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https://github.com/idris-lang/Idris2.git
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182 lines
5.3 KiB
Idris
182 lines
5.3 KiB
Idris
module Data.Fin
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import public Data.Maybe
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import Data.Nat
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import Decidable.Equality
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||| Numbers strictly less than some bound. The name comes from "finite sets".
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||| It's probably not a good idea to use `Fin` for arithmetic, and they will be
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||| exceedingly inefficient at run time.
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||| @ n the upper bound
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public export
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data Fin : (n : Nat) -> Type where
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FZ : Fin (S k)
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FS : Fin k -> Fin (S k)
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export
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implementation Uninhabited (Fin Z) where
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uninhabited FZ impossible
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uninhabited (FS f) impossible
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export
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FSInjective : (m : Fin k) -> (n : Fin k) -> FS m = FS n -> m = n
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FSInjective left _ Refl = Refl
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export
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implementation Eq (Fin n) where
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(==) FZ FZ = True
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(==) (FS k) (FS k') = k == k'
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(==) _ _ = False
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||| There are no elements of `Fin Z`
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export
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FinZAbsurd : Fin Z -> Void
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FinZAbsurd FZ impossible
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export
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FinZElim : Fin Z -> a
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FinZElim x = void (FinZAbsurd x)
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||| Convert a Fin to a Nat
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public export
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finToNat : Fin n -> Nat
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finToNat FZ = Z
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finToNat (FS k) = S (finToNat k)
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||| `finToNat` is injective
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export
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finToNatInjective : (fm : Fin k) -> (fn : Fin k) -> (finToNat fm) = (finToNat fn) -> fm = fn
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finToNatInjective (FS m) FZ Refl impossible
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finToNatInjective FZ (FS n) Refl impossible
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finToNatInjective (FS m) (FS n) prf =
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cong FS (finToNatInjective m n (succInjective (finToNat m) (finToNat n) prf))
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finToNatInjective FZ FZ Refl = Refl
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export
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implementation Cast (Fin n) Nat where
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cast x = finToNat x
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||| Convert a Fin to an Integer
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public export
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finToInteger : Fin n -> Integer
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finToInteger FZ = 0
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finToInteger (FS k) = 1 + finToInteger k
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export
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implementation Cast (Fin n) Integer where
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cast x = finToInteger x
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||| Weaken the bound on a Fin by 1
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public export
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weaken : Fin n -> Fin (S n)
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weaken FZ = FZ
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weaken (FS k) = FS (weaken k)
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||| Weaken the bound on a Fin by some amount
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public export
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weakenN : (n : Nat) -> Fin m -> Fin (m + n)
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weakenN n FZ = FZ
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weakenN n (FS f) = FS (weakenN n f)
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||| Weaken the bound on a Fin using a constructive comparison
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public export
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weakenLTE : Fin n -> LTE n m -> Fin m
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weakenLTE FZ LTEZero impossible
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weakenLTE (FS _) LTEZero impossible
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weakenLTE FZ (LTESucc y) = FZ
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weakenLTE (FS x) (LTESucc y) = FS $ weakenLTE x y
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||| Attempt to tighten the bound on a Fin.
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||| Return `Left` if the bound could not be tightened, or `Right` if it could.
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export
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strengthen : {n : _} -> Fin (S n) -> Either (Fin (S n)) (Fin n)
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strengthen {n = S k} FZ = Right FZ
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strengthen {n = S k} (FS i) with (strengthen i)
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strengthen (FS i) | Left x = Left (FS x)
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strengthen (FS i) | Right x = Right (FS x)
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strengthen f = Left f
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||| Add some natural number to a Fin, extending the bound accordingly
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||| @ n the previous bound
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||| @ m the number to increase the Fin by
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public export
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shift : (m : Nat) -> Fin n -> Fin (m + n)
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shift Z f = f
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shift {n=n} (S m) f = FS {k = (m + n)} (shift m f)
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||| The largest element of some Fin type
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public export
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last : {n : _} -> Fin (S n)
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last {n=Z} = FZ
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last {n=S _} = FS last
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public export total
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FSinjective : {f : Fin n} -> {f' : Fin n} -> (FS f = FS f') -> f = f'
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FSinjective Refl = Refl
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export
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implementation Ord (Fin n) where
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compare FZ FZ = EQ
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compare FZ (FS _) = LT
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compare (FS _) FZ = GT
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compare (FS x) (FS y) = compare x y
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-- Construct a Fin from an integer literal which must fit in the given Fin
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public export
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natToFin : Nat -> (n : Nat) -> Maybe (Fin n)
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natToFin Z (S j) = Just FZ
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natToFin (S k) (S j)
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= case natToFin k j of
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Just k' => Just (FS k')
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Nothing => Nothing
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natToFin _ _ = Nothing
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||| Convert an `Integer` to a `Fin`, provided the integer is within bounds.
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||| @n The upper bound of the Fin
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public export
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integerToFin : Integer -> (n : Nat) -> Maybe (Fin n)
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integerToFin x Z = Nothing -- make sure 'n' is concrete, to save reduction!
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integerToFin x n = if x >= 0 then natToFin (fromInteger x) n else Nothing
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||| Allow overloading of Integer literals for Fin.
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||| @ x the Integer that the user typed
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||| @ prf an automatically-constructed proof that `x` is in bounds
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public export
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fromInteger : (x : Integer) -> {n : Nat} ->
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{auto prf : (IsJust (integerToFin x n))} ->
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Fin n
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fromInteger {n} x {prf} with (integerToFin x n)
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fromInteger {n} x {prf = ItIsJust} | Just y = y
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||| Convert an Integer to a Fin in the required bounds/
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||| This is essentially a composition of `mod` and `fromInteger`
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public export
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restrict : (n : Nat) -> Integer -> Fin (S n)
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restrict n val = let val' = assert_total (abs (mod val (cast (S n)))) in
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-- reasoning about primitives, so we need the
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-- 'believe_me'. It's fine because val' must be
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-- in the right range
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fromInteger {n = S n} val'
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{prf = believe_me {a=IsJust (Just val')} ItIsJust}
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--------------------------------------------------------------------------------
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-- DecEq
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--------------------------------------------------------------------------------
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export total
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FZNotFS : {f : Fin n} -> FZ {k = n} = FS f -> Void
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FZNotFS Refl impossible
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export
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implementation DecEq (Fin n) where
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decEq FZ FZ = Yes Refl
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decEq FZ (FS f) = No FZNotFS
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decEq (FS f) FZ = No $ negEqSym FZNotFS
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decEq (FS f) (FS f')
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= case decEq f f' of
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Yes p => Yes $ cong FS p
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No p => No $ \h => p $ FSinjective {f = f} {f' = f'} h
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