# System FC: equality constraints and coercions

For many years, GHC's intermediate language was essentially:

• System Fw, plus
• algebraic data types (including existentials)

But that is inadequate to describe GADTs and associated types. So in 2006 we extended GHC to support System FC, which adds

• equality constraints and coercions

You can find a full description of FC in the paper https://ghc.haskell.org/trac/ghc/wiki/ReadingList#TypeEqualities; note that GHC uses the system described in post-publication Appendix C, not the system in the main body of the paper. The notes that follow sketch the implementation of FC in GHC, but without duplicating the contents of the paper.

A coercion c, is a type-level term, with a kind of the form T1 :=: T2. (c :: T1 :=: T2) is a proof that a term of type T1 can be coerced to type T2. Coercions are classified by a new sort of kind (with the form T1 :=: T2). Most of the coercion construction and manipulation functions are found in the Coercion module, compiler/types/Coercion.hs.

Coercions appear in Core in the form of Cast expressions: if t :: T1 and c :: T1:=:T2, then (t `cast` c) :: T2. See Commentary/Compiler/CoreSynType.

## Coercions and Coercion Kinds

The syntax of coercions extends the syntax of types (and the type Coercion is just a synonym for Type). By representing coercion evidence on the type level, we can take advantage of the existing erasure mechanism and keep non-termination out of coercion proofs (which is necessary to keep the system sound). The syntax of coercions and types also overlaps a lot. A normal type is evidence for the reflexive coercion, i.e.,

Int :: Int :=: Int

Coercion variables are used to abstract over evidence of type equality, as in

(/\c::(a :=: Bool). \x::a. if (x `cast` c) then 0 else 1) :: (a :=: Bool) => a -> Int

There are also coercion constants that are introduced by the compiler to implement some source language features (newtypes for now, associated types soon and probably more in the future). Coercion constants are represented as TyCons made with the constructor CoercionTyCon.

Coercions are type level terms and can have normal type constructors applied to them. The action of type constructors on coercions is much like in a logical relation. So if c1 :: T1 :=: T2 then

[c1] :: [T1] :=: [T2]

and if c2 :: S1 :=: S2 then

c1 -> c2 :: (T1 -> S1 :=: T2 -> S2)

The sharing of syntax means that a normal type can be looked at as either a type or as coercion evidence, so we use two different kinding relations, one to find type-kinds (implemented in Type as `typeKind :: Type -> Kind`) and one to find coercion-kinds (implemented in Coercion as coercionKind :: Coercion -> Kind).

Coercion variables are distinguished from type variables, and non-coercion type variables (just like any normal type) can be used as the reflexive coercion, while coercion variables have a particular coercion kind which need not be reflexive.

The internal representation of GADTs is as regular algebraic datatypes that carry coercion evidence as arguments. A declaration like

data T a b where
T1 :: a -> b -> T [a] (a,b)

would result in a data constructor with type

T1 :: forall a b. forall a1 b1. (a :=: [a1], b :=: (a1, b1)) => a1 -> b1 -> T a b

This means that (unlike in the previous intermediate language) all data constructor return types have the form T a1 ... an where a1 through an are the parameters of the datatype.

However, we also generate wrappers for GADT data constructors which have the expected user-defined type, in this case

\$wT1 = /\a b. \x y. T1 [a] (a,b) a b [a] (a,b) x y

Where the 4th and 5th arguments given to T1 are the reflexive coercions

[a]   :: [a] :=: [a]
(a,b) :: (a,b) :=: (a,b)

## Representation of coercion assumptions

In most of the compiler, as in the FC paper, coercions are abstracted using ForAllTy cv ty where cv is a coercion variable, with a kind of the form PredTy (EqPred T1 T2). However, during type inference it is convenient to treat such coercion qualifiers in the same way other class membership or implicit parameter qualifiers are treated. So functions like tcSplitForAllTy and tcSplitPhiTy and tcSplitSigmaTy, treat ForAllTy cv ty as if it were FunTy (PredTy (EqPred T1 T2)) ty (where PredTy (EqPred T1 T2) is the kind of cv). Also, several of the dataConXXX functions treat coercion members of the data constructor as if they were dictionary predicates (i.e. they return the PredTy (EqPred T1 T2) with the theta).

## Newtypes are coerced types

The implementation of newtypes has changed to include explicit type coercions in the place of the previously used ad-hoc mechanism. For a newtype declared by

newtype T a = MkT (a -> a)

the NewTyCon for T will contain nt_co = CoT where:

CoT t : (T t :=: t -> t)

This TyCon is a CoercionTyCon, so it does not have a kind on its own; it basically has its own typing rule for the fully-applied version. If the newtype T has k type variables, then CoT has arity at most k. In the case that the right hand side is a type application ending with the same type variables as the left hand side, we "eta-contract" the coercion. So if we had

newtype S a = MkT [a]

then we would generate the arity 0 coercion CoS : S :=: []. The primary reason we do this is to make newtype deriving cleaner. If the coercion cannot be reduced in this fashion, then it has the same arity as the tycon.

In the paper we'd write

axiom CoT : (forall t. T t) :=: (forall t. [t])

and then when we used CoT at a particular type, s, we'd say

CoT @ s

which encodes as (TyConApp instCoercionTyCon [TyConApp CoT [], s])

But in GHC we instead make CoT into a new piece of type syntax (like instCoercionTyCon, symCoercionTyCon etc), which must always be saturated, but which encodes as

TyConApp CoT [s]

In the vocabulary of the paper it's as if we had axiom declarations like

axiom CoT t :  T t :=: [t]

The newtype coercion is used to wrap and unwrap newtypes whenever the constructor or case is used in the Haskell source code.

Such coercions are always used when the newtype is recursive and are optional for non-recursive newtypes. Whether or not they are used can be easily changed by altering the function mkNewTyConRhs in iface/BuildTyCl.lhs.

## Roles

Roles specify what nature of equality a coercion is proving. See Roles and RolesImplementation.

## Simplification

• exprIsConApp_maybe
• simplExpr
Last modified on Jul 23, 2017 5:40:01 AM