8.07/3.60	YES
10.32/4.20	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.32/4.20	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.32/4.20	
10.32/4.20	
10.32/4.20	H-Termination with start terms of the given HASKELL could be proven:
10.32/4.20	
10.32/4.20	(0) HASKELL
10.32/4.20	(1) LR [EQUIVALENT, 0 ms]
10.32/4.20	(2) HASKELL
10.32/4.20	(3) BR [EQUIVALENT, 0 ms]
10.32/4.20	(4) HASKELL
10.32/4.20	(5) COR [EQUIVALENT, 0 ms]
10.32/4.20	(6) HASKELL
10.32/4.20	(7) LetRed [EQUIVALENT, 0 ms]
10.32/4.20	(8) HASKELL
10.32/4.20	(9) NumRed [SOUND, 4 ms]
10.32/4.20	(10) HASKELL
10.32/4.20	(11) Narrow [SOUND, 0 ms]
10.32/4.20	(12) QDP
10.32/4.20	(13) TransformationProof [EQUIVALENT, 0 ms]
10.32/4.20	(14) QDP
10.32/4.20	(15) DependencyGraphProof [EQUIVALENT, 0 ms]
10.32/4.20	(16) QDP
10.32/4.20	(17) TransformationProof [EQUIVALENT, 0 ms]
10.32/4.20	(18) QDP
10.32/4.20	(19) DependencyGraphProof [EQUIVALENT, 0 ms]
10.32/4.20	(20) QDP
10.32/4.20	(21) UsableRulesProof [EQUIVALENT, 0 ms]
10.32/4.20	(22) QDP
10.32/4.20	(23) QReductionProof [EQUIVALENT, 0 ms]
10.32/4.20	(24) QDP
10.32/4.20	(25) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.32/4.20	(26) YES
10.32/4.20	
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(0)
10.32/4.20	Obligation:
10.32/4.20	mainModule Main 
10.32/4.20	module Main where {
10.32/4.20	import qualified Prelude;
10.32/4.20	}
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(1) LR (EQUIVALENT)
10.32/4.20	Lambda Reductions:
10.32/4.20	The following Lambda expression
10.32/4.20	"\(_,xs'')->xs''"
10.32/4.20	is transformed to
10.32/4.20	"xs''0 (_,xs'') = xs'';
10.32/4.20	"
10.32/4.20	The following Lambda expression
10.32/4.20	"\(xs',_)->xs'"
10.32/4.20	is transformed to
10.32/4.20	"xs'0 (xs',_) = xs';
10.32/4.20	"
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(2)
10.32/4.20	Obligation:
10.32/4.20	mainModule Main 
10.32/4.20	module Main where {
10.32/4.20	import qualified Prelude;
10.32/4.20	}
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(3) BR (EQUIVALENT)
10.32/4.20	Replaced joker patterns by fresh variables and removed binding patterns.
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(4)
10.32/4.20	Obligation:
10.32/4.20	mainModule Main 
10.32/4.20	module Main where {
10.32/4.20	import qualified Prelude;
10.32/4.20	}
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(5) COR (EQUIVALENT)
10.32/4.20	Cond Reductions:
10.32/4.20	The following Function with conditions
10.32/4.20	"splitAt n xs|n <= 0([],xs);
10.32/4.20	splitAt vw [] = ([],[]);
10.32/4.20	splitAt n (x : xs) = (x : xs',xs'') where {
10.32/4.20	vu42  = splitAt (n - 1) xs;
10.32/4.20	;
10.32/4.20	xs'  = xs'0 vu42;
10.32/4.20	;
10.32/4.20	xs''  = xs''0 vu42;
10.32/4.20	;
10.32/4.20	xs''0 (vx,xs'') = xs'';
10.32/4.20	;
10.32/4.20	xs'0 (xs',vy) = xs';
10.32/4.20	} 
10.32/4.20	;
10.32/4.20	"
10.32/4.20	is transformed to
10.32/4.20	"splitAt n xs = splitAt3 n xs;
10.32/4.20	splitAt vw [] = splitAt1 vw [];
10.32/4.20	splitAt n (x : xs) = splitAt0 n (x : xs);
10.32/4.20	"
10.32/4.20	"splitAt0 n (x : xs) = (x : xs',xs'') where {
10.32/4.20	vu42  = splitAt (n - 1) xs;
10.32/4.20	;
10.32/4.20	xs'  = xs'0 vu42;
10.32/4.20	;
10.32/4.20	xs''  = xs''0 vu42;
10.32/4.20	;
10.32/4.20	xs''0 (vx,xs'') = xs'';
10.32/4.20	;
10.32/4.20	xs'0 (xs',vy) = xs';
10.32/4.20	} 
10.32/4.20	;
10.32/4.20	"
10.32/4.20	"splitAt1 vw [] = ([],[]);
10.32/4.20	splitAt1 ww wx = splitAt0 ww wx;
10.32/4.20	"
10.32/4.20	"splitAt2 n xs True = ([],xs);
10.32/4.20	splitAt2 n xs False = splitAt1 n xs;
10.32/4.20	"
10.32/4.20	"splitAt3 n xs = splitAt2 n xs (n <= 0);
10.32/4.20	splitAt3 wy wz = splitAt1 wy wz;
10.32/4.20	"
10.32/4.20	The following Function with conditions
