7.76/3.56	YES
9.36/4.00	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.36/4.00	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.36/4.00	
9.36/4.00	
9.36/4.00	H-Termination with start terms of the given HASKELL could be proven:
9.36/4.00	
9.36/4.00	(0) HASKELL
9.36/4.00	(1) BR [EQUIVALENT, 0 ms]
9.36/4.00	(2) HASKELL
9.36/4.00	(3) COR [EQUIVALENT, 0 ms]
9.36/4.00	(4) HASKELL
9.36/4.00	(5) Narrow [SOUND, 0 ms]
9.36/4.00	(6) AND
9.36/4.00	    (7) QDP
9.36/4.00	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.36/4.00	        (9) YES
9.36/4.00	    (10) QDP
9.36/4.00	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.36/4.00	        (12) YES
9.36/4.00	    (13) QDP
9.36/4.00	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.36/4.00	        (15) YES
9.36/4.00	
9.36/4.00	
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(0)
9.36/4.00	Obligation:
9.36/4.00	mainModule Main 
9.36/4.00	module Main where {
9.36/4.00	import qualified Prelude;
9.36/4.00	}
9.36/4.00	
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(1) BR (EQUIVALENT)
9.36/4.00	Replaced joker patterns by fresh variables and removed binding patterns.
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(2)
9.36/4.00	Obligation:
9.36/4.00	mainModule Main 
9.36/4.00	module Main where {
9.36/4.00	import qualified Prelude;
9.36/4.00	}
9.36/4.00	
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(3) COR (EQUIVALENT)
9.36/4.00	Cond Reductions:
9.36/4.00	The following Function with conditions
9.36/4.00	"undefined |Falseundefined;
9.36/4.00	"
9.36/4.00	is transformed to
9.36/4.00	"undefined  = undefined1;
9.36/4.00	"
9.36/4.00	"undefined0 True = undefined;
9.36/4.00	"
9.36/4.00	"undefined1  = undefined0 False;
9.36/4.00	"
9.36/4.00	
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(4)
9.36/4.00	Obligation:
9.36/4.00	mainModule Main 
9.36/4.00	module Main where {
9.36/4.00	import qualified Prelude;
9.36/4.00	}
9.36/4.00	
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(5) Narrow (SOUND)
9.36/4.00	Haskell To QDPs
9.36/4.00	
9.36/4.00	digraph dp_graph {
9.36/4.00	node [outthreshold=100, inthreshold=100];1[label="(-)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.36/4.00	3[label="(-) vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.36/4.00	4[label="(-) vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.36/4.00	5[label="primMinusFloat vx3 vx4",fontsize=16,color="burlywood",shape="box"];92[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];5 -> 92[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	92 -> 6[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	6[label="primMinusFloat (Float vx30 vx31) vx4",fontsize=16,color="burlywood",shape="box"];93[label="vx4/Float vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];6 -> 93[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	93 -> 7[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	7[label="primMinusFloat (Float vx30 vx31) (Float vx40 vx41)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.36/4.00	8[label="Float (vx30 * vx41 - vx40 * vx31) (vx31 * vx41)",fontsize=16,color="green",shape="box"];8 -> 9[label="",style="dashed", color="green", weight=3];
9.36/4.00	8 -> 10[label="",style="dashed", color="green", weight=3];
9.36/4.00	9[label="vx30 * vx41 - vx40 * vx31",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.36/4.00	10[label="vx31 * vx41",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3];
9.36/4.00	11 -> 13[label="",style="dashed", color="red", weight=0];
9.36/4.00	11[label="primMinusInt (vx30 * vx41) (vx40 * vx31)",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	11 -> 15[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	12[label="primMulInt vx31 vx41",fontsize=16,color="burlywood",shape="box"];94[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];12 -> 94[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	94 -> 16[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	95[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];12 -> 95[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	95 -> 17[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	14 -> 10[label="",style="dashed", color="red", weight=0];
9.36/4.00	14[label="vx30 * vx41",fontsize=16,color="magenta"];14 -> 18[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	15 -> 10[label="",style="dashed", color="red", weight=0];
9.36/4.00	15[label="vx40 * vx31",fontsize=16,color="magenta"];15 -> 19[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	15 -> 20[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	13[label="primMinusInt vx6 vx5",fontsize=16,color="burlywood",shape="triangle"];96[label="vx6/Pos vx60",fontsize=10,color="white",style="solid",shape="box"];13 -> 96[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	96 -> 21[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	97[label="vx6/Neg vx60",fontsize=10,color="white",style="solid",shape="box"];13 -> 97[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	97 -> 22[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	16[label="primMulInt (Pos vx310) vx41",fontsize=16,color="burlywood",shape="box"];98[label="vx41/Pos vx410",fontsize=10,color="white",style="solid",shape="box"];16 -> 98[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	98 -> 23[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	99[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];16 -> 