Types have an algebra very analogous to the algebra of ordinary numbers (video). This is the basic table of correspondences. Code with all the extensions available here.

type a * b = (a,b) type a + b = Either a b type b ^ a = a -> b type O = Void type I = () -- check out typelits implementation -- http://hackage.haskell.org/package/base-4.12.0.0/docs/src/GHC.TypeNats.html#%2A infixl 6 + -- ((a + b) + c) associates to the left infixl 7 * infixr 8 ^ -- derived definitions type Succ a = I + a type Two = Succ I type Three = Succ Two type Four = Succ Three

One way to see that this makes sense is by counting the cardinality of types built out of these combinators. Unit is the type with 1 inhabitant. Void has 0 inhabitants. If a has and b has possible values, then Either a b has inhabitants, (a,b) has and there are possible tabulations of a->b. We’re gonna stick to just polynomials for the rest of this, ignoring a->b.

Another way of looking at this is if two finitely inhabited types have the same number of inhabitants, then the types can be put into an isomorphism with each other. In other words, types modulo isomorphisms can be thought as representing the natural numbers. Because of this, we can build a curious proof system for the natural numbers using ordinary type manipulation.

In addition, we also get a natural way of expressing and manipulating polynomials.Polymorphic types can be seen as being very similar to polynomial expressions with natural coefficients N[x] . The polymorphic type variables have the ability to be instantiated to any value, corresponding to evaluating the polynomial for some number.

type ExamplePoly a = I + a + a * a * Three

The Lens ecosystem gives some interesting combinators for manipulating this algebra. The type Iso' a b contains isomorphisms. Since we’re only considering types up to isomorphism, this Iso' represents equality. We can give identity isomorphisms, compose isomorphisms and reverse isomorphisms.

type a ~~ b = Iso' a b refl :: a ~~ a refl = id compose :: (a ~~ b) -> (b ~~ c) -> (a ~~ c) compose = (.) rev :: a ~~ b -> b ~~ a rev = from

We can already form a very simple proof.

-- a very simple proof. Holds by definition oneonetwo :: (I + I) ~~ Two oneonetwo = id

Now we’ll add some more combinators, basically the axioms that the types mod isos are a commutative semiring. Semirings have an addition and multiplication operator that distribute over each other. It is interesting to note that I believe all of these Iso' actually are guaranteed to be isomorphisms ( to . from = id and from . to = id ) because of parametricity. from and to are unique ignoring any issues with bottoms because the polymorphic type signature is so constraining. This is not usually guaranteed to be true in Haskell just from saying it is an Iso' . If I give you an Iso' Bool Bool it might actually be the iso (const True) (const True) for example, which is not an isomorphism.

-- now we start to state our axioms -- interestingly, I believe the Iso and Lens laws to follow actually follow via parametricity. -- associativity + a identity object make mul and plus a monoid plus_assoc :: Iso' (a + (b + c)) ((a + b) + c) plus_assoc = iso assoc unassoc where assoc (Left a) = Left (Left a) assoc (Right (Left b)) = Left (Right b) assoc (Right (Right c)) = Right c unassoc (Left (Left a)) = Left a unassoc (Left (Right b)) = Right (Left b) unassoc (Right c) = (Right (Right c)) mul_assoc :: Iso' (a * (b * c)) ((a * b) * c) mul_assoc = iso (\(a,(b,c)) -> ((a,b),c)) (\((a,b),c) -> (a,(b,c))) -- use `absurd` from Data.Void for empty case. id_plus :: Iso' (a + O) a id_plus = iso (\case Left x -> x Right x -> absurd x) Left id_mul :: Iso' (a * I) a id_mul = iso (\(x,_) -> x) (\x -> (x,())) -- they are also commutative -- specialized version of swapped from Control.Lens.Iso for future clarity comm_plus :: Iso' (a + b) (b + a) comm_plus = swapped comm_mul :: Iso' (a * b) (b * a) comm_mul = swapped -- I don't see this one in Lens. Maybe it is there though? -- distributive property rdist :: Iso' (a * (b + c)) (a * b + a * c) rdist = iso (\(a,bc) -> (a,) +++ (a,) $ bc) (\case Left (a,b) -> (a, Left b) Right (a,c) -> (a, Right c)) mul_zero :: Iso' (a,O) O mul_zero = iso (\(_,y) -> absurd y) absurd

There are also combinators for lifting isomorphisms into bifunctors: firsting , seconding , and bimapping . These are important for indexing into subexpressions of our types in a point-free style.