10.32/4.20	"undefined |Falseundefined;
10.32/4.20	"
10.32/4.20	is transformed to
10.32/4.20	"undefined  = undefined1;
10.32/4.20	"
10.32/4.20	"undefined0 True = undefined;
10.32/4.20	"
10.32/4.20	"undefined1  = undefined0 False;
10.32/4.20	"
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(6)
10.32/4.20	Obligation:
10.32/4.20	mainModule Main 
10.32/4.20	module Main where {
10.32/4.20	import qualified Prelude;
10.32/4.20	}
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(7) LetRed (EQUIVALENT)
10.32/4.20	Let/Where Reductions:
10.32/4.20	The bindings of the following Let/Where expression
10.32/4.20	"(x : xs',xs'') where {
10.32/4.20	vu42  = splitAt (n - 1) xs;
10.32/4.20	;
10.32/4.20	xs'  = xs'0 vu42;
10.32/4.20	;
10.32/4.20	xs''  = xs''0 vu42;
10.32/4.20	;
10.32/4.20	xs''0 (vx,xs'') = xs'';
10.32/4.20	;
10.32/4.20	xs'0 (xs',vy) = xs';
10.32/4.20	} 
10.32/4.20	"
10.32/4.20	are unpacked to the following functions on top level
10.32/4.20	"splitAt0Xs''0 xu xv (vx,xs'') = xs'';
10.32/4.20	"
10.32/4.20	"splitAt0Xs'0 xu xv (xs',vy) = xs';
10.32/4.20	"
10.32/4.20	"splitAt0Vu42 xu xv = splitAt (xu - 1) xv;
10.32/4.20	"
10.32/4.20	"splitAt0Xs' xu xv = splitAt0Xs'0 xu xv (splitAt0Vu42 xu xv);
10.32/4.20	"
10.32/4.20	"splitAt0Xs'' xu xv = splitAt0Xs''0 xu xv (splitAt0Vu42 xu xv);
10.32/4.20	"
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(8)
10.32/4.20	Obligation:
10.32/4.20	mainModule Main 
10.32/4.20	module Main where {
10.32/4.20	import qualified Prelude;
10.32/4.20	}
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(9) NumRed (SOUND)
10.32/4.20	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(10)
10.32/4.20	Obligation:
10.32/4.20	mainModule Main 
10.32/4.20	module Main where {
10.32/4.20	import qualified Prelude;
10.32/4.20	}
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(11) Narrow (SOUND)
10.32/4.20	Haskell To QDPs
10.32/4.20	
10.32/4.20	digraph dp_graph {
10.32/4.20	node [outthreshold=100, inthreshold=100];1[label="splitAt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.32/4.20	3[label="splitAt xw3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.32/4.20	4[label="splitAt xw3 xw4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.32/4.20	5[label="splitAt3 xw3 xw4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.32/4.20	6[label="splitAt2 xw3 xw4 (xw3 <= Pos Zero)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.32/4.20	7[label="splitAt2 xw3 xw4 (compare xw3 (Pos Zero) /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
10.32/4.20	8[label="splitAt2 xw3 xw4 (not (compare xw3 (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
10.32/4.20	9[label="splitAt2 xw3 xw4 (not (primCmpInt xw3 (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];64[label="xw3/Pos xw30",fontsize=10,color="white",style="solid",shape="box"];9 -> 64[label="",style="solid", color="burlywood", weight=9];
10.32/4.20	64 -> 10[label="",style="solid", color="burlywood", weight=3];
10.32/4.20	65[label="xw3/Neg xw30",fontsize=10,color="white",style="solid",shape="box"];9 -> 65[label="",style="solid", color="burlywood", weight=9];
10.32/4.20	65 -> 11[label="",style="solid", color="burlywood", weight=3];
10.32/4.20	10[label="splitAt2 (Pos xw30) xw4 (not (primCmpInt (Pos xw30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];66[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];10 -> 66[label="",style="solid", color="burlywood", weight=9];
10.32/4.20	66 -> 12[label="",style="solid", color="burlywood", weight=3];
10.32/4.20	67[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 67[label="",style="solid", color="burlywood", weight=9];
10.32/4.20	67 -> 13[label="",style="solid", color="burlywood", weight=3];
