Cardinalities of finite types

Main declarations

• Fintype.card α: Cardinality of a fintype. Equal to Finset.univ.card.
• Fintype.truncEquivFin: A fintype α is computably equivalent to Fin (card α). The Trunc-free, noncomputable version is Fintype.equivFin.
• Fintype.truncEquivOfCardEq Fintype.equivOfCardEq: Two fintypes of same cardinality are equivalent. See above.
• Fin.equiv_iff_eq: Fin m ≃ Fin n iff m = n.
• Infinite.natEmbedding: An embedding of ℕ into an infinite type.

We also provide the following versions of the pigeonholes principle.

• Fintype.exists_ne_map_eq_of_card_lt and isEmpty_of_card_lt: Finitely many pigeons and pigeonholes. Weak formulation.
• Finite.exists_ne_map_eq_of_infinite: Infinitely many pigeons in finitely many pigeonholes. Weak formulation.
• Finite.exists_infinite_fiber: Infinitely many pigeons in finitely many pigeonholes. Strong formulation.

Some more pigeonhole-like statements can be found in Data.Fintype.CardEmbedding.

Types which have an injection from/a surjection to an Infinite type are themselves Infinite. See Infinite.of_injective and Infinite.of_surjective.

Instances

We provide Infinite instances for

• specific types: ℕ, ℤ, String
• type constructors: Multiset α, List α

def Fintype.card (α : Type u_4) [Fintype α] : ℕ

card α is the number of elements in α, defined when α is a fintype.

Equations

• Fintype.card α = Finset.univ.card

def Fintype.truncEquivFin (α : Type u_4) [DecidableEq α] [Fintype α] : Trunc (α ≃ Fin (Fintype.card α))

There is (computably) an equivalence between α and Fin (card α).

Since it is not unique and depends on which permutation of the universe list is used, the equivalence is wrapped in Trunc to preserve computability.

See Fintype.equivFin for the noncomputable version, and Fintype.truncEquivFinOfCardEq and Fintype.equivFinOfCardEq for an equiv α ≃ Fin n given Fintype.card α = n.

See Fintype.truncFinBijection for a version without [DecidableEq α].

Equations

• One or more equations did not get rendered due to their size.

noncomputable def Fintype.equivFin (α : Type u_4) [Fintype α] : α ≃ Fin (Fintype.card α)

There is (noncomputably) an equivalence between α and Fin (card α).

See Fintype.truncEquivFin for the computable version, and Fintype.truncEquivFinOfCardEq and Fintype.equivFinOfCardEq for an equiv α ≃ Fin n given Fintype.card α = n.

Equations

• Fintype.equivFin α = (Fintype.truncEquivFin α).out

def Fintype.truncFinBijection (α : Type u_4) [Fintype α] : Trunc { f : Fin (Fintype.card α) → α // Function.Bijective f }

There is (computably) a bijection between Fin (card α) and α.

Since it is not unique and depends on which permutation of the universe list is used, the bijection is wrapped in Trunc to preserve computability.

See Fintype.truncEquivFin for a version that gives an equivalence given [DecidableEq α].

Equations

• One or more equations did not get rendered due to their size.

theorem Fintype.subtype_card {α : Type u_1} {p : α → Prop} (s : Finset α) (H : ∀ (x : α), x ∈ s ↔ p x) : Fintype.card { x : α // p x } = s.card

theorem Fintype.card_of_subtype {α : Type u_1} {p : α → Prop} (s : Finset α) (H : ∀ (x : α), x ∈ s ↔ p x) [Fintype { x : α // p x }] : Fintype.card { x : α // p x } = s.card

@[simp]

theorem Fintype.card_ofFinset {α : Type u_1} {p : Set α} (s : Finset α) (H : ∀ (x : α), x ∈ s ↔ x ∈ p) : Fintype.card ↑p = s.card

theorem Fintype.card_of_finset' {α : Type u_1} {p : Set α} (s : Finset α) (H : ∀ (x : α), x ∈ s ↔ x ∈ p) [Fintype ↑p] : Fintype.card ↑p = s.card

theorem Fintype.ofEquiv_card {α : Type u_1} {β : Type u_2} [Fintype α] (f : α ≃ β) : Fintype.card β = Fintype.card α

theorem Fintype.card_congr {α : Type u_4} {β : Type u_5} [Fintype α] [Fintype β] (f : α ≃ β) : Fintype.card α = Fintype.card β

theorem Fintype.card_congr' {α β : Type u_4} [Fintype α] [Fintype β] (h : α = β) : Fintype.card α = Fintype.card β

def Fintype.truncEquivFinOfCardEq {α : Type u_1} [Fintype α] [DecidableEq α] {n : ℕ} (h : Fintype.card α = n) : Trunc (α ≃ Fin n)

If the cardinality of α is n, there is computably a bijection between α and Fin n.

See Fintype.equivFinOfCardEq for the noncomputable definition, and Fintype.truncEquivFin and Fintype.equivFin for the bijection α ≃ Fin (card α).

Equations

• Fintype.truncEquivFinOfCardEq h = Trunc.map (fun (e : α ≃ Fin (Fintype.card α)) => e.trans (finCongr h)) (Fintype.truncEquivFin α)

noncomputable def Fintype.equivFinOfCardEq {α : Type u_1} [Fintype α] {n : ℕ} (h : Fintype.card α = n) : α ≃ Fin n

If the cardinality of α is n, there is noncomputably a bijection between α and Fin n.

