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https://github.com/idris-lang/Idris2.git
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100 lines
4.2 KiB
Idris
100 lines
4.2 KiB
Idris
module Data.Fin.Extra
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import Data.Fin
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import Data.Nat
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%default total
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||| Proof that an element **n** of Fin **m** , when converted to Nat is smaller than the bound **m**.
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export
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elemSmallerThanBound : (n : Fin m) -> LT (finToNat n) m
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elemSmallerThanBound FZ = LTESucc LTEZero
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elemSmallerThanBound (FS x) = LTESucc (elemSmallerThanBound x)
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||| Proof that application of finToNat the last element of Fin **S n** gives **n**.
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export
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finToNatLastIsBound : {n : Nat} -> finToNat (Fin.last {n}) = n
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finToNatLastIsBound {n=Z} = Refl
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finToNatLastIsBound {n=S k} = rewrite finToNatLastIsBound {n=k} in Refl
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||| Proof that an element **n** of Fin **m** , when converted to Nat is smaller than the bound **m**.
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export
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finToNatWeakenNeutral : {m : Nat} -> {n : Fin m} -> finToNat (weaken n) = finToNat n
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finToNatWeakenNeutral {n=FZ} = Refl
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finToNatWeakenNeutral {m=S (S _)} {n=FS _} = cong S finToNatWeakenNeutral
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-- ||| Proof that it's possible to strengthen a weakened element of Fin **m**.
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-- export
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-- strengthenWeakenNeutral : {m : Nat} -> (n : Fin m) -> strengthen (weaken n) = Right n
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-- strengthenWeakenNeutral {m=S _} FZ = Refl
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-- strengthenWeakenNeutral {m=S (S _)} (FS k) = rewrite strengthenWeakenNeutral k in Refl
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||| Proof that it's not possible to strengthen the last element of Fin **n**.
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export
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strengthenLastIsLeft : {n : Nat} -> strengthen (Fin.last {n}) = Left (Fin.last {n})
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strengthenLastIsLeft {n=Z} = Refl
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strengthenLastIsLeft {n=S k} = rewrite strengthenLastIsLeft {n=k} in Refl
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||| Enumerate elements of Fin **n** backwards.
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export
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invFin : {n : Nat} -> Fin n -> Fin n
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invFin FZ = last
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invFin {n=S (S _)} (FS k) = weaken (invFin k)
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||| Proof that an inverse of a weakened element of Fin **n** is a successive of an inverse of an original element.
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export
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invWeakenIsSucc : {n : Nat} -> (m : Fin n) -> invFin (weaken m) = FS (invFin m)
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invWeakenIsSucc FZ = Refl
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invWeakenIsSucc {n=S (S _)} (FS k) = rewrite invWeakenIsSucc k in Refl
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||| Proof that double inversion of Fin **n** gives the original.
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export
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doubleInvFinSame : {n : Nat} -> (m : Fin n) -> invFin (invFin m) = m
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doubleInvFinSame {n=S Z} FZ = Refl
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doubleInvFinSame {n=S (S k)} FZ = rewrite doubleInvFinSame {n=S k} FZ in Refl
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doubleInvFinSame {n=S (S _)} (FS x) = trans (invWeakenIsSucc $ invFin x) (cong FS $ doubleInvFinSame x)
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||| Proof that an inverse of the last element of Fin (S **n**) in FZ.
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export
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invLastIsFZ : {n : Nat} -> invFin (Fin.last {n}) = FZ
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invLastIsFZ {n=Z} = Refl
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invLastIsFZ {n=S k} = rewrite invLastIsFZ {n=k} in Refl
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-- ||| Proof that it's possible to strengthen an inverse of a succesive element of Fin **n**.
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-- export
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-- strengthenNotLastIsRight : (m : Fin (S n)) -> strengthen (invFin (FS m)) = Right (invFin m)
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-- strengthenNotLastIsRight m = strengthenWeakenNeutral (invFin m)
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--
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||| Either tightens the bound on a Fin or proves that it's the last.
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export
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strengthen' : {n : Nat} -> (m : Fin (S n)) -> Either (m = Fin.last) (m' : Fin n ** finToNat m = finToNat m')
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strengthen' {n = Z} FZ = Left Refl
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strengthen' {n = S k} FZ = Right (FZ ** Refl)
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strengthen' {n = S k} (FS m) = case strengthen' m of
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Left eq => Left $ cong FS eq
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Right (m' ** eq) => Right (FS m' ** cong S eq)
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||| A view of Nat as a quotient of some number and a finite remainder.
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public export
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data FractionView : (n : Nat) -> (d : Nat) -> Type where
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Fraction : (n : Nat) -> (d : Nat) -> {auto ok: GT d Z} ->
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(q : Nat) -> (r : Fin d) ->
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q * d + finToNat r = n -> FractionView n d
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||| Converts Nat to the fractional view with a non-zero divisor.
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export
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divMod : (n, d : Nat) -> {auto ok: GT d Z} -> FractionView n d
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divMod Z (S d) = Fraction Z (S d) Z FZ Refl
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divMod {ok=_} (S n) (S d) =
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let Fraction {ok=ok} n (S d) q r eq = divMod n (S d) in
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case strengthen' r of
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Left eq' => Fraction {ok=ok} (S n) (S d) (S q) FZ $
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rewrite sym eq in
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rewrite trans (cong finToNat eq') finToNatLastIsBound in
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cong S $ trans
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(plusZeroRightNeutral (d + q * S d))
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(plusCommutative d (q * S d))
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Right (r' ** eq') => Fraction {ok=ok} (S n) (S d) q (FS r') $
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rewrite sym $ plusSuccRightSucc (q * S d) (finToNat r') in
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cong S $ trans (sym $ cong (plus (q * S d)) eq') eq
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