Documentation

Mathlib.Logic.Small.Defs

Small types #

A type is w-small if there exists an equivalence to some S : Type w.

We provide a noncomputable model Shrink α : Type w, and equivShrink α : α ≃ Shrink α.

A subsingleton type is w-small for any w.

If α ≃ β, then Small.{w} α ↔ Small.{w} β.

See Mathlib.Logic.Small.Basic for further instances and theorems.

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class Small (α : Type v) :
Prop

A type is Small.{w} if there exists an equivalence to some S : Type w.

  • equiv_small : ∃ (S : Type w), Nonempty (α ≃ S)

    If a type is Small.{w}, then there exists an equivalence with some S : Type w

Instances
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theorem small_iff (α : Type v) :
Small.{w, v} α ↔ ∃ (S : Type w), Nonempty (α ≃ S)
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theorem Small.mk' {α : Type v} {S : Type w} (e : α ≃ S) :
Small.{w, v} α

Constructor for Small α from an explicit witness type and equivalence.

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def Shrink (α : Type v) [Small.{w, v} α] :
Type w

An arbitrarily chosen model in Type w for a w-small type.

Equations
Instances For
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noncomputable def equivShrink (α : Type v) [Small.{w, v} α] :
α ≃ Shrink.{w, v} α

The noncomputable equivalence between a w-small type and a model.

Equations
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theorem Shrink.ext {α : Type v} [Small.{w, v} α] {x y : Shrink.{w, v} α} (w : (equivShrink α).symm x = (equivShrink α).symm y) :
x = y
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noncomputable def Shrink.rec {α : Type u_1} [Small.{w, u_1} α] {F : Shrink.{w, u_1} α → Sort v} (h : (X : α) → F ((equivShrink α) X)) (X : Shrink.{w, u_1} α) :
F X
Equations
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instance small_self (α : Type v) :
Small.{v, v} α
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theorem small_map {α : Type u_1} {β : Type u_2} [hβ : Small.{w, u_2} β] (e : α ≃ β) :
Small.{w, u_1} α
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theorem small_lift (α : Type u) [hα : Small.{v, u} α] :
Small.{max v w, u} α
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theorem small_max (α : Type v) :
Small.{max w v, v} α

Due to https://github.com/leanprover/lean4/issues/2297, this is useless as an instance.

See however Logic.UnivLE, whose API is able to indirectly provide this instance.

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instance small_zero (α : Type) :
Small.{w, 0} α
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@[instance 100]
instance small_succ (α : Type v) :
Small.{v + 1, v} α
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instance small_ulift (α : Type u) [Small.{v, u} α] :
Small.{v, max w u} (ULift.{w, u} α)
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theorem small_type :
Small.{max (u + 1) v, u + 1} (Type u)
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theorem small_congr {α : Type u_1} {β : Type u_2} (e : α ≃ β) :
Small.{w, u_1} α ↔ Small.{w, u_2} β
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instance small_sigma {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)
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theorem not_small_type :
¬Small.{u, max (u + 1) (v + 1)} (Type (max u v))