A Coercion is concrete evidence of the equality/convertibility of two types.

Essentially a typed Name, that may also contain some additional information about the Var and it's use sites.

Essentially funResultTy on kinds

Find the result Kind of a type synonym, after applying it to its arity number of type variables Actually this function works fine on data types too, but they'd always return *, so we never need to ask

Essentially splitFunTys on kinds

Essentially splitFunTysN on kinds

See Type for details of the distinction between these Kinds

Is this a kind (i.e. a type-of-types)?

Is this a super-kind (i.e. a type-of-kinds)?

Given two kinds k1 and k2, creates the Kind k1 -> k2

Iterated application of mkArrowKind

True of any sub-kind of ArgTypeKind

True of any sub-kind of OpenTypeKind (i.e. anything except arrow)

k1 `isSubKind` k2 checks that k1 <: k2

Used when generalising: default kind ? and ?? to *. See Type for more information on what that means

kc1 `isSubKindCon` kc2 checks that kc1 <: kc2

Makes a coercion type from two types: the types whose equality is proven by the relevant Coercion

If it is the case that

 c :: (t1 ~ t2)

i.e. the kind of c relates t1 and t2, then coercionKind c = Pair t1 t2.

Apply coercionKind to multiple Coercions

Create a symmetric version of the given Coercion that asserts equality between the same types but in the other direction, so a kind of t1 ~ t2 becomes the kind t2 ~ t1.

Create a new Coercion by composing the two given Coercions transitively.

Instantiates a Coercion with a Type argument.

Apply a Coercion to another Coercion.

Apply a type constructor to a list of coercions.

Make a function Coercion between two other Coercions

Make a Coercion which binds a variable within an inner Coercion

Manufacture a coercion from thin air. Needless to say, this is not usually safe, but it is used when we know we are dealing with bottom, which is one case in which it is safe. This is also used to implement the unsafeCoerce# primitive. Optimise by pushing down through type constructors.

Create a coercion constructor (axiom) suitable for the given newtype TyCon. The Name should be that of a new coercion CoAxiom, the TyVars the arguments expected by the newtype and the type the appropriate right hand side of the newtype, with the free variables a subset of those TyVars.

Create a coercion identifying a data, newtype or type representation type and its family instance. It has the form Co tvs :: F ts ~ R tvs, where Co is the coercion constructor built here, F the family tycon and R the (derived) representation tycon.

Sometimes we want to look through a newtype and get its associated coercion. This function only strips *one layer* of newtype off, so the caller will usually call itself recursively. Furthermore, this function should only be applied to types of kind *, hence the newtype is always saturated. If co : ty ~ ty' then:

 splitNewTypeRepCo_maybe ty = Just (ty', co)

The function returns Nothing for non-newtypes or fully-transparent newtypes.

If co :: T ts ~ rep_ty then:

 instNewTyCon_maybe T ts = Just (rep_ty, co)

This breaks a Coercion with type T A B C ~ T D E F into a list of Coercions of kinds A ~ D, B ~ E and E ~ F. Hence:

 decomposeCo 3 c = [nth 0 c, nth 1 c, nth 2 c]

Attempts to obtain the type variable underlying a Coercion

Attempts to tease a coercion apart into a type constructor and the application of a number of coercion arguments to that constructor

Attempt to take a coercion application apart.

A substitution of Coercions for CoVars (OR TyVars, when doing a "lifting" substitution)

Substitute within a Coercion

Substitute within several Coercions

liftCoMatch is sort of inverse to liftCoSubst. In particular, if liftCoMatch vars ty co == Just s, then tyCoSubst s ty == co. That is, it matches a type against a coercion of the same shape, and returns a lifting substitution which could have been used to produce the given coercion from the given type.

The "lifting" operation which substitutes coercions for type variables in a type to produce a coercion.

For the inverse operation, see liftCoMatch

Determines syntactic equality of coercions