module Data.Fin.Strong

import Data.Fin

public export
data SFin : Nat -> Type where
  FZ : SFin (S n)
  FS : SFin (S n) -> SFin (S (S n))

public export
0 indexPred : (0 x : Fin n) -> Nat
indexPred {n = S n} x = n

public export
0 indexPredPrf : (0 x : Fin n) -> n = S (indexPred x)
indexPredPrf {n = S n} x = Refl

public export
0 indexPred' : (0 x : SFin n) -> Nat
indexPred' {n = S n} x = n

public export
0 indexPred'Prf : (0 x : SFin n) -> n = S (indexPred' x)
indexPred'Prf {n = S n} x = Refl

public export
finToStrong : Fin n -> SFin n
finToStrong FZ = FZ
finToStrong (FS x) =
  rewrite indexPredPrf x in
  FS (replace {p = SFin} (indexPredPrf x) (finToStrong x))

public export
strongToFin : SFin n -> Fin n
strongToFin FZ = FZ
strongToFin (FS x) = FS (strongToFin x)

export
finToStrongToFin : (x : Fin n) -> strongToFin (finToStrong x) = x
finToStrongToFin FZ = Refl
finToStrongToFin (FS FZ) = Refl
finToStrongToFin (FS (FS x)) = cong FS $ finToStrongToFin (FS x)

export
strongToFinToStrong : (x : SFin n) -> finToStrong (strongToFin x) = x
strongToFinToStrong FZ = Refl
strongToFinToStrong (FS x) = cong FS (strongToFinToStrong x)

%inline
export
FS' : SFin n -> SFin (S n)
FS' = finToStrong . FS . strongToFin

export
Injective (FS {n}) where
  injective Refl = Refl

export
Injective (FS' {n}) where
  injective {x = FZ, y = FZ} prf = Refl
  injective {x = FS x, y = FS y} prf = cong FS $ injective {f = FS'} $ injective prf

export
Uninhabited (SFin 0) where uninhabited x impossible

export
Uninhabited (Strong.FS x = FZ) where uninhabited prf impossible

export
sucNonZero : (x : SFin n) -> Not (FS' x = FZ)
sucNonZero FZ prf = absurd prf
sucNonZero (FS x) prf = absurd prf

export
sucIsFS : (x : SFin (S n)) -> FS' x = FS x
sucIsFS x = rewrite strongToFinToStrong x in Refl