7.93/3.54	YES
9.65/3.99	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.65/3.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.65/3.99	
9.65/3.99	
9.65/3.99	H-Termination with start terms of the given HASKELL could be proven:
9.65/3.99	
9.65/3.99	(0) HASKELL
9.65/3.99	(1) BR [EQUIVALENT, 0 ms]
9.65/3.99	(2) HASKELL
9.65/3.99	(3) COR [EQUIVALENT, 0 ms]
9.65/3.99	(4) HASKELL
9.65/3.99	(5) Narrow [SOUND, 0 ms]
9.65/3.99	(6) AND
9.65/3.99	    (7) QDP
9.65/3.99	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.65/3.99	        (9) YES
9.65/3.99	    (10) QDP
9.65/3.99	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.65/3.99	        (12) YES
9.65/3.99	    (13) QDP
9.65/3.99	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.65/3.99	        (15) YES
9.65/3.99	
9.65/3.99	
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(0)
9.65/3.99	Obligation:
9.65/3.99	mainModule Main 
9.65/3.99	module Main where {
9.65/3.99	import qualified Prelude;
9.65/3.99	}
9.65/3.99	
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(1) BR (EQUIVALENT)
9.65/3.99	Replaced joker patterns by fresh variables and removed binding patterns.
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(2)
9.65/3.99	Obligation:
9.65/3.99	mainModule Main 
9.65/3.99	module Main where {
9.65/3.99	import qualified Prelude;
9.65/3.99	}
9.65/3.99	
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(3) COR (EQUIVALENT)
9.65/3.99	Cond Reductions:
9.65/3.99	The following Function with conditions
9.65/3.99	"undefined |Falseundefined;
9.65/3.99	"
9.65/3.99	is transformed to
9.65/3.99	"undefined  = undefined1;
9.65/3.99	"
9.65/3.99	"undefined0 True = undefined;
9.65/3.99	"
9.65/3.99	"undefined1  = undefined0 False;
9.65/3.99	"
9.65/3.99	
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(4)
9.65/3.99	Obligation:
9.65/3.99	mainModule Main 
9.65/3.99	module Main where {
9.65/3.99	import qualified Prelude;
9.65/3.99	}
9.65/3.99	
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(5) Narrow (SOUND)
9.65/3.99	Haskell To QDPs
9.65/3.99	
9.65/3.99	digraph dp_graph {
9.65/3.99	node [outthreshold=100, inthreshold=100];1[label="(+)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.65/3.99	3[label="(+) vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.65/3.99	4[label="(+) vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.65/3.99	5[label="primPlusFloat 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];
9.65/3.99	92 -> 6[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	6[label="primPlusFloat (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];
9.65/3.99	93 -> 7[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	7[label="primPlusFloat (Float vx30 vx31) (Float vx40 vx41)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.65/3.99	8[label="Float (vx30 * vx41 + vx40 * vx31) (vx31 * vx41)",fontsize=16,color="green",shape="box"];8 -> 9[label="",style="dashed", color="green", weight=3];
9.65/3.99	8 -> 10[label="",style="dashed", color="green", weight=3];
9.65/3.99	9[label="vx30 * vx41 + vx40 * vx31",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.65/3.99	10[label="vx31 * vx41",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3];
9.65/3.99	11 -> 13[label="",style="dashed", color="red", weight=0];
9.65/3.99	11[label="primPlusInt (vx30 * vx41) (vx40 * vx31)",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	11 -> 15[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	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];
9.65/3.99	94 -> 16[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	95[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];12 -> 95[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	95 -> 17[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	14 -> 10[label="",style="dashed", color="red", weight=0];
9.65/3.99	14[label="vx30 * vx41",fontsize=16,color="magenta"];14 -> 18[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	15 -> 10[label="",style="dashed", color="red", weight=0];
9.65/3.99	15[label="vx40 * vx31",fontsize=16,color="magenta"];15 -> 19[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	15 -> 20[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	13[label="primPlusInt 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];
9.65/3.99	96 -> 21[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	97[label="vx6/Neg vx60",fontsize=10,color="white",style="solid",shape="box"];13 -> 97[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	97 -> 22[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	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];
9.65/3.99	98 -> 23[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	99[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];16 -> 99[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	99 -> 24[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	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];
