{
(c) The University of Glasgow 2006
(c) The GRASP/AQUA Project, Glasgow University, 1998
\section[GHC.Core.TyCo.Rep]{Type and Coercion  friends' interface}
Note [The Typerelated module hierarchy]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
GHC.Core.Class
GHC.Core.Coercion.Axiom
GHC.Core.TyCon imports GHC.Core.{Class, Coercion.Axiom}
GHC.Core.TyCo.Rep imports GHC.Core.{Class, Coercion.Axiom, TyCon}
GHC.Core.TyCo.Ppr imports GHC.Core.TyCo.Rep
GHC.Core.TyCo.FVs imports GHC.Core.TyCo.Rep
GHC.Core.TyCo.Subst imports GHC.Core.TyCo.{Rep, FVs, Ppr}
GHC.Core.TyCo.Tidy imports GHC.Core.TyCo.{Rep, FVs}
GHC.Builtin.Types.Prim imports GHC.Core.TyCo.Rep ( including mkTyConTy )
GHC.Core.Coercion imports GHC.Core.Type
}
 We expose the relevant stuff from this module via the Type module
{# OPTIONS_HADDOCK nothome #}
{# LANGUAGE CPP, MultiWayIf, PatternSynonyms, BangPatterns, DeriveDataTypeable #}
module GHC.Core.TyCo.Rep (
TyThing(..), tyThingCategory, pprTyThingCategory, pprShortTyThing,
 * Types
Type(..),
TyLit(..),
KindOrType, Kind,
KnotTied,
PredType, ThetaType,  Synonyms
ArgFlag(..), AnonArgFlag(..),
 * Coercions
Coercion(..),
UnivCoProvenance(..),
CoercionHole(..), BlockSubstFlag(..), coHoleCoVar, setCoHoleCoVar,
CoercionN, CoercionR, CoercionP, KindCoercion,
MCoercion(..), MCoercionR, MCoercionN,
 * Functions over types
mkTyConTy, mkTyVarTy, mkTyVarTys,
mkTyCoVarTy, mkTyCoVarTys,
mkFunTy, mkVisFunTy, mkInvisFunTy, mkVisFunTys,
mkForAllTy, mkForAllTys, mkInvisForAllTys,
mkPiTy, mkPiTys,
mkFunTyMany,
mkScaledFunTy,
mkVisFunTyMany, mkVisFunTysMany,
mkInvisFunTyMany, mkInvisFunTysMany,
mkTyConApp,
 * Functions over binders
TyCoBinder(..), TyCoVarBinder, TyBinder,
binderVar, binderVars, binderType, binderArgFlag,
delBinderVar,
isInvisibleArgFlag, isVisibleArgFlag,
isInvisibleBinder, isVisibleBinder,
isTyBinder, isNamedBinder,
 * Functions over coercions
pickLR,
 ** Analyzing types
TyCoFolder(..), foldTyCo,
 * Sizes
typeSize, coercionSize, provSize,
 * Multiplicities
Scaled(..), scaledMult, scaledThing, mapScaledType, Mult
) where
#include "HsVersions.h"
import GHC.Prelude
import {# SOURCE #} GHC.Core.TyCo.Ppr ( pprType, pprCo, pprTyLit )
 Transitively pulls in a LOT of stuff, better to break the loop
import {# SOURCE #} GHC.Core.ConLike ( ConLike(..), conLikeName )
 friends:
import GHC.Iface.Type
import GHC.Types.Var
import GHC.Types.Var.Set
import GHC.Types.Name hiding ( varName )
import GHC.Core.TyCon
import GHC.Core.Coercion.Axiom
 others
import GHC.Builtin.Names ( liftedTypeKindTyConKey, manyDataConKey )
import {# SOURCE #} GHC.Builtin.Types ( liftedTypeKindTyCon, manyDataConTy )
import GHC.Types.Basic ( LeftOrRight(..), pickLR )
import GHC.Types.Unique ( hasKey )
import GHC.Utils.Outputable
import GHC.Data.FastString
import GHC.Utils.Misc
import GHC.Utils.Panic
 libraries
import qualified Data.Data as Data hiding ( TyCon )
import Data.IORef ( IORef )  for CoercionHole
{
%************************************************************************
%* *
TyThing
%* *
%************************************************************************
Despite the fact that DataCon has to be imported via a hiboot route,
this module seems the right place for TyThing, because it's needed for
funTyCon and all the types in GHC.Builtin.Types.Prim.
It is also SOURCEimported into "GHC.Types.Name"
Note [ATyCon for classes]
~~~~~~~~~~~~~~~~~~~~~~~~~
Both classes and type constructors are represented in the type environment
as ATyCon. You can tell the difference, and get to the class, with
isClassTyCon :: TyCon > Bool
tyConClass_maybe :: TyCon > Maybe Class
The Class and its associated TyCon have the same Name.
}
  A global typecheckablething, essentially anything that has a name.
 Not to be confused with a 'TcTyThing', which is also a typecheckable
 thing but in the *local* context. See "GHC.Tc.Utils.Env" for how to retrieve
 a 'TyThing' given a 'Name'.
data TyThing
= AnId Id
 AConLike ConLike
 ATyCon TyCon  TyCons and classes; see Note [ATyCon for classes]
 ACoAxiom (CoAxiom Branched)
instance Outputable TyThing where
ppr = pprShortTyThing
instance NamedThing TyThing where  Can't put this with the type
getName (AnId id) = getName id  decl, because the DataCon instance
getName (ATyCon tc) = getName tc  isn't visible there
getName (ACoAxiom cc) = getName cc
getName (AConLike cl) = conLikeName cl
pprShortTyThing :: TyThing > SDoc
 c.f. GHC.Core.Ppr.TyThing.pprTyThing, which prints all the details
pprShortTyThing thing
= pprTyThingCategory thing <+> quotes (ppr (getName thing))
pprTyThingCategory :: TyThing > SDoc
pprTyThingCategory = text . capitalise . tyThingCategory
tyThingCategory :: TyThing > String
tyThingCategory (ATyCon tc)
 isClassTyCon tc = "class"
 otherwise = "type constructor"
tyThingCategory (ACoAxiom _) = "coercion axiom"
tyThingCategory (AnId _) = "identifier"
tyThingCategory (AConLike (RealDataCon _)) = "data constructor"
tyThingCategory (AConLike (PatSynCon _)) = "pattern synonym"
{ **********************************************************************
* *
Type
* *
********************************************************************** }
  The key representation of types within the compiler
type KindOrType = Type  See Note [Arguments to type constructors]
  The key type representing kinds in the compiler.
type Kind = Type
 If you edit this type, you may need to update the GHC formalism
 See Note [GHC Formalism] in GHC.Core.Lint
data Type
 See Note [Nontrivial definitional equality]
= TyVarTy Var  ^ Vanilla type or kind variable (*never* a coercion variable)
 AppTy
Type
Type  ^ Type application to something other than a 'TyCon'. Parameters:

 1) Function: must /not/ be a 'TyConApp' or 'CastTy',
 must be another 'AppTy', or 'TyVarTy'
 See Note [Respecting definitional equality] \(EQ1) about the
 no 'CastTy' requirement

 2) Argument type
 TyConApp
TyCon
[KindOrType]  ^ Application of a 'TyCon', including newtypes /and/ synonyms.
 Invariant: saturated applications of 'FunTyCon' must
 use 'FunTy' and saturated synonyms must use their own
 constructors. However, /unsaturated/ 'FunTyCon's
 do appear as 'TyConApp's.
 Parameters:

 1) Type constructor being applied to.

 2) Type arguments. Might not have enough type arguments
 here to saturate the constructor.
 Even type synonyms are not necessarily saturated;
 for example unsaturated type synonyms
 can appear as the right hand side of a type synonym.
 ForAllTy
{# UNPACK #} !TyCoVarBinder
Type  ^ A Π type.
 INVARIANT: If the binder is a coercion variable, it must
 be mentioned in the Type. See
 Note [Unused coercion variable in ForAllTy]
 FunTy  ^ FUN m t1 t2 Very common, so an important special case
 See Note [Function types]
{ ft_af :: AnonArgFlag  Is this (>) or (=>)?
, ft_mult :: Mult  Multiplicity
, ft_arg :: Type  Argument type
, ft_res :: Type }  Result type
 LitTy TyLit  ^ Type literals are similar to type constructors.
 CastTy
Type
KindCoercion  ^ A kind cast. The coercion is always nominal.
 INVARIANT: The cast is never reflexive
 INVARIANT: The Type is not a CastTy (use TransCo instead)
 INVARIANT: The Type is not a ForAllTy over a type variable
 See Note [Respecting definitional equality] \(EQ2), (EQ3), (EQ4)
 CoercionTy
Coercion  ^ Injection of a Coercion into a type
 This should only ever be used in the RHS of an AppTy,
 in the list of a TyConApp, when applying a promoted
 GADT data constructor
deriving Data.Data
instance Outputable Type where
ppr = pprType
 NOTE: Other parts of the code assume that type literals do not contain
 types or type variables.
data TyLit
= NumTyLit Integer
 StrTyLit FastString
deriving (Eq, Data.Data)
instance Ord TyLit where
compare (NumTyLit _) (StrTyLit _) = LT
compare (StrTyLit _) (NumTyLit _) = GT
compare (NumTyLit x) (NumTyLit y) = compare x y
compare (StrTyLit x) (StrTyLit y) = uniqCompareFS x y
instance Outputable TyLit where
ppr = pprTyLit
{ Note [Function types]
~~~~~~~~~~~~~~~~~~~~~~~~
FFunTy is the constructor for a function type. Lots of things to say
about it!
