Finite Cardinality Functions

Main Definitions

• Nat.card α is the cardinality of α as a natural number. If α is infinite, Nat.card α = 0.
• ENat.card α is the cardinality of α as an extended natural number. If α is infinite, ENat.card α = ⊤.

noncomputable def Nat.card (α : Type u_3) : ℕ

Nat.card α is the cardinality of α as a natural number. If α is infinite, Nat.card α = 0.

@[simp]

theorem Nat.card_eq_fintype_card {α : Type u_1} [Fintype α] : Nat.card α = Fintype.card α

theorem Fintype.card_eq_nat_card {α : Type u_1} {x✝ : Fintype α} : card α = Nat.card α

Because this theorem takes Fintype α as a non-instance argument, it can be used in particular when Fintype.card ends up with different instance than the one found by inference

theorem Nat.card_eq_finsetCard {α : Type u_1} (s : Finset α) : Nat.card ↥s = s.card

theorem Nat.card_eq_card_toFinset {α : Type u_1} (s : Set α) [Fintype ↑s] : Nat.card ↑s = s.toFinset.card

theorem Nat.card_eq_card_finite_toFinset {α : Type u_1} {s : Set α} (hs : s.Finite) : Nat.card ↑s = hs.toFinset.card

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

@[simp]

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

@[simp]

theorem Nat.card_eq_zero_of_infinite {α : Type u_1} [Infinite α] : Nat.card α = 0

theorem Nat.cast_card {α : Type u_1} [Finite α] : ↑(Nat.card α) = Cardinal.mk α

theorem Set.Infinite.card_eq_zero {α : Type u_1} {s : Set α} (hs : s.Infinite) : Nat.card ↑s = 0

theorem Nat.card_eq_zero {α : Type u_1} : Nat.card α = 0 ↔ IsEmpty α ∨ Infinite α

theorem Nat.card_ne_zero {α : Type u_1} : Nat.card α ≠ 0 ↔ Nonempty α ∧ Finite α

theorem Nat.card_pos_iff {α : Type u_1} : 0 < Nat.card α ↔ Nonempty α ∧ Finite α

@[simp]

theorem Nat.card_pos {α : Type u_1} [Nonempty α] [Finite α] : 0 < Nat.card α

theorem Nat.finite_of_card_ne_zero {α : Type u_1} (h : Nat.card α ≠ 0) : Finite α

theorem Nat.card_congr {α : Type u_1} {β : Type u_2} (f : α ≃ β) : Nat.card α = Nat.card β

theorem Nat.card_le_card_of_injective {α : Type u} {β : Type v} [Finite β] (f : α → β) (hf : Function.Injective f) : Nat.card α ≤ Nat.card β

theorem Nat.card_le_card_of_surjective {α : Type u} {β : Type v} [Finite α] (f : α → β) (hf : Function.Surjective f) : Nat.card β ≤ Nat.card α

theorem Nat.card_eq_of_bijective {α : Type u_1} {β : Type u_2} (f : α → β) (hf : Function.Bijective f) : Nat.card α = Nat.card β

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

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

theorem Function.Injective.bijective_of_nat_card_le {α : Type u_1} {β : Type u_2} [Finite β] {f : α → β} (inj : Injective f) (hc : Nat.card β ≤ Nat.card α) : Bijective f

theorem Function.Surjective.bijective_of_nat_card_le {α : Type u_1} {β : Type u_2} [Finite α] {f : α → β} (surj : Surjective f) (hc : Nat.card α ≤ Nat.card β) : Bijective f

theorem Nat.card_eq_of_equiv_fin {α : Type u_3} {n : ℕ} (f : α ≃ Fin n) : Nat.card α = n

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

theorem Nat.card_mono {α : Type u_1} {s t : Set α} (ht : t.Finite) (h : s ⊆ t) : Nat.card ↑s ≤ Nat.card ↑t

theorem Nat.card_image_le {α : Type u_1} {β : Type u_2} {s : Set α} {f : α → β} (hs : s.Finite) : Nat.card ↑(f '' s) ≤ Nat.card ↑s

theorem Nat.card_image_of_injOn {α : Type u_1} {β : Type u_2} {s : Set α} {f : α → β} (hf : Set.InjOn f s) : Nat.card ↑(f '' s) = Nat.card ↑s

theorem Nat.card_image_of_injective {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Injective f) (s : Set α) : Nat.card ↑(f '' s) = Nat.card ↑s

theorem Nat.card_image_equiv {α : Type u_1} {β : Type u_2} {s : Set α} (e : α ≃ β) : Nat.card ↑(⇑e '' s) = Nat.card ↑s

