8.98/3.93	YES
10.78/4.41	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.78/4.41	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.78/4.41	
10.78/4.41	
10.78/4.41	H-Termination with start terms of the given HASKELL could be proven:
10.78/4.41	
10.78/4.41	(0) HASKELL
10.78/4.41	(1) BR [EQUIVALENT, 0 ms]
10.78/4.41	(2) HASKELL
10.78/4.41	(3) COR [EQUIVALENT, 0 ms]
10.78/4.41	(4) HASKELL
10.78/4.41	(5) Narrow [SOUND, 0 ms]
10.78/4.41	(6) AND
10.78/4.41	    (7) QDP
10.78/4.41	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.78/4.41	        (9) YES
10.78/4.41	    (10) QDP
10.78/4.41	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.78/4.41	        (12) YES
10.78/4.41	
10.78/4.41	
10.78/4.41	----------------------------------------
10.78/4.41	
10.78/4.41	(0)
10.78/4.41	Obligation:
10.78/4.41	mainModule Main 
10.78/4.41	module Main where {
10.78/4.41	import qualified Prelude;
10.78/4.41	}
10.78/4.41	
10.78/4.41	----------------------------------------
10.78/4.41	
10.78/4.41	(1) BR (EQUIVALENT)
10.78/4.41	Replaced joker patterns by fresh variables and removed binding patterns.
10.78/4.41	----------------------------------------
10.78/4.41	
10.78/4.41	(2)
10.78/4.41	Obligation:
10.78/4.41	mainModule Main 
10.78/4.41	module Main where {
10.78/4.41	import qualified Prelude;
10.78/4.41	}
10.78/4.41	
10.78/4.41	----------------------------------------
10.78/4.41	
10.78/4.41	(3) COR (EQUIVALENT)
10.78/4.41	Cond Reductions:
10.78/4.41	The following Function with conditions
10.78/4.41	"max x y|x <= yy|otherwisex;
10.78/4.41	"
10.78/4.41	is transformed to
10.78/4.41	"max x y = max2 x y;
10.78/4.41	"
10.78/4.41	"max0 x y True = x;
10.78/4.41	"
10.78/4.41	"max1 x y True = y;
10.78/4.41	max1 x y False = max0 x y otherwise;
10.78/4.41	"
10.78/4.41	"max2 x y = max1 x y (x <= y);
10.78/4.41	"
10.78/4.41	The following Function with conditions
10.78/4.41	"undefined |Falseundefined;
10.78/4.41	"
10.78/4.41	is transformed to
10.78/4.41	"undefined  = undefined1;
10.78/4.41	"
10.78/4.41	"undefined0 True = undefined;
10.78/4.41	"
10.78/4.41	"undefined1  = undefined0 False;
10.78/4.41	"
10.78/4.41	
10.78/4.41	----------------------------------------
10.78/4.41	
10.78/4.41	(4)
10.78/4.41	Obligation:
10.78/4.41	mainModule Main 
10.78/4.41	module Main where {
10.78/4.41	import qualified Prelude;
10.78/4.41	}
10.78/4.41	
10.78/4.41	----------------------------------------
10.78/4.41	
10.78/4.41	(5) Narrow (SOUND)
10.78/4.41	Haskell To QDPs
10.78/4.41	
10.78/4.41	digraph dp_graph {
10.78/4.41	node [outthreshold=100, inthreshold=100];1[label="max",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.78/4.41	3[label="max vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.78/4.41	4[label="max vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.78/4.41	5[label="max2 vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.78/4.41	6[label="max1 vx3 vx4 (vx3 <= vx4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.78/4.41	7[label="max1 vx3 vx4 (compare vx3 vx4 /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
10.78/4.41	8[label="max1 vx3 vx4 (not (compare vx3 vx4 == GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
10.78/4.41	9[label="max1 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];
10.78/4.41	591 -> 10[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	592[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];9 -> 592[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	592 -> 11[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	10[label="max1 (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];
10.78/4.41	593 -> 12[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	594[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 594[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	594 -> 13[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	11[label="max1 (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];
10.78/4.41	595 -> 14[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	596[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 596[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	596 -> 15[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	12[label="max1 (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];
10.78/4.41	597 -> 16[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	598[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 598[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	598 -> 17[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	13[label="max1 (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];