* FFunTy is the data constructor, meaning "full function type".
* The function type constructor (>) has kind
(>) :: forall {r1} {r2}. TYPE r1 > TYPE r2 > Type LiftedRep
mkTyConApp ensure that we convert a saturated application
TyConApp (>) [r1,r2,t1,t2] into FunTy t1 t2
dropping the 'r1' and 'r2' arguments; they are easily recovered
from 't1' and 't2'.
* For the time being its RuntimeRep quantifiers are left
inferred. This is to allow for it to evolve.
* The ft_af field says whether or not this is an invisible argument
VisArg: t1 > t2 Ordinary function type
InvisArg: t1 => t2 t1 is guaranteed to be a predicate type,
i.e. t1 :: Constraint
See Note [Types for coercions, predicates, and evidence]
This visibility info makes no difference in Core; it matters
only when we regard the type as a Haskell source type.
* FunTy is a (unidirectional) pattern synonym that allows
positional pattern matching (FunTy arg res), ignoring the
ArgFlag.
}
{ 
Commented out until the pattern match
checker can handle it; see #16185
For now we use the CPP macro #define FunTy FFunTy _
(see HsVersions.h) to allow pattern matching on a
(positional) FunTy constructor.
{# COMPLETE FunTy, TyVarTy, AppTy, TyConApp
, ForAllTy, LitTy, CastTy, CoercionTy :: Type #}
  'FunTy' is a (unidirectional) pattern synonym for the common
 case where we want to match on the argument/result type, but
 ignoring the AnonArgFlag
pattern FunTy :: Type > Type > Type
pattern FunTy arg res < FFunTy { ft_arg = arg, ft_res = res }
End of commented out block
 }
{ Note [Types for coercions, predicates, and evidence]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We treat differently:
(a) Predicate types
Test: isPredTy
Binders: DictIds
Kind: Constraint
Examples: (Eq a), and (a ~ b)
(b) Coercion types are primitive, unboxed equalities
Test: isCoVarTy
Binders: CoVars (can appear in coercions)
Kind: TYPE (TupleRep [])
Examples: (t1 ~# t2) or (t1 ~R# t2)
(c) Evidence types is the type of evidence manipulated by
the type constraint solver.
Test: isEvVarType
Binders: EvVars
Kind: Constraint or TYPE (TupleRep [])
Examples: all coercion types and predicate types
Coercion types and predicate types are mutually exclusive,
but evidence types are a superset of both.
When treated as a user type,
 Predicates (of kind Constraint) are invisible and are
implicitly instantiated
 Coercion types, and nonpred evidence types (i.e. not
of kind Constrain), are just regular old types, are
visible, and are not implicitly instantiated.
In a FunTy { ft_af = InvisArg }, the argument type is always
a Predicate type.
Note [Constraints in kinds]
~~~~~~~~~~~~~~~~~~~~~~~~~~~
Do we allow a type constructor to have a kind like
S :: Eq a => a > Type
No, we do not. Doing so would mean would need a TyConApp like
S @k @(d :: Eq k) (ty :: k)
and we have no way to build, or decompose, evidence like
(d :: Eq k) at the type level.
But we admit one exception: equality. We /do/ allow, say,
MkT :: (a ~ b) => a > b > Type a b
Why? Because we can, without much difficulty. Moreover
we can promote a GADT data constructor (see TyCon
Note [Promoted data constructors]), like
data GT a b where
MkGT : a > a > GT a a
so programmers might reasonably expect to be able to
promote MkT as well.
How does this work?
* In GHC.Tc.Validity.checkConstraintsOK we reject kinds that
have constraints other than (a~b) and (a~~b).
* In Inst.tcInstInvisibleTyBinder we instantiate a call
of MkT by emitting
[W] co :: alpha ~# beta
and producing the elaborated term
MkT @alpha @beta (Eq# alpha beta co)
We don't generate a boxed "Wanted"; we generate only a
regular old /unboxed/ primitiveequality Wanted, and build
the box on the spot.
* How can we get such a MkT? By promoting a GADTstyle data
constructor
data T a b where
MkT :: (a~b) => a > b > T a b
See DataCon.mkPromotedDataCon
and Note [Promoted data constructors] in GHC.Core.TyCon
* We support both homogeneous (~) and heterogeneous (~~)
equality. (See Note [The equality types story]
in GHC.Builtin.Types.Prim for a primer on these equality types.)
* How do we prevent a MkT having an illegal constraint like
Eq a? We check for this at usesites; see GHC.Tc.Gen.HsType.tcTyVar,
specifically dc_theta_illegal_constraint.
* Notice that nothing special happens if
K :: (a ~# b) => blah
because (a ~# b) is not a predicate type, and is never
implicitly instantiated. (Mind you, it's not clear how you
could creates a type constructor with such a kind.) See
Note [Types for coercions, predicates, and evidence]
* The existence of promoted MkT with an equalityconstraint
argument is the (only) reason that the AnonTCB constructor
of TyConBndrVis carries an AnonArgFlag (VisArg/InvisArg).
For example, when we promote the data constructor
MkT :: forall a b. (a~b) => a > b > T a b
we get a PromotedDataCon with tyConBinders
Bndr (a :: Type) (NamedTCB Inferred)
Bndr (b :: Type) (NamedTCB Inferred)
Bndr (_ :: a ~ b) (AnonTCB InvisArg)
Bndr (_ :: a) (AnonTCB VisArg))
Bndr (_ :: b) (AnonTCB VisArg))
* One might reasonably wonder who *unpacks* these boxes once they are
made. After all, there is no typelevel `case` construct. The
surprising answer is that no one ever does. Instead, if a GADT
constructor is used on the lefthand side of a type family equation,
that occurrence forces GHC to unify the types in question. For
example:
data G a where
MkG :: G Bool
type family F (x :: G a) :: a where
F MkG = False
When checking the LHS `F MkG`, GHC sees the MkG constructor and then must
unify F's implicit parameter `a` with Bool. This succeeds, making the equation
F Bool (MkG @Bool ) = False
Note that we never need unpack the coercion. This is because type
family equations are *not* parametric in their kind variables. That
is, we could have just said
type family H (x :: G a) :: a where
H _ = False
The presence of False on the RHS also forces `a` to become Bool,
giving us
H Bool _ = False
The fact that any of this works stems from the lack of phase
separation between types and kinds (unlike the very present phase
separation between terms and types).
Once we have the ability to patternmatch on types below toplevel,
this will no longer cut it, but it seems fine for now.
Note [Arguments to type constructors]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Because of kind polymorphism, in addition to type application we now
have kind instantiation. We reuse the same notations to do so.
For example:
Just (* > *) Maybe
Right * Nat Zero
are represented by:
TyConApp (PromotedDataCon Just) [* > *, Maybe]
TyConApp (PromotedDataCon Right) [*, Nat, (PromotedDataCon Zero)]
Important note: Nat is used as a *kind* and not as a type. This can be
confusing, since typelevel Nat and kindlevel Nat are identical. We
use the kind of (PromotedDataCon Right) to know if its arguments are
kinds or types.
This kind instantiation only happens in TyConApp currently.
Note [Nontrivial definitional equality]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Is Int > <*> the same as Int? YES! In order to reduce headaches,
we decide that any reflexive casts in types are just ignored.
(Indeed they must be. See Note [Respecting definitional equality].)
More generally, the `eqType` function, which defines Core's type equality
relation, ignores casts and coercion arguments, as long as the
two types have the same kind. This allows us to be a little sloppier
in keeping track of coercions, which is a good thing. It also means
that eqType does not depend on eqCoercion, which is also a good thing.
Why is this sensible? That is, why is something different than αequivalence
appropriate for the implementation of eqType?
Anything smaller than ~ and homogeneous is an appropriate definition for
equality. The type safety of FC depends only on ~. Let's say η : τ ~ σ. Any
expression of type τ can be transmuted to one of type σ at any point by
casting. The same is true of expressions of type σ. So in some sense, τ and σ
are interchangeable.
But let's be more precise. If we examine the typing rules of FC (say, those in
https://cs.brynmawr.edu/~rae/papers/2015/equalities/equalities.pdf)
there are several places where the same metavariable is used in two different
premises to a rule. (For example, see Ty_App.) There is an implicit equality
check here. What definition of equality should we use? By convention, we use
αequivalence. Take any rule with one (or more) of these implicit equality
checks. Then there is an admissible rule that uses ~ instead of the implicit
check, adding in casts as appropriate.
The only problem here is that ~ is heterogeneous. To make the kinds work out
in the admissible rule that uses ~, it is necessary to homogenize the
coercions. That is, if we have η : (τ : κ1) ~ (σ : κ2), then we don't use η;
we use η > kind η, which is homogeneous.
The effect of this all is that eqType, the implementation of the implicit
equality check, can use any homogeneous relation that is smaller than ~, as
those rules must also be admissible.
A more drawn out argument around all of this is presented in Section 7.2 of
Richard E's thesis (http://cs.brynmawr.edu/~rae/papers/2016/thesis/eisenbergthesis.pdf).
What would go wrong if we insisted on the casts matching? See the beginning of
Section 8 in the unpublished paper above. Theoretically, nothing at all goes
wrong. But in practical terms, getting the coercions right proved to be
nightmarish. And types would explode: during kindchecking, we often produce
reflexive kind coercions. When we try to cast by these, mkCastTy just discards
them. But if we used an eqType that distinguished between Int and Int > <*>,
then we couldn't discard  the output of kindchecking would be enormous,
and we would need enormous casts with lots of CoherenceCo's to straighten
them out.
