8.09/3.55	YES
9.42/3.97	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.42/3.97	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.42/3.97	
9.42/3.97	
9.42/3.97	H-Termination with start terms of the given HASKELL could be proven:
9.42/3.97	
9.42/3.97	(0) HASKELL
9.42/3.97	(1) BR [EQUIVALENT, 0 ms]
9.42/3.97	(2) HASKELL
9.42/3.97	(3) COR [EQUIVALENT, 0 ms]
9.42/3.97	(4) HASKELL
9.42/3.97	(5) Narrow [SOUND, 0 ms]
9.42/3.97	(6) AND
9.42/3.97	    (7) QDP
9.42/3.97	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.42/3.97	        (9) YES
9.42/3.97	    (10) QDP
9.42/3.97	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.42/3.97	        (12) YES
9.42/3.97	    (13) QDP
9.42/3.97	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.42/3.97	        (15) YES
9.42/3.97	
9.42/3.97	
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(0)
9.42/3.97	Obligation:
9.42/3.97	mainModule Main 
9.42/3.97	module Main where {
9.42/3.97	import qualified Prelude;
9.42/3.97	}
9.42/3.97	
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(1) BR (EQUIVALENT)
9.42/3.97	Replaced joker patterns by fresh variables and removed binding patterns.
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(2)
9.42/3.97	Obligation:
9.42/3.97	mainModule Main 
9.42/3.97	module Main where {
9.42/3.97	import qualified Prelude;
9.42/3.97	}
9.42/3.97	
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(3) COR (EQUIVALENT)
9.42/3.97	Cond Reductions:
9.42/3.97	The following Function with conditions
9.42/3.97	"undefined |Falseundefined;
9.42/3.97	"
9.42/3.97	is transformed to
9.42/3.97	"undefined  = undefined1;
9.42/3.97	"
9.42/3.97	"undefined0 True = undefined;
9.42/3.97	"
9.42/3.97	"undefined1  = undefined0 False;
9.42/3.97	"
9.42/3.97	
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(4)
9.42/3.97	Obligation:
9.42/3.97	mainModule Main 
9.42/3.97	module Main where {
9.42/3.97	import qualified Prelude;
9.42/3.97	}
9.42/3.97	
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(5) Narrow (SOUND)
9.42/3.97	Haskell To QDPs
9.42/3.97	
9.42/3.97	digraph dp_graph {
9.42/3.97	node [outthreshold=100, inthreshold=100];1[label="subtract",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.42/3.97	3[label="subtract vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.42/3.97	4[label="subtract vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.42/3.97	5[label="flip (-) vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.42/3.97	6[label="(-) vx4 vx3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.42/3.97	7[label="primMinusFloat vx4 vx3",fontsize=16,color="burlywood",shape="box"];94[label="vx4/Float vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];7 -> 94[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	94 -> 8[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	8[label="primMinusFloat (Float vx40 vx41) vx3",fontsize=16,color="burlywood",shape="box"];95[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];8 -> 95[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	95 -> 9[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	9[label="primMinusFloat (Float vx40 vx41) (Float vx30 vx31)",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
9.42/3.97	10[label="Float (vx40 * vx31 - vx30 * vx41) (vx41 * vx31)",fontsize=16,color="green",shape="box"];10 -> 11[label="",style="dashed", color="green", weight=3];
9.42/3.97	10 -> 12[label="",style="dashed", color="green", weight=3];
9.42/3.97	11[label="vx40 * vx31 - vx30 * vx41",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.42/3.97	12[label="vx41 * vx31",fontsize=16,color="black",shape="triangle"];12 -> 14[label="",style="solid", color="black", weight=3];
9.42/3.97	13 -> 15[label="",style="dashed", color="red", weight=0];
9.42/3.97	13[label="primMinusInt (vx40 * vx31) (vx30 * vx41)",fontsize=16,color="magenta"];13 -> 16[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	13 -> 17[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	14[label="primMulInt vx41 vx31",fontsize=16,color="burlywood",shape="box"];96[label="vx41/Pos vx410",fontsize=10,color="white",style="solid",shape="box"];14 -> 96[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	96 -> 18[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	97[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];14 -> 97[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	97 -> 19[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	16 -> 12[label="",style="dashed", color="red", weight=0];
