-- |
-- Module      : Conjure.Conjurable
-- Copyright   : (c) 2021-2024 Rudy Matela
-- License     : 3-Clause BSD  (see the file LICENSE)
-- Maintainer  : Rudy Matela <rudy@matela.com.br>
--
-- This module is part of "Conjure".
--
-- This defines the 'Conjurable' typeclass
-- and utilities involving it.
--
-- You are probably better off importing "Conjure".
module Conjure.Conjurable
  ( Reification1
  , Reification
  , Conjurable (..)
  , conjureType
  , reifyTiers
  , reifyEquality
  , reifyExpress
  , conjureApplication
  , conjureVarApplication
  , conjurePats
  , conjureHoles
  , conjureTiersFor
  , conjureListFor
  , conjureSizeFor
  , conjureGrounds
  , conjureAreEqual
  , conjureMkEquation
  , A, B, C, D, E, F
  , conjureIsUnbreakable
  , conjureReification
  , conjureReification1
  , conjureDynamicEq
  , conjureIsNumeric
  , cevaluate
  , ceval
  , cevl
  , Name (..)
  , Express (..)
  , conjureArgumentPats
  , conjureMostGeneralCanonicalVariation
  )
where

import Test.LeanCheck
import Test.LeanCheck.Utils
import Test.LeanCheck.Error (errorToFalse)
import Conjure.Expr hiding (application)
import Conjure.Defn
import Data.Functor ((<$>))
import Control.Applicative ((<*>))
import Data.Dynamic
import Data.Express

import Data.Int     -- for instances
import Data.Word    -- for instances
import Data.Ratio   -- for instance
import Data.Complex -- for instance


-- | Single reification of some functions over a type as 'Expr's.
--
-- This is a sixtuple, in order:
--
-- 1. a hole encoded as an 'Expr';
-- 2. the '==' function encoded as an 'Expr' when available;
-- 3. 'tiers' of enumerated test values encoded as 'Expr's when available;
-- 4. infinite list of potential variable names;
-- 5. boolean indicating whether the type is atomic;
-- 6. the 'conjureSize' function encoded as an 'Expr'.
type Reification1  =  (Expr, Maybe Expr, Maybe [[Expr]], [String], Bool, Expr)

-- | A reification over a collection of types.
--
-- Represented as a transformation of a list to a list.
type Reification  =  [Reification1] -> [Reification1]


-- | A primtive expression (paired with instance reification).
type Prim  =  (Expr, Reification)


-- | Provides a primitive value to Conjure.
--   To be used on 'Show' instances.
--   (cf. 'prim')
pr :: (Conjurable a, Show a) => a -> Prim
pr x  =  (val x, conjureType x)


-- | Provides a primitive value to Conjure.
--   To be used on values that are not 'Show' instances
--   such as functions.
--   (cf. 'pr')
prim :: Conjurable a => String -> a -> Prim
prim s x  =  (value s x, conjureType x)


-- | Class of 'Conjurable' types.
-- Functions are 'Conjurable'
-- if all their arguments are 'Conjurable', 'Listable' and 'Show'able.
--
-- For atomic types that are 'Listable',
-- instances are defined as:
--
-- > instance Conjurable Atomic where
-- >   conjureTiers  =  reifyTiers
--
-- For atomic types that are both 'Listable' and 'Eq',
-- instances are defined as:
--
-- > instance Conjurable Atomic where
-- >   conjureTiers     =  reifyTiers
-- >   conjureEquality  =  reifyEquality
--
-- For types with subtypes,
-- instances are defined as:
--
-- > instance Conjurable Composite where
-- >   conjureTiers     =  reifyTiers
-- >   conjureEquality  =  reifyEquality
-- >   conjureSubTypes x  =  conjureType y
-- >                      .  conjureType z
-- >                      .  conjureType w
-- >     where
-- >     (Composite ... y ... z ... w ...)  =  x
--
-- Above @x@, @y@, @z@ and @w@ are just proxies.
-- The @Proxy@ type was avoided for backwards compatibility.
--
-- Please see the source code of "Conjure.Conjurable" for more examples.
--
-- (cf. 'reifyTiers', 'reifyEquality', 'conjureType')
class (Typeable a, Name a) => Conjurable a where
  conjureArgumentHoles :: a -> [Expr]
  conjureArgumentHoles _  =  []

  -- | Returns 'Just' the '==' function encoded as an 'Expr' when available
  --   or 'Nothing' otherwise.
  --
  -- Use 'reifyEquality' when defining this.
  conjureEquality :: a -> Maybe Expr
  conjureEquality _  =  Nothing

  -- | Returns 'Just' 'tiers' of values encoded as 'Expr's when possible
  --   or 'Nothing' otherwise.
  --
  -- Use 'reifyTiers' when defining this.
  conjureTiers :: a -> Maybe [[Expr]]
  conjureTiers _  =  Nothing

  conjureSubTypes :: a -> Reification
  conjureSubTypes _  =  id

  -- | Returns an if-function encoded as an 'Expr'.
  conjureIf :: a -> Expr
  conjureIf   =  ifFor

  -- | Returns a top-level case breakdown.
  conjureCases :: a -> [Expr]
  conjureCases _  =  []

  conjureArgumentCases :: a -> [[Expr]]
  conjureArgumentCases _  =  []

  -- | Returns the (recursive) size of the given value.
  conjureSize :: a -> Int
  conjureSize _  =  0

  -- | Returns a function that deeply reencodes an expression when possible.
  --   ('id' when not available.)
  --
  -- Use 'reifyExpress' when defining this.
  conjureExpress :: a -> Expr -> Expr

  conjureEvaluate :: (Expr->Expr) -> Int -> Defn -> Expr -> Maybe a
  conjureEvaluate  =  devaluate


-- | To be used in the implementation of 'conjureSubTypes'.
--
-- > instance ... => Conjurable <Type> where
-- >   ...
-- >   conjureSubTypes x  =  conjureType (field1 x)
-- >                      .  conjureType (field2 x)
-- >                      .  ...
-- >                      .  conjureType (fieldN x)
-- >   ...
conjureType :: Conjurable a => a -> Reification
conjureType x ms  =
  if hole x `elem` [h | (h,_,_,_,_,_) <- ms]
  then ms
  else conjureSubTypes x $ conjureReification1 x : ms

-- | like 'conjureType' but without type repetitions
nubConjureType :: Conjurable a => a -> Reification
nubConjureType x  =  nubOn (\(eh,_,_,_,_,_) -> eh) . conjureType x
-- The use of nubOn above is O(n^2).
-- So long as there is not a huge number of subtypes of a, so we're fine.