See Fintype.truncEquivFinOfCardEq for the computable definition, and Fintype.truncEquivFin and Fintype.equivFin for the bijection α ≃ Fin (card α).

Equations

• Fintype.equivFinOfCardEq h = (Fintype.truncEquivFinOfCardEq h).out

def Fintype.truncEquivOfCardEq {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β] (h : Fintype.card α = Fintype.card β) : Trunc (α ≃ β)

Two Fintypes with the same cardinality are (computably) in bijection.

See Fintype.equivOfCardEq for the noncomputable version, and Fintype.truncEquivFinOfCardEq and Fintype.equivFinOfCardEq for the specialization to Fin.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def Fintype.equivOfCardEq {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (h : Fintype.card α = Fintype.card β) : α ≃ β

Two Fintypes with the same cardinality are (noncomputably) in bijection.

See Fintype.truncEquivOfCardEq for the computable version, and Fintype.truncEquivFinOfCardEq and Fintype.equivFinOfCardEq for the specialization to Fin.

Equations

• Fintype.equivOfCardEq h = (Fintype.truncEquivOfCardEq h).out

theorem Fintype.card_eq {α : Type u_4} {β : Type u_5} [_F : Fintype α] [_G : Fintype β] : Fintype.card α = Fintype.card β ↔ Nonempty (α ≃ β)

@[simp]

theorem Fintype.card_ofSubsingleton {α : Type u_1} (a : α) [Subsingleton α] : Fintype.card α = 1

Note: this lemma is specifically about Fintype.ofSubsingleton. For a statement about arbitrary Fintype instances, use either Fintype.card_le_one_iff_subsingleton or Fintype.card_unique.

@[simp]

theorem Fintype.card_unique {α : Type u_1} [Unique α] [h : Fintype α] : Fintype.card α = 1

@[simp]

theorem Fintype.card_ofIsEmpty {α : Type u_1} [IsEmpty α] : Fintype.card α = 0

Note: this lemma is specifically about Fintype.ofIsEmpty. For a statement about arbitrary Fintype instances, use Fintype.card_eq_zero.

@[simp]

theorem Set.toFinset_card {α : Type u_4} (s : Set α) [Fintype ↑s] : s.toFinset.card = Fintype.card ↑s

@[simp]

theorem Finset.card_univ {α : Type u_1} [Fintype α] : Finset.univ.card = Fintype.card α

theorem Finset.eq_univ_of_card {α : Type u_1} [Fintype α] (s : Finset α) (hs : s.card = Fintype.card α) : s = Finset.univ

theorem Finset.card_eq_iff_eq_univ {α : Type u_1} [Fintype α] (s : Finset α) : s.card = Fintype.card α ↔ s = Finset.univ

theorem Finset.card_le_univ {α : Type u_1} [Fintype α] (s : Finset α) : s.card ≤ Fintype.card α

theorem Finset.card_lt_univ_of_not_mem {α : Type u_1} [Fintype α] {s : Finset α} {x : α} (hx : x ∉ s) : s.card < Fintype.card α

theorem Finset.card_lt_iff_ne_univ {α : Type u_1} [Fintype α] (s : Finset α) : s.card < Fintype.card α ↔ s ≠ Finset.univ

theorem Finset.card_compl_lt_iff_nonempty {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) : sᶜ.card < Fintype.card α ↔ s.Nonempty

theorem Finset.card_univ_diff {α : Type u_1} [DecidableEq α] [Fintype α] (s : Finset α) : (Finset.univ \ s).card = Fintype.card α - s.card

theorem Finset.card_compl {α : Type u_1} [DecidableEq α] [Fintype α] (s : Finset α) : sᶜ.card = Fintype.card α - s.card

@[simp]

theorem Finset.card_add_card_compl {α : Type u_1} [DecidableEq α] [Fintype α] (s : Finset α) : s.card + sᶜ.card = Fintype.card α

@[simp]

theorem Finset.card_compl_add_card {α : Type u_1} [DecidableEq α] [Fintype α] (s : Finset α) : sᶜ.card + s.card = Fintype.card α

theorem Fintype.card_compl_set {α : Type u_1} [Fintype α] (s : Set α) [Fintype ↑s] [Fintype ↑sᶜ] : Fintype.card ↑sᶜ = Fintype.card α - Fintype.card ↑s

@[simp]

theorem Fintype.card_fin (n : ℕ) : Fintype.card (Fin n) = n

theorem Fintype.card_fin_lt_of_le {m n : ℕ} (h : m ≤ n) : Fintype.card { i : Fin n // ↑i < m } = m

theorem Finset.card_fin (n : ℕ) : Finset.univ.card = n

theorem fin_injective : Function.Injective Fin

Fin as a map from ℕ to Type is injective. Note that since this is a statement about equality of types, using it should be avoided if possible.

theorem Fin.val_eq_val_of_heq {k l : ℕ} {i : Fin k} {j : Fin l} (h : HEq i j) : ↑i = ↑j

theorem Fin.cast_eq_cast' {n m : ℕ} (h : Fin n = Fin m) : cast h = Fin.cast ⋯

A reversed version of Fin.cast_eq_cast that is easier to rewrite with.

