/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) DependencyGraphProof [EQUIVALENT, 0 ms]
        (9) AND
            (10) QDP
                (11) TransformationProof [EQUIVALENT, 5 ms]
                (12) QDP
                (13) UsableRulesProof [EQUIVALENT, 0 ms]
                (14) QDP
                (15) QReductionProof [EQUIVALENT, 0 ms]
                (16) QDP
                (17) TransformationProof [EQUIVALENT, 0 ms]
                (18) QDP
                (19) UsableRulesProof [EQUIVALENT, 0 ms]
                (20) QDP
                (21) QReductionProof [EQUIVALENT, 0 ms]
                (22) QDP
                (23) TransformationProof [EQUIVALENT, 0 ms]
                (24) QDP
                (25) TransformationProof [EQUIVALENT, 0 ms]
                (26) QDP
                (27) DependencyGraphProof [EQUIVALENT, 0 ms]
                (28) QDP
                (29) UsableRulesProof [EQUIVALENT, 0 ms]
                (30) QDP
                (31) QReductionProof [EQUIVALENT, 0 ms]
                (32) QDP
                (33) TransformationProof [EQUIVALENT, 0 ms]
                (34) QDP
                (35) UsableRulesProof [EQUIVALENT, 0 ms]
                (36) QDP
                (37) QReductionProof [EQUIVALENT, 0 ms]
                (38) QDP
                (39) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (40) YES
            (41) QDP
                (42) QDPOrderProof [EQUIVALENT, 48 ms]
                (43) QDP
                (44) DependencyGraphProof [EQUIVALENT, 0 ms]
                (45) QDP
                (46) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (47) YES
    (48) QDP
        (49) DependencyGraphProof [EQUIVALENT, 0 ms]
        (50) AND
            (51) QDP
                (52) MRRProof [EQUIVALENT, 0 ms]
                (53) QDP
                (54) PisEmptyProof [EQUIVALENT, 0 ms]
                (55) YES
            (56) QDP
                (57) QDPOrderProof [EQUIVALENT, 5 ms]
                (58) QDP
                (59) DependencyGraphProof [EQUIVALENT, 0 ms]
                (60) QDP
                (61) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (62) YES
    (63) QDP
        (64) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (65) 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 ;

data Ratio a = CnPc a a ;

data Tup2 b a = Tup2 b a ;

error :: a;
error = stop MyTrue;

fst :: Tup2 a b  ->  a;
fst (Tup2 x _) = x;

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;

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

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

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

primQrmInt :: MyInt  ->  MyInt  ->  Tup2 MyInt MyInt;
primQrmInt x y = Tup2 (primQuotInt x y) (primRemInt 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 vx vy = Main.error;

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

properFraction :: Ratio MyInt  ->  Tup2 MyInt (Ratio MyInt);
properFraction (CnPc x y) = Tup2 (qProperFraction x y) (CnPc (rProperFraction x y) y);

qProperFraction :: MyInt  ->  MyInt  ->  MyInt;
qProperFraction x y = fst (quotRemMyInt x y);

quotRemMyInt :: MyInt  ->  MyInt  ->  Tup2 MyInt MyInt;
quotRemMyInt = primQrmInt;

rProperFraction :: MyInt  ->  MyInt  ->  MyInt;
rProperFraction x y = snd (quotRemMyInt x y);

snd :: Tup2 a b  ->  b;
snd (Tup2 _ y) = y;

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 ;

data Ratio a = CnPc a a ;

data Tup2 b a = Tup2 b a ;

error :: a;
error = stop MyTrue;

fst :: Tup2 b a  ->  b;
fst (Tup2 x wv) = x;

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;

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

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

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

primQrmInt :: MyInt  ->  MyInt  ->  Tup2 MyInt MyInt;
primQrmInt x y = Tup2 (primQuotInt x y) (primRemInt 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 vx vy = Main.error;

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

properFraction :: Ratio MyInt  ->  Tup2 MyInt (Ratio MyInt);
properFraction (CnPc x y) = Tup2 (qProperFraction x y) (CnPc (rProperFraction x y) y);

qProperFraction :: MyInt  ->  MyInt  ->  MyInt;
qProperFraction x y = fst (quotRemMyInt x y);

quotRemMyInt :: MyInt  ->  MyInt  ->  Tup2 MyInt MyInt;
quotRemMyInt = primQrmInt;

rProperFraction :: MyInt  ->  MyInt  ->  MyInt;
rProperFraction x y = snd (quotRemMyInt x y);

snd :: Tup2 b a  ->  a;
snd (Tup2 ww y) = y;

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 ;

data Ratio a = CnPc a a ;

data Tup2 a b = Tup2 a b ;

error :: a;
error = stop MyTrue;

fst :: Tup2 b a  ->  b;
fst (Tup2 x wv) = x;

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;

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

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

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

primQrmInt :: MyInt  ->  MyInt  ->  Tup2 MyInt MyInt;
primQrmInt x y = Tup2 (primQuotInt x y) (primRemInt 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 vx vy = Main.error;

