# Documentation

## Instances #

We provide the Fintype instance for the sum of two fintypes.

instance instFintypeSum (α : Type u) (β : Type v) [] [] :
Fintype (α β)
Equations
@[simp]
theorem Finset.univ_disjSum_univ {α : Type u_3} {β : Type u_4} [] [] :
Finset.disjSum Finset.univ Finset.univ = Finset.univ
@[simp]
theorem Fintype.card_sum {α : Type u_1} {β : Type u_2} [] [] :
Fintype.card (α β) =
def fintypeOfFintypeNe {α : Type u_1} (a : α) (h : Fintype { b : α // b a }) :

If the subtype of all-but-one elements is a Fintype then the type itself is a Fintype.

Equations
Instances For
theorem image_subtype_ne_univ_eq_image_erase {α : Type u_1} {β : Type u_2} [] [] (k : β) (b : αβ) :
Finset.image (fun (i : { a : α // b a k }) => b i) Finset.univ = Finset.erase (Finset.image b Finset.univ) k
theorem image_subtype_univ_ssubset_image_univ {α : Type u_1} {β : Type u_2} [] [] (k : β) (b : αβ) (hk : k Finset.image b Finset.univ) (p : βProp) [] (hp : ¬p k) :
Finset.image (fun (i : { a : α // p (b a) }) => b i) Finset.univ Finset.image b Finset.univ
theorem Finset.exists_equiv_extend_of_card_eq {α : Type u_1} {β : Type u_2} [] [] {t : } (hαt : = t.card) {s : } {f : αβ} (hfst : t) (hfs : Set.InjOn f s) :
∃ (g : α { x : β // x t }), is, (g i) = f i

Any injection from a finset s in a fintype α to a finset t of the same cardinality as α can be extended to a bijection between α and t.

theorem Set.MapsTo.exists_equiv_extend_of_card_eq {α : Type u_1} {β : Type u_2} [] {t : } (hαt : = t.card) {s : Set α} {f : αβ} (hfst : Set.MapsTo f s t) (hfs : ) :
∃ (g : α { x : β // x t }), is, (g i) = f i

Any injection from a set s in a fintype α to a finset t of the same cardinality as α can be extended to a bijection between α and t.

theorem Fintype.card_subtype_or {α : Type u_1} (p : αProp) (q : αProp) [Fintype { x : α // p x }] [Fintype { x : α // q x }] [Fintype { x : α // p x q x }] :
Fintype.card { x : α // p x q x } Fintype.card { x : α // p x } + Fintype.card { x : α // q x }
theorem Fintype.card_subtype_or_disjoint {α : Type u_1} (p : αProp) (q : αProp) (h : Disjoint p q) [Fintype { x : α // p x }] [Fintype { x : α // q x }] [Fintype { x : α // p x q x }] :
Fintype.card { x : α // p x q x } = Fintype.card { x : α // p x } + Fintype.card { x : α // q x }
@[simp]
theorem infinite_sum {α : Type u_1} {β : Type u_2} :
Infinite (α β)