{-# LANGUAGE LambdaCase #-}

module TcTypeNats
  ( typeNatTyCons
  , typeNatCoAxiomRules
  , BuiltInSynFamily(..)

  , typeNatAddTyCon
  , typeNatMulTyCon
  , typeNatExpTyCon
  , typeNatLeqTyCon
  , typeNatSubTyCon
  , typeNatCmpTyCon
  , typeSymbolCmpTyCon
  , typeSymbolAppendTyCon
  ) where

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
                  , typeNatCmpTyFamNameKey
                  , typeSymbolCmpTyFamNameKey
                  , typeSymbolAppendFamNameKey
                  )
import FastString ( FastString
                  , fsLit, nilFS, nullFS, unpackFS, mkFastString, appendFS
                  )
import qualified Data.Map as Map
import Data.Maybe ( isJust )
import Data.List  ( isPrefixOf, isSuffixOf )

{-------------------------------------------------------------------------------
Built-in type constructors for functions on type-level nats
-}

typeNatTyCons :: [TyCon]
typeNatTyCons =
  [ typeNatAddTyCon
  , typeNatMulTyCon
  , typeNatExpTyCon
  , typeNatLeqTyCon
  , typeNatSubTyCon
  , 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

typeNatExpTyCon :: TyCon
typeNatExpTyCon = mkTypeNatFunTyCon2 name
  BuiltInSynFamily
    { sfMatchFam      = matchFamExp
    , sfInteractTop   = interactTopExp
    , sfInteractInert = interactInertExp
    }
  where
  name = mkWiredInTyConName UserSyntax gHC_TYPENATS (fsLit "^")
                typeNatExpTyFamNameKey typeNatExpTyCon

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 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.
axAddDef
  , axMulDef
  , axExpDef
  , axLeqDef
  , axCmpNatDef
  , axCmpSymbolDef
  , axAppendSymbolDef
  , axAdd0L
  , axAdd0R
  , axMul0L
  , axMul0R
  , axMul1L
  , axMul1R
  , axExp1L
  , axExp0R
  , axExp1R
  , axLeqRefl
  , axCmpNatRefl
  , axCmpSymbolRefl
  , axLeq0L
  , axSubDef
  , axSub0R
  , axAppendSymbol0R
  , axAppendSymbol0L
  :: 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)

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
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

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
  , axAppendSymbol0R
  , axAppendSymbol0L
  ]



{-------------------------------------------------------------------------------
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]

(.^.) :: 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




-- 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

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

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 _ _ = []

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 _ _ = []

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 _ _ _ _ = []

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 _ _ _ _ = []


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 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)