-- | Conjures a 'Reification1' for a 'Conjurable' type.
--
-- This is used in the implementation of 'conjureReification'.
conjureReification1 :: Conjurable a => a -> Reification1
conjureReification1 x  =  (hole x, conjureEquality x, conjureTiers x, names x, null $ conjureCases x, value "conjureSize" (conjureSize -:> x))

-- | Conjures a list of 'Reification1'
--   for a 'Conjurable' type, its subtypes and 'Bool'.
--
-- This is used in the implementation of
-- 'conjureHoles',
-- 'conjureMkEquation',
-- 'conjureAreEqual',
-- 'conjureTiersFor',
-- 'conjureNamesFor',
-- 'conjureIsUnbreakable',
-- etc.
conjureReification :: Conjurable a => a -> [Reification1]
conjureReification x  =  nubConjureType x [conjureReification1 bool]
  where
  bool :: Bool
  bool  =  error "conjureReification: evaluated proxy boolean value (definitely a bug)"

-- | Reifies equality '==' in a 'Conjurable' type instance.
--
-- This is to be used
-- in the definition of 'conjureEquality'
-- of 'Conjurable' typeclass instances:
--
-- > instance ... => Conjurable <Type> where
-- >   ...
-- >   conjureEquality  =  reifyEquality
-- >   ...
reifyEquality :: (Eq a, Typeable a) => a -> Maybe Expr
reifyEquality  =  Just . head . reifyEq

-- | Reifies equality to be used in a conjurable type.
--
-- This is to be used
-- in the definition of 'conjureTiers'
-- of 'Conjurable' typeclass instances:
--
-- > instance ... => Conjurable <Type> where
-- >   ...
-- >   conjureTiers  =  reifyTiers
-- >   ...
reifyTiers :: (Listable a, Show a, Typeable a) => a -> Maybe [[Expr]]
reifyTiers  =  Just . mkExprTiers

-- | Reifies the 'expr' function in a 'Conjurable' type instance.
--
-- This is to be used
-- in the definition of 'conjureExpress'
-- of 'Conjurable' typeclass instances.
--
-- > instance ... => Conjurable <Type> where
-- >   ...
-- >   conjureExpress  =  reifyExpress
-- >   ...
reifyExpress :: (Express a, Show a) => a -> Expr -> Expr
reifyExpress a e  =  case exprE $$ e of
  Nothing -> e -- identity needed for types such as functions
  Just e' -> eval (error $ "reifyExpress: cannot eval " ++ show e') e'
  where
  exprE  =  value "expr" (expr -:> a)

mkExprTiers :: (Listable a, Show a, Typeable a) => a -> [[Expr]]
mkExprTiers a  =  mapT val (tiers -: [[a]])

-- | Computes a list of holes encoded as 'Expr's
--   from a 'Conjurable' functional value.
--
-- (cf. 'Conjure.Prim.cjHoles')
conjureHoles :: Conjurable f => f -> [Expr]
conjureHoles f  =  [eh | (eh,_,Just _,_,_,_) <- conjureReification f]

-- | Computes a function that makes an equation between two expressions.
conjureMkEquation :: Conjurable f => f -> Expr -> Expr -> Expr
conjureMkEquation f  =  mkEquation [eq | (_,Just eq,_,_,_,_) <- conjureReification f]

conjureDynamicEq :: Conjurable f => f -> Dynamic
conjureDynamicEq f  =  case conjureMkEquation f efxs efxs of
                       (Value "==" deq :$ _ :$ _) -> deq
                       _ -> error "conjureDynamicEq: expected an == but found something else.  Bug!"
  where
  efxs  =  conjureApplication "f" f

-- | Given a 'Conjurable' functional value,
--   computes a function that checks whether two 'Expr's are equal
--   up to a given number of tests.
conjureAreEqual :: Conjurable f => f -> Int -> Expr -> Expr -> Bool
conjureAreEqual f maxTests  =  (===)
  where
  (-==-)  =  conjureMkEquation f
  e1 === e2  =  isTrue $ e1 -==- e2
  isTrue  =  all (errorToFalse . eval False) . gs
  gs  =  take maxTests . conjureGrounds f

-- | Compute 'tiers' of values encoded as 'Expr's
--   of the type of the given 'Expr'.
conjureTiersFor :: Conjurable f => f -> Expr -> [[Expr]]
conjureTiersFor f e  =  tf allTiers
  where
  allTiers :: [ [[Expr]] ]
  allTiers  =  [etiers | (_,_,Just etiers,_,_,_) <- conjureReification f]
  tf []  =  [[e]] -- no tiers found, keep variable
  tf (etiers:etc)  =  case etiers of
                      ((e':_):_) | typ e' == typ e -> etiers
                      _                            -> tf etc

conjureGrounds :: Conjurable f => f -> Expr -> [Expr]
conjureGrounds  =  grounds . conjureTiersFor

-- | Compure a 'list' of values encoded as 'Expr's
--   of the type of the given 'Expr'.
conjureListFor :: Conjurable f => f -> Expr -> [Expr]
conjureListFor f  =  concat . conjureTiersFor f

conjureIsNumeric :: Conjurable f => f -> Expr -> Bool
conjureIsNumeric f e  =  case conjureListFor f e of
                         -- We assume tiers of numeric values start with 0
                         -- not so unfair...
                         (Value "0" _):_ -> True
                         _ -> False

