{
(c) The University of Glasgow 2006
(c) The GRASP/AQUA Project, Glasgow University, 1998
\section[TyCoRep]{Type and Coercion  friends' interface}
Note [The Typerelated module hierarchy]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Class
CoAxiom
TyCon imports Class, CoAxiom
TyCoRep imports Class, CoAxiom, TyCon
TysPrim imports TyCoRep ( including mkTyConTy )
Kind imports TysPrim ( mainly for primitive kinds )
Type imports Kind
Coercion imports Type
}
 We expose the relevant stuff from this module via the Type module
{# OPTIONS_HADDOCK nothome #}
{# LANGUAGE CPP, DeriveDataTypeable, MultiWayIf #}
module TyCoRep (
TyThing(..), tyThingCategory, pprTyThingCategory, pprShortTyThing,
 * Types
Type(..),
TyLit(..),
KindOrType, Kind,
KnotTied,
PredType, ThetaType,  Synonyms
ArgFlag(..),
 * Coercions
Coercion(..),
UnivCoProvenance(..),
CoercionHole(..), coHoleCoVar, setCoHoleCoVar,
CoercionN, CoercionR, CoercionP, KindCoercion,
MCoercion(..), MCoercionR, MCoercionN,
 * Functions over types
mkTyConTy, mkTyVarTy, mkTyVarTys,
mkTyCoVarTy, mkTyCoVarTys,
mkFunTy, mkFunTys, mkTyCoForAllTy, mkForAllTys,
mkForAllTy,
mkTyCoPiTy, mkTyCoPiTys,
mkPiTys,
kindRep_maybe, kindRep,
isLiftedTypeKind, isUnliftedTypeKind,
isLiftedRuntimeRep, isUnliftedRuntimeRep,
isRuntimeRepTy, isRuntimeRepVar,
sameVis,
 * Functions over binders
TyCoBinder(..), TyCoVarBinder, TyBinder,
binderVar, binderVars, binderType, binderArgFlag,
delBinderVar,
isInvisibleArgFlag, isVisibleArgFlag,
isInvisibleBinder, isVisibleBinder,
isTyBinder, isNamedBinder,
tyCoBinderArgFlag,
 * Functions over coercions
pickLR,
 * Prettyprinting
pprType, pprParendType, pprPrecType, pprPrecTypeX,
pprTypeApp, pprTCvBndr, pprTCvBndrs,
pprSigmaType,
pprTheta, pprParendTheta, pprForAll, pprUserForAll,
pprTyVar, pprTyVars,
pprThetaArrowTy, pprClassPred,
pprKind, pprParendKind, pprTyLit,
PprPrec(..), topPrec, sigPrec, opPrec, funPrec, appPrec, maybeParen,
pprDataCons, pprWithExplicitKindsWhen,
pprCo, pprParendCo,
debugPprType,
 * Free variables
tyCoVarsOfType, tyCoVarsOfTypeDSet, tyCoVarsOfTypes, tyCoVarsOfTypesDSet,
tyCoFVsBndr, tyCoFVsVarBndr, tyCoFVsVarBndrs,
tyCoFVsOfType, tyCoVarsOfTypeList,
tyCoFVsOfTypes, tyCoVarsOfTypesList,
coVarsOfType, coVarsOfTypes,
coVarsOfCo, coVarsOfCos,
tyCoVarsOfCo, tyCoVarsOfCos,
tyCoVarsOfCoDSet,
tyCoFVsOfCo, tyCoFVsOfCos,
tyCoVarsOfCoList, tyCoVarsOfProv,
almostDevoidCoVarOfCo,
injectiveVarsOfType, tyConAppNeedsKindSig,
noFreeVarsOfType, noFreeVarsOfCo,
 * Substitutions
TCvSubst(..), TvSubstEnv, CvSubstEnv,
emptyTvSubstEnv, emptyCvSubstEnv, composeTCvSubstEnv, composeTCvSubst,
emptyTCvSubst, mkEmptyTCvSubst, isEmptyTCvSubst,
mkTCvSubst, mkTvSubst, mkCvSubst,
getTvSubstEnv,
getCvSubstEnv, getTCvInScope, getTCvSubstRangeFVs,
isInScope, notElemTCvSubst,
setTvSubstEnv, setCvSubstEnv, zapTCvSubst,
extendTCvInScope, extendTCvInScopeList, extendTCvInScopeSet,
extendTCvSubst, extendTCvSubstWithClone,
extendCvSubst, extendCvSubstWithClone,
extendTvSubst, extendTvSubstBinderAndInScope, extendTvSubstWithClone,
extendTvSubstList, extendTvSubstAndInScope,
extendTCvSubstList,
unionTCvSubst, zipTyEnv, zipCoEnv, mkTyCoInScopeSet,
zipTvSubst, zipCvSubst,
zipTCvSubst,
mkTvSubstPrs,
substTyWith, substTyWithCoVars, substTysWith, substTysWithCoVars,
substCoWith,
substTy, substTyAddInScope,
substTyUnchecked, substTysUnchecked, substThetaUnchecked,
substTyWithUnchecked,
substCoUnchecked, substCoWithUnchecked,
substTyWithInScope,
substTys, substTheta,
lookupTyVar,
substCo, substCos, substCoVar, substCoVars, lookupCoVar,
cloneTyVarBndr, cloneTyVarBndrs,
substVarBndr, substVarBndrs,
substTyVarBndr, substTyVarBndrs,
substCoVarBndr,
substTyVar, substTyVars, substTyCoVars,
substForAllCoBndr,
substVarBndrUsing, substForAllCoBndrUsing,
checkValidSubst, isValidTCvSubst,
 * Tidying type related things up for printing
tidyType, tidyTypes,
tidyOpenType, tidyOpenTypes,
tidyOpenKind,
tidyVarBndr, tidyVarBndrs, tidyFreeTyCoVars, avoidNameClashes,
tidyOpenTyCoVar, tidyOpenTyCoVars,
tidyTyCoVarOcc,
tidyTopType,
tidyKind,
tidyCo, tidyCos,
tidyTyCoVarBinder, tidyTyCoVarBinders,
 * Sizes
typeSize, coercionSize, provSize
) where
#include "HsVersions.h"
import GhcPrelude
import {# SOURCE #} DataCon( dataConFullSig
, dataConUserTyVarBinders
, DataCon )
import {# SOURCE #} Type( isPredTy, isCoercionTy, mkAppTy, mkCastTy
, tyCoVarsOfTypeWellScoped
, tyCoVarsOfTypesWellScoped
, scopedSort
, coreView )
 Transitively pulls in a LOT of stuff, better to break the loop
import {# SOURCE #} Coercion
import {# SOURCE #} ConLike ( ConLike(..), conLikeName )
import {# SOURCE #} ToIface( toIfaceTypeX, toIfaceTyLit, toIfaceForAllBndr
, toIfaceTyCon, toIfaceTcArgs, toIfaceCoercionX )
 friends:
import IfaceType
import Var
import VarEnv
import VarSet
import Name hiding ( varName )
import TyCon
import Class
import CoAxiom
import FV
 others
import BasicTypes ( LeftOrRight(..), PprPrec(..), topPrec, sigPrec, opPrec
, funPrec, appPrec, maybeParen, pickLR )
import PrelNames
import Outputable
import DynFlags
import FastString
import Pair
import UniqSupply
import Util
import UniqFM
import UniqSet
 libraries
import qualified Data.Data as Data hiding ( TyCon )
import Data.List
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 TysPrim.
It is also SOURCEimported into Name.hs
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 'TcEnv' 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. PprTyThing.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 coreSyn/CoreLint.hs
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.
 FunTy Type Type  ^ t1 > t2 Very common, so an important special case
 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 refl.
 INVARIANT: The Type is not a CastTy (use TransCo instead)
 See Note [Respecting definitional equality] (EQ2) and (EQ3)
 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
 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, Ord, Data.Data)
{ 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.
Lastly, 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)).
In sum, in order to uphold (EQ), we need the following three 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] in Type.
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 mkTyCoForAllTy, we check whether the variable is a coercion
variable and whether it is used 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.
}
  A type labeled 'KnotTied' might have knottied tycons in it. See
 Note [Type checking recursive type and class declarations] in
 TcTyClsDecls
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 Type  A termlambda binder. Type here can be CoercionTy.
 Visibility is determined by the type (Constraint vs. *)
deriving Data.Data
  '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 ty) = isPredTy ty
  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.
isTyBinder :: TyCoBinder > Bool
isTyBinder (Named bnd) = isTyVarBinder bnd
isTyBinder _ = True
tyCoBinderArgFlag :: TyCoBinder > ArgFlag
tyCoBinderArgFlag (Named (Bndr _ flag)) = flag
tyCoBinderArgFlag (Anon ty)
 isPredTy ty = Inferred
 otherwise = Required
{ 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/Specified. Function signature with inferred kind polymorphism.
f4 :: a b > Int
So 'f4' gets the type f4 :: 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
f5 :: a (b :: k) > Int
This time 'k' is Specified, because it is mentioned explicitly,
so we get f5 :: 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 TcTyClsDecls
 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: with fprintexplicitforalls, Inferred binders are written
in braces:
f :: forall {k} (a:k). S k a > Int
Otherwise, they are printed like Specified binders.
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 TcExpr.
}
{ **********************************************************************
* *
PredType
* *
********************************************************************** }
  A type of the form @p@ of kind @Constraint@ 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 TysPrim,
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`  Associates to the right
  Make an arrow type
mkFunTy :: Type > Type > Type
mkFunTy arg res = FunTy arg res
  Make nested arrow types
mkFunTys :: [Type] > Type > Type
mkFunTys tys ty = foldr mkFunTy ty tys
  If tv is a coercion variable and it is not used in the body, returns
 a FunTy, otherwise makes a forall type.
 See Note [Unused coercion variable in ForAllTy]
mkTyCoForAllTy :: TyCoVar > ArgFlag > Type > Type
mkTyCoForAllTy tv vis ty
 isCoVar tv
, not (tv `elemVarSet` tyCoVarsOfType ty)
= ASSERT( vis == Inferred )
mkFunTy (varType tv) ty
 otherwise
= ForAllTy (Bndr tv vis) ty
  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
mkTyCoPiTy :: TyCoBinder > Type > Type
mkTyCoPiTy (Anon ty1) ty2 = FunTy ty1 ty2
mkTyCoPiTy (Named (Bndr tv vis)) ty = mkTyCoForAllTy tv vis ty
  Like 'mkTyCoPiTy', but does not check the occurrence of the binder
mkPiTy:: TyCoBinder > Type > Type
mkPiTy (Anon ty1) ty2 = FunTy ty1 ty2
mkPiTy (Named (Bndr tv vis)) ty = mkForAllTy tv vis ty
mkTyCoPiTys :: [TyCoBinder] > Type > Type
mkTyCoPiTys tbs ty = foldr mkTyCoPiTy ty tbs
  Like 'mkTyCoPiTys', but does not check the occurrence of the binder
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 []
{
Some basic functions, put here to break loops eg with the pretty printer
}
  Extract the RuntimeRep classifier of a type from its kind. For example,
 @kindRep * = LiftedRep@; Panics if this is not possible.
 Treats * and Constraint as the same
kindRep :: HasDebugCallStack => Kind > Type
kindRep k = case kindRep_maybe k of
Just r > r
Nothing > pprPanic "kindRep" (ppr k)
  Given a kind (TYPE rr), extract its RuntimeRep classifier rr.
 For example, @kindRep_maybe * = Just LiftedRep@
 Returns 'Nothing' if the kind is not of form (TYPE rr)
 Treats * and Constraint as the same
kindRep_maybe :: HasDebugCallStack => Kind > Maybe Type
kindRep_maybe kind
 Just kind' < coreView kind = kindRep_maybe kind'
 TyConApp tc [arg] < kind
, tc `hasKey` tYPETyConKey = Just arg
 otherwise = Nothing
  This version considers Constraint to be the same as *. Returns True
 if the argument is equivalent to Type/Constraint and False otherwise.
 See Note [Kind Constraint and kind Type]
isLiftedTypeKind :: Kind > Bool
isLiftedTypeKind kind
= case kindRep_maybe kind of
Just rep > isLiftedRuntimeRep rep
Nothing > False
  Returns True if the kind classifies unlifted types and False otherwise.
 Note that this returns False for levitypolymorphic kinds, which may
 be specialized to a kind that classifies unlifted types.
isUnliftedTypeKind :: Kind > Bool
isUnliftedTypeKind kind
= case kindRep_maybe kind of
Just rep > isUnliftedRuntimeRep rep
Nothing > False
isLiftedRuntimeRep, isUnliftedRuntimeRep :: Type > Bool
 isLiftedRuntimeRep is true of LiftedRep :: RuntimeRep
 Similarly isUnliftedRuntimeRep
isLiftedRuntimeRep rep
 Just rep' < coreView rep = isLiftedRuntimeRep rep'
 TyConApp rr_tc args < rep
, rr_tc `hasKey` liftedRepDataConKey = ASSERT( null args ) True
 otherwise = False
isUnliftedRuntimeRep rep
 Just rep' < coreView rep = isUnliftedRuntimeRep rep'
 TyConApp rr_tc args < rep
, isUnliftedRuntimeRepTyCon rr_tc = ASSERT( null args ) True
 otherwise = False
isUnliftedRuntimeRepTyCon :: TyCon > Bool
isUnliftedRuntimeRepTyCon rr_tc
= elem (getUnique rr_tc) unliftedRepDataConKeys
  Is this the type 'RuntimeRep'?
isRuntimeRepTy :: Type > Bool
isRuntimeRepTy ty  Just ty' < coreView ty = isRuntimeRepTy ty'
isRuntimeRepTy (TyConApp tc []) = tc `hasKey` runtimeRepTyConKey
isRuntimeRepTy _ = False
  Is a tyvar of type 'RuntimeRep'?
isRuntimeRepVar :: TyVar > Bool
isRuntimeRepVar = isRuntimeRepTy . tyVarKind
{
%************************************************************************
%* *
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 coreSyn/CoreLint.hs
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
 AppCo Coercion CoercionN  lift AppTy
 AppCo :: e > N > e
 See Note [Forall coercions]
 ForAllCo TyCoVar KindCoercion Coercion
 ForAllCo :: _ > N > e > e
 FunCo Role Coercion Coercion  lift FunTy
 FunCo :: "e" > e > e > e
 These are special
 CoVarCo CoVar  :: _ > (N or R)
 result role depends on the tycon of the variable's type
 AxiomInstCo :: e > _ > [N] > 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]
 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
  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 consequencs 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 Trac #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 Trac #1496. The full story is in docs/corespec/corespec.pdf. Also, see