-- a specialized version of firsting and seconding for clarity -- firsting and seconding feel to me more like words for tuples lefting :: (a ~~ b) -> (a + c) ~~ (b + c) lefting = firsting righting :: (a ~~ b) -> (c + a) ~~ (c + b) righting = seconding

Here is a slightly more complicated proof now available to us.

-- a more complicated proof twotwofour :: Iso' (Two + Two) Four twotwofour = rev plus_assoc

We can attempt a more interesting and difficult proof. I developed this iteratively using . _ hole expressions so that GHC would tell me what I had manipulated my type to at that point in my proof.

ldist :: ((b + c) * a) ~~ (b * a + c * a) ldist = comm_mul . rdist . (lefting comm_mul) . (righting comm_mul) -- very painful. Using holes _ and error messages absolutely essential factorexample :: ((a + I) * (a + I)) ~~ (a * a + Two * a + I) factorexample = rdist . -- distribute out the right side (lefting (comm_mul . rdist)) . -- in the left sum term distribute out (righting (comm_mul . rdist)) . -- in the right sum term distribute out plus_assoc . -- reassociate plus to the left (lefting (lefting (righting comm_mul))) . -- turn a * I term to I * a (lefting (rev plus_assoc)) . -- associate the two a*I terms together into an (a * I + a * I) term (lefting (righting (rev ldist))) . -- factor together that subterm into Two * a (righting id_mul) -- convert last I * I term into I

Artwork Courtesy of David. Sorry for any motion sickness.

The proof here is actually pretty trivial and can be completely automated away. We’ll get to that later.

If Iso' is equality, what about the other members of the Lens family? Justin Le says that Lens s a are witness to the isomorphism of a type s to the tuple of something and a . Prism witness a similar thing for sums. Once we are only considering types mod isos, if you think about it, these are expressions of two familiar relations on the natural numbers: the inequality relation and the divisibility relation

type a >= b = Prism' a b type a .| b = Lens' a b

Mathematically, these relations can be composed with equalities, just like in the lens hierarchy Lens and Prism can be composed with `Iso`. Both form a category, since they both have id and (.) .

{- the core combinators from the lens library for manipulating these are _1 :: (a * b) .| a _2 :: (a * b) .| b _Left :: (a + b) >= a _Right :: (a + b) >= b For example: -} twodiv4 :: (Two * Two) .| Two twodiv4 = _1 onelesstwo :: Two >= I onelesstwo = _Left threedivthree :: Three .| Three threedivthree = id

Here are a couple identities that we can’t derive from these basic combinators. There are probably others. Woah-ho, my bad. These are totally derivable using id_mul, id_plus, mul_zero, _1, _2, _Left, _Right.

-- once again, these are true via parametricity, more or less one_div :: a .| I -- one_div = lens (const ()) const one_div = (rev id_mul) . _2 zero_lte :: a >= O -- zero_lte = prism' absurd (const Nothing) zero_lte = (rev id_plus) . _Right zero_div :: O .| a -- zero_div = lens absurd const zero_div = (rev mul_zero) . _1

Pretty neat! Random thoughts and questions before we get into the slog of automation:

Traversal is the “is polynomial in” relation, which seems a rather weak statement on it’s own.

Implementing automatic polynomial division is totally possible and interesting

What is the deal with infinite types like [a] ? Fix . I suppose this is a theory of formal power series / combinatorial species. Fun combinatorics, generatingfunctionology. Brent Yorgey did his dissertation on this stuff. Wow. I’ve never really read this. It is way more relevant than I realized.

