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


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


YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) DependencyGraphProof [EQUIVALENT, 0 ms]
        (12) AND
            (13) QDP
                (14) QDPSizeChangeProof [EQUIVALENT, 27 ms]
                (15) YES
            (16) QDP
                (17) MRRProof [EQUIVALENT, 0 ms]
                (18) QDP
                (19) PisEmptyProof [EQUIVALENT, 0 ms]
                (20) YES


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

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

data MyBool = MyTrue  | MyFalse ;

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

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

data Tup2 b a = Tup2 b a ;

error :: a;
error = stop MyTrue;

floatProperFractionFloat (Float wy wz) = Tup2 (fromIntMyInt (quotMyInt wy wz)) (msFloat (Float wy wz) (fromIntFloat (quotMyInt wy wz)));

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

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

msFloat :: Float  ->  Float  ->  Float;
msFloat = primMinusFloat;

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

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;

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;

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

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

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;

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;

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);

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 ww wx = Main.error;

properFractionFloat :: Float  ->  Tup2 MyInt Float;
properFractionFloat = floatProperFractionFloat;

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

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

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

truncateFloat :: Float  ->  MyInt;
truncateFloat x = truncateM x;

truncateM xu = truncateM0 xu (truncateVu6 xu);

truncateM0 xu (Tup2 m vv) = m;

truncateVu6 xu = properFractionFloat xu;

}

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

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

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

data MyBool = MyTrue  | MyFalse ;

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

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

data Tup2 a b = Tup2 a b ;

error :: a;
error = stop MyTrue;

floatProperFractionFloat (Float wy wz) = Tup2 (fromIntMyInt (quotMyInt wy wz)) (msFloat (Float wy wz) (fromIntFloat (quotMyInt wy wz)));

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

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

msFloat :: Float  ->  Float  ->  Float;
msFloat = primMinusFloat;

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

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;

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;

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

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

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;

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;

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);

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 ww wx = Main.error;

properFractionFloat :: Float  ->  Tup2 MyInt Float;
properFractionFloat = floatProperFractionFloat;

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

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

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

truncateFloat :: Float  ->  MyInt;
truncateFloat x = truncateM x;

truncateM xu = truncateM0 xu (truncateVu6 xu);

truncateM0 xu (Tup2 m vv) = m;

truncateVu6 xu = properFractionFloat xu;

}

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

(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 Float = Float MyInt MyInt ;

data MyBool = MyTrue  | MyFalse ;

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

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

data Tup2 b a = Tup2 b a ;

error :: a;
error = stop MyTrue;

floatProperFractionFloat (Float wy wz) = Tup2 (fromIntMyInt (quotMyInt wy wz)) (msFloat (Float wy wz) (fromIntFloat (quotMyInt wy wz)));

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

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

msFloat :: Float  ->  Float  ->  Float;
msFloat = primMinusFloat;

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

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;

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;

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

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

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;

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;

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);

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 ww wx = Main.error;

properFractionFloat :: Float  ->  Tup2 MyInt Float;
properFractionFloat = floatProperFractionFloat;

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

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

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

truncateFloat :: Float  ->  MyInt;
truncateFloat x = truncateM x;

truncateM xu = truncateM0 xu (truncateVu6 xu);

truncateM0 xu (Tup2 m vv) = m;