Would anything go wrong if eqType respected type families? No, not at all. But
that makes eqType rather hard to implement.
Thus, the guideline for eqType is that it should be the largest
easytoimplement relation that is still smaller than ~ and homogeneous. The
precise choice of relation is somewhat incidental, as long as the smart
constructors and destructors in Type respect whatever relation is chosen.
Another helpful principle with eqType is this:
(EQ) If (t1 `eqType` t2) then I can replace t1 by t2 anywhere.
This principle also tells us that eqType must relate only types with the
same kinds.
Note [Respecting definitional equality]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Note [Nontrivial definitional equality] introduces the property (EQ).
How is this upheld?
Any function that pattern matches on all the constructors will have to
consider the possibility of CastTy. Presumably, those functions will handle
CastTy appropriately and we'll be OK.
More dangerous are the splitXXX functions. Let's focus on splitTyConApp.
We don't want it to fail on (T a b c > co). Happily, if we have
(T a b c > co) `eqType` (T d e f)
then co must be reflexive. Why? eqType checks that the kinds are equal, as
well as checking that (a `eqType` d), (b `eqType` e), and (c `eqType` f).
By the kind check, we know that (T a b c > co) and (T d e f) have the same
kind. So the only way that co could be nonreflexive is for (T a b c) to have
a different kind than (T d e f). But because T's kind is closed (all tycon kinds
are closed), the only way for this to happen is that one of the arguments has
to differ, leading to a contradiction. Thus, co is reflexive.
Accordingly, by eliminating reflexive casts, splitTyConApp need not worry
about outermost casts to uphold (EQ). Eliminating reflexive casts is done
in mkCastTy.
Unforunately, that's not the end of the story. Consider comparing
(T a b c) =? (T a b > (co > )) (c > co)
These two types have the same kind (Type), but the left type is a TyConApp
while the right type is not. To handle this case, we say that the righthand
type is illformed, requiring an AppTy never to have a casted TyConApp
on its left. It is easy enough to pull around the coercions to maintain
this invariant, as done in Type.mkAppTy. In the example above, trying to
form the righthand type will instead yield (T a b (c > co > sym co) > ).
Both the casts there are reflexive and will be dropped. Huzzah.
This idea of pulling coercions to the right works for splitAppTy as well.
However, there is one hiccup: it's possible that a coercion doesn't relate two
Pitypes. For example, if we have @type family Fun a b where Fun a b = a > b@,
then we might have (T :: Fun Type Type) and (T > axFun) Int. That axFun can't
be pulled to the right. But we don't need to pull it: (T > axFun) Int is not
`eqType` to any proper TyConApp  thus, leaving it where it is doesn't violate
our (EQ) property.
In order to detect reflexive casts reliably, we must make sure not
to have nested casts: we update (t > co1 > co2) to (t > (co1 `TransCo` co2)).
One other troublesome case is ForAllTy. See Note [Weird typing rule for ForAllTy].
The kind of the body is the same as the kind of the ForAllTy. Accordingly,
ForAllTy tv (ty > co) and (ForAllTy tv ty) > co
are `eqType`. But only the first can be split by splitForAllTy. So we forbid
the second form, instead pushing the coercion inside to get the first form.
This is done in mkCastTy.
In sum, in order to uphold (EQ), we need the following invariants:
(EQ1) No decomposable CastTy to the left of an AppTy, where a decomposable
cast is one that relates either a FunTy to a FunTy or a
ForAllTy to a ForAllTy.
(EQ2) No reflexive casts in CastTy.
(EQ3) No nested CastTys.
(EQ4) No CastTy over (ForAllTy (Bndr tyvar vis) body).
See Note [Weird typing rule for ForAllTy]
These invariants are all documented above, in the declaration for Type.
Note [Unused coercion variable in ForAllTy]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Suppose we have
\(co:t1 ~ t2). e
What type should we give to this expression?
(1) forall (co:t1 ~ t2) > t
(2) (t1 ~ t2) > t
If co is used in t, (1) should be the right choice.
if co is not used in t, we would like to have (1) and (2) equivalent.
However, we want to keep eqType simple and don't want eqType (1) (2) to return
True in any case.
We decide to always construct (2) if co is not used in t.
Thus in mkLamType, we check whether the variable is a coercion
variable (of type (t1 ~# t2), and whether it is unused in the
body. If so, it returns a FunTy instead of a ForAllTy.
There are cases we want to skip the check. For example, the check is
unnecessary when it is known from the context that the input variable
is a type variable. In those cases, we use mkForAllTy.
Note [Weird typing rule for ForAllTy]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Here is the (truncated) typing rule for the dependent ForAllTy:
inner : TYPE r
tyvar is not free in r

ForAllTy (Bndr tyvar vis) inner : TYPE r
Note that the kind of `inner` is the kind of the overall ForAllTy. This is
necessary because every ForAllTy over a type variable is erased at runtime.
Thus the runtime representation of a ForAllTy (as encoded, via TYPE rep, in
the kind) must be the same as the representation of the body. We must check
for skolemescape, though. The skolemescape would prevent a definition like
undefined :: forall (r :: RuntimeRep) (a :: TYPE r). a
because the type's kind (TYPE r) mentions the outofscope r. Luckily, the real
type of undefined is
undefined :: forall (r :: RuntimeRep) (a :: TYPE r). HasCallStack => a
and that HasCallStack constraint neatly sidesteps the potential skolemescape
problem.
If the bound variable is a coercion variable:
inner : TYPE r
covar is free in inner

ForAllTy (Bndr covar vis) inner : Type
Here, the kind of the ForAllTy is just Type, because coercion abstractions
are *not* erased. The "covar is free in inner" premise is solely to maintain
the representation invariant documented in
Note [Unused coercion variable in ForAllTy]. Though there is surface similarity
between this freevar check and the one in the tyvar rule, these two restrictions
are truly unrelated.
}
  A type labeled 'KnotTied' might have knottied tycons in it. See
 Note [Type checking recursive type and class declarations] in
 "GHC.Tc.TyCl"
type KnotTied ty = ty
{ **********************************************************************
* *
TyCoBinder and ArgFlag
* *
********************************************************************** }
  A 'TyCoBinder' represents an argument to a function. TyCoBinders can be
 dependent ('Named') or nondependent ('Anon'). They may also be visible or
 not. See Note [TyCoBinders]
data TyCoBinder
= Named TyCoVarBinder  A typelambda binder
 Anon AnonArgFlag (Scaled Type)  A termlambda binder. Type here can be CoercionTy.
 Visibility is determined by the AnonArgFlag
deriving Data.Data
instance Outputable TyCoBinder where
ppr (Anon af ty) = ppr af <+> ppr ty
ppr (Named (Bndr v Required)) = ppr v
 See Note [Explicit Case Statement for Specificity]
ppr (Named (Bndr v (Invisible spec))) = case spec of
SpecifiedSpec > char '@' <> ppr v
InferredSpec > braces (ppr v)
  'TyBinder' is like 'TyCoBinder', but there can only be 'TyVarBinder'
 in the 'Named' field.
type TyBinder = TyCoBinder
  Remove the binder's variable from the set, if the binder has
 a variable.
delBinderVar :: VarSet > TyCoVarBinder > VarSet
delBinderVar vars (Bndr tv _) = vars `delVarSet` tv
  Does this binder bind an invisible argument?
isInvisibleBinder :: TyCoBinder > Bool
isInvisibleBinder (Named (Bndr _ vis)) = isInvisibleArgFlag vis
isInvisibleBinder (Anon InvisArg _) = True
isInvisibleBinder (Anon VisArg _) = False
  Does this binder bind a visible argument?
isVisibleBinder :: TyCoBinder > Bool
isVisibleBinder = not . isInvisibleBinder
isNamedBinder :: TyCoBinder > Bool
isNamedBinder (Named {}) = True
isNamedBinder (Anon {}) = False
  If its a named binder, is the binder a tyvar?
 Returns True for nondependent binder.
 This check that we're really returning a *Ty*Binder (as opposed to a
 coercion binder). That way, if/when we allow coercion quantification
 in more places, we'll know we missed updating some function.
isTyBinder :: TyCoBinder > Bool
isTyBinder (Named bnd) = isTyVarBinder bnd
isTyBinder _ = True
{ Note [TyCoBinders]
~~~~~~~~~~~~~~~~~~~
A ForAllTy contains a TyCoVarBinder. But a type can be decomposed
to a telescope consisting of a [TyCoBinder]
A TyCoBinder represents the type of binders  that is, the type of an
argument to a Pitype. GHC Core currently supports two different
Pitypes:
* A nondependent function type,
written with >, e.g. ty1 > ty2
represented as FunTy ty1 ty2. These are
lifted to Coercions with the corresponding FunCo.
* A dependent compiletimeonly polytype,
written with forall, e.g. forall (a:*). ty
represented as ForAllTy (Bndr a v) ty
Both Pitypes classify terms/types that take an argument. In other
words, if `x` is either a function or a polytype, `x arg` makes sense
(for an appropriate `arg`).
Note [VarBndrs, TyCoVarBinders, TyConBinders, and visibility]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
* A ForAllTy (used for both types and kinds) contains a TyCoVarBinder.
Each TyCoVarBinder
Bndr a tvis
is equipped with tvis::ArgFlag, which says whether or not arguments
for this binder should be visible (explicit) in source Haskell.
