9.15/3.76	YES
11.66/4.44	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.66/4.44	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.66/4.44	
11.66/4.44	
11.66/4.44	H-Termination with start terms of the given HASKELL could be proven:
11.66/4.44	
11.66/4.44	(0) HASKELL
11.66/4.44	(1) LR [EQUIVALENT, 0 ms]
11.66/4.44	(2) HASKELL
11.66/4.44	(3) IFR [EQUIVALENT, 0 ms]
11.66/4.44	(4) HASKELL
11.66/4.44	(5) BR [EQUIVALENT, 0 ms]
11.66/4.44	(6) HASKELL
11.66/4.44	(7) COR [EQUIVALENT, 0 ms]
11.66/4.44	(8) HASKELL
11.66/4.44	(9) LetRed [EQUIVALENT, 11 ms]
11.66/4.44	(10) HASKELL
11.66/4.44	(11) NumRed [SOUND, 0 ms]
11.66/4.44	(12) HASKELL
11.66/4.44	(13) Narrow [SOUND, 0 ms]
11.66/4.44	(14) AND
11.66/4.44	    (15) QDP
11.66/4.44	        (16) DependencyGraphProof [EQUIVALENT, 0 ms]
11.66/4.44	        (17) AND
11.66/4.44	            (18) QDP
11.66/4.44	                (19) MRRProof [EQUIVALENT, 21 ms]
11.66/4.44	                (20) QDP
11.66/4.44	                (21) PisEmptyProof [EQUIVALENT, 0 ms]
11.66/4.44	                (22) YES
11.66/4.44	            (23) QDP
11.66/4.44	                (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.66/4.44	                (25) YES
11.66/4.44	    (26) QDP
11.66/4.44	        (27) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.66/4.44	        (28) YES
11.66/4.44	
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(0)
11.66/4.44	Obligation:
11.66/4.44	mainModule Main 
11.66/4.44	module Main where {
11.66/4.44	import qualified Prelude;
11.66/4.44	}
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(1) LR (EQUIVALENT)
11.66/4.44	Lambda Reductions:
11.66/4.44	The following Lambda expression
11.66/4.44	"\(m,_)->m"
11.66/4.44	is transformed to
11.66/4.44	"m0 (m,_) = m;
11.66/4.44	"
11.66/4.44	The following Lambda expression
11.66/4.44	"\(q,_)->q"
11.66/4.44	is transformed to
11.66/4.44	"q1 (q,_) = q;
11.66/4.44	"
11.66/4.44	The following Lambda expression
11.66/4.44	"\(_,r)->r"
11.66/4.44	is transformed to
11.66/4.44	"r0 (_,r) = r;
11.66/4.44	"
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(2)
11.66/4.44	Obligation:
11.66/4.44	mainModule Main 
11.66/4.44	module Main where {
11.66/4.44	import qualified Prelude;
11.66/4.44	}
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(3) IFR (EQUIVALENT)
11.66/4.44	If Reductions:
11.66/4.44	The following If expression
11.66/4.44	"if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero"
11.66/4.44	is transformed to
11.66/4.44	"primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y));
11.66/4.44	primDivNatS0 x y False = Zero;
11.66/4.44	"
11.66/4.44	The following If expression
11.66/4.44	"if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x"
11.66/4.44	is transformed to
11.66/4.44	"primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y);
11.66/4.44	primModNatS0 x y False = Succ x;
11.66/4.44	"
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(4)
11.66/4.44	Obligation:
11.66/4.44	mainModule Main 
11.66/4.44	module Main where {
11.66/4.44	import qualified Prelude;
11.66/4.44	}
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(5) BR (EQUIVALENT)
11.66/4.44	Replaced joker patterns by fresh variables and removed binding patterns.