99[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	99 -> 24[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	17[label="primMulInt (Neg vx310) vx41",fontsize=16,color="burlywood",shape="box"];100[label="vx41/Pos vx410",fontsize=10,color="white",style="solid",shape="box"];17 -> 100[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	100 -> 25[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	101[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];17 -> 101[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	101 -> 26[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	18[label="vx30",fontsize=16,color="green",shape="box"];19[label="vx31",fontsize=16,color="green",shape="box"];20[label="vx40",fontsize=16,color="green",shape="box"];21[label="primMinusInt (Pos vx60) vx5",fontsize=16,color="burlywood",shape="box"];102[label="vx5/Pos vx50",fontsize=10,color="white",style="solid",shape="box"];21 -> 102[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	102 -> 27[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	103[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];21 -> 103[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	103 -> 28[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	22[label="primMinusInt (Neg vx60) vx5",fontsize=16,color="burlywood",shape="box"];104[label="vx5/Pos vx50",fontsize=10,color="white",style="solid",shape="box"];22 -> 104[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	104 -> 29[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	105[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];22 -> 105[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	105 -> 30[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	23[label="primMulInt (Pos vx310) (Pos vx410)",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
9.36/4.00	24[label="primMulInt (Pos vx310) (Neg vx410)",fontsize=16,color="black",shape="box"];24 -> 32[label="",style="solid", color="black", weight=3];
9.36/4.00	25[label="primMulInt (Neg vx310) (Pos vx410)",fontsize=16,color="black",shape="box"];25 -> 33[label="",style="solid", color="black", weight=3];
9.36/4.00	26[label="primMulInt (Neg vx310) (Neg vx410)",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
9.36/4.00	27[label="primMinusInt (Pos vx60) (Pos vx50)",fontsize=16,color="black",shape="box"];27 -> 35[label="",style="solid", color="black", weight=3];
9.36/4.00	28[label="primMinusInt (Pos vx60) (Neg vx50)",fontsize=16,color="black",shape="box"];28 -> 36[label="",style="solid", color="black", weight=3];
9.36/4.00	29[label="primMinusInt (Neg vx60) (Pos vx50)",fontsize=16,color="black",shape="box"];29 -> 37[label="",style="solid", color="black", weight=3];
9.36/4.00	30[label="primMinusInt (Neg vx60) (Neg vx50)",fontsize=16,color="black",shape="box"];30 -> 38[label="",style="solid", color="black", weight=3];
9.36/4.00	31[label="Pos (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];31 -> 39[label="",style="dashed", color="green", weight=3];
9.36/4.00	32[label="Neg (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];32 -> 40[label="",style="dashed", color="green", weight=3];
9.36/4.00	33[label="Neg (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];33 -> 41[label="",style="dashed", color="green", weight=3];
9.36/4.00	34[label="Pos (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];34 -> 42[label="",style="dashed", color="green", weight=3];
9.36/4.00	35[label="primMinusNat vx60 vx50",fontsize=16,color="burlywood",shape="triangle"];106[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];35 -> 106[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	106 -> 43[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	107[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 107[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	107 -> 44[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	36[label="Pos (primPlusNat vx60 vx50)",fontsize=16,color="green",shape="box"];36 -> 45[label="",style="dashed", color="green", weight=3];
9.36/4.00	37[label="Neg (primPlusNat vx60 vx50)",fontsize=16,color="green",shape="box"];37 -> 46[label="",style="dashed", color="green", weight=3];
9.36/4.00	38 -> 35[label="",style="dashed", color="red", weight=0];
9.36/4.00	38[label="primMinusNat vx50 vx60",fontsize=16,color="magenta"];38 -> 47[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	38 -> 48[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	39[label="primMulNat vx310 vx410",fontsize=16,color="burlywood",shape="triangle"];108[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];39 -> 108[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	108 -> 49[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	109[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];39 -> 109[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	109 -> 50[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	40 -> 39[label="",style="dashed", color="red", weight=0];
9.36/4.00	40[label="primMulNat vx310 vx410",fontsize=16,color="magenta"];40 -> 51[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	41 -> 39[label="",style="dashed", color="red", weight=0];
9.36/4.00	41[label="primMulNat vx310 vx410",fontsize=16,color="magenta"];41 -> 52[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	42 -> 39[label="",style="dashed", color="red", weight=0];
