/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) IFR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) AND
    (9) QDP
        (10) DependencyGraphProof [EQUIVALENT, 0 ms]
        (11) AND
            (12) QDP
                (13) MRRProof [EQUIVALENT, 4 ms]
                (14) QDP
                (15) PisEmptyProof [EQUIVALENT, 0 ms]
                (16) YES
            (17) QDP
                (18) QDPSizeChangeProof [EQUIVALENT, 13 ms]
                (19) YES
    (20) QDP
        (21) DependencyGraphProof [EQUIVALENT, 0 ms]
        (22) AND
            (23) QDP
                (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (25) YES
            (26) QDP
                (27) MRRProof [EQUIVALENT, 0 ms]
                (28) QDP
                (29) PisEmptyProof [EQUIVALENT, 0 ms]
                (30) YES
    (31) QDP
        (32) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (33) YES


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(1) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x"
is transformed to
"primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y);
primModNatS0 x y False = Succ x;
"
The following If expression
"if primGEqNatS x y then primModNatP (primMinusNatS x y) (Succ y) else primMinusNatS y x"
is transformed to
"primModNatP0 x y True = primModNatP (primMinusNatS x y) (Succ y);
primModNatP0 x y False = primMinusNatS y x;
"

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

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

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

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

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

(6)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="mod",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="mod vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="mod 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"];679[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 679[label="",style="solid", color="burlywood", weight=9];
679 -> 6[label="",style="solid", color="burlywood", weight=3];
680[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 680[label="",style="solid", color="burlywood", weight=9];
680 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="primModInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];681[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 681[label="",style="solid", color="burlywood", weight=9];
681 -> 8[label="",style="solid", color="burlywood", weight=3];
682[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 682[label="",style="solid", color="burlywood", weight=9];
682 -> 9[label="",style="solid", color="burlywood", weight=3];
7[label="primModInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];683[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 683[label="",style="solid", color="burlywood", weight=9];
683 -> 10[label="",style="solid", color="burlywood", weight=3];
684[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 684[label="",style="solid", color="burlywood", weight=9];
684 -> 11[label="",style="solid", color="burlywood", weight=3];
8[label="primModInt (Pos vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];685[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 685[label="",style="solid", color="burlywood", weight=9];
685 -> 12[label="",style="solid", color="burlywood", weight=3];
686[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 686[label="",style="solid", color="burlywood", weight=9];
686 -> 13[label="",style="solid", color="burlywood", weight=3];
9[label="primModInt (Pos vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];687[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 687[label="",style="solid", color="burlywood", weight=9];
687 -> 14[label="",style="solid", color="burlywood", weight=3];
688[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 688[label="",style="solid", color="burlywood", weight=9];
688 -> 15[label="",style="solid", color="burlywood", weight=3];
10[label="primModInt (Neg vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];689[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 689[label="",style="solid", color="burlywood", weight=9];
689 -> 16[label="",style="solid", color="burlywood", weight=3];
690[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 690[label="",style="solid", color="burlywood", weight=9];
690 -> 17[label="",style="solid", color="burlywood", weight=3];
11[label="primModInt (Neg vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];691[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 691[label="",style="solid", color="burlywood", weight=9];
691 -> 18[label="",style="solid", color="burlywood", weight=3];
692[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 692[label="",style="solid", color="burlywood", weight=9];
692 -> 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"];693[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 693[label="",style="solid", color="burlywood", weight=9];
693 -> 33[label="",style="solid", color="burlywood", weight=3];
694[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 694[label="",style="solid", color="burlywood", weight=9];
694 -> 34[label="",style="solid", color="burlywood", weight=3];
