/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) DependencyGraphProof [EQUIVALENT, 0 ms]
        (9) AND
            (10) QDP
                (11) QDPSizeChangeProof [EQUIVALENT, 7 ms]
                (12) YES
            (13) QDP
                (14) QDPOrderProof [EQUIVALENT, 49 ms]
                (15) QDP
                (16) DependencyGraphProof [EQUIVALENT, 0 ms]
                (17) QDP
                (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (19) YES
    (20) QDP
        (21) DependencyGraphProof [EQUIVALENT, 0 ms]
        (22) AND
            (23) QDP
                (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (25) YES
            (26) QDP
                (27) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (28) YES
    (29) QDP
        (30) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (31) YES


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data MyBool = MyTrue  | MyFalse ;

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

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

error :: a;
error = stop MyTrue;

modMyInt :: MyInt  ->  MyInt  ->  MyInt;
modMyInt = primModInt;

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;

primModInt :: MyInt  ->  MyInt  ->  MyInt;
primModInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
primModInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatP x (Main.Succ y));
primModInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatP x (Main.Succ y));
primModInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
primModInt vv vw = Main.error;

primModNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primModNatP Main.Zero Main.Zero = Main.error;
primModNatP Main.Zero (Main.Succ x) = Main.Zero;
primModNatP (Main.Succ x) Main.Zero = Main.error;
primModNatP (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
primModNatP (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatP0 x y (primGEqNatS x y);

primModNatP0 x y MyTrue = primModNatP (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
primModNatP0 x y MyFalse = primMinusNatS (Main.Succ y) 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;

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 MyBool = MyTrue  | MyFalse ;

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

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

error :: a;
error = stop MyTrue;

modMyInt :: MyInt  ->  MyInt  ->  MyInt;
modMyInt = primModInt;

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;

primModInt :: MyInt  ->  MyInt  ->  MyInt;
primModInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
primModInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatP x (Main.Succ y));
primModInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatP x (Main.Succ y));
primModInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
primModInt vv vw = Main.error;

primModNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primModNatP Main.Zero Main.Zero = Main.error;
primModNatP Main.Zero (Main.Succ x) = Main.Zero;
primModNatP (Main.Succ x) Main.Zero = Main.error;
primModNatP (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
primModNatP (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatP0 x y (primGEqNatS x y);

primModNatP0 x y MyTrue = primModNatP (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
primModNatP0 x y MyFalse = primMinusNatS (Main.Succ y) 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;

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 MyBool = MyTrue  | MyFalse ;

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

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

error :: a;
error = stop MyTrue;

modMyInt :: MyInt  ->  MyInt  ->  MyInt;
modMyInt = primModInt;

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;

primModInt :: MyInt  ->  MyInt  ->  MyInt;
primModInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
primModInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatP x (Main.Succ y));
primModInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatP x (Main.Succ y));
primModInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
primModInt vv vw = Main.error;

primModNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primModNatP Main.Zero Main.Zero = Main.error;
primModNatP Main.Zero (Main.Succ x) = Main.Zero;
primModNatP (Main.Succ x) Main.Zero = Main.error;
primModNatP (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
primModNatP (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatP0 x y (primGEqNatS x y);

primModNatP0 x y MyTrue = primModNatP (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
primModNatP0 x y MyFalse = primMinusNatS (Main.Succ y) 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;