theorem card_finset_fin_le {n : ℕ} (s : Finset (Fin n)) : s.card ≤ n

theorem Fintype.card_subtype_eq {α : Type u_1} (y : α) [Fintype { x : α // x = y }] : Fintype.card { x : α // x = y } = 1

theorem Fintype.card_subtype_eq' {α : Type u_1} (y : α) [Fintype { x : α // y = x }] : Fintype.card { x : α // y = x } = 1

theorem Fintype.card_empty : Fintype.card Empty = 0

theorem Fintype.card_pempty : Fintype.card PEmpty.{u_4 + 1} = 0

theorem Fintype.card_unit : Fintype.card Unit = 1

@[simp]

theorem Fintype.card_punit : Fintype.card PUnit.{u_4 + 1} = 1

@[simp]

theorem Fintype.card_bool : Fintype.card Bool = 2

@[simp]

theorem Fintype.card_ulift (α : Type u_4) [Fintype α] : Fintype.card (ULift.{u_5, u_4} α) = Fintype.card α

@[simp]

theorem Fintype.card_plift (α : Type u_4) [Fintype α] : Fintype.card (PLift α) = Fintype.card α

@[simp]

theorem Fintype.card_orderDual (α : Type u_4) [Fintype α] : Fintype.card αᵒᵈ = Fintype.card α

@[simp]

theorem Fintype.card_lex (α : Type u_4) [Fintype α] : Fintype.card (Lex α) = Fintype.card α

@[simp]

theorem Fintype.card_multiplicative (α : Type u_4) [Fintype α] : Fintype.card (Multiplicative α) = Fintype.card α

@[simp]

theorem Fintype.card_additive (α : Type u_4) [Fintype α] : Fintype.card (Additive α) = Fintype.card α

noncomputable def Fintype.sumLeft {α : Type u_4} {β : Type u_5} [Fintype (α ⊕ β)] : Fintype α

Given that α ⊕ β is a fintype, α is also a fintype. This is non-computable as it uses that Sum.inl is an injection, but there's no clear inverse if α is empty.

Equations

• Fintype.sumLeft = Fintype.ofInjective Sum.inl ⋯

noncomputable def Fintype.sumRight {α : Type u_4} {β : Type u_5} [Fintype (α ⊕ β)] : Fintype β

Given that α ⊕ β is a fintype, β is also a fintype. This is non-computable as it uses that Sum.inr is an injection, but there's no clear inverse if β is empty.

Equations

• Fintype.sumRight = Fintype.ofInjective Sum.inr ⋯

Relation to Finite

In this section we prove that α : Type* is Finite if and only if Fintype α is nonempty.

theorem Fintype.finite {α : Type u_4} (_inst : Fintype α) : Finite α

@[instance 900]

instance Finite.of_fintype (α : Type u_4) [Fintype α] : Finite α

For efficiency reasons, we want Finite instances to have higher priority than ones coming from Fintype instances.

Equations

• ⋯ = ⋯

theorem finite_iff_nonempty_fintype (α : Type u_4) : Finite α ↔ Nonempty (Fintype α)

theorem nonempty_fintype (α : Type u_4) [Finite α] : Nonempty (Fintype α)

See also nonempty_encodable, nonempty_denumerable.

noncomputable def Fintype.ofFinite (α : Type u_4) [Finite α] : Fintype α

Noncomputably get a Fintype instance from a Finite instance. This is not an instance because we want Fintype instances to be useful for computations.

Equations

• Fintype.ofFinite α = ⋯.some

theorem Finite.of_injective {α : Sort u_4} {β : Sort u_5} [Finite β] (f : α → β) (H : Function.Injective f) : Finite α

@[instance 100]

instance Finite.of_subsingleton {α : Sort u_4} [Subsingleton α] : Finite α

Equations

• ⋯ = ⋯

instance prop (p : Prop) : Finite p

Equations

• ⋯ = ⋯

instance Subtype.finite {α : Sort u_4} [Finite α] {p : α → Prop} : Finite { x : α // p x }

This instance also provides [Finite s] for s : Set α.

Equations

• ⋯ = ⋯

theorem Finite.of_surjective {α : Sort u_4} {β : Sort u_5} [Finite α] (f : α → β) (H : Function.Surjective f) : Finite β

instance Quot.finite {α : Sort u_4} [Finite α] (r : α → α → Prop) : Finite (Quot r)

Equations

• ⋯ = ⋯

instance Quotient.finite {α : Sort u_4} [Finite α] (s : Setoid α) : Finite (Quotient s)

Equations

• ⋯ = ⋯

theorem Finite.exists_univ_list (α : Type u_4) [Finite α] : ∃ (l : List α), l.Nodup ∧ ∀ (x : α), x ∈ l

theorem List.Nodup.length_le_card {α : Type u_4} [Fintype α] {l : List α} (h : l.Nodup) : l.length ≤ Fintype.card α

theorem Fintype.card_le_of_injective {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (f : α → β) (hf : Function.Injective f) : Fintype.card α ≤ Fintype.card β

theorem Fintype.card_le_of_embedding {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (f : α ↪ β) : Fintype.card α ≤ Fintype.card β

theorem Fintype.card_lt_of_injective_of_not_mem {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (f : α → β) (h : Function.Injective f) {b : β} (w : b ∉ Set.range f) : Fintype.card α < Fintype.card β

theorem Fintype.card_lt_of_injective_not_surjective {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (f : α → β) (h : Function.Injective f) (h' : ¬Function.Surjective f) : Fintype.card α < Fintype.card β

theorem Fintype.card_le_of_surjective {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (f : α → β) (h : Function.Surjective f) : Fintype.card β ≤ Fintype.card α

theorem Fintype.card_range_le {α : Type u_4} {β : Type u_5} (f : α → β) [Fintype α] [Fintype ↑(Set.range f)] : Fintype.card ↑(Set.range f) ≤ Fintype.card α

theorem Fintype.card_range {α : Type u_4} {β : Type u_5} {F : Type u_6} [FunLike F α β] [EmbeddingLike F α β] (f : F) [Fintype α] [Fintype ↑(Set.range ⇑f)] : Fintype.card ↑(Set.range ⇑f) = Fintype.card α

theorem Fintype.exists_ne_map_eq_of_card_lt {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (f : α → β) (h : Fintype.card β < Fintype.card α) : ∃ (x : α) (y : α), x ≠ y ∧ f x = f y

