8.30/3.72	YES
10.41/4.24	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.41/4.24	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.41/4.24	
10.41/4.24	
10.41/4.24	H-Termination with start terms of the given HASKELL could be proven:
10.41/4.24	
10.41/4.24	(0) HASKELL
10.41/4.24	(1) IFR [EQUIVALENT, 0 ms]
10.41/4.24	(2) HASKELL
10.41/4.24	(3) BR [EQUIVALENT, 0 ms]
10.41/4.24	(4) HASKELL
10.41/4.24	(5) COR [EQUIVALENT, 0 ms]
10.41/4.24	(6) HASKELL
10.41/4.24	(7) Narrow [SOUND, 0 ms]
10.41/4.24	(8) AND
10.41/4.24	    (9) QDP
10.41/4.24	        (10) DependencyGraphProof [EQUIVALENT, 0 ms]
10.41/4.24	        (11) AND
10.41/4.24	            (12) QDP
10.41/4.24	                (13) MRRProof [EQUIVALENT, 66 ms]
10.41/4.24	                (14) QDP
10.41/4.24	                (15) PisEmptyProof [EQUIVALENT, 0 ms]
10.41/4.24	                (16) YES
10.41/4.24	            (17) QDP
10.41/4.24	                (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.41/4.24	                (19) YES
10.41/4.24	    (20) QDP
10.41/4.24	        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.41/4.24	        (22) YES
10.41/4.24	
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(0)
10.41/4.24	Obligation:
10.41/4.24	mainModule Main 
10.41/4.24	module Main where {
10.41/4.24	import qualified Prelude;
10.41/4.24	}
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(1) IFR (EQUIVALENT)
10.41/4.24	If Reductions:
10.41/4.24	The following If expression
10.41/4.24	"if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x"
10.41/4.24	is transformed to
10.41/4.24	"primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y);
10.41/4.24	primModNatS0 x y False = Succ x;
10.41/4.24	"
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(2)
10.41/4.24	Obligation:
10.41/4.24	mainModule Main 
10.41/4.24	module Main where {
10.41/4.24	import qualified Prelude;
10.41/4.24	}
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(3) BR (EQUIVALENT)
10.41/4.24	Replaced joker patterns by fresh variables and removed binding patterns.
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(4)
10.41/4.24	Obligation:
10.41/4.24	mainModule Main 
10.41/4.24	module Main where {
10.41/4.24	import qualified Prelude;
10.41/4.24	}
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(5) COR (EQUIVALENT)
10.41/4.24	Cond Reductions:
10.41/4.24	The following Function with conditions
10.41/4.24	"undefined |Falseundefined;
10.41/4.24	"
10.41/4.24	is transformed to
10.41/4.24	"undefined  = undefined1;
10.41/4.24	"
10.41/4.24	"undefined0 True = undefined;
10.41/4.24	"
10.41/4.24	"undefined1  = undefined0 False;
10.41/4.24	"
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(6)
10.41/4.24	Obligation:
10.41/4.24	mainModule Main 
10.41/4.24	module Main where {
10.41/4.24	import qualified Prelude;
10.41/4.24	}
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(7) Narrow (SOUND)
10.41/4.24	Haskell To QDPs
10.41/4.24	
10.41/4.24	digraph dp_graph {
10.41/4.24	node [outthreshold=100, inthreshold=100];1[label="rem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.41/4.24	3[label="rem vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.41/4.24	4[label="rem vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.41/4.24	5[label="primRemInt vz3 vz4",fontsize=16,color="burlywood",shape="box"];264[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 264[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	264 -> 6[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	265[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 265[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	265 -> 7[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	6[label="primRemInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];266[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 266[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	266 -> 8[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	267[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 267[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	267 -> 9[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	7[label="primRemInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];268[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 268[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	268 -> 10[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	269[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 269[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	269 -> 11[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	8[label="primRemInt (Pos vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];270[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 270[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	270 -> 12[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	271[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 271[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	271 -> 13[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	9[label="primRemInt (Pos vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];272[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 272[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	272 -> 14[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	273[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 273[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	273 -> 15[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	10[label="primRemInt (Neg vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];274[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 274[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	274 -> 16[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	275[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 275[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	275 -> 17[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	11[label="primRemInt (Neg vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];276[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 276[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	276 -> 18[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	277[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 277[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	277 -> 19[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	12[label="primRemInt (Pos vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
10.41/4.24	13[label="primRemInt (Pos vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
10.41/4.24	14[label="primRemInt (Pos vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
10.41/4.24	15[label="primRemInt (Pos vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
10.41/4.24	16[label="primRemInt (Neg vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
10.41/4.24	17[label="primRemInt (Neg vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
10.41/4.24	18[label="primRemInt (Neg vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
10.41/4.24	19[label="primRemInt (Neg vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
10.41/4.24	20[label="Pos (primModNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3];
10.41/4.24	21[label="error []",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3];
10.41/4.24	22[label="Pos (primModNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];22 -> 30[label="",style="dashed", color="green", weight=3];
10.41/4.24	23 -> 21[label="",style="dashed", color="red", weight=0];
10.41/4.24	23[label="error []",fontsize=16,color="magenta"];24[label="Neg (primModNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];24 -> 31[label="",style="dashed", color="green", weight=3];
10.41/4.24	25 -> 21[label="",style="dashed", color="red", weight=0];
10.41/4.24	25[label="error []",fontsize=16,color="magenta"];26[label="Neg (primModNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];26 -> 32[label="",style="dashed", color="green", weight=3];
10.41/4.24	27 -> 21[label="",style="dashed", color="red", weight=0];
10.41/4.24	27[label="error []",fontsize=16,color="magenta"];28[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];278[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 278[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	278 -> 33[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	279[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 279[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	279 -> 34[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	29[label="error []",fontsize=16,color="red",shape="box"];30 -> 28[label="",style="dashed", color="red", weight=0];
10.41/4.24	30[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];30 -> 35[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	31 -> 28[label="",style="dashed", color="red", weight=0];
10.41/4.24	31[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];31 -> 36[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	32 -> 28[label="",style="dashed", color="red", weight=0];
10.41/4.24	32[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];32 -> 37[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	32 -> 38[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	33[label="primModNatS (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];33 -> 39[label="",style="solid", color="black", weight=3];
10.41/4.24	34[label="primModNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 40[label="",style="solid", color="black", weight=3];
10.41/4.24	35[label="vz400",fontsize=16,color="green",shape="box"];36[label="vz30",fontsize=16,color="green",shape="box"];37[label="vz400",fontsize=16,color="green",shape="box"];38[label="vz30",fontsize=16,color="green",shape="box"];39[label="primModNatS0 vz300 vz400 (primGEqNatS vz300 vz400)",fontsize=16,color="burlywood",shape="box"];280[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];39 -> 280[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	280 -> 41[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	281[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];39 -> 281[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	281 -> 42[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	40[label="Zero",fontsize=16,color="green",shape="box"];41[label="primModNatS0 (Succ vz3000) vz400 (primGEqNatS (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];282[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];41 -> 282[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	282 -> 43[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	283[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 283[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	283 -> 44[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	42[label="primModNatS0 Zero vz400 (primGEqNatS Zero vz400)",fontsize=16,color="burlywood",shape="box"];284[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];42 -> 284[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	284 -> 45[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	285[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 285[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	285 -> 46[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	43[label="primModNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];43 -> 47[label="",style="solid", color="black", weight=3];
