Conor McBride has posed an interesting problem:
implement constructors P v for embedding pure values v O for holes f :$ a for application, left-associative and an interpreting function emmental such that emmental (P (+) :$ (P (*) :$ O :$ P 10) :$ O) 4 2 = 42
I believe one can do better. There is no need for the :$ operation, and there is no need for of lifting non-functional constants. The following code demonstrates a more economical notation. The code essentially implements Hypothetical Reasoning, considering an expression with holes as an (intuitionistic) sequent. Each hole is one formula in the antecedent. The formulas in the antecedent are ordered, so our logic is actually the relevance, substructural intuitionistic logic. The function emmental takes the form of the Deduction Theorem (and its implementation is the proof of the theorem). The test takes the form
test_McBride = dt ((ih *! (10::Int)) +! ih) 4 2 where ih = holet (undefined::Int)
and gives the predictable answer. Another test, which shows that we can pile up holes (aka hypotheses) at will, and they all line up:
test7 = let exp = l1 (map succ) ((1::Int) !: ih !: ih !: ih !: [5]) ih = holet (undefined::Int) deduced = dt exp in deduced 11 12 13 -- [2,12,13,14,6]
test_order = dt( (ih `p` ih) `p` (ih `p` ih) ) 1 2 3 4 where ih = holet (undefined::Int) -- ((1,2),(3,4))
It is clear from our examples that our holes are monomorphically typed; the lifted functions don't have to be. We could have implemented things the other way around. We could have played with more complex inferences when applying polymorphic functions to polymorphic sequents. At present, we chose the simplest solution. Here it is. {-# OPTIONS -fglasgow-exts #-} {-# OPTIONS -fallow-undecidable-instances #-} {-# OPTIONS -fallow-overlapping-instances #-} -- the extensions are merely for the sake of IsHypothetical below -- Lightweight sequent calculus: computations on sequents module SEQ where -- Hypothetical value: one formula in the antecedent newtype H a = H a -- The primitive sequent: a |- a hole :: H a -> a hole (H x) = x -- The version of the above to make it easy to assign a specific type to it holet :: a -> H a -> a holet _ = hole -- This class proves, by construction, the Deduction Theorem -- x |- b ==> x -> b class DeductionTh seq r | seq -> r where dt :: seq -> r instance (IsHypothetical seq f, DeductionTh' f seq r) => DeductionTh seq r where dt = dt' (undefined::f) class DeductionTh' f seq r | f seq -> r where dt' :: f -> seq -> r instance DeductionTh y r => DeductionTh' HTrue (H x -> y) (x->r) where dt' _ hf = \x -> dt (hf (H x)) instance DeductionTh' HFalse a a where dt' _ = id -- Lifting an a->b function to sequents class Lift1 a b arg res | a b arg -> res where l1 :: (a->b) -> (arg->res) instance (IsHypothetical arg f, Lift1' f a b arg res) => Lift1 a b arg res where l1 = l1' (undefined::f) class Lift1' f a b arg res | f a b arg -> res where l1' :: f -> (a->b) -> (arg->res) instance Lift1 a b y b' => Lift1' HTrue a b (H x -> y) (H x -> b') where l1' _ fn arg = \hx -> l1 fn (arg hx) -- here we assert a local functional dependency instance TypeCast arg a => Lift1' HFalse a b arg b where l1' _ fn arg = fn (typeCast arg) test1 = dt True test2 :: Int -> Int test2 = dt hole test2' = dt (holet (1::Int)) test3 = dt (l1 not True) test4 = dt (l1 not (holet False)) -- expected type error -- test4' = dt (l1 not (holet (1::Int))) test4'' = test4 False -- Lifting an a->b->c function to two sequents -- we should keep the order of formulas in the combined antecedent -- The following is just a funnily written `append' of two sequences class Lift2 a b c arg1 arg2 res | a b c arg1 arg2 -> res where l2 :: (a->b->c) -> (arg1->arg2->res) instance (IsHypothetical arg1 f1, IsHypothetical arg2 f2, Lift2' f1 f2 a b c arg1 arg2 res) => Lift2 a b c arg1 arg2 res where l2 = l2' (undefined::f1) (undefined::f2) class Lift2' f1 f2 a b c arg1 arg2 res | f1 f2 a b c arg1 arg2 -> res where l2' :: f1 -> f2 -> (a->b->c) -> (arg1->arg2->res) -- First, drain the antecedent formulas from arg1 instance Lift2 a b c y1 arg2 r => Lift2' HTrue f2 a b c (H x1 -> y1) arg2 (H x1 -> r) where l2' _ _ fn arg1 arg2 = \hx1 -> l2 fn (arg1 hx1) arg2 -- now, add the antecedent formulas from arg2, if any instance Lift2 a b c x1 y2 r => Lift2' HFalse HTrue a b c x1 (H x2 -> y2) (H x2 -> r) where l2' _ _ fn arg1 arg2 = \hx2 -> l2 fn arg1 (arg2 hx2) instance (TypeCast x1 a, TypeCast x2 b) => Lift2' HFalse HFalse a b c x1 x2 c where l2' _ _ fn arg1 arg2 = fn (typeCast arg1) (typeCast arg2) -- Time for tests infixl 6 +! infixl 7 *! infixr 5 !: x +! y = l2 (+) x y x *! y = l2 (*) x y x `p` y = l2 (,) x y x !: y = l2 (:) x y test5 = dt ((1::Int) +! (2::Int)) test51 = dt (holet (1::Int) +! (2::Int)) test51' = test51 10 -- 12 test52 = dt (holet (undefined::Int) `p` holet True) test52' = test52 5 False -- (5,False) test_McBride = dt ((ih *! (10::Int)) +! ih) 4 2 where ih = holet (undefined::Int) test6 = dt( l1 sum (ih !: ih !: ih !: [5]) ) 1 2 3 where ih = holet (undefined::Int) -- 11 test7 = let exp = l1 (map succ) ((1::Int) !: ih !: ih !: ih !: [5]) ih = holet (undefined::Int) deduced = dt exp in deduced 11 12 13 -- [2,12,13,14,6] -- Testing the proper ordering of formulas in the antecedent test_order = dt( (ih `p` ih) `p` (ih `p` ih) ) 1 2 3 4 where ih = holet (undefined::Int) -- ((1,2),(3,4)) -- see http://okmij.org/ftp/Haskell/typecast.html#deep-join data HTrue data HFalse class IsHypothetical a b | a -> b instance TypeCast f HTrue => IsHypothetical (H x->y) f instance TypeCast f HFalse => IsHypothetical a f class TypeCast a b | a -> b, b->a where typeCast :: a -> b class TypeCast' t a b | t a -> b, t b -> a where typeCast' :: t->a->b class TypeCast'' t a b | t a -> b, t b -> a where typeCast'' :: t->a->b instance TypeCast' () a b => TypeCast a b where typeCast x = typeCast' () x instance TypeCast'' t a b => TypeCast' t a b where typeCast' = typeCast'' instance TypeCast'' () a a where typeCast'' _ x = x