The pigeonhole principle for finitely many pigeons and pigeonholes. This is the Fintype version of Finset.exists_ne_map_eq_of_card_lt_of_maps_to.

theorem Fintype.card_eq_one_iff {α : Type u_1} [Fintype α] : Fintype.card α = 1 ↔ ∃ (x : α), ∀ (y : α), y = x

theorem Fintype.card_eq_zero_iff {α : Type u_1} [Fintype α] : Fintype.card α = 0 ↔ IsEmpty α

@[simp]

theorem Fintype.card_eq_zero {α : Type u_1} [Fintype α] [IsEmpty α] : Fintype.card α = 0

theorem Fintype.card_of_isEmpty {α : Type u_1} [Fintype α] [IsEmpty α] : Fintype.card α = 0

Alias of Fintype.card_eq_zero.

theorem Fintype.card_eq_one_iff_nonempty_unique {α : Type u_1} [Fintype α] : Fintype.card α = 1 ↔ Nonempty (Unique α)

def Fintype.cardEqZeroEquivEquivEmpty {α : Type u_1} [Fintype α] : Fintype.card α = 0 ≃ (α ≃ Empty)

A Fintype with cardinality zero is equivalent to Empty.

Equations

• Fintype.cardEqZeroEquivEquivEmpty = (Equiv.ofIff ⋯).trans (Equiv.equivEmptyEquiv α).symm

theorem Fintype.card_pos_iff {α : Type u_1} [Fintype α] : 0 < Fintype.card α ↔ Nonempty α

theorem Fintype.card_pos {α : Type u_1} [Fintype α] [h : Nonempty α] : 0 < Fintype.card α

@[simp]

theorem Fintype.card_ne_zero {α : Type u_1} [Fintype α] [Nonempty α] : Fintype.card α ≠ 0

instance Fintype.instNeZeroNatCardOfNonempty {α : Type u_1} [Fintype α] [Nonempty α] : NeZero (Fintype.card α)

Equations

• ⋯ = ⋯

theorem Fintype.card_le_one_iff {α : Type u_1} [Fintype α] : Fintype.card α ≤ 1 ↔ ∀ (a b : α), a = b

theorem Fintype.card_le_one_iff_subsingleton {α : Type u_1} [Fintype α] : Fintype.card α ≤ 1 ↔ Subsingleton α

theorem Fintype.one_lt_card_iff_nontrivial {α : Type u_1} [Fintype α] : 1 < Fintype.card α ↔ Nontrivial α

theorem Fintype.exists_ne_of_one_lt_card {α : Type u_1} [Fintype α] (h : 1 < Fintype.card α) (a : α) : ∃ (b : α), b ≠ a

theorem Fintype.exists_pair_of_one_lt_card {α : Type u_1} [Fintype α] (h : 1 < Fintype.card α) : ∃ (a : α) (b : α), a ≠ b

theorem Fintype.card_eq_one_of_forall_eq {α : Type u_1} [Fintype α] {i : α} (h : ∀ (j : α), j = i) : Fintype.card α = 1

theorem Fintype.exists_unique_iff_card_one {α : Type u_4} [Fintype α] (p : α → Prop) [DecidablePred p] : (∃! a : α, p a) ↔ (Finset.filter (fun (x : α) => p x) Finset.univ).card = 1

theorem Fintype.one_lt_card {α : Type u_1} [Fintype α] [h : Nontrivial α] : 1 < Fintype.card α

theorem Fintype.one_lt_card_iff {α : Type u_1} [Fintype α] : 1 < Fintype.card α ↔ ∃ (a : α) (b : α), a ≠ b

theorem Fintype.two_lt_card_iff {α : Type u_1} [Fintype α] : 2 < Fintype.card α ↔ ∃ (a : α) (b : α) (c : α), a ≠ b ∧ a ≠ c ∧ b ≠ c

theorem Fintype.card_of_bijective {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] {f : α → β} (hf : Function.Bijective f) : Fintype.card α = Fintype.card β

theorem Finite.surjective_of_injective {α : Type u_1} [Finite α] {f : α → α} (hinj : Function.Injective f) : Function.Surjective f

theorem Finite.injective_iff_surjective {α : Type u_1} [Finite α] {f : α → α} : Function.Injective f ↔ Function.Surjective f

theorem Finite.injective_iff_bijective {α : Type u_1} [Finite α] {f : α → α} : Function.Injective f ↔ Function.Bijective f

theorem Finite.surjective_iff_bijective {α : Type u_1} [Finite α] {f : α → α} : Function.Surjective f ↔ Function.Bijective f

theorem Finite.injective_iff_surjective_of_equiv {α : Type u_1} {β : Type u_2} [Finite α] {f : α → β} (e : α ≃ β) : Function.Injective f ↔ Function.Surjective f

theorem Function.Injective.bijective_of_finite {α : Type u_1} [Finite α] {f : α → α} : Function.Injective f → Function.Bijective f

