10.88/4.55	YES
12.75/5.08	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.75/5.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.75/5.08	
12.75/5.08	
12.75/5.08	H-Termination with start terms of the given HASKELL could be proven:
12.75/5.08	
12.75/5.08	(0) HASKELL
12.75/5.08	(1) BR [EQUIVALENT, 0 ms]
12.75/5.08	(2) HASKELL
12.75/5.08	(3) COR [EQUIVALENT, 1 ms]
12.75/5.08	(4) HASKELL
12.75/5.08	(5) NumRed [SOUND, 4 ms]
12.75/5.08	(6) HASKELL
12.75/5.08	(7) Narrow [SOUND, 0 ms]
12.75/5.08	(8) QDP
12.75/5.08	(9) TransformationProof [EQUIVALENT, 0 ms]
12.75/5.08	(10) QDP
12.75/5.08	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
12.75/5.08	(12) QDP
12.75/5.08	(13) UsableRulesProof [EQUIVALENT, 0 ms]
12.75/5.08	(14) QDP
12.75/5.08	(15) QReductionProof [EQUIVALENT, 0 ms]
12.75/5.08	(16) QDP
12.75/5.08	(17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.75/5.08	(18) YES
12.75/5.08	
12.75/5.08	
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(0)
12.75/5.08	Obligation:
12.75/5.08	mainModule Main 
12.75/5.08	module Maybe where {
12.75/5.08	import qualified List;
12.75/5.08	import qualified Main;
12.75/5.08	import qualified Prelude;
12.75/5.08	}
12.75/5.08	module List where {
12.75/5.08	import qualified Main;
12.75/5.08	import qualified Maybe;
12.75/5.08	import qualified Prelude;
12.75/5.08	genericTake :: Integral a => a  ->  [b]  ->  [b];
12.75/5.08	genericTake 0 _ = [];
12.75/5.08	genericTake _ [] = [];
12.75/5.08	genericTake n (x : xs)  | n > 0 = x : genericTake (n - 1) xs;
12.75/5.08	genericTake _ _ = error [];
12.75/5.08	
12.75/5.08	}
12.75/5.08	module Main where {
12.75/5.08	import qualified List;
12.75/5.08	import qualified Maybe;
12.75/5.08	import qualified Prelude;
12.75/5.08	}
12.75/5.08	
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(1) BR (EQUIVALENT)
12.75/5.08	Replaced joker patterns by fresh variables and removed binding patterns.
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(2)
12.75/5.08	Obligation:
12.75/5.08	mainModule Main 
12.75/5.08	module Maybe where {
12.75/5.08	import qualified List;
12.75/5.08	import qualified Main;
12.75/5.08	import qualified Prelude;
12.75/5.08	}
12.75/5.08	module List where {
12.75/5.08	import qualified Main;
12.75/5.08	import qualified Maybe;
12.75/5.08	import qualified Prelude;
12.75/5.08	genericTake :: Integral b => b  ->  [a]  ->  [a];
12.75/5.08	genericTake 0 xw = [];
12.75/5.08	genericTake xx [] = [];
12.75/5.08	genericTake n (x : xs)  | n > 0 = x : genericTake (n - 1) xs;
12.75/5.08	genericTake xy xz = error [];
12.75/5.08	
12.75/5.08	}
12.75/5.08	module Main where {
12.75/5.08	import qualified List;
12.75/5.08	import qualified Maybe;
12.75/5.08	import qualified Prelude;
12.75/5.08	}
12.75/5.08	
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(3) COR (EQUIVALENT)
12.75/5.08	Cond Reductions:
12.75/5.08	The following Function with conditions
12.75/5.08	"undefined |Falseundefined;
12.75/5.08	"
12.75/5.08	is transformed to
12.75/5.08	"undefined  = undefined1;
12.75/5.08	"
12.75/5.08	"undefined0 True = undefined;
12.75/5.08	"
12.75/5.08	"undefined1  = undefined0 False;
12.75/5.08	"
12.75/5.08	The following Function with conditions
12.75/5.08	"genericTake 0 xw = [];
12.75/5.08	genericTake xx [] = [];
12.75/5.08	genericTake n (x : xs)|n > 0x : genericTake (n - 1) xs;
12.75/5.08	genericTake xy xz = error [];
12.75/5.08	"
12.75/5.08	is transformed to
12.75/5.08	"genericTake zu xw = genericTake5 zu xw;
12.75/5.08	genericTake xx [] = genericTake3 xx [];
12.75/5.08	genericTake n (x : xs) = genericTake2 n (x : xs);
12.75/5.08	genericTake xy xz = genericTake0 xy xz;
12.75/5.08	"
12.75/5.08	"genericTake0 xy xz = error [];
12.75/5.08	"
12.75/5.08	"genericTake1 n x xs True = x : genericTake (n - 1) xs;
