Built-in things
---------------

- Where are they defined?

  + Haskell-module?
    Not so nice. Hard to find out what's predefined.

  + Agda2 prelude
    Nicer, but what would this look like?

- How are they accessed?

  + Problem in Agda 1: Bool already defined.
  + import Prelude / import Builtin ?

- Could we use a more general FFI?

  + Maybe.. but requires more work (hs-plugins, or the like)

Literals/sugar
--------------

- What does sugar expand to? When?

- What is the type of a literal?

  + Where is it specified?

  + Pragmas?

    {-# LITERAL NATURAL is PRIMITIVE Integer #-}
    {-# LITERAL LIST is SUGAR nil, (::)  #-}

    Nice, because we can allow either sugar or builtin for some types (like
    strings or naturals):

      {-# LITERAL NUMBER is PRIMITIVE Integer #-} or
      {-# LITERAL NUMBER is SUGAR FOR zero, suc #-}

    Builtin: NATURAL, FLOAT, CHAR, STRING

Possible solution
-----------------

- Add a primitive keyword:

  + primitive integerPlus : Integer -> Integer -> Integer

- Add "primitive" definitions:

  + data Defn = ... | Primitive Arity ([Term] -> TC Term)

  + The function is responsible for normalising its arguments if needed.

- Primitive functions are defined in TypeChecking.Primitive

  + primitives :: Map String (Arity, [Term] -> TC Term)
    primitives = Map.fromList
      [ "integerPlus", (2, integerPlus)
      , ...
      ]

    integerPlus :: [Term] -> TC Term
    integerPlus [x, y] = do
      (x,y) <- normalise (x,y)
      case (x,y) of
	(LitInt n, LitInt m) -> return $ LitInt $ n + m
	_		     -> ...

    integerEquals (Lit n) (Lit m)
      | n == m	= primTrue

    primTrue :: TC Term
    primTrue = lookupPrim "TRUE"

- Define a prelude/builtin module

  {-# BUILTIN NATURAL Integer #-}
  {-# BUILTIN FLOAT   Float   #-}
  {-# BUILTIN CHAR    Char    #-}
  postulate Integer : Set
	    Float   : Set
	    Char    : Set

  {-# SUGAR LIST nil :: #-}
  data List (A:Set) : Set where
    nil  : List A
    (::) : A -> List A -> List A

  {-# SUGAR STRING nil, (::) #-}
  String : Set
  String = List Char

  {-# BUILTIN FALSE false #-}
  {-# BUILTIN TRUE  true  #-}
  data Bool : Set where
    false : Bool
    true  : Bool

  primitive
    integerPlus  : Integer -> Integer -> Integer
    integerEqual : Integer -> Integer -> Bool

  postulate
    integerPlusAssoc : (x,y,z:Integer) -> 

Built-in things and Parameterised Modules
-----------------------------------------

What is the type of 1 in the following example:

  module Int (I:Set) where
    {-# BUILTIN INTEGER I #-}

  postulate Int1 : Set
	    Int2 : Set

  module Int1 = Int Int1
  module Int2 = Int Int2

Possible solution: don't allow BUILTIN in parameterised modules.

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