GHC/Core/TyCo/Rep.hs

    {-# LANGUAGE CPP                #-}
    {-# LANGUAGE DeriveDataTypeable #-}
    {-# LANGUAGE MultiWayIf         #-}

    {-# OPTIONS_HADDOCK not-home #-}

    {-
    (c) The University of Glasgow 2006
    (c) The GRASP/AQUA Project, Glasgow University, 1998
    \section[GHC.Core.TyCo.Rep]{Type and Coercion - friends' interface}

    Note [The Type-related module hierarchy]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
      GHC.Core.Class
      GHC.Core.Coercion.Axiom
      GHC.Core.TyCon           imports GHC.Core.{Class, Coercion.Axiom}
      GHC.Core.TyCo.Rep        imports GHC.Core.{Class, Coercion.Axiom, TyCon}
      GHC.Core.TyCo.Ppr        imports GHC.Core.TyCo.Rep
      GHC.Core.TyCo.FVs        imports GHC.Core.TyCo.Rep
      GHC.Core.TyCo.Subst      imports GHC.Core.TyCo.{Rep, FVs, Ppr}
      GHC.Core.TyCo.Tidy       imports GHC.Core.TyCo.{Rep, FVs}
      GHC.Builtin.Types.Prim   imports GHC.Core.TyCo.Rep ( including mkTyConTy )
      GHC.Core.Coercion        imports GHC.Core.Type
    -}

    -- We expose the relevant stuff from this module via the Type module
    module GHC.Core.TyCo.Rep (

            -- * Types
            Type(..),

            TyLit(..),
            KindOrType, Kind,
            KnotTied,
            PredType, ThetaType,      -- Synonyms
            ArgFlag(..), AnonArgFlag(..),

            -- * Coercions
            Coercion(..),
            UnivCoProvenance(..),
            CoercionHole(..), coHoleCoVar, setCoHoleCoVar,
            CoercionN, CoercionR, CoercionP, KindCoercion,
            MCoercion(..), MCoercionR, MCoercionN,

            -- * Functions over types
            mkTyConTy_, mkTyVarTy, mkTyVarTys,
            mkTyCoVarTy, mkTyCoVarTys,
            mkFunTy, mkVisFunTy, mkInvisFunTy, mkVisFunTys,
            mkForAllTy, mkForAllTys, mkInvisForAllTys,
            mkPiTy, mkPiTys,
            mkFunTyMany,
            mkScaledFunTy,
            mkVisFunTyMany, mkVisFunTysMany,
            mkInvisFunTyMany, mkInvisFunTysMany,
            nonDetCmpTyLit, cmpTyLit,

            -- * Functions over binders
            TyCoBinder(..), TyCoVarBinder, TyBinder,
            binderVar, binderVars, binderType, binderArgFlag,
            delBinderVar,
            isInvisibleArgFlag, isVisibleArgFlag,
            isInvisibleBinder, isVisibleBinder,
            isTyBinder, isNamedBinder,

            -- * Functions over coercions
            pickLR,

            -- ** Analyzing types
            TyCoFolder(..), foldTyCo,

            -- * Sizes
            typeSize, coercionSize, provSize,

            -- * Multiplicities
            Scaled(..), scaledMult, scaledThing, mapScaledType, Mult
        ) where

    #include "HsVersions.h"

    import GHC.Prelude

    import {-# SOURCE #-} GHC.Core.TyCo.Ppr ( pprType, pprCo, pprTyLit )

       -- Transitively pulls in a LOT of stuff, better to break the loop

    -- friends:
    import GHC.Iface.Type
    import GHC.Types.Var
    import GHC.Types.Var.Set
    import GHC.Core.TyCon
    import GHC.Core.Coercion.Axiom

    -- others
    import {-# SOURCE #-} GHC.Builtin.Types ( manyDataConTy )
    import GHC.Types.Basic ( LeftOrRight(..), pickLR )
    import GHC.Types.Unique ( Uniquable(..) )
    import GHC.Utils.Outputable
    import GHC.Data.FastString
    import GHC.Utils.Misc
    import GHC.Utils.Panic

    -- libraries
    import qualified Data.Data as Data hiding ( TyCon )
    import Data.IORef ( IORef )   -- for CoercionHole

    {- **********************************************************************
    *                                                                       *
                            Type
    *                                                                       *
    ********************************************************************** -}

    -- | The key representation of types within the compiler

    type KindOrType = Type -- See Note [Arguments to type constructors]

    -- | The key type representing kinds in the compiler.
    type Kind = Type

    -- If you edit this type, you may need to update the GHC formalism
    -- See Note [GHC Formalism] in GHC.Core.Lint
    data Type
      -- See Note [Non-trivial definitional equality]
      = TyVarTy Var -- ^ Vanilla type or kind variable (*never* a coercion variable)

      | AppTy
            Type
            Type            -- ^ Type application to something other than a 'TyCon'. Parameters:
                            --
                            --  1) Function: must /not/ be a 'TyConApp' or 'CastTy',
                            --     must be another 'AppTy', or 'TyVarTy'
                            --     See Note [Respecting definitional equality] \(EQ1) about the
                            --     no 'CastTy' requirement
                            --
                            --  2) Argument type

      | TyConApp
            TyCon
            [KindOrType]    -- ^ Application of a 'TyCon', including newtypes /and/ synonyms.
                            -- Invariant: saturated applications of 'FunTyCon' must
                            -- use 'FunTy' and saturated synonyms must use their own
                            -- constructors. However, /unsaturated/ 'FunTyCon's
                            -- do appear as 'TyConApp's.
                            -- Parameters:
                            --
                            -- 1) Type constructor being applied to.
                            --
                            -- 2) Type arguments. Might not have enough type arguments
                            --    here to saturate the constructor.
                            --    Even type synonyms are not necessarily saturated;
                            --    for example unsaturated type synonyms
                            --    can appear as the right hand side of a type synonym.

      | ForAllTy
            {-# UNPACK #-} !TyCoVarBinder
            Type            -- ^ A Π type.
                 -- INVARIANT: If the binder is a coercion variable, it must
                 -- be mentioned in the Type. See
                 -- Note [Unused coercion variable in ForAllTy]

      | FunTy      -- ^ FUN m t1 t2   Very common, so an important special case
                    -- See Note [Function types]
         { ft_af  :: AnonArgFlag    -- Is this (->) or (=>)?
         , ft_mult :: Mult          -- Multiplicity
         , ft_arg :: Type           -- Argument type
         , ft_res :: Type }         -- Result type

      | LitTy TyLit     -- ^ Type literals are similar to type constructors.

      | CastTy
            Type
            KindCoercion  -- ^ A kind cast. The coercion is always nominal.
                          -- INVARIANT: The cast is never reflexive \(EQ2)
                          -- INVARIANT: The Type is not a CastTy (use TransCo instead) \(EQ3)
                          -- INVARIANT: The Type is not a ForAllTy over a tyvar \(EQ4)
                          -- See Note [Respecting definitional equality]

      | CoercionTy
            Coercion    -- ^ Injection of a Coercion into a type
                        -- This should only ever be used in the RHS of an AppTy,
                        -- in the list of a TyConApp, when applying a promoted
                        -- GADT data constructor

      deriving Data.Data

    instance Outputable Type where
      ppr = pprType

    -- NOTE:  Other parts of the code assume that type literals do not contain
    -- types or type variables.
    data TyLit
      = NumTyLit Integer
      | StrTyLit FastString
      | CharTyLit Char
      deriving (Eq, Data.Data)

    -- Non-determinism arises due to uniqCompareFS
    nonDetCmpTyLit :: TyLit -> TyLit -> Ordering
    nonDetCmpTyLit = cmpTyLitWith NonDetFastString

    -- Slower than nonDetCmpTyLit but deterministic
    cmpTyLit :: TyLit -> TyLit -> Ordering
    cmpTyLit = cmpTyLitWith LexicalFastString

    {-# INLINE cmpTyLitWith #-}
    cmpTyLitWith :: Ord r => (FastString -> r) -> TyLit -> TyLit -> Ordering
    cmpTyLitWith _ (NumTyLit  x) (NumTyLit  y) = compare x y
    cmpTyLitWith w (StrTyLit  x) (StrTyLit  y) = compare (w x) (w y)
    cmpTyLitWith _ (CharTyLit x) (CharTyLit y) = compare x y
    cmpTyLitWith _ a b = compare (tag a) (tag b)
      where
        tag :: TyLit -> Int
        tag NumTyLit{}  = 0
        tag StrTyLit{}  = 1
        tag CharTyLit{} = 2

    instance Outputable TyLit where
       ppr = pprTyLit

    {- Note [Function types]
    ~~~~~~~~~~~~~~~~~~~~~~~~
    FFunTy is the constructor for a function type.  Lots of things to say
    about it!