12.75/5.08	genericTake1 n x xs False = genericTake0 n (x : xs);
12.75/5.08	"
12.75/5.08	"genericTake2 n (x : xs) = genericTake1 n x xs (n > 0);
12.75/5.08	genericTake2 yv yw = genericTake0 yv yw;
12.75/5.08	"
12.75/5.08	"genericTake3 xx [] = [];
12.75/5.08	genericTake3 yy yz = genericTake2 yy yz;
12.75/5.08	"
12.75/5.08	"genericTake4 True zu xw = [];
12.75/5.08	genericTake4 zv zw zx = genericTake3 zw zx;
12.75/5.08	"
12.75/5.08	"genericTake5 zu xw = genericTake4 (zu == 0) zu xw;
12.75/5.08	genericTake5 zy zz = genericTake3 zy zz;
12.75/5.08	"
12.75/5.08	
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(4)
12.75/5.08	Obligation:
12.75/5.08	mainModule Main 
12.75/5.08	module Maybe where {
12.75/5.08	import qualified List;
12.75/5.08	import qualified Main;
12.75/5.08	import qualified Prelude;
12.75/5.08	}
12.75/5.08	module List where {
12.75/5.08	import qualified Main;
12.75/5.08	import qualified Maybe;
12.75/5.08	import qualified Prelude;
12.75/5.08	genericTake :: Integral a => a  ->  [b]  ->  [b];
12.75/5.08	genericTake zu xw = genericTake5 zu xw;
12.75/5.08	genericTake xx [] = genericTake3 xx [];
12.75/5.08	genericTake n (x : xs) = genericTake2 n (x : xs);
12.75/5.08	genericTake xy xz = genericTake0 xy xz;
12.75/5.08	
12.75/5.08	genericTake0 xy xz = error [];
12.75/5.08	
12.75/5.08	genericTake1 n x xs True = x : genericTake (n - 1) xs;
12.75/5.08	genericTake1 n x xs False = genericTake0 n (x : xs);
12.75/5.08	
12.75/5.08	genericTake2 n (x : xs) = genericTake1 n x xs (n > 0);
12.75/5.08	genericTake2 yv yw = genericTake0 yv yw;
12.75/5.08	
12.75/5.08	genericTake3 xx [] = [];
12.75/5.08	genericTake3 yy yz = genericTake2 yy yz;
12.75/5.08	
12.75/5.08	genericTake4 True zu xw = [];
12.75/5.08	genericTake4 zv zw zx = genericTake3 zw zx;
12.75/5.08	
12.75/5.08	genericTake5 zu xw = genericTake4 (zu == 0) zu xw;
12.75/5.08	genericTake5 zy zz = genericTake3 zy zz;
12.75/5.08	
12.75/5.08	}
12.75/5.08	module Main where {
12.75/5.08	import qualified List;
12.75/5.08	import qualified Maybe;
12.75/5.08	import qualified Prelude;
12.75/5.08	}
12.75/5.08	
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(5) NumRed (SOUND)
12.75/5.08	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(6)
12.75/5.08	Obligation:
12.75/5.08	mainModule Main 
12.75/5.08	module Maybe where {
12.75/5.08	import qualified List;
12.75/5.08	import qualified Main;
12.75/5.08	import qualified Prelude;
12.75/5.08	}
12.75/5.08	module List where {
12.75/5.08	import qualified Main;
12.75/5.08	import qualified Maybe;
12.75/5.08	import qualified Prelude;
12.75/5.08	genericTake :: Integral a => a  ->  [b]  ->  [b];
12.75/5.08	genericTake zu xw = genericTake5 zu xw;
12.75/5.08	genericTake xx [] = genericTake3 xx [];
12.75/5.08	genericTake n (x : xs) = genericTake2 n (x : xs);
12.75/5.08	genericTake xy xz = genericTake0 xy xz;
12.75/5.08	
12.75/5.08	genericTake0 xy xz = error [];
12.75/5.08	
12.75/5.08	genericTake1 n x xs True = x : genericTake (n - fromInt (Pos (Succ Zero))) xs;
12.75/5.08	genericTake1 n x xs False = genericTake0 n (x : xs);
12.75/5.08	
12.75/5.08	genericTake2 n (x : xs) = genericTake1 n x xs (n > fromInt (Pos Zero));
12.75/5.08	genericTake2 yv yw = genericTake0 yv yw;
12.75/5.08	
12.75/5.08	genericTake3 xx [] = [];
12.75/5.08	genericTake3 yy yz = genericTake2 yy yz;
12.75/5.08	
12.75/5.08	genericTake4 True zu xw = [];
12.75/5.08	genericTake4 zv zw zx = genericTake3 zw zx;
12.75/5.08	
12.75/5.08	genericTake5 zu xw = genericTake4 (zu == fromInt (Pos Zero)) zu xw;
12.75/5.08	genericTake5 zy zz = genericTake3 zy zz;
12.75/5.08	
12.75/5.08	}
12.75/5.08	module Main where {
12.75/5.08	import qualified List;
12.75/5.08	import qualified Maybe;
12.75/5.08	import qualified Prelude;
12.75/5.08	}
12.75/5.08	
