Function types of a given arity

This provides FunctionOfArity, such that OfArity α β 2 = α → α → β. Note that it is often preferable to use (Fin n → α) → β in place of OfArity n α β.

Main definitions

• Function.OfArity α β n: n-ary function α → α → ... → β. Defined inductively.
• Function.OfArity.const α b n: n-ary constant function equal to b.

def Function.OfArity (α : Type u) (β : Type u) : ℕ → Type u

The type of n-ary functions α → α → ... → β.

Note that this is not universe polymorphic, as this would require that when n=0 we produce either Unit → β or ULift β.

Equations

• Function.OfArity α β 0 = β
• Function.OfArity α β (Nat.succ n) = (α → Function.OfArity α β n)

@[simp]

theorem Function.ofArity_zero (α : Type u) (β : Type u) : Function.OfArity α β 0 = β

@[simp]

theorem Function.ofArity_succ (α : Type u) (β : Type u) (n : ℕ) : Function.OfArity α β (Nat.succ n) = (α → Function.OfArity α β n)

def Function.OfArity.const (α : Type u) {β : Type u} (b : β) (n : ℕ) : Function.OfArity α β n

Constant n-ary function with value a.

Equations

• Function.OfArity.const α b 0 = b
• Function.OfArity.const α b (Nat.succ n) = fun (x : α) => Function.OfArity.const α b n

@[simp]

theorem Function.OfArity.const_zero (α : Type u) {β : Type u} (b : β) : Function.OfArity.const α b 0 = b

@[simp]

theorem Function.OfArity.const_succ (α : Type u) {β : Type u} (b : β) (n : ℕ) : Function.OfArity.const α b (Nat.succ n) = fun (x : α) => Function.OfArity.const α b n

theorem Function.OfArity.const_succ_apply (α : Type u) {β : Type u} (b : β) (n : ℕ) (x : α) : Function.OfArity.const α b (Nat.succ n) x = Function.OfArity.const α b n

instance Function.OfArity.OfArity.inhabited {α : Type u_1} {β : Type u_1} {n : ℕ} [Inhabited β] : Inhabited (Function.OfArity α β n)

Equations

• Function.OfArity.OfArity.inhabited = { default := Function.OfArity.const α default n }