29[label="error []",fontsize=16,color="red",shape="box"];30[label="primModNatP vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];695[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];30 -> 695[label="",style="solid", color="burlywood", weight=9];
695 -> 35[label="",style="solid", color="burlywood", weight=3];
696[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 696[label="",style="solid", color="burlywood", weight=9];
696 -> 36[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 -> 37[label="",style="dashed", color="magenta", weight=3];
31 -> 38[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 -> 39[label="",style="dashed", color="magenta", weight=3];
32 -> 40[label="",style="dashed", color="magenta", weight=3];
33[label="primModNatS (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];33 -> 41[label="",style="solid", color="black", weight=3];
34[label="primModNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 42[label="",style="solid", color="black", weight=3];
35[label="primModNatP (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];35 -> 43[label="",style="solid", color="black", weight=3];
36[label="primModNatP Zero (Succ vz400)",fontsize=16,color="black",shape="box"];36 -> 44[label="",style="solid", color="black", weight=3];
37[label="vz30",fontsize=16,color="green",shape="box"];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="primModNatS0 vz300 vz400 (primGEqNatS vz300 vz400)",fontsize=16,color="burlywood",shape="box"];697[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];41 -> 697[label="",style="solid", color="burlywood", weight=9];
697 -> 45[label="",style="solid", color="burlywood", weight=3];
698[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 698[label="",style="solid", color="burlywood", weight=9];
698 -> 46[label="",style="solid", color="burlywood", weight=3];
42[label="Zero",fontsize=16,color="green",shape="box"];43[label="primModNatP0 vz300 vz400 (primGEqNatS vz300 vz400)",fontsize=16,color="burlywood",shape="box"];699[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];43 -> 699[label="",style="solid", color="burlywood", weight=9];
699 -> 47[label="",style="solid", color="burlywood", weight=3];
700[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];43 -> 700[label="",style="solid", color="burlywood", weight=9];
700 -> 48[label="",style="solid", color="burlywood", weight=3];
44[label="Zero",fontsize=16,color="green",shape="box"];45[label="primModNatS0 (Succ vz3000) vz400 (primGEqNatS (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];701[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];45 -> 701[label="",style="solid", color="burlywood", weight=9];
701 -> 49[label="",style="solid", color="burlywood", weight=3];
702[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 702[label="",style="solid", color="burlywood", weight=9];
702 -> 50[label="",style="solid", color="burlywood", weight=3];
46[label="primModNatS0 Zero vz400 (primGEqNatS Zero vz400)",fontsize=16,color="burlywood",shape="box"];703[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];46 -> 703[label="",style="solid", color="burlywood", weight=9];
703 -> 51[label="",style="solid", color="burlywood", weight=3];
704[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 704[label="",style="solid", color="burlywood", weight=9];
704 -> 52[label="",style="solid", color="burlywood", weight=3];
47[label="primModNatP0 (Succ vz3000) vz400 (primGEqNatS (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];705[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];47 -> 705[label="",style="solid", color="burlywood", weight=9];
705 -> 53[label="",style="solid", color="burlywood", weight=3];
706[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 706[label="",style="solid", color="burlywood", weight=9];
706 -> 54[label="",style="solid", color="burlywood", weight=3];
48[label="primModNatP0 Zero vz400 (primGEqNatS Zero vz400)",fontsize=16,color="burlywood",shape="box"];707[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];48 -> 707[label="",style="solid", color="burlywood", weight=9];
707 -> 55[label="",style="solid", color="burlywood", weight=3];
708[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 708[label="",style="solid", color="burlywood", weight=9];
708 -> 56[label="",style="solid", color="burlywood", weight=3];
49[label="primModNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];49 -> 57[label="",style="solid", color="black", weight=3];
50[label="primModNatS0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];50 -> 58[label="",style="solid", color="black", weight=3];
51[label="primModNatS0 Zero (Succ vz4000) (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];51 -> 59[label="",style="solid", color="black", weight=3];
52[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];52 -> 60[label="",style="solid", color="black", weight=3];
53[label="primModNatP0 (Succ vz3000) (Succ vz4000) (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];53 -> 61[label="",style="solid", color="black", weight=3];
54[label="primModNatP0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];54 -> 62[label="",style="solid", color="black", weight=3];
55[label="primModNatP0 Zero (Succ vz4000) (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];55 -> 63[label="",style="solid", color="black", weight=3];