Alias of the forward direction of Finite.injective_iff_bijective.

theorem Function.Surjective.bijective_of_finite {α : Type u_1} [Finite α] {f : α → α} : Function.Surjective f → Function.Bijective f

Alias of the forward direction of Finite.surjective_iff_bijective.

theorem Function.Injective.surjective_of_fintype {α : Type u_1} {β : Type u_2} [Finite α] {f : α → β} (e : α ≃ β) : Function.Injective f → Function.Surjective f

Alias of the forward direction of Finite.injective_iff_surjective_of_equiv.

theorem Function.Surjective.injective_of_fintype {α : Type u_1} {β : Type u_2} [Finite α] {f : α → β} (e : α ≃ β) : Function.Surjective f → Function.Injective f

Alias of the reverse direction of Finite.injective_iff_surjective_of_equiv.

theorem Fintype.bijective_iff_injective_and_card {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (f : α → β) : Function.Bijective f ↔ Function.Injective f ∧ Fintype.card α = Fintype.card β

theorem Fintype.bijective_iff_surjective_and_card {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (f : α → β) : Function.Bijective f ↔ Function.Surjective f ∧ Fintype.card α = Fintype.card β

theorem Function.LeftInverse.rightInverse_of_card_le {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] {f : α → β} {g : β → α} (hfg : Function.LeftInverse f g) (hcard : Fintype.card α ≤ Fintype.card β) : Function.RightInverse f g

theorem Function.RightInverse.leftInverse_of_card_le {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] {f : α → β} {g : β → α} (hfg : Function.RightInverse f g) (hcard : Fintype.card β ≤ Fintype.card α) : Function.LeftInverse f g

def Equiv.ofLeftInverseOfCardLE {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (hβα : Fintype.card β ≤ Fintype.card α) (f : α → β) (g : β → α) (h : Function.LeftInverse g f) : α ≃ β

Construct an equivalence from functions that are inverse to each other.

Equations

• Equiv.ofLeftInverseOfCardLE hβα f g h = { toFun := f, invFun := g, left_inv := h, right_inv := ⋯ }

@[simp]

theorem Equiv.ofLeftInverseOfCardLE_symm_apply {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (hβα : Fintype.card β ≤ Fintype.card α) (f : α → β) (g : β → α) (h : Function.LeftInverse g f) (a✝ : β) : (Equiv.ofLeftInverseOfCardLE hβα f g h).symm a✝ = g a✝

@[simp]

theorem Equiv.ofLeftInverseOfCardLE_apply {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (hβα : Fintype.card β ≤ Fintype.card α) (f : α → β) (g : β → α) (h : Function.LeftInverse g f) (a✝ : α) : (Equiv.ofLeftInverseOfCardLE hβα f g h) a✝ = f a✝

def Equiv.ofRightInverseOfCardLE {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (hαβ : Fintype.card α ≤ Fintype.card β) (f : α → β) (g : β → α) (h : Function.RightInverse g f) : α ≃ β

Construct an equivalence from functions that are inverse to each other.

Equations

• Equiv.ofRightInverseOfCardLE hαβ f g h = { toFun := f, invFun := g, left_inv := ⋯, right_inv := h }

@[simp]

theorem Equiv.ofRightInverseOfCardLE_symm_apply {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (hαβ : Fintype.card α ≤ Fintype.card β) (f : α → β) (g : β → α) (h : Function.RightInverse g f) (a✝ : β) : (Equiv.ofRightInverseOfCardLE hαβ f g h).symm a✝ = g a✝

@[simp]

theorem Equiv.ofRightInverseOfCardLE_apply {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (hαβ : Fintype.card α ≤ Fintype.card β) (f : α → β) (g : β → α) (h : Function.RightInverse g f) (a✝ : α) : (Equiv.ofRightInverseOfCardLE hαβ f g h) a✝ = f a✝

@[simp]

theorem Fintype.card_coe {α : Type u_1} (s : Finset α) [Fintype { x : α // x ∈ s }] : Fintype.card { x : α // x ∈ s } = s.card

noncomputable def Finset.equivFin {α : Type u_1} (s : Finset α) : { x : α // x ∈ s } ≃ Fin s.card

Noncomputable equivalence between a finset s coerced to a type and Fin #s.

Equations

• s.equivFin = Fintype.equivFinOfCardEq ⋯

noncomputable def Finset.equivFinOfCardEq {α : Type u_1} {s : Finset α} {n : ℕ} (h : s.card = n) : { x : α // x ∈ s } ≃ Fin n

Noncomputable equivalence between a finset s as a fintype and Fin n, when there is a proof that #s = n.

Equations

• Finset.equivFinOfCardEq h = Fintype.equivFinOfCardEq ⋯

theorem Finset.card_eq_of_equiv_fin {α : Type u_1} {s : Finset α} {n : ℕ} (i : { x : α // x ∈ s } ≃ Fin n) : s.card = n

theorem Finset.card_eq_of_equiv_fintype {α : Type u_1} {β : Type u_2} {s : Finset α} [Fintype β] (i : { x : α // x ∈ s } ≃ β) : s.card = Fintype.card β

noncomputable def Finset.equivOfCardEq {α : Type u_1} {β : Type u_2} {s : Finset α} {t : Finset β} (h : s.card = t.card) : { x : α // x ∈ s } ≃ { x : β // x ∈ t }

Noncomputable equivalence between two finsets s and t as fintypes when there is a proof that #s = #t.