http://ghc.haskell.org/trac/ghc/wiki/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 Trac #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 UnsafeCoerceProv, 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
= UnsafeCoerceProv  ^ From @unsafeCoerce#@. These are unsound.
 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 UnsafeCoerceProv = text "(unsafeCoerce#)"
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_ref :: IORef (Maybe Coercion)
}
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)
{ 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 TysPrim 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 TcRnTypes.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
dicionaries 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 Trac #14584, and Note [What prevents a
constraint from floating] in TcSimplify, 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, ]
%************************************************************************
%* *
Free variables of types and coercions
%* *
%************************************************************************
}
{ Note [Free variables of types]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The family of functions tyCoVarsOfType, tyCoVarsOfTypes etc, returns
a VarSet that is closed over the types of its variables. More precisely,
if S = tyCoVarsOfType( t )
and (a:k) is in S
then tyCoVarsOftype( k ) is a subset of S
Example: The tyCoVars of this ((a:* > k) Int) is {a, k}.
We could /not/ close over the kinds of the variable occurrences, and
instead do so at call sites, but it seems that we always want to do
so, so it's easiest to do it here.
It turns out that getting the free variables of types is performance critical,
so we profiled several versions, exploring different implementation strategies.
1. Baseline version: uses FV naively. Essentially:
tyCoVarsOfType ty = fvVarSet $ tyCoFVsOfType ty
This is not nice, because FV introduces some overhead to implement
determinism, and throught its "interesting var" function, neither of which
we need here, so they are a complete waste.
2. UnionVarSet version: instead of reusing the FVbased code, we simply used
VarSets directly, trying to avoid the overhead of FV. E.g.:
 FV version:
tyCoFVsOfType (AppTy fun arg) a b c = (tyCoFVsOfType fun `unionFV` tyCoFVsOfType arg) a b c
 UnionVarSet version:
tyCoVarsOfType (AppTy fun arg) = (tyCoVarsOfType fun `unionVarSet` tyCoVarsOfType arg)
This looks deceptively similar, but while FV internally builds a list and
setgenerating function, the VarSet functions manipulate sets directly, and
the latter peforms a lot worse than the naive FV version.
3. Accumulatorstyle VarSet version: this is what we use now. We do use VarSet
as our data structure, but delegate the actual work to a new
ty_co_vars_of_... family of functions, which use accumulator style and the
"inscope set" filter found in the internals of FV, but without the
determinism overhead.
See Trac #14880.
Note [Closing over free variable kinds]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
tyCoVarsOfType and tyCoFVsOfType, while traversing a type, will also close over
free variable kinds. In previous GHC versions, this happened naively: whenever
we would encounter an occurrence of a free type variable, we would close over
its kind. This, however is wrong for two reasons (see Trac #14880):
1. Efficiency. If we have Proxy (a::k) > Proxy (a::k) > Proxy (a::k), then
we don't want to have to traverse k more than once.
2. Correctness. Imagine we have forall k. b > k, where b has
kind k, for some k bound in an outer scope. If we look at b's kind inside
the forall, we'll collect that k is free and then remove k from the set of
free variables. This is plain wrong. We must instead compute that b is free
and then conclude that b's kind is free.
An obvious first approach is to move the closingoverkinds from the
occurrences of a type variable to after finding the free vars  however, this
turns out to introduce performance regressions, and isn't even entirely
correct.
In fact, it isn't even important *when* we close over kinds; what matters is
that we handle each type var exactly once, and that we do it in the right
context.
So the next approach we tried was to use the "inscope set" part of FV or the
equivalent argument in the accumulatorstyle `ty_co_vars_of_type` function, to
say "don't bother with variables we have already closed over". This should work
fine in theory, but the code is complicated and doesn't perform well.
But there is a simpler way, which is implemented here. Consider the two points
above:
1. Efficiency: we now have an accumulator, so the second time we encounter 'a',
we'll ignore it, certainly not looking at its kind  this is why
prechecking set membership before inserting ends up not only being faster,
but also being correct.
2. Correctness: we have an "inscope set" (I think we should call it it a
"boundvar set"), specifying variables that are bound by a forall in the type
we are traversing; we simply ignore these variables, certainly not looking at
their kind.
So now consider:
forall k. b > k
where b :: k>Type is free; but of course, it's a different k! When looking at
b > k we'll have k in the boundvar set. So we'll ignore the k. But suppose
this is our first encounter with b; we want the free vars of its kind. But we
want to behave as if we took the free vars of its kind at the end; that is,
with no bound vars in scope.
So the solution is easy. The old code was this:
ty_co_vars_of_type (TyVarTy v) is acc
 v `elemVarSet` is = acc
 v `elemVarSet` acc = acc
 otherwise = ty_co_vars_of_type (tyVarKind v) is (extendVarSet acc v)
Now all we need to do is take the free vars of tyVarKind v *with an empty
boundvar set*, thus:
ty_co_vars_of_type (TyVarTy v) is acc
 v `elemVarSet` is = acc
 v `elemVarSet` acc = acc
 otherwise = ty_co_vars_of_type (tyVarKind v) emptyVarSet (extendVarSet acc v)
^^^^^^^^^^^
And that's it.
}
tyCoVarsOfType :: Type > TyCoVarSet
 See Note [Free variables of types]
tyCoVarsOfType ty = ty_co_vars_of_type ty emptyVarSet emptyVarSet
tyCoVarsOfTypes :: [Type] > TyCoVarSet
tyCoVarsOfTypes tys = ty_co_vars_of_types tys emptyVarSet emptyVarSet
ty_co_vars_of_type :: Type > TyCoVarSet > TyCoVarSet > TyCoVarSet
ty_co_vars_of_type (TyVarTy v) is acc
 v `elemVarSet` is = acc
 v `elemVarSet` acc = acc
 otherwise = ty_co_vars_of_type (tyVarKind v)
emptyVarSet  See Note [Closing over free variable kinds]
(extendVarSet acc v)
ty_co_vars_of_type (TyConApp _ tys) is acc = ty_co_vars_of_types tys is acc
ty_co_vars_of_type (LitTy {}) _ acc = acc
ty_co_vars_of_type (AppTy fun arg) is acc = ty_co_vars_of_type fun is (ty_co_vars_of_type arg is acc)
ty_co_vars_of_type (FunTy arg res) is acc = ty_co_vars_of_type arg is (ty_co_vars_of_type res is acc)
ty_co_vars_of_type (ForAllTy (Bndr tv _) ty) is acc = ty_co_vars_of_type (varType tv) is $
ty_co_vars_of_type ty (extendVarSet is tv) acc
ty_co_vars_of_type (CastTy ty co) is acc = ty_co_vars_of_type ty is (ty_co_vars_of_co co is acc)
ty_co_vars_of_type (CoercionTy co) is acc = ty_co_vars_of_co co is acc
ty_co_vars_of_types :: [Type] > TyCoVarSet > TyCoVarSet > TyCoVarSet
ty_co_vars_of_types [] _ acc = acc
ty_co_vars_of_types (ty:tys) is acc = ty_co_vars_of_type ty is (ty_co_vars_of_types tys is acc)
tyCoVarsOfCo :: Coercion > TyCoVarSet
 See Note [Free variables of types]
tyCoVarsOfCo co = ty_co_vars_of_co co emptyVarSet emptyVarSet
tyCoVarsOfCos :: [Coercion] > TyCoVarSet
tyCoVarsOfCos cos = ty_co_vars_of_cos cos emptyVarSet emptyVarSet
ty_co_vars_of_co :: Coercion > TyCoVarSet > TyCoVarSet > TyCoVarSet
ty_co_vars_of_co (Refl ty) is acc = ty_co_vars_of_type ty is acc
ty_co_vars_of_co (GRefl _ ty mco) is acc = ty_co_vars_of_type ty is $
ty_co_vars_of_mco mco is acc
ty_co_vars_of_co (TyConAppCo _ _ cos) is acc = ty_co_vars_of_cos cos is acc
ty_co_vars_of_co (AppCo co arg) is acc = ty_co_vars_of_co co is $
ty_co_vars_of_co arg is acc
ty_co_vars_of_co (ForAllCo tv kind_co co) is acc = ty_co_vars_of_co kind_co is $
ty_co_vars_of_co co (extendVarSet is tv) acc
ty_co_vars_of_co (FunCo _ co1 co2) is acc = ty_co_vars_of_co co1 is $
ty_co_vars_of_co co2 is acc
ty_co_vars_of_co (CoVarCo v) is acc = ty_co_vars_of_co_var v is acc
ty_co_vars_of_co (HoleCo h) is acc = ty_co_vars_of_co_var (coHoleCoVar h) is acc
 See Note [CoercionHoles and coercion free variables]
ty_co_vars_of_co (AxiomInstCo _ _ cos) is acc = ty_co_vars_of_cos cos is acc
ty_co_vars_of_co (UnivCo p _ t1 t2) is acc = ty_co_vars_of_prov p is $
ty_co_vars_of_type t1 is $
ty_co_vars_of_type t2 is acc
ty_co_vars_of_co (SymCo co) is acc = ty_co_vars_of_co co is acc
ty_co_vars_of_co (TransCo co1 co2) is acc = ty_co_vars_of_co co1 is $
ty_co_vars_of_co co2 is acc
ty_co_vars_of_co (NthCo _ _ co) is acc = ty_co_vars_of_co co is acc
ty_co_vars_of_co (LRCo _ co) is acc = ty_co_vars_of_co co is acc
ty_co_vars_of_co (InstCo co arg) is acc = ty_co_vars_of_co co is $
ty_co_vars_of_co arg is acc
ty_co_vars_of_co (KindCo co) is acc = ty_co_vars_of_co co is acc
ty_co_vars_of_co (SubCo co) is acc = ty_co_vars_of_co co is acc
ty_co_vars_of_co (AxiomRuleCo _ cs) is acc = ty_co_vars_of_cos cs is acc
ty_co_vars_of_mco :: MCoercion > TyCoVarSet > TyCoVarSet > TyCoVarSet
ty_co_vars_of_mco MRefl _is acc = acc
ty_co_vars_of_mco (MCo co) is acc = ty_co_vars_of_co co is acc
ty_co_vars_of_co_var :: CoVar > TyCoVarSet > TyCoVarSet > TyCoVarSet
ty_co_vars_of_co_var v is acc
 v `elemVarSet` is = acc
 v `elemVarSet` acc = acc
 otherwise = ty_co_vars_of_type (varType v)
emptyVarSet  See Note [Closing over free variable kinds]
(extendVarSet acc v)
ty_co_vars_of_cos :: [Coercion] > TyCoVarSet > TyCoVarSet > TyCoVarSet
ty_co_vars_of_cos [] _ acc = acc
ty_co_vars_of_cos (co:cos) is acc = ty_co_vars_of_co co is (ty_co_vars_of_cos cos is acc)
tyCoVarsOfProv :: UnivCoProvenance > TyCoVarSet
tyCoVarsOfProv prov = ty_co_vars_of_prov prov emptyVarSet emptyVarSet
ty_co_vars_of_prov :: UnivCoProvenance > TyCoVarSet > TyCoVarSet > TyCoVarSet
ty_co_vars_of_prov (PhantomProv co) is acc = ty_co_vars_of_co co is acc
ty_co_vars_of_prov (ProofIrrelProv co) is acc = ty_co_vars_of_co co is acc
ty_co_vars_of_prov UnsafeCoerceProv _ acc = acc
ty_co_vars_of_prov (PluginProv _) _ acc = acc
  Generates an inscope set from the free variables in a list of types
 and a list of coercions
mkTyCoInScopeSet :: [Type] > [Coercion] > InScopeSet
mkTyCoInScopeSet tys cos
= mkInScopeSet (ty_co_vars_of_types tys emptyVarSet $
ty_co_vars_of_cos cos emptyVarSet emptyVarSet)
  `tyCoFVsOfType` that returns free variables of a type in a deterministic
 set. For explanation of why using `VarSet` is not deterministic see
 Note [Deterministic FV] in FV.
tyCoVarsOfTypeDSet :: Type > DTyCoVarSet
 See Note [Free variables of types]
tyCoVarsOfTypeDSet ty = fvDVarSet $ tyCoFVsOfType ty
  `tyCoFVsOfType` that returns free variables of a type in deterministic
 order. For explanation of why using `VarSet` is not deterministic see
 Note [Deterministic FV] in FV.
tyCoVarsOfTypeList :: Type > [TyCoVar]
 See Note [Free variables of types]
tyCoVarsOfTypeList ty = fvVarList $ tyCoFVsOfType ty
  Returns free variables of types, including kind variables as
 a nondeterministic set. For type synonyms it does /not/ expand the
 synonym.
tyCoVarsOfTypesSet :: TyVarEnv Type > TyCoVarSet
 See Note [Free variables of types]
tyCoVarsOfTypesSet tys = tyCoVarsOfTypes $ nonDetEltsUFM tys
 It's OK to use nonDetEltsUFM here because we immediately forget the
 ordering by returning a set
  Returns free variables of types, including kind variables as
 a deterministic set. For type synonyms it does /not/ expand the
 synonym.
tyCoVarsOfTypesDSet :: [Type] > DTyCoVarSet
 See Note [Free variables of types]
tyCoVarsOfTypesDSet tys = fvDVarSet $ tyCoFVsOfTypes tys
  Returns free variables of types, including kind variables as
 a deterministically ordered list. For type synonyms it does /not/ expand the
 synonym.
tyCoVarsOfTypesList :: [Type] > [TyCoVar]
 See Note [Free variables of types]
tyCoVarsOfTypesList tys = fvVarList $ tyCoFVsOfTypes tys
  The worker for `tyCoFVsOfType` and `tyCoFVsOfTypeList`.
 The previous implementation used `unionVarSet` which is O(n+m) and can
 make the function quadratic.
 It's exported, so that it can be composed with
 other functions that compute free variables.
 See Note [FV naming conventions] in FV.

