Finite types #

This file defines a typeclass to state that a type is finite.

Main declarations #

Fintype α: Typeclass saying that a type is finite. It takes as fields a Finset and a proof that all terms of type α are in it.
Finset.univ: The finset of all elements of a fintype.

See Data.Fintype.Card for the cardinality of a fintype, the equivalence with Fin (Fintype.card α), and pigeonhole principles.

Instances #

Instances for Fintype for

{x // p x} are in this file as Fintype.subtype
Option α are in Data.Fintype.Option
α × β are in Data.Fintype.Prod
α ⊕ β are in Data.Fintype.Sum
Σ (a : α), β a are in Data.Fintype.Sigma

These files also contain appropriate Infinite instances for these types.

Infinite instances for ℕ, ℤ, Multiset α, and List α are in Data.Fintype.Lattice.

Types which have a surjection from/an injection to a Fintype are themselves fintypes. See Fintype.ofInjective and Fintype.ofSurjective.

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class Fintype (α : Type u_4) :

Type u_4

elems : Finset α
The Finset containing all elements of a Fintype
complete : ∀ (x : α), x ∈ Fintype.elems
A proof that elems contains every element of the type

Fintype α means that α is finite, i.e. there are only finitely many distinct elements of type α. The evidence of this is a finset elems (a list up to permutation without duplicates), together with a proof that everything of type α is in the list.

Instances

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def Finset.univ {α : Type u_1} [Fintype α] :

Finset α

univ is the universal finite set of type Finset α implied from the assumption Fintype α.

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@[simp]

theorem Finset.mem_univ {α : Type u_1} [Fintype α] (x : α) :

x ∈ Finset.univ

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theorem Finset.mem_univ_val {α : Type u_1} [Fintype α] (x : α) :

x ∈ Finset.univ.val

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theorem Finset.eq_univ_iff_forall {α : Type u_1} [Fintype α] {s : Finset α} :

s = Finset.univ ↔ ∀ (x : α), x ∈ s

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theorem Finset.eq_univ_of_forall {α : Type u_1} [Fintype α] {s : Finset α} :

(∀ (x : α), x ∈ s) → s = Finset.univ

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@[simp]

theorem Finset.coe_univ {α : Type u_1} [Fintype α] :

↑Finset.univ = Set.univ

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@[simp]

theorem Finset.coe_eq_univ {α : Type u_1} [Fintype α] {s : Finset α} :

↑s = Set.univ ↔ s = Finset.univ

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theorem Finset.Nonempty.eq_univ {α : Type u_1} [Fintype α] {s : Finset α} [Subsingleton α] :

Finset.Nonempty s → s = Finset.univ

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theorem Finset.univ_nonempty_iff {α : Type u_1} [Fintype α] :

Finset.Nonempty Finset.univ ↔ Nonempty α

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theorem Finset.univ_nonempty {α : Type u_1} [Fintype α] [Nonempty α] :

Finset.Nonempty Finset.univ

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theorem Finset.univ_eq_empty_iff {α : Type u_1} [Fintype α] :

Finset.univ = ∅ ↔ IsEmpty α

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@[simp]

theorem Finset.univ_eq_empty {α : Type u_1} [Fintype α] [IsEmpty α] :

Finset.univ = ∅

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@[simp]

theorem Finset.univ_unique {α : Type u_1} [Fintype α] [Unique α] :

Finset.univ = {default}

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@[simp]

theorem Finset.subset_univ {α : Type u_1} [Fintype α] (s : Finset α) :

s ⊆ Finset.univ

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instance Finset.boundedOrder {α : Type u_1} [Fintype α] :

BoundedOrder (Finset α)

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@[simp]

theorem Finset.top_eq_univ {α : Type u_1} [Fintype α] :

⊤ = Finset.univ

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theorem Finset.ssubset_univ_iff {α : Type u_1} [Fintype α] {s : Finset α} :

s ⊂ Finset.univ ↔ s ≠ Finset.univ

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theorem Finset.codisjoint_left {α : Type u_1} [Fintype α] {s : Finset α} {t : Finset α} :

Codisjoint s t ↔ ∀ ⦃a : α⦄, ¬a ∈ s → a ∈ t

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theorem Finset.codisjoint_right {α : Type u_1} [Fintype α] {s : Finset α} {t : Finset α} :

Codisjoint s t ↔ ∀ ⦃a : α⦄, ¬a ∈ t → a ∈ s

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instance Finset.booleanAlgebra {α : Type u_1} [Fintype α] [DecidableEq α] :

BooleanAlgebra (Finset α)

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theorem Finset.sdiff_eq_inter_compl {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) (t : Finset α) :

s \ t = s ∩ tᶜ

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theorem Finset.compl_eq_univ_sdiff {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) :

sᶜ = Finset.univ \ s

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@[simp]

theorem Finset.mem_compl {α : Type u_1} [Fintype α] {s : Finset α} [DecidableEq α] {a : α} :

a ∈ sᶜ ↔ ¬a ∈ s

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theorem Finset.not_mem_compl {α : Type u_1} [Fintype α] {s : Finset α} [DecidableEq α] {a : α} :

¬a ∈ sᶜ ↔ a ∈ s

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@[simp]

theorem Finset.coe_compl {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) :

↑sᶜ = (↑s)ᶜ

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@[simp]

theorem Finset.compl_subset_compl {α : Type u_1} [Fintype α] {s : Finset α} {t : Finset α} [DecidableEq α] :

sᶜ ⊆ tᶜ ↔ t ⊆ s

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@[simp]

theorem Finset.compl_ssubset_compl {α : Type u_1} [Fintype α] {s : Finset α} {t : Finset α} [DecidableEq α] :

sᶜ ⊂ tᶜ ↔ t ⊂ s

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@[simp]

theorem Finset.compl_empty {α : Type u_1} [Fintype α] [DecidableEq α] :

∅ᶜ = Finset.univ

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@[simp]

theorem Finset.compl_univ {α : Type u_1} [Fintype α] [DecidableEq α] :