* A TyCon contains a list of TyConBinders. Each TyConBinder
Bndr a cvis
is equipped with cvis::TyConBndrVis, which says whether or not type
and kind arguments for this TyCon should be visible (explicit) in
source Haskell.
This table summarises the visibility rules:

 Occurrences look like this
 GHC displays type as in Haskell source code

 Bndr a tvis :: TyCoVarBinder, in the binder of ForAllTy for a term
 tvis :: ArgFlag
 tvis = Inferred: f :: forall {a}. type Arg not allowed: f
f :: forall {co}. type Arg not allowed: f
 tvis = Specified: f :: forall a. type Arg optional: f or f @Int
 tvis = Required: T :: forall k > type Arg required: T *
 This last form is illegal in terms: See Note [No Required TyCoBinder in terms]

 Bndr k cvis :: TyConBinder, in the TyConBinders of a TyCon
 cvis :: TyConBndrVis
 cvis = AnonTCB: T :: kind > kind Required: T *
 cvis = NamedTCB Inferred: T :: forall {k}. kind Arg not allowed: T
 T :: forall {co}. kind Arg not allowed: T
 cvis = NamedTCB Specified: T :: forall k. kind Arg not allowed[1]: T
 cvis = NamedTCB Required: T :: forall k > kind Required: T *

[1] In types, in the Specified case, it would make sense to allow
optional kind applications, thus (T @*), but we have not
yet implemented that
 In term declarations 
* Inferred. Function defn, with no signature: f1 x = x
We infer f1 :: forall {a}. a > a, with 'a' Inferred
It's Inferred because it doesn't appear in any
userwritten signature for f1
* Specified. Function defn, with signature (implicit forall):
f2 :: a > a; f2 x = x
So f2 gets the type f2 :: forall a. a > a, with 'a' Specified
even though 'a' is not bound in the source code by an explicit forall
* Specified. Function defn, with signature (explicit forall):
f3 :: forall a. a > a; f3 x = x
So f3 gets the type f3 :: forall a. a > a, with 'a' Specified
* Inferred. Function defn, with signature (explicit forall), marked as inferred:
f4 :: forall {a}. a > a; f4 x = x
So f4 gets the type f4 :: forall {a}. a > a, with 'a' Inferred
It's Inferred because the user marked it as such, even though it does appear
in the userwritten signature for f4
* Inferred/Specified. Function signature with inferred kind polymorphism.
f5 :: a b > Int
So 'f5' gets the type f5 :: forall {k} (a:k>*) (b:k). a b > Int
Here 'k' is Inferred (it's not mentioned in the type),
but 'a' and 'b' are Specified.
* Specified. Function signature with explicit kind polymorphism
f6 :: a (b :: k) > Int
This time 'k' is Specified, because it is mentioned explicitly,
so we get f6 :: forall (k:*) (a:k>*) (b:k). a b > Int
* Similarly pattern synonyms:
Inferred  from inferred types (e.g. no pattern type signature)
 or from inferred kind polymorphism
 In type declarations 
* Inferred (k)
data T1 a b = MkT1 (a b)
Here T1's kind is T1 :: forall {k:*}. (k>*) > k > *
The kind variable 'k' is Inferred, since it is not mentioned
Note that 'a' and 'b' correspond to /Anon/ TyCoBinders in T1's kind,
and Anon binders don't have a visibility flag. (Or you could think
of Anon having an implicit Required flag.)
* Specified (k)
data T2 (a::k>*) b = MkT (a b)
Here T's kind is T :: forall (k:*). (k>*) > k > *
The kind variable 'k' is Specified, since it is mentioned in
the signature.
* Required (k)
data T k (a::k>*) b = MkT (a b)
Here T's kind is T :: forall k:* > (k>*) > k > *
The kind is Required, since it bound in a positional way in T's declaration
Every use of T must be explicitly applied to a kind
* Inferred (k1), Specified (k)
data T a b (c :: k) = MkT (a b) (Proxy c)
Here T's kind is T :: forall {k1:*} (k:*). (k1>*) > k1 > k > *
So 'k' is Specified, because it appears explicitly,
but 'k1' is Inferred, because it does not
Generally, in the list of TyConBinders for a TyCon,
* Inferred arguments always come first
* Specified, Anon and Required can be mixed
e.g.
data Foo (a :: Type) :: forall b. (a > b > Type) > Type where ...
Here Foo's TyConBinders are
[Required 'a', Specified 'b', Anon]
and its kind prints as
Foo :: forall a > forall b. (a > b > Type) > Type
See also Note [Required, Specified, and Inferred for types] in GHC.Tc.TyCl
 Printing 
We print forall types with enough syntax to tell you their visibility
flag. But this is not source Haskell, and these types may not all
be parsable.
Specified: a list of Specified binders is written between `forall` and `.`:
const :: forall a b. a > b > a
Inferred: like Specified, but every binder is written in braces:
f :: forall {k} (a:k). S k a > Int
Required: binders are put between `forall` and `>`:
T :: forall k > *
 Other points 
* In classic Haskell, all named binders (that is, the type variables in
a polymorphic function type f :: forall a. a > a) have been Inferred.
* Inferred variables correspond to "generalized" variables from the
Visible Type Applications paper (ESOP'16).
Note [No Required TyCoBinder in terms]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We don't allow Required foralls for term variables, including pattern
synonyms and data constructors. Why? Because then an application
would need a /compulsory/ type argument (possibly without an "@"?),
thus (f Int); and we don't have concrete syntax for that.
We could change this decision, but Required, Named TyCoBinders are rare
anyway. (Most are Anons.)
However the type of a term can (just about) have a required quantifier;
see Note [Required quantifiers in the type of a term] in GHC.Tc.Gen.Expr.
}
{ **********************************************************************
* *
PredType
* *
********************************************************************** }
  A type of the form @p@ of constraint kind represents a value whose type is
 the Haskell predicate @p@, where a predicate is what occurs before
 the @=>@ in a Haskell type.

 We use 'PredType' as documentation to mark those types that we guarantee to
 have this kind.

 It can be expanded into its representation, but:

 * The type checker must treat it as opaque

 * The rest of the compiler treats it as transparent

 Consider these examples:

 > f :: (Eq a) => a > Int
 > g :: (?x :: Int > Int) => a > Int
 > h :: (r\l) => {r} => {l::Int  r}

 Here the @Eq a@ and @?x :: Int > Int@ and @r\l@ are all called \"predicates\"
type PredType = Type
  A collection of 'PredType's
type ThetaType = [PredType]
{
(We don't support TREX records yet, but the setup is designed
to expand to allow them.)
A Haskell qualified type, such as that for f,g,h above, is
represented using
* a FunTy for the double arrow
* with a type of kind Constraint as the function argument
The predicate really does turn into a real extra argument to the
function. If the argument has type (p :: Constraint) then the predicate p is
represented by evidence of type p.
%************************************************************************
%* *
Simple constructors
%* *
%************************************************************************
These functions are here so that they can be used by GHC.Builtin.Types.Prim,
which in turn is imported by Type
}
mkTyVarTy :: TyVar > Type
mkTyVarTy v = ASSERT2( isTyVar v, ppr v <+> dcolon <+> ppr (tyVarKind v) )
TyVarTy v
mkTyVarTys :: [TyVar] > [Type]
mkTyVarTys = map mkTyVarTy  a common use of mkTyVarTy
mkTyCoVarTy :: TyCoVar > Type
mkTyCoVarTy v
 isTyVar v
= TyVarTy v
 otherwise
= CoercionTy (CoVarCo v)
mkTyCoVarTys :: [TyCoVar] > [Type]
mkTyCoVarTys = map mkTyCoVarTy
infixr 3 `mkFunTy`, `mkVisFunTy`, `mkInvisFunTy`, `mkVisFunTyMany`,
`mkInvisFunTyMany`  Associates to the right
mkFunTy :: AnonArgFlag > Mult > Type > Type > Type
mkFunTy af mult arg res = FunTy { ft_af = af
, ft_mult = mult
, ft_arg = arg
, ft_res = res }
mkScaledFunTy :: AnonArgFlag > Scaled Type > Type > Type
mkScaledFunTy af (Scaled mult arg) res = mkFunTy af mult arg res
mkVisFunTy, mkInvisFunTy :: Mult > Type > Type > Type
mkVisFunTy = mkFunTy VisArg
mkInvisFunTy = mkFunTy InvisArg
mkFunTyMany :: AnonArgFlag > Type > Type > Type
mkFunTyMany af = mkFunTy af manyDataConTy
  Special, common, case: Arrow type with mult Many
mkVisFunTyMany :: Type > Type > Type
mkVisFunTyMany = mkVisFunTy manyDataConTy
mkInvisFunTyMany :: Type > Type > Type
mkInvisFunTyMany = mkInvisFunTy manyDataConTy
  Make nested arrow types
mkVisFunTys :: [Scaled Type] > Type > Type
mkVisFunTys tys ty = foldr (mkScaledFunTy VisArg) ty tys
mkVisFunTysMany :: [Type] > Type > Type
mkVisFunTysMany tys ty = foldr mkVisFunTyMany ty tys
mkInvisFunTysMany :: [Type] > Type > Type
mkInvisFunTysMany tys ty = foldr mkInvisFunTyMany ty tys
  Like 'mkTyCoForAllTy', but does not check the occurrence of the binder
 See Note [Unused coercion variable in ForAllTy]
mkForAllTy :: TyCoVar > ArgFlag > Type > Type
mkForAllTy tv vis ty = ForAllTy (Bndr tv vis) ty
  Wraps foralls over the type using the provided 'TyCoVar's from left to right
mkForAllTys :: [TyCoVarBinder] > Type > Type
mkForAllTys tyvars ty = foldr ForAllTy ty tyvars
  Wraps foralls over the type using the provided 'InvisTVBinder's from left to right
mkInvisForAllTys :: [InvisTVBinder] > Type > Type
mkInvisForAllTys tyvars ty = foldr ForAllTy ty $ tyVarSpecToBinders tyvars
mkPiTy :: TyCoBinder > Type > Type
mkPiTy (Anon af ty1) ty2 = mkScaledFunTy af ty1 ty2
mkPiTy (Named (Bndr tv vis)) ty = mkForAllTy tv vis ty
mkPiTys :: [TyCoBinder] > Type > Type
mkPiTys tbs ty = foldr mkPiTy ty tbs
  Create the plain type constructor type which has been applied to no type arguments at all.
