Coercion.hs 76.7 KB
Newer Older
1
-- (c) The University of Glasgow 2006
2

3 4
{-# LANGUAGE CPP, DeriveDataTypeable #-}

5
-- | Module for (a) type kinds and (b) type coercions,
6
-- as used in System FC. See 'CoreSyn.Expr' for
batterseapower's avatar
batterseapower committed
7 8
-- more on System FC and how coercions fit into it.
--
9
module Coercion (
batterseapower's avatar
batterseapower committed
10
        -- * Main data type
11
        Coercion(..), Var, CoVar,
12
        LeftOrRight(..), pickLR,
13
        Role(..), ltRole,
14

dreixel's avatar
dreixel committed
15
        -- ** Functions over coercions
16
        coVarKind, coVarRole,
17
        coercionType, coercionKind, coercionKinds, isReflCo,
18
        isReflCo_maybe, coercionRole, coercionKindRole,
batterseapower's avatar
batterseapower committed
19
        mkCoercionType,
20

21
        -- ** Constructing coercions
22
        mkReflCo, mkCoVarCo,
23
        mkAxInstCo, mkUnbranchedAxInstCo, mkAxInstLHS, mkAxInstRHS,
24
        mkUnbranchedAxInstRHS,
25
        mkPiCo, mkPiCos, mkCoCast,
26
        mkSymCo, mkTransCo, mkNthCo, mkNthCoRole, mkLRCo,
27
        mkInstCo, mkAppCo, mkAppCoFlexible, mkTyConAppCo, mkFunCo,
28
        mkForAllCo, mkUnsafeCo, mkUnivCo, mkSubCo, mkPhantomCo,
29
        mkNewTypeCo, downgradeRole,
30
        mkAxiomRuleCo,
TomSchrijvers's avatar
TomSchrijvers committed
31

32
        -- ** Decomposition
33
        instNewTyCon_maybe,
eir@cis.upenn.edu's avatar
eir@cis.upenn.edu committed
34 35 36 37

        NormaliseStepper, NormaliseStepResult(..), composeSteppers,
        modifyStepResultCo, unwrapNewTypeStepper,
        topNormaliseNewType_maybe, topNormaliseTypeX_maybe,
38

39
        decomposeCo, getCoVar_maybe,
40 41
        splitAppCo_maybe,
        splitForAllCo_maybe,
42
        nthRole, tyConRolesX,
eir@cis.upenn.edu's avatar
eir@cis.upenn.edu committed
43
        setNominalRole_maybe,
44

45 46
        -- ** Coercion variables
        mkCoVar, isCoVar, isCoVarType, coVarName, setCoVarName, setCoVarUnique,
47 48 49

        -- ** Free variables
        tyCoVarsOfCo, tyCoVarsOfCos, coVarsOfCo, coercionSize,
50

51
        -- ** Substitution
52
        CvSubstEnv, emptyCvSubstEnv,
53 54
        CvSubst(..), emptyCvSubst, Coercion.lookupTyVar, lookupCoVar,
        isEmptyCvSubst, zapCvSubstEnv, getCvInScope,
55
        substCo, substCos, substCoVar, substCoVars,
56
        substCoWithTy, substCoWithTys,
57
        cvTvSubst, tvCvSubst, mkCvSubst, zipOpenCvSubst,
58 59
        substTy, extendTvSubst,
        extendCvSubstAndInScope, extendTvSubstAndInScope,
60
        substTyVarBndr, substCoVarBndr,
61

62 63
        -- ** Lifting
        liftCoMatch, liftCoSubstTyVar, liftCoSubstWith,
64

batterseapower's avatar
batterseapower committed
65
        -- ** Comparison
66
        coreEqCoercion, coreEqCoercion2,
67

68 69
        -- ** Forcing evaluation of coercions
        seqCo,
70

71
        -- * Pretty-printing
72 73
        pprCo, pprParendCo,
        pprCoAxiom, pprCoAxBranch, pprCoAxBranchHdr,
Simon Peyton Jones's avatar
Simon Peyton Jones committed
74 75 76

        -- * Tidying
        tidyCo, tidyCos,
TomSchrijvers's avatar
TomSchrijvers committed
77

78
        -- * Other
eir@cis.upenn.edu's avatar
eir@cis.upenn.edu committed
79
        applyCo,
80
       ) where
81 82 83

#include "HsVersions.h"

84
import Unify    ( MatchEnv(..), matchList )
85
import TypeRep
86 87
import qualified Type
import Type hiding( substTy, substTyVarBndr, extendTvSubst )
88
import TyCon
89
import CoAxiom
90
import Var
91
import VarEnv
simonpj@microsoft.com's avatar
simonpj@microsoft.com committed
92
import VarSet
93
import Binary
94
import Maybes   ( orElse )
95 96
import Name     ( Name, NamedThing(..), nameUnique, nameModule, getSrcSpan )
import OccName  ( parenSymOcc )
97 98
import Util
import BasicTypes
99
import Outputable
100 101
import Unique
import Pair
Simon Peyton Jones's avatar
Simon Peyton Jones committed
102
import SrcLoc
103
import PrelNames        ( funTyConKey, eqPrimTyConKey, eqReprPrimTyConKey )
eir@cis.upenn.edu's avatar
eir@cis.upenn.edu committed
104
import Control.Applicative hiding ( empty )
105
#if __GLASGOW_HASKELL__ < 709
106
import Data.Traversable (traverse, sequenceA)
107
#endif
108
import FastString
109
import ListSetOps
110 111

import qualified Data.Data as Data hiding ( TyCon )
112
import Control.Arrow ( first )
113

114 115 116
{-
************************************************************************
*                                                                      *
117
            Coercions
118 119 120
*                                                                      *
************************************************************************
-}
121

122 123
-- | A 'Coercion' is concrete evidence of the equality/convertibility
-- of two types.
124

125 126
-- If you edit this type, you may need to update the GHC formalism
-- See Note [GHC Formalism] in coreSyn/CoreLint.lhs
127
data Coercion
128 129 130 131 132 133 134
  -- 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.

