TcCanonical.hs 82.2 KB
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{-# LANGUAGE CPP #-}

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module TcCanonical(
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     canonicalize,
     unifyDerived,
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     makeSuperClasses, maybeSym,
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     StopOrContinue(..), stopWith, continueWith,
     solveCallStack    -- For TcSimplify
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  ) where
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#include "HsVersions.h"

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import GhcPrelude

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import TcRnTypes
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import TcUnify( swapOverTyVars, metaTyVarUpdateOK )
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import TcType
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import Type
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import TcFlatten
import TcSMonad
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import TcEvidence
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import Class
import TyCon
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import TyCoRep   -- cleverly decomposes types, good for completeness checking
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import Coercion
import FamInstEnv ( FamInstEnvs )
import FamInst ( tcTopNormaliseNewTypeTF_maybe )
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import Var
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import VarEnv( mkInScopeSet )
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import Outputable
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import DynFlags( DynFlags )
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import NameSet
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import RdrName
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import HsTypes( HsIPName(..) )
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import Pair
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import Util
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import Bag
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import MonadUtils
import Control.Monad
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import Data.Maybe ( isJust )
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import Data.List  ( zip4, foldl' )
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import BasicTypes
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import Data.Bifunctor ( bimap )

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{-
************************************************************************
*                                                                      *
*                      The Canonicaliser                               *
*                                                                      *
************************************************************************
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Note [Canonicalization]
~~~~~~~~~~~~~~~~~~~~~~~
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Canonicalization converts a simple constraint to a canonical form. It is
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unary (i.e. treats individual constraints one at a time).

Constraints originating from user-written code come into being as
CNonCanonicals (except for CHoleCans, arising from holes). We know nothing
about these constraints. So, first:

     Classify CNonCanoncal constraints, depending on whether they
     are equalities, class predicates, or other.

Then proceed depending on the shape of the constraint. Generally speaking,
each constraint gets flattened and then decomposed into one of several forms
(see type Ct in TcRnTypes).

When an already-canonicalized constraint gets kicked out of the inert set,
it must be recanonicalized. But we know a bit about its shape from the
last time through, so we can skip the classification step.

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-}
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-- Top-level canonicalization
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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canonicalize :: Ct -> TcS (StopOrContinue Ct)
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canonicalize (CNonCanonical { cc_ev = ev })
  = {-# SCC "canNC" #-}
    case classifyPredType (ctEvPred ev) of
      ClassPred cls tys     -> do traceTcS "canEvNC:cls" (ppr cls <+> ppr tys)
                                  canClassNC ev cls tys
      EqPred eq_rel ty1 ty2 -> do traceTcS "canEvNC:eq" (ppr ty1 $$ ppr ty2)
                                  canEqNC    ev eq_rel ty1 ty2
      IrredPred {}          -> do traceTcS "canEvNC:irred" (ppr (ctEvPred ev))
                                  canIrred   ev

canonicalize (CIrredCan { cc_ev = ev })
  = canIrred ev
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canonicalize (CDictCan { cc_ev = ev, cc_class  = cls
                       , cc_tyargs = xis, cc_pend_sc = pend_sc })
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  = {-# SCC "canClass" #-}
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    canClass ev cls xis pend_sc

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canonicalize (CTyEqCan { cc_ev = ev
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                       , cc_tyvar  = tv
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                       , cc_rhs    = xi
                       , cc_eq_rel = eq_rel })
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  = {-# SCC "canEqLeafTyVarEq" #-}
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    canEqNC ev eq_rel (mkTyVarTy tv) xi
      -- NB: Don't use canEqTyVar because that expects flattened types,
      -- and tv and xi may not be flat w.r.t. an updated inert set
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canonicalize (CFunEqCan { cc_ev = ev
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                        , cc_fun    = fn
                        , cc_tyargs = xis1
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                        , cc_fsk    = fsk })
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  = {-# SCC "canEqLeafFunEq" #-}
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    canCFunEqCan ev fn xis1 fsk
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canonicalize (CHoleCan { cc_ev = ev, cc_hole = hole })
  = canHole ev hole
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{-
************************************************************************
*                                                                      *
*                      Class Canonicalization
*                                                                      *
************************************************************************
-}
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canClassNC :: CtEvidence -> Class -> [Type] -> TcS (StopOrContinue Ct)
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-- "NC" means "non-canonical"; that is, we have got here
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-- from a NonCanonical constraint, not from a CDictCan
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-- Precondition: EvVar is class evidence
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canClassNC ev cls tys
  | isGiven ev  -- See Note [Eagerly expand given superclasses]
  = do { sc_cts <- mkStrictSuperClasses ev cls tys
       ; emitWork sc_cts
       ; canClass ev cls tys False }
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  | isWanted ev
  , Just ip_name <- isCallStackPred cls tys
  , OccurrenceOf func <- ctLocOrigin loc
  -- If we're given a CallStack constraint that arose from a function
  -- call, we need to push the current call-site onto the stack instead
  -- of solving it directly from a given.
  -- See Note [Overview of implicit CallStacks] in TcEvidence
  -- and Note [Solving CallStack constraints] in TcSMonad
  = do { -- First we emit a new constraint that will capture the
         -- given CallStack.
       ; let new_loc = setCtLocOrigin loc (IPOccOrigin (HsIPName ip_name))
                            -- We change the origin to IPOccOrigin so
                            -- this rule does not fire again.
                            -- See Note [Overview of implicit CallStacks]

       ; new_ev <- newWantedEvVarNC new_loc pred

         -- Then we solve the wanted by pushing the call-site
         -- onto the newly emitted CallStack
       ; let ev_cs = EvCsPushCall func (ctLocSpan loc) (ctEvTerm new_ev)
       ; solveCallStack ev ev_cs

       ; canClass new_ev cls tys False }

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  | otherwise
  = canClass ev cls tys (has_scs cls)
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  where
    has_scs cls = not (null (classSCTheta cls))
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    loc  = ctEvLoc ev
    pred = ctEvPred ev

solveCallStack :: CtEvidence -> EvCallStack -> TcS ()
-- Also called from TcSimplify when defaulting call stacks
solveCallStack ev ev_cs = do
  -- We're given ev_cs :: CallStack, but the evidence term should be a
  -- dictionary, so we have to coerce ev_cs to a dictionary for
  -- `IP ip CallStack`. See Note [Overview of implicit CallStacks]
  let ev_tm = mkEvCast (EvCallStack ev_cs) (wrapIP (ctEvPred ev))
  setWantedEvBind (ctEvId ev) ev_tm
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canClass :: CtEvidence
         -> Class -> [Type]
         -> Bool            -- True <=> un-explored superclasses
         -> TcS (StopOrContinue Ct)
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-- Precondition: EvVar is class evidence
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canClass ev cls tys pend_sc
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  =   -- all classes do *nominal* matching
