See also Set.Finite.Fintype and Set.Finite.nonempty_fintype.

noncomputable def Set.Finite.fintype {α : Type u} {s : Set α} (h : s.Finite) :
Fintype s

A finite set coerced to a type is a Fintype. This is the Fintype projection for a Set.Finite.

Note that because Finite isn't a typeclass, this definition will not fire if it is made into an instance

Equations
• h.fintype = .some
Instances For
noncomputable def Set.Finite.toFinset {α : Type u} {s : Set α} (h : s.Finite) :

Using choice, get the Finset that represents this Set.

Equations
• h.toFinset = s.toFinset
Instances For
theorem Set.Finite.toFinset_eq_toFinset {α : Type u} {s : Set α} [Fintype s] (h : s.Finite) :
h.toFinset = s.toFinset
@[simp]
theorem Set.toFinite_toFinset {α : Type u} (s : Set α) [Fintype s] :
.toFinset = s.toFinset
theorem Set.Finite.exists_finset {α : Type u} {s : Set α} (h : s.Finite) :
∃ (s' : ), ∀ (a : α), a s' a s
theorem Set.Finite.exists_finset_coe {α : Type u} {s : Set α} (h : s.Finite) :
∃ (s' : ), s' = s
instance Set.instCanLiftFinsetToSetFinite {α : Type u} :
CanLift (Set α) (Finset α) Finset.toSet Set.Finite

Finite sets can be lifted to finsets.

Equations
• =
def Set.Infinite {α : Type u} (s : Set α) :

A set is infinite if it is not finite.

This is protected so that it does not conflict with global Infinite.

Equations
• s.Infinite = ¬s.Finite
Instances For
@[simp]
theorem Set.not_infinite {α : Type u} {s : Set α} :
¬s.Infinite s.Finite
@[simp]
theorem Set.Finite.not_infinite {α : Type u} {s : Set α} :
s.Finite¬s.Infinite

Alias of the reverse direction of Set.not_infinite.

theorem Set.finite_or_infinite {α : Type u} (s : Set α) :
s.Finite s.Infinite

See also finite_or_infinite, fintypeOrInfinite.

theorem Set.infinite_or_finite {α : Type u} (s : Set α) :
s.Infinite s.Finite

Basic properties of Set.Finite.toFinset#

@[simp]
theorem Set.Finite.mem_toFinset {α : Type u} {s : Set α} {a : α} (hs : s.Finite) :
a hs.toFinset a s
@[simp]
theorem Set.Finite.coe_toFinset {α : Type u} {s : Set α} (hs : s.Finite) :
hs.toFinset = s
@[simp]
theorem Set.Finite.toFinset_nonempty {α : Type u} {s : Set α} (hs : s.Finite) :
hs.toFinset.Nonempty s.Nonempty
theorem Set.Finite.coeSort_toFinset {α : Type u} {s : Set α} (hs : s.Finite) :
{ x : α // x hs.toFinset } = s

Note that this is an equality of types not holding definitionally. Use wisely.

@[simp]
theorem Set.Finite.subtypeEquivToFinset_apply_coe {α : Type u} {s : Set α} (hs : s.Finite) (a : { a : α // a s }) :
(hs.subtypeEquivToFinset a) = a
@[simp]
theorem Set.Finite.subtypeEquivToFinset_symm_apply_coe {α : Type u} {s : Set α} (hs : s.Finite) (b : { b : α // b hs.toFinset }) :
(hs.subtypeEquivToFinset.symm b) = b
def Set.Finite.subtypeEquivToFinset {α : Type u} {s : Set α} (hs : s.Finite) :
{ x : α // x s } { x : α // x hs.toFinset }

The identity map, bundled as an equivalence between the subtypes of s : Set α and of h.toFinset : Finset α, where h is a proof of finiteness of s.

Equations
• hs.subtypeEquivToFinset = (Equiv.refl α).subtypeEquiv
Instances For
@[simp]
theorem Set.Finite.toFinset_inj {α : Type u} {s : Set α} {t : Set α} {hs : s.Finite} {ht : t.Finite} :
hs.toFinset = ht.toFinset s = t
@[simp]
theorem Set.Finite.toFinset_subset {α : Type u} {s : Set α} {hs : s.Finite} {t : } :
hs.toFinset t s t
@[simp]
theorem Set.Finite.toFinset_ssubset {α : Type u} {s : Set α} {hs : s.Finite} {t : } :
hs.toFinset t s t
@[simp]
theorem Set.Finite.subset_toFinset {α : Type u} {t : Set α} {ht : t.Finite} {s : } :
s ht.toFinset s t
@[simp]
theorem Set.Finite.ssubset_toFinset {α : Type u} {t : Set α} {ht : t.Finite} {s : } :
s ht.toFinset s t
theorem Set.Finite.toFinset_subset_toFinset {α : Type u} {s : Set α} {t : Set α} {hs : s.Finite} {ht : t.Finite} :
hs.toFinset ht.toFinset s t
theorem Set.Finite.toFinset_ssubset_toFinset {α : Type u} {s : Set α} {t : Set α} {hs : s.Finite} {ht : t.Finite} :
hs.toFinset ht.toFinset s t
theorem Set.Finite.toFinset_mono {α : Type u} {s : Set α} {t : Set α} {hs : s.Finite} {ht : t.Finite} :
s ths.toFinset ht.toFinset

Alias of the reverse direction of Set.Finite.toFinset_subset_toFinset.

theorem Set.Finite.toFinset_strictMono {α : Type u} {s : Set α} {t : Set α} {hs : s.Finite} {ht : t.Finite} :
s ths.toFinset ht.toFinset

Alias of the reverse direction of Set.Finite.toFinset_ssubset_toFinset.

@[simp]
theorem Set.Finite.toFinset_setOf {α : Type u} [] (p : αProp) [] (h : {x : α | p x}.Finite) :
h.toFinset = Finset.filter p Finset.univ
@[simp]
theorem Set.Finite.disjoint_toFinset {α : Type u} {s : Set α} {t : Set α} {hs : s.Finite} {ht : t.Finite} :
Disjoint hs.toFinset ht.toFinset Disjoint s t
theorem Set.Finite.toFinset_inter {α : Type u} {s : Set α} {t : Set α} [] (hs : s.Finite) (ht : t.Finite) (h : (s t).Finite) :
h.toFinset = hs.toFinset ht.toFinset
theorem Set.Finite.toFinset_union {α : Type u} {s : Set α} {t : Set α} [] (hs : s.Finite) (ht : t.Finite) (h : (s t).Finite) :
h.toFinset = hs.toFinset ht.toFinset
theorem Set.Finite.toFinset_diff {α : Type u} {s : Set α} {t : Set α} [] (hs : s.Finite) (ht : t.Finite) (h : (s \ t).Finite) :
h.toFinset = hs.toFinset \ ht.toFinset
theorem Set.Finite.toFinset_symmDiff {α : Type u} {s : Set α} {t : Set α} [] (hs : s.Finite) (ht : t.Finite) (h : (symmDiff s t).Finite) :
h.toFinset = symmDiff hs.toFinset ht.toFinset
theorem Set.Finite.toFinset_compl {α : Type u} {s : Set α} [] [] (hs : s.Finite) (h : s.Finite) :
h.toFinset = hs.toFinset
theorem Set.Finite.toFinset_univ {α : Type u} [] (h : Set.univ.Finite) :
h.toFinset = Finset.univ
@[simp]
theorem Set.Finite.toFinset_eq_empty {α : Type u} {s : Set α} {h : s.Finite} :
h.toFinset = s =
theorem Set.Finite.toFinset_empty {α : Type u} (h : .Finite) :
h.toFinset =
@[simp]
theorem Set.Finite.toFinset_eq_univ {α : Type u} {s : Set α} [] {h : s.Finite} :
h.toFinset = Finset.univ s = Set.univ
theorem Set.Finite.toFinset_image {α : Type u} {β : Type v} {s : Set α} [] (f : αβ) (hs : s.Finite) (h : (f '' s).Finite) :
h.toFinset = Finset.image f hs.toFinset
theorem Set.Finite.toFinset_range {α : Type u} {β : Type v} [] [] (f : βα) (h : (Set.range f).Finite) :
h.toFinset = Finset.image f Finset.univ

