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Add and use liftC and liftC2 #2

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19 changes: 13 additions & 6 deletions RankNTypes/Church.hs
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Original file line number Diff line number Diff line change
Expand Up @@ -59,6 +59,14 @@ unchurch_bool = (\a -> \b -> \c -> c a b) True False
type ChurchNum = forall a. (a -> a) -> a -> a
newtype Church = Church { unChurch :: ChurchNum }

-- Convenience functions for the Church type wrapper,
-- analogous to liftM2 for monads.
liftC :: (ChurchNum -> ChurchNum) -> Church -> Church
liftC f (Church x) = Church (f x)

liftC2 :: (ChurchNum -> ChurchNum -> ChurchNum) -> Church -> Church -> Church
liftC2 f (Church x) (Church y) = Church (f x y)

-- Church Numeral: 0
-- λf.λx.x
zero :: Church
Expand Down Expand Up @@ -129,18 +137,17 @@ ynr f = (\h -> h $ Mu h) (\x -> f . (\(Mu g) -> g) x $ x)
-- λn.λf.λx.f (n f x)
succ :: Church -> Church

succ = \n -> Church $ \f -> \x -> f (unChurch n f x)
succ = liftC $ \n -> \f -> \x -> f (n f x)

-- Church Predecessor
-- λn.λf.λx.n (λg.λh.h (g f)) (λu.x) (λu.u)
pred :: Church -> Church
pred = \n -> Church $
\f -> \x -> unChurch n (\g -> \h -> h (g f)) (\u -> x) (\u -> u)
pred = liftC $ \n -> \f -> \x -> n (\g -> \h -> h (g f)) (\u -> x) (\u -> u)

-- Church Addition
-- λm.λn.λf.λx.m f (n f x)
add :: Church -> Church -> Church
add = \m -> \n -> Church $ \f -> \x -> unChurch m f (unChurch n f x)
add = liftC2 $ \m -> \n -> \f -> \x -> m f (n f x)

-- Church Subtraction
-- λm.λn. n pred m
Expand All @@ -150,7 +157,7 @@ sub = \m -> \n -> unChurch n pred m
-- Church Multiplication
-- λm.λn.λf.m (n f)
mult :: Church -> Church -> Church
mult = \m -> \n -> Church $ \f -> unChurch m (unChurch n f)
mult = liftC2 $ \m -> \n -> \f -> m (n f)

-- Church Division (gets the floor if divides to a fraction)
-- λd n m.ifelse (geq n m) (succ (d (sub n m) m)) zero
Expand All @@ -160,7 +167,7 @@ div = y (\d n m -> ifelse (geq n m) (succ (d (sub n m) m)) zero)
-- Church Exponentiation
-- λm.λn.n m
exp :: Church -> Church -> Church
exp = \m -> \n -> Church $ (unChurch n) (unChurch m)
exp = liftC2 $ \m -> \n -> n m

-- Church Factorial
-- λf n.ifelse (is_zero n) one (mult n (fac (pred n)))
Expand Down