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


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


YES
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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 be proven:

(0) HASKELL
(1) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) AND
    (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (12) YES
    (13) QDP
        (14) DependencyGraphProof [EQUIVALENT, 0 ms]
        (15) QDP
        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (17) YES


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

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

data Float = Float MyInt MyInt ;

data Integer = Integer 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 ;

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

doubleToFloat :: Double  ->  Float;
doubleToFloat (Double x y) = Float x y;

encodeFloatFloat :: Integer  ->  MyInt  ->  Float;
encodeFloatFloat = primFloatEncode;

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;

fromDoubleFloat :: Double  ->  Float;
fromDoubleFloat = doubleToFloat;

fromIntFloat :: MyInt  ->  Float;
fromIntFloat = primIntToFloat;

fromIntegerFloat :: Integer  ->  Float;
fromIntegerFloat = primIntegerToFloat;

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

fsFloat :: Float  ->  Float  ->  Float;
fsFloat = primDivFloat;

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

msMyInt :: MyInt  ->  MyInt  ->  MyInt;
msMyInt = primMinusInt;

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

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

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;

primDivFloat :: Float  ->  Float  ->  Float;
primDivFloat (Float x1 x2) (Float y1 y2) = Float (srMyInt x1 y2) (srMyInt x2 y1);

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 vx vy = 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;

primFloatEncode :: Integer  ->  MyInt  ->  Float;
primFloatEncode x y = srFloat (fromIntegerFloat x) (primFloatEncodePower (Main.Pos (Main.Succ (Main.Succ Main.Zero))) y);

primFloatEncodePower :: MyInt  ->  MyInt  ->  Float;
primFloatEncodePower vv wy = primFloatEncodePower4 vv wy;
primFloatEncodePower x vw = primFloatEncodePower2 x vw;
primFloatEncodePower x y = primFloatEncodePower0 x y;

primFloatEncodePower0 x y = fsFloat (fromDoubleFloat (Double (Main.Pos (Main.Succ Main.Zero)) (Main.Pos (Main.Succ Main.Zero)))) (primFloatEncodePower x (negateMyInt y));

primFloatEncodePower1 MyTrue x vw = srFloat (fromIntFloat x) (primFloatEncodePower x (msMyInt vw (Main.Pos (Main.Succ Main.Zero))));
primFloatEncodePower1 vz wu wv = primFloatEncodePower0 wu wv;

primFloatEncodePower2 x vw = primFloatEncodePower1 (gtEsMyInt vw (Main.Pos (Main.Succ Main.Zero))) x vw;
primFloatEncodePower2 ww wx = primFloatEncodePower0 ww wx;

primFloatEncodePower3 MyTrue vv wy = fromDoubleFloat (Double (Main.Pos (Main.Succ Main.Zero)) (Main.Pos (Main.Succ Main.Zero)));
primFloatEncodePower3 wz xu xv = primFloatEncodePower2 xu xv;

primFloatEncodePower4 vv wy = primFloatEncodePower3 (esEsMyInt wy (Main.Pos Main.Zero)) vv wy;
primFloatEncodePower4 xw xx = primFloatEncodePower2 xw xx;

primIntToFloat :: MyInt  ->  Float;
primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));

primIntegerToFloat :: Integer  ->  Float;
primIntegerToFloat (Integer x) = primIntToFloat x;

primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;

primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;

primMulFloat :: Float  ->  Float  ->  Float;
primMulFloat (Float x1 x2) (Float y1 y2) = Float (srMyInt x1 y1) (srMyInt x2 y2);

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));

srFloat :: Float  ->  Float  ->  Float;
srFloat = primMulFloat;

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

}

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

(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 Float = Float MyInt MyInt ;

data Integer = Integer 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 ;

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

doubleToFloat :: Double  ->  Float;
doubleToFloat (Double x y) = Float x y;

encodeFloatFloat :: Integer  ->  MyInt  ->  Float;
encodeFloatFloat = primFloatEncode;

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;

fromDoubleFloat :: Double  ->  Float;
fromDoubleFloat = doubleToFloat;

fromIntFloat :: MyInt  ->  Float;
fromIntFloat = primIntToFloat;

fromIntegerFloat :: Integer  ->  Float;
fromIntegerFloat = primIntegerToFloat;

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

fsFloat :: Float  ->  Float  ->  Float;
fsFloat = primDivFloat;