? . I suppose this is a theory of formal power series / combinatorial species. Fun combinatorics, generatingfunctionology. Brent Yorgey did his dissertation on this stuff. Wow. I’ve never really read this. It is way more relevant than I realized. Multivariate polynomial algorithms would also be interesting to explore (Grobner basis, multivariate division)

Derivatives of types and zippers – Conor McBride

Negative Numbers and Fractions?

Lifting to rank-1 types. Define Negative and Fractions via Galois connection?

-- partially applied + type P n = (Either n) -- partially applied * type M n = (,) n

Edit: /u/carette (wonder who that is 😉 ) writes:

“You should dig into

J Carette, A Sabry Computing with semirings and weak rig groupoids, in Proceedings of ESOP 2016, p. 123-148. Agda code in https://github.com/JacquesCarette/pi-dual/tree/master/Univalence. A lot of the algebra you develop is there too.

If you hunt around in my repos, you’ll also find things about lenses, exploring some of the same things you mention here.”

Similar ideas taken further and with more sophistication. Very interesting. Check it out.

Automation

Our factor example above was quite painful, yet the theorem was exceedingly obvious by expansion of the left and right sides. Can we automate that? Yes we can!

Here’s the battle plan:

Distribute out all expressions like so that all multiplication nodes appear at the bottom of the tree.

so that all multiplication nodes appear at the bottom of the tree. Reduce the expression by absorbing all stupid , , terms.

, , terms. Reassociate everything to the right, giving a list like format

Sort the multiplicative terms by power of the variable

Once we have these operations, we’ll combine them into a canonical operation. From there, most equality proofs can be performed using the rewrite operation, which merely puts both sides into canonical form

-- | The main combinator to expand out any polynomial into canonical form with terms sorted canonical :: (Dist a b, Absorb b c, RightAssoc c d, SortTerm d e) => a ~~ e canonical = dist . absorb . rightAssoc . sortTerm rewrite :: forall a' a b c d e e' d' c' b'. (Dist a b, Absorb b c, RightAssoc c d, SortTerm d e, e ~ e', Dist a' b', Absorb b' c', RightAssoc c' d', SortTerm d' e') => a ~~ a' rewrite = canonical . (rev canonical)

Once we have those, the painful theorem above and harder ones becomes trivial.

ex9 :: (a ~ V a') => ((a + I) * (I + a)) ~~ (a * a + Two * a + I) ex9 = rewrite ex10 :: (a ~ V a') => ((a + I) * (I + a) * (Two + a)) ~~ (a * a * a + a * a * Four + Two + Four * a + a) ex10 = rewrite

Now we’ll build the Typeclasses necessary to achieve each of these aims in order. The Typeclass system is perfect for what we want to do, as it builds terms by inspecting types. It isn’t perfect in the sense that typeclass pattern matching needs to be tricked into doing what we need. I have traded in cleverness and elegance with verbosity.

In order to make our lives easier, we’ll need to tag every variable name with a newtype wrapper. Otherwise we won’t know when we’ve hit a leaf node that is a variable. I’ve used this trick before here in an early version of my faking Compiling to Categories series. These wrappers are easily automatically stripped.

-- For working with Variables -- a newtype to mark variable position newtype V a = V a -- Suggested usage, bind the V in a unification predicate to keep expression looking clean -- (V a' ~ a) => (a + I) * a -- multinomials. to be implemented some other day -- a phantom l labelled newtype for variable ordering. newtype VL l a = VL a

A common pattern I exploit is to use a type family to drive complicated recursion. Closed type families allow more overlap and default patterns which is very useful for programming. However, type families do not carry values, so we need to flip flop between the typeclass and type family system to achieve our ends.