    * FFunTy is the data constructor, meaning "full function type".

    * The function type constructor (->) has kind
         (->) :: forall {r1} {r2}. TYPE r1 -> TYPE r2 -> Type LiftedRep
      mkTyConApp ensure that we convert a saturated application
        TyConApp (->) [r1,r2,t1,t2] into FunTy t1 t2
      dropping the 'r1' and 'r2' arguments; they are easily recovered
      from 't1' and 't2'.

    * For the time being its RuntimeRep quantifiers are left
      inferred. This is to allow for it to evolve.

    * The ft_af field says whether or not this is an invisible argument
         VisArg:   t1 -> t2    Ordinary function type
         InvisArg: t1 => t2    t1 is guaranteed to be a predicate type,
                               i.e. t1 :: Constraint
      See Note [Types for coercions, predicates, and evidence]

      This visibility info makes no difference in Core; it matters
      only when we regard the type as a Haskell source type.

    * FunTy is a (unidirectional) pattern synonym that allows
      positional pattern matching (FunTy arg res), ignoring the
      ArgFlag.
    -}

    {- -----------------------
          Commented out until the pattern match
          checker can handle it; see #16185

          For now we use the CPP macro #define FunTy FFunTy _
          (see HsVersions.h) to allow pattern matching on a
          (positional) FunTy constructor.

    {-# COMPLETE FunTy, TyVarTy, AppTy, TyConApp
               , ForAllTy, LitTy, CastTy, CoercionTy :: Type #-}

    -- | 'FunTy' is a (uni-directional) pattern synonym for the common
    -- case where we want to match on the argument/result type, but
    -- ignoring the AnonArgFlag
    pattern FunTy :: Type -> Type -> Type
    pattern FunTy arg res <- FFunTy { ft_arg = arg, ft_res = res }

           End of commented out block
    ---------------------------------- -}

    {- Note [Types for coercions, predicates, and evidence]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We treat differently:

      (a) Predicate types
            Test: isPredTy
            Binders: DictIds
            Kind: Constraint
            Examples: (Eq a), and (a ~ b)

      (b) Coercion types are primitive, unboxed equalities
            Test: isCoVarTy
            Binders: CoVars (can appear in coercions)
            Kind: TYPE (TupleRep [])
            Examples: (t1 ~# t2) or (t1 ~R# t2)

      (c) Evidence types is the type of evidence manipulated by
          the type constraint solver.
            Test: isEvVarType
            Binders: EvVars
            Kind: Constraint or TYPE (TupleRep [])
            Examples: all coercion types and predicate types

    Coercion types and predicate types are mutually exclusive,
    but evidence types are a superset of both.

    When treated as a user type,

      - Predicates (of kind Constraint) are invisible and are
        implicitly instantiated

      - Coercion types, and non-pred evidence types (i.e. not
        of kind Constrain), are just regular old types, are
        visible, and are not implicitly instantiated.

    In a FunTy { ft_af = InvisArg }, the argument type is always
    a Predicate type.

    Note [Weird typing rule for ForAllTy]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Here are the typing rules for ForAllTy:

    tyvar : Type
    inner : TYPE r
    tyvar does not occur in r
    ------------------------------------
    ForAllTy (Bndr tyvar vis) inner : TYPE r

    inner : TYPE r
    ------------------------------------
    ForAllTy (Bndr covar vis) inner : Type

    Note that the kind of the result depends on whether the binder is a
    tyvar or a covar. The kind of a forall-over-tyvar is the same as
    the kind of the inner type. This is because quantification over types
    is erased before runtime. By contrast, the kind of a forall-over-covar
    is always Type, because a forall-over-covar is compiled into a function
    taking a 0-bit-wide erased coercion argument.

    Because the tyvar form above includes r in its result, we must
    be careful not to let any variables escape -- thus the last premise
    of the rule above.

    Note [Constraints in kinds]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Do we allow a type constructor to have a kind like
       S :: Eq a => a -> Type

    No, we do not.  Doing so would mean would need a TyConApp like
       S @k @(d :: Eq k) (ty :: k)
     and we have no way to build, or decompose, evidence like
     (d :: Eq k) at the type level.

    But we admit one exception: equality.  We /do/ allow, say,
       MkT :: (a ~ b) => a -> b -> Type a b

    Why?  Because we can, without much difficulty.  Moreover
    we can promote a GADT data constructor (see TyCon
    Note [Promoted data constructors]), like
      data GT a b where
        MkGT : a -> a -> GT a a
    so programmers might reasonably expect to be able to
    promote MkT as well.

    How does this work?

    * In GHC.Tc.Validity.checkConstraintsOK we reject kinds that
      have constraints other than (a~b) and (a~~b).

    * In Inst.tcInstInvisibleTyBinder we instantiate a call
      of MkT by emitting
         [W] co :: alpha ~# beta
      and producing the elaborated term
         MkT @alpha @beta (Eq# alpha beta co)
      We don't generate a boxed "Wanted"; we generate only a
      regular old /unboxed/ primitive-equality Wanted, and build
      the box on the spot.

    * How can we get such a MkT?  By promoting a GADT-style data
      constructor
         data T a b where
           MkT :: (a~b) => a -> b -> T a b
      See DataCon.mkPromotedDataCon
      and Note [Promoted data constructors] in GHC.Core.TyCon

    * We support both homogeneous (~) and heterogeneous (~~)
      equality.  (See Note [The equality types story]
      in GHC.Builtin.Types.Prim for a primer on these equality types.)

    * How do we prevent a MkT having an illegal constraint like
      Eq a?  We check for this at use-sites; see GHC.Tc.Gen.HsType.tcTyVar,
      specifically dc_theta_illegal_constraint.

    * Notice that nothing special happens if
        K :: (a ~# b) => blah
      because (a ~# b) is not a predicate type, and is never
      implicitly instantiated. (Mind you, it's not clear how you
      could creates a type constructor with such a kind.) See
      Note [Types for coercions, predicates, and evidence]

    * The existence of promoted MkT with an equality-constraint
      argument is the (only) reason that the AnonTCB constructor
      of TyConBndrVis carries an AnonArgFlag (VisArg/InvisArg).
      For example, when we promote the data constructor
         MkT :: forall a b. (a~b) => a -> b -> T a b
      we get a PromotedDataCon with tyConBinders
          Bndr (a :: Type)  (NamedTCB Inferred)
          Bndr (b :: Type)  (NamedTCB Inferred)
          Bndr (_ :: a ~ b) (AnonTCB InvisArg)
          Bndr (_ :: a)     (AnonTCB VisArg))
          Bndr (_ :: b)     (AnonTCB VisArg))

    * One might reasonably wonder who *unpacks* these boxes once they are
      made. After all, there is no type-level `case` construct. The
      surprising answer is that no one ever does. Instead, if a GADT
      constructor is used on the left-hand side of a type family equation,
      that occurrence forces GHC to unify the types in question. For
      example:

      data G a where
        MkG :: G Bool

      type family F (x :: G a) :: a where
        F MkG = False

      When checking the LHS `F MkG`, GHC sees the MkG constructor and then must
      unify F's implicit parameter `a` with Bool. This succeeds, making the equation

        F Bool (MkG @Bool <Bool>) = False

      Note that we never need unpack the coercion. This is because type
      family equations are *not* parametric in their kind variables. That
      is, we could have just said

      type family H (x :: G a) :: a where
        H _ = False

      The presence of False on the RHS also forces `a` to become Bool,
      giving us

        H Bool _ = False

      The fact that any of this works stems from the lack of phase
      separation between types and kinds (unlike the very present phase
      separation between terms and types).

      Once we have the ability to pattern-match on types below top-level,
      this will no longer cut it, but it seems fine for now.