10.32/4.20	11[label="splitAt2 (Neg xw30) xw4 (not (primCmpInt (Neg xw30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];68[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];11 -> 68[label="",style="solid", color="burlywood", weight=9];
10.32/4.20	68 -> 14[label="",style="solid", color="burlywood", weight=3];
10.32/4.20	69[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 69[label="",style="solid", color="burlywood", weight=9];
10.32/4.20	69 -> 15[label="",style="solid", color="burlywood", weight=3];
10.32/4.20	12[label="splitAt2 (Pos (Succ xw300)) xw4 (not (primCmpInt (Pos (Succ xw300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
10.32/4.20	13[label="splitAt2 (Pos Zero) xw4 (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
10.32/4.20	14[label="splitAt2 (Neg (Succ xw300)) xw4 (not (primCmpInt (Neg (Succ xw300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
10.32/4.20	15[label="splitAt2 (Neg Zero) xw4 (not (primCmpInt (Neg Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
10.32/4.20	16[label="splitAt2 (Pos (Succ xw300)) xw4 (not (primCmpNat (Succ xw300) Zero == GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
10.32/4.20	17[label="splitAt2 (Pos Zero) xw4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
10.32/4.20	18[label="splitAt2 (Neg (Succ xw300)) xw4 (not (LT == GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
10.32/4.20	19[label="splitAt2 (Neg Zero) xw4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
10.32/4.20	20[label="splitAt2 (Pos (Succ xw300)) xw4 (not (GT == GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
10.32/4.20	21[label="splitAt2 (Pos Zero) xw4 (not False)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
10.32/4.20	22[label="splitAt2 (Neg (Succ xw300)) xw4 (not False)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
10.32/4.20	23[label="splitAt2 (Neg Zero) xw4 (not False)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
10.32/4.20	24[label="splitAt2 (Pos (Succ xw300)) xw4 (not True)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
10.32/4.20	25[label="splitAt2 (Pos Zero) xw4 True",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
10.32/4.20	26[label="splitAt2 (Neg (Succ xw300)) xw4 True",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
10.32/4.20	27[label="splitAt2 (Neg Zero) xw4 True",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
10.32/4.20	28[label="splitAt2 (Pos (Succ xw300)) xw4 False",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
10.32/4.20	29[label="([],xw4)",fontsize=16,color="green",shape="box"];30[label="([],xw4)",fontsize=16,color="green",shape="box"];31[label="([],xw4)",fontsize=16,color="green",shape="box"];32[label="splitAt1 (Pos (Succ xw300)) xw4",fontsize=16,color="burlywood",shape="box"];70[label="xw4/xw40 : xw41",fontsize=10,color="white",style="solid",shape="box"];32 -> 70[label="",style="solid", color="burlywood", weight=9];
10.32/4.20	70 -> 33[label="",style="solid", color="burlywood", weight=3];
10.32/4.20	71[label="xw4/[]",fontsize=10,color="white",style="solid",shape="box"];32 -> 71[label="",style="solid", color="burlywood", weight=9];
10.32/4.20	71 -> 34[label="",style="solid", color="burlywood", weight=3];
10.32/4.20	33[label="splitAt1 (Pos (Succ xw300)) (xw40 : xw41)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
10.32/4.20	34[label="splitAt1 (Pos (Succ xw300)) []",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
10.32/4.20	35[label="splitAt0 (Pos (Succ xw300)) (xw40 : xw41)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
