8.08/3.53	YES
9.71/3.96	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.71/3.96	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.71/3.96	
9.71/3.96	
9.71/3.96	H-Termination with start terms of the given HASKELL could be proven:
9.71/3.96	
9.71/3.96	(0) HASKELL
9.71/3.96	(1) BR [EQUIVALENT, 0 ms]
9.71/3.96	(2) HASKELL
9.71/3.96	(3) COR [EQUIVALENT, 0 ms]
9.71/3.96	(4) HASKELL
9.71/3.96	(5) Narrow [SOUND, 0 ms]
9.71/3.96	(6) AND
9.71/3.96	    (7) QDP
9.71/3.96	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.71/3.96	        (9) YES
9.71/3.96	    (10) QDP
9.71/3.96	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.71/3.96	        (12) YES
9.71/3.96	
9.71/3.96	
9.71/3.96	----------------------------------------
9.71/3.96	
9.71/3.96	(0)
9.71/3.96	Obligation:
9.71/3.96	mainModule Main 
9.71/3.96	module Main where {
9.71/3.96	import qualified Prelude;
9.71/3.96	}
9.71/3.96	
9.71/3.96	----------------------------------------
9.71/3.96	
9.71/3.96	(1) BR (EQUIVALENT)
9.71/3.96	Replaced joker patterns by fresh variables and removed binding patterns.
9.71/3.96	----------------------------------------
9.71/3.96	
9.71/3.96	(2)
9.71/3.96	Obligation:
9.71/3.96	mainModule Main 
9.71/3.96	module Main where {
9.71/3.96	import qualified Prelude;
9.71/3.96	}
9.71/3.96	
9.71/3.96	----------------------------------------
9.71/3.96	
9.71/3.96	(3) COR (EQUIVALENT)
9.71/3.96	Cond Reductions:
9.71/3.96	The following Function with conditions
9.71/3.96	"undefined |Falseundefined;
9.71/3.96	"
9.71/3.96	is transformed to
9.71/3.96	"undefined  = undefined1;
9.71/3.96	"
9.71/3.96	"undefined0 True = undefined;
9.71/3.96	"
9.71/3.96	"undefined1  = undefined0 False;
9.71/3.96	"
9.71/3.96	
9.71/3.96	----------------------------------------
9.71/3.96	
9.71/3.96	(4)
9.71/3.96	Obligation:
9.71/3.96	mainModule Main 
9.71/3.96	module Main where {
9.71/3.96	import qualified Prelude;
9.71/3.96	}
9.71/3.96	
9.71/3.96	----------------------------------------
9.71/3.96	
9.71/3.96	(5) Narrow (SOUND)
9.71/3.96	Haskell To QDPs
9.71/3.96	
9.71/3.96	digraph dp_graph {
9.71/3.96	node [outthreshold=100, inthreshold=100];1[label="elem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.71/3.96	3[label="elem vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.71/3.96	4[label="elem vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.71/3.96	5[label="any . (==)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.71/3.96	6[label="any ((==) vz3) vz4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.71/3.96	7[label="or . map ((==) vz3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.71/3.96	8[label="or (map ((==) vz3) vz4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.71/3.96	9[label="foldr (||) False (map ((==) vz3) vz4)",fontsize=16,color="burlywood",shape="triangle"];76[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];9 -> 76[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	76 -> 10[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	77[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 77[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	77 -> 11[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	10[label="foldr (||) False (map ((==) vz3) (vz40 : vz41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.71/3.96	11[label="foldr (||) False (map ((==) vz3) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.71/3.96	12[label="foldr (||) False (((==) vz3 vz40) : map ((==) vz3) vz41)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.71/3.96	13[label="foldr (||) False []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.71/3.96	14 -> 16[label="",style="dashed", color="red", weight=0];
9.71/3.96	14[label="(||) (==) vz3 vz40 foldr (||) False (map ((==) vz3) vz41)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
9.71/3.96	15[label="False",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
9.71/3.96	17[label="foldr (||) False (map ((==) vz3) vz41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
9.71/3.96	16[label="(||) (==) vz3 vz40 vz5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