10.78/4.41	599 -> 18[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	600[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 600[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	600 -> 19[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	14[label="max1 (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];
10.78/4.41	601 -> 20[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	602[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];14 -> 602[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	602 -> 21[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	15[label="max1 (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];
10.78/4.41	603 -> 22[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	604[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];15 -> 604[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	604 -> 23[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	16[label="max1 (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];
10.78/4.41	17[label="max1 (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];
10.78/4.41	18[label="max1 (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];
10.78/4.41	605 -> 26[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	606[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 606[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	606 -> 27[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	19[label="max1 (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];
10.78/4.41	607 -> 28[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	608[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 608[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	608 -> 29[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	20[label="max1 (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];
10.78/4.41	21[label="max1 (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];
10.78/4.41	22[label="max1 (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];
10.78/4.41	609 -> 32[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	610[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 610[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	610 -> 33[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	23[label="max1 (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];
10.78/4.41	611 -> 34[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	612[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 612[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	612 -> 35[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	24[label="max1 (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];
10.78/4.41	613 -> 36[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	614[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 614[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	614 -> 37[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	25[label="max1 (Pos (Succ vx300)) (Neg vx40) (not (GT == GT))",fontsize=16,color="black",shape="box"];25 -> 38[label="",style="solid", color="black", weight=3];
10.78/4.41	26[label="max1 (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];
10.78/4.41	27[label="max1 (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];
10.78/4.41	28[label="max1 (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];
10.78/4.41	29[label="max1 (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];
10.78/4.41	30[label="max1 (Neg (Succ vx300)) (Pos vx40) (not (LT == GT))",fontsize=16,color="black",shape="box"];30 -> 43[label="",style="solid", color="black", weight=3];
10.78/4.41	31[label="max1 (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];
10.78/4.41	615 -> 44[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	616[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];31 -> 616[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	616 -> 45[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	32[label="max1 (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];
10.78/4.41	33[label="max1 (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];
10.78/4.41	34[label="max1 (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];
10.78/4.41	35[label="max1 (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];
10.78/4.41	36[label="max1 (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];
10.78/4.41	37[label="max1 (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];
10.78/4.41	38[label="max1 (Pos (Succ vx300)) (Neg vx40) (not True)",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