 Etaexpanded because that makes it run faster (apparently)
 See Note [FV eta expansion] in FV for explanation.
tyCoFVsOfType :: Type > FV
 See Note [Free variables of types]
tyCoFVsOfType (TyVarTy v) f bound_vars (acc_list, acc_set)
 not (f v) = (acc_list, acc_set)
 v `elemVarSet` bound_vars = (acc_list, acc_set)
 v `elemVarSet` acc_set = (acc_list, acc_set)
 otherwise = tyCoFVsOfType (tyVarKind v) f
emptyVarSet  See Note [Closing over free variable kinds]
(v:acc_list, extendVarSet acc_set v)
tyCoFVsOfType (TyConApp _ tys) f bound_vars acc = tyCoFVsOfTypes tys f bound_vars acc
tyCoFVsOfType (LitTy {}) f bound_vars acc = emptyFV f bound_vars acc
tyCoFVsOfType (AppTy fun arg) f bound_vars acc = (tyCoFVsOfType fun `unionFV` tyCoFVsOfType arg) f bound_vars acc
tyCoFVsOfType (FunTy arg res) f bound_vars acc = (tyCoFVsOfType arg `unionFV` tyCoFVsOfType res) f bound_vars acc
tyCoFVsOfType (ForAllTy bndr ty) f bound_vars acc = tyCoFVsBndr bndr (tyCoFVsOfType ty) f bound_vars acc
tyCoFVsOfType (CastTy ty co) f bound_vars acc = (tyCoFVsOfType ty `unionFV` tyCoFVsOfCo co) f bound_vars acc
tyCoFVsOfType (CoercionTy co) f bound_vars acc = tyCoFVsOfCo co f bound_vars acc
tyCoFVsBndr :: TyCoVarBinder > FV > FV
 Free vars of (forall b. )
tyCoFVsBndr (Bndr tv _) fvs = tyCoFVsVarBndr tv fvs
tyCoFVsVarBndrs :: [Var] > FV > FV
tyCoFVsVarBndrs vars fvs = foldr tyCoFVsVarBndr fvs vars
tyCoFVsVarBndr :: Var > FV > FV
tyCoFVsVarBndr var fvs
= tyCoFVsOfType (varType var)  Free vars of its type/kind
`unionFV` delFV var fvs  Delete it from the thinginside
tyCoFVsOfTypes :: [Type] > FV
 See Note [Free variables of types]
tyCoFVsOfTypes (ty:tys) fv_cand in_scope acc = (tyCoFVsOfType ty `unionFV` tyCoFVsOfTypes tys) fv_cand in_scope acc
tyCoFVsOfTypes [] fv_cand in_scope acc = emptyFV fv_cand in_scope acc
  Get a deterministic set of the vars free in a coercion
tyCoVarsOfCoDSet :: Coercion > DTyCoVarSet
 See Note [Free variables of types]
tyCoVarsOfCoDSet co = fvDVarSet $ tyCoFVsOfCo co
tyCoVarsOfCoList :: Coercion > [TyCoVar]
 See Note [Free variables of types]
tyCoVarsOfCoList co = fvVarList $ tyCoFVsOfCo co
tyCoFVsOfMCo :: MCoercion > FV
tyCoFVsOfMCo MRefl = emptyFV
tyCoFVsOfMCo (MCo co) = tyCoFVsOfCo co
tyCoVarsOfCosSet :: CoVarEnv Coercion > TyCoVarSet
tyCoVarsOfCosSet cos = tyCoVarsOfCos $ nonDetEltsUFM cos
 It's OK to use nonDetEltsUFM here because we immediately forget the
 ordering by returning a set
tyCoFVsOfCo :: Coercion > FV
 Extracts type and coercion variables from a coercion
 See Note [Free variables of types]
tyCoFVsOfCo (Refl ty) fv_cand in_scope acc
= tyCoFVsOfType ty fv_cand in_scope acc
tyCoFVsOfCo (GRefl _ ty mco) fv_cand in_scope acc
= (tyCoFVsOfType ty `unionFV` tyCoFVsOfMCo mco) fv_cand in_scope acc
tyCoFVsOfCo (TyConAppCo _ _ cos) fv_cand in_scope acc = tyCoFVsOfCos cos fv_cand in_scope acc
tyCoFVsOfCo (AppCo co arg) fv_cand in_scope acc
= (tyCoFVsOfCo co `unionFV` tyCoFVsOfCo arg) fv_cand in_scope acc
tyCoFVsOfCo (ForAllCo tv kind_co co) fv_cand in_scope acc
= (tyCoFVsVarBndr tv (tyCoFVsOfCo co) `unionFV` tyCoFVsOfCo kind_co) fv_cand in_scope acc
tyCoFVsOfCo (FunCo _ co1 co2) fv_cand in_scope acc
= (tyCoFVsOfCo co1 `unionFV` tyCoFVsOfCo co2) fv_cand in_scope acc
tyCoFVsOfCo (CoVarCo v) fv_cand in_scope acc
= tyCoFVsOfCoVar v fv_cand in_scope acc
tyCoFVsOfCo (HoleCo h) fv_cand in_scope acc
= tyCoFVsOfCoVar (coHoleCoVar h) fv_cand in_scope acc
 See Note [CoercionHoles and coercion free variables]
tyCoFVsOfCo (AxiomInstCo _ _ cos) fv_cand in_scope acc = tyCoFVsOfCos cos fv_cand in_scope acc
tyCoFVsOfCo (UnivCo p _ t1 t2) fv_cand in_scope acc
= (tyCoFVsOfProv p `unionFV` tyCoFVsOfType t1
`unionFV` tyCoFVsOfType t2) fv_cand in_scope acc
tyCoFVsOfCo (SymCo co) fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
tyCoFVsOfCo (TransCo co1 co2) fv_cand in_scope acc = (tyCoFVsOfCo co1 `unionFV` tyCoFVsOfCo co2) fv_cand in_scope acc
tyCoFVsOfCo (NthCo _ _ co) fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
tyCoFVsOfCo (LRCo _ co) fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
tyCoFVsOfCo (InstCo co arg) fv_cand in_scope acc = (tyCoFVsOfCo co `unionFV` tyCoFVsOfCo arg) fv_cand in_scope acc
tyCoFVsOfCo (KindCo co) fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
tyCoFVsOfCo (SubCo co) fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
tyCoFVsOfCo (AxiomRuleCo _ cs) fv_cand in_scope acc = tyCoFVsOfCos cs fv_cand in_scope acc
tyCoFVsOfCoVar :: CoVar > FV
tyCoFVsOfCoVar v fv_cand in_scope acc
= (unitFV v `unionFV` tyCoFVsOfType (varType v)) fv_cand in_scope acc
tyCoFVsOfProv :: UnivCoProvenance > FV
tyCoFVsOfProv UnsafeCoerceProv fv_cand in_scope acc = emptyFV fv_cand in_scope acc
tyCoFVsOfProv (PhantomProv co) fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
tyCoFVsOfProv (ProofIrrelProv co) fv_cand in_scope acc = tyCoFVsOfCo co fv_cand in_scope acc
tyCoFVsOfProv (PluginProv _) fv_cand in_scope acc = emptyFV fv_cand in_scope acc
tyCoFVsOfCos :: [Coercion] > FV
tyCoFVsOfCos [] fv_cand in_scope acc = emptyFV fv_cand in_scope acc
tyCoFVsOfCos (co:cos) fv_cand in_scope acc = (tyCoFVsOfCo co `unionFV` tyCoFVsOfCos cos) fv_cand in_scope acc
 Extracting the CoVars of a type or coercion 
{
Note [CoVarsOfX and the InterestingVarFun]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The coVarsOfType, coVarsOfTypes, coVarsOfCo, and coVarsOfCos functions are
implemented in terms of the respective FV equivalents (tyCoFVsOf...), rather
than the VarSetbased flavors (tyCoVarsOf...), despite the performance
considerations outlined in Note [Free variables of types].
This is because FV includes the InterestingVarFun, which is useful here,
because we can cleverly use it to restrict our calculations to CoVars  this
is what getCoVarSet achieves.
See Trac #14880.
}
getCoVarSet :: FV > CoVarSet
getCoVarSet fv = snd (fv isCoVar emptyVarSet ([], emptyVarSet))
coVarsOfType :: Type > CoVarSet
coVarsOfType ty = getCoVarSet (tyCoFVsOfType ty)
coVarsOfTypes :: [Type] > TyCoVarSet
coVarsOfTypes tys = getCoVarSet (tyCoFVsOfTypes tys)
coVarsOfCo :: Coercion > CoVarSet
coVarsOfCo co = getCoVarSet (tyCoFVsOfCo co)
coVarsOfCos :: [Coercion] > CoVarSet
coVarsOfCos cos = getCoVarSet (tyCoFVsOfCos cos)
 Whether a covar is /Almost Devoid/ in a type or coercion 
  Given a covar and a coercion, returns True if covar is almost devoid in
 the coercion. That is, covar can only appear in Refl and GRefl.
 See last wrinkle in Note [Unused coercion variable in ForAllCo] in Coercion
almostDevoidCoVarOfCo :: CoVar > Coercion > Bool
almostDevoidCoVarOfCo cv co =
almost_devoid_co_var_of_co co cv
almost_devoid_co_var_of_co :: Coercion > CoVar > Bool
almost_devoid_co_var_of_co (Refl {}) _ = True  covar is allowed in Refl and
almost_devoid_co_var_of_co (GRefl {}) _ = True  GRefl, so we don't look into
 the coercions
almost_devoid_co_var_of_co (TyConAppCo _ _ cos) cv
= almost_devoid_co_var_of_cos cos cv
almost_devoid_co_var_of_co (AppCo co arg) cv
= almost_devoid_co_var_of_co co cv
&& almost_devoid_co_var_of_co arg cv
almost_devoid_co_var_of_co (ForAllCo v kind_co co) cv
= almost_devoid_co_var_of_co kind_co cv
&& (v == cv  almost_devoid_co_var_of_co co cv)
almost_devoid_co_var_of_co (FunCo _ co1 co2) cv
= almost_devoid_co_var_of_co co1 cv
&& almost_devoid_co_var_of_co co2 cv
almost_devoid_co_var_of_co (CoVarCo v) cv = v /= cv
almost_devoid_co_var_of_co (HoleCo h) cv = (coHoleCoVar h) /= cv
almost_devoid_co_var_of_co (AxiomInstCo _ _ cos) cv
= almost_devoid_co_var_of_cos cos cv
almost_devoid_co_var_of_co (UnivCo p _ t1 t2) cv
= almost_devoid_co_var_of_prov p cv
&& almost_devoid_co_var_of_type t1 cv
&& almost_devoid_co_var_of_type t2 cv
almost_devoid_co_var_of_co (SymCo co) cv
= almost_devoid_co_var_of_co co cv
almost_devoid_co_var_of_co (TransCo co1 co2) cv
= almost_devoid_co_var_of_co co1 cv
&& almost_devoid_co_var_of_co co2 cv
almost_devoid_co_var_of_co (NthCo _ _ co) cv
= almost_devoid_co_var_of_co co cv
almost_devoid_co_var_of_co (LRCo _ co) cv
= almost_devoid_co_var_of_co co cv
almost_devoid_co_var_of_co (InstCo co arg) cv
= almost_devoid_co_var_of_co co cv
&& almost_devoid_co_var_of_co arg cv
almost_devoid_co_var_of_co (KindCo co) cv
= almost_devoid_co_var_of_co co cv
almost_devoid_co_var_of_co (SubCo co) cv
= almost_devoid_co_var_of_co co cv
almost_devoid_co_var_of_co (AxiomRuleCo _ cs) cv
= almost_devoid_co_var_of_cos cs cv
almost_devoid_co_var_of_cos :: [Coercion] > CoVar > Bool
almost_devoid_co_var_of_cos [] _ = True
almost_devoid_co_var_of_cos (co:cos) cv
= almost_devoid_co_var_of_co co cv
&& almost_devoid_co_var_of_cos cos cv
almost_devoid_co_var_of_prov :: UnivCoProvenance > CoVar > Bool
almost_devoid_co_var_of_prov (PhantomProv co) cv
= almost_devoid_co_var_of_co co cv
almost_devoid_co_var_of_prov (ProofIrrelProv co) cv
= almost_devoid_co_var_of_co co cv
almost_devoid_co_var_of_prov UnsafeCoerceProv _ = True
almost_devoid_co_var_of_prov (PluginProv _) _ = True
almost_devoid_co_var_of_type :: Type > CoVar > Bool
almost_devoid_co_var_of_type (TyVarTy _) _ = True
almost_devoid_co_var_of_type (TyConApp _ tys) cv
= almost_devoid_co_var_of_types tys cv
almost_devoid_co_var_of_type (LitTy {}) _ = True
almost_devoid_co_var_of_type (AppTy fun arg) cv
= almost_devoid_co_var_of_type fun cv
&& almost_devoid_co_var_of_type arg cv
almost_devoid_co_var_of_type (FunTy arg res) cv
= almost_devoid_co_var_of_type arg cv
&& almost_devoid_co_var_of_type res cv
almost_devoid_co_var_of_type (ForAllTy (Bndr v _) ty) cv
= almost_devoid_co_var_of_type (varType v) cv
&& (v == cv  almost_devoid_co_var_of_type ty cv)
almost_devoid_co_var_of_type (CastTy ty co) cv
= almost_devoid_co_var_of_type ty cv
&& almost_devoid_co_var_of_co co cv
almost_devoid_co_var_of_type (CoercionTy co) cv
= almost_devoid_co_var_of_co co cv
almost_devoid_co_var_of_types :: [Type] > CoVar > Bool
almost_devoid_co_var_of_types [] _ = True
almost_devoid_co_var_of_types (ty:tys) cv
= almost_devoid_co_var_of_type ty cv
&& almost_devoid_co_var_of_types tys cv
 Injective free vars 
  Returns the free variables of a 'Type' that are in injective positions.
 For example, if @F@ is a noninjective type family, then:

