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


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

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


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

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

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

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

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

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

(3) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"min x y|x <= yx|otherwisey;
"
is transformed to
"min x y = min2 x y;
"
"min0 x y True = y;
"
"min1 x y True = x;
min1 x y False = min0 x y otherwise;
"
"min2 x y = min1 x y (x <= y);
"
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;
}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="min",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="min vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="min vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="min2 vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="min1 vx3 vx4 (vx3 <= vx4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="min1 vx3 vx4 (compare vx3 vx4 /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="min1 vx3 vx4 (not (compare vx3 vx4 == GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="min1 vx3 vx4 (not (primCmpInt vx3 vx4 == GT))",fontsize=16,color="burlywood",shape="box"];591[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];9 -> 591[label="",style="solid", color="burlywood", weight=9];
591 -> 10[label="",style="solid", color="burlywood", weight=3];
592[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];9 -> 592[label="",style="solid", color="burlywood", weight=9];
592 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="min1 (Pos vx30) vx4 (not (primCmpInt (Pos vx30) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];593[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];10 -> 593[label="",style="solid", color="burlywood", weight=9];
593 -> 12[label="",style="solid", color="burlywood", weight=3];
594[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 594[label="",style="solid", color="burlywood", weight=9];
594 -> 13[label="",style="solid", color="burlywood", weight=3];
11[label="min1 (Neg vx30) vx4 (not (primCmpInt (Neg vx30) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];595[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];11 -> 595[label="",style="solid", color="burlywood", weight=9];
595 -> 14[label="",style="solid", color="burlywood", weight=3];
596[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 596[label="",style="solid", color="burlywood", weight=9];
596 -> 15[label="",style="solid", color="burlywood", weight=3];
12[label="min1 (Pos (Succ vx300)) vx4 (not (primCmpInt (Pos (Succ vx300)) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];597[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 597[label="",style="solid", color="burlywood", weight=9];
597 -> 16[label="",style="solid", color="burlywood", weight=3];
598[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 598[label="",style="solid", color="burlywood", weight=9];
598 -> 17[label="",style="solid", color="burlywood", weight=3];
13[label="min1 (Pos Zero) vx4 (not (primCmpInt (Pos Zero) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];599[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 599[label="",style="solid", color="burlywood", weight=9];
599 -> 18[label="",style="solid", color="burlywood", weight=3];
600[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 600[label="",style="solid", color="burlywood", weight=9];
600 -> 19[label="",style="solid", color="burlywood", weight=3];
14[label="min1 (Neg (Succ vx300)) vx4 (not (primCmpInt (Neg (Succ vx300)) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];601[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];14 -> 601[label="",style="solid", color="burlywood", weight=9];
601 -> 20[label="",style="solid", color="burlywood", weight=3];
602[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];14 -> 602[label="",style="solid", color="burlywood", weight=9];
602 -> 21[label="",style="solid", color="burlywood", weight=3];