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(7) Narrow (SOUND)
12.75/5.08	Haskell To QDPs
12.75/5.08	
12.75/5.08	digraph dp_graph {
12.75/5.08	node [outthreshold=100, inthreshold=100];1[label="List.genericTake",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.75/5.08	3[label="List.genericTake vuu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.75/5.08	4[label="List.genericTake vuu3 vuu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.75/5.08	5[label="List.genericTake5 vuu3 vuu4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
12.75/5.08	6[label="List.genericTake4 (vuu3 == fromInt (Pos Zero)) vuu3 vuu4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
12.75/5.08	7[label="List.genericTake4 (primEqInt vuu3 (fromInt (Pos Zero))) vuu3 vuu4",fontsize=16,color="burlywood",shape="box"];62[label="vuu3/Pos vuu30",fontsize=10,color="white",style="solid",shape="box"];7 -> 62[label="",style="solid", color="burlywood", weight=9];
12.75/5.08	62 -> 8[label="",style="solid", color="burlywood", weight=3];
12.75/5.08	63[label="vuu3/Neg vuu30",fontsize=10,color="white",style="solid",shape="box"];7 -> 63[label="",style="solid", color="burlywood", weight=9];
12.75/5.08	63 -> 9[label="",style="solid", color="burlywood", weight=3];
12.75/5.08	8[label="List.genericTake4 (primEqInt (Pos vuu30) (fromInt (Pos Zero))) (Pos vuu30) vuu4",fontsize=16,color="burlywood",shape="box"];64[label="vuu30/Succ vuu300",fontsize=10,color="white",style="solid",shape="box"];8 -> 64[label="",style="solid", color="burlywood", weight=9];
12.75/5.08	64 -> 10[label="",style="solid", color="burlywood", weight=3];
12.75/5.08	65[label="vuu30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 65[label="",style="solid", color="burlywood", weight=9];
12.75/5.08	65 -> 11[label="",style="solid", color="burlywood", weight=3];
12.75/5.08	9[label="List.genericTake4 (primEqInt (Neg vuu30) (fromInt (Pos Zero))) (Neg vuu30) vuu4",fontsize=16,color="burlywood",shape="box"];66[label="vuu30/Succ vuu300",fontsize=10,color="white",style="solid",shape="box"];9 -> 66[label="",style="solid", color="burlywood", weight=9];
12.75/5.08	66 -> 12[label="",style="solid", color="burlywood", weight=3];
12.75/5.08	67[label="vuu30/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 67[label="",style="solid", color="burlywood", weight=9];
12.75/5.08	67 -> 13[label="",style="solid", color="burlywood", weight=3];
12.75/5.08	10[label="List.genericTake4 (primEqInt (Pos (Succ vuu300)) (fromInt (Pos Zero))) (Pos (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
12.75/5.08	11[label="List.genericTake4 (primEqInt (Pos Zero) (fromInt (Pos Zero))) (Pos Zero) vuu4",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3];
12.75/5.08	12[label="List.genericTake4 (primEqInt (Neg (Succ vuu300)) (fromInt (Pos Zero))) (Neg (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
12.75/5.08	13[label="List.genericTake4 (primEqInt (Neg Zero) (fromInt (Pos Zero))) (Neg Zero) vuu4",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
12.75/5.08	14[label="List.genericTake4 (primEqInt (Pos (Succ vuu300)) (Pos Zero)) (Pos (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
12.75/5.08	15[label="List.genericTake4 (primEqInt (Pos Zero) (Pos Zero)) (Pos Zero) vuu4",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
12.75/5.08	16[label="List.genericTake4 (primEqInt (Neg (Succ vuu300)) (Pos Zero)) (Neg (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
12.75/5.08	17[label="List.genericTake4 (primEqInt (Neg Zero) (Pos Zero)) (Neg Zero) vuu4",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
12.75/5.08	18[label="List.genericTake4 False (Pos (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