    ASSERT2( ctEvRole ev == Nominal, ppr ev $$ ppr cls $$ ppr tys )
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    do { (xis, cos) <- flattenManyNom ev tys
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       ; let co = mkTcTyConAppCo Nominal (classTyCon cls) cos
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             xi = mkClassPred cls xis
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             mk_ct new_ev = CDictCan { cc_ev = new_ev
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                                     , cc_tyargs = xis
                                     , cc_class = cls
                                     , cc_pend_sc = pend_sc }
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       ; mb <- rewriteEvidence ev xi co
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       ; traceTcS "canClass" (vcat [ ppr ev
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                                   , ppr xi, ppr mb ])
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       ; return (fmap mk_ct mb) }
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{- Note [The superclass story]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We need to add superclass constraints for two reasons:

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* For givens [G], they give us a route to to proof.  E.g.
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    f :: Ord a => a -> Bool
    f x = x == x
  We get a Wanted (Eq a), which can only be solved from the superclass
  of the Given (Ord a).

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* For wanteds [W], and deriveds [WD], [D], they may give useful
  functional dependencies.  E.g.
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     class C a b | a -> b where ...
     class C a b => D a b where ...
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  Now a [W] constraint (D Int beta) has (C Int beta) as a superclass
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  and that might tell us about beta, via C's fundeps.  We can get this
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  by generating a [D] (C Int beta) constraint.  It's derived because
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  we don't actually have to cough up any evidence for it; it's only there
  to generate fundep equalities.

See Note [Why adding superclasses can help].

For these reasons we want to generate superclass constraints for both
Givens and Wanteds. But:

* (Minor) they are often not needed, so generating them aggressively
  is a waste of time.

* (Major) if we want recursive superclasses, there would be an infinite
  number of them.  Here is a real-life example (Trac #10318);

     class (Frac (Frac a) ~ Frac a,
            Fractional (Frac a),
            IntegralDomain (Frac a))
         => IntegralDomain a where
      type Frac a :: *

  Notice that IntegralDomain has an associated type Frac, and one
  of IntegralDomain's superclasses is another IntegralDomain constraint.

So here's the plan:

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1. Eagerly generate superclasses for given (but not wanted)
   constraints; see Note [Eagerly expand given superclasses].
   This is done in canClassNC, when we take a non-canonical constraint
   and cannonicalise it.

   However stop if you encounter the same class twice.  That is,
   expand eagerly, but have a conservative termination condition: see
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   Note [Expanding superclasses] in TcType.

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2. Solve the wanteds as usual, but do no further expansion of
   superclasses for canonical CDictCans in solveSimpleGivens or
   solveSimpleWanteds; Note [Danger of adding superclasses during solving]

   However, /do/ continue to eagerly expand superlasses for /given/
   non-canonical constraints (canClassNC does this).  As Trac #12175
   showed, a type-family application can expand to a class constraint,
   and we want to see its superclasses for just the same reason as
   Note [Eagerly expand given superclasses].
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3. If we have any remaining unsolved wanteds
        (see Note [When superclasses help] in TcRnTypes)
   try harder: take both the Givens and Wanteds, and expand
   superclasses again.  This may succeed in generating (a finite
   number of) extra Givens, and extra Deriveds. Both may help the
   proof.  This is done in TcSimplify.expandSuperClasses.
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4. Go round to (2) again.  This loop (2,3,4) is implemented
   in TcSimplify.simpl_loop.

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The cc_pend_sc flag in a CDictCan records whether the superclasses of
this constraint have been expanded.  Specifically, in Step 3 we only
expand superclasses for constraints with cc_pend_sc set to true (i.e.
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isPendingScDict holds).

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Why do we do this?  Two reasons:

* To avoid repeated work, by repeatedly expanding the superclasses of
  same constraint,

* To terminate the above loop, at least in the -XNoRecursiveSuperClasses
  case.  If there are recursive superclasses we could, in principle,
  expand forever, always encountering new constraints.

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When we take a CNonCanonical or CIrredCan, but end up classifying it
as a CDictCan, we set the cc_pend_sc flag to False.

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Note [Eagerly expand given superclasses]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In step (1) of Note [The superclass story], why do we eagerly expand
Given superclasses by one layer?  Mainly because of some very obscure
cases like this:
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   instance Bad a => Eq (T a)

   f :: (Ord (T a)) => blah
   f x = ....needs Eq (T a), Ord (T a)....

Here if we can't satisfy (Eq (T a)) from the givens we'll use the
instance declaration; but then we are stuck with (Bad a).  Sigh.
This is really a case of non-confluent proofs, but to stop our users
complaining we expand one layer in advance.

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Note [Instance and Given overlap] in TcInteract.

We also want to do this if we have

   f :: F (T a) => blah

where
   type instance F (T a) = Ord (T a)

So we may need to do a little work on the givens to expose the
class that has the superclasses.  That's why the superclass
expansion for Givens happens in canClassNC.

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Note [Why adding superclasses can help]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Examples of how adding superclasses can help:
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    --- Example 1
        class C a b | a -> b
    Suppose we want to solve
         [G] C a b
         [W] C a beta
    Then adding [D] beta~b will let us solve it.

    -- Example 2 (similar but using a type-equality superclass)
        class (F a ~ b) => C a b
    And try to sllve:
         [G] C a b
         [W] C a beta
    Follow the superclass rules to add
         [G] F a ~ b
         [D] F a ~ beta
    Now we we get [D] beta ~ b, and can solve that.

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    -- Example (tcfail138)
      class L a b | a -> b
      class (G a, L a b) => C a b

      instance C a b' => G (Maybe a)
      instance C a b  => C (Maybe a) a
      instance L (Maybe a) a

    When solving the superclasses of the (C (Maybe a) a) instance, we get
      [G] C a b, and hance by superclasses, [G] G a, [G] L a b
      [W] G (Maybe a)
    Use the instance decl to get
      [W] C a beta
    Generate its derived superclass
      [D] L a beta.  Now using fundeps, combine with [G] L a b to get
      [D] beta ~ b
    which is what we want.

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Note [Danger of adding superclasses during solving]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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Here's a serious, but now out-dated example, from Trac #4497:
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   class Num (RealOf t) => Normed t
   type family RealOf x

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Assume the generated wanted constraint is:
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   [W] RealOf e ~ e
   [W] Normed e

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If we were to be adding the superclasses during simplification we'd get:
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   [W] RealOf e ~ e
   [W] Normed e
   [D] RealOf e ~ fuv
   [D] Num fuv
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==>
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   e := fuv, Num fuv, Normed fuv, RealOf fuv ~ fuv
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While looks exactly like our original constraint. If we add the
superclass of (Normed fuv) again we'd loop.  By adding superclasses
definitely only once, during canonicalisation, this situation can't
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happen.
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Mind you, now that Wanteds cannot rewrite Derived, I think this particular
situation can't happen.
  -}
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makeSuperClasses :: [Ct] -> TcS [Ct]
-- Returns strict superclasses, transitively, see Note [The superclasses story]
-- See Note [The superclass story]
-- The loop-breaking here follows Note [Expanding superclasses] in TcType
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-- Specifically, for an incoming (C t) constraint, we return all of (C t)'s
--    superclasses, up to /and including/ the first repetition of C
--
-- Example:  class D a => C a
--           class C [a] => D a
-- makeSuperClasses (C x) will return (D x, C [x])
--