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

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="modMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="modMyInt vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="modMyInt vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="primModInt vz3 vz4",fontsize=16,color="burlywood",shape="box"];626[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 626[label="",style="solid", color="burlywood", weight=9];
626 -> 6[label="",style="solid", color="burlywood", weight=3];
627[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 627[label="",style="solid", color="burlywood", weight=9];
627 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="primModInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];628[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 628[label="",style="solid", color="burlywood", weight=9];
628 -> 8[label="",style="solid", color="burlywood", weight=3];
629[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 629[label="",style="solid", color="burlywood", weight=9];
629 -> 9[label="",style="solid", color="burlywood", weight=3];
7[label="primModInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];630[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 630[label="",style="solid", color="burlywood", weight=9];
630 -> 10[label="",style="solid", color="burlywood", weight=3];
631[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 631[label="",style="solid", color="burlywood", weight=9];
631 -> 11[label="",style="solid", color="burlywood", weight=3];
8[label="primModInt (Pos vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];632[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 632[label="",style="solid", color="burlywood", weight=9];
632 -> 12[label="",style="solid", color="burlywood", weight=3];
633[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 633[label="",style="solid", color="burlywood", weight=9];
633 -> 13[label="",style="solid", color="burlywood", weight=3];
9[label="primModInt (Pos vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];634[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 634[label="",style="solid", color="burlywood", weight=9];
634 -> 14[label="",style="solid", color="burlywood", weight=3];
635[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 635[label="",style="solid", color="burlywood", weight=9];
635 -> 15[label="",style="solid", color="burlywood", weight=3];
10[label="primModInt (Neg vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];636[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 636[label="",style="solid", color="burlywood", weight=9];
636 -> 16[label="",style="solid", color="burlywood", weight=3];
637[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 637[label="",style="solid", color="burlywood", weight=9];
637 -> 17[label="",style="solid", color="burlywood", weight=3];
11[label="primModInt (Neg vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];638[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 638[label="",style="solid", color="burlywood", weight=9];
638 -> 18[label="",style="solid", color="burlywood", weight=3];
639[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 639[label="",style="solid", color="burlywood", weight=9];
639 -> 19[label="",style="solid", color="burlywood", weight=3];
12[label="primModInt (Pos vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
13[label="primModInt (Pos vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
14[label="primModInt (Pos vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
15[label="primModInt (Pos vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
16[label="primModInt (Neg vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
17[label="primModInt (Neg vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
18[label="primModInt (Neg vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
19[label="primModInt (Neg vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
20[label="Pos (primModNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3];
21[label="error",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3];
22[label="Neg (primModNatP vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];22 -> 30[label="",style="dashed", color="green", weight=3];
23 -> 21[label="",style="dashed", color="red", weight=0];
23[label="error",fontsize=16,color="magenta"];24[label="Pos (primModNatP vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];24 -> 31[label="",style="dashed", color="green", weight=3];
25 -> 21[label="",style="dashed", color="red", weight=0];
25[label="error",fontsize=16,color="magenta"];26[label="Neg (primModNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];26 -> 32[label="",style="dashed", color="green", weight=3];
27 -> 21[label="",style="dashed", color="red", weight=0];
27[label="error",fontsize=16,color="magenta"];28[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];640[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 640[label="",style="solid", color="burlywood", weight=9];
640 -> 33[label="",style="solid", color="burlywood", weight=3];
641[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 641[label="",style="solid", color="burlywood", weight=9];
641 -> 34[label="",style="solid", color="burlywood", weight=3];
29[label="stop MyTrue",fontsize=16,color="black",shape="box"];29 -> 35[label="",style="solid", color="black", weight=3];
30[label="primModNatP vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];642[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];30 -> 642[label="",style="solid", color="burlywood", weight=9];
642 -> 36[label="",style="solid", color="burlywood", weight=3];
643[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 643[label="",style="solid", color="burlywood", weight=9];
643 -> 37[label="",style="solid", color="burlywood", weight=3];
