8.57/3.87	YES
10.24/4.35	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.24/4.35	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.24/4.35	
10.24/4.35	
10.24/4.35	H-Termination with start terms of the given HASKELL could be proven:
10.24/4.35	
10.24/4.35	(0) HASKELL
10.24/4.35	(1) BR [EQUIVALENT, 0 ms]
10.24/4.35	(2) HASKELL
10.24/4.35	(3) COR [EQUIVALENT, 0 ms]
10.24/4.35	(4) HASKELL
10.24/4.35	(5) Narrow [SOUND, 0 ms]
10.24/4.35	(6) AND
10.24/4.35	    (7) QDP
10.24/4.35	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.24/4.35	        (9) YES
10.24/4.35	    (10) QDP
10.24/4.35	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.24/4.35	        (12) YES
10.24/4.35	    (13) QDP
10.24/4.35	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.24/4.35	        (15) YES
10.24/4.35	
10.24/4.35	
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(0)
10.24/4.35	Obligation:
10.24/4.35	mainModule Main 
10.24/4.35	module Main where {
10.24/4.35	import qualified Prelude;
10.24/4.35	}
10.24/4.35	
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(1) BR (EQUIVALENT)
10.24/4.35	Replaced joker patterns by fresh variables and removed binding patterns.
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(2)
10.24/4.35	Obligation:
10.24/4.35	mainModule Main 
10.24/4.35	module Main where {
10.24/4.35	import qualified Prelude;
10.24/4.35	}
10.24/4.35	
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(3) COR (EQUIVALENT)
10.24/4.35	Cond Reductions:
10.24/4.35	The following Function with conditions
10.24/4.35	"min x y|x <= yx|otherwisey;
10.24/4.35	"
10.24/4.35	is transformed to
10.24/4.35	"min x y = min2 x y;
10.24/4.35	"
10.24/4.35	"min0 x y True = y;
10.24/4.35	"
10.24/4.35	"min1 x y True = x;
10.24/4.35	min1 x y False = min0 x y otherwise;
10.24/4.35	"
10.24/4.35	"min2 x y = min1 x y (x <= y);
10.24/4.35	"
10.24/4.35	The following Function with conditions
10.24/4.35	"undefined |Falseundefined;
10.24/4.35	"
10.24/4.35	is transformed to
10.24/4.35	"undefined  = undefined1;
10.24/4.35	"
10.24/4.35	"undefined0 True = undefined;
10.24/4.35	"
10.24/4.35	"undefined1  = undefined0 False;
10.24/4.35	"
10.24/4.35	
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(4)
10.24/4.35	Obligation:
10.24/4.35	mainModule Main 
10.24/4.35	module Main where {
10.24/4.35	import qualified Prelude;
10.24/4.35	}
10.24/4.35	
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(5) Narrow (SOUND)
10.24/4.35	Haskell To QDPs
10.24/4.35	
10.24/4.35	digraph dp_graph {
10.24/4.35	node [outthreshold=100, inthreshold=100];1[label="minimum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.24/4.35	3[label="minimum vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.24/4.35	4[label="foldl1 min vx3",fontsize=16,color="burlywood",shape="box"];601[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 601[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	601 -> 5[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	602[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 602[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	602 -> 6[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	5[label="foldl1 min (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
10.24/4.35	6[label="foldl1 min []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.24/4.35	7[label="foldl min vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];603[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 603[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	603 -> 9[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	604[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 604[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	604 -> 10[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl min vx30 (vx310 : vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10.24/4.35	10[label="foldl min vx30 []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
10.24/4.35	11 -> 7[label="",style="dashed", color="red", weight=0];
10.24/4.35	11[label="foldl min (min vx30 vx310) vx311",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	11 -> 14[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	12[label="vx30",fontsize=16,color="green",shape="box"];13[label="vx311",fontsize=16,color="green",shape="box"];14[label="min vx30 vx310",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
10.24/4.35	15[label="min2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
10.24/4.35	16[label="min1 vx30 vx310 (vx30 <= vx310)",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
10.24/4.35	17[label="min1 vx30 vx310 (compare vx30 vx310 /= GT)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
10.24/4.35	18[label="min1 vx30 vx310 (not (compare vx30 vx310 == GT))",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