9.36/4.00	42[label="primMulNat vx310 vx410",fontsize=16,color="magenta"];42 -> 53[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	42 -> 54[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	43[label="primMinusNat (Succ vx600) vx50",fontsize=16,color="burlywood",shape="box"];110[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];43 -> 110[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	110 -> 55[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	111[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];43 -> 111[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	111 -> 56[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	44[label="primMinusNat Zero vx50",fontsize=16,color="burlywood",shape="box"];112[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];44 -> 112[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	112 -> 57[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	113[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];44 -> 113[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	113 -> 58[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	45[label="primPlusNat vx60 vx50",fontsize=16,color="burlywood",shape="triangle"];114[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];45 -> 114[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	114 -> 59[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	115[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 115[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	115 -> 60[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	46 -> 45[label="",style="dashed", color="red", weight=0];
9.36/4.00	46[label="primPlusNat vx60 vx50",fontsize=16,color="magenta"];46 -> 61[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	46 -> 62[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	47[label="vx50",fontsize=16,color="green",shape="box"];48[label="vx60",fontsize=16,color="green",shape="box"];49[label="primMulNat (Succ vx3100) vx410",fontsize=16,color="burlywood",shape="box"];116[label="vx410/Succ vx4100",fontsize=10,color="white",style="solid",shape="box"];49 -> 116[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	116 -> 63[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	117[label="vx410/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 117[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	117 -> 64[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	50[label="primMulNat Zero vx410",fontsize=16,color="burlywood",shape="box"];118[label="vx410/Succ vx4100",fontsize=10,color="white",style="solid",shape="box"];50 -> 118[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	118 -> 65[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	119[label="vx410/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 119[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	119 -> 66[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	51[label="vx410",fontsize=16,color="green",shape="box"];52[label="vx310",fontsize=16,color="green",shape="box"];53[label="vx310",fontsize=16,color="green",shape="box"];54[label="vx410",fontsize=16,color="green",shape="box"];55[label="primMinusNat (Succ vx600) (Succ vx500)",fontsize=16,color="black",shape="box"];55 -> 67[label="",style="solid", color="black", weight=3];
9.36/4.00	56[label="primMinusNat (Succ vx600) Zero",fontsize=16,color="black",shape="box"];56 -> 68[label="",style="solid", color="black", weight=3];
9.36/4.00	57[label="primMinusNat Zero (Succ vx500)",fontsize=16,color="black",shape="box"];57 -> 69[label="",style="solid", color="black", weight=3];
9.36/4.00	58[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];58 -> 70[label="",style="solid", color="black", weight=3];
9.36/4.00	59[label="primPlusNat (Succ vx600) vx50",fontsize=16,color="burlywood",shape="box"];120[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];59 -> 120[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	120 -> 71[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	121[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 121[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	121 -> 72[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	60[label="primPlusNat Zero vx50",fontsize=16,color="burlywood",shape="box"];122[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];60 -> 122[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	122 -> 73[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	123[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];60 -> 123[label="",style="solid", color="burlywood", weight=9];
9.36/4.00	123 -> 74[label="",style="solid", color="burlywood", weight=3];
9.36/4.00	61[label="vx50",fontsize=16,color="green",shape="box"];62[label="vx60",fontsize=16,color="green",shape="box"];63[label="primMulNat (Succ vx3100) (Succ vx4100)",fontsize=16,color="black",shape="box"];63 -> 75[label="",style="solid", color="black", weight=3];
9.36/4.00	64[label="primMulNat (Succ vx3100) Zero",fontsize=16,color="black",shape="box"];64 -> 76[label="",style="solid", color="black", weight=3];
9.36/4.00	65[label="primMulNat Zero (Succ vx4100)",fontsize=16,color="black",shape="box"];65 -> 77[label="",style="solid", color="black", weight=3];
9.36/4.00	66[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];66 -> 78[label="",style="solid", color="black", weight=3];
9.36/4.00	67 -> 35[label="",style="dashed", color="red", weight=0];