135
  -- These ones mirror the shape of types
136 137
  = -- Refl :: "e" -> _ -> e
    Refl Role Type  -- See Note [Refl invariant]
138 139 140 141 142 143 144 145 146 147
          -- Invariant: applications of (Refl T) to a bunch of identity coercions
          --            always show up as Refl.
          -- For example  (Refl T) (Refl a) (Refl b) shows up as (Refl (T a b)).

          -- Applications of (Refl T) to some coercions, at least one of
          -- which is NOT the identity, show up as TyConAppCo.
          -- (They may not be fully saturated however.)
          -- ConAppCo coercions (like all coercions other than Refl)
          -- are NEVER the identity.

148 149
          -- Use (Refl Representational _), not (SubCo (Refl Nominal _))

150
  -- These ones simply lift the correspondingly-named
151
  -- Type constructors into Coercions
152

153 154
  -- TyConAppCo :: "e" -> _ -> ?? -> e
  -- See Note [TyConAppCo roles]
155
  | TyConAppCo Role TyCon [Coercion]    -- lift TyConApp
156 157 158
               -- The TyCon is never a synonym;
               -- we expand synonyms eagerly
               -- But it can be a type function
159 160

  | AppCo Coercion Coercion        -- lift AppTy
161
          -- AppCo :: e -> N -> e
162 163 164

  -- See Note [Forall coercions]
  | ForAllCo TyVar Coercion       -- forall a. g
165
         -- :: _ -> e -> e
166 167

  -- These are special
168 169 170 171
  | CoVarCo CoVar      -- :: _ -> (N or R)
                       -- result role depends on the tycon of the variable's type

    -- AxiomInstCo :: e -> _ -> [N] -> e
172
  | AxiomInstCo (CoAxiom Branched) BranchIndex [Coercion]
173
     -- See also [CoAxiom index]
174
     -- The coercion arguments always *precisely* saturate
175
     -- arity of (that branch of) the CoAxiom.  If there are
176
     -- any left over, we use AppCo.  See
177 178
     -- See [Coercion axioms applied to coercions]

179 180 181 182
         -- see Note [UnivCo]
  | UnivCo Role Type Type      -- :: "e" -> _ -> _ -> e
  | SymCo Coercion             -- :: e -> e
  | TransCo Coercion Coercion  -- :: e -> e -> e
183

184 185 186 187
    -- The number of types and coercions should match exactly the expectations
    -- of the CoAxiomRule (i.e., the rule is fully saturated).
  | AxiomRuleCo CoAxiomRule [Type] [Coercion]

188
  -- These are destructors
189

190
  | NthCo  Int         Coercion     -- Zero-indexed; decomposes (T t0 ... tn)
191
    -- :: _ -> e -> ?? (inverse of TyConAppCo, see Note [TyConAppCo roles])
192
  | LRCo   LeftOrRight Coercion     -- Decomposes (t_left t_right)
193
    -- :: _ -> N -> N
194
  | InstCo Coercion Type
195 196 197 198
    -- :: e -> _ -> e

  | SubCo Coercion                  -- Turns a ~N into a ~R
    -- :: N -> R
199
  deriving (Data.Data, Data.Typeable)
200

201 202
-- If you edit this type, you may need to update the GHC formalism
-- See Note [GHC Formalism] in coreSyn/CoreLint.lhs
203
data LeftOrRight = CLeft | CRight
204 205
                 deriving( Eq, Data.Data, Data.Typeable )

206 207 208 209 210 211 212 213 214
instance Binary LeftOrRight where
   put_ bh CLeft  = putByte bh 0
   put_ bh CRight = putByte bh 1

   get bh = do { h <- getByte bh
               ; case h of
                   0 -> return CLeft
                   _ -> return CRight }

215 216 217
pickLR :: LeftOrRight -> (a,a) -> a
pickLR CLeft  (l,_) = l
pickLR CRight (_,r) = r
218

219
{-
220 221
Note [Refl invariant]
~~~~~~~~~~~~~~~~~~~~~
222 223
Coercions have the following invariant
     Refl is always lifted as far as possible.
224 225 226 227 228 229 230 231 232 233 234 235 236 237 238

You might think that a consequencs is:
     Every identity coercions has Refl at the root

But that's not quite true because of coercion variables.  Consider
     g         where g :: Int~Int
     Left h    where h :: Maybe Int ~ Maybe Int
etc.  So the consequence is only true of coercions that
have no coercion variables.

Note [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
239
expose further opportunities for optimization.
240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274

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.

275 276 277 278 279 280 281 282 283 284 285 286
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.
287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304

Note [Forall coercions]
~~~~~~~~~~~~~~~~~~~~~~~
Constructing coercions between forall-types can be a bit tricky.
Currently, the situation is as follows:

  ForAllCo TyVar Coercion

represents a coercion between polymorphic types, with the rule

           v : k       g : t1 ~ t2
  ----------------------------------------------
  ForAllCo v g : (all v:k . t1) ~ (all v:k . t2)

Note that it's only necessary to coerce between polymorphic types
where the type variables have identical kinds, because equality on
kinds is trivial.