 @
 injectiveTyVarsOf( Either c (Maybe (a, F b c)) ) = {a,c}
 @

 If @'injectiveVarsOfType' ty = itvs@, then knowing @ty@ fixes @itvs@.
 More formally, if
 @a@ is in @'injectiveVarsOfType' ty@
 and @S1(ty) ~ S2(ty)@,
 then @S1(a) ~ S2(a)@,
 where @S1@ and @S2@ are arbitrary substitutions.

 See @Note [When does a tycon application need an explicit kind signature?]@.
injectiveVarsOfType :: Type > FV
injectiveVarsOfType = go
where
go ty  Just ty' < coreView ty
= go ty'
go (TyVarTy v) = unitFV v `unionFV` go (tyVarKind v)
go (AppTy f a) = go f `unionFV` go a
go (FunTy ty1 ty2) = go ty1 `unionFV` go ty2
go (TyConApp tc tys) =
case tyConInjectivityInfo tc of
NotInjective > emptyFV
Injective inj > mapUnionFV go $
filterByList (inj ++ repeat True) tys
 Oversaturated arguments to a tycon are
 always injective, hence the repeat True
go (ForAllTy tvb ty) = tyCoFVsBndr tvb $ go ty
go LitTy{} = emptyFV
go (CastTy ty _) = go ty
go CoercionTy{} = emptyFV
  Does a 'TyCon' (that is applied to some number of arguments) need to be
 ascribed with an explicit kind signature to resolve ambiguity if rendered as
 a sourcesyntax type?
 (See @Note [When does a tycon application need an explicit kind signature?]@
 for a full explanation of what this function checks for.)
 Morally, this function ought to belong in TyCon.hs, not TyCoRep.hs, but
 accomplishing this requires a fair deal of futzing aruond with .hsboot
 files.
tyConAppNeedsKindSig
:: Bool  ^ Should specified binders count towards injective positions in
 the kind of the TyCon?
> TyCon
> Int  ^ The number of args the 'TyCon' is applied to.
> Bool  ^ Does @T t_1 ... t_n@ need a kind signature? (Where @n@ is the
 number of arguments)
tyConAppNeedsKindSig spec_inj_pos tc n_args
 LT < listLengthCmp tc_binders n_args
= False
 otherwise
= let (dropped_binders, remaining_binders)
= splitAt n_args tc_binders
result_kind = mkTyConKind remaining_binders tc_res_kind
result_vars = tyCoVarsOfType result_kind
dropped_vars = fvVarSet $
mapUnionFV (injective_vars_of_binder spec_inj_pos)
dropped_binders
in not (subVarSet result_vars dropped_vars)
where
tc_binders = tyConBinders tc
tc_res_kind = tyConResKind tc
 Returns the variables that would be fixed by knowing a TyConBinder. See
 Note [When does a tycon application need an explicit kind signature?]
 for a more detailed explanation of what this function does.
injective_vars_of_binder
:: Bool  Should specified binders count towards injective positions?
 (If you're using visible kind applications, then you want True
 here.)
> TyConBinder > FV
injective_vars_of_binder spec_inj_pos (Bndr tv vis) =
case vis of
AnonTCB > injectiveVarsOfType (varType tv)
NamedTCB argf
 (argf == Required)
 (spec_inj_pos && (argf == Specified))
> unitFV tv `unionFV` injectiveVarsOfType (varType tv)
 otherwise
> emptyFV
{
Note [When does a tycon application need an explicit kind signature?]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
There are a couple of places in GHC where we convert Core Types into forms that
more closely resemble userwritten syntax. These include:
1. Template Haskell Type reification (see, for instance, TcSplice.reify_tc_app)
2. Converting Types to LHsTypes (in HsUtils.typeToLHsType, or in Haddock)
This conversion presents a challenge: how do we ensure that the resulting type
has enough kind information so as not to be ambiguous? To better motivate this
question, consider the following Core type:
 Foo :: Type > Type
type Foo = Proxy Type
There is nothing ambiguous about the RHS of Foo in Core. But if we were to,
say, reify it into a TH Type, then it's tempting to just drop the invisible
Type argument and simply return `Proxy`. But now we've lost crucial kind
information: we don't know if we're dealing with `Proxy Type` or `Proxy Bool`
or `Proxy Int` or something else! We've inadvertently introduced ambiguity.
Unlike in other situations in GHC, we can't just turn on
fprintexplicitkinds, as we need to produce something which has the same
structure as a sourcesyntax type. Moreover, we can't rely on visible kind
application, since the first kind argument to Proxy is inferred, not specified.
Our solution is to annotate certain tycons with their kinds whenever they
appear in applied form in order to resolve the ambiguity. For instance, we
would reify the RHS of Foo like so:
type Foo = (Proxy :: Type > Type)
We need to devise an algorithm that determines precisely which tycons need
these explicit kind signatures. We certainly don't want to annotate _every_
tycon with a kind signature, or else we might end up with horribly bloated
types like the following:
(Either :: Type > Type > Type) (Int :: Type) (Char :: Type)
We only want to annotate tycons that absolutely require kind signatures in
order to resolve some sort of ambiguity, and nothing more.
Suppose we have a tycon application (T ty_1 ... ty_n). Why might this type
require a kind signature? It might require it when we need to fill in any of
T's omitted arguments. By "omitted argument", we mean one that is dropped when
reifying ty_1 ... ty_n. Sometimes, the omitted arguments are inferred and
specified arguments (e.g., TH reification in TcSplice), and sometimes the
omitted arguments are only the inferred ones (e.g., in HsUtils.typeToLHsType,
which reifies specified arguments through visible kind application).
Regardless, the key idea is that _some_ arguments are going to be omitted after
reification, and the only mechanism we have at our disposal for filling them in
is through explicit kind signatures.
What do we mean by "fill in"? Let's consider this small example:
T :: forall {k}. Type > (k > Type) > k
Moreover, we have this application of T:
T @{j} Int aty
When we reify this type, we omit the inferred argument @{j}. Is it fixed by the
other (noninferred) arguments? Yes! If we know the kind of (aty :: blah), then
we'll generate an equality constraint (kappa > Type) and, assuming we can
solve it, that will fix `kappa`. (Here, `kappa` is the unification variable
that we instantiate `k` with.)
Therefore, for any application of a tycon T to some arguments, the Question We
Must Answer is:
* Given the first n arguments of T, do the kinds of the nonomitted arguments
fill in the omitted arguments?
(This is still a bit handwavey, but we'll refine this question incrementally
as we explain more of the machinery underlying this process.)
Answering this question is precisely the role that the `injectiveVarsOfType`
and `injective_vars_of_binder` functions exist to serve. If an omitted argument
`a` appears in the set returned by `injectiveVarsOfType ty`, then knowing
`ty` determines (i.e., fills in) `a`. (More on `injective_vars_of_binder` in a
bit.)
More formally, if
`a` is in `injectiveVarsOfType ty`
and S1(ty) ~ S2(ty),
then S1(a) ~ S2(a),
where S1 and S2 are arbitrary substitutions.
For example, is `F` is a noninjective type family, then
injectiveVarsOfType(Either c (Maybe (a, F b c))) = {a, c}
Now that we know what this function does, here is a second attempt at the
Question We Must Answer:
* Given the first n arguments of T (ty_1 ... ty_n), consider the binders
of T that are instantiated by nonomitted arguments. Do the injective
variables of these binders fill in the remainder of T's kind?
Alright, we're getting closer. Next, we need to clarify what the injective
variables of a tycon binder are. This the role that the
`injective_vars_of_binder` function serves. Here is what this function does for
each form of tycon binder:
* Anonymous binders are injective positions. For example, in the promoted data
constructor '(:):
'(:) :: forall a. a > [a] > [a]
The second and third tyvar binders (of kinds `a` and `[a]`) are both
anonymous, so if we had '(:) 'True '[], then the kinds of 'True and
'[] would contribute to the kind of '(:) 'True '[]. Therefore,
injective_vars_of_binder(_ :: a) = injectiveVarsOfType(a) = {a}.
(Similarly, injective_vars_of_binder(_ :: [a]) = {a}.)
* Named binders:
 Inferred binders are never injective positions. For example, in this data
type:
data Proxy a
Proxy :: forall {k}. k > Type
If we had Proxy 'True, then the kind of 'True would not contribute to the
kind of Proxy 'True. Therefore,
injective_vars_of_binder(forall {k}. ...) = {}.
 Required binders are injective positions. For example, in this data type:
data Wurble k (a :: k) :: k
Wurble :: forall k > k > k
The first tyvar binder (of kind `forall k`) has required visibility, so if
we had Wurble (Maybe a) Nothing, then the kind of Maybe a would
contribute to the kind of Wurble (Maybe a) Nothing. Hence,
injective_vars_of_binder(forall a > ...) = {a}.
 Specified binders /might/ be injective positions, depending on how you
approach things. Continuing the '(:) example:
'(:) :: forall a. a > [a] > [a]
Normally, the (forall a. ...) tyvar binder wouldn't contribute to the kind
of '(:) 'True '[], since it's not explicitly instantiated by the user. But
if visible kind application is enabled, then this is possible, since the
user can write '(:) @Bool 'True '[]. (In that case,
injective_vars_of_binder(forall a. ...) = {a}.)
There are some situations where using visible kind application is appropriate
(e.g., HsUtils.typeToLHsType) and others where it is not (e.g., TH
reification), so the `injective_vars_of_binder` function is parametrized by
a Bool which decides if specified binders should be counted towards
injective positions or not.
Now that we've defined injective_vars_of_binder, we can refine the Question We
Must Answer once more:
* Given the first n arguments of T (ty_1 ... ty_n), consider the binders
of T that are instantiated by nonomitted arguments. For each such binder
b_i, take the union of all injective_vars_of_binder(b_i). Is this set a
superset of the free variables of the remainder of T's kind?
If the answer to this question is "no", then (T ty_1 ... ty_n) needs an
explicit kind signature, since T's kind has kind variables leftover that
aren't fixed by the nonomitted arguments.
One last sticking point: what does "the remainder of T's kind" mean? You might
be tempted to think that it corresponds to all of the arguments in the kind of
T that would normally be instantiated by omitted arguments. But this isn't
quite right, strictly speaking. Consider the following (silly) example:
S :: forall {k}. Type > Type
And suppose we have this application of S:
S Int Bool
The Int argument would be omitted, and
injective_vars_of_binder(_ :: Type) = {}. This is not a superset of {k}, which
might suggest that (S Bool) needs an explicit kind signature. But
(S Bool :: Type) doesn't actually fix `k`! This is because the kind signature
only affects the /result/ of the application, not all of the individual
arguments. So adding a kind signature here won't make a difference. Therefore,
the fourth (and final) iteration of the Question We Must Answer is:
* Given the first n arguments of T (ty_1 ... ty_n), consider the binders
of T that are instantiated by nonomitted arguments. For each such binder
b_i, take the union of all injective_vars_of_binder(b_i). Is this set a
superset of the free variables of the kind of (T ty_1 ... ty_n)?
Phew, that was a lot of work!
How can be sure that this is correct? That is, how can we be sure that in the
event that we leave off a kind annotation, that one could infer the kind of the
tycon application from its arguments? It's essentially a proof by induction: if
we can infer the kinds of every subtree of a type, then the whole tycon
application will have an inferrable kindunless, of course, the remainder of
the tycon application's kind has uninstantiated kind variables.
What happens if T is oversaturated? That is, if T's kind has fewer than n
arguments, in the case that the concrete application instantiates a result
kind variable with an arrow kind? If we run out of arguments, we do not attach
a kind annotation. This should be a rare case, indeed. Here is an example:
data T1 :: k1 > k2 > *
data T2 :: k1 > k2 > *
type family G (a :: k) :: k
type instance G T1 = T2
type instance F Char = (G T1 Bool :: (* > *) > *)  F from above
Here G's kind is (forall k. k > k), and the desugared RHS of that last
instance of F is (G (* > (* > *) > *) (T1 * (* > *)) Bool). According to
the algorithm above, there are 3 arguments to G so we should peel off 3
arguments in G's kind. But G's kind has only two arguments. This is the
rare special case, and we choose not to annotate the application of G with
a kind signature. After all, we needn't do this, since that instance would
be reified as:
type instance F Char = G (T1 :: * > (* > *) > *) Bool
So the kind of G isn't ambiguous anymore due to the explicit kind annotation
on its argument. See #8953 and test th/T8953.
}
 No free vars 
  Returns True if this type has no free variables. Should be the same as
 isEmptyVarSet . tyCoVarsOfType, but faster in the nonforall case.
noFreeVarsOfType :: Type > Bool
noFreeVarsOfType (TyVarTy _) = False
noFreeVarsOfType (AppTy t1 t2) = noFreeVarsOfType t1 && noFreeVarsOfType t2
noFreeVarsOfType (TyConApp _ tys) = all noFreeVarsOfType tys
noFreeVarsOfType ty@(ForAllTy {}) = isEmptyVarSet (tyCoVarsOfType ty)
noFreeVarsOfType (FunTy t1 t2) = noFreeVarsOfType t1 && noFreeVarsOfType t2
noFreeVarsOfType (LitTy _) = True
noFreeVarsOfType (CastTy ty co) = noFreeVarsOfType ty && noFreeVarsOfCo co
noFreeVarsOfType (CoercionTy co) = noFreeVarsOfCo co
noFreeVarsOfMCo :: MCoercion > Bool
noFreeVarsOfMCo MRefl = True
noFreeVarsOfMCo (MCo co) = noFreeVarsOfCo co
noFreeVarsOfTypes :: [Type] > Bool
noFreeVarsOfTypes = all noFreeVarsOfType
  Returns True if this coercion has no free variables. Should be the same as
 isEmptyVarSet . tyCoVarsOfCo, but faster in the nonforall case.
noFreeVarsOfCo :: Coercion > Bool
noFreeVarsOfCo (Refl ty) = noFreeVarsOfType ty
noFreeVarsOfCo (GRefl _ ty co) = noFreeVarsOfType ty && noFreeVarsOfMCo co
noFreeVarsOfCo (TyConAppCo _ _ args) = all noFreeVarsOfCo args
noFreeVarsOfCo (AppCo c1 c2) = noFreeVarsOfCo c1 && noFreeVarsOfCo c2
noFreeVarsOfCo co@(ForAllCo {}) = isEmptyVarSet (tyCoVarsOfCo co)
noFreeVarsOfCo (FunCo _ c1 c2) = noFreeVarsOfCo c1 && noFreeVarsOfCo c2
noFreeVarsOfCo (CoVarCo _) = False
noFreeVarsOfCo (HoleCo {}) = True  I'm unsure; probably never happens
noFreeVarsOfCo (AxiomInstCo _ _ args) = all noFreeVarsOfCo args
noFreeVarsOfCo (UnivCo p _ t1 t2) = noFreeVarsOfProv p &&
noFreeVarsOfType t1 &&
noFreeVarsOfType t2
noFreeVarsOfCo (SymCo co) = noFreeVarsOfCo co
noFreeVarsOfCo (TransCo co1 co2) = noFreeVarsOfCo co1 && noFreeVarsOfCo co2
noFreeVarsOfCo (NthCo _ _ co) = noFreeVarsOfCo co
noFreeVarsOfCo (LRCo _ co) = noFreeVarsOfCo co
noFreeVarsOfCo (InstCo co1 co2) = noFreeVarsOfCo co1 && noFreeVarsOfCo co2
noFreeVarsOfCo (KindCo co) = noFreeVarsOfCo co
noFreeVarsOfCo (SubCo co) = noFreeVarsOfCo co
noFreeVarsOfCo (AxiomRuleCo _ cs) = all noFreeVarsOfCo cs
  Returns True if this UnivCoProv has no free variables. Should be the same as
 isEmptyVarSet . tyCoVarsOfProv, but faster in the nonforall case.
noFreeVarsOfProv :: UnivCoProvenance > Bool
noFreeVarsOfProv UnsafeCoerceProv = True
noFreeVarsOfProv (PhantomProv co) = noFreeVarsOfCo co
noFreeVarsOfProv (ProofIrrelProv co) = noFreeVarsOfCo co
noFreeVarsOfProv (PluginProv {}) = True
{
%************************************************************************
%* *
Substitutions
Data type defined here to avoid unnecessary mutual recursion
%* *
%************************************************************************
}
  Type & coercion substitution

 #tcvsubst_invariant#
 The following invariants must hold of a 'TCvSubst':