9.42/3.97	16[label="vx40 * vx31",fontsize=16,color="magenta"];16 -> 20[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	17 -> 12[label="",style="dashed", color="red", weight=0];
9.42/3.97	17[label="vx30 * vx41",fontsize=16,color="magenta"];17 -> 21[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	17 -> 22[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	15[label="primMinusInt vx6 vx5",fontsize=16,color="burlywood",shape="triangle"];98[label="vx6/Pos vx60",fontsize=10,color="white",style="solid",shape="box"];15 -> 98[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	98 -> 23[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	99[label="vx6/Neg vx60",fontsize=10,color="white",style="solid",shape="box"];15 -> 99[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	99 -> 24[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	18[label="primMulInt (Pos vx410) vx31",fontsize=16,color="burlywood",shape="box"];100[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];18 -> 100[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	100 -> 25[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	101[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];18 -> 101[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	101 -> 26[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	19[label="primMulInt (Neg vx410) vx31",fontsize=16,color="burlywood",shape="box"];102[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];19 -> 102[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	102 -> 27[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	103[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];19 -> 103[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	103 -> 28[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	20[label="vx40",fontsize=16,color="green",shape="box"];21[label="vx41",fontsize=16,color="green",shape="box"];22[label="vx30",fontsize=16,color="green",shape="box"];23[label="primMinusInt (Pos vx60) vx5",fontsize=16,color="burlywood",shape="box"];104[label="vx5/Pos vx50",fontsize=10,color="white",style="solid",shape="box"];23 -> 104[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	104 -> 29[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	105[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];23 -> 105[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	105 -> 30[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	24[label="primMinusInt (Neg vx60) vx5",fontsize=16,color="burlywood",shape="box"];106[label="vx5/Pos vx50",fontsize=10,color="white",style="solid",shape="box"];24 -> 106[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	106 -> 31[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	107[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];24 -> 107[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	107 -> 32[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	25[label="primMulInt (Pos vx410) (Pos vx310)",fontsize=16,color="black",shape="box"];25 -> 33[label="",style="solid", color="black", weight=3];
9.42/3.97	26[label="primMulInt (Pos vx410) (Neg vx310)",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
9.42/3.97	27[label="primMulInt (Neg vx410) (Pos vx310)",fontsize=16,color="black",shape="box"];27 -> 35[label="",style="solid", color="black", weight=3];
9.42/3.97	28[label="primMulInt (Neg vx410) (Neg vx310)",fontsize=16,color="black",shape="box"];28 -> 36[label="",style="solid", color="black", weight=3];
9.42/3.97	29[label="primMinusInt (Pos vx60) (Pos vx50)",fontsize=16,color="black",shape="box"];29 -> 37[label="",style="solid", color="black", weight=3];
9.42/3.97	30[label="primMinusInt (Pos vx60) (Neg vx50)",fontsize=16,color="black",shape="box"];30 -> 38[label="",style="solid", color="black", weight=3];
9.42/3.97	31[label="primMinusInt (Neg vx60) (Pos vx50)",fontsize=16,color="black",shape="box"];31 -> 39[label="",style="solid", color="black", weight=3];
9.42/3.97	32[label="primMinusInt (Neg vx60) (Neg vx50)",fontsize=16,color="black",shape="box"];32 -> 40[label="",style="solid", color="black", weight=3];
9.42/3.97	33[label="Pos (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];33 -> 41[label="",style="dashed", color="green", weight=3];
