10.72/4.52	YES
12.92/5.12	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.92/5.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.92/5.12	
12.92/5.12	
12.92/5.12	H-Termination with start terms of the given HASKELL could be proven:
12.92/5.12	
12.92/5.12	(0) HASKELL
12.92/5.12	(1) BR [EQUIVALENT, 0 ms]
12.92/5.12	(2) HASKELL
12.92/5.12	(3) COR [EQUIVALENT, 6 ms]
12.92/5.12	(4) HASKELL
12.92/5.12	(5) NumRed [SOUND, 0 ms]
12.92/5.12	(6) HASKELL
12.92/5.12	(7) Narrow [SOUND, 0 ms]
12.92/5.12	(8) QDP
12.92/5.12	(9) TransformationProof [EQUIVALENT, 0 ms]
12.92/5.12	(10) QDP
12.92/5.12	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
12.92/5.12	(12) QDP
12.92/5.12	(13) UsableRulesProof [EQUIVALENT, 0 ms]
12.92/5.12	(14) QDP
12.92/5.12	(15) QReductionProof [EQUIVALENT, 0 ms]
12.92/5.12	(16) QDP
12.92/5.12	(17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.92/5.12	(18) YES
12.92/5.12	
12.92/5.12	
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(0)
12.92/5.12	Obligation:
12.92/5.12	mainModule Main 
12.92/5.12	module Maybe where {
12.92/5.12	import qualified List;
12.92/5.12	import qualified Main;
12.92/5.12	import qualified Prelude;
12.92/5.12	}
12.92/5.12	module List where {
12.92/5.12	import qualified Main;
12.92/5.12	import qualified Maybe;
12.92/5.12	import qualified Prelude;
12.92/5.12	genericIndex :: Integral b => [a]  ->  b  ->  a;
12.92/5.12	genericIndex (x : _) 0 = x;
12.92/5.12	genericIndex (_ : xs) n  | n > 0 = genericIndex xs (n - 1)
12.92/5.12	 | otherwise = error [];
12.92/5.12	genericIndex _ _ = error [];
12.92/5.12	
12.92/5.12	}
12.92/5.12	module Main where {
12.92/5.12	import qualified List;
12.92/5.12	import qualified Maybe;
12.92/5.12	import qualified Prelude;
12.92/5.12	}
12.92/5.12	
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(1) BR (EQUIVALENT)
12.92/5.12	Replaced joker patterns by fresh variables and removed binding patterns.
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(2)
12.92/5.12	Obligation:
12.92/5.12	mainModule Main 
12.92/5.12	module Maybe where {
12.92/5.12	import qualified List;
12.92/5.12	import qualified Main;
12.92/5.12	import qualified Prelude;
12.92/5.12	}
12.92/5.12	module List where {
12.92/5.12	import qualified Main;
12.92/5.12	import qualified Maybe;
12.92/5.12	import qualified Prelude;
12.92/5.12	genericIndex :: Integral b => [a]  ->  b  ->  a;
12.92/5.12	genericIndex (x : xw) 0 = x;
12.92/5.12	genericIndex (xx : xs) n  | n > 0 = genericIndex xs (n - 1)
12.92/5.12	 | otherwise = error [];
12.92/5.12	genericIndex xy xz = error [];
12.92/5.12	
12.92/5.12	}
12.92/5.12	module Main where {
12.92/5.12	import qualified List;
12.92/5.12	import qualified Maybe;
12.92/5.12	import qualified Prelude;
12.92/5.12	}
12.92/5.12	
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(3) COR (EQUIVALENT)
12.92/5.12	Cond Reductions:
12.92/5.12	The following Function with conditions
12.92/5.12	"undefined |Falseundefined;
12.92/5.12	"
12.92/5.12	is transformed to
12.92/5.12	"undefined  = undefined1;
12.92/5.12	"
12.92/5.12	"undefined0 True = undefined;
12.92/5.12	"
12.92/5.12	"undefined1  = undefined0 False;
12.92/5.12	"
12.92/5.12	The following Function with conditions
12.92/5.12	"genericIndex (x : xw) 0 = x;
12.92/5.12	genericIndex (xx : xs) n|n > 0genericIndex xs (n - 1)|otherwiseerror [];
12.92/5.12	genericIndex xy xz = error [];
12.92/5.12	"
12.92/5.12	is transformed to
12.92/5.12	"genericIndex (x : xw) yy = genericIndex5 (x : xw) yy;
12.92/5.12	genericIndex (xx : xs) n = genericIndex3 (xx : xs) n;
12.92/5.12	genericIndex xy xz = genericIndex0 xy xz;
12.92/5.12	"
12.92/5.12	"genericIndex0 xy xz = error [];
12.92/5.12	"