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

properFraction :: Ratio MyInt  ->  Tup2 MyInt (Ratio MyInt);
properFraction (CnPc x y) = Tup2 (qProperFraction x y) (CnPc (rProperFraction x y) y);

qProperFraction :: MyInt  ->  MyInt  ->  MyInt;
qProperFraction x y = fst (quotRemMyInt x y);

quotRemMyInt :: MyInt  ->  MyInt  ->  Tup2 MyInt MyInt;
quotRemMyInt = primQrmInt;

rProperFraction :: MyInt  ->  MyInt  ->  MyInt;
rProperFraction x y = snd (quotRemMyInt x y);

snd :: Tup2 b a  ->  a;
snd (Tup2 ww y) = y;

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

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="properFraction",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="properFraction wx3",fontsize=16,color="burlywood",shape="triangle"];666[label="wx3/CnPc wx30 wx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 666[label="",style="solid", color="burlywood", weight=9];
666 -> 4[label="",style="solid", color="burlywood", weight=3];
4[label="properFraction (CnPc wx30 wx31)",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="Tup2 (qProperFraction wx30 wx31) (CnPc (rProperFraction wx30 wx31) wx31)",fontsize=16,color="green",shape="box"];5 -> 6[label="",style="dashed", color="green", weight=3];
5 -> 7[label="",style="dashed", color="green", weight=3];
6[label="qProperFraction wx30 wx31",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="rProperFraction wx30 wx31",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="fst (quotRemMyInt wx30 wx31)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="snd (quotRemMyInt wx30 wx31)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="fst (primQrmInt wx30 wx31)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="snd (primQrmInt wx30 wx31)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="fst (Tup2 (primQuotInt wx30 wx31) (primRemInt wx30 wx31))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="snd (Tup2 (primQuotInt wx30 wx31) (primRemInt wx30 wx31))",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="primQuotInt wx30 wx31",fontsize=16,color="burlywood",shape="box"];667[label="wx30/Pos wx300",fontsize=10,color="white",style="solid",shape="box"];14 -> 667[label="",style="solid", color="burlywood", weight=9];
667 -> 16[label="",style="solid", color="burlywood", weight=3];
668[label="wx30/Neg wx300",fontsize=10,color="white",style="solid",shape="box"];14 -> 668[label="",style="solid", color="burlywood", weight=9];
668 -> 17[label="",style="solid", color="burlywood", weight=3];
15[label="primRemInt wx30 wx31",fontsize=16,color="burlywood",shape="box"];669[label="wx30/Pos wx300",fontsize=10,color="white",style="solid",shape="box"];15 -> 669[label="",style="solid", color="burlywood", weight=9];
669 -> 18[label="",style="solid", color="burlywood", weight=3];
670[label="wx30/Neg wx300",fontsize=10,color="white",style="solid",shape="box"];15 -> 670[label="",style="solid", color="burlywood", weight=9];
670 -> 19[label="",style="solid", color="burlywood", weight=3];
16[label="primQuotInt (Pos wx300) wx31",fontsize=16,color="burlywood",shape="box"];671[label="wx31/Pos wx310",fontsize=10,color="white",style="solid",shape="box"];16 -> 671[label="",style="solid", color="burlywood", weight=9];
671 -> 20[label="",style="solid", color="burlywood", weight=3];
672[label="wx31/Neg wx310",fontsize=10,color="white",style="solid",shape="box"];16 -> 672[label="",style="solid", color="burlywood", weight=9];
672 -> 21[label="",style="solid", color="burlywood", weight=3];
17[label="primQuotInt (Neg wx300) wx31",fontsize=16,color="burlywood",shape="box"];673[label="wx31/Pos wx310",fontsize=10,color="white",style="solid",shape="box"];17 -> 673[label="",style="solid", color="burlywood", weight=9];
673 -> 22[label="",style="solid", color="burlywood", weight=3];
674[label="wx31/Neg wx310",fontsize=10,color="white",style="solid",shape="box"];17 -> 674[label="",style="solid", color="burlywood", weight=9];
674 -> 23[label="",style="solid", color="burlywood", weight=3];
18[label="primRemInt (Pos wx300) wx31",fontsize=16,color="burlywood",shape="box"];675[label="wx31/Pos wx310",fontsize=10,color="white",style="solid",shape="box"];18 -> 675[label="",style="solid", color="burlywood", weight=9];
675 -> 24[label="",style="solid", color="burlywood", weight=3];
676[label="wx31/Neg wx310",fontsize=10,color="white",style="solid",shape="box"];18 -> 676[label="",style="solid", color="burlywood", weight=9];
676 -> 25[label="",style="solid", color="burlywood", weight=3];
19[label="primRemInt (Neg wx300) wx31",fontsize=16,color="burlywood",shape="box"];677[label="wx31/Pos wx310",fontsize=10,color="white",style="solid",shape="box"];19 -> 677[label="",style="solid", color="burlywood", weight=9];