15[label="min1 (Neg Zero) vx4 (not (primCmpInt (Neg Zero) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];603[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];15 -> 603[label="",style="solid", color="burlywood", weight=9];
603 -> 22[label="",style="solid", color="burlywood", weight=3];
604[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];15 -> 604[label="",style="solid", color="burlywood", weight=9];
604 -> 23[label="",style="solid", color="burlywood", weight=3];
16[label="min1 (Pos (Succ vx300)) (Pos vx40) (not (primCmpInt (Pos (Succ vx300)) (Pos vx40) == GT))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
17[label="min1 (Pos (Succ vx300)) (Neg vx40) (not (primCmpInt (Pos (Succ vx300)) (Neg vx40) == GT))",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
18[label="min1 (Pos Zero) (Pos vx40) (not (primCmpInt (Pos Zero) (Pos vx40) == GT))",fontsize=16,color="burlywood",shape="box"];605[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];18 -> 605[label="",style="solid", color="burlywood", weight=9];
605 -> 26[label="",style="solid", color="burlywood", weight=3];
606[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 606[label="",style="solid", color="burlywood", weight=9];
606 -> 27[label="",style="solid", color="burlywood", weight=3];
19[label="min1 (Pos Zero) (Neg vx40) (not (primCmpInt (Pos Zero) (Neg vx40) == GT))",fontsize=16,color="burlywood",shape="box"];607[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];19 -> 607[label="",style="solid", color="burlywood", weight=9];
607 -> 28[label="",style="solid", color="burlywood", weight=3];
608[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 608[label="",style="solid", color="burlywood", weight=9];
608 -> 29[label="",style="solid", color="burlywood", weight=3];
20[label="min1 (Neg (Succ vx300)) (Pos vx40) (not (primCmpInt (Neg (Succ vx300)) (Pos vx40) == GT))",fontsize=16,color="black",shape="box"];20 -> 30[label="",style="solid", color="black", weight=3];
21[label="min1 (Neg (Succ vx300)) (Neg vx40) (not (primCmpInt (Neg (Succ vx300)) (Neg vx40) == GT))",fontsize=16,color="black",shape="box"];21 -> 31[label="",style="solid", color="black", weight=3];
22[label="min1 (Neg Zero) (Pos vx40) (not (primCmpInt (Neg Zero) (Pos vx40) == GT))",fontsize=16,color="burlywood",shape="box"];609[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];22 -> 609[label="",style="solid", color="burlywood", weight=9];
609 -> 32[label="",style="solid", color="burlywood", weight=3];
610[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 610[label="",style="solid", color="burlywood", weight=9];
610 -> 33[label="",style="solid", color="burlywood", weight=3];
23[label="min1 (Neg Zero) (Neg vx40) (not (primCmpInt (Neg Zero) (Neg vx40) == GT))",fontsize=16,color="burlywood",shape="box"];611[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];23 -> 611[label="",style="solid", color="burlywood", weight=9];
611 -> 34[label="",style="solid", color="burlywood", weight=3];
612[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 612[label="",style="solid", color="burlywood", weight=9];
612 -> 35[label="",style="solid", color="burlywood", weight=3];
24[label="min1 (Pos (Succ vx300)) (Pos vx40) (not (primCmpNat (Succ vx300) vx40 == GT))",fontsize=16,color="burlywood",shape="box"];613[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];24 -> 613[label="",style="solid", color="burlywood", weight=9];
613 -> 36[label="",style="solid", color="burlywood", weight=3];
614[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 614[label="",style="solid", color="burlywood", weight=9];
614 -> 37[label="",style="solid", color="burlywood", weight=3];
25[label="min1 (Pos (Succ vx300)) (Neg vx40) (not (GT == GT))",fontsize=16,color="black",shape="box"];25 -> 38[label="",style="solid", color="black", weight=3];
26[label="min1 (Pos Zero) (Pos (Succ vx400)) (not (primCmpInt (Pos Zero) (Pos (Succ vx400)) == GT))",fontsize=16,color="black",shape="box"];26 -> 39[label="",style="solid", color="black", weight=3];
27[label="min1 (Pos Zero) (Pos Zero) (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];27 -> 40[label="",style="solid", color="black", weight=3];