theorem Nat.card_preimage_of_injOn {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set β} (hf : Set.InjOn f (f ⁻¹' s)) (hsf : s ⊆ Set.range f) : Nat.card ↑(f ⁻¹' s) = Nat.card ↑s

theorem Nat.card_preimage_of_injective {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set β} (hf : Function.Injective f) (hsf : s ⊆ Set.range f) : Nat.card ↑(f ⁻¹' s) = Nat.card ↑s

theorem Nat.card_univ {α : Type u_1} : Nat.card ↑Set.univ = Nat.card α

theorem Nat.card_range_of_injective {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Injective f) : Nat.card ↑(Set.range f) = Nat.card α

noncomputable def Nat.equivFinOfCardPos {α : Type u_3} (h : Nat.card α ≠ 0) : α ≃ Fin (Nat.card α)

If the cardinality is positive, that means it is a finite type, so there is an equivalence between α and Fin (Nat.card α). See also Finite.equivFin.

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

theorem Nat.card_eq_one_iff_unique {α : Type u_1} : Nat.card α = 1 ↔ Subsingleton α ∧ Nonempty α

@[simp]

theorem Nat.card_unique {α : Type u_1} [Nonempty α] [Subsingleton α] : Nat.card α = 1

theorem Nat.card_eq_one_iff_exists {α : Type u_1} : Nat.card α = 1 ↔ ∃ (x : α), ∀ (y : α), y = x

theorem Nat.card_eq_two_iff {α : Type u_1} : Nat.card α = 2 ↔ ∃ (x : α) (y : α), x ≠ y ∧ {x, y} = Set.univ

theorem Nat.card_eq_two_iff' {α : Type u_1} (x : α) : Nat.card α = 2 ↔ ∃! y : α, y ≠ x

@[simp]

theorem Nat.card_subtype_true {α : Type u_1} : Nat.card { _a : α // True } = Nat.card α

@[simp]

theorem Nat.card_sum {α : Type u_1} {β : Type u_2} [Finite α] [Finite β] : Nat.card (α ⊕ β) = Nat.card α + Nat.card β

@[simp]

theorem Nat.card_prod (α : Type u_3) (β : Type u_4) : Nat.card (α × β) = Nat.card α * Nat.card β

@[simp]

theorem Nat.card_ulift (α : Type u_3) : Nat.card (ULift.{u_4, u_3} α) = Nat.card α

@[simp]

theorem Nat.card_plift (α : Type u_3) : Nat.card (PLift α) = Nat.card α

theorem Nat.card_sigma {α : Type u_1} {β : α → Type u_3} [Fintype α] [∀ (a : α), Finite (β a)] : Nat.card (Sigma β) = ∑ a : α, Nat.card (β a)

theorem Nat.card_pi {α : Type u_1} {β : α → Type u_3} [Fintype α] : Nat.card ((a : α) → β a) = ∏ a : α, Nat.card (β a)

theorem Nat.card_fun {α : Type u_1} {β : Type u_2} [Finite α] : Nat.card (α → β) = Nat.card β ^ Nat.card α

@[simp]

theorem Nat.card_zmod (n : ℕ) : Nat.card (ZMod n) = n

theorem Set.card_singleton_prod {α : Type u_1} {β : Type u_2} (a : α) (t : Set β) : Nat.card ↑({a} ×ˢ t) = Nat.card ↑t

theorem Set.card_prod_singleton {α : Type u_1} {β : Type u_2} (s : Set α) (b : β) : Nat.card ↑(s ×ˢ {b}) = Nat.card ↑s

theorem Set.natCard_pos {α : Type u_1} {s : Set α} (hs : s.Finite) : 0 < Nat.card ↑s ↔ s.Nonempty

theorem Set.Nonempty.natCard_pos {α : Type u_1} {s : Set α} (hs : s.Finite) : s.Nonempty → 0 < Nat.card ↑s

Alias of the reverse direction of Set.natCard_pos.

theorem Set.natCard_graphOn {α : Type u_1} {β : Type u_2} (s : Set α) (f : α → β) : Nat.card ↑(graphOn f s) = Nat.card ↑s

noncomputable def ENat.card (α : Type u_3) : ℕ∞

ENat.card α is the cardinality of α as an extended natural number. If α is infinite, ENat.card α = ⊤.