-- NB: the incoming constraints have had their cc_pend_sc flag already
--     flipped to False, by isPendingScDict, so we are /obliged/ to at
--     least produce the immediate superclasses
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makeSuperClasses cts = concatMapM go cts
  where
    go (CDictCan { cc_ev = ev, cc_class = cls, cc_tyargs = tys })
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          = mkStrictSuperClasses ev cls tys
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    go ct = pprPanic "makeSuperClasses" (ppr ct)

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mkStrictSuperClasses :: CtEvidence -> Class -> [Type] -> TcS [Ct]
-- Return constraints for the strict superclasses of (c tys)
mkStrictSuperClasses ev cls tys
  = mk_strict_superclasses (unitNameSet (className cls)) ev cls tys

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mk_superclasses :: NameSet -> CtEvidence -> TcS [Ct]
-- Return this constraint, plus its superclasses, if any
mk_superclasses rec_clss ev
  | ClassPred cls tys <- classifyPredType (ctEvPred ev)
  = mk_superclasses_of rec_clss ev cls tys

  | otherwise   -- Superclass is not a class predicate
  = return [mkNonCanonical ev]

mk_superclasses_of :: NameSet -> CtEvidence -> Class -> [Type] -> TcS [Ct]
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-- Always return this class constraint,
-- and expand its superclasses
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mk_superclasses_of rec_clss ev cls tys
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  | loop_found = do { traceTcS "mk_superclasses_of: loop" (ppr cls <+> ppr tys)
                    ; return [this_ct] }  -- cc_pend_sc of this_ct = True
  | otherwise  = do { traceTcS "mk_superclasses_of" (vcat [ ppr cls <+> ppr tys
                                                          , ppr (isCTupleClass cls)
                                                          , ppr rec_clss
                                                          ])
                    ; sc_cts <- mk_strict_superclasses rec_clss' ev cls tys
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                    ; return (this_ct : sc_cts) }
                                   -- cc_pend_sc of this_ct = False
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  where
    cls_nm     = className cls
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    loop_found = not (isCTupleClass cls) && cls_nm `elemNameSet` rec_clss
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                 -- Tuples never contribute to recursion, and can be nested
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    rec_clss'  = rec_clss `extendNameSet` cls_nm
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    this_ct    = CDictCan { cc_ev = ev, cc_class = cls, cc_tyargs = tys
                          , cc_pend_sc = loop_found }
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                 -- NB: If there is a loop, we cut off, so we have not
                 --     added the superclasses, hence cc_pend_sc = True
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mk_strict_superclasses :: NameSet -> CtEvidence -> Class -> [Type] -> TcS [Ct]
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-- Always return the immediate superclasses of (cls tys);
-- and expand their superclasses, provided none of them are in rec_clss
-- nor are repeated
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mk_strict_superclasses rec_clss ev cls tys
  | CtGiven { ctev_evar = evar, ctev_loc = loc } <- ev
  = do { sc_evs <- newGivenEvVars (mk_given_loc loc)
                                  (mkEvScSelectors (EvId evar) cls tys)
       ; concatMapM (mk_superclasses rec_clss) sc_evs }
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  | all noFreeVarsOfType tys
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  = return [] -- Wanteds with no variables yield no deriveds.
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              -- See Note [Improvement from Ground Wanteds]
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  | otherwise -- Wanted/Derived case, just add Derived superclasses
              -- that can lead to improvement.
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  = do { let loc = ctEvLoc ev
       ; sc_evs <- mapM (newDerivedNC loc) (immSuperClasses cls tys)
       ; concatMapM (mk_superclasses rec_clss) sc_evs }
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  where
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    size = sizeTypes tys
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    mk_given_loc loc
       | isCTupleClass cls
       = loc   -- For tuple predicates, just take them apart, without
               -- adding their (large) size into the chain.  When we
               -- get down to a base predicate, we'll include its size.
               -- Trac #10335

       | GivenOrigin skol_info <- ctLocOrigin loc
         -- See Note [Solving superclass constraints] in TcInstDcls
         -- for explantation of this transformation for givens
       = case skol_info of
            InstSkol -> loc { ctl_origin = GivenOrigin (InstSC size) }
            InstSC n -> loc { ctl_origin = GivenOrigin (InstSC (n `max` size)) }
            _        -> loc

       | otherwise  -- Probably doesn't happen, since this function
       = loc        -- is only used for Givens, but does no harm
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{-
************************************************************************
*                                                                      *
*                      Irreducibles canonicalization
*                                                                      *
************************************************************************
-}
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canIrred :: CtEvidence -> TcS (StopOrContinue Ct)
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-- Precondition: ty not a tuple and no other evidence form
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canIrred old_ev
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  = do { let old_ty = ctEvPred old_ev
       ; traceTcS "can_pred" (text "IrredPred = " <+> ppr old_ty)
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       ; (xi,co) <- flatten FM_FlattenAll old_ev old_ty -- co :: xi ~ old_ty
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       ; rewriteEvidence old_ev xi co `andWhenContinue` \ new_ev ->
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    do { -- Re-classify, in case flattening has improved its shape
       ; case classifyPredType (ctEvPred new_ev) of
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           ClassPred cls tys     -> canClassNC new_ev cls tys
           EqPred eq_rel ty1 ty2 -> canEqNC new_ev eq_rel ty1 ty2
           _                     -> continueWith $
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                                    mkIrredCt new_ev } }
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canHole :: CtEvidence -> Hole -> TcS (StopOrContinue Ct)
canHole ev hole
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  = do { let ty = ctEvPred ev
       ; (xi,co) <- flatten FM_SubstOnly ev ty -- co :: xi ~ ty
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       ; rewriteEvidence ev xi co `andWhenContinue` \ new_ev ->
    do { emitInsoluble (CHoleCan { cc_ev = new_ev
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                                 , cc_hole = hole })
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       ; stopWith new_ev "Emit insoluble hole" } }
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{-
************************************************************************
*                                                                      *
*        Equalities
*                                                                      *
************************************************************************
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Note [Canonicalising equalities]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In order to canonicalise an equality, we look at the structure of the
two types at hand, looking for similarities. A difficulty is that the
types may look dissimilar before flattening but similar after flattening.
However, we don't just want to jump in and flatten right away, because
this might be wasted effort. So, after looking for similarities and failing,
we flatten and then try again. Of course, we don't want to loop, so we
track whether or not we've already flattened.

It is conceivable to do a better job at tracking whether or not a type
is flattened, but this is left as future work. (Mar '15)
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Note [FunTy and decomposing tycon applications]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

When can_eq_nc' attempts to decompose a tycon application we haven't yet zonked.
This means that we may very well have a FunTy containing a type of some unknown
kind. For instance, we may have,

    FunTy (a :: k) Int

Where k is a unification variable. tcRepSplitTyConApp_maybe panics in the event
that it sees such a type as it cannot determine the RuntimeReps which the (->)
is applied to. Consequently, it is vital that we instead use
tcRepSplitTyConApp_maybe', which simply returns Nothing in such a case.

When this happens can_eq_nc' will fail to decompose, zonk, and try again.
Zonking should fill the variable k, meaning that decomposition will succeed the
second time around.
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-}
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canEqNC :: CtEvidence -> EqRel -> Type -> Type -> TcS (StopOrContinue Ct)
canEqNC ev eq_rel ty1 ty2
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  = do { result <- zonk_eq_types ty1 ty2
       ; case result of
           Left (Pair ty1' ty2') -> can_eq_nc False ev eq_rel ty1' ty1 ty2' ty2
