module Data.Term.Occurs

import Data.Fin.Occurs
import Data.Maybe.Properties
import Data.Term
import Data.Term.Zipper

import Syntax.PreorderReasoning

-- Check -----------------------------------------------------------------------

export
check : Fin k -> Term sig k -> Maybe (Term sig (pred k))
checkAll : Fin k -> Vect n (Term sig k) -> Maybe (Vect n (Term sig (pred k)))

check x (Var y) = Var <$> thick x y
check x (Op op ts) = Op op <$> checkAll x ts

checkAll x [] = Just []
checkAll x (t :: ts) = [| check x t :: checkAll x ts |]

-- Properties

export
checkOccurs : (x : Fin k) -> (zip : Zipper sig k) -> IsNothing (check x (zip + Var x))
checkAllOccurs :
  (x : Fin k) ->
  (i : Fin (S n)) ->
  (ts : Vect n (Term sig k)) ->
  (zip : Zipper sig k) ->
  IsNothing (checkAll x (insertAt i (zip + Var x) ts))

checkOccurs x Top = mapNothing Var (thickRefl x)
checkOccurs x (Op op i ts zip) = mapNothing (Op op) (checkAllOccurs x i ts zip)

checkAllOccurs x FZ ts zip =
  appLeftNothingIsNothing
    (appRightNothingIsNothing (Just (::)) (checkOccurs x zip))
    (checkAll x ts)
checkAllOccurs x (FS i) (t :: ts) zip =
  appRightNothingIsNothing
    (Just (::) <*> check x t)
    (checkAllOccurs x i ts zip)

export
checkNothing :
  (x : Fin k) ->
  (t : Term sig k) ->
  (0 _ : IsNothing (check x t)) ->
  (zip : Zipper sig k ** t = zip + Var x)
checkAllNothing :
  (x : Fin k) ->
  (t : Term sig k) ->
  (ts : Vect n (Term sig k)) ->
  (0 _ : IsNothing (checkAll x (t :: ts))) ->
  (i ** ts' ** zip : Zipper sig k ** t :: ts = insertAt i (zip + Var x) ts')

checkNothing x (Var y) prf =
  (Top ** sym (cong Var (thickNothing x y (mapNothingInverse Var (thick x y) prf))))
checkNothing x (Op op (t :: ts)) prf =
  let (i ** ts' ** zip ** prf) = checkAllNothing x t ts (mapNothingInverse (Op op) _ prf) in
  (Op op i ts' zip ** cong (Op op) prf)

checkAllNothing x t ts prf with (appNothingInverse (Just (::) <*> check x t) (checkAll x ts) prf)
  _ | Left prf' = case appNothingInverse (Just (::)) (check x t) prf' of
      Right prf =>
        let (zip ** prf) = checkNothing x t prf in
        (FZ ** ts ** zip ** cong (:: ts) prf)
  checkAllNothing x t (t' :: ts) prf | Right prf' =
    let (i ** ts ** zip ** prf) = checkAllNothing x t' ts prf' in
    (FS i ** t :: ts ** zip ** cong (t ::) prf)

export
checkThin : (x : Fin k) -> (t : Term sig (pred k)) -> IsJust (check x (pure (thin x) <$> t))
checkAllThin :
  (x : Fin k) ->
  (ts : Vect n (Term sig (pred k))) ->
  IsJust (checkAll x (map (pure (thin x) <$>) ts))

checkThin x (Var y) = mapIsJust Var (thickNeq x (thin x y) (\prf => thinSkips x y $ sym prf))
checkThin x (Op op ts) = mapIsJust (Op op) (checkAllThin x ts)

checkAllThin x [] = ItIsJust
checkAllThin x (t :: ts) =
  appIsJust
    (appIsJust ItIsJust (checkThin x t))
    (checkAllThin x ts)

export
checkJust :
  (x : Fin k) ->
  (t : Term sig k) ->
  (0 _ : check x t = Just u) ->
  t = pure (thin x) <$> u
checkAllJust :
  (x : Fin k) ->
  (ts : Vect n (Term sig k)) ->
  (0 _ : checkAll x ts = Just us) ->
  ts = map (pure (thin x) <$>) us

checkJust x (Var y) prf =
  let 0 z = mapJustInverse Var (thick x y) prf in
  let 0 prf = thickJust x y (fst z.snd) in
  sym $ Calc $
    |~ pure (thin x) <$> u
    ~~ pure (thin x) <$> Var z.fst ...(cong (pure (thin x) <$>) $ snd z.snd)
    ~~ Var y                       ...(cong Var prf)
checkJust x (Op op ts) prf =
  let 0 z = mapJustInverse (Op op) (checkAll x ts) prf in
  let 0 prf = checkAllJust x ts (fst z.snd) in
  Calc $
    |~ Op op ts
    ~~ pure (thin x) <$> Op op z.fst ...(cong (Op op) prf)
    ~~ pure (thin x) <$> u           ...(sym $ cong (pure (thin x) <$>) $ snd z.snd)

checkAllJust x [] Refl = Refl
checkAllJust x (t :: ts) prf =
  let 0 z = appJustInverse (Just (::) <*> check x t) (checkAll x ts) prf in
  let 0 w = appJustInverse (Just (::)) (check x t) (fst z.snd.snd) in
  Calc $
    |~ t :: ts
    ~~ map (pure (thin x) <$>) (w.snd.fst :: z.snd.fst)    ...(cong2 (::) (checkJust x t (fst $ snd w.snd.snd)) (checkAllJust x ts (fst $ snd z.snd.snd)))
    ~~ map (pure (thin x) <$>) (w.fst w.snd.fst z.snd.fst) ...(cong (\f => map (pure (thin x) <$>) (f w.snd.fst z.snd.fst)) $ injective $ fst w.snd.snd)
    ~~ map (pure (thin x) <$>) (z.fst z.snd.fst)           ...(sym $ cong (\f => map (pure (thin x) <$>) (f z.snd.fst)) $ snd $ snd w.snd.snd)
    ~~ map (pure (thin x) <$>) us                          ...(sym $ cong (map (pure (thin x) <$>)) $ snd $ snd z.snd.snd)