10.32/4.20	36[label="([],[])",fontsize=16,color="green",shape="box"];37[label="(xw40 : splitAt0Xs' (Pos (Succ xw300)) xw41,splitAt0Xs'' (Pos (Succ xw300)) xw41)",fontsize=16,color="green",shape="box"];37 -> 38[label="",style="dashed", color="green", weight=3];
10.32/4.20	37 -> 39[label="",style="dashed", color="green", weight=3];
10.32/4.20	38[label="splitAt0Xs' (Pos (Succ xw300)) xw41",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
10.32/4.20	39[label="splitAt0Xs'' (Pos (Succ xw300)) xw41",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
10.32/4.20	40 -> 44[label="",style="dashed", color="red", weight=0];
10.32/4.20	40[label="splitAt0Xs'0 (Pos (Succ xw300)) xw41 (splitAt0Vu42 (Pos (Succ xw300)) xw41)",fontsize=16,color="magenta"];40 -> 45[label="",style="dashed", color="magenta", weight=3];
10.32/4.20	41 -> 49[label="",style="dashed", color="red", weight=0];
10.32/4.20	41[label="splitAt0Xs''0 (Pos (Succ xw300)) xw41 (splitAt0Vu42 (Pos (Succ xw300)) xw41)",fontsize=16,color="magenta"];41 -> 50[label="",style="dashed", color="magenta", weight=3];
10.32/4.20	45[label="splitAt0Vu42 (Pos (Succ xw300)) xw41",fontsize=16,color="black",shape="triangle"];45 -> 47[label="",style="solid", color="black", weight=3];
10.32/4.20	44[label="splitAt0Xs'0 (Pos (Succ xw300)) xw41 xw5",fontsize=16,color="burlywood",shape="triangle"];72[label="xw5/(xw50,xw51)",fontsize=10,color="white",style="solid",shape="box"];44 -> 72[label="",style="solid", color="burlywood", weight=9];
10.32/4.20	72 -> 48[label="",style="solid", color="burlywood", weight=3];
10.32/4.20	50 -> 45[label="",style="dashed", color="red", weight=0];
10.32/4.20	50[label="splitAt0Vu42 (Pos (Succ xw300)) xw41",fontsize=16,color="magenta"];49[label="splitAt0Xs''0 (Pos (Succ xw300)) xw41 xw6",fontsize=16,color="burlywood",shape="triangle"];73[label="xw6/(xw60,xw61)",fontsize=10,color="white",style="solid",shape="box"];49 -> 73[label="",style="solid", color="burlywood", weight=9];
10.32/4.20	73 -> 52[label="",style="solid", color="burlywood", weight=3];
10.32/4.20	47 -> 4[label="",style="dashed", color="red", weight=0];
10.32/4.20	47[label="splitAt (Pos (Succ xw300) - Pos (Succ Zero)) xw41",fontsize=16,color="magenta"];47 -> 53[label="",style="dashed", color="magenta", weight=3];
10.32/4.20	47 -> 54[label="",style="dashed", color="magenta", weight=3];
10.32/4.20	48[label="splitAt0Xs'0 (Pos (Succ xw300)) xw41 (xw50,xw51)",fontsize=16,color="black",shape="box"];48 -> 55[label="",style="solid", color="black", weight=3];
10.32/4.20	52[label="splitAt0Xs''0 (Pos (Succ xw300)) xw41 (xw60,xw61)",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
10.32/4.20	53[label="Pos (Succ xw300) - Pos (Succ Zero)",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
10.32/4.20	54[label="xw41",fontsize=16,color="green",shape="box"];55[label="xw50",fontsize=16,color="green",shape="box"];56[label="xw61",fontsize=16,color="green",shape="box"];57[label="primMinusInt (Pos (Succ xw300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];57 -> 58[label="",style="solid", color="black", weight=3];
10.32/4.20	58[label="primMinusNat (Succ xw300) (Succ Zero)",fontsize=16,color="black",shape="box"];58 -> 59[label="",style="solid", color="black", weight=3];
10.32/4.20	59[label="primMinusNat xw300 Zero",fontsize=16,color="burlywood",shape="box"];74[label="xw300/Succ xw3000",fontsize=10,color="white",style="solid",shape="box"];59 -> 74[label="",style="solid", color="burlywood", weight=9];
10.32/4.20	74 -> 60[label="",style="solid", color="burlywood", weight=3];
10.32/4.20	75[label="xw300/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 75[label="",style="solid", color="burlywood", weight=9];
10.32/4.20	75 -> 61[label="",style="solid", color="burlywood", weight=3];
