{-# LANGUAGE LambdaCase #-} module TcTypeNats ( typeNatTyCons , typeNatCoAxiomRules , BuiltInSynFamily(..) -- If you define a new built-in type family, make sure to export its TyCon -- from here as well. -- See Note [Adding built-in type families] , typeNatAddTyCon , typeNatMulTyCon , typeNatExpTyCon , typeNatLeqTyCon , typeNatSubTyCon , typeNatDivTyCon , typeNatModTyCon , typeNatLogTyCon , typeNatCmpTyCon , typeSymbolCmpTyCon , typeSymbolAppendTyCon ) where import GhcPrelude import Type import Pair import TcType ( TcType, tcEqType ) import TyCon ( TyCon, FamTyConFlav(..), mkFamilyTyCon , Injectivity(..) ) import Coercion ( Role(..) ) import TcRnTypes ( Xi ) import CoAxiom ( CoAxiomRule(..), BuiltInSynFamily(..), TypeEqn ) import Name ( Name, BuiltInSyntax(..) ) import TysWiredIn import TysPrim ( mkTemplateAnonTyConBinders ) import PrelNames ( gHC_TYPELITS , gHC_TYPENATS , typeNatAddTyFamNameKey , typeNatMulTyFamNameKey , typeNatExpTyFamNameKey , typeNatLeqTyFamNameKey , typeNatSubTyFamNameKey , typeNatDivTyFamNameKey , typeNatModTyFamNameKey , typeNatLogTyFamNameKey , typeNatCmpTyFamNameKey , typeSymbolCmpTyFamNameKey , typeSymbolAppendFamNameKey ) import FastString ( FastString , fsLit, nilFS, nullFS, unpackFS, mkFastString, appendFS ) import qualified Data.Map as Map import Data.Maybe ( isJust ) import Control.Monad ( guard ) import Data.List ( isPrefixOf, isSuffixOf ) {- Note [Type-level literals] ~~~~~~~~~~~~~~~~~~~~~~~~~~ There are currently two forms of type-level literals: natural numbers, and symbols (even though this module is named TcTypeNats, it covers both). Type-level literals are supported by CoAxiomRules (conditional axioms), which power the built-in type families (see Note [Adding built-in type families]). Currently, all built-in type families are for the express purpose of supporting type-level literals. See also the Wiki page: https://ghc.haskell.org/trac/ghc/wiki/TypeNats Note [Adding built-in type families] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ There are a few steps to adding a built-in type family: * Adding a unique for the type family TyCon These go in PrelNames. It will likely be of the form @myTyFamNameKey = mkPreludeTyConUnique xyz@, where @xyz@ is a number that has not been chosen before in PrelNames. There are several examples already in PrelNames—see, for instance, typeNatAddTyFamNameKey. * Adding the type family TyCon itself This goes in TcTypeNats. There are plenty of examples of how to define these—see, for instance, typeNatAddTyCon. Once your TyCon has been defined, be sure to: - Export it from TcTypeNats. (Not doing so caused #14632.) - Include it in the typeNatTyCons list, defined in TcTypeNats. * Exposing associated type family axioms When defining the type family TyCon, you will need to define an axiom for the type family in general (see, for instance, axAddDef), and perhaps other auxiliary axioms for special cases of the type family (see, for instance, axAdd0L and axAdd0R). After you have defined all of these axioms, be sure to include them in the typeNatCoAxiomRules list, defined in TcTypeNats. (Not doing so caused #14934.) * Define the type family somewhere Finally, you will need to define the type family somewhere, likely in @base@. Currently, all of the built-in type families are defined in GHC.TypeLits or GHC.TypeNats, so those are likely candidates. Since the behavior of your built-in type family is specified in TcTypeNats, you should give an open type family definition with no instances, like so: type family MyTypeFam (m :: Nat) (n :: Nat) :: Nat Changing the argument and result kinds as appropriate. * Update the relevant test cases The GHC test suite will likely need to be updated after you add your built-in type family. For instance: - The T9181 test prints the :browse contents of GHC.TypeLits, so if you added a test there, the expected output of T9181 will need to change. - The TcTypeNatSimple and TcTypeSymbolSimple tests have compile-time unit tests, as well as TcTypeNatSimpleRun and TcTypeSymbolSimpleRun, which have runtime unit tests. Consider adding further unit tests to those if your built-in type family deals with Nats or Symbols, respectively. -} {------------------------------------------------------------------------------- Built-in type constructors for functions on type-level nats -} -- The list of built-in type family TyCons that GHC uses. -- If you define a built-in type family, make sure to add it to this list. -- See Note [Adding built-in type families] typeNatTyCons :: [TyCon] typeNatTyCons = [ typeNatAddTyCon , typeNatMulTyCon , typeNatExpTyCon , typeNatLeqTyCon , typeNatSubTyCon , typeNatDivTyCon , typeNatModTyCon , typeNatLogTyCon , typeNatCmpTyCon , typeSymbolCmpTyCon , typeSymbolAppendTyCon ] typeNatAddTyCon :: TyCon typeNatAddTyCon = mkTypeNatFunTyCon2 name BuiltInSynFamily { sfMatchFam = matchFamAdd , sfInteractTop = interactTopAdd , sfInteractInert = interactInertAdd } where name = mkWiredInTyConName UserSyntax gHC_TYPENATS (fsLit "+") typeNatAddTyFamNameKey typeNatAddTyCon typeNatSubTyCon :: TyCon typeNatSubTyCon = mkTypeNatFunTyCon2 name BuiltInSynFamily { sfMatchFam = matchFamSub , sfInteractTop = interactTopSub , sfInteractInert = interactInertSub } where name = mkWiredInTyConName UserSyntax gHC_TYPENATS (fsLit "-") typeNatSubTyFamNameKey typeNatSubTyCon typeNatMulTyCon :: TyCon typeNatMulTyCon = mkTypeNatFunTyCon2 name BuiltInSynFamily { sfMatchFam = matchFamMul , sfInteractTop = interactTopMul , sfInteractInert = interactInertMul } where name = mkWiredInTyConName UserSyntax gHC_TYPENATS (fsLit "*") typeNatMulTyFamNameKey typeNatMulTyCon typeNatDivTyCon :: TyCon typeNatDivTyCon = mkTypeNatFunTyCon2 name BuiltInSynFamily { sfMatchFam = matchFamDiv , sfInteractTop = interactTopDiv , sfInteractInert = interactInertDiv } where name = mkWiredInTyConName UserSyntax gHC_TYPENATS (fsLit "Div") typeNatDivTyFamNameKey typeNatDivTyCon typeNatModTyCon :: TyCon typeNatModTyCon = mkTypeNatFunTyCon2 name BuiltInSynFamily { sfMatchFam = matchFamMod , sfInteractTop = interactTopMod , sfInteractInert = interactInertMod } where name = mkWiredInTyConName UserSyntax gHC_TYPENATS (fsLit "Mod") typeNatModTyFamNameKey typeNatModTyCon typeNatExpTyCon :: TyCon typeNatExpTyCon = mkTypeNatFunTyCon2 name BuiltInSynFamily { sfMatchFam = matchFamExp , sfInteractTop = interactTopExp , sfInteractInert = interactInertExp } where name = mkWiredInTyConName UserSyntax gHC_TYPENATS (fsLit "^") typeNatExpTyFamNameKey typeNatExpTyCon typeNatLogTyCon :: TyCon typeNatLogTyCon = mkTypeNatFunTyCon1 name BuiltInSynFamily { sfMatchFam = matchFamLog , sfInteractTop = interactTopLog , sfInteractInert = interactInertLog } where name = mkWiredInTyConName UserSyntax gHC_TYPENATS (fsLit "Log2") typeNatLogTyFamNameKey typeNatLogTyCon typeNatLeqTyCon :: TyCon typeNatLeqTyCon = mkFamilyTyCon name (mkTemplateAnonTyConBinders [ typeNatKind, typeNatKind ]) boolTy Nothing (BuiltInSynFamTyCon ops) Nothing NotInjective where name = mkWiredInTyConName UserSyntax gHC_TYPENATS (fsLit "<=?") typeNatLeqTyFamNameKey typeNatLeqTyCon ops = BuiltInSynFamily { sfMatchFam = matchFamLeq , sfInteractTop = interactTopLeq , sfInteractInert = interactInertLeq } typeNatCmpTyCon :: TyCon typeNatCmpTyCon = mkFamilyTyCon name (mkTemplateAnonTyConBinders [ typeNatKind, typeNatKind ]) orderingKind Nothing (BuiltInSynFamTyCon ops) Nothing NotInjective where name = mkWiredInTyConName UserSyntax gHC_TYPENATS (fsLit "CmpNat") typeNatCmpTyFamNameKey typeNatCmpTyCon ops = BuiltInSynFamily { sfMatchFam = matchFamCmpNat , sfInteractTop = interactTopCmpNat , sfInteractInert = \_ _ _ _ -> [] } typeSymbolCmpTyCon :: TyCon typeSymbolCmpTyCon = mkFamilyTyCon name (mkTemplateAnonTyConBinders [ typeSymbolKind, typeSymbolKind ]) orderingKind Nothing (BuiltInSynFamTyCon ops) Nothing NotInjective where name = mkWiredInTyConName UserSyntax gHC_TYPELITS (fsLit "CmpSymbol") typeSymbolCmpTyFamNameKey typeSymbolCmpTyCon ops = BuiltInSynFamily { sfMatchFam = matchFamCmpSymbol , sfInteractTop = interactTopCmpSymbol , sfInteractInert = \_ _ _ _ -> [] } typeSymbolAppendTyCon :: TyCon typeSymbolAppendTyCon = mkTypeSymbolFunTyCon2 name BuiltInSynFamily { sfMatchFam = matchFamAppendSymbol , sfInteractTop = interactTopAppendSymbol , sfInteractInert = interactInertAppendSymbol } where name = mkWiredInTyConName UserSyntax gHC_TYPELITS (fsLit "AppendSymbol") typeSymbolAppendFamNameKey typeSymbolAppendTyCon -- Make a unary built-in constructor of kind: Nat -> Nat mkTypeNatFunTyCon1 :: Name -> BuiltInSynFamily -> TyCon mkTypeNatFunTyCon1 op tcb = mkFamilyTyCon op (mkTemplateAnonTyConBinders [ typeNatKind ]) typeNatKind Nothing (BuiltInSynFamTyCon tcb) Nothing NotInjective -- Make a binary built-in constructor of kind: Nat -> Nat -> Nat mkTypeNatFunTyCon2 :: Name -> BuiltInSynFamily -> TyCon mkTypeNatFunTyCon2 op tcb = mkFamilyTyCon op (mkTemplateAnonTyConBinders [ typeNatKind, typeNatKind ]) typeNatKind Nothing (BuiltInSynFamTyCon tcb) Nothing NotInjective -- Make a binary built-in constructor of kind: Symbol -> Symbol -> Symbol mkTypeSymbolFunTyCon2 :: Name -> BuiltInSynFamily -> TyCon mkTypeSymbolFunTyCon2 op tcb = mkFamilyTyCon op (mkTemplateAnonTyConBinders [ typeSymbolKind, typeSymbolKind ]) typeSymbolKind Nothing (BuiltInSynFamTyCon tcb) Nothing NotInjective {------------------------------------------------------------------------------- Built-in rules axioms -------------------------------------------------------------------------------} -- If you add additional rules, please remember to add them to -- `typeNatCoAxiomRules` also. -- See Note [Adding built-in type families] axAddDef , axMulDef , axExpDef , axLeqDef , axCmpNatDef , axCmpSymbolDef , axAppendSymbolDef , axAdd0L , axAdd0R , axMul0L , axMul0R , axMul1L , axMul1R , axExp1L , axExp0R , axExp1R , axLeqRefl , axCmpNatRefl , axCmpSymbolRefl , axLeq0L , axSubDef , axSub0R , axAppendSymbol0R , axAppendSymbol0L , axDivDef , axDiv1 , axModDef , axMod1 , axLogDef :: CoAxiomRule axAddDef = mkBinAxiom "AddDef" typeNatAddTyCon $ \x y -> Just $ num (x + y) axMulDef = mkBinAxiom "MulDef" typeNatMulTyCon $ \x y -> Just $ num (x * y) axExpDef = mkBinAxiom "ExpDef" typeNatExpTyCon $ \x y -> Just $ num (x ^ y) axLeqDef = mkBinAxiom "LeqDef" typeNatLeqTyCon $ \x y -> Just $ bool (x <= y) axCmpNatDef = mkBinAxiom "CmpNatDef" typeNatCmpTyCon $ \x y -> Just $ ordering (compare x y) axCmpSymbolDef = CoAxiomRule { coaxrName = fsLit "CmpSymbolDef" , coaxrAsmpRoles = [Nominal, Nominal] , coaxrRole = Nominal , coaxrProves = \cs -> do [Pair s1 s2, Pair t1 t2] <- return cs s2' <- isStrLitTy s2 t2' <- isStrLitTy t2 return (mkTyConApp typeSymbolCmpTyCon [s1,t1] === ordering (compare s2' t2')) } axAppendSymbolDef = CoAxiomRule { coaxrName = fsLit "AppendSymbolDef" , coaxrAsmpRoles = [Nominal, Nominal] , coaxrRole = Nominal , coaxrProves = \cs -> do [Pair s1 s2, Pair t1 t2] <- return cs s2' <- isStrLitTy s2 t2' <- isStrLitTy t2 let z = mkStrLitTy (appendFS s2' t2') return (mkTyConApp typeSymbolAppendTyCon [s1, t1] === z) } axSubDef = mkBinAxiom "SubDef" typeNatSubTyCon $ \x y -> fmap num (minus x y) axDivDef = mkBinAxiom "DivDef" typeNatDivTyCon $ \x y -> do guard (y /= 0) return (num (div x y)) axModDef = mkBinAxiom "ModDef" typeNatModTyCon $ \x y -> do guard (y /= 0) return (num (mod x y)) axLogDef = mkUnAxiom "LogDef" typeNatLogTyCon $ \x -> do (a,_) <- genLog x 2 return (num a) axAdd0L = mkAxiom1 "Add0L" $ \(Pair s t) -> (num 0 .+. s) === t axAdd0R = mkAxiom1 "Add0R" $ \(Pair s t) -> (s .+. num 0) === t axSub0R = mkAxiom1 "Sub0R" $ \(Pair s t) -> (s .-. num 0) === t axMul0L = mkAxiom1 "Mul0L" $ \(Pair s _) -> (num 0 .*. s) === num 0 axMul0R = mkAxiom1 "Mul0R" $ \(Pair s _) -> (s .*. num 0) === num 0 axMul1L = mkAxiom1 "Mul1L" $ \(Pair s t) -> (num 1 .*. s) === t axMul1R = mkAxiom1 "Mul1R" $ \(Pair s t) -> (s .*. num 1) === t axDiv1 = mkAxiom1 "Div1" $ \(Pair s t) -> (tDiv s (num 1) === t) axMod1 = mkAxiom1 "Mod1" $ \(Pair s _) -> (tMod s (num 1) === num 0) -- XXX: Shouldn't we check that _ is 0? axExp1L = mkAxiom1 "Exp1L" $ \(Pair s _) -> (num 1 .^. s) === num 1 axExp0R = mkAxiom1 "Exp0R" $ \(Pair s _) -> (s .^. num 0) === num 1 axExp1R = mkAxiom1 "Exp1R" $ \(Pair s t) -> (s .^. num 1) === t axLeqRefl = mkAxiom1 "LeqRefl" $ \(Pair s _) -> (s <== s) === bool True axCmpNatRefl = mkAxiom1 "CmpNatRefl" $ \(Pair s _) -> (cmpNat s s) === ordering EQ axCmpSymbolRefl = mkAxiom1 "CmpSymbolRefl" $ \(Pair s _) -> (cmpSymbol s s) === ordering EQ axLeq0L = mkAxiom1 "Leq0L" $ \(Pair s _) -> (num 0 <== s) === bool True axAppendSymbol0R = mkAxiom1 "Concat0R" $ \(Pair s t) -> (mkStrLitTy nilFS `appendSymbol` s) === t axAppendSymbol0L = mkAxiom1 "Concat0L" $ \(Pair s t) -> (s `appendSymbol` mkStrLitTy nilFS) === t -- The list of built-in type family axioms that GHC uses. -- If you define new axioms, make sure to include them in this list. -- See Note [Adding built-in type families] typeNatCoAxiomRules :: Map.Map FastString CoAxiomRule typeNatCoAxiomRules = Map.fromList $ map (\x -> (coaxrName x, x)) [ axAddDef , axMulDef , axExpDef , axLeqDef , axCmpNatDef , axCmpSymbolDef , axAppendSymbolDef , axAdd0L , axAdd0R , axMul0L , axMul0R , axMul1L , axMul1R , axExp1L , axExp0R , axExp1R , axLeqRefl , axCmpNatRefl , axCmpSymbolRefl , axLeq0L , axSubDef , axSub0R , axAppendSymbol0R , axAppendSymbol0L , axDivDef , axDiv1 , axModDef , axMod1 , axLogDef ] {------------------------------------------------------------------------------- Various utilities for making axioms and types -------------------------------------------------------------------------------} (.+.) :: Type -> Type -> Type s .+. t = mkTyConApp typeNatAddTyCon [s,t] (.-.) :: Type -> Type -> Type s .-. t = mkTyConApp typeNatSubTyCon [s,t] (.*.) :: Type -> Type -> Type s .*. t = mkTyConApp typeNatMulTyCon [s,t] tDiv :: Type -> Type -> Type tDiv s t = mkTyConApp typeNatDivTyCon [s,t] tMod :: Type -> Type -> Type tMod s t = mkTyConApp typeNatModTyCon [s,t] (.^.) :: Type -> Type -> Type s .