305 306 307 308 309 310 311 312 313
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?
314

315 316
Answer: we simply treat (~) as an ordinary type constructor, so these
types really look like
317

318 319
   ((~) [c] a) -> [a] -> c
   ((~) [c] b) -> [b] -> c
320

321
So the coercion between the two is obviously
322

323
   ((~) [c] g) -> [g] -> c
324

325 326
Another way to see this to say that we simply collapse predicates to
their representation type (see Type.coreView and Type.predTypeRep).
327

328
This collapse is done by mkPredCo; there is no PredCo constructor
329
in Coercion.  This is important because we need Nth to work on
330 331 332
predicates too:
    Nth 1 ((~) [c] g) = g
See Simplify.simplCoercionF, which generates such selections.
333

dreixel's avatar
dreixel committed
334 335 336 337 338 339 340 341 342 343 344 345 346
Note [Kind coercions]
~~~~~~~~~~~~~~~~~~~~~
Suppose T :: * -> *, and g :: A ~ B
Then the coercion
   TyConAppCo T [g]      T g : T A ~ T B

Now suppose S :: forall k. k -> *, and g :: A ~ B
Then the coercion
   TyConAppCo S [Refl *, g]   T <*> g : T * A ~ T * B

Notice that the arguments to TyConAppCo are coercions, but the first
represents a *kind* coercion. Now, we don't allow any non-trivial kind
coercions, so it's an invariant that any such kind coercions are Refl.
347
Lint checks this.
dreixel's avatar
dreixel committed
348 349 350

However it's inconvenient to insist that these kind coercions are always
*structurally* (Refl k), because the key function exprIsConApp_maybe
351
pushes coercions into constructor arguments, so
dreixel's avatar
dreixel committed
352 353 354 355 356
       C k ty e |> g
may turn into
       C (Nth 0 g) ....
Now (Nth 0 g) will optimise to Refl, but perhaps not instantly.

357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492
Note [Roles]
~~~~~~~~~~~~
Roles are a solution to the GeneralizedNewtypeDeriving problem, articulated
in Trac #1496. The full story is in docs/core-spec/core-spec.pdf. Also, see
http://ghc.haskell.org/trac/ghc/wiki/RolesImplementation

Here is one way to phrase the problem:

Given:
newtype Age = MkAge Int
type family F x
type instance F Age = Bool
type instance F Int = Char

This compiles down to:
axAge :: Age ~ Int
axF1 :: F Age ~ Bool
axF2 :: F Int ~ Char

Then, we can make:
(sym (axF1) ; F axAge ; axF2) :: Bool ~ Char

Yikes!