-- | Compute variable names for the given 'Expr' type.
conjureNamesFor :: Conjurable f => f -> Expr -> [String]
conjureNamesFor f e  =  head
                     $  [ns | (eh, _, _, ns, _, _) <- conjureReification f, typ e == typ eh]
                     ++ [names (undefined :: Int)] -- use [Int] on lists

conjureMostGeneralCanonicalVariation :: Conjurable f => f -> Expr -> Expr
conjureMostGeneralCanonicalVariation f  =  canonicalizeWith (conjureNamesFor f)
                                        .  fastMostGeneralVariation


-- | Conjures an 'Expr'-encoded size function for the given expression type.
--
-- > > conjureSizeFor (undefined :: [Int] -> [Bool]) i_
-- > conjureSize :: Int -> Int
--
-- > > conjureSizeFor (undefined :: [Int] -> [Bool]) is_
-- > conjureSize :: [Int] -> Int
--
-- > > conjureSizeFor (undefined :: [Int] -> [Bool]) bs_
-- > conjureSize :: [Bool] -> Int
conjureSizeFor :: Conjurable f => f -> Expr -> Expr
conjureSizeFor f eh  =
  case [esz | (_,_,_,_,_,esz) <- conjureReification f, isWellTyped (esz :$ eh)] of
  (esz:_) -> esz
  _ -> error $ "Conjure.conjureSizeFor: could not find size for " ++ show eh

-- | Checks if an 'Expr' is of an unbreakable type.
conjureIsUnbreakable :: Conjurable f => f -> Expr -> Bool
conjureIsUnbreakable f e  =  head
  [is | (h,_,_,_,is,_) <- conjureReification f, typ h == typ e]

instance Conjurable () where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureCases _   =  [val ()]

instance Conjurable Bool where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureCases _   =  [val False, val True]

instance Conjurable Int where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  size            where  size x | x < 0      =  0
                                                    | otherwise  =  x

-- allows easy modification of the global size function for integer values
-- duplicated above in the Int instance for performance reasons
integralSize :: Integral a => a -> Int
integralSize  =  fromIntegral . size  where  size x | x < 0      =  0
                                                    | otherwise  =  x

instance Conjurable Integer where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable Char where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers

-- bind equality to the given argument type
(==:) :: (a -> a -> Bool) -> a -> (a -> a -> Bool)
(==:)  =  const

-- the reconstruction of equality functions for polymorphic types
-- such as [a], (a,b), Maybe a, Either a b
-- is only needed so we don't impose an Eq restriction on the type context.

instance (Conjurable a, Listable a, Express a, Show a) => Conjurable [a] where
  conjureExpress   =  reifyExpress
  conjureSubTypes xs  =  conjureType (head xs)
  conjureTiers     =  reifyTiers
  conjureSize      =  length
  conjureCases xs  =  [ val ([] -: xs)
                      , value ":" ((:) ->>: xs) :$ hole x :$ hole xs
                      ]  where  x  =  head xs
  conjureEquality xs  =  from <$> conjureEquality x
    where
    x  =  head xs
    from e  =  value "==" (==)
      where
      (.==.)  =  evl e ==: x
      []     == []     = True
      (x:xs) == []     = False
      []     == (y:ys) = False
      (x:xs) == (y:ys) = x .==. y && xs == ys

instance ( Conjurable a, Listable a, Show a, Express a
         , Conjurable b, Listable b, Show b, Express b
         ) => Conjurable (a,b) where
  conjureExpress   =  reifyExpress
  conjureTiers     =  reifyTiers
  conjureSubTypes xy  =  conjureType (fst xy)
                      .  conjureType (snd xy)
  conjureCases xy  =  [value "," ((,) ->>: xy) :$ hole x :$ hole y]
    where
    (x,y) = (undefined,undefined) -: xy
  conjureEquality xy  =  from <$> conjureEquality x <*> conjureEquality y
    where
    (x,y)  =  xy
    from e1 e2  =  value "==" (==)
      where
      (==.)  =  evl e1 ==: x
      (.==)  =  evl e2 ==: y
      (x1,y1) == (x2,y2)  =  x1 ==. x2 && y1 .== y2


instance ( Conjurable a, Listable a, Show a, Express a
         , Conjurable b, Listable b, Show b, Express b
         , Conjurable c, Listable c, Show c, Express c
         ) => Conjurable (a,b,c) where
  conjureExpress   =  reifyExpress
  conjureTiers     =  reifyTiers
  conjureSubTypes xyz =  conjureType x
                      .  conjureType y
                      .  conjureType z
                      where (x,y,z) = xyz
  conjureCases xyz  =  [value ",," ((,,) ->>>: xyz) :$ hole x :$ hole y :$ hole z]
    where
    (x,y,z) = (undefined,undefined,undefined) -: xyz
  conjureEquality xyz  =  from
                      <$> conjureEquality x
                      <*> conjureEquality y
                      <*> conjureEquality z
    where
    (x,y,z)  =  xyz
    from e1 e2 e3  =  value "==" (==)
      where
      (==..)  =  evl e1 ==: x
      (.==.)  =  evl e2 ==: y
      (..==)  =  evl e3 ==: z
      (x1,y1,z1) == (x2,y2,z2)  =  x1 ==.. x2
                                && y1 .==. y2
                                && z1 ..== z2

instance (Conjurable a, Listable a, Show a, Express a) => Conjurable (Maybe a) where
  conjureExpress   =  reifyExpress
  conjureTiers     =  reifyTiers
  conjureSubTypes mx  =  conjureType (fromJust mx)
  conjureCases mx  =  [ value "Nothing" (Nothing -: mx)
                      , value "Just" (Just ->: mx) :$ hole x
                      ]
    where
    Just x  =  undefined -: mx
  conjureEquality mx  =  from <$> conjureEquality x
    where
    x  =  fromJust mx
    from e  =  value "==" (==)
      where
      (.==.)  =  evl e ==: x
      Nothing  == Nothing   =  True
      Nothing  == (Just _)  =  False
      (Just _) == Nothing   =  False
      (Just x) == (Just y)  =  x .==. y