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

msMyInt :: MyInt  ->  MyInt  ->  MyInt;
msMyInt = primMinusInt;

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

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

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;

primDivFloat :: Float  ->  Float  ->  Float;
primDivFloat (Float x1 x2) (Float y1 y2) = Float (srMyInt x1 y2) (srMyInt x2 y1);

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 vx vy = 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;

primFloatEncode :: Integer  ->  MyInt  ->  Float;
primFloatEncode x y = srFloat (fromIntegerFloat x) (primFloatEncodePower (Main.Pos (Main.Succ (Main.Succ Main.Zero))) y);

primFloatEncodePower :: MyInt  ->  MyInt  ->  Float;
primFloatEncodePower vv wy = primFloatEncodePower4 vv wy;
primFloatEncodePower x vw = primFloatEncodePower2 x vw;
primFloatEncodePower x y = primFloatEncodePower0 x y;

primFloatEncodePower0 x y = fsFloat (fromDoubleFloat (Double (Main.Pos (Main.Succ Main.Zero)) (Main.Pos (Main.Succ Main.Zero)))) (primFloatEncodePower x (negateMyInt y));

primFloatEncodePower1 MyTrue x vw = srFloat (fromIntFloat x) (primFloatEncodePower x (msMyInt vw (Main.Pos (Main.Succ Main.Zero))));
primFloatEncodePower1 vz wu wv = primFloatEncodePower0 wu wv;

primFloatEncodePower2 x vw = primFloatEncodePower1 (gtEsMyInt vw (Main.Pos (Main.Succ Main.Zero))) x vw;
primFloatEncodePower2 ww wx = primFloatEncodePower0 ww wx;

primFloatEncodePower3 MyTrue vv wy = fromDoubleFloat (Double (Main.Pos (Main.Succ Main.Zero)) (Main.Pos (Main.Succ Main.Zero)));
primFloatEncodePower3 wz xu xv = primFloatEncodePower2 xu xv;

primFloatEncodePower4 vv wy = primFloatEncodePower3 (esEsMyInt wy (Main.Pos Main.Zero)) vv wy;
primFloatEncodePower4 xw xx = primFloatEncodePower2 xw xx;

primIntToFloat :: MyInt  ->  Float;
primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));

primIntegerToFloat :: Integer  ->  Float;
primIntegerToFloat (Integer x) = primIntToFloat x;

primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;

primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;

primMulFloat :: Float  ->  Float  ->  Float;
primMulFloat (Float x1 x2) (Float y1 y2) = Float (srMyInt x1 y1) (srMyInt x2 y2);

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));

srFloat :: Float  ->  Float  ->  Float;
srFloat = primMulFloat;

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

}

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

(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 Float = Float MyInt MyInt ;

data Integer = Integer 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 ;

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

doubleToFloat :: Double  ->  Float;
doubleToFloat (Double x y) = Float x y;

encodeFloatFloat :: Integer  ->  MyInt  ->  Float;
encodeFloatFloat = primFloatEncode;

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;

fromDoubleFloat :: Double  ->  Float;
fromDoubleFloat = doubleToFloat;

fromIntFloat :: MyInt  ->  Float;
fromIntFloat = primIntToFloat;

fromIntegerFloat :: Integer  ->  Float;
fromIntegerFloat = primIntegerToFloat;

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

fsFloat :: Float  ->  Float  ->  Float;
fsFloat = primDivFloat;

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

msMyInt :: MyInt  ->  MyInt  ->  MyInt;
msMyInt = primMinusInt;

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

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

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;

primDivFloat :: Float  ->  Float  ->  Float;
primDivFloat (Float x1 x2) (Float y1 y2) = Float (srMyInt x1 y2) (srMyInt x2 y1);

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 vx vy = 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;

primFloatEncode :: Integer  ->  MyInt  ->  Float;
primFloatEncode x y = srFloat (fromIntegerFloat x) (primFloatEncodePower (Main.Pos (Main.Succ (Main.Succ Main.Zero))) y);

primFloatEncodePower :: MyInt  ->  MyInt  ->  Float;
primFloatEncodePower vv wy = primFloatEncodePower4 vv wy;
primFloatEncodePower x vw = primFloatEncodePower2 x vw;
primFloatEncodePower x y = primFloatEncodePower0 x y;