9.71/3.96	18[label="vz41",fontsize=16,color="green",shape="box"];19[label="(||) primEqInt vz3 vz40 vz5",fontsize=16,color="burlywood",shape="box"];78[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];19 -> 78[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	78 -> 20[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	79[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];19 -> 79[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	79 -> 21[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	20[label="(||) primEqInt (Pos vz30) vz40 vz5",fontsize=16,color="burlywood",shape="box"];80[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];20 -> 80[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	80 -> 22[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	81[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 81[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	81 -> 23[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	21[label="(||) primEqInt (Neg vz30) vz40 vz5",fontsize=16,color="burlywood",shape="box"];82[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];21 -> 82[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	82 -> 24[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	83[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 83[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	83 -> 25[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	22[label="(||) primEqInt (Pos (Succ vz300)) vz40 vz5",fontsize=16,color="burlywood",shape="box"];84[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];22 -> 84[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	84 -> 26[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	85[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];22 -> 85[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	85 -> 27[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	23[label="(||) primEqInt (Pos Zero) vz40 vz5",fontsize=16,color="burlywood",shape="box"];86[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];23 -> 86[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	86 -> 28[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	87[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];23 -> 87[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	87 -> 29[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	24[label="(||) primEqInt (Neg (Succ vz300)) vz40 vz5",fontsize=16,color="burlywood",shape="box"];88[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];24 -> 88[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	88 -> 30[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	89[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];24 -> 89[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	89 -> 31[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	25[label="(||) primEqInt (Neg Zero) vz40 vz5",fontsize=16,color="burlywood",shape="box"];90[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];25 -> 90[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	90 -> 32[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	91[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];25 -> 91[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	91 -> 33[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	26[label="(||) primEqInt (Pos (Succ vz300)) (Pos vz400) vz5",fontsize=16,color="burlywood",shape="box"];92[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 92[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	92 -> 34[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	93[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];26 -> 93[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	93 -> 35[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	27[label="(||) primEqInt (Pos (Succ vz300)) (Neg vz400) vz5",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
9.71/3.96	28[label="(||) primEqInt (Pos Zero) (Pos vz400) vz5",fontsize=16,color="burlywood",shape="box"];94[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 94[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	94 -> 37[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	95[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 95[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	95 -> 38[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	29[label="(||) primEqInt (Pos Zero) (Neg vz400) vz5",fontsize=16,color="burlywood",shape="box"];96[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 96[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	96 -> 39[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	97[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 97[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	97 -> 40[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	30[label="(||) primEqInt (Neg (Succ vz300)) (Pos vz400) vz5",fontsize=16,color="black",shape="box"];30 -> 41[label="",style="solid", color="black", weight=3];
