Documentation

Mathlib.SetTheory.Cardinal.Finite

Finite Cardinality Functions #

Main Definitions #

def Nat.card (α : Type u_3) :

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

Equations
Instances For
    @[simp]
    theorem Fintype.card_eq_nat_card {α : Type u_1} :
    ∀ {x : Fintype α}, Fintype.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 { x : α // x 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
    @[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 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} :
    theorem Nat.card_ne_zero {α : Type u_1} :
    theorem Nat.card_pos_iff {α : Type u_1} :
    @[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) :
    theorem Nat.card_congr {α : Type u_1} {β : Type u_2} (f : α β) :
    theorem Nat.card_le_card_of_injective {α : Type u} {β : Type v} [Finite β] (f : αβ) (hf : Function.Injective f) :
    theorem Nat.card_le_card_of_surjective {α : Type u} {β : Type v} [Finite α] (f : αβ) (hf : Function.Surjective f) :
    theorem Nat.card_eq_of_bijective {α : Type u_1} {β : Type u_2} (f : αβ) (hf : Function.Bijective f) :
    theorem Function.Injective.bijective_of_nat_card_le {α : Type u_1} {β : Type u_2} [Finite β] {f : αβ} (inj : Function.Injective f) (hc : Nat.card β Nat.card α) :
    theorem Function.Surjective.bijective_of_nat_card_le {α : Type u_1} {β : Type u_2} [Finite α] {f : αβ} (surj : Function.Surjective f) (hc : Nat.card α Nat.card β) :
    theorem Nat.card_eq_of_equiv_fin {α : Type u_3} {n : } (f : α Fin n) :
    Nat.card α = n
    theorem Nat.card_mono {α : Type u_1} {s : Set α} {t : Set α} (ht : t.Finite) (h : s t) :
    theorem Nat.card_image_le {α : Type u_1} {β : Type u_2} {s : Set α} {f : αβ} (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 : αβ} (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 : αβ} (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 : αβ} {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 : αβ} {s : Set β} (hf : Function.Injective f) (hsf : s Set.range f) :
    Nat.card (f ⁻¹' s) = Nat.card s
    @[simp]
    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 : αβ} (hf : Function.Injective f) :
    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.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      theorem Nat.card_of_subsingleton {α : Type u_1} (a : α) [Subsingleton α] :
      Nat.card α = 1
      @[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_sum {α : Type u_1} {β : Type u_2} [Finite α] [Finite 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]
      @[simp]
      theorem Nat.card_plift (α : Type u_3) :
      theorem Nat.card_pi {α : Type u_1} {β : αType 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 (αβ) = Nat.card β ^ Nat.card α
      @[simp]
      theorem Nat.card_zmod (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
      def PartENat.card (α : Type u_3) :

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

      Equations
      Instances For
        @[simp]
        theorem PartENat.card_sum (α : Type u_3) (β : Type u_4) :
        theorem PartENat.card_congr {α : Type u_3} {β : Type u_4} (f : α β) :
        @[simp]
        theorem PartENat.card_image_of_injOn {α : Type u} {β : Type v} {f : αβ} {s : Set α} (h : Set.InjOn f s) :
        theorem PartENat.card_image_of_injective {α : Type u} {β : Type v} (f : αβ) (s : Set α) (h : Function.Injective f) :
        @[simp]
        theorem Cardinal.natCast_le_toPartENat_iff {n : } {c : Cardinal.{u_3}} :
        n Cardinal.toPartENat c n c
        @[simp]
        theorem Cardinal.toPartENat_le_natCast_iff {c : Cardinal.{u_3}} {n : } :
        Cardinal.toPartENat c n c n
        @[simp]
        theorem Cardinal.natCast_eq_toPartENat_iff {n : } {c : Cardinal.{u_3}} :
        n = Cardinal.toPartENat c n = c
        @[simp]
        theorem Cardinal.toPartENat_eq_natCast_iff {c : Cardinal.{u_3}} {n : } :
        Cardinal.toPartENat c = n c = n
        @[simp]
        theorem Cardinal.natCast_lt_toPartENat_iff {n : } {c : Cardinal.{u_3}} :
        n < Cardinal.toPartENat c n < c
        @[simp]
        theorem Cardinal.toPartENat_lt_natCast_iff {n : } {c : Cardinal.{u_3}} :
        Cardinal.toPartENat c < n c < n