11.66/4.44	
11.66/4.44	Binding Reductions:
11.66/4.44	The bind variable of the following binding Pattern
11.66/4.44	"frac@(Float vz wu)"
11.66/4.44	is replaced by the following term
11.66/4.44	"Float vz wu"
11.66/4.44	The bind variable of the following binding Pattern
11.66/4.44	"frac@(Double xu xv)"
11.66/4.44	is replaced by the following term
11.66/4.44	"Double xu xv"
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(6)
11.66/4.44	Obligation:
11.66/4.44	mainModule Main 
11.66/4.44	module Main where {
11.66/4.44	import qualified Prelude;
11.66/4.44	}
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(7) COR (EQUIVALENT)
11.66/4.44	Cond Reductions:
11.66/4.44	The following Function with conditions
11.66/4.44	"undefined |Falseundefined;
11.66/4.44	"
11.66/4.44	is transformed to
11.66/4.44	"undefined  = undefined1;
11.66/4.44	"
11.66/4.44	"undefined0 True = undefined;
11.66/4.44	"
11.66/4.44	"undefined1  = undefined0 False;
11.66/4.44	"
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(8)
11.66/4.44	Obligation:
11.66/4.44	mainModule Main 
11.66/4.44	module Main where {
11.66/4.44	import qualified Prelude;
11.66/4.44	}
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(9) LetRed (EQUIVALENT)
11.66/4.44	Let/Where Reductions:
11.66/4.44	The bindings of the following Let/Where expression
11.66/4.44	"m where {
11.66/4.44	m  = m0 vu6;
11.66/4.44	;
11.66/4.44	m0 (m,vv) = m;
11.66/4.44	;
11.66/4.44	vu6  = properFraction x;
11.66/4.44	} 
11.66/4.44	"
11.66/4.44	are unpacked to the following functions on top level
11.66/4.44	"truncateVu6 xw = properFraction xw;
11.66/4.44	"
11.66/4.44	"truncateM xw = truncateM0 xw (truncateVu6 xw);
11.66/4.44	"
11.66/4.44	"truncateM0 xw (m,vv) = m;
11.66/4.44	"
11.66/4.44	The bindings of the following Let/Where expression
11.66/4.44	"(fromIntegral q,r :% y) where {
11.66/4.44	q  = q1 vu30;
11.66/4.44	;
11.66/4.44	q1 (q,vw) = q;
11.66/4.44	;
11.66/4.44	r  = r0 vu30;
11.66/4.44	;
11.66/4.44	r0 (vx,r) = r;
11.66/4.44	;
11.66/4.44	vu30  = quotRem x y;
11.66/4.44	} 
11.66/4.44	"
11.66/4.44	are unpacked to the following functions on top level
11.66/4.44	"properFractionR0 xx xy (vx,r) = r;
11.66/4.44	"
11.66/4.44	"properFractionQ xx xy = properFractionQ1 xx xy (properFractionVu30 xx xy);
11.66/4.44	"
11.66/4.44	"properFractionQ1 xx xy (q,vw) = q;
11.66/4.44	"
11.66/4.44	"properFractionVu30 xx xy = quotRem xx xy;
11.66/4.44	"
11.66/4.44	"properFractionR xx xy = properFractionR0 xx xy (properFractionVu30 xx xy);
11.66/4.44	"
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(10)
11.66/4.44	Obligation:
11.66/4.44	mainModule Main 
11.66/4.44	module Main where {
11.66/4.44	import qualified Prelude;
11.66/4.44	}
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(11) NumRed (SOUND)
11.66/4.44	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(12)
11.66/4.44	Obligation:
11.66/4.44	mainModule Main 
11.66/4.44	module Main where {
11.66/4.44	import qualified Prelude;
11.66/4.44	}
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(13) Narrow (SOUND)
11.66/4.44	Haskell To QDPs
11.66/4.44	
11.66/4.44	digraph dp_graph {
11.66/4.44	node [outthreshold=100, inthreshold=100];1[label="truncate",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.66/4.44	3[label="truncate xz3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
11.66/4.44	4[label="truncateM xz3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
11.66/4.44	5[label="truncateM0 xz3 (truncateVu6 xz3)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
11.66/4.44	6[label="truncateM0 xz3 (properFraction xz3)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
11.66/4.44	7[label="truncateM0 xz3 (floatProperFractionFloat xz3)",fontsize=16,color="burlywood",shape="box"];281[label="xz3/Float xz30 xz31",fontsize=10,color="white",style="solid",shape="box"];7 -> 281[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	281 -> 8[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	8[label="truncateM0 (Float xz30 xz31) (floatProperFractionFloat (Float xz30 xz31))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
11.66/4.44	9[label="truncateM0 (Float xz30 xz31) (fromInt (xz30 `quot` xz31),Float xz30 xz31 - fromInt (xz30 `quot` xz31))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
11.66/4.44	10[label="fromInt (xz30 `quot` xz31)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11.66/4.44	11[label="xz30 `quot` xz31",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