Fintype instances #

Every instance here should have a corresponding Set.Finite constructor in the next section.

instance Set.fintypeUniv {α : Type u} [] :
Fintype Set.univ
Equations
instance Set.fintypeTop {α : Type u} [] :
Equations
noncomputable def Set.fintypeOfFiniteUniv {α : Type u} (H : Set.univ.Finite) :

If (Set.univ : Set α) is finite then α is a finite type.

Equations
Instances For
instance Set.fintypeUnion {α : Type u} [] (s : Set α) (t : Set α) [Fintype s] [Fintype t] :
Fintype (s t)
Equations
instance Set.fintypeSep {α : Type u} (s : Set α) (p : αProp) [Fintype s] [] :
Fintype {a : α | a s p a}
Equations
instance Set.fintypeInter {α : Type u} (s : Set α) (t : Set α) [] [Fintype s] [Fintype t] :
Fintype (s t)
Equations
instance Set.fintypeInterOfLeft {α : Type u} (s : Set α) (t : Set α) [Fintype s] [DecidablePred fun (x : α) => x t] :
Fintype (s t)

A Fintype instance for set intersection where the left set has a Fintype instance.

Equations
instance Set.fintypeInterOfRight {α : Type u} (s : Set α) (t : Set α) [Fintype t] [DecidablePred fun (x : α) => x s] :
Fintype (s t)

A Fintype instance for set intersection where the right set has a Fintype instance.

Equations
def Set.fintypeSubset {α : Type u} (s : Set α) {t : Set α} [Fintype s] [DecidablePred fun (x : α) => x t] (h : t s) :
Fintype t

A Fintype structure on a set defines a Fintype structure on its subset.

Equations
• s.fintypeSubset h = .mpr (s.fintypeInterOfLeft t)
Instances For
instance Set.fintypeDiff {α : Type u} [] (s : Set α) (t : Set α) [Fintype s] [Fintype t] :
Fintype (s \ t)
Equations
instance Set.fintypeDiffLeft {α : Type u} (s : Set α) (t : Set α) [Fintype s] [DecidablePred fun (x : α) => x t] :
Fintype (s \ t)
Equations
• s.fintypeDiffLeft t = s.fintypeSep fun (x : α) => x t
instance Set.fintypeiUnion {α : Type u} {ι : Sort w} [] [Fintype (PLift ι)] (f : ιSet α) [(i : ι) → Fintype (f i)] :
Fintype (⋃ (i : ι), f i)
Equations
instance Set.fintypesUnion {α : Type u} [] {s : Set (Set α)} [Fintype s] [H : (t : s) → Fintype t] :
Equations
def Set.fintypeBiUnion {α : Type u} [] {ι : Type u_1} (s : Set ι) [Fintype s] (t : ιSet α) (H : (i : ι) → i sFintype (t i)) :
Fintype (⋃ xs, t x)

A union of sets with Fintype structure over a set with Fintype structure has a Fintype structure.

Equations
• s.fintypeBiUnion t H = Fintype.ofFinset (s.toFinset.attach.biUnion fun (x : { x : ι // x s.toFinset }) => (t x).toFinset)
Instances For
instance Set.fintypeBiUnion' {α : Type u} [] {ι : Type u_1} (s : Set ι) [Fintype s] (t : ιSet α) [(i : ι) → Fintype (t i)] :
Fintype (⋃ xs, t x)
Equations
• s.fintypeBiUnion' t = Fintype.ofFinset (s.toFinset.biUnion fun (x : ι) => (t x).toFinset)
theorem Set.toFinset_iUnion {α : Type u} {β : Type v} [] [] (f : βSet α) [(w : β) → Fintype (f w)] :
(⋃ (x : β), f x).toFinset = Finset.univ.biUnion fun (x : β) => (f x).toFinset
def Set.fintypeBind {α : Type u_1} {β : Type u_1} [] (s : Set α) [Fintype s] (f : αSet β) (H : (a : α) → a sFintype (f a)) :
Fintype (s >>= f)

If s : Set α is a set with Fintype instance and f : α → Set β is a function such that each f a, a ∈ s, has a Fintype structure, then s >>= f has a Fintype structure.

Equations
• s.fintypeBind f H = s.fintypeBiUnion f H
Instances For
instance Set.fintypeBind' {α : Type u_1} {β : Type u_1} [] (s : Set α) [Fintype s] (f : αSet β) [(a : α) → Fintype (f a)] :
Fintype (s >>= f)
Equations
• s.fintypeBind' f = s.fintypeBiUnion' f
instance Set.fintypeEmpty {α : Type u} :
Equations
• Set.fintypeEmpty =
instance Set.fintypeSingleton {α : Type u} (a : α) :
Fintype {a}
Equations
instance Set.fintypePure {α : Type u} (a : α) :
Fintype (pure a)
Equations
• Set.fintypePure = Set.fintypeSingleton
instance Set.fintypeInsert {α : Type u} (a : α) (s : Set α) [] [Fintype s] :
Fintype (insert a s)

A Fintype instance for inserting an element into a Set using the corresponding insert function on Finset. This requires DecidableEq α. There is also Set.fintypeInsert' when a ∈ s is decidable.

Equations
def Set.fintypeInsertOfNotMem {α : Type u} {a : α} (s : Set α) [Fintype s] (h : as) :
Fintype (insert a s)

A Fintype structure on insert a s when inserting a new element.

Equations
Instances For
def Set.fintypeInsertOfMem {α : Type u} {a : α} (s : Set α) [Fintype s] (h : a s) :
Fintype (insert a s)

A Fintype structure on insert a s when inserting a pre-existing element.

Equations
Instances For
@[instance 100]
instance Set.fintypeInsert' {α : Type u} (a : α) (s : Set α) [Decidable (a s)] [Fintype s] :
Fintype (insert a s)

The Set.fintypeInsert instance requires decidable equality, but when a ∈ s is decidable for this particular a we can still get a Fintype instance by using Set.fintypeInsertOfNotMem or Set.fintypeInsertOfMem.

This instance pre-dates Set.fintypeInsert, and it is less efficient. When Set.decidableMemOfFintype is made a local instance, then this instance would override Set.fintypeInsert if not for the fact that its priority has been adjusted. See Note [lower instance priority].