10.24/4.35	19[label="min1 vx30 vx310 (not (primCmpInt vx30 vx310 == GT))",fontsize=16,color="burlywood",shape="box"];605[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];19 -> 605[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	605 -> 20[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	606[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];19 -> 606[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	606 -> 21[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	20[label="min1 (Pos vx300) vx310 (not (primCmpInt (Pos vx300) vx310 == GT))",fontsize=16,color="burlywood",shape="box"];607[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];20 -> 607[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	607 -> 22[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	608[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 608[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	608 -> 23[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	21[label="min1 (Neg vx300) vx310 (not (primCmpInt (Neg vx300) vx310 == GT))",fontsize=16,color="burlywood",shape="box"];609[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];21 -> 609[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	609 -> 24[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	610[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 610[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	610 -> 25[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	22[label="min1 (Pos (Succ vx3000)) vx310 (not (primCmpInt (Pos (Succ vx3000)) vx310 == GT))",fontsize=16,color="burlywood",shape="box"];611[label="vx310/Pos vx3100",fontsize=10,color="white",style="solid",shape="box"];22 -> 611[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	611 -> 26[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	612[label="vx310/Neg vx3100",fontsize=10,color="white",style="solid",shape="box"];22 -> 612[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	612 -> 27[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	23[label="min1 (Pos Zero) vx310 (not (primCmpInt (Pos Zero) vx310 == GT))",fontsize=16,color="burlywood",shape="box"];613[label="vx310/Pos vx3100",fontsize=10,color="white",style="solid",shape="box"];23 -> 613[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	613 -> 28[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	614[label="vx310/Neg vx3100",fontsize=10,color="white",style="solid",shape="box"];23 -> 614[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	614 -> 29[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	24[label="min1 (Neg (Succ vx3000)) vx310 (not (primCmpInt (Neg (Succ vx3000)) vx310 == GT))",fontsize=16,color="burlywood",shape="box"];615[label="vx310/Pos vx3100",fontsize=10,color="white",style="solid",shape="box"];24 -> 615[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	615 -> 30[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	616[label="vx310/Neg vx3100",fontsize=10,color="white",style="solid",shape="box"];24 -> 616[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	616 -> 31[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	25[label="min1 (Neg Zero) vx310 (not (primCmpInt (Neg Zero) vx310 == GT))",fontsize=16,color="burlywood",shape="box"];617[label="vx310/Pos vx3100",fontsize=10,color="white",style="solid",shape="box"];25 -> 617[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	617 -> 32[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	618[label="vx310/Neg vx3100",fontsize=10,color="white",style="solid",shape="box"];25 -> 618[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	618 -> 33[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	26[label="min1 (Pos (Succ vx3000)) (Pos vx3100) (not (primCmpInt (Pos (Succ vx3000)) (Pos vx3100) == GT))",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
10.24/4.35	27[label="min1 (Pos (Succ vx3000)) (Neg vx3100) (not (primCmpInt (Pos (Succ vx3000)) (Neg vx3100) == GT))",fontsize=16,color="black",shape="box"];27 -> 35[label="",style="solid", color="black", weight=3];
10.24/4.35	28[label="min1 (Pos Zero) (Pos vx3100) (not (primCmpInt (Pos Zero) (Pos vx3100) == GT))",fontsize=16,color="burlywood",shape="box"];619[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];28 -> 619[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	619 -> 36[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	620[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 620[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	620 -> 37[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	29[label="min1 (Pos Zero) (Neg vx3100) (not (primCmpInt (Pos Zero) (Neg vx3100) == GT))",fontsize=16,color="burlywood",shape="box"];621[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];29 -> 621[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	621 -> 38[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	622[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 622[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	622 -> 39[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	30[label="min1 (Neg (Succ vx3000)) (Pos vx3100) (not (primCmpInt (Neg (Succ vx3000)) (Pos vx3100) == GT))",fontsize=16,color="black",shape="box"];30 -> 40[label="",style="solid", color="black", weight=3];