Finset.univᶜ = ∅

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@[simp]

theorem Finset.compl_eq_empty_iff {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) :

sᶜ = ∅ ↔ s = Finset.univ

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@[simp]

theorem Finset.compl_eq_univ_iff {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) :

sᶜ = Finset.univ ↔ s = ∅

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@[simp]

theorem Finset.union_compl {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) :

s ∪ sᶜ = Finset.univ

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@[simp]

theorem Finset.inter_compl {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) :

s ∩ sᶜ = ∅

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@[simp]

theorem Finset.compl_union {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) (t : Finset α) :

(s ∪ t)ᶜ = sᶜ ∩ tᶜ

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@[simp]

theorem Finset.compl_inter {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) (t : Finset α) :

(s ∩ t)ᶜ = sᶜ ∪ tᶜ

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@[simp]

theorem Finset.compl_erase {α : Type u_1} [Fintype α] {s : Finset α} [DecidableEq α] {a : α} :

(Finset.erase s a)ᶜ = insert a sᶜ

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@[simp]

theorem Finset.compl_insert {α : Type u_1} [Fintype α] {s : Finset α} [DecidableEq α] {a : α} :

(insert a s)ᶜ = Finset.erase sᶜ a

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theorem Finset.insert_compl_insert {α : Type u_1} [Fintype α] {s : Finset α} [DecidableEq α] {a : α} (ha : ¬a ∈ s) :

insert a (insert a s)ᶜ = sᶜ

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@[simp]

theorem Finset.insert_compl_self {α : Type u_1} [Fintype α] [DecidableEq α] (x : α) :

insert x {x}ᶜ = Finset.univ

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@[simp]

theorem Finset.compl_filter {α : Type u_1} [Fintype α] [DecidableEq α] (p : α → Prop) [DecidablePred p] [(x : α) → Decidable ¬p x] :

(Finset.filter p Finset.univ)ᶜ = Finset.filter (fun x => ¬p x) Finset.univ

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theorem Finset.compl_ne_univ_iff_nonempty {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) :

sᶜ ≠ Finset.univ ↔ Finset.Nonempty s

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theorem Finset.compl_singleton {α : Type u_1} [Fintype α] [DecidableEq α] (a : α) :

{a}ᶜ = Finset.erase Finset.univ a

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theorem Finset.insert_inj_on' {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) :

Set.InjOn (fun a => insert a s) ↑sᶜ

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theorem Finset.image_univ_of_surjective {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq α] [Fintype β] {f : β → α} (hf : Function.Surjective f) :

Finset.image f Finset.univ = Finset.univ

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@[simp]

theorem Finset.image_univ_equiv {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq α] [Fintype β] (f : β ≃ α) :

Finset.image (↑f) Finset.univ = Finset.univ

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@[simp]

theorem Finset.univ_inter {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) :

Finset.univ ∩ s = s

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@[simp]

theorem Finset.inter_univ {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) :

s ∩ Finset.univ = s

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@[simp]

theorem Finset.inter_eq_univ {α : Type u_1} [Fintype α] {s : Finset α} {t : Finset α} [DecidableEq α] :

s ∩ t = Finset.univ ↔ s = Finset.univ ∧ t = Finset.univ

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theorem Finset.map_univ_of_surjective {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] {f : β ↪ α} (hf : Function.Surjective ↑f) :

Finset.map f Finset.univ = Finset.univ

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@[simp]

theorem Finset.map_univ_equiv {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (f : β ≃ α) :

Finset.map (Equiv.toEmbedding f) Finset.univ = Finset.univ

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@[simp]

theorem Finset.piecewise_univ {α : Type u_1} [Fintype α] [(i : α) → Decidable (i ∈ Finset.univ)] {δ : α → Sort u_4} (f : (i : α) → δ i) (g : (i : α) → δ i) :

Finset.piecewise Finset.univ f g = f

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theorem Finset.piecewise_compl {α : Type u_1} [Fintype α] [DecidableEq α] (s : Finset α) [(i : α) → Decidable (i ∈ s)] [(i : α) → Decidable (i ∈ sᶜ)] {δ : α → Sort u_4} (f : (i : α) → δ i) (g : (i : α) → δ i) :

Finset.piecewise sᶜ f g = Finset.piecewise s g f

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@[simp]

theorem Finset.piecewise_erase_univ {α : Type u_1} [Fintype α] {δ : α → Sort u_4} [DecidableEq α] (a : α) (f : (a : α) → δ a) (g : (a : α) → δ a) :

Finset.piecewise (Finset.erase Finset.univ a) f g = Function.update f a (g a)

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theorem Finset.univ_map_equiv_to_embedding {α : Type u_4} {β : Type u_5} [Fintype α] [Fintype β] (e : α ≃ β) :

Finset.map (Equiv.toEmbedding e) Finset.univ = Finset.univ

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@[simp]

theorem Finset.univ_filter_exists {α : Type u_1} {β : Type u_2} [Fintype α] (f : α → β) [Fintype β] [DecidablePred fun y => ∃ x, f x = y] [DecidableEq β] :

Finset.filter (fun y => ∃ x, f x = y) Finset.univ = Finset.image f Finset.univ

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theorem Finset.univ_filter_mem_range {α : Type u_1} {β : Type u_2} [Fintype α] (f : α → β) [Fintype β] [DecidablePred fun y => y ∈ Set.range f] [DecidableEq β] :

Finset.filter (fun y => y ∈ Set.range f) Finset.univ = Finset.image f Finset.univ

Note this is a special case of (Finset.image_preimage f univ _).symm.