Equations
• = if h : a s then s.fintypeInsertOfMem h else s.fintypeInsertOfNotMem h
instance Set.fintypeImage {α : Type u} {β : Type v} [] (s : Set α) (f : αβ) [Fintype s] :
Fintype (f '' s)
Equations
def Set.fintypeOfFintypeImage {α : Type u} {β : Type v} (s : Set α) {f : αβ} {g : β} (I : ) [Fintype (f '' s)] :
Fintype s

If a function f has a partial inverse and sends a set s to a set with [Fintype] instance, then s has a Fintype structure as well.

Equations
Instances For
instance Set.fintypeRange {α : Type u} {ι : Sort w} [] (f : ια) [Fintype (PLift ι)] :
Equations
instance Set.fintypeMap {α : Type u_1} {β : Type u_1} [] (s : Set α) (f : αβ) [Fintype s] :
Fintype (f <$> s) Equations • Set.fintypeMap = Set.fintypeImage instance Set.fintypeLTNat (n : ) : Fintype {i : | i < n} Equations instance Set.fintypeLENat (n : ) : Fintype {i : | i n} Equations This is not an instance so that it does not conflict with the one in Mathlib/Order/LocallyFinite.lean. Equations Instances For instance Set.fintypeProd {α : Type u} {β : Type v} (s : Set α) (t : Set β) [Fintype s] [Fintype t] : Fintype (s ×ˢ t) Equations instance Set.fintypeOffDiag {α : Type u} [] (s : Set α) [Fintype s] : Fintype s.offDiag Equations instance Set.fintypeImage2 {α : Type u} {β : Type v} {γ : Type x} [] (f : αβγ) (s : Set α) (t : Set β) [hs : Fintype s] [ht : Fintype t] : Fintype (Set.image2 f s t) image2 f s t is Fintype if s and t are. Equations • = .mpr ((s ×ˢ t).fintypeImage fun (x : α × β) => f x.1 x.2) instance Set.fintypeSeq {α : Type u} {β : Type v} [] (f : Set (αβ)) (s : Set α) [Fintype f] [Fintype s] : Fintype (f.seq s) Equations • f.fintypeSeq s = .mpr (f.fintypeBiUnion' fun (x : αβ) => x '' s) instance Set.fintypeSeq' {α : Type u} {β : Type u} [] (f : Set (αβ)) (s : Set α) [Fintype f] [Fintype s] : Fintype (Seq.seq f fun (x : Unit) => s) Equations • f.fintypeSeq' s = f.fintypeSeq s instance Set.fintypeMemFinset {α : Type u} (s : ) : Fintype {a : α | a s} Equations • = s.fintypeCoeSort theorem Equiv.set_finite_iff {α : Type u} {β : Type v} {s : Set α} {t : Set β} (hst : s t) : s.Finite t.Finite Finset # @[simp] theorem Finset.finite_toSet {α : Type u} (s : ) : (↑s).Finite Gives a Set.Finite for the Finset coerced to a Set. This is a wrapper around Set.toFinite. theorem Finset.finite_toSet_toFinset {α : Type u} (s : ) : .toFinset = s @[simp] theorem Multiset.finite_toSet {α : Type u} (s : ) : {x : α | x s}.Finite @[simp] theorem Multiset.finite_toSet_toFinset {α : Type u} [] (s : ) : .toFinset = s.toFinset @[simp] theorem List.finite_toSet {α : Type u} (l : List α) : {x : α | x l}.Finite Finite instances # There is seemingly some overlap between the following instances and the Fintype instances in Data.Set.Finite. While every Fintype instance gives a Finite instance, those instances that depend on Fintype or Decidable instances need an additional Finite instance to be able to generally apply. Some set instances do not appear here since they are consequences of others, for example Subtype.Finite for subsets of a finite type. instance Finite.Set.finite_union {α : Type u} (s : Set α) (t : Set α) [Finite s] [Finite t] : Finite (s t) Equations • = instance Finite.Set.finite_sep {α : Type u} (s : Set α) (p : αProp) [Finite s] : Finite {a : α | a s p a} Equations • = theorem Finite.Set.subset {α : Type u} (s : Set α) {t : Set α} [Finite s] (h : t s) : Finite t instance Finite.Set.finite_inter_of_right {α : Type u} (s : Set α) (t : Set α) [Finite t] : Finite (s t) Equations • = instance Finite.Set.finite_inter_of_left {α : Type u} (s : Set α) (t : Set α) [Finite s] : Finite (s t) Equations • = instance Finite.Set.finite_diff {α : Type u} (s : Set α) (t : Set α) [Finite s] : Finite (s \ t) Equations • = instance Finite.Set.finite_range {α : Type u} {ι : Sort w} (f : ια) [] : Equations • = instance Finite.Set.finite_iUnion {α : Type u} {ι : Sort w} [] (f : ιSet α) [∀ (i : ι), Finite (f i)] : Finite (⋃ (i : ι), f i) Equations • = instance Finite.Set.finite_sUnion {α : Type u} {s : Set (Set α)} [Finite s] [H : ∀ (t : s), Finite t] : Finite (⋃₀ s) Equations • = theorem Finite.Set.finite_biUnion {α : Type u} {ι : Type u_1} (s : Set ι) [Finite s] (t : ιSet α) (H : is, Finite (t i)) : Finite (⋃ xs, t x) instance Finite.Set.finite_biUnion' {α : Type u} {ι : Type u_1} (s : Set ι) [Finite s] (t : ιSet α) [∀ (i : ι), Finite (t i)] : Finite (⋃ xs, t x) Equations • = instance Finite.Set.finite_biUnion'' {α : Type u} {ι : Type u_1} (p : ιProp) [h : Finite {x : ι | p x}] (t : ιSet α) [∀ (i : ι), Finite (t i)] : Finite (⋃ (x : ι), ⋃ (_ : p x), t x) Example: Finite (⋃ (i < n), f i) where f : ℕ → Set α and [∀ i, Finite (f i)] (when given instances from Order.Interval.Finset.Nat). Equations • = instance Finite.Set.finite_iInter {α : Type u} {ι : Sort u_1} [] (t : ιSet α) [∀ (i : ι), Finite (t i)] : Finite (⋂ (i : ι), t i) Equations • = instance Finite.Set.finite_insert {α : Type u} (a : α) (s : Set α) [Finite s] : Finite (insert a s) Equations • = instance Finite.Set.finite_image {α : Type u} {β : Type v} (s : Set α) (f : αβ) [Finite s] : Finite (f '' s) Equations • = instance Finite.Set.finite_replacement {α : Type u} {β : Type v} [] (f : αβ) : Finite {x : β | ∃ (x_1 : α), f x_1 = x} Equations • = instance Finite.Set.finite_prod {α : Type u} {β : Type v} (s : Set α) (t : Set β) [Finite s] [Finite t] : Finite (s ×ˢ t) Equations • = instance Finite.Set.finite_image2 {α : Type u} {β : Type v} {γ : Type x} (f : αβγ) (s : Set α) (t : Set β) [Finite s] [Finite t] : Finite (Set.image2 f s t) Equations • = instance Finite.Set.finite_seq {α : Type u} {β : Type v} (f : Set (αβ)) (s : Set α) [Finite f] [Finite s] : Finite (f.seq s) Equations • = Constructors for Set.Finite# Every constructor here should have a corresponding Fintype instance in the previous section (or in the Fintype module). The implementation of these constructors ideally should be no more than Set.toFinite, after possibly setting up some Fintype and classical Decidable instances. theorem Set.Finite.of_subsingleton {α : Type u} [] (s : Set α) : s.Finite theorem Set.finite_univ {α : Type u} [] : Set.univ.Finite theorem Set.finite_univ_iff {α : Type u} : Set.univ.Finite theorem Finite.of_finite_univ {α : Type u} : Set.univ.Finite Alias of the forward direction of Set.finite_univ_iff. theorem Set.Finite.subset {α : Type u} {s : Set α} (hs : s.Finite) {t : Set α} (ht : t s) : t.Finite theorem Set.Finite.union {α : Type u} {s : Set α} {t : Set α} (hs : s.Finite) (ht : t.Finite) : (s t).Finite theorem