9.36/4.00	67[label="primMinusNat vx600 vx500",fontsize=16,color="magenta"];67 -> 79[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	67 -> 80[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	68[label="Pos (Succ vx600)",fontsize=16,color="green",shape="box"];69[label="Neg (Succ vx500)",fontsize=16,color="green",shape="box"];70[label="Pos Zero",fontsize=16,color="green",shape="box"];71[label="primPlusNat (Succ vx600) (Succ vx500)",fontsize=16,color="black",shape="box"];71 -> 81[label="",style="solid", color="black", weight=3];
9.36/4.00	72[label="primPlusNat (Succ vx600) Zero",fontsize=16,color="black",shape="box"];72 -> 82[label="",style="solid", color="black", weight=3];
9.36/4.00	73[label="primPlusNat Zero (Succ vx500)",fontsize=16,color="black",shape="box"];73 -> 83[label="",style="solid", color="black", weight=3];
9.36/4.00	74[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];74 -> 84[label="",style="solid", color="black", weight=3];
9.36/4.00	75 -> 45[label="",style="dashed", color="red", weight=0];
9.36/4.00	75[label="primPlusNat (primMulNat vx3100 (Succ vx4100)) (Succ vx4100)",fontsize=16,color="magenta"];75 -> 85[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	75 -> 86[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	76[label="Zero",fontsize=16,color="green",shape="box"];77[label="Zero",fontsize=16,color="green",shape="box"];78[label="Zero",fontsize=16,color="green",shape="box"];79[label="vx600",fontsize=16,color="green",shape="box"];80[label="vx500",fontsize=16,color="green",shape="box"];81[label="Succ (Succ (primPlusNat vx600 vx500))",fontsize=16,color="green",shape="box"];81 -> 87[label="",style="dashed", color="green", weight=3];
9.36/4.00	82[label="Succ vx600",fontsize=16,color="green",shape="box"];83[label="Succ vx500",fontsize=16,color="green",shape="box"];84[label="Zero",fontsize=16,color="green",shape="box"];85[label="Succ vx4100",fontsize=16,color="green",shape="box"];86 -> 39[label="",style="dashed", color="red", weight=0];
9.36/4.00	86[label="primMulNat vx3100 (Succ vx4100)",fontsize=16,color="magenta"];86 -> 88[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	86 -> 89[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	87 -> 45[label="",style="dashed", color="red", weight=0];
9.36/4.00	87[label="primPlusNat vx600 vx500",fontsize=16,color="magenta"];87 -> 90[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	87 -> 91[label="",style="dashed", color="magenta", weight=3];
9.36/4.00	88[label="vx3100",fontsize=16,color="green",shape="box"];89[label="Succ vx4100",fontsize=16,color="green",shape="box"];90[label="vx500",fontsize=16,color="green",shape="box"];91[label="vx600",fontsize=16,color="green",shape="box"];}
9.36/4.00	
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(6)
9.36/4.00	Complex Obligation (AND)
9.36/4.00	
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(7)
9.36/4.00	Obligation:
9.36/4.00	Q DP problem:
9.36/4.00	The TRS P consists of the following rules:
9.36/4.00	
9.36/4.00	   new_primMulNat(Succ(vx3100), Succ(vx4100)) -> new_primMulNat(vx3100, Succ(vx4100))
9.36/4.00	
9.36/4.00	R is empty.
9.36/4.00	Q is empty.
9.36/4.00	We have to consider all minimal (P,Q,R)-chains.
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(8) QDPSizeChangeProof (EQUIVALENT)
9.36/4.00	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. 
9.36/4.00	
9.36/4.00	From the DPs we obtained the following set of size-change graphs:
9.36/4.00	*new_primMulNat(Succ(vx3100), Succ(vx4100)) -> new_primMulNat(vx3100, Succ(vx4100))
9.36/4.00	The graph contains the following edges 1 > 1, 2 >= 2
9.36/4.00	
9.36/4.00	
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(9)
9.36/4.00	YES
9.36/4.00	
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(10)
9.36/4.00	Obligation:
9.36/4.00	Q DP problem:
9.36/4.00	The TRS P consists of the following rules:
9.36/4.00	
9.36/4.00	   new_primPlusNat(Succ(vx600), Succ(vx500)) -> new_primPlusNat(vx600, vx500)
9.36/4.00	
9.36/4.00	R is empty.
9.36/4.00	Q is empty.
9.36/4.00	We have to consider all minimal (P,Q,R)-chains.
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(11) QDPSizeChangeProof (EQUIVALENT)
9.36/4.00	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. 
9.36/4.00	
9.36/4.00	From the DPs we obtained the following set of size-change graphs:
9.36/4.00	*new_primPlusNat(Succ(vx600), Succ(vx500)) -> new_primPlusNat(vx600, vx500)
9.36/4.00	The graph contains the following edges 1 > 1, 2 > 2
9.36/4.00	
9.36/4.00	
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(12)
9.36/4.00	YES
9.36/4.00	
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(13)
9.36/4.00	Obligation:
9.36/4.00	Q DP problem:
9.36/4.00	The TRS P consists of the following rules:
9.36/4.00	
9.36/4.00	   new_primMinusNat(Succ(vx600), Succ(vx500)) -> new_primMinusNat(vx600, vx500)
9.36/4.00	
9.36/4.00	R is empty.
9.36/4.00	Q is empty.
9.36/4.00	We have to consider all minimal (P,Q,R)-chains.
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(14) QDPSizeChangeProof (EQUIVALENT)
9.36/4.00	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. 
9.36/4.00	
9.36/4.00	From the DPs we obtained the following set of size-change graphs:
9.36/4.00	*new_primMinusNat(Succ(vx600), Succ(vx500)) -> new_primMinusNat(vx600, vx500)
9.36/4.00	The graph contains the following edges 1 > 1, 2 > 2
9.36/4.00	
9.36/4.00	
9.36/4.00	----------------------------------------
9.36/4.00	
9.36/4.00	(15)
9.36/4.00	YES
9.36/4.05	EOF