curry3.lean4

class uncurry_ty (α) where
  ty : (α → Sort _) → Sort _
  uncurry : (∀ x, β x) → ty β

open uncurry_ty

instance : uncurry_ty α where
  ty β := ∀ x, β x
  uncurry := id

instance [uncurry_ty α'] : uncurry_ty (α × α') where
  ty β := ∀ x, ty λ x' => β ⟨x, x'⟩
  uncurry f x := uncurry λ x' => f ⟨x, x'⟩

instance [∀ x, uncurry_ty (α' x)] : uncurry_ty (Sigma α') where
  ty β := ∀ x, ty λ x' => β ⟨x, x'⟩
  uncurry f x := uncurry λ x' => f ⟨x, x'⟩

def sum₃ : Nat × Nat × Nat → Nat :=
λ (x, y, z) => x + y + z

def head : (Σ n, Fin (n + 1) → Nat) → Nat :=
λ ⟨n, f⟩ => f 0

#eval uncurry sum₃ 1 2 3
#eval uncurry head 1 λ _ => 42

def usum₃ : Nat → Nat → Nat → Nat := uncurry sum₃
def uhead : ∀ n, (Fin (n + 1) → Nat) → Nat := uncurry head

class uncurry_ty' (α) where
  ty' : _
  uncurry' : α → ty'

open uncurry_ty'

instance : uncurry_ty' α where
  ty' := α
  uncurry' := id

instance [uncurry_ty α] {β : α → _} : uncurry_ty' (∀ x, β x) where
  ty' := ty β
  uncurry' := uncurry

class uncurry_ty'' (α) where
  ty'' : (β : α → Sort _) → [∀ x, uncurry_ty' (β x)] → Sort _
  uncurry'' : [∀ x, uncurry_ty' (β x)] → (∀ x, β x) → ty'' β

open uncurry_ty''

instance [uncurry_ty α] : uncurry_ty'' α where
  ty'' β := ty λ x => ty' (β x)
  uncurry'' f := uncurry λ x => uncurry' (f x)

def sum₃' : Nat → Nat × Nat → Nat :=
λ x (y, z) => x + y + z

def head' : ∀ {α : Type _}, (Σ n, Fin (n + 1) → α) → α :=
λ ⟨n, f⟩ => f 0

#eval uncurry'' sum₃ 1 2 3
#eval uncurry'' head 1 λ _ => 42

def usum₃' : Nat → Nat → Nat → Nat := uncurry'' sum₃'
def uhead' : ∀ {α : Type _} n, (Fin (n + 1) → α) → α := uncurry'' head'

#eval uncurry'' sum₃' 1 2 3
#eval uncurry'' head' 1 λ _ => 42