Documentation

Mathlib.Logic.Small.Basic

Instances and theorems for Small. #

In particular we prove small_of_injective and small_of_surjective.

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instance small_subtype (α : Type v) [Small.{w, v} α] (P : α → Prop) :
Small.{w, v} { x : α // P x }
Equations
  • ⋯ = ⋯
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theorem small_of_injective {α : Type v} {β : Type w} [Small.{u, w} β] {f : α → β} (hf : Function.Injective f) :
Small.{u, v} α
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theorem small_of_surjective {α : Type v} {β : Type w} [Small.{u, v} α] {f : α → β} (hf : Function.Surjective f) :
Small.{u, w} β
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instance small_subsingleton (α : Type v) [Subsingleton α] :
Small.{w, v} α
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  • ⋯ = ⋯
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theorem small_of_injective_of_exists {α : Type v} {β : Type w} {γ : Type v'} [Small.{u, v} α] (f : α → γ) {g : β → γ} (hg : Function.Injective g) (h : ∀ (b : β), ∃ (a : α), f a = g b) :
Small.{u, w} β

This can be seen as a version of small_of_surjective in which the function f doesn't actually land in β but in some larger type γ related to β via an injective function g.

We don't define Countable.toSmall in this file, to keep imports to Logic to a minimum.

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instance small_Pi {α : Type u_2} (β : α → Type u_1) [Small.{w, u_2} α] [∀ (a : α), Small.{w, u_1} (β a)] :
Small.{w, max u_1 u_2} ((a : α) → β a)
Equations
  • ⋯ = ⋯
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instance small_prod {α : Type u_1} {β : Type u_2} [Small.{w, u_1} α] [Small.{w, u_2} β] :
Small.{w, max u_2 u_1} (α × β)
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  • ⋯ = ⋯
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instance small_sum {α : Type u_1} {β : Type u_2} [Small.{w, u_1} α] [Small.{w, u_2} β] :
Small.{w, max u_2 u_1} (α ⊕ β)
Equations
  • ⋯ = ⋯
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instance small_set {α : Type u_1} [Small.{w, u_1} α] :
Small.{w, u_1} (Set α)
Equations
  • ⋯ = ⋯