10.24/4.35	31[label="min1 (Neg (Succ vx3000)) (Neg vx3100) (not (primCmpInt (Neg (Succ vx3000)) (Neg vx3100) == GT))",fontsize=16,color="black",shape="box"];31 -> 41[label="",style="solid", color="black", weight=3];
10.24/4.35	32[label="min1 (Neg Zero) (Pos vx3100) (not (primCmpInt (Neg Zero) (Pos vx3100) == GT))",fontsize=16,color="burlywood",shape="box"];623[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];32 -> 623[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	623 -> 42[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	624[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 624[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	624 -> 43[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	33[label="min1 (Neg Zero) (Neg vx3100) (not (primCmpInt (Neg Zero) (Neg vx3100) == GT))",fontsize=16,color="burlywood",shape="box"];625[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];33 -> 625[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	625 -> 44[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	626[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 626[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	626 -> 45[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	34[label="min1 (Pos (Succ vx3000)) (Pos vx3100) (not (primCmpNat (Succ vx3000) vx3100 == GT))",fontsize=16,color="burlywood",shape="box"];627[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];34 -> 627[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	627 -> 46[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	628[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];34 -> 628[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	628 -> 47[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	35[label="min1 (Pos (Succ vx3000)) (Neg vx3100) (not (GT == GT))",fontsize=16,color="black",shape="box"];35 -> 48[label="",style="solid", color="black", weight=3];
10.24/4.35	36[label="min1 (Pos Zero) (Pos (Succ vx31000)) (not (primCmpInt (Pos Zero) (Pos (Succ vx31000)) == GT))",fontsize=16,color="black",shape="box"];36 -> 49[label="",style="solid", color="black", weight=3];
10.24/4.35	37[label="min1 (Pos Zero) (Pos Zero) (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];37 -> 50[label="",style="solid", color="black", weight=3];
10.24/4.35	38[label="min1 (Pos Zero) (Neg (Succ vx31000)) (not (primCmpInt (Pos Zero) (Neg (Succ vx31000)) == GT))",fontsize=16,color="black",shape="box"];38 -> 51[label="",style="solid", color="black", weight=3];
10.24/4.35	39[label="min1 (Pos Zero) (Neg Zero) (not (primCmpInt (Pos Zero) (Neg Zero) == GT))",fontsize=16,color="black",shape="box"];39 -> 52[label="",style="solid", color="black", weight=3];
10.24/4.35	40[label="min1 (Neg (Succ vx3000)) (Pos vx3100) (not (LT == GT))",fontsize=16,color="black",shape="box"];40 -> 53[label="",style="solid", color="black", weight=3];
10.24/4.35	41[label="min1 (Neg (Succ vx3000)) (Neg vx3100) (not (primCmpNat vx3100 (Succ vx3000) == GT))",fontsize=16,color="burlywood",shape="box"];629[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];41 -> 629[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	629 -> 54[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	630[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 630[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	630 -> 55[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	42[label="min1 (Neg Zero) (Pos (Succ vx31000)) (not (primCmpInt (Neg Zero) (Pos (Succ vx31000)) == GT))",fontsize=16,color="black",shape="box"];42 -> 56[label="",style="solid", color="black", weight=3];
10.24/4.35	43[label="min1 (Neg Zero) (Pos Zero) (not (primCmpInt (Neg Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];43 -> 57[label="",style="solid", color="black", weight=3];
10.24/4.35	44[label="min1 (Neg Zero) (Neg (Succ vx31000)) (not (primCmpInt (Neg Zero) (Neg (Succ vx31000)) == GT))",fontsize=16,color="black",shape="box"];44 -> 58[label="",style="solid", color="black", weight=3];
10.24/4.35	45[label="min1 (Neg Zero) (Neg Zero) (not (primCmpInt (Neg Zero) (Neg Zero) == GT))",fontsize=16,color="black",shape="box"];45 -> 59[label="",style="solid", color="black", weight=3];
10.24/4.35	46[label="min1 (Pos (Succ vx3000)) (Pos (Succ vx31000)) (not (primCmpNat (Succ vx3000) (Succ vx31000) == GT))",fontsize=16,color="black",shape="box"];46 -> 60[label="",style="solid", color="black", weight=3];
10.24/4.35	47[label="min1 (Pos (Succ vx3000)) (Pos Zero) (not (primCmpNat (Succ vx3000) Zero == GT))",fontsize=16,color="black",shape="box"];47 -> 61[label="",style="solid", color="black", weight=3];
10.24/4.35	48[label="min1 (Pos (Succ vx3000)) (Neg vx3100) (not True)",fontsize=16,color="black",shape="box"];48 -> 62[label="",style="solid", color="black", weight=3];