The solution is _roles_, as articulated in "Generative Type Abstraction and
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 [UnivCo]
~~~~~~~~~~~~~
The UnivCo ("universal coercion") serves two rather separate functions:
 - the implementation for unsafeCoerce#
 - placeholder for phantom parameters in a TyConAppCo

At Representational, it asserts that two (possibly unrelated)
types have the same representation and can be casted to one another.
This form is necessary for unsafeCoerce#.

For optimisation purposes, it is convenient to allow UnivCo to appear
at Nominal role. If we have

data Foo a = MkFoo (F a)   -- F is a type family

and we want an unsafe coercion from Foo Int to Foo Bool, then it would
be nice to have (TyConAppCo Foo (UnivCo Nominal Int Bool)). So, we allow
Nominal UnivCo's.

At Phantom role, it is used as an argument to TyConAppCo in the place
of a phantom parameter (a type parameter unused in the type definition).

For example:

data Q a = MkQ Int

We want a coercion for (Q Bool) ~R (Q Char).

(TyConAppCo Representational Q [UnivCo Phantom Bool Char]) does the trick.

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:

newtype Age = MkAge Int

Nominal: All arguments must have role Nominal. Why? So that Foo Age ~N Foo Int
does *not* hold.

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.

Phantom: All arguments must have role Phantom. This one isn't strictly
necessary for soundness, but this choice removes ambiguity.



The rules here also dictate what the parameters to mkTyConAppCo.

493 494
************************************************************************
*                                                                      *
495
\subsection{Coercion variables}
496 497 498
*                                                                      *
************************************************************************
-}
499 500 501 502 503 504 505 506 507 508 509 510 511 512

coVarName :: CoVar -> Name
coVarName = varName

setCoVarUnique :: CoVar -> Unique -> CoVar
setCoVarUnique = setVarUnique

setCoVarName :: CoVar -> Name -> CoVar
setCoVarName   = setVarName

isCoVar :: Var -> Bool
isCoVar v = isCoVarType (varType v)

isCoVarType :: Type -> Bool
513
isCoVarType ty      -- Tests for t1 ~# t2, the unboxed equality
514
  = case splitTyConApp_maybe ty of
515 516
      Just (tc,tys) -> (tc `hasKey` eqPrimTyConKey || tc `hasKey` eqReprPrimTyConKey)
                       && tys `lengthAtLeast` 2
517
      Nothing       -> False
518 519 520

tyCoVarsOfCo :: Coercion -> VarSet
-- Extracts type and coercion variables from a coercion
521 522 523 524 525
tyCoVarsOfCo (Refl _ ty)           = tyVarsOfType ty
tyCoVarsOfCo (TyConAppCo _ _ cos)  = tyCoVarsOfCos cos
tyCoVarsOfCo (AppCo co1 co2)       = tyCoVarsOfCo co1 `unionVarSet` tyCoVarsOfCo co2
tyCoVarsOfCo (ForAllCo tv co)      = tyCoVarsOfCo co `delVarSet` tv
tyCoVarsOfCo (CoVarCo v)           = unitVarSet v
526
tyCoVarsOfCo (AxiomInstCo _ _ cos) = tyCoVarsOfCos cos
527 528 529 530 531 532 533
tyCoVarsOfCo (UnivCo _ ty1 ty2)    = tyVarsOfType ty1 `unionVarSet` tyVarsOfType ty2
tyCoVarsOfCo (SymCo co)            = tyCoVarsOfCo co
tyCoVarsOfCo (TransCo co1 co2)     = tyCoVarsOfCo co1 `unionVarSet` tyCoVarsOfCo co2
tyCoVarsOfCo (NthCo _ co)          = tyCoVarsOfCo co
tyCoVarsOfCo (LRCo _ co)           = tyCoVarsOfCo co
tyCoVarsOfCo (InstCo co ty)        = tyCoVarsOfCo co `unionVarSet` tyVarsOfType ty
tyCoVarsOfCo (SubCo co)            = tyCoVarsOfCo co
534
tyCoVarsOfCo (AxiomRuleCo _ ts cs) = tyVarsOfTypes ts `unionVarSet` tyCoVarsOfCos cs
535 536

tyCoVarsOfCos :: [Coercion] -> VarSet
537
tyCoVarsOfCos = mapUnionVarSet tyCoVarsOfCo
538 539 540

coVarsOfCo :: Coercion -> VarSet
-- Extract *coerction* variables only.  Tiresome to repeat the code, but easy.
541 542 543 544 545
coVarsOfCo (Refl _ _)            = emptyVarSet
coVarsOfCo (TyConAppCo _ _ cos)  = coVarsOfCos cos
coVarsOfCo (AppCo co1 co2)       = coVarsOfCo co1 `unionVarSet` coVarsOfCo co2
coVarsOfCo (ForAllCo _ co)       = coVarsOfCo co
coVarsOfCo (CoVarCo v)           = unitVarSet v
546
coVarsOfCo (AxiomInstCo _ _ cos) = coVarsOfCos cos
547 548 549 550 551 552 553
coVarsOfCo (UnivCo _ _ _)        = emptyVarSet
coVarsOfCo (SymCo co)            = coVarsOfCo co
coVarsOfCo (TransCo co1 co2)     = coVarsOfCo co1 `unionVarSet` coVarsOfCo co2
coVarsOfCo (NthCo _ co)          = coVarsOfCo co
coVarsOfCo (LRCo _ co)           = coVarsOfCo co
coVarsOfCo (InstCo co _)         = coVarsOfCo co
coVarsOfCo (SubCo co)            = coVarsOfCo co
554
coVarsOfCo (AxiomRuleCo _ _ cos) = coVarsOfCos cos
555 556

coVarsOfCos :: [Coercion] -> VarSet
557
coVarsOfCos = mapUnionVarSet coVarsOfCo
558 559

coercionSize :: Coercion -> Int
560 561 562 563 564
coercionSize (Refl _ ty)           = typeSize ty
coercionSize (TyConAppCo _ _ cos)  = 1 + sum (map coercionSize cos)
coercionSize (AppCo co1 co2)       = coercionSize co1 + coercionSize co2
coercionSize (ForAllCo _ co)       = 1 + coercionSize co
coercionSize (CoVarCo _)           = 1
565