primFloatEncodePower0 x y = fsFloat (fromDoubleFloat (Double (Main.Pos (Main.Succ Main.Zero)) (Main.Pos (Main.Succ Main.Zero)))) (primFloatEncodePower x (negateMyInt y));

primFloatEncodePower1 MyTrue x vw = srFloat (fromIntFloat x) (primFloatEncodePower x (msMyInt vw (Main.Pos (Main.Succ Main.Zero))));
primFloatEncodePower1 vz wu wv = primFloatEncodePower0 wu wv;

primFloatEncodePower2 x vw = primFloatEncodePower1 (gtEsMyInt vw (Main.Pos (Main.Succ Main.Zero))) x vw;
primFloatEncodePower2 ww wx = primFloatEncodePower0 ww wx;

primFloatEncodePower3 MyTrue vv wy = fromDoubleFloat (Double (Main.Pos (Main.Succ Main.Zero)) (Main.Pos (Main.Succ Main.Zero)));
primFloatEncodePower3 wz xu xv = primFloatEncodePower2 xu xv;

primFloatEncodePower4 vv wy = primFloatEncodePower3 (esEsMyInt wy (Main.Pos Main.Zero)) vv wy;
primFloatEncodePower4 xw xx = primFloatEncodePower2 xw xx;

primIntToFloat :: MyInt  ->  Float;
primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));

primIntegerToFloat :: Integer  ->  Float;
primIntegerToFloat (Integer x) = primIntToFloat x;

primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;

primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;

primMulFloat :: Float  ->  Float  ->  Float;
primMulFloat (Float x1 x2) (Float y1 y2) = Float (srMyInt x1 y1) (srMyInt x2 y2);

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));

srFloat :: Float  ->  Float  ->  Float;
srFloat = primMulFloat;