31 -> 30[label="",style="dashed", color="red", weight=0];
31[label="primModNatP vz30 (Succ vz400)",fontsize=16,color="magenta"];31 -> 38[label="",style="dashed", color="magenta", weight=3];
31 -> 39[label="",style="dashed", color="magenta", weight=3];
32 -> 28[label="",style="dashed", color="red", weight=0];
32[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];32 -> 40[label="",style="dashed", color="magenta", weight=3];
32 -> 41[label="",style="dashed", color="magenta", weight=3];
33[label="primModNatS (Succ vz300) (Succ vz400)",fontsize=16,color="burlywood",shape="box"];644[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];33 -> 644[label="",style="solid", color="burlywood", weight=9];
644 -> 42[label="",style="solid", color="burlywood", weight=3];
645[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 645[label="",style="solid", color="burlywood", weight=9];
645 -> 43[label="",style="solid", color="burlywood", weight=3];
34[label="primModNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 44[label="",style="solid", color="black", weight=3];
35[label="error []",fontsize=16,color="red",shape="box"];36[label="primModNatP (Succ vz300) (Succ vz400)",fontsize=16,color="burlywood",shape="box"];646[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];36 -> 646[label="",style="solid", color="burlywood", weight=9];
646 -> 45[label="",style="solid", color="burlywood", weight=3];
647[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 647[label="",style="solid", color="burlywood", weight=9];
647 -> 46[label="",style="solid", color="burlywood", weight=3];
37[label="primModNatP Zero (Succ vz400)",fontsize=16,color="black",shape="box"];37 -> 47[label="",style="solid", color="black", weight=3];
38[label="vz400",fontsize=16,color="green",shape="box"];39[label="vz30",fontsize=16,color="green",shape="box"];40[label="vz400",fontsize=16,color="green",shape="box"];41[label="vz30",fontsize=16,color="green",shape="box"];42[label="primModNatS (Succ vz300) (Succ (Succ vz4000))",fontsize=16,color="black",shape="box"];42 -> 48[label="",style="solid", color="black", weight=3];
43[label="primModNatS (Succ vz300) (Succ Zero)",fontsize=16,color="black",shape="box"];43 -> 49[label="",style="solid", color="black", weight=3];
44[label="Zero",fontsize=16,color="green",shape="box"];45[label="primModNatP (Succ vz300) (Succ (Succ vz4000))",fontsize=16,color="black",shape="box"];45 -> 50[label="",style="solid", color="black", weight=3];
46[label="primModNatP (Succ vz300) (Succ Zero)",fontsize=16,color="black",shape="box"];46 -> 51[label="",style="solid", color="black", weight=3];
47[label="Zero",fontsize=16,color="green",shape="box"];48[label="primModNatS0 vz300 vz4000 (primGEqNatS vz300 (Succ vz4000))",fontsize=16,color="burlywood",shape="box"];648[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];48 -> 648[label="",style="solid", color="burlywood", weight=9];
648 -> 52[label="",style="solid", color="burlywood", weight=3];
649[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 649[label="",style="solid", color="burlywood", weight=9];
649 -> 53[label="",style="solid", color="burlywood", weight=3];
49[label="Zero",fontsize=16,color="green",shape="box"];50[label="primModNatP0 vz300 vz4000 (primGEqNatS vz300 vz4000)",fontsize=16,color="burlywood",shape="box"];650[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];50 -> 650[label="",style="solid", color="burlywood", weight=9];
650 -> 54[label="",style="solid", color="burlywood", weight=3];
651[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 651[label="",style="solid", color="burlywood", weight=9];
651 -> 55[label="",style="solid", color="burlywood", weight=3];
51[label="Zero",fontsize=16,color="green",shape="box"];52[label="primModNatS0 (Succ vz3000) vz4000 (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="primModNatS0 Zero vz4000 (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
54[label="primModNatP0 (Succ vz3000) vz4000 (primGEqNatS (Succ vz3000) vz4000)",fontsize=16,color="burlywood",shape="box"];652[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];54 -> 652[label="",style="solid", color="burlywood", weight=9];
652 -> 58[label="",style="solid", color="burlywood", weight=3];
653[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];54 -> 653[label="",style="solid", color="burlywood", weight=9];
653 -> 59[label="",style="solid", color="burlywood", weight=3];
55[label="primModNatP0 Zero vz4000 (primGEqNatS Zero vz4000)",fontsize=16,color="burlywood",shape="box"];654[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];55 -> 654[label="",style="solid", color="burlywood", weight=9];
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68[label="primModNatP0 Zero Zero MyTrue",fontsize=16,color="black",shape="box"];68 -> 77[label="",style="solid", color="black", weight=3];
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80[label="primModNatS0 (Succ Zero) (Succ vz40000) MyFalse",fontsize=16,color="black",shape="box"];80 -> 94[label="",style="solid", color="black", weight=3];
81[label="primModNatS0 (Succ Zero) Zero MyTrue",fontsize=16,color="black",shape="box"];81 -> 95[label="",style="solid", color="black", weight=3];
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665[label="vz42/Zero",fontsize=10,color="white",style="solid",shape="box"];523 -> 665[label="",style="solid", color="burlywood", weight=9];
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667 -> 532[label="",style="solid", color="burlywood", weight=3];
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534[label="vz40000",fontsize=16,color="green",shape="box"];535[label="vz30000",fontsize=16,color="green",shape="box"];536[label="vz40000",fontsize=16,color="green",shape="box"];537[label="Succ vz30000",fontsize=16,color="green",shape="box"];533[label="primModNatS0 (Succ vz44) (Succ vz45) (primGEqNatS vz46 vz47)",fontsize=16,color="burlywood",shape="triangle"];668[label="vz46/Succ vz460",fontsize=10,color="white",style="solid",shape="box"];533 -> 668[label="",style="solid", color="burlywood", weight=9];