instance ( Conjurable a, Listable a, Show a, Express a
         , Conjurable b, Listable b, Show b, Express b
         ) => Conjurable (Either a b) where
  conjureExpress   =  reifyExpress
  conjureTiers     =  reifyTiers
  conjureSubTypes elr  =  conjureType l . conjureType r
    where
    Left l   =  elr
    Right r  =  elr
  conjureCases exy  =  [ value "Left" (Left ->: exy) :$ hole x
                       , value "Right" (Right ->: exy) :$ hole y
                       ]
    where
    Left x  =  Left undefined -: exy
    Right y  =  Right undefined -: exy
  conjureEquality elr  =  from <$> conjureEquality l <*> conjureEquality r
    where
    Left l   =  elr
    Right r  =  elr
    from el er  =  value "==" (==)
      where
      (==.)  =  evl el ==: l
      (.==)  =  evl er ==: r
      (Left x)  == (Left y)   =  x ==. y
      (Left _)  == (Right _)  =  False
      (Right _) == (Left _)   =  False
      (Right x) == (Right y)  =  x .== y

instance (Conjurable a, Conjurable b) => Conjurable (a -> b) where
  conjureArgumentHoles f  =  hole (argTy f) : conjureArgumentHoles (f undefined)
  conjureSubTypes f  =  conjureType (argTy f) . conjureType (resTy f)
  conjureIf f  =  conjureIf (f undefined)
  conjureArgumentCases f  =  conjureCases (argTy f) : conjureArgumentCases (f undefined)
  conjureExpress f e
    | typ e == typeOf (argTy f)  =  conjureExpress (argTy f) e
    | otherwise                  =  conjureExpress (f undefined) e
  conjureEvaluate exprExpr mx defn ef  =  mf
    where
    ce  =  conjureEvaluate exprExpr mx defn
    mf  =  case ce (holeAsTypeOf ef :$ hole x) -: Just (f x) of
           Nothing -> Nothing
           Just _  -> Just $ \x -> fromMaybe err . ce $ ef :$ exprExpr (value "" x)
    f  =  undefined -: fromJust mf
    x  =  argTy f
    err  =  error "conjureEvaluate (a->b): BUG!  This should never be evaluated as it is protected by the outer case."

argTy :: (a -> b) -> a
argTy _  =  undefined

resTy :: (a -> b) -> b
resTy _  =  undefined

-- | Evaluates a 'Defn' into a regular Haskell value
--   returning 'Nothing' when there's a type mismatch.
--
-- The integer argument indicates the limit of recursive evaluations.
cevaluate :: Conjurable f => Int -> Defn -> Maybe f
cevaluate mx defn  =  mr
  where
  mr  =  conjureEvaluate exprExpr mx defn ef'
  exprExpr  =  conjureExpress $ fromJust mr
  (ef':_)  =  unfoldApp . fst $ head defn

-- | Evaluates a 'Defn' into a regular Haskell value
--   returning the given default value when there's a type mismatch.
--
-- The integer argument indicates the limit of recursive evaluations.
ceval :: Conjurable f => Int -> f -> Defn -> f
ceval mx z  =  fromMaybe z . cevaluate mx

-- | Evaluates a 'Defn' into a regular Haskell value
--   raising an error there's a type mismatch.
--
-- The integer argument indicates the limit of recursive evaluations.
cevl :: Conjurable f => Int -> Defn -> f
cevl mx  =  ceval mx err
  where
  err  =  error "cevl: type mismatch"

-- | Computes a complete application for the given function.
--
-- > > conjureApplication "not" not
-- > not p :: Bool
--
-- > > conjureApplication "+" ((+) :: Int -> Int -> Int)
-- > x + y :: Int
--
-- (cf. 'conjureVarApplication')
conjureApplication :: Conjurable f => String -> f -> Expr
conjureApplication  =  conjureWhatApplication value

-- | Computes a complete application for a variable
--   of the same type of the given function.
--
-- > > conjureVarApplication "not" not
-- > not p :: Bool
--
-- > > conjureVarApplication "+" ((+) :: Int -> Int -> Int)
-- > x + y :: Int
--
-- (cf. 'conjureApplication')
conjureVarApplication :: Conjurable f => String -> f -> Expr
conjureVarApplication  =  conjureWhatApplication var

-- | Used in the implementation of 'conjureApplication' and 'conjureVarApplication'.
conjureWhatApplication :: Conjurable f => (String -> f -> Expr) -> String -> f -> Expr
conjureWhatApplication what nm f  =  mostGeneralCanonicalVariation . foldApp
                                  $  what nf f : zipWith varAsTypeOf nas (conjureArgumentHoles f)
  where
  (nf:nas)  =  words nm ++ repeat ""


-- | Computes tiers of sets of patterns for the given function.
--
-- > > conjurePats [zero] "f" (undefined :: Int -> Int)
-- > [[[f x :: Int]],[[f 0 :: Int,f x :: Int]]]
conjurePats :: Conjurable f => [Expr] -> String -> f -> [[ [Expr] ]]
conjurePats  es nm f  =  mapT (map mkApp)
                      $  combinePatternOptions
                      $  conjureArgumentPats es f
  where
  mkApp  =  foldApp . (ef:)
         .  unfold
         .  conjureMostGeneralCanonicalVariation f
         .  fold
  ef  =  var (head $ words nm) f  -- TODO: take the tail into account

  -- What a horrible enumeration hack below...
  -- Good luck to anyone who plans to refactor this.

  -- after application of mkApp, we end up w/  [[ [Pat   ] ]]
  combinePatternOptions :: [ [[ [Expr] ]] ] -> [[ [[Expr]] ]]
  combinePatternOptions []            =  [[[[]]]]
  combinePatternOptions (esss:essss)  =  concatPrefixesWithT esss
                                      $  combinePatternOptions essss

  -- The three functions below are all transformations over the same type
  -- [[ [[Expr]] ]]
  -- That's tiers of complete LHS of function definitions.
  -- We proceed right to left building all possible patterns.