10.32/4.20	60[label="primMinusNat (Succ xw3000) Zero",fontsize=16,color="black",shape="box"];60 -> 62[label="",style="solid", color="black", weight=3];
10.32/4.20	61[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];61 -> 63[label="",style="solid", color="black", weight=3];
10.32/4.20	62[label="Pos (Succ xw3000)",fontsize=16,color="green",shape="box"];63[label="Pos Zero",fontsize=16,color="green",shape="box"];}
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(12)
10.32/4.20	Obligation:
10.32/4.20	Q DP problem:
10.32/4.20	The TRS P consists of the following rules:
10.32/4.20	
10.32/4.20	   new_splitAt0Vu42(xw300, xw41, h) -> new_splitAt(new_primMinusNat(xw300), xw41, h)
10.32/4.20	   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
10.32/4.20	   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt(new_primMinusNat(xw300), xw41, h)
10.32/4.20	
10.32/4.20	The TRS R consists of the following rules:
10.32/4.20	
10.32/4.20	   new_primMinusNat(Succ(xw3000)) -> Pos(Succ(xw3000))
10.32/4.20	   new_primMinusNat(Zero) -> Pos(Zero)
10.32/4.20	
10.32/4.20	The set Q consists of the following terms:
10.32/4.20	
10.32/4.20	   new_primMinusNat(Succ(x0))
10.32/4.20	   new_primMinusNat(Zero)
10.32/4.20	
10.32/4.20	We have to consider all minimal (P,Q,R)-chains.
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(13) TransformationProof (EQUIVALENT)
10.32/4.20	By narrowing [LPAR04] the rule new_splitAt0Vu42(xw300, xw41, h) -> new_splitAt(new_primMinusNat(xw300), xw41, h) at position [0] we obtained the following new rules [LPAR04]:
10.32/4.20	
10.32/4.20	   (new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2),new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2))
10.32/4.20	   (new_splitAt0Vu42(Zero, y1, y2) -> new_splitAt(Pos(Zero), y1, y2),new_splitAt0Vu42(Zero, y1, y2) -> new_splitAt(Pos(Zero), y1, y2))
10.32/4.20	
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(14)
10.32/4.20	Obligation:
10.32/4.20	Q DP problem:
10.32/4.20	The TRS P consists of the following rules:
10.32/4.20	
10.32/4.20	   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
10.32/4.20	   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt(new_primMinusNat(xw300), xw41, h)
10.32/4.20	   new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
10.32/4.20	   new_splitAt0Vu42(Zero, y1, y2) -> new_splitAt(Pos(Zero), y1, y2)
10.32/4.20	
10.32/4.20	The TRS R consists of the following rules:
10.32/4.20	
10.32/4.20	   new_primMinusNat(Succ(xw3000)) -> Pos(Succ(xw3000))
10.32/4.20	   new_primMinusNat(Zero) -> Pos(Zero)
10.32/4.20	
10.32/4.20	The set Q consists of the following terms:
10.32/4.20	
10.32/4.20	   new_primMinusNat(Succ(x0))
10.32/4.20	   new_primMinusNat(Zero)
10.32/4.20	
10.32/4.20	We have to consider all minimal (P,Q,R)-chains.
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(15) DependencyGraphProof (EQUIVALENT)
10.32/4.20	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(16)
10.32/4.20	Obligation:
10.32/4.20	Q DP problem:
10.32/4.20	The TRS P consists of the following rules:
10.32/4.20	
10.32/4.20	   new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
10.32/4.20	   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
10.32/4.20	   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt(new_primMinusNat(xw300), xw41, h)
10.32/4.20	
10.32/4.20	The TRS R consists of the following rules:
10.32/4.20	
10.32/4.20	   new_primMinusNat(Succ(xw3000)) -> Pos(Succ(xw3000))
10.32/4.20	   new_primMinusNat(Zero) -> Pos(Zero)
10.32/4.20	
10.32/4.20	The set Q consists of the following terms:
10.32/4.20	
10.32/4.20	   new_primMinusNat(Succ(x0))
10.32/4.20	   new_primMinusNat(Zero)
10.32/4.20	
10.32/4.20	We have to consider all minimal (P,Q,R)-chains.
