Documentation

Mathlib.SetTheory.Cardinal.Finite

Finite Cardinality Functions #

Main Definitions #

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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.

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

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    theorem Nat.card_eq_finsetCard {α : Type u_1} (s : Finset α) :
    Nat.card { x : α // x ∈ s } = s.card
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    theorem Nat.card_eq_card_toFinset {α : Type u_1} (s : Set α) [Fintype ↑s] :
    Nat.card ↑s = s.toFinset.card
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    theorem Nat.card_eq_card_finite_toFinset {α : Type u_1} {s : Set α} (hs : s.Finite) :
    Nat.card ↑s = hs.toFinset.card
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    @[simp]
    theorem Nat.card_of_isEmpty {α : Type u_1} [IsEmpty α] :
    Nat.card α = 0
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    @[simp]
    theorem Nat.card_eq_zero_of_infinite {α : Type u_1} [Infinite α] :
    Nat.card α = 0
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    theorem Nat.cast_card {α : Type u_1} [Finite α] :
    ↑(Nat.card α) = Cardinal.mk α
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    theorem Set.Infinite.card_eq_zero {α : Type u_1} {s : Set α} (hs : s.Infinite) :
    Nat.card ↑s = 0
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    theorem Nat.card_eq_zero {α : Type u_1} :
    Nat.card α = 0 ↔ IsEmpty α ∨ Infinite α
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    theorem Nat.card_ne_zero {α : Type u_1} :
    Nat.card α ≠ 0 ↔ Nonempty α ∧ Finite α
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    theorem Nat.card_pos_iff {α : Type u_1} :
    0 < Nat.card α ↔ Nonempty α ∧ Finite α
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    @[simp]
    theorem Nat.card_pos {α : Type u_1} [Nonempty α] [Finite α] :
    0 < Nat.card α
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    theorem Nat.finite_of_card_ne_zero {α : Type u_1} (h : Nat.card α ≠ 0) :
    Finite α
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    theorem Nat.card_congr {α : Type u_1} {β : Type u_2} (f : α ≃ β) :
    Nat.card α = Nat.card β
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    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
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    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
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    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
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    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
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    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
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    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
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    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
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    theorem Nat.card_eq_of_equiv_fin {α : Type u_3} {n : ℕ} (f : α ≃ Fin n) :
    Nat.card α = n
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    theorem Nat.card_mono {α : Type u_1} {s t : Set α} (ht : t.Finite) (h : s ⊆ t) :
    Nat.card ↑s ≤ Nat.card ↑t
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    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
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    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
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    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
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    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
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    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
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    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
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    @[simp]
    theorem Nat.card_univ {α : Type u_1} :
    Nat.card ↑Set.univ = Nat.card α
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    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
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    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.

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    • One or more equations did not get rendered due to their size.
    Instances For
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      theorem Nat.card_of_subsingleton {α : Type u_1} (a : α) [Subsingleton α] :
      Nat.card α = 1
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      theorem Nat.card_eq_one_iff_unique {α : Type u_1} :
      Nat.card α = 1 ↔ Subsingleton α ∧ Nonempty α
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      @[simp]
      theorem Nat.card_unique {α : Type u_1} [Nonempty α] [Subsingleton α] :
      Nat.card α = 1
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      theorem Nat.card_eq_one_iff_exists {α : Type u_1} :
      Nat.card α = 1 ↔ ∃ (x : α), ∀ (y : α), y = x
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      theorem Nat.card_eq_two_iff {α : Type u_1} :
      Nat.card α = 2 ↔ ∃ (x : α) (y : α), x ≠ y ∧ {x, y} = Set.univ
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      theorem Nat.card_eq_two_iff' {α : Type u_1} (x : α) :
      Nat.card α = 2 ↔ ∃! y : α, y ≠ x
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      @[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
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      @[simp]
      theorem Nat.card_prod (α : Type u_3) (β : Type u_4) :
      Nat.card (α × β) = Nat.card α * Nat.card β
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      @[simp]
      theorem Nat.card_ulift (α : Type u_3) :
      Nat.card (ULift.{u_4, u_3} α) = Nat.card α
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      @[simp]
      theorem Nat.card_plift (α : Type u_3) :
      Nat.card (PLift α) = Nat.card α
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      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)
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      theorem Nat.card_fun {α : Type u_1} {β : Type u_2} [Finite α] :
      Nat.card (α → β) = Nat.card β ^ Nat.card α
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      @[simp]
      theorem Nat.card_zmod (n : ℕ) :
      Nat.card (ZMod n) = n
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      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
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      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
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      theorem Set.natCard_pos {α : Type u_1} {s : Set α} (hs : s.Finite) :
      0 < Nat.card ↑s ↔ s.Nonempty
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      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.

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      @[simp]
      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
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      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 = ⊤.

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        @[simp]
        theorem ENat.card_eq_coe_fintype_card {α : Type u_1} [Fintype α] :
        card α = ↑(Fintype.card α)
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        @[simp]
        theorem ENat.card_eq_top_of_infinite {α : Type u_1} [Infinite α] :
        card α = ⊤
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        @[simp]
        theorem ENat.card_sum (α : Type u_3) (β : Type u_4) :
        card (α ⊕ β) = card α + card β
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        theorem ENat.card_congr {α : Type u_3} {β : Type u_4} (f : α ≃ β) :
        card α = card β
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        @[simp]
        theorem ENat.card_ulift (α : Type u_3) :
        card (ULift.{u_4, u_3} α) = card α
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        @[simp]
        theorem ENat.card_plift (α : Type u_3) :
        card (PLift α) = card α
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        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
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        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
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        @[simp]
        theorem Cardinal.natCast_le_toENat_iff {n : ℕ} {c : Cardinal.{u_3}} :
        ↑n ≤ toENat c ↔ ↑n ≤ c
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        theorem Cardinal.toENat_le_natCast_iff {c : Cardinal.{u_3}} {n : ℕ} :
        toENat c ≤ ↑n ↔ c ≤ ↑n
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        @[simp]
        theorem Cardinal.natCast_eq_toENat_iff {n : ℕ} {c : Cardinal.{u_3}} :
        ↑n = toENat c ↔ ↑n = c
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        theorem Cardinal.toENat_eq_natCast_iff {c : Cardinal.{u_3}} {n : ℕ} :
        toENat c = ↑n ↔ c = ↑n
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        @[simp]
        theorem Cardinal.natCast_lt_toENat_iff {n : ℕ} {c : Cardinal.{u_3}} :
        ↑n < toENat c ↔ ↑n < c
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        @[simp]
        theorem Cardinal.toENat_lt_natCast_iff {n : ℕ} {c : Cardinal.{u_3}} :
        toENat c < ↑n ↔ c < ↑n
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        theorem ENat.card_eq_zero_iff_empty (α : Type u_3) :
        card α = 0 ↔ IsEmpty α
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        theorem ENat.card_le_one_iff_subsingleton (α : Type u_3) :
        card α ≤ 1 ↔ Subsingleton α
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        theorem ENat.one_lt_card_iff_nontrivial (α : Type u_3) :
        1 < card α ↔ Nontrivial α