9.42/3.97	34[label="Neg (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];34 -> 42[label="",style="dashed", color="green", weight=3];
9.42/3.97	35[label="Neg (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];35 -> 43[label="",style="dashed", color="green", weight=3];
9.42/3.97	36[label="Pos (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];36 -> 44[label="",style="dashed", color="green", weight=3];
9.42/3.97	37[label="primMinusNat vx60 vx50",fontsize=16,color="burlywood",shape="triangle"];108[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];37 -> 108[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	108 -> 45[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	109[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];37 -> 109[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	109 -> 46[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	38[label="Pos (primPlusNat vx60 vx50)",fontsize=16,color="green",shape="box"];38 -> 47[label="",style="dashed", color="green", weight=3];
9.42/3.97	39[label="Neg (primPlusNat vx60 vx50)",fontsize=16,color="green",shape="box"];39 -> 48[label="",style="dashed", color="green", weight=3];
9.42/3.97	40 -> 37[label="",style="dashed", color="red", weight=0];
9.42/3.97	40[label="primMinusNat vx50 vx60",fontsize=16,color="magenta"];40 -> 49[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	40 -> 50[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	41[label="primMulNat vx410 vx310",fontsize=16,color="burlywood",shape="triangle"];110[label="vx410/Succ vx4100",fontsize=10,color="white",style="solid",shape="box"];41 -> 110[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	110 -> 51[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	111[label="vx410/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 111[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	111 -> 52[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	42 -> 41[label="",style="dashed", color="red", weight=0];
9.42/3.97	42[label="primMulNat vx410 vx310",fontsize=16,color="magenta"];42 -> 53[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	43 -> 41[label="",style="dashed", color="red", weight=0];
9.42/3.97	43[label="primMulNat vx410 vx310",fontsize=16,color="magenta"];43 -> 54[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	44 -> 41[label="",style="dashed", color="red", weight=0];
9.42/3.97	44[label="primMulNat vx410 vx310",fontsize=16,color="magenta"];44 -> 55[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	44 -> 56[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	45[label="primMinusNat (Succ vx600) vx50",fontsize=16,color="burlywood",shape="box"];112[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];45 -> 112[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	112 -> 57[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	113[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 113[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	113 -> 58[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	46[label="primMinusNat Zero vx50",fontsize=16,color="burlywood",shape="box"];114[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];46 -> 114[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	114 -> 59[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	115[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 115[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	115 -> 60[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	47[label="primPlusNat vx60 vx50",fontsize=16,color="burlywood",shape="triangle"];116[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];47 -> 116[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	116 -> 61[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	117[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 117[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	117 -> 62[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	48 -> 47[label="",style="dashed", color="red", weight=0];