truncateVu6 xu = properFractionFloat xu;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="truncateFloat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="truncateFloat vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="truncateM vy3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="truncateM0 vy3 (truncateVu6 vy3)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="truncateM0 vy3 (properFractionFloat vy3)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="truncateM0 vy3 (floatProperFractionFloat vy3)",fontsize=16,color="burlywood",shape="box"];282[label="vy3/Float vy30 vy31",fontsize=10,color="white",style="solid",shape="box"];7 -> 282[label="",style="solid", color="burlywood", weight=9];
282 -> 8[label="",style="solid", color="burlywood", weight=3];
8[label="truncateM0 (Float vy30 vy31) (floatProperFractionFloat (Float vy30 vy31))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="truncateM0 (Float vy30 vy31) (Tup2 (fromIntMyInt (quotMyInt vy30 vy31)) (msFloat (Float vy30 vy31) (fromIntFloat (quotMyInt vy30 vy31))))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="fromIntMyInt (quotMyInt vy30 vy31)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="quotMyInt vy30 vy31",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="primQuotInt vy30 vy31",fontsize=16,color="burlywood",shape="box"];283[label="vy30/Pos vy300",fontsize=10,color="white",style="solid",shape="box"];12 -> 283[label="",style="solid", color="burlywood", weight=9];
283 -> 13[label="",style="solid", color="burlywood", weight=3];
284[label="vy30/Neg vy300",fontsize=10,color="white",style="solid",shape="box"];12 -> 284[label="",style="solid", color="burlywood", weight=9];
284 -> 14[label="",style="solid", color="burlywood", weight=3];
13[label="primQuotInt (Pos vy300) vy31",fontsize=16,color="burlywood",shape="box"];285[label="vy31/Pos vy310",fontsize=10,color="white",style="solid",shape="box"];13 -> 285[label="",style="solid", color="burlywood", weight=9];
285 -> 15[label="",style="solid", color="burlywood", weight=3];
286[label="vy31/Neg vy310",fontsize=10,color="white",style="solid",shape="box"];13 -> 286[label="",style="solid", color="burlywood", weight=9];
286 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="primQuotInt (Neg vy300) vy31",fontsize=16,color="burlywood",shape="box"];287[label="vy31/Pos vy310",fontsize=10,color="white",style="solid",shape="box"];14 -> 287[label="",style="solid", color="burlywood", weight=9];
287 -> 17[label="",style="solid", color="burlywood", weight=3];
288[label="vy31/Neg vy310",fontsize=10,color="white",style="solid",shape="box"];14 -> 288[label="",style="solid", color="burlywood", weight=9];
288 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="primQuotInt (Pos vy300) (Pos vy310)",fontsize=16,color="burlywood",shape="box"];289[label="vy310/Succ vy3100",fontsize=10,color="white",style="solid",shape="box"];15 -> 289[label="",style="solid", color="burlywood", weight=9];
289 -> 19[label="",style="solid", color="burlywood", weight=3];
290[label="vy310/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 290[label="",style="solid", color="burlywood", weight=9];
290 -> 20[label="",style="solid", color="burlywood", weight=3];
16[label="primQuotInt (Pos vy300) (Neg vy310)",fontsize=16,color="burlywood",shape="box"];291[label="vy310/Succ vy3100",fontsize=10,color="white",style="solid",shape="box"];16 -> 291[label="",style="solid", color="burlywood", weight=9];
291 -> 21[label="",style="solid", color="burlywood", weight=3];
292[label="vy310/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 292[label="",style="solid", color="burlywood", weight=9];
292 -> 22[label="",style="solid", color="burlywood", weight=3];
17[label="primQuotInt (Neg vy300) (Pos vy310)",fontsize=16,color="burlywood",shape="box"];293[label="vy310/Succ vy3100",fontsize=10,color="white",style="solid",shape="box"];17 -> 293[label="",style="solid", color="burlywood", weight=9];
