8.51/3.71	YES
10.25/4.23	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
10.25/4.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.25/4.23	
10.25/4.23	
10.25/4.23	H-Termination with start terms of the given HASKELL could be proven:
10.25/4.23	
10.25/4.23	(0) HASKELL
10.25/4.23	(1) IFR [EQUIVALENT, 0 ms]
10.25/4.23	(2) HASKELL
10.25/4.23	(3) BR [EQUIVALENT, 0 ms]
10.25/4.23	(4) HASKELL
10.25/4.23	(5) COR [EQUIVALENT, 0 ms]
10.25/4.23	(6) HASKELL
10.25/4.23	(7) Narrow [SOUND, 0 ms]
10.25/4.23	(8) AND
10.25/4.23	    (9) QDP
10.25/4.23	        (10) DependencyGraphProof [EQUIVALENT, 0 ms]
10.25/4.23	        (11) AND
10.25/4.23	            (12) QDP
10.25/4.23	                (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.25/4.23	                (14) YES
10.25/4.23	            (15) QDP
10.25/4.23	                (16) TransformationProof [EQUIVALENT, 0 ms]
10.25/4.23	                (17) QDP
10.25/4.23	                (18) TransformationProof [EQUIVALENT, 0 ms]
10.25/4.23	                (19) QDP
10.25/4.23	                (20) QDPSizeChangeProof [EQUIVALENT, 3 ms]
10.25/4.23	                (21) YES
10.25/4.23	    (22) QDP
10.25/4.23	        (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.25/4.23	        (24) YES
10.25/4.23	    (25) QDP
10.25/4.23	        (26) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.25/4.23	        (27) YES
10.25/4.23	
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(0)
10.25/4.23	Obligation:
10.25/4.23	mainModule Main 
10.25/4.23	module Main where {
10.25/4.23	import qualified Prelude;
10.25/4.23	}
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(1) IFR (EQUIVALENT)
10.25/4.23	If Reductions:
10.25/4.23	The following If expression
10.25/4.23	"if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero"
10.25/4.23	is transformed to
10.25/4.23	"primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y));
10.25/4.23	primDivNatS0 x y False = Zero;
10.25/4.23	"
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(2)
10.25/4.23	Obligation:
10.25/4.23	mainModule Main 
10.25/4.23	module Main where {
10.25/4.23	import qualified Prelude;
10.25/4.23	}
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(3) BR (EQUIVALENT)
10.25/4.23	Replaced joker patterns by fresh variables and removed binding patterns.
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(4)
10.25/4.23	Obligation:
10.25/4.23	mainModule Main 
10.25/4.23	module Main where {
10.25/4.23	import qualified Prelude;
10.25/4.23	}
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(5) COR (EQUIVALENT)
10.25/4.23	Cond Reductions:
10.25/4.23	The following Function with conditions
10.25/4.23	"undefined |Falseundefined;
10.25/4.23	"
10.25/4.23	is transformed to
10.25/4.23	"undefined  = undefined1;
10.25/4.23	"
10.25/4.23	"undefined0 True = undefined;
10.25/4.23	"
10.25/4.23	"undefined1  = undefined0 False;
10.25/4.23	"
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(6)
10.25/4.23	Obligation:
10.25/4.23	mainModule Main 
10.25/4.23	module Main where {
10.25/4.23	import qualified Prelude;
10.25/4.23	}
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(7) Narrow (SOUND)
10.25/4.23	Haskell To QDPs
10.25/4.23	
10.25/4.23	digraph dp_graph {
10.25/4.23	node [outthreshold=100, inthreshold=100];1[label="div",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.25/4.23	3[label="div vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.25/4.23	4[label="div vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.25/4.23	5[label="primDivInt vz3 vz4",fontsize=16,color="burlywood",shape="box"];288[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 288[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	288 -> 6[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	289[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 289[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	289 -> 7[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	6[label="primDivInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];290[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 290[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	290 -> 8[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	291[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 291[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	291 -> 9[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	7[label="primDivInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];292[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 292[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	292 -> 10[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	293[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 293[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	293 -> 11[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	8[label="primDivInt (Pos vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];294[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 294[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	294 -> 12[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	295[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 295[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	295 -> 13[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	9[label="primDivInt (Pos vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];296[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 296[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	296 -> 14[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	297[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 297[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	297 -> 15[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	10[label="primDivInt (Neg vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];298[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 298[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	298 -> 16[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	299[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 299[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	299 -> 17[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	11[label="primDivInt (Neg vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];300[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 300[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	300 -> 18[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	301[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 301[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	301 -> 19[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	12[label="primDivInt (Pos vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