9.65/3.99	100 -> 25[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	101[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];17 -> 101[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	101 -> 26[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	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="primPlusInt (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];
9.65/3.99	102 -> 27[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	103[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];21 -> 103[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	103 -> 28[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	22[label="primPlusInt (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];
9.65/3.99	104 -> 29[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	105[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];22 -> 105[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	105 -> 30[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	23[label="primMulInt (Pos vx310) (Pos vx410)",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
9.65/3.99	24[label="primMulInt (Pos vx310) (Neg vx410)",fontsize=16,color="black",shape="box"];24 -> 32[label="",style="solid", color="black", weight=3];
9.65/3.99	25[label="primMulInt (Neg vx310) (Pos vx410)",fontsize=16,color="black",shape="box"];25 -> 33[label="",style="solid", color="black", weight=3];
9.65/3.99	26[label="primMulInt (Neg vx310) (Neg vx410)",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
9.65/3.99	27[label="primPlusInt (Pos vx60) (Pos vx50)",fontsize=16,color="black",shape="box"];27 -> 35[label="",style="solid", color="black", weight=3];
9.65/3.99	28[label="primPlusInt (Pos vx60) (Neg vx50)",fontsize=16,color="black",shape="box"];28 -> 36[label="",style="solid", color="black", weight=3];
9.65/3.99	29[label="primPlusInt (Neg vx60) (Pos vx50)",fontsize=16,color="black",shape="box"];29 -> 37[label="",style="solid", color="black", weight=3];
9.65/3.99	30[label="primPlusInt (Neg vx60) (Neg vx50)",fontsize=16,color="black",shape="box"];30 -> 38[label="",style="solid", color="black", weight=3];
9.65/3.99	31[label="Pos (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];31 -> 39[label="",style="dashed", color="green", weight=3];
9.65/3.99	32[label="Neg (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];32 -> 40[label="",style="dashed", color="green", weight=3];
9.65/3.99	33[label="Neg (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];33 -> 41[label="",style="dashed", color="green", weight=3];
9.65/3.99	34[label="Pos (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];34 -> 42[label="",style="dashed", color="green", weight=3];
9.65/3.99	35[label="Pos (primPlusNat vx60 vx50)",fontsize=16,color="green",shape="box"];35 -> 43[label="",style="dashed", color="green", weight=3];
9.65/3.99	36[label="primMinusNat vx60 vx50",fontsize=16,color="burlywood",shape="triangle"];106[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];36 -> 106[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	106 -> 44[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	107[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 107[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	107 -> 45[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	37 -> 36[label="",style="dashed", color="red", weight=0];
9.65/3.99	37[label="primMinusNat vx50 vx60",fontsize=16,color="magenta"];37 -> 46[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	37 -> 47[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	38[label="Neg (primPlusNat vx60 vx50)",fontsize=16,color="green",shape="box"];38 -> 48[label="",style="dashed", color="green", weight=3];
9.65/3.99	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];
9.65/3.99	108 -> 49[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	109[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];39 -> 109[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	109 -> 50[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	40 -> 39[label="",style="dashed", color="red", weight=0];
9.65/3.99	40[label="primMulNat vx310 vx410",fontsize=16,color="magenta"];40 -> 51[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	41 -> 39[label="",style="dashed", color="red", weight=0];
9.65/3.99	41[label="primMulNat vx310 vx410",fontsize=16,color="magenta"];41 -> 52[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	42 -> 39[label="",style="dashed", color="red", weight=0];
