# Builtins

data UPair : (A : AnyType) -> (B : AnyType) -> AnyType

  The non-dependent pair type, also known as conjunction, usable with
  UniqueTypes.

  A

    the type of the left elements in the pair

  B

    the type of the right elements in the pair

  MkUPair : (a : A) -> (b : B) -> UPair A B

    A pair of elements

    a

      the left element of the pair

    b

      the right element of the pair

Sigma : (a : Type) -> (P : a -> Type) -> Type
data Pair : (A : Type) -> (B : Type) -> Type

  The non-dependent pair type, also known as conjunction.

  A

    the type of the left elements in the pair

  B

    the type of the right elements in the pair

  MkPair : (a : A) -> (b : B) -> (A, B)

    A pair of elements

    a

      the left element of the pair

    b

      the right element of the pair

MkSigma : (x : a) -> (prf : P x) -> DPair a P
data DPair : (a : Type) -> (P : a -> Type) -> Type

  Dependent pairs aid in the construction of dependent types by
  providing evidence that some value resides in the type.

  Formally, speaking, dependent pairs represent existential
  quantification - they consist of a witness for the existential
  claim and a proof that the property holds for it.

  a

    the value to place in the type.

  P

    the dependent type that requires the value.

  MkDPair : (x : a) -> (pf : P x) -> DPair a P