mkTyConTy :: TyCon > Type
mkTyConTy tycon = TyConApp tycon []
  A key function: builds a 'TyConApp' or 'FunTy' as appropriate to
 its arguments. Applies its arguments to the constructor from left to right.
mkTyConApp :: TyCon > [Type] > Type
mkTyConApp tycon tys
 isFunTyCon tycon
, [w, _rep1,_rep2,ty1,ty2] < tys
 The FunTyCon (>) is always a visible one
= FunTy { ft_af = VisArg, ft_mult = w, ft_arg = ty1, ft_res = ty2 }
 Note [mkTyConApp and Type]
 tycon `hasKey` liftedTypeKindTyConKey
= ASSERT2( null tys, ppr tycon $$ ppr tys )
liftedTypeKindTyConApp
 tycon `hasKey` manyDataConKey
 There are a lot of occurrences of 'Many' so it's a small optimisation to
 avoid reboxing every time `mkTyConApp` is called.
= ASSERT2( null tys, ppr tycon $$ ppr tys )
manyDataConTy
 otherwise
= TyConApp tycon tys
 This is a single, global definition of the type `Type`
 Defined here so it is only allocated once.
 See Note [mkTyConApp and Type]
liftedTypeKindTyConApp :: Type
liftedTypeKindTyConApp = TyConApp liftedTypeKindTyCon []
{
Note [mkTyConApp and Type]
~~~~~~~~~~~~~~~~~~~~~~~~~~
Whilst benchmarking it was observed in #17292 that GHC allocated a lot
of `TyConApp` constructors. Upon further inspection a large number of these
TyConApp constructors were all duplicates of `Type` applied to no arguments.
```
(From a sample of 100000 TyConApp closures)
0x45f3523  28732  `Type`
0x420b840702  9629  generic type constructors
0x42055b7e46  9596
0x420559b582  9511
0x420bb15a1e  9509
0x420b86c6ba  9501
0x42055bac1e  9496
0x45e68fd  538  `TYPE ...`
```
Therefore in `mkTyConApp` we have a special case for `Type` to ensure that
only one `TyConApp 'Type []` closure is allocated during the course of
compilation. In order to avoid a potentially expensive series of checks in
`mkTyConApp` only this egregious case is special cased at the moment.
}
{
%************************************************************************
%* *
Coercions
%* *
%************************************************************************
}
  A 'Coercion' is concrete evidence of the equality/convertibility
 of two types.
 If you edit this type, you may need to update the GHC formalism
 See Note [GHC Formalism] in GHC.Core.Lint
data Coercion
 Each constructor has a "role signature", indicating the way roles are
 propagated through coercions.
  P, N, and R stand for coercions of the given role
  e stands for a coercion of a specific unknown role
 (think "role polymorphism")
  "e" stands for an explicit role parameter indicating role e.
  _ stands for a parameter that is not a Role or Coercion.
 These ones mirror the shape of types
=  Refl :: _ > N
Refl Type  See Note [Refl invariant]
 Invariant: applications of (Refl T) to a bunch of identity coercions
 always show up as Refl.
 For example (Refl T) (Refl a) (Refl b) shows up as (Refl (T a b)).
 Applications of (Refl T) to some coercions, at least one of
 which is NOT the identity, show up as TyConAppCo.
 (They may not be fully saturated however.)
 ConAppCo coercions (like all coercions other than Refl)
 are NEVER the identity.
 Use (GRefl Representational ty MRefl), not (SubCo (Refl ty))
 GRefl :: "e" > _ > Maybe N > e
 See Note [Generalized reflexive coercion]
 GRefl Role Type MCoercionN  See Note [Refl invariant]
 Use (Refl ty), not (GRefl Nominal ty MRefl)
 Use (GRefl Representational _ _), not (SubCo (GRefl Nominal _ _))
 These ones simply lift the correspondinglynamed
 Type constructors into Coercions
 TyConAppCo :: "e" > _ > ?? > e
 See Note [TyConAppCo roles]
 TyConAppCo Role TyCon [Coercion]  lift TyConApp
 The TyCon is never a synonym;
 we expand synonyms eagerly
 But it can be a type function
 TyCon is never a saturated (>); use FunCo instead
 AppCo Coercion CoercionN  lift AppTy
 AppCo :: e > N > e
 See Note [Forall coercions]
 ForAllCo TyCoVar KindCoercion Coercion
 ForAllCo :: _ > N > e > e
 FunCo Role CoercionN Coercion Coercion  lift FunTy
 FunCo :: "e" > N > e > e > e
 Note: why doesn't FunCo have a AnonArgFlag, like FunTy?
 Because the AnonArgFlag has no impact on Core; it is only
 there to guide implicit instantiation of Haskell source
 types, and that is irrelevant for coercions, which are
 Coreonly.
 These are special
 CoVarCo CoVar  :: _ > (N or R)
 result role depends on the tycon of the variable's type
 AxiomInstCo :: e > _ > ?? > e
 AxiomInstCo (CoAxiom Branched) BranchIndex [Coercion]
 See also [CoAxiom index]
 The coercion arguments always *precisely* saturate
 arity of (that branch of) the CoAxiom. If there are
 any left over, we use AppCo.
 See [Coercion axioms applied to coercions]
 The roles of the argument coercions are determined
 by the cab_roles field of the relevant branch of the CoAxiom
 AxiomRuleCo CoAxiomRule [Coercion]
 AxiomRuleCo is very like AxiomInstCo, but for a CoAxiomRule
 The number coercions should match exactly the expectations
 of the CoAxiomRule (i.e., the rule is fully saturated).
 UnivCo UnivCoProvenance Role Type Type
 :: _ > "e" > _ > _ > e
 SymCo Coercion  :: e > e
 TransCo Coercion Coercion  :: e > e > e
 NthCo Role Int Coercion  Zeroindexed; decomposes (T t0 ... tn)
 :: "e" > _ > e0 > e (inverse of TyConAppCo, see Note [TyConAppCo roles])
 Using NthCo on a ForAllCo gives an N coercion always
 See Note [NthCo and newtypes]

 Invariant: (NthCo r i co), it is always the case that r = role of (Nth i co)
 That is: the role of the entire coercion is redundantly cached here.
 See Note [NthCo Cached Roles]
 LRCo LeftOrRight CoercionN  Decomposes (t_left t_right)
 :: _ > N > N
 InstCo Coercion CoercionN
 :: e > N > e
 See Note [InstCo roles]
 Extract a kind coercion from a (heterogeneous) type coercion
 NB: all kind coercions are Nominal
 KindCo Coercion
 :: e > N
 SubCo CoercionN  Turns a ~N into a ~R
 :: N > R
 HoleCo CoercionHole  ^ See Note [Coercion holes]
 Only present during typechecking
deriving Data.Data
type CoercionN = Coercion  always nominal
type CoercionR = Coercion  always representational
type CoercionP = Coercion  always phantom
type KindCoercion = CoercionN  always nominal
instance Outputable Coercion where
ppr = pprCo
  A semantically more meaningful type to represent what may or may not be a
 useful 'Coercion'.
data MCoercion
= MRefl
 A trivial Reflexivity coercion
 MCo Coercion
 Other coercions
deriving Data.Data
type MCoercionR = MCoercion
type MCoercionN = MCoercion
instance Outputable MCoercion where
ppr MRefl = text "MRefl"
ppr (MCo co) = text "MCo" <+> ppr co
{
Note [Refl invariant]
~~~~~~~~~~~~~~~~~~~~~
Invariant 1:
Coercions have the following invariant
Refl (similar for GRefl r ty MRefl) is always lifted as far as possible.
You might think that a consequences is:
Every identity coercions has Refl at the root
But that's not quite true because of coercion variables. Consider
g where g :: Int~Int
Left h where h :: Maybe Int ~ Maybe Int
etc. So the consequence is only true of coercions that
have no coercion variables.
Note [Generalized reflexive coercion]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
GRefl is a generalized reflexive coercion (see #15192). It wraps a kind
coercion, which might be reflexive (MRefl) or any coercion (MCo co). The typing
rules for GRefl:
ty : k1

GRefl r ty MRefl: ty ~r ty
ty : k1 co :: k1 ~ k2

GRefl r ty (MCo co) : ty ~r ty > co
Consider we have
g1 :: s ~r t
s :: k1
g2 :: k1 ~ k2
and we want to construct a coercions co which has type
(s > g2) ~r t
We can define
co = Sym (GRefl r s g2) ; g1
It is easy to see that
Refl == GRefl Nominal ty MRefl :: ty ~n ty
A nominal reflexive coercion is quite common, so we keep the special form Refl to
save allocation.