    Note [Arguments to type constructors]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Because of kind polymorphism, in addition to type application we now
    have kind instantiation. We reuse the same notations to do so.

    For example:

      Just (* -> *) Maybe
      Right * Nat Zero

    are represented by:

      TyConApp (PromotedDataCon Just) [* -> *, Maybe]
      TyConApp (PromotedDataCon Right) [*, Nat, (PromotedDataCon Zero)]

    Important note: Nat is used as a *kind* and not as a type. This can be
    confusing, since type-level Nat and kind-level 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 [Non-trivial 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 α-equivalence
    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://richarde.dev/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
    α-equivalence. 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://richarde.dev/papers/2016/thesis/eisenberg-thesis.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 kind-checking, 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 kind-checking would be enormous,
    and we would need enormous casts with lots of CoherenceCo's to straighten
    them out.

    Would anything go wrong if eqType looked through 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
    easy-to-implement 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.

    Interestingly, it must be the case that the free variables of t1 and t2
    might be different, even if t1 `eqType` t2. A simple example of this is
    if we have both cv1 :: k1 ~ k2 and cv2 :: k1 ~ k2 in the environment.
    Then t1 = t |> cv1 and t2 = t |> cv2 are eqType; yet cv1 is in the free
    vars of t1 and cv2 is in the free vars of t2. Unless we choose to implement
    eqType to be just α-equivalence, this wrinkle around free variables
    remains.

    Yet not all is lost: we can say that any two equal types share the same
    *relevant* free variables. Here, a relevant variable is a shallow
    free variable (see Note [Shallow and deep free variables] in GHC.Core.TyCo.FVs)
    that does not appear within a coercion. Note that type variables can
    appear within coercions (in, say, a Refl node), but that coercion variables
    cannot appear outside a coercion. We do not (yet) have a function to
    extract relevant free variables, but it would not be hard to write if
    the need arises.

    Besides eqType, another equality relation that upholds the (EQ) property above
    is /typechecker equality/, which is implemented as
    GHC.Tc.Utils.TcType.tcEqType. See
    Note [Typechecker equality vs definitional equality] in GHC.Tc.Utils.TcType for
    what the difference between eqType and tcEqType is.

    Note [Respecting definitional equality]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Note [Non-trivial 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 non-reflexive 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. This is (EQ1) below.

    Unforunately, that's not the end of the story. Consider comparing
      (T a b c)      =?       (T a b |> (co -> <Type>)) (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 right-hand
    type is ill-formed, 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 right-hand type will instead yield (T a b (c |> co |> sym co) |> <Type>).
    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
    Pi-types. For example, if we have @type family Fun a b where Fun a b = a -> b@,
    then we might have (T :: Fun Type Type) and (T |> axFun) Int. That axFun can't
    be pulled to the right. But we don't need to pull it: (T |> axFun) Int is not
    `eqType` to any proper TyConApp -- thus, leaving it where it is doesn't violate
    our (EQ) property.

    In order to detect reflexive casts reliably, we must make sure not
    to have nested casts: we update (t |> co1 |> co2) to (t |> (co1 `TransCo` co2)).
    This is (EQ2) below.

    One other troublesome case is ForAllTy. See Note [Weird typing rule for ForAllTy].
    The kind of the body is the same as the kind of the ForAllTy. Accordingly,

      ForAllTy tv (ty |> co)     and     (ForAllTy tv ty) |> co

    are `eqType`. But only the first can be split by splitForAllTy. So we forbid
    the second form, instead pushing the coercion inside to get the first form.
    This is done in mkCastTy.

    In sum, in order to uphold (EQ), we need the following invariants:

      (EQ1) No decomposable CastTy to the left of an AppTy, where a decomposable
            cast is one that relates either a FunTy to a FunTy or a
            ForAllTy to a ForAllTy.
      (EQ2) No reflexive casts in CastTy.
      (EQ3) No nested CastTys.
      (EQ4) No CastTy over (ForAllTy (Bndr tyvar vis) body).
            See Note [Weird typing rule for ForAllTy]

    These invariants are all documented above, in the declaration for Type.

    Note [Equality on FunTys]
    ~~~~~~~~~~~~~~~~~~~~~~~~~
    A (FunTy vis mult arg res) is just an abbreviation for a
      TyConApp funTyCon [mult, arg_rep, res_rep, arg, res]
    where
      arg :: TYPE arg_rep
      res :: TYPE res_rep
    Note that the vis field of a FunTy appears nowhere in the
    equivalent TyConApp. In Core, this is OK, because we no longer
    care about the visibility of the argument in a FunTy
    (the vis distinguishes between arg -> res and arg => res).
    In the type-checker, we are careful not to decompose FunTys
    with an invisible argument. See also Note [Decomposing fat arrow c=>t]
    in GHC.Core.Type.

    In order to compare FunTys while respecting how they could
    expand into TyConApps, we must check
    the kinds of the arg and the res.

    Note [Unused coercion variable in ForAllTy]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Suppose we have
      \(co:t1 ~ t2). e

    What type should we give to this expression?
      (1) forall (co:t1 ~ t2) -> t
      (2) (t1 ~ t2) -> t

    If co is used in t, (1) should be the right choice.
    if co is not used in t, we would like to have (1) and (2) equivalent.

    However, we want to keep eqType simple and don't want eqType (1) (2) to return
    True in any case.

    We decide to always construct (2) if co is not used in t.

    Thus in mkLamType, we check whether the variable is a coercion
    variable (of type (t1 ~# t2), and whether it is un-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.

    Note [Weird typing rule for ForAllTy]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Here is the (truncated) typing rule for the dependent ForAllTy:

      inner : TYPE r
      tyvar is not free in r
      ----------------------------------------
      ForAllTy (Bndr tyvar vis) inner : TYPE r

    Note that the kind of `inner` is the kind of the overall ForAllTy. This is
    necessary because every ForAllTy over a type variable is erased at runtime.
    Thus the runtime representation of a ForAllTy (as encoded, via TYPE rep, in
    the kind) must be the same as the representation of the body. We must check
    for skolem-escape, though. The skolem-escape would prevent a definition like

      undefined :: forall (r :: RuntimeRep) (a :: TYPE r). a

    because the type's kind (TYPE r) mentions the out-of-scope r. Luckily, the real
    type of undefined is

      undefined :: forall (r :: RuntimeRep) (a :: TYPE r). HasCallStack => a

    and that HasCallStack constraint neatly sidesteps the potential skolem-escape
    problem.

    If the bound variable is a coercion variable:

      inner : TYPE r
      covar is free in inner
      ------------------------------------
      ForAllTy (Bndr covar vis) inner : Type

    Here, the kind of the ForAllTy is just Type, because coercion abstractions
    are *not* erased. The "covar is free in inner" premise is solely to maintain
    the representation invariant documented in
    Note [Unused coercion variable in ForAllTy]. Though there is surface similarity
    between this free-var check and the one in the tyvar rule, these two restrictions
    are truly unrelated.

    -}

    -- | A type labeled 'KnotTied' might have knot-tied tycons in it. See
    -- Note [Type checking recursive type and class declarations] in
    -- "GHC.Tc.TyCl"
    type KnotTied ty = ty

    {- **********************************************************************
    *                                                                       *
                      TyCoBinder and ArgFlag
    *                                                                       *
    ********************************************************************** -}

    -- | A 'TyCoBinder' represents an argument to a function. TyCoBinders can be
    -- dependent ('Named') or nondependent ('Anon'). They may also be visible or
    -- not. See Note [TyCoBinders]
    data TyCoBinder
      = Named TyCoVarBinder    -- A type-lambda binder
      | Anon AnonArgFlag (Scaled Type)  -- A term-lambda binder. Type here can be CoercionTy.
                                        -- Visibility is determined by the AnonArgFlag
      deriving Data.Data

    instance Outputable TyCoBinder where
      ppr (Anon af ty) = ppr af <+> ppr ty
      ppr (Named (Bndr v Required))  = ppr v
      -- See Note [Explicit Case Statement for Specificity]
      ppr (Named (Bndr v (Invisible spec))) = case spec of
        SpecifiedSpec -> char '@' <> ppr v
        InferredSpec  -> braces (ppr v)

    -- | 'TyBinder' is like 'TyCoBinder', but there can only be 'TyVarBinder'
    -- in the 'Named' field.
    type TyBinder = TyCoBinder