 1. The inscope set is needed /only/ to
 guide the generation of fresh uniques

 2. In particular, the /kind/ of the type variables in
 the inscope set is not relevant

 3. The substitution is only applied ONCE! This is because
 in general such application will not reach a fixed point.
data TCvSubst
= TCvSubst InScopeSet  The inscope type and kind variables
TvSubstEnv  Substitutes both type and kind variables
CvSubstEnv  Substitutes coercion variables
 See Note [Substitutions apply only once]
 and Note [Extending the TvSubstEnv]
 and Note [Substituting types and coercions]
 and Note [The substitution invariant]
  A substitution of 'Type's for 'TyVar's
 and 'Kind's for 'KindVar's
type TvSubstEnv = TyVarEnv Type
 NB: A TvSubstEnv is used
 both inside a TCvSubst (with the applyonce invariant
 discussed in Note [Substitutions apply only once],
 and also independently in the middle of matching,
 and unification (see Types.Unify).
 So you have to look at the context to know if it's idempotent or
 applyonce or whatever
  A substitution of 'Coercion's for 'CoVar's
type CvSubstEnv = CoVarEnv Coercion
{ Note [The substitution invariant]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
When calling (substTy subst ty) it should be the case that
the inscope set in the substitution is a superset of both:
(SIa) The free vars of the range of the substitution
(SIb) The free vars of ty minus the domain of the substitution
The same rules apply to other substitutions (notably CoreSubst.Subst)
* Reason for (SIa). Consider
substTy [a :> Maybe b] (forall b. b>a)
we must rename the forall b, to get
forall b2. b2 > Maybe b
Making 'b' part of the inscope set forces this renaming to
take place.
* Reason for (SIb). Consider
substTy [a :> Maybe b] (forall b. (a,b,x))
Then if we use the inscope set {b}, satisfying (SIa), there is
a danger we will rename the forall'd variable to 'x' by mistake,
getting this:
forall x. (Maybe b, x, x)
Breaking (SIb) caused the bug from #11371.
Note: if the free vars of the range of the substitution are freshly created,
then the problems of (SIa) can't happen, and so it would be sound to
ignore (SIa).
Note [Substitutions apply only once]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We use TCvSubsts to instantiate things, and we might instantiate
forall a b. ty
with the types
[a, b], or [b, a].
So the substitution might go [a>b, b>a]. A similar situation arises in Core
when we find a beta redex like
(/\ a /\ b > e) b a
Then we also end up with a substitution that permutes type variables. Other
variations happen to; for example [a > (a, b)].
********************************************************
*** So a substitution must be applied precisely once ***
********************************************************
A TCvSubst is not idempotent, but, unlike the nonidempotent substitution
we use during unifications, it must not be repeatedly applied.
Note [Extending the TvSubstEnv]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
See #tcvsubst_invariant# for the invariants that must hold.
This invariant allows a shortcut when the subst envs are empty:
if the TvSubstEnv and CvSubstEnv are empty  i.e. (isEmptyTCvSubst subst)
holds  then (substTy subst ty) does nothing.
For example, consider:
(/\a. /\b:(a~Int). ...b..) Int
We substitute Int for 'a'. The Unique of 'b' does not change, but
nevertheless we add 'b' to the TvSubstEnv, because b's kind does change
This invariant has several crucial consequences:
* In substVarBndr, we need extend the TvSubstEnv
 if the unique has changed
 or if the kind has changed
* In substTyVar, we do not need to consult the inscope set;
the TvSubstEnv is enough
* In substTy, substTheta, we can shortcircuit when the TvSubstEnv is empty
Note [Substituting types and coercions]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Types and coercions are mutually recursive, and either may have variables
"belonging" to the other. Thus, every time we wish to substitute in a
type, we may also need to substitute in a coercion, and vice versa.
However, the constructor used to create type variables is distinct from
that of coercion variables, so we carry two VarEnvs in a TCvSubst. Note
that it would be possible to use the CoercionTy constructor to combine
these environments, but that seems like a false economy.
Note that the TvSubstEnv should *never* map a CoVar (built with the Id
constructor) and the CvSubstEnv should *never* map a TyVar. Furthermore,
the range of the TvSubstEnv should *never* include a type headed with
CoercionTy.
}
emptyTvSubstEnv :: TvSubstEnv
emptyTvSubstEnv = emptyVarEnv
emptyCvSubstEnv :: CvSubstEnv
emptyCvSubstEnv = emptyVarEnv
composeTCvSubstEnv :: InScopeSet
> (TvSubstEnv, CvSubstEnv)
> (TvSubstEnv, CvSubstEnv)
> (TvSubstEnv, CvSubstEnv)
 ^ @(compose env1 env2)(x)@ is @env1(env2(x))@; i.e. apply @env2@ then @env1@.
 It assumes that both are idempotent.
 Typically, @env1@ is the refinement to a base substitution @env2@
composeTCvSubstEnv in_scope (tenv1, cenv1) (tenv2, cenv2)
= ( tenv1 `plusVarEnv` mapVarEnv (substTy subst1) tenv2
, cenv1 `plusVarEnv` mapVarEnv (substCo subst1) cenv2 )
 First apply env1 to the range of env2
 Then combine the two, making sure that env1 loses if
 both bind the same variable; that's why env1 is the
 *left* argument to plusVarEnv, because the right arg wins
where
subst1 = TCvSubst in_scope tenv1 cenv1
  Composes two substitutions, applying the second one provided first,
 like in function composition.
composeTCvSubst :: TCvSubst > TCvSubst > TCvSubst
composeTCvSubst (TCvSubst is1 tenv1 cenv1) (TCvSubst is2 tenv2 cenv2)
= TCvSubst is3 tenv3 cenv3
where
is3 = is1 `unionInScope` is2
(tenv3, cenv3) = composeTCvSubstEnv is3 (tenv1, cenv1) (tenv2, cenv2)
emptyTCvSubst :: TCvSubst
emptyTCvSubst = TCvSubst emptyInScopeSet emptyTvSubstEnv emptyCvSubstEnv
mkEmptyTCvSubst :: InScopeSet > TCvSubst
mkEmptyTCvSubst is = TCvSubst is emptyTvSubstEnv emptyCvSubstEnv
isEmptyTCvSubst :: TCvSubst > Bool
 See Note [Extending the TvSubstEnv]
isEmptyTCvSubst (TCvSubst _ tenv cenv) = isEmptyVarEnv tenv && isEmptyVarEnv cenv
mkTCvSubst :: InScopeSet > (TvSubstEnv, CvSubstEnv) > TCvSubst
mkTCvSubst in_scope (tenv, cenv) = TCvSubst in_scope tenv cenv
mkTvSubst :: InScopeSet > TvSubstEnv > TCvSubst
 ^ Make a TCvSubst with specified tyvar subst and empty covar subst
mkTvSubst in_scope tenv = TCvSubst in_scope tenv emptyCvSubstEnv
mkCvSubst :: InScopeSet > CvSubstEnv > TCvSubst
 ^ Make a TCvSubst with specified covar subst and empty tyvar subst
mkCvSubst in_scope cenv = TCvSubst in_scope emptyTvSubstEnv cenv
getTvSubstEnv :: TCvSubst > TvSubstEnv
getTvSubstEnv (TCvSubst _ env _) = env
getCvSubstEnv :: TCvSubst > CvSubstEnv
getCvSubstEnv (TCvSubst _ _ env) = env
getTCvInScope :: TCvSubst > InScopeSet
getTCvInScope (TCvSubst in_scope _ _) = in_scope
  Returns the free variables of the types in the range of a substitution as
 a nondeterministic set.
getTCvSubstRangeFVs :: TCvSubst > VarSet
getTCvSubstRangeFVs (TCvSubst _ tenv cenv)
= unionVarSet tenvFVs cenvFVs
where
tenvFVs = tyCoVarsOfTypesSet tenv
cenvFVs = tyCoVarsOfCosSet cenv
isInScope :: Var > TCvSubst > Bool
isInScope v (TCvSubst in_scope _ _) = v `elemInScopeSet` in_scope
notElemTCvSubst :: Var > TCvSubst > Bool
notElemTCvSubst v (TCvSubst _ tenv cenv)
 isTyVar v
= not (v `elemVarEnv` tenv)
 otherwise
= not (v `elemVarEnv` cenv)
setTvSubstEnv :: TCvSubst > TvSubstEnv > TCvSubst
setTvSubstEnv (TCvSubst in_scope _ cenv) tenv = TCvSubst in_scope tenv cenv
setCvSubstEnv :: TCvSubst > CvSubstEnv > TCvSubst
setCvSubstEnv (TCvSubst in_scope tenv _) cenv = TCvSubst in_scope tenv cenv
zapTCvSubst :: TCvSubst > TCvSubst
zapTCvSubst (TCvSubst in_scope _ _) = TCvSubst in_scope emptyVarEnv emptyVarEnv
extendTCvInScope :: TCvSubst > Var > TCvSubst
extendTCvInScope (TCvSubst in_scope tenv cenv) var
= TCvSubst (extendInScopeSet in_scope var) tenv cenv
extendTCvInScopeList :: TCvSubst > [Var] > TCvSubst
extendTCvInScopeList (TCvSubst in_scope tenv cenv) vars
= TCvSubst (extendInScopeSetList in_scope vars) tenv cenv
extendTCvInScopeSet :: TCvSubst > VarSet > TCvSubst
extendTCvInScopeSet (TCvSubst in_scope tenv cenv) vars
= TCvSubst (extendInScopeSetSet in_scope vars) tenv cenv
extendTCvSubst :: TCvSubst > TyCoVar > Type > TCvSubst
extendTCvSubst subst v ty
 isTyVar v
= extendTvSubst subst v ty
 CoercionTy co < ty
= extendCvSubst subst v co
 otherwise
= pprPanic "extendTCvSubst" (ppr v <+> text ">" <+> ppr ty)
extendTCvSubstWithClone :: TCvSubst > TyCoVar > TyCoVar > TCvSubst
extendTCvSubstWithClone subst tcv
 isTyVar tcv = extendTvSubstWithClone subst tcv
 otherwise = extendCvSubstWithClone subst tcv
extendTvSubst :: TCvSubst > TyVar > Type > TCvSubst
extendTvSubst (TCvSubst in_scope tenv cenv) tv ty
= TCvSubst in_scope (extendVarEnv tenv tv ty) cenv
extendTvSubstBinderAndInScope :: TCvSubst > TyCoBinder > Type > TCvSubst
extendTvSubstBinderAndInScope subst (Named (Bndr v _)) ty
= ASSERT( isTyVar v )
extendTvSubstAndInScope subst v ty
extendTvSubstBinderAndInScope subst (Anon _) _
= subst
extendTvSubstWithClone :: TCvSubst > TyVar > TyVar > TCvSubst
 Adds a new tv > tv mapping, /and/ extends the inscope set
extendTvSubstWithClone (TCvSubst in_scope tenv cenv) tv tv'
= TCvSubst (extendInScopeSetSet in_scope new_in_scope)
(extendVarEnv tenv tv (mkTyVarTy tv'))
cenv
where
new_in_scope = tyCoVarsOfType (tyVarKind tv') `extendVarSet` tv'
extendCvSubst :: TCvSubst > CoVar > Coercion > TCvSubst
extendCvSubst (TCvSubst in_scope tenv cenv) v co
= TCvSubst in_scope tenv (extendVarEnv cenv v co)
extendCvSubstWithClone :: TCvSubst > CoVar > CoVar > TCvSubst
extendCvSubstWithClone (TCvSubst in_scope tenv cenv) cv cv'
= TCvSubst (extendInScopeSetSet in_scope new_in_scope)
tenv
(extendVarEnv cenv cv (mkCoVarCo cv'))
where
new_in_scope = tyCoVarsOfType (varType cv') `extendVarSet` cv'
extendTvSubstAndInScope :: TCvSubst > TyVar > Type > TCvSubst
 Also extends the inscope set
extendTvSubstAndInScope (TCvSubst in_scope tenv cenv) tv ty
= TCvSubst (in_scope `extendInScopeSetSet` tyCoVarsOfType ty)
(extendVarEnv tenv tv ty)
cenv
extendTvSubstList :: TCvSubst > [Var] > [Type] > TCvSubst
extendTvSubstList subst tvs tys
= foldl2 extendTvSubst subst tvs tys
extendTCvSubstList :: TCvSubst > [Var] > [Type] > TCvSubst
extendTCvSubstList subst tvs tys
= foldl2 extendTCvSubst subst tvs tys
unionTCvSubst :: TCvSubst > TCvSubst > TCvSubst
 Works when the ranges are disjoint
unionTCvSubst (TCvSubst in_scope1 tenv1 cenv1) (TCvSubst in_scope2 tenv2 cenv2)
= ASSERT( not (tenv1 `intersectsVarEnv` tenv2)
&& not (cenv1 `intersectsVarEnv` cenv2) )
TCvSubst (in_scope1 `unionInScope` in_scope2)
(tenv1 `plusVarEnv` tenv2)
(cenv1 `plusVarEnv` cenv2)
 mkTvSubstPrs and zipTvSubst generate the inscope set from
 the types given; but it's just a thunk so with a bit of luck
 it'll never be evaluated
  Generates the inscope set for the 'TCvSubst' from the types in the incoming
 environment. No CoVars, please!
zipTvSubst :: HasDebugCallStack => [TyVar] > [Type] > TCvSubst
zipTvSubst tvs tys
= mkTvSubst (mkInScopeSet (tyCoVarsOfTypes tys)) tenv
where
tenv = zipTyEnv tvs tys
  Generates the inscope set for the 'TCvSubst' from the types in the incoming
 environment. No TyVars, please!
zipCvSubst :: HasDebugCallStack => [CoVar] > [Coercion] > TCvSubst
zipCvSubst cvs cos
= TCvSubst (mkInScopeSet (tyCoVarsOfCos cos)) emptyTvSubstEnv cenv
where
cenv = zipCoEnv cvs cos
zipTCvSubst :: HasDebugCallStack => [TyCoVar] > [Type] > TCvSubst
zipTCvSubst tcvs tys
= zip_tcvsubst tcvs tys (mkEmptyTCvSubst $ mkInScopeSet (tyCoVarsOfTypes tys))
where zip_tcvsubst :: [TyCoVar] > [Type] > TCvSubst > TCvSubst
zip_tcvsubst (tv:tvs) (ty:tys) subst
= zip_tcvsubst tvs tys (extendTCvSubst subst tv ty)
zip_tcvsubst [] [] subst = subst  empty case
zip_tcvsubst _ _ _ = pprPanic "zipTCvSubst: length mismatch"
(ppr tcvs <+> ppr tys)
  Generates the inscope set for the 'TCvSubst' from the types in the
 incoming environment. No CoVars, please!
mkTvSubstPrs :: [(TyVar, Type)] > TCvSubst
mkTvSubstPrs prs =
ASSERT2( onlyTyVarsAndNoCoercionTy, text "prs" <+> ppr prs )
mkTvSubst in_scope tenv
where tenv = mkVarEnv prs
in_scope = mkInScopeSet $ tyCoVarsOfTypes $ map snd prs
onlyTyVarsAndNoCoercionTy =
and [ isTyVar tv && not (isCoercionTy ty)
 (tv, ty) < prs ]
zipTyEnv :: HasDebugCallStack => [TyVar] > [Type] > TvSubstEnv
zipTyEnv tyvars tys
 debugIsOn
, not (all isTyVar tyvars)
= pprPanic "zipTyEnv" (ppr tyvars <+> ppr tys)
 otherwise
= ASSERT( all (not . isCoercionTy) tys )
mkVarEnv (zipEqual "zipTyEnv" tyvars tys)
 There used to be a special case for when
 ty == TyVarTy tv
 (a notuncommon case) in which case the substitution was dropped.
 But the typetidier changes the printname of a type variable without
 changing the unique, and that led to a bug. Why? Pretidying, we had
 a type {Foo t}, where Foo is a onemethod class. So Foo is really a newtype.
 And it happened that t was the type variable of the class. Posttiding,
 it got turned into {Foo t2}. The extcore printer expanded this using
 sourceTypeRep, but that said "Oh, t == t2" because they have the same unique,
 and so generated a rep type mentioning t not t2.