Note [Coercion axioms applied to coercions]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The reason coercion axioms can be applied to coercions and not just
types is to allow for better optimization. There are some cases where
we need to be able to "push transitivity inside" an axiom in order to
expose further opportunities for optimization.
For example, suppose we have
C a : t[a] ~ F a
g : b ~ c
and we want to optimize
sym (C b) ; t[g] ; C c
which has the kind
F b ~ F c
(stopping through t[b] and t[c] along the way).
We'd like to optimize this to just F g  but how? The key is
that we need to allow axioms to be instantiated by *coercions*,
not just by types. Then we can (in certain cases) push
transitivity inside the axiom instantiations, and then react
oppositepolarity instantiations of the same axiom. In this
case, e.g., we match t[g] against the LHS of (C c)'s kind, to
obtain the substitution a > g (note this operation is sort
of the dual of lifting!) and hence end up with
C g : t[b] ~ F c
which indeed has the same kind as t[g] ; C c.
Now we have
sym (C b) ; C g
which can be optimized to F g.
Note [CoAxiom index]
~~~~~~~~~~~~~~~~~~~~
A CoAxiom has 1 or more branches. Each branch has contains a list
of the free type variables in that branch, the LHS type patterns,
and the RHS type for that branch. When we apply an axiom to a list
of coercions, we must choose which branch of the axiom we wish to
use, as the different branches may have different numbers of free
type variables. (The number of type patterns is always the same
among branches, but that doesn't quite concern us here.)
The Int in the AxiomInstCo constructor is the 0indexed number
of the chosen branch.
Note [Forall coercions]
~~~~~~~~~~~~~~~~~~~~~~~
Constructing coercions between foralltypes can be a bit tricky,
because the kinds of the bound tyvars can be different.
The typing rule is:
kind_co : k1 ~ k2
tv1:k1  co : t1 ~ t2

ForAllCo tv1 kind_co co : all tv1:k1. t1 ~
all tv1:k2. (t2[tv1 > tv1 > sym kind_co])
First, the TyCoVar stored in a ForAllCo is really an optimisation: this field
should be a Name, as its kind is redundant. Thinking of the field as a Name
is helpful in understanding what a ForAllCo means.
The kind of TyCoVar always matches the lefthand kind of the coercion.
The idea is that kind_co gives the two kinds of the tyvar. See how, in the
conclusion, tv1 is assigned kind k1 on the left but kind k2 on the right.
Of course, a type variable can't have different kinds at the same time. So,
we arbitrarily prefer the first kind when using tv1 in the inner coercion
co, which shows that t1 equals t2.
The last wrinkle is that we need to fix the kinds in the conclusion. In
t2, tv1 is assumed to have kind k1, but it has kind k2 in the conclusion of
the rule. So we do a kindfixing substitution, replacing (tv1:k1) with
(tv1:k2) > sym kind_co. This substitution is slightly bizarre, because it
mentions the same name with different kinds, but it *is* wellkinded, noting
that `(tv1:k2) > sym kind_co` has kind k1.
This all really would work storing just a Name in the ForAllCo. But we can't
add Names to, e.g., VarSets, and there generally is just an impedance mismatch
in a bunch of places. So we use tv1. When we need tv2, we can use
setTyVarKind.
Note [Predicate coercions]
~~~~~~~~~~~~~~~~~~~~~~~~~~
Suppose we have
g :: a~b
How can we coerce between types
([c]~a) => [a] > c
and
([c]~b) => [b] > c
where the equality predicate *itself* differs?
Answer: we simply treat (~) as an ordinary type constructor, so these
types really look like
((~) [c] a) > [a] > c
((~) [c] b) > [b] > c
So the coercion between the two is obviously
((~) [c] g) > [g] > c
Another way to see this to say that we simply collapse predicates to
their representation type (see Type.coreView and Type.predTypeRep).
This collapse is done by mkPredCo; there is no PredCo constructor
in Coercion. This is important because we need Nth to work on
predicates too:
Nth 1 ((~) [c] g) = g
See Simplify.simplCoercionF, which generates such selections.
Note [Roles]
~~~~~~~~~~~~
Roles are a solution to the GeneralizedNewtypeDeriving problem, articulated
in #1496. The full story is in docs/corespec/corespec.pdf. Also, see
https://gitlab.haskell.org/ghc/ghc/wikis/rolesimplementation
Here is one way to phrase the problem:
Given:
newtype Age = MkAge Int
type family F x
type instance F Age = Bool
type instance F Int = Char
This compiles down to:
axAge :: Age ~ Int
axF1 :: F Age ~ Bool
axF2 :: F Int ~ Char
Then, we can make:
(sym (axF1) ; F axAge ; axF2) :: Bool ~ Char
Yikes!
The solution is _roles_, as articulated in "Generative Type Abstraction and
Typelevel Computation" (POPL 2010), available at
http://www.seas.upenn.edu/~sweirich/papers/popl163afweirich.pdf
The specification for roles has evolved somewhat since that paper. For the
current full details, see the documentation in docs/corespec. Here are some
highlights.
We label every equality with a notion of type equivalence, of which there are
three options: Nominal, Representational, and Phantom. A ground type is
nominally equivalent only with itself. A newtype (which is considered a ground
type in Haskell) is representationally equivalent to its representation.
Anything is "phantomly" equivalent to anything else. We use "N", "R", and "P"
to denote the equivalences.
The axioms above would be:
axAge :: Age ~R Int
axF1 :: F Age ~N Bool
axF2 :: F Age ~N Char
Then, because transitivity applies only to coercions proving the same notion
of equivalence, the above construction is impossible.
However, there is still an escape hatch: we know that any two types that are
nominally equivalent are representationally equivalent as well. This is what
the form SubCo proves  it "demotes" a nominal equivalence into a
representational equivalence. So, it would seem the following is possible:
sub (sym axF1) ; F axAge ; sub axF2 :: Bool ~R Char  WRONG
What saves us here is that the arguments to a type function F, lifted into a
coercion, *must* prove nominal equivalence. So, (F axAge) is illformed, and
we are safe.
Roles are attached to parameters to TyCons. When lifting a TyCon into a
coercion (through TyConAppCo), we need to ensure that the arguments to the
TyCon respect their roles. For example:
data T a b = MkT a (F b)
If we know that a1 ~R a2, then we know (T a1 b) ~R (T a2 b). But, if we know
that b1 ~R b2, we know nothing about (T a b1) and (T a b2)! This is because
the type function F branches on b's *name*, not representation. So, we say
that 'a' has role Representational and 'b' has role Nominal. The third role,
Phantom, is for parameters not used in the type's definition. Given the
following definition
data Q a = MkQ Int
the Phantom role allows us to say that (Q Bool) ~R (Q Char), because we
can construct the coercion Bool ~P Char (using UnivCo).
See the paper cited above for more examples and information.
Note [TyConAppCo roles]
~~~~~~~~~~~~~~~~~~~~~~~
The TyConAppCo constructor has a role parameter, indicating the role at
which the coercion proves equality. The choice of this parameter affects
the required roles of the arguments of the TyConAppCo. To help explain
it, assume the following definition:
type instance F Int = Bool  Axiom axF : F Int ~N Bool
newtype Age = MkAge Int  Axiom axAge : Age ~R Int
data Foo a = MkFoo a  Role on Foo's parameter is Representational
TyConAppCo Nominal Foo axF : Foo (F Int) ~N Foo Bool
For (TyConAppCo Nominal) all arguments must have role Nominal. Why?
So that Foo Age ~N Foo Int does *not* hold.
TyConAppCo Representational Foo (SubCo axF) : Foo (F Int) ~R Foo Bool
TyConAppCo Representational Foo axAge : Foo Age ~R Foo Int
For (TyConAppCo Representational), all arguments must have the roles
corresponding to the result of tyConRoles on the TyCon. This is the
whole point of having roles on the TyCon to begin with. So, we can
have Foo Age ~R Foo Int, if Foo's parameter has role R.
If a Representational TyConAppCo is oversaturated (which is otherwise fine),
the spillover arguments must all be at Nominal. This corresponds to the
behavior for AppCo.
TyConAppCo Phantom Foo (UnivCo Phantom Int Bool) : Foo Int ~P Foo Bool
All arguments must have role Phantom. This one isn't strictly
necessary for soundness, but this choice removes ambiguity.
The rules here dictate the roles of the parameters to mkTyConAppCo
(should be checked by Lint).
Note [NthCo and newtypes]
~~~~~~~~~~~~~~~~~~~~~~~~~
Suppose we have
newtype N a = MkN Int
type role N representational
This yields axiom
NTCo:N :: forall a. N a ~R Int
We can then build
co :: forall a b. N a ~R N b
co = NTCo:N a ; sym (NTCo:N b)
for any `a` and `b`. Because of the role annotation on N, if we use
NthCo, we'll get out a representational coercion. That is:
NthCo r 0 co :: forall a b. a ~R b
Yikes! Clearly, this is terrible. The solution is simple: forbid
NthCo to be used on newtypes if the internal coercion is representational.
This is not just some corner case discovered by a segfault somewhere;
it was discovered in the proof of soundness of roles and described
in the "Safe Coercions" paper (ICFP '14).
Note [NthCo Cached Roles]
~~~~~~~~~~~~~~~~~~~~~~~~~
Why do we cache the role of NthCo in the NthCo constructor?
Because computing role(Nth i co) involves figuring out that
co :: T tys1 ~ T tys2
using coercionKind, and finding (coercionRole co), and then looking
at the tyConRoles of T. Avoiding bad asymptotic behaviour here means
we have to compute the kind and role of a coercion simultaneously,
which makes the code complicated and inefficient.