677 -> 26[label="",style="solid", color="burlywood", weight=3];
678[label="wx31/Neg wx310",fontsize=10,color="white",style="solid",shape="box"];19 -> 678[label="",style="solid", color="burlywood", weight=9];
678 -> 27[label="",style="solid", color="burlywood", weight=3];
20[label="primQuotInt (Pos wx300) (Pos wx310)",fontsize=16,color="burlywood",shape="box"];679[label="wx310/Succ wx3100",fontsize=10,color="white",style="solid",shape="box"];20 -> 679[label="",style="solid", color="burlywood", weight=9];
679 -> 28[label="",style="solid", color="burlywood", weight=3];
680[label="wx310/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 680[label="",style="solid", color="burlywood", weight=9];
680 -> 29[label="",style="solid", color="burlywood", weight=3];
21[label="primQuotInt (Pos wx300) (Neg wx310)",fontsize=16,color="burlywood",shape="box"];681[label="wx310/Succ wx3100",fontsize=10,color="white",style="solid",shape="box"];21 -> 681[label="",style="solid", color="burlywood", weight=9];
681 -> 30[label="",style="solid", color="burlywood", weight=3];
682[label="wx310/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 682[label="",style="solid", color="burlywood", weight=9];
682 -> 31[label="",style="solid", color="burlywood", weight=3];
22[label="primQuotInt (Neg wx300) (Pos wx310)",fontsize=16,color="burlywood",shape="box"];683[label="wx310/Succ wx3100",fontsize=10,color="white",style="solid",shape="box"];22 -> 683[label="",style="solid", color="burlywood", weight=9];
683 -> 32[label="",style="solid", color="burlywood", weight=3];
684[label="wx310/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 684[label="",style="solid", color="burlywood", weight=9];
684 -> 33[label="",style="solid", color="burlywood", weight=3];
23[label="primQuotInt (Neg wx300) (Neg wx310)",fontsize=16,color="burlywood",shape="box"];685[label="wx310/Succ wx3100",fontsize=10,color="white",style="solid",shape="box"];23 -> 685[label="",style="solid", color="burlywood", weight=9];
685 -> 34[label="",style="solid", color="burlywood", weight=3];
686[label="wx310/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 686[label="",style="solid", color="burlywood", weight=9];
686 -> 35[label="",style="solid", color="burlywood", weight=3];
24[label="primRemInt (Pos wx300) (Pos wx310)",fontsize=16,color="burlywood",shape="box"];687[label="wx310/Succ wx3100",fontsize=10,color="white",style="solid",shape="box"];24 -> 687[label="",style="solid", color="burlywood", weight=9];
687 -> 36[label="",style="solid", color="burlywood", weight=3];
688[label="wx310/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 688[label="",style="solid", color="burlywood", weight=9];
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638[label="primDivNatS0 (Succ wx42) (Succ wx43) MyFalse",fontsize=16,color="black",shape="box"];638 -> 647[label="",style="solid", color="black", weight=3];
639 -> 637[label="",style="dashed", color="red", weight=0];
639[label="primDivNatS0 (Succ wx42) (Succ wx43) MyTrue",fontsize=16,color="magenta"];138[label="Succ wx30000",fontsize=16,color="green",shape="box"];139[label="Zero",fontsize=16,color="green",shape="box"];594[label="wx310000",fontsize=16,color="green",shape="box"];595[label="wx310000",fontsize=16,color="green",shape="box"];596[label="wx300000",fontsize=16,color="green",shape="box"];597[label="Succ wx300000",fontsize=16,color="green",shape="box"];593[label="primModNatS0 (Succ wx47) (Succ wx48) (primGEqNatS wx49 wx50)",fontsize=16,color="burlywood",shape="triangle"];721[label="wx49/Succ wx490",fontsize=10,color="white",style="solid",shape="box"];593 -> 721[label="",style="solid", color="burlywood", weight=9];
721 -> 634[label="",style="solid", color="burlywood", weight=3];
722[label="wx49/Zero",fontsize=10,color="white",style="solid",shape="box"];593 -> 722[label="",style="solid", color="burlywood", weight=9];
722 -> 635[label="",style="solid", color="burlywood", weight=3];
142 -> 65[label="",style="dashed", color="red", weight=0];
142[label="primModNatS (primMinusNatS (Succ (Succ wx300000)) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];142 -> 155[label="",style="dashed", color="magenta", weight=3];
142 -> 156[label="",style="dashed", color="magenta", weight=3];
400[label="Zero",fontsize=16,color="green",shape="box"];401[label="wx310000",fontsize=16,color="green",shape="box"];399[label="primModNatS0 (Succ wx33) (Succ wx34) MyFalse",fontsize=16,color="black",shape="triangle"];399 -> 414[label="",style="solid", color="black", weight=3];
144 -> 65[label="",style="dashed", color="red", weight=0];