 Simplest fix is to nuke the "optimisation"
zipCoEnv :: HasDebugCallStack => [CoVar] > [Coercion] > CvSubstEnv
zipCoEnv cvs cos
 debugIsOn
, not (all isCoVar cvs)
= pprPanic "zipCoEnv" (ppr cvs <+> ppr cos)
 otherwise
= mkVarEnv (zipEqual "zipCoEnv" cvs cos)
instance Outputable TCvSubst where
ppr (TCvSubst ins tenv cenv)
= brackets $ sep[ text "TCvSubst",
nest 2 (text "In scope:" <+> ppr ins),
nest 2 (text "Type env:" <+> ppr tenv),
nest 2 (text "Co env:" <+> ppr cenv) ]
{
%************************************************************************
%* *
Performing type or kind substitutions
%* *
%************************************************************************
Note [Sym and ForAllCo]
~~~~~~~~~~~~~~~~~~~~~~~
In OptCoercion, we try to push "sym" out to the leaves of a coercion. But,
how do we push sym into a ForAllCo? It's a little ugly.
Here is the typing rule:
h : k1 ~# k2
(tv : k1)  g : ty1 ~# ty2

ForAllCo tv h g : (ForAllTy (tv : k1) ty1) ~#
(ForAllTy (tv : k2) (ty2[tv > tv > sym h]))
Here is what we want:
ForAllCo tv h' g' : (ForAllTy (tv : k2) (ty2[tv > tv > sym h])) ~#
(ForAllTy (tv : k1) ty1)
Because the kinds of the type variables to the right of the colon are the kinds
coerced by h', we know (h' : k2 ~# k1). Thus, (h' = sym h).
Now, we can rewrite ty1 to be (ty1[tv > tv > sym h' > h']). We thus want
ForAllCo tv h' g' :
(ForAllTy (tv : k2) (ty2[tv > tv > h'])) ~#
(ForAllTy (tv : k1) (ty1[tv > tv > h'][tv > tv > sym h']))
We thus see that we want
g' : ty2[tv > tv > h'] ~# ty1[tv > tv > h']
and thus g' = sym (g[tv > tv > h']).
Putting it all together, we get this:
sym (ForAllCo tv h g)
==>
ForAllCo tv (sym h) (sym g[tv > tv > sym h])
Note [Substituting in a coercion hole]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
It seems highly suspicious to be substituting in a coercion that still
has coercion holes. Yet, this can happen in a situation like this:
f :: forall k. k :~: Type > ()
f Refl = let x :: forall (a :: k). [a] > ...
x = ...
When we check x's type signature, we require that k ~ Type. We indeed
know this due to the Refl pattern match, but the eager unifier can't
make use of givens. So, when we're done looking at x's type, a coercion
hole will remain. Then, when we're checking x's definition, we skolemise
x's type (in order to, e.g., bring the scoped type variable `a` into scope).
This requires performing a substitution for the fresh skolem variables.
This subsitution needs to affect the kind of the coercion hole, too 
otherwise, the kind will have an outofscope variable in it. More problematically
in practice (we won't actually notice the outofscope variable ever), skolems
in the kind might have too high a level, triggering a failure to uphold the
invariant that no free variables in a type have a higher level than the
ambient level in the type checker. In the event of having free variables in the
hole's kind, I'm pretty sure we'll always have an erroneous program, so we
don't need to worry what will happen when the hole gets filled in. After all,
a hole relating a locallybound type variable will be unable to be solved. This
is why it's OK not to look through the IORef of a coercion hole during
substitution.
}
  Type substitution, see 'zipTvSubst'
substTyWith :: HasCallStack => [TyVar] > [Type] > Type > Type
 Works only if the domain of the substitution is a
 superset of the type being substituted into
substTyWith tvs tys = {#SCC "substTyWith" #}
ASSERT( tvs `equalLength` tys )
substTy (zipTvSubst tvs tys)
  Type substitution, see 'zipTvSubst'. Disables sanity checks.
 The problems that the sanity checks in substTy catch are described in
 Note [The substitution invariant].
 The goal of #11371 is to migrate all the calls of substTyUnchecked to
 substTy and remove this function. Please don't use in new code.
substTyWithUnchecked :: [TyVar] > [Type] > Type > Type
substTyWithUnchecked tvs tys
= ASSERT( tvs `equalLength` tys )
substTyUnchecked (zipTvSubst tvs tys)
  Substitute tyvars within a type using a known 'InScopeSet'.
 Precondition: the 'in_scope' set should satisfy Note [The substitution
 invariant]; specifically it should include the free vars of 'tys',
 and of 'ty' minus the domain of the subst.
substTyWithInScope :: InScopeSet > [TyVar] > [Type] > Type > Type
substTyWithInScope in_scope tvs tys ty =
ASSERT( tvs `equalLength` tys )
substTy (mkTvSubst in_scope tenv) ty
where tenv = zipTyEnv tvs tys
  Coercion substitution, see 'zipTvSubst'
substCoWith :: HasCallStack => [TyVar] > [Type] > Coercion > Coercion
substCoWith tvs tys = ASSERT( tvs `equalLength` tys )
substCo (zipTvSubst tvs tys)
  Coercion substitution, see 'zipTvSubst'. Disables sanity checks.
 The problems that the sanity checks in substCo catch are described in
 Note [The substitution invariant].
 The goal of #11371 is to migrate all the calls of substCoUnchecked to
 substCo and remove this function. Please don't use in new code.
substCoWithUnchecked :: [TyVar] > [Type] > Coercion > Coercion
substCoWithUnchecked tvs tys
= ASSERT( tvs `equalLength` tys )
substCoUnchecked (zipTvSubst tvs tys)
  Substitute covars within a type
substTyWithCoVars :: [CoVar] > [Coercion] > Type > Type
substTyWithCoVars cvs cos = substTy (zipCvSubst cvs cos)
  Type substitution, see 'zipTvSubst'
substTysWith :: [TyVar] > [Type] > [Type] > [Type]
substTysWith tvs tys = ASSERT( tvs `equalLength` tys )
substTys (zipTvSubst tvs tys)
  Type substitution, see 'zipTvSubst'
substTysWithCoVars :: [CoVar] > [Coercion] > [Type] > [Type]
substTysWithCoVars cvs cos = ASSERT( cvs `equalLength` cos )
substTys (zipCvSubst cvs cos)
  Substitute within a 'Type' after adding the free variables of the type
 to the inscope set. This is useful for the case when the free variables
 aren't already in the inscope set or easily available.
 See also Note [The substitution invariant].
substTyAddInScope :: TCvSubst > Type > Type
substTyAddInScope subst ty =
substTy (extendTCvInScopeSet subst $ tyCoVarsOfType ty) ty
  When calling `substTy` it should be the case that the inscope set in
 the substitution is a superset of the free vars of the range of the
 substitution.
 See also Note [The substitution invariant].
isValidTCvSubst :: TCvSubst > Bool
isValidTCvSubst (TCvSubst in_scope tenv cenv) =
(tenvFVs `varSetInScope` in_scope) &&
(cenvFVs `varSetInScope` in_scope)
where
tenvFVs = tyCoVarsOfTypesSet tenv
cenvFVs = tyCoVarsOfCosSet cenv
  This checks if the substitution satisfies the invariant from
 Note [The substitution invariant].
checkValidSubst :: HasCallStack => TCvSubst > [Type] > [Coercion] > a > a
checkValidSubst subst@(TCvSubst in_scope tenv cenv) tys cos a
 TODO (RAE): Change back to ASSERT
= WARN( not (isValidTCvSubst subst),
text "in_scope" <+> ppr in_scope $$
text "tenv" <+> ppr tenv $$
text "tenvFVs" <+> ppr (tyCoVarsOfTypesSet tenv) $$
text "cenv" <+> ppr cenv $$
text "cenvFVs" <+> ppr (tyCoVarsOfCosSet cenv) $$
text "tys" <+> ppr tys $$
text "cos" <+> ppr cos )
WARN( not tysCosFVsInScope,
text "in_scope" <+> ppr in_scope $$
text "tenv" <+> ppr tenv $$
text "cenv" <+> ppr cenv $$
text "tys" <+> ppr tys $$
text "cos" <+> ppr cos $$
text "needInScope" <+> ppr needInScope )
a
where
substDomain = nonDetKeysUFM tenv ++ nonDetKeysUFM cenv
 It's OK to use nonDetKeysUFM here, because we only use this list to
 remove some elements from a set
needInScope = (tyCoVarsOfTypes tys `unionVarSet` tyCoVarsOfCos cos)
`delListFromUniqSet_Directly` substDomain
tysCosFVsInScope = needInScope `varSetInScope` in_scope
  Substitute within a 'Type'
 The substitution has to satisfy the invariants described in
 Note [The substitution invariant].
substTy :: HasCallStack => TCvSubst > Type > Type
substTy subst ty
 isEmptyTCvSubst subst = ty
 otherwise = checkValidSubst subst [ty] [] $
subst_ty subst ty
  Substitute within a 'Type' disabling the sanity checks.
 The problems that the sanity checks in substTy catch are described in
 Note [The substitution invariant].
 The goal of #11371 is to migrate all the calls of substTyUnchecked to
 substTy and remove this function. Please don't use in new code.
substTyUnchecked :: TCvSubst > Type > Type
substTyUnchecked subst ty
 isEmptyTCvSubst subst = ty
 otherwise = subst_ty subst ty
  Substitute within several 'Type's
 The substitution has to satisfy the invariants described in
 Note [The substitution invariant].
substTys :: HasCallStack => TCvSubst > [Type] > [Type]
substTys subst tys
 isEmptyTCvSubst subst = tys
 otherwise = checkValidSubst subst tys [] $ map (subst_ty subst) tys
  Substitute within several 'Type's disabling the sanity checks.
 The problems that the sanity checks in substTys catch are described in
 Note [The substitution invariant].
 The goal of #11371 is to migrate all the calls of substTysUnchecked to
 substTys and remove this function. Please don't use in new code.
substTysUnchecked :: TCvSubst > [Type] > [Type]
substTysUnchecked subst tys
 isEmptyTCvSubst subst = tys
 otherwise = map (subst_ty subst) tys
  Substitute within a 'ThetaType'
 The substitution has to satisfy the invariants described in
 Note [The substitution invariant].
substTheta :: HasCallStack => TCvSubst > ThetaType > ThetaType
substTheta = substTys
  Substitute within a 'ThetaType' disabling the sanity checks.
 The problems that the sanity checks in substTys catch are described in
 Note [The substitution invariant].
 The goal of #11371 is to migrate all the calls of substThetaUnchecked to
 substTheta and remove this function. Please don't use in new code.
substThetaUnchecked :: TCvSubst > ThetaType > ThetaType
substThetaUnchecked = substTysUnchecked
subst_ty :: TCvSubst > Type > Type
 subst_ty is the main workhorse for type substitution

 Note that the in_scope set is poked only if we hit a forall
 so it may often never be fully computed
subst_ty subst ty
= go ty
where
go (TyVarTy tv) = substTyVar subst tv
go (AppTy fun arg) = mkAppTy (go fun) $! (go arg)
 The mkAppTy smart constructor is important
 we might be replacing (a Int), represented with App
 by [Int], represented with TyConApp
go (TyConApp tc tys) = let args = map go tys
in args `seqList` TyConApp tc args
go (FunTy arg res) = (FunTy $! go arg) $! go res
go (ForAllTy (Bndr tv vis) ty)
= case substVarBndrUnchecked subst tv of
(subst', tv') >
(ForAllTy $! ((Bndr $! tv') vis)) $!
(subst_ty subst' ty)
go (LitTy n) = LitTy $! n
go (CastTy ty co) = (mkCastTy $! (go ty)) $! (subst_co subst co)
go (CoercionTy co) = CoercionTy $! (subst_co subst co)
substTyVar :: TCvSubst > TyVar > Type
substTyVar (TCvSubst _ tenv _) tv
= ASSERT( isTyVar tv )
case lookupVarEnv tenv tv of
Just ty > ty
Nothing > TyVarTy tv
substTyVars :: TCvSubst > [TyVar] > [Type]
substTyVars subst = map $ substTyVar subst
substTyCoVars :: TCvSubst > [TyCoVar] > [Type]
substTyCoVars subst = map $ substTyCoVar subst
substTyCoVar :: TCvSubst > TyCoVar > Type
substTyCoVar subst tv
 isTyVar tv = substTyVar subst tv
 otherwise = CoercionTy $ substCoVar subst tv
lookupTyVar :: TCvSubst > TyVar > Maybe Type
 See Note [Extending the TCvSubst]
lookupTyVar (TCvSubst _ tenv _) tv
= ASSERT( isTyVar tv )
lookupVarEnv tenv tv
  Substitute within a 'Coercion'
 The substitution has to satisfy the invariants described in
 Note [The substitution invariant].
substCo :: HasCallStack => TCvSubst > Coercion > Coercion
substCo subst co
 isEmptyTCvSubst subst = co
 otherwise = checkValidSubst subst [] [co] $ subst_co subst co
  Substitute within a 'Coercion' disabling sanity checks.
 The problems that the sanity checks in substCo catch are described in
 Note [The substitution invariant].
 The goal of #11371 is to migrate all the calls of substCoUnchecked to
 substCo and remove this function. Please don't use in new code.
substCoUnchecked :: TCvSubst > Coercion > Coercion
substCoUnchecked subst co
 isEmptyTCvSubst subst = co
 otherwise = subst_co subst co
  Substitute within several 'Coercion's
 The substitution has to satisfy the invariants described in
 Note [The substitution invariant].
substCos :: HasCallStack => TCvSubst > [Coercion] > [Coercion]
substCos subst cos
 isEmptyTCvSubst subst = cos
 otherwise = checkValidSubst subst [] cos $ map (subst_co subst) cos
subst_co :: TCvSubst > Coercion > Coercion
subst_co subst co
= go co
where
go_ty :: Type > Type
go_ty = subst_ty subst
go_mco :: MCoercion > MCoercion
go_mco MRefl = MRefl
go_mco (MCo co) = MCo (go co)
go :: Coercion > Coercion
go (Refl ty) = mkNomReflCo $! (go_ty ty)
go (GRefl r ty mco) = (mkGReflCo r $! (go_ty ty)) $! (go_mco mco)
go (TyConAppCo r tc args)= let args' = map go args
in args' `seqList` mkTyConAppCo r tc args'
go (AppCo co arg) = (mkAppCo $! go co) $! go arg
go (ForAllCo tv kind_co co)
= case substForAllCoBndrUnchecked subst tv kind_co of
(subst', tv', kind_co') >
((mkForAllCo $! tv') $! kind_co') $! subst_co subst' co
go (FunCo r co1 co2) = (mkFunCo r $! go co1) $! go co2
go (CoVarCo cv) = substCoVar subst cv
go (AxiomInstCo con ind cos) = mkAxiomInstCo con ind $! map go cos
go (UnivCo p r t1 t2) = (((mkUnivCo $! go_prov p) $! r) $!
(go_ty t1)) $! (go_ty t2)
go (SymCo co) = mkSymCo $! (go co)
go (TransCo co1 co2) = (mkTransCo $! (go co1)) $! (go co2)
go (NthCo r d co) = mkNthCo r d $! (go co)
go (LRCo lr co) = mkLRCo lr $! (go co)
go (InstCo co arg) = (mkInstCo $! (go co)) $! go arg
go (KindCo co) = mkKindCo $! (go co)
go (SubCo co) = mkSubCo $! (go co)
go (AxiomRuleCo c cs) = let cs1 = map go cs
in cs1 `seqList` AxiomRuleCo c cs1
go (HoleCo h) = HoleCo $! go_hole h
go_prov UnsafeCoerceProv = UnsafeCoerceProv
go_prov (PhantomProv kco) = PhantomProv (go kco)
go_prov (ProofIrrelProv kco) = ProofIrrelProv (go kco)
go_prov p@(PluginProv _) = p
 See Note [Substituting in a coercion hole]
go_hole h@(CoercionHole { ch_co_var = cv })
= h { ch_co_var = updateVarType go_ty cv }
substForAllCoBndr :: TCvSubst > TyCoVar > KindCoercion
> (TCvSubst, TyCoVar, Coercion)
substForAllCoBndr subst
= substForAllCoBndrUsing False (substCo subst) subst
  Like 'substForAllCoBndr', but disables sanity checks.
 The problems that the sanity checks in substCo catch are described in
 Note [The substitution invariant].
 The goal of #11371 is to migrate all the calls of substCoUnchecked to
 substCo and remove this function. Please don't use in new code.
substForAllCoBndrUnchecked :: TCvSubst > TyCoVar > KindCoercion
> (TCvSubst, TyCoVar, Coercion)
substForAllCoBndrUnchecked subst
= substForAllCoBndrUsing False (substCoUnchecked subst) subst
 See Note [Sym and ForAllCo]
substForAllCoBndrUsing :: Bool  apply sym to binder?
> (Coercion > Coercion)  transformation to kind co
> TCvSubst > TyCoVar > KindCoercion
> (TCvSubst, TyCoVar, KindCoercion)
substForAllCoBndrUsing sym sco subst old_var
 isTyVar old_var = substForAllCoTyVarBndrUsing sym sco subst old_var
 otherwise = substForAllCoCoVarBndrUsing sym sco subst old_var
substForAllCoTyVarBndrUsing :: Bool  apply sym to binder?