9.42/3.97	48[label="primPlusNat vx60 vx50",fontsize=16,color="magenta"];48 -> 63[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	48 -> 64[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	49[label="vx50",fontsize=16,color="green",shape="box"];50[label="vx60",fontsize=16,color="green",shape="box"];51[label="primMulNat (Succ vx4100) vx310",fontsize=16,color="burlywood",shape="box"];118[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];51 -> 118[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	118 -> 65[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	119[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];51 -> 119[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	119 -> 66[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	52[label="primMulNat Zero vx310",fontsize=16,color="burlywood",shape="box"];120[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];52 -> 120[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	120 -> 67[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	121[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 121[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	121 -> 68[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	53[label="vx310",fontsize=16,color="green",shape="box"];54[label="vx410",fontsize=16,color="green",shape="box"];55[label="vx310",fontsize=16,color="green",shape="box"];56[label="vx410",fontsize=16,color="green",shape="box"];57[label="primMinusNat (Succ vx600) (Succ vx500)",fontsize=16,color="black",shape="box"];57 -> 69[label="",style="solid", color="black", weight=3];
9.42/3.97	58[label="primMinusNat (Succ vx600) Zero",fontsize=16,color="black",shape="box"];58 -> 70[label="",style="solid", color="black", weight=3];
9.42/3.97	59[label="primMinusNat Zero (Succ vx500)",fontsize=16,color="black",shape="box"];59 -> 71[label="",style="solid", color="black", weight=3];
9.42/3.97	60[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];60 -> 72[label="",style="solid", color="black", weight=3];
9.42/3.97	61[label="primPlusNat (Succ vx600) vx50",fontsize=16,color="burlywood",shape="box"];122[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];61 -> 122[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	122 -> 73[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	123[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];61 -> 123[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	123 -> 74[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	62[label="primPlusNat Zero vx50",fontsize=16,color="burlywood",shape="box"];124[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];62 -> 124[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	124 -> 75[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	125[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 125[label="",style="solid", color="burlywood", weight=9];
9.42/3.97	125 -> 76[label="",style="solid", color="burlywood", weight=3];
9.42/3.97	63[label="vx60",fontsize=16,color="green",shape="box"];64[label="vx50",fontsize=16,color="green",shape="box"];65[label="primMulNat (Succ vx4100) (Succ vx3100)",fontsize=16,color="black",shape="box"];65 -> 77[label="",style="solid", color="black", weight=3];
9.42/3.97	66[label="primMulNat (Succ vx4100) Zero",fontsize=16,color="black",shape="box"];66 -> 78[label="",style="solid", color="black", weight=3];
9.42/3.97	67[label="primMulNat Zero (Succ vx3100)",fontsize=16,color="black",shape="box"];67 -> 79[label="",style="solid", color="black", weight=3];
9.42/3.97	68[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];68 -> 80[label="",style="solid", color="black", weight=3];
9.42/3.97	69 -> 37[label="",style="dashed", color="red", weight=0];
9.42/3.97	69[label="primMinusNat vx600 vx500",fontsize=16,color="magenta"];69 -> 81[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	69 -> 82[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	70[label="Pos (Succ vx600)",fontsize=16,color="green",shape="box"];71[label="Neg (Succ vx500)",fontsize=16,color="green",shape="box"];72[label="Pos Zero",fontsize=16,color="green",shape="box"];73[label="primPlusNat (Succ vx600) (Succ vx500)",fontsize=16,color="black",shape="box"];73 -> 83[label="",style="solid", color="black", weight=3];
9.42/3.97	74[label="primPlusNat (Succ vx600) Zero",fontsize=16,color="black",shape="box"];74 -> 84[label="",style="solid", color="black", weight=3];
9.42/3.97	75[label="primPlusNat Zero (Succ vx500)",fontsize=16,color="black",shape="box"];75 -> 85[label="",style="solid", color="black", weight=3];
9.42/3.97	76[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];76 -> 86[label="",style="solid", color="black", weight=3];
9.42/3.97	77 -> 47[label="",style="dashed", color="red", weight=0];