10.24/4.35	49[label="min1 (Pos Zero) (Pos (Succ vx31000)) (not (primCmpNat Zero (Succ vx31000) == GT))",fontsize=16,color="black",shape="box"];49 -> 63[label="",style="solid", color="black", weight=3];
10.24/4.35	50[label="min1 (Pos Zero) (Pos Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];50 -> 64[label="",style="solid", color="black", weight=3];
10.24/4.35	51[label="min1 (Pos Zero) (Neg (Succ vx31000)) (not (GT == GT))",fontsize=16,color="black",shape="box"];51 -> 65[label="",style="solid", color="black", weight=3];
10.24/4.35	52[label="min1 (Pos Zero) (Neg Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];52 -> 66[label="",style="solid", color="black", weight=3];
10.24/4.35	53[label="min1 (Neg (Succ vx3000)) (Pos vx3100) (not False)",fontsize=16,color="black",shape="box"];53 -> 67[label="",style="solid", color="black", weight=3];
10.24/4.35	54[label="min1 (Neg (Succ vx3000)) (Neg (Succ vx31000)) (not (primCmpNat (Succ vx31000) (Succ vx3000) == GT))",fontsize=16,color="black",shape="box"];54 -> 68[label="",style="solid", color="black", weight=3];
10.24/4.35	55[label="min1 (Neg (Succ vx3000)) (Neg Zero) (not (primCmpNat Zero (Succ vx3000) == GT))",fontsize=16,color="black",shape="box"];55 -> 69[label="",style="solid", color="black", weight=3];
10.24/4.35	56[label="min1 (Neg Zero) (Pos (Succ vx31000)) (not (LT == GT))",fontsize=16,color="black",shape="box"];56 -> 70[label="",style="solid", color="black", weight=3];
10.24/4.35	57[label="min1 (Neg Zero) (Pos Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];57 -> 71[label="",style="solid", color="black", weight=3];
10.24/4.35	58[label="min1 (Neg Zero) (Neg (Succ vx31000)) (not (primCmpNat (Succ vx31000) Zero == GT))",fontsize=16,color="black",shape="box"];58 -> 72[label="",style="solid", color="black", weight=3];
10.24/4.35	59[label="min1 (Neg Zero) (Neg Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];59 -> 73[label="",style="solid", color="black", weight=3];
10.24/4.35	60 -> 463[label="",style="dashed", color="red", weight=0];
10.24/4.35	60[label="min1 (Pos (Succ vx3000)) (Pos (Succ vx31000)) (not (primCmpNat vx3000 vx31000 == GT))",fontsize=16,color="magenta"];60 -> 464[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	60 -> 465[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	60 -> 466[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	60 -> 467[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	61[label="min1 (Pos (Succ vx3000)) (Pos Zero) (not (GT == GT))",fontsize=16,color="black",shape="box"];61 -> 76[label="",style="solid", color="black", weight=3];
10.24/4.35	62[label="min1 (Pos (Succ vx3000)) (Neg vx3100) False",fontsize=16,color="black",shape="box"];62 -> 77[label="",style="solid", color="black", weight=3];
10.24/4.35	63[label="min1 (Pos Zero) (Pos (Succ vx31000)) (not (LT == GT))",fontsize=16,color="black",shape="box"];63 -> 78[label="",style="solid", color="black", weight=3];
10.24/4.35	64[label="min1 (Pos Zero) (Pos Zero) (not False)",fontsize=16,color="black",shape="box"];64 -> 79[label="",style="solid", color="black", weight=3];
10.24/4.35	65[label="min1 (Pos Zero) (Neg (Succ vx31000)) (not True)",fontsize=16,color="black",shape="box"];65 -> 80[label="",style="solid", color="black", weight=3];
10.24/4.35	66[label="min1 (Pos Zero) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];66 -> 81[label="",style="solid", color="black", weight=3];
10.24/4.35	67[label="min1 (Neg (Succ vx3000)) (Pos vx3100) True",fontsize=16,color="black",shape="box"];67 -> 82[label="",style="solid", color="black", weight=3];
10.24/4.35	68 -> 533[label="",style="dashed", color="red", weight=0];
10.24/4.35	68[label="min1 (Neg (Succ vx3000)) (Neg (Succ vx31000)) (not (primCmpNat vx31000 vx3000 == GT))",fontsize=16,color="magenta"];68 -> 534[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	68 -> 535[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	68 -> 536[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	68 -> 537[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	69[label="min1 (Neg (Succ vx3000)) (Neg Zero) (not (LT == GT))",fontsize=16,color="black",shape="box"];69 -> 85[label="",style="solid", color="black", weight=3];
10.24/4.35	70[label="min1 (Neg Zero) (Pos (Succ vx31000)) (not False)",fontsize=16,color="black",shape="box"];70 -> 86[label="",style="solid", color="black", weight=3];
10.24/4.35	71[label="min1 (Neg Zero) (Pos Zero) (not False)",fontsize=16,color="black",shape="box"];71 -> 87[label="",style="solid", color="black", weight=3];
10.24/4.35	72[label="min1 (Neg Zero) (Neg (Succ vx31000)) (not (GT == GT))",fontsize=16,color="black",shape="box"];72 -> 88[label="",style="solid", color="black", weight=3];
10.24/4.35	73[label="min1 (Neg Zero) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];73 -> 89[label="",style="solid", color="black", weight=3];
