Garbage Can Compiling to Categories with Inspectable Lambdas
There are a couple kinds of functions that we can turn into totally inspectable data.
Linear functions can be reconstituted into a matrix if you give a basis of vectors.
Functions from enumerable types can be turned into a lookup table
Sufficiently polymorphic functions are another example though. forall a. a> a is commonly known to only be id. The same goes for fst = forall a b. (a,b)>a and snd and swap and all the nesting of . These functions have exactly one inhabiting value (excluding internal churning and the possibility of going into an infinite loop).
So the type directly tells us the implementation
forall a. (a,a)>a is similar. It can only be fst or snd. Types that reuse a type parameter in the input can only be permutations.
I’ve been trying to find a way to take a written lambda and convert it to data automatically and have been having trouble.
An opaque type that we have hidden the contructors to is the same (T,T)>T can only be fst or snd specialized to T since we can’t possibly destruct on T.
We can figure out which one by giving a labeled example to that function and then inspecting a single output. This gives the permutation and duplication that was done.
Similarly for T > Either T T
Once we have this, we can (Hopefully) reinterpret this lambda in terms of a monoidal category.
{# LANGUAGE RankNTypes, GADTs, FlexibleInstances,
DataKinds, TypeFamilies,MultiParamTypeClasses,
FlexibleContexts,
ScopedTypeVariables,
FunctionalDependencies,
GADTs,
TypeOperators
#}
AllowAmbiguousTypes,
 OverlappingInstances,
 UndecidableInstances,
import Data.Proxy
import Unsafe.Coerce
data Tag = Tag Int deriving Show
type family (MonoMorphTag a) :: * where
MonoMorphTag (a,b) = (MonoMorphTag a, MonoMorphTag b)
MonoMorphTag (a>b) = (MonoMorphTag a) > (MonoMorphTag b)
MonoMorphTag Int = Int
MonoMorphTag [a] = [a]
MonoMorphTag (a,b,c) = (a,b,c)
MonoMorphTag (a,b,c,d) = (a,b,c,d)
MonoMorphTag Double = Double
MonoMorphTag () = ()
MonoMorphTag Char = Char
MonoMorphTag _ = Tag
unsafeMonoTag :: a > MonoMorphTag a
unsafeMonoTag = unsafeCoerce
 unsafeTagLeaves :: forall a. MonoMorphTag a > Tag
 unsafeTagLeaves = unsafeCoerce
type T = Tag
class GetVal a where
val :: Int > Proxy a > (a, Int)
instance GetVal Tag where
val n _ = (Tag n, n+1)
instance (GetVal a, GetVal b) => GetVal (a,b) where
val n _ = ((v1, v2), n'') where
(v1 , n') = val n (Proxy :: Proxy a)
(v2 , n'') = val n' (Proxy :: Proxy b)
data TagTree a = Node (TagTree a) (TagTree a)  Leaf a deriving Show   Apply (k a b) TagTree
class Treeify a b where
treeify :: a > TagTree b
instance Treeify Tag Tag where
treeify x = Leaf x
instance (Treeify a Tag, Treeify b Tag) => Treeify (a,b) Tag where
treeify (a,b) = Node (treeify a) (treeify b)
class MonoMorph a where
type Mono a :: *
instance MonoMorph (a,b) where
type Mono (a,b) = (Mono a, Mono b)
{
instance MonoMorph (MonoMorphTag a) where
type Mono a = Tag
}
{
 Hmm I'm not sure how to monomorhpize this.
fst' :: (TagTup a) => (a, b) > a
fst' = fst
}
{
class AutoCurry a b  a > b where
autocurry :: a > b
instance AutoCurry (a>b>Tag) ((a,b)>Tag) where
autocurry f = uncurry f
instance AutoCurry c (a>c') => AutoCurry (b>c) ((b,a) > c') where
autocurry f = uncurry (\b > autocurry (f b))
}
data Monoidal = Dup  Mon Monoidal Monoidal  Par Monoidal Monoidal  Fst  Snd  Id  Comp Monoidal Monoidal deriving Show
data Monoidal' a b where
Id' :: Monoidal' a a
Dup' :: Monoidal' a (a,a)
Fst' :: Monoidal' (a,b) a
Snd' :: Monoidal' (a,b) b
Comp' :: Monoidal' b c > Monoidal' a b > Monoidal' a c
Mon' :: Monoidal' a a' > Monoidal' b b' > Monoidal' (a,b) (a',b')
data FunData = FunData {inval :: TagTree Tag, outval :: TagTree Tag} deriving Show
class TestIdea a b where
works :: (a > b) > (a, b)
instance (GetVal a) => TestIdea a b where
works f = (inval, f inval) where inval = fst $ val 0 (Proxy :: Proxy a)  fst $ val 0 (Proxy :: Proxy b)
fuckmyshitup :: (GetVal a, Treeify a Tag, Treeify b Tag) => (a > b) > FunData
fuckmyshitup f = let (a, b) = works f in FunData ((treeify a) :: TagTree Tag) ((treeify b):: TagTree Tag)
ccc :: FunData > Monoidal
ccc (FunData x (Node y z)) = Mon (ccc $ FunData x y) (ccc $ FunData x z)
ccc (FunData (Leaf _) (Leaf _)) = Id
ccc (FunData (Node x y) z@(Leaf (Tag n))) = if inleft n x then Comp Fst (ccc (FunData x z)) else Comp Snd (ccc (FunData y z))
ineither :: Int > TagTree Tag > Bool
ineither n (Node x y) = (ineither n x)  (ineither n y)
ineither n (Leaf (Tag n')) = n == n'
inleft :: Int > TagTree Tag > Bool
inleft n (Node l r) = ineither n l
inleft n (Leaf (Tag n')) = n == n'
inright :: Int > TagTree Tag > Bool
inright n (Node l r) = ineither n r
inright n (Leaf (Tag n')) = n == n'
 Then we can compile to categories. Replacing the entire structure with dup and par and
 fst, snd, etc.