    -- | Remove the binder's variable from the set, if the binder has
    -- a variable.
    delBinderVar :: VarSet -> TyCoVarBinder -> VarSet
    delBinderVar vars (Bndr tv _) = vars `delVarSet` tv

    -- | Does this binder bind an invisible argument?
    isInvisibleBinder :: TyCoBinder -> Bool
    isInvisibleBinder (Named (Bndr _ vis)) = isInvisibleArgFlag vis
    isInvisibleBinder (Anon InvisArg _)    = True
    isInvisibleBinder (Anon VisArg   _)    = False

    -- | Does this binder bind a visible argument?
    isVisibleBinder :: TyCoBinder -> Bool
    isVisibleBinder = not . isInvisibleBinder

    isNamedBinder :: TyCoBinder -> Bool
    isNamedBinder (Named {}) = True
    isNamedBinder (Anon {})  = False

    -- | If its a named binder, is the binder a tyvar?
    -- Returns True for nondependent binder.
    -- This check that we're really returning a *Ty*Binder (as opposed to a
    -- coercion binder). That way, if/when we allow coercion quantification
    -- in more places, we'll know we missed updating some function.
    isTyBinder :: TyCoBinder -> Bool
    isTyBinder (Named bnd) = isTyVarBinder bnd
    isTyBinder _ = True

    {- Note [TyCoBinders]
    ~~~~~~~~~~~~~~~~~~~
    A ForAllTy contains a TyCoVarBinder.  But a type can be decomposed
    to a telescope consisting of a [TyCoBinder]

    A TyCoBinder represents the type of binders -- that is, the type of an
    argument to a Pi-type. GHC Core currently supports two different
    Pi-types:

     * A non-dependent function type,
       written with ->, e.g. ty1 -> ty2
       represented as FunTy ty1 ty2. These are
       lifted to Coercions with the corresponding FunCo.

     * A dependent compile-time-only polytype,
       written with forall, e.g.  forall (a:*). ty
       represented as ForAllTy (Bndr a v) ty

    Both Pi-types 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
      user-written signature for f1

    * Specified.  Function defn, with signature (implicit forall):
         f2 :: a -> a; f2 x = x
      So f2 gets the type f2 :: forall a. a -> a, with 'a' Specified
      even though 'a' is not bound in the source code by an explicit forall

    * Specified.  Function defn, with signature (explicit forall):
         f3 :: forall a. a -> a; f3 x = x
      So f3 gets the type f3 :: forall a. a -> a, with 'a' Specified

    * Inferred.  Function defn, with signature (explicit forall), marked as inferred:
         f4 :: forall {a}. a -> a; f4 x = x
      So f4 gets the type f4 :: forall {a}. a -> a, with 'a' Inferred
      It's Inferred because the user marked it as such, even though it does appear
      in the user-written signature for f4

    * Inferred/Specified.  Function signature with inferred kind polymorphism.
         f5 :: a b -> Int
      So 'f5' gets the type f5 :: forall {k} (a:k->*) (b:k). a b -> Int
      Here 'k' is Inferred (it's not mentioned in the type),
      but 'a' and 'b' are Specified.

    * Specified.  Function signature with explicit kind polymorphism
         f6 :: a (b :: k) -> Int
      This time 'k' is Specified, because it is mentioned explicitly,
      so we get f6 :: forall (k:*) (a:k->*) (b:k). a b -> Int

    * Similarly pattern synonyms:
      Inferred - from inferred types (e.g. no pattern type signature)
               - or from inferred kind polymorphism

    ---- In type declarations ----

    * Inferred (k)
         data T1 a b = MkT1 (a b)
      Here T1's kind is  T1 :: forall {k:*}. (k->*) -> k -> *
      The kind variable 'k' is Inferred, since it is not mentioned

      Note that 'a' and 'b' correspond to /Anon/ TyCoBinders in T1's kind,
      and Anon binders don't have a visibility flag. (Or you could think
      of Anon having an implicit Required flag.)

    * Specified (k)
         data T2 (a::k->*) b = MkT (a b)
      Here T's kind is  T :: forall (k:*). (k->*) -> k -> *
      The kind variable 'k' is Specified, since it is mentioned in
      the signature.

    * Required (k)
         data T k (a::k->*) b = MkT (a b)
      Here T's kind is  T :: forall k:* -> (k->*) -> k -> *
      The kind is Required, since it bound in a positional way in T's declaration
      Every use of T must be explicitly applied to a kind

    * Inferred (k1), Specified (k)
         data T a b (c :: k) = MkT (a b) (Proxy c)
      Here T's kind is  T :: forall {k1:*} (k:*). (k1->*) -> k1 -> k -> *
      So 'k' is Specified, because it appears explicitly,
      but 'k1' is Inferred, because it does not

    Generally, in the list of TyConBinders for a TyCon,

    * Inferred arguments always come first
    * Specified, Anon and Required can be mixed

    e.g.
      data Foo (a :: Type) :: forall b. (a -> b -> Type) -> Type where ...

    Here Foo's TyConBinders are
       [Required 'a', Specified 'b', Anon]
    and its kind prints as
       Foo :: forall a -> forall b. (a -> b -> Type) -> Type

    See also Note [Required, Specified, and Inferred for types] in GHC.Tc.TyCl

    ---- Printing -----

     We print forall types with enough syntax to tell you their visibility
     flag.  But this is not source Haskell, and these types may not all
     be parsable.

     Specified: a list of Specified binders is written between `forall` and `.`:
                   const :: forall a b. a -> b -> a

     Inferred: like Specified, but every binder is written in braces:
                   f :: forall {k} (a:k). S k a -> Int

     Required: binders are put between `forall` and `->`:
                  T :: forall k -> *

    ---- Other points -----

    * In classic Haskell, all named binders (that is, the type variables in
      a polymorphic function type f :: forall a. a -> a) have been Inferred.

    * Inferred variables correspond to "generalized" variables from the
      Visible Type Applications paper (ESOP'16).

    Note [No Required TyCoBinder in terms]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We don't allow Required foralls for term variables, including pattern
    synonyms and data constructors.  Why?  Because then an application
    would need a /compulsory/ type argument (possibly without an "@"?),
    thus (f Int); and we don't have concrete syntax for that.

    We could change this decision, but Required, Named TyCoBinders are rare
    anyway.  (Most are Anons.)

    However the type of a term can (just about) have a required quantifier;
    see Note [Required quantifiers in the type of a term] in GHC.Tc.Gen.Expr.
    -}

    {- **********************************************************************
    *                                                                       *
                            PredType
    *                                                                       *
    ********************************************************************** -}

    -- | A type of the form @p@ of constraint kind represents a value whose type is
    -- the Haskell predicate @p@, where a predicate is what occurs before
    -- the @=>@ in a Haskell type.
    --
    -- We use 'PredType' as documentation to mark those types that we guarantee to
    -- have this kind.
    --
    -- It can be expanded into its representation, but:
    --
    -- * The type checker must treat it as opaque
    --
    -- * The rest of the compiler treats it as transparent
    --
    -- Consider these examples:
    --
    -- > f :: (Eq a) => a -> Int
    -- > g :: (?x :: Int -> Int) => a -> Int
    -- > h :: (r\l) => {r} => {l::Int | r}
    --
    -- Here the @Eq a@ and @?x :: Int -> Int@ and @r\l@ are all called \"predicates\"
    type PredType = Type

    -- | A collection of 'PredType's
    type ThetaType = [PredType]

    {-
    (We don't support TREX records yet, but the setup is designed
    to expand to allow them.)

    A Haskell qualified type, such as that for f,g,h above, is
    represented using
            * a FunTy for the double arrow
            * with a type of kind Constraint as the function argument

    The predicate really does turn into a real extra argument to the
    function.  If the argument has type (p :: Constraint) then the predicate p is
    represented by evidence of type p.