9.71/3.96	31[label="(||) primEqInt (Neg (Succ vz300)) (Neg vz400) vz5",fontsize=16,color="burlywood",shape="box"];98[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];31 -> 98[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	98 -> 42[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	99[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];31 -> 99[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	99 -> 43[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	32[label="(||) primEqInt (Neg Zero) (Pos vz400) vz5",fontsize=16,color="burlywood",shape="box"];100[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];32 -> 100[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	100 -> 44[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	101[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 101[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	101 -> 45[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	33[label="(||) primEqInt (Neg Zero) (Neg vz400) vz5",fontsize=16,color="burlywood",shape="box"];102[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];33 -> 102[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	102 -> 46[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	103[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 103[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	103 -> 47[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	34[label="(||) primEqInt (Pos (Succ vz300)) (Pos (Succ vz4000)) vz5",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
9.71/3.96	35[label="(||) primEqInt (Pos (Succ vz300)) (Pos Zero) vz5",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
9.71/3.96	36[label="(||) False vz5",fontsize=16,color="black",shape="triangle"];36 -> 50[label="",style="solid", color="black", weight=3];
9.71/3.96	37[label="(||) primEqInt (Pos Zero) (Pos (Succ vz4000)) vz5",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3];
9.71/3.96	38[label="(||) primEqInt (Pos Zero) (Pos Zero) vz5",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
9.71/3.96	39[label="(||) primEqInt (Pos Zero) (Neg (Succ vz4000)) vz5",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3];
9.71/3.96	40[label="(||) primEqInt (Pos Zero) (Neg Zero) vz5",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3];
9.71/3.96	41 -> 36[label="",style="dashed", color="red", weight=0];
9.71/3.96	41[label="(||) False vz5",fontsize=16,color="magenta"];42[label="(||) primEqInt (Neg (Succ vz300)) (Neg (Succ vz4000)) vz5",fontsize=16,color="black",shape="box"];42 -> 55[label="",style="solid", color="black", weight=3];
9.71/3.96	43[label="(||) primEqInt (Neg (Succ vz300)) (Neg Zero) vz5",fontsize=16,color="black",shape="box"];43 -> 56[label="",style="solid", color="black", weight=3];
9.71/3.96	44[label="(||) primEqInt (Neg Zero) (Pos (Succ vz4000)) vz5",fontsize=16,color="black",shape="box"];44 -> 57[label="",style="solid", color="black", weight=3];
9.71/3.96	45[label="(||) primEqInt (Neg Zero) (Pos Zero) vz5",fontsize=16,color="black",shape="box"];45 -> 58[label="",style="solid", color="black", weight=3];
9.71/3.96	46[label="(||) primEqInt (Neg Zero) (Neg (Succ vz4000)) vz5",fontsize=16,color="black",shape="box"];46 -> 59[label="",style="solid", color="black", weight=3];
9.71/3.96	47[label="(||) primEqInt (Neg Zero) (Neg Zero) vz5",fontsize=16,color="black",shape="box"];47 -> 60[label="",style="solid", color="black", weight=3];
9.71/3.96	48[label="(||) primEqNat vz300 vz4000 vz5",fontsize=16,color="burlywood",shape="triangle"];104[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];48 -> 104[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	104 -> 61[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	105[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 105[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	105 -> 62[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	49 -> 36[label="",style="dashed", color="red", weight=0];
9.71/3.96	49[label="(||) False vz5",fontsize=16,color="magenta"];50[label="vz5",fontsize=16,color="green",shape="box"];51 -> 36[label="",style="dashed", color="red", weight=0];
9.71/3.96	51[label="(||) False vz5",fontsize=16,color="magenta"];52[label="(||) True vz5",fontsize=16,color="black",shape="triangle"];52 -> 63[label="",style="solid", color="black", weight=3];
9.71/3.96	53 -> 36[label="",style="dashed", color="red", weight=0];
9.71/3.96	53[label="(||) False vz5",fontsize=16,color="magenta"];54 -> 52[label="",style="dashed", color="red", weight=0];
9.71/3.96	54[label="(||) True vz5",fontsize=16,color="magenta"];55 -> 48[label="",style="dashed", color="red", weight=0];
9.71/3.96	55[label="(||) primEqNat vz300 vz4000 vz5",fontsize=16,color="magenta"];55 -> 64[label="",style="dashed", color="magenta", weight=3];
9.71/3.96	55 -> 65[label="",style="dashed", color="magenta", weight=3];
9.71/3.96	56 -> 36[label="",style="dashed", color="red", weight=0];
9.71/3.96	56[label="(||) False vz5",fontsize=16,color="magenta"];57 -> 36[label="",style="dashed", color="red", weight=0];
9.71/3.96	57[label="(||) False vz5",fontsize=16,color="magenta"];58 -> 52[label="",style="dashed", color="red", weight=0];