Set.Finite.finite_of_compl {α : Type u} {s : Set α} (hs : s.Finite) (hsc : s.Finite) : theorem Set.Finite.sup {α : Type u} {s : Set α} {t : Set α} : s.Finitet.Finite(s t).Finite theorem Set.Finite.sep {α : Type u} {s : Set α} (hs : s.Finite) (p : αProp) : {a : α | a s p a}.Finite theorem Set.Finite.inter_of_left {α : Type u} {s : Set α} (hs : s.Finite) (t : Set α) : (s t).Finite theorem Set.Finite.inter_of_right {α : Type u} {s : Set α} (hs : s.Finite) (t : Set α) : (t s).Finite theorem Set.Finite.inf_of_left {α : Type u} {s : Set α} (h : s.Finite) (t : Set α) : (s t).Finite theorem Set.Finite.inf_of_right {α : Type u} {s : Set α} (h : s.Finite) (t : Set α) : (t s).Finite theorem Set.Infinite.mono {α : Type u} {s : Set α} {t : Set α} (h : s t) : s.Infinitet.Infinite theorem Set.Finite.diff {α : Type u} {s : Set α} (hs : s.Finite) (t : Set α) : (s \ t).Finite theorem Set.Finite.of_diff {α : Type u} {s : Set α} {t : Set α} (hd : (s \ t).Finite) (ht : t.Finite) : s.Finite theorem Set.finite_iUnion {α : Type u} {ι : Sort w} [] {f : ιSet α} (H : ∀ (i : ι), (f i).Finite) : (⋃ (i : ι), f i).Finite theorem Set.Finite.biUnion' {α : Type u} {ι : Type u_1} {s : Set ι} (hs : s.Finite) {t : (i : ι) → i sSet α} (ht : ∀ (i : ι) (hi : i s), (t i hi).Finite) : (⋃ (i : ι), ⋃ (h : i s), t i h).Finite Dependent version of Finite.biUnion. theorem Set.Finite.biUnion {α : Type u} {ι : Type u_1} {s : Set ι} (hs : s.Finite) {t : ιSet α} (ht : is, (t i).Finite) : (⋃ is, t i).Finite theorem Set.Finite.sUnion {α : Type u} {s : Set (Set α)} (hs : s.Finite) (H : ts, t.Finite) : (⋃₀ s).Finite theorem Set.Finite.sInter {α : Type u_1} {s : Set (Set α)} {t : Set α} (ht : t s) (hf : t.Finite) : (⋂₀ s).Finite theorem Set.Finite.iUnion {α : Type u} {ι : Type u_1} {s : ιSet α} {t : Set ι} (ht : t.Finite) (hs : it, (s i).Finite) (he : it, s i = ) : (⋃ (i : ι), s i).Finite If sets s i are finite for all i from a finite set t and are empty for i ∉ t, then the union ⋃ i, s i is a finite set. theorem Set.Finite.bind {α : Type u_1} {β : Type u_1} {s : Set α} {f : αSet β} (h : s.Finite) (hf : as, (f a).Finite) : (s >>= f).Finite @[simp] theorem Set.finite_empty {α : Type u} : .Finite theorem Set.Infinite.nonempty {α : Type u} {s : Set α} (h : s.Infinite) : s.Nonempty @[simp] theorem Set.finite_singleton {α : Type u} (a : α) : {a}.Finite theorem Set.finite_pure {α : Type u} (a : α) : (pure a).Finite @[simp] theorem Set.Finite.insert {α : Type u} (a : α) {s : Set α} (hs : s.Finite) : (insert a s).Finite theorem Set.Finite.image {α : Type u} {β : Type v} {s : Set α} (f : αβ) (hs : s.Finite) : (f '' s).Finite theorem Set.finite_range {α : Type u} {ι : Sort w} (f : ια) [] : (Set.range f).Finite theorem Set.Finite.of_surjOn {α : Type u} {β : Type v} {s : Set α} {t : Set β} (f : αβ) (hf : Set.SurjOn f s t) (hs : s.Finite) : t.Finite theorem Set.Finite.dependent_image {α : Type u} {β : Type v} {s : Set α} (hs : s.Finite) (F : (i : α) → i sβ) : {y : β | ∃ (x : α) (hx : x s), F x hx = y}.Finite theorem Set.Finite.map {α : Type u_1} {β : Type u_1} {s : Set α} (f : αβ) : s.Finite(f <$> s).Finite
theorem Set.Finite.of_finite_image {α : Type u} {β : Type v} {s : Set α} {f : αβ} (h : (f '' s).Finite) (hi : ) :
s.Finite
theorem Set.finite_of_finite_preimage {α : Type u} {β : Type v} {f : αβ} {s : Set β} (h : (f ⁻¹' s).Finite) (hs : s ) :
s.Finite
theorem Set.Finite.of_preimage {α : Type u} {β : Type v} {f : αβ} {s : Set β} (h : (f ⁻¹' s).Finite) (hf : ) :
s.Finite
theorem Set.Finite.preimage {α : Type u} {β : Type v} {f : αβ} {s : Set β} (I : Set.InjOn f (f ⁻¹' s)) (h : s.Finite) :
(f ⁻¹' s).Finite
theorem Set.Finite.preimage' {α : Type u} {β : Type v} {f : αβ} {s : Set β} (h : s.Finite) (hf : bs, (f ⁻¹' {b}).Finite) :
(f ⁻¹' s).Finite
theorem Set.Infinite.preimage {α : Type u} {β : Type v} {f : αβ} {s : Set β} (hs : s.Infinite) (hf : s ) :
(f ⁻¹' s).Infinite
theorem Set.Infinite.preimage' {α : Type u} {β : Type v} {f : αβ} {s : Set β} (hs : (s ).Infinite) :
(f ⁻¹' s).Infinite
theorem Set.Finite.preimage_embedding {α : Type u} {β : Type v} {s : Set β} (f : α β) (h : s.Finite) :
(f ⁻¹' s).Finite
theorem Set.finite_lt_nat (n : ) :
{i : | i < n}.Finite
theorem Set.finite_le_nat (n : ) :
{i : | i n}.Finite
theorem Set.Finite.surjOn_iff_bijOn_of_mapsTo {α : Type u} {s : Set α} {f : αα} (hs : s.Finite) (hm : Set.MapsTo f s s) :
theorem Set.Finite.injOn_iff_bijOn_of_mapsTo {α : Type u} {s : Set α} {f : αα} (hs : s.Finite) (hm : Set.MapsTo f s s) :
Set.BijOn f s s
theorem Set.Finite.prod {α : Type u} {β : Type v} {s : Set α} {t : Set β} (hs : s.Finite) (ht : t.Finite) :
(s ×ˢ t).Finite
theorem Set.Finite.of_prod_left {α : Type u} {β : Type v} {s : Set α} {t : Set β} (h : (s ×ˢ t).Finite) :
t.Nonemptys.Finite
theorem Set.Finite.of_prod_right {α : Type u} {β : Type v} {s : Set α} {t : Set β} (h : (s ×ˢ t).Finite) :
s.Nonemptyt.Finite
theorem Set.Infinite.prod_left {α : Type u} {β : Type v} {s : Set α} {t : Set β} (hs : s.Infinite) (ht : t.Nonempty) :
(s ×ˢ t).Infinite
theorem Set.Infinite.prod_right {α : Type u} {β : Type v} {s : Set α} {t : Set β} (ht : t.Infinite) (hs : s.Nonempty) :
(s ×ˢ t).Infinite
theorem Set.infinite_prod {α : Type u} {β : Type v} {s : Set α} {t : Set β} :
(s ×ˢ t).Infinite s.Infinite t.Nonempty t.Infinite s.Nonempty
theorem Set.finite_prod {α : Type u} {β : Type v} {s : Set α} {t : Set β} :
(s ×ˢ t).Finite (s.Finite t = ) (t.Finite s = )
theorem Set.Finite.offDiag {α : Type u} {s : Set α} (hs : s.Finite) :
s.offDiag.Finite
theorem Set.Finite.image2 {α : Type u} {β : Type v} {γ : Type x} {s : Set α} {t : Set β} (f : αβγ) (hs : s.Finite) (ht : t.Finite) :
(Set.image2 f s t).Finite
theorem Set.Finite.seq {α : Type u} {β : Type v} {f : Set (αβ)} {s : Set α} (hf : f.Finite) (hs : s.Finite) :
(f.seq s).Finite
theorem Set.Finite.seq' {α : Type u} {β : Type u} {f : Set (αβ)} {s : Set α} (hf : f.Finite) (hs : s.Finite) :
(Seq.seq f fun (x : Unit) => s).Finite
theorem Set.finite_mem_finset {α : Type u} (s : ) :
{a : α | a s}.Finite
theorem Set.Subsingleton.finite {α : Type u} {s : Set α} (h : s.Subsingleton) :
s.Finite
theorem Set.Infinite.nontrivial {α : Type u} {s : Set α} (hs : s.Infinite) :
s.Nontrivial
theorem Set.finite_preimage_inl_and_inr {α : Type u} {β : Type v} {s : Set (α β)} :
(Sum.inl ⁻¹' s).Finite (Sum.inr ⁻¹' s).Finite s.Finite
theorem Set.exists_finite_iff_finset {α : Type u} {p : Set αProp} :
(∃ (s : Set α), s.Finite p s) ∃ (s : ), p s
theorem Set.Finite.finite_subsets {α : Type u} {a : Set α} (h : a.Finite) :
{b : Set α | b a}.Finite