10.24/4.35	464[label="vx3000",fontsize=16,color="green",shape="box"];465[label="vx3000",fontsize=16,color="green",shape="box"];466[label="vx31000",fontsize=16,color="green",shape="box"];467[label="vx31000",fontsize=16,color="green",shape="box"];463[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat vx42 vx43 == GT))",fontsize=16,color="burlywood",shape="triangle"];631[label="vx42/Succ vx420",fontsize=10,color="white",style="solid",shape="box"];463 -> 631[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	631 -> 500[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	632[label="vx42/Zero",fontsize=10,color="white",style="solid",shape="box"];463 -> 632[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	632 -> 501[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	76[label="min1 (Pos (Succ vx3000)) (Pos Zero) (not True)",fontsize=16,color="black",shape="box"];76 -> 94[label="",style="solid", color="black", weight=3];
10.24/4.35	77[label="min0 (Pos (Succ vx3000)) (Neg vx3100) otherwise",fontsize=16,color="black",shape="box"];77 -> 95[label="",style="solid", color="black", weight=3];
10.24/4.35	78[label="min1 (Pos Zero) (Pos (Succ vx31000)) (not False)",fontsize=16,color="black",shape="box"];78 -> 96[label="",style="solid", color="black", weight=3];
10.24/4.35	79[label="min1 (Pos Zero) (Pos Zero) True",fontsize=16,color="black",shape="box"];79 -> 97[label="",style="solid", color="black", weight=3];
10.24/4.35	80[label="min1 (Pos Zero) (Neg (Succ vx31000)) False",fontsize=16,color="black",shape="box"];80 -> 98[label="",style="solid", color="black", weight=3];
10.24/4.35	81[label="min1 (Pos Zero) (Neg Zero) True",fontsize=16,color="black",shape="box"];81 -> 99[label="",style="solid", color="black", weight=3];
10.24/4.35	82[label="Neg (Succ vx3000)",fontsize=16,color="green",shape="box"];534[label="vx3000",fontsize=16,color="green",shape="box"];535[label="vx3000",fontsize=16,color="green",shape="box"];536[label="vx31000",fontsize=16,color="green",shape="box"];537[label="vx31000",fontsize=16,color="green",shape="box"];533[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat vx51 vx52 == GT))",fontsize=16,color="burlywood",shape="triangle"];633[label="vx51/Succ vx510",fontsize=10,color="white",style="solid",shape="box"];533 -> 633[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	633 -> 574[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	634[label="vx51/Zero",fontsize=10,color="white",style="solid",shape="box"];533 -> 634[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	634 -> 575[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	85[label="min1 (Neg (Succ vx3000)) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];85 -> 104[label="",style="solid", color="black", weight=3];
10.24/4.35	86[label="min1 (Neg Zero) (Pos (Succ vx31000)) True",fontsize=16,color="black",shape="box"];86 -> 105[label="",style="solid", color="black", weight=3];
10.24/4.35	87[label="min1 (Neg Zero) (Pos Zero) True",fontsize=16,color="black",shape="box"];87 -> 106[label="",style="solid", color="black", weight=3];
10.24/4.35	88[label="min1 (Neg Zero) (Neg (Succ vx31000)) (not True)",fontsize=16,color="black",shape="box"];88 -> 107[label="",style="solid", color="black", weight=3];
10.24/4.35	89[label="min1 (Neg Zero) (Neg Zero) True",fontsize=16,color="black",shape="box"];89 -> 108[label="",style="solid", color="black", weight=3];
10.24/4.35	500[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat (Succ vx420) vx43 == GT))",fontsize=16,color="burlywood",shape="box"];635[label="vx43/Succ vx430",fontsize=10,color="white",style="solid",shape="box"];500 -> 635[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	635 -> 506[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	636[label="vx43/Zero",fontsize=10,color="white",style="solid",shape="box"];500 -> 636[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	636 -> 507[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	501[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat Zero vx43 == GT))",fontsize=16,color="burlywood",shape="box"];637[label="vx43/Succ vx430",fontsize=10,color="white",style="solid",shape="box"];501 -> 637[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	637 -> 508[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	638[label="vx43/Zero",fontsize=10,color="white",style="solid",shape="box"];501 -> 638[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	638 -> 509[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	94[label="min1 (Pos (Succ vx3000)) (Pos Zero) False",fontsize=16,color="black",shape="box"];94 -> 113[label="",style="solid", color="black", weight=3];
10.24/4.35	95[label="min0 (Pos (Succ vx3000)) (Neg vx3100) True",fontsize=16,color="black",shape="box"];95 -> 114[label="",style="solid", color="black", weight=3];
10.24/4.35	96[label="min1 (Pos Zero) (Pos (Succ vx31000)) True",fontsize=16,color="black",shape="box"];96 -> 115[label="",style="solid", color="black", weight=3];
10.24/4.35	97[label="Pos Zero",fontsize=16,color="green",shape="box"];98[label="min0 (Pos Zero) (Neg (Succ vx31000)) otherwise",fontsize=16,color="black",shape="box"];98 -> 116[label="",style="solid", color="black", weight=3];