  -- concatenates all of the possibilities of prefixing
  -- from tiers of possibilities
  concatPrefixesWithT :: [[[Expr]]] -> [[ [[Expr]] ]] -> [[ [[Expr]] ]]
  concatPrefixesWithT esss r  =  concatMapT (`concatPrefixesWith` r) esss

  -- concatenates the possibilities of prefixing from a list of prefixes
  concatPrefixesWith :: [Expr] -> [[ [[Expr]] ]] -> [[ [[Expr]] ]]
  concatPrefixesWith es r  =  mapT concat $ products [prefixWith e r | e <- es]

  -- prefixes with the given expression
  prefixWith :: Expr -> [[ [[Expr]] ]] -> [[ [[Expr]] ]]
  prefixWith e  =  mapT (map (e:))

  -- this was useful in figuring out the implementations
  -- of the local functions above
  --
  -- > prefixWith2 :: Expr -> Expr -> [[Bs]] -> [[Bs]]
  -- > prefixWith2 e1 e2 r  =  productWith (++) (mapT (map (e1:)) r)
  -- >                                          (mapT (map (e2:)) r)
  --
  -- A prefixWith3 would follow similarly.

  -- To debug the above function use:
  -- > import Test.LeanCheck.Tiers (printTiers)
  -- > printTiers 100 $ newConjurePats [zero] "f" (undefined :: Int -> Int -> Int)


-- | Returns a list of tiers of possible patterns for each argument.
--
-- The outer list has the same number of elements as the number of arguments
-- of the given function.
--
-- This function is internal and only used in the implementation of 'conjurePats'.
-- It may be removed from the API without further notice.
-- It has been temporarily promoted to public to help refactor 'conjurePats'.
conjureArgumentPats :: Conjurable f => [Expr] -> f -> [ [[ [Expr] ]] ]
conjureArgumentPats es f = zipWith mk (conjureArgumentHoles f) (conjureArgumentCases f)
  where
  -- deal with types that have no cases such as ints.
  mk h []  =  mapT (++ [h]) $ setsOf [[e] | e <- es, typ e == typ h]
  -- deal with types that have cases, such as lists, maybes, etc.
  mk h cs  =  [[[h]], [cs]]


prods :: [[a]] -> [[a]]
prods  =  foldr (productWith (:)) [[]]
  where
  productWith (?) xs ys  =  [x ? y | x <- xs, y <- ys]


-- -- -- other Conjurable instances -- -- --

instance Conjurable Ordering where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers

instance Conjurable Float where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  round

instance Conjurable Double where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  round

instance Conjurable Int8 where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable Int16 where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable Int32 where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable Int64 where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable Word where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable Word8 where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable Word16 where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable Word32 where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable Word64 where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance (Integral a, Conjurable a, Listable a, Show a, Eq a, Express a) => Conjurable (Ratio a) where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize q    =  conjureSize (numerator q) + conjureSize (denominator q)
  conjureSubTypes q  =  conjureType (numerator q)
  conjureCases q  =  [value "%" ((%) ->>: q) :$ hole n :$ hole d]
    where
    n  =  numerator q
    d  =  denominator q

instance (RealFloat a, Conjurable a, Listable a, Show a, Eq a, Express a) => Conjurable (Complex a) where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize x    =  conjureSize (realPart x) + conjureSize (imagPart x)
  conjureSubTypes x  =  conjureType (realPart x)


-- Conjurable helper types --
instance Conjurable A where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable B where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable C where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable D where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable E where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize

instance Conjurable F where
  conjureExpress   =  reifyExpress
  conjureEquality  =  reifyEquality
  conjureTiers     =  reifyTiers
  conjureSize      =  integralSize


-- Conjurable tuples --

instance ( Conjurable a, Listable a, Show a, Express a
         , Conjurable b, Listable b, Show b, Express b
         , Conjurable c, Listable c, Show c, Express c
         , Conjurable d, Listable d, Show d, Express d
         ) => Conjurable (a,b,c,d) where
  conjureExpress   =  reifyExpress
  conjureTiers     =  reifyTiers
  conjureSubTypes xyzw =  conjureType x
                       .  conjureType y
                       .  conjureType z
                       .  conjureType w
                       where (x,y,z,w) = xyzw
  conjureEquality xyzw  =  from
                       <$> conjureEquality x
                       <*> conjureEquality y
                       <*> conjureEquality z
                       <*> conjureEquality w
    where
    (x,y,z,w)  =  xyzw
    from e1 e2 e3 e4  =  value "==" (==)
      where
      (==...)  =  evl e1 ==: x
      (.==..)  =  evl e2 ==: y
      (..==.)  =  evl e3 ==: z
      (...==)  =  evl e4 ==: w
      (x1,y1,z1,w1) == (x2,y2,z2,w2)  =  x1 ==... x2
                                      && y1 .==.. y2
                                      && z1 ..==. z2
                                      && w1 ...== w2

instance ( Conjurable a, Listable a, Show a, Express a
         , Conjurable b, Listable b, Show b, Express b
         , Conjurable c, Listable c, Show c, Express c
         , Conjurable d, Listable d, Show d, Express d
         , Conjurable e, Listable e, Show e, Express e
         ) => Conjurable (a,b,c,d,e) where
  conjureExpress   =  reifyExpress
  conjureTiers     =  reifyTiers
  conjureSubTypes xyzwv =  conjureType x
                        .  conjureType y
                        .  conjureType z
                        .  conjureType w
                        .  conjureType v
                        where (x,y,z,w,v) = xyzwv
  conjureEquality xyzwv  =  from
                        <$> conjureEquality x
                        <*> conjureEquality y
                        <*> conjureEquality z
                        <*> conjureEquality w
                        <*> conjureEquality v
    where
    (x,y,z,w,v)  =  xyzwv
    from e1 e2 e3 e4 e5  =  value "==" (==)
      where
      (==....)  =  evl e1 ==: x
      (.==...)  =  evl e2 ==: y
      (..==..)  =  evl e3 ==: z
      (...==.)  =  evl e4 ==: w
      (....==)  =  evl e5 ==: v
      (x1,y1,z1,w1,v1) == (x2,y2,z2,w2,v2)  =  x1 ==.... x2
                                            && y1 .==... y2
                                            && z1 ..==.. z2
                                            && w1 ...==. w2
                                            && v1 ....== v2

