/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


MAYBE
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


H-Termination with start terms of the given HASKELL could not be shown:

(0) HASKELL
(1) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL


----------------------------------------

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data Double = Double MyInt MyInt ;

data MyBool = MyTrue  | MyFalse ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

data Ordering = LT  | EQ  | GT ;

data Ratio a = CnPc a a ;

absMyInt :: MyInt  ->  MyInt;
absMyInt = absReal;

absReal x = absReal2 x;

absReal0 x MyTrue = negateMyInt x;

absReal1 x MyTrue = x;
absReal1 x MyFalse = absReal0 x otherwise;

absReal2 x = absReal1 x (gtEsMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
compareMyInt = primCmpInt;

doubleToRatio (Double x y) = pc (fromIntMyInt x) (fromIntMyInt y);

error :: a;
error = stop MyTrue;

esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
esEsMyInt = primEqInt;

esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
esEsOrdering LT LT = MyTrue;
esEsOrdering LT EQ = MyFalse;
esEsOrdering LT GT = MyFalse;
esEsOrdering EQ LT = MyFalse;
esEsOrdering EQ EQ = MyTrue;
esEsOrdering EQ GT = MyFalse;
esEsOrdering GT LT = MyFalse;
esEsOrdering GT EQ = MyFalse;
esEsOrdering GT GT = MyTrue;

fromDoubleRatio :: Double  ->  Ratio MyInt;
fromDoubleRatio = doubleToRatio;

fromIntMyInt :: MyInt  ->  MyInt;
fromIntMyInt x = x;

fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
fsEsOrdering x y = not (esEsOrdering x y);

gcd yx yy = gcd3 yx yy;
gcd x y = gcd0 x y;

gcd0 x y = gcd0Gcd' (absMyInt x) (absMyInt y);

gcd0Gcd' x xx = gcd0Gcd'2 x xx;
gcd0Gcd' x y = gcd0Gcd'0 x y;

gcd0Gcd'0 x y = gcd0Gcd' y (remMyInt x y);

gcd0Gcd'1 MyTrue x xx = x;
gcd0Gcd'1 xy xz yu = gcd0Gcd'0 xz yu;

gcd0Gcd'2 x xx = gcd0Gcd'1 (esEsMyInt xx (fromIntMyInt (Main.Pos Main.Zero))) x xx;
gcd0Gcd'2 yv yw = gcd0Gcd'0 yv yw;

gcd1 MyTrue yx yy = Main.error;
gcd1 yz zu zv = gcd0 zu zv;

gcd2 MyTrue yx yy = gcd1 (esEsMyInt yy (fromIntMyInt (Main.Pos Main.Zero))) yx yy;
gcd2 zw zx zy = gcd0 zx zy;

gcd3 yx yy = gcd2 (esEsMyInt yx (fromIntMyInt (Main.Pos Main.Zero))) yx yy;
gcd3 zz vuu = gcd0 zz vuu;

gtEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
gtEsMyInt x y = fsEsOrdering (compareMyInt x y) LT;

gtMyInt :: MyInt  ->  MyInt  ->  MyBool;
gtMyInt x y = esEsOrdering (compareMyInt x y) GT;

negateMyInt :: MyInt  ->  MyInt;
negateMyInt = primNegInt;

not :: MyBool  ->  MyBool;
not MyTrue = MyFalse;
not MyFalse = MyTrue;

otherwise :: MyBool;
otherwise = MyTrue;

pc x y = reduce (srMyInt x (signumMyInt y)) (absMyInt y);

primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
primCmpInt (Main.Pos x) (Main.Neg y) = GT;
primCmpInt (Main.Neg x) (Main.Pos y) = LT;
primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;

primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
primCmpNat Main.Zero Main.Zero = EQ;
primCmpNat Main.Zero (Main.Succ y) = LT;
primCmpNat (Main.Succ x) Main.Zero = GT;
primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;

primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primDivNatS Main.Zero Main.Zero = Main.error;
primDivNatS (Main.Succ x) Main.Zero = Main.error;
primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
primDivNatS Main.Zero (Main.Succ x) = Main.Zero;

primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
primDivNatS0 x y MyFalse = Main.Zero;

primEqInt :: MyInt  ->  MyInt  ->  MyBool;
primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt xv xw = MyFalse;

primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
primEqNat Main.Zero Main.Zero = MyTrue;
primEqNat Main.Zero (Main.Succ y) = MyFalse;
primEqNat (Main.Succ x) Main.Zero = MyFalse;
primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;

primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
primGEqNatS Main.Zero Main.Zero = MyTrue;

primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
primMinusNatS x Main.Zero = x;

primModNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primModNatS Main.Zero Main.Zero = Main.error;
primModNatS Main.Zero (Main.Succ x) = Main.Zero;
primModNatS (Main.Succ x) Main.Zero = Main.error;
primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y));

primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
primModNatS0 x y MyFalse = Main.Succ x;

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primMulNat Main.Zero Main.Zero = Main.Zero;
primMulNat Main.Zero (Main.Succ y) = Main.Zero;
primMulNat (Main.Succ x) Main.Zero = Main.Zero;
primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);

primNegInt :: MyInt  ->  MyInt;
primNegInt (Main.Pos x) = Main.Neg x;
primNegInt (Main.Neg x) = Main.Pos x;

primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primPlusNat Main.Zero Main.Zero = Main.Zero;
primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));

primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt wz xu = Main.error;

primRemInt :: MyInt  ->  MyInt  ->  MyInt;
primRemInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
primRemInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
primRemInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
primRemInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
primRemInt wx wy = Main.error;

quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
quotMyInt = primQuotInt;

reduce x y = reduce2 x y;

reduce2 x y = reduce2Reduce1 x y x y (esEsMyInt y (fromIntMyInt (Main.Pos Main.Zero)));

reduce2D vuv vuw = gcd vuv vuw;

reduce2Reduce0 vuv vuw x y MyTrue = CnPc (quotMyInt x (reduce2D vuv vuw)) (quotMyInt y (reduce2D vuv vuw));

reduce2Reduce1 vuv vuw x y MyTrue = Main.error;
reduce2Reduce1 vuv vuw x y MyFalse = reduce2Reduce0 vuv vuw x y otherwise;

remMyInt :: MyInt  ->  MyInt  ->  MyInt;
remMyInt = primRemInt;

signumMyInt :: MyInt  ->  MyInt;
signumMyInt = signumReal;

signumReal x = signumReal3 x;

signumReal0 x MyTrue = fromIntMyInt (Main.Neg (Main.Succ Main.Zero));

signumReal1 x MyTrue = fromIntMyInt (Main.Pos (Main.Succ Main.Zero));
signumReal1 x MyFalse = signumReal0 x otherwise;

signumReal2 x MyTrue = fromIntMyInt (Main.Pos Main.Zero);
signumReal2 x MyFalse = signumReal1 x (gtMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

signumReal3 x = signumReal2 x (esEsMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

srMyInt :: MyInt  ->  MyInt  ->  MyInt;
srMyInt = primMulInt;

stop :: MyBool  ->  a;
stop MyFalse = stop MyFalse;

}

----------------------------------------

(1) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data Double = Double MyInt MyInt ;

data MyBool = MyTrue  | MyFalse ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

data Ordering = LT  | EQ  | GT ;

data Ratio a = CnPc a a ;

absMyInt :: MyInt  ->  MyInt;
absMyInt = absReal;

absReal x = absReal2 x;

absReal0 x MyTrue = negateMyInt x;

absReal1 x MyTrue = x;
absReal1 x MyFalse = absReal0 x otherwise;

absReal2 x = absReal1 x (gtEsMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
compareMyInt = primCmpInt;

doubleToRatio (Double x y) = pc (fromIntMyInt x) (fromIntMyInt y);

error :: a;
error = stop MyTrue;

esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
esEsMyInt = primEqInt;

esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
esEsOrdering LT LT = MyTrue;
esEsOrdering LT EQ = MyFalse;
esEsOrdering LT GT = MyFalse;
esEsOrdering EQ LT = MyFalse;
esEsOrdering EQ EQ = MyTrue;
esEsOrdering EQ GT = MyFalse;
esEsOrdering GT LT = MyFalse;
esEsOrdering GT EQ = MyFalse;
esEsOrdering GT GT = MyTrue;

fromDoubleRatio :: Double  ->  Ratio MyInt;
fromDoubleRatio = doubleToRatio;

