Note [Role inference]

GHC/Tc/TyCl/Utils.hs:353 compiler

The role inference algorithm datatype definitions to infer the roles on the
parameters. Although these roles are stored in the tycons, we can perform this
algorithm on the built tycons, as long as we don't peek at an as-yet-unknown
roles field! Ah, the magic of laziness.

First, we choose appropriate initial roles. For families and classes, roles
(including initial roles) are N. For datatypes, we start with the role in the
role annotation (if any), or otherwise use Phantom. This is done in
initialRoleEnv1.

The function irGroup then propagates role information until it reaches a
fixpoint, preferring N over (R or P) and R over P. To aid in this, we have a
monad RoleM, which is a combination reader and state monad. In its state are
the current RoleEnv, which gets updated by role propagation, and an update
bit, which we use to know whether or not we've reached the fixpoint. The
environment of RoleM contains the tycon whose parameters we are inferring, and
a VarEnv from parameters to their positions, so we can update the RoleEnv.
Between tycons, this reader information is missing; it is added by
addRoleInferenceInfo.

There are two kinds of tycons to consider: algebraic ones (excluding classes)
and type synonyms. (Remember, families don't participate -- all their parameters
are N.) An algebraic tycon processes each of its datacons, in turn. Note that
a datacon's universally quantified parameters might be different from the parent
tycon's parameters, so we use the datacon's univ parameters in the mapping from
vars to positions. Note also that we don't want to infer roles for existentials
(they're all at N, too), so we put them in the set of local variables. As an
optimisation, we skip any tycons whose roles are already all Nominal, as there
nowhere else for them to go. For synonyms, we just analyse their right-hand sides.

irType walks through a type, looking for uses of a variable of interest and
propagating role information. Because anything used under a phantom position
is at phantom and anything used under a nominal position is at nominal, the
irType function can assume that anything it sees is at representational. (The
other possibilities are pruned when they're encountered.)

The rest of the code is just plumbing.

How do we know that this algorithm is correct? It should meet the following
specification:

Let Z be a role context -- a mapping from variables to roles. The following
rules define the property (Z |- t : r), where t is a type and r is a role:

Z(a) = r'        r' <= r
RCVar
Z |- a : r

RCConst
Z |- T : r               -- T is a type constructor

Z |- t1 : r
Z |- t2 : N
RCApp
Z |- t1 t2 : r

forall i<=n. (r_i is R or N) implies Z |- t_i : r_i
roles(T) = r_1 .. r_n
RCDApp
Z |- T t_1 .. t_n : R

Z, a:N |- t : r
RCAll
Z |- forall a:k.t : r


We also have the following rules:

For all datacon_i in type T, where a_1 .. a_n are universally quantified
and b_1 .. b_m are existentially quantified, and the arguments are t_1 .. t_p,
then if forall j<=p, a_1 : r_1 .. a_n : r_n, b_1 : N .. b_m : N |- t_j : R,
then roles(T) = r_1 .. r_n

roles(->) = R, R
roles(~#) = N, N

With -dcore-lint on, the output of this algorithm is checked in checkValidRoles,
called from checkValidTycon.

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