144[label="primModNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];144 -> 157[label="",style="dashed", color="magenta", weight=3];
144 -> 158[label="",style="dashed", color="magenta", weight=3];
644[label="wx450",fontsize=16,color="green",shape="box"];645[label="wx440",fontsize=16,color="green",shape="box"];646[label="Succ (primDivNatS (primMinusNatS (Succ wx42) (Succ wx43)) (Succ (Succ wx43)))",fontsize=16,color="green",shape="box"];646 -> 652[label="",style="dashed", color="green", weight=3];
647[label="Zero",fontsize=16,color="green",shape="box"];634[label="primModNatS0 (Succ wx47) (Succ wx48) (primGEqNatS (Succ wx490) wx50)",fontsize=16,color="burlywood",shape="box"];723[label="wx50/Succ wx500",fontsize=10,color="white",style="solid",shape="box"];634 -> 723[label="",style="solid", color="burlywood", weight=9];
723 -> 640[label="",style="solid", color="burlywood", weight=3];
724[label="wx50/Zero",fontsize=10,color="white",style="solid",shape="box"];634 -> 724[label="",style="solid", color="burlywood", weight=9];
724 -> 641[label="",style="solid", color="burlywood", weight=3];
635[label="primModNatS0 (Succ wx47) (Succ wx48) (primGEqNatS Zero wx50)",fontsize=16,color="burlywood",shape="box"];725[label="wx50/Succ wx500",fontsize=10,color="white",style="solid",shape="box"];635 -> 725[label="",style="solid", color="burlywood", weight=9];
725 -> 642[label="",style="solid", color="burlywood", weight=3];
726[label="wx50/Zero",fontsize=10,color="white",style="solid",shape="box"];635 -> 726[label="",style="solid", color="burlywood", weight=9];
726 -> 643[label="",style="solid", color="burlywood", weight=3];
155 -> 440[label="",style="dashed", color="red", weight=0];
155[label="primMinusNatS (Succ (Succ wx300000)) (Succ Zero)",fontsize=16,color="magenta"];155 -> 441[label="",style="dashed", color="magenta", weight=3];
155 -> 442[label="",style="dashed", color="magenta", weight=3];
156[label="Succ Zero",fontsize=16,color="green",shape="box"];414[label="Succ (Succ wx33)",fontsize=16,color="green",shape="box"];157 -> 440[label="",style="dashed", color="red", weight=0];
157[label="primMinusNatS (Succ Zero) (Succ Zero)",fontsize=16,color="magenta"];157 -> 443[label="",style="dashed", color="magenta", weight=3];
157 -> 444[label="",style="dashed", color="magenta", weight=3];
158[label="Succ Zero",fontsize=16,color="green",shape="box"];652 -> 60[label="",style="dashed", color="red", weight=0];
652[label="primDivNatS (primMinusNatS (Succ wx42) (Succ wx43)) (Succ (Succ wx43))",fontsize=16,color="magenta"];652 -> 658[label="",style="dashed", color="magenta", weight=3];
652 -> 659[label="",style="dashed", color="magenta", weight=3];
640[label="primModNatS0 (Succ wx47) (Succ wx48) (primGEqNatS (Succ wx490) (Succ wx500))",fontsize=16,color="black",shape="box"];640 -> 648[label="",style="solid", color="black", weight=3];
641[label="primModNatS0 (Succ wx47) (Succ wx48) (primGEqNatS (Succ wx490) Zero)",fontsize=16,color="black",shape="box"];641 -> 649[label="",style="solid", color="black", weight=3];
642[label="primModNatS0 (Succ wx47) (Succ wx48) (primGEqNatS Zero (Succ wx500))",fontsize=16,color="black",shape="box"];642 -> 650[label="",style="solid", color="black", weight=3];
643[label="primModNatS0 (Succ wx47) (Succ wx48) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];643 -> 651[label="",style="solid", color="black", weight=3];
441[label="Zero",fontsize=16,color="green",shape="box"];442[label="Succ wx300000",fontsize=16,color="green",shape="box"];440[label="primMinusNatS (Succ wx36) (Succ wx37)",fontsize=16,color="black",shape="triangle"];440 -> 475[label="",style="solid", color="black", weight=3];
443[label="Zero",fontsize=16,color="green",shape="box"];444[label="Zero",fontsize=16,color="green",shape="box"];658 -> 475[label="",style="dashed", color="red", weight=0];
658[label="primMinusNatS (Succ wx42) (Succ wx43)",fontsize=16,color="magenta"];658 -> 662[label="",style="dashed", color="magenta", weight=3];
658 -> 663[label="",style="dashed", color="magenta", weight=3];
659[label="Succ wx43",fontsize=16,color="green",shape="box"];648 -> 593[label="",style="dashed", color="red", weight=0];
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648 -> 654[label="",style="dashed", color="magenta", weight=3];
649[label="primModNatS0 (Succ wx47) (Succ wx48) MyTrue",fontsize=16,color="black",shape="triangle"];649 -> 655[label="",style="solid", color="black", weight=3];
650 -> 399[label="",style="dashed", color="red", weight=0];
650[label="primModNatS0 (Succ wx47) (Succ wx48) MyFalse",fontsize=16,color="magenta"];650 -> 656[label="",style="dashed", color="magenta", weight=3];
650 -> 657[label="",style="dashed", color="magenta", weight=3];
651 -> 649[label="",style="dashed", color="red", weight=0];