This only happens for NthCo. Caching the role solves the problem, and
allows coercionKind and coercionRole to be simple.
See #11735
Note [InstCo roles]
~~~~~~~~~~~~~~~~~~~
Here is (essentially) the typing rule for InstCo:
g :: (forall a. t1) ~r (forall a. t2)
w :: s1 ~N s2
 InstCo
InstCo g w :: (t1 [a > s1]) ~r (t2 [a > s2])
Note that the Coercion w *must* be nominal. This is necessary
because the variable a might be used in a "nominal position"
(that is, a place where role inference would require a nominal
role) in t1 or t2. If we allowed w to be representational, we
could get bogus equalities.
A more nuanced treatment might be able to relax this condition
somewhat, by checking if t1 and/or t2 use their bound variables
in nominal ways. If not, having w be representational is OK.
%************************************************************************
%* *
UnivCoProvenance
%* *
%************************************************************************
A UnivCo is a coercion whose proof does not directly express its role
and kind (indeed for some UnivCos, like PluginProv, there /is/ no proof).
The different kinds of UnivCo are described by UnivCoProvenance. Really
each is entirely separate, but they all share the need to represent their
role and kind, which is done in the UnivCo constructor.
}
  For simplicity, we have just one UnivCo that represents a coercion from
 some type to some other type, with (in general) no restrictions on the
 type. The UnivCoProvenance specifies more exactly what the coercion really
 is and why a program should (or shouldn't!) trust the coercion.
 It is reasonable to consider each constructor of 'UnivCoProvenance'
 as a totally independent coercion form; their only commonality is
 that they don't tell you what types they coercion between. (That info
 is in the 'UnivCo' constructor of 'Coercion'.
data UnivCoProvenance
= PhantomProv KindCoercion  ^ See Note [Phantom coercions]. Only in Phantom
 roled coercions
 ProofIrrelProv KindCoercion  ^ From the fact that any two coercions are
 considered equivalent. See Note [ProofIrrelProv].
 Can be used in Nominal or Representational coercions
 PluginProv String  ^ From a plugin, which asserts that this coercion
 is sound. The string is for the use of the plugin.
deriving Data.Data
instance Outputable UnivCoProvenance where
ppr (PhantomProv _) = text "(phantom)"
ppr (ProofIrrelProv _) = text "(proof irrel.)"
ppr (PluginProv str) = parens (text "plugin" <+> brackets (text str))
  A coercion to be filled in by the typechecker. See Note [Coercion holes]
data CoercionHole
= CoercionHole { ch_co_var :: CoVar
 See Note [CoercionHoles and coercion free variables]
, ch_blocker :: BlockSubstFlag  should this hole block substitution?
 See (2a) in TcCanonical
 Note [Equalities with incompatible kinds]
, ch_ref :: IORef (Maybe Coercion)
}
data BlockSubstFlag = YesBlockSubst
 NoBlockSubst
coHoleCoVar :: CoercionHole > CoVar
coHoleCoVar = ch_co_var
setCoHoleCoVar :: CoercionHole > CoVar > CoercionHole
setCoHoleCoVar h cv = h { ch_co_var = cv }
instance Data.Data CoercionHole where
 don't traverse?
toConstr _ = abstractConstr "CoercionHole"
gunfold _ _ = error "gunfold"
dataTypeOf _ = mkNoRepType "CoercionHole"
instance Outputable CoercionHole where
ppr (CoercionHole { ch_co_var = cv }) = braces (ppr cv)
instance Outputable BlockSubstFlag where
ppr YesBlockSubst = text "YesBlockSubst"
ppr NoBlockSubst = text "NoBlockSubst"
{ Note [Phantom coercions]
~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider
data T a = T1  T2
Then we have
T s ~R T t
for any old s,t. The witness for this is (TyConAppCo T Rep co),
where (co :: s ~P t) is a phantom coercion built with PhantomProv.
The role of the UnivCo is always Phantom. The Coercion stored is the
(nominal) kind coercion between the types
kind(s) ~N kind (t)
Note [Coercion holes]
~~~~~~~~~~~~~~~~~~~~~~~~
During typechecking, constraint solving for type classes works by
 Generate an evidence Id, d7 :: Num a
 Wrap it in a Wanted constraint, [W] d7 :: Num a
 Use the evidence Id where the evidence is needed
 Solve the constraint later
 When solved, add an enclosing letbinding let d7 = .... in ....
which actually binds d7 to the (Num a) evidence
For equality constraints we use a different strategy. See Note [The
equality types story] in GHC.Builtin.Types.Prim for background on equality constraints.
 For /boxed/ equality constraints, (t1 ~N t2) and (t1 ~R t2), it's just
like type classes above. (Indeed, boxed equality constraints *are* classes.)
 But for /unboxed/ equality constraints (t1 ~R# t2) and (t1 ~N# t2)
we use a different plan
For unboxed equalities:
 Generate a CoercionHole, a mutable variable just like a unification
variable
 Wrap the CoercionHole in a Wanted constraint; see GHC.Tc.Utils.TcEvDest
 Use the CoercionHole in a Coercion, via HoleCo
 Solve the constraint later
 When solved, fill in the CoercionHole by side effect, instead of
doing the letbinding thing
The main reason for all this is that there may be no good place to letbind
the evidence for unboxed equalities:
 We emit constraints for kind coercions, to be used to cast a
type's kind. These coercions then must be used in types. Because
they might appear in a toplevel type, there is no place to bind
these (unlifted) coercions in the usual way.
 A coercion for (forall a. t1) ~ (forall a. t2) will look like
forall a. (coercion for t1~t2)
But the coercion for (t1~t2) may mention 'a', and we don't have
letbindings within coercions. We could add them, but coercion
holes are easier.
 Moreover, nothing is lost from the lack of letbindings. For
dictionaries want to achieve sharing to avoid recomoputing the
dictionary. But coercions are entirely erased, so there's little
benefit to sharing. Indeed, even if we had a letbinding, we
always inline types and coercions at every use site and drop the
binding.
Other notes about HoleCo:
* INVARIANT: CoercionHole and HoleCo are used only during type checking,
and should never appear in Core. Just like unification variables; a Type
can contain a TcTyVar, but only during type checking. If, one day, we
use typelevel information to separate out forms that can appear during
typechecking vs forms that can appear in core proper, holes in Core will
be ruled out.
* See Note [CoercionHoles and coercion free variables]
* Coercion holes can be compared for equality like other coercions:
by looking at the types coerced.
Note [CoercionHoles and coercion free variables]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Why does a CoercionHole contain a CoVar, as well as reference to
fill in? Because we want to treat that CoVar as a free variable of
the coercion. See #14584, and Note [What prevents a
constraint from floating] in GHC.Tc.Solver, item (4):
forall k. [W] co1 :: t1 ~# t2 > co2
[W] co2 :: k ~# *
Here co2 is a CoercionHole. But we /must/ know that it is free in
co1, because that's all that stops it floating outside the
implication.
Note [ProofIrrelProv]
~~~~~~~~~~~~~~~~~~~~~
A ProofIrrelProv is a coercion between coercions. For example:
data G a where
MkG :: G Bool
In core, we get
G :: * > *
MkG :: forall (a :: *). (a ~ Bool) > G a
Now, consider 'MkG  that is, MkG used in a type  and suppose we want
a proof that ('MkG a1 co1) ~ ('MkG a2 co2). This will have to be
TyConAppCo Nominal MkG [co3, co4]
where
co3 :: co1 ~ co2
co4 :: a1 ~ a2
Note that
co1 :: a1 ~ Bool
co2 :: a2 ~ Bool
Here,
co3 = UnivCo (ProofIrrelProv co5) Nominal (CoercionTy co1) (CoercionTy co2)
where
co5 :: (a1 ~ Bool) ~ (a2 ~ Bool)
co5 = TyConAppCo Nominal (~#) [<*>, <*>, co4, ]
}
{ *********************************************************************
* *
foldType and foldCoercion
* *
********************************************************************* }
{ Note [foldType]
~~~~~~~~~~~~~~~~~~
foldType is a bit more powerful than perhaps it looks:
* You can fold with an accumulating parameter, via
TyCoFolder env (Endo a)
Recall newtype Endo a = Endo (a>a)
* You can fold monadically with a monad M, via
TyCoFolder env (M a)
provided you have
instance .. => Monoid (M a)
Note [mapType vs foldType]
~~~~~~~~~~~~~~~~~~~~~~~~~~
We define foldType here, but mapType in module Type. Why?
* foldType is used in GHC.Core.TyCo.FVs for finding free variables.
It's a very simple function that analyses a type,
but does not construct one.
* mapType constructs new types, and so it needs to call
the "smart constructors", mkAppTy, mkCastTy, and so on.
These are sophisticated functions, and can't be defined
here in GHC.Core.TyCo.Rep.
Note [Specialising foldType]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We inline foldType at every call site (there are not many), so that it
becomes specialised for the particular monoid *and* TyCoFolder at
that site. This is just for efficiency, but walking over types is
done a *lot* in GHC, so worth optimising.
We were worried that
TyCoFolder env (Endo a)
might not etaexpand. Recall newtype Endo a = Endo (a>a).
In particular, given
fvs :: Type > TyCoVarSet
fvs ty = appEndo (foldType tcf emptyVarSet ty) emptyVarSet
tcf :: TyCoFolder enf (Endo a)
tcf = TyCoFolder { tcf_tyvar = do_tv, ... }
where
do_tvs is tv = Endo do_it
where
do_it acc  tv `elemVarSet` is = acc
 tv `elemVarSet` acc = acc
 otherwise = acc `extendVarSet` tv
we want to end up with
fvs ty = go emptyVarSet ty emptyVarSet
where
go env (TyVarTy tv) acc = acc `extendVarSet` tv
..etc..