293 -> 23[label="",style="solid", color="burlywood", weight=3];
294[label="vy310/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 294[label="",style="solid", color="burlywood", weight=9];
294 -> 24[label="",style="solid", color="burlywood", weight=3];
18[label="primQuotInt (Neg vy300) (Neg vy310)",fontsize=16,color="burlywood",shape="box"];295[label="vy310/Succ vy3100",fontsize=10,color="white",style="solid",shape="box"];18 -> 295[label="",style="solid", color="burlywood", weight=9];
295 -> 25[label="",style="solid", color="burlywood", weight=3];
296[label="vy310/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 296[label="",style="solid", color="burlywood", weight=9];
296 -> 26[label="",style="solid", color="burlywood", weight=3];
19[label="primQuotInt (Pos vy300) (Pos (Succ vy3100))",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
20[label="primQuotInt (Pos vy300) (Pos Zero)",fontsize=16,color="black",shape="box"];20 -> 28[label="",style="solid", color="black", weight=3];
21[label="primQuotInt (Pos vy300) (Neg (Succ vy3100))",fontsize=16,color="black",shape="box"];21 -> 29[label="",style="solid", color="black", weight=3];
22[label="primQuotInt (Pos vy300) (Neg Zero)",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3];
23[label="primQuotInt (Neg vy300) (Pos (Succ vy3100))",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
24[label="primQuotInt (Neg vy300) (Pos Zero)",fontsize=16,color="black",shape="box"];24 -> 32[label="",style="solid", color="black", weight=3];
25[label="primQuotInt (Neg vy300) (Neg (Succ vy3100))",fontsize=16,color="black",shape="box"];25 -> 33[label="",style="solid", color="black", weight=3];
26[label="primQuotInt (Neg vy300) (Neg Zero)",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
27[label="Pos (primDivNatS vy300 (Succ vy3100))",fontsize=16,color="green",shape="box"];27 -> 35[label="",style="dashed", color="green", weight=3];
28[label="error",fontsize=16,color="black",shape="triangle"];28 -> 36[label="",style="solid", color="black", weight=3];
29[label="Neg (primDivNatS vy300 (Succ vy3100))",fontsize=16,color="green",shape="box"];29 -> 37[label="",style="dashed", color="green", weight=3];
30 -> 28[label="",style="dashed", color="red", weight=0];
30[label="error",fontsize=16,color="magenta"];31[label="Neg (primDivNatS vy300 (Succ vy3100))",fontsize=16,color="green",shape="box"];31 -> 38[label="",style="dashed", color="green", weight=3];
32 -> 28[label="",style="dashed", color="red", weight=0];
32[label="error",fontsize=16,color="magenta"];33[label="Pos (primDivNatS vy300 (Succ vy3100))",fontsize=16,color="green",shape="box"];33 -> 39[label="",style="dashed", color="green", weight=3];
34 -> 28[label="",style="dashed", color="red", weight=0];
34[label="error",fontsize=16,color="magenta"];35[label="primDivNatS vy300 (Succ vy3100)",fontsize=16,color="burlywood",shape="triangle"];297[label="vy300/Succ vy3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 297[label="",style="solid", color="burlywood", weight=9];
297 -> 40[label="",style="solid", color="burlywood", weight=3];
298[label="vy300/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 298[label="",style="solid", color="burlywood", weight=9];
298 -> 41[label="",style="solid", color="burlywood", weight=3];
36[label="stop MyTrue",fontsize=16,color="black",shape="box"];36 -> 42[label="",style="solid", color="black", weight=3];
37 -> 35[label="",style="dashed", color="red", weight=0];
37[label="primDivNatS vy300 (Succ vy3100)",fontsize=16,color="magenta"];37 -> 43[label="",style="dashed", color="magenta", weight=3];
38 -> 35[label="",style="dashed", color="red", weight=0];
38[label="primDivNatS vy300 (Succ vy3100)",fontsize=16,color="magenta"];38 -> 44[label="",style="dashed", color="magenta", weight=3];
39 -> 35[label="",style="dashed", color="red", weight=0];
39[label="primDivNatS vy300 (Succ vy3100)",fontsize=16,color="magenta"];39 -> 45[label="",style="dashed", color="magenta", weight=3];