9.65/3.99	42[label="primMulNat vx310 vx410",fontsize=16,color="magenta"];42 -> 53[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	42 -> 54[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	43[label="primPlusNat vx60 vx50",fontsize=16,color="burlywood",shape="triangle"];110[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];43 -> 110[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	110 -> 55[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	111[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];43 -> 111[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	111 -> 56[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	44[label="primMinusNat (Succ vx600) 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];
9.65/3.99	112 -> 57[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	113[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];44 -> 113[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	113 -> 58[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	45[label="primMinusNat Zero vx50",fontsize=16,color="burlywood",shape="box"];114[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];45 -> 114[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	114 -> 59[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	115[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 115[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	115 -> 60[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	46[label="vx60",fontsize=16,color="green",shape="box"];47[label="vx50",fontsize=16,color="green",shape="box"];48 -> 43[label="",style="dashed", color="red", weight=0];
9.65/3.99	48[label="primPlusNat vx60 vx50",fontsize=16,color="magenta"];48 -> 61[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	48 -> 62[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	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];
9.65/3.99	116 -> 63[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	117[label="vx410/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 117[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	117 -> 64[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	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];
9.65/3.99	118 -> 65[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	119[label="vx410/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 119[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	119 -> 66[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	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="primPlusNat (Succ vx600) vx50",fontsize=16,color="burlywood",shape="box"];120[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];55 -> 120[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	120 -> 67[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	121[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];55 -> 121[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	121 -> 68[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	56[label="primPlusNat Zero vx50",fontsize=16,color="burlywood",shape="box"];122[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];56 -> 122[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	122 -> 69[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	123[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];56 -> 123[label="",style="solid", color="burlywood", weight=9];
9.65/3.99	123 -> 70[label="",style="solid", color="burlywood", weight=3];
9.65/3.99	57[label="primMinusNat (Succ vx600) (Succ vx500)",fontsize=16,color="black",shape="box"];57 -> 71[label="",style="solid", color="black", weight=3];
9.65/3.99	58[label="primMinusNat (Succ vx600) Zero",fontsize=16,color="black",shape="box"];58 -> 72[label="",style="solid", color="black", weight=3];
9.65/3.99	59[label="primMinusNat Zero (Succ vx500)",fontsize=16,color="black",shape="box"];59 -> 73[label="",style="solid", color="black", weight=3];
9.65/3.99	60[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];60 -> 74[label="",style="solid", color="black", weight=3];
9.65/3.99	61[label="vx60",fontsize=16,color="green",shape="box"];62[label="vx50",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];
9.65/3.99	64[label="primMulNat (Succ vx3100) Zero",fontsize=16,color="black",shape="box"];64 -> 76[label="",style="solid", color="black", weight=3];
9.65/3.99	65[label="primMulNat Zero (Succ vx4100)",fontsize=16,color="black",shape="box"];65 -> 77[label="",style="solid", color="black", weight=3];
9.65/3.99	66[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];66 -> 78[label="",style="solid", color="black", weight=3];
9.65/3.99	67[label="primPlusNat (Succ vx600) (Succ vx500)",fontsize=16,color="black",shape="box"];67 -> 79[label="",style="solid", color="black", weight=3];
9.65/3.99	68[label="primPlusNat (Succ vx600) Zero",fontsize=16,color="black",shape="box"];68 -> 80[label="",style="solid", color="black", weight=3];
9.65/3.99	69[label="primPlusNat Zero (Succ vx500)",fontsize=16,color="black",shape="box"];69 -> 81[label="",style="solid", color="black", weight=3];
9.65/3.99	70[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];70 -> 82[label="",style="solid", color="black", weight=3];