instance ( Conjurable a, Listable a, Show a, Express a
         , Conjurable b, Listable b, Show b, Express b
         , Conjurable c, Listable c, Show c, Express c
         , Conjurable d, Listable d, Show d, Express d
         , Conjurable e, Listable e, Show e, Express e
         , Conjurable f, Listable f, Show f, Express f
         ) => Conjurable (a,b,c,d,e,f) where
  conjureExpress   =  reifyExpress
  conjureTiers     =  reifyTiers
  conjureSubTypes xyzwvu =  conjureType x
                         .  conjureType y
                         .  conjureType z
                         .  conjureType w
                         .  conjureType v
                         .  conjureType u
                         where (x,y,z,w,v,u) = xyzwvu
  conjureEquality xyzwvu  =  from
                         <$> conjureEquality x
                         <*> conjureEquality y
                         <*> conjureEquality z
                         <*> conjureEquality w
                         <*> conjureEquality v
                         <*> conjureEquality u
    where
    (x,y,z,w,v,u)  =  xyzwvu
    from e1 e2 e3 e4 e5 e6  =  value "==" (==)
      where
      (==.....)  =  evl e1 ==: x
      (.==....)  =  evl e2 ==: y
      (..==...)  =  evl e3 ==: z
      (...==..)  =  evl e4 ==: w
      (....==.)  =  evl e5 ==: v
      (.....==)  =  evl e6 ==: u
      (x1,y1,z1,w1,v1,u1) == (x2,y2,z2,w2,v2,u2)  =  x1 ==..... x2
                                                  && y1 .==.... y2
                                                  && z1 ..==... z2
                                                  && w1 ...==.. w2
                                                  && v1 ....==. v2
                                                  && u1 .....== u2

instance ( Conjurable a, Listable a, Show a, Express a
         , Conjurable b, Listable b, Show b, Express b
         , Conjurable c, Listable c, Show c, Express c
         , Conjurable d, Listable d, Show d, Express d
         , Conjurable e, Listable e, Show e, Express e
         , Conjurable f, Listable f, Show f, Express f
         , Conjurable g, Listable g, Show g, Express g
         ) => Conjurable (a,b,c,d,e,f,g) where
  conjureExpress   =  reifyExpress
  conjureTiers     =  reifyTiers
  conjureSubTypes xyzwvut =  conjureType x
                          .  conjureType y
                          .  conjureType z
                          .  conjureType w
                          .  conjureType v
                          .  conjureType u
                          .  conjureType t
                          where (x,y,z,w,v,u,t) = xyzwvut
  conjureEquality xyzwvut  =  from
                          <$> conjureEquality x
                          <*> conjureEquality y
                          <*> conjureEquality z
                          <*> conjureEquality w
                          <*> conjureEquality v
                          <*> conjureEquality u
                          <*> conjureEquality t
    where
    (x,y,z,w,v,u,t)  =  xyzwvut
    from e1 e2 e3 e4 e5 e6 e7  =  value "==" (==)
      where
      (==......)  =  evl e1 ==: x
      (.==.....)  =  evl e2 ==: y
      (..==....)  =  evl e3 ==: z
      (...==...)  =  evl e4 ==: w
      (....==..)  =  evl e5 ==: v
      (.....==.)  =  evl e6 ==: u
      (......==)  =  evl e7 ==: t
      (x1,y1,z1,w1,v1,u1,t1) == (x2,y2,z2,w2,v2,u2,t2)  =  x1 ==...... x2
                                                        && y1 .==..... y2
                                                        && z1 ..==.... z2
                                                        && w1 ...==... w2
                                                        && v1 ....==.. v2
                                                        && u1 .....==. u2
                                                        && t1 ......== t2

instance ( Conjurable a, Listable a, Show a, Express a
         , Conjurable b, Listable b, Show b, Express b
         , Conjurable c, Listable c, Show c, Express c
         , Conjurable d, Listable d, Show d, Express d
         , Conjurable e, Listable e, Show e, Express e
         , Conjurable f, Listable f, Show f, Express f
         , Conjurable g, Listable g, Show g, Express g
         , Conjurable h, Listable h, Show h, Express h
         ) => Conjurable (a,b,c,d,e,f,g,h) where
  conjureExpress   =  reifyExpress
  conjureTiers     =  reifyTiers
  conjureSubTypes xyzwvuts  =  conjureType x
                            .  conjureType y
                            .  conjureType z
                            .  conjureType w
                            .  conjureType v
                            .  conjureType u
                            .  conjureType t
                            .  conjureType s
                            where (x,y,z,w,v,u,t,s) = xyzwvuts
  conjureEquality xyzwvuts  =  from
                           <$> conjureEquality x
                           <*> conjureEquality y
                           <*> conjureEquality z
                           <*> conjureEquality w
                           <*> conjureEquality v
                           <*> conjureEquality u
                           <*> conjureEquality t
                           <*> conjureEquality s
    where
    (x,y,z,w,v,u,t,s)  =  xyzwvuts
    from e1 e2 e3 e4 e5 e6 e7 e8  =  value "==" (==)
      where
      (==.......)  =  evl e1 ==: x
      (.==......)  =  evl e2 ==: y
      (..==.....)  =  evl e3 ==: z
      (...==....)  =  evl e4 ==: w
      (....==...)  =  evl e5 ==: v
      (.....==..)  =  evl e6 ==: u
      (......==.)  =  evl e7 ==: t
      (.......==)  =  evl e8 ==: s
      (x1,y1,z1,w1,v1,u1,t1,s1) == (x2,y2,z2,w2,v2,u2,t2,s2)  =  x1 ==....... x2
                                                              && y1 .==...... y2
                                                              && z1 ..==..... z2
                                                              && w1 ...==.... w2
                                                              && v1 ....==... v2
                                                              && u1 .....==.. u2
                                                              && t1 ......==. t2
                                                              && s1 .......== s2