fromIntMyInt :: MyInt  ->  MyInt;
fromIntMyInt x = x;

fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
fsEsOrdering x y = not (esEsOrdering x y);

gcd yx yy = gcd3 yx yy;
gcd x y = gcd0 x y;

gcd0 x y = gcd0Gcd' (absMyInt x) (absMyInt y);

gcd0Gcd' x xx = gcd0Gcd'2 x xx;
gcd0Gcd' x y = gcd0Gcd'0 x y;

gcd0Gcd'0 x y = gcd0Gcd' y (remMyInt x y);

gcd0Gcd'1 MyTrue x xx = x;
gcd0Gcd'1 xy xz yu = gcd0Gcd'0 xz yu;

gcd0Gcd'2 x xx = gcd0Gcd'1 (esEsMyInt xx (fromIntMyInt (Main.Pos Main.Zero))) x xx;
gcd0Gcd'2 yv yw = gcd0Gcd'0 yv yw;

gcd1 MyTrue yx yy = Main.error;
gcd1 yz zu zv = gcd0 zu zv;

gcd2 MyTrue yx yy = gcd1 (esEsMyInt yy (fromIntMyInt (Main.Pos Main.Zero))) yx yy;
gcd2 zw zx zy = gcd0 zx zy;

gcd3 yx yy = gcd2 (esEsMyInt yx (fromIntMyInt (Main.Pos Main.Zero))) yx yy;
gcd3 zz vuu = gcd0 zz vuu;

gtEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
gtEsMyInt x y = fsEsOrdering (compareMyInt x y) LT;

gtMyInt :: MyInt  ->  MyInt  ->  MyBool;
gtMyInt x y = esEsOrdering (compareMyInt x y) GT;

negateMyInt :: MyInt  ->  MyInt;
negateMyInt = primNegInt;

not :: MyBool  ->  MyBool;
not MyTrue = MyFalse;
not MyFalse = MyTrue;

otherwise :: MyBool;
otherwise = MyTrue;

pc x y = reduce (srMyInt x (signumMyInt y)) (absMyInt y);

primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
primCmpInt (Main.Pos x) (Main.Neg y) = GT;
primCmpInt (Main.Neg x) (Main.Pos y) = LT;
primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;

primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
primCmpNat Main.Zero Main.Zero = EQ;
primCmpNat Main.Zero (Main.Succ y) = LT;
primCmpNat (Main.Succ x) Main.Zero = GT;
primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;

primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primDivNatS Main.Zero Main.Zero = Main.error;
primDivNatS (Main.Succ x) Main.Zero = Main.error;
primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
primDivNatS Main.Zero (Main.Succ x) = Main.Zero;

primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
primDivNatS0 x y MyFalse = Main.Zero;

primEqInt :: MyInt  ->  MyInt  ->  MyBool;
primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt xv xw = MyFalse;

primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
primEqNat Main.Zero Main.Zero = MyTrue;
primEqNat Main.Zero (Main.Succ y) = MyFalse;
primEqNat (Main.Succ x) Main.Zero = MyFalse;
primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;

primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
primGEqNatS Main.Zero Main.Zero = MyTrue;

primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
primMinusNatS x Main.Zero = x;

primModNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primModNatS Main.Zero Main.Zero = Main.error;
primModNatS Main.Zero (Main.Succ x) = Main.Zero;
primModNatS (Main.Succ x) Main.Zero = Main.error;
primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y));

primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
primModNatS0 x y MyFalse = Main.Succ x;

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primMulNat Main.Zero Main.Zero = Main.Zero;
primMulNat Main.Zero (Main.Succ y) = Main.Zero;
primMulNat (Main.Succ x) Main.Zero = Main.Zero;
primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);

primNegInt :: MyInt  ->  MyInt;
primNegInt (Main.Pos x) = Main.Neg x;
primNegInt (Main.Neg x) = Main.Pos x;

primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primPlusNat Main.Zero Main.Zero = Main.Zero;
primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));

primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt wz xu = Main.error;

primRemInt :: MyInt  ->  MyInt  ->  MyInt;
primRemInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
primRemInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
primRemInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
primRemInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
primRemInt wx wy = Main.error;

quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
quotMyInt = primQuotInt;

reduce x y = reduce2 x y;