^. t = mkTyConApp typeNatExpTyCon [s,t] (<==) :: Type -> Type -> Type s <== t = mkTyConApp typeNatLeqTyCon [s,t] cmpNat :: Type -> Type -> Type cmpNat s t = mkTyConApp typeNatCmpTyCon [s,t] cmpSymbol :: Type -> Type -> Type cmpSymbol s t = mkTyConApp typeSymbolCmpTyCon [s,t] appendSymbol :: Type -> Type -> Type appendSymbol s t = mkTyConApp typeSymbolAppendTyCon [s, t] (===) :: Type -> Type -> Pair Type x === y = Pair x y num :: Integer -> Type num = mkNumLitTy bool :: Bool -> Type bool b = if b then mkTyConApp promotedTrueDataCon [] else mkTyConApp promotedFalseDataCon [] isBoolLitTy :: Type -> Maybe Bool isBoolLitTy tc = do (tc,[]) <- splitTyConApp_maybe tc case () of _ | tc == promotedFalseDataCon -> return False | tc == promotedTrueDataCon -> return True | otherwise -> Nothing orderingKind :: Kind orderingKind = mkTyConApp orderingTyCon [] ordering :: Ordering -> Type ordering o = case o of LT -> mkTyConApp promotedLTDataCon [] EQ -> mkTyConApp promotedEQDataCon [] GT -> mkTyConApp promotedGTDataCon [] isOrderingLitTy :: Type -> Maybe Ordering isOrderingLitTy tc = do (tc1,[]) <- splitTyConApp_maybe tc case () of _ | tc1 == promotedLTDataCon -> return LT | tc1 == promotedEQDataCon -> return EQ | tc1 == promotedGTDataCon -> return GT | otherwise -> Nothing known :: (Integer -> Bool) -> TcType -> Bool known p x = case isNumLitTy x of Just a -> p a Nothing -> False mkUnAxiom :: String -> TyCon -> (Integer -> Maybe Type) -> CoAxiomRule mkUnAxiom str tc f = CoAxiomRule { coaxrName = fsLit str , coaxrAsmpRoles = [Nominal] , coaxrRole = Nominal , coaxrProves = \cs -> do [Pair s1 s2] <- return cs s2' <- isNumLitTy s2 z <- f s2' return (mkTyConApp tc [s1] === z) } -- For the definitional axioms mkBinAxiom :: String -> TyCon -> (Integer -> Integer -> Maybe Type) -> CoAxiomRule mkBinAxiom str tc f = CoAxiomRule { coaxrName = fsLit str , coaxrAsmpRoles = [Nominal, Nominal] , coaxrRole = Nominal , coaxrProves = \cs -> do [Pair s1 s2, Pair t1 t2] <- return cs s2' <- isNumLitTy s2 t2' <- isNumLitTy t2 z <- f s2' t2' return (mkTyConApp tc [s1,t1] === z) } mkAxiom1 :: String -> (TypeEqn -> TypeEqn) -> CoAxiomRule mkAxiom1 str f = CoAxiomRule { coaxrName = fsLit str , coaxrAsmpRoles = [Nominal] , coaxrRole = Nominal , coaxrProves = \case [eqn] -> Just (f eqn) _ -> Nothing } {------------------------------------------------------------------------------- Evaluation -------------------------------------------------------------------------------} matchFamAdd :: [Type] -> Maybe (CoAxiomRule, [Type], Type) matchFamAdd [s,t] | Just 0 <- mbX = Just (axAdd0L, [t], t) | Just 0 <- mbY = Just (axAdd0R, [s], s) | Just x <- mbX, Just y <- mbY = Just (axAddDef, [s,t], num (x + y)) where mbX = isNumLitTy s mbY = isNumLitTy t matchFamAdd _ = Nothing matchFamSub :: [Type] -> Maybe (CoAxiomRule, [Type], Type) matchFamSub [s,t] | Just 0 <- mbY = Just (axSub0R, [s], s) | Just x <- mbX, Just y <- mbY, Just z <- minus x y = Just (axSubDef, [s,t], num z) where mbX = isNumLitTy s mbY = isNumLitTy t matchFamSub _ = Nothing matchFamMul :: [Type] -> Maybe (CoAxiomRule, [Type], Type) matchFamMul [s,t] | Just 0 <- mbX = Just (axMul0L, [t], num 0) | Just 0 <- mbY = Just (axMul0R, [s], num 0) | Just 1 <- mbX = Just (axMul1L, [t], t) | Just 1 <- mbY = Just (axMul1R, [s], s) | Just x <- mbX, Just y <- mbY = Just (axMulDef, [s,t], num (x * y)) where mbX = isNumLitTy s mbY = isNumLitTy t matchFamMul _ = Nothing matchFamDiv :: [Type] -> Maybe (CoAxiomRule, [Type], Type) matchFamDiv [s,t] | Just 1 <- mbY = Just (axDiv1, [s], s) | Just x <- mbX, Just y <- mbY, y /= 0 = Just (axDivDef, [s,t], num (div x y)) where mbX = isNumLitTy s mbY = isNumLitTy t matchFamDiv _ = Nothing matchFamMod :: [Type] -> Maybe (CoAxiomRule, [Type], Type) matchFamMod [s,t] | Just 1 <- mbY = Just (axMod1, [s], num 0) | Just x <- mbX, Just y <- mbY, y /= 0 = Just (axModDef, [s,t], num (mod x y)) where mbX = isNumLitTy s mbY = isNumLitTy t matchFamMod _ = Nothing matchFamExp :: [Type] -> Maybe (CoAxiomRule, [Type], Type) matchFamExp [s,t] | Just 0 <- mbY = Just (axExp0R, [s], num 1) | Just 1 <- mbX = Just (axExp1L, [t], num 1) | Just 1 <- mbY = Just (axExp1R, [s], s) | Just x <- mbX, Just y <- mbY = Just (axExpDef, [s,t], num (x ^ y)) where mbX = isNumLitTy s mbY = isNumLitTy t matchFamExp _ = Nothing matchFamLog :: [Type] -> Maybe (CoAxiomRule, [Type], Type) matchFamLog [s] | Just x <- mbX, Just (n,_) <- genLog x 2 = Just (axLogDef, [s], num n) where mbX = isNumLitTy s matchFamLog _ = Nothing matchFamLeq :: [Type] -> Maybe (CoAxiomRule, [Type], Type) matchFamLeq [s,t] | Just 0 <- mbX = Just (axLeq0L, [t], bool True) | Just x <- mbX, Just y <- mbY = Just (axLeqDef, [s,t], bool (x <= y)) | tcEqType s t = Just (axLeqRefl, [s], bool True) where mbX = isNumLitTy s mbY = isNumLitTy t matchFamLeq _ = Nothing matchFamCmpNat :: [Type] -> Maybe (CoAxiomRule, [Type], Type) matchFamCmpNat [s,t] | Just x <- mbX, Just y <- mbY = Just (axCmpNatDef, [s,t], ordering (compare x y)) | tcEqType s t = Just (axCmpNatRefl, [s], ordering EQ) where mbX = isNumLitTy s mbY = isNumLitTy t matchFamCmpNat _ = Nothing matchFamCmpSymbol :: [Type] -> Maybe (CoAxiomRule, [Type], Type) matchFamCmpSymbol [s,t] | Just x <- mbX, Just y <- mbY = Just (axCmpSymbolDef, [s,t], ordering (compare x y)) | tcEqType s t = Just (axCmpSymbolRefl, [s], ordering EQ) where mbX = isStrLitTy s mbY = isStrLitTy t matchFamCmpSymbol _ = Nothing matchFamAppendSymbol :: [Type] -> Maybe (CoAxiomRule, [Type], Type) matchFamAppendSymbol [s,t] | Just x <- mbX, nullFS x = Just (axAppendSymbol0R, [t], t) | Just y <- mbY, nullFS y = Just (axAppendSymbol0L, [s], s) | Just x <- mbX, Just y <- mbY = Just (axAppendSymbolDef, [s,t], mkStrLitTy (appendFS x y)) where mbX = isStrLitTy s mbY = isStrLitTy t matchFamAppendSymbol _ = Nothing {------------------------------------------------------------------------------- Interact with axioms -------------------------------------------------------------------------------} interactTopAdd :: [Xi] -> Xi -> [Pair Type] interactTopAdd [s,t] r | Just 0 <- mbZ = [ s === num 0, t === num 0 ] -- (s + t ~ 0) => (s ~ 0, t ~ 0) | Just x <- mbX, Just z <- mbZ, Just y <- minus z x = [t === num y] -- (5 + t ~ 8) => (t ~ 3) | Just y <- mbY, Just z <- mbZ, Just x <- minus z y = [s === num x] -- (s + 5 ~ 8) => (s ~ 3) where mbX = isNumLitTy s mbY = isNumLitTy t mbZ = isNumLitTy r interactTopAdd _ _ = [] {- Note [Weakened interaction rule for subtraction] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ A simpler interaction here might be: `s - t ~ r` --> `t + r ~ s` This would enable us to reuse all the code for addition. Unfortunately, this works a little too well at the moment. Consider the following example: 0 - 5 ~ r --> 5 + r ~ 0 --> (5 = 0, r = 0) This (correctly) spots that the constraint cannot be solved. However, this may be a problem if the constraint did not need to be solved in the first place! Consider the following example: f :: Proxy (If (5 <=? 0) (0 - 5) (5 - 0)) -> Proxy 5 f = id Currently, GHC is strict while evaluating functions, so this does not work, because even though the `If` should evaluate to `5 - 0`, we also evaluate the "then" branch which generates the constraint `0 - 5 ~ r`, which fails. So, for the time being, we only add an improvement when the RHS is a constant, which happens to work OK for the moment, although clearly we need to do something more general. -} interactTopSub :: [Xi] -> Xi -> [Pair Type] interactTopSub [s,t] r | Just z <- mbZ = [ s === (num z .