Equations

• Finset.equivOfCardEq h = Fintype.equivOfCardEq ⋯

theorem Finset.card_eq_of_equiv {α : Type u_1} {β : Type u_2} {s : Finset α} {t : Finset β} (i : { x : α // x ∈ s } ≃ { x : β // x ∈ t }) : s.card = t.card

theorem Finset.exists_superset_card_eq {α : Type u_1} [Fintype α] {n : ℕ} {s : Finset α} (hsn : s.card ≤ n) (hnα : n ≤ Fintype.card α) : ∃ (t : Finset α), s ⊆ t ∧ t.card = n

We can inflate a set s to any bigger size.

@[simp]

theorem Fintype.card_prop : Fintype.card Prop = 2

theorem set_fintype_card_le_univ {α : Type u_1} [Fintype α] (s : Set α) [Fintype ↑s] : Fintype.card ↑s ≤ Fintype.card α

theorem set_fintype_card_eq_univ_iff {α : Type u_1} [Fintype α] (s : Set α) [Fintype ↑s] : Fintype.card ↑s = Fintype.card α ↔ s = Set.univ

noncomputable def Function.Embedding.equivOfFintypeSelfEmbedding {α : Type u_1} [Finite α] (e : α ↪ α) : α ≃ α

An embedding from a Fintype to itself can be promoted to an equivalence.

Equations

• e.equivOfFintypeSelfEmbedding = Equiv.ofBijective ⇑e ⋯

@[simp]

theorem Function.Embedding.equiv_of_fintype_self_embedding_to_embedding {α : Type u_1} [Finite α] (e : α ↪ α) : e.equivOfFintypeSelfEmbedding.toEmbedding = e

@[simp]

theorem Function.Embedding.isEmpty_of_card_lt {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (h : Fintype.card β < Fintype.card α) : IsEmpty (α ↪ β)

If ‖β‖ < ‖α‖ there are no embeddings α ↪ β. This is a formulation of the pigeonhole principle.

Note this cannot be an instance as it needs h.

def Function.Embedding.truncOfCardLE {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β] (h : Fintype.card α ≤ Fintype.card β) : Trunc (α ↪ β)

A constructive embedding of a fintype α in another fintype β when card α ≤ card β.

Equations

• One or more equations did not get rendered due to their size.

theorem Function.Embedding.nonempty_of_card_le {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (h : Fintype.card α ≤ Fintype.card β) : Nonempty (α ↪ β)

theorem Function.Embedding.nonempty_iff_card_le {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] : Nonempty (α ↪ β) ↔ Fintype.card α ≤ Fintype.card β

theorem Function.Embedding.exists_of_card_le_finset {α : Type u_1} {β : Type u_2} [Fintype α] {s : Finset β} (h : Fintype.card α ≤ s.card) : ∃ (f : α ↪ β), Set.range ⇑f ⊆ ↑s

@[simp]

theorem Finset.univ_map_embedding {α : Type u_4} [Fintype α] (e : α ↪ α) : Finset.map e Finset.univ = Finset.univ

theorem Fintype.card_lt_of_surjective_not_injective {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (f : α → β) (h : Function.Surjective f) (h' : ¬Function.Injective f) : Fintype.card β < Fintype.card α

theorem Fintype.card_subtype_le {α : Type u_1} [Fintype α] (p : α → Prop) [DecidablePred p] : Fintype.card { x : α // p x } ≤ Fintype.card α

theorem Fintype.card_subtype_lt {α : Type u_1} [Fintype α] {p : α → Prop} [DecidablePred p] {x : α} (hx : ¬p x) : Fintype.card { x : α // p x } < Fintype.card α

theorem Fintype.card_subtype {α : Type u_1} [Fintype α] (p : α → Prop) [DecidablePred p] : Fintype.card { x : α // p x } = (Finset.filter (fun (x : α) => p x) Finset.univ).card

@[simp]

theorem Fintype.card_subtype_compl {α : Type u_1} [Fintype α] (p : α → Prop) [Fintype { x : α // p x }] [Fintype { x : α // ¬p x }] : Fintype.card { x : α // ¬p x } = Fintype.card α - Fintype.card { x : α // p x }

theorem Fintype.card_subtype_mono {α : Type u_1} (p q : α → Prop) (h : p ≤ q) [Fintype { x : α // p x }] [Fintype { x : α // q x }] : Fintype.card { x : α // p x } ≤ Fintype.card { x : α // q x }

theorem Fintype.card_compl_eq_card_compl {α : Type u_1} [Finite α] (p q : α → Prop) [Fintype { x : α // p x }] [Fintype { x : α // ¬p x }] [Fintype { x : α // q x }] [Fintype { x : α // ¬q x }] (h : Fintype.card { x : α // p x } = Fintype.card { x : α // q x }) : Fintype.card { x : α // ¬p x } = Fintype.card { x : α // ¬q x }

If two subtypes of a fintype have equal cardinality, so do their complements.