12.92/5.12	"genericIndex1 xx xs n True = error [];
12.92/5.12	genericIndex1 xx xs n False = genericIndex0 (xx : xs) n;
12.92/5.12	"
12.92/5.12	"genericIndex2 xx xs n True = genericIndex xs (n - 1);
12.92/5.12	genericIndex2 xx xs n False = genericIndex1 xx xs n otherwise;
12.92/5.12	"
12.92/5.12	"genericIndex3 (xx : xs) n = genericIndex2 xx xs n (n > 0);
12.92/5.12	genericIndex3 yv yw = genericIndex0 yv yw;
12.92/5.12	"
12.92/5.12	"genericIndex4 True (x : xw) yy = x;
12.92/5.12	genericIndex4 yz zu zv = genericIndex3 zu zv;
12.92/5.12	"
12.92/5.12	"genericIndex5 (x : xw) yy = genericIndex4 (yy == 0) (x : xw) yy;
12.92/5.12	genericIndex5 zw zx = genericIndex3 zw zx;
12.92/5.12	"
12.92/5.12	
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(4)
12.92/5.12	Obligation:
12.92/5.12	mainModule Main 
12.92/5.12	module Maybe where {
12.92/5.12	import qualified List;
12.92/5.12	import qualified Main;
12.92/5.12	import qualified Prelude;
12.92/5.12	}
12.92/5.12	module List where {
12.92/5.12	import qualified Main;
12.92/5.12	import qualified Maybe;
12.92/5.12	import qualified Prelude;
12.92/5.12	genericIndex :: Integral a => [b]  ->  a  ->  b;
12.92/5.12	genericIndex (x : xw) yy = genericIndex5 (x : xw) yy;
12.92/5.12	genericIndex (xx : xs) n = genericIndex3 (xx : xs) n;
12.92/5.12	genericIndex xy xz = genericIndex0 xy xz;
12.92/5.12	
12.92/5.12	genericIndex0 xy xz = error [];
12.92/5.12	
12.92/5.12	genericIndex1 xx xs n True = error [];
12.92/5.12	genericIndex1 xx xs n False = genericIndex0 (xx : xs) n;
12.92/5.12	
12.92/5.12	genericIndex2 xx xs n True = genericIndex xs (n - 1);
12.92/5.12	genericIndex2 xx xs n False = genericIndex1 xx xs n otherwise;
12.92/5.12	
12.92/5.12	genericIndex3 (xx : xs) n = genericIndex2 xx xs n (n > 0);
12.92/5.12	genericIndex3 yv yw = genericIndex0 yv yw;
12.92/5.12	
12.92/5.12	genericIndex4 True (x : xw) yy = x;
12.92/5.12	genericIndex4 yz zu zv = genericIndex3 zu zv;
12.92/5.12	
12.92/5.12	genericIndex5 (x : xw) yy = genericIndex4 (yy == 0) (x : xw) yy;
12.92/5.12	genericIndex5 zw zx = genericIndex3 zw zx;
12.92/5.12	
12.92/5.12	}
12.92/5.12	module Main where {
12.92/5.12	import qualified List;
12.92/5.12	import qualified Maybe;
12.92/5.12	import qualified Prelude;
12.92/5.12	}
12.92/5.12	
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(5) NumRed (SOUND)
12.92/5.12	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(6)
12.92/5.12	Obligation:
12.92/5.12	mainModule Main 
12.92/5.12	module Maybe where {
12.92/5.12	import qualified List;
12.92/5.12	import qualified Main;
12.92/5.12	import qualified Prelude;
12.92/5.12	}
12.92/5.12	module List where {
12.92/5.12	import qualified Main;
12.92/5.12	import qualified Maybe;
12.92/5.12	import qualified Prelude;
12.92/5.12	genericIndex :: Integral b => [a]  ->  b  ->  a;
12.92/5.12	genericIndex (x : xw) yy = genericIndex5 (x : xw) yy;
12.92/5.12	genericIndex (xx : xs) n = genericIndex3 (xx : xs) n;
12.92/5.12	genericIndex xy xz = genericIndex0 xy xz;
12.92/5.12	
12.92/5.12	genericIndex0 xy xz = error [];
12.92/5.12	
12.92/5.12	genericIndex1 xx xs n True = error [];
12.92/5.12	genericIndex1 xx xs n False = genericIndex0 (xx : xs) n;
12.92/5.12	
12.92/5.12	genericIndex2 xx xs n True = genericIndex xs (n - fromInt (Pos (Succ Zero)));
12.92/5.12	genericIndex2 xx xs n False = genericIndex1 xx xs n otherwise;
12.92/5.12	
12.92/5.12	genericIndex3 (xx : xs) n = genericIndex2 xx xs n (n > fromInt (Pos Zero));
12.92/5.12	genericIndex3 yv yw = genericIndex0 yv yw;
12.92/5.12	
12.92/5.12	genericIndex4 True (x : xw) yy = x;
12.92/5.12	genericIndex4 yz zu zv = genericIndex3 zu zv;
12.92/5.12	