10.25/4.23	13[label="primDivInt (Pos vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
10.25/4.23	14[label="primDivInt (Pos vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
10.25/4.23	15[label="primDivInt (Pos vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
10.25/4.23	16[label="primDivInt (Neg vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
10.25/4.23	17[label="primDivInt (Neg vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
10.25/4.23	18[label="primDivInt (Neg vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
10.25/4.23	19[label="primDivInt (Neg vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
10.25/4.23	20[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3];
10.25/4.23	21[label="error []",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3];
10.25/4.23	22[label="Neg (primDivNatP vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];22 -> 30[label="",style="dashed", color="green", weight=3];
10.25/4.23	23 -> 21[label="",style="dashed", color="red", weight=0];
10.25/4.23	23[label="error []",fontsize=16,color="magenta"];24[label="Neg (primDivNatP vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];24 -> 31[label="",style="dashed", color="green", weight=3];
10.25/4.23	25 -> 21[label="",style="dashed", color="red", weight=0];
10.25/4.23	25[label="error []",fontsize=16,color="magenta"];26[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];26 -> 32[label="",style="dashed", color="green", weight=3];
10.25/4.23	27 -> 21[label="",style="dashed", color="red", weight=0];
10.25/4.23	27[label="error []",fontsize=16,color="magenta"];28[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];302[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 302[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	302 -> 33[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	303[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 303[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	303 -> 34[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	29[label="error []",fontsize=16,color="red",shape="box"];30[label="primDivNatP vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];304[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];30 -> 304[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	304 -> 35[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	305[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 305[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	305 -> 36[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	31 -> 30[label="",style="dashed", color="red", weight=0];
10.25/4.23	31[label="primDivNatP vz30 (Succ vz400)",fontsize=16,color="magenta"];31 -> 37[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	31 -> 38[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	32 -> 28[label="",style="dashed", color="red", weight=0];
10.25/4.23	32[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];32 -> 39[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	32 -> 40[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	33[label="primDivNatS (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];33 -> 41[label="",style="solid", color="black", weight=3];
10.25/4.23	34[label="primDivNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 42[label="",style="solid", color="black", weight=3];
10.25/4.23	35[label="primDivNatP (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];35 -> 43[label="",style="solid", color="black", weight=3];
10.25/4.23	36[label="primDivNatP Zero (Succ vz400)",fontsize=16,color="black",shape="box"];36 -> 44[label="",style="solid", color="black", weight=3];
10.25/4.23	37[label="vz30",fontsize=16,color="green",shape="box"];38[label="vz400",fontsize=16,color="green",shape="box"];39[label="vz30",fontsize=16,color="green",shape="box"];40[label="vz400",fontsize=16,color="green",shape="box"];41[label="primDivNatS0 vz300 vz400 (primGEqNatS vz300 vz400)",fontsize=16,color="burlywood",shape="box"];306[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];41 -> 306[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	306 -> 45[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	307[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 307[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	307 -> 46[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	42[label="Zero",fontsize=16,color="green",shape="box"];43[label="Succ (primDivNatP (primMinusNatS vz300 vz400) (Succ vz400))",fontsize=16,color="green",shape="box"];43 -> 47[label="",style="dashed", color="green", weight=3];
10.25/4.23	44[label="Zero",fontsize=16,color="green",shape="box"];45[label="primDivNatS0 (Succ vz3000) vz400 (primGEqNatS (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];308[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];45 -> 308[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	308 -> 48[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	309[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 309[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	309 -> 49[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	46[label="primDivNatS0 Zero vz400 (primGEqNatS Zero vz400)",fontsize=16,color="burlywood",shape="box"];310[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];46 -> 310[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	310 -> 50[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	311[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 311[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	311 -> 51[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	47 -> 30[label="",style="dashed", color="red", weight=0];