           Right ty              -> canEqReflexive ev eq_rel ty }
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can_eq_nc
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   :: Bool            -- True => both types are flat
   -> CtEvidence
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   -> EqRel
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   -> Type -> Type    -- LHS, after and before type-synonym expansion, resp
   -> Type -> Type    -- RHS, after and before type-synonym expansion, resp
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   -> TcS (StopOrContinue Ct)
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can_eq_nc flat ev eq_rel ty1 ps_ty1 ty2 ps_ty2
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  = do { traceTcS "can_eq_nc" $
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         vcat [ ppr flat, ppr ev, ppr eq_rel, ppr ty1, ppr ps_ty1, ppr ty2, ppr ps_ty2 ]
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       ; rdr_env <- getGlobalRdrEnvTcS
       ; fam_insts <- getFamInstEnvs
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       ; can_eq_nc' flat rdr_env fam_insts ev eq_rel ty1 ps_ty1 ty2 ps_ty2 }
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can_eq_nc'
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   :: Bool           -- True => both input types are flattened
   -> GlobalRdrEnv   -- needed to see which newtypes are in scope
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   -> FamInstEnvs    -- needed to unwrap data instances
   -> CtEvidence
   -> EqRel
   -> Type -> Type    -- LHS, after and before type-synonym expansion, resp
   -> Type -> Type    -- RHS, after and before type-synonym expansion, resp
   -> TcS (StopOrContinue Ct)
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-- Expand synonyms first; see Note [Type synonyms and canonicalization]
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can_eq_nc' flat _rdr_env _envs ev eq_rel ty1 ps_ty1 ty2 ps_ty2
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  | Just ty1' <- tcView ty1 = can_eq_nc flat ev eq_rel ty1' ps_ty1 ty2  ps_ty2
  | Just ty2' <- tcView ty2 = can_eq_nc flat ev eq_rel ty1  ps_ty1 ty2' ps_ty2
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-- need to check for reflexivity in the ReprEq case.
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-- See Note [Eager reflexivity check]
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-- Check only when flat because the zonk_eq_types check in canEqNC takes
-- care of the non-flat case.
can_eq_nc' True _rdr_env _envs ev ReprEq ty1 _ ty2 _
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  | ty1 `tcEqType` ty2
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  = canEqReflexive ev ReprEq ty1

-- When working with ReprEq, unwrap newtypes.
can_eq_nc' _flat rdr_env envs ev ReprEq ty1 _ ty2 ps_ty2
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  | Just stuff1 <- tcTopNormaliseNewTypeTF_maybe envs rdr_env ty1
  = can_eq_newtype_nc ev NotSwapped ty1 stuff1 ty2 ps_ty2
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can_eq_nc' _flat rdr_env envs ev ReprEq ty1 ps_ty1 ty2 _
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  | Just stuff2 <- tcTopNormaliseNewTypeTF_maybe envs rdr_env ty2
  = can_eq_newtype_nc ev IsSwapped  ty2 stuff2 ty1 ps_ty1
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-- Now, check for tyvars. This must happen before CastTy because we need
-- to catch casted tyvars, as the flattener might produce these,
-- due to the fact that flattened types have flattened kinds.
-- See Note [Flattening].
-- Note that there can be only one cast on the tyvar because this will
-- run after the "get rid of casts" case of can_eq_nc' function on the
-- not-yet-flattened types.
-- NB: pattern match on True: we want only flat types sent to canEqTyVar.
-- See also Note [No top-level newtypes on RHS of representational equalities]
can_eq_nc' True _rdr_env _envs ev eq_rel ty1 ps_ty1 ty2 ps_ty2
  | Just (tv1, co1) <- getCastedTyVar_maybe ty1
  = canEqTyVar ev eq_rel NotSwapped tv1 co1 ps_ty1 ty2 ps_ty2
can_eq_nc' True _rdr_env _envs ev eq_rel ty1 ps_ty1 ty2 ps_ty2
  | Just (tv2, co2) <- getCastedTyVar_maybe ty2
  = canEqTyVar ev eq_rel IsSwapped tv2 co2 ps_ty2 ty1 ps_ty1

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-- Then, get rid of casts
can_eq_nc' flat _rdr_env _envs ev eq_rel (CastTy ty1 co1) _ ty2 ps_ty2
  = canEqCast flat ev eq_rel NotSwapped ty1 co1 ty2 ps_ty2
can_eq_nc' flat _rdr_env _envs ev eq_rel ty1 ps_ty1 (CastTy ty2 co2) _
  = canEqCast flat ev eq_rel IsSwapped ty2 co2 ty1 ps_ty1

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----------------------
-- Otherwise try to decompose
----------------------

-- Literals
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can_eq_nc' _flat _rdr_env _envs ev eq_rel ty1@(LitTy l1) _ (LitTy l2) _
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 | l1 == l2
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  = do { setEqIfWanted ev (mkReflCo (eqRelRole eq_rel) ty1)
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       ; stopWith ev "Equal LitTy" }
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-- Try to decompose type constructor applications
-- Including FunTy (s -> t)
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can_eq_nc' _flat _rdr_env _envs ev eq_rel ty1 _ ty2 _
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    --- See Note [FunTy and decomposing type constructor applications].
  | Just (tc1, tys1) <- tcRepSplitTyConApp_maybe' ty1
  , Just (tc2, tys2) <- tcRepSplitTyConApp_maybe' ty2
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  , not (isTypeFamilyTyCon tc1)
  , not (isTypeFamilyTyCon tc2)
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  = canTyConApp ev eq_rel tc1 tys1 tc2 tys2
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can_eq_nc' _flat _rdr_env _envs ev eq_rel
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           s1@(ForAllTy {}) _ s2@(ForAllTy {}) _
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  = can_eq_nc_forall ev eq_rel s1 s2
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-- See Note [Canonicalising type applications] about why we require flat types
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can_eq_nc' True _rdr_env _envs ev eq_rel (AppTy t1 s1) _ ty2 _
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  | Just (t2, s2) <- tcSplitAppTy_maybe ty2
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  = can_eq_app ev eq_rel t1 s1 t2 s2
can_eq_nc' True _rdr_env _envs ev eq_rel ty1 _ (AppTy t2 s2) _
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  | Just (t1, s1) <- tcSplitAppTy_maybe ty1
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  = can_eq_app ev eq_rel t1 s1 t2 s2
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-- No similarity in type structure detected. Flatten and try again.
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can_eq_nc' False rdr_env envs ev eq_rel _ ps_ty1 _ ps_ty2
  = do { (xi1, co1) <- flatten FM_FlattenAll ev ps_ty1
       ; (xi2, co2) <- flatten FM_FlattenAll ev ps_ty2
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       ; rewriteEqEvidence ev NotSwapped xi1 xi2 co1 co2
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         `andWhenContinue` \ new_ev ->
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         can_eq_nc' True rdr_env envs new_ev eq_rel xi1 xi1 xi2 xi2 }

-- We've flattened and the types don't match. Give up.
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can_eq_nc' True _rdr_env _envs ev _eq_rel _ ps_ty1 _ ps_ty2
  = do { traceTcS "can_eq_nc' catch-all case" (ppr ps_ty1 $$ ppr ps_ty2)
       ; canEqHardFailure ev ps_ty1 ps_ty2 }

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---------------------------------
can_eq_nc_forall :: CtEvidence -> EqRel
                 -> Type -> Type    -- LHS and RHS
                 -> TcS (StopOrContinue Ct)
-- (forall as. phi1) ~ (forall bs. phi2)
-- Check for length match of as, bs
-- Then build an implication constraint: forall as. phi1 ~ phi2[as/bs]
-- But remember also to unify the kinds of as and bs
--  (this is the 'go' loop), and actually substitute phi2[as |> cos / bs]
-- Remember also that we might have forall z (a:z). blah
--  so we must proceed one binder at a time (Trac #13879)

can_eq_nc_forall ev eq_rel s1 s2
 | CtWanted { ctev_loc = loc, ctev_dest = orig_dest } <- ev
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 = do { let free_tvs       = tyCoVarsOfTypes [s1,s2]
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            (bndrs1, phi1) = tcSplitForAllTyVarBndrs s1
            (bndrs2, phi2) = tcSplitForAllTyVarBndrs s2
      ; if not (equalLength bndrs1 bndrs2)
        then do { traceTcS "Forall failure" $
                     vcat [ ppr s1, ppr s2, ppr bndrs1, ppr bndrs2
                          , ppr (map binderArgFlag bndrs1)
                          , ppr (map binderArgFlag bndrs2) ]
                ; canEqHardFailure ev s1 s2 }
        else
   do { traceTcS "Creating implication for polytype equality" $ ppr ev
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      ; let empty_subst1 = mkEmptyTCvSubst $ mkInScopeSet free_tvs
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      ; (subst1, skol_tvs) <- tcInstSkolTyVarsX empty_subst1 $
                              binderVars bndrs1

      ; let skol_info = UnifyForAllSkol phi1
            phi1' = substTy subst1 phi1

            -- Unify the kinds, extend the substitution
            go (skol_tv:skol_tvs) subst (bndr2:bndrs2)
              = do { let tv2 = binderVar bndr2
                   ; kind_co <- unifyWanted loc Nominal
                                            (tyVarKind skol_tv)
                                            (substTy subst (tyVarKind tv2))
                   ; let subst' = extendTvSubst subst tv2
                                       (mkCastTy (mkTyVarTy skol_tv) kind_co)
                   ; co <- go skol_tvs subst' bndrs2