12.75/5.08	19[label="List.genericTake4 True (Pos Zero) vuu4",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
12.75/5.08	20[label="List.genericTake4 False (Neg (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
12.75/5.08	21[label="List.genericTake4 True (Neg Zero) vuu4",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
12.75/5.08	22[label="List.genericTake3 (Pos (Succ vuu300)) vuu4",fontsize=16,color="burlywood",shape="box"];68[label="vuu4/vuu40 : vuu41",fontsize=10,color="white",style="solid",shape="box"];22 -> 68[label="",style="solid", color="burlywood", weight=9];
12.75/5.08	68 -> 26[label="",style="solid", color="burlywood", weight=3];
12.75/5.08	69[label="vuu4/[]",fontsize=10,color="white",style="solid",shape="box"];22 -> 69[label="",style="solid", color="burlywood", weight=9];
12.75/5.08	69 -> 27[label="",style="solid", color="burlywood", weight=3];
12.75/5.08	23[label="[]",fontsize=16,color="green",shape="box"];24[label="List.genericTake3 (Neg (Succ vuu300)) vuu4",fontsize=16,color="burlywood",shape="box"];70[label="vuu4/vuu40 : vuu41",fontsize=10,color="white",style="solid",shape="box"];24 -> 70[label="",style="solid", color="burlywood", weight=9];
12.75/5.08	70 -> 28[label="",style="solid", color="burlywood", weight=3];
12.75/5.08	71[label="vuu4/[]",fontsize=10,color="white",style="solid",shape="box"];24 -> 71[label="",style="solid", color="burlywood", weight=9];
12.75/5.08	71 -> 29[label="",style="solid", color="burlywood", weight=3];
12.75/5.08	25[label="[]",fontsize=16,color="green",shape="box"];26[label="List.genericTake3 (Pos (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
12.75/5.08	27[label="List.genericTake3 (Pos (Succ vuu300)) []",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
12.75/5.08	28[label="List.genericTake3 (Neg (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
12.75/5.08	29[label="List.genericTake3 (Neg (Succ vuu300)) []",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
12.75/5.08	30[label="List.genericTake2 (Pos (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
12.75/5.08	31[label="[]",fontsize=16,color="green",shape="box"];32[label="List.genericTake2 (Neg (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3];
12.75/5.08	33[label="[]",fontsize=16,color="green",shape="box"];34[label="List.genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 (Pos (Succ vuu300) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
12.75/5.08	35[label="List.genericTake1 (Neg (Succ vuu300)) vuu40 vuu41 (Neg (Succ vuu300) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
12.75/5.08	36[label="List.genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 (compare (Pos (Succ vuu300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
12.75/5.08	37[label="List.genericTake1 (Neg (Succ vuu300)) vuu40 vuu41 (compare (Neg (Succ vuu300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
12.75/5.08	38[label="List.genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 (primCmpInt (Pos (Succ vuu300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
12.75/5.08	39[label="List.genericTake1 (Neg (Succ vuu300)) vuu40 vuu41 (primCmpInt (Neg (Succ vuu300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
12.75/5.08	40[label="List.genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 (primCmpInt (Pos (Succ vuu300)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];40 -> 42[label="",style="solid", color="black", weight=3];
12.75/5.08	41[label="List.genericTake1 (Neg (Succ vuu300)) vuu40 vuu41 (primCmpInt (Neg (Succ vuu300)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
12.75/5.08	42[label="List.genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 (primCmpNat (Succ vuu300) Zero == GT)",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