10.25/4.23	47[label="primDivNatP (primMinusNatS vz300 vz400) (Succ vz400)",fontsize=16,color="magenta"];47 -> 52[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	48[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];48 -> 53[label="",style="solid", color="black", weight=3];
10.25/4.23	49[label="primDivNatS0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];49 -> 54[label="",style="solid", color="black", weight=3];
10.25/4.23	50[label="primDivNatS0 Zero (Succ vz4000) (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3];
10.25/4.23	51[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];51 -> 56[label="",style="solid", color="black", weight=3];
10.25/4.23	52[label="primMinusNatS vz300 vz400",fontsize=16,color="burlywood",shape="triangle"];312[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];52 -> 312[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	312 -> 57[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	313[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 313[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	313 -> 58[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	53 -> 236[label="",style="dashed", color="red", weight=0];
10.25/4.23	53[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)",fontsize=16,color="magenta"];53 -> 237[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	53 -> 238[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	53 -> 239[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	53 -> 240[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	54[label="primDivNatS0 (Succ vz3000) Zero True",fontsize=16,color="black",shape="box"];54 -> 61[label="",style="solid", color="black", weight=3];
10.25/4.23	55[label="primDivNatS0 Zero (Succ vz4000) False",fontsize=16,color="black",shape="box"];55 -> 62[label="",style="solid", color="black", weight=3];
10.25/4.23	56[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];56 -> 63[label="",style="solid", color="black", weight=3];
10.25/4.23	57[label="primMinusNatS (Succ vz3000) vz400",fontsize=16,color="burlywood",shape="box"];314[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];57 -> 314[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	314 -> 64[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	315[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];57 -> 315[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	315 -> 65[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	58[label="primMinusNatS Zero vz400",fontsize=16,color="burlywood",shape="box"];316[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];58 -> 316[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	316 -> 66[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	317[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 317[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	317 -> 67[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	237[label="vz4000",fontsize=16,color="green",shape="box"];238[label="vz3000",fontsize=16,color="green",shape="box"];239[label="vz4000",fontsize=16,color="green",shape="box"];240[label="vz3000",fontsize=16,color="green",shape="box"];236[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz23 vz24)",fontsize=16,color="burlywood",shape="triangle"];318[label="vz23/Succ vz230",fontsize=10,color="white",style="solid",shape="box"];236 -> 318[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	318 -> 269[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	319[label="vz23/Zero",fontsize=10,color="white",style="solid",shape="box"];236 -> 319[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	319 -> 270[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	61[label="Succ (primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];61 -> 72[label="",style="dashed", color="green", weight=3];
10.25/4.23	62[label="Zero",fontsize=16,color="green",shape="box"];63[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];63 -> 73[label="",style="dashed", color="green", weight=3];
10.25/4.23	64[label="primMinusNatS (Succ vz3000) (Succ vz4000)",fontsize=16,color="black",shape="box"];64 -> 74[label="",style="solid", color="black", weight=3];
10.25/4.23	65[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="black",shape="box"];65 -> 75[label="",style="solid", color="black", weight=3];
10.25/4.23	66[label="primMinusNatS Zero (Succ vz4000)",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3];
10.25/4.23	67[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];67 -> 77[label="",style="solid", color="black", weight=3];
10.25/4.23	269[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) vz24)",fontsize=16,color="burlywood",shape="box"];320[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];269 -> 320[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	320 -> 271[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	321[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];269 -> 321[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	321 -> 272[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	270[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero vz24)",fontsize=16,color="burlywood",shape="box"];322[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];270 -> 322[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	322 -> 273[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	323[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];270 -> 323[label="",style="solid", color="burlywood", weight=9];
10.25/4.23	323 -> 274[label="",style="solid", color="burlywood", weight=3];
10.25/4.23	72 -> 28[label="",style="dashed", color="red", weight=0];