    %************************************************************************
    %*                                                                      *
                Simple constructors
    %*                                                                      *
    %************************************************************************

    These functions are here so that they can be used by GHC.Builtin.Types.Prim,
    which in turn is imported by Type
    -}

    mkTyVarTy  :: TyVar   -> Type
    mkTyVarTy v = ASSERT2( isTyVar v, ppr v <+> dcolon <+> ppr (tyVarKind v) )
                  TyVarTy v

    mkTyVarTys :: [TyVar] -> [Type]
    mkTyVarTys = map mkTyVarTy -- a common use of mkTyVarTy

    mkTyCoVarTy :: TyCoVar -> Type
    mkTyCoVarTy v
      | isTyVar v
      = TyVarTy v
      | otherwise
      = CoercionTy (CoVarCo v)

    mkTyCoVarTys :: [TyCoVar] -> [Type]
    mkTyCoVarTys = map mkTyCoVarTy

    infixr 3 `mkFunTy`, `mkVisFunTy`, `mkInvisFunTy`, `mkVisFunTyMany`,
             `mkInvisFunTyMany`      -- Associates to the right

    mkFunTy :: AnonArgFlag -> Mult -> Type -> Type -> Type
    mkFunTy af mult arg res = FunTy { ft_af = af
                                    , ft_mult = mult
                                    , ft_arg = arg
                                    , ft_res = res }

    mkScaledFunTy :: AnonArgFlag -> Scaled Type -> Type -> Type
    mkScaledFunTy af (Scaled mult arg) res = mkFunTy af mult arg res

    mkVisFunTy, mkInvisFunTy :: Mult -> Type -> Type -> Type
    mkVisFunTy   = mkFunTy VisArg
    mkInvisFunTy = mkFunTy InvisArg

    mkFunTyMany :: AnonArgFlag -> Type -> Type -> Type
    mkFunTyMany af = mkFunTy af manyDataConTy

    -- | Special, common, case: Arrow type with mult Many
    mkVisFunTyMany :: Type -> Type -> Type
    mkVisFunTyMany = mkVisFunTy manyDataConTy

    mkInvisFunTyMany :: Type -> Type -> Type
    mkInvisFunTyMany = mkInvisFunTy manyDataConTy

    -- | Make nested arrow types
    mkVisFunTys :: [Scaled Type] -> Type -> Type
    mkVisFunTys tys ty = foldr (mkScaledFunTy VisArg) ty tys

    mkVisFunTysMany :: [Type] -> Type -> Type
    mkVisFunTysMany tys ty = foldr mkVisFunTyMany ty tys

    mkInvisFunTysMany :: [Type] -> Type -> Type
    mkInvisFunTysMany tys ty = foldr mkInvisFunTyMany ty tys

    -- | Like 'mkTyCoForAllTy', but does not check the occurrence of the binder
    -- See Note [Unused coercion variable in ForAllTy]
    mkForAllTy :: TyCoVar -> ArgFlag -> Type -> Type
    mkForAllTy tv vis ty = ForAllTy (Bndr tv vis) ty

    -- | Wraps foralls over the type using the provided 'TyCoVar's from left to right
    mkForAllTys :: [TyCoVarBinder] -> Type -> Type
    mkForAllTys tyvars ty = foldr ForAllTy ty tyvars

    -- | Wraps foralls over the type using the provided 'InvisTVBinder's from left to right
    mkInvisForAllTys :: [InvisTVBinder] -> Type -> Type
    mkInvisForAllTys tyvars ty = foldr ForAllTy ty $ tyVarSpecToBinders tyvars

    mkPiTy :: TyCoBinder -> Type -> Type
    mkPiTy (Anon af ty1) ty2        = mkScaledFunTy af ty1 ty2
    mkPiTy (Named (Bndr tv vis)) ty = mkForAllTy tv vis ty

    mkPiTys :: [TyCoBinder] -> Type -> Type
    mkPiTys tbs ty = foldr mkPiTy ty tbs

    -- | Create a nullary 'TyConApp'. In general you should rather use
    -- 'GHC.Core.Type.mkTyConTy'. This merely exists to break the import cycle
    -- between 'GHC.Core.TyCon' and this module.
    mkTyConTy_ :: TyCon -> Type
    mkTyConTy_ tycon = TyConApp tycon []

    {-
    %************************************************************************
    %*                                                                      *
                Coercions
    %*                                                                      *
    %************************************************************************
    -}

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

    -- If you edit this type, you may need to update the GHC formalism
    -- See Note [GHC Formalism] in GHC.Core.Lint
    data Coercion
      -- Each constructor has a "role signature", indicating the way roles are
      -- propagated through coercions.
      --    -  P, N, and R stand for coercions of the given role
      --    -  e stands for a coercion of a specific unknown role
      --           (think "role polymorphism")
      --    -  "e" stands for an explicit role parameter indicating role e.
      --    -   _ stands for a parameter that is not a Role or Coercion.

      -- These ones mirror the shape of types
      = -- Refl :: _ -> N
        -- A special case reflexivity for a very common case: Nominal reflexivity
        -- If you need Representational, use (GRefl Representational ty MRefl)
        --                               not (SubCo (Refl ty))
        Refl Type  -- See Note [Refl invariant]

      -- 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 correspondingly-named
      -- Type constructors into Coercions

      -- TyConAppCo :: "e" -> _ -> ?? -> e
      -- See Note [TyConAppCo roles]
      | TyConAppCo Role TyCon [Coercion]    -- lift TyConApp
                   -- The TyCon is never a synonym;
                   -- we expand synonyms eagerly
                   -- But it can be a type function
                   -- TyCon is never a saturated (->); use FunCo instead

      | AppCo Coercion CoercionN             -- lift AppTy
              -- AppCo :: e -> N -> e

      -- See Note [Forall coercions]
      | ForAllCo TyCoVar KindCoercion Coercion
             -- ForAllCo :: _ -> N -> e -> e

      | FunCo Role CoercionN Coercion Coercion         -- lift FunTy
             -- FunCo :: "e" -> N -> e -> e -> e
             -- Note: why doesn't FunCo have a AnonArgFlag, like FunTy?
             -- Because the AnonArgFlag has no impact on Core; it is only
             -- there to guide implicit instantiation of Haskell source
             -- types, and that is irrelevant for coercions, which are
             -- Core-only.

      -- These are special
      | CoVarCo CoVar      -- :: _ -> (N or R)
                           -- result role depends on the tycon of the variable's type

        -- AxiomInstCo :: e -> _ -> ?? -> e
      | AxiomInstCo (CoAxiom Branched) BranchIndex [Coercion]
         -- See also [CoAxiom index]
         -- The coercion arguments always *precisely* saturate
         -- arity of (that branch of) the CoAxiom. If there are
         -- any left over, we use AppCo.
         -- See [Coercion axioms applied to coercions]
         -- The roles of the argument coercions are determined
         -- by the cab_roles field of the relevant branch of the CoAxiom

      | AxiomRuleCo CoAxiomRule [Coercion]
        -- AxiomRuleCo is very like AxiomInstCo, but for a CoAxiomRule
        -- The number coercions should match exactly the expectations
        -- of the CoAxiomRule (i.e., the rule is fully saturated).

      | UnivCo UnivCoProvenance Role Type Type
          -- :: _ -> "e" -> _ -> _ -> e

      | SymCo Coercion             -- :: e -> e
      | TransCo Coercion Coercion  -- :: e -> e -> e

      | NthCo  Role Int Coercion     -- Zero-indexed; decomposes (T t0 ... tn)
        -- :: "e" -> _ -> e0 -> e (inverse of TyConAppCo, see Note [TyConAppCo roles])
        -- Using NthCo on a ForAllCo gives an N coercion always
        -- See Note [NthCo and newtypes]
        --
        -- Invariant:  (NthCo r i co), it is always the case that r = role of (Nth i co)
        -- That is: the role of the entire coercion is redundantly cached here.
        -- See Note [NthCo Cached Roles]

      | LRCo   LeftOrRight CoercionN     -- Decomposes (t_left t_right)
        -- :: _ -> N -> N
      | InstCo Coercion CoercionN
        -- :: e -> N -> e
        -- See Note [InstCo roles]

      -- Extract a kind coercion from a (heterogeneous) type coercion
      -- NB: all kind coercions are Nominal
      | KindCo Coercion
         -- :: e -> N

      | SubCo CoercionN                  -- Turns a ~N into a ~R
        -- :: N -> R

      | HoleCo CoercionHole              -- ^ See Note [Coercion holes]
                                         -- Only present during typechecking
      deriving Data.Data

    type CoercionN = Coercion       -- always nominal
    type CoercionR = Coercion       -- always representational
    type CoercionP = Coercion       -- always phantom
    type KindCoercion = CoercionN   -- always nominal

    instance Outputable Coercion where
      ppr = pprCo

    -- | A semantically more meaningful type to represent what may or may not be a
    -- useful 'Coercion'.
    data MCoercion
      = MRefl
        -- A trivial Reflexivity coercion
      | MCo Coercion
        -- Other coercions
      deriving Data.Data
    type MCoercionR = MCoercion
    type MCoercionN = MCoercion

    instance Outputable MCoercion where
      ppr MRefl    = text "MRefl"
      ppr (MCo co) = text "MCo" <+> ppr co

    {- Note [Refl invariant]
    ~~~~~~~~~~~~~~~~~~~~~~~~
    Invariant 1: Refl lifting
            Refl (similar for GRefl r ty MRefl) is always lifted as far as possible.
        For example
            (Refl T) (Refl a) (Refl b) is normalised (by mkAPpCo) to  (Refl (T a b)).