10.41/4.24	44[label="primModNatS0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3];
10.41/4.24	45[label="primModNatS0 Zero (Succ vz4000) (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3];
10.41/4.24	46[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3];
10.41/4.24	47 -> 202[label="",style="dashed", color="red", weight=0];
10.41/4.24	47[label="primModNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)",fontsize=16,color="magenta"];47 -> 203[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	47 -> 204[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	47 -> 205[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	47 -> 206[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	48[label="primModNatS0 (Succ vz3000) Zero True",fontsize=16,color="black",shape="box"];48 -> 53[label="",style="solid", color="black", weight=3];
10.41/4.24	49[label="primModNatS0 Zero (Succ vz4000) False",fontsize=16,color="black",shape="box"];49 -> 54[label="",style="solid", color="black", weight=3];
10.41/4.24	50[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3];
10.41/4.24	203[label="vz4000",fontsize=16,color="green",shape="box"];204[label="vz3000",fontsize=16,color="green",shape="box"];205[label="vz4000",fontsize=16,color="green",shape="box"];206[label="vz3000",fontsize=16,color="green",shape="box"];202[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz23 vz24)",fontsize=16,color="burlywood",shape="triangle"];286[label="vz23/Succ vz230",fontsize=10,color="white",style="solid",shape="box"];202 -> 286[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	286 -> 235[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	287[label="vz23/Zero",fontsize=10,color="white",style="solid",shape="box"];202 -> 287[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	287 -> 236[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	53 -> 28[label="",style="dashed", color="red", weight=0];
10.41/4.24	53[label="primModNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];53 -> 60[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	53 -> 61[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	54[label="Succ Zero",fontsize=16,color="green",shape="box"];55 -> 28[label="",style="dashed", color="red", weight=0];
10.41/4.24	55[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];55 -> 62[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	55 -> 63[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	235[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) vz24)",fontsize=16,color="burlywood",shape="box"];288[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];235 -> 288[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	288 -> 237[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	289[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];235 -> 289[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	289 -> 238[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	236[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero vz24)",fontsize=16,color="burlywood",shape="box"];290[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];236 -> 290[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	290 -> 239[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	291[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];236 -> 291[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	291 -> 240[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	60[label="Zero",fontsize=16,color="green",shape="box"];61[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="black",shape="triangle"];61 -> 68[label="",style="solid", color="black", weight=3];
10.41/4.24	62[label="Zero",fontsize=16,color="green",shape="box"];63[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];63 -> 69[label="",style="solid", color="black", weight=3];
10.41/4.24	237[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) (Succ vz240))",fontsize=16,color="black",shape="box"];237 -> 241[label="",style="solid", color="black", weight=3];
10.41/4.24	238[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) Zero)",fontsize=16,color="black",shape="box"];238 -> 242[label="",style="solid", color="black", weight=3];
10.41/4.24	239[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero (Succ vz240))",fontsize=16,color="black",shape="box"];239 -> 243[label="",style="solid", color="black", weight=3];
10.41/4.24	240[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];240 -> 244[label="",style="solid", color="black", weight=3];
10.41/4.24	68[label="Succ vz3000",fontsize=16,color="green",shape="box"];69[label="Zero",fontsize=16,color="green",shape="box"];241 -> 202[label="",style="dashed", color="red", weight=0];
10.41/4.24	241[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz230 vz240)",fontsize=16,color="magenta"];241 -> 245[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	241 -> 246[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	242[label="primModNatS0 (Succ vz21) (Succ vz22) True",fontsize=16,color="black",shape="triangle"];242 -> 247[label="",style="solid", color="black", weight=3];