56[label="primModNatP0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];56 -> 64[label="",style="solid", color="black", weight=3];
57 -> 548[label="",style="dashed", color="red", weight=0];
57[label="primModNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)",fontsize=16,color="magenta"];57 -> 549[label="",style="dashed", color="magenta", weight=3];
57 -> 550[label="",style="dashed", color="magenta", weight=3];
57 -> 551[label="",style="dashed", color="magenta", weight=3];
57 -> 552[label="",style="dashed", color="magenta", weight=3];
58[label="primModNatS0 (Succ vz3000) Zero True",fontsize=16,color="black",shape="box"];58 -> 67[label="",style="solid", color="black", weight=3];
59[label="primModNatS0 Zero (Succ vz4000) False",fontsize=16,color="black",shape="box"];59 -> 68[label="",style="solid", color="black", weight=3];
60[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];60 -> 69[label="",style="solid", color="black", weight=3];
61 -> 591[label="",style="dashed", color="red", weight=0];
61[label="primModNatP0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)",fontsize=16,color="magenta"];61 -> 592[label="",style="dashed", color="magenta", weight=3];
61 -> 593[label="",style="dashed", color="magenta", weight=3];
61 -> 594[label="",style="dashed", color="magenta", weight=3];
61 -> 595[label="",style="dashed", color="magenta", weight=3];
62[label="primModNatP0 (Succ vz3000) Zero True",fontsize=16,color="black",shape="box"];62 -> 72[label="",style="solid", color="black", weight=3];
63[label="primModNatP0 Zero (Succ vz4000) False",fontsize=16,color="black",shape="box"];63 -> 73[label="",style="solid", color="black", weight=3];
64[label="primModNatP0 Zero Zero True",fontsize=16,color="black",shape="box"];64 -> 74[label="",style="solid", color="black", weight=3];
549[label="vz4000",fontsize=16,color="green",shape="box"];550[label="vz4000",fontsize=16,color="green",shape="box"];551[label="vz3000",fontsize=16,color="green",shape="box"];552[label="vz3000",fontsize=16,color="green",shape="box"];548[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS vz52 vz53)",fontsize=16,color="burlywood",shape="triangle"];709[label="vz52/Succ vz520",fontsize=10,color="white",style="solid",shape="box"];548 -> 709[label="",style="solid", color="burlywood", weight=9];
709 -> 589[label="",style="solid", color="burlywood", weight=3];
710[label="vz52/Zero",fontsize=10,color="white",style="solid",shape="box"];548 -> 710[label="",style="solid", color="burlywood", weight=9];
710 -> 590[label="",style="solid", color="burlywood", weight=3];
67 -> 28[label="",style="dashed", color="red", weight=0];
67[label="primModNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];67 -> 79[label="",style="dashed", color="magenta", weight=3];
67 -> 80[label="",style="dashed", color="magenta", weight=3];
68[label="Succ Zero",fontsize=16,color="green",shape="box"];69 -> 28[label="",style="dashed", color="red", weight=0];
69[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];69 -> 81[label="",style="dashed", color="magenta", weight=3];
69 -> 82[label="",style="dashed", color="magenta", weight=3];
592[label="vz4000",fontsize=16,color="green",shape="box"];593[label="vz3000",fontsize=16,color="green",shape="box"];594[label="vz3000",fontsize=16,color="green",shape="box"];595[label="vz4000",fontsize=16,color="green",shape="box"];591[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS vz57 vz58)",fontsize=16,color="burlywood",shape="triangle"];711[label="vz57/Succ vz570",fontsize=10,color="white",style="solid",shape="box"];591 -> 711[label="",style="solid", color="burlywood", weight=9];
711 -> 632[label="",style="solid", color="burlywood", weight=3];
712[label="vz57/Zero",fontsize=10,color="white",style="solid",shape="box"];591 -> 712[label="",style="solid", color="burlywood", weight=9];
712 -> 633[label="",style="solid", color="burlywood", weight=3];
72 -> 30[label="",style="dashed", color="red", weight=0];
72[label="primModNatP (primMinusNatS (Succ vz3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];72 -> 87[label="",style="dashed", color="magenta", weight=3];
72 -> 88[label="",style="dashed", color="magenta", weight=3];
73[label="primMinusNatS (Succ vz4000) Zero",fontsize=16,color="black",shape="triangle"];73 -> 89[label="",style="solid", color="black", weight=3];
74 -> 30[label="",style="dashed", color="red", weight=0];
74[label="primModNatP (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];74 -> 90[label="",style="dashed", color="magenta", weight=3];
74 -> 91[label="",style="dashed", color="magenta", weight=3];
589[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS (Succ vz520) vz53)",fontsize=16,color="burlywood",shape="box"];713[label="vz53/Succ vz530",fontsize=10,color="white",style="solid",shape="box"];589 -> 713[label="",style="solid", color="burlywood", weight=9];
713 -> 634[label="",style="solid", color="burlywood", weight=3];
714[label="vz53/Zero",fontsize=10,color="white",style="solid",shape="box"];589 -> 714[label="",style="solid", color="burlywood", weight=9];
714 -> 635[label="",style="solid", color="burlywood", weight=3];