Here is the implementation of the distributor Dist . We make RDist and LDist typeclasses that make a sweep of the entire tree, using ldist and rdist as makes sense. It was convenient to separate these into two classes for my mental sanity. I am not convinced even now that I have every case. Then the master control class Dist runs these classes until any node that has a (*) in it has no nodes with (+) underneath, as checked by the HasPlus type family.

class RDist a b | a -> b where rdist' :: Iso' a b instance RDist a a' => RDist (a * I) (a' * I) where rdist' = firsting rdist' instance RDist a a' => RDist (a * O) (a' * O) where rdist' = firsting rdist' instance RDist a a' => RDist (a * (V b)) (a' * (V b)) where rdist' = firsting rdist' instance (RDist (a * b) ab, RDist (a * c) ac) => RDist (a * (b + c)) (ab + ac) where rdist' = rdist . (bimapping rdist' rdist') instance (RDist a a', RDist (b * c) bc) => RDist (a * (b * c)) (a' * bc) where rdist' = (bimapping rdist' rdist') instance (RDist a a', RDist b b') => RDist (a + b) (a' + b') where rdist' = bimapping rdist' rdist' instance RDist O O where rdist' = id instance RDist I I where rdist' = id instance RDist (V a) (V a) where rdist' = id -- can derive ldist from swapped . rdist' . swapped? class LDist a b | a -> b where ldist' :: Iso' a b instance (LDist (b * a) ab, LDist (c * a) ac) => LDist ((b + c) * a) (ab + ac) where ldist' = ldist . (bimapping ldist' ldist') instance (LDist (b * c) bc, LDist a a') => LDist ((b * c) * a) (bc * a') where ldist' = bimapping ldist' ldist' instance LDist a a' => LDist (I * a) (I * a') where ldist' = seconding ldist' instance LDist a a' => LDist (O * a) (O * a') where ldist' = seconding ldist' instance LDist a a' => LDist ((V b) * a) ((V b) * a') where ldist' = seconding ldist' instance (LDist a a', LDist b b') => LDist (a + b) (a' + b') where ldist' = bimapping ldist' ldist' instance LDist O O where ldist' = id instance LDist I I where ldist' = id instance LDist (V a) (V a) where ldist' = id type family HasPlus a where HasPlus (a + b) = 'True HasPlus (a * b) = (HasPlus a) || (HasPlus b) HasPlus I = 'False HasPlus O = 'False HasPlus (V _) = 'False class Dist' f a b | f a -> b where dist' :: a ~~ b instance (f ~ HasPlus ab', LDist (a * b) ab, RDist ab ab', Dist' f ab' ab'') => Dist' 'True (a * b) ab'' where dist' = ldist' . rdist' . (dist' @f) instance Dist' 'False (a * b) (a * b) where dist' = id instance (HasPlus a ~ fa, HasPlus b ~ fb, Dist' fa a a', Dist' fb b b') => Dist' x (a + b) (a' + b') where dist' = bimapping (dist' @fa) (dist' @fb) -- is that enough though? only dist if instance Dist' x I I where dist' = id instance Dist' x O O where dist' = id instance Dist' x (V a) (V a) where dist' = id class Dist a b | a -> b where dist :: a ~~ b -- dist' should distributed out all multiplications instance (f ~ HasPlus a, Dist' f a b) => Dist a b where dist = dist' @f

Next is the Absorb type class. It is arranged somewhat similarly to the above. Greedily absorb, and keep doing it until no absorptions are left. I think that works.