theorem Fintype.card_quotient_le {α : Type u_1} [Fintype α] (s : Setoid α) [DecidableRel fun (x1 x2 : α) => x1 ≈ x2] : Fintype.card (Quotient s) ≤ Fintype.card α

theorem Fintype.card_quotient_lt {α : Type u_1} [Fintype α] {s : Setoid α} [DecidableRel fun (x1 x2 : α) => x1 ≈ x2] {x y : α} (h1 : x ≠ y) (h2 : x ≈ y) : Fintype.card (Quotient s) < Fintype.card α

theorem univ_eq_singleton_of_card_one {α : Type u_4} [Fintype α] (x : α) (h : Fintype.card α = 1) : Finset.univ = {x}

theorem Finite.wellFounded_of_trans_of_irrefl {α : Type u_1} [Finite α] (r : α → α → Prop) [IsTrans α r] [IsIrrefl α r] : WellFounded r

@[instance 100]

instance Finite.to_wellFoundedLT {α : Type u_1} [Finite α] [Preorder α] : WellFoundedLT α

Equations

• ⋯ = ⋯

@[instance 100]

instance Finite.to_wellFoundedGT {α : Type u_1} [Finite α] [Preorder α] : WellFoundedGT α

Equations

• ⋯ = ⋯

instance Bool.instWellFoundedLT : WellFoundedLT Bool

Equations

• Bool.instWellFoundedLT = ⋯

instance Bool.instWellFoundedGT : WellFoundedGT Bool

Equations

• Bool.instWellFoundedGT = ⋯

instance Prop.instWellFoundedLT : WellFoundedLT Prop

Equations

• «Prop».instWellFoundedLT = ⋯

instance Prop.instWellFoundedGT : WellFoundedGT Prop

Equations

• «Prop».instWellFoundedGT = ⋯

theorem Fintype.false {α : Type u_1} [Infinite α] (_h : Fintype α) : False

@[simp]

theorem isEmpty_fintype {α : Type u_4} : IsEmpty (Fintype α) ↔ Infinite α

noncomputable def fintypeOfNotInfinite {α : Type u_4} (h : ¬Infinite α) : Fintype α

A non-infinite type is a fintype.

Equations

• fintypeOfNotInfinite h = Fintype.ofFinite α

noncomputable def fintypeOrInfinite (α : Type u_4) : Fintype α ⊕' Infinite α

Any type is (classically) either a Fintype, or Infinite.

One can obtain the relevant typeclasses via cases fintypeOrInfinite α.

Equations

• fintypeOrInfinite α = if h : Infinite α then PSum.inr h else PSum.inl (fintypeOfNotInfinite h)

theorem Finset.exists_minimal {α : Type u_4} [Preorder α] (s : Finset α) (h : s.Nonempty) : ∃ m ∈ s, ∀ x ∈ s, ¬x < m

theorem Finset.exists_maximal {α : Type u_4} [Preorder α] (s : Finset α) (h : s.Nonempty) : ∃ m ∈ s, ∀ x ∈ s, ¬m < x

theorem Infinite.of_not_fintype {α : Type u_1} (h : Fintype α → False) : Infinite α

theorem Infinite.of_injective_to_set {α : Type u_1} {s : Set α} (hs : s ≠ Set.univ) {f : α → ↑s} (hf : Function.Injective f) : Infinite α

If s : Set α is a proper subset of α and f : α → s is injective, then α is infinite.

theorem Infinite.of_surjective_from_set {α : Type u_1} {s : Set α} (hs : s ≠ Set.univ) {f : ↑s → α} (hf : Function.Surjective f) : Infinite α

If s : Set α is a proper subset of α and f : s → α is surjective, then α is infinite.

theorem Infinite.exists_not_mem_finset {α : Type u_1} [Infinite α] (s : Finset α) : ∃ (x : α), x ∉ s

@[instance 100]

instance Infinite.instNontrivial (α : Type u_4) [Infinite α] : Nontrivial α

Equations

• ⋯ = ⋯

theorem Infinite.nonempty (α : Type u_4) [Infinite α] : Nonempty α

theorem Infinite.of_injective {α : Sort u_4} {β : Sort u_5} [Infinite β] (f : β → α) (hf : Function.Injective f) : Infinite α

theorem Infinite.of_surjective {α : Sort u_4} {β : Sort u_5} [Infinite β] (f : α → β) (hf : Function.Surjective f) : Infinite α

instance Infinite.instSigmaOfNonempty {α : Type u_1} {β : α → Type u_4} [Infinite α] [∀ (a : α), Nonempty (β a)] : Infinite ((a : α) × β a)

Equations

• ⋯ = ⋯

theorem Infinite.sigma_of_right {α : Type u_1} {β : α → Type u_4} {a : α} [Infinite (β a)] : Infinite ((a : α) × β a)

instance Infinite.instSigmaOfNonempty_1 {α : Type u_1} {β : α → Type u_4} [Nonempty α] [∀ (a : α), Infinite (β a)] : Infinite ((a : α) × β a)

Equations

• ⋯ = ⋯

instance instInfiniteNat : Infinite ℕ

Equations

• instInfiniteNat = ⋯

instance Int.infinite : Infinite ℤ

Equations

• Int.infinite = ⋯

instance instInfiniteMultisetOfNonempty {α : Type u_1} [Nonempty α] : Infinite (Multiset α)

Equations

• ⋯ = ⋯

instance instInfiniteListOfNonempty {α : Type u_1} [Nonempty α] : Infinite (List α)

Equations

• ⋯ = ⋯

instance String.infinite : Infinite String

Equations

• String.infinite = ⋯

instance Infinite.set {α : Type u_1} [Infinite α] : Infinite (Set α)

Equations

• ⋯ = ⋯

instance instInfiniteFinset {α : Type u_1} [Infinite α] : Infinite (Finset α)