590[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS Zero vz53)",fontsize=16,color="burlywood",shape="box"];715[label="vz53/Succ vz530",fontsize=10,color="white",style="solid",shape="box"];590 -> 715[label="",style="solid", color="burlywood", weight=9];
715 -> 636[label="",style="solid", color="burlywood", weight=3];
716[label="vz53/Zero",fontsize=10,color="white",style="solid",shape="box"];590 -> 716[label="",style="solid", color="burlywood", weight=9];
716 -> 637[label="",style="solid", color="burlywood", weight=3];
79 -> 73[label="",style="dashed", color="red", weight=0];
79[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="magenta"];79 -> 96[label="",style="dashed", color="magenta", weight=3];
80[label="Zero",fontsize=16,color="green",shape="box"];81[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];81 -> 97[label="",style="solid", color="black", weight=3];
82[label="Zero",fontsize=16,color="green",shape="box"];632[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS (Succ vz570) vz58)",fontsize=16,color="burlywood",shape="box"];717[label="vz58/Succ vz580",fontsize=10,color="white",style="solid",shape="box"];632 -> 717[label="",style="solid", color="burlywood", weight=9];
717 -> 638[label="",style="solid", color="burlywood", weight=3];
718[label="vz58/Zero",fontsize=10,color="white",style="solid",shape="box"];632 -> 718[label="",style="solid", color="burlywood", weight=9];
718 -> 639[label="",style="solid", color="burlywood", weight=3];
633[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS Zero vz58)",fontsize=16,color="burlywood",shape="box"];719[label="vz58/Succ vz580",fontsize=10,color="white",style="solid",shape="box"];633 -> 719[label="",style="solid", color="burlywood", weight=9];
719 -> 640[label="",style="solid", color="burlywood", weight=3];
720[label="vz58/Zero",fontsize=10,color="white",style="solid",shape="box"];633 -> 720[label="",style="solid", color="burlywood", weight=9];
720 -> 641[label="",style="solid", color="burlywood", weight=3];
87 -> 73[label="",style="dashed", color="red", weight=0];
87[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="magenta"];87 -> 102[label="",style="dashed", color="magenta", weight=3];
88[label="Zero",fontsize=16,color="green",shape="box"];89[label="Succ vz4000",fontsize=16,color="green",shape="box"];90 -> 81[label="",style="dashed", color="red", weight=0];
90[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];91[label="Zero",fontsize=16,color="green",shape="box"];634[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS (Succ vz520) (Succ vz530))",fontsize=16,color="black",shape="box"];634 -> 642[label="",style="solid", color="black", weight=3];
635[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS (Succ vz520) Zero)",fontsize=16,color="black",shape="box"];635 -> 643[label="",style="solid", color="black", weight=3];
636[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS Zero (Succ vz530))",fontsize=16,color="black",shape="box"];636 -> 644[label="",style="solid", color="black", weight=3];
637[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];637 -> 645[label="",style="solid", color="black", weight=3];
96[label="vz3000",fontsize=16,color="green",shape="box"];97[label="Zero",fontsize=16,color="green",shape="box"];638[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS (Succ vz570) (Succ vz580))",fontsize=16,color="black",shape="box"];638 -> 646[label="",style="solid", color="black", weight=3];
639[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS (Succ vz570) Zero)",fontsize=16,color="black",shape="box"];639 -> 647[label="",style="solid", color="black", weight=3];
640[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS Zero (Succ vz580))",fontsize=16,color="black",shape="box"];640 -> 648[label="",style="solid", color="black", weight=3];
641[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];641 -> 649[label="",style="solid", color="black", weight=3];
102[label="vz3000",fontsize=16,color="green",shape="box"];642 -> 548[label="",style="dashed", color="red", weight=0];
642[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS vz520 vz530)",fontsize=16,color="magenta"];642 -> 650[label="",style="dashed", color="magenta", weight=3];
642 -> 651[label="",style="dashed", color="magenta", weight=3];
643[label="primModNatS0 (Succ vz50) (Succ vz51) True",fontsize=16,color="black",shape="triangle"];643 -> 652[label="",style="solid", color="black", weight=3];
644[label="primModNatS0 (Succ vz50) (Succ vz51) False",fontsize=16,color="black",shape="box"];644 -> 653[label="",style="solid", color="black", weight=3];
645 -> 643[label="",style="dashed", color="red", weight=0];
645[label="primModNatS0 (Succ vz50) (Succ vz51) True",fontsize=16,color="magenta"];646 -> 591[label="",style="dashed", color="red", weight=0];
646[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS vz570 vz580)",fontsize=16,color="magenta"];646 -> 654[label="",style="dashed", color="magenta", weight=3];
646 -> 655[label="",style="dashed", color="magenta", weight=3];
647[label="primModNatP0 (Succ vz55) (Succ vz56) True",fontsize=16,color="black",shape="triangle"];647 -> 656[label="",style="solid", color="black", weight=3];
648[label="primModNatP0 (Succ vz55) (Succ vz56) False",fontsize=16,color="black",shape="box"];648 -> 657[label="",style="solid", color="black", weight=3];