12.92/5.12	genericIndex5 (x : xw) yy = genericIndex4 (yy == fromInt (Pos Zero)) (x : xw) yy;
12.92/5.12	genericIndex5 zw zx = genericIndex3 zw zx;
12.92/5.12	
12.92/5.12	}
12.92/5.12	module Main where {
12.92/5.12	import qualified List;
12.92/5.12	import qualified Maybe;
12.92/5.12	import qualified Prelude;
12.92/5.12	}
12.92/5.12	
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(7) Narrow (SOUND)
12.92/5.12	Haskell To QDPs
12.92/5.12	
12.92/5.12	digraph dp_graph {
12.92/5.12	node [outthreshold=100, inthreshold=100];1[label="List.genericIndex",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.92/5.12	3[label="List.genericIndex zy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.92/5.12	4[label="List.genericIndex zy3 zy4",fontsize=16,color="burlywood",shape="triangle"];58[label="zy3/zy30 : zy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 58[label="",style="solid", color="burlywood", weight=9];
12.92/5.12	58 -> 5[label="",style="solid", color="burlywood", weight=3];
12.92/5.12	59[label="zy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 59[label="",style="solid", color="burlywood", weight=9];
12.92/5.12	59 -> 6[label="",style="solid", color="burlywood", weight=3];
12.92/5.12	5[label="List.genericIndex (zy30 : zy31) zy4",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
12.92/5.12	6[label="List.genericIndex [] zy4",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.92/5.12	7[label="List.genericIndex5 (zy30 : zy31) zy4",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.92/5.12	8[label="List.genericIndex0 [] zy4",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
12.92/5.12	9[label="List.genericIndex4 (zy4 == fromInt (Pos Zero)) (zy30 : zy31) zy4",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
12.92/5.12	10[label="error []",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3];
12.92/5.12	11[label="List.genericIndex4 (primEqInt zy4 (fromInt (Pos Zero))) (zy30 : zy31) zy4",fontsize=16,color="burlywood",shape="box"];60[label="zy4/Pos zy40",fontsize=10,color="white",style="solid",shape="box"];11 -> 60[label="",style="solid", color="burlywood", weight=9];
12.92/5.12	60 -> 13[label="",style="solid", color="burlywood", weight=3];
12.92/5.12	61[label="zy4/Neg zy40",fontsize=10,color="white",style="solid",shape="box"];11 -> 61[label="",style="solid", color="burlywood", weight=9];
12.92/5.12	61 -> 14[label="",style="solid", color="burlywood", weight=3];
12.92/5.12	12[label="error []",fontsize=16,color="red",shape="box"];13[label="List.genericIndex4 (primEqInt (Pos zy40) (fromInt (Pos Zero))) (zy30 : zy31) (Pos zy40)",fontsize=16,color="burlywood",shape="box"];62[label="zy40/Succ zy400",fontsize=10,color="white",style="solid",shape="box"];13 -> 62[label="",style="solid", color="burlywood", weight=9];
12.92/5.12	62 -> 15[label="",style="solid", color="burlywood", weight=3];
12.92/5.12	63[label="zy40/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 63[label="",style="solid", color="burlywood", weight=9];
12.92/5.12	63 -> 16[label="",style="solid", color="burlywood", weight=3];
12.92/5.12	14[label="List.genericIndex4 (primEqInt (Neg zy40) (fromInt (Pos Zero))) (zy30 : zy31) (Neg zy40)",fontsize=16,color="burlywood",shape="box"];64[label="zy40/Succ zy400",fontsize=10,color="white",style="solid",shape="box"];14 -> 64[label="",style="solid", color="burlywood", weight=9];
12.92/5.12	64 -> 17[label="",style="solid", color="burlywood", weight=3];
12.92/5.12	65[label="zy40/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 65[label="",style="solid", color="burlywood", weight=9];
12.92/5.12	65 -> 18[label="",style="solid", color="burlywood", weight=3];