28[label="min1 (Pos Zero) (Neg (Succ vx400)) (not (primCmpInt (Pos Zero) (Neg (Succ vx400)) == GT))",fontsize=16,color="black",shape="box"];28 -> 41[label="",style="solid", color="black", weight=3];
29[label="min1 (Pos Zero) (Neg Zero) (not (primCmpInt (Pos Zero) (Neg Zero) == GT))",fontsize=16,color="black",shape="box"];29 -> 42[label="",style="solid", color="black", weight=3];
30[label="min1 (Neg (Succ vx300)) (Pos vx40) (not (LT == GT))",fontsize=16,color="black",shape="box"];30 -> 43[label="",style="solid", color="black", weight=3];
31[label="min1 (Neg (Succ vx300)) (Neg vx40) (not (primCmpNat vx40 (Succ vx300) == GT))",fontsize=16,color="burlywood",shape="box"];615[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];31 -> 615[label="",style="solid", color="burlywood", weight=9];
615 -> 44[label="",style="solid", color="burlywood", weight=3];
616[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];31 -> 616[label="",style="solid", color="burlywood", weight=9];
616 -> 45[label="",style="solid", color="burlywood", weight=3];
32[label="min1 (Neg Zero) (Pos (Succ vx400)) (not (primCmpInt (Neg Zero) (Pos (Succ vx400)) == GT))",fontsize=16,color="black",shape="box"];32 -> 46[label="",style="solid", color="black", weight=3];
33[label="min1 (Neg Zero) (Pos Zero) (not (primCmpInt (Neg Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];33 -> 47[label="",style="solid", color="black", weight=3];
34[label="min1 (Neg Zero) (Neg (Succ vx400)) (not (primCmpInt (Neg Zero) (Neg (Succ vx400)) == GT))",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
35[label="min1 (Neg Zero) (Neg Zero) (not (primCmpInt (Neg Zero) (Neg Zero) == GT))",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
36[label="min1 (Pos (Succ vx300)) (Pos (Succ vx400)) (not (primCmpNat (Succ vx300) (Succ vx400) == GT))",fontsize=16,color="black",shape="box"];36 -> 50[label="",style="solid", color="black", weight=3];
37[label="min1 (Pos (Succ vx300)) (Pos Zero) (not (primCmpNat (Succ vx300) Zero == GT))",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3];
38[label="min1 (Pos (Succ vx300)) (Neg vx40) (not True)",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
39[label="min1 (Pos Zero) (Pos (Succ vx400)) (not (primCmpNat Zero (Succ vx400) == GT))",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3];
40[label="min1 (Pos Zero) (Pos Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3];
41[label="min1 (Pos Zero) (Neg (Succ vx400)) (not (GT == GT))",fontsize=16,color="black",shape="box"];41 -> 55[label="",style="solid", color="black", weight=3];
42[label="min1 (Pos Zero) (Neg Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];42 -> 56[label="",style="solid", color="black", weight=3];
43[label="min1 (Neg (Succ vx300)) (Pos vx40) (not False)",fontsize=16,color="black",shape="box"];43 -> 57[label="",style="solid", color="black", weight=3];
44[label="min1 (Neg (Succ vx300)) (Neg (Succ vx400)) (not (primCmpNat (Succ vx400) (Succ vx300) == GT))",fontsize=16,color="black",shape="box"];44 -> 58[label="",style="solid", color="black", weight=3];
45[label="min1 (Neg (Succ vx300)) (Neg Zero) (not (primCmpNat Zero (Succ vx300) == GT))",fontsize=16,color="black",shape="box"];45 -> 59[label="",style="solid", color="black", weight=3];
46[label="min1 (Neg Zero) (Pos (Succ vx400)) (not (LT == GT))",fontsize=16,color="black",shape="box"];46 -> 60[label="",style="solid", color="black", weight=3];
47[label="min1 (Neg Zero) (Pos Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];47 -> 61[label="",style="solid", color="black", weight=3];
48[label="min1 (Neg Zero) (Neg (Succ vx400)) (not (primCmpNat (Succ vx400) Zero == GT))",fontsize=16,color="black",shape="box"];48 -> 62[label="",style="solid", color="black", weight=3];
49[label="min1 (Neg Zero) (Neg Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];49 -> 63[label="",style="solid", color="black", weight=3];
50 -> 453[label="",style="dashed", color="red", weight=0];
50[label="min1 (Pos (Succ vx300)) (Pos (Succ vx400)) (not (primCmpNat vx300 vx400 == GT))",fontsize=16,color="magenta"];50 -> 454[label="",style="dashed", color="magenta", weight=3];
50 -> 455[label="",style="dashed", color="magenta", weight=3];
50 -> 456[label="",style="dashed", color="magenta", weight=3];
