formal/lean/mathlib/logic/small.lean

/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import data.vector.basic

/-!
# 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 `equiv_shrink α : α ≃ shrink α`.

A subsingleton type is `w`-small for any `w`.

If `α ≃ β`, then `small.{w} α ↔ small.{w} β`.
-/

universes u w v

/--
A type is `small.{w}` if there exists an equivalence to some `S : Type w`.
-/
class small (α : Type v) : Prop :=
(equiv_small : ∃ (S : Type w), nonempty (α ≃ S))

/--
Constructor for `small α` from an explicit witness type and equivalence.
-/
lemma small.mk' {α : Type v} {S : Type w} (e : α ≃ S) : small.{w} α :=
⟨⟨S, ⟨e⟩⟩⟩

/--
An arbitrarily chosen model in `Type w` for a `w`-small type.
-/
@[nolint has_nonempty_instance]
def shrink (α : Type v) [small.{w} α] : Type w :=
classical.some (@small.equiv_small α _)

/--
The noncomputable equivalence between a `w`-small type and a model.
-/
noncomputable
def equiv_shrink (α : Type v) [small.{w} α] : α ≃ shrink α :=
nonempty.some (classical.some_spec (@small.equiv_small α _))

@[priority 100]
instance small_self (α : Type v) : small.{v} α :=
small.mk' $ equiv.refl α

theorem small_map {α : Type*} {β : Type*} [hβ : small.{w} β] (e : α ≃ β) : small.{w} α :=
let ⟨γ, ⟨f⟩⟩ := hβ.equiv_small in small.mk' (e.trans f)

theorem small_lift (α : Type u) [hα : small.{v} α] : small.{max v w} α :=
let ⟨⟨γ, ⟨f⟩⟩⟩ := hα in small.mk' $ f.trans equiv.ulift.symm

@[priority 100]
instance small_max (α : Type v) : small.{max w v} α :=
small_lift.{v w} α

instance small_ulift (α : Type u) [small.{v} α] : small.{v} (ulift.{w} α) :=
small_map equiv.ulift

theorem small_type : small.{max (u+1) v} (Type u) := small_max.{max (u+1) v} _

section
open_locale classical

theorem small_congr {α : Type*} {β : Type*} (e : α ≃ β) : small.{w} α ↔ small.{w} β :=
⟨λ h, @small_map _ _ h e.symm, λ h, @small_map _ _ h e⟩

instance small_subtype (α : Type v) [small.{w} α] (P : α → Prop) : small.{w} { x // P x } :=
small_map (equiv_shrink α).subtype_equiv_of_subtype'

theorem small_of_injective {α : Type v} {β : Type w} [small.{u} β] {f : α → β}
  (hf : function.injective f) : small.{u} α :=
small_map (equiv.of_injective f hf)

theorem small_of_surjective {α : Type v} {β : Type w} [small.{u} α] {f : α → β}
  (hf : function.surjective f) : small.{u} β :=
small_of_injective (function.injective_surj_inv hf)

theorem small_subset {α : Type v} {s t : set α} (hts : t ⊆ s) [small.{u} s] : small.{u} t :=
let f : t → s := λ x, ⟨x, hts x.prop⟩ in
  @small_of_injective _ _ _ f (λ x y hxy, subtype.ext (subtype.mk.inj hxy))

@[priority 100]
instance small_subsingleton (α : Type v) [subsingleton α] : small.{w} α :=
begin
  rcases is_empty_or_nonempty α; resetI,
  { apply small_map (equiv.equiv_pempty α) },
  { apply small_map equiv.punit_of_nonempty_of_subsingleton, assumption' },
end

/-!
We don't define `small_of_fintype` or `small_of_countable` in this file,
to keep imports to `logic` to a minimum.
-/

instance small_Pi {α} (β : α → Type*) [small.{w} α] [∀ a, small.{w} (β a)] :
  small.{w} (Π a, β a) :=
⟨⟨Π a' : shrink α, shrink (β ((equiv_shrink α).symm a')),
  ⟨equiv.Pi_congr (equiv_shrink α) (λ a, by simpa using equiv_shrink (β a))⟩⟩⟩

instance small_sigma {α} (β : α → Type*) [small.{w} α] [∀ a, small.{w} (β a)] :
  small.{w} (Σ a, β a) :=
⟨⟨Σ a' : shrink α, shrink (β ((equiv_shrink α).symm a')),
  ⟨equiv.sigma_congr (equiv_shrink α) (λ a, by simpa using equiv_shrink (β a))⟩⟩⟩

instance small_prod {α β} [small.{w} α] [small.{w} β] : small.{w} (α × β) :=
⟨⟨shrink α × shrink β,
  ⟨equiv.prod_congr (equiv_shrink α) (equiv_shrink β)⟩⟩⟩

instance small_sum {α β} [small.{w} α] [small.{w} β] : small.{w} (α ⊕ β) :=
⟨⟨shrink α ⊕ shrink β,
  ⟨equiv.sum_congr (equiv_shrink α) (equiv_shrink β)⟩⟩⟩

instance small_set {α} [small.{w} α] : small.{w} (set α) :=
⟨⟨set (shrink α), ⟨equiv.set.congr (equiv_shrink α)⟩⟩⟩

instance small_range {α : Type v} {β : Type w} (f : α → β) [small.{u} α] :
  small.{u} (set.range f) :=
small_of_surjective set.surjective_onto_range

instance small_image {α : Type v} {β : Type w} (f : α → β) (S : set α) [small.{u} S] :
  small.{u} (f '' S) :=
small_of_surjective set.surjective_onto_image

theorem not_small_type : ¬ small.{u} (Type (max u v))
| ⟨⟨S, ⟨e⟩⟩⟩ := @function.cantor_injective (Σ α, e.symm α)
  (λ a, ⟨_, cast (e.3 _).symm a⟩)
  (λ a b e, (cast_inj _).1 $ eq_of_heq (sigma.mk.inj e).2)

instance small_vector {α : Type v} {n : ℕ} [small.{u} α] :
  small.{u} (vector α n) :=
small_of_injective (equiv.vector_equiv_fin α n).injective

instance small_list {α : Type v} [small.{u} α] :
  small.{u} (list α) :=
begin
  let e : (Σ n, vector α n) ≃ list α := equiv.sigma_fiber_equiv list.length,
  exact small_of_surjective e.surjective,
end

end