10.41/4.24	243[label="primModNatS0 (Succ vz21) (Succ vz22) False",fontsize=16,color="black",shape="box"];243 -> 248[label="",style="solid", color="black", weight=3];
10.41/4.24	244 -> 242[label="",style="dashed", color="red", weight=0];
10.41/4.24	244[label="primModNatS0 (Succ vz21) (Succ vz22) True",fontsize=16,color="magenta"];245[label="vz240",fontsize=16,color="green",shape="box"];246[label="vz230",fontsize=16,color="green",shape="box"];247 -> 28[label="",style="dashed", color="red", weight=0];
10.41/4.24	247[label="primModNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22))",fontsize=16,color="magenta"];247 -> 249[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	247 -> 250[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	248[label="Succ (Succ vz21)",fontsize=16,color="green",shape="box"];249[label="Succ vz22",fontsize=16,color="green",shape="box"];250[label="primMinusNatS (Succ vz21) (Succ vz22)",fontsize=16,color="black",shape="box"];250 -> 251[label="",style="solid", color="black", weight=3];
10.41/4.24	251[label="primMinusNatS vz21 vz22",fontsize=16,color="burlywood",shape="triangle"];292[label="vz21/Succ vz210",fontsize=10,color="white",style="solid",shape="box"];251 -> 292[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	292 -> 252[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	293[label="vz21/Zero",fontsize=10,color="white",style="solid",shape="box"];251 -> 293[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	293 -> 253[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	252[label="primMinusNatS (Succ vz210) vz22",fontsize=16,color="burlywood",shape="box"];294[label="vz22/Succ vz220",fontsize=10,color="white",style="solid",shape="box"];252 -> 294[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	294 -> 254[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	295[label="vz22/Zero",fontsize=10,color="white",style="solid",shape="box"];252 -> 295[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	295 -> 255[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	253[label="primMinusNatS Zero vz22",fontsize=16,color="burlywood",shape="box"];296[label="vz22/Succ vz220",fontsize=10,color="white",style="solid",shape="box"];253 -> 296[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	296 -> 256[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	297[label="vz22/Zero",fontsize=10,color="white",style="solid",shape="box"];253 -> 297[label="",style="solid", color="burlywood", weight=9];
10.41/4.24	297 -> 257[label="",style="solid", color="burlywood", weight=3];
10.41/4.24	254[label="primMinusNatS (Succ vz210) (Succ vz220)",fontsize=16,color="black",shape="box"];254 -> 258[label="",style="solid", color="black", weight=3];
10.41/4.24	255[label="primMinusNatS (Succ vz210) Zero",fontsize=16,color="black",shape="box"];255 -> 259[label="",style="solid", color="black", weight=3];
10.41/4.24	256[label="primMinusNatS Zero (Succ vz220)",fontsize=16,color="black",shape="box"];256 -> 260[label="",style="solid", color="black", weight=3];
10.41/4.24	257[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];257 -> 261[label="",style="solid", color="black", weight=3];
10.41/4.24	258 -> 251[label="",style="dashed", color="red", weight=0];
10.41/4.24	258[label="primMinusNatS vz210 vz220",fontsize=16,color="magenta"];258 -> 262[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	258 -> 263[label="",style="dashed", color="magenta", weight=3];
10.41/4.24	259[label="Succ vz210",fontsize=16,color="green",shape="box"];260[label="Zero",fontsize=16,color="green",shape="box"];261[label="Zero",fontsize=16,color="green",shape="box"];262[label="vz220",fontsize=16,color="green",shape="box"];263[label="vz210",fontsize=16,color="green",shape="box"];}
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(8)
10.41/4.24	Complex Obligation (AND)
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(9)
10.41/4.24	Obligation:
10.41/4.24	Q DP problem:
10.41/4.24	The TRS P consists of the following rules:
10.41/4.24	
10.41/4.24	   new_primModNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
10.41/4.24	   new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
10.41/4.24	   new_primModNatS00(vz21, vz22) -> new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
10.41/4.24	   new_primModNatS(Succ(Zero), Zero) -> new_primModNatS(new_primMinusNatS2, Zero)
10.41/4.24	   new_primModNatS(Succ(Succ(vz3000)), Zero) -> new_primModNatS(new_primMinusNatS1(vz3000), Zero)
10.41/4.24	   new_primModNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primModNatS0(vz21, vz22, vz230, vz240)
10.41/4.24	   new_primModNatS0(vz21, vz22, Zero, Zero) -> new_primModNatS00(vz21, vz22)
10.41/4.24	
10.41/4.24	The TRS R consists of the following rules:
10.41/4.24	
10.41/4.24	   new_primMinusNatS0(Zero, Succ(vz220)) -> Zero
10.41/4.24	   new_primMinusNatS0(Zero, Zero) -> Zero
10.41/4.24	   new_primMinusNatS1(vz3000) -> Succ(vz3000)
10.41/4.24	   new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220)
10.41/4.24	   new_primMinusNatS2 -> Zero
10.41/4.24	   new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210)
10.41/4.24	
10.41/4.24	The set Q consists of the following terms:
10.41/4.24	
10.41/4.24	   new_primMinusNatS0(Succ(x0), Zero)
10.41/4.24	   new_primMinusNatS0(Zero, Zero)
10.41/4.24	   new_primMinusNatS2
10.41/4.24	   new_primMinusNatS0(Succ(x0), Succ(x1))
10.41/4.24	   new_primMinusNatS0(Zero, Succ(x0))
10.41/4.24	   new_primMinusNatS1(x0)
10.41/4.24	
10.41/4.24	We have to consider all minimal (P,Q,R)-chains.