                   ; return (mkForAllCo skol_tv kind_co co) }

            -- Done: unify phi1 ~ phi2
            go [] subst bndrs2
              = ASSERT( null bndrs2 )
                unifyWanted loc (eqRelRole eq_rel)
                            phi1' (substTy subst phi2)

            go _ _ _ = panic "cna_eq_nc_forall"  -- case (s:ss) []

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            empty_subst2 = mkEmptyTCvSubst (getTCvInScope subst1)
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      ; (implic, _ev_binds, all_co) <- buildImplication skol_info skol_tvs [] $
                                       go skol_tvs empty_subst2 bndrs2
           -- We have nowhere to put these bindings
           -- but TcSimplify.setImplicationStatus
           -- checks that we don't actually use them
           -- when skol_info = UnifyForAllSkol

      ; updWorkListTcS (extendWorkListImplic implic)
      ; setWantedEq orig_dest all_co
      ; stopWith ev "Deferred polytype equality" } }

 | otherwise
 = do { traceTcS "Omitting decomposition of given polytype equality" $
        pprEq s1 s2    -- See Note [Do not decompose given polytype equalities]
      ; stopWith ev "Discard given polytype equality" }

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---------------------------------
-- | Compare types for equality, while zonking as necessary. Gives up
-- as soon as it finds that two types are not equal.
-- This is quite handy when some unification has made two
-- types in an inert wanted to be equal. We can discover the equality without
-- flattening, which is sometimes very expensive (in the case of type functions).
-- In particular, this function makes a ~20% improvement in test case
-- perf/compiler/T5030.
--
-- Returns either the (partially zonked) types in the case of
-- inequality, or the one type in the case of equality. canEqReflexive is
-- a good next step in the 'Right' case. Returning 'Left' is always safe.
--
-- NB: This does *not* look through type synonyms. In fact, it treats type
-- synonyms as rigid constructors. In the future, it might be convenient
-- to look at only those arguments of type synonyms that actually appear
-- in the synonym RHS. But we're not there yet.
zonk_eq_types :: TcType -> TcType -> TcS (Either (Pair TcType) TcType)
zonk_eq_types = go
  where
    go (TyVarTy tv1) (TyVarTy tv2) = tyvar_tyvar tv1 tv2
    go (TyVarTy tv1) ty2           = tyvar NotSwapped tv1 ty2
    go ty1 (TyVarTy tv2)           = tyvar IsSwapped  tv2 ty1

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    -- We handle FunTys explicitly here despite the fact that they could also be
    -- treated as an application. Why? Well, for one it's cheaper to just look
    -- at two types (the argument and result types) than four (the argument,
    -- result, and their RuntimeReps). Also, we haven't completely zonked yet,
    -- so we may run into an unzonked type variable while trying to compute the
    -- RuntimeReps of the argument and result types. This can be observed in
    -- testcase tc269.
    go ty1 ty2
      | Just (arg1, res1) <- split1
      , Just (arg2, res2) <- split2
      = do { res_a <- go arg1 arg2
           ; res_b <- go res1 res2
           ; return $ combine_rev mkFunTy res_b res_a
           }
      | isJust split1 || isJust split2
      = bale_out ty1 ty2
      where
        split1 = tcSplitFunTy_maybe ty1
        split2 = tcSplitFunTy_maybe ty2

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    go ty1 ty2
      | Just (tc1, tys1) <- tcRepSplitTyConApp_maybe ty1
      , Just (tc2, tys2) <- tcRepSplitTyConApp_maybe ty2
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      = if tc1 == tc2 && tys1 `equalLength` tys2
          -- Crucial to check for equal-length args, because
          -- we cannot assume that the two args to 'go' have
          -- the same kind.  E.g go (Proxy *      (Maybe Int))
          --                        (Proxy (*->*) Maybe)
          -- We'll call (go (Maybe Int) Maybe)
          -- See Trac #13083
        then tycon tc1 tys1 tys2
        else bale_out ty1 ty2
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    go ty1 ty2
      | Just (ty1a, ty1b) <- tcRepSplitAppTy_maybe ty1
      , Just (ty2a, ty2b) <- tcRepSplitAppTy_maybe ty2
      = do { res_a <- go ty1a ty2a
           ; res_b <- go ty1b ty2b
           ; return $ combine_rev mkAppTy res_b res_a }

    go ty1@(LitTy lit1) (LitTy lit2)
      | lit1 == lit2
      = return (Right ty1)

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    go ty1 ty2 = bale_out ty1 ty2
      -- We don't handle more complex forms here

    bale_out ty1 ty2 = return $ Left (Pair ty1 ty2)
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    tyvar :: SwapFlag -> TcTyVar -> TcType
          -> TcS (Either (Pair TcType) TcType)
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      -- Try to do as little as possible, as anything we do here is redundant
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      -- with flattening. In particular, no need to zonk kinds. That's why
      -- we don't use the already-defined zonking functions
    tyvar swapped tv ty
      = case tcTyVarDetails tv of
          MetaTv { mtv_ref = ref }
            -> do { cts <- readTcRef ref
                  ; case cts of
                      Flexi        -> give_up
                      Indirect ty' -> unSwap swapped go ty' ty }
          _ -> give_up
      where
        give_up = return $ Left $ unSwap swapped Pair (mkTyVarTy tv) ty

    tyvar_tyvar tv1 tv2
      | tv1 == tv2 = return (Right (mkTyVarTy tv1))
      | otherwise  = do { (ty1', progress1) <- quick_zonk tv1
                        ; (ty2', progress2) <- quick_zonk tv2
                        ; if progress1 || progress2
                          then go ty1' ty2'
                          else return $ Left (Pair (TyVarTy tv1) (TyVarTy tv2)) }

    quick_zonk tv = case tcTyVarDetails tv of
      MetaTv { mtv_ref = ref }
        -> do { cts <- readTcRef ref
              ; case cts of
                  Flexi        -> return (TyVarTy tv, False)
                  Indirect ty' -> return (ty', True) }
      _ -> return (TyVarTy tv, False)

      -- This happens for type families, too. But recall that failure
      -- here just means to try harder, so it's OK if the type function
      -- isn't injective.
    tycon :: TyCon -> [TcType] -> [TcType]
          -> TcS (Either (Pair TcType) TcType)
    tycon tc tys1 tys2
      = do { results <- zipWithM go tys1 tys2
           ; return $ case combine_results results of
               Left tys  -> Left (mkTyConApp tc <$> tys)
               Right tys -> Right (mkTyConApp tc tys) }

    combine_results :: [Either (Pair TcType) TcType]
                    -> Either (Pair [TcType]) [TcType]
    combine_results = bimap (fmap reverse) reverse .
                      foldl' (combine_rev (:)) (Right [])

      -- combine (in reverse) a new result onto an already-combined result
    combine_rev :: (a -> b -> c)
                -> Either (Pair b) b
                -> Either (Pair a) a
                -> Either (Pair c) c
    combine_rev f (Left list) (Left elt) = Left (f <$> elt     <*> list)
    combine_rev f (Left list) (Right ty) = Left (f <$> pure ty <*> list)
    combine_rev f (Right tys) (Left elt) = Left (f <$> elt     <*> pure tys)
    combine_rev f (Right tys) (Right ty) = Right (f ty tys)
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{-
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Note [Newtypes can blow the stack]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Suppose we have

  newtype X = MkX (Int -> X)
  newtype Y = MkY (Int -> Y)

and now wish to prove

  [W] X ~R Y

This Wanted will loop, expanding out the newtypes ever deeper looking
for a solid match or a solid discrepancy. Indeed, there is something
appropriate to this looping, because X and Y *do* have the same representation,
in the limit -- they're both (Fix ((->) Int)). However, no finitely-sized
coercion will ever witness it. This loop won't actually cause GHC to hang,
though, because we check our depth when unwrapping newtypes.

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Note [Eager reflexivity check]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Suppose we have

  newtype X = MkX (Int -> X)

and

  [W] X ~R X
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Naively, we would start unwrapping X and end up in a loop. Instead,
we do this eager reflexivity check. This is necessary only for representational
equality because the flattener technology deals with the similar case
(recursive type families) for nominal equality.

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Note that this check does not catch all cases, but it will catch the cases
we're most worried about, types like X above that are actually inhabited.
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Here's another place where this reflexivity check is key:
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Consider trying to prove (f a) ~R (f a). The AppTys in there can't
be decomposed, because representational equality isn't congruent with respect
to AppTy. So, when canonicalising the equality above, we get stuck and
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would normally produce a CIrredCan. However, we really do want to
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be able to solve (f a) ~R (f a). So, in the representational case only,
we do a reflexivity check.

(This would be sound in the nominal case, but unnecessary, and I [Richard
E.] am worried that it would slow down the common case.)
-}

------------------------
-- | We're able to unwrap a newtype. Update the bits accordingly.
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can_eq_newtype_nc :: CtEvidence           -- ^ :: ty1 ~ ty2
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                  -> SwapFlag
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                  -> TcType                                    -- ^ ty1