10.78/4.41	39[label="max1 (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];
10.78/4.41	40[label="max1 (Pos Zero) (Pos Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3];
10.78/4.41	41[label="max1 (Pos Zero) (Neg (Succ vx400)) (not (GT == GT))",fontsize=16,color="black",shape="box"];41 -> 55[label="",style="solid", color="black", weight=3];
10.78/4.41	42[label="max1 (Pos Zero) (Neg Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];42 -> 56[label="",style="solid", color="black", weight=3];
10.78/4.41	43[label="max1 (Neg (Succ vx300)) (Pos vx40) (not False)",fontsize=16,color="black",shape="box"];43 -> 57[label="",style="solid", color="black", weight=3];
10.78/4.41	44[label="max1 (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];
10.78/4.41	45[label="max1 (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];
10.78/4.41	46[label="max1 (Neg Zero) (Pos (Succ vx400)) (not (LT == GT))",fontsize=16,color="black",shape="box"];46 -> 60[label="",style="solid", color="black", weight=3];
10.78/4.41	47[label="max1 (Neg Zero) (Pos Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];47 -> 61[label="",style="solid", color="black", weight=3];
10.78/4.41	48[label="max1 (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];
10.78/4.41	49[label="max1 (Neg Zero) (Neg Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];49 -> 63[label="",style="solid", color="black", weight=3];
10.78/4.41	50 -> 453[label="",style="dashed", color="red", weight=0];
10.78/4.41	50[label="max1 (Pos (Succ vx300)) (Pos (Succ vx400)) (not (primCmpNat vx300 vx400 == GT))",fontsize=16,color="magenta"];50 -> 454[label="",style="dashed", color="magenta", weight=3];
10.78/4.41	50 -> 455[label="",style="dashed", color="magenta", weight=3];
10.78/4.41	50 -> 456[label="",style="dashed", color="magenta", weight=3];
10.78/4.41	50 -> 457[label="",style="dashed", color="magenta", weight=3];
10.78/4.41	51[label="max1 (Pos (Succ vx300)) (Pos Zero) (not (GT == GT))",fontsize=16,color="black",shape="box"];51 -> 66[label="",style="solid", color="black", weight=3];
10.78/4.41	52[label="max1 (Pos (Succ vx300)) (Neg vx40) False",fontsize=16,color="black",shape="box"];52 -> 67[label="",style="solid", color="black", weight=3];
10.78/4.41	53[label="max1 (Pos Zero) (Pos (Succ vx400)) (not (LT == GT))",fontsize=16,color="black",shape="box"];53 -> 68[label="",style="solid", color="black", weight=3];
10.78/4.41	54[label="max1 (Pos Zero) (Pos Zero) (not False)",fontsize=16,color="black",shape="box"];54 -> 69[label="",style="solid", color="black", weight=3];
10.78/4.41	55[label="max1 (Pos Zero) (Neg (Succ vx400)) (not True)",fontsize=16,color="black",shape="box"];55 -> 70[label="",style="solid", color="black", weight=3];
10.78/4.41	56[label="max1 (Pos Zero) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];56 -> 71[label="",style="solid", color="black", weight=3];
10.78/4.41	57[label="max1 (Neg (Succ vx300)) (Pos vx40) True",fontsize=16,color="black",shape="box"];57 -> 72[label="",style="solid", color="black", weight=3];
10.78/4.41	58 -> 523[label="",style="dashed", color="red", weight=0];
10.78/4.41	58[label="max1 (Neg (Succ vx300)) (Neg (Succ vx400)) (not (primCmpNat vx400 vx300 == GT))",fontsize=16,color="magenta"];58 -> 524[label="",style="dashed", color="magenta", weight=3];
10.78/4.41	58 -> 525[label="",style="dashed", color="magenta", weight=3];
10.78/4.41	58 -> 526[label="",style="dashed", color="magenta", weight=3];
10.78/4.41	58 -> 527[label="",style="dashed", color="magenta", weight=3];
10.78/4.41	59[label="max1 (Neg (Succ vx300)) (Neg Zero) (not (LT == GT))",fontsize=16,color="black",shape="box"];59 -> 75[label="",style="solid", color="black", weight=3];
10.78/4.41	60[label="max1 (Neg Zero) (Pos (Succ vx400)) (not False)",fontsize=16,color="black",shape="box"];60 -> 76[label="",style="solid", color="black", weight=3];
10.78/4.41	61[label="max1 (Neg Zero) (Pos Zero) (not False)",fontsize=16,color="black",shape="box"];61 -> 77[label="",style="solid", color="black", weight=3];
10.78/4.41	62[label="max1 (Neg Zero) (Neg (Succ vx400)) (not (GT == GT))",fontsize=16,color="black",shape="box"];62 -> 78[label="",style="solid", color="black", weight=3];
10.78/4.41	63[label="max1 (Neg Zero) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];63 -> 79[label="",style="solid", color="black", weight=3];
10.78/4.41	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="max1 (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];