@[simp]

theorem ENat.card_eq_coe_fintype_card {α : Type u_1} [Fintype α] : card α = ↑(Fintype.card α)

@[simp]

theorem ENat.card_eq_top_of_infinite {α : Type u_1} [Infinite α] : card α = ⊤

@[simp]

theorem ENat.card_eq_top {α : Type u_1} : card α = ⊤ ↔ Infinite α

@[simp]

theorem ENat.card_lt_top_of_finite {α : Type u_1} [Finite α] : card α < ⊤

@[simp]

theorem ENat.card_lt_top {α : Type u_1} : card α < ⊤ ↔ Finite α

@[simp]

theorem ENat.card_sum (α : Type u_3) (β : Type u_4) : card (α ⊕ β) = card α + card β

theorem ENat.card_congr {α : Type u_3} {β : Type u_4} (f : α ≃ β) : card α = card β

@[simp]

theorem ENat.card_ulift (α : Type u_3) : card (ULift.{u_4, u_3} α) = card α

@[simp]

theorem ENat.card_plift (α : Type u_3) : card (PLift α) = card α

theorem ENat.card_image_of_injOn {α : Type u_3} {β : Type u_4} {f : α → β} {s : Set α} (h : Set.InjOn f s) : card ↑(f '' s) = card ↑s

theorem ENat.card_image_of_injective {α : Type u_3} {β : Type u_4} (f : α → β) (s : Set α) (h : Function.Injective f) : card ↑(f '' s) = card ↑s

theorem ENat.card_le_card_of_injective {α : Type u_3} {β : Type u_4} {f : α → β} (hf : Function.Injective f) : card α ≤ card β

@[deprecated Cardinal.natCast_le_toENat (since := "2026-02-17")]

theorem Cardinal.natCast_le_toENat_iff {n : ℕ} {c : Cardinal.{u_3}} : ↑n ≤ toENat c ↔ ↑n ≤ c

@[deprecated Cardinal.toENat_le_natCast (since := "2026-02-17")]

theorem Cardinal.toENat_le_natCast_iff {c : Cardinal.{u_3}} {n : ℕ} : toENat c ≤ ↑n ↔ c ≤ ↑n

@[deprecated Cardinal.natCast_eq_toENat (since := "2026-02-17")]

theorem Cardinal.natCast_eq_toENat_iff {n : ℕ} {c : Cardinal.{u_3}} : ↑n = toENat c ↔ ↑n = c

@[deprecated Cardinal.toENat_eq_natCast (since := "2026-02-17")]

theorem Cardinal.toENat_eq_natCast_iff {c : Cardinal.{u_3}} {n : ℕ} : toENat c = ↑n ↔ c = ↑n

@[deprecated Cardinal.natCast_lt_toENat (since := "2026-02-17")]

theorem Cardinal.natCast_lt_toENat_iff {n : ℕ} {c : Cardinal.{u_3}} : ↑n < toENat c ↔ ↑n < c

@[deprecated Cardinal.toENat_lt_natCast (since := "2026-02-17")]

theorem Cardinal.toENat_lt_natCast_iff {n : ℕ} {c : Cardinal.{u_3}} : toENat c < ↑n ↔ c < ↑n

theorem ENat.card_eq_zero_iff_empty (α : Type u_3) : card α = 0 ↔ IsEmpty α

theorem ENat.card_ne_zero_iff_nonempty (α : Type u_3) : card α ≠ 0 ↔ Nonempty α

theorem ENat.one_le_card_iff_nonempty (α : Type u_3) : 1 ≤ card α ↔ Nonempty α

@[simp]

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

theorem ENat.card_pos_iff_nonempty {α : Type u_1} : 0 < card α ↔ Nonempty α

theorem ENat.card_le_one_iff_subsingleton (α : Type u_3) : card α ≤ 1 ↔ Subsingleton α

@[simp]

theorem ENat.card_le_one {α : Type u_1} [Subsingleton α] : card α ≤ 1

theorem ENat.card_eq_one_iff_unique {α : Type u_3} : card α = 1 ↔ Nonempty (Unique α)

theorem ENat.one_lt_card_iff_nontrivial (α : Type u_3) : 1 < card α ↔ Nontrivial α

@[simp]

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

theorem ENat.exists_ne_ne_of_three_le {α : Type u_1} (h : 3 ≤ card α) (x y : α) : ∃ (z : α), z ≠ x ∧ z ≠ y

@[simp]

theorem ENat.card_prod (α : Type u_3) (β : Type u_4) : card (α × β) = card α * card β

@[simp]

theorem ENat.card_fun {α : Type u_3} {β : Type u_4} : card (α → β) = card β ^ card α