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theorem Finset.coe_filter_univ {α : Type u_1} [Fintype α] (p : α → Prop) [DecidablePred p] :

↑(Finset.filter p Finset.univ) = {x | p x}

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@[simp]

theorem Finset.subtype_eq_univ {α : Type u_1} {s : Finset α} {p : α → Prop} [DecidablePred p] [Fintype { a // p a }] :

Finset.subtype p s = Finset.univ ↔ ∀ ⦃a : α⦄, p a → a ∈ s

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@[simp]

theorem Finset.subtype_univ {α : Type u_1} [Fintype α] (p : α → Prop) [DecidablePred p] [Fintype { a // p a }] :

Finset.subtype p Finset.univ = Finset.univ

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instance Fintype.decidablePiFintype {α : Type u_5} {β : α → Type u_4} [(a : α) → DecidableEq (β a)] [Fintype α] :

DecidableEq ((a : α) → β a)

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instance Fintype.decidableForallFintype {α : Type u_1} {p : α → Prop} [DecidablePred p] [Fintype α] :

Decidable ((a : α) → p a)

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instance Fintype.decidableExistsFintype {α : Type u_1} {p : α → Prop} [DecidablePred p] [Fintype α] :

Decidable (∃ a, p a)

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instance Fintype.decidableMemRangeFintype {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (f : α → β) :

DecidablePred fun x => x ∈ Set.range f

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instance Fintype.decidableEqEquivFintype {α : Type u_1} {β : Type u_2} [DecidableEq β] [Fintype α] :

DecidableEq (α ≃ β)

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instance Fintype.decidableEqEmbeddingFintype {α : Type u_1} {β : Type u_2} [DecidableEq β] [Fintype α] :

DecidableEq (α ↪ β)

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theorem Fintype.decidableEqZeroHomFintype.proof_1 {α : Type u_1} {β : Type u_2} [Zero α] [Zero β] (a : ZeroHom α β) (b : ZeroHom α β) :

↑a = ↑b ↔ a = b

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instance Fintype.decidableEqZeroHomFintype {α : Type u_1} {β : Type u_2} [DecidableEq β] [Fintype α] [Zero α] [Zero β] :

DecidableEq (ZeroHom α β)

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instance Fintype.decidableEqOneHomFintype {α : Type u_1} {β : Type u_2} [DecidableEq β] [Fintype α] [One α] [One β] :

DecidableEq (OneHom α β)

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theorem Fintype.decidableEqAddHomFintype.proof_1 {α : Type u_1} {β : Type u_2} [Add α] [Add β] (a : AddHom α β) (b : AddHom α β) :

↑a = ↑b ↔ a = b

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instance Fintype.decidableEqAddHomFintype {α : Type u_1} {β : Type u_2} [DecidableEq β] [Fintype α] [Add α] [Add β] :

DecidableEq (AddHom α β)

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instance Fintype.decidableEqMulHomFintype {α : Type u_1} {β : Type u_2} [DecidableEq β] [Fintype α] [Mul α] [Mul β] :

DecidableEq (α →ₙ* β)

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instance Fintype.decidableEqAddMonoidHomFintype {α : Type u_1} {β : Type u_2} [DecidableEq β] [Fintype α] [AddZeroClass α] [AddZeroClass β] :

DecidableEq (α →+ β)

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theorem Fintype.decidableEqAddMonoidHomFintype.proof_1 {α : Type u_1} {β : Type u_2} [AddZeroClass α] [AddZeroClass β] (a : α →+ β) (b : α →+ β) :

↑a = ↑b ↔ a = b

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instance Fintype.decidableEqMonoidHomFintype {α : Type u_1} {β : Type u_2} [DecidableEq β] [Fintype α] [MulOneClass α] [MulOneClass β] :

DecidableEq (α →* β)

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instance Fintype.decidableEqMonoidWithZeroHomFintype {α : Type u_1} {β : Type u_2} [DecidableEq β] [Fintype α] [MulZeroOneClass α] [MulZeroOneClass β] :

DecidableEq (α →*₀ β)

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instance Fintype.decidableEqRingHomFintype {α : Type u_1} {β : Type u_2} [DecidableEq β] [Fintype α] [Semiring α] [Semiring β] :

DecidableEq (α →+* β)

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instance Fintype.decidableInjectiveFintype {α : Type u_1} {β : Type u_2} [DecidableEq α] [DecidableEq β] [Fintype α] :

DecidablePred Function.Injective

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instance Fintype.decidableSurjectiveFintype {α : Type u_1} {β : Type u_2} [DecidableEq β] [Fintype α] [Fintype β] :

DecidablePred Function.Surjective

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instance Fintype.decidableBijectiveFintype {α : Type u_1} {β : Type u_2} [DecidableEq α] [DecidableEq β] [Fintype α] [Fintype β] :

DecidablePred Function.Bijective

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instance Fintype.decidableRightInverseFintype {α : Type u_1} {β : Type u_2} [DecidableEq α] [Fintype α] (f : α → β) (g : β → α) :

Decidable (Function.RightInverse f g)

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instance Fintype.decidableLeftInverseFintype {α : Type u_1} {β : Type u_2} [DecidableEq β] [Fintype β] (f : α → β) (g : β → α) :

Decidable (Function.LeftInverse f g)

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def Fintype.ofMultiset {α : Type u_1} [DecidableEq α] (s : Multiset α) (H : ∀ (x : α), x ∈ s) :

Fintype α

Construct a proof of Fintype α from a universal multiset

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def Fintype.ofList {α : Type u_1} [DecidableEq α] (l : List α) (H : ∀ (x : α), x ∈ l) :

Fintype α

Construct a proof of Fintype α from a universal list

Instances For

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instance Fintype.subsingleton (α : Type u_4) :

Subsingleton (Fintype α)

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def Fintype.subtype {α : Type u_1} {p : α → Prop} (s : Finset α) (H : ∀ (x : α), x ∈ s ↔ p x) :

Fintype { x // p x }

Given a predicate that can be represented by a finset, the subtype associated to the predicate is a fintype.

Instances For

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def Fintype.ofFinset {α : Type u_1} {p : Set α} (s : Finset α) (H : ∀ (x : α), x ∈ s ↔ x ∈ p) :

Fintype ↑p

Construct a fintype from a finset with the same elements.