coercionSize (AxiomInstCo _ _ cos) = 1 + sum (map coercionSize cos)
566 567 568 569 570 571 572
coercionSize (UnivCo _ ty1 ty2)  = typeSize ty1 + typeSize ty2
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 ty)        = 1 + coercionSize co + typeSize ty
coercionSize (SubCo co)            = 1 + coercionSize co
573 574
coercionSize (AxiomRuleCo _ tys cos) = 1 + sum (map typeSize tys)
                                         + sum (map coercionSize cos)
575

576 577 578
{-
************************************************************************
*                                                                      *
Simon Peyton Jones's avatar
Simon Peyton Jones committed
579
                            Tidying coercions
580 581 582
*                                                                      *
************************************************************************
-}
Simon Peyton Jones's avatar
Simon Peyton Jones committed
583 584 585 586 587

tidyCo :: TidyEnv -> Coercion -> Coercion
tidyCo env@(_, subst) co
  = go co
  where
588 589 590 591 592 593 594 595 596 597
    go (Refl r ty)            = Refl r (tidyType env ty)
    go (TyConAppCo r tc cos)  = let args = map go cos
                                in args `seqList` TyConAppCo r tc args
    go (AppCo co1 co2)        = (AppCo $! go co1) $! go co2
    go (ForAllCo tv co)       = ForAllCo tvp $! (tidyCo envp co)
                                where
                                  (envp, tvp) = tidyTyVarBndr env tv
    go (CoVarCo cv)           = case lookupVarEnv subst cv of
                                  Nothing  -> CoVarCo cv
                                  Just cv' -> CoVarCo cv'
Simon Peyton Jones's avatar
Simon Peyton Jones committed
598
    go (AxiomInstCo con ind cos) = let args = tidyCos env cos
599 600 601 602 603 604 605 606
                                   in args `seqList` AxiomInstCo con ind args
    go (UnivCo r ty1 ty2)     = (UnivCo r $! tidyType env ty1) $! tidyType env ty2
    go (SymCo co)             = SymCo $! go co
    go (TransCo co1 co2)      = (TransCo $! go co1) $! go co2
    go (NthCo d co)           = NthCo d $! go co
    go (LRCo lr co)           = LRCo lr $! go co
    go (InstCo co ty)         = (InstCo $! go co) $! tidyType env ty
    go (SubCo co)             = SubCo $! go co
Simon Peyton Jones's avatar
Simon Peyton Jones committed
607

608 609 610 611 612 613
    go (AxiomRuleCo ax tys cos) = let tys1 = map (tidyType env) tys
                                      cos1 = tidyCos env cos
                                  in tys1 `seqList` cos1 `seqList`
                                     AxiomRuleCo ax tys1 cos1


Simon Peyton Jones's avatar
Simon Peyton Jones committed
614 615 616
tidyCos :: TidyEnv -> [Coercion] -> [Coercion]
tidyCos env = map (tidyCo env)

617 618 619
{-
************************************************************************
*                                                                      *
620
                   Pretty-printing coercions
621 622
*                                                                      *
************************************************************************
623

624 625 626 627 628
@pprCo@ is the standard @Coercion@ printer; the overloaded @ppr@
function is defined to use this.  @pprParendCo@ is the same, except it
puts parens around the type, except for the atomic cases.
@pprParendCo@ works just by setting the initial context precedence
very high.
629
-}
630

631 632 633 634 635 636 637
instance Outputable Coercion where
  ppr = pprCo

pprCo, pprParendCo :: Coercion -> SDoc
pprCo       co = ppr_co TopPrec   co
pprParendCo co = ppr_co TyConPrec co

638
ppr_co :: TyPrec -> Coercion -> SDoc
639
ppr_co _ (Refl r ty) = angleBrackets (ppr ty) <> ppr_role r
640

641
ppr_co p co@(TyConAppCo _ tc [_,_])
642
  | tc `hasKey` funTyConKey = ppr_fun_co p co
643

644 645 646 647 648
ppr_co _ (TyConAppCo r tc cos)  = pprTcApp TyConPrec ppr_co tc cos <> ppr_role r
ppr_co p (AppCo co1 co2)        = maybeParen p TyConPrec $
                                  pprCo co1 <+> ppr_co TyConPrec co2
ppr_co p co@(ForAllCo {})       = ppr_forall_co p co
ppr_co _ (CoVarCo cv)           = parenSymOcc (getOccName cv) (ppr cv)
649
ppr_co p (AxiomInstCo con index cos)
650 651
  = pprPrefixApp p (ppr (getName con) <> brackets (ppr index))
                   (map (ppr_co TyConPrec) cos)
652

653 654 655 656 657
ppr_co p co@(TransCo {}) = maybeParen p FunPrec $
                           case trans_co_list co [] of
                             [] -> panic "ppr_co"
                             (co:cos) -> sep ( ppr_co FunPrec co
                                             : [ char ';' <+> ppr_co FunPrec co | co <- cos])
658 659 660
ppr_co p (InstCo co ty) = maybeParen p TyConPrec $
                          pprParendCo co <> ptext (sLit "@") <> pprType ty

661
ppr_co p (UnivCo r ty1 ty2) = pprPrefixApp p (ptext (sLit "UnivCo") <+> ppr r)
662
                                           [pprParendType ty1, pprParendType ty2]
663
ppr_co p (SymCo co)         = pprPrefixApp p (ptext (sLit "Sym")) [pprParendCo co]