-- RAbsorb matches only on the right hand side of binary operators -- matching on both sides is ungainly to write class RAbsorb a b | a -> b where rabsorb :: a ~~ b -- obvious absorptions instance RAbsorb x x' => RAbsorb (x + O) x' where rabsorb = id_plus . rabsorb instance RAbsorb x x' => RAbsorb (x + I) (x' + I) where rabsorb = firsting rabsorb instance RAbsorb x x' => RAbsorb (x * I) x' where rabsorb = id_mul . rabsorb instance RAbsorb (x * O) O where rabsorb = mul_zero instance RAbsorb x x' => RAbsorb (x * (V a)) (x' * (V a)) where rabsorb = firsting rabsorb instance RAbsorb x x' => RAbsorb (x + (V a)) (x' + (V a)) where rabsorb = lefting rabsorb -- recursion steps instance (RAbsorb (y * z) yz, RAbsorb x x') => RAbsorb (x * (y * z)) (x' * yz) where rabsorb = bimapping rabsorb rabsorb instance (RAbsorb (y + z) yz, RAbsorb x x') => RAbsorb (x * (y + z)) (x' * yz) where rabsorb = bimapping rabsorb rabsorb instance (RAbsorb (y + z) yz, RAbsorb x x') => RAbsorb (x + (y + z)) (x' + yz) where rabsorb = bimapping rabsorb rabsorb instance (RAbsorb (y * z) yz, RAbsorb x x') => RAbsorb (x + (y * z)) (x' + yz) where rabsorb = bimapping rabsorb rabsorb -- base cases instance RAbsorb O O where rabsorb = id instance RAbsorb I I where rabsorb = id instance RAbsorb (V a) (V a) where rabsorb = id -- mirror of RAbsorb class LAbsorb a b | a -> b where labsorb :: a ~~ b instance LAbsorb x x' => LAbsorb (O + x) x' where labsorb = comm_plus . id_plus . labsorb instance LAbsorb x x' => LAbsorb (I + x) (I + x') where labsorb = righting labsorb instance LAbsorb x x' => LAbsorb (I * x) x' where labsorb = comm_mul . id_mul . labsorb instance LAbsorb (O * x) O where labsorb = comm_mul . mul_zero instance LAbsorb x x' => LAbsorb ((V a) + x) ((V a) + x') where labsorb = righting labsorb instance LAbsorb x x' => LAbsorb ((V a) * x) ((V a) * x') where labsorb = seconding labsorb instance (LAbsorb (y * z) yz, LAbsorb x x') => LAbsorb ((y * z) * x) (yz * x') where labsorb = bimapping labsorb labsorb instance (LAbsorb (y + z) yz, LAbsorb x x') => LAbsorb ((y + z) * x) (yz * x') where labsorb = bimapping labsorb labsorb instance (LAbsorb (y + z) yz, LAbsorb x x') => LAbsorb ((y + z) + x) (yz + x') where labsorb = bimapping labsorb labsorb instance (LAbsorb (y * z) yz, LAbsorb x x') => LAbsorb ((y * z) + x) (yz + x') where labsorb = bimapping labsorb labsorb instance LAbsorb O O where labsorb = id instance LAbsorb I I where labsorb = id instance LAbsorb (V a) (V a) where labsorb = id -- labsorb :: (Swapped p, RAbsorb (p b a) (p b' a')) => (p a b) ~~ (p a' b') -- labsorb = swapped . rabsorb . swapped -- a test function to see if an expression is properly absorbed type family Absorbed a where Absorbed (O + a) = 'False Absorbed (a + O) = 'False Absorbed (a * I) = 'False Absorbed (I * a) = 'False Absorbed (a * O) = 'False Absorbed (O * a) = 'False Absorbed (a + b) = (Absorbed a) && (Absorbed b) Absorbed (a * b) = (Absorbed a) && (Absorbed b) Absorbed I = 'True Absorbed O = 'True Absorbed (V a) = 'True -- iteratively rabsorbs and leftabsorbs until tree is properly absorbed. class Absorb' f a b | f a -> b where absorb' :: a ~~ b instance Absorb' 'True a a where absorb' = id instance (LAbsorb a a', RAbsorb a' a'', f ~ Absorbed a'', Absorb' f a'' a''') => Absorb' 'False a a''' where absorb' = labsorb . rabsorb . (absorb' @f) -- wrapper class to avoid the flag. class Absorb a b | a -> b where absorb :: a ~~ b instance (f ~ Absorbed a, Absorb' f a b) => Absorb a b where absorb = absorb' @f

The Associators are a little simpler. You basically just look for the wrong association pattern and call plus_assoc or mul_assoc until they don’t occur anymore, then recurse. We can be assured we’re always making progress if we either switch some association structure or recurse into subparts.