instance ( Conjurable a, Listable a, Show a, Express a
         , Conjurable b, Listable b, Show b, Express b
         , Conjurable c, Listable c, Show c, Express c
         , Conjurable d, Listable d, Show d, Express d
         , Conjurable e, Listable e, Show e, Express e
         , Conjurable f, Listable f, Show f, Express f
         , Conjurable g, Listable g, Show g, Express g
         , Conjurable h, Listable h, Show h, Express h
         , Conjurable i, Listable i, Show i, Express i
         ) => Conjurable (a,b,c,d,e,f,g,h,i) where
  conjureExpress   =  reifyExpress
  conjureTiers     =  reifyTiers
  conjureSubTypes xyzwvutsr  =  conjureType x
                             .  conjureType y
                             .  conjureType z
                             .  conjureType w
                             .  conjureType v
                             .  conjureType u
                             .  conjureType t
                             .  conjureType s
                             .  conjureType r
                             where (x,y,z,w,v,u,t,s,r) = xyzwvutsr
  conjureEquality xyzwvutsr  =  from
                            <$> conjureEquality x
                            <*> conjureEquality y
                            <*> conjureEquality z
                            <*> conjureEquality w
                            <*> conjureEquality v
                            <*> conjureEquality u
                            <*> conjureEquality t
                            <*> conjureEquality s
                            <*> conjureEquality r
    where
    (x,y,z,w,v,u,t,s,r)  =  xyzwvutsr
    from e1 e2 e3 e4 e5 e6 e7 e8 e9  =  value "==" (==)
      where
      (==........)  =  evl e1 ==: x
      (.==.......)  =  evl e2 ==: y
      (..==......)  =  evl e3 ==: z
      (...==.....)  =  evl e4 ==: w
      (....==....)  =  evl e5 ==: v
      (.....==...)  =  evl e6 ==: u
      (......==..)  =  evl e7 ==: t
      (.......==.)  =  evl e8 ==: s
      (........==)  =  evl e9 ==: r
      (x1,y1,z1,w1,v1,u1,t1,s1,r1) == (x2,y2,z2,w2,v2,u2,t2,s2,r2)  =  x1 ==........ x2
                                                                    && y1 .==....... y2
                                                                    && z1 ..==...... z2
                                                                    && w1 ...==..... w2
                                                                    && v1 ....==.... v2
                                                                    && u1 .....==... u2
                                                                    && t1 ......==.. t2
                                                                    && s1 .......==. s2
                                                                    && r1 ........== r2

instance ( Conjurable a, Listable a, Show a, Express a
         , Conjurable b, Listable b, Show b, Express b
         , Conjurable c, Listable c, Show c, Express c
         , Conjurable d, Listable d, Show d, Express d
         , Conjurable e, Listable e, Show e, Express e
         , Conjurable f, Listable f, Show f, Express f
         , Conjurable g, Listable g, Show g, Express g
         , Conjurable h, Listable h, Show h, Express h
         , Conjurable i, Listable i, Show i, Express i
         , Conjurable j, Listable j, Show j, Express j
         ) => Conjurable (a,b,c,d,e,f,g,h,i,j) where
  conjureExpress   =  reifyExpress
  conjureTiers     =  reifyTiers
  conjureSubTypes xyzwvutsrq  =  conjureType x
                              .  conjureType y
                              .  conjureType z
                              .  conjureType w
                              .  conjureType v
                              .  conjureType u
                              .  conjureType t
                              .  conjureType s
                              .  conjureType r
                              .  conjureType q
                              where (x,y,z,w,v,u,t,s,r,q) = xyzwvutsrq
  conjureEquality xyzwvutsrq  =  from
                             <$> conjureEquality x
                             <*> conjureEquality y
                             <*> conjureEquality z
                             <*> conjureEquality w
                             <*> conjureEquality v
                             <*> conjureEquality u
                             <*> conjureEquality t
                             <*> conjureEquality s
                             <*> conjureEquality r
                             <*> conjureEquality q
    where
    (x,y,z,w,v,u,t,s,r,q)  =  xyzwvutsrq
    from e1 e2 e3 e4 e5 e6 e7 e8 e9 e10  =  value "==" (==)
      where
      (==.........)  =  evl e1  ==: x
      (.==........)  =  evl e2  ==: y
      (..==.......)  =  evl e3  ==: z
      (...==......)  =  evl e4  ==: w
      (....==.....)  =  evl e5  ==: v
      (.....==....)  =  evl e6  ==: u
      (......==...)  =  evl e7  ==: t
      (.......==..)  =  evl e8  ==: s
      (........==.)  =  evl e9  ==: r
      (.........==)  =  evl e10 ==: q
      (x1,y1,z1,w1,v1,u1,t1,s1,r1,q1) == (x2,y2,z2,w2,v2,u2,t2,s2,r2,q2)  =  x1 ==......... x2
                                                                          && y1 .==........ y2
                                                                          && z1 ..==....... z2
                                                                          && w1 ...==...... w2
                                                                          && v1 ....==..... v2
                                                                          && u1 .....==.... u2
                                                                          && t1 ......==... t2
                                                                          && s1 .......==.. s2
                                                                          && r1 ........==. r2
                                                                          && q1 .........== q2