 Make an infix operator $'
data Apply k a b c = Apply (FreeCat k a b) c
type ($$) = Apply
 No, don't need getval.
 We'll just need it for treeify?
{instance GetVal c => GetVal (Apply k a b c) where
val n _ = where x, n' = val n Proxy c
}
 Another Option
data A
data B
data C
 This is basically a lambda calculus
 I could probably finitely enumerate through all the typeclasses for all the variables
example = Proxy :: Proxy ((A,B) > B)
 Hmm this would allow you to force duplicate input types though.
{
class (Tagify a ~ a, Tagify b ~ b) => TestIdea a b where
works :: (a > b) > (a, b)
instance (GetVal a) => TestIdea a b where
works f = (inval, f inval) where inval = fst $ val 0 (Proxy :: Proxy a)  fst $ val 0 (Proxy :: Proxy b)
}
thisworks :: String
thisworks = works id
 fst . (val 0)
{
instance (F a ~ flag, GetVal' flag a) => GetVal a where
val = val' (Proxy :: Proxy flag)
class GetVal' (flag :: Bool) a where
val' :: Proxy flag > a > Tagify a
instance (GetVal a, GetVal b) => GetVal' 'True (a,b) where
val' _ (x,y) = (val x, val y)
instance GetVal' 'False a where
val' _ x = Tag 0
}
What about TH? Also the new quantified constraints extensions might be helpful?
Ok. A Different approach. This works much better to what I had in mind. you can write aribatrary ((x,y,) > (y,x)) tuple like lambdas and it will convert them to a category. I really had to hack around to get the thing to compile. Like that Pick typeclass, what the heck? Why can I get defaults values in type families but not in typeclasses?
It is all decidedly not typesafe. You can get totally nonsensical things to compile to something. However if you stick to lambdas, you’ll be ok. Maybe.
No on further review this does not work. I got tricked that the type seemed ok at a certain point. A couple problems arise upon actual application. Since the idea is to drive the form based on the type variables upon actual application to something that has types of the same form it gets all screwed up. Also tons of instances are overlapping, although I think this is fixable.
Maybe what I need is existential types that can’t ever unify together accidentally.
A couple thought on typelevel programming principles:

Typeclasses are hard to get default cases. You want to use type families if that is what you want

Typeclasses need unique stuff to appear on the right hand side. Only 1 pattern should match. You might need to add extra parameters to match to which you can force on the left hand side of the instance

~ type equality is real useful
An alternative to using lambda is to use an explicit Proxy. The type variables are basically just as good for syntactic purposes (a touch more noisy).
{# LANGUAGE RankNTypes, GADTs, FlexibleInstances,
DataKinds, TypeFamilies,MultiParamTypeClasses,
ImpredicativeTypes,
FlexibleContexts,
ScopedTypeVariables,
FunctionalDependencies,
UndecidableInstances,
GADTs,
TypeOperators
#}
 OverlappingInstances, NoImplicitPrelude

UndecidableInstances,
OverlappingInstances,
import Data.Type.Bool
import Data.Proxy
import Control.Category
import GHC.Base hiding (id,(.))
class IsId a where
val :: a > a
 toCat
instance forall a. IsId (a > a) where
val _ = id
{
class Catable f a b  f > a,b where
toCat :: forall k. CartesianCategory k => k a b
instance forall a b. Catable ((a,b)>a) (a,b) a where
toCat = fst
}
class Fst ab a  ab > a where
 toCat :: forall k. k ab a
instance forall a b. Fst (a,b) a
class Anything b where
fun :: b > b
class Stringly a where
stringly :: a > String
instance (Stringly a, Stringly b) => Stringly (a,b) where
stringly (x,y) = "(" ++ (stringly x) ++ "," ++ (stringly y) ++ ")"
{
instance (Stringly a, Stringly b) => Stringly (a > b) where
stringly f = "(" ++ (stringly x) ++ ">" ++ (stringly y) ++ ")"
}
class Category k where
dot' :: k b c > k a b > k a c
id' :: k a a
instance Category (>) where
dot' = (.)
id' = id
class Category k => CartesianCat k where
fst' :: k (a,b) a
snd' :: k (a,b) b
join' :: k a b > k a c > k a (b,c)
instance CartesianCat (>) where
fst' = fst
snd' = snd
join' = join''
class Catable a b where
toCat :: CartesianCat k => (a > b) > (k a b)
 toCat (\x > ((x,x),x)) . id
 it's not INSANE to just list out a finite list of possibilities ((a,b),c) etc.