651[label="primModNatS0 (Succ wx47) (Succ wx48) MyTrue",fontsize=16,color="magenta"];475[label="primMinusNatS wx36 wx37",fontsize=16,color="burlywood",shape="triangle"];727[label="wx36/Succ wx360",fontsize=10,color="white",style="solid",shape="box"];475 -> 727[label="",style="solid", color="burlywood", weight=9];
727 -> 516[label="",style="solid", color="burlywood", weight=3];
728[label="wx36/Zero",fontsize=10,color="white",style="solid",shape="box"];475 -> 728[label="",style="solid", color="burlywood", weight=9];
728 -> 517[label="",style="solid", color="burlywood", weight=3];
662[label="Succ wx43",fontsize=16,color="green",shape="box"];663[label="Succ wx42",fontsize=16,color="green",shape="box"];653[label="wx500",fontsize=16,color="green",shape="box"];654[label="wx490",fontsize=16,color="green",shape="box"];655 -> 65[label="",style="dashed", color="red", weight=0];
655[label="primModNatS (primMinusNatS (Succ wx47) (Succ (Succ wx48))) (Succ (Succ (Succ wx48)))",fontsize=16,color="magenta"];655 -> 660[label="",style="dashed", color="magenta", weight=3];
655 -> 661[label="",style="dashed", color="magenta", weight=3];
656[label="wx47",fontsize=16,color="green",shape="box"];657[label="wx48",fontsize=16,color="green",shape="box"];516[label="primMinusNatS (Succ wx360) wx37",fontsize=16,color="burlywood",shape="box"];729[label="wx37/Succ wx370",fontsize=10,color="white",style="solid",shape="box"];516 -> 729[label="",style="solid", color="burlywood", weight=9];
729 -> 530[label="",style="solid", color="burlywood", weight=3];
730[label="wx37/Zero",fontsize=10,color="white",style="solid",shape="box"];516 -> 730[label="",style="solid", color="burlywood", weight=9];
730 -> 531[label="",style="solid", color="burlywood", weight=3];
517[label="primMinusNatS Zero wx37",fontsize=16,color="burlywood",shape="box"];731[label="wx37/Succ wx370",fontsize=10,color="white",style="solid",shape="box"];517 -> 731[label="",style="solid", color="burlywood", weight=9];
731 -> 532[label="",style="solid", color="burlywood", weight=3];
732[label="wx37/Zero",fontsize=10,color="white",style="solid",shape="box"];517 -> 732[label="",style="solid", color="burlywood", weight=9];
732 -> 533[label="",style="solid", color="burlywood", weight=3];
660 -> 475[label="",style="dashed", color="red", weight=0];
660[label="primMinusNatS (Succ wx47) (Succ (Succ wx48))",fontsize=16,color="magenta"];660 -> 664[label="",style="dashed", color="magenta", weight=3];
660 -> 665[label="",style="dashed", color="magenta", weight=3];
661[label="Succ (Succ wx48)",fontsize=16,color="green",shape="box"];530[label="primMinusNatS (Succ wx360) (Succ wx370)",fontsize=16,color="black",shape="box"];530 -> 544[label="",style="solid", color="black", weight=3];
531[label="primMinusNatS (Succ wx360) Zero",fontsize=16,color="black",shape="box"];531 -> 545[label="",style="solid", color="black", weight=3];
532[label="primMinusNatS Zero (Succ wx370)",fontsize=16,color="black",shape="box"];532 -> 546[label="",style="solid", color="black", weight=3];
533[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];533 -> 547[label="",style="solid", color="black", weight=3];
664[label="Succ (Succ wx48)",fontsize=16,color="green",shape="box"];665[label="Succ wx47",fontsize=16,color="green",shape="box"];544 -> 475[label="",style="dashed", color="red", weight=0];
544[label="primMinusNatS wx360 wx370",fontsize=16,color="magenta"];544 -> 591[label="",style="dashed", color="magenta", weight=3];
544 -> 592[label="",style="dashed", color="magenta", weight=3];
545[label="Succ wx360",fontsize=16,color="green",shape="box"];546[label="Zero",fontsize=16,color="green",shape="box"];547[label="Zero",fontsize=16,color="green",shape="box"];591[label="wx370",fontsize=16,color="green",shape="box"];592[label="wx360",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(wx47, wx48, Main.Zero, Main.Zero) -> new_primModNatS00(wx47, wx48)
   new_primModNatS0(wx47, wx48, Main.Succ(wx490), Main.Zero) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx47), Main.Succ(Main.Succ(wx48))), Main.Succ(Main.Succ(wx48)))
   new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS1, Main.Succ(Main.Zero))
   new_primModNatS0(wx47, wx48, Main.Succ(wx490), Main.Succ(wx500)) -> new_primModNatS0(wx47, wx48, wx490, wx500)
   new_primModNatS00(wx47, wx48) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx47), Main.Succ(Main.Succ(wx48))), Main.Succ(Main.Succ(wx48)))
   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS0(wx300000), Main.Succ(Main.Zero))
   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Succ(wx310000))) -> new_primModNatS0(Main.Succ(wx300000), wx310000, wx300000, wx310000)