9.65/3.99	71 -> 36[label="",style="dashed", color="red", weight=0];
9.65/3.99	71[label="primMinusNat vx600 vx500",fontsize=16,color="magenta"];71 -> 83[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	71 -> 84[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	72[label="Pos (Succ vx600)",fontsize=16,color="green",shape="box"];73[label="Neg (Succ vx500)",fontsize=16,color="green",shape="box"];74[label="Pos Zero",fontsize=16,color="green",shape="box"];75 -> 43[label="",style="dashed", color="red", weight=0];
9.65/3.99	75[label="primPlusNat (primMulNat vx3100 (Succ vx4100)) (Succ vx4100)",fontsize=16,color="magenta"];75 -> 85[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	75 -> 86[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	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="Succ (Succ (primPlusNat vx600 vx500))",fontsize=16,color="green",shape="box"];79 -> 87[label="",style="dashed", color="green", weight=3];
9.65/3.99	80[label="Succ vx600",fontsize=16,color="green",shape="box"];81[label="Succ vx500",fontsize=16,color="green",shape="box"];82[label="Zero",fontsize=16,color="green",shape="box"];83[label="vx500",fontsize=16,color="green",shape="box"];84[label="vx600",fontsize=16,color="green",shape="box"];85 -> 39[label="",style="dashed", color="red", weight=0];
9.65/3.99	85[label="primMulNat vx3100 (Succ vx4100)",fontsize=16,color="magenta"];85 -> 88[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	85 -> 89[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	86[label="Succ vx4100",fontsize=16,color="green",shape="box"];87 -> 43[label="",style="dashed", color="red", weight=0];
9.65/3.99	87[label="primPlusNat vx600 vx500",fontsize=16,color="magenta"];87 -> 90[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	87 -> 91[label="",style="dashed", color="magenta", weight=3];
9.65/3.99	88[label="vx3100",fontsize=16,color="green",shape="box"];89[label="Succ vx4100",fontsize=16,color="green",shape="box"];90[label="vx600",fontsize=16,color="green",shape="box"];91[label="vx500",fontsize=16,color="green",shape="box"];}
9.65/3.99	
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(6)
9.65/3.99	Complex Obligation (AND)
9.65/3.99	
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(7)
9.65/3.99	Obligation:
9.65/3.99	Q DP problem:
9.65/3.99	The TRS P consists of the following rules:
9.65/3.99	
9.65/3.99	   new_primMulNat(Succ(vx3100), Succ(vx4100)) -> new_primMulNat(vx3100, Succ(vx4100))
9.65/3.99	
9.65/3.99	R is empty.
9.65/3.99	Q is empty.
9.65/3.99	We have to consider all minimal (P,Q,R)-chains.
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(8) QDPSizeChangeProof (EQUIVALENT)
9.65/3.99	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. 
9.65/3.99	
9.65/3.99	From the DPs we obtained the following set of size-change graphs:
9.65/3.99	*new_primMulNat(Succ(vx3100), Succ(vx4100)) -> new_primMulNat(vx3100, Succ(vx4100))
9.65/3.99	The graph contains the following edges 1 > 1, 2 >= 2
9.65/3.99	
9.65/3.99	
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(9)
9.65/3.99	YES
9.65/3.99	
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(10)
9.65/3.99	Obligation:
9.65/3.99	Q DP problem:
9.65/3.99	The TRS P consists of the following rules:
9.65/3.99	
9.65/3.99	   new_primMinusNat(Succ(vx600), Succ(vx500)) -> new_primMinusNat(vx600, vx500)
9.65/3.99	
9.65/3.99	R is empty.
9.65/3.99	Q is empty.
9.65/3.99	We have to consider all minimal (P,Q,R)-chains.
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(11) QDPSizeChangeProof (EQUIVALENT)
9.65/3.99	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. 
9.65/3.99	
9.65/3.99	From the DPs we obtained the following set of size-change graphs:
9.65/3.99	*new_primMinusNat(Succ(vx600), Succ(vx500)) -> new_primMinusNat(vx600, vx500)
9.65/3.99	The graph contains the following edges 1 > 1, 2 > 2
9.65/3.99	
9.65/3.99	
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(12)
9.65/3.99	YES
9.65/3.99	
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(13)
9.65/3.99	Obligation:
9.65/3.99	Q DP problem:
9.65/3.99	The TRS P consists of the following rules:
9.65/3.99	
9.65/3.99	   new_primPlusNat(Succ(vx600), Succ(vx500)) -> new_primPlusNat(vx600, vx500)
9.65/3.99	
9.65/3.99	R is empty.
9.65/3.99	Q is empty.
9.65/3.99	We have to consider all minimal (P,Q,R)-chains.
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(14) QDPSizeChangeProof (EQUIVALENT)
9.65/3.99	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. 
9.65/3.99	
9.65/3.99	From the DPs we obtained the following set of size-change graphs:
9.65/3.99	*new_primPlusNat(Succ(vx600), Succ(vx500)) -> new_primPlusNat(vx600, vx500)
9.65/3.99	The graph contains the following edges 1 > 1, 2 > 2
9.65/3.99	
9.65/3.99	
9.65/3.99	----------------------------------------
9.65/3.99	
9.65/3.99	(15)
9.65/3.99	YES
9.69/4.06	EOF