{-# LANGUAGE UndecidableInstances #-}
{-# OPTIONS_GHC -fno-warn-orphans #-}
-- | Symantic for '(->)'.
module Language.Symantic.Lib.Function where

import Prelude hiding (const, flip, id)
import qualified Data.Function as Fun
import qualified Data.MonoTraversable as MT

import Language.Symantic

-- * Class 'Sym_Function'
type instance Sym (->) = Sym_Function
class Sym_Function term where
	comp  :: term (b -> c) -> term (a -> b) -> term (a -> c); infixr 9 `comp`
	const :: term a -> term b -> term a
	flip  :: term (a -> b -> c) -> term (b -> a -> c)
	id    :: term a -> term a
	default comp  :: Sym_Function (UnT term) => Trans term => term (b -> c) -> term (a -> b) -> term (a -> c)
	default const :: Sym_Function (UnT term) => Trans term => term a -> term b -> term a
	default flip  :: Sym_Function (UnT term) => Trans term => term (a -> b -> c) -> term (b -> a -> c)
	default id    :: Sym_Function (UnT term) => Trans term => term a -> term a
	comp  = trans2 comp
	const = trans2 const
	flip  = trans1 flip
	id    = trans1 id

-- Interpreting
instance Sym_Function Eval where
	comp  = eval2 (Fun..)
	const = eval2 Fun.const
	flip  = eval1 Fun.flip
	id    = eval1 Fun.id
instance Sym_Function View where
	comp  = viewInfix "." (infixR 9)
	const = view2 "const"
	flip  = view1 "flip"
	id    = view1 "id"
instance (Sym_Function r1, Sym_Function r2) => Sym_Function (Dup r1 r2) where
	comp  = dup2 @Sym_Function comp
	const = dup2 @Sym_Function const
	flip  = dup1 @Sym_Function flip
	id    = dup1 @Sym_Function id

-- Transforming
instance (Sym_Function term, Sym_Lambda term) => Sym_Function (BetaT term)

-- Typing
instance ClassInstancesFor (->) where
	proveConstraintFor _c (TyApp _ q (TyApp _ z _r))
	 | Just HRefl <- proj_ConstKiTy @_ @(->) z
	 = case () of
		 _ | Just HRefl <- proj_ConstKiTy @_ @Functor     q -> Just Dict
		   | Just HRefl <- proj_ConstKiTy @_ @Applicative q -> Just Dict
		   | Just HRefl <- proj_ConstKiTy @_ @Monad       q -> Just Dict
		 _ -> Nothing
	proveConstraintFor _c (TyApp _ q (TyApp _ (TyApp _ z _a) b))
	 | Just HRefl <- proj_ConstKiTy @_ @(->) z
	 = case () of
		 _ | Just HRefl <- proj_ConstKiTy @_ @Monoid q
		   , Just Dict  <- proveConstraint (q `tyApp` b) -> Just Dict
		   | Just HRefl <- proj_ConstKiTy @_ @MT.MonoFunctor q -> Just Dict
		 _ -> Nothing
	proveConstraintFor _c _q = Nothing
instance TypeInstancesFor (->)

-- Compiling
instance Gram_Term_AtomsFor src ss g (->)
instance (Source src, SymInj ss (->)) => ModuleFor src ss (->) where
	moduleFor = ["Function"] `moduleWhere`
	 [ "const" := teFunction_const
	 , "flip"  := teFunction_flip
	 , "id"    := teFunction_id
	 , "." `withInfixR` 9 := teFunction_compose
	 -- , "$" `withInfixR` 0 := teFunction_app
	 ]

-- ** 'Type's
tyFun :: Source src => LenInj vs => Type src vs (->)
tyFun = tyConst @(K (->)) @(->)

a0 :: Source src => LenInj vs => KindInj (K a) =>
     Type src (Proxy a ': vs) a
a0 = tyVar "a" varZ

b1 :: Source src => LenInj vs => KindInj (K b) =>
     Type src (a ': Proxy b ': vs) b
b1 = tyVar "b" $ VarS varZ

c2 :: Source src => LenInj vs => KindInj (K c) =>
     Type src (a ': b ': Proxy c ': vs) c
c2 = tyVar "c" $ VarS $ VarS varZ

-- ** 'Term's
teFunction_compose :: TermDef (->) '[Proxy a, Proxy b, Proxy c] (() #> ((b -> c) -> (a -> b) -> (a -> c)))
teFunction_compose = Term noConstraint ((b1 ~> c2) ~> (a0 ~> b1) ~> (a0 ~> c2)) $ teSym @(->) $ lam2 comp

teFunction_const :: TermDef (->) '[Proxy a, Proxy b] (() #> (a -> b -> a))
teFunction_const = Term noConstraint (a0 ~> b1 ~> a0) $ teSym @(->) $ lam2 const

teFunction_flip :: TermDef (->) '[Proxy a, Proxy b, Proxy c] (() #> ((a -> b -> c) -> (b -> a -> c)))
teFunction_flip = Term noConstraint ((a0 ~> b1 ~> c2) ~> (b1 ~> a0 ~> c2)) $ teSym @(->) $ lam1 flip

teFunction_id :: TermDef (->) '[Proxy a] (() #> (a -> a))
teFunction_id = Term noConstraint (a0 ~> a0) $ teSym @(->) $ lam1 id