10.78/4.41	617 -> 490[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	618[label="vx43/Zero",fontsize=10,color="white",style="solid",shape="box"];453 -> 618[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	618 -> 491[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	66[label="max1 (Pos (Succ vx300)) (Pos Zero) (not True)",fontsize=16,color="black",shape="box"];66 -> 84[label="",style="solid", color="black", weight=3];
10.78/4.41	67[label="max0 (Pos (Succ vx300)) (Neg vx40) otherwise",fontsize=16,color="black",shape="box"];67 -> 85[label="",style="solid", color="black", weight=3];
10.78/4.41	68[label="max1 (Pos Zero) (Pos (Succ vx400)) (not False)",fontsize=16,color="black",shape="box"];68 -> 86[label="",style="solid", color="black", weight=3];
10.78/4.41	69[label="max1 (Pos Zero) (Pos Zero) True",fontsize=16,color="black",shape="box"];69 -> 87[label="",style="solid", color="black", weight=3];
10.78/4.41	70[label="max1 (Pos Zero) (Neg (Succ vx400)) False",fontsize=16,color="black",shape="box"];70 -> 88[label="",style="solid", color="black", weight=3];
10.78/4.41	71[label="max1 (Pos Zero) (Neg Zero) True",fontsize=16,color="black",shape="box"];71 -> 89[label="",style="solid", color="black", weight=3];
10.78/4.41	72[label="Pos vx40",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="max1 (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];
10.78/4.41	619 -> 564[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	620[label="vx52/Zero",fontsize=10,color="white",style="solid",shape="box"];523 -> 620[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	620 -> 565[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	75[label="max1 (Neg (Succ vx300)) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];75 -> 94[label="",style="solid", color="black", weight=3];
10.78/4.41	76[label="max1 (Neg Zero) (Pos (Succ vx400)) True",fontsize=16,color="black",shape="box"];76 -> 95[label="",style="solid", color="black", weight=3];
10.78/4.41	77[label="max1 (Neg Zero) (Pos Zero) True",fontsize=16,color="black",shape="box"];77 -> 96[label="",style="solid", color="black", weight=3];
10.78/4.41	78[label="max1 (Neg Zero) (Neg (Succ vx400)) (not True)",fontsize=16,color="black",shape="box"];78 -> 97[label="",style="solid", color="black", weight=3];
10.78/4.41	79[label="max1 (Neg Zero) (Neg Zero) True",fontsize=16,color="black",shape="box"];79 -> 98[label="",style="solid", color="black", weight=3];
10.78/4.41	490[label="max1 (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];
10.78/4.41	621 -> 496[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	622[label="vx44/Zero",fontsize=10,color="white",style="solid",shape="box"];490 -> 622[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	622 -> 497[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	491[label="max1 (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];
10.78/4.41	623 -> 498[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	624[label="vx44/Zero",fontsize=10,color="white",style="solid",shape="box"];491 -> 624[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	624 -> 499[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	84[label="max1 (Pos (Succ vx300)) (Pos Zero) False",fontsize=16,color="black",shape="box"];84 -> 103[label="",style="solid", color="black", weight=3];
10.78/4.41	85[label="max0 (Pos (Succ vx300)) (Neg vx40) True",fontsize=16,color="black",shape="box"];85 -> 104[label="",style="solid", color="black", weight=3];
10.78/4.41	86[label="max1 (Pos Zero) (Pos (Succ vx400)) True",fontsize=16,color="black",shape="box"];86 -> 105[label="",style="solid", color="black", weight=3];
10.78/4.41	87[label="Pos Zero",fontsize=16,color="green",shape="box"];88[label="max0 (Pos Zero) (Neg (Succ vx400)) otherwise",fontsize=16,color="black",shape="box"];88 -> 106[label="",style="solid", color="black", weight=3];
10.78/4.41	89[label="Neg Zero",fontsize=16,color="green",shape="box"];564[label="max1 (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];
10.78/4.41	625 -> 568[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	626[label="vx53/Zero",fontsize=10,color="white",style="solid",shape="box"];564 -> 626[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	626 -> 569[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	565[label="max1 (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];
10.78/4.41	627 -> 570[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	628[label="vx53/Zero",fontsize=10,color="white",style="solid",shape="box"];565 -> 628[label="",style="solid", color="burlywood", weight=9];
10.78/4.41	628 -> 571[label="",style="solid", color="burlywood", weight=3];