9.42/3.97	77[label="primPlusNat (primMulNat vx4100 (Succ vx3100)) (Succ vx3100)",fontsize=16,color="magenta"];77 -> 87[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	77 -> 88[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	78[label="Zero",fontsize=16,color="green",shape="box"];79[label="Zero",fontsize=16,color="green",shape="box"];80[label="Zero",fontsize=16,color="green",shape="box"];81[label="vx600",fontsize=16,color="green",shape="box"];82[label="vx500",fontsize=16,color="green",shape="box"];83[label="Succ (Succ (primPlusNat vx600 vx500))",fontsize=16,color="green",shape="box"];83 -> 89[label="",style="dashed", color="green", weight=3];
9.42/3.97	84[label="Succ vx600",fontsize=16,color="green",shape="box"];85[label="Succ vx500",fontsize=16,color="green",shape="box"];86[label="Zero",fontsize=16,color="green",shape="box"];87 -> 41[label="",style="dashed", color="red", weight=0];
9.42/3.97	87[label="primMulNat vx4100 (Succ vx3100)",fontsize=16,color="magenta"];87 -> 90[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	87 -> 91[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	88[label="Succ vx3100",fontsize=16,color="green",shape="box"];89 -> 47[label="",style="dashed", color="red", weight=0];
9.42/3.97	89[label="primPlusNat vx600 vx500",fontsize=16,color="magenta"];89 -> 92[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	89 -> 93[label="",style="dashed", color="magenta", weight=3];
9.42/3.97	90[label="Succ vx3100",fontsize=16,color="green",shape="box"];91[label="vx4100",fontsize=16,color="green",shape="box"];92[label="vx600",fontsize=16,color="green",shape="box"];93[label="vx500",fontsize=16,color="green",shape="box"];}
9.42/3.97	
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(6)
9.42/3.97	Complex Obligation (AND)
9.42/3.97	
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(7)
9.42/3.97	Obligation:
9.42/3.97	Q DP problem:
9.42/3.97	The TRS P consists of the following rules:
9.42/3.97	
9.42/3.97	   new_primMulNat(Succ(vx4100), Succ(vx3100)) -> new_primMulNat(vx4100, Succ(vx3100))
9.42/3.97	
9.42/3.97	R is empty.
9.42/3.97	Q is empty.
9.42/3.97	We have to consider all minimal (P,Q,R)-chains.
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(8) QDPSizeChangeProof (EQUIVALENT)
9.42/3.97	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. 
9.42/3.97	
9.42/3.97	From the DPs we obtained the following set of size-change graphs:
9.42/3.97	*new_primMulNat(Succ(vx4100), Succ(vx3100)) -> new_primMulNat(vx4100, Succ(vx3100))
9.42/3.97	The graph contains the following edges 1 > 1, 2 >= 2
9.42/3.97	
9.42/3.97	
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(9)
9.42/3.97	YES
9.42/3.97	
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(10)
9.42/3.97	Obligation:
9.42/3.97	Q DP problem:
9.42/3.97	The TRS P consists of the following rules:
9.42/3.97	
9.42/3.97	   new_primPlusNat(Succ(vx600), Succ(vx500)) -> new_primPlusNat(vx600, vx500)
9.42/3.97	
9.42/3.97	R is empty.
9.42/3.97	Q is empty.
9.42/3.97	We have to consider all minimal (P,Q,R)-chains.
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(11) QDPSizeChangeProof (EQUIVALENT)
9.42/3.97	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. 
9.42/3.97	
9.42/3.97	From the DPs we obtained the following set of size-change graphs:
9.42/3.97	*new_primPlusNat(Succ(vx600), Succ(vx500)) -> new_primPlusNat(vx600, vx500)
9.42/3.97	The graph contains the following edges 1 > 1, 2 > 2
9.42/3.97	
9.42/3.97	
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(12)
9.42/3.97	YES
9.42/3.97	
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(13)
9.42/3.97	Obligation:
9.42/3.97	Q DP problem:
9.42/3.97	The TRS P consists of the following rules:
9.42/3.97	
9.42/3.97	   new_primMinusNat(Succ(vx600), Succ(vx500)) -> new_primMinusNat(vx600, vx500)
9.42/3.97	
9.42/3.97	R is empty.
9.42/3.97	Q is empty.
9.42/3.97	We have to consider all minimal (P,Q,R)-chains.
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(14) QDPSizeChangeProof (EQUIVALENT)
9.42/3.97	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. 
9.42/3.97	
9.42/3.97	From the DPs we obtained the following set of size-change graphs:
9.42/3.97	*new_primMinusNat(Succ(vx600), Succ(vx500)) -> new_primMinusNat(vx600, vx500)
9.42/3.97	The graph contains the following edges 1 > 1, 2 > 2
9.42/3.97	
9.42/3.97	
9.42/3.97	----------------------------------------
9.42/3.97	
9.42/3.97	(15)
9.42/3.97	YES
9.74/4.01	EOF