10.25/4.23	72[label="primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];72 -> 82[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	72 -> 83[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	73 -> 28[label="",style="dashed", color="red", weight=0];
10.25/4.23	73[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];73 -> 84[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	73 -> 85[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	74 -> 52[label="",style="dashed", color="red", weight=0];
10.25/4.23	74[label="primMinusNatS vz3000 vz4000",fontsize=16,color="magenta"];74 -> 86[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	74 -> 87[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	75[label="Succ vz3000",fontsize=16,color="green",shape="box"];76[label="Zero",fontsize=16,color="green",shape="box"];77[label="Zero",fontsize=16,color="green",shape="box"];271[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) (Succ vz240))",fontsize=16,color="black",shape="box"];271 -> 275[label="",style="solid", color="black", weight=3];
10.25/4.23	272[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) Zero)",fontsize=16,color="black",shape="box"];272 -> 276[label="",style="solid", color="black", weight=3];
10.25/4.23	273[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero (Succ vz240))",fontsize=16,color="black",shape="box"];273 -> 277[label="",style="solid", color="black", weight=3];
10.25/4.23	274[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];274 -> 278[label="",style="solid", color="black", weight=3];
10.25/4.23	82 -> 52[label="",style="dashed", color="red", weight=0];
10.25/4.23	82[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="magenta"];82 -> 93[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	82 -> 94[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	83[label="Zero",fontsize=16,color="green",shape="box"];84 -> 52[label="",style="dashed", color="red", weight=0];
10.25/4.23	84[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];84 -> 95[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	84 -> 96[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	85[label="Zero",fontsize=16,color="green",shape="box"];86[label="vz4000",fontsize=16,color="green",shape="box"];87[label="vz3000",fontsize=16,color="green",shape="box"];275 -> 236[label="",style="dashed", color="red", weight=0];
10.25/4.23	275[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz230 vz240)",fontsize=16,color="magenta"];275 -> 279[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	275 -> 280[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	276[label="primDivNatS0 (Succ vz21) (Succ vz22) True",fontsize=16,color="black",shape="triangle"];276 -> 281[label="",style="solid", color="black", weight=3];
10.25/4.23	277[label="primDivNatS0 (Succ vz21) (Succ vz22) False",fontsize=16,color="black",shape="box"];277 -> 282[label="",style="solid", color="black", weight=3];
10.25/4.23	278 -> 276[label="",style="dashed", color="red", weight=0];
10.25/4.23	278[label="primDivNatS0 (Succ vz21) (Succ vz22) True",fontsize=16,color="magenta"];93[label="Zero",fontsize=16,color="green",shape="box"];94[label="Succ vz3000",fontsize=16,color="green",shape="box"];95[label="Zero",fontsize=16,color="green",shape="box"];96[label="Zero",fontsize=16,color="green",shape="box"];279[label="vz240",fontsize=16,color="green",shape="box"];280[label="vz230",fontsize=16,color="green",shape="box"];281[label="Succ (primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22)))",fontsize=16,color="green",shape="box"];281 -> 283[label="",style="dashed", color="green", weight=3];
10.25/4.23	282[label="Zero",fontsize=16,color="green",shape="box"];283 -> 28[label="",style="dashed", color="red", weight=0];
10.25/4.23	283[label="primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22))",fontsize=16,color="magenta"];283 -> 284[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	283 -> 285[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	284 -> 52[label="",style="dashed", color="red", weight=0];
10.25/4.23	284[label="primMinusNatS (Succ vz21) (Succ vz22)",fontsize=16,color="magenta"];284 -> 286[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	284 -> 287[label="",style="dashed", color="magenta", weight=3];
10.25/4.23	285[label="Succ vz22",fontsize=16,color="green",shape="box"];286[label="Succ vz22",fontsize=16,color="green",shape="box"];287[label="Succ vz21",fontsize=16,color="green",shape="box"];}
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(8)
10.25/4.23	Complex Obligation (AND)
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(9)
10.25/4.23	Obligation:
10.25/4.23	Q DP problem:
10.25/4.23	The TRS P consists of the following rules:
10.25/4.23	
10.25/4.23	   new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
10.25/4.23	   new_primDivNatS0(vz21, vz22, Zero, Zero) -> new_primDivNatS00(vz21, vz22)
10.25/4.23	   new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22))
10.25/4.23	   new_primDivNatS(Succ(Succ(vz3000)), Zero) -> new_primDivNatS(new_primMinusNatS0(Succ(vz3000), Zero), Zero)
10.25/4.23	   new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240)
10.25/4.23	   new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22))
10.25/4.23	   new_primDivNatS(Succ(Zero), Zero) -> new_primDivNatS(new_primMinusNatS0(Zero, Zero), Zero)
10.25/4.23	
10.25/4.23	The TRS R consists of the following rules:
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Succ(vz4000)) -> Zero
10.25/4.23	   new_primMinusNatS0(Zero, Zero) -> Zero
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Zero) -> Succ(vz3000)
10.25/4.23	
10.25/4.23	The set Q consists of the following terms:
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Zero)
10.25/4.23	   new_primMinusNatS0(Succ(x0), Zero)
10.25/4.23	   new_primMinusNatS0(Succ(x0), Succ(x1))
10.25/4.23	   new_primMinusNatS0(Zero, Succ(x0))
10.25/4.23	
10.25/4.23	We have to consider all minimal (P,Q,R)-chains.