                  -> ((Bag GlobalRdrElt, TcCoercion), TcType)  -- ^ :: ty1 ~ ty1'
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                  -> TcType               -- ^ ty2
                  -> TcType               -- ^ ty2, with type synonyms
                  -> TcS (StopOrContinue Ct)
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can_eq_newtype_nc ev swapped ty1 ((gres, co), ty1') ty2 ps_ty2
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  = do { traceTcS "can_eq_newtype_nc" $
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         vcat [ ppr ev, ppr swapped, ppr co, ppr gres, ppr ty1', ppr ty2 ]
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         -- check for blowing our stack:
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         -- See Note [Newtypes can blow the stack]
       ; checkReductionDepth (ctEvLoc ev) ty1
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       ; addUsedGREs (bagToList gres)
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           -- we have actually used the newtype constructor here, so
           -- make sure we don't warn about importing it!

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       ; rewriteEqEvidence ev swapped ty1' ps_ty2
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                           (mkTcSymCo co) (mkTcReflCo Representational ps_ty2)
         `andWhenContinue` \ new_ev ->
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         can_eq_nc False new_ev ReprEq ty1' ty1' ty2 ps_ty2 }
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---------
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-- ^ Decompose a type application.
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-- All input types must be flat. See Note [Canonicalising type applications]
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can_eq_app :: CtEvidence       -- :: s1 t1 ~r s2 t2
           -> EqRel            -- r
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           -> Xi -> Xi         -- s1 t1
           -> Xi -> Xi         -- s2 t2
           -> TcS (StopOrContinue Ct)
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-- AppTys only decompose for nominal equality, so this case just leads
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-- to an irreducible constraint; see typecheck/should_compile/T10494
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-- See Note [Decomposing equality], note {4}
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can_eq_app ev ReprEq _ _ _ _
  = do { traceTcS "failing to decompose representational AppTy equality" (ppr ev)
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       ; continueWith (mkIrredCt ev) }
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          -- no need to call canEqFailure, because that flattens, and the
          -- types involved here are already flat
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can_eq_app ev NomEq s1 t1 s2 t2
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  | CtDerived { ctev_loc = loc } <- ev
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  = do { unifyDeriveds loc [Nominal, Nominal] [s1, t1] [s2, t2]
       ; stopWith ev "Decomposed [D] AppTy" }
  | CtWanted { ctev_dest = dest, ctev_loc = loc } <- ev
  = do { co_s <- unifyWanted loc Nominal s1 s2
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       ; let arg_loc
               | isNextArgVisible s1 = loc
               | otherwise           = updateCtLocOrigin loc toInvisibleOrigin
       ; co_t <- unifyWanted arg_loc Nominal t1 t2
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       ; let co = mkAppCo co_s co_t
       ; setWantedEq dest co
       ; stopWith ev "Decomposed [W] AppTy" }
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  | CtGiven { ctev_evar = evar, ctev_loc = loc } <- ev
  = do { let co   = mkTcCoVarCo evar
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             co_s = mkTcLRCo CLeft  co
             co_t = mkTcLRCo CRight co
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       ; evar_s <- newGivenEvVar loc ( mkTcEqPredLikeEv ev s1 s2
                                     , EvCoercion co_s )
       ; evar_t <- newGivenEvVar loc ( mkTcEqPredLikeEv ev t1 t2
                                     , EvCoercion co_t )
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       ; emitWorkNC [evar_t]
       ; canEqNC evar_s NomEq s1 s2 }
  | otherwise  -- Can't happen
  = error "can_eq_app"
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-----------------------
-- | Break apart an equality over a casted type
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-- looking like   (ty1 |> co1) ~ ty2   (modulo a swap-flag)
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canEqCast :: Bool         -- are both types flat?
          -> CtEvidence
          -> EqRel
          -> SwapFlag
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          -> TcType -> Coercion   -- LHS (res. RHS), ty1 |> co1
          -> TcType -> TcType     -- RHS (res. LHS), ty2 both normal and pretty
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          -> TcS (StopOrContinue Ct)
canEqCast flat ev eq_rel swapped ty1 co1 ty2 ps_ty2
  = do { traceTcS "Decomposing cast" (vcat [ ppr ev
                                           , ppr ty1 <+> text "|>" <+> ppr co1
                                           , ppr ps_ty2 ])
       ; rewriteEqEvidence ev swapped ty1 ps_ty2
                           (mkTcReflCo role ty1
                              `mkTcCoherenceRightCo` co1)
                           (mkTcReflCo role ps_ty2)
         `andWhenContinue` \ new_ev ->
         can_eq_nc flat new_ev eq_rel ty1 ty1 ty2 ps_ty2 }
  where
    role = eqRelRole eq_rel

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------------------------
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canTyConApp :: CtEvidence -> EqRel
            -> TyCon -> [TcType]
            -> TyCon -> [TcType]
            -> TcS (StopOrContinue Ct)
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-- See Note [Decomposing TyConApps]
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canTyConApp ev eq_rel tc1 tys1 tc2 tys2
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  | tc1 == tc2
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  , tys1 `equalLength` tys2
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  = do { inerts <- getTcSInerts
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       ; if can_decompose inerts
         then do { traceTcS "canTyConApp"
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                       (ppr ev $$ ppr eq_rel $$ ppr tc1 $$ ppr tys1 $$ ppr tys2)
                 ; canDecomposableTyConAppOK ev eq_rel tc1 tys1 tys2
                 ; stopWith ev "Decomposed TyConApp" }
         else canEqFailure ev eq_rel ty1 ty2 }
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  -- See Note [Skolem abstract data] (at tyConSkolem)
  | tyConSkolem tc1 || tyConSkolem tc2
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  = do { traceTcS "canTyConApp: skolem abstract" (ppr tc1 $$ ppr tc2)
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       ; continueWith (mkIrredCt ev) }
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  -- Fail straight away for better error messages
  -- See Note [Use canEqFailure in canDecomposableTyConApp]
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  | eq_rel == ReprEq && not (isGenerativeTyCon tc1 Representational &&
                             isGenerativeTyCon tc2 Representational)
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  = canEqFailure ev eq_rel ty1 ty2
  | otherwise
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  = canEqHardFailure ev ty1 ty2
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  where
    ty1 = mkTyConApp tc1 tys1
    ty2 = mkTyConApp tc2 tys2

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    loc  = ctEvLoc ev
    pred = ctEvPred ev

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     -- See Note [Decomposing equality]
    can_decompose inerts
      =  isInjectiveTyCon tc1 (eqRelRole eq_rel)
      || (ctEvFlavour ev /= Given && isEmptyBag (matchableGivens loc pred inerts))

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{-
Note [Use canEqFailure in canDecomposableTyConApp]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We must use canEqFailure, not canEqHardFailure here, because there is
the possibility of success if working with a representational equality.
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Here is one case:
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  type family TF a where TF Char = Bool
  data family DF a
  newtype instance DF Bool = MkDF Int

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Suppose we are canonicalising (Int ~R DF (TF a)), where we don't yet
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know `a`. This is *not* a hard failure, because we might soon learn
that `a` is, in fact, Char, and then the equality succeeds.
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Here is another case:

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  [G] Age ~R Int
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where Age's constructor is not in scope. We don't want to report
an "inaccessible code" error in the context of this Given!

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For example, see typecheck/should_compile/T10493, repeated here:

  import Data.Ord (Down)  -- no constructor

  foo :: Coercible (Down Int) Int => Down Int -> Int
  foo = coerce

That should compile, but only because we use canEqFailure and not
canEqHardFailure.

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Note [Decomposing equality]
~~~~~~~~~~~~~~~~~~~~~~~~~~~
If we have a constraint (of any flavour and role) that looks like
T tys1 ~ T tys2, what can we conclude about tys1 and tys2? The answer,
of course, is "it depends". This Note spells it all out.