50 -> 457[label="",style="dashed", color="magenta", weight=3];
51[label="min1 (Pos (Succ vx300)) (Pos Zero) (not (GT == GT))",fontsize=16,color="black",shape="box"];51 -> 66[label="",style="solid", color="black", weight=3];
52[label="min1 (Pos (Succ vx300)) (Neg vx40) False",fontsize=16,color="black",shape="box"];52 -> 67[label="",style="solid", color="black", weight=3];
53[label="min1 (Pos Zero) (Pos (Succ vx400)) (not (LT == GT))",fontsize=16,color="black",shape="box"];53 -> 68[label="",style="solid", color="black", weight=3];
54[label="min1 (Pos Zero) (Pos Zero) (not False)",fontsize=16,color="black",shape="box"];54 -> 69[label="",style="solid", color="black", weight=3];
55[label="min1 (Pos Zero) (Neg (Succ vx400)) (not True)",fontsize=16,color="black",shape="box"];55 -> 70[label="",style="solid", color="black", weight=3];
56[label="min1 (Pos Zero) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];56 -> 71[label="",style="solid", color="black", weight=3];
57[label="min1 (Neg (Succ vx300)) (Pos vx40) True",fontsize=16,color="black",shape="box"];57 -> 72[label="",style="solid", color="black", weight=3];
58 -> 523[label="",style="dashed", color="red", weight=0];
58[label="min1 (Neg (Succ vx300)) (Neg (Succ vx400)) (not (primCmpNat vx400 vx300 == GT))",fontsize=16,color="magenta"];58 -> 524[label="",style="dashed", color="magenta", weight=3];
58 -> 525[label="",style="dashed", color="magenta", weight=3];
58 -> 526[label="",style="dashed", color="magenta", weight=3];
58 -> 527[label="",style="dashed", color="magenta", weight=3];
59[label="min1 (Neg (Succ vx300)) (Neg Zero) (not (LT == GT))",fontsize=16,color="black",shape="box"];59 -> 75[label="",style="solid", color="black", weight=3];
60[label="min1 (Neg Zero) (Pos (Succ vx400)) (not False)",fontsize=16,color="black",shape="box"];60 -> 76[label="",style="solid", color="black", weight=3];
61[label="min1 (Neg Zero) (Pos Zero) (not False)",fontsize=16,color="black",shape="box"];61 -> 77[label="",style="solid", color="black", weight=3];
62[label="min1 (Neg Zero) (Neg (Succ vx400)) (not (GT == GT))",fontsize=16,color="black",shape="box"];62 -> 78[label="",style="solid", color="black", weight=3];
63[label="min1 (Neg Zero) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];63 -> 79[label="",style="solid", color="black", weight=3];
454[label="vx300",fontsize=16,color="green",shape="box"];455[label="vx400",fontsize=16,color="green",shape="box"];456[label="vx300",fontsize=16,color="green",shape="box"];457[label="vx400",fontsize=16,color="green",shape="box"];453[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat vx43 vx44 == GT))",fontsize=16,color="burlywood",shape="triangle"];617[label="vx43/Succ vx430",fontsize=10,color="white",style="solid",shape="box"];453 -> 617[label="",style="solid", color="burlywood", weight=9];
617 -> 490[label="",style="solid", color="burlywood", weight=3];
618[label="vx43/Zero",fontsize=10,color="white",style="solid",shape="box"];453 -> 618[label="",style="solid", color="burlywood", weight=9];
618 -> 491[label="",style="solid", color="burlywood", weight=3];
66[label="min1 (Pos (Succ vx300)) (Pos Zero) (not True)",fontsize=16,color="black",shape="box"];66 -> 84[label="",style="solid", color="black", weight=3];
67[label="min0 (Pos (Succ vx300)) (Neg vx40) otherwise",fontsize=16,color="black",shape="box"];67 -> 85[label="",style="solid", color="black", weight=3];
68[label="min1 (Pos Zero) (Pos (Succ vx400)) (not False)",fontsize=16,color="black",shape="box"];68 -> 86[label="",style="solid", color="black", weight=3];
69[label="min1 (Pos Zero) (Pos Zero) True",fontsize=16,color="black",shape="box"];69 -> 87[label="",style="solid", color="black", weight=3];
70[label="min1 (Pos Zero) (Neg (Succ vx400)) False",fontsize=16,color="black",shape="box"];70 -> 88[label="",style="solid", color="black", weight=3];
71[label="min1 (Pos Zero) (Neg Zero) True",fontsize=16,color="black",shape="box"];71 -> 89[label="",style="solid", color="black", weight=3];