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(10) DependencyGraphProof (EQUIVALENT)
10.25/4.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(11)
10.25/4.23	Complex Obligation (AND)
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(12)
10.25/4.23	Obligation:
10.25/4.23	Q DP problem:
10.25/4.23	The TRS P consists of the following rules:
10.25/4.23	
10.25/4.23	   new_primDivNatS(Succ(Succ(vz3000)), Zero) -> new_primDivNatS(new_primMinusNatS0(Succ(vz3000), Zero), Zero)
10.25/4.23	
10.25/4.23	The TRS R consists of the following rules:
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Succ(vz4000)) -> Zero
10.25/4.23	   new_primMinusNatS0(Zero, Zero) -> Zero
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Zero) -> Succ(vz3000)
10.25/4.23	
10.25/4.23	The set Q consists of the following terms:
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Zero)
10.25/4.23	   new_primMinusNatS0(Succ(x0), Zero)
10.25/4.23	   new_primMinusNatS0(Succ(x0), Succ(x1))
10.25/4.23	   new_primMinusNatS0(Zero, Succ(x0))
10.25/4.23	
10.25/4.23	We have to consider all minimal (P,Q,R)-chains.
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(13) QDPSizeChangeProof (EQUIVALENT)
10.25/4.23	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
10.25/4.23	
10.25/4.23	Order:Polynomial interpretation [POLO]:
10.25/4.23	
10.25/4.23	   POL(Succ(x_1)) = 1 + x_1
10.25/4.23	   POL(Zero) = 1
10.25/4.23	   POL(new_primMinusNatS0(x_1, x_2)) = x_1
10.25/4.23	
10.25/4.23	
10.25/4.23	
10.25/4.23	
10.25/4.23	From the DPs we obtained the following set of size-change graphs:
10.25/4.23	*new_primDivNatS(Succ(Succ(vz3000)), Zero) -> new_primDivNatS(new_primMinusNatS0(Succ(vz3000), Zero), Zero) (allowed arguments on rhs = {1, 2})
10.25/4.23	The graph contains the following edges 1 > 1, 2 >= 2
10.25/4.23	
10.25/4.23	
10.25/4.23	
10.25/4.23	We oriented the following set of usable rules [AAECC05,FROCOS05].
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Zero) -> Succ(vz3000)
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(14)
10.25/4.23	YES
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(15)
10.25/4.23	Obligation:
10.25/4.23	Q DP problem:
10.25/4.23	The TRS P consists of the following rules:
10.25/4.23	
10.25/4.23	   new_primDivNatS0(vz21, vz22, Zero, Zero) -> new_primDivNatS00(vz21, vz22)
10.25/4.23	   new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22))
10.25/4.23	   new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
10.25/4.23	   new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22))
10.25/4.23	   new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240)
10.25/4.23	
10.25/4.23	The TRS R consists of the following rules:
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Succ(vz4000)) -> Zero
10.25/4.23	   new_primMinusNatS0(Zero, Zero) -> Zero
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Zero) -> Succ(vz3000)
10.25/4.23	
10.25/4.23	The set Q consists of the following terms:
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Zero)
10.25/4.23	   new_primMinusNatS0(Succ(x0), Zero)
10.25/4.23	   new_primMinusNatS0(Succ(x0), Succ(x1))
10.25/4.23	   new_primMinusNatS0(Zero, Succ(x0))
10.25/4.23	
10.25/4.23	We have to consider all minimal (P,Q,R)-chains.