10.24/4.35	99[label="Pos Zero",fontsize=16,color="green",shape="box"];574[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat (Succ vx510) vx52 == GT))",fontsize=16,color="burlywood",shape="box"];639[label="vx52/Succ vx520",fontsize=10,color="white",style="solid",shape="box"];574 -> 639[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	639 -> 578[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	640[label="vx52/Zero",fontsize=10,color="white",style="solid",shape="box"];574 -> 640[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	640 -> 579[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	575[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat Zero vx52 == GT))",fontsize=16,color="burlywood",shape="box"];641[label="vx52/Succ vx520",fontsize=10,color="white",style="solid",shape="box"];575 -> 641[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	641 -> 580[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	642[label="vx52/Zero",fontsize=10,color="white",style="solid",shape="box"];575 -> 642[label="",style="solid", color="burlywood", weight=9];
10.24/4.35	642 -> 581[label="",style="solid", color="burlywood", weight=3];
10.24/4.35	104[label="min1 (Neg (Succ vx3000)) (Neg Zero) True",fontsize=16,color="black",shape="box"];104 -> 121[label="",style="solid", color="black", weight=3];
10.24/4.35	105[label="Neg Zero",fontsize=16,color="green",shape="box"];106[label="Neg Zero",fontsize=16,color="green",shape="box"];107[label="min1 (Neg Zero) (Neg (Succ vx31000)) False",fontsize=16,color="black",shape="box"];107 -> 122[label="",style="solid", color="black", weight=3];
10.24/4.35	108[label="Neg Zero",fontsize=16,color="green",shape="box"];506[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat (Succ vx420) (Succ vx430) == GT))",fontsize=16,color="black",shape="box"];506 -> 514[label="",style="solid", color="black", weight=3];
10.24/4.35	507[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat (Succ vx420) Zero == GT))",fontsize=16,color="black",shape="box"];507 -> 515[label="",style="solid", color="black", weight=3];
10.24/4.35	508[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat Zero (Succ vx430) == GT))",fontsize=16,color="black",shape="box"];508 -> 516[label="",style="solid", color="black", weight=3];
10.24/4.35	509[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];509 -> 517[label="",style="solid", color="black", weight=3];
10.24/4.35	113[label="min0 (Pos (Succ vx3000)) (Pos Zero) otherwise",fontsize=16,color="black",shape="box"];113 -> 128[label="",style="solid", color="black", weight=3];
10.24/4.35	114[label="Neg vx3100",fontsize=16,color="green",shape="box"];115[label="Pos Zero",fontsize=16,color="green",shape="box"];116[label="min0 (Pos Zero) (Neg (Succ vx31000)) True",fontsize=16,color="black",shape="box"];116 -> 129[label="",style="solid", color="black", weight=3];
10.24/4.35	578[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat (Succ vx510) (Succ vx520) == GT))",fontsize=16,color="black",shape="box"];578 -> 584[label="",style="solid", color="black", weight=3];
10.24/4.35	579[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat (Succ vx510) Zero == GT))",fontsize=16,color="black",shape="box"];579 -> 585[label="",style="solid", color="black", weight=3];
10.24/4.35	580[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat Zero (Succ vx520) == GT))",fontsize=16,color="black",shape="box"];580 -> 586[label="",style="solid", color="black", weight=3];
10.24/4.35	581[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];581 -> 587[label="",style="solid", color="black", weight=3];
10.24/4.35	121[label="Neg (Succ vx3000)",fontsize=16,color="green",shape="box"];122[label="min0 (Neg Zero) (Neg (Succ vx31000)) otherwise",fontsize=16,color="black",shape="box"];122 -> 135[label="",style="solid", color="black", weight=3];
10.24/4.35	514 -> 463[label="",style="dashed", color="red", weight=0];
10.24/4.35	514[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat vx420 vx430 == GT))",fontsize=16,color="magenta"];514 -> 522[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	514 -> 523[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	515[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (GT == GT))",fontsize=16,color="black",shape="box"];515 -> 524[label="",style="solid", color="black", weight=3];
10.24/4.35	516[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (LT == GT))",fontsize=16,color="black",shape="box"];516 -> 525[label="",style="solid", color="black", weight=3];
10.24/4.35	517[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];517 -> 526[label="",style="solid", color="black", weight=3];
10.24/4.35	128[label="min0 (Pos (Succ vx3000)) (Pos Zero) True",fontsize=16,color="black",shape="box"];128 -> 143[label="",style="solid", color="black", weight=3];
10.24/4.35	129[label="Neg (Succ vx31000)",fontsize=16,color="green",shape="box"];584 -> 533[label="",style="dashed", color="red", weight=0];