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def Fintype.ofBijective {α : Type u_1} {β : Type u_2} [Fintype α] (f : α → β) (H : Function.Bijective f) :

Fintype β

If f : α → β is a bijection and α is a fintype, then β is also a fintype.

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def Fintype.ofSurjective {α : Type u_1} {β : Type u_2} [DecidableEq β] [Fintype α] (f : α → β) (H : Function.Surjective f) :

Fintype β

If f : α → β is a surjection and α is a fintype, then β is also a fintype.

Instances For

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instance Finset.decidableCodisjoint {α : Type u_1} [Fintype α] [DecidableEq α] {s : Finset α} {t : Finset α} :

Decidable (Codisjoint s t)

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instance Finset.decidableIsCompl {α : Type u_1} [Fintype α] [DecidableEq α] {s : Finset α} {t : Finset α} :

Decidable (IsCompl s t)

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def Function.Injective.invOfMemRange {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] {f : α → β} (hf : Function.Injective f) :

↑(Set.range f) → α

The inverse of an hf : injective function f : α → β, of the type ↥(Set.range f) → α. This is the computable version of Function.invFun that requires Fintype α and DecidableEq β, or the function version of applying (Equiv.ofInjective f hf).symm. This function should not usually be used for actual computation because for most cases, an explicit inverse can be stated that has better computational properties. This function computes by checking all terms a : α to find the f a = b, so it is O(N) where N = Fintype.card α.

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theorem Function.Injective.left_inv_of_invOfMemRange {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] {f : α → β} (hf : Function.Injective f) (b : ↑(Set.range f)) :

f (Function.Injective.invOfMemRange hf b) = ↑b

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@[simp]

theorem Function.Injective.right_inv_of_invOfMemRange {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] {f : α → β} (hf : Function.Injective f) (a : α) :

Function.Injective.invOfMemRange hf { val := f a, property := (_ : f a ∈ Set.range f) } = a

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theorem Function.Injective.invFun_restrict {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] {f : α → β} (hf : Function.Injective f) [Nonempty α] :

Set.restrict (Set.range f) (Function.invFun f) = Function.Injective.invOfMemRange hf

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theorem Function.Injective.invOfMemRange_surjective {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] {f : α → β} (hf : Function.Injective f) :

Function.Surjective (Function.Injective.invOfMemRange hf)

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def Function.Embedding.invOfMemRange {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (f : α ↪ β) (b : ↑(Set.range ↑f)) :

The inverse of an embedding f : α ↪ β, of the type ↥(Set.range f) → α. This is the computable version of Function.invFun that requires Fintype α and DecidableEq β, or the function version of applying (Equiv.ofInjective f f.injective).symm. This function should not usually be used for actual computation because for most cases, an explicit inverse can be stated that has better computational properties. This function computes by checking all terms a : α to find the f a = b, so it is O(N) where N = Fintype.card α.

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@[simp]

theorem Function.Embedding.left_inv_of_invOfMemRange {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (f : α ↪ β) (b : ↑(Set.range ↑f)) :

↑f (Function.Embedding.invOfMemRange f b) = ↑b

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@[simp]

theorem Function.Embedding.right_inv_of_invOfMemRange {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (f : α ↪ β) (a : α) :

Function.Embedding.invOfMemRange f { val := ↑f a, property := (_ : ↑f a ∈ Set.range ↑f) } = a

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theorem Function.Embedding.invFun_restrict {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (f : α ↪ β) [Nonempty α] :

Set.restrict (Set.range ↑f) (Function.invFun ↑f) = Function.Embedding.invOfMemRange f

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theorem Function.Embedding.invOfMemRange_surjective {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (f : α ↪ β) :

Function.Surjective (Function.Embedding.invOfMemRange f)

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noncomputable def Fintype.ofInjective {α : Type u_1} {β : Type u_2} [Fintype β] (f : α → β) (H : Function.Injective f) :

Fintype α

Given an injective function to a fintype, the domain is also a fintype. This is noncomputable because injectivity alone cannot be used to construct preimages.

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def Fintype.ofEquiv {β : Type u_2} (α : Type u_4) [Fintype α] (f : α ≃ β) :

Fintype β

If f : α ≃ β and α is a fintype, then β is also a fintype.

Instances For

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def Fintype.ofSubsingleton {α : Type u_1} (a : α) [Subsingleton α] :

Fintype α

Any subsingleton type with a witness is a fintype (with one term).

Instances For

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@[simp]

theorem Fintype.univ_ofSubsingleton {α : Type u_1} (a : α) [Subsingleton α] :

Finset.univ = {a}

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instance Fintype.ofIsEmpty {α : Type u_1} [IsEmpty α] :

Fintype α

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@[simp]

theorem Fintype.univ_of_isEmpty {α : Type u_1} [IsEmpty α] :

Finset.univ = ∅

Note: this lemma is specifically about Fintype.of_isEmpty. For a statement about arbitrary Fintype instances, use Finset.univ_eq_empty.

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def Set.toFinset {α : Type u_1} (s : Set α) [Fintype ↑s] :

Finset α

Construct a finset enumerating a set s, given a Fintype instance.

Instances For

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theorem Set.toFinset_congr {α : Type u_1} {s : Set α} {t : Set α} [Fintype ↑s] [Fintype ↑t] (h : s = t) :

Set.toFinset s = Set.toFinset t

source

@[simp]

theorem Set.mem_toFinset {α : Type u_1} {s : Set α} [Fintype ↑s] {a : α} :

a ∈ Set.toFinset s ↔ a ∈ s

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theorem Set.toFinset_ofFinset {α : Type u_1} {p : Set α} (s : Finset α) (H : ∀ (x : α), x ∈ s ↔ x ∈ p) :

Set.toFinset p = s

Many Fintype instances for sets are defined using an extensionally equal Finset. Rewriting s.toFinset with Set.toFinset_ofFinset replaces the term with such a Finset.

source

def Set.decidableMemOfFintype {α : Type u_1} [DecidableEq α] (s : Set α) [Fintype ↑s] (a : α) :

Decidable (a ∈ s)

Membership of a set with a Fintype instance is decidable.