10.78/4.41	94[label="max1 (Neg (Succ vx300)) (Neg Zero) True",fontsize=16,color="black",shape="box"];94 -> 111[label="",style="solid", color="black", weight=3];
10.78/4.41	95[label="Pos (Succ vx400)",fontsize=16,color="green",shape="box"];96[label="Pos Zero",fontsize=16,color="green",shape="box"];97[label="max1 (Neg Zero) (Neg (Succ vx400)) False",fontsize=16,color="black",shape="box"];97 -> 112[label="",style="solid", color="black", weight=3];
10.78/4.41	98[label="Neg Zero",fontsize=16,color="green",shape="box"];496[label="max1 (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];
10.78/4.41	497[label="max1 (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];
10.78/4.41	498[label="max1 (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];
10.78/4.41	499[label="max1 (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];
10.78/4.41	103[label="max0 (Pos (Succ vx300)) (Pos Zero) otherwise",fontsize=16,color="black",shape="box"];103 -> 118[label="",style="solid", color="black", weight=3];
10.78/4.41	104[label="Pos (Succ vx300)",fontsize=16,color="green",shape="box"];105[label="Pos (Succ vx400)",fontsize=16,color="green",shape="box"];106[label="max0 (Pos Zero) (Neg (Succ vx400)) True",fontsize=16,color="black",shape="box"];106 -> 119[label="",style="solid", color="black", weight=3];
10.78/4.41	568[label="max1 (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];
10.78/4.41	569[label="max1 (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];
10.78/4.41	570[label="max1 (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];
10.78/4.41	571[label="max1 (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];
10.78/4.41	111[label="Neg Zero",fontsize=16,color="green",shape="box"];112[label="max0 (Neg Zero) (Neg (Succ vx400)) otherwise",fontsize=16,color="black",shape="box"];112 -> 125[label="",style="solid", color="black", weight=3];
10.78/4.41	504 -> 453[label="",style="dashed", color="red", weight=0];
10.78/4.41	504[label="max1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat vx430 vx440 == GT))",fontsize=16,color="magenta"];504 -> 512[label="",style="dashed", color="magenta", weight=3];
10.78/4.41	504 -> 513[label="",style="dashed", color="magenta", weight=3];
10.78/4.41	505[label="max1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (GT == GT))",fontsize=16,color="black",shape="box"];505 -> 514[label="",style="solid", color="black", weight=3];
10.78/4.41	506[label="max1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (LT == GT))",fontsize=16,color="black",shape="box"];506 -> 515[label="",style="solid", color="black", weight=3];
10.78/4.41	507[label="max1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];507 -> 516[label="",style="solid", color="black", weight=3];
10.78/4.41	118[label="max0 (Pos (Succ vx300)) (Pos Zero) True",fontsize=16,color="black",shape="box"];118 -> 133[label="",style="solid", color="black", weight=3];
10.78/4.41	119[label="Pos Zero",fontsize=16,color="green",shape="box"];574 -> 523[label="",style="dashed", color="red", weight=0];
10.78/4.41	574[label="max1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat vx520 vx530 == GT))",fontsize=16,color="magenta"];574 -> 579[label="",style="dashed", color="magenta", weight=3];
10.78/4.41	574 -> 580[label="",style="dashed", color="magenta", weight=3];
10.78/4.41	575[label="max1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (GT == GT))",fontsize=16,color="black",shape="box"];575 -> 581[label="",style="solid", color="black", weight=3];
10.78/4.41	576[label="max1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (LT == GT))",fontsize=16,color="black",shape="box"];576 -> 582[label="",style="solid", color="black", weight=3];
10.78/4.41	577[label="max1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];577 -> 583[label="",style="solid", color="black", weight=3];
10.78/4.41	125[label="max0 (Neg Zero) (Neg (Succ vx400)) True",fontsize=16,color="black",shape="box"];125 -> 141[label="",style="solid", color="black", weight=3];
10.78/4.41	512[label="vx430",fontsize=16,color="green",shape="box"];513[label="vx440",fontsize=16,color="green",shape="box"];514[label="max1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not True)",fontsize=16,color="black",shape="box"];514 -> 566[label="",style="solid", color="black", weight=3];
10.78/4.41	515[label="max1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not False)",fontsize=16,color="black",shape="triangle"];515 -> 567[label="",style="solid", color="black", weight=3];
10.78/4.41	516 -> 515[label="",style="dashed", color="red", weight=0];