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(16) TransformationProof (EQUIVALENT)
10.25/4.23	By rewriting [LPAR04] the rule new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22)) at position [0] we obtained the following new rules [LPAR04]:
10.25/4.23	
10.25/4.23	   (new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)),new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)))
10.25/4.23	
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(17)
10.25/4.23	Obligation:
10.25/4.23	Q DP problem:
10.25/4.23	The TRS P consists of the following rules:
10.25/4.23	
10.25/4.23	   new_primDivNatS0(vz21, vz22, Zero, Zero) -> new_primDivNatS00(vz21, vz22)
10.25/4.23	   new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
10.25/4.23	   new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22))
10.25/4.23	   new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240)
10.25/4.23	   new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
10.25/4.23	
10.25/4.23	The TRS R consists of the following rules:
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Succ(vz4000)) -> Zero
10.25/4.23	   new_primMinusNatS0(Zero, Zero) -> Zero
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Zero) -> Succ(vz3000)
10.25/4.23	
10.25/4.23	The set Q consists of the following terms:
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Zero)
10.25/4.23	   new_primMinusNatS0(Succ(x0), Zero)
10.25/4.23	   new_primMinusNatS0(Succ(x0), Succ(x1))
10.25/4.23	   new_primMinusNatS0(Zero, Succ(x0))
10.25/4.23	
10.25/4.23	We have to consider all minimal (P,Q,R)-chains.
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(18) TransformationProof (EQUIVALENT)
10.25/4.23	By rewriting [LPAR04] the rule new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22)) at position [0] we obtained the following new rules [LPAR04]:
10.25/4.23	
10.25/4.23	   (new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)),new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)))
10.25/4.23	
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(19)
10.25/4.23	Obligation:
10.25/4.23	Q DP problem:
10.25/4.23	The TRS P consists of the following rules:
10.25/4.23	
10.25/4.23	   new_primDivNatS0(vz21, vz22, Zero, Zero) -> new_primDivNatS00(vz21, vz22)
10.25/4.23	   new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
10.25/4.23	   new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240)
10.25/4.23	   new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
10.25/4.23	   new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
10.25/4.23	
10.25/4.23	The TRS R consists of the following rules:
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Succ(vz4000)) -> Zero
10.25/4.23	   new_primMinusNatS0(Zero, Zero) -> Zero
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Zero) -> Succ(vz3000)
10.25/4.23	
10.25/4.23	The set Q consists of the following terms:
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Zero)
10.25/4.23	   new_primMinusNatS0(Succ(x0), Zero)
10.25/4.23	   new_primMinusNatS0(Succ(x0), Succ(x1))
10.25/4.23	   new_primMinusNatS0(Zero, Succ(x0))
10.25/4.23	
10.25/4.23	We have to consider all minimal (P,Q,R)-chains.
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(20) QDPSizeChangeProof (EQUIVALENT)
10.25/4.23	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
10.25/4.23	
10.25/4.23	Order:Polynomial interpretation [POLO]:
10.25/4.23	
10.25/4.23	   POL(Succ(x_1)) = 1 + x_1
10.25/4.23	   POL(Zero) = 1
10.25/4.23	   POL(new_primMinusNatS0(x_1, x_2)) = x_1
10.25/4.23	
10.25/4.23	
10.25/4.23	
10.25/4.23	
10.25/4.23	From the DPs we obtained the following set of size-change graphs:
10.25/4.23	*new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) (allowed arguments on rhs = {1, 2})
10.25/4.23	The graph contains the following edges 1 >= 1
10.25/4.23	
10.25/4.23	
10.25/4.23	*new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) (allowed arguments on rhs = {1, 2, 3, 4})
10.25/4.23	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
10.25/4.23	
10.25/4.23	
10.25/4.23	*new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) (allowed arguments on rhs = {1, 2, 3, 4})
10.25/4.23	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
10.25/4.23	
10.25/4.23	
10.25/4.23	*new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) (allowed arguments on rhs = {1, 2})
10.25/4.23	The graph contains the following edges 1 >= 1
10.25/4.23	
10.25/4.23	
10.25/4.23	*new_primDivNatS0(vz21, vz22, Zero, Zero) -> new_primDivNatS00(vz21, vz22) (allowed arguments on rhs = {1, 2})
10.25/4.23	The graph contains the following edges 1 >= 1, 2 >= 2
10.25/4.23	
10.25/4.23	
10.25/4.23	
10.25/4.23	We oriented the following set of usable rules [AAECC05,FROCOS05].