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

}

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

(5) Narrow (SOUND)
Haskell To QDPs

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185[label="primMulNat (Succ yu3000) (Succ yu6000)",fontsize=16,color="black",shape="box"];185 -> 191[label="",style="solid", color="black", weight=3];
186[label="primMulNat (Succ yu3000) Zero",fontsize=16,color="black",shape="box"];186 -> 192[label="",style="solid", color="black", weight=3];
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198 -> 170[label="",style="dashed", color="red", weight=0];
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198 -> 202[label="",style="dashed", color="magenta", weight=3];
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288 -> 203[label="",style="solid", color="burlywood", weight=3];
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291 -> 206[label="",style="solid", color="burlywood", weight=3];
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292 -> 217[label="",style="solid", color="burlywood", weight=3];
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293 -> 218[label="",style="solid", color="burlywood", weight=3];
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294 -> 224[label="",style="solid", color="burlywood", weight=3];
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295 -> 225[label="",style="solid", color="burlywood", weight=3];
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220[label="primFloatEncodePower1 MyTrue (Pos (Succ (Succ Zero))) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];220 -> 229[label="",style="solid", color="black", weight=3];
222 -> 160[label="",style="dashed", color="red", weight=0];
222[label="fromDoubleFloat (Double (Pos (Succ Zero)) (Pos (Succ Zero)))",fontsize=16,color="magenta"];223 -> 88[label="",style="dashed", color="red", weight=0];
223[label="primFloatEncodePower (Pos (Succ (Succ Zero))) (negateMyInt (Neg (Succ yu400)))",fontsize=16,color="magenta"];223 -> 230[label="",style="dashed", color="magenta", weight=3];
221[label="fsFloat yu9 yu8",fontsize=16,color="black",shape="triangle"];221 -> 231[label="",style="solid", color="black", weight=3];
224[label="primPlusNat (Succ yu700) (Succ yu60000)",fontsize=16,color="black",shape="box"];224 -> 232[label="",style="solid", color="black", weight=3];
225[label="primPlusNat (Succ yu700) Zero",fontsize=16,color="black",shape="box"];225 -> 233[label="",style="solid", color="black", weight=3];
226[label="primPlusNat Zero (Succ yu60000)",fontsize=16,color="black",shape="box"];226 -> 234[label="",style="solid", color="black", weight=3];
227[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];227 -> 235[label="",style="solid", color="black", weight=3];
228 -> 236[label="",style="dashed", color="red", weight=0];
228[label="srFloat (fromIntFloat (Pos (Succ (Succ Zero)))) (primFloatEncodePower (Pos (Succ (Succ Zero))) (msMyInt (Pos (Succ (Succ yu4000))) (Pos (Succ Zero))))",fontsize=16,color="magenta"];228 -> 237[label="",style="dashed", color="magenta", weight=3];
229 -> 236[label="",style="dashed", color="red", weight=0];
229[label="srFloat (fromIntFloat (Pos (Succ (Succ Zero)))) (primFloatEncodePower (Pos (Succ (Succ Zero))) (msMyInt (Pos (Succ Zero)) (Pos (Succ Zero))))",fontsize=16,color="magenta"];229 -> 238[label="",style="dashed", color="magenta", weight=3];
230[label="negateMyInt (Neg (Succ yu400))",fontsize=16,color="black",shape="box"];230 -> 239[label="",style="solid", color="black", weight=3];
231[label="primDivFloat yu9 yu8",fontsize=16,color="burlywood",shape="box"];298[label="yu9/Float yu90 yu91",fontsize=10,color="white",style="solid",shape="box"];231 -> 298[label="",style="solid", color="burlywood", weight=9];
298 -> 240[label="",style="solid", color="burlywood", weight=3];
232[label="Succ (Succ (primPlusNat yu700 yu60000))",fontsize=16,color="green",shape="box"];232 -> 241[label="",style="dashed", color="green", weight=3];
233[label="Succ yu700",fontsize=16,color="green",shape="box"];234[label="Succ yu60000",fontsize=16,color="green",shape="box"];235[label="Zero",fontsize=16,color="green",shape="box"];237 -> 88[label="",style="dashed", color="red", weight=0];
237[label="primFloatEncodePower (Pos (Succ (Succ Zero))) (msMyInt (Pos (Succ (Succ yu4000))) (Pos (Succ Zero)))",fontsize=16,color="magenta"];237 -> 242[label="",style="dashed", color="magenta", weight=3];
236[label="srFloat (fromIntFloat (Pos (Succ (Succ Zero)))) yu10",fontsize=16,color="black",shape="triangle"];236 -> 243[label="",style="solid", color="black", weight=3];
238 -> 88[label="",style="dashed", color="red", weight=0];
238[label="primFloatEncodePower (Pos (Succ (Succ Zero))) (msMyInt (Pos (Succ Zero)) (Pos (Succ Zero)))",fontsize=16,color="magenta"];238 -> 244[label="",style="dashed", color="magenta", weight=3];
239[label="primNegInt (Neg (Succ yu400))",fontsize=16,color="black",shape="box"];239 -> 245[label="",style="solid", color="black", weight=3];
240[label="primDivFloat (Float yu90 yu91) yu8",fontsize=16,color="burlywood",shape="box"];299[label="yu8/Float yu80 yu81",fontsize=10,color="white",style="solid",shape="box"];240 -> 299[label="",style="solid", color="burlywood", weight=9];
299 -> 246[label="",style="solid", color="burlywood", weight=3];
241 -> 213[label="",style="dashed", color="red", weight=0];
241[label="primPlusNat yu700 yu60000",fontsize=16,color="magenta"];241 -> 247[label="",style="dashed", color="magenta", weight=3];
241 -> 248[label="",style="dashed", color="magenta", weight=3];
242[label="msMyInt (Pos (Succ (Succ yu4000))) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];242 -> 249[label="",style="solid", color="black", weight=3];
243[label="primMulFloat (fromIntFloat (Pos (Succ (Succ Zero)))) yu10",fontsize=16,color="black",shape="box"];243 -> 250[label="",style="solid", color="black", weight=3];
244[label="msMyInt (Pos (Succ Zero)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];244 -> 251[label="",style="solid", color="black", weight=3];
245[label="Pos (Succ yu400)",fontsize=16,color="green",shape="box"];246[label="primDivFloat (Float yu90 yu91) (Float yu80 yu81)",fontsize=16,color="black",shape="box"];246 -> 252[label="",style="solid", color="black", weight=3];
247[label="yu700",fontsize=16,color="green",shape="box"];248[label="yu60000",fontsize=16,color="green",shape="box"];249[label="primMinusInt (Pos (Succ (Succ yu4000))) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];249 -> 253[label="",style="solid", color="black", weight=3];
250[label="primMulFloat (primIntToFloat (Pos (Succ (Succ Zero)))) yu10",fontsize=16,color="black",shape="box"];250 -> 254[label="",style="solid", color="black", weight=3];
251[label="primMinusInt (Pos (Succ Zero)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];251 -> 255[label="",style="solid", color="black", weight=3];
252[label="Float (srMyInt yu90 yu81) (srMyInt yu91 yu80)",fontsize=16,color="green",shape="box"];252 -> 256[label="",style="dashed", color="green", weight=3];
252 -> 257[label="",style="dashed", color="green", weight=3];
253[label="primMinusNat (Succ (Succ yu4000)) (Succ Zero)",fontsize=16,color="black",shape="box"];253 -> 258[label="",style="solid", color="black", weight=3];
254 -> 87[label="",style="dashed", color="red", weight=0];
254[label="primMulFloat (Float (Pos (Succ (Succ Zero))) (Pos (Succ Zero))) yu10",fontsize=16,color="magenta"];254 -> 259[label="",style="dashed", color="magenta", weight=3];
254 -> 260[label="",style="dashed", color="magenta", weight=3];
255[label="primMinusNat (Succ Zero) (Succ Zero)",fontsize=16,color="black",shape="box"];255 -> 261[label="",style="solid", color="black", weight=3];
256 -> 138[label="",style="dashed", color="red", weight=0];
256[label="srMyInt yu90 yu81",fontsize=16,color="magenta"];256 -> 262[label="",style="dashed", color="magenta", weight=3];
256 -> 263[label="",style="dashed", color="magenta", weight=3];
257 -> 138[label="",style="dashed", color="red", weight=0];
257[label="srMyInt yu91 yu80",fontsize=16,color="magenta"];257 -> 264[label="",style="dashed", color="magenta", weight=3];
257 -> 265[label="",style="dashed", color="magenta", weight=3];
258[label="primMinusNat (Succ yu4000) Zero",fontsize=16,color="black",shape="box"];258 -> 266[label="",style="solid", color="black", weight=3];
259[label="Pos (Succ (Succ Zero))",fontsize=16,color="green",shape="box"];260[label="yu10",fontsize=16,color="green",shape="box"];261[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];261 -> 267[label="",style="solid", color="black", weight=3];
262[label="yu90",fontsize=16,color="green",shape="box"];263[label="yu81",fontsize=16,color="green",shape="box"];264[label="yu91",fontsize=16,color="green",shape="box"];265[label="yu80",fontsize=16,color="green",shape="box"];266[label="Pos (Succ yu4000)",fontsize=16,color="green",shape="box"];267[label="Pos Zero",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primMulNat(Main.Succ(yu3000), Main.Succ(yu6000)) -> new_primMulNat(yu3000, Main.Succ(yu6000))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(8) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_primMulNat(Main.Succ(yu3000), Main.Succ(yu6000)) -> new_primMulNat(yu3000, Main.Succ(yu6000))
The graph contains the following edges 1 > 1, 2 >= 2