reduce2 x y = reduce2Reduce1 x y x y (esEsMyInt y (fromIntMyInt (Main.Pos Main.Zero)));

reduce2D vuv vuw = gcd vuv vuw;

reduce2Reduce0 vuv vuw x y MyTrue = CnPc (quotMyInt x (reduce2D vuv vuw)) (quotMyInt y (reduce2D vuv vuw));

reduce2Reduce1 vuv vuw x y MyTrue = Main.error;
reduce2Reduce1 vuv vuw x y MyFalse = reduce2Reduce0 vuv vuw x y otherwise;

remMyInt :: MyInt  ->  MyInt  ->  MyInt;
remMyInt = primRemInt;

signumMyInt :: MyInt  ->  MyInt;
signumMyInt = signumReal;

signumReal x = signumReal3 x;

signumReal0 x MyTrue = fromIntMyInt (Main.Neg (Main.Succ Main.Zero));

signumReal1 x MyTrue = fromIntMyInt (Main.Pos (Main.Succ Main.Zero));
signumReal1 x MyFalse = signumReal0 x otherwise;

signumReal2 x MyTrue = fromIntMyInt (Main.Pos Main.Zero);
signumReal2 x MyFalse = signumReal1 x (gtMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

signumReal3 x = signumReal2 x (esEsMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

srMyInt :: MyInt  ->  MyInt  ->  MyInt;
srMyInt = primMulInt;

stop :: MyBool  ->  a;
stop MyFalse = stop MyFalse;

}

----------------------------------------

(3) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

----------------------------------------

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data Double = Double MyInt MyInt ;

data MyBool = MyTrue  | MyFalse ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

data Ordering = LT  | EQ  | GT ;

data Ratio a = CnPc a a ;

absMyInt :: MyInt  ->  MyInt;
absMyInt = absReal;

absReal x = absReal2 x;

absReal0 x MyTrue = negateMyInt x;

absReal1 x MyTrue = x;
absReal1 x MyFalse = absReal0 x otherwise;

absReal2 x = absReal1 x (gtEsMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
compareMyInt = primCmpInt;

doubleToRatio (Double x y) = pc (fromIntMyInt x) (fromIntMyInt y);

error :: a;
error = stop MyTrue;

esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
esEsMyInt = primEqInt;

esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
esEsOrdering LT LT = MyTrue;
esEsOrdering LT EQ = MyFalse;
esEsOrdering LT GT = MyFalse;
esEsOrdering EQ LT = MyFalse;
esEsOrdering EQ EQ = MyTrue;
esEsOrdering EQ GT = MyFalse;
esEsOrdering GT LT = MyFalse;
esEsOrdering GT EQ = MyFalse;
esEsOrdering GT GT = MyTrue;

fromDoubleRatio :: Double  ->  Ratio MyInt;
fromDoubleRatio = doubleToRatio;

fromIntMyInt :: MyInt  ->  MyInt;
fromIntMyInt x = x;

fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
fsEsOrdering x y = not (esEsOrdering x y);

gcd yx yy = gcd3 yx yy;
gcd x y = gcd0 x y;

gcd0 x y = gcd0Gcd' (absMyInt x) (absMyInt y);

gcd0Gcd' x xx = gcd0Gcd'2 x xx;
gcd0Gcd' x y = gcd0Gcd'0 x y;

gcd0Gcd'0 x y = gcd0Gcd' y (remMyInt x y);

gcd0Gcd'1 MyTrue x xx = x;
gcd0Gcd'1 xy xz yu = gcd0Gcd'0 xz yu;

gcd0Gcd'2 x xx = gcd0Gcd'1 (esEsMyInt xx (fromIntMyInt (Main.Pos Main.Zero))) x xx;
gcd0Gcd'2 yv yw = gcd0Gcd'0 yv yw;

gcd1 MyTrue yx yy = Main.error;
gcd1 yz zu zv = gcd0 zu zv;

gcd2 MyTrue yx yy = gcd1 (esEsMyInt yy (fromIntMyInt (Main.Pos Main.Zero))) yx yy;
gcd2 zw zx zy = gcd0 zx zy;

gcd3 yx yy = gcd2 (esEsMyInt yx (fromIntMyInt (Main.Pos Main.Zero))) yx yy;
gcd3 zz vuu = gcd0 zz vuu;

gtEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
gtEsMyInt x y = fsEsOrdering (compareMyInt x y) LT;

gtMyInt :: MyInt  ->  MyInt  ->  MyBool;
gtMyInt x y = esEsOrdering (compareMyInt x y) GT;

negateMyInt :: MyInt  ->  MyInt;
negateMyInt = primNegInt;

not :: MyBool  ->  MyBool;
not MyTrue = MyFalse;
not MyFalse = MyTrue;

otherwise :: MyBool;
otherwise = MyTrue;

pc x y = reduce (srMyInt x (signumMyInt y)) (absMyInt y);

primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
primCmpInt (Main.Pos x) (Main.Neg y) = GT;
primCmpInt (Main.Neg x) (Main.Pos y) = LT;
primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;

primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
primCmpNat Main.Zero Main.Zero = EQ;
primCmpNat Main.Zero (Main.Succ y) = LT;
primCmpNat (Main.Succ x) Main.Zero = GT;
primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;

primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primDivNatS Main.Zero Main.Zero = Main.error;
primDivNatS (Main.Succ x) Main.Zero = Main.error;
primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
primDivNatS Main.Zero (Main.Succ x) = Main.Zero;

primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
primDivNatS0 x y MyFalse = Main.Zero;

primEqInt :: MyInt  ->  MyInt  ->  MyBool;
primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt xv xw = MyFalse;

primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
primEqNat Main.Zero Main.Zero = MyTrue;
primEqNat Main.Zero (Main.Succ y) = MyFalse;
primEqNat (Main.Succ x) Main.Zero = MyFalse;
primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;

primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
primGEqNatS Main.Zero Main.Zero = MyTrue;

primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
primMinusNatS x Main.Zero = x;

primModNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primModNatS Main.Zero Main.Zero = Main.error;
primModNatS Main.Zero (Main.Succ x) = Main.Zero;
primModNatS (Main.Succ x) Main.Zero = Main.error;
primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y));

primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
primModNatS0 x y MyFalse = Main.Succ x;

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primMulNat Main.Zero Main.Zero = Main.Zero;
primMulNat Main.Zero (Main.Succ y) = Main.Zero;
primMulNat (Main.Succ x) Main.Zero = Main.Zero;
primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);

primNegInt :: MyInt  ->  MyInt;
primNegInt (Main.Pos x) = Main.Neg x;
primNegInt (Main.Neg x) = Main.Pos x;

primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primPlusNat Main.Zero Main.Zero = Main.Zero;
primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));

primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt wz xu = Main.error;

primRemInt :: MyInt  ->  MyInt  ->  MyInt;
primRemInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
primRemInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
primRemInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
primRemInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
primRemInt wx wy = Main.error;

quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
quotMyInt = primQuotInt;

reduce x y = reduce2 x y;

reduce2 x y = reduce2Reduce1 x y x y (esEsMyInt y (fromIntMyInt (Main.Pos Main.Zero)));

reduce2D vuv vuw = gcd vuv vuw;

reduce2Reduce0 vuv vuw x y MyTrue = CnPc (quotMyInt x (reduce2D vuv vuw)) (quotMyInt y (reduce2D vuv vuw));

reduce2Reduce1 vuv vuw x y MyTrue = Main.error;
reduce2Reduce1 vuv vuw x y MyFalse = reduce2Reduce0 vuv vuw x y otherwise;

remMyInt :: MyInt  ->  MyInt  ->  MyInt;
remMyInt = primRemInt;

signumMyInt :: MyInt  ->  MyInt;
signumMyInt = signumReal;

signumReal x = signumReal3 x;

signumReal0 x MyTrue = fromIntMyInt (Main.Neg (Main.Succ Main.Zero));

signumReal1 x MyTrue = fromIntMyInt (Main.Pos (Main.Succ Main.Zero));
signumReal1 x MyFalse = signumReal0 x otherwise;

signumReal2 x MyTrue = fromIntMyInt (Main.Pos Main.Zero);
signumReal2 x MyFalse = signumReal1 x (gtMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

signumReal3 x = signumReal2 x (esEsMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

srMyInt :: MyInt  ->  MyInt  ->  MyInt;
srMyInt = primMulInt;

stop :: MyBool  ->  a;
stop MyFalse = stop MyFalse;

}