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(10) DependencyGraphProof (EQUIVALENT)
10.41/4.24	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(11)
10.41/4.24	Complex Obligation (AND)
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(12)
10.41/4.24	Obligation:
10.41/4.24	Q DP problem:
10.41/4.24	The TRS P consists of the following rules:
10.41/4.24	
10.41/4.24	   new_primModNatS(Succ(Succ(vz3000)), Zero) -> new_primModNatS(new_primMinusNatS1(vz3000), Zero)
10.41/4.24	
10.41/4.24	The TRS R consists of the following rules:
10.41/4.24	
10.41/4.24	   new_primMinusNatS0(Zero, Succ(vz220)) -> Zero
10.41/4.24	   new_primMinusNatS0(Zero, Zero) -> Zero
10.41/4.24	   new_primMinusNatS1(vz3000) -> Succ(vz3000)
10.41/4.24	   new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220)
10.41/4.24	   new_primMinusNatS2 -> Zero
10.41/4.24	   new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210)
10.41/4.24	
10.41/4.24	The set Q consists of the following terms:
10.41/4.24	
10.41/4.24	   new_primMinusNatS0(Succ(x0), Zero)
10.41/4.24	   new_primMinusNatS0(Zero, Zero)
10.41/4.24	   new_primMinusNatS2
10.41/4.24	   new_primMinusNatS0(Succ(x0), Succ(x1))
10.41/4.24	   new_primMinusNatS0(Zero, Succ(x0))
10.41/4.24	   new_primMinusNatS1(x0)
10.41/4.24	
10.41/4.24	We have to consider all minimal (P,Q,R)-chains.
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(13) MRRProof (EQUIVALENT)
10.41/4.24	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
10.41/4.24	
10.41/4.24	Strictly oriented dependency pairs:
10.41/4.24	
10.41/4.24	   new_primModNatS(Succ(Succ(vz3000)), Zero) -> new_primModNatS(new_primMinusNatS1(vz3000), Zero)
10.41/4.24	
10.41/4.24	Strictly oriented rules of the TRS R:
10.41/4.24	
10.41/4.24	   new_primMinusNatS0(Zero, Succ(vz220)) -> Zero
10.41/4.24	   new_primMinusNatS0(Zero, Zero) -> Zero
10.41/4.24	   new_primMinusNatS1(vz3000) -> Succ(vz3000)
10.41/4.24	   new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220)
10.41/4.24	   new_primMinusNatS2 -> Zero
10.41/4.24	   new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210)
10.41/4.24	
10.41/4.24	Used ordering: Polynomial interpretation [POLO]:
10.41/4.24	
10.41/4.24	   POL(Succ(x_1)) = 1 + 2*x_1
10.41/4.24	   POL(Zero) = 1
10.41/4.24	   POL(new_primMinusNatS0(x_1, x_2)) = x_1 + x_2
10.41/4.24	   POL(new_primMinusNatS1(x_1)) = 2 + 2*x_1
10.41/4.24	   POL(new_primMinusNatS2) = 2
10.41/4.24	   POL(new_primModNatS(x_1, x_2)) = x_1 + x_2
10.41/4.24	
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(14)
10.41/4.24	Obligation:
10.41/4.24	Q DP problem:
10.41/4.24	P is empty.
10.41/4.24	R is empty.
10.41/4.24	The set Q consists of the following terms:
10.41/4.24	
10.41/4.24	   new_primMinusNatS0(Succ(x0), Zero)
10.41/4.24	   new_primMinusNatS0(Zero, Zero)
10.41/4.24	   new_primMinusNatS2
10.41/4.24	   new_primMinusNatS0(Succ(x0), Succ(x1))
10.41/4.24	   new_primMinusNatS0(Zero, Succ(x0))
10.41/4.24	   new_primMinusNatS1(x0)
10.41/4.24	
10.41/4.24	We have to consider all minimal (P,Q,R)-chains.
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(15) PisEmptyProof (EQUIVALENT)
10.41/4.24	The TRS P is empty. Hence, there is no (P,Q,R) chain.
