module Quox.Syntax.Term.Reduce

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

%default total


mutual
  ||| true if a term has a closure or dimension closure at the top level,
  ||| or is `E` applied to such an elimination
  public export %inline
  topCloT : Term d n -> Bool
  topCloT (CloT  _ _) = True
  topCloT (DCloT _ _) = True
  topCloT (E e)       = topCloE e
  topCloT _           = False

  ||| true if an elimination has a closure or dimension closure at the top level
  public export %inline
  topCloE : Elim d n -> Bool
  topCloE (CloE  _ _) = True
  topCloE (DCloE _ _) = True
  topCloE _           = False


public export IsNotCloT : Term d n -> Type
IsNotCloT = So . not . topCloT

||| a term which is not a top level closure
public export NotCloTerm : Nat -> Nat -> Type
NotCloTerm d n = Subset (Term d n) IsNotCloT

public export IsNotCloE : Elim d n -> Type
IsNotCloE = So . not . topCloE

||| an elimination which is not a top level closure
public export NotCloElim : Nat -> Nat -> Type
NotCloElim d n = Subset (Elim d n) IsNotCloE

public export %inline
ncloT : (t : Term d n) -> (0 _ : IsNotCloT t) => NotCloTerm d n
ncloT t @{p} = Element t p

public export %inline
ncloE : (t : Elim d n) -> (0 _ : IsNotCloE t) => NotCloElim d n
ncloE e @{p} = Element e p



mutual
  ||| if the input term has any top-level closures, push them under one layer of
  ||| syntax
  export %inline
  pushSubstsT : Term d n -> NotCloTerm d n
  pushSubstsT s = pushSubstsTWith id id s

  ||| if the input elimination has any top-level closures, push them under one
  ||| layer of syntax
  export %inline
  pushSubstsE : Elim d n -> NotCloElim d n
  pushSubstsE e = pushSubstsEWith id id e

  export
  pushSubstsTWith : DSubst dfrom dto -> TSubst dto from to ->
                    Term dfrom from -> NotCloTerm dto to
  pushSubstsTWith th ph (TYPE l) =
    ncloT $ TYPE l
  pushSubstsTWith th ph (Pi qty x a body) =
    ncloT $ Pi qty x (subs a th ph) (subs body th ph)
  pushSubstsTWith th ph (Lam x body) =
    ncloT $ Lam x $ subs body th ph
  pushSubstsTWith th ph (E e) =
    let Element e _ = pushSubstsEWith th ph e in ncloT $ E e
  pushSubstsTWith th ph (CloT  s ps) =
    pushSubstsTWith th (comp' th ps ph) s
  pushSubstsTWith th ph (DCloT s ps) =
    pushSubstsTWith (ps . th) ph s

  export
  pushSubstsEWith : DSubst dfrom dto -> TSubst dto from to ->
                    Elim dfrom from -> NotCloElim dto to
  pushSubstsEWith th ph (F x) =
    ncloE $ F x
  pushSubstsEWith th ph (B i) =
    assert_total pushSubstsE $ ph !! i
  pushSubstsEWith th ph (f :@ s) =
    ncloE $ subs f th ph :@ subs s th ph
  pushSubstsEWith th ph (s :# a) =
    ncloE $ subs s th ph :# subs a th ph
  pushSubstsEWith th ph (CloE  e ps) =
    pushSubstsEWith th (comp' th ps ph) e
  pushSubstsEWith th ph (DCloE e ps) =
    pushSubstsEWith (ps . th) ph e


parameters (th : DSubst dfrom dto) (ph : TSubst dto from to)
  public export %inline
  pushSubstsTWith' : Term dfrom from -> Term dto to
  pushSubstsTWith' s = (pushSubstsTWith th ph s).fst

  public export %inline
  pushSubstsEWith' : Elim dfrom from -> Elim dto to
  pushSubstsEWith' e = (pushSubstsEWith th ph e).fst


public export %inline
pushSubstsT' : Term d n -> Term d n
pushSubstsT' s = (pushSubstsT s).fst

public export %inline
pushSubstsE' : Elim d n -> Elim d n
pushSubstsE' e = (pushSubstsE e).fst


mutual
  -- tightening a term/elim also causes substitutions to be pushed through.
  -- this is because otherwise a variable in an unused part of the subst
  -- would cause it to incorrectly fail

  export covering
  Tighten (Term d) where
    tighten p (TYPE l) =
      pure $ TYPE l
    tighten p (Pi qty x arg res) =
      Pi qty x <$> tighten p arg
               <*> tighten p res
    tighten p (Lam x body) =
      Lam x <$> tighten p body
    tighten p (E e) =
      E <$> tighten p e
    tighten p (CloT tm th) =
      tighten p $ pushSubstsTWith' id th tm
    tighten p (DCloT tm th) =
      tighten p $ pushSubstsTWith' th id tm

  export covering
  Tighten (Elim d) where
    tighten p (F x) =
      pure $ F x
    tighten p (B i) =
      B <$> tighten p i
    tighten p (fun :@ arg) =
      [|tighten p fun :@ tighten p arg|]
    tighten p (tm :# ty) =
      [|tighten p tm :# tighten p ty|]
    tighten p (CloE el th) =
      tighten p $ pushSubstsEWith' id th el
    tighten p (DCloE el th) =
      tighten p $ pushSubstsEWith' th id el

  export covering
  Tighten (ScopeTerm d) where
    tighten p (TUsed   body) = TUsed   <$> tighten (Keep p) body
    tighten p (TUnused body) = TUnused <$> tighten p body


public export %inline
weakT : Term d n -> Term d (S n)
weakT t = t //. shift 1

public export %inline
weakE : Elim d n -> Elim d (S n)
weakE t = t //. shift 1