169 lines
4.8 KiB
Idris
169 lines
4.8 KiB
Idris
module Quox.NatExtra
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import public Data.Nat
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import public Data.Nat.Properties
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import public Data.Nat.Division
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import Data.DPair
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import Data.SnocList
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import Data.Vect
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%default total
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public export
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data LTE' : Nat -> Nat -> Type where
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LTERefl : LTE' n n
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LTESuccR : LTE' m n -> LTE' m (S n)
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%builtin Natural LTE'
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public export %hint
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lteZero' : {n : Nat} -> LTE' 0 n
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lteZero' {n = 0} = LTERefl
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lteZero' {n = S n} = LTESuccR lteZero'
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public export %hint
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lteSucc' : LTE' m n -> LTE' (S m) (S n)
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lteSucc' LTERefl = LTERefl
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lteSucc' (LTESuccR p) = LTESuccR $ lteSucc' p
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public export
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fromLte : {n : Nat} -> LTE m n -> LTE' m n
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fromLte LTEZero = lteZero'
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fromLte (LTESucc p) = lteSucc' $ fromLte p
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public export
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toLte : {n : Nat} -> m `LTE'` n -> m `LTE` n
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toLte LTERefl = reflexive
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toLte (LTESuccR p) = lteSuccRight (toLte p)
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private
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0 baseNZ : n `GTE` 2 => NonZero n
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baseNZ @{LTESucc _} = SIsNonZero
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parameters {base : Nat} {auto 0 _ : base `GTE` 2} (chars : Vect base Char)
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private
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showAtBase' : List Char -> Nat -> List Char
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showAtBase' acc 0 = acc
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showAtBase' acc k =
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let dig = natToFinLT (modNatNZ k base baseNZ) @{boundModNatNZ {}} in
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showAtBase' (index dig chars :: acc)
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(assert_smaller k $ divNatNZ k base baseNZ)
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export
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showAtBase : Nat -> String
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showAtBase = pack . showAtBase' []
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export
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showHex : Nat -> String
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showHex = showAtBase $ fromList $ unpack "0123456789ABCDEF"
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n2i = natToInteger
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i2n = fromInteger {ty = Nat}
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private partial %inline
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divNatViaInteger : (m, n : Nat) -> Nat
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divNatViaInteger m n = i2n $ n2i m `div` n2i n
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%transform "divNat" divNat = divNatViaInteger
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private %inline
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divNatViaIntegerNZ : (m, n : Nat) -> (0 _ : NonZero n) -> Nat
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divNatViaIntegerNZ m n _ = assert_total divNatViaInteger m n
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%transform "divNatNZ" divNatNZ = divNatViaIntegerNZ
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private partial %inline
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modNatViaInteger : (m, n : Nat) -> Nat
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modNatViaInteger m n = i2n $ n2i m `mod` n2i n
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%transform "modNat" modNat = modNatViaInteger
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private %inline
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modNatViaIntegerNZ : (m, n : Nat) -> (0 _ : NonZero n) -> Nat
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modNatViaIntegerNZ m n _ = assert_total modNatViaInteger m n
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%transform "modNatNZ" modNatNZ = modNatViaIntegerNZ
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public export
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data ViewLsb : Nat -> Type where
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Lsb0 : (n : Nat) -> (0 eq : n' = 2 * n) -> ViewLsb n'
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Lsb1 : (n : Nat) -> (0 eq : n' = S (2 * n)) -> ViewLsb n'
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%name ViewLsb p, q
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public export data IsLsb0 : ViewLsb n -> Type where ItIsLsb0 : IsLsb0 (Lsb0 n eq)
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public export data IsLsb1 : ViewLsb n -> Type where ItIsLsb1 : IsLsb1 (Lsb1 n eq)
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export
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0 lsbMutex : n = (2 * a) -> n = S (2 * b) -> Void
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lsbMutex ev od {n = 0} impossible
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lsbMutex ev od {n = 1} {a = S a} {b = 0} =
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let ev = injective ev in
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let s = sym $ plusSuccRightSucc a (a + 0) in
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absurd $ trans ev s
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lsbMutex ev od {n = S (S n)} {a = S a} {b = S b} =
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let ev = injective $
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trans (injective ev) (sym $ plusSuccRightSucc a (a + 0)) in
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let od = trans (injective $ injective od)
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(sym $ plusSuccRightSucc b (b + 0)) in
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lsbMutex ev od
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public export %hint
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doubleIsLsb0 : (p : ViewLsb (2 * n)) -> IsLsb0 p
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doubleIsLsb0 (Lsb0 k eq) = ItIsLsb0
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doubleIsLsb0 (Lsb1 k eq) = void $ absurd $ lsbMutex Refl eq
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public export %hint
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sDoubleIsLsb1 : (p : ViewLsb (S (2 * n))) -> IsLsb1 p
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sDoubleIsLsb1 (Lsb1 k eq) = ItIsLsb1
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sDoubleIsLsb1 (Lsb0 k eq) = void $ absurd $ lsbMutex Refl (sym eq)
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export
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fromLsb0 : (p : ViewLsb n) -> (0 _ : IsLsb0 p) =>
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Subset Nat (\n' => n = 2 * n')
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fromLsb0 (Lsb0 n' eq) @{ItIsLsb0} = Element n' eq
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export
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fromLsb1 : (p : ViewLsb n) -> (0 _ : IsLsb1 p) =>
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Subset Nat (\n' => n = S (2 * n'))
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fromLsb1 (Lsb1 n' eq) @{ItIsLsb1} = Element n' eq
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private
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viewLsb' : (m, d : Nat) -> (0 _ : m `LT` 2) -> ViewLsb (m + 2 * d)
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viewLsb' 0 d p = Lsb0 d Refl
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viewLsb' 1 d p = Lsb1 d Refl
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viewLsb' (S (S _)) _ (LTESucc p) = void $ absurd p
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export
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viewLsb : (n : Nat) -> ViewLsb n
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viewLsb n =
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let 0 nz : NonZero 2 = %search in
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rewrite DivisionTheorem n 2 nz nz in
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rewrite multCommutative (divNatNZ n 2 nz) 2 in
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viewLsb' (modNatNZ n 2 nz) (divNatNZ n 2 nz) (boundModNatNZ n 2 nz)
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export
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0 doubleInj : {m, n : Nat} -> 2 * m = 2 * n -> m = n
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doubleInj eq =
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multRightCancel m n 2 %search $
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trans (multCommutative m 2) $ trans eq (multCommutative 2 n)
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export
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0 sDoubleInj : {m, n : Nat} -> S (2 * m) = S (2 * n) -> m = n
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sDoubleInj eq = doubleInj $ injective eq
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export
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0 lsbOdd : (n : Nat) -> (eq ** viewLsb (S (2 * n)) = Lsb1 n eq)
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lsbOdd n with (viewLsb (S (2 * n)))
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_ | Lsb0 _ eq = void $ absurd $ lsbMutex Refl (sym eq)
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_ | Lsb1 n' eq with (sDoubleInj eq)
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lsbOdd n | (Lsb1 n eq) | Refl = (eq ** Refl)
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export
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0 lsbEven : (n : Nat) -> (eq ** viewLsb (2 * n) = Lsb0 n eq)
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lsbEven n with (viewLsb (2 * n))
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_ | Lsb1 _ eq = void $ absurd $ lsbMutex Refl eq
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_ | Lsb0 n' eq with (doubleInj eq)
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lsbEven n | Lsb0 n eq | Refl = (eq ** Refl)
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