# Data.Fin

FSInjective : (m : Fin k) -> (n : Fin k) -> (FS m = FS n) -> m = n
FSinjective : (FS f = FS f') -> f = f'
FZNotFS : (FZ = FS f) -> Void
data Fin : (n : Nat) -> Type

  Numbers strictly less than some bound. The name comes from "finite sets".

  It's probably not a good idea to use `Fin` for arithmetic, and they will be
  exceedingly inefficient at run time.

  n

    the upper bound

  FZ : Fin (S k)
  FS : Fin k -> Fin (S k)
FinZAbsurd : Fin 0 -> Void

  There are no elements of `Fin Z`

FinZElim : Fin 0 -> a
addFin : (x : Fin n) -> (y : Fin m) -> Fin (finToNat x + m)

  Add two Fins

finFromIntegerErrors : Err -> Maybe (List ErrorReportPart)
finToInteger : Fin n -> Integer

  Convert a Fin to an Integer

finToNat : Fin n -> Nat

  Convert a Fin to a Nat

finToNatInjective : (fm : Fin k) -> (fn : Fin k) -> (finToNat fm = finToNat fn) -> fm = fn

  `finToNat` is injective

fromInteger : (x : Integer) -> {default ItIsJust prf : IsJust (integerToFin x n)} -> Fin n

  Allow overloading of Integer literals for Fin.

  x

    the Integer that the user typed

  prf

    an automatically-constructed proof that `x` is in bounds

integerToFin : Integer -> (n : Nat) -> Maybe (Fin n)

  Convert an `Integer` to a `Fin`, provided the integer is within bounds.

  n

    The upper bound of the Fin

last : Fin (S n)

  The largest element of some Fin type

natToFin : Nat -> (n : Nat) -> Maybe (Fin n)
restrict : (n : Nat) -> Integer -> Fin (S n)

  Convert an Integer to a Fin in the required bounds/
  This is essentially a composition of `mod` and `fromInteger`

shift : (m : Nat) -> Fin n -> Fin (m + n)

  Add some natural number to a Fin, extending the bound accordingly

  n

    the previous bound

  m

    the number to increase the Fin by

strengthen : Fin (S n) -> Either (Fin (S n)) (Fin n)

  Attempt to tighten the bound on a Fin.
  Return `Left` if the bound could not be tightened, or `Right` if it could.

weaken : Fin n -> Fin (S n)

  Weaken the bound on a Fin by 1

weakenLTE : LTE n m -> Fin n -> Fin m

  Weaken the bound on a Fin using an inequality proof

weakenN : (n : Nat) -> Fin m -> Fin (m + n)

  Weaken the bound on a Fin by some amount