39 -> 46[label="",style="dashed", color="magenta", weight=3];
40[label="primDivNatS (Succ vy3000) (Succ vy3100)",fontsize=16,color="black",shape="box"];40 -> 47[label="",style="solid", color="black", weight=3];
41[label="primDivNatS Zero (Succ vy3100)",fontsize=16,color="black",shape="box"];41 -> 48[label="",style="solid", color="black", weight=3];
42[label="error []",fontsize=16,color="red",shape="box"];43[label="vy3100",fontsize=16,color="green",shape="box"];44[label="vy300",fontsize=16,color="green",shape="box"];45[label="vy3100",fontsize=16,color="green",shape="box"];46[label="vy300",fontsize=16,color="green",shape="box"];47[label="primDivNatS0 vy3000 vy3100 (primGEqNatS vy3000 vy3100)",fontsize=16,color="burlywood",shape="box"];299[label="vy3000/Succ vy30000",fontsize=10,color="white",style="solid",shape="box"];47 -> 299[label="",style="solid", color="burlywood", weight=9];
299 -> 49[label="",style="solid", color="burlywood", weight=3];
300[label="vy3000/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 300[label="",style="solid", color="burlywood", weight=9];
300 -> 50[label="",style="solid", color="burlywood", weight=3];
48[label="Zero",fontsize=16,color="green",shape="box"];49[label="primDivNatS0 (Succ vy30000) vy3100 (primGEqNatS (Succ vy30000) vy3100)",fontsize=16,color="burlywood",shape="box"];301[label="vy3100/Succ vy31000",fontsize=10,color="white",style="solid",shape="box"];49 -> 301[label="",style="solid", color="burlywood", weight=9];
301 -> 51[label="",style="solid", color="burlywood", weight=3];
302[label="vy3100/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 302[label="",style="solid", color="burlywood", weight=9];
302 -> 52[label="",style="solid", color="burlywood", weight=3];
50[label="primDivNatS0 Zero vy3100 (primGEqNatS Zero vy3100)",fontsize=16,color="burlywood",shape="box"];303[label="vy3100/Succ vy31000",fontsize=10,color="white",style="solid",shape="box"];50 -> 303[label="",style="solid", color="burlywood", weight=9];
303 -> 53[label="",style="solid", color="burlywood", weight=3];
304[label="vy3100/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 304[label="",style="solid", color="burlywood", weight=9];
304 -> 54[label="",style="solid", color="burlywood", weight=3];
51[label="primDivNatS0 (Succ vy30000) (Succ vy31000) (primGEqNatS (Succ vy30000) (Succ vy31000))",fontsize=16,color="black",shape="box"];51 -> 55[label="",style="solid", color="black", weight=3];
52[label="primDivNatS0 (Succ vy30000) Zero (primGEqNatS (Succ vy30000) Zero)",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="primDivNatS0 Zero (Succ vy31000) (primGEqNatS Zero (Succ vy31000))",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
54[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3];
55 -> 219[label="",style="dashed", color="red", weight=0];
55[label="primDivNatS0 (Succ vy30000) (Succ vy31000) (primGEqNatS vy30000 vy31000)",fontsize=16,color="magenta"];55 -> 220[label="",style="dashed", color="magenta", weight=3];
55 -> 221[label="",style="dashed", color="magenta", weight=3];
55 -> 222[label="",style="dashed", color="magenta", weight=3];
55 -> 223[label="",style="dashed", color="magenta", weight=3];
56[label="primDivNatS0 (Succ vy30000) Zero MyTrue",fontsize=16,color="black",shape="box"];56 -> 61[label="",style="solid", color="black", weight=3];
57[label="primDivNatS0 Zero (Succ vy31000) MyFalse",fontsize=16,color="black",shape="box"];57 -> 62[label="",style="solid", color="black", weight=3];
58[label="primDivNatS0 Zero Zero MyTrue",fontsize=16,color="black",shape="box"];58 -> 63[label="",style="solid", color="black", weight=3];
220[label="vy31000",fontsize=16,color="green",shape="box"];221[label="vy30000",fontsize=16,color="green",shape="box"];222[label="vy30000",fontsize=16,color="green",shape="box"];223[label="vy31000",fontsize=16,color="green",shape="box"];219[label="primDivNatS0 (Succ vy20) (Succ vy21) (primGEqNatS vy22 vy23)",fontsize=16,color="burlywood",shape="triangle"];305[label="vy22/Succ vy220",fontsize=10,color="white",style="solid",shape="box"];219 -> 305[label="",style="solid", color="burlywood", weight=9];