72[label="Neg (Succ vx300)",fontsize=16,color="green",shape="box"];524[label="vx300",fontsize=16,color="green",shape="box"];525[label="vx400",fontsize=16,color="green",shape="box"];526[label="vx300",fontsize=16,color="green",shape="box"];527[label="vx400",fontsize=16,color="green",shape="box"];523[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat vx52 vx53 == GT))",fontsize=16,color="burlywood",shape="triangle"];619[label="vx52/Succ vx520",fontsize=10,color="white",style="solid",shape="box"];523 -> 619[label="",style="solid", color="burlywood", weight=9];
619 -> 564[label="",style="solid", color="burlywood", weight=3];
620[label="vx52/Zero",fontsize=10,color="white",style="solid",shape="box"];523 -> 620[label="",style="solid", color="burlywood", weight=9];
620 -> 565[label="",style="solid", color="burlywood", weight=3];
75[label="min1 (Neg (Succ vx300)) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];75 -> 94[label="",style="solid", color="black", weight=3];
76[label="min1 (Neg Zero) (Pos (Succ vx400)) True",fontsize=16,color="black",shape="box"];76 -> 95[label="",style="solid", color="black", weight=3];
77[label="min1 (Neg Zero) (Pos Zero) True",fontsize=16,color="black",shape="box"];77 -> 96[label="",style="solid", color="black", weight=3];
78[label="min1 (Neg Zero) (Neg (Succ vx400)) (not True)",fontsize=16,color="black",shape="box"];78 -> 97[label="",style="solid", color="black", weight=3];
79[label="min1 (Neg Zero) (Neg Zero) True",fontsize=16,color="black",shape="box"];79 -> 98[label="",style="solid", color="black", weight=3];
490[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat (Succ vx430) vx44 == GT))",fontsize=16,color="burlywood",shape="box"];621[label="vx44/Succ vx440",fontsize=10,color="white",style="solid",shape="box"];490 -> 621[label="",style="solid", color="burlywood", weight=9];
621 -> 496[label="",style="solid", color="burlywood", weight=3];
622[label="vx44/Zero",fontsize=10,color="white",style="solid",shape="box"];490 -> 622[label="",style="solid", color="burlywood", weight=9];
622 -> 497[label="",style="solid", color="burlywood", weight=3];
491[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat Zero vx44 == GT))",fontsize=16,color="burlywood",shape="box"];623[label="vx44/Succ vx440",fontsize=10,color="white",style="solid",shape="box"];491 -> 623[label="",style="solid", color="burlywood", weight=9];
623 -> 498[label="",style="solid", color="burlywood", weight=3];
624[label="vx44/Zero",fontsize=10,color="white",style="solid",shape="box"];491 -> 624[label="",style="solid", color="burlywood", weight=9];
624 -> 499[label="",style="solid", color="burlywood", weight=3];
84[label="min1 (Pos (Succ vx300)) (Pos Zero) False",fontsize=16,color="black",shape="box"];84 -> 103[label="",style="solid", color="black", weight=3];
85[label="min0 (Pos (Succ vx300)) (Neg vx40) True",fontsize=16,color="black",shape="box"];85 -> 104[label="",style="solid", color="black", weight=3];
86[label="min1 (Pos Zero) (Pos (Succ vx400)) True",fontsize=16,color="black",shape="box"];86 -> 105[label="",style="solid", color="black", weight=3];
87[label="Pos Zero",fontsize=16,color="green",shape="box"];88[label="min0 (Pos Zero) (Neg (Succ vx400)) otherwise",fontsize=16,color="black",shape="box"];88 -> 106[label="",style="solid", color="black", weight=3];
89[label="Pos Zero",fontsize=16,color="green",shape="box"];564[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat (Succ vx520) vx53 == GT))",fontsize=16,color="burlywood",shape="box"];625[label="vx53/Succ vx530",fontsize=10,color="white",style="solid",shape="box"];564 -> 625[label="",style="solid", color="burlywood", weight=9];
625 -> 568[label="",style="solid", color="burlywood", weight=3];
626[label="vx53/Zero",fontsize=10,color="white",style="solid",shape="box"];564 -> 626[label="",style="solid", color="burlywood", weight=9];
626 -> 569[label="",style="solid", color="burlywood", weight=3];
565[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat Zero vx53 == GT))",fontsize=16,color="burlywood",shape="box"];627[label="vx53/Succ vx530",fontsize=10,color="white",style="solid",shape="box"];565 -> 627[label="",style="solid", color="burlywood", weight=9];
627 -> 570[label="",style="solid", color="burlywood", weight=3];
628[label="vx53/Zero",fontsize=10,color="white",style="solid",shape="box"];565 -> 628[label="",style="solid", color="burlywood", weight=9];
628 -> 571[label="",style="solid", color="burlywood", weight=3];
