Idris2/libs/base/Data/Fin.idr

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module Data.Fin
import Control.Function
import public Control.Ord
import Data.List1
import public Data.Maybe
import public Data.Nat
import public Data.So
import Decidable.Equality.Core
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%default total
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||| 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
public export
data Fin : (n : Nat) -> Type where
FZ : Fin (S k)
FS : Fin k -> Fin (S k)
||| Coerce between Fins with equal indices
public export
coerce : {n : Nat} -> (0 eq : m = n) -> Fin m -> Fin n -- TODO: make linear
coerce {n = S _} eq FZ = FZ
coerce {n = Z} eq FZ impossible
coerce {n = S _} eq (FS k) = FS (coerce (injective eq) k)
coerce {n = Z} eq (FS k) impossible
%transform "coerce-identity" coerce eq k = replace {p = Fin} eq k
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export
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Uninhabited (Fin Z) where
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uninhabited FZ impossible
uninhabited (FS f) impossible
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export
Uninhabited (FZ = FS k) where
uninhabited Refl impossible
export
Uninhabited (FS k = FZ) where
uninhabited Refl impossible
export
Uninhabited (n = m) => Uninhabited (FS n = FS m) where
uninhabited Refl @{nm} = uninhabited Refl @{nm}
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export
Injective FS where
injective Refl = Refl
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export
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Eq (Fin n) where
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(==) FZ FZ = True
(==) (FS k) (FS k') = k == k'
(==) _ _ = False
||| Convert a Fin to a Nat
public export
finToNat : Fin n -> Nat
finToNat FZ = Z
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finToNat (FS k) = S $ finToNat k
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finToNatInjective : (fm : Fin k) -> (fn : Fin k) -> (finToNat fm) = (finToNat fn) -> fm = fn
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finToNatInjective FZ FZ _ = Refl
finToNatInjective (FS _) FZ Refl impossible
finToNatInjective FZ (FS _) Refl impossible
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finToNatInjective (FS m) (FS n) prf =
cong FS $ finToNatInjective m n $ injective prf
||| `finToNat` is injective
export
Injective Fin.finToNat where
injective = (finToNatInjective _ _)
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export
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Cast (Fin n) Nat where
cast = finToNat
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||| Convert a Fin to an Integer
public export
finToInteger : Fin n -> Integer
finToInteger FZ = 0
finToInteger (FS k) = 1 + finToInteger k
-- %builtin NaturalToInteger Data.Fin.finToInteger
export
Show (Fin n) where
show = show . finToInteger
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export
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Cast (Fin n) Integer where
cast = finToInteger
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||| Weaken the bound on a Fin by 1
public export
weaken : Fin n -> Fin (S n)
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
public export
weakenN : (0 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
public export
weakenLTE : Fin n -> LTE n m -> Fin m
weakenLTE FZ LTEZero impossible
weakenLTE (FS _) LTEZero impossible
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weakenLTE FZ (LTESucc _) = 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.
||| Return the tightened bound if there is one, else nothing.
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export
strengthen : {n : _} -> Fin (S n) -> Maybe (Fin n)
strengthen {n = S _} FZ = Just FZ
strengthen {n = S _} (FS p) with (strengthen p)
strengthen (FS _) | Nothing = Nothing
strengthen (FS _) | Just q = Just $ FS q
strengthen _ = Nothing
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||| Add some natural number to a Fin, extending the bound accordingly
||| @ n the previous bound
||| @ m the number to increase the Fin by
public export
shift : (m : Nat) -> Fin n -> Fin (m + n)
shift Z f = f
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shift (S m) f = FS $ shift m f
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||| Increment a Fin, wrapping on overflow
public export
finS : {n : Nat} -> Fin n -> Fin n
finS {n = S _} x = case strengthen x of
Nothing => FZ
Just y => FS y
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||| The largest element of some Fin type
public export
last : {n : _} -> Fin (S n)
last {n=Z} = FZ
last {n=S _} = FS last
||| The finite complement of some Fin.
||| The number as far along as the input, but starting from the other end.