Using this as an instance leads to potential loops with Subtype.fintype under certain decidability assumptions, so it should only be declared a local instance.

Instances For

source

@[simp]

theorem Set.coe_toFinset {α : Type u_1} (s : Set α) [Fintype ↑s] :

↑(Set.toFinset s) = s

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@[simp]

theorem Set.toFinset_nonempty {α : Type u_1} {s : Set α} [Fintype ↑s] :

Finset.Nonempty (Set.toFinset s) ↔ Set.Nonempty s

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@[simp]

theorem Set.toFinset_inj {α : Type u_1} {s : Set α} {t : Set α} [Fintype ↑s] [Fintype ↑t] :

Set.toFinset s = Set.toFinset t ↔ s = t

source

theorem Set.toFinset_subset_toFinset {α : Type u_1} {s : Set α} {t : Set α} [Fintype ↑s] [Fintype ↑t] :

Set.toFinset s ⊆ Set.toFinset t ↔ s ⊆ t

source

@[simp]

theorem Set.toFinset_ssubset {α : Type u_1} {s : Set α} [Fintype ↑s] {t : Finset α} :

Set.toFinset s ⊂ t ↔ s ⊂ ↑t

source

@[simp]

theorem Set.subset_toFinset {α : Type u_1} {t : Set α} {s : Finset α} [Fintype ↑t] :

s ⊆ Set.toFinset t ↔ ↑s ⊆ t

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@[simp]

theorem Set.ssubset_toFinset {α : Type u_1} {t : Set α} {s : Finset α} [Fintype ↑t] :

s ⊂ Set.toFinset t ↔ ↑s ⊂ t

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theorem Set.toFinset_ssubset_toFinset {α : Type u_1} {s : Set α} {t : Set α} [Fintype ↑s] [Fintype ↑t] :

Set.toFinset s ⊂ Set.toFinset t ↔ s ⊂ t

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@[simp]

theorem Set.toFinset_subset {α : Type u_1} {s : Set α} [Fintype ↑s] {t : Finset α} :

Set.toFinset s ⊆ t ↔ s ⊆ ↑t

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theorem Set.toFinset_mono {α : Type u_1} {s : Set α} {t : Set α} [Fintype ↑s] [Fintype ↑t] :

s ⊆ t → Set.toFinset s ⊆ Set.toFinset t

Alias of the reverse direction of Set.toFinset_subset_toFinset.

source

theorem Set.toFinset_strict_mono {α : Type u_1} {s : Set α} {t : Set α} [Fintype ↑s] [Fintype ↑t] :

s ⊂ t → Set.toFinset s ⊂ Set.toFinset t

Alias of the reverse direction of Set.toFinset_ssubset_toFinset.

source

@[simp]

theorem Set.disjoint_toFinset {α : Type u_1} {s : Set α} {t : Set α} [Fintype ↑s] [Fintype ↑t] :

Disjoint (Set.toFinset s) (Set.toFinset t) ↔ Disjoint s t

source

@[simp]

theorem Set.toFinset_inter {α : Type u_1} (s : Set α) (t : Set α) [DecidableEq α] [Fintype ↑s] [Fintype ↑t] [Fintype ↑(s ∩ t)] :

Set.toFinset (s ∩ t) = Set.toFinset s ∩ Set.toFinset t

source

@[simp]

theorem Set.toFinset_union {α : Type u_1} (s : Set α) (t : Set α) [DecidableEq α] [Fintype ↑s] [Fintype ↑t] [Fintype ↑(s ∪ t)] :

Set.toFinset (s ∪ t) = Set.toFinset s ∪ Set.toFinset t

source

@[simp]

theorem Set.toFinset_diff {α : Type u_1} (s : Set α) (t : Set α) [DecidableEq α] [Fintype ↑s] [Fintype ↑t] [Fintype ↑(s \ t)] :

Set.toFinset (s \ t) = Set.toFinset s \ Set.toFinset t

source

@[simp]

theorem Set.toFinset_symmDiff {α : Type u_1} (s : Set α) (t : Set α) [DecidableEq α] [Fintype ↑s] [Fintype ↑t] [Fintype ↑(s ∆ t)] :

Set.toFinset (s ∆ t) = Set.toFinset s ∆ Set.toFinset t

source

@[simp]

theorem Set.toFinset_compl {α : Type u_1} (s : Set α) [DecidableEq α] [Fintype ↑s] [Fintype α] [Fintype ↑sᶜ] :

Set.toFinset sᶜ = (Set.toFinset s)ᶜ

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@[simp]

theorem Set.toFinset_empty {α : Type u_1} [Fintype ↑∅] :

Set.toFinset ∅ = ∅

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@[simp]

theorem Set.toFinset_univ {α : Type u_1} [Fintype α] [Fintype ↑Set.univ] :

Set.toFinset Set.univ = Finset.univ

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@[simp]

theorem Set.toFinset_eq_empty {α : Type u_1} {s : Set α} [Fintype ↑s] :

Set.toFinset s = ∅ ↔ s = ∅

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@[simp]

theorem Set.toFinset_eq_univ {α : Type u_1} {s : Set α} [Fintype α] [Fintype ↑s] :

Set.toFinset s = Finset.univ ↔ s = Set.univ

source

@[simp]

theorem Set.toFinset_setOf {α : Type u_1} [Fintype α] (p : α → Prop) [DecidablePred p] [Fintype ↑{x | p x}] :

Set.toFinset {x | p x} = Finset.filter p Finset.univ

source

theorem Set.toFinset_ssubset_univ {α : Type u_1} [Fintype α] {s : Set α} [Fintype ↑s] :