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100[label="primMinusNatS vz3000 Zero",fontsize=16,color="burlywood",shape="box"];670[label="vz3000/Succ vz30000",fontsize=10,color="white",style="solid",shape="box"];100 -> 670[label="",style="solid", color="burlywood", weight=9];
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671[label="vz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];100 -> 671[label="",style="solid", color="burlywood", weight=9];
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673[label="vz47/Zero",fontsize=10,color="white",style="solid",shape="box"];574 -> 673[label="",style="solid", color="burlywood", weight=9];
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578[label="primModNatP0 (Succ vz39) (Succ vz40) MyFalse",fontsize=16,color="black",shape="box"];578 -> 587[label="",style="solid", color="black", weight=3];
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579[label="primModNatP0 (Succ vz39) (Succ vz40) MyTrue",fontsize=16,color="magenta"];115[label="primMinusNatS (Succ vz30000) Zero",fontsize=16,color="black",shape="box"];115 -> 132[label="",style="solid", color="black", weight=3];
116[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];116 -> 133[label="",style="solid", color="black", weight=3];
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587[label="primMinusNatS (Succ (Succ vz40)) (Succ vz39)",fontsize=16,color="black",shape="box"];587 -> 594[label="",style="solid", color="black", weight=3];
132[label="Succ vz30000",fontsize=16,color="green",shape="box"];133[label="Zero",fontsize=16,color="green",shape="box"];588 -> 533[label="",style="dashed", color="red", weight=0];
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590[label="primModNatS0 (Succ vz44) (Succ vz45) MyFalse",fontsize=16,color="black",shape="box"];590 -> 598[label="",style="solid", color="black", weight=3];
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597 -> 603[label="",style="dashed", color="magenta", weight=3];
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679[label="vz39/Zero",fontsize=10,color="white",style="solid",shape="box"];599 -> 679[label="",style="solid", color="burlywood", weight=9];
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601[label="primMinusNatS (Succ vz40) Zero",fontsize=16,color="black",shape="box"];601 -> 607[label="",style="solid", color="black", weight=3];
602[label="Succ (Succ vz45)",fontsize=16,color="green",shape="box"];603 -> 599[label="",style="dashed", color="red", weight=0];
603[label="primMinusNatS (Succ vz44) (Succ (Succ vz45))",fontsize=16,color="magenta"];603 -> 608[label="",style="dashed", color="magenta", weight=3];
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605[label="primMinusNatS Zero (Succ vz40)",fontsize=16,color="black",shape="box"];605 -> 611[label="",style="solid", color="black", weight=3];
606[label="primMinusNatS vz40 vz390",fontsize=16,color="burlywood",shape="triangle"];680[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];606 -> 680[label="",style="solid", color="burlywood", weight=9];
680 -> 612[label="",style="solid", color="burlywood", weight=3];
681[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];606 -> 681[label="",style="solid", color="burlywood", weight=9];
681 -> 613[label="",style="solid", color="burlywood", weight=3];
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610[label="primMinusNatS vz390 vz40",fontsize=16,color="magenta"];610 -> 614[label="",style="dashed", color="magenta", weight=3];
610 -> 615[label="",style="dashed", color="magenta", weight=3];
611[label="Zero",fontsize=16,color="green",shape="box"];612[label="primMinusNatS (Succ vz400) vz390",fontsize=16,color="burlywood",shape="box"];682[label="vz390/Succ vz3900",fontsize=10,color="white",style="solid",shape="box"];612 -> 682[label="",style="solid", color="burlywood", weight=9];
682 -> 616[label="",style="solid", color="burlywood", weight=3];
683[label="vz390/Zero",fontsize=10,color="white",style="solid",shape="box"];612 -> 683[label="",style="solid", color="burlywood", weight=9];
683 -> 617[label="",style="solid", color="burlywood", weight=3];
613[label="primMinusNatS Zero vz390",fontsize=16,color="burlywood",shape="box"];684[label="vz390/Succ vz3900",fontsize=10,color="white",style="solid",shape="box"];613 -> 684[label="",style="solid", color="burlywood", weight=9];
684 -> 618[label="",style="solid", color="burlywood", weight=3];
685[label="vz390/Zero",fontsize=10,color="white",style="solid",shape="box"];613 -> 685[label="",style="solid", color="burlywood", weight=9];
685 -> 619[label="",style="solid", color="burlywood", weight=3];
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617[label="primMinusNatS (Succ vz400) Zero",fontsize=16,color="black",shape="box"];617 -> 621[label="",style="solid", color="black", weight=3];
618[label="primMinusNatS Zero (Succ vz3900)",fontsize=16,color="black",shape="box"];618 -> 622[label="",style="solid", color="black", weight=3];
619[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];619 -> 623[label="",style="solid", color="black", weight=3];
620 -> 606[label="",style="dashed", color="red", weight=0];
620[label="primMinusNatS vz400 vz3900",fontsize=16,color="magenta"];620 -> 624[label="",style="dashed", color="magenta", weight=3];
620 -> 625[label="",style="dashed", color="magenta", weight=3];
621[label="Succ vz400",fontsize=16,color="green",shape="box"];622[label="Zero",fontsize=16,color="green",shape="box"];623[label="Zero",fontsize=16,color="green",shape="box"];624[label="vz400",fontsize=16,color="green",shape="box"];625[label="vz3900",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