664 665
ppr_co p (NthCo n co)       = pprPrefixApp p (ptext (sLit "Nth:") <> int n) [pprParendCo co]
ppr_co p (LRCo sel co)      = pprPrefixApp p (ppr sel) [pprParendCo co]
666
ppr_co p (SubCo co)         = pprPrefixApp p (ptext (sLit "Sub")) [pprParendCo co]
667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684
ppr_co p (AxiomRuleCo co ts cs) = maybeParen p TopPrec $
                                  ppr_axiom_rule_co co ts cs

ppr_axiom_rule_co :: CoAxiomRule -> [Type] -> [Coercion] -> SDoc
ppr_axiom_rule_co co ts ps = ppr (coaxrName co) <> ppTs ts $$ nest 2 (ppPs ps)
  where
  ppTs []   = Outputable.empty
  ppTs [t]  = ptext (sLit "@") <> ppr_type TopPrec t
  ppTs ts   = ptext (sLit "@") <>
                parens (hsep $ punctuate comma $ map pprType ts)

  ppPs []   = Outputable.empty
  ppPs [p]  = pprParendCo p
  ppPs (p : ps) = ptext (sLit "(") <+> pprCo p $$
                  vcat [ ptext (sLit ",") <+> pprCo q | q <- ps ] $$
                  ptext (sLit ")")


685 686

ppr_role :: Role -> SDoc
687 688 689 690 691
ppr_role r = underscore <> pp_role
  where pp_role = case r of
                    Nominal          -> char 'N'
                    Representational -> char 'R'
                    Phantom          -> char 'P'
692

693 694 695 696
trans_co_list :: Coercion -> [Coercion] -> [Coercion]
trans_co_list (TransCo co1 co2) cos = trans_co_list co1 (trans_co_list co2 cos)
trans_co_list co                cos = co : cos

697 698 699
instance Outputable LeftOrRight where
  ppr CLeft    = ptext (sLit "Left")
  ppr CRight   = ptext (sLit "Right")
700

701
ppr_fun_co :: TyPrec -> Coercion -> SDoc
702 703
ppr_fun_co p co = pprArrowChain p (split co)
  where
704
    split :: Coercion -> [SDoc]
705
    split (TyConAppCo _ f [arg,res])
706 707 708 709
      | f `hasKey` funTyConKey
      = ppr_co FunPrec arg : split res
    split co = [ppr_co TopPrec co]

710
ppr_forall_co :: TyPrec -> Coercion -> SDoc
711 712
ppr_forall_co p ty
  = maybeParen p FunPrec $
713
    sep [pprForAll tvs, ppr_co TopPrec rho]
714 715 716 717 718
  where
    (tvs,  rho) = split1 [] ty
    split1 tvs (ForAllCo tv ty) = split1 (tv:tvs) ty
    split1 tvs ty               = (reverse tvs, ty)

719 720 721 722 723 724
pprCoAxiom :: CoAxiom br -> SDoc
pprCoAxiom ax@(CoAxiom { co_ax_tc = tc, co_ax_branches = branches })
  = hang (ptext (sLit "axiom") <+> ppr ax <+> dcolon)
       2 (vcat (map (pprCoAxBranch tc) $ fromBranchList branches))

pprCoAxBranch :: TyCon -> CoAxBranch -> SDoc
725 726
pprCoAxBranch fam_tc (CoAxBranch { cab_tvs = tvs
                                 , cab_lhs = lhs
Simon Peyton Jones's avatar
Simon Peyton Jones committed
727
                                 , cab_rhs = rhs })
728
  = hang (pprUserForAll tvs)
729
       2 (hang (pprTypeApp fam_tc lhs) 2 (equals <+> (ppr rhs)))
Simon Peyton Jones's avatar
Simon Peyton Jones committed
730 731 732 733 734 735 736 737 738 739

pprCoAxBranchHdr :: CoAxiom br -> BranchIndex -> SDoc
pprCoAxBranchHdr ax@(CoAxiom { co_ax_tc = fam_tc, co_ax_name = name }) index
  | CoAxBranch { cab_lhs = tys, cab_loc = loc } <- coAxiomNthBranch ax index
  = hang (pprTypeApp fam_tc tys)
       2 (ptext (sLit "-- Defined") <+> ppr_loc loc)
  where
        ppr_loc loc
          | isGoodSrcSpan loc
          = ptext (sLit "at") <+> ppr (srcSpanStart loc)
740

Simon Peyton Jones's avatar
Simon Peyton Jones committed
741 742 743
          | otherwise
          = ptext (sLit "in") <+>
              quotes (ppr (nameModule name))
744

745 746 747
{-
************************************************************************
*                                                                      *
748
        Functions over Kinds
749 750 751
*                                                                      *
************************************************************************
-}
batterseapower's avatar
batterseapower committed
752

753
-- | This breaks a 'Coercion' with type @T A B C ~ T D E F@ into
754
-- a list of 'Coercion's of kinds @A ~ D@, @B ~ E@ and @E ~ F@. Hence:
batterseapower's avatar
batterseapower committed
755
--
756
-- > decomposeCo 3 c = [nth 0 c, nth 1 c, nth 2 c]
757
decomposeCo :: Arity -> Coercion -> [Coercion]
758
decomposeCo arity co
759 760
  = [mkNthCo n co | n <- [0..(arity-1)] ]
           -- Remember, Nth is zero-indexed
761 762 763

-- | Attempts to obtain the type variable underlying a 'Coercion'
getCoVar_maybe :: Coercion -> Maybe CoVar
764
getCoVar_maybe (CoVarCo cv) = Just cv
765 766
getCoVar_maybe _            = Nothing

767
-- first result has role equal to input; second result is Nominal
768 769 770
splitAppCo_maybe :: Coercion -> Maybe (Coercion, Coercion)
-- ^ Attempt to take a coercion application apart.
splitAppCo_maybe (AppCo co1 co2) = Just (co1, co2)
771
splitAppCo_maybe (TyConAppCo r tc cos)
772
  | isDecomposableTyCon tc || cos `lengthExceeds` tyConArity tc
773
  , Just (cos', co') <- snocView cos
774
  , Just co'' <- setNominalRole_maybe co'
775
  = Just (mkTyConAppCo r tc cos', co'') -- Never create unsaturated type family apps!