305 -> 252[label="",style="solid", color="burlywood", weight=3];
306[label="vy22/Zero",fontsize=10,color="white",style="solid",shape="box"];219 -> 306[label="",style="solid", color="burlywood", weight=9];
306 -> 253[label="",style="solid", color="burlywood", weight=3];
61[label="Succ (primDivNatS (primMinusNatS (Succ vy30000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];61 -> 68[label="",style="dashed", color="green", weight=3];
62[label="Zero",fontsize=16,color="green",shape="box"];63[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];63 -> 69[label="",style="dashed", color="green", weight=3];
252[label="primDivNatS0 (Succ vy20) (Succ vy21) (primGEqNatS (Succ vy220) vy23)",fontsize=16,color="burlywood",shape="box"];307[label="vy23/Succ vy230",fontsize=10,color="white",style="solid",shape="box"];252 -> 307[label="",style="solid", color="burlywood", weight=9];
307 -> 254[label="",style="solid", color="burlywood", weight=3];
308[label="vy23/Zero",fontsize=10,color="white",style="solid",shape="box"];252 -> 308[label="",style="solid", color="burlywood", weight=9];
308 -> 255[label="",style="solid", color="burlywood", weight=3];
253[label="primDivNatS0 (Succ vy20) (Succ vy21) (primGEqNatS Zero vy23)",fontsize=16,color="burlywood",shape="box"];309[label="vy23/Succ vy230",fontsize=10,color="white",style="solid",shape="box"];253 -> 309[label="",style="solid", color="burlywood", weight=9];
309 -> 256[label="",style="solid", color="burlywood", weight=3];
310[label="vy23/Zero",fontsize=10,color="white",style="solid",shape="box"];253 -> 310[label="",style="solid", color="burlywood", weight=9];
310 -> 257[label="",style="solid", color="burlywood", weight=3];
68 -> 35[label="",style="dashed", color="red", weight=0];
68[label="primDivNatS (primMinusNatS (Succ vy30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];68 -> 74[label="",style="dashed", color="magenta", weight=3];
68 -> 75[label="",style="dashed", color="magenta", weight=3];
69 -> 35[label="",style="dashed", color="red", weight=0];
69[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];69 -> 76[label="",style="dashed", color="magenta", weight=3];
69 -> 77[label="",style="dashed", color="magenta", weight=3];
254[label="primDivNatS0 (Succ vy20) (Succ vy21) (primGEqNatS (Succ vy220) (Succ vy230))",fontsize=16,color="black",shape="box"];254 -> 258[label="",style="solid", color="black", weight=3];
255[label="primDivNatS0 (Succ vy20) (Succ vy21) (primGEqNatS (Succ vy220) Zero)",fontsize=16,color="black",shape="box"];255 -> 259[label="",style="solid", color="black", weight=3];
256[label="primDivNatS0 (Succ vy20) (Succ vy21) (primGEqNatS Zero (Succ vy230))",fontsize=16,color="black",shape="box"];256 -> 260[label="",style="solid", color="black", weight=3];
257[label="primDivNatS0 (Succ vy20) (Succ vy21) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];257 -> 261[label="",style="solid", color="black", weight=3];
74[label="Zero",fontsize=16,color="green",shape="box"];75[label="primMinusNatS (Succ vy30000) Zero",fontsize=16,color="black",shape="triangle"];75 -> 83[label="",style="solid", color="black", weight=3];
76[label="Zero",fontsize=16,color="green",shape="box"];77[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];77 -> 84[label="",style="solid", color="black", weight=3];
258 -> 219[label="",style="dashed", color="red", weight=0];
258[label="primDivNatS0 (Succ vy20) (Succ vy21) (primGEqNatS vy220 vy230)",fontsize=16,color="magenta"];258 -> 262[label="",style="dashed", color="magenta", weight=3];
258 -> 263[label="",style="dashed", color="magenta", weight=3];
259[label="primDivNatS0 (Succ vy20) (Succ vy21) MyTrue",fontsize=16,color="black",shape="triangle"];259 -> 264[label="",style="solid", color="black", weight=3];