There are finitely many subsets of a given finite set

theorem Set.Finite.powerset {α : Type u} {s : Set α} (h : s.Finite) :
(𝒫 s).Finite
theorem Set.exists_subset_image_finite_and {α : Type u} {β : Type v} {f : αβ} {s : Set α} {p : Set βProp} :
(∃ tf '' s, t.Finite p t) ts, t.Finite p (f '' t)
theorem Set.Finite.pi {ι : Type u_1} [] {κ : ιType u_2} {t : (i : ι) → Set (κ i)} (ht : ∀ (i : ι), (t i).Finite) :
(Set.univ.pi t).Finite

Finite product of finite sets is finite

theorem Set.Finite.pi' {ι : Type u_1} [] {κ : ιType u_2} {t : (i : ι) → Set (κ i)} (ht : ∀ (i : ι), (t i).Finite) :
{f : (i : ι) → κ i | ∀ (i : ι), f i t i}.Finite

Finite product of finite sets is finite. Note this is a variant of Set.Finite.pi without the extra i ∈ univ binder.

theorem Set.union_finset_finite_of_range_finite {α : Type u} {β : Type v} (f : α) (h : (Set.range f).Finite) :
(⋃ (a : α), (f a)).Finite

A finite union of finsets is finite.

theorem Set.finite_range_ite {α : Type u} {β : Type v} {p : αProp} [] {f : αβ} {g : αβ} (hf : (Set.range f).Finite) (hg : (Set.range g).Finite) :
(Set.range fun (x : α) => if p x then f x else g x).Finite
theorem Set.finite_range_const {α : Type u} {β : Type v} {c : β} :
(Set.range fun (x : α) => c).Finite

Properties #

instance Set.Finite.inhabited {α : Type u} :
Inhabited { s : Set α // s.Finite }
Equations
• Set.Finite.inhabited = { default := , }
@[simp]
theorem Set.finite_union {α : Type u} {s : Set α} {t : Set α} :
(s t).Finite s.Finite t.Finite
theorem Set.finite_image_iff {α : Type u} {β : Type v} {s : Set α} {f : αβ} (hi : ) :
(f '' s).Finite s.Finite
theorem Set.univ_finite_iff_nonempty_fintype {α : Type u} :
Set.univ.Finite Nonempty (Fintype α)
theorem Set.Finite.toFinset_singleton {α : Type u} {a : α} (ha : optParam {a}.Finite ) :
= {a}
@[simp]
theorem Set.Finite.toFinset_insert {α : Type u} [] {s : Set α} {a : α} (hs : (insert a s).Finite) :
hs.toFinset = insert a .toFinset
theorem Set.Finite.toFinset_insert' {α : Type u} [] {a : α} {s : Set α} (hs : s.Finite) :
.toFinset = insert a hs.toFinset
theorem Set.Finite.toFinset_prod {α : Type u} {β : Type v} {s : Set α} {t : Set β} (hs : s.Finite) (ht : t.Finite) :
hs.toFinset ×ˢ ht.toFinset = .toFinset
theorem Set.Finite.toFinset_offDiag {α : Type u} {s : Set α} [] (hs : s.Finite) :
.toFinset = hs.toFinset.offDiag
theorem Set.Finite.fin_embedding {α : Type u} {s : Set α} (h : s.Finite) :
∃ (n : ) (f : Fin n α), = s
theorem Set.Finite.fin_param {α : Type u} {s : Set α} (h : s.Finite) :
∃ (n : ) (f : Fin nα),
theorem Set.finite_option {α : Type u} {s : Set (Option α)} :
s.Finite {x : α | some x s}.Finite
theorem Set.finite_image_fst_and_snd_iff {α : Type u} {β : Type v} {s : Set (α × β)} :
(Prod.fst '' s).Finite (Prod.snd '' s).Finite s.Finite
theorem Set.forall_finite_image_eval_iff {δ : Type u_1} [] {κ : δType u_2} {s : Set ((d : δ) → κ d)} :
(∀ (d : δ), ( '' s).Finite) s.Finite
theorem Set.finite_subset_iUnion {α : Type u} {s : Set α} (hs : s.Finite) {ι : Type u_1} {t : ιSet α} (h : s ⋃ (i : ι), t i) :
∃ (I : Set ι), I.Finite s iI, t i
theorem Set.eq_finite_iUnion_of_finite_subset_iUnion {α : Type u} {ι : Type u_1} {s : ιSet α} {t : Set α} (tfin : t.Finite) (h : t ⋃ (i : ι), s i) :
∃ (I : Set ι), I.Finite ∃ (σ : {i : ι | i I}Set α), (∀ (i : {i : ι | i I}), (σ i).Finite) (∀ (i : {i : ι | i I}), σ i s i) t = ⋃ (i : {i : ι | i I}), σ i
theorem Set.Finite.induction_on {α : Type u} {C : Set αProp} {s : Set α} (h : s.Finite) (H0 : C ) (H1 : ∀ {a : α} {s : Set α}, ass.FiniteC sC (insert a s)) :
C s
theorem Set.Finite.induction_on' {α : Type u} {C : Set αProp} {S : Set α} (h : S.Finite) (H0 : C ) (H1 : ∀ {a : α} {s : Set α}, a Ss SasC sC (insert a s)) :
C S

