id p = p p && p = p p || p = p p && q = q && p p || q = q || p not (not p) = p p && True = p True && p = p p && False = False False && p = False p || True = True True || p = True p || False = p False || p = p p && not p = False p || not p = True not p && p = False not p || p = True (p && q) && r = p && (q && r) (p || q) || r = p || (q || r) not (p && q) = not p || not q not (p || q) = not p && not q not False = True not True = False not (not p && not q) = p || q not (not p || not q) = p && q p && not (p && q) = p && not q id x = x abs (abs x) = abs x x + 0 = x 0 + x = x x * 1 = x 1 * x = x x * 0 = 0 0 * x = 0 x + y = y + x x * y = y * x (x + y) + z = x + (y + z) (x * y) * z = x * (y * z) (x + x) * y = x * (y + y) x * (y + 1) = x + x * y (x + 1) * y = x + x * y x * (y + z) = x * y + x * z (x + y) * z = x * z + y * z negate (negate x) = x x + negate x = 0 negate x + x = 0 abs (negate x) = abs x 2 * x = x + x x * 2 = x + x 3 * x = x + (x + x) x * 3 = x + (x + x) 4 * x = x + (x + (x + x)) x * 4 = x + (x + (x + x)) abs (x * x) = x * x abs x * abs y = abs (x * y) abs x * abs x = abs (x + x) abs (abs x + abs y) = abs x + abs y abs (x + x) * y = abs x * y + abs x * y abs x * signum x = x signum x * abs x = x id xs = xs head (x:xs) = x tail (x:xs) = xs xs ++ [] = xs [] ++ xs = xs [x] ++ xs = x:xs (x:xs) ++ ys = x:(xs ++ ys) (xs ++ ys) ++ zs = xs ++ (ys ++ zs) elem x (sort xs) = elem x xs elem x (insert y xs) = elem x (y:xs) sort (sort xs) = sort xs insert x [] = [x] sort (xs ++ ys) = sort (ys ++ xs) sort (insert x xs) = insert x (sort xs) sort (x:xs) = insert x (sort xs) sort (xs ++ sort ys) = sort (xs ++ ys) sort (sort xs ++ ys) = sort (xs ++ ys) insert x (insert y xs) = insert y (insert x xs) insert x (x:xs) = x:x:xs insert x [y] = insert y [x] length (x:xs) = length (y:xs) length (xs ++ ys) = length (ys ++ xs) length (x:y:xs) = length (z:x':xs) length (x:(xs ++ ys)) = length (y:(ys ++ xs)) length (xs ++ (ys ++ zs)) = length (xs ++ (zs ++ ys)) not (odd x) = even x not (even x) = odd x x == x = True x /= x = False p == p = True p /= p = False f (g x) = (f . g) x map id xs = xs map (f . g) xs = map f (map g xs) f . id = f id . f = f (f . g) . h = f . (g . h) not . not = id (_ :: Int,[],_ :: Int) (x :: Int,[],x :: Int) (0 :: Int,[],0 :: Int) (id _ :: Int,[],id _ :: Int) (_ :: Bool,[],_ :: Bool) (_ :: Char,[],_ :: Char) (_ :: [Int],[],_ :: [Int]) (y :: Int,[],y :: Int) (1 :: Int,[],1 :: Int) (id x :: Int,[],id x :: Int) (id 0 :: Int,[],id 0 :: Int) (id (id _) :: Int,[],id (id _) :: Int) (negate _ :: Int,[],negate _ :: Int) (abs _ :: Int,[],abs _ :: Int) (_ + _ :: Int,[],_ + _ :: Int) (head _ :: Int,[],head _ :: Int) (ord _ :: Int,[],ord _ :: Int) (p :: Bool,[],p :: Bool) (False :: Bool,[],False :: Bool) (True :: Bool,[],True :: Bool) (not _ :: Bool,[],not _ :: Bool) (c :: Char,[],c :: Char) ('a' :: Char,[],'a' :: Char) (xs :: [Int],[],xs :: [Int]) ([] :: [Int],[],[] :: [Int]) (_:_ :: [Int],[],_:_ :: [Int]) (id :: Int -> Int,[],id :: Int -> Int) ((+) :: Int -> Int -> Int,[],(+) :: Int -> Int -> Int) (z :: Int,[],z :: Int) (-1 :: Int,[],-1 :: Int) (id y :: Int,[],id