12.92/5.12	15[label="List.genericIndex4 (primEqInt (Pos (Succ zy400)) (fromInt (Pos Zero))) (zy30 : zy31) (Pos (Succ zy400))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
12.92/5.12	16[label="List.genericIndex4 (primEqInt (Pos Zero) (fromInt (Pos Zero))) (zy30 : zy31) (Pos Zero)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
12.92/5.12	17[label="List.genericIndex4 (primEqInt (Neg (Succ zy400)) (fromInt (Pos Zero))) (zy30 : zy31) (Neg (Succ zy400))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
12.92/5.12	18[label="List.genericIndex4 (primEqInt (Neg Zero) (fromInt (Pos Zero))) (zy30 : zy31) (Neg Zero)",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
12.92/5.12	19[label="List.genericIndex4 (primEqInt (Pos (Succ zy400)) (Pos Zero)) (zy30 : zy31) (Pos (Succ zy400))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
12.92/5.12	20[label="List.genericIndex4 (primEqInt (Pos Zero) (Pos Zero)) (zy30 : zy31) (Pos Zero)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
12.92/5.12	21[label="List.genericIndex4 (primEqInt (Neg (Succ zy400)) (Pos Zero)) (zy30 : zy31) (Neg (Succ zy400))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
12.92/5.12	22[label="List.genericIndex4 (primEqInt (Neg Zero) (Pos Zero)) (zy30 : zy31) (Neg Zero)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
12.92/5.12	23[label="List.genericIndex4 False (zy30 : zy31) (Pos (Succ zy400))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
12.92/5.12	24[label="List.genericIndex4 True (zy30 : zy31) (Pos Zero)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
12.92/5.12	25[label="List.genericIndex4 False (zy30 : zy31) (Neg (Succ zy400))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
12.92/5.12	26[label="List.genericIndex4 True (zy30 : zy31) (Neg Zero)",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
12.92/5.12	27[label="List.genericIndex3 (zy30 : zy31) (Pos (Succ zy400))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
12.92/5.12	28[label="zy30",fontsize=16,color="green",shape="box"];29[label="List.genericIndex3 (zy30 : zy31) (Neg (Succ zy400))",fontsize=16,color="black",shape="box"];29 -> 32[label="",style="solid", color="black", weight=3];
12.92/5.12	30[label="zy30",fontsize=16,color="green",shape="box"];31[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) (Pos (Succ zy400) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
12.92/5.12	32[label="List.genericIndex2 zy30 zy31 (Neg (Succ zy400)) (Neg (Succ zy400) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3];
12.92/5.12	33[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) (compare (Pos (Succ zy400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
12.92/5.12	34[label="List.genericIndex2 zy30 zy31 (Neg (Succ zy400)) (compare (Neg (Succ zy400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
12.92/5.12	35[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) (primCmpInt (Pos (Succ zy400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
12.92/5.12	36[label="List.genericIndex2 zy30 zy31 (Neg (Succ zy400)) (primCmpInt (Neg (Succ zy400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
12.92/5.12	37[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) (primCmpInt (Pos (Succ zy400)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
12.92/5.12	38[label="List.genericIndex2 zy30 zy31 (Neg (Succ zy400)) (primCmpInt (Neg (Succ zy400)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
12.92/5.12	39[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) (primCmpNat (Succ zy400) Zero == GT)",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