11.66/4.44	12[label="primQuotInt xz30 xz31",fontsize=16,color="burlywood",shape="box"];282[label="xz30/Pos xz300",fontsize=10,color="white",style="solid",shape="box"];12 -> 282[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	282 -> 13[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	283[label="xz30/Neg xz300",fontsize=10,color="white",style="solid",shape="box"];12 -> 283[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	283 -> 14[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	13[label="primQuotInt (Pos xz300) xz31",fontsize=16,color="burlywood",shape="box"];284[label="xz31/Pos xz310",fontsize=10,color="white",style="solid",shape="box"];13 -> 284[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	284 -> 15[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	285[label="xz31/Neg xz310",fontsize=10,color="white",style="solid",shape="box"];13 -> 285[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	285 -> 16[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	14[label="primQuotInt (Neg xz300) xz31",fontsize=16,color="burlywood",shape="box"];286[label="xz31/Pos xz310",fontsize=10,color="white",style="solid",shape="box"];14 -> 286[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	286 -> 17[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	287[label="xz31/Neg xz310",fontsize=10,color="white",style="solid",shape="box"];14 -> 287[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	287 -> 18[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	15[label="primQuotInt (Pos xz300) (Pos xz310)",fontsize=16,color="burlywood",shape="box"];288[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];15 -> 288[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	288 -> 19[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	289[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 289[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	289 -> 20[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	16[label="primQuotInt (Pos xz300) (Neg xz310)",fontsize=16,color="burlywood",shape="box"];290[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];16 -> 290[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	290 -> 21[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	291[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 291[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	291 -> 22[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	17[label="primQuotInt (Neg xz300) (Pos xz310)",fontsize=16,color="burlywood",shape="box"];292[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];17 -> 292[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	292 -> 23[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	293[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 293[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	293 -> 24[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	18[label="primQuotInt (Neg xz300) (Neg xz310)",fontsize=16,color="burlywood",shape="box"];294[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];18 -> 294[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	294 -> 25[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	295[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 295[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	295 -> 26[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	19[label="primQuotInt (Pos xz300) (Pos (Succ xz3100))",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
11.66/4.44	20[label="primQuotInt (Pos xz300) (Pos Zero)",fontsize=16,color="black",shape="box"];20 -> 28[label="",style="solid", color="black", weight=3];
11.66/4.44	21[label="primQuotInt (Pos xz300) (Neg (Succ xz3100))",fontsize=16,color="black",shape="box"];21 -> 29[label="",style="solid", color="black", weight=3];
11.66/4.44	22[label="primQuotInt (Pos xz300) (Neg Zero)",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3];
11.66/4.44	23[label="primQuotInt (Neg xz300) (Pos (Succ xz3100))",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
11.66/4.44	24[label="primQuotInt (Neg xz300) (Pos Zero)",fontsize=16,color="black",shape="box"];24 -> 32[label="",style="solid", color="black", weight=3];
11.66/4.44	25[label="primQuotInt (Neg xz300) (Neg (Succ xz3100))",fontsize=16,color="black",shape="box"];25 -> 33[label="",style="solid", color="black", weight=3];
11.66/4.44	26[label="primQuotInt (Neg xz300) (Neg Zero)",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
11.66/4.44	27[label="Pos (primDivNatS xz300 (Succ xz3100))",fontsize=16,color="green",shape="box"];27 -> 35[label="",style="dashed", color="green", weight=3];
11.66/4.44	28[label="error []",fontsize=16,color="black",shape="triangle"];28 -> 36[label="",style="solid", color="black", weight=3];
11.66/4.44	29[label="Neg (primDivNatS xz300 (Succ xz3100))",fontsize=16,color="green",shape="box"];29 -> 37[label="",style="dashed", color="green", weight=3];