And indeed this happens.
 Selections from 'tcf' are done at compile time
 'go' is nicely etaexpanded.
We were also worried about
deep_fvs :: Type > TyCoVarSet
deep_fvs ty = appEndo (foldType deep_tcf emptyVarSet ty) emptyVarSet
deep_tcf :: TyCoFolder enf (Endo a)
deep_tcf = TyCoFolder { tcf_tyvar = do_tv, ... }
where
do_tvs is tv = Endo do_it
where
do_it acc  tv `elemVarSet` is = acc
 tv `elemVarSet` acc = acc
 otherwise = deep_fvs (varType tv)
`unionVarSet` acc
`extendVarSet` tv
Here deep_fvs and deep_tcf are mutually recursive, unlike fvs and tcf.
But, amazingly, we get good code here too. GHC is careful not to makr
TyCoFolder data constructor for deep_tcf as a loop breaker, so the
record selections still cancel. And eta expansion still happens too.
}
data TyCoFolder env a
= TyCoFolder
{ tcf_view :: Type > Maybe Type  Optional "view" function
 E.g. expand synonyms
, tcf_tyvar :: env > TyVar > a
, tcf_covar :: env > CoVar > a
, tcf_hole :: env > CoercionHole > a
 ^ What to do with coercion holes.
 See Note [Coercion holes] in "GHC.Core.TyCo.Rep".
, tcf_tycobinder :: env > TyCoVar > ArgFlag > env
 ^ The returned env is used in the extended scope
}
{# INLINE foldTyCo #}  See Note [Specialising foldType]
foldTyCo :: Monoid a => TyCoFolder env a > env
> (Type > a, [Type] > a, Coercion > a, [Coercion] > a)
foldTyCo (TyCoFolder { tcf_view = view
, tcf_tyvar = tyvar
, tcf_tycobinder = tycobinder
, tcf_covar = covar
, tcf_hole = cohole }) env
= (go_ty env, go_tys env, go_co env, go_cos env)
where
go_ty env ty  Just ty' < view ty = go_ty env ty'
go_ty env (TyVarTy tv) = tyvar env tv
go_ty env (AppTy t1 t2) = go_ty env t1 `mappend` go_ty env t2
go_ty _ (LitTy {}) = mempty
go_ty env (CastTy ty co) = go_ty env ty `mappend` go_co env co
go_ty env (CoercionTy co) = go_co env co
go_ty env (FunTy _ w arg res) = go_ty env w `mappend` go_ty env arg `mappend` go_ty env res
go_ty env (TyConApp _ tys) = go_tys env tys
go_ty env (ForAllTy (Bndr tv vis) inner)
= let !env' = tycobinder env tv vis  Avoid building a thunk here
in go_ty env (varType tv) `mappend` go_ty env' inner
 Explicit recursion becuase using foldr builds a local
 loop (with env free) and I'm not confident it'll be
 lambda lifted in the end
go_tys _ [] = mempty
go_tys env (t:ts) = go_ty env t `mappend` go_tys env ts
go_cos _ [] = mempty
go_cos env (c:cs) = go_co env c `mappend` go_cos env cs
go_co env (Refl ty) = go_ty env ty
go_co env (GRefl _ ty MRefl) = go_ty env ty
go_co env (GRefl _ ty (MCo co)) = go_ty env ty `mappend` go_co env co
go_co env (TyConAppCo _ _ args) = go_cos env args
go_co env (AppCo c1 c2) = go_co env c1 `mappend` go_co env c2
go_co env (FunCo _ cw c1 c2) = go_co env cw `mappend`
go_co env c1 `mappend`
go_co env c2
go_co env (CoVarCo cv) = covar env cv
go_co env (AxiomInstCo _ _ args) = go_cos env args
go_co env (HoleCo hole) = cohole env hole
go_co env (UnivCo p _ t1 t2) = go_prov env p `mappend` go_ty env t1
`mappend` go_ty env t2
go_co env (SymCo co) = go_co env co
go_co env (TransCo c1 c2) = go_co env c1 `mappend` go_co env c2
go_co env (AxiomRuleCo _ cos) = go_cos env cos
go_co env (NthCo _ _ co) = go_co env co
go_co env (LRCo _ co) = go_co env co
go_co env (InstCo co arg) = go_co env co `mappend` go_co env arg
go_co env (KindCo co) = go_co env co
go_co env (SubCo co) = go_co env co
go_co env (ForAllCo tv kind_co co)
= go_co env kind_co `mappend` go_ty env (varType tv)
`mappend` go_co env' co
where
env' = tycobinder env tv Inferred
go_prov env (PhantomProv co) = go_co env co
go_prov env (ProofIrrelProv co) = go_co env co
go_prov _ (PluginProv _) = mempty
{ *********************************************************************
* *
typeSize, coercionSize
* *
********************************************************************* }
 NB: We put typeSize/coercionSize here because they are mutually
 recursive, and have the CPR property. If we have mutual
 recursion across a hiboot file, we don't get the CPR property
 and these functions allocate a tremendous amount of rubbish.
 It's not critical (because typeSize is really only used in
 debug mode, but I tripped over an example (T5642) in which
 typeSize was one of the biggest single allocators in all of GHC.
 And it's easy to fix, so I did.
 NB: typeSize does not respect `eqType`, in that two types that
 are `eqType` may return different sizes. This is OK, because this
 function is used only in reporting, not decisionmaking.
typeSize :: Type > Int
typeSize (LitTy {}) = 1
typeSize (TyVarTy {}) = 1
typeSize (AppTy t1 t2) = typeSize t1 + typeSize t2
typeSize (FunTy _ _ t1 t2) = typeSize t1 + typeSize t2
typeSize (ForAllTy (Bndr tv _) t) = typeSize (varType tv) + typeSize t
typeSize (TyConApp _ ts) = 1 + sum (map typeSize ts)
typeSize (CastTy ty co) = typeSize ty + coercionSize co
typeSize (CoercionTy co) = coercionSize co
coercionSize :: Coercion > Int
coercionSize (Refl ty) = typeSize ty
coercionSize (GRefl _ ty MRefl) = typeSize ty
coercionSize (GRefl _ ty (MCo co)) = 1 + typeSize ty + coercionSize co
coercionSize (TyConAppCo _ _ args) = 1 + sum (map coercionSize args)
coercionSize (AppCo co arg) = coercionSize co + coercionSize arg
coercionSize (ForAllCo _ h co) = 1 + coercionSize co + coercionSize h
coercionSize (FunCo _ w co1 co2) = 1 + coercionSize co1 + coercionSize co2
+ coercionSize w
coercionSize (CoVarCo _) = 1
coercionSize (HoleCo _) = 1
coercionSize (AxiomInstCo _ _ args) = 1 + sum (map coercionSize args)
coercionSize (UnivCo p _ t1 t2) = 1 + provSize p + typeSize t1 + typeSize t2
coercionSize (SymCo co) = 1 + coercionSize co
coercionSize (TransCo co1 co2) = 1 + coercionSize co1 + coercionSize co2
coercionSize (NthCo _ _ co) = 1 + coercionSize co
coercionSize (LRCo _ co) = 1 + coercionSize co
coercionSize (InstCo co arg) = 1 + coercionSize co + coercionSize arg
coercionSize (KindCo co) = 1 + coercionSize co
coercionSize (SubCo co) = 1 + coercionSize co
coercionSize (AxiomRuleCo _ cs) = 1 + sum (map coercionSize cs)
provSize :: UnivCoProvenance > Int
provSize (PhantomProv co) = 1 + coercionSize co
provSize (ProofIrrelProv co) = 1 + coercionSize co
provSize (PluginProv _) = 1
{
************************************************************************
* *
Multiplicities
* *
************************************************************************
These definitions are here to avoid module loops, and to keep
GHC.Core.Multiplicity above this module.
}
  A shorthand for data with an attached 'Mult' element (the multiplicity).
data Scaled a = Scaled Mult a
deriving (Data.Data)
 You might think that this would be a natural candidate for
 Functor, Traversable but Krzysztof says (!3674) "it was too easy
 to accidentally lift functions (substitutions, zonking etc.) from
 Type > Type to Scaled Type > Scaled Type, ignoring
 multiplicities and causing bugs". So we don't.
instance (Outputable a) => Outputable (Scaled a) where
ppr (Scaled _cnt t) = ppr t
 Do not print the multiplicity here because it tends to be too verbose
scaledMult :: Scaled a > Mult
scaledMult (Scaled m _) = m
scaledThing :: Scaled a > a
scaledThing (Scaled _ t) = t
  Apply a function to both the Mult and the Type in a 'Scaled Type'
mapScaledType :: (Type > Type) > Scaled Type > Scaled Type
mapScaledType f (Scaled m t) = Scaled (f m) (f t)
{ 
Mult is a type alias for Type.
Mult must contain Type because multiplicity variables are mere type variables
(of kind Multiplicity) in Haskell. So the simplest implementation is to make
Mult be Type.
Multiplicities can be formed with:
 One: GHC.Types.One (= oneDataCon)
 Many: GHC.Types.Many (= manyDataCon)
 Multiplication: GHC.Types.MultMul (= multMulTyCon)
So that Mult feels a bit more structured, we provide pattern synonyms and smart
constructors for these.
}
type Mult = Type