        You might think that a consequences is:
             Every identity coercion 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.

    Invariant 2: TyConAppCo
       An application of (Refl T) to some coercions, at least one of which is
       NOT the identity, is normalised to TyConAppCo.  (They may not be
       fully saturated however.)  TyConAppCo coercions (like all coercions
       other than Refl) are NEVER the identity.

    Note [Generalized reflexive coercion]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    GRefl is a generalized reflexive coercion (see #15192). It wraps a kind
    coercion, which might be reflexive (MRefl) or any coercion (MCo co). The typing
    rules for GRefl:

      ty : k1
      ------------------------------------
      GRefl r ty MRefl: ty ~r ty

      ty : k1       co :: k1 ~ k2
      ------------------------------------
      GRefl r ty (MCo co) : ty ~r ty |> co

    Consider we have

       g1 :: s ~r t
       s  :: k1
       g2 :: k1 ~ k2

    and we want to construct a coercions co which has type

       (s |> g2) ~r t

    We can define

       co = Sym (GRefl r s g2) ; g1

    It is easy to see that

       Refl == GRefl Nominal ty MRefl :: ty ~n ty

    A nominal reflexive coercion is quite common, so we keep the special form Refl to
    save allocation.

    Note [Coercion axioms applied to coercions]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    The reason coercion axioms can be applied to coercions and not just
    types is to allow for better optimization.  There are some cases where
    we need to be able to "push transitivity inside" an axiom in order to
    expose further opportunities for optimization.

    For example, suppose we have

      C a : t[a] ~ F a
      g   : b ~ c

    and we want to optimize

      sym (C b) ; t[g] ; C c

    which has the kind

      F b ~ F c

    (stopping through t[b] and t[c] along the way).

    We'd like to optimize this to just F g -- but how?  The key is
    that we need to allow axioms to be instantiated by *coercions*,
    not just by types.  Then we can (in certain cases) push
    transitivity inside the axiom instantiations, and then react
    opposite-polarity 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 0-indexed number
    of the chosen branch.

    Note [Forall coercions]
    ~~~~~~~~~~~~~~~~~~~~~~~
    Constructing coercions between forall-types 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 left-hand 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 kind-fixing 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* well-kinded, noting
    that `(tv1:k2) |> sym kind_co` has kind k1.

    This all really would work storing just a Name in the ForAllCo. But we can't
    add Names to, e.g., VarSets, and there generally is just an impedance mismatch
    in a bunch of places. So we use tv1. When we need tv2, we can use
    setTyVarKind.

    Note [Predicate coercions]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~
    Suppose we have
       g :: a~b
    How can we coerce between types
       ([c]~a) => [a] -> c
    and
       ([c]~b) => [b] -> c
    where the equality predicate *itself* differs?

    Answer: we simply treat (~) as an ordinary type constructor, so these
    types really look like

       ((~) [c] a) -> [a] -> c
       ((~) [c] b) -> [b] -> c

    So the coercion between the two is obviously

       ((~) [c] g) -> [g] -> c

    Another way to see this to say that we simply collapse predicates to
    their representation type (see Type.coreView and Type.predTypeRep).

    This collapse is done by mkPredCo; there is no PredCo constructor
    in Coercion.  This is important because we need Nth to work on
    predicates too:
        Nth 1 ((~) [c] g) = g
    See Simplify.simplCoercionF, which generates such selections.

    Note [Roles]
    ~~~~~~~~~~~~
    Roles are a solution to the GeneralizedNewtypeDeriving problem, articulated
    in #1496. The full story is in docs/core-spec/core-spec.pdf. Also, see
    https://gitlab.haskell.org/ghc/ghc/wikis/roles-implementation

    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
    Type-level Computation" (POPL 2010), available at
    http://www.seas.upenn.edu/~sweirich/papers/popl163af-weirich.pdf

    The specification for roles has evolved somewhat since that paper. For the
    current full details, see the documentation in docs/core-spec. 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 ill-formed, 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 over-saturated (which is otherwise fine),
      the spill-over arguments must all be at Nominal. This corresponds to the
      behavior for AppCo.

    TyConAppCo Phantom Foo (UnivCo Phantom Int Bool) : Foo Int ~P Foo Bool
      All arguments must have role Phantom. This one isn't strictly
      necessary for soundness, but this choice removes ambiguity.

    The rules here dictate the roles of the parameters to mkTyConAppCo
    (should be checked by Lint).

    Note [NthCo and newtypes]
    ~~~~~~~~~~~~~~~~~~~~~~~~~
    Suppose we have

      newtype N a = MkN Int
      type role N representational

    This yields axiom

      NTCo:N :: forall a. N a ~R Int

    We can then build

      co :: forall a b. N a ~R N b
      co = NTCo:N a ; sym (NTCo:N b)

    for any `a` and `b`. Because of the role annotation on N, if we use
    NthCo, we'll get out a representational coercion. That is:

      NthCo r 0 co :: forall a b. a ~R b

    Yikes! Clearly, this is terrible. The solution is simple: forbid
    NthCo to be used on newtypes if the internal coercion is representational.

    This is not just some corner case discovered by a segfault somewhere;
    it was discovered in the proof of soundness of roles and described
    in the "Safe Coercions" paper (ICFP '14).

    Note [NthCo Cached Roles]
    ~~~~~~~~~~~~~~~~~~~~~~~~~
    Why do we cache the role of NthCo in the NthCo constructor?
    Because computing role(Nth i co) involves figuring out that

      co :: T tys1 ~ T tys2

    using coercionKind, and finding (coercionRole co), and then looking
    at the tyConRoles of T. Avoiding bad asymptotic behaviour here means
    we have to compute the kind and role of a coercion simultaneously,
    which makes the code complicated and inefficient.

    This only happens for NthCo. Caching the role solves the problem, and
    allows coercionKind and coercionRole to be simple.

    See #11735

    Note [InstCo roles]
    ~~~~~~~~~~~~~~~~~~~
    Here is (essentially) the typing rule for InstCo:

    g :: (forall a. t1) ~r (forall a. t2)
    w :: s1 ~N s2
    ------------------------------- InstCo
    InstCo g w :: (t1 [a |-> s1]) ~r (t2 [a |-> s2])

    Note that the Coercion w *must* be nominal. This is necessary
    because the variable a might be used in a "nominal position"
    (that is, a place where role inference would require a nominal
    role) in t1 or t2. If we allowed w to be representational, we
    could get bogus equalities.

    A more nuanced treatment might be able to relax this condition
    somewhat, by checking if t1 and/or t2 use their bound variables
    in nominal ways. If not, having w be representational is OK.

    %************************************************************************
    %*                                                                      *
                    UnivCoProvenance
    %*                                                                      *
    %************************************************************************

    A UnivCo is a coercion whose proof does not directly express its role
    and kind (indeed for some UnivCos, like PluginProv, there /is/ no proof).

    The different kinds of UnivCo are described by UnivCoProvenance.  Really
    each is entirely separate, but they all share the need to represent their
    role and kind, which is done in the UnivCo constructor.