12.75/5.08	43[label="List.genericTake1 (Neg (Succ vuu300)) vuu40 vuu41 (LT == GT)",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
12.75/5.08	44[label="List.genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 (GT == GT)",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
12.75/5.08	45[label="List.genericTake1 (Neg (Succ vuu300)) vuu40 vuu41 False",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3];
12.75/5.08	46[label="List.genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 True",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
12.75/5.08	47[label="List.genericTake0 (Neg (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
12.75/5.08	48[label="vuu40 : List.genericTake (Pos (Succ vuu300) - fromInt (Pos (Succ Zero))) vuu41",fontsize=16,color="green",shape="box"];48 -> 50[label="",style="dashed", color="green", weight=3];
12.75/5.08	49[label="error []",fontsize=16,color="black",shape="box"];49 -> 51[label="",style="solid", color="black", weight=3];
12.75/5.08	50 -> 4[label="",style="dashed", color="red", weight=0];
12.75/5.08	50[label="List.genericTake (Pos (Succ vuu300) - fromInt (Pos (Succ Zero))) vuu41",fontsize=16,color="magenta"];50 -> 52[label="",style="dashed", color="magenta", weight=3];
12.75/5.08	50 -> 53[label="",style="dashed", color="magenta", weight=3];
12.75/5.08	51[label="error []",fontsize=16,color="red",shape="box"];52[label="Pos (Succ vuu300) - fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];52 -> 54[label="",style="solid", color="black", weight=3];
12.75/5.08	53[label="vuu41",fontsize=16,color="green",shape="box"];54[label="primMinusInt (Pos (Succ vuu300)) (fromInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];54 -> 55[label="",style="solid", color="black", weight=3];
12.75/5.08	55[label="primMinusInt (Pos (Succ vuu300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];55 -> 56[label="",style="solid", color="black", weight=3];
12.75/5.08	56[label="primMinusNat (Succ vuu300) (Succ Zero)",fontsize=16,color="black",shape="box"];56 -> 57[label="",style="solid", color="black", weight=3];
12.75/5.08	57[label="primMinusNat vuu300 Zero",fontsize=16,color="burlywood",shape="box"];72[label="vuu300/Succ vuu3000",fontsize=10,color="white",style="solid",shape="box"];57 -> 72[label="",style="solid", color="burlywood", weight=9];
12.75/5.08	72 -> 58[label="",style="solid", color="burlywood", weight=3];
12.75/5.08	73[label="vuu300/Zero",fontsize=10,color="white",style="solid",shape="box"];57 -> 73[label="",style="solid", color="burlywood", weight=9];
12.75/5.08	73 -> 59[label="",style="solid", color="burlywood", weight=3];
12.75/5.08	58[label="primMinusNat (Succ vuu3000) Zero",fontsize=16,color="black",shape="box"];58 -> 60[label="",style="solid", color="black", weight=3];
12.75/5.08	59[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];59 -> 61[label="",style="solid", color="black", weight=3];
12.75/5.08	60[label="Pos (Succ vuu3000)",fontsize=16,color="green",shape="box"];61[label="Pos Zero",fontsize=16,color="green",shape="box"];}
12.75/5.08	
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(8)
12.75/5.08	Obligation:
12.75/5.08	Q DP problem:
12.75/5.08	The TRS P consists of the following rules:
12.75/5.08	
12.75/5.08	   new_genericTake(Pos(Succ(vuu300)), :(vuu40, vuu41), ba) -> new_genericTake(new_primMinusNat(vuu300), vuu41, ba)
12.75/5.08	
12.75/5.08	The TRS R consists of the following rules:
12.75/5.08	
12.75/5.08	   new_primMinusNat(Succ(vuu3000)) -> Pos(Succ(vuu3000))
12.75/5.08	   new_primMinusNat(Zero) -> Pos(Zero)
12.75/5.08	
12.75/5.08	The set Q consists of the following terms:
12.75/5.08	
12.75/5.08	   new_primMinusNat(Succ(x0))
12.75/5.08	   new_primMinusNat(Zero)
12.75/5.08	
12.75/5.08	We have to consider all minimal (P,Q,R)-chains.