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

   new_primModNatS0(vz44, vz45, Main.Zero, Main.Zero) -> new_primModNatS00(vz44, vz45)
   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Succ(vz470)) -> new_primModNatS0(vz44, vz45, vz460, vz470)
   new_primModNatS00(vz44, vz45) -> new_primModNatS(new_primMinusNatS2(Main.Succ(vz44), Main.Succ(vz45)), Main.Succ(Main.Succ(vz45)))
   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS0(Main.Succ(vz30000)), Main.Succ(Main.Zero))
   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Succ(vz40000))) -> new_primModNatS0(Main.Succ(vz30000), vz40000, vz30000, vz40000)
   new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS0(Main.Zero), Main.Succ(Main.Zero))
   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Zero) -> new_primModNatS(new_primMinusNatS2(Main.Succ(vz44), Main.Succ(vz45)), Main.Succ(Main.Succ(vz45)))

The TRS R consists of the following rules:

   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
   new_primMinusNatS0(Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0))
   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
   new_primMinusNatS2(Main.Zero, x0)
   new_primMinusNatS0(Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), x1)
   new_primMinusNatS3(Main.Zero, Main.Zero)

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

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

(9)
Complex Obligation (AND)

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

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

   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS0(Main.Succ(vz30000)), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
   new_primMinusNatS0(Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0))
   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
   new_primMinusNatS2(Main.Zero, x0)
   new_primMinusNatS0(Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), x1)
   new_primMinusNatS3(Main.Zero, Main.Zero)

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

(11) 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_1




From the DPs we obtained the following set of size-change graphs:
*new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS0(Main.Succ(vz30000)), Main.Succ(Main.Zero)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 >= 2



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

   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)