776 777
       -- Use mkTyConAppCo to preserve the invariant
       --  that identity coercions are always represented by Refl
778 779
splitAppCo_maybe (Refl r ty)
  | Just (ty1, ty2) <- splitAppTy_maybe ty
780
  = Just (Refl r ty1, Refl Nominal ty2)
781 782 783 784 785
splitAppCo_maybe _ = Nothing

splitForAllCo_maybe :: Coercion -> Maybe (TyVar, Coercion)
splitForAllCo_maybe (ForAllCo tv co) = Just (tv, co)
splitForAllCo_maybe _                = Nothing
786 787 788 789

-------------------------------------------------------
-- and some coercion kind stuff

790
coVarKind :: CoVar -> (Type,Type)
791 792
coVarKind cv
 | Just (tc, [_kind,ty1,ty2]) <- splitTyConApp_maybe (varType cv)
793
 = ASSERT(tc `hasKey` eqPrimTyConKey || tc `hasKey` eqReprPrimTyConKey)
794 795 796
   (ty1,ty2)
 | otherwise = panic "coVarKind, non coercion variable"

797 798 799 800 801 802 803 804 805 806 807 808 809 810
coVarRole :: CoVar -> Role
coVarRole cv
  | tc `hasKey` eqPrimTyConKey
  = Nominal
  | tc `hasKey` eqReprPrimTyConKey
  = Representational
  | otherwise
  = pprPanic "coVarRole: unknown tycon" (ppr cv)

  where
    tc = case tyConAppTyCon_maybe (varType cv) of
           Just tc0 -> tc0
           Nothing  -> pprPanic "coVarRole: not tyconapp" (ppr cv)

811
-- | Makes a coercion type from two types: the types whose equality
812
-- is proven by the relevant 'Coercion'
813 814 815 816
mkCoercionType :: Role -> Type -> Type -> Type
mkCoercionType Nominal          = mkPrimEqPred
mkCoercionType Representational = mkReprPrimEqPred
mkCoercionType Phantom          = panic "mkCoercionType"
817

818
isReflCo :: Coercion -> Bool
819 820
isReflCo (Refl {})         = True
isReflCo _                 = False
821 822

isReflCo_maybe :: Coercion -> Maybe Type
823 824
isReflCo_maybe (Refl _ ty)       = Just ty
isReflCo_maybe _                 = Nothing
825

826 827 828
{-
************************************************************************
*                                                                      *
829
            Building coercions
830 831
*                                                                      *
************************************************************************
832

833 834 835 836 837 838 839 840 841
Note [Role twiddling functions]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

There are a plethora of functions for twiddling roles:

mkSubCo: Requires a nominal input coercion and always produces a
representational output. This is used when you (the programmer) are sure you
know exactly that role you have and what you want.

842
downgradeRole_maybe: This function takes both the input role and the output role
843 844 845 846 847 848 849 850
as parameters. (The *output* role comes first!) It can only *downgrade* a
role -- that is, change it from N to R or P, or from R to P. This one-way
behavior is why there is the "_maybe". If an upgrade is requested, this
function produces Nothing. This is used when you need to change the role of a
coercion, but you're not sure (as you're writing the code) of which roles are
involved.

This function could have been written using coercionRole to ascertain the role
851
of the input. But, that function is recursive, and the caller of downgradeRole_maybe
852 853
often knows the input role. So, this is more efficient.

854
downgradeRole: This is just like downgradeRole_maybe, but it panics if the conversion
855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877
isn't a downgrade.

setNominalRole_maybe: This is the only function that can *upgrade* a coercion. The result
(if it exists) is always Nominal. The input can be at any role. It works on a
"best effort" basis, as it should never be strictly necessary to upgrade a coercion
during compilation. It is currently only used within GHC in splitAppCo_maybe. In order
to be a proper inverse of mkAppCo, the second coercion that splitAppCo_maybe returns
must be nominal. But, it's conceivable that splitAppCo_maybe is operating over a
TyConAppCo that uses a representational coercion. Hence the need for setNominalRole_maybe.
splitAppCo_maybe, in turn, is used only within coercion optimization -- thus, it is
not absolutely critical that setNominalRole_maybe be complete.

Note that setNominalRole_maybe will never upgrade a phantom UnivCo. Phantom
UnivCos are perfectly type-safe, whereas representational and nominal ones are
not. Indeed, `unsafeCoerce` is implemented via a representational UnivCo.
(Nominal ones are no worse than representational ones, so this function *will*
change a UnivCo Representational to a UnivCo Nominal.)

Conal Elliott also came across a need for this function while working with the GHC
API, as he was decomposing Core casts. The Core casts use representational coercions,
as they must, but his use case required nominal coercions (he was building a GADT).
So, that's why this function is exported from this module.

878
One might ask: shouldn't downgradeRole_maybe just use setNominalRole_maybe as appropriate?
879 880