{
data HeldApply k a b = HeldApply (k a b) a
($$) :: Category k => k a b > b > HeldApply k a b
f $$ x = HeldApply f
instance Catable a (HeldApply a b) where
toCat
Doesn't seem to work. We don't have an a get get the heldapply out of the function
Maybe we could pass in the approriate function as a a lambda \f x > Apply f x
instance ExponentialCategory k where
apply :: k (k a b, a) b
}
instance Catable a a where
toCat _ = id'
 why is this okay? should these be covered by the other cases?
instance Catable (a,b) a where
toCat _ = fst'
instance Catable (a,b) b where
toCat _ = snd'
dup x = (x,x)
{
instance Catable a (a,a) where
toCat _ = dup
}
join'' f g x = (f x, g x)
 iterates down through the output
instance (Catable a b, Catable a c) => Catable a (b,c) where
toCat f = join' (toCat (fst . f)) (toCat (snd . f))
{
instance (InL c (a,b), Catable a c) => Catable (a,b) c where
toCat f = (toCat (f . fst))
instance (InR c (a,b), Catable a c) => Catable (a,b) c where
toCat f = (toCat (f . snd))
}
instance (Catable a c, Catable b c, Pick' c (a,b) (In a c)) => Catable (a,b) c where
toCat f = pick' (Proxy :: Proxy (In a c))
{
instance (Catable a c, Catable b c, Pick c (a,b) (In a c)) => Catable (a,b) c where
toCat f = (toCat (pick (Proxy :: Proxy (In a c))))
}
{
class In a c where
find :: c > a
instance In a a
find = id
instance In a b => In a (b,c)
find = find . fst
instance In a c => In a (b,c)
find = find . snd
}
{
type family (LorR a c) :: Nat where
LorR a (a,_) = 1
LorR a (_,a) = 2
LorR a ((b,c),d) = (LorR a (b,c)) + (LorR a d)
LorR a (d,(b,c)) = (LorR a (b,c)) + (LorR a d)
LorR a _ = 0
}
type family (In a c) :: Bool where
In a a = 'True
In a (a,_) = 'True
In a (_,a) = 'True
In a ((b,c),d) = In a (b,c)  In a d
In a (d,(b,c)) = In a (b,c)  In a d
In a _ = 'False
{
type Snd = forall a b. (a,b) > b
type family (FstSnd a) :: * where
FstSnd 'True = Snd
FstSnd 'False = Snd
}
class Pick a c (d :: Bool) where
pick :: Proxy d > c > a
instance (Pick a (e,f) (In a e), (e,f) ~ b) => Pick a (b,c) 'True where
pick _ = (pick (Proxy :: Proxy (In a e))) . fst
instance (Pick a (e,f) (In a e), (e,f) ~ c) => Pick a (b,c) 'False where
pick _ = (pick (Proxy :: Proxy (In a e))) . snd
instance Pick a (a,b) 'True where
pick _ = fst
instance Pick a (b,a) 'False where
pick _ = snd
instance Pick a a d where
pick _ = id
 The bool is true if in the left branch
class Pick' a c (d :: Bool) where
pick' :: CartesianCat k => Proxy d > k c a
instance (Pick' a (e,f) (In a e), (e,f) ~ b) => Pick' a (b,c) 'True where
pick' _ = dot' (pick' (Proxy :: Proxy (In a e))) fst'
instance (Pick' a (e,f) (In a e), (e,f) ~ c) => Pick' a (b,c) 'False where
pick' _ = dot' (pick' (Proxy :: Proxy (In a e))) snd'
instance Pick' a (a,b) 'True where
pick' _ = fst'
instance Pick' a (b,a) 'False where
pick' _ = snd'
instance Pick' a a d where
pick' _ = id'
{
class InL a c where
instance InL a a
instance In a b => InL a (b,c)
class InR a c
instance InR a a
instance In a b => InR a (c,b)
}
{
instance (Catable a c, Catable b c) => Catable (a,b) c where
toCat f =
instance (Catable a c, Catable b c) => Catable a (b,c) where
toCat f =
}
{
instance (Stringly a, Stringly b, (a,b) ~ c, IsTup c ~ 'True) => Stringly c where
stringly (x,y) = "(" ++ (stringly x) ++ "," ++ (stringly y) ++ ")"
}
instance (IsTup a ~ 'False, IsArr a ~ 'False) => Stringly a where
 stringly _ = "_"
instance forall a. Anything a where
fun = id
example :: a > a
example = val id
type family (IsTup a) :: Bool where
IsTup (a,b) = 'True
IsTup _ = 'False
type family (IsArr a) :: Bool where
IsArr (a>b) = 'True
IsArr _ = 'False