Equations

• ⋯ = ⋯

instance instInfiniteOption {α : Type u_1} [Infinite α] : Infinite (Option α)

Equations

• ⋯ = ⋯

instance Sum.infinite_of_left {α : Type u_1} {β : Type u_2} [Infinite α] : Infinite (α ⊕ β)

Equations

• ⋯ = ⋯

instance Sum.infinite_of_right {α : Type u_1} {β : Type u_2} [Infinite β] : Infinite (α ⊕ β)

Equations

• ⋯ = ⋯

instance Prod.infinite_of_right {α : Type u_1} {β : Type u_2} [Nonempty α] [Infinite β] : Infinite (α × β)

Equations

• ⋯ = ⋯

instance Prod.infinite_of_left {α : Type u_1} {β : Type u_2} [Infinite α] [Nonempty β] : Infinite (α × β)

Equations

• ⋯ = ⋯

instance instInfiniteProdSubtypeCommute {α : Type u_1} [Mul α] [Infinite α] : Infinite { p : α × α // Commute p.1 p.2 }

Equations

• ⋯ = ⋯

noncomputable def Infinite.natEmbedding (α : Type u_4) [Infinite α] : ℕ ↪ α

Embedding of ℕ into an infinite type.

Equations

• Infinite.natEmbedding α = { toFun := Infinite.natEmbeddingAux α, inj' := ⋯ }

theorem Infinite.exists_subset_card_eq (α : Type u_4) [Infinite α] (n : ℕ) : ∃ (s : Finset α), s.card = n

See Infinite.exists_superset_card_eq for a version that, for an s : Finset α, provides a superset t : Finset α, s ⊆ t such that #t is fixed.

theorem Infinite.exists_superset_card_eq {α : Type u_1} [Infinite α] (s : Finset α) (n : ℕ) (hn : s.card ≤ n) : ∃ (t : Finset α), s ⊆ t ∧ t.card = n

See Infinite.exists_subset_card_eq for a version that provides an arbitrary s : Finset α for any cardinality.

noncomputable def fintypeOfFinsetCardLe {ι : Type u_4} (n : ℕ) (w : ∀ (s : Finset ι), s.card ≤ n) : Fintype ι

If every finset in a type has bounded cardinality, that type is finite.

Equations

• fintypeOfFinsetCardLe n w = fintypeOfNotInfinite ⋯

theorem not_injective_infinite_finite {α : Sort u_4} {β : Sort u_5} [Infinite α] [Finite β] (f : α → β) : ¬Function.Injective f

theorem Finite.exists_ne_map_eq_of_infinite {α : Sort u_4} {β : Sort u_5} [Infinite α] [Finite β] (f : α → β) : ∃ (x : α) (y : α), x ≠ y ∧ f x = f y

The pigeonhole principle for infinitely many pigeons in finitely many pigeonholes. If there are infinitely many pigeons in finitely many pigeonholes, then there are at least two pigeons in the same pigeonhole.

See also: Fintype.exists_ne_map_eq_of_card_lt, Finite.exists_infinite_fiber.

instance Function.Embedding.is_empty {α : Sort u_4} {β : Sort u_5} [Infinite α] [Finite β] : IsEmpty (α ↪ β)

Equations

• ⋯ = ⋯

theorem Finite.exists_infinite_fiber {α : Type u_1} {β : Type u_2} [Infinite α] [Finite β] (f : α → β) : ∃ (y : β), Infinite ↑(f ⁻¹' {y})

The strong pigeonhole principle for infinitely many pigeons in finitely many pigeonholes. If there are infinitely many pigeons in finitely many pigeonholes, then there is a pigeonhole with infinitely many pigeons.

See also: Finite.exists_ne_map_eq_of_infinite

theorem not_surjective_finite_infinite {α : Sort u_4} {β : Sort u_5} [Finite α] [Infinite β] (f : α → β) : ¬Function.Surjective f

theorem List.exists_pw_disjoint_with_card {α : Type u_4} [Fintype α] {c : List ℕ} (hc : c.sum ≤ Fintype.card α) : ∃ (o : List (List α)), List.map List.length o = c ∧ (∀ s ∈ o, s.Nodup) ∧ List.Pairwise List.Disjoint o

For any c : List ℕ whose sum is at most Fintype.card α, we can find o : List (List α) whose members have no duplicate, whose lengths given by c, and which are pairwise disjoint

def truncOfCardPos {α : Type u_4} [Fintype α] (h : 0 < Fintype.card α) : Trunc α

A Fintype with positive cardinality constructively contains an element.

Equations

• truncOfCardPos h = truncOfNonemptyFintype α

theorem Fintype.induction_subsingleton_or_nontrivial {P : (α : Type u_4) → [inst : Fintype α] → Prop} (α : Type u_4) [Fintype α] (hbase : ∀ (α : Type u_4) [inst : Fintype α] [inst_1 : Subsingleton α], P α) (hstep : ∀ (α : Type u_4) [inst : Fintype α] [inst_1 : Nontrivial α], (∀ (β : Type u_4) [inst_2 : Fintype β], Fintype.card β < Fintype.card α → P β) → P α) : P α

A custom induction principle for fintypes. The base case is a subsingleton type, and the induction step is for non-trivial types, and one can assume the hypothesis for smaller types (via Fintype.card).

The major premise is Fintype α, so to use this with the induction tactic you have to give a name to that instance and use that name.