    -}

    -- | For simplicity, we have just one UnivCo that represents a coercion from
    -- some type to some other type, with (in general) no restrictions on the
    -- type. The UnivCoProvenance specifies more exactly what the coercion really
    -- is and why a program should (or shouldn't!) trust the coercion.
    -- It is reasonable to consider each constructor of 'UnivCoProvenance'
    -- as a totally independent coercion form; their only commonality is
    -- that they don't tell you what types they coercion between. (That info
    -- is in the 'UnivCo' constructor of 'Coercion'.
    data UnivCoProvenance
      = PhantomProv KindCoercion -- ^ See Note [Phantom coercions]. Only in Phantom
                                 -- roled coercions

      | ProofIrrelProv KindCoercion  -- ^ From the fact that any two coercions are
                                     --   considered equivalent. See Note [ProofIrrelProv].
                                     -- Can be used in Nominal or Representational coercions

      | PluginProv String  -- ^ From a plugin, which asserts that this coercion
                           --   is sound. The string is for the use of the plugin.

      | CorePrepProv       -- See Note [Unsafe coercions] in GHC.Core.CoreToStg.Prep
          Bool   -- True  <=> the UnivCo must be homogeneously kinded
                 -- False <=> allow hetero-kinded, e.g. Int ~ Int#

      deriving Data.Data

    instance Outputable UnivCoProvenance where
      ppr (PhantomProv _)    = text "(phantom)"
      ppr (ProofIrrelProv _) = text "(proof irrel.)"
      ppr (PluginProv str)   = parens (text "plugin" <+> brackets (text str))
      ppr (CorePrepProv _)   = text "(CorePrep)"

    -- | A coercion to be filled in by the type-checker. 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)

    instance Uniquable CoercionHole where
      getUnique (CoercionHole { ch_co_var = cv }) = getUnique 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 let-binding  let d7 = .... in ....
        which actually binds d7 to the (Num a) evidence

    For equality constraints we use a different strategy.  See Note [The
    equality types story] in GHC.Builtin.Types.Prim for background on equality constraints.
      - For /boxed/ equality constraints, (t1 ~N t2) and (t1 ~R t2), it's just
        like type classes above. (Indeed, boxed equality constraints *are* classes.)
      - But for /unboxed/ equality constraints (t1 ~R# t2) and (t1 ~N# t2)
        we use a different plan

    For unboxed equalities:
      - Generate a CoercionHole, a mutable variable just like a unification
        variable
      - Wrap the CoercionHole in a Wanted constraint; see GHC.Tc.Utils.TcEvDest
      - Use the CoercionHole in a Coercion, via HoleCo
      - Solve the constraint later
      - When solved, fill in the CoercionHole by side effect, instead of
        doing the let-binding thing

    The main reason for all this is that there may be no good place to let-bind
    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 top-level 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
        let-bindings within coercions.  We could add them, but coercion
        holes are easier.

      - Moreover, nothing is lost from the lack of let-bindings. For
        dictionaries want to achieve sharing to avoid recomoputing the
        dictionary.  But coercions are entirely erased, so there's little
        benefit to sharing. Indeed, even if we had a let-binding, 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 type-level information to separate out forms that can appear during
       type-checking vs forms that can appear in core proper, holes in Core will
       be ruled out.

     * See Note [CoercionHoles and coercion free variables]

     * Coercion holes can be compared for equality like other coercions:
       by looking at the types coerced.

    Note [CoercionHoles and coercion free variables]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Why does a CoercionHole contain a CoVar, as well as reference to
    fill in?  Because we want to treat that CoVar as a free variable of
    the coercion.  See #14584, and Note [What prevents a
    constraint from floating] in GHC.Tc.Solver, item (4):

            forall k. [W] co1 :: t1 ~# t2 |> co2
                      [W] co2 :: k ~# *

    Here co2 is a CoercionHole. But we /must/ know that it is free in
    co1, because that's all that stops it floating outside the
    implication.

    Note [ProofIrrelProv]
    ~~~~~~~~~~~~~~~~~~~~~
    A ProofIrrelProv is a coercion between coercions. For example:

      data G a where
        MkG :: G Bool

    In core, we get

      G :: * -> *
      MkG :: forall (a :: *). (a ~ Bool) -> G a

    Now, consider 'MkG -- that is, MkG used in a type -- and suppose we want
    a proof that ('MkG a1 co1) ~ ('MkG a2 co2). This will have to be

      TyConAppCo Nominal MkG [co3, co4]
      where
        co3 :: co1 ~ co2
        co4 :: a1 ~ a2

    Note that
      co1 :: a1 ~ Bool
      co2 :: a2 ~ Bool

    Here,
      co3 = UnivCo (ProofIrrelProv co5) Nominal (CoercionTy co1) (CoercionTy co2)
      where
        co5 :: (a1 ~ Bool) ~ (a2 ~ Bool)
        co5 = TyConAppCo Nominal (~#) [<*>, <*>, co4, <Bool>]
    -}

    {- *********************************************************************
    *                                                                      *
                    foldType  and   foldCoercion
    *                                                                      *
    ********************************************************************* -}

    {- Note [foldType]
    ~~~~~~~~~~~~~~~~~~
    foldType is a bit more powerful than perhaps it looks:

    * You can fold with an accumulating parameter, via
         TyCoFolder env (Endo a)
      Recall newtype Endo a = Endo (a->a)

    * You can fold monadically with a monad M, via
         TyCoFolder env (M a)
      provided you have
         instance ..  => Monoid (M a)

    Note [mapType vs foldType]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~
    We define foldType here, but mapType in module Type. Why?

    * foldType is used in GHC.Core.TyCo.FVs for finding free variables.
      It's a very simple function that analyses a type,
      but does not construct one.

    * mapType constructs new types, and so it needs to call
      the "smart constructors", mkAppTy, mkCastTy, and so on.
      These are sophisticated functions, and can't be defined
      here in GHC.Core.TyCo.Rep.

    Note [Specialising foldType]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We inline foldType at every call site (there are not many), so that it
    becomes specialised for the particular monoid *and* TyCoFolder at
    that site.  This is just for efficiency, but walking over types is
    done a *lot* in GHC, so worth optimising.

    We were worried that
        TyCoFolder env (Endo a)
    might not eta-expand.  Recall newtype Endo a = Endo (a->a).

    In particular, given
       fvs :: Type -> TyCoVarSet
       fvs ty = appEndo (foldType tcf emptyVarSet ty) emptyVarSet

       tcf :: TyCoFolder enf (Endo a)
       tcf = TyCoFolder { tcf_tyvar = do_tv, ... }
          where
            do_tvs is tv = Endo do_it
               where
                 do_it acc | tv `elemVarSet` is  = acc
                           | tv `elemVarSet` acc = acc
                           | otherwise = acc `extendVarSet` tv

    we want to end up with
       fvs ty = go emptyVarSet ty emptyVarSet
         where
           go env (TyVarTy tv) acc = acc `extendVarSet` tv
           ..etc..

    And indeed this happens.
      - Selections from 'tcf' are done at compile time
      - 'go' is nicely eta-expanded.

    We were also worried about
       deep_fvs :: Type -> TyCoVarSet
       deep_fvs ty = appEndo (foldType deep_tcf emptyVarSet ty) emptyVarSet

       deep_tcf :: TyCoFolder enf (Endo a)
       deep_tcf = TyCoFolder { tcf_tyvar = do_tv, ... }
          where
            do_tvs is tv = Endo do_it
               where
                 do_it acc | tv `elemVarSet` is  = acc
                           | tv `elemVarSet` acc = acc
                           | otherwise = deep_fvs (varType tv)
                                         `unionVarSet` acc
                                         `extendVarSet` tv

    Here deep_fvs and deep_tcf are mutually recursive, unlike fvs and tcf.
    But, amazingly, we get good code here too. GHC is careful not to makr
    TyCoFolder data constructor for deep_tcf as a loop breaker, so the
    record selections still cancel.  And eta expansion still happens too.
    -}

    data TyCoFolder env a
      = TyCoFolder
          { tcf_view  :: Type -> Maybe Type   -- Optional "view" function
                                              -- E.g. expand synonyms
          , tcf_tyvar :: env -> TyVar -> a
          , tcf_covar :: env -> CoVar -> a
          , tcf_hole  :: env -> CoercionHole -> a
              -- ^ What to do with coercion holes.
              -- See Note [Coercion holes] in "GHC.Core.TyCo.Rep".