649 -> 647[label="",style="dashed", color="red", weight=0];
649[label="primModNatP0 (Succ vz55) (Succ vz56) True",fontsize=16,color="magenta"];650[label="vz530",fontsize=16,color="green",shape="box"];651[label="vz520",fontsize=16,color="green",shape="box"];652 -> 28[label="",style="dashed", color="red", weight=0];
652[label="primModNatS (primMinusNatS (Succ vz50) (Succ vz51)) (Succ (Succ vz51))",fontsize=16,color="magenta"];652 -> 658[label="",style="dashed", color="magenta", weight=3];
652 -> 659[label="",style="dashed", color="magenta", weight=3];
653[label="Succ (Succ vz50)",fontsize=16,color="green",shape="box"];654[label="vz570",fontsize=16,color="green",shape="box"];655[label="vz580",fontsize=16,color="green",shape="box"];656 -> 30[label="",style="dashed", color="red", weight=0];
656[label="primModNatP (primMinusNatS (Succ vz55) (Succ vz56)) (Succ (Succ vz56))",fontsize=16,color="magenta"];656 -> 660[label="",style="dashed", color="magenta", weight=3];
656 -> 661[label="",style="dashed", color="magenta", weight=3];
657[label="primMinusNatS (Succ vz56) (Succ vz55)",fontsize=16,color="black",shape="triangle"];657 -> 662[label="",style="solid", color="black", weight=3];
658 -> 657[label="",style="dashed", color="red", weight=0];
658[label="primMinusNatS (Succ vz50) (Succ vz51)",fontsize=16,color="magenta"];658 -> 663[label="",style="dashed", color="magenta", weight=3];
658 -> 664[label="",style="dashed", color="magenta", weight=3];
659[label="Succ vz51",fontsize=16,color="green",shape="box"];660 -> 657[label="",style="dashed", color="red", weight=0];
660[label="primMinusNatS (Succ vz55) (Succ vz56)",fontsize=16,color="magenta"];660 -> 665[label="",style="dashed", color="magenta", weight=3];
660 -> 666[label="",style="dashed", color="magenta", weight=3];
661[label="Succ vz56",fontsize=16,color="green",shape="box"];662[label="primMinusNatS vz56 vz55",fontsize=16,color="burlywood",shape="triangle"];721[label="vz56/Succ vz560",fontsize=10,color="white",style="solid",shape="box"];662 -> 721[label="",style="solid", color="burlywood", weight=9];
721 -> 667[label="",style="solid", color="burlywood", weight=3];
722[label="vz56/Zero",fontsize=10,color="white",style="solid",shape="box"];662 -> 722[label="",style="solid", color="burlywood", weight=9];
722 -> 668[label="",style="solid", color="burlywood", weight=3];
663[label="vz50",fontsize=16,color="green",shape="box"];664[label="vz51",fontsize=16,color="green",shape="box"];665[label="vz55",fontsize=16,color="green",shape="box"];666[label="vz56",fontsize=16,color="green",shape="box"];667[label="primMinusNatS (Succ vz560) vz55",fontsize=16,color="burlywood",shape="box"];723[label="vz55/Succ vz550",fontsize=10,color="white",style="solid",shape="box"];667 -> 723[label="",style="solid", color="burlywood", weight=9];
723 -> 669[label="",style="solid", color="burlywood", weight=3];
724[label="vz55/Zero",fontsize=10,color="white",style="solid",shape="box"];667 -> 724[label="",style="solid", color="burlywood", weight=9];
724 -> 670[label="",style="solid", color="burlywood", weight=3];
668[label="primMinusNatS Zero vz55",fontsize=16,color="burlywood",shape="box"];725[label="vz55/Succ vz550",fontsize=10,color="white",style="solid",shape="box"];668 -> 725[label="",style="solid", color="burlywood", weight=9];
725 -> 671[label="",style="solid", color="burlywood", weight=3];
726[label="vz55/Zero",fontsize=10,color="white",style="solid",shape="box"];668 -> 726[label="",style="solid", color="burlywood", weight=9];
726 -> 672[label="",style="solid", color="burlywood", weight=3];
669[label="primMinusNatS (Succ vz560) (Succ vz550)",fontsize=16,color="black",shape="box"];669 -> 673[label="",style="solid", color="black", weight=3];
670[label="primMinusNatS (Succ vz560) Zero",fontsize=16,color="black",shape="box"];670 -> 674[label="",style="solid", color="black", weight=3];
671[label="primMinusNatS Zero (Succ vz550)",fontsize=16,color="black",shape="box"];671 -> 675[label="",style="solid", color="black", weight=3];
672[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];672 -> 676[label="",style="solid", color="black", weight=3];
673 -> 662[label="",style="dashed", color="red", weight=0];
673[label="primMinusNatS vz560 vz550",fontsize=16,color="magenta"];673 -> 677[label="",style="dashed", color="magenta", weight=3];
673 -> 678[label="",style="dashed", color="magenta", weight=3];
674[label="Succ vz560",fontsize=16,color="green",shape="box"];675[label="Zero",fontsize=16,color="green",shape="box"];676[label="Zero",fontsize=16,color="green",shape="box"];677[label="vz560",fontsize=16,color="green",shape="box"];678[label="vz550",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS00(vz50, vz51) -> new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
   new_primModNatS(Succ(Succ(vz3000)), Zero) -> new_primModNatS(new_primMinusNatS0(vz3000), Zero)
   new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
   new_primModNatS(Succ(Zero), Zero) -> new_primModNatS(new_primMinusNatS1, Zero)
   new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) -> new_primModNatS0(vz50, vz51, vz520, vz530)
   new_primModNatS0(vz50, vz51, Succ(vz520), Zero) -> new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
   new_primModNatS0(vz50, vz51, Zero, Zero) -> new_primModNatS00(vz50, vz51)