class LeftAssoc a b | a -> b where leftAssoc :: Iso' a b instance LeftAssoc a a' => LeftAssoc (a + I) (a' + I) where leftAssoc = firsting leftAssoc instance LeftAssoc a a' => LeftAssoc (a + O) (a' + O) where leftAssoc = firsting leftAssoc instance LeftAssoc a a' => LeftAssoc (a * I) (a' * I) where leftAssoc = firsting leftAssoc instance LeftAssoc a a' => LeftAssoc (a * O) (a' * O) where leftAssoc = firsting leftAssoc instance LeftAssoc a a' => LeftAssoc (a * (V b)) (a' * (V b)) where leftAssoc = firsting leftAssoc instance LeftAssoc a a' => LeftAssoc (a + (V b)) (a' + (V b)) where leftAssoc = firsting leftAssoc instance (LeftAssoc ((a + b) + c) abc') => LeftAssoc (a + (b + c)) abc' where leftAssoc = plus_assoc . leftAssoc instance (LeftAssoc ((a * b) * c) abc') => LeftAssoc (a * (b * c)) abc' where leftAssoc = mul_assoc . leftAssoc instance (LeftAssoc (b * c) bc, LeftAssoc a a') => LeftAssoc (a + (b * c)) (a' + bc) where leftAssoc = bimapping leftAssoc leftAssoc -- a * (b + c) -> a * b + a * c -- This case won't happen if we've already distribute out. instance (LeftAssoc (b + c) bc, LeftAssoc a a') => LeftAssoc (a * (b + c)) (a' * bc) where leftAssoc = bimapping leftAssoc leftAssoc instance LeftAssoc O O where leftAssoc = id instance LeftAssoc I I where leftAssoc = id instance LeftAssoc (V a) (V a) where leftAssoc = id -- right assoc will completely associate strings of + or -. Mixed terms are not associated. -- cases on left hand side of binary expression -- always makes progress by either reassociating or recursing class RightAssoc a b | a -> b where rightAssoc :: Iso' a b instance (RightAssoc (a + (b + c)) abc') => RightAssoc ((a + b) + c) abc' where rightAssoc = (rev plus_assoc) . rightAssoc instance (RightAssoc (a * (b * c)) abc') => RightAssoc ((a * b) * c) abc' where rightAssoc = (rev mul_assoc) . rightAssoc instance RightAssoc a a' => RightAssoc (I + a) (I + a') where rightAssoc = seconding rightAssoc instance RightAssoc a a' => RightAssoc (O + a) (O + a') where rightAssoc = seconding rightAssoc instance RightAssoc a a' => RightAssoc (I * a) (I * a') where rightAssoc = seconding rightAssoc instance RightAssoc a a' => RightAssoc (O * a) (O * a') where rightAssoc = seconding rightAssoc instance RightAssoc a a' => RightAssoc ((V b) + a) ((V b) + a') where rightAssoc = seconding rightAssoc instance RightAssoc a a' => RightAssoc ((V b) * a) ((V b) * a') where rightAssoc = seconding rightAssoc instance (RightAssoc (b * c) bc, RightAssoc a a') => RightAssoc ((b * c) + a) (bc + a') where rightAssoc = bimapping rightAssoc rightAssoc instance (RightAssoc (b + c) bc, RightAssoc a a') => RightAssoc ((b + c) * a) (bc * a') where rightAssoc = bimapping rightAssoc rightAssoc instance RightAssoc O O where rightAssoc = id instance RightAssoc I I where rightAssoc = id instance RightAssoc (V a) (V a) where rightAssoc = id

Finally, the SortTerm routine. SortTerm is a bubble sort. The typeclass Bubble does a single sweep of swapping down the type level list-like structure we’ve built. The SortTerm uses the Sorted type family to check if it is finished. If it isn’t, it call Bubble again.