260[label="primDivNatS0 (Succ vy20) (Succ vy21) MyFalse",fontsize=16,color="black",shape="box"];260 -> 265[label="",style="solid", color="black", weight=3];
261 -> 259[label="",style="dashed", color="red", weight=0];
261[label="primDivNatS0 (Succ vy20) (Succ vy21) MyTrue",fontsize=16,color="magenta"];83[label="Succ vy30000",fontsize=16,color="green",shape="box"];84[label="Zero",fontsize=16,color="green",shape="box"];262[label="vy230",fontsize=16,color="green",shape="box"];263[label="vy220",fontsize=16,color="green",shape="box"];264[label="Succ (primDivNatS (primMinusNatS (Succ vy20) (Succ vy21)) (Succ (Succ vy21)))",fontsize=16,color="green",shape="box"];264 -> 266[label="",style="dashed", color="green", weight=3];
265[label="Zero",fontsize=16,color="green",shape="box"];266 -> 35[label="",style="dashed", color="red", weight=0];
266[label="primDivNatS (primMinusNatS (Succ vy20) (Succ vy21)) (Succ (Succ vy21))",fontsize=16,color="magenta"];266 -> 267[label="",style="dashed", color="magenta", weight=3];
266 -> 268[label="",style="dashed", color="magenta", weight=3];
267[label="Succ vy21",fontsize=16,color="green",shape="box"];268[label="primMinusNatS (Succ vy20) (Succ vy21)",fontsize=16,color="black",shape="box"];268 -> 269[label="",style="solid", color="black", weight=3];
269[label="primMinusNatS vy20 vy21",fontsize=16,color="burlywood",shape="triangle"];311[label="vy20/Succ vy200",fontsize=10,color="white",style="solid",shape="box"];269 -> 311[label="",style="solid", color="burlywood", weight=9];
311 -> 270[label="",style="solid", color="burlywood", weight=3];
312[label="vy20/Zero",fontsize=10,color="white",style="solid",shape="box"];269 -> 312[label="",style="solid", color="burlywood", weight=9];
312 -> 271[label="",style="solid", color="burlywood", weight=3];
270[label="primMinusNatS (Succ vy200) vy21",fontsize=16,color="burlywood",shape="box"];313[label="vy21/Succ vy210",fontsize=10,color="white",style="solid",shape="box"];270 -> 313[label="",style="solid", color="burlywood", weight=9];
313 -> 272[label="",style="solid", color="burlywood", weight=3];
314[label="vy21/Zero",fontsize=10,color="white",style="solid",shape="box"];270 -> 314[label="",style="solid", color="burlywood", weight=9];
314 -> 273[label="",style="solid", color="burlywood", weight=3];
271[label="primMinusNatS Zero vy21",fontsize=16,color="burlywood",shape="box"];315[label="vy21/Succ vy210",fontsize=10,color="white",style="solid",shape="box"];271 -> 315[label="",style="solid", color="burlywood", weight=9];
315 -> 274[label="",style="solid", color="burlywood", weight=3];
316[label="vy21/Zero",fontsize=10,color="white",style="solid",shape="box"];271 -> 316[label="",style="solid", color="burlywood", weight=9];
316 -> 275[label="",style="solid", color="burlywood", weight=3];
272[label="primMinusNatS (Succ vy200) (Succ vy210)",fontsize=16,color="black",shape="box"];272 -> 276[label="",style="solid", color="black", weight=3];
273[label="primMinusNatS (Succ vy200) Zero",fontsize=16,color="black",shape="box"];273 -> 277[label="",style="solid", color="black", weight=3];
274[label="primMinusNatS Zero (Succ vy210)",fontsize=16,color="black",shape="box"];274 -> 278[label="",style="solid", color="black", weight=3];
275[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];275 -> 279[label="",style="solid", color="black", weight=3];
276 -> 269[label="",style="dashed", color="red", weight=0];
276[label="primMinusNatS vy200 vy210",fontsize=16,color="magenta"];276 -> 280[label="",style="dashed", color="magenta", weight=3];
276 -> 281[label="",style="dashed", color="magenta", weight=3];
277[label="Succ vy200",fontsize=16,color="green",shape="box"];278[label="Zero",fontsize=16,color="green",shape="box"];279[label="Zero",fontsize=16,color="green",shape="box"];280[label="vy200",fontsize=16,color="green",shape="box"];281[label="vy210",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

