Type class for finitely enumerable types. The property is stronger than Fintype in that it assigns each element a rank in a finite enumeration.

class FinEnum (α : Sort u_1) : Sort (max 1 u_1)

• card : ℕ

  FinEnum.card is the cardinality of the FinEnum

• equiv : α ≃ Fin (FinEnum.card α)

  FinEnum.Equiv states that type α is in bijection with Fin card, the size of the FinEnum

• decEq : DecidableEq α

FinEnum α means that α is finite and can be enumerated in some order, i.e. α has an explicit bijection with Fin n for some n.

def FinEnum.ofEquiv (α : Sort u_1) {β : Sort u_2} [FinEnum α] (h : β ≃ α) : FinEnum β

transport a FinEnum instance across an equivalence

def FinEnum.ofNodupList {α : Type u} [DecidableEq α] (xs : List α) (h : ∀ (x : α), x ∈ xs) (h' : List.Nodup xs) : FinEnum α

create a FinEnum instance from an exhaustive list without duplicates

def FinEnum.ofList {α : Type u} [DecidableEq α] (xs : List α) (h : ∀ (x : α), x ∈ xs) : FinEnum α

create a FinEnum instance from an exhaustive list; duplicates are removed

def FinEnum.toList (α : Type u_1) [FinEnum α] : List α

create an exhaustive list of the values of a given type

@[simp]

theorem FinEnum.mem_toList {α : Type u} [FinEnum α] (x : α) : x ∈ FinEnum.toList α

@[simp]

theorem FinEnum.nodup_toList {α : Type u} [FinEnum α] : List.Nodup (FinEnum.toList α)

def FinEnum.ofSurjective {α : Type u} {β : Type u_1} (f : β → α) [DecidableEq α] [FinEnum β] (h : Function.Surjective f) : FinEnum α

create a FinEnum instance using a surjection

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

create a FinEnum instance using an injection

instance FinEnum.pempty : FinEnum PEmpty.{u_1 + 1}

instance FinEnum.empty : FinEnum Empty

instance FinEnum.punit : FinEnum PUnit.{u_1 + 1}

instance FinEnum.prod {α : Type u} {β : Type u_1} [FinEnum α] [FinEnum β] : FinEnum (α × β)

instance FinEnum.sum {α : Type u} {β : Type u_1} [FinEnum α] [FinEnum β] : FinEnum (α ⊕ β)

instance FinEnum.fin {n : ℕ} : FinEnum (Fin n)

instance FinEnum.Quotient.enum {α : Type u} [FinEnum α] (s : Setoid α) [DecidableRel fun x x_1 => x ≈ x_1] : FinEnum (Quotient s)

def FinEnum.Finset.enum {α : Type u} [DecidableEq α] : List α → List (Finset α)

enumerate all finite sets of a given type

Equations

• FinEnum.Finset.enum [] = [∅]
• FinEnum.Finset.enum (x_1 :: xs) = do let r ← FinEnum.Finset.enum xs [r, {x_1} ∪ r]

@[simp]

theorem FinEnum.Finset.mem_enum {α : Type u} [DecidableEq α] (s : Finset α) (xs : List α) : s ∈ FinEnum.Finset.enum xs ↔ ∀ (x : α), x ∈ s → x ∈ xs

instance FinEnum.Finset.finEnum {α : Type u} [FinEnum α] : FinEnum (Finset α)

instance FinEnum.Subtype.finEnum {α : Type u} [FinEnum α] (p : α → Prop) [DecidablePred p] : FinEnum { x // p x }

instance FinEnum.instFinEnumSigma {α : Type u} (β : α → Type v) [FinEnum α] [(a : α) → FinEnum (β a)] : FinEnum (Sigma β)

instance FinEnum.PSigma.finEnum {α : Type u} {β : α → Type v} [FinEnum α] [(a : α) → FinEnum (β a)] : FinEnum ((a : α) ×' β a)

instance FinEnum.PSigma.finEnumPropLeft {α : Prop} {β : α → Type v} [(a : α) → FinEnum (β a)] [Decidable α] : FinEnum ((a : α) ×' β a)

instance FinEnum.PSigma.finEnumPropRight {α : Type u} {β : α → Prop} [FinEnum α] [(a : α) → Decidable (β a)] : FinEnum ((a : α) ×' β a)

instance FinEnum.PSigma.finEnumPropProp {α : Prop} {β : α → Prop} [Decidable α] [(a : α) → Decidable (β a)] : FinEnum ((a : α) ×' β a)

instance FinEnum.instFintype {α : Type u} [FinEnum α] : Fintype α

def FinEnum.Pi.cons {α : Type u} {β : α → Type v} [DecidableEq α] (x : α) (xs : List α) (y : β x) (f : (a : α) → a ∈ xs → β a) (a : α) : a ∈ x :: xs → β a

For Pi.cons x xs y f create a function where every i ∈ xs is mapped to f i and x is mapped to y

def FinEnum.Pi.tail {α : Type u} {β : α → Type v} {x : α} {xs : List α} (f : (a : α) → a ∈ x :: xs → β a) (a : α) : a ∈ xs → β a

Given f a function whose domain is x :: xs, produce a function whose domain is restricted to xs.

def FinEnum.pi {α : Type u} {β : α → Type (max u v)} [DecidableEq α] (xs : List α) : ((a : α) → List (β a)) → List ((a : α) → a ∈ xs → β a)

pi xs f creates the list of functions g such that, for x ∈ xs, g x ∈ f x

Equations

• FinEnum.pi [] x = [fun x h => False.elim (_ : False)]
• FinEnum.pi (x_2 :: xs) x = Seq.seq (FinEnum.Pi.cons x_2 xs <$> x x_2) fun x => FinEnum.pi xs x

theorem FinEnum.mem_pi {α : Type u} {β : α → Type (max u u_1)} [FinEnum α] [(a : α) → FinEnum (β a)] (xs : List α) (f : (a : α) → a ∈ xs → β a) : f ∈ FinEnum.pi xs fun x => FinEnum.toList (β x)

def FinEnum.pi.enum {α : Type u} (β : α → Type (max u v)) [FinEnum α] [(a : α) → FinEnum (β a)] : List ((a : α) → β a)

enumerate all functions whose domain and range are finitely enumerable

theorem FinEnum.pi.mem_enum {α : Type u} {β : α → Type (max u v)} [FinEnum α] [(a : α) → FinEnum (β a)] (f : (a : α) → β a) : f ∈ FinEnum.pi.enum β

instance FinEnum.pi.finEnum {α : Type u} {β : α → Type (max u v)} [FinEnum α] [(a : α) → FinEnum (β a)] : FinEnum ((a : α) → β a)

instance FinEnum.pfunFinEnum (p : Prop) [Decidable p] (α : p → Type) [(hp : p) → FinEnum (α hp)] : FinEnum ((hp : p) → α hp)