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS1 -> new_primMinusNatS3(Main.Zero, Main.Zero)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS0(wx300000) -> new_primMinusNatS3(Main.Succ(wx300000), Main.Zero)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

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

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.
----------------------------------------

(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(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS0(wx300000), Main.Succ(Main.Zero))
   new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS1, Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS1 -> new_primMinusNatS3(Main.Zero, Main.Zero)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS0(wx300000) -> new_primMinusNatS3(Main.Succ(wx300000), Main.Zero)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

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

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

(11) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS0(wx300000), Main.Succ(Main.Zero)) at position [0] we obtained the following new rules [LPAR04]:

   (new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero)),new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero)))


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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS1, Main.Succ(Main.Zero))
   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS1 -> new_primMinusNatS3(Main.Zero, Main.Zero)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS0(wx300000) -> new_primMinusNatS3(Main.Succ(wx300000), Main.Zero)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

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

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

(13) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS1, Main.Succ(Main.Zero))
   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero
   new_primMinusNatS1 -> new_primMinusNatS3(Main.Zero, Main.Zero)

The set Q consists of the following terms:

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

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

(15) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primMinusNatS0(x0)


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

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

   new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS1, Main.Succ(Main.Zero))
   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero
   new_primMinusNatS1 -> new_primMinusNatS3(Main.Zero, Main.Zero)

The set Q consists of the following terms:

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

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

(17) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS1, Main.Succ(Main.Zero)) at position [0] we obtained the following new rules [LPAR04]:

   (new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Zero, Main.Zero), Main.Succ(Main.Zero)),new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Zero, Main.Zero), Main.Succ(Main.Zero)))


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

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

   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero))
   new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Zero, Main.Zero), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero
   new_primMinusNatS1 -> new_primMinusNatS3(Main.Zero, Main.Zero)

The set Q consists of the following terms:

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

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

(19) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

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

   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero))
   new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Zero, Main.Zero), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

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

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

(21) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primMinusNatS1


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

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero))
   new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Zero, Main.Zero), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

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

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

(23) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero)) at position [0] we obtained the following new rules [LPAR04]:

   (new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero)),new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero)))


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

(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Zero, Main.Zero), Main.Succ(Main.Zero))
   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

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

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

(25) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS3(Main.Zero, Main.Zero), Main.Succ(Main.Zero)) at position [0] we obtained the following new rules [LPAR04]:

   (new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS2(Main.Zero, Main.Zero), Main.Succ(Main.Zero)),new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS2(Main.Zero, Main.Zero), Main.Succ(Main.Zero)))


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

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

   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero))
   new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS2(Main.Zero, Main.Zero), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

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

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

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

(28)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

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

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

(29) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(30)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)

The set Q consists of the following terms:

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

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

(31) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primMinusNatS3(x0, x1)


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

(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)

The set Q consists of the following terms:

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

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

(33) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx300000), Main.Zero), Main.Succ(Main.Zero)) at position [0] we obtained the following new rules [LPAR04]:

   (new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(Main.Succ(wx300000), Main.Succ(Main.Zero)),new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(Main.Succ(wx300000), Main.Succ(Main.Zero)))


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

(34)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(Main.Succ(wx300000), Main.Succ(Main.Zero))

The TRS R consists of the following rules:

   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)

The set Q consists of the following terms:

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

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

(35) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(36)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(Main.Succ(wx300000), Main.Succ(Main.Zero))

R is empty.
The set Q consists of the following terms:

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

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

(37) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

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


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

(38)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(Main.Succ(wx300000), Main.Succ(Main.Zero))

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

(39) 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_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Zero)) -> new_primModNatS(Main.Succ(wx300000), Main.Succ(Main.Zero))
The graph contains the following edges 1 > 1, 2 >= 2


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

(40)
YES

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

(41)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS00(wx47, wx48) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx47), Main.Succ(Main.Succ(wx48))), Main.Succ(Main.Succ(wx48)))
   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(wx300000))), Main.Succ(Main.Succ(wx310000))) -> new_primModNatS0(Main.Succ(wx300000), wx310000, wx300000, wx310000)
   new_primModNatS0(wx47, wx48, Main.Zero, Main.Zero) -> new_primModNatS00(wx47, wx48)
   new_primModNatS0(wx47, wx48, Main.Succ(wx490), Main.Zero) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx47), Main.Succ(Main.Succ(wx48))), Main.Succ(Main.Succ(wx48)))
   new_primModNatS0(wx47, wx48, Main.Succ(wx490), Main.Succ(wx500)) -> new_primModNatS0(wx47, wx48, wx490, wx500)