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(16)
10.41/4.24	YES
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(17)
10.41/4.24	Obligation:
10.41/4.24	Q DP problem:
10.41/4.24	The TRS P consists of the following rules:
10.41/4.24	
10.41/4.24	   new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
10.41/4.24	   new_primModNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
10.41/4.24	   new_primModNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primModNatS0(vz21, vz22, vz230, vz240)
10.41/4.24	   new_primModNatS0(vz21, vz22, Zero, Zero) -> new_primModNatS00(vz21, vz22)
10.41/4.24	   new_primModNatS00(vz21, vz22) -> new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
10.41/4.24	
10.41/4.24	The TRS R consists of the following rules:
10.41/4.24	
10.41/4.24	   new_primMinusNatS0(Zero, Succ(vz220)) -> Zero
10.41/4.24	   new_primMinusNatS0(Zero, Zero) -> Zero
10.41/4.24	   new_primMinusNatS1(vz3000) -> Succ(vz3000)
10.41/4.24	   new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220)
10.41/4.24	   new_primMinusNatS2 -> Zero
10.41/4.24	   new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210)
10.41/4.24	
10.41/4.24	The set Q consists of the following terms:
10.41/4.24	
10.41/4.24	   new_primMinusNatS0(Succ(x0), Zero)
10.41/4.24	   new_primMinusNatS0(Zero, Zero)
10.41/4.24	   new_primMinusNatS2
10.41/4.24	   new_primMinusNatS0(Succ(x0), Succ(x1))
10.41/4.24	   new_primMinusNatS0(Zero, Succ(x0))
10.41/4.24	   new_primMinusNatS1(x0)
10.41/4.24	
10.41/4.24	We have to consider all minimal (P,Q,R)-chains.
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(18) QDPSizeChangeProof (EQUIVALENT)
10.41/4.24	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
10.41/4.24	
10.41/4.24	Order:Polynomial interpretation [POLO]:
10.41/4.24	
10.41/4.24	   POL(Succ(x_1)) = 1 + x_1
10.41/4.24	   POL(Zero) = 1
10.41/4.24	   POL(new_primMinusNatS0(x_1, x_2)) = x_1
10.41/4.24	
10.41/4.24	
10.41/4.24	
10.41/4.24	
10.41/4.24	From the DPs we obtained the following set of size-change graphs:
10.41/4.24	*new_primModNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) (allowed arguments on rhs = {1, 2})
10.41/4.24	The graph contains the following edges 1 >= 1
10.41/4.24	
10.41/4.24	
10.41/4.24	*new_primModNatS00(vz21, vz22) -> new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) (allowed arguments on rhs = {1, 2})
10.41/4.24	The graph contains the following edges 1 >= 1
10.41/4.24	
10.41/4.24	
10.41/4.24	*new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatS0(vz3000, vz4000, vz3000, vz4000) (allowed arguments on rhs = {1, 2, 3, 4})
10.41/4.24	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
10.41/4.24	
10.41/4.24	
10.41/4.24	*new_primModNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primModNatS0(vz21, vz22, vz230, vz240) (allowed arguments on rhs = {1, 2, 3, 4})
10.41/4.24	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
10.41/4.24	
10.41/4.24	
10.41/4.24	*new_primModNatS0(vz21, vz22, Zero, Zero) -> new_primModNatS00(vz21, vz22) (allowed arguments on rhs = {1, 2})
10.41/4.24	The graph contains the following edges 1 >= 1, 2 >= 2
10.41/4.24	
10.41/4.24	
10.41/4.24	
10.41/4.24	We oriented the following set of usable rules [AAECC05,FROCOS05].
10.41/4.24	
10.41/4.24	   new_primMinusNatS0(Zero, Zero) -> Zero
10.41/4.24	   new_primMinusNatS0(Zero, Succ(vz220)) -> Zero
10.41/4.24	   new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210)
10.41/4.24	   new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220)
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(19)
10.41/4.24	YES
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(20)
10.41/4.24	Obligation:
10.41/4.24	Q DP problem:
10.41/4.24	The TRS P consists of the following rules:
10.41/4.24	
10.41/4.24	   new_primMinusNatS(Succ(vz210), Succ(vz220)) -> new_primMinusNatS(vz210, vz220)
10.41/4.24	
10.41/4.24	R is empty.
10.41/4.24	Q is empty.
10.41/4.24	We have to consider all minimal (P,Q,R)-chains.
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(21) QDPSizeChangeProof (EQUIVALENT)
10.41/4.24	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. 
10.41/4.24	
10.41/4.24	From the DPs we obtained the following set of size-change graphs:
10.41/4.24	*new_primMinusNatS(Succ(vz210), Succ(vz220)) -> new_primMinusNatS(vz210, vz220)
10.41/4.24	The graph contains the following edges 1 > 1, 2 > 2
10.41/4.24	
10.41/4.24	
10.41/4.24	----------------------------------------
10.41/4.24	
10.41/4.24	(22)
10.41/4.24	YES
10.41/4.29	EOF