In this Note, "decomposition" refers to taking the constraint
  [fl] (T tys1 ~X T tys2)
(for some flavour fl and some role X) and replacing it with
  [fls'] (tys1 ~Xs' tys2)
where that notation indicates a list of new constraints, where the
new constraints may have different flavours and different roles.

The key property to consider is injectivity. When decomposing a Given the
decomposition is sound if and only if T is injective in all of its type
arguments. When decomposing a Wanted, the decomposition is sound (assuming the
correct roles in the produced equality constraints), but it may be a guess --
that is, an unforced decision by the constraint solver. Decomposing Wanteds
over injective TyCons does not entail guessing. But sometimes we want to
decompose a Wanted even when the TyCon involved is not injective! (See below.)

So, in broad strokes, we want this rule:

(*) Decompose a constraint (T tys1 ~X T tys2) if and only if T is injective
at role X.

Pursuing the details requires exploring three axes:
* Flavour: Given vs. Derived vs. Wanted
* Role: Nominal vs. Representational
* TyCon species: datatype vs. newtype vs. data family vs. type family vs. type variable

(So a type variable isn't a TyCon, but it's convenient to put the AppTy case
in the same table.)

Right away, we can say that Derived behaves just as Wanted for the purposes
of decomposition. The difference between Derived and Wanted is the handling of
evidence. Since decomposition in these cases isn't a matter of soundness but of
guessing, we want the same behavior regardless of evidence.

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Here is a table (discussion following) detailing where decomposition of
   (T s1 ... sn) ~r (T t1 .. tn)
is allowed.  The first four lines (Data types ... type family) refer
to TyConApps with various TyCons T; the last line is for AppTy, where
there is presumably a type variable at the head, so it's actually
   (s s1 ... sn) ~r (t t1 .. tn)
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NOMINAL               GIVEN                       WANTED

Datatype               YES                         YES
Newtype                YES                         YES
Data family            YES                         YES
Type family            YES, in injective args{1}   YES, in injective args{1}
Type variable          YES                         YES

REPRESENTATIONAL      GIVEN                       WANTED

Datatype               YES                         YES
Newtype                NO{2}                      MAYBE{2}
Data family            NO{3}                      MAYBE{3}
Type family             NO                          NO
Type variable          NO{4}                       NO{4}

{1}: Type families can be injective in some, but not all, of their arguments,
so we want to do partial decomposition. This is quite different than the way
other decomposition is done, where the decomposed equalities replace the original
one. We thus proceed much like we do with superclasses: emitting new Givens
when "decomposing" a partially-injective type family Given and new Deriveds
when "decomposing" a partially-injective type family Wanted. (As of the time of
writing, 13 June 2015, the implementation of injective type families has not
been merged, but it should be soon. Please delete this parenthetical if the
implementation is indeed merged.)