Set.toFinset s ⊂ Finset.univ ↔ s ⊂ Set.univ

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@[simp]

theorem Set.toFinset_image {α : Type u_1} {β : Type u_2} [DecidableEq β] (f : α → β) (s : Set α) [Fintype ↑s] [Fintype ↑(f '' s)] :

Set.toFinset (f '' s) = Finset.image f (Set.toFinset s)

source

@[simp]

theorem Set.toFinset_range {α : Type u_1} {β : Type u_2} [DecidableEq α] [Fintype β] (f : β → α) [Fintype ↑(Set.range f)] :

Set.toFinset (Set.range f) = Finset.image f Finset.univ

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@[simp]

theorem Set.toFinset_singleton {α : Type u_1} (a : α) [Fintype ↑{a}] :

Set.toFinset {a} = {a}

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@[simp]

theorem Set.toFinset_insert {α : Type u_1} [DecidableEq α] {a : α} {s : Set α} [Fintype ↑(insert a s)] [Fintype ↑s] :

Set.toFinset (insert a s) = insert a (Set.toFinset s)

source

theorem Set.filter_mem_univ_eq_toFinset {α : Type u_1} [Fintype α] (s : Set α) [Fintype ↑s] [DecidablePred fun x => x ∈ s] :

Finset.filter (fun x => x ∈ s) Finset.univ = Set.toFinset s

source

@[simp]

theorem Finset.toFinset_coe {α : Type u_1} (s : Finset α) [Fintype ↑↑s] :

Set.toFinset ↑s = s

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instance Fin.fintype (n : ℕ) :

Fintype (Fin n)

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theorem Fin.univ_def (n : ℕ) :

Finset.univ = { val := ↑(List.finRange n), nodup := (_ : List.Nodup (List.finRange n)) }

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@[simp]

theorem List.toFinset_finRange (n : ℕ) :

List.toFinset (List.finRange n) = Finset.univ

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@[simp]

theorem Fin.image_succAbove_univ {n : ℕ} (i : Fin (n + 1)) :

Finset.image (Fin.succAbove i) Finset.univ = {i}ᶜ

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@[simp]

theorem Fin.image_succ_univ (n : ℕ) :

Finset.image Fin.succ Finset.univ = {0}ᶜ

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@[simp]

theorem Fin.image_castSucc (n : ℕ) :

Finset.image Fin.castSucc Finset.univ = {Fin.last n}ᶜ

source

theorem Fin.univ_succ (n : ℕ) :

Finset.univ = Finset.cons 0 (Finset.map { toFun := Fin.succ, inj' := (_ : Function.Injective Fin.succ) } Finset.univ) (_ : ¬0 ∈ Finset.map { toFun := Fin.succ, inj' := (_ : Function.Injective Fin.succ) } Finset.univ)

Embed Fin n into Fin (n + 1) by prepending zero to the univ

source

theorem Fin.univ_castSuccEmb (n : ℕ) :

Finset.univ = Finset.cons (Fin.last n) (Finset.map Fin.castSuccEmb.toEmbedding Finset.univ) (_ : ¬Fin.last n ∈ Finset.map { toFun := Fin.castSucc, inj' := (_ : ∀ (x x_1 : Fin n), Fin.castSucc x = Fin.castSucc x_1 → x = x_1) } Finset.univ)

Embed Fin n into Fin (n + 1) by appending a new Fin.last n to the univ

source

theorem Fin.univ_succAbove (n : ℕ) (p : Fin (n + 1)) :

Finset.univ = Finset.cons p (Finset.map (Fin.succAboveEmb p).toEmbedding Finset.univ) (_ : ¬p ∈ Finset.map { toFun := Fin.succAbove p, inj' := (_ : ∀ (x x_1 : Fin n), Fin.succAbove p x = Fin.succAbove p x_1 → x = x_1) } Finset.univ)

Embed Fin n into Fin (n + 1) by inserting around a specified pivot p : Fin (n + 1) into the univ

source

instance Unique.fintype {α : Type u_4} [Unique α] :

Fintype α

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instance Fintype.subtypeEq {α : Type u_1} (y : α) :

Fintype { x // x = y }

Short-circuit instance to decrease search for Unique.fintype, since that relies on a subsingleton elimination for Unique.

source

instance Fintype.subtypeEq' {α : Type u_1} (y : α) :

Fintype { x // y = x }

Short-circuit instance to decrease search for Unique.fintype, since that relies on a subsingleton elimination for Unique.

source

theorem Fintype.univ_empty :

Finset.univ = ∅

source

theorem Fintype.univ_pempty :

Finset.univ = ∅

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instance Unit.fintype :

Fintype Unit

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theorem Fintype.univ_unit :

Finset.univ = {()}

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instance PUnit.fintype :

Fintype PUnit.{u_4 + 1}

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theorem Fintype.univ_punit :

Finset.univ = {PUnit.unit}

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instance Bool.fintype :

Fintype Bool

source

@[simp]

theorem Fintype.univ_bool :

Finset.univ = {true, false}

source

instance Additive.fintype {α : Type u_1} [Fintype α] :

Fintype (Additive α)

source

instance Multiplicative.fintype {α : Type u_1} [Fintype α] :

Fintype (Multiplicative α)

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def Fintype.prodLeft {α : Type u_4} {β : Type u_5} [DecidableEq α] [Fintype (α × β)] [Nonempty β] :

Fintype α

Given that α × β is a fintype, α is also a fintype.

Instances For

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def Fintype.prodRight {α : Type u_4} {β : Type u_5} [DecidableEq β] [Fintype (α × β)] [Nonempty α] :

Fintype β

Given that α × β is a fintype, β is also a fintype.