public export
complement : {n : Nat} -> Fin n -> Fin n
complement {n = S _} FZ = last
complement {n = S _} (FS x) = weaken $ complement x
||| All of the Fin elements
public export
allFins : (n : Nat) -> List1 (Fin (S n))
allFins Z = FZ ::: []
allFins (S n) = FZ ::: map FS (forget (allFins n))
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export
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Ord (Fin n) where
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compare FZ FZ = EQ
compare FZ (FS _) = LT
compare (FS _) FZ = GT
compare (FS x) (FS y) = compare x y
namespace Monoid
public export
[Minimum] {n : Nat} -> Monoid (Fin $ S n) using Semigroup.Minimum where
neutral = last
public export
[Maximum] Monoid (Fin $ S n) using Semigroup.Maximum where
neutral = FZ
public export
natToFinLT : (x : Nat) -> {0 n : Nat} ->
{auto 0 prf : x `LT` n} ->
Fin n
natToFinLT Z {prf = LTESucc _} = FZ
natToFinLT (S k) {prf = LTESucc _} = FS $ natToFinLT k
public export
natToFinLt : (x : Nat) -> {n : Nat} ->
{auto 0 prf : So (x < n)} ->
Fin n
natToFinLt Z {n = S _} = FZ
natToFinLt (S k) {n = S _} = FS $ natToFinLt k
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public export
natToFin : Nat -> (n : Nat) -> Maybe (Fin n)
natToFin x n = case isLT x n of
Yes prf => Just $ natToFinLT x
No contra => Nothing
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||| Convert an `Integer` to a `Fin`, provided the integer is within bounds.
||| @n The upper bound of the Fin
public export
integerToFin : Integer -> (n : Nat) -> Maybe (Fin n)
integerToFin x Z = Nothing -- make sure 'n' is concrete, to save reduction!
integerToFin x n = if x >= 0 then natToFin (fromInteger x) n else Nothing
public export
maybeLTE : (x : Nat) -> (y : Nat) -> Maybe (x `LTE` y)
maybeLTE Z _ = Just LTEZero
maybeLTE (S x) (S y) = LTESucc <$> maybeLTE x y
maybeLTE _ _ = Nothing
public export
maybeLT : (x : Nat) -> (y : Nat) -> Maybe (x `LT` y)
maybeLT x y = maybeLTE (S x) y
public export
finFromInteger : (x : Integer) -> {n : Nat} ->
{auto 0 prf : So (fromInteger x < n)} ->
Fin n
finFromInteger x = natToFinLt (fromInteger x)
-- Direct comparison between `Integer` and `Nat`.
-- We cannot convert the `Nat` to `Integer`, because that will not work with open terms;
-- but we cannot convert the `Integer` to `Nat` either, because that slows down typechecking
-- even when the function is unused (issue #2032)
public export
integerLessThanNat : Integer -> Nat -> Bool
integerLessThanNat x n with (x < the Integer 0)
integerLessThanNat _ _ | True = True -- if `x < 0` then `x < n` for any `n : Nat`
integerLessThanNat x (S m) | False = integerLessThanNat (x-1) m -- recursive case
integerLessThanNat x Z | False = False -- `x >= 0` contradicts `x < Z`
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||| Allow overloading of Integer literals for Fin.
||| @ x the Integer that the user typed
||| @ prf an automatically-constructed proof that `x` is in bounds
public export
fromInteger : (x : Integer) -> {n : Nat} ->
{auto 0 prf : So (integerLessThanNat x n)} ->
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Fin n
fromInteger x = finFromInteger x {prf = lemma prf} where
-- to be minimally invasive, we just call the previous implementation.
-- however, having a different proof obligation resolves #2032
0 lemma : {x : Integer} -> {n : Nat} -> So (integerLessThanNat x n) -> So (fromInteger {ty=Nat} x < n)
lemma oh = believe_me oh
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-- %builtin IntegerToNatural Data.Fin.fromInteger
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||| Convert an Integer to a Fin in the required bounds/
||| This is essentially a composition of `mod` and `fromInteger`
public export
restrict : (n : Nat) -> Integer -> Fin (S n)
restrict n val = let val' = assert_total (abs (mod val (cast (S n)))) in
-- reasoning about primitives, so we need the
-- 'believe_me'. It's fine because val' must be
-- in the right range
fromInteger {n = S n} val'
{prf = believe_me {a=IsJust (Just val')} ItIsJust}
--------------------------------------------------------------------------------
-- Num
--------------------------------------------------------------------------------
public export
{n : Nat} -> Num (Fin (S n)) where
FZ + y = y
(+) {n = S _} (FS x) y = finS $ assert_smaller x (weaken x) + y
FZ * y = FZ
(FS x) * y = y + (assert_smaller x $ weaken x) * y
fromInteger = restrict n
public export
{n : Nat} -> Neg (Fin (S n)) where
negate = finS . complement
x - y = x + (negate y)
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--------------------------------------------------------------------------------
-- DecEq
--------------------------------------------------------------------------------
public export
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DecEq (Fin n) where
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decEq FZ FZ = Yes Refl
decEq (FS f) (FS f') = decEqCong $ decEq f f'
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decEq FZ (FS f) = No absurd
decEq (FS f) FZ = No absurd
namespace Equality
||| Pointwise equality of Fins
||| It is sometimes complicated to prove equalities on type-changing
||| operations on Fins.