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

(9)
YES

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

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primPlusNat(Main.Succ(yu700), Main.Succ(yu60000)) -> new_primPlusNat(yu700, yu60000)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_primPlusNat(Main.Succ(yu700), Main.Succ(yu60000)) -> new_primPlusNat(yu700, yu60000)
The graph contains the following edges 1 > 1, 2 > 2


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

(12)
YES

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

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primFloatEncodePower(Main.Neg(Main.Succ(yu400))) -> new_primFloatEncodePower(Main.Pos(Main.Succ(yu400)))
   new_primFloatEncodePower(Main.Pos(Main.Succ(Main.Succ(yu4000)))) -> new_primFloatEncodePower(Main.Pos(Main.Succ(yu4000)))
   new_primFloatEncodePower(Main.Pos(Main.Succ(Main.Zero))) -> new_primFloatEncodePower(Main.Pos(Main.Zero))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(14) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primFloatEncodePower(Main.Pos(Main.Succ(Main.Succ(yu4000)))) -> new_primFloatEncodePower(Main.Pos(Main.Succ(yu4000)))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(16) QDPSizeChangeProof (EQUIVALENT)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 

Order:EMB rules!

AFS: 
Main.Pos(x1)  =  x1

Main.Succ(x1)  =  Main.Succ(x1)





From the DPs we obtained the following set of size-change graphs:
*new_primFloatEncodePower(Main.Pos(Main.Succ(Main.Succ(yu4000)))) -> new_primFloatEncodePower(Main.Pos(Main.Succ(yu4000))) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1



We oriented the following set of usable rules [AAECC05,FROCOS05].
none

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

(17)
YES