9.71/3.96	58[label="(||) True vz5",fontsize=16,color="magenta"];59 -> 36[label="",style="dashed", color="red", weight=0];
9.71/3.96	59[label="(||) False vz5",fontsize=16,color="magenta"];60 -> 52[label="",style="dashed", color="red", weight=0];
9.71/3.96	60[label="(||) True vz5",fontsize=16,color="magenta"];61[label="(||) primEqNat (Succ vz3000) vz4000 vz5",fontsize=16,color="burlywood",shape="box"];106[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];61 -> 106[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	106 -> 66[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	107[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];61 -> 107[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	107 -> 67[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	62[label="(||) primEqNat Zero vz4000 vz5",fontsize=16,color="burlywood",shape="box"];108[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];62 -> 108[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	108 -> 68[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	109[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 109[label="",style="solid", color="burlywood", weight=9];
9.71/3.96	109 -> 69[label="",style="solid", color="burlywood", weight=3];
9.71/3.96	63[label="True",fontsize=16,color="green",shape="box"];64[label="vz4000",fontsize=16,color="green",shape="box"];65[label="vz300",fontsize=16,color="green",shape="box"];66[label="(||) primEqNat (Succ vz3000) (Succ vz40000) vz5",fontsize=16,color="black",shape="box"];66 -> 70[label="",style="solid", color="black", weight=3];
9.71/3.96	67[label="(||) primEqNat (Succ vz3000) Zero vz5",fontsize=16,color="black",shape="box"];67 -> 71[label="",style="solid", color="black", weight=3];
9.71/3.96	68[label="(||) primEqNat Zero (Succ vz40000) vz5",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3];
9.71/3.96	69[label="(||) primEqNat Zero Zero vz5",fontsize=16,color="black",shape="box"];69 -> 73[label="",style="solid", color="black", weight=3];
9.71/3.96	70 -> 48[label="",style="dashed", color="red", weight=0];
9.71/3.96	70[label="(||) primEqNat vz3000 vz40000 vz5",fontsize=16,color="magenta"];70 -> 74[label="",style="dashed", color="magenta", weight=3];
9.71/3.96	70 -> 75[label="",style="dashed", color="magenta", weight=3];
9.71/3.96	71 -> 36[label="",style="dashed", color="red", weight=0];
9.71/3.96	71[label="(||) False vz5",fontsize=16,color="magenta"];72 -> 36[label="",style="dashed", color="red", weight=0];
9.71/3.96	72[label="(||) False vz5",fontsize=16,color="magenta"];73 -> 52[label="",style="dashed", color="red", weight=0];
9.71/3.96	73[label="(||) True vz5",fontsize=16,color="magenta"];74[label="vz40000",fontsize=16,color="green",shape="box"];75[label="vz3000",fontsize=16,color="green",shape="box"];}
9.71/3.96	
9.71/3.96	----------------------------------------
9.71/3.96	
9.71/3.96	(6)
9.71/3.96	Complex Obligation (AND)
9.71/3.96	
9.71/3.96	----------------------------------------
9.71/3.96	
9.71/3.96	(7)
9.71/3.96	Obligation:
9.71/3.96	Q DP problem:
9.71/3.96	The TRS P consists of the following rules:
9.71/3.96	
9.71/3.96	   new_foldr(vz3, :(vz40, vz41)) -> new_foldr(vz3, vz41)
9.71/3.96	
9.71/3.96	R is empty.
9.71/3.96	Q is empty.
9.71/3.96	We have to consider all minimal (P,Q,R)-chains.
9.71/3.96	----------------------------------------
9.71/3.96	
9.71/3.96	(8) QDPSizeChangeProof (EQUIVALENT)
9.71/3.96	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. 
9.71/3.96	
9.71/3.96	From the DPs we obtained the following set of size-change graphs:
9.71/3.96	*new_foldr(vz3, :(vz40, vz41)) -> new_foldr(vz3, vz41)
9.71/3.96	The graph contains the following edges 1 >= 1, 2 > 2
9.71/3.96	
9.71/3.96	
9.71/3.96	----------------------------------------
9.71/3.96	
9.71/3.96	(9)
9.71/3.96	YES
9.71/3.96	
9.71/3.96	----------------------------------------
9.71/3.96	
9.71/3.96	(10)
9.71/3.96	Obligation:
9.71/3.96	Q DP problem:
9.71/3.96	The TRS P consists of the following rules:
9.71/3.96	
9.71/3.96	   new_pePe(Succ(vz3000), Succ(vz40000), vz5) -> new_pePe(vz3000, vz40000, vz5)
9.71/3.96	
9.71/3.96	R is empty.
9.71/3.96	Q is empty.
9.71/3.96	We have to consider all minimal (P,Q,R)-chains.
9.71/3.96	----------------------------------------
9.71/3.96	
9.71/3.96	(11) QDPSizeChangeProof (EQUIVALENT)
9.71/3.96	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. 
9.71/3.96	
9.71/3.96	From the DPs we obtained the following set of size-change graphs:
9.71/3.96	*new_pePe(Succ(vz3000), Succ(vz40000), vz5) -> new_pePe(vz3000, vz40000, vz5)
9.71/3.96	The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
9.71/3.96	
9.71/3.96	
9.71/3.96	----------------------------------------
9.71/3.96	
9.71/3.96	(12)
9.71/3.96	YES
9.71/3.99	EOF