94[label="min1 (Neg (Succ vx300)) (Neg Zero) True",fontsize=16,color="black",shape="box"];94 -> 111[label="",style="solid", color="black", weight=3];
95[label="Neg Zero",fontsize=16,color="green",shape="box"];96[label="Neg Zero",fontsize=16,color="green",shape="box"];97[label="min1 (Neg Zero) (Neg (Succ vx400)) False",fontsize=16,color="black",shape="box"];97 -> 112[label="",style="solid", color="black", weight=3];
98[label="Neg Zero",fontsize=16,color="green",shape="box"];496[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat (Succ vx430) (Succ vx440) == GT))",fontsize=16,color="black",shape="box"];496 -> 504[label="",style="solid", color="black", weight=3];
497[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat (Succ vx430) Zero == GT))",fontsize=16,color="black",shape="box"];497 -> 505[label="",style="solid", color="black", weight=3];
498[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat Zero (Succ vx440) == GT))",fontsize=16,color="black",shape="box"];498 -> 506[label="",style="solid", color="black", weight=3];
499[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];499 -> 507[label="",style="solid", color="black", weight=3];
103[label="min0 (Pos (Succ vx300)) (Pos Zero) otherwise",fontsize=16,color="black",shape="box"];103 -> 118[label="",style="solid", color="black", weight=3];
104[label="Neg vx40",fontsize=16,color="green",shape="box"];105[label="Pos Zero",fontsize=16,color="green",shape="box"];106[label="min0 (Pos Zero) (Neg (Succ vx400)) True",fontsize=16,color="black",shape="box"];106 -> 119[label="",style="solid", color="black", weight=3];
568[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat (Succ vx520) (Succ vx530) == GT))",fontsize=16,color="black",shape="box"];568 -> 574[label="",style="solid", color="black", weight=3];
569[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat (Succ vx520) Zero == GT))",fontsize=16,color="black",shape="box"];569 -> 575[label="",style="solid", color="black", weight=3];
570[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat Zero (Succ vx530) == GT))",fontsize=16,color="black",shape="box"];570 -> 576[label="",style="solid", color="black", weight=3];
571[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];571 -> 577[label="",style="solid", color="black", weight=3];
111[label="Neg (Succ vx300)",fontsize=16,color="green",shape="box"];112[label="min0 (Neg Zero) (Neg (Succ vx400)) otherwise",fontsize=16,color="black",shape="box"];112 -> 125[label="",style="solid", color="black", weight=3];
504 -> 453[label="",style="dashed", color="red", weight=0];
504[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat vx430 vx440 == GT))",fontsize=16,color="magenta"];504 -> 512[label="",style="dashed", color="magenta", weight=3];
504 -> 513[label="",style="dashed", color="magenta", weight=3];
505[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (GT == GT))",fontsize=16,color="black",shape="box"];505 -> 514[label="",style="solid", color="black", weight=3];
506[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (LT == GT))",fontsize=16,color="black",shape="box"];506 -> 515[label="",style="solid", color="black", weight=3];
507[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];507 -> 516[label="",style="solid", color="black", weight=3];
118[label="min0 (Pos (Succ vx300)) (Pos Zero) True",fontsize=16,color="black",shape="box"];118 -> 133[label="",style="solid", color="black", weight=3];
119[label="Neg (Succ vx400)",fontsize=16,color="green",shape="box"];574 -> 523[label="",style="dashed", color="red", weight=0];
574[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat vx520 vx530 == GT))",fontsize=16,color="magenta"];574 -> 579[label="",style="dashed", color="magenta", weight=3];
574 -> 580[label="",style="dashed", color="magenta", weight=3];
575[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (GT == GT))",fontsize=16,color="black",shape="box"];575 -> 581[label="",style="solid", color="black", weight=3];
576[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (LT == GT))",fontsize=16,color="black",shape="box"];576 -> 582[label="",style="solid", color="black", weight=3];
577[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];577 -> 583[label="",style="solid", color="black", weight=3];