Analogous to Finset.induction_on'.

theorem Set.Finite.dinduction_on {α : Type u} {C : (s : Set α) → s.FiniteProp} (s : Set α) (h : s.Finite) (H0 : C ) (H1 : ∀ {a : α} {s : Set α}, as∀ (h : s.Finite), C s hC (insert a s) ) :
C s h
theorem Set.Finite.induction_to {α : Type u} {C : Set αProp} {S : Set α} (h : S.Finite) (S0 : Set α) (hS0 : S0 S) (H0 : C S0) (H1 : sS, C saS \ s, C (insert a s)) :
C S

Induction up to a finite set S.

theorem Set.Finite.induction_to_univ {α : Type u} [] {C : Set αProp} (S0 : Set α) (H0 : C S0) (H1 : ∀ (S : Set α), S Set.univC SaS, C (insert a S)) :
C Set.univ

Induction up to univ.

theorem Set.seq_of_forall_finite_exists {γ : Type u_1} {P : γSet γProp} (h : ∀ (t : Set γ), t.Finite∃ (c : γ), P c t) :
∃ (u : γ), ∀ (n : ), P (u n) (u '' )

If P is some relation between terms of γ and sets in γ, such that every finite set t : Set γ has some c : γ related to it, then there is a recursively defined sequence u in γ so u n is related to the image of {0, 1, ..., n-1} under u.

(We use this later to show sequentially compact sets are totally bounded.)

Cardinality #

theorem Set.empty_card {α : Type u} :
theorem Set.empty_card' {α : Type u} {h : } :
theorem Set.card_fintypeInsertOfNotMem {α : Type u} {a : α} (s : Set α) [Fintype s] (h : as) :
Fintype.card (insert a s) = + 1
@[simp]
theorem Set.card_insert {α : Type u} {a : α} (s : Set α) [Fintype s] (h : as) {d : Fintype (insert a s)} :
Fintype.card (insert a s) = + 1
theorem Set.card_image_of_inj_on {α : Type u} {β : Type v} {s : Set α} [Fintype s] {f : αβ} [Fintype (f '' s)] (H : xs, ys, f x = f yx = y) :
Fintype.card (f '' s) =
theorem Set.card_image_of_injective {α : Type u} {β : Type v} (s : Set α) [Fintype s] {f : αβ} [Fintype (f '' s)] (H : ) :
Fintype.card (f '' s) =
@[simp]
theorem Set.card_singleton {α : Type u} (a : α) :
Fintype.card {a} = 1
theorem Set.card_lt_card {α : Type u} {s : Set α} {t : Set α} [Fintype s] [Fintype t] (h : s t) :
theorem Set.card_le_card {α : Type u} {s : Set α} {t : Set α} [Fintype s] [Fintype t] (hsub : s t) :
theorem Set.eq_of_subset_of_card_le {α : Type u} {s : Set α} {t : Set α} [Fintype s] [Fintype t] (hsub : s t) (hcard : ) :
s = t
theorem Set.card_range_of_injective {α : Type u} {β : Type v} [] {f : αβ} (hf : ) [Fintype (Set.range f)] :
theorem Set.Finite.card_toFinset {α : Type u} {s : Set α} [Fintype s] (h : s.Finite) :
h.toFinset.card =
theorem Set.card_ne_eq {α : Type u} [] (a : α) [Fintype {x : α | x a}] :
Fintype.card {x : α | x a} =

Infinite sets #

theorem Set.infinite_univ_iff {α : Type u} :
Set.univ.Infinite
theorem Set.infinite_univ {α : Type u} [h : ] :
Set.univ.Infinite
theorem Set.infinite_coe_iff {α : Type u} {s : Set α} :
Infinite s s.Infinite
theorem Set.Infinite.to_subtype {α : Type u} {s : Set α} :
s.InfiniteInfinite s

Alias of the reverse direction of Set.infinite_coe_iff.

theorem Set.Infinite.exists_not_mem_finite {α : Type u} {s : Set α} {t : Set α} (hs : s.Infinite) (ht : t.Finite) :
as, at
theorem Set.Infinite.exists_not_mem_finset {α : Type u} {s : Set α} (hs : s.Infinite) (t : ) :
as, at
theorem Set.Finite.exists_not_mem {α : Type u} {s : Set α} [] (hs : s.Finite) :
∃ (a : α), as
theorem Finset.exists_not_mem {α : Type u} [] (s : ) :
∃ (a : α), as
noncomputable def Set.Infinite.natEmbedding {α : Type u} (s : Set α) (h : s.Infinite) :
s

Embedding of ℕ into an infinite set.

Equations
Instances For
theorem Set.Infinite.exists_subset_card_eq {α : Type u} {s : Set α} (hs : s.Infinite) (n : ) :
∃ (t : ), t s t.card = n
theorem Set.infinite_of_finite_compl {α : Type u} [] {s : Set α} (hs : s.Finite) :
s.Infinite
theorem Set.Finite.infinite_compl {α : Type u} [] {s : Set α} (hs : s.Finite) :
s.Infinite
theorem Set.Infinite.diff {α : Type u} {s : Set α} {t : Set α} (hs : s.Infinite) (ht : t.Finite) :
(s \ t).Infinite
@[simp]
theorem Set.infinite_union {α : Type u} {s : Set α} {t : Set α} :
(s t).Infinite s.Infinite t.Infinite
theorem Set.Infinite.of_image {α : Type u} {β : Type v} (f : αβ) {s : Set α} (hs : (f '' s).Infinite) :
s.Infinite
theorem Set.infinite_image_iff {α : Type u} {β : Type v} {s : Set α} {f : αβ} (hi : ) :
(f '' s).Infinite s.Infinite
theorem Set.infinite_range_iff {α : Type u} {β : Type v} {f : αβ} (hi : ) :
(Set.range f).Infinite
theorem Set.Infinite.image {α : Type u} {β : Type v} {s : Set α} {f : αβ} (hi : ) :
s.Infinite(f '' s).Infinite

Alias of the reverse direction of Set.infinite_image_iff.