11.66/4.44	30 -> 28[label="",style="dashed", color="red", weight=0];
11.66/4.44	30[label="error []",fontsize=16,color="magenta"];31[label="Neg (primDivNatS xz300 (Succ xz3100))",fontsize=16,color="green",shape="box"];31 -> 38[label="",style="dashed", color="green", weight=3];
11.66/4.44	32 -> 28[label="",style="dashed", color="red", weight=0];
11.66/4.44	32[label="error []",fontsize=16,color="magenta"];33[label="Pos (primDivNatS xz300 (Succ xz3100))",fontsize=16,color="green",shape="box"];33 -> 39[label="",style="dashed", color="green", weight=3];
11.66/4.44	34 -> 28[label="",style="dashed", color="red", weight=0];
11.66/4.44	34[label="error []",fontsize=16,color="magenta"];35[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="burlywood",shape="triangle"];296[label="xz300/Succ xz3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 296[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	296 -> 40[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	297[label="xz300/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 297[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	297 -> 41[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	36[label="error []",fontsize=16,color="red",shape="box"];37 -> 35[label="",style="dashed", color="red", weight=0];
11.66/4.44	37[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="magenta"];37 -> 42[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	38 -> 35[label="",style="dashed", color="red", weight=0];
11.66/4.44	38[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="magenta"];38 -> 43[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	39 -> 35[label="",style="dashed", color="red", weight=0];
11.66/4.44	39[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="magenta"];39 -> 44[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	39 -> 45[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	40[label="primDivNatS (Succ xz3000) (Succ xz3100)",fontsize=16,color="black",shape="box"];40 -> 46[label="",style="solid", color="black", weight=3];
11.66/4.44	41[label="primDivNatS Zero (Succ xz3100)",fontsize=16,color="black",shape="box"];41 -> 47[label="",style="solid", color="black", weight=3];
11.66/4.44	42[label="xz3100",fontsize=16,color="green",shape="box"];43[label="xz300",fontsize=16,color="green",shape="box"];44[label="xz300",fontsize=16,color="green",shape="box"];45[label="xz3100",fontsize=16,color="green",shape="box"];46[label="primDivNatS0 xz3000 xz3100 (primGEqNatS xz3000 xz3100)",fontsize=16,color="burlywood",shape="box"];298[label="xz3000/Succ xz30000",fontsize=10,color="white",style="solid",shape="box"];46 -> 298[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	298 -> 48[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	299[label="xz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 299[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	299 -> 49[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	47[label="Zero",fontsize=16,color="green",shape="box"];48[label="primDivNatS0 (Succ xz30000) xz3100 (primGEqNatS (Succ xz30000) xz3100)",fontsize=16,color="burlywood",shape="box"];300[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];48 -> 300[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	300 -> 50[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	301[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 301[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	301 -> 51[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	49[label="primDivNatS0 Zero xz3100 (primGEqNatS Zero xz3100)",fontsize=16,color="burlywood",shape="box"];302[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];49 -> 302[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	302 -> 52[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	303[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 303[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	303 -> 53[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	50[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS (Succ xz30000) (Succ xz31000))",fontsize=16,color="black",shape="box"];50 -> 54[label="",style="solid", color="black", weight=3];
11.66/4.44	51[label="primDivNatS0 (Succ xz30000) Zero (primGEqNatS (Succ xz30000) Zero)",fontsize=16,color="black",shape="box"];51 -> 55[label="",style="solid", color="black", weight=3];
11.66/4.44	52[label="primDivNatS0 Zero (Succ xz31000) (primGEqNatS Zero (Succ xz31000))",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
11.66/4.44	53[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
11.66/4.44	54 -> 218[label="",style="dashed", color="red", weight=0];