10.24/4.35	584[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat vx510 vx520 == GT))",fontsize=16,color="magenta"];584 -> 589[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	584 -> 590[label="",style="dashed", color="magenta", weight=3];
10.24/4.35	585[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (GT == GT))",fontsize=16,color="black",shape="box"];585 -> 591[label="",style="solid", color="black", weight=3];
10.24/4.35	586[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (LT == GT))",fontsize=16,color="black",shape="box"];586 -> 592[label="",style="solid", color="black", weight=3];
10.24/4.35	587[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];587 -> 593[label="",style="solid", color="black", weight=3];
10.24/4.35	135[label="min0 (Neg Zero) (Neg (Succ vx31000)) True",fontsize=16,color="black",shape="box"];135 -> 151[label="",style="solid", color="black", weight=3];
10.24/4.35	522[label="vx420",fontsize=16,color="green",shape="box"];523[label="vx430",fontsize=16,color="green",shape="box"];524[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not True)",fontsize=16,color="black",shape="box"];524 -> 576[label="",style="solid", color="black", weight=3];
10.24/4.35	525[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not False)",fontsize=16,color="black",shape="triangle"];525 -> 577[label="",style="solid", color="black", weight=3];
10.24/4.35	526 -> 525[label="",style="dashed", color="red", weight=0];
10.24/4.35	526[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not False)",fontsize=16,color="magenta"];143[label="Pos Zero",fontsize=16,color="green",shape="box"];589[label="vx520",fontsize=16,color="green",shape="box"];590[label="vx510",fontsize=16,color="green",shape="box"];591[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not True)",fontsize=16,color="black",shape="box"];591 -> 595[label="",style="solid", color="black", weight=3];
10.24/4.35	592[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not False)",fontsize=16,color="black",shape="triangle"];592 -> 596[label="",style="solid", color="black", weight=3];
10.24/4.35	593 -> 592[label="",style="dashed", color="red", weight=0];
10.24/4.35	593[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not False)",fontsize=16,color="magenta"];151[label="Neg (Succ vx31000)",fontsize=16,color="green",shape="box"];576[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) False",fontsize=16,color="black",shape="box"];576 -> 582[label="",style="solid", color="black", weight=3];
10.24/4.35	577[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) True",fontsize=16,color="black",shape="box"];577 -> 583[label="",style="solid", color="black", weight=3];
10.24/4.35	595[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) False",fontsize=16,color="black",shape="box"];595 -> 597[label="",style="solid", color="black", weight=3];
10.24/4.35	596[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) True",fontsize=16,color="black",shape="box"];596 -> 598[label="",style="solid", color="black", weight=3];
10.24/4.35	582[label="min0 (Pos (Succ vx40)) (Pos (Succ vx41)) otherwise",fontsize=16,color="black",shape="box"];582 -> 588[label="",style="solid", color="black", weight=3];
10.24/4.35	583[label="Pos (Succ vx40)",fontsize=16,color="green",shape="box"];597[label="min0 (Neg (Succ vx49)) (Neg (Succ vx50)) otherwise",fontsize=16,color="black",shape="box"];597 -> 599[label="",style="solid", color="black", weight=3];
10.24/4.35	598[label="Neg (Succ vx49)",fontsize=16,color="green",shape="box"];588[label="min0 (Pos (Succ vx40)) (Pos (Succ vx41)) True",fontsize=16,color="black",shape="box"];588 -> 594[label="",style="solid", color="black", weight=3];
10.24/4.35	599[label="min0 (Neg (Succ vx49)) (Neg (Succ vx50)) True",fontsize=16,color="black",shape="box"];599 -> 600[label="",style="solid", color="black", weight=3];
10.24/4.35	594[label="Pos (Succ vx41)",fontsize=16,color="green",shape="box"];600[label="Neg (Succ vx50)",fontsize=16,color="green",shape="box"];}
10.24/4.35	
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(6)
10.24/4.35	Complex Obligation (AND)
10.24/4.35	
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(7)
10.24/4.35	Obligation:
10.24/4.35	Q DP problem:
10.24/4.35	The TRS P consists of the following rules:
10.24/4.35	
10.24/4.35	   new_foldl(vx30, :(vx310, vx311)) -> new_foldl(new_min11(vx30, vx310), vx311)
10.24/4.35	
10.24/4.35	The TRS R consists of the following rules:
10.24/4.35	
10.24/4.35	   new_min11(Neg(Zero), Neg(Zero)) -> Neg(Zero)
10.24/4.35	   new_min11(Neg(Zero), Pos(Zero)) -> Neg(Zero)
10.24/4.35	   new_min11(Neg(Succ(vx3000)), Neg(Zero)) -> Neg(Succ(vx3000))
10.24/4.35	   new_min11(Neg(Zero), Neg(Succ(vx31000))) -> Neg(Succ(vx31000))
10.24/4.35	   new_min11(Pos(Zero), Neg(Zero)) -> Pos(Zero)
10.24/4.35	   new_min13(vx40, vx41, Succ(vx420), Zero) -> Pos(Succ(vx41))
10.24/4.35	   new_min11(Pos(Zero), Pos(Zero)) -> Pos(Zero)