theorem Set.Infinite.image2_left {α : Type u} {β : Type v} {γ : Type x} {f : αβγ} {s : Set α} {t : Set β} {b : β} (hs : s.Infinite) (hb : b t) (hf : Set.InjOn (fun (a : α) => f a b) s) :
(Set.image2 f s t).Infinite
theorem Set.Infinite.image2_right {α : Type u} {β : Type v} {γ : Type x} {f : αβγ} {s : Set α} {t : Set β} {a : α} (ht : t.Infinite) (ha : a s) (hf : Set.InjOn (f a) t) :
(Set.image2 f s t).Infinite
theorem Set.infinite_image2 {α : Type u} {β : Type v} {γ : Type x} {f : αβγ} {s : Set α} {t : Set β} (hfs : bt, Set.InjOn (fun (a : α) => f a b) s) (hft : as, Set.InjOn (f a) t) :
(Set.image2 f s t).Infinite s.Infinite t.Nonempty t.Infinite s.Nonempty
theorem Set.finite_image2 {α : Type u} {β : Type v} {γ : Type x} {f : αβγ} {s : Set α} {t : Set β} (hfs : bt, Set.InjOn (fun (x : α) => f x b) s) (hft : as, Set.InjOn (f a) t) :
(Set.image2 f s t).Finite s.Finite t.Finite s = t =
theorem Set.infinite_of_injOn_mapsTo {α : Type u} {β : Type v} {s : Set α} {t : Set β} {f : αβ} (hi : ) (hm : Set.MapsTo f s t) (hs : s.Infinite) :
t.Infinite
theorem Set.Infinite.exists_ne_map_eq_of_mapsTo {α : Type u} {β : Type v} {s : Set α} {t : Set β} {f : αβ} (hs : s.Infinite) (hf : Set.MapsTo f s t) (ht : t.Finite) :
xs, ys, x y f x = f y
theorem Set.infinite_range_of_injective {α : Type u} {β : Type v} [] {f : αβ} (hi : ) :
(Set.range f).Infinite
theorem Set.infinite_of_injective_forall_mem {α : Type u} {β : Type v} [] {s : Set β} {f : αβ} (hi : ) (hf : ∀ (x : α), f x s) :
s.Infinite
theorem Set.not_injOn_infinite_finite_image {α : Type u} {β : Type v} {f : αβ} {s : Set α} (h_inf : s.Infinite) (h_fin : (f '' s).Finite) :
theorem Set.infinite_iUnion {α : Type u} {ι : Type u_1} [] {s : ιSet α} (hs : ) :
(⋃ (i : ι), s i).Infinite
theorem Set.Infinite.biUnion {α : Type u} {ι : Type u_1} {s : ιSet α} {a : Set ι} (ha : a.Infinite) (hs : ) :
(⋃ ia, s i).Infinite
theorem Set.Infinite.sUnion {α : Type u} {s : Set (Set α)} (hs : s.Infinite) :
(⋃₀ s).Infinite

Order properties #

theorem Set.infinite_of_forall_exists_gt {α : Type u} [] [] {s : Set α} (h : ∀ (a : α), bs, a < b) :
s.Infinite
theorem Set.infinite_of_forall_exists_lt {α : Type u} [] [] {s : Set α} (h : ∀ (a : α), bs, b < a) :
s.Infinite
theorem Set.finite_isTop (α : Type u_1) [] :
{x : α | }.Finite
theorem Set.finite_isBot (α : Type u_1) [] :
{x : α | }.Finite
theorem Set.Infinite.exists_lt_map_eq_of_mapsTo {α : Type u} {β : Type v} [] {s : Set α} {t : Set β} {f : αβ} (hs : s.Infinite) (hf : Set.MapsTo f s t) (ht : t.Finite) :
xs, ys, x < y f x = f y
theorem Set.Finite.exists_lt_map_eq_of_forall_mem {α : Type u} {β : Type v} [] [] {t : Set β} {f : αβ} (hf : ∀ (a : α), f a t) (ht : t.Finite) :
∃ (a : α) (b : α), a < b f a = f b
theorem Set.exists_min_image {α : Type u} {β : Type v} [] (s : Set α) (f : αβ) (h1 : s.Finite) :
s.Nonemptyas, bs, f a f b
theorem Set.exists_max_image {α : Type u} {β : Type v} [] (s : Set α) (f : αβ) (h1 : s.Finite) :
s.Nonemptyas, bs, f b f a
theorem Set.exists_lower_bound_image {α : Type u} {β : Type v} [] [] (s : Set α) (f : αβ) (h : s.Finite) :
∃ (a : α), bs, f a f b
theorem Set.exists_upper_bound_image {α : Type u} {β : Type v} [] [] (s : Set α) (f : αβ) (h : s.Finite) :
∃ (a : α), bs, f b f a
theorem Set.Finite.iSup_biInf_of_monotone {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [Preorder ι'] [Nonempty ι'] [IsDirected ι' fun (x1 x2 : ι') => x1 x2] [] {s : Set ι} (hs : s.Finite) {f : ιι'α} (hf : is, Monotone (f i)) :
⨆ (j : ι'), is, f i j = is, ⨆ (j : ι'), f i j
theorem Set.Finite.iSup_biInf_of_antitone {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [Preorder ι'] [Nonempty ι'] [IsDirected ι' (Function.swap fun (x1 x2 : ι') => x1 x2)] [] {s : Set ι} (hs : s.Finite) {f : ιι'α} (hf : is, Antitone (f i)) :
⨆ (j : ι'), is, f i j = is, ⨆ (j : ι'), f i j
theorem Set.Finite.iInf_biSup_of_monotone {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [Preorder ι'] [Nonempty ι'] [IsDirected ι' (Function.swap fun (x1 x2 : ι') => x1 x2)] [] {s : Set ι} (hs : s.Finite) {f : ιι'α} (hf : is, Monotone (f i)) :
⨅ (j : ι'), is, f i j = is, ⨅ (j : ι'), f i j
theorem Set.Finite.iInf_biSup_of_antitone {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [Preorder ι'] [Nonempty ι'] [IsDirected ι' fun (x1 x2 : ι') => x1 x2] [] {s : Set ι} (hs : s.Finite) {f : ιι'α} (hf : is, Antitone (f i)) :
⨅ (j : ι'), is, f i j = is, ⨅ (j : ι'), f i j
theorem Set.iSup_iInf_of_monotone {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [] [Preorder ι'] [Nonempty ι'] [IsDirected ι' fun (x1 x2 : ι') => x1 x2] [] {f : ιι'α} (hf : ∀ (i : ι), Monotone (f i)) :
⨆ (j : ι'), ⨅ (i : ι), f i j = ⨅ (i : ι), ⨆ (j : ι'), f i j
theorem Set.iSup_iInf_of_antitone {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [] [Preorder ι'] [Nonempty ι'] [IsDirected ι' (Function.swap fun (x1 x2 : ι') => x1 x2)] [] {f : ιι'α} (hf : ∀ (i : ι), Antitone (f i)) :
⨆ (j : ι'), ⨅ (i : ι), f i j = ⨅ (i : ι), ⨆ (j : ι'), f i j
theorem Set.iInf_iSup_of_monotone {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [] [Preorder ι'] [Nonempty ι'] [IsDirected ι' (Function.swap fun (x1 x2 : ι') => x1 x2)] [] {f : ιι'α} (hf : ∀ (i : ι), Monotone (f i)) :
⨅ (j : ι'), ⨆ (i : ι), f i j = ⨆ (i : ι), ⨅ (j : ι'), f i j
theorem Set.iInf_iSup_of_antitone {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [] [Preorder ι'] [Nonempty ι'] [IsDirected ι' fun (x1 x2 : ι') => x1 x2] [] {f : ιι'α} (hf : ∀ (i : ι), Antitone (f i)) :
⨅ (j : ι'), ⨆ (i : ι), f i j = ⨆ (i : ι), ⨅ (j : ι'), f i j
theorem Set.iUnion_iInter_of_monotone {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [] [Preorder ι'] [IsDirected ι' fun (x1 x2 : ι') => x1 x2] [Nonempty ι'] {s : ιι'Set α} (hs : ∀ (i : ι), Monotone (s i)) :
⋃ (j : ι'), ⋂ (i : ι), s i j = ⋂ (i : ι), ⋃ (j : ι'), s i j