125[label="min0 (Neg Zero) (Neg (Succ vx400)) True",fontsize=16,color="black",shape="box"];125 -> 141[label="",style="solid", color="black", weight=3];
512[label="vx430",fontsize=16,color="green",shape="box"];513[label="vx440",fontsize=16,color="green",shape="box"];514[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not True)",fontsize=16,color="black",shape="box"];514 -> 566[label="",style="solid", color="black", weight=3];
515[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not False)",fontsize=16,color="black",shape="triangle"];515 -> 567[label="",style="solid", color="black", weight=3];
516 -> 515[label="",style="dashed", color="red", weight=0];
516[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not False)",fontsize=16,color="magenta"];133[label="Pos Zero",fontsize=16,color="green",shape="box"];579[label="vx530",fontsize=16,color="green",shape="box"];580[label="vx520",fontsize=16,color="green",shape="box"];581[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not True)",fontsize=16,color="black",shape="box"];581 -> 585[label="",style="solid", color="black", weight=3];
582[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not False)",fontsize=16,color="black",shape="triangle"];582 -> 586[label="",style="solid", color="black", weight=3];
583 -> 582[label="",style="dashed", color="red", weight=0];
583[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not False)",fontsize=16,color="magenta"];141[label="Neg (Succ vx400)",fontsize=16,color="green",shape="box"];566[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) False",fontsize=16,color="black",shape="box"];566 -> 572[label="",style="solid", color="black", weight=3];
567[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) True",fontsize=16,color="black",shape="box"];567 -> 573[label="",style="solid", color="black", weight=3];
585[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) False",fontsize=16,color="black",shape="box"];585 -> 587[label="",style="solid", color="black", weight=3];
586[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) True",fontsize=16,color="black",shape="box"];586 -> 588[label="",style="solid", color="black", weight=3];
572[label="min0 (Pos (Succ vx41)) (Pos (Succ vx42)) otherwise",fontsize=16,color="black",shape="box"];572 -> 578[label="",style="solid", color="black", weight=3];
573[label="Pos (Succ vx41)",fontsize=16,color="green",shape="box"];587[label="min0 (Neg (Succ vx50)) (Neg (Succ vx51)) otherwise",fontsize=16,color="black",shape="box"];587 -> 589[label="",style="solid", color="black", weight=3];
588[label="Neg (Succ vx50)",fontsize=16,color="green",shape="box"];578[label="min0 (Pos (Succ vx41)) (Pos (Succ vx42)) True",fontsize=16,color="black",shape="box"];578 -> 584[label="",style="solid", color="black", weight=3];
589[label="min0 (Neg (Succ vx50)) (Neg (Succ vx51)) True",fontsize=16,color="black",shape="box"];589 -> 590[label="",style="solid", color="black", weight=3];
584[label="Pos (Succ vx42)",fontsize=16,color="green",shape="box"];590[label="Neg (Succ vx51)",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

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

   new_min1(vx50, vx51, Succ(vx520), Succ(vx530)) -> new_min1(vx50, vx51, vx520, vx530)

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

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

From the DPs we obtained the following set of size-change graphs:
*new_min1(vx50, vx51, Succ(vx520), Succ(vx530)) -> new_min1(vx50, vx51, vx520, vx530)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


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

(9)
YES

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

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

   new_min10(vx41, vx42, Succ(vx430), Succ(vx440)) -> new_min10(vx41, vx42, vx430, vx440)

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

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

From the DPs we obtained the following set of size-change graphs:
*new_min10(vx41, vx42, Succ(vx430), Succ(vx440)) -> new_min10(vx41, vx42, vx430, vx440)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


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

(12)
YES