Instances For

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instance ULift.fintype (α : Type u_4) [Fintype α] :

Fintype (ULift.{u_5, u_4} α)

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instance PLift.fintype (α : Type u_4) [Fintype α] :

Fintype (PLift α)

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instance OrderDual.fintype (α : Type u_4) [Fintype α] :

Fintype αᵒᵈ

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instance OrderDual.finite (α : Type u_4) [Finite α] :

Finite αᵒᵈ

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instance Lex.fintype (α : Type u_4) [Fintype α] :

Fintype (Lex α)

`Fintype (s : Finset α)` #

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instance Finset.fintypeCoeSort {α : Type u} (s : Finset α) :

Fintype { x // x ∈ s }

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@[simp]

theorem Finset.univ_eq_attach {α : Type u} (s : Finset α) :

Finset.univ = Finset.attach s

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theorem Fintype.coe_image_univ {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] {f : α → β} :

↑(Finset.image f Finset.univ) = Set.range f

source

instance List.Subtype.fintype {α : Type u_1} [DecidableEq α] (l : List α) :

Fintype { x // x ∈ l }

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instance Multiset.Subtype.fintype {α : Type u_1} [DecidableEq α] (s : Multiset α) :

Fintype { x // x ∈ s }

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instance Finset.Subtype.fintype {α : Type u_1} (s : Finset α) :

Fintype { x // x ∈ s }

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instance FinsetCoe.fintype {α : Type u_1} (s : Finset α) :

Fintype ↑↑s

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theorem Finset.attach_eq_univ {α : Type u_1} {s : Finset α} :

Finset.attach s = Finset.univ

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instance PLift.fintypeProp (p : Prop) [Decidable p] :

Fintype (PLift p)

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instance Prop.fintype :

Fintype Prop

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@[simp]

theorem Fintype.univ_Prop :

Finset.univ = {True, False}

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instance Subtype.fintype {α : Type u_1} (p : α → Prop) [DecidablePred p] [Fintype α] :

Fintype { x // p x }

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def setFintype {α : Type u_1} [Fintype α] (s : Set α) [DecidablePred fun x => x ∈ s] :

Fintype ↑s

A set on a fintype, when coerced to a type, is a fintype.

Instances For

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@[simp]

theorem val_inv_unitsEquivProdSubtype_symm_apply (α : Type u_1) [Monoid α] (p : { p // p.fst * p.snd = 1 ∧ p.snd * p.fst = 1 }) :

↑(↑(unitsEquivProdSubtype α).symm p)⁻¹ = (↑p).snd

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@[simp]

theorem val_unitsEquivProdSubtype_symm_apply (α : Type u_1) [Monoid α] (p : { p // p.fst * p.snd = 1 ∧ p.snd * p.fst = 1 }) :

↑(↑(unitsEquivProdSubtype α).symm p) = (↑p).fst

source

@[simp]

theorem unitsEquivProdSubtype_apply_coe (α : Type u_1) [Monoid α] (u : αˣ) :

↑(↑(unitsEquivProdSubtype α) u) = (↑u, ↑u⁻¹)

source

def unitsEquivProdSubtype (α : Type u_1) [Monoid α] :

αˣ ≃ { p // p.fst * p.snd = 1 ∧ p.snd * p.fst = 1 }

The αˣ type is equivalent to a subtype of α × α.

Instances For

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@[simp]

theorem unitsEquivNeZero_apply_coe (α : Type u_1) [GroupWithZero α] (a : αˣ) :

↑(↑(unitsEquivNeZero α) a) = ↑a

source

@[simp]

theorem unitsEquivNeZero_symm_apply (α : Type u_1) [GroupWithZero α] (a : { a // a ≠ 0 }) :

↑(unitsEquivNeZero α).symm a = Units.mk0 ↑a (_ : ↑a ≠ 0)

source

def unitsEquivNeZero (α : Type u_1) [GroupWithZero α] :

αˣ ≃ { a // a ≠ 0 }

In a GroupWithZero α, the unit group αˣ is equivalent to the subtype of nonzero elements.

Instances For

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noncomputable def Fintype.finsetEquivSet {α : Type u_1} [Fintype α] :

Finset α ≃ Set α

Given Fintype α, finsetEquivSet is the equiv between Finset α and Set α. (All sets on a finite type are finite.)

Instances For

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@[simp]

theorem Fintype.coe_finsetEquivSet {α : Type u_1} [Fintype α] :

↑Fintype.finsetEquivSet = Finset.toSet

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@[simp]

theorem Fintype.finsetEquivSet_apply {α : Type u_1} [Fintype α] (s : Finset α) :

↑Fintype.finsetEquivSet s = ↑s

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@[simp]

theorem Fintype.finsetEquivSet_symm_apply {α : Type u_1} [Fintype α] (s : Set α) [Fintype ↑s] :

↑Fintype.finsetEquivSet.symm s = Set.toFinset s

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@[simp]

theorem Fintype.finsetOrderIsoSet_toEquiv {α : Type u_1} [Fintype α] :

Fintype.finsetOrderIsoSet.toEquiv = Fintype.finsetEquivSet

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noncomputable def Fintype.finsetOrderIsoSet {α : Type u_1} [Fintype α] :

Finset α ≃o Set α

Given a fintype α, finsetOrderIsoSet is the order isomorphism between Finset α and Set α (all sets on a finite type are finite).

Instances For

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@[simp]

theorem Fintype.coe_finsetOrderIsoSet {α : Type u_1} [Fintype α] :

↑Fintype.finsetOrderIsoSet = Finset.toSet

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@[simp]

theorem Fintype.coe_finsetOrderIsoSet_symm {α : Type u_1} [Fintype α] :

↑(OrderIso.symm Fintype.finsetOrderIsoSet) = ↑Fintype.finsetEquivSet.symm

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instance Quotient.fintype {α : Type u_1} [Fintype α] (s : Setoid α) [DecidableRel fun x x_1 => x ≈ x_1] :

Fintype (Quotient s)

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instance PSigma.fintypePropLeft {α : Prop} {β : α → Type u_4} [Decidable α] [(a : α) → Fintype (β a)] :

Fintype ((a : α) ×' β a)

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instance PSigma.fintypePropRight {α : Type u_4} {β : α → Prop} [(a : α) → Decidable (β a)] [Fintype α] :

