module Quox.Syntax.Term.Split

import Quox.Syntax.Term.Base
import Quox.Syntax.Term.Subst

import Quox.No
import Data.Vect

%default total


public export %inline
isLam : Term {} -> Bool
isLam (Lam {}) = True
isLam _        = False

public export
0 NotLam : Pred $ Term {}
NotLam = No . isLam


public export %inline
isDLam : Term {} -> Bool
isDLam (DLam {}) = True
isDLam _         = False

public export
0 NotDLam : Pred $ Term {}
NotDLam = No . isDLam


public export %inline
isPair : Term {} -> Bool
isPair (Pair {}) = True
isPair _         = False

public export
0 NotPair : Pred $ Term {}
NotPair = No . isPair


public export %inline
isApp : Elim {} -> Bool
isApp (_ :@ _) = True
isApp _        = False

public export
0 NotApp : Pred $ Elim {}
NotApp = No . isApp


public export %inline
isDApp : Elim {} -> Bool
isDApp (_ :% _) = True
isDApp _        = False

public export
0 NotDApp : Pred $ Elim {}
NotDApp = No . isDApp


infixl 9 :@@
||| apply multiple arguments at once
public export %inline
(:@@) : Elim q d n -> List (Term q d n) -> Elim q d n
f :@@ ss = foldl (:@) f ss

public export
record GetArgs q d n where
  constructor GotArgs
  fun      : Elim q d n
  args     : List (Term q d n)
  0 notApp : NotApp fun

export
getArgs' : Elim q d n -> List (Term q d n) -> GetArgs q d n
getArgs' fun args = case nchoose $ isApp fun of
  Left  y => let f :@ a = fun in getArgs' f (a :: args)
  Right n => GotArgs {fun, args, notApp = n}

||| splits an application into its head and arguments. if it's not an
||| application then the list is just empty
export %inline
getArgs : Elim q d n -> GetArgs q d n
getArgs e = getArgs' e []


infixl 9 :%%
||| apply multiple dimension arguments at once
public export %inline
(:%%) : Elim q d n -> List (Dim d) -> Elim q d n
f :%% ss = foldl (:%) f ss

public export
record GetDArgs q d n where
  constructor GotDArgs
  fun       : Elim q d n
  args      : List (Dim d)
  0 notDApp : NotDApp fun

export
getDArgs' : Elim q d n -> List (Dim d) -> GetDArgs q d n
getDArgs' fun args = case nchoose $ isDApp fun of
  Left  y => let f :% d = fun in getDArgs' f (d :: args)
  Right n => GotDArgs {fun, args, notDApp = n}

||| splits a dimension application into its head and arguments. if it's not an
||| d application then the list is just empty
export %inline
getDArgs : Elim q d n -> GetDArgs q d n
getDArgs e = getDArgs' e []


infixr 1 :\\
public export
absN : Vect m Name -> Term q d (m + n) -> Term q d n
absN {m = 0}   []        s = s
absN {m = S m} (x :: xs) s = Lam x $ TUsed $ absN xs $
                             rewrite sym $ plusSuccRightSucc m n in s

public export %inline
(:\\) : Vect m Name -> Term q d (m + n) -> Term q d n
(:\\) = absN


infixr 1 :\\%
public export
dabsN : Vect m Name -> Term q (m + d) n -> Term q d n
dabsN {m = 0}   []        s = s
dabsN {m = S m} (x :: xs) s = DLam x $ DUsed $ dabsN xs $
                              rewrite sym $ plusSuccRightSucc m d in s

public export %inline
(:\\%) : Vect m Name -> Term q (m + d) n -> Term q d n
(:\\%) = dabsN


public export
record GetLams q d n where
  constructor GotLams
  {0 lams, rest : Nat}
  names    : Vect lams Name
  body     : Term q d rest
  0 eq     : lams + n = rest
  0 notLam : NotLam body

export
getLams' : forall lams, rest.
           Vect lams Name -> Term q d rest -> (0 eq : lams + n = rest) ->
           GetLams q d n
getLams' xs s Refl = case nchoose $ isLam s of
  Left  y => let Lam x body = s in
             getLams' (x :: xs) (assert_smaller s body.term) Refl
  Right n => GotLams xs s Refl n

export %inline
getLams : Term q d n -> GetLams q d n
getLams s = getLams' [] s Refl


public export
record GetDLams q d n where
  constructor GotDLams
  {0 lams, rest : Nat}
  names     : Vect lams Name
  body      : Term q rest n
  0 eq      : lams + d = rest
  0 notDLam : NotDLam body

export
getDLams' : forall lams, rest.
            Vect lams Name -> Term q rest n -> (0 eq : lams + d = rest) ->
            GetDLams q d n
getDLams' is s Refl = case nchoose $ isDLam s of
  Left  y => let DLam i body = s in
             getDLams' (i :: is) (assert_smaller s body.term) Refl
  Right n => GotDLams is s Refl n

export %inline
getDLams : Term q d n -> GetDLams q d n
getDLams s = getDLams' [] s Refl


public export
record GetPairs q d n where
  constructor GotPairs
  init : List $ Term q d n
  last : Term q d n
  notPair : NotPair last

export
getPairs : Term q d n -> GetPairs q d n
getPairs t = case nchoose $ isPair t of
  Left y =>
    let Pair s t = t; GotPairs ts t np = getPairs t in
    GotPairs (s :: ts) t np
  Right n => GotPairs [] t n