12.92/5.12	40[label="List.genericIndex2 zy30 zy31 (Neg (Succ zy400)) (LT == GT)",fontsize=16,color="black",shape="box"];40 -> 42[label="",style="solid", color="black", weight=3];
12.92/5.12	41[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) (GT == GT)",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
12.92/5.12	42[label="List.genericIndex2 zy30 zy31 (Neg (Succ zy400)) False",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
12.92/5.12	43[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) True",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
12.92/5.12	44[label="List.genericIndex1 zy30 zy31 (Neg (Succ zy400)) otherwise",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
12.92/5.12	45 -> 4[label="",style="dashed", color="red", weight=0];
12.92/5.12	45[label="List.genericIndex zy31 (Pos (Succ zy400) - fromInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];45 -> 47[label="",style="dashed", color="magenta", weight=3];
12.92/5.12	45 -> 48[label="",style="dashed", color="magenta", weight=3];
12.92/5.12	46[label="List.genericIndex1 zy30 zy31 (Neg (Succ zy400)) True",fontsize=16,color="black",shape="box"];46 -> 49[label="",style="solid", color="black", weight=3];
12.92/5.12	47[label="Pos (Succ zy400) - fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];47 -> 50[label="",style="solid", color="black", weight=3];
12.92/5.12	48[label="zy31",fontsize=16,color="green",shape="box"];49 -> 10[label="",style="dashed", color="red", weight=0];
12.92/5.12	49[label="error []",fontsize=16,color="magenta"];50[label="primMinusInt (Pos (Succ zy400)) (fromInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];50 -> 51[label="",style="solid", color="black", weight=3];
12.92/5.12	51[label="primMinusInt (Pos (Succ zy400)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];51 -> 52[label="",style="solid", color="black", weight=3];
12.92/5.12	52[label="primMinusNat (Succ zy400) (Succ Zero)",fontsize=16,color="black",shape="box"];52 -> 53[label="",style="solid", color="black", weight=3];
12.92/5.12	53[label="primMinusNat zy400 Zero",fontsize=16,color="burlywood",shape="box"];66[label="zy400/Succ zy4000",fontsize=10,color="white",style="solid",shape="box"];53 -> 66[label="",style="solid", color="burlywood", weight=9];
12.92/5.12	66 -> 54[label="",style="solid", color="burlywood", weight=3];
12.92/5.12	67[label="zy400/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 67[label="",style="solid", color="burlywood", weight=9];
12.92/5.12	67 -> 55[label="",style="solid", color="burlywood", weight=3];
12.92/5.12	54[label="primMinusNat (Succ zy4000) Zero",fontsize=16,color="black",shape="box"];54 -> 56[label="",style="solid", color="black", weight=3];
12.92/5.12	55[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];55 -> 57[label="",style="solid", color="black", weight=3];
12.92/5.12	56[label="Pos (Succ zy4000)",fontsize=16,color="green",shape="box"];57[label="Pos Zero",fontsize=16,color="green",shape="box"];}
12.92/5.12	
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(8)
12.92/5.12	Obligation:
12.92/5.12	Q DP problem:
12.92/5.12	The TRS P consists of the following rules:
12.92/5.12	
12.92/5.12	   new_genericIndex(:(zy30, zy31), Pos(Succ(zy400)), ba) -> new_genericIndex(zy31, new_primMinusNat(zy400), ba)
12.92/5.12	
12.92/5.12	The TRS R consists of the following rules:
12.92/5.12	
12.92/5.12	   new_primMinusNat(Succ(zy4000)) -> Pos(Succ(zy4000))
12.92/5.12	   new_primMinusNat(Zero) -> Pos(Zero)
12.92/5.12	
12.92/5.12	The set Q consists of the following terms:
12.92/5.12	
12.92/5.12	   new_primMinusNat(Succ(x0))
12.92/5.12	   new_primMinusNat(Zero)
12.92/5.12	
12.92/5.12	We have to consider all minimal (P,Q,R)-chains.