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Zero) -> Zero
10.25/4.23	   new_primMinusNatS0(Zero, Succ(vz4000)) -> Zero
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Zero) -> Succ(vz3000)
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(21)
10.25/4.23	YES
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(22)
10.25/4.23	Obligation:
10.25/4.23	Q DP problem:
10.25/4.23	The TRS P consists of the following rules:
10.25/4.23	
10.25/4.23	   new_primMinusNatS(Succ(vz3000), Succ(vz4000)) -> new_primMinusNatS(vz3000, vz4000)
10.25/4.23	
10.25/4.23	R is empty.
10.25/4.23	Q is empty.
10.25/4.23	We have to consider all minimal (P,Q,R)-chains.
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(23) QDPSizeChangeProof (EQUIVALENT)
10.25/4.23	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. 
10.25/4.23	
10.25/4.23	From the DPs we obtained the following set of size-change graphs:
10.25/4.23	*new_primMinusNatS(Succ(vz3000), Succ(vz4000)) -> new_primMinusNatS(vz3000, vz4000)
10.25/4.23	The graph contains the following edges 1 > 1, 2 > 2
10.25/4.23	
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(24)
10.25/4.23	YES
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(25)
10.25/4.23	Obligation:
10.25/4.23	Q DP problem:
10.25/4.23	The TRS P consists of the following rules:
10.25/4.23	
10.25/4.23	   new_primDivNatP(Succ(vz300), vz400) -> new_primDivNatP(new_primMinusNatS0(vz300, vz400), vz400)
10.25/4.23	
10.25/4.23	The TRS R consists of the following rules:
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Succ(vz4000)) -> Zero
10.25/4.23	   new_primMinusNatS0(Zero, Zero) -> Zero
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Zero) -> Succ(vz3000)
10.25/4.23	
10.25/4.23	The set Q consists of the following terms:
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Zero)
10.25/4.23	   new_primMinusNatS0(Succ(x0), Zero)
10.25/4.23	   new_primMinusNatS0(Succ(x0), Succ(x1))
10.25/4.23	   new_primMinusNatS0(Zero, Succ(x0))
10.25/4.23	
10.25/4.23	We have to consider all minimal (P,Q,R)-chains.
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(26) QDPSizeChangeProof (EQUIVALENT)
10.25/4.23	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
10.25/4.23	
10.25/4.23	Order:Polynomial interpretation [POLO]:
10.25/4.23	
10.25/4.23	   POL(Succ(x_1)) = 1 + x_1
10.25/4.23	   POL(Zero) = 0
10.25/4.23	   POL(new_primMinusNatS0(x_1, x_2)) = x_1
10.25/4.23	
10.25/4.23	
10.25/4.23	
10.25/4.23	
10.25/4.23	From the DPs we obtained the following set of size-change graphs:
10.25/4.23	*new_primDivNatP(Succ(vz300), vz400) -> new_primDivNatP(new_primMinusNatS0(vz300, vz400), vz400) (allowed arguments on rhs = {1, 2})
10.25/4.23	The graph contains the following edges 1 > 1, 2 >= 2
10.25/4.23	
10.25/4.23	
10.25/4.23	
10.25/4.23	We oriented the following set of usable rules [AAECC05,FROCOS05].
10.25/4.23	
10.25/4.23	   new_primMinusNatS0(Zero, Zero) -> Zero
10.25/4.23	   new_primMinusNatS0(Zero, Succ(vz4000)) -> Zero
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Zero) -> Succ(vz3000)
10.25/4.23	   new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.25/4.23	
10.25/4.23	----------------------------------------
10.25/4.23	
10.25/4.23	(27)
10.25/4.23	YES
10.51/4.27	EOF