The TRS R consists of the following rules:

   new_primMinusNatS1 -> Zero
   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
   new_primMinusNatS3(Zero, Zero) -> Zero
   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
   new_primMinusNatS0(vz4000) -> Succ(vz4000)
   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero

The set Q consists of the following terms:

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

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

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

(11)
Complex Obligation (AND)

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

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

   new_primModNatS(Succ(Succ(vz3000)), Zero) -> new_primModNatS(new_primMinusNatS0(vz3000), Zero)

The TRS R consists of the following rules:

   new_primMinusNatS1 -> Zero
   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
   new_primMinusNatS3(Zero, Zero) -> Zero
   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
   new_primMinusNatS0(vz4000) -> Succ(vz4000)
   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero

The set Q consists of the following terms:

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

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

(13) 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_primModNatS(Succ(Succ(vz3000)), Zero) -> new_primModNatS(new_primMinusNatS0(vz3000), Zero)

Strictly oriented rules of the TRS R:

   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
   new_primMinusNatS3(Zero, Zero) -> Zero
   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero

Used ordering: Polynomial interpretation [POLO]:

   POL(Succ(x_1)) = 1 + x_1
   POL(Zero) = 2
   POL(new_primMinusNatS0(x_1)) = 1 + x_1
   POL(new_primMinusNatS1) = 2
   POL(new_primMinusNatS2(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(new_primMinusNatS3(x_1, x_2)) = 1 + 2*x_1 + x_2
   POL(new_primModNatS(x_1, x_2)) = x_1 + x_2


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

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

   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
   new_primMinusNatS0(vz4000) -> Succ(vz4000)

The set Q consists of the following terms:

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

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

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

(16)
YES

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

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

   new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
   new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) -> new_primModNatS0(vz50, vz51, vz520, vz530)
   new_primModNatS0(vz50, vz51, Succ(vz520), Zero) -> new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
   new_primModNatS0(vz50, vz51, Zero, Zero) -> new_primModNatS00(vz50, vz51)
   new_primModNatS00(vz50, vz51) -> new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))

The TRS R consists of the following rules:

   new_primMinusNatS1 -> Zero
   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
   new_primMinusNatS3(Zero, Zero) -> Zero
   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
   new_primMinusNatS0(vz4000) -> Succ(vz4000)
   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero

The set Q consists of the following terms:

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

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

(18) 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(Succ(x_1)) = 1 + x_1
   POL(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_primModNatS0(vz50, vz51, Succ(vz520), Zero) -> new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1


*new_primModNatS00(vz50, vz51) -> new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1


*new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatS0(vz3000, vz4000, vz3000, vz4000) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4


*new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) -> new_primModNatS0(vz50, vz51, vz520, vz530) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


*new_primModNatS0(vz50, vz51, Zero, Zero) -> new_primModNatS00(vz50, vz51) (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(Zero, Zero) -> Zero
   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero
   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)

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

(19)
YES

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

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

   new_primModNatP(Succ(Succ(vz3000)), Zero) -> new_primModNatP(new_primMinusNatS0(vz3000), Zero)
   new_primModNatP00(vz55, vz56) -> new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
   new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatP0(vz3000, vz4000, vz3000, vz4000)
   new_primModNatP0(vz55, vz56, Succ(vz570), Zero) -> new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
   new_primModNatP(Succ(Zero), Zero) -> new_primModNatP(new_primMinusNatS1, Zero)
   new_primModNatP0(vz55, vz56, Zero, Zero) -> new_primModNatP00(vz55, vz56)
   new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) -> new_primModNatP0(vz55, vz56, vz570, vz580)

The TRS R consists of the following rules:

   new_primMinusNatS1 -> Zero
   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
   new_primMinusNatS3(Zero, Zero) -> Zero
   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
   new_primMinusNatS0(vz4000) -> Succ(vz4000)
   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero

The set Q consists of the following terms:

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

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(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatP0(vz3000, vz4000, vz3000, vz4000)
   new_primModNatP0(vz55, vz56, Succ(vz570), Zero) -> new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
   new_primModNatP0(vz55, vz56, Zero, Zero) -> new_primModNatP00(vz55, vz56)
   new_primModNatP00(vz55, vz56) -> new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
   new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) -> new_primModNatP0(vz55, vz56, vz570, vz580)

The TRS R consists of the following rules:

   new_primMinusNatS1 -> Zero
   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
   new_primMinusNatS3(Zero, Zero) -> Zero
   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
   new_primMinusNatS0(vz4000) -> Succ(vz4000)
   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero

The set Q consists of the following terms:

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

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(Succ(x_1)) = 1 + x_1
   POL(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(vz55, vz56, Succ(vz570), Zero) -> new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1


*new_primModNatP00(vz55, vz56) -> new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1


*new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatP0(vz3000, vz4000, vz3000, vz4000) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4


*new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) -> new_primModNatP0(vz55, vz56, vz570, vz580) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


*new_primModNatP0(vz55, vz56, Zero, Zero) -> new_primModNatP00(vz55, vz56) (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(Zero, Zero) -> Zero
   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero
   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)

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

(25)
YES

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

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

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

The TRS R consists of the following rules:

   new_primMinusNatS1 -> Zero
   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
   new_primMinusNatS3(Zero, Zero) -> Zero
   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
   new_primMinusNatS0(vz4000) -> Succ(vz4000)
   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero

The set Q consists of the following terms:

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

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

(27) 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_primModNatP(Succ(Succ(vz3000)), Zero) -> new_primModNatP(new_primMinusNatS0(vz3000), Zero)

Strictly oriented rules of the TRS R:

   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
   new_primMinusNatS3(Zero, Zero) -> Zero
   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero

Used ordering: Polynomial interpretation [POLO]:

   POL(Succ(x_1)) = 1 + x_1
   POL(Zero) = 2
   POL(new_primMinusNatS0(x_1)) = 1 + x_1
   POL(new_primMinusNatS1) = 2
   POL(new_primMinusNatS2(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(new_primMinusNatS3(x_1, x_2)) = 1 + 2*x_1 + x_2
   POL(new_primModNatP(x_1, x_2)) = x_1 + x_2


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

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

   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
   new_primMinusNatS0(vz4000) -> Succ(vz4000)

The set Q consists of the following terms:

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

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

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

(30)
YES

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

(31)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primMinusNatS(Succ(vz560), Succ(vz550)) -> new_primMinusNatS(vz560, vz550)

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

(32) 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(Succ(vz560), Succ(vz550)) -> new_primMinusNatS(vz560, vz550)
The graph contains the following edges 1 > 1, 2 > 2


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

(33)
YES