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS1 -> new_primMinusNatS3(Main.Zero, Main.Zero)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS0(wx300000) -> new_primMinusNatS3(Main.Succ(wx300000), Main.Zero)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

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

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

(42) 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(wx300000))), Main.Succ(Main.Succ(wx310000))) -> new_primModNatS0(Main.Succ(wx300000), wx310000, wx300000, wx310000)
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_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(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero


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

(43)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS00(wx47, wx48) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx47), Main.Succ(Main.Succ(wx48))), Main.Succ(Main.Succ(wx48)))
   new_primModNatS0(wx47, wx48, Main.Zero, Main.Zero) -> new_primModNatS00(wx47, wx48)
   new_primModNatS0(wx47, wx48, Main.Succ(wx490), Main.Zero) -> new_primModNatS(new_primMinusNatS2(Main.Succ(wx47), Main.Succ(Main.Succ(wx48))), Main.Succ(Main.Succ(wx48)))
   new_primModNatS0(wx47, wx48, Main.Succ(wx490), Main.Succ(wx500)) -> new_primModNatS0(wx47, wx48, wx490, wx500)

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS1 -> new_primMinusNatS3(Main.Zero, Main.Zero)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS0(wx300000) -> new_primMinusNatS3(Main.Succ(wx300000), Main.Zero)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

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

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

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

(45)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS0(wx47, wx48, Main.Succ(wx490), Main.Succ(wx500)) -> new_primModNatS0(wx47, wx48, wx490, wx500)

The TRS R consists of the following rules:

   new_primMinusNatS3(wx36, wx37) -> new_primMinusNatS2(wx36, wx37)
   new_primMinusNatS1 -> new_primMinusNatS3(Main.Zero, Main.Zero)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS0(wx300000) -> new_primMinusNatS3(Main.Succ(wx300000), Main.Zero)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

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

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

(46) 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(wx47, wx48, Main.Succ(wx490), Main.Succ(wx500)) -> new_primModNatS0(wx47, wx48, wx490, wx500)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


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

(47)
YES

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

(48)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS0(wx42, wx43, Main.Succ(wx440), Main.Succ(wx450)) -> new_primDivNatS0(wx42, wx43, wx440, wx450)
   new_primDivNatS00(wx42, wx43) -> new_primDivNatS(new_primMinusNatS2(Main.Succ(wx42), Main.Succ(wx43)), Main.Succ(wx43))
   new_primDivNatS(Main.Succ(Main.Zero), Main.Zero) -> new_primDivNatS(new_primMinusNatS5, Main.Zero)
   new_primDivNatS(Main.Succ(Main.Succ(wx30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS4(wx30000), Main.Zero)
   new_primDivNatS0(wx42, wx43, Main.Zero, Main.Zero) -> new_primDivNatS00(wx42, wx43)
   new_primDivNatS(Main.Succ(Main.Succ(wx30000)), Main.Succ(wx31000)) -> new_primDivNatS0(wx30000, wx31000, wx30000, wx31000)
   new_primDivNatS0(wx42, wx43, Main.Succ(wx440), Main.Zero) -> new_primDivNatS(new_primMinusNatS2(Main.Succ(wx42), Main.Succ(wx43)), Main.Succ(wx43))

The TRS R consists of the following rules:

   new_primMinusNatS5 -> Main.Zero
   new_primMinusNatS4(wx30000) -> Main.Succ(wx30000)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS4(x0)
   new_primMinusNatS2(Main.Zero, Main.Succ(x0))
   new_primMinusNatS2(Main.Zero, Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), Main.Zero)
   new_primMinusNatS5
   new_primMinusNatS2(Main.Succ(x0), Main.Succ(x1))

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

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

(50)
Complex Obligation (AND)

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

(51)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS(Main.Succ(Main.Succ(wx30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS4(wx30000), Main.Zero)

The TRS R consists of the following rules:

   new_primMinusNatS5 -> Main.Zero
   new_primMinusNatS4(wx30000) -> Main.Succ(wx30000)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS4(x0)
   new_primMinusNatS2(Main.Zero, Main.Succ(x0))
   new_primMinusNatS2(Main.Zero, Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), Main.Zero)
   new_primMinusNatS5
   new_primMinusNatS2(Main.Succ(x0), Main.Succ(x1))

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

(52) 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(wx30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS4(wx30000), Main.Zero)

Strictly oriented rules of the TRS R:

   new_primMinusNatS4(wx30000) -> Main.Succ(wx30000)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

Used ordering: Polynomial interpretation [POLO]:

   POL(Main.Succ(x_1)) = 1 + 2*x_1
   POL(Main.Zero) = 2
   POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2
   POL(new_primMinusNatS2(x_1, x_2)) = x_1 + 2*x_2
   POL(new_primMinusNatS4(x_1)) = 2 + 2*x_1
   POL(new_primMinusNatS5) = 2


