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# Coercions in GHC's core language

Ever since coercions were introduced into GHC's Core language I have treated

- Coercions like types
- Coercion variables like type variables

In particular, casts, coercion applications, and coercion abstractoins are all erased before we generate code.

I now think that this is the wrong approach. This note describes why.

## Difficulties with the current approach

Ther are two problems with the current approach

- Equality evidence variables ("type variables") are treated differently to dictionary evidence variables ("term varaibles"). This leads to lots of tiresome non-uniformities.
- In an abstraction
`/\a\x:a.e`the type variable`a`can appear in the type of a term-variable binder`x`. In contrast`x`can't appear in the type of another binder. Coercion binders behave exactly like term binders in this way, and quite unlike type binders. - More seeriously, we don't have a decent way to handle superclass equalities.

The last problem is the one that triggered this note, and needs a bit more explanation. Consider

class (F a ~ b, Eq a) => C a b where op :: a -> b

The dictionary for C looks like this:

data C a b where MkC :: (F a ~ b, Num a) => (a->b) -> C a b

Now imagine typechecking a function like this

f :: C a b => a -> a f x = x + 1

The Core program we generate looks something like this:

f = /\a b. \(d:C a b). let (nd : Num a) = case d of { MkC _ d _ -> d } in (+) nd x (fromInteger nd 1)

The `nd` binding extracts the `Num` superclass dictionary from the
`C` dictionary; the case expression is called a *superclass selector*.

Now suppose that we needed to use the equality superclass rather than
the `Num` superclass:

g :: C a b => [F a] -> [b] g xs = xs

The obvious translation would look like this:

g = /\ab. \(d:C a b). let (eq : F a ~ b) = case d of { MkC eq _ _ -> eq } in xs |> [eq]

But Core doesn't (currently) have a let-binding form that binds a coercion variable, and whose right-hand side is a term (in this example, a case expression) rather than a literal coercion! So the current plan is to generate this instead:

g = /\ab. \(d:C a b). case d of { MkC eq _ _ -> in xs |> [eq] }

This non-uniformity of equality and dictionary evidence is extremely awkward in the desugarer. Moreover, it means that we can't abstract the superclass selector; we'd really like to have:

g = /\ab. \(d:C a b). let (eq : F a ~ b) = sc_sel1 d in xs |> [eq]

And it interacts poorly with the class-op rules that GHC uses to simplify dictinary selectors. Imagine the call

dIB :: C Int Bool dIB g Int Bool d

...unfinished...

## Main proposal

Recall our basic types

type Id = Var -- in Var.lhs type TyVar = Var data CoreExpr -- in CoreSyn.lhs = Var Var | Lit Lit | Type Type | Coercion Coercion | App CoreExpr CoreExpr | Lam Var CoreExpr | Cast CoreExpr Coercion | Let CoreBind CoreExpr | Case... | Note ... data CoreBind = NonRec Var CoreExpr | Rec [(Id,CoreExpr)] data Type -- in TypeRep.lhs = TyVar TyVar | AppTy Type Type | FunTy Type Type | ForAllTy Var Type | PredTy PredType | TyConApp TyCon [Type] data PredType = EqPred Type Type | ClassP Class [Type] | IParam Name Type

Note that

`Var`can be a type variable, coercion variable, or term variable. You can tell which with a dynamic test (e.g.`isId :: Var -> Bool`).

`Lam`is used for type abstractions, coercion abstractions, and value abstractions. The`Var`can tell you which.

- Type applications (in a term) look like
`(App f (Type t))`. The`(Type t)`part must literally appear there, with no intervening junk. This is not statically enforced, but it turns out to be much more convenient than having a constructor`TyApp CoreExpr Type`.

OK now the new proposal is to *treat equality evidence just like any other sort of evidence*.

- A coercion variable is treated like term-level identifier, not a type-level identifier. (More on what that means below.)

- A coercion is an
`CoreExpr`, of form`Coercion g`, whose type is`(s ~ t)`, of form`PredTy (EqPred s t)`.

- Unlike type applications, coercion applications are not required to have a
`(Coercion g)`as the argument. For example, suppose we havef :: forall a. (a~Int) => a -> Int id :: forall b. b->b c :: x~Int

Then the term `(f x (id (x~Int) c))` would be fine. Notice that the coercion argument is an appplication of the identity function. (Yes it's a bit contrived.) In `CoreExpr` form it would look like:

App (App (Var f) (Type x)) (App (App (Var id) (Type (PredTy (EqPred x Int)))) (Var c))

- Similarly a let-binding can bind a coercion
Let (NonRec c (...a coercion-valued term..)) (...body...)

- Coercion application is call-by value. Ditto let-bindings. You must have the evidence before calling the function.

- So it doesn't make sense to have recursive coercion bindings.

- If we see
`Let (NonRec c (Coercion g)) e`we can substitute`(Coercion g)`for any term-level occurrences of`c`in the term`e`, and`g`for`c`in any occurrences of`c`in coercions inside`e`. (This seems a bit messy.)