instance ( Conjurable a, Listable a, Show a, Express a
         , Conjurable b, Listable b, Show b, Express b
         , Conjurable c, Listable c, Show c, Express c
         , Conjurable d, Listable d, Show d, Express d
         , Conjurable e, Listable e, Show e, Express e
         , Conjurable f, Listable f, Show f, Express f
         , Conjurable g, Listable g, Show g, Express g
         , Conjurable h, Listable h, Show h, Express h
         , Conjurable i, Listable i, Show i, Express i
         , Conjurable j, Listable j, Show j, Express j
         , Conjurable k, Listable k, Show k, Express k
         ) => Conjurable (a,b,c,d,e,f,g,h,i,j,k) where
  conjureExpress   =  reifyExpress
  conjureTiers     =  reifyTiers
  conjureSubTypes xyzwvutsrqp  =  conjureType x
                               .  conjureType y
                               .  conjureType z
                               .  conjureType w
                               .  conjureType v
                               .  conjureType u
                               .  conjureType t
                               .  conjureType s
                               .  conjureType r
                               .  conjureType q
                               .  conjureType p
                               where (x,y,z,w,v,u,t,s,r,q,p) = xyzwvutsrqp
  conjureEquality xyzwvutsrqp  =  from
                              <$> conjureEquality x
                              <*> conjureEquality y
                              <*> conjureEquality z
                              <*> conjureEquality w
                              <*> conjureEquality v
                              <*> conjureEquality u
                              <*> conjureEquality t
                              <*> conjureEquality s
                              <*> conjureEquality r
                              <*> conjureEquality q
                              <*> conjureEquality p
    where
    (x,y,z,w,v,u,t,s,r,q,p)  =  xyzwvutsrqp
    from e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11  =  value "==" (==)
      where
      (==..........)  =  evl e1  ==: x
      (.==.........)  =  evl e2  ==: y
      (..==........)  =  evl e3  ==: z
      (...==.......)  =  evl e4  ==: w
      (....==......)  =  evl e5  ==: v
      (.....==.....)  =  evl e6  ==: u
      (......==....)  =  evl e7  ==: t
      (.......==...)  =  evl e8  ==: s
      (........==..)  =  evl e9  ==: r
      (.........==.)  =  evl e10 ==: q
      (..........==)  =  evl e11 ==: p
      (x1,y1,z1,w1,v1,u1,t1,s1,r1,q1,p1) == (x2,y2,z2,w2,v2,u2,t2,s2,r2,q2,p2)  =  x1 ==.......... x2
                                                                                && y1 .==......... y2
                                                                                && z1 ..==........ z2
                                                                                && w1 ...==....... w2
                                                                                && v1 ....==...... v2
                                                                                && u1 .....==..... u2
                                                                                && t1 ......==.... t2
                                                                                && s1 .......==... s2
                                                                                && r1 ........==.. r2
                                                                                && q1 .........==. q2
                                                                                && p1 ..........== p2


instance ( Conjurable a, Listable a, Show a, Express a
         , Conjurable b, Listable b, Show b, Express b
         , Conjurable c, Listable c, Show c, Express c
         , Conjurable d, Listable d, Show d, Express d
         , Conjurable e, Listable e, Show e, Express e
         , Conjurable f, Listable f, Show f, Express f
         , Conjurable g, Listable g, Show g, Express g
         , Conjurable h, Listable h, Show h, Express h
         , Conjurable i, Listable i, Show i, Express i
         , Conjurable j, Listable j, Show j, Express j
         , Conjurable k, Listable k, Show k, Express k
         , Conjurable l, Listable l, Show l, Express l
         ) => Conjurable (a,b,c,d,e,f,g,h,i,j,k,l) where
  conjureExpress   =  reifyExpress
  conjureTiers     =  reifyTiers
  conjureSubTypes xyzwvutsrqpo  =  conjureType x
                                .  conjureType y
                                .  conjureType z
                                .  conjureType w
                                .  conjureType v
                                .  conjureType u
                                .  conjureType t
                                .  conjureType s
                                .  conjureType r
                                .  conjureType q
                                .  conjureType p
                                .  conjureType o
                                where (x,y,z,w,v,u,t,s,r,q,p,o) = xyzwvutsrqpo
  conjureEquality xyzwvutsrqpo  =  from
                               <$> conjureEquality x
                               <*> conjureEquality y
                               <*> conjureEquality z
                               <*> conjureEquality w
                               <*> conjureEquality v
                               <*> conjureEquality u
                               <*> conjureEquality t
                               <*> conjureEquality s
                               <*> conjureEquality r
                               <*> conjureEquality q
                               <*> conjureEquality p
                               <*> conjureEquality o
    where
    (x,y,z,w,v,u,t,s,r,q,p,o)  =  xyzwvutsrqpo
    from e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12  =  value "==" (==)
      where
      (==...........)  =  evl e1  ==: x
      (.==..........)  =  evl e2  ==: y
      (..==.........)  =  evl e3  ==: z
      (...==........)  =  evl e4  ==: w
      (....==.......)  =  evl e5  ==: v
      (.....==......)  =  evl e6  ==: u
      (......==.....)  =  evl e7  ==: t
      (.......==....)  =  evl e8  ==: s
      (........==...)  =  evl e9  ==: r
      (.........==..)  =  evl e10 ==: q
      (..........==.)  =  evl e11 ==: p
      (...........==)  =  evl e12 ==: o
      (x1,y1,z1,w1,v1,u1,t1,s1,r1,q1,p1,o1) == (x2,y2,z2,w2,v2,u2,t2,s2,r2,q2,p2,o2)  =  x1 ==........... x2
                                                                                      && y1 .==.......... y2
                                                                                      && z1 ..==......... z2
                                                                                      && w1 ...==........ w2
                                                                                      && v1 ....==....... v2
                                                                                      && u1 .....==...... u2
                                                                                      && t1 ......==..... t2
                                                                                      && s1 .......==.... s2
                                                                                      && r1 ........==... r2
                                                                                      && q1 .........==.. q2
                                                                                      && p1 ..........==. p2
                                                                                      && o1 ...........== o2


instance Name A
instance Name B
instance Name C
instance Name D
instance Name E
instance Name F