{-# language RankNTypes #-}
{-# language PatternGuards #-}
{-# language ViewPatterns #-}
{-# language PatternSynonyms #-}
--------------------------------------------------------------------------------
-- |
-- Module      :  Numeric.Natural.Lens
-- Copyright   :  (C) 2017 Edward Kmett
-- License     :  BSD-style (see the file LICENSE)
-- Maintainer  :  Edward Kmett <ekmett@gmail.com>
-- Stability   :  provisional
-- Portability :  portable
--
-- Useful tools for Gödel numbering.
-------------------------------------------------------------------------------
module Numeric.Natural.Lens
  ( _Pair
  , _Sum
  , _Naturals
  , pattern Pair
  , pattern Sum
  , pattern Naturals
  ) where

import Control.Lens
import Numeric.Natural

-- | The natural numbers are isomorphic to the product of the natural numbers with itself.
--
-- @N = N*N@
_Pair :: Iso' Natural (Natural, Natural)
_Pair :: Iso' Natural (Natural, Natural)
_Pair = forall s a b t. (s -> a) -> (b -> t) -> Iso s t a b
iso forall {b}. Integral b => b -> (b, b)
hither (forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry forall {t}. Integral t => t -> t -> t
yon) where
  yon :: t -> t -> t
yon t
0 t
0 = t
0
  yon t
m t
n = case forall a. Integral a => a -> a -> (a, a)
quotRem t
m t
2 of
    (t
q,t
r) -> t
r forall a. Num a => a -> a -> a
+ t
2 forall a. Num a => a -> a -> a
* t -> t -> t
yon t
n t
q -- rotation

  hither :: b -> (b, b)
hither b
0 = (b
0,b
0)
  hither b
n = case forall a. Integral a => a -> a -> (a, a)
quotRem b
n b
2 of
   (b
p,b
r) -> case b -> (b, b)
hither b
p of
     (b
x,b
y) -> (b
rforall a. Num a => a -> a -> a
+b
2forall a. Num a => a -> a -> a
*b
y,b
x) -- rotation

-- | The natural numbers are isomorphic to disjoint sums of natural numbers embedded as
-- evens or odds.
--
-- @N = 2*N@
_Sum :: Iso' Natural (Either Natural Natural)
_Sum :: Iso' Natural (Either Natural Natural)
_Sum = forall s a b t. (s -> a) -> (b -> t) -> Iso s t a b
iso forall {b}. Integral b => b -> Either b b
hither forall {a}. Num a => Either a a -> a
yon where
  hither :: b -> Either b b
hither b
p = case forall a. Integral a => a -> a -> (a, a)
quotRem b
p b
2 of
    (b
q,b
0) -> forall a b. a -> Either a b
Left b
q
    (b
q,b
1) -> forall a b. b -> Either a b
Right b
q
    (b, b)
_     -> forall a. HasCallStack => [Char] -> a
error [Char]
"_Sum: impossible"
  yon :: Either a a -> a
yon (Left a
q)  = a
2forall a. Num a => a -> a -> a
*a
q
  yon (Right a
q) = a
2forall a. Num a => a -> a -> a
*a
qforall a. Num a => a -> a -> a
+a
1

-- | The natural numbers are isomorphic to lists of natural numbers
_Naturals :: Iso' Natural [Natural]
_Naturals :: Iso' Natural [Natural]
_Naturals = forall s a b t. (s -> a) -> (b -> t) -> Iso s t a b
iso Natural -> [Natural]
hither [Natural] -> Natural
yon where
  hither :: Natural -> [Natural]
hither Natural
0 = []
  hither Natural
n | (Natural
h, Natural
t) <- (Natural
nforall a. Num a => a -> a -> a
-Natural
1)forall s a. s -> Getting a s a -> a
^.Iso' Natural (Natural, Natural)
_Pair = Natural
h forall a. a -> [a] -> [a]
: Natural -> [Natural]
hither Natural
t
  yon :: [Natural] -> Natural
yon [] = Natural
0
  yon (Natural
x:[Natural]
xs) = Natural
1 forall a. Num a => a -> a -> a
+ forall b (m :: * -> *) t. MonadReader b m => AReview t b -> m t
review Iso' Natural (Natural, Natural)
_Pair (Natural
x, [Natural] -> Natural
yon [Natural]
xs)

-- |
-- interleaves the bits of two natural numbers
pattern Pair :: Natural -> Natural -> Natural
pattern $bPair :: Natural -> Natural -> Natural
$mPair :: forall {r}.
Natural -> (Natural -> Natural -> r) -> ((# #) -> r) -> r
Pair x y <- (view _Pair -> (x,y)) where
  Pair Natural
x Natural
y = forall b (m :: * -> *) t. MonadReader b m => AReview t b -> m t
review Iso' Natural (Natural, Natural)
_Pair (Natural
x,Natural
y)

-- |
-- @
-- Sum (Left q) = 2*q
-- Sum (Right q) = 2*q+1
-- @
pattern Sum :: Either Natural Natural -> Natural
pattern $bSum :: Either Natural Natural -> Natural
$mSum :: forall {r}.
Natural -> (Either Natural Natural -> r) -> ((# #) -> r) -> r
Sum s <- (view _Sum -> s) where
  Sum Either Natural Natural
s = forall b (m :: * -> *) t. MonadReader b m => AReview t b -> m t
review Iso' Natural (Either Natural Natural)
_Sum Either Natural Natural
s

-- |
-- @
-- Naturals [] = 0
-- Naturals (h:t) = 1 + Pair h (Naturals t)
-- @
pattern Naturals :: [Natural] -> Natural
pattern $bNaturals :: [Natural] -> Natural
$mNaturals :: forall {r}. Natural -> ([Natural] -> r) -> ((# #) -> r) -> r
Naturals xs <- (view _Naturals -> xs) where
  Naturals [Natural]
xs = forall b (m :: * -> *) t. MonadReader b m => AReview t b -> m t
review Iso' Natural [Natural]
_Naturals [Natural]
xs