+. t) ] -- (s - t ~ 5) => (5 + t ~ s) where mbZ = isNumLitTy r interactTopSub _ _ = [] interactTopMul :: [Xi] -> Xi -> [Pair Type] interactTopMul [s,t] r | Just 1 <- mbZ = [ s === num 1, t === num 1 ] -- (s * t ~ 1) => (s ~ 1, t ~ 1) | Just x <- mbX, Just z <- mbZ, Just y <- divide z x = [t === num y] -- (3 * t ~ 15) => (t ~ 5) | Just y <- mbY, Just z <- mbZ, Just x <- divide z y = [s === num x] -- (s * 3 ~ 15) => (s ~ 5) where mbX = isNumLitTy s mbY = isNumLitTy t mbZ = isNumLitTy r interactTopMul _ _ = [] interactTopDiv :: [Xi] -> Xi -> [Pair Type] interactTopDiv _ _ = [] -- I can't think of anything... interactTopMod :: [Xi] -> Xi -> [Pair Type] interactTopMod _ _ = [] -- I can't think of anything... interactTopExp :: [Xi] -> Xi -> [Pair Type] interactTopExp [s,t] r | Just 0 <- mbZ = [ s === num 0 ] -- (s ^ t ~ 0) => (s ~ 0) | Just x <- mbX, Just z <- mbZ, Just y <- logExact z x = [t === num y] -- (2 ^ t ~ 8) => (t ~ 3) | Just y <- mbY, Just z <- mbZ, Just x <- rootExact z y = [s === num x] -- (s ^ 2 ~ 9) => (s ~ 3) where mbX = isNumLitTy s mbY = isNumLitTy t mbZ = isNumLitTy r interactTopExp _ _ = [] interactTopLog :: [Xi] -> Xi -> [Pair Type] interactTopLog _ _ = [] -- I can't think of anything... interactTopLeq :: [Xi] -> Xi -> [Pair Type] interactTopLeq [s,t] r | Just 0 <- mbY, Just True <- mbZ = [ s === num 0 ] -- (s <= 0) => (s ~ 0) where mbY = isNumLitTy t mbZ = isBoolLitTy r interactTopLeq _ _ = [] interactTopCmpNat :: [Xi] -> Xi -> [Pair Type] interactTopCmpNat [s,t] r | Just EQ <- isOrderingLitTy r = [ s === t ] interactTopCmpNat _ _ = [] interactTopCmpSymbol :: [Xi] -> Xi -> [Pair Type] interactTopCmpSymbol [s,t] r | Just EQ <- isOrderingLitTy r = [ s === t ] interactTopCmpSymbol _ _ = [] interactTopAppendSymbol :: [Xi] -> Xi -> [Pair Type] interactTopAppendSymbol [s,t] r -- (AppendSymbol a b ~ "") => (a ~ "", b ~ "") | Just z <- mbZ, nullFS z = [s === mkStrLitTy nilFS, t === mkStrLitTy nilFS ] -- (AppendSymbol "foo" b ~ "foobar") => (b ~ "bar") | Just x <- fmap unpackFS mbX, Just z <- fmap unpackFS mbZ, x `isPrefixOf` z = [ t === mkStrLitTy (mkFastString $ drop (length x) z) ] -- (AppendSymbol f "bar" ~ "foobar") => (f ~ "foo") | Just y <- fmap unpackFS mbY, Just z <- fmap unpackFS mbZ, y `isSuffixOf` z = [ t === mkStrLitTy (mkFastString $ take (length z - length y) z) ] where mbX = isStrLitTy s mbY = isStrLitTy t mbZ = isStrLitTy r interactTopAppendSymbol _ _ = [] {------------------------------------------------------------------------------- Interaction with inerts -------------------------------------------------------------------------------} interactInertAdd :: [Xi] -> Xi -> [Xi] -> Xi -> [Pair Type] interactInertAdd [x1,y1] z1 [x2,y2] z2 | sameZ && tcEqType x1 x2 = [ y1 === y2 ] | sameZ && tcEqType y1 y2 = [ x1 === x2 ] where sameZ = tcEqType z1 z2 interactInertAdd _ _ _ _ = [] interactInertSub :: [Xi] -> Xi -> [Xi] -> Xi -> [Pair Type] interactInertSub [x1,y1] z1 [x2,y2] z2 | sameZ && tcEqType x1 x2 = [ y1 === y2 ] | sameZ && tcEqType y1 y2 = [ x1 === x2 ] where sameZ = tcEqType z1 z2 interactInertSub _ _ _ _ = [] interactInertMul :: [Xi] -> Xi -> [Xi] -> Xi -> [Pair Type] interactInertMul [x1,y1] z1 [x2,y2] z2 | sameZ && known (/= 0) x1 && tcEqType x1 x2 = [ y1 === y2 ] | sameZ && known (/= 0) y1 && tcEqType y1 y2 = [ x1 === x2 ] where sameZ = tcEqType z1 z2 interactInertMul _ _ _ _ = [] interactInertDiv :: [Xi] -> Xi -> [Xi] -> Xi -> [Pair Type] interactInertDiv _ _ _ _ = [] interactInertMod :: [Xi] -> Xi -> [Xi] -> Xi -> [Pair Type] interactInertMod _ _ _ _ = [] interactInertExp :: [Xi] -> Xi -> [Xi] -> Xi -> [Pair Type] interactInertExp [x1,y1] z1 [x2,y2] z2 | sameZ && known (> 1) x1 && tcEqType x1 x2 = [ y1 === y2 ] | sameZ && known (> 0) y1 && tcEqType y1 y2 = [ x1 === x2 ] where sameZ = tcEqType z1 z2 interactInertExp _ _ _ _ = [] interactInertLog :: [Xi] -> Xi -> [Xi] -> Xi -> [Pair Type] interactInertLog _ _ _ _ = [] interactInertLeq :: [Xi] -> Xi -> [Xi] -> Xi -> [Pair Type] interactInertLeq [x1,y1] z1 [x2,y2] z2 | bothTrue && tcEqType x1 y2 && tcEqType y1 x2 = [ x1 === y1 ] | bothTrue && tcEqType y1 x2 = [ (x1 <== y2) === bool True ] | bothTrue && tcEqType y2 x1 = [ (x2 <== y1) === bool True ] where bothTrue = isJust $ do True <- isBoolLitTy z1 True <- isBoolLitTy z2 return () interactInertLeq _ _ _ _ = [] interactInertAppendSymbol :: [Xi] -> Xi -> [Xi] -> Xi -> [Pair Type] interactInertAppendSymbol [x1,y1] z1 [x2,y2] z2 | sameZ && tcEqType x1 x2 = [ y1 === y2 ] | sameZ && tcEqType y1 y2 = [ x1 === x2 ] where sameZ = tcEqType z1 z2 interactInertAppendSymbol _ _ _ _ = [] {- ----------------------------------------------------------------------------- These inverse functions are used for simplifying propositions using concrete natural numbers. ----------------------------------------------------------------------------- -} -- | Subtract two natural numbers. minus :: Integer -> Integer -> Maybe Integer minus x y = if x >= y then Just (x - y) else Nothing -- | Compute the exact logarithm of a natural number. -- The logarithm base is the second argument. logExact :: Integer -> Integer -> Maybe Integer logExact x y = do (z,True) <- genLog x y return z -- | Divide two natural numbers. divide :: Integer -> Integer -> Maybe Integer divide _ 0 = Nothing divide x y = case divMod x y of (a,0) -> Just a _ -> Nothing -- | Compute the exact root of a natural number. -- The second argument specifies which root we are computing. rootExact :: Integer -> Integer -> Maybe Integer rootExact x y = do (z,True) <- genRoot x y return z {- | Compute the n-th root of a natural number, rounded down to the closest natural number. The boolean indicates if the result is exact (i.e., True means no rounding was done, False means rounded down). The second argument specifies which root we are computing. -} genRoot :: Integer -> Integer -> Maybe (Integer, Bool) genRoot _ 0 = Nothing genRoot x0 1 = Just (x0, True) genRoot x0 root = Just (search 0 (x0+1)) where search from to = let x = from + div (to - from) 2 a = x ^ root in case compare a x0 of EQ -> (x, True) LT | x /= from -> search x to | otherwise -> (from, False) GT | x /= to -> search from x | otherwise -> (from, False) {- | Compute the logarithm of a number in the given base, rounded down to the closest integer. The boolean indicates if we the result is exact (i.e., True means no rounding happened, False means we rounded down). The logarithm base is the second argument. -} genLog :: Integer -> Integer -> Maybe (Integer, Bool) genLog x 0 = if x == 1 then Just (0, True) else Nothing genLog _ 1 = Nothing genLog 0 _ = Nothing genLog x base = Just (exactLoop 0 x) where exactLoop s i | i == 1 = (s,True) | i < base = (s,False) | otherwise = let s1 = s + 1 in s1 `seq` case divMod i base of (j,r) | r == 0 -> exactLoop s1 j | otherwise -> (underLoop s1 j, False) underLoop s i | i < base = s | otherwise = let s1 = s + 1 in s1 `seq` underLoop s1 (div i base)