||| This inductive definition can be used to simplify proof. We can
||| recover proofs of equalities by using `homoPointwiseIsEqual`.
public export
data Pointwise : Fin m -> Fin n -> Type where
FZ : Pointwise FZ FZ
FS : Pointwise k l -> Pointwise (FS k) (FS l)
infix 6 ~~~
||| Convenient infix notation for the notion of pointwise equality of Fins
public export
(~~~) : Fin m -> Fin n -> Type
(~~~) = Pointwise
||| Pointwise equality is reflexive
export
reflexive : {k : Fin m} -> k ~~~ k
reflexive {k = FZ} = FZ
reflexive {k = FS k} = FS reflexive
||| Pointwise equality is symmetric
export
symmetric : k ~~~ l -> l ~~~ k
symmetric FZ = FZ
symmetric (FS prf) = FS (symmetric prf)
||| Pointwise equality is transitive
export
transitive : j ~~~ k -> k ~~~ l -> j ~~~ l
transitive FZ FZ = FZ
transitive (FS prf) (FS prg) = FS (transitive prf prg)
||| Pointwise equality is compatible with coerce
export
coerceEq : {k : Fin m} -> (0 eq : m = n) -> coerce eq k ~~~ k
coerceEq {k = FZ} Refl = FZ
coerceEq {k = FS k} Refl = FS (coerceEq _)
||| The actual proof used by coerce is irrelevant
export
congCoerce : {0 n, q : Nat} -> {k : Fin m} -> {l : Fin p} ->
{0 eq1 : m = n} -> {0 eq2 : p = q} ->
k ~~~ l ->
coerce eq1 k ~~~ coerce eq2 l
congCoerce eq
= transitive (coerceEq _)
$ transitive eq $ symmetric $ coerceEq _
||| Last is congruent wrt index equality
export
congLast : {m : Nat} -> (0 _ : m = n) -> last {n=m} ~~~ last {n}
congLast Refl = reflexive
export
congShift : (m : Nat) -> k ~~~ l -> shift m k ~~~ shift m l
congShift Z prf = prf
congShift (S m) prf = FS (congShift m prf)
||| WeakenN is congruent wrt pointwise equality
export
congWeakenN : k ~~~ l -> weakenN n k ~~~ weakenN n l
congWeakenN FZ = FZ
congWeakenN (FS prf) = FS (congWeakenN prf)
||| Pointwise equality is propositional equality on Fins that have the same type
export
homoPointwiseIsEqual : {0 k, l : Fin m} -> k ~~~ l -> k === l
homoPointwiseIsEqual FZ = Refl
homoPointwiseIsEqual (FS prf) = cong FS (homoPointwiseIsEqual prf)
||| Pointwise equality is propositional equality modulo transport on Fins that
||| have provably equal types
export
hetPointwiseIsTransport :
{0 k : Fin m} -> {0 l : Fin n} ->
(0 eq : m === n) -> k ~~~ l -> k === rewrite eq in l
hetPointwiseIsTransport Refl = homoPointwiseIsEqual
export
finToNatQuotient : k ~~~ l -> finToNat k === finToNat l
finToNatQuotient FZ = Refl
finToNatQuotient (FS prf) = cong S (finToNatQuotient prf)
||| Propositional equality on `finToNat`s implies pointwise equality on the `Fin`s themselves
export
finToNatEqualityAsPointwise : (k : Fin m) ->
(l : Fin n) ->
finToNat k = finToNat l ->
k ~~~ l
finToNatEqualityAsPointwise FZ FZ _ = FZ
finToNatEqualityAsPointwise FZ (FS _) prf = absurd prf
finToNatEqualityAsPointwise (FS _) FZ prf = absurd prf
finToNatEqualityAsPointwise (FS k) (FS l) prf = FS $ finToNatEqualityAsPointwise k l (injective prf)
export
weakenNeutral : (k : Fin n) -> weaken k ~~~ k
weakenNeutral FZ = FZ
weakenNeutral (FS k) = FS (weakenNeutral k)
export
weakenNNeutral : (0 m : Nat) -> (k : Fin n) -> weakenN m k ~~~ k
weakenNNeutral m FZ = FZ
weakenNNeutral m (FS k) = FS (weakenNNeutral m k)