type family (a :: k) == (b :: k) :: Bool where f a == g b = f == g && a == b a == a = 'True _ == _ = 'False type family SortedTerm a :: Bool where SortedTerm (a + (b + c)) = (((CmpTerm a b) == 'EQ) || ((CmpTerm a b) == 'GT)) && (SortedTerm (b + c)) SortedTerm (a + b) = ((CmpTerm a b) == 'EQ) || ((CmpTerm a b) == 'GT) --SortedTerm a = 'True SortedTerm I = 'True SortedTerm O = 'True SortedTerm (V a) = 'True -- higher powers of V are bigger. -- CmpTerm compares TimesLists. type family CmpTerm a b where CmpTerm ((V a) * b) ((V a) * c) = CmpTerm b c CmpTerm ((V a) * b) (V a) = 'GT CmpTerm (V a) ((V a) * b) = 'LT CmpTerm I ((V a) * b) = 'LT CmpTerm ((V a) * b) I = 'GT CmpTerm (V a) (V a) = 'EQ CmpTerm I (V a) = 'LT CmpTerm (V a) I = 'GT CmpTerm I I = 'EQ -- Maybe this is all uneccessary since we'll expand out and abosrb to a*a + a*a + a + a + a + a -- type a == b = TEq.(==) a b -- Head and Tail of PlusLists type family PlusHead a where PlusHead (x + y) = x PlusHead x = x type family PlusTail a where PlusTail (x + y) = y -- bubble assume a plusList of multiplicative terms. I.E. fully distributed, fully rightassociated , fully absorbed -- does one pass of a bubble sort class Bubble f a b | f a -> b where bubble :: a ~~ b -- more to go instance (f ~ CmpTerm b (PlusHead c), Bubble f (b + c) bc) => Bubble 'EQ (a + (b + c)) (a + bc) where bubble = righting (bubble @f) instance (f ~ CmpTerm b (PlusHead c), Bubble f (b + c) bc) => Bubble 'GT (a + (b + c)) (a + bc) where bubble = righting (bubble @f) instance (f ~ CmpTerm a (PlusHead c), Bubble f (a + c) ac) => Bubble 'LT (a + (b + c)) (b + ac) where bubble = plus_assoc . (lefting comm_plus) . (rev plus_assoc) . righting (bubble @f) -- The times, or constants shows that we're at the end of our + list. instance Bubble 'EQ (a + (b * c)) (a + (b * c)) where bubble = id instance Bubble 'GT (a + (b * c)) (a + (b * c)) where bubble = id instance Bubble 'LT (a + (b * c)) ((b * c) + a) where bubble = comm_plus instance Bubble 'EQ (a + I) (a + I) where bubble = id instance Bubble 'GT (a + I) (a + I) where bubble = id instance Bubble 'LT (a + I) (I + a) where bubble = comm_plus instance Bubble 'EQ (a + O) (a + O) where bubble = id instance Bubble 'GT (a + O) (a + O) where bubble = id instance Bubble 'LT (a + O) (O + a) where -- shouldn't happen bubble = comm_plus instance Bubble 'EQ (a + (V b)) (a + (V b)) where bubble = id instance Bubble 'GT (a + (V b)) (a + (V b)) where bubble = id instance Bubble 'LT (a + (V b)) ((V b) + a) where bubble = comm_plus -- goofy base cases in case bubble gets called on a single element instance Bubble x O O where bubble = id instance Bubble x I I where bubble = id instance Bubble x (V a) (V a) where bubble = id -- sort term assumes rightassociated, fully distributed, fully I O absorbed expressions class SortTerm' f a b | f a -> b where -- f is flag whether PlusList is sorted. sortTerm' :: a ~~ b instance SortTerm' 'True a a where sortTerm' = id -- a single term with no plus shouldn't get here. That is why PlusTail is ok. instance (f ~ CmpTerm (PlusHead a) (PlusHead (PlusTail a)), Bubble f a a', f' ~ SortedTerm a', SortTerm' f' a' b) => SortTerm' 'False a b where sortTerm' = (bubble @f) . (sortTerm' @f') class SortTerm a b | a -> b where -- f is flag whether PlusList is sorted. sortTerm :: a ~~ b instance (f ~ SortedTerm a, SortTerm' f a a') => SortTerm a a' where sortTerm = sortTerm' @f

Hope you thought this was neat!