y :: Int) (id 1 :: Int,[],id 1 :: Int) (id (id x) :: Int,[],id (id x) :: Int) (id (id 0) :: Int,[],id (id 0) :: Int) (id (id (id _)) :: Int,[],id (id (id _)) :: Int) (id (negate _) :: Int,[],id (negate _) :: Int) (id (abs _) :: Int,[],id (abs _) :: Int) (id (_ + _) :: Int,[],id (_ + _) :: Int) (id (head _) :: Int,[],id (head _) :: Int) (id (ord _) :: Int,[],id (ord _) :: Int) (negate x :: Int,[],negate x :: Int) (negate 0 :: Int,[],negate 0 :: Int) (negate (id _) :: Int,[],negate (id _) :: Int) (abs x :: Int,[],abs x :: Int) (abs 0 :: Int,[],abs 0 :: Int) (abs (id _) :: Int,[],abs (id _) :: Int) (_ + x :: Int,[],_ + x :: Int) (_ + 0 :: Int,[],_ + 0 :: Int) (_ + id _ :: Int,[],_ + id _ :: Int) (x + _ :: Int,[],x + _ :: Int) (0 + _ :: Int,[],0 + _ :: Int) (id _ + _ :: Int,[],id _ + _ :: Int) (_ * _ :: Int,[],_ * _ :: Int) (f _ :: Int,[],f _ :: Int) (head xs :: Int,[],head xs :: Int) (head [] :: Int,[],head [] :: Int) (head (_:_) :: Int,[],head (_:_) :: Int) (ord c :: Int,[],ord c :: Int) (ord 'a' :: Int,[],ord 'a' :: Int) (q :: Bool,[],q :: Bool) (not p :: 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_ + _) :: Int) (id (_ * _) :: Int,[],id (_ * _) :: Int) (id (f _) :: Int,[],id (f _) :: Int) (id (head xs) :: Int,[],id (head xs) :: Int) (id (head []) :: Int,[],id (head []) :: Int) (id (head (_:_)) :: Int,[],id (head (_:_)) :: Int) (id (ord c) :: Int,[],id (ord c) :: Int) (id (ord 'a') :: Int,[],id (ord 'a') :: Int) (negate y :: Int,[],negate y :: Int) (negate 1 :: Int,[],negate 1 :: Int) (negate (id x) :: Int,[],negate (id x) :: Int) (negate (id 0) :: Int,[],negate (id 0) :: Int) (negate (id (id _)) :: Int,[],negate (id (id _)) :: Int) (negate (negate _) :: Int,[],negate (negate _) :: Int) (negate (abs _) :: Int,[],negate (abs _) :: Int) (negate (_ + _) :: Int,[],negate (_ + _) :: Int) (negate (head _) :: Int,[],negate (head _) :: Int) (negate (ord _) :: Int,[],negate (ord _) :: Int) (abs y :: Int,[],abs y :: Int) (abs 1 :: Int,[],abs 1 :: Int) (abs (id x) :: Int,[],abs (id x) :: Int) (abs (id 0) :: Int,[],abs (id 0) :: Int) (abs (id (id _)) :: Int,[],abs (id (id _)) :: Int) (abs (negate _) :: Int,[],abs (negate _) :: Int) (abs (abs _) :: Int,[],abs (abs _) :: Int) (abs (_ + _) :: Int,[],abs (_ + _) :: Int) (abs (head _) :: Int,[],abs (head _) :: Int) (abs (ord _) :: Int,[],abs (ord _) :: Int) (_ + y :: Int,[],_ + y :: Int) (_ + 1 :: Int,[],_ + 1 :: Int) (_ + id x :: Int,[],_ + id x :: Int) (_ + id 0 :: Int,[],_ + id 0 :: Int) (_ + id (id _) :: Int,[],_ + id (id _) :: Int) (_ + negate _ :: Int,[],_ + negate _ :: Int) (_ + abs _ :: Int,[],_ + abs _ :: Int) (_ + (_ + _) :: Int,[],_ + (_ + _) :: Int) (_ + head _ :: Int,[],_ + head _ :: Int) (_ + ord _ :: Int,[],_ + ord _ :: Int) (x + x :: Int,[],x + x :: Int) (x + 0 :: Int,[],x + 0 :: Int) (x + id _ :: Int,[],x + id _ :: Int) (0 + x :: Int,[],0 + x :: Int) (0 + 0 :: Int,[],0 + 0 :: Int) (0 + id _ :: Int,[],0 + id _ :: Int) (id _ + x :: Int,[],id _ + x :: Int) (id _ + 0 :: Int,[],id _ + 0 :: Int) (id _ + id _ :: Int,[],id _ + id _ :: Int) (_ * x :: Int,[],_ * x :: Int) (_ * 0 :: Int,[],_ * 0 :: Int) (_ * id _ :: 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Int) (head (_ ++ _) :: Int,[],head (_ ++ _) :: Int) (ord d :: Int,[],ord d :: Int) (ord ' ' :: Int,[],ord ' ' :: Int) (r :: Bool,[],r :: Bool) (not q :: Bool,[],not q :: Bool) (not (not p) :: Bool,[],not (not p) :: Bool) (not (not False) :: Bool,[],not (not False) :: Bool) (not (not True) :: Bool,[],not (not True) :: Bool) (not (not (not _)) :: Bool,[],not (not (not _)) :: Bool) (not (_ || _) :: Bool,[],not (_ || _) :: Bool) (_ || p :: Bool,[],_ || p :: Bool) (_ || False :: Bool,[],_ || False :: Bool) (_ || True :: Bool,[],_ || True :: Bool) (_ || not _ :: Bool,[],_ || not _ :: Bool) (p || _ :: Bool,[],p || _ :: Bool) (False || _ :: Bool,[],False || _ :: Bool) (True || _ :: Bool,[],True || _ :: Bool) (not _ || _ :: Bool,[],not _ || _ :: Bool) (_ && _ :: Bool,[],_ && _ :: Bool) (_ == _ :: Bool,[],_ == _ :: Bool) (_ == _ :: Bool,[],_ == _ :: Bool) (odd _ :: Bool,[],odd _ :: Bool) (even _ :: Bool,[],even _ :: Bool) (elem _ _ :: Bool,[],elem _ _ :: Bool) (e :: Char,[],e :: Char) ('b' :: Char,[],'b' :: Char) (zs :: [Int],[],zs :: [Int]) ([0,0] :: [Int],[],[0,0] :: [Int]) ([1] :: [Int],[],[1] :: [Int]) (_:ys :: [Int],[],_:ys :: [Int]) ([_,0] :: [Int],[],[_,0] :: [Int]) (_:_:xs :: [Int],[],_:_:xs :: [Int]) ([_,_] :: [Int],[],[_,_] :: [Int]) (_:_:_:_ :: [Int],[],_:_:_:_ :: [Int]) (_:x:_ :: [Int],[],_:x:_ :: [Int]) (_:0:_ :: [Int],[],_:0:_ :: [Int]) (_:id _:_ :: [Int],[],_:id _:_ :: [Int]) (_:tail _ :: [Int],[],_:tail _ :: [Int]) (_:(_ ++ _) :: [Int],[],_:(_ ++ _) :: [Int]) (x:xs :: [Int],[],x:xs :: [Int]) ([x] :: [Int],[],[x] :: [Int]) (x:_:_ :: [Int],[],x:_:_ :: [Int]) (0:xs :: [Int],[],0:xs :: [Int]) ([0] :: [Int],[],[0] :: [Int]) (0:_:_ :: [Int],[],0:_:_ :: [Int]) (id _:xs :: [Int],[],id _:xs :: [Int]) ([id _] :: [Int],[],[id _] :: [Int]) (id _:_:_ :: [Int],[],id _:_:_ :: [Int]) (y:_ :: [Int],[],y:_ :: [Int]) (1:_ :: [Int],[],1:_ :: [Int]) (id x:_ :: [Int],[],id x:_ :: [Int]) (id 0:_ :: [Int],[],id 0:_ :: [Int]) (id (id _):_ :: [Int],[],id (id _):_ :: [Int]) (negate _:_ :: [Int],[],negate _:_ :: [Int]) (abs _:_ :: [Int],[],abs _:_ :: [Int]) (_ + _:_ :: [Int],[],_ + _:_ :: [Int]) (head _:_ :: [Int],[],head _:_ :: [Int]) (ord _:_ :: [Int],[],ord _:_ :: [Int]) (tail xs :: [Int],[],tail xs :: [Int]) (tail [] :: [Int],[],tail [] :: [Int]) (tail (_:_) :: [Int],[],tail (_:_) :: [Int]) (_ ++ xs :: [Int],[],_ ++ xs :: [Int]) (_ ++ [] :: [Int],[],_ ++ [] :: [Int]) (_ ++ (_:_) :: [Int],[],_ ++ (_:_) :: [Int]) (xs ++ _ :: [Int],[],xs ++ _ :: [Int]) ([] ++ _ :: [Int],[],[] ++ _ :: [Int]) ((_:_) ++ _ :: [Int],[],(_:_) ++ _ :: [Int]) (sort _ :: [Int],[],sort _ :: [Int]) (insert _ _ :: [Int],[],insert _ _ :: [Int]) ((x +) :: Int -> Int,[],(x +) :: Int -> Int) ((0 +) :: Int -> Int,[],(0 +) :: Int -> Int) ((id _ +) :: Int -> Int,[],(id _ +) :: Int -> Int) ((_ *) :: Int -> Int,[],(_ *) :: Int -> Int) (f :: Int -> Int,[],f :: Int -> Int) ((_ ||) :: Bool -> Bool,[],(_ ||) :: Bool -> Bool) ((&&) :: Bool -> Bool -> Bool,[],(&&) :: Bool -> Bool -> Bool) (y' :: Int,[],y' 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