          , tcf_tycobinder :: env -> TyCoVar -> ArgFlag -> env
              -- ^ The returned env is used in the extended scope
          }

    {-# INLINE foldTyCo  #-}  -- See Note [Specialising foldType]
    foldTyCo :: Monoid a => TyCoFolder env a -> env
             -> (Type -> a, [Type] -> a, Coercion -> a, [Coercion] -> a)
    foldTyCo (TyCoFolder { tcf_view       = view
                         , tcf_tyvar      = tyvar
                         , tcf_tycobinder = tycobinder
                         , tcf_covar      = covar
                         , tcf_hole       = cohole }) env
      = (go_ty env, go_tys env, go_co env, go_cos env)
      where
        go_ty env ty | Just ty' <- view ty = go_ty env ty'
        go_ty env (TyVarTy tv)      = tyvar env tv
        go_ty env (AppTy t1 t2)     = go_ty env t1 `mappend` go_ty env t2
        go_ty _   (LitTy {})        = mempty
        go_ty env (CastTy ty co)    = go_ty env ty `mappend` go_co env co
        go_ty env (CoercionTy co)   = go_co env co
        go_ty env (FunTy _ w arg res) = go_ty env w `mappend` go_ty env arg `mappend` go_ty env res
        go_ty env (TyConApp _ tys)  = go_tys env tys
        go_ty env (ForAllTy (Bndr tv vis) inner)
          = let !env' = tycobinder env tv vis  -- Avoid building a thunk here
            in go_ty env (varType tv) `mappend` go_ty env' inner

        -- Explicit recursion because using foldr builds a local
        -- loop (with env free) and I'm not confident it'll be
        -- lambda lifted in the end
        go_tys _   []     = mempty
        go_tys env (t:ts) = go_ty env t `mappend` go_tys env ts

        go_cos _   []     = mempty
        go_cos env (c:cs) = go_co env c `mappend` go_cos env cs

        go_co env (Refl ty)               = go_ty env ty
        go_co env (GRefl _ ty MRefl)      = go_ty env ty
        go_co env (GRefl _ ty (MCo co))   = go_ty env ty `mappend` go_co env co
        go_co env (TyConAppCo _ _ args)   = go_cos env args
        go_co env (AppCo c1 c2)           = go_co env c1 `mappend` go_co env c2
        go_co env (FunCo _ cw c1 c2)      = go_co env cw `mappend`
                                            go_co env c1 `mappend`
                                            go_co env c2
        go_co env (CoVarCo cv)            = covar env cv
        go_co env (AxiomInstCo _ _ args)  = go_cos env args
        go_co env (HoleCo hole)           = cohole env hole
        go_co env (UnivCo p _ t1 t2)      = go_prov env p `mappend` go_ty env t1
                                                          `mappend` go_ty env t2
        go_co env (SymCo co)              = go_co env co
        go_co env (TransCo c1 c2)         = go_co env c1 `mappend` go_co env c2
        go_co env (AxiomRuleCo _ cos)     = go_cos env cos
        go_co env (NthCo _ _ co)          = go_co env co
        go_co env (LRCo _ co)             = go_co env co
        go_co env (InstCo co arg)         = go_co env co `mappend` go_co env arg
        go_co env (KindCo co)             = go_co env co
        go_co env (SubCo co)              = go_co env co
        go_co env (ForAllCo tv kind_co co)
          = go_co env kind_co `mappend` go_ty env (varType tv)
                              `mappend` go_co env' co
          where
            env' = tycobinder env tv Inferred

        go_prov env (PhantomProv co)    = go_co env co
        go_prov env (ProofIrrelProv co) = go_co env co
        go_prov _   (PluginProv _)      = mempty
        go_prov _   (CorePrepProv _)    = mempty

    {- *********************************************************************
    *                                                                      *
                       typeSize, coercionSize
    *                                                                      *
    ********************************************************************* -}

    -- NB: We put typeSize/coercionSize here because they are mutually
    --     recursive, and have the CPR property.  If we have mutual
    --     recursion across a hi-boot 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 decision-making.

    typeSize :: Type -> Int
    typeSize (LitTy {})                 = 1
    typeSize (TyVarTy {})               = 1
    typeSize (AppTy t1 t2)              = typeSize t1 + typeSize t2
    typeSize (FunTy _ _ t1 t2)          = typeSize t1 + typeSize t2
    typeSize (ForAllTy (Bndr tv _) t)   = typeSize (varType tv) + typeSize t
    typeSize (TyConApp _ ts)            = 1 + sum (map typeSize ts)
    typeSize (CastTy ty co)             = typeSize ty + coercionSize co
    typeSize (CoercionTy co)            = coercionSize co

    coercionSize :: Coercion -> Int
    coercionSize (Refl ty)             = typeSize ty
    coercionSize (GRefl _ ty MRefl)    = typeSize ty
    coercionSize (GRefl _ ty (MCo co)) = 1 + typeSize ty + coercionSize co
    coercionSize (TyConAppCo _ _ args) = 1 + sum (map coercionSize args)
    coercionSize (AppCo co arg)      = coercionSize co + coercionSize arg
    coercionSize (ForAllCo _ h co)   = 1 + coercionSize co + coercionSize h
    coercionSize (FunCo _ w co1 co2) = 1 + coercionSize co1 + coercionSize co2
                                                            + coercionSize w
    coercionSize (CoVarCo _)         = 1
    coercionSize (HoleCo _)          = 1
    coercionSize (AxiomInstCo _ _ args) = 1 + sum (map coercionSize args)
    coercionSize (UnivCo p _ t1 t2)  = 1 + provSize p + typeSize t1 + typeSize t2
    coercionSize (SymCo co)          = 1 + coercionSize co
    coercionSize (TransCo co1 co2)   = 1 + coercionSize co1 + coercionSize co2
    coercionSize (NthCo _ _ co)      = 1 + coercionSize co
    coercionSize (LRCo  _ co)        = 1 + coercionSize co
    coercionSize (InstCo co arg)     = 1 + coercionSize co + coercionSize arg
    coercionSize (KindCo co)         = 1 + coercionSize co
    coercionSize (SubCo co)          = 1 + coercionSize co
    coercionSize (AxiomRuleCo _ cs)  = 1 + sum (map coercionSize cs)

    provSize :: UnivCoProvenance -> Int
    provSize (PhantomProv co)    = 1 + coercionSize co
    provSize (ProofIrrelProv co) = 1 + coercionSize co
    provSize (PluginProv _)      = 1
    provSize (CorePrepProv _)    = 1

    {-
    ************************************************************************
    *                                                                      *
                        Multiplicities
    *                                                                      *
    ************************************************************************

    These definitions are here to avoid module loops, and to keep
    GHC.Core.Multiplicity above this module.

    -}

    -- | A shorthand for data with an attached 'Mult' element (the multiplicity).
    data Scaled a = Scaled !Mult a
      deriving (Data.Data)
      -- You might think that this would be a natural candidate for
      -- Functor, Traversable but Krzysztof says (!3674) "it was too easy
      -- to accidentally lift functions (substitutions, zonking etc.) from
      -- Type -> Type to Scaled Type -> Scaled Type, ignoring
      -- multiplicities and causing bugs".  So we don't.
      --
      -- Being strict in a is worse for performance, so we are only strict on the
      -- Mult part of scaled.

    instance (Outputable a) => Outputable (Scaled a) where
       ppr (Scaled _cnt t) = ppr t
         -- Do not print the multiplicity here because it tends to be too verbose

    scaledMult :: Scaled a -> Mult
    scaledMult (Scaled m _) = m

    scaledThing :: Scaled a -> a
    scaledThing (Scaled _ t) = t

    -- | Apply a function to both the Mult and the Type in a 'Scaled Type'
    mapScaledType :: (Type -> Type) -> Scaled Type -> Scaled Type
    mapScaledType f (Scaled m t) = Scaled (f m) (f t)

    {- |
    Mult is a type alias for Type.

    Mult must contain Type because multiplicity variables are mere type variables
    (of kind Multiplicity) in Haskell. So the simplest implementation is to make
    Mult be Type.

    Multiplicities can be formed with:
    - One: GHC.Types.One (= oneDataCon)
    - Many: GHC.Types.Many (= manyDataCon)
    - Multiplication: GHC.Types.MultMul (= multMulTyCon)

    So that Mult feels a bit more structured, we provide pattern synonyms and smart
    constructors for these.
    -}
    type Mult = Type