10.24/4.35	   new_min11(Neg(Zero), Pos(Succ(vx31000))) -> Neg(Zero)
10.24/4.35	   new_min12(vx49, vx50) -> Neg(Succ(vx49))
10.24/4.35	   new_min11(Pos(Zero), Neg(Succ(vx31000))) -> Neg(Succ(vx31000))
10.24/4.35	   new_min15(vx49, vx50, Zero, Zero) -> new_min12(vx49, vx50)
10.24/4.35	   new_min11(Pos(Succ(vx3000)), Pos(Succ(vx31000))) -> new_min13(vx3000, vx31000, vx3000, vx31000)
10.24/4.35	   new_min15(vx49, vx50, Succ(vx510), Succ(vx520)) -> new_min15(vx49, vx50, vx510, vx520)
10.24/4.35	   new_min13(vx40, vx41, Zero, Succ(vx430)) -> new_min14(vx40, vx41)
10.24/4.35	   new_min11(Pos(Succ(vx3000)), Pos(Zero)) -> Pos(Zero)
10.24/4.35	   new_min11(Pos(Zero), Pos(Succ(vx31000))) -> Pos(Zero)
10.24/4.35	   new_min11(Neg(Succ(vx3000)), Pos(vx3100)) -> Neg(Succ(vx3000))
10.24/4.35	   new_min14(vx40, vx41) -> Pos(Succ(vx40))
10.24/4.35	   new_min11(Neg(Succ(vx3000)), Neg(Succ(vx31000))) -> new_min15(vx3000, vx31000, vx31000, vx3000)
10.24/4.35	   new_min13(vx40, vx41, Zero, Zero) -> new_min14(vx40, vx41)
10.24/4.35	   new_min15(vx49, vx50, Succ(vx510), Zero) -> Neg(Succ(vx50))
10.24/4.35	   new_min15(vx49, vx50, Zero, Succ(vx520)) -> new_min12(vx49, vx50)
10.24/4.35	   new_min11(Pos(Succ(vx3000)), Neg(vx3100)) -> Neg(vx3100)
10.24/4.35	   new_min13(vx40, vx41, Succ(vx420), Succ(vx430)) -> new_min13(vx40, vx41, vx420, vx430)
10.24/4.35	
10.24/4.35	The set Q consists of the following terms:
10.24/4.35	
10.24/4.35	   new_min15(x0, x1, Succ(x2), Succ(x3))
10.24/4.35	   new_min11(Pos(Succ(x0)), Pos(Zero))
10.24/4.35	   new_min12(x0, x1)
10.24/4.35	   new_min11(Neg(Zero), Neg(Succ(x0)))
10.24/4.35	   new_min11(Neg(Zero), Pos(Zero))
10.24/4.35	   new_min11(Pos(Zero), Neg(Zero))
10.24/4.35	   new_min14(x0, x1)
10.24/4.35	   new_min11(Neg(Succ(x0)), Neg(Zero))
10.24/4.35	   new_min11(Neg(Zero), Neg(Zero))
10.24/4.35	   new_min13(x0, x1, Zero, Succ(x2))
10.24/4.35	   new_min11(Neg(Succ(x0)), Neg(Succ(x1)))
10.24/4.35	   new_min11(Neg(Zero), Pos(Succ(x0)))
10.24/4.35	   new_min11(Pos(Zero), Neg(Succ(x0)))
10.24/4.35	   new_min13(x0, x1, Succ(x2), Succ(x3))
10.24/4.35	   new_min15(x0, x1, Zero, Zero)
10.24/4.35	   new_min11(Neg(Succ(x0)), Pos(x1))
10.24/4.35	   new_min11(Pos(Zero), Pos(Zero))
10.24/4.35	   new_min15(x0, x1, Succ(x2), Zero)
10.24/4.35	   new_min11(Pos(Succ(x0)), Neg(x1))
10.24/4.35	   new_min13(x0, x1, Zero, Zero)
10.24/4.35	   new_min13(x0, x1, Succ(x2), Zero)
10.24/4.35	   new_min11(Pos(Succ(x0)), Pos(Succ(x1)))
10.24/4.35	   new_min15(x0, x1, Zero, Succ(x2))
10.24/4.35	   new_min11(Pos(Zero), Pos(Succ(x0)))
10.24/4.35	
10.24/4.35	We have to consider all minimal (P,Q,R)-chains.
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(8) QDPSizeChangeProof (EQUIVALENT)
10.24/4.35	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. 
10.24/4.35	
10.24/4.35	From the DPs we obtained the following set of size-change graphs:
10.24/4.35	*new_foldl(vx30, :(vx310, vx311)) -> new_foldl(new_min11(vx30, vx310), vx311)
10.24/4.35	The graph contains the following edges 2 > 2
10.24/4.35	
10.24/4.35	
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(9)
10.24/4.35	YES
10.24/4.35	
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(10)
10.24/4.35	Obligation:
10.24/4.35	Q DP problem:
10.24/4.35	The TRS P consists of the following rules:
10.24/4.35	
10.24/4.35	   new_min1(vx49, vx50, Succ(vx510), Succ(vx520)) -> new_min1(vx49, vx50, vx510, vx520)
10.24/4.35	
10.24/4.35	R is empty.
10.24/4.35	Q is empty.
10.24/4.35	We have to consider all minimal (P,Q,R)-chains.
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(11) QDPSizeChangeProof (EQUIVALENT)
10.24/4.35	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. 
10.24/4.35	
10.24/4.35	From the DPs we obtained the following set of size-change graphs:
10.24/4.35	*new_min1(vx49, vx50, Succ(vx510), Succ(vx520)) -> new_min1(vx49, vx50, vx510, vx520)
10.24/4.35	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
10.24/4.35	
10.24/4.35	
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(12)
10.24/4.35	YES
10.24/4.35	
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(13)
10.24/4.35	Obligation:
10.24/4.35	Q DP problem:
10.24/4.35	The TRS P consists of the following rules:
10.24/4.35	
10.24/4.35	   new_min10(vx40, vx41, Succ(vx420), Succ(vx430)) -> new_min10(vx40, vx41, vx420, vx430)
10.24/4.35	
10.24/4.35	R is empty.
10.24/4.35	Q is empty.
10.24/4.35	We have to consider all minimal (P,Q,R)-chains.
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(14) QDPSizeChangeProof (EQUIVALENT)
10.24/4.35	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. 
10.24/4.35	
10.24/4.35	From the DPs we obtained the following set of size-change graphs:
10.24/4.35	*new_min10(vx40, vx41, Succ(vx420), Succ(vx430)) -> new_min10(vx40, vx41, vx420, vx430)
10.24/4.35	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
10.24/4.35	
10.24/4.35	
10.24/4.35	----------------------------------------
10.24/4.35	
10.24/4.35	(15)
10.24/4.35	YES
10.56/4.42	EOF