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data Mu : (pattern : Type -> Type) -> Type where | |
In : {f: Type -> Type} -> f (Mu f) -> Mu f | |
data TwoOps x a = Val x | Addunit | Mulunit | Add a a | Mul a a | |
Functor (TwoOps x) where | |
map f (Val y) = Val y | |
map f (Addunit) = Addunit | |
map f (Mulunit) = Mulunit | |
map f (Add a1 a2) = Add (f a1) (f a2) | |
map f (Mul a1 a2) = Mul (f a1) (f a2) | |
Freesemiring : Type -> Type | |
Freesemiring x = Mu (TwoOps x) | |
Algebra : (Type -> Type) -> Type -> Type | |
Algebra f a = f a -> a | |
--universal property of the initial algebra | |
cata : Functor f => Algebra f a -> Mu f -> a | |
cata alg (In op) = alg (map (cata alg) op) | |
nats : Algebra (TwoOps Nat) Nat | |
nats (Val x) = x | |
nats (Addunit) = 0 | |
nats (Mulunit) = 1 | |
nats ((Add x y)) = x + y | |
nats ((Mul x y)) = x * y | |
cata' : Algebra (TwoOps a) a -> Algebra Freesemiring a | |
cata' = cata | |
-- use the universal property to freely extend nats | |
freenats = cata' nats | |
--min plus semiring | |
trop : Algebra (TwoOps (Maybe Double)) (Maybe Double) | |
trop (Val x) = x | |
trop Addunit = Nothing | |
trop Mulunit = Just 0 | |
trop (Add Nothing y) = y | |
trop (Add x Nothing) = x | |
trop (Add (Just x) (Just y)) = Just (min x y) | |
trop (Mul Nothing y) = Nothing | |
trop (Mul x Nothing) = Nothing | |
trop (Mul (Just x) (Just y)) = Just (x + y) | |
-- the free extension | |
freetrop = cata' trop | |
data Fin : Nat -> Type where | |
Zero : Fin (S n) | |
Suc : (i : Fin n) -> Fin (S n) | |
-- the 2-rig of types | |
Tworig : Algebra (TwoOps Type) Type | |
Tworig (Val a) = a | |
Tworig Addunit = Void | |
Tworig Mulunit = Fin 1 | |
Tworig (Add a b) = Either a b | |
Tworig (Mul a b) = (a,b) | |
--and its free extension | |
freetypes= cata' Tworig | |
--example | |
term : Freesemiring Type | |
term = In $ Mul (In (Val (Fin 4))) (In Addunit) | |
bigtype = freetypes term | |
-- did i just write a types compiler? |
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