mirror of
https://github.com/idris-lang/Idris2.git
synced 2024-12-24 04:09:10 +03:00
282 lines
8.2 KiB
Idris
282 lines
8.2 KiB
Idris
module Data.Fin
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import Data.List1
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import public Data.Maybe
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import Data.Nat
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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".
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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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||| Cast between Fins with equal indices
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public export
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cast : {n : Nat} -> (0 eq : m = n) -> Fin m -> Fin n
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cast {n = S _} eq FZ = FZ
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cast {n = Z} eq FZ impossible
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cast {n = S _} eq (FS k) = FS (cast (succInjective _ _ eq) k)
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cast {n = Z} eq (FS k) impossible
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export
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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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Uninhabited (FZ = FS k) where
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uninhabited Refl impossible
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export
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Uninhabited (FS k = FZ) where
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uninhabited Refl impossible
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export
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fsInjective : FS m = FS n -> m = n
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fsInjective Refl = Refl
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export
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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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||| 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 FZ FZ _ = Refl
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finToNatInjective (FS _) FZ Refl impossible
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finToNatInjective FZ (FS _) 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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export
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Cast (Fin n) Nat where
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cast = finToNat
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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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Cast (Fin n) Integer where
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cast = finToInteger
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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 : (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
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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 _) = 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 _} FZ = Right FZ
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strengthen {n = S _} (FS i) with (strengthen i)
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strengthen (FS _) | Left x = Left $ FS x
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strengthen (FS _) | 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 (S m) f = FS $ 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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||| All of the Fin elements
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public export
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allFins : (n : Nat) -> List1 (Fin (S n))
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allFins Z = FZ ::: []
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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
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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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public export
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natToFin : Nat -> (n : Nat) -> Maybe (Fin n)
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natToFin Z (S _) = 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 0 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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public export
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DecEq (Fin n) where
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decEq FZ FZ = Yes Refl
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decEq FZ (FS f) = No absurd
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decEq (FS f) FZ = No absurd
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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 $ p . fsInjective
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namespace Equality
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||| Pointwise equality of Fins
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||| It is sometimes complicated to prove equalities on type-changing
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||| operations on Fins.
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||| This inductive definition can be used to simplify proof. We can
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||| recover proofs of equalities by using `homoPointwiseIsEqual`.
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public export
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data Pointwise : Fin m -> Fin n -> Type where
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FZ : Pointwise FZ FZ
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FS : Pointwise k l -> Pointwise (FS k) (FS l)
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infix 6 ~~~
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||| Convenient infix notation for the notion of pointwise equality of Fins
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public export
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(~~~) : Fin m -> Fin n -> Type
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(~~~) = Pointwise
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||| Pointwise equality is reflexive
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export
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reflexive : {k : Fin m} -> k ~~~ k
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reflexive {k = FZ} = FZ
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reflexive {k = FS k} = FS reflexive
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||| Pointwise equality is symmetric
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export
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symmetric : k ~~~ l -> l ~~~ k
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symmetric FZ = FZ
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symmetric (FS prf) = FS (symmetric prf)
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||| Pointwise equality is transitive
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export
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transitive : j ~~~ k -> k ~~~ l -> j ~~~ l
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transitive FZ FZ = FZ
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transitive (FS prf) (FS prg) = FS (transitive prf prg)
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||| Pointwise equality is compatible with cast
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export
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castEq : {k : Fin m} -> (0 eq : m = n) -> cast eq k ~~~ k
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castEq {k = FZ} Refl = FZ
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castEq {k = FS k} Refl = FS (castEq _)
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||| The actual proof used by cast is irrelevant
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export
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congCast : {0 n, q : Nat} -> {k : Fin m} -> {l : Fin p} ->
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{0 eq1 : m = n} -> {0 eq2 : p = q} ->
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k ~~~ l ->
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cast eq1 k ~~~ cast eq2 l
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congCast eq = transitive (castEq _) (transitive eq $ symmetric $ castEq _)
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||| Last is congruent wrt index equality
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export
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congLast : {m : Nat} -> (0 _ : m = n) -> last {n=m} ~~~ last {n}
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congLast Refl = reflexive
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export
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congShift : (m : Nat) -> k ~~~ l -> shift m k ~~~ shift m l
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congShift Z prf = prf
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congShift (S m) prf = FS (congShift m prf)
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||| WeakenN is congruent wrt pointwise equality
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export
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congWeakenN : k ~~~ l -> weakenN n k ~~~ weakenN n l
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congWeakenN FZ = FZ
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congWeakenN (FS prf) = FS (congWeakenN prf)
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||| Pointwise equality is propositional equality on Fins that have the same type
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export
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homoPointwiseIsEqual : {0 k, l : Fin m} -> k ~~~ l -> k === l
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homoPointwiseIsEqual FZ = Refl
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homoPointwiseIsEqual (FS prf) = cong FS (homoPointwiseIsEqual prf)
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||| Pointwise equality is propositional equality modulo transport on Fins that
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||| have provably equal types
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export
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hetPointwiseIsTransport :
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{0 k : Fin m} -> {0 l : Fin n} ->
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(0 eq : m === n) -> k ~~~ l -> k === rewrite eq in l
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hetPointwiseIsTransport Refl = homoPointwiseIsEqual
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export
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finToNatQuotient : k ~~~ l -> finToNat k === finToNat l
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finToNatQuotient FZ = Refl
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finToNatQuotient (FS prf) = cong S (finToNatQuotient prf)
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export
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weakenNeutral : (k : Fin n) -> weaken k ~~~ k
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weakenNeutral FZ = FZ
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weakenNeutral (FS k) = FS (weakenNeutral k)
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export
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weakenNNeutral : (0 m : Nat) -> (k : Fin n) -> weakenN m k ~~~ k
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weakenNNeutral m FZ = FZ
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weakenNNeutral m (FS k) = FS (weakenNNeutral m k)
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