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{2}: See Note [Decomposing newtypes at representational role]
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{3}: Because of the possibility of newtype instances, we must treat
data families like newtypes. See also Note [Decomposing newtypes at
representational role]. See #10534 and test case
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typecheck/should_fail/T10534.
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{4}: Because type variables can stand in for newtypes, we conservatively do not
decompose AppTys over representational equality.

In the implementation of can_eq_nc and friends, we don't directly pattern
match using lines like in the tables above, as those tables don't cover
all cases (what about PrimTyCon? tuples?). Instead we just ask about injectivity,
boiling the tables above down to rule (*). The exceptions to rule (*) are for
injective type families, which are handled separately from other decompositions,
and the MAYBE entries above.

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Note [Decomposing newtypes at representational role]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This note discusses the 'newtype' line in the REPRESENTATIONAL table
in Note [Decomposing equality]. (At nominal role, newtypes are fully
decomposable.)

Here is a representative example of why representational equality over
newtypes is tricky:

  newtype Nt a = Mk Bool         -- NB: a is not used in the RHS,
  type role Nt representational  -- but the user gives it an R role anyway

If we have [W] Nt alpha ~R Nt beta, we *don't* want to decompose to
[W] alpha ~R beta, because it's possible that alpha and beta aren't
representationally equal. Here's another example.

  newtype Nt a = MkNt (Id a)
  type family Id a where Id a = a

  [W] Nt Int ~R Nt Age

Because of its use of a type family, Nt's parameter will get inferred to have
a nominal role. Thus, decomposing the wanted will yield [W] Int ~N Age, which
is unsatisfiable. Unwrapping, though, leads to a solution.

Conclusion:
 * Unwrap newtypes before attempting to decompose them.
   This is done in can_eq_nc'.

It all comes from the fact that newtypes aren't necessarily injective
w.r.t. representational equality.

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Furthermore, as explained in Note [NthCo and newtypes] in TyCoRep, we can't use
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NthCo on representational coercions over newtypes. NthCo comes into play
only when decomposing givens.

Conclusion:
 * Do not decompose [G] N s ~R N t

Is it sensible to decompose *Wanted* constraints over newtypes?  Yes!
It's the only way we could ever prove (IO Int ~R IO Age), recalling
that IO is a newtype.

However we must be careful.  Consider

  type role Nt representational

  [G] Nt a ~R Nt b       (1)
  [W] NT alpha ~R Nt b   (2)
  [W] alpha ~ a          (3)

If we focus on (3) first, we'll substitute in (2), and now it's
identical to the given (1), so we succeed.  But if we focus on (2)
first, and decompose it, we'll get (alpha ~R b), which is not soluble.
This is exactly like the question of overlapping Givens for class
constraints: see Note [Instance and Given overlap] in TcInteract.

Conclusion:
  * Decompose [W] N s ~R N t  iff there no given constraint that could
    later solve it.
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-}

canDecomposableTyConAppOK :: CtEvidence -> EqRel
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                          -> TyCon -> [TcType] -> [TcType]
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                          -> TcS ()
-- Precondition: tys1 and tys2 are the same length, hence "OK"
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canDecomposableTyConAppOK ev eq_rel tc tys1 tys2
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  = case ev of
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     CtDerived {}
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        -> unifyDeriveds loc tc_roles tys1 tys2
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     CtWanted { ctev_dest = dest }
        -> do { cos <- zipWith4M unifyWanted new_locs tc_roles tys1 tys2
              ; setWantedEq dest (mkTyConAppCo role tc cos) }
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     CtGiven { ctev_evar = evar }
        -> do { let ev_co = mkCoVarCo evar
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              ; given_evs <- newGivenEvVars loc $
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                             [ ( mkPrimEqPredRole r ty1 ty2
                               , EvCoercion (mkNthCo i ev_co) )
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                             | (r, ty1, ty2, i) <- zip4 tc_roles tys1 tys2 [0..]
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                             , r /= Phantom
                             , not (isCoercionTy ty1) && not (isCoercionTy ty2) ]
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              ; emitWorkNC given_evs }
  where
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    loc        = ctEvLoc ev
    role       = eqRelRole eq_rel
    tc_roles   = tyConRolesX role tc

      -- the following makes a better distinction between "kind" and "type"
      -- in error messages
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    bndrs      = tyConBinders tc
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    is_kinds   = map isNamedTyConBinder bndrs
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    is_viss    = map isVisibleTyConBinder bndrs
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    kind_xforms = map (\is_kind -> if is_kind then toKindLoc else id) is_kinds
    vis_xforms  = map (\is_vis  -> if is_vis  then id
                                   else flip updateCtLocOrigin toInvisibleOrigin)
                      is_viss

    -- zipWith3 (.) composes its first two arguments and applies it to the third
    new_locs = zipWith3 (.) kind_xforms vis_xforms (repeat loc)
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-- | Call when canonicalizing an equality fails, but if the equality is
-- representational, there is some hope for the future.
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-- Examples in Note [Use canEqFailure in canDecomposableTyConApp]
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canEqFailure :: CtEvidence -> EqRel
             -> TcType -> TcType -> TcS (StopOrContinue Ct)
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canEqFailure ev NomEq ty1 ty2
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  = canEqHardFailure ev ty1 ty2
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canEqFailure ev ReprEq ty1 ty2
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  = do { (xi1, co1) <- flatten FM_FlattenAll ev ty1
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       ; (xi2, co2) <- flatten FM_FlattenAll ev ty2
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            -- We must flatten the types before putting them in the
            -- inert set, so that we are sure to kick them out when
            -- new equalities become available
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       ; traceTcS "canEqFailure with ReprEq" $
         vcat [ ppr ev, ppr ty1, ppr ty2, ppr xi1, ppr xi2 ]
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       ; rewriteEqEvidence ev NotSwapped xi1 xi2 co1 co2
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         `andWhenContinue` \ new_ev ->
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         continueWith (mkIrredCt new_ev) }
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-- | Call when canonicalizing an equality fails with utterly no hope.
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canEqHardFailure :: CtEvidence
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                 -> TcType -> TcType -> TcS (StopOrContinue Ct)
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-- See Note [Make sure that insolubles are fully rewritten]
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canEqHardFailure ev ty1 ty2
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  = do { (s1, co1) <- flatten FM_SubstOnly ev ty1
       ; (s2, co2) <- flatten FM_SubstOnly ev ty2
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       ; rewriteEqEvidence ev NotSwapped s1 s2 co1 co2
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         `andWhenContinue` \ new_ev ->
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    do { emitInsoluble (mkInsolubleCt new_ev)
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       ; stopWith new_ev "Definitely not equal" }}
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{-
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Note [Decomposing TyConApps]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
If we see (T s1 t1 ~ T s2 t2), then we can just decompose to
  (s1 ~ s2, t1 ~ t2)
and push those back into the work list.  But if
  s1 = K k1    s2 = K k2
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then we will just decomopose s1~s2, and it might be better to
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do so on the spot.  An important special case is where s1=s2,
and we get just Refl.

So canDecomposableTyCon is a fast-path decomposition that uses
unifyWanted etc to short-cut that work.

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Note [Canonicalising type applications]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Given (s1 t1) ~ ty2, how should we proceed?
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The simple things is to see if ty2 is of form (s2 t2), and
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decompose.  By this time s1 and s2 can't be saturated type