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(17) TransformationProof (EQUIVALENT)
10.32/4.20	By narrowing [LPAR04] the rule new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt(new_primMinusNat(xw300), xw41, h) at position [0] we obtained the following new rules [LPAR04]:
10.32/4.20	
10.32/4.20	   (new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3),new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3))
10.32/4.20	   (new_splitAt(Pos(Succ(Zero)), :(y1, y2), y3) -> new_splitAt(Pos(Zero), y2, y3),new_splitAt(Pos(Succ(Zero)), :(y1, y2), y3) -> new_splitAt(Pos(Zero), y2, y3))
10.32/4.20	
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(18)
10.32/4.20	Obligation:
10.32/4.20	Q DP problem:
10.32/4.20	The TRS P consists of the following rules:
10.32/4.20	
10.32/4.20	   new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
10.32/4.20	   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
10.32/4.20	   new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3)
10.32/4.20	   new_splitAt(Pos(Succ(Zero)), :(y1, y2), y3) -> new_splitAt(Pos(Zero), y2, y3)
10.32/4.20	
10.32/4.20	The TRS R consists of the following rules:
10.32/4.20	
10.32/4.20	   new_primMinusNat(Succ(xw3000)) -> Pos(Succ(xw3000))
10.32/4.20	   new_primMinusNat(Zero) -> Pos(Zero)
10.32/4.20	
10.32/4.20	The set Q consists of the following terms:
10.32/4.20	
10.32/4.20	   new_primMinusNat(Succ(x0))
10.32/4.20	   new_primMinusNat(Zero)
10.32/4.20	
10.32/4.20	We have to consider all minimal (P,Q,R)-chains.
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(19) DependencyGraphProof (EQUIVALENT)
10.32/4.20	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(20)
10.32/4.20	Obligation:
10.32/4.20	Q DP problem:
10.32/4.20	The TRS P consists of the following rules:
10.32/4.20	
10.32/4.20	   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
10.32/4.20	   new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
10.32/4.20	   new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3)
10.32/4.20	
10.32/4.20	The TRS R consists of the following rules:
10.32/4.20	
10.32/4.20	   new_primMinusNat(Succ(xw3000)) -> Pos(Succ(xw3000))
10.32/4.20	   new_primMinusNat(Zero) -> Pos(Zero)
10.32/4.20	
10.32/4.20	The set Q consists of the following terms:
10.32/4.20	
10.32/4.20	   new_primMinusNat(Succ(x0))
10.32/4.20	   new_primMinusNat(Zero)
10.32/4.20	
10.32/4.20	We have to consider all minimal (P,Q,R)-chains.
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(21) UsableRulesProof (EQUIVALENT)
10.32/4.20	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(22)
10.32/4.20	Obligation:
10.32/4.20	Q DP problem:
10.32/4.20	The TRS P consists of the following rules:
10.32/4.20	
10.32/4.20	   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
10.32/4.20	   new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
10.32/4.20	   new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3)
10.32/4.20	
10.32/4.20	R is empty.
10.32/4.20	The set Q consists of the following terms:
10.32/4.20	
10.32/4.20	   new_primMinusNat(Succ(x0))
10.32/4.20	   new_primMinusNat(Zero)
10.32/4.20	
10.32/4.20	We have to consider all minimal (P,Q,R)-chains.
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(23) QReductionProof (EQUIVALENT)
10.32/4.20	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
10.32/4.20	
10.32/4.20	   new_primMinusNat(Succ(x0))
10.32/4.20	   new_primMinusNat(Zero)
10.32/4.20	
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(24)
10.32/4.20	Obligation:
10.32/4.20	Q DP problem:
10.32/4.20	The TRS P consists of the following rules:
10.32/4.20	
10.32/4.20	   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
10.32/4.20	   new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
10.32/4.20	   new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3)
10.32/4.20	
10.32/4.20	R is empty.
10.32/4.20	Q is empty.
10.32/4.20	We have to consider all minimal (P,Q,R)-chains.
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(25) QDPSizeChangeProof (EQUIVALENT)
10.32/4.20	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
10.32/4.20	
10.32/4.20	From the DPs we obtained the following set of size-change graphs:
10.32/4.20	*new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
10.32/4.20	The graph contains the following edges 2 >= 2, 3 >= 3
10.32/4.20	
10.32/4.20	
10.32/4.20	*new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3)
10.32/4.20	The graph contains the following edges 2 > 2, 3 >= 3
10.32/4.20	
10.32/4.20	
10.32/4.20	*new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
10.32/4.20	The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
10.32/4.20	
10.32/4.20	
10.32/4.20	----------------------------------------
10.32/4.20	
10.32/4.20	(26)
10.32/4.20	YES
10.59/4.24	EOF