> (Coercion > Coercion)  transformation to kind co
> TCvSubst > TyVar > KindCoercion
> (TCvSubst, TyVar, KindCoercion)
substForAllCoTyVarBndrUsing sym sco (TCvSubst in_scope tenv cenv) old_var old_kind_co
= ASSERT( isTyVar old_var )
( TCvSubst (in_scope `extendInScopeSet` new_var) new_env cenv
, new_var, new_kind_co )
where
new_env  no_change && not sym = delVarEnv tenv old_var
 sym = extendVarEnv tenv old_var $
TyVarTy new_var `CastTy` new_kind_co
 otherwise = extendVarEnv tenv old_var (TyVarTy new_var)
no_kind_change = noFreeVarsOfCo old_kind_co
no_change = no_kind_change && (new_var == old_var)
new_kind_co  no_kind_change = old_kind_co
 otherwise = sco old_kind_co
Pair new_ki1 _ = coercionKind new_kind_co
 We could do substitution to (tyVarKind old_var). We don't do so because
 we already substituted new_kind_co, which contains the kind information
 we want. We don't want to do substitution once more. Also, in most cases,
 new_kind_co is a Refl, in which case coercionKind is really fast.
new_var = uniqAway in_scope (setTyVarKind old_var new_ki1)
substForAllCoCoVarBndrUsing :: Bool  apply sym to binder?
> (Coercion > Coercion)  transformation to kind co
> TCvSubst > CoVar > KindCoercion
> (TCvSubst, CoVar, KindCoercion)
substForAllCoCoVarBndrUsing sym sco (TCvSubst in_scope tenv cenv)
old_var old_kind_co
= ASSERT( isCoVar old_var )
( TCvSubst (in_scope `extendInScopeSet` new_var) tenv new_cenv
, new_var, new_kind_co )
where
new_cenv  no_change && not sym = delVarEnv cenv old_var
 otherwise = extendVarEnv cenv old_var (mkCoVarCo new_var)
no_kind_change = noFreeVarsOfCo old_kind_co
no_change = no_kind_change && (new_var == old_var)
new_kind_co  no_kind_change = old_kind_co
 otherwise = sco old_kind_co
Pair h1 h2 = coercionKind new_kind_co
new_var = uniqAway in_scope $ mkCoVar (varName old_var) new_var_type
new_var_type  sym = h2
 otherwise = h1
substCoVar :: TCvSubst > CoVar > Coercion
substCoVar (TCvSubst _ _ cenv) cv
= case lookupVarEnv cenv cv of
Just co > co
Nothing > CoVarCo cv
substCoVars :: TCvSubst > [CoVar] > [Coercion]
substCoVars subst cvs = map (substCoVar subst) cvs
lookupCoVar :: TCvSubst > Var > Maybe Coercion
lookupCoVar (TCvSubst _ _ cenv) v = lookupVarEnv cenv v
substTyVarBndr :: HasCallStack => TCvSubst > TyVar > (TCvSubst, TyVar)
substTyVarBndr = substTyVarBndrUsing substTy
substTyVarBndrs :: HasCallStack => TCvSubst > [TyVar] > (TCvSubst, [TyVar])
substTyVarBndrs = mapAccumL substTyVarBndr
substVarBndr :: HasCallStack => TCvSubst > TyCoVar > (TCvSubst, TyCoVar)
substVarBndr = substVarBndrUsing substTy
substVarBndrs :: HasCallStack => TCvSubst > [TyCoVar] > (TCvSubst, [TyCoVar])
substVarBndrs = mapAccumL substVarBndr
substCoVarBndr :: HasCallStack => TCvSubst > CoVar > (TCvSubst, CoVar)
substCoVarBndr = substCoVarBndrUsing substTy
  Like 'substVarBndr', but disables sanity checks.
 The problems that the sanity checks in substTy catch are described in
 Note [The substitution invariant].
 The goal of #11371 is to migrate all the calls of substTyUnchecked to
 substTy and remove this function. Please don't use in new code.
substVarBndrUnchecked :: TCvSubst > TyCoVar > (TCvSubst, TyCoVar)
substVarBndrUnchecked = substVarBndrUsing substTyUnchecked
substVarBndrUsing :: (TCvSubst > Type > Type)
> TCvSubst > TyCoVar > (TCvSubst, TyCoVar)
substVarBndrUsing subst_fn subst v
 isTyVar v = substTyVarBndrUsing subst_fn subst v
 otherwise = substCoVarBndrUsing subst_fn subst v
  Substitute a tyvar in a binding position, returning an
 extended subst and a new tyvar.
 Use the supplied function to substitute in the kind
substTyVarBndrUsing
:: (TCvSubst > Type > Type)  ^ Use this to substitute in the kind
> TCvSubst > TyVar > (TCvSubst, TyVar)
substTyVarBndrUsing subst_fn subst@(TCvSubst in_scope tenv cenv) old_var
= ASSERT2( _no_capture, pprTyVar old_var $$ pprTyVar new_var $$ ppr subst )
ASSERT( isTyVar old_var )
(TCvSubst (in_scope `extendInScopeSet` new_var) new_env cenv, new_var)
where
new_env  no_change = delVarEnv tenv old_var
 otherwise = extendVarEnv tenv old_var (TyVarTy new_var)
_no_capture = not (new_var `elemVarSet` tyCoVarsOfTypesSet tenv)
 Assertion check that we are not capturing something in the substitution
old_ki = tyVarKind old_var
no_kind_change = noFreeVarsOfType old_ki  verify that kind is closed
no_change = no_kind_change && (new_var == old_var)
 no_change means that the new_var is identical in
 all respects to the old_var (same unique, same kind)
 See Note [Extending the TCvSubst]

 In that case we don't need to extend the substitution
 to map old to new. But instead we must zap any
 current substitution for the variable. For example:
 (\x.e) with id_subst = [x > e']
 Here we must simply zap the substitution for x
new_var  no_kind_change = uniqAway in_scope old_var
 otherwise = uniqAway in_scope $
setTyVarKind old_var (subst_fn subst old_ki)
 The uniqAway part makes sure the new variable is not already in scope
  Substitute a covar in a binding position, returning an
 extended subst and a new covar.
 Use the supplied function to substitute in the kind
substCoVarBndrUsing
:: (TCvSubst > Type > Type)
> TCvSubst > CoVar > (TCvSubst, CoVar)
substCoVarBndrUsing subst_fn subst@(TCvSubst in_scope tenv cenv) old_var
= ASSERT( isCoVar old_var )
(TCvSubst (in_scope `extendInScopeSet` new_var) tenv new_cenv, new_var)
where
new_co = mkCoVarCo new_var
no_kind_change = noFreeVarsOfTypes [t1, t2]
no_change = new_var == old_var && no_kind_change
new_cenv  no_change = delVarEnv cenv old_var
 otherwise = extendVarEnv cenv old_var new_co
new_var = uniqAway in_scope subst_old_var
subst_old_var = mkCoVar (varName old_var) new_var_type
(_, _, t1, t2, role) = coVarKindsTypesRole old_var
t1' = subst_fn subst t1
t2' = subst_fn subst t2
new_var_type = mkCoercionType role t1' t2'
 It's important to do the substitution for coercions,
 because they can have free type variables
cloneTyVarBndr :: TCvSubst > TyVar > Unique > (TCvSubst, TyVar)
cloneTyVarBndr subst@(TCvSubst in_scope tv_env cv_env) tv uniq
= ASSERT2( isTyVar tv, ppr tv )  I think it's only called on TyVars
(TCvSubst (extendInScopeSet in_scope tv')
(extendVarEnv tv_env tv (mkTyVarTy tv')) cv_env, tv')
where
old_ki = tyVarKind tv
no_kind_change = noFreeVarsOfType old_ki  verify that kind is closed
tv1  no_kind_change = tv
 otherwise = setTyVarKind tv (substTy subst old_ki)
tv' = setVarUnique tv1 uniq
cloneTyVarBndrs :: TCvSubst > [TyVar] > UniqSupply > (TCvSubst, [TyVar])
cloneTyVarBndrs subst [] _usupply = (subst, [])
cloneTyVarBndrs subst (t:ts) usupply = (subst'', tv:tvs)
where
(uniq, usupply') = takeUniqFromSupply usupply
(subst' , tv ) = cloneTyVarBndr subst t uniq
(subst'', tvs) = cloneTyVarBndrs subst' ts usupply'
{
%************************************************************************
%* *
Prettyprinting types
Defined very early because of debug printing in assertions
%* *
%************************************************************************
@pprType@ is the standard @Type@ printer; the overloaded @ppr@ function is
defined to use this. @pprParendType@ is the same, except it puts
parens around the type, except for the atomic cases. @pprParendType@
works just by setting the initial context precedence very high.
Note that any function which prettyprints a @Type@ first converts the @Type@
to an @IfaceType@. See Note [IfaceType and prettyprinting] in IfaceType.
See Note [Precedence in types] in BasicTypes.
}

 When prettyprinting types, we convert to IfaceType,
 and prettyprint that.
 See Note [Pretty printing via IfaceSyn] in PprTyThing

pprType, pprParendType :: Type > SDoc
pprType = pprPrecType topPrec
pprParendType = pprPrecType appPrec
pprPrecType :: PprPrec > Type > SDoc
pprPrecType = pprPrecTypeX emptyTidyEnv
pprPrecTypeX :: TidyEnv > PprPrec > Type > SDoc
pprPrecTypeX env prec ty
= getPprStyle $ \sty >
if debugStyle sty  Use debugPprType when in
then debug_ppr_ty prec ty  when in debugstyle
else pprPrecIfaceType prec (tidyToIfaceTypeStyX env ty sty)
pprTyLit :: TyLit > SDoc
pprTyLit = pprIfaceTyLit . toIfaceTyLit
pprKind, pprParendKind :: Kind > SDoc
pprKind = pprType
pprParendKind = pprParendType
tidyToIfaceTypeStyX :: TidyEnv > Type > PprStyle > IfaceType
tidyToIfaceTypeStyX env ty sty
 userStyle sty = tidyToIfaceTypeX env ty
 otherwise = toIfaceTypeX (tyCoVarsOfType ty) ty
 in latter case, don't tidy, as we'll be printing uniques.
tidyToIfaceType :: Type > IfaceType
tidyToIfaceType = tidyToIfaceTypeX emptyTidyEnv
tidyToIfaceTypeX :: TidyEnv > Type > IfaceType
 It's vital to tidy before converting to an IfaceType
 or nested binders will become indistinguishable!

 Also for the free type variables, tell toIfaceTypeX to
 leave them as IfaceFreeTyVar. This is superimportant
 for debug printing.
tidyToIfaceTypeX env ty = toIfaceTypeX (mkVarSet free_tcvs) (tidyType env' ty)
where
env' = tidyFreeTyCoVars env free_tcvs
free_tcvs = tyCoVarsOfTypeWellScoped ty

pprCo, pprParendCo :: Coercion > SDoc
pprCo co = getPprStyle $ \ sty > pprIfaceCoercion (tidyToIfaceCoSty co sty)
pprParendCo co = getPprStyle $ \ sty > pprParendIfaceCoercion (tidyToIfaceCoSty co sty)
tidyToIfaceCoSty :: Coercion > PprStyle > IfaceCoercion
tidyToIfaceCoSty co sty
 userStyle sty = tidyToIfaceCo co
 otherwise = toIfaceCoercionX (tyCoVarsOfCo co) co
 in latter case, don't tidy, as we'll be printing uniques.
tidyToIfaceCo :: Coercion > IfaceCoercion
 It's vital to tidy before converting to an IfaceType
 or nested binders will become indistinguishable!

 Also for the free type variables, tell toIfaceCoercionX to
 leave them as IfaceFreeCoVar. This is superimportant
 for debug printing.
tidyToIfaceCo co = toIfaceCoercionX (mkVarSet free_tcvs) (tidyCo env co)
where
env = tidyFreeTyCoVars emptyTidyEnv free_tcvs
free_tcvs = scopedSort $ tyCoVarsOfCoList co

pprClassPred :: Class > [Type] > SDoc
pprClassPred clas tys = pprTypeApp (classTyCon clas) tys

pprTheta :: ThetaType > SDoc
pprTheta = pprIfaceContext topPrec . map tidyToIfaceType
pprParendTheta :: ThetaType > SDoc
pprParendTheta = pprIfaceContext appPrec . map tidyToIfaceType
pprThetaArrowTy :: ThetaType > SDoc
pprThetaArrowTy = pprIfaceContextArr . map tidyToIfaceType

instance Outputable Type where
ppr ty = pprType ty
instance Outputable TyLit where
ppr = pprTyLit

pprSigmaType :: Type > SDoc
pprSigmaType = pprIfaceSigmaType ShowForAllWhen . tidyToIfaceType
pprForAll :: [TyCoVarBinder] > SDoc
pprForAll tvs = pprIfaceForAll (map toIfaceForAllBndr tvs)
  Print a userlevel forall; see Note [When to print foralls]
pprUserForAll :: [TyCoVarBinder] > SDoc
pprUserForAll = pprUserIfaceForAll . map toIfaceForAllBndr
pprTCvBndrs :: [TyCoVarBinder] > SDoc
pprTCvBndrs tvs = sep (map pprTCvBndr tvs)
pprTCvBndr :: TyCoVarBinder > SDoc
pprTCvBndr = pprTyVar . binderVar
pprTyVars :: [TyVar] > SDoc
pprTyVars tvs = sep (map pprTyVar tvs)
pprTyVar :: TyVar > SDoc
 Print a type variable binder with its kind (but not if *)
 Here we do not go via IfaceType, because the duplication with
 pprIfaceTvBndr is minimal, and the loss of uniques etc in
 debug printing is disastrous
pprTyVar tv
 isLiftedTypeKind kind = ppr tv
 otherwise = parens (ppr tv <+> dcolon <+> ppr kind)
where
kind = tyVarKind tv
instance Outputable TyCoBinder where
ppr (Anon ty) = text "[anon]" <+> ppr ty
ppr (Named (Bndr v Required)) = ppr v
ppr (Named (Bndr v Specified)) = char '@' <> ppr v
ppr (Named (Bndr v Inferred)) = braces (ppr v)

instance Outputable Coercion where  defined here to avoid orphans
ppr = pprCo
debugPprType :: Type > SDoc
 ^ debugPprType is a simple pretty printer that prints a type
 without going through IfaceType. It does not format as prettily
 as the normal route, but it's much more direct, and that can
 be useful for debugging. E.g. with dpprdebug it prints the
 kind on typevariable /occurrences/ which the normal route
 fundamentally cannot do.
debugPprType ty = debug_ppr_ty topPrec ty
debug_ppr_ty :: PprPrec > Type > SDoc
debug_ppr_ty _ (LitTy l)
= ppr l
debug_ppr_ty _ (TyVarTy tv)
= ppr tv  With dpprdebug we get (tv :: kind)
debug_ppr_ty prec (FunTy arg res)
= maybeParen prec funPrec $
sep [debug_ppr_ty funPrec arg, arrow <+> debug_ppr_ty prec res]
debug_ppr_ty prec (TyConApp tc tys)
 null tys = ppr tc
 otherwise = maybeParen prec appPrec $
hang (ppr tc) 2 (sep (map (debug_ppr_ty appPrec) tys))
debug_ppr_ty _ (AppTy t1 t2)
= hang (debug_ppr_ty appPrec t1)  Print parens so we see ((a b) c)
2 (debug_ppr_ty appPrec t2)  so that we can distinguish
 TyConApp from AppTy
debug_ppr_ty prec (CastTy ty co)
= maybeParen prec topPrec $
hang (debug_ppr_ty topPrec ty)
2 (text ">" <+> ppr co)
debug_ppr_ty _ (CoercionTy co)
= parens (text "CO" <+> ppr co)
debug_ppr_ty prec ty@(ForAllTy {})
 (tvs, body) < split ty
= maybeParen prec funPrec $
hang (text "forall" <+> fsep (map ppr tvs) <> dot)
 The (map ppr tvs) will print kindannotated
 tvs, because we are (usually) in debugstyle
2 (ppr body)
where
split ty  ForAllTy tv ty' < ty
, (tvs, body) < split ty'
= (tv:tvs, body)
 otherwise
= ([], ty)
{
Note [When to print foralls]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Mostly we want to print toplevel foralls when (and only when) the user specifies
fprintexplicitforalls. But when kind polymorphism is at work, that suppresses
too much information; see Trac #9018.
So I'm trying out this rule: print explicit foralls if
a) User specifies fprintexplicitforalls, or
b) Any of the quantified type variables has a kind
that mentions a kind variable
This catches common situations, such as a type siguature
f :: m a
which means
f :: forall k. forall (m :: k>*) (a :: k). m a
We really want to see both the "forall k" and the kind signatures
on m and a. The latter comes from pprTCvBndr.
Note [Infix type variables]
~~~~~~~~~~~~~~~~~~~~~~~~~~~
With TypeOperators you can say
f :: (a ~> b) > b
and the (~>) is considered a type variable. However, the type
prettyprinter in this module will just see (a ~> b) as
App (App (TyVarTy "~>") (TyVarTy "a")) (TyVarTy "b")
So it'll print the type in prefix form. To avoid confusion we must
remember to parenthesise the operator, thus
(~>) a b > b
See Trac #2766.
}
pprDataCons :: TyCon > SDoc
pprDataCons = sepWithVBars . fmap pprDataConWithArgs . tyConDataCons
where
sepWithVBars [] = empty
sepWithVBars docs = sep (punctuate (space <> vbar) docs)
pprDataConWithArgs :: DataCon > SDoc
pprDataConWithArgs dc = sep [forAllDoc, thetaDoc, ppr dc <+> argsDoc]
where
(_univ_tvs, _ex_tvs, _eq_spec, theta, arg_tys, _res_ty) = dataConFullSig dc
user_bndrs = dataConUserTyVarBinders dc
forAllDoc = pprUserForAll user_bndrs
thetaDoc = pprThetaArrowTy theta
argsDoc = hsep (fmap pprParendType arg_tys)
pprTypeApp :: TyCon > [Type] > SDoc
pprTypeApp tc tys
= pprIfaceTypeApp topPrec (toIfaceTyCon tc)
(toIfaceTcArgs tc tys)
 TODO: toIfaceTcArgs seems rather wasteful here