10.78/4.41	516[label="max1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not False)",fontsize=16,color="magenta"];133[label="Pos (Succ vx300)",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="max1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not True)",fontsize=16,color="black",shape="box"];581 -> 585[label="",style="solid", color="black", weight=3];
10.78/4.41	582[label="max1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not False)",fontsize=16,color="black",shape="triangle"];582 -> 586[label="",style="solid", color="black", weight=3];
10.78/4.41	583 -> 582[label="",style="dashed", color="red", weight=0];
10.78/4.41	583[label="max1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not False)",fontsize=16,color="magenta"];141[label="Neg Zero",fontsize=16,color="green",shape="box"];566[label="max1 (Pos (Succ vx41)) (Pos (Succ vx42)) False",fontsize=16,color="black",shape="box"];566 -> 572[label="",style="solid", color="black", weight=3];
10.78/4.41	567[label="max1 (Pos (Succ vx41)) (Pos (Succ vx42)) True",fontsize=16,color="black",shape="box"];567 -> 573[label="",style="solid", color="black", weight=3];
10.78/4.41	585[label="max1 (Neg (Succ vx50)) (Neg (Succ vx51)) False",fontsize=16,color="black",shape="box"];585 -> 587[label="",style="solid", color="black", weight=3];
10.78/4.41	586[label="max1 (Neg (Succ vx50)) (Neg (Succ vx51)) True",fontsize=16,color="black",shape="box"];586 -> 588[label="",style="solid", color="black", weight=3];
10.78/4.41	572[label="max0 (Pos (Succ vx41)) (Pos (Succ vx42)) otherwise",fontsize=16,color="black",shape="box"];572 -> 578[label="",style="solid", color="black", weight=3];
10.78/4.41	573[label="Pos (Succ vx42)",fontsize=16,color="green",shape="box"];587[label="max0 (Neg (Succ vx50)) (Neg (Succ vx51)) otherwise",fontsize=16,color="black",shape="box"];587 -> 589[label="",style="solid", color="black", weight=3];
10.78/4.41	588[label="Neg (Succ vx51)",fontsize=16,color="green",shape="box"];578[label="max0 (Pos (Succ vx41)) (Pos (Succ vx42)) True",fontsize=16,color="black",shape="box"];578 -> 584[label="",style="solid", color="black", weight=3];
10.78/4.41	589[label="max0 (Neg (Succ vx50)) (Neg (Succ vx51)) True",fontsize=16,color="black",shape="box"];589 -> 590[label="",style="solid", color="black", weight=3];
10.78/4.41	584[label="Pos (Succ vx41)",fontsize=16,color="green",shape="box"];590[label="Neg (Succ vx50)",fontsize=16,color="green",shape="box"];}
10.78/4.41	
10.78/4.41	----------------------------------------
10.78/4.41	
10.78/4.41	(6)
10.78/4.41	Complex Obligation (AND)
10.78/4.41	
10.78/4.41	----------------------------------------
10.78/4.41	
10.78/4.41	(7)
10.78/4.41	Obligation:
10.78/4.41	Q DP problem:
10.78/4.41	The TRS P consists of the following rules:
10.78/4.41	
10.78/4.41	   new_max1(vx50, vx51, Succ(vx520), Succ(vx530)) -> new_max1(vx50, vx51, vx520, vx530)
10.78/4.41	
10.78/4.41	R is empty.
10.78/4.41	Q is empty.
10.78/4.41	We have to consider all minimal (P,Q,R)-chains.
10.78/4.41	----------------------------------------
10.78/4.41	
10.78/4.41	(8) QDPSizeChangeProof (EQUIVALENT)
10.78/4.41	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. 
10.78/4.41	
10.78/4.41	From the DPs we obtained the following set of size-change graphs:
10.78/4.41	*new_max1(vx50, vx51, Succ(vx520), Succ(vx530)) -> new_max1(vx50, vx51, vx520, vx530)
10.78/4.41	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
10.78/4.41	
10.78/4.41	
10.78/4.41	----------------------------------------
10.78/4.41	
10.78/4.41	(9)
10.78/4.41	YES
10.78/4.41	
10.78/4.41	----------------------------------------
10.78/4.41	
10.78/4.41	(10)
10.78/4.41	Obligation:
10.78/4.41	Q DP problem:
10.78/4.41	The TRS P consists of the following rules:
10.78/4.41	
10.78/4.41	   new_max10(vx41, vx42, Succ(vx430), Succ(vx440)) -> new_max10(vx41, vx42, vx430, vx440)
10.78/4.41	
10.78/4.41	R is empty.
10.78/4.41	Q is empty.
10.78/4.41	We have to consider all minimal (P,Q,R)-chains.
10.78/4.41	----------------------------------------
10.78/4.41	
10.78/4.41	(11) QDPSizeChangeProof (EQUIVALENT)
10.78/4.41	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. 
10.78/4.41	
10.78/4.41	From the DPs we obtained the following set of size-change graphs:
10.78/4.41	*new_max10(vx41, vx42, Succ(vx430), Succ(vx440)) -> new_max10(vx41, vx42, vx430, vx440)
10.78/4.41	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
10.78/4.41	
10.78/4.41	
10.78/4.41	----------------------------------------
10.78/4.41	
10.78/4.41	(12)
10.78/4.41	YES
10.78/4.45	EOF