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

   new_primMinusNatS(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS(vy200, vy210)

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_primMinusNatS(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS(vy200, vy210)
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_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vy30000), Main.Zero)
   new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Succ(vy230)) -> new_primDivNatS0(vy20, vy21, vy220, vy230)
   new_primDivNatS00(vy20, vy21) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21))
   new_primDivNatS0(vy20, vy21, Main.Zero, Main.Zero) -> new_primDivNatS00(vy20, vy21)
   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Succ(vy31000)) -> new_primDivNatS0(vy30000, vy31000, vy30000, vy31000)
   new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21))
   new_primDivNatS(Main.Succ(Main.Zero), Main.Zero) -> new_primDivNatS(new_primMinusNatS2, Main.Zero)

The TRS R consists of the following rules:

   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
   new_primMinusNatS1(vy30000) -> Main.Succ(vy30000)
   new_primMinusNatS2 -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Zero, Main.Zero)
   new_primMinusNatS2
   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
   new_primMinusNatS1(x0)
   new_primMinusNatS0(Main.Zero, Main.Succ(x0))

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

(11) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
----------------------------------------

(12)
Complex Obligation (AND)

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

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

   new_primDivNatS0(vy20, vy21, Main.Zero, Main.Zero) -> new_primDivNatS00(vy20, vy21)
   new_primDivNatS00(vy20, vy21) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21))
   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Succ(vy31000)) -> new_primDivNatS0(vy30000, vy31000, vy30000, vy31000)
   new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Succ(vy230)) -> new_primDivNatS0(vy20, vy21, vy220, vy230)
   new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21))

The TRS R consists of the following rules:

   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
   new_primMinusNatS1(vy30000) -> Main.Succ(vy30000)
   new_primMinusNatS2 -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Zero, Main.Zero)
   new_primMinusNatS2
   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
   new_primMinusNatS1(x0)
   new_primMinusNatS0(Main.Zero, Main.Succ(x0))

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

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

Order:Polynomial interpretation [POLO]:

   POL(Main.Succ(x_1)) = 1 + x_1
   POL(Main.Zero) = 1
   POL(new_primMinusNatS0(x_1, x_2)) = x_1




From the DPs we obtained the following set of size-change graphs:
*new_primDivNatS00(vy20, vy21) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1


*new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Succ(vy31000)) -> new_primDivNatS0(vy30000, vy31000, vy30000, vy31000) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4


*new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Succ(vy230)) -> new_primDivNatS0(vy20, vy21, vy220, vy230) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


*new_primDivNatS0(vy20, vy21, Main.Zero, Main.Zero) -> new_primDivNatS00(vy20, vy21) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1, 2 >= 2


*new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1



We oriented the following set of usable rules [AAECC05,FROCOS05].

   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)

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

(15)
YES

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

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vy30000), Main.Zero)

The TRS R consists of the following rules:

   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
   new_primMinusNatS1(vy30000) -> Main.Succ(vy30000)
   new_primMinusNatS2 -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Zero, Main.Zero)
   new_primMinusNatS2
   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
   new_primMinusNatS1(x0)
   new_primMinusNatS0(Main.Zero, Main.Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
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(17) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented dependency pairs:

   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vy30000), Main.Zero)

Strictly oriented rules of the TRS R:

   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero

Used ordering: Polynomial interpretation [POLO]:

   POL(Main.Succ(x_1)) = 1 + x_1
   POL(Main.Zero) = 2
   POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2
   POL(new_primMinusNatS0(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(new_primMinusNatS1(x_1)) = 1 + x_1
   POL(new_primMinusNatS2) = 2


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(18)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   new_primMinusNatS1(vy30000) -> Main.Succ(vy30000)
   new_primMinusNatS2 -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Zero, Main.Zero)
   new_primMinusNatS2
   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
   new_primMinusNatS1(x0)
   new_primMinusNatS0(Main.Zero, Main.Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
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(19) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(20)
YES