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(9) TransformationProof (EQUIVALENT)
12.92/5.12	By narrowing [LPAR04] the rule new_genericIndex(:(zy30, zy31), Pos(Succ(zy400)), ba) -> new_genericIndex(zy31, new_primMinusNat(zy400), ba) at position [1] we obtained the following new rules [LPAR04]:
12.92/5.12	
12.92/5.12	   (new_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3),new_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3))
12.92/5.12	   (new_genericIndex(:(y0, y1), Pos(Succ(Zero)), y3) -> new_genericIndex(y1, Pos(Zero), y3),new_genericIndex(:(y0, y1), Pos(Succ(Zero)), y3) -> new_genericIndex(y1, Pos(Zero), y3))
12.92/5.12	
12.92/5.12	
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(10)
12.92/5.12	Obligation:
12.92/5.12	Q DP problem:
12.92/5.12	The TRS P consists of the following rules:
12.92/5.12	
12.92/5.12	   new_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3)
12.92/5.12	   new_genericIndex(:(y0, y1), Pos(Succ(Zero)), y3) -> new_genericIndex(y1, Pos(Zero), y3)
12.92/5.12	
12.92/5.12	The TRS R consists of the following rules:
12.92/5.12	
12.92/5.12	   new_primMinusNat(Succ(zy4000)) -> Pos(Succ(zy4000))
12.92/5.12	   new_primMinusNat(Zero) -> Pos(Zero)
12.92/5.12	
12.92/5.12	The set Q consists of the following terms:
12.92/5.12	
12.92/5.12	   new_primMinusNat(Succ(x0))
12.92/5.12	   new_primMinusNat(Zero)
12.92/5.12	
12.92/5.12	We have to consider all minimal (P,Q,R)-chains.
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(11) DependencyGraphProof (EQUIVALENT)
12.92/5.12	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(12)
12.92/5.12	Obligation:
12.92/5.12	Q DP problem:
12.92/5.12	The TRS P consists of the following rules:
12.92/5.12	
12.92/5.12	   new_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3)
12.92/5.12	
12.92/5.12	The TRS R consists of the following rules:
12.92/5.12	
12.92/5.12	   new_primMinusNat(Succ(zy4000)) -> Pos(Succ(zy4000))
12.92/5.12	   new_primMinusNat(Zero) -> Pos(Zero)
12.92/5.12	
12.92/5.12	The set Q consists of the following terms:
12.92/5.12	
12.92/5.12	   new_primMinusNat(Succ(x0))
12.92/5.12	   new_primMinusNat(Zero)
12.92/5.12	
12.92/5.12	We have to consider all minimal (P,Q,R)-chains.
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(13) UsableRulesProof (EQUIVALENT)
12.92/5.12	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.
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(14)
12.92/5.12	Obligation:
12.92/5.12	Q DP problem:
12.92/5.12	The TRS P consists of the following rules:
12.92/5.12	
12.92/5.12	   new_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3)
12.92/5.12	
12.92/5.12	R is empty.
12.92/5.12	The set Q consists of the following terms:
12.92/5.12	
12.92/5.12	   new_primMinusNat(Succ(x0))
12.92/5.12	   new_primMinusNat(Zero)
12.92/5.12	
12.92/5.12	We have to consider all minimal (P,Q,R)-chains.
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(15) QReductionProof (EQUIVALENT)
12.92/5.12	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
12.92/5.12	
12.92/5.12	   new_primMinusNat(Succ(x0))
12.92/5.12	   new_primMinusNat(Zero)
12.92/5.12	
12.92/5.12	
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(16)
12.92/5.12	Obligation:
12.92/5.12	Q DP problem:
12.92/5.12	The TRS P consists of the following rules:
12.92/5.12	
12.92/5.12	   new_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3)
12.92/5.12	
12.92/5.12	R is empty.
12.92/5.12	Q is empty.
12.92/5.12	We have to consider all minimal (P,Q,R)-chains.
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(17) QDPSizeChangeProof (EQUIVALENT)
12.92/5.12	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. 
12.92/5.12	
12.92/5.12	From the DPs we obtained the following set of size-change graphs:
12.92/5.12	*new_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3)
12.92/5.12	The graph contains the following edges 1 > 1, 3 >= 3
12.92/5.12	
12.92/5.12	
12.92/5.12	----------------------------------------
12.92/5.12	
12.92/5.12	(18)
12.92/5.12	YES
13.15/5.16	EOF