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

(12)
YES

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

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

   new_primModNatS00(vz44, vz45) -> new_primModNatS(new_primMinusNatS2(Main.Succ(vz44), Main.Succ(vz45)), Main.Succ(Main.Succ(vz45)))
   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Succ(vz40000))) -> new_primModNatS0(Main.Succ(vz30000), vz40000, vz30000, vz40000)
   new_primModNatS0(vz44, vz45, Main.Zero, Main.Zero) -> new_primModNatS00(vz44, vz45)
   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Succ(vz470)) -> new_primModNatS0(vz44, vz45, vz460, vz470)
   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Zero) -> new_primModNatS(new_primMinusNatS2(Main.Succ(vz44), Main.Succ(vz45)), Main.Succ(Main.Succ(vz45)))

The TRS R consists of the following rules:

   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
   new_primMinusNatS0(Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0))
   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
   new_primMinusNatS2(Main.Zero, x0)
   new_primMinusNatS0(Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), x1)
   new_primMinusNatS3(Main.Zero, Main.Zero)

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

(14) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Succ(vz40000))) -> new_primModNatS0(Main.Succ(vz30000), vz40000, vz30000, vz40000)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(Main.Succ(x_1)) = 1 + x_1
   POL(Main.Zero) = 0
   POL(new_primMinusNatS2(x_1, x_2)) = x_1
   POL(new_primMinusNatS3(x_1, x_2)) = 1 + x_1
   POL(new_primModNatS(x_1, x_2)) = x_1
   POL(new_primModNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1
   POL(new_primModNatS00(x_1, x_2)) = 1 + x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)


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

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

   new_primModNatS00(vz44, vz45) -> new_primModNatS(new_primMinusNatS2(Main.Succ(vz44), Main.Succ(vz45)), Main.Succ(Main.Succ(vz45)))
   new_primModNatS0(vz44, vz45, Main.Zero, Main.Zero) -> new_primModNatS00(vz44, vz45)
   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Succ(vz470)) -> new_primModNatS0(vz44, vz45, vz460, vz470)
   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Zero) -> new_primModNatS(new_primMinusNatS2(Main.Succ(vz44), Main.Succ(vz45)), Main.Succ(Main.Succ(vz45)))

The TRS R consists of the following rules:

   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
   new_primMinusNatS0(Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0))
   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
   new_primMinusNatS2(Main.Zero, x0)
   new_primMinusNatS0(Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), x1)
   new_primMinusNatS3(Main.Zero, Main.Zero)

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

(16) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
----------------------------------------

(17)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Succ(vz470)) -> new_primModNatS0(vz44, vz45, vz460, vz470)

The TRS R consists of the following rules:

   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
   new_primMinusNatS0(Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0))
   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
   new_primMinusNatS2(Main.Zero, x0)
   new_primMinusNatS0(Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), x1)
   new_primMinusNatS3(Main.Zero, Main.Zero)

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

(18) 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_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Succ(vz470)) -> new_primModNatS0(vz44, vz45, vz460, vz470)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


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

(19)
YES

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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatP(Main.Succ(Main.Zero), Main.Succ(Main.Zero)) -> new_primModNatP(new_primMinusNatS1, Main.Succ(Main.Zero))
   new_primModNatP00(vz39, vz40) -> new_primModNatP(new_primMinusNatS2(vz39, vz40), Main.Succ(Main.Succ(vz40)))
   new_primModNatP0(vz39, vz40, Main.Zero, Main.Zero) -> new_primModNatP00(vz39, vz40)
   new_primModNatP0(vz39, vz40, Main.Succ(vz410), Main.Succ(vz420)) -> new_primModNatP0(vz39, vz40, vz410, vz420)
   new_primModNatP0(vz39, vz40, Main.Succ(vz410), Main.Zero) -> new_primModNatP(new_primMinusNatS2(vz39, vz40), Main.Succ(Main.Succ(vz40)))
   new_primModNatP(Main.Succ(Main.Succ(vz3000)), Main.Succ(Main.Zero)) -> new_primModNatP(new_primMinusNatS0(vz3000), Main.Succ(Main.Zero))
   new_primModNatP(Main.Succ(Main.Succ(vz3000)), Main.Succ(Main.Succ(vz40000))) -> new_primModNatP0(vz3000, vz40000, vz3000, vz40000)