An increasing union distributes over finite intersection.

theorem Set.iUnion_iInter_of_antitone {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [] [Preorder ι'] [IsDirected ι' (Function.swap fun (x1 x2 : ι') => x1 x2)] [Nonempty ι'] {s : ιι'Set α} (hs : ∀ (i : ι), Antitone (s i)) :
⋃ (j : ι'), ⋂ (i : ι), s i j = ⋂ (i : ι), ⋃ (j : ι'), s i j

A decreasing union distributes over finite intersection.

theorem Set.iInter_iUnion_of_monotone {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [] [Preorder ι'] [IsDirected ι' (Function.swap fun (x1 x2 : ι') => x1 x2)] [Nonempty ι'] {s : ιι'Set α} (hs : ∀ (i : ι), Monotone (s i)) :
⋂ (j : ι'), ⋃ (i : ι), s i j = ⋃ (i : ι), ⋂ (j : ι'), s i j

An increasing intersection distributes over finite union.

theorem Set.iInter_iUnion_of_antitone {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [] [Preorder ι'] [IsDirected ι' fun (x1 x2 : ι') => x1 x2] [Nonempty ι'] {s : ιι'Set α} (hs : ∀ (i : ι), Antitone (s i)) :
⋂ (j : ι'), ⋃ (i : ι), s i j = ⋃ (i : ι), ⋂ (j : ι'), s i j

A decreasing intersection distributes over finite union.

theorem Set.iUnion_pi_of_monotone {ι : Type u_1} {ι' : Type u_2} [] [Nonempty ι'] {α : ιType u_3} {I : Set ι} {s : (i : ι) → ι'Set (α i)} (hI : I.Finite) (hs : iI, Monotone (s i)) :
(⋃ (j : ι'), I.pi fun (i : ι) => s i j) = I.pi fun (i : ι) => ⋃ (j : ι'), s i j
theorem Set.iUnion_univ_pi_of_monotone {ι : Type u_1} {ι' : Type u_2} [] [Nonempty ι'] [] {α : ιType u_3} {s : (i : ι) → ι'Set (α i)} (hs : ∀ (i : ι), Monotone (s i)) :
(⋃ (j : ι'), Set.univ.pi fun (i : ι) => s i j) = Set.univ.pi fun (i : ι) => ⋃ (j : ι'), s i j
theorem Set.finite_range_findGreatest {α : Type u} {P : α} [(x : α) → DecidablePred (P x)] {b : } :
(Set.range fun (x : α) => Nat.findGreatest (P x) b).Finite
theorem Set.Finite.exists_maximal_wrt {α : Type u} {β : Type v} [] (f : αβ) (s : Set α) (h : s.Finite) (hs : s.Nonempty) :
as, a's, f a f a'f a = f a'
theorem Set.Finite.exists_maximal_wrt' {α : Type u} {β : Type v} [] (f : αβ) (s : Set α) (h : (f '' s).Finite) (hs : s.Nonempty) :
as, a's, f a f a'f a = f a'

A version of Finite.exists_maximal_wrt with the (weaker) hypothesis that the image of s is finite rather than s itself.

theorem Set.Finite.exists_minimal_wrt {α : Type u} {β : Type v} [] (f : αβ) (s : Set α) (h : s.Finite) (hs : s.Nonempty) :
as, a's, f a' f af a = f a'
theorem Set.Finite.exists_minimal_wrt' {α : Type u} {β : Type v} [] (f : αβ) (s : Set α) (h : (f '' s).Finite) (hs : s.Nonempty) :
as, a's, f a' f af a = f a'

A version of Finite.exists_minimal_wrt with the (weaker) hypothesis that the image of s is finite rather than s itself.

theorem Set.Finite.bddAbove {α : Type u} [] [IsDirected α fun (x1 x2 : α) => x1 x2] [] {s : Set α} (hs : s.Finite) :

A finite set is bounded above.

theorem Set.Finite.bddAbove_biUnion {α : Type u} {β : Type v} [] [IsDirected α fun (x1 x2 : α) => x1 x2] [] {I : Set β} {S : βSet α} (H : I.Finite) :
BddAbove (⋃ iI, S i) iI, BddAbove (S i)

A finite union of sets which are all bounded above is still bounded above.

theorem Set.infinite_of_not_bddAbove {α : Type u} [] [IsDirected α fun (x1 x2 : α) => x1 x2] [] {s : Set α} :
s.Infinite
theorem Set.Finite.bddBelow {α : Type u} [] [IsDirected α fun (x1 x2 : α) => x1 x2] [] {s : Set α} (hs : s.Finite) :

A finite set is bounded below.

theorem Set.Finite.bddBelow_biUnion {α : Type u} {β : Type v} [] [IsDirected α fun (x1 x2 : α) => x1 x2] [] {I : Set β} {S : βSet α} (H : I.Finite) :
BddBelow (⋃ iI, S i) iI, BddBelow (S i)

A finite union of sets which are all bounded below is still bounded below.

theorem Set.infinite_of_not_bddBelow {α : Type u} [] [IsDirected α fun (x1 x2 : α) => x1 x2] [] {s : Set α} :
s.Infinite
theorem Finset.exists_card_eq {α : Type u} [] (n : ) :
∃ (s : ), s.card = n
theorem Finset.bddAbove {α : Type u} [] [] (s : ) :

A finset is bounded above.

theorem Finset.bddBelow {α : Type u} [] [] (s : ) :

A finset is bounded below.

theorem Finite.of_forall_not_lt_lt {α : Type u} [] (h : ∀ ⦃x y z : α⦄, x < yy < zFalse) :

If a linear order does not contain any triple of elements x < y < z, then this type is finite.

theorem Set.finite_of_forall_not_lt_lt {α : Type u} [] {s : Set α} (h : xs, ys, zs, x < yy < zFalse) :
s.Finite

If a set s does not contain any triple of elements x < y < z, then s is finite.

theorem Set.finite_diff_iUnion_Ioo {α : Type u} [] (s : Set α) :
(s \ xs, ys, Set.Ioo x y).Finite
theorem Set.finite_diff_iUnion_Ioo' {α : Type u} [] (s : Set α) :
(s \ ⋃ (x : s × s), Set.Ioo x.1 x.2).Finite
theorem Directed.exists_mem_subset_of_finset_subset_biUnion {α : Type u_1} {ι : Type u_2} [] {f : ιSet α} (h : Directed (fun (x1 x2 : Set α) => x1 x2) f) {s : } (hs : s ⋃ (i : ι), f i) :
∃ (i : ι), s f i
theorem DirectedOn.exists_mem_subset_of_finset_subset_biUnion {α : Type u_1} {ι : Type u_2} {f : ιSet α} {c : Set ι} (hn : c.Nonempty) (hc : DirectedOn (fun (i j : ι) => f i f j) c) {s : } (hs : s ic, f i) :
ic, s f i