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

(53)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   new_primMinusNatS5 -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS4(x0)
   new_primMinusNatS2(Main.Zero, Main.Succ(x0))
   new_primMinusNatS2(Main.Zero, Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), Main.Zero)
   new_primMinusNatS5
   new_primMinusNatS2(Main.Succ(x0), Main.Succ(x1))

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

(54) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(55)
YES

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

(56)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS0(wx42, wx43, Main.Zero, Main.Zero) -> new_primDivNatS00(wx42, wx43)
   new_primDivNatS00(wx42, wx43) -> new_primDivNatS(new_primMinusNatS2(Main.Succ(wx42), Main.Succ(wx43)), Main.Succ(wx43))
   new_primDivNatS(Main.Succ(Main.Succ(wx30000)), Main.Succ(wx31000)) -> new_primDivNatS0(wx30000, wx31000, wx30000, wx31000)
   new_primDivNatS0(wx42, wx43, Main.Succ(wx440), Main.Succ(wx450)) -> new_primDivNatS0(wx42, wx43, wx440, wx450)
   new_primDivNatS0(wx42, wx43, Main.Succ(wx440), Main.Zero) -> new_primDivNatS(new_primMinusNatS2(Main.Succ(wx42), Main.Succ(wx43)), Main.Succ(wx43))

The TRS R consists of the following rules:

   new_primMinusNatS5 -> Main.Zero
   new_primMinusNatS4(wx30000) -> Main.Succ(wx30000)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS4(x0)
   new_primMinusNatS2(Main.Zero, Main.Succ(x0))
   new_primMinusNatS2(Main.Zero, Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), Main.Zero)
   new_primMinusNatS5
   new_primMinusNatS2(Main.Succ(x0), Main.Succ(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   new_primDivNatS(Main.Succ(Main.Succ(wx30000)), Main.Succ(wx31000)) -> new_primDivNatS0(wx30000, wx31000, wx30000, wx31000)
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) = 1
   POL(new_primDivNatS(x_1, x_2)) = x_1
   POL(new_primDivNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1
   POL(new_primDivNatS00(x_1, x_2)) = 1 + x_1
   POL(new_primMinusNatS2(x_1, x_2)) = x_1

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

   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero


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

(58)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS0(wx42, wx43, Main.Zero, Main.Zero) -> new_primDivNatS00(wx42, wx43)
   new_primDivNatS00(wx42, wx43) -> new_primDivNatS(new_primMinusNatS2(Main.Succ(wx42), Main.Succ(wx43)), Main.Succ(wx43))
   new_primDivNatS0(wx42, wx43, Main.Succ(wx440), Main.Succ(wx450)) -> new_primDivNatS0(wx42, wx43, wx440, wx450)
   new_primDivNatS0(wx42, wx43, Main.Succ(wx440), Main.Zero) -> new_primDivNatS(new_primMinusNatS2(Main.Succ(wx42), Main.Succ(wx43)), Main.Succ(wx43))

The TRS R consists of the following rules:

   new_primMinusNatS5 -> Main.Zero
   new_primMinusNatS4(wx30000) -> Main.Succ(wx30000)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS4(x0)
   new_primMinusNatS2(Main.Zero, Main.Succ(x0))
   new_primMinusNatS2(Main.Zero, Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), Main.Zero)
   new_primMinusNatS5
   new_primMinusNatS2(Main.Succ(x0), Main.Succ(x1))

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

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

(60)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS0(wx42, wx43, Main.Succ(wx440), Main.Succ(wx450)) -> new_primDivNatS0(wx42, wx43, wx440, wx450)

The TRS R consists of the following rules:

   new_primMinusNatS5 -> Main.Zero
   new_primMinusNatS4(wx30000) -> Main.Succ(wx30000)
   new_primMinusNatS2(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS2(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS2(wx360, wx370)
   new_primMinusNatS2(Main.Succ(wx360), Main.Zero) -> Main.Succ(wx360)
   new_primMinusNatS2(Main.Zero, Main.Succ(wx370)) -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS4(x0)
   new_primMinusNatS2(Main.Zero, Main.Succ(x0))
   new_primMinusNatS2(Main.Zero, Main.Zero)
   new_primMinusNatS2(Main.Succ(x0), Main.Zero)
   new_primMinusNatS5
   new_primMinusNatS2(Main.Succ(x0), Main.Succ(x1))

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

(61) 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_primDivNatS0(wx42, wx43, Main.Succ(wx440), Main.Succ(wx450)) -> new_primDivNatS0(wx42, wx43, wx440, wx450)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


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

(62)
YES

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

(63)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primMinusNatS(Main.Succ(wx360), Main.Succ(wx370)) -> new_primMinusNatS(wx360, wx370)

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

(64) 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(wx360), Main.Succ(wx370)) -> new_primMinusNatS(wx360, wx370)
The graph contains the following edges 1 > 1, 2 > 2


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

(65)
YES