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(9) TransformationProof (EQUIVALENT)
12.75/5.08	By narrowing [LPAR04] the rule new_genericTake(Pos(Succ(vuu300)), :(vuu40, vuu41), ba) -> new_genericTake(new_primMinusNat(vuu300), vuu41, ba) at position [0] we obtained the following new rules [LPAR04]:
12.75/5.08	
12.75/5.08	   (new_genericTake(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericTake(Pos(Succ(x0)), y2, y3),new_genericTake(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericTake(Pos(Succ(x0)), y2, y3))
12.75/5.08	   (new_genericTake(Pos(Succ(Zero)), :(y1, y2), y3) -> new_genericTake(Pos(Zero), y2, y3),new_genericTake(Pos(Succ(Zero)), :(y1, y2), y3) -> new_genericTake(Pos(Zero), y2, y3))
12.75/5.08	
12.75/5.08	
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(10)
12.75/5.08	Obligation:
12.75/5.08	Q DP problem:
12.75/5.08	The TRS P consists of the following rules:
12.75/5.08	
12.75/5.08	   new_genericTake(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericTake(Pos(Succ(x0)), y2, y3)
12.75/5.08	   new_genericTake(Pos(Succ(Zero)), :(y1, y2), y3) -> new_genericTake(Pos(Zero), y2, y3)
12.75/5.08	
12.75/5.08	The TRS R consists of the following rules:
12.75/5.08	
12.75/5.08	   new_primMinusNat(Succ(vuu3000)) -> Pos(Succ(vuu3000))
12.75/5.08	   new_primMinusNat(Zero) -> Pos(Zero)
12.75/5.08	
12.75/5.08	The set Q consists of the following terms:
12.75/5.08	
12.75/5.08	   new_primMinusNat(Succ(x0))
12.75/5.08	   new_primMinusNat(Zero)
12.75/5.08	
12.75/5.08	We have to consider all minimal (P,Q,R)-chains.
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(11) DependencyGraphProof (EQUIVALENT)
12.75/5.08	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(12)
12.75/5.08	Obligation:
12.75/5.08	Q DP problem:
12.75/5.08	The TRS P consists of the following rules:
12.75/5.08	
12.75/5.08	   new_genericTake(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericTake(Pos(Succ(x0)), y2, y3)
12.75/5.08	
12.75/5.08	The TRS R consists of the following rules:
12.75/5.08	
12.75/5.08	   new_primMinusNat(Succ(vuu3000)) -> Pos(Succ(vuu3000))
12.75/5.08	   new_primMinusNat(Zero) -> Pos(Zero)
12.75/5.08	
12.75/5.08	The set Q consists of the following terms:
12.75/5.08	
12.75/5.08	   new_primMinusNat(Succ(x0))
12.75/5.08	   new_primMinusNat(Zero)
12.75/5.08	
12.75/5.08	We have to consider all minimal (P,Q,R)-chains.
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(13) UsableRulesProof (EQUIVALENT)
12.75/5.08	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.
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(14)
12.75/5.08	Obligation:
12.75/5.08	Q DP problem:
12.75/5.08	The TRS P consists of the following rules:
12.75/5.08	
12.75/5.08	   new_genericTake(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericTake(Pos(Succ(x0)), y2, y3)
12.75/5.08	
12.75/5.08	R is empty.
12.75/5.08	The set Q consists of the following terms:
12.75/5.08	
12.75/5.08	   new_primMinusNat(Succ(x0))
12.75/5.08	   new_primMinusNat(Zero)
12.75/5.08	
12.75/5.08	We have to consider all minimal (P,Q,R)-chains.
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(15) QReductionProof (EQUIVALENT)
12.75/5.08	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
12.75/5.08	
12.75/5.08	   new_primMinusNat(Succ(x0))
12.75/5.08	   new_primMinusNat(Zero)
12.75/5.08	
12.75/5.08	
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(16)
12.75/5.08	Obligation:
12.75/5.08	Q DP problem:
12.75/5.08	The TRS P consists of the following rules:
12.75/5.08	
12.75/5.08	   new_genericTake(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericTake(Pos(Succ(x0)), y2, y3)
12.75/5.08	
12.75/5.08	R is empty.
12.75/5.08	Q is empty.
12.75/5.08	We have to consider all minimal (P,Q,R)-chains.
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(17) QDPSizeChangeProof (EQUIVALENT)
12.75/5.08	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. 
12.75/5.08	
12.75/5.08	From the DPs we obtained the following set of size-change graphs:
12.75/5.08	*new_genericTake(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericTake(Pos(Succ(x0)), y2, y3)
12.75/5.08	The graph contains the following edges 2 > 2, 3 >= 3
12.75/5.08	
12.75/5.08	
12.75/5.08	----------------------------------------
12.75/5.08	
12.75/5.08	(18)
12.75/5.08	YES
12.99/5.13	EOF