Fintype ((a : α) ×' β a)

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instance PSigma.fintypePropProp {α : Prop} {β : α → Prop} [Decidable α] [(a : α) → Decidable (β a)] :

Fintype ((a : α) ×' β a)

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instance pfunFintype (p : Prop) [Decidable p] (α : p → Type u_4) [(hp : p) → Fintype (α hp)] :

Fintype ((hp : p) → α hp)

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theorem mem_image_univ_iff_mem_range {α : Type u_4} {β : Type u_5} [Fintype α] [DecidableEq β] {f : α → β} {b : β} :

b ∈ Finset.image f Finset.univ ↔ b ∈ Set.range f

source

def Fintype.chooseX {α : Type u_1} [Fintype α] (p : α → Prop) [DecidablePred p] (hp : ∃! a, p a) :

{ a // p a }

Given a fintype α and a predicate p, associate to a proof that there is a unique element of α satisfying p this unique element, as an element of the corresponding subtype.

Instances For

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def Fintype.choose {α : Type u_1} [Fintype α] (p : α → Prop) [DecidablePred p] (hp : ∃! a, p a) :

Given a fintype α and a predicate p, associate to a proof that there is a unique element of α satisfying p this unique element, as an element of α.

Instances For

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theorem Fintype.choose_spec {α : Type u_1} [Fintype α] (p : α → Prop) [DecidablePred p] (hp : ∃! a, p a) :

p (Fintype.choose p hp)

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theorem Fintype.choose_subtype_eq {α : Type u_4} (p : α → Prop) [Fintype { a // p a }] [DecidableEq α] (x : { a // p a }) (h : optParam (∃! a, ↑a = ↑x) (_ : ∃ x, (fun a => ↑a = ↑x) x ∧ ∀ (y : { a // p a }), (fun a => ↑a = ↑x) y → y = x)) :

Fintype.choose (fun y => ↑y = ↑x) h = x

source

def Fintype.bijInv {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] {f : α → β} (f_bij : Function.Bijective f) (b : β) :

bijInv f is the unique inverse to a bijection f. This acts as a computable alternative to Function.invFun.

Instances For

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theorem Fintype.leftInverse_bijInv {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] {f : α → β} (f_bij : Function.Bijective f) :

Function.LeftInverse (Fintype.bijInv f_bij) f

source

theorem Fintype.rightInverse_bijInv {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] {f : α → β} (f_bij : Function.Bijective f) :

Function.RightInverse (Fintype.bijInv f_bij) f

source

theorem Fintype.bijective_bijInv {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] {f : α → β} (f_bij : Function.Bijective f) :

Function.Bijective (Fintype.bijInv f_bij)

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def truncOfMultisetExistsMem {α : Type u_4} (s : Multiset α) :

(∃ x, x ∈ s) → Trunc α

For s : Multiset α, we can lift the existential statement that ∃ x, x ∈ s to a Trunc α.

Instances For

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def truncOfNonemptyFintype (α : Type u_4) [Nonempty α] [Fintype α] :

Trunc α

A Nonempty Fintype constructively contains an element.

Instances For

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def truncSigmaOfExists {α : Type u_4} [Fintype α] {P : α → Prop} [DecidablePred P] (h : ∃ a, P a) :

Trunc ((a : α) ×' P a)

By iterating over the elements of a fintype, we can lift an existential statement ∃ a, P a to Trunc (Σ' a, P a), containing data.

Instances For

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@[simp]

theorem Multiset.count_univ {α : Type u_1} [Fintype α] [DecidableEq α] (a : α) :

Multiset.count a Finset.univ.val = 1

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@[simp]

theorem Multiset.map_univ_val_equiv {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (e : α ≃ β) :

Multiset.map (↑e) Finset.univ.val = Finset.univ.val

source

noncomputable def seqOfForallFinsetExistsAux {α : Type u_4} [DecidableEq α] (P : α → Prop) (r : α → α → Prop) (h : ∀ (s : Finset α), ∃ y, ((x : α) → x ∈ s → P x) → P y ∧ ((x : α) → x ∈ s → r x y)) :

ℕ → α

Auxiliary definition to show exists_seq_of_forall_finset_exists.

Equations

seqOfForallFinsetExistsAux P r h x = Classical.choose (h (Finset.image (fun i => seqOfForallFinsetExistsAux P r h ↑i) Finset.univ))

Instances For

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theorem exists_seq_of_forall_finset_exists {α : Type u_4} (P : α → Prop) (r : α → α → Prop) (h : ∀ (s : Finset α), ((x : α) → x ∈ s → P x) → ∃ y, P y ∧ ((x : α) → x ∈ s → r x y)) :

∃ f, ((n : ℕ) → P (f n)) ∧ ((m n : ℕ) → m < n → r (f m) (f n))

Induction principle to build a sequence, by adding one point at a time satisfying a given relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set s, one can find another point satisfying some relation r with respect to all the points in s. Then one may construct a function f : ℕ → α such that r (f m) (f n) holds whenever m < n. We also ensure that all constructed points satisfy a given predicate P.

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theorem exists_seq_of_forall_finset_exists' {α : Type u_4} (P : α → Prop) (r : α → α → Prop) [IsSymm α r] (h : ∀ (s : Finset α), ((x : α) → x ∈ s → P x) → ∃ y, P y ∧ ((x : α) → x ∈ s → r x y)) :

∃ f, ((n : ℕ) → P (f n)) ∧ ((m n : ℕ) → m ≠ n → r (f m) (f n))

Induction principle to build a sequence, by adding one point at a time satisfying a given symmetric relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set s, one can find another point satisfying some relation r with respect to all the points in s. Then one may construct a function f : ℕ → α such that r (f m) (f n) holds whenever m ≠ n. We also ensure that all constructed points satisfy a given predicate P.

Finite types #

Main declarations #

Instances #

Fintype (s : Finset α) #

`Fintype (s : Finset α)` #