The TRS R consists of the following rules:

   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
   new_primMinusNatS0(Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
   new_primMinusNatS1 -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0))
   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
   new_primMinusNatS2(Main.Zero, x0)
   new_primMinusNatS0(Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), x1)
   new_primMinusNatS1
   new_primMinusNatS3(Main.Zero, Main.Zero)

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

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

(22)
Complex Obligation (AND)

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

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatP(Main.Succ(Main.Succ(vz3000)), Main.Succ(Main.Succ(vz40000))) -> new_primModNatP0(vz3000, vz40000, vz3000, vz40000)
   new_primModNatP0(vz39, vz40, Main.Zero, Main.Zero) -> new_primModNatP00(vz39, vz40)
   new_primModNatP00(vz39, vz40) -> new_primModNatP(new_primMinusNatS2(vz39, vz40), Main.Succ(Main.Succ(vz40)))
   new_primModNatP0(vz39, vz40, Main.Succ(vz410), Main.Succ(vz420)) -> new_primModNatP0(vz39, vz40, vz410, vz420)
   new_primModNatP0(vz39, vz40, Main.Succ(vz410), Main.Zero) -> new_primModNatP(new_primMinusNatS2(vz39, vz40), Main.Succ(Main.Succ(vz40)))

The TRS R consists of the following rules:

   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
   new_primMinusNatS0(Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
   new_primMinusNatS1 -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0))
   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
   new_primMinusNatS2(Main.Zero, x0)
   new_primMinusNatS0(Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), x1)
   new_primMinusNatS1
   new_primMinusNatS3(Main.Zero, Main.Zero)

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

(24) 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_primMinusNatS2(x_1, x_2)) = x_1
   POL(new_primMinusNatS3(x_1, x_2)) = x_1




From the DPs we obtained the following set of size-change graphs:
*new_primModNatP0(vz39, vz40, Main.Succ(vz410), Main.Zero) -> new_primModNatP(new_primMinusNatS2(vz39, vz40), Main.Succ(Main.Succ(vz40))) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1


*new_primModNatP00(vz39, vz40) -> new_primModNatP(new_primMinusNatS2(vz39, vz40), Main.Succ(Main.Succ(vz40))) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1


*new_primModNatP(Main.Succ(Main.Succ(vz3000)), Main.Succ(Main.Succ(vz40000))) -> new_primModNatP0(vz3000, vz40000, vz3000, vz40000) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4


*new_primModNatP0(vz39, vz40, Main.Succ(vz410), Main.Succ(vz420)) -> new_primModNatP0(vz39, vz40, vz410, vz420) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


*new_primModNatP0(vz39, vz40, Main.Zero, Main.Zero) -> new_primModNatP00(vz39, vz40) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1, 2 >= 2



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

   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)

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

(25)
YES

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

(26)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatP(Main.Succ(Main.Succ(vz3000)), Main.Succ(Main.Zero)) -> new_primModNatP(new_primMinusNatS0(vz3000), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
   new_primMinusNatS0(Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
   new_primMinusNatS1 -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0))
   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
   new_primMinusNatS2(Main.Zero, x0)
   new_primMinusNatS0(Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), x1)
   new_primMinusNatS1
   new_primMinusNatS3(Main.Zero, Main.Zero)

We have to consider all minimal (P,Q,R)-chains.
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(27) 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_1




From the DPs we obtained the following set of size-change graphs:
*new_primModNatP(Main.Succ(Main.Succ(vz3000)), Main.Succ(Main.Zero)) -> new_primModNatP(new_primMinusNatS0(vz3000), Main.Succ(Main.Zero)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 >= 2



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

   new_primMinusNatS0(Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)

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(28)
YES

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(29)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primMinusNatS(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS(vz400, vz3900)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(30) 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(vz400), Main.Succ(vz3900)) -> new_primMinusNatS(vz400, vz3900)
The graph contains the following edges 1 > 1, 2 > 2


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(31)
YES