I (Richard E.) have decided not to do this, because upgrading a role is bizarre and
a caller should have to ask for this behavior explicitly.
881
-}
882

883
mkCoVarCo :: CoVar -> Coercion
884
-- cv :: s ~# t
885
mkCoVarCo cv
886
  | ty1 `eqType` ty2 = Refl (coVarRole cv) ty1
887 888 889
  | otherwise        = CoVarCo cv
  where
    (ty1, ty2) = ASSERT( isCoVar cv ) coVarKind cv
890

891
mkReflCo :: Role -> Type -> Coercion
892
mkReflCo = Refl
893

894
mkAxInstCo :: Role -> CoAxiom br -> BranchIndex -> [Type] -> Coercion
895
-- mkAxInstCo can legitimately be called over-staturated;
896
-- i.e. with more type arguments than the coercion requires
897
mkAxInstCo role ax index tys
898
  | arity == n_tys = downgradeRole role ax_role $ AxiomInstCo ax_br index rtys
899
  | otherwise      = ASSERT( arity < n_tys )
900
                     downgradeRole role ax_role $
901
                     foldl AppCo (AxiomInstCo ax_br index (take arity rtys))
902 903
                                 (drop arity rtys)
  where
904 905 906 907 908 909 910
    n_tys     = length tys
    ax_br     = toBranchedAxiom ax
    branch    = coAxiomNthBranch ax_br index
    arity     = length $ coAxBranchTyVars branch
    arg_roles = coAxBranchRoles branch
    rtys      = zipWith mkReflCo (arg_roles ++ repeat Nominal) tys
    ax_role   = coAxiomRole ax
911 912

-- to be used only with unbranched axioms
913 914 915
mkUnbranchedAxInstCo :: Role -> CoAxiom Unbranched -> [Type] -> Coercion
mkUnbranchedAxInstCo role ax tys
  = mkAxInstCo role ax 0 tys
916

917
mkAxInstLHS, mkAxInstRHS :: CoAxiom br -> BranchIndex -> [Type] -> Type
918 919
-- Instantiate the axiom with specified types,
-- returning the instantiated RHS
920
-- A companion to mkAxInstCo:
921
--    mkAxInstRhs ax index tys = snd (coercionKind (mkAxInstCo ax index tys))
922 923 924
mkAxInstLHS ax index tys
  | CoAxBranch { cab_tvs = tvs, cab_lhs = lhs } <- coAxiomNthBranch ax index
  , (tys1, tys2) <- splitAtList tvs tys
925
  = ASSERT( tvs `equalLength` tys1 )
926 927 928 929 930
    mkTyConApp (coAxiomTyCon ax) (substTysWith tvs tys1 lhs ++ tys2)

mkAxInstRHS ax index tys
  | CoAxBranch { cab_tvs = tvs, cab_rhs = rhs } <- coAxiomNthBranch ax index
  , (tys1, tys2) <- splitAtList tvs tys
931
  = ASSERT( tvs `equalLength` tys1 )
932
    mkAppTys (substTyWith tvs tys1 rhs) tys2
933 934 935

mkUnbranchedAxInstRHS :: CoAxiom Unbranched -> [Type] -> Type
mkUnbranchedAxInstRHS ax = mkAxInstRHS ax 0
936

937
-- | Apply a 'Coercion' to another 'Coercion'.
938 939
-- The second coercion must be Nominal, unless the first is Phantom.
-- If the first is Phantom, then the second can be either Phantom or Nominal.
940
mkAppCo :: Coercion -> Coercion -> Coercion
941 942 943 944 945 946 947 948 949 950
mkAppCo co1 co2 = mkAppCoFlexible co1 Nominal co2
-- Note, mkAppCo is careful to maintain invariants regarding
-- where Refl constructors appear; see the comments in the definition
-- of Coercion and the Note [Refl invariant] in types/TypeRep.lhs.

-- | Apply a 'Coercion' to another 'Coercion'.
-- The second 'Coercion's role is given, making this more flexible than
-- 'mkAppCo'.
mkAppCoFlexible :: Coercion -> Role -> Coercion -> Coercion
mkAppCoFlexible (Refl r ty1) _ (Refl _ ty2)
951
  = Refl r (mkAppTy ty1 ty2)
952 953 954
mkAppCoFlexible (Refl r ty1) r2 co2
  | Just (tc, tys) <- splitTyConApp_maybe ty1
    -- Expand type synonyms; a TyConAppCo can't have a type synonym (Trac #9102)
955 956
  = TyConAppCo r tc (zip_roles (tyConRolesX r tc) tys)
  where
957
    zip_roles (r1:_)  []        = [downgradeRole r1 r2 co2]
958 959
    zip_roles (r1:rs) (ty1:tys) = mkReflCo r1 ty1 : zip_roles rs tys
    zip_roles _       _         = panic "zip_roles" -- but the roles are infinite...
960
mkAppCoFlexible (TyConAppCo r tc cos) r2 co
961
  = case r of
962 963
      Nominal          -> ASSERT( r2 == Nominal )
                          TyConAppCo Nominal tc (cos ++ [co])
964 965
      Representational -> TyConAppCo Representational tc (cos ++ [co'])
        where new_role = (tyConRolesX Representational tc) !! (length cos)
966
              co'      = downgradeRole new_role r2 co
967 968
      Phantom          -> TyConAppCo Phantom tc (cos ++ [mkPhantomCo co])

969 970 971
mkAppCoFlexible co1 _r2 co2 = ASSERT( _r2 == Nominal )
                              AppCo co1 co2

batterseapower's avatar
batterseapower committed
972 973

-- | Applies multiple 'Coercion's to another 'Coercion', from left to right.
974
-- See also 'mkAppCo'.
975
mkAppCos :: Coercion -> [Coercion] -> Coercion
976
mkAppCos co1 cos = foldl mkAppCo co1 cos
977

978 979 980 981
-- | Apply a type constructor to a list of coercions. It is the
-- caller's responsibility to get the roles correct on argument coercions.
mkTyConAppCo :: Role -> TyCon -> [Coercion] -> Coercion
mkTyConAppCo r tc cos
982
               -- Expand type synonyms
983
  | Just (tv_co_prs, rhs_ty, leftover_cos) <- tcExpandTyCon_maybe tc cos
984
  = mkAppCos (liftCoSubst r tv_co_prs rhs_ty) leftover_cos
985

986
  | Just tys <- traverse isReflCo_maybe cos
987
  = Refl r (