  Display all kind information (with @fprintexplicitkinds@) when the
 provided 'Bool' argument is 'True'.
 See @Note [Kind arguments in error messages]@ in "TcErrors".
pprWithExplicitKindsWhen :: Bool > SDoc > SDoc
pprWithExplicitKindsWhen b
= updSDocDynFlags $ \dflags >
if b then gopt_set dflags Opt_PrintExplicitKinds
else dflags
{
%************************************************************************
%* *
\subsection{TidyType}
%* *
%************************************************************************
}
  This tidies up a type for printing in an error message, or in
 an interface file.

 It doesn't change the uniques at all, just the print names.
tidyVarBndrs :: TidyEnv > [TyCoVar] > (TidyEnv, [TyCoVar])
tidyVarBndrs tidy_env tvs
= mapAccumL tidyVarBndr (avoidNameClashes tvs tidy_env) tvs
tidyVarBndr :: TidyEnv > TyCoVar > (TidyEnv, TyCoVar)
tidyVarBndr tidy_env@(occ_env, subst) var
= case tidyOccName occ_env (getHelpfulOccName var) of
(occ_env', occ') > ((occ_env', subst'), var')
where
subst' = extendVarEnv subst var var'
var' = setVarType (setVarName var name') type'
type' = tidyType tidy_env (varType var)
name' = tidyNameOcc name occ'
name = varName var
avoidNameClashes :: [TyCoVar] > TidyEnv > TidyEnv
 Seed the occ_env with clashes among the names, see
 Node [Tidying multiple names at once] in OccName
avoidNameClashes tvs (occ_env, subst)
= (avoidClashesOccEnv occ_env occs, subst)
where
occs = map getHelpfulOccName tvs
getHelpfulOccName :: TyCoVar > OccName
 A TcTyVar with a System Name is probably a
 unification variable; when we tidy them we give them a trailing
 "0" (or 1 etc) so that they don't take precedence for the
 unmodified name. Plus, indicating a unification variable in
 this way is a helpful clue for users
getHelpfulOccName tv
 isSystemName name, isTcTyVar tv
= mkTyVarOcc (occNameString occ ++ "0")
 otherwise
= occ
where
name = varName tv
occ = getOccName name
tidyTyCoVarBinder :: TidyEnv > VarBndr TyCoVar vis
> (TidyEnv, VarBndr TyCoVar vis)
tidyTyCoVarBinder tidy_env (Bndr tv vis)
= (tidy_env', Bndr tv' vis)
where
(tidy_env', tv') = tidyVarBndr tidy_env tv
tidyTyCoVarBinders :: TidyEnv > [VarBndr TyCoVar vis]
> (TidyEnv, [VarBndr TyCoVar vis])
tidyTyCoVarBinders = mapAccumL tidyTyCoVarBinder

tidyFreeTyCoVars :: TidyEnv > [TyCoVar] > TidyEnv
 ^ Add the free 'TyVar's to the env in tidy form,
 so that we can tidy the type they are free in
tidyFreeTyCoVars (full_occ_env, var_env) tyvars
= fst (tidyOpenTyCoVars (full_occ_env, var_env) tyvars)

tidyOpenTyCoVars :: TidyEnv > [TyCoVar] > (TidyEnv, [TyCoVar])
tidyOpenTyCoVars env tyvars = mapAccumL tidyOpenTyCoVar env tyvars

tidyOpenTyCoVar :: TidyEnv > TyCoVar > (TidyEnv, TyCoVar)
 ^ Treat a new 'TyCoVar' as a binder, and give it a fresh tidy name
 using the environment if one has not already been allocated. See
 also 'tidyVarBndr'
tidyOpenTyCoVar env@(_, subst) tyvar
= case lookupVarEnv subst tyvar of
Just tyvar' > (env, tyvar')  Already substituted
Nothing >
let env' = tidyFreeTyCoVars env (tyCoVarsOfTypeList (tyVarKind tyvar))
in tidyVarBndr env' tyvar  Treat it as a binder

tidyTyCoVarOcc :: TidyEnv > TyCoVar > TyCoVar
tidyTyCoVarOcc env@(_, subst) tv
= case lookupVarEnv subst tv of
Nothing > updateVarType (tidyType env) tv
Just tv' > tv'

tidyTypes :: TidyEnv > [Type] > [Type]
tidyTypes env tys = map (tidyType env) tys

tidyType :: TidyEnv > Type > Type
tidyType _ (LitTy n) = LitTy n
tidyType env (TyVarTy tv) = TyVarTy (tidyTyCoVarOcc env tv)
tidyType env (TyConApp tycon tys) = let args = tidyTypes env tys
in args `seqList` TyConApp tycon args
tidyType env (AppTy fun arg) = (AppTy $! (tidyType env fun)) $! (tidyType env arg)
tidyType env (FunTy fun arg) = (FunTy $! (tidyType env fun)) $! (tidyType env arg)
tidyType env (ty@(ForAllTy{})) = mkForAllTys' (zip tvs' vis) $! tidyType env' body_ty
where
(tvs, vis, body_ty) = splitForAllTys' ty
(env', tvs') = tidyVarBndrs env tvs
tidyType env (CastTy ty co) = (CastTy $! tidyType env ty) $! (tidyCo env co)
tidyType env (CoercionTy co) = CoercionTy $! (tidyCo env co)
 The following two functions differ from mkForAllTys and splitForAllTys in that
 they expect/preserve the ArgFlag argument. Thes belong to types/Type.hs, but
 how should they be named?
mkForAllTys' :: [(TyCoVar, ArgFlag)] > Type > Type
mkForAllTys' tvvs ty = foldr strictMkForAllTy ty tvvs
where
strictMkForAllTy (tv,vis) ty = (ForAllTy $! ((Bndr $! tv) $! vis)) $! ty
splitForAllTys' :: Type > ([TyCoVar], [ArgFlag], Type)
splitForAllTys' ty = go ty [] []
where
go (ForAllTy (Bndr tv vis) ty) tvs viss = go ty (tv:tvs) (vis:viss)
go ty tvs viss = (reverse tvs, reverse viss, ty)

  Grabs the free type variables, tidies them
 and then uses 'tidyType' to work over the type itself
tidyOpenTypes :: TidyEnv > [Type] > (TidyEnv, [Type])
tidyOpenTypes env tys
= (env', tidyTypes (trimmed_occ_env, var_env) tys)
where
(env'@(_, var_env), tvs') = tidyOpenTyCoVars env $
tyCoVarsOfTypesWellScoped tys
trimmed_occ_env = initTidyOccEnv (map getOccName tvs')
 The idea here was that we restrict the new TidyEnv to the
 _free_ vars of the types, so that we don't gratuitously rename
 the _bound_ variables of the types.

tidyOpenType :: TidyEnv > Type > (TidyEnv, Type)
tidyOpenType env ty = let (env', [ty']) = tidyOpenTypes env [ty] in
(env', ty')

  Calls 'tidyType' on a toplevel type (i.e. with an empty tidying environment)
tidyTopType :: Type > Type
tidyTopType ty = tidyType emptyTidyEnv ty

tidyOpenKind :: TidyEnv > Kind > (TidyEnv, Kind)
tidyOpenKind = tidyOpenType
tidyKind :: TidyEnv > Kind > Kind
tidyKind = tidyType

tidyCo :: TidyEnv > Coercion > Coercion
tidyCo env@(_, subst) co
= go co
where
go_mco MRefl = MRefl
go_mco (MCo co) = MCo (go co)
go (Refl ty) = Refl (tidyType env ty)
go (GRefl r ty mco) = GRefl r (tidyType env ty) $! go_mco mco
go (TyConAppCo r tc cos) = let args = map go cos
in args `seqList` TyConAppCo r tc args
go (AppCo co1 co2) = (AppCo $! go co1) $! go co2
go (ForAllCo tv h co) = ((ForAllCo $! tvp) $! (go h)) $! (tidyCo envp co)
where (envp, tvp) = tidyVarBndr env tv
 the case above duplicates a bit of work in tidying h and the kind
 of tv. But the alternative is to use coercionKind, which seems worse.
go (FunCo r co1 co2) = (FunCo r $! go co1) $! go co2
go (CoVarCo cv) = case lookupVarEnv subst cv of
Nothing > CoVarCo cv
Just cv' > CoVarCo cv'
go (HoleCo h) = HoleCo h
go (AxiomInstCo con ind cos) = let args = map go cos
in args `seqList` AxiomInstCo con ind args
go (UnivCo p r t1 t2) = (((UnivCo $! (go_prov p)) $! r) $!
tidyType env t1) $! tidyType env t2
go (SymCo co) = SymCo $! go co
go (TransCo co1 co2) = (TransCo $! go co1) $! go co2
go (NthCo r d co) = NthCo r d $! go co
go (LRCo lr co) = LRCo lr $! go co
go (InstCo co ty) = (InstCo $! go co) $! go ty
go (KindCo co) = KindCo $! go co
go (SubCo co) = SubCo $! go co
go (AxiomRuleCo ax cos) = let cos1 = tidyCos env cos
in cos1 `seqList` AxiomRuleCo ax cos1
go_prov UnsafeCoerceProv = UnsafeCoerceProv
go_prov (PhantomProv co) = PhantomProv (go co)
go_prov (ProofIrrelProv co) = ProofIrrelProv (go co)
go_prov p@(PluginProv _) = p
tidyCos :: TidyEnv > [Coercion] > [Coercion]
tidyCos env = map (tidyCo env)
{ *********************************************************************
* *
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 _ co1 co2) = 1 + coercionSize co1 + coercionSize co2
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 UnsafeCoerceProv = 1
provSize (PhantomProv co) = 1 + coercionSize co
provSize (ProofIrrelProv co) = 1 + coercionSize co
provSize (PluginProv _) = 1