11.66/4.44	54[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS xz30000 xz31000)",fontsize=16,color="magenta"];54 -> 219[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	54 -> 220[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	54 -> 221[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	54 -> 222[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	55[label="primDivNatS0 (Succ xz30000) Zero True",fontsize=16,color="black",shape="box"];55 -> 60[label="",style="solid", color="black", weight=3];
11.66/4.44	56[label="primDivNatS0 Zero (Succ xz31000) False",fontsize=16,color="black",shape="box"];56 -> 61[label="",style="solid", color="black", weight=3];
11.66/4.44	57[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];57 -> 62[label="",style="solid", color="black", weight=3];
11.66/4.44	219[label="xz31000",fontsize=16,color="green",shape="box"];220[label="xz30000",fontsize=16,color="green",shape="box"];221[label="xz31000",fontsize=16,color="green",shape="box"];222[label="xz30000",fontsize=16,color="green",shape="box"];218[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS xz22 xz23)",fontsize=16,color="burlywood",shape="triangle"];304[label="xz22/Succ xz220",fontsize=10,color="white",style="solid",shape="box"];218 -> 304[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	304 -> 251[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	305[label="xz22/Zero",fontsize=10,color="white",style="solid",shape="box"];218 -> 305[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	305 -> 252[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	60[label="Succ (primDivNatS (primMinusNatS (Succ xz30000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];60 -> 67[label="",style="dashed", color="green", weight=3];
11.66/4.44	61[label="Zero",fontsize=16,color="green",shape="box"];62[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];62 -> 68[label="",style="dashed", color="green", weight=3];
11.66/4.44	251[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS (Succ xz220) xz23)",fontsize=16,color="burlywood",shape="box"];306[label="xz23/Succ xz230",fontsize=10,color="white",style="solid",shape="box"];251 -> 306[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	306 -> 253[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	307[label="xz23/Zero",fontsize=10,color="white",style="solid",shape="box"];251 -> 307[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	307 -> 254[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	252[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero xz23)",fontsize=16,color="burlywood",shape="box"];308[label="xz23/Succ xz230",fontsize=10,color="white",style="solid",shape="box"];252 -> 308[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	308 -> 255[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	309[label="xz23/Zero",fontsize=10,color="white",style="solid",shape="box"];252 -> 309[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	309 -> 256[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	67 -> 35[label="",style="dashed", color="red", weight=0];
11.66/4.44	67[label="primDivNatS (primMinusNatS (Succ xz30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];67 -> 73[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	67 -> 74[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	68 -> 35[label="",style="dashed", color="red", weight=0];
11.66/4.44	68[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];68 -> 75[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	68 -> 76[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	253[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS (Succ xz220) (Succ xz230))",fontsize=16,color="black",shape="box"];253 -> 257[label="",style="solid", color="black", weight=3];
11.66/4.44	254[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS (Succ xz220) Zero)",fontsize=16,color="black",shape="box"];254 -> 258[label="",style="solid", color="black", weight=3];
11.66/4.44	255[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero (Succ xz230))",fontsize=16,color="black",shape="box"];255 -> 259[label="",style="solid", color="black", weight=3];
11.66/4.44	256[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];256 -> 260[label="",style="solid", color="black", weight=3];
11.66/4.44	73[label="primMinusNatS (Succ xz30000) Zero",fontsize=16,color="black",shape="triangle"];73 -> 82[label="",style="solid", color="black", weight=3];
11.66/4.44	74[label="Zero",fontsize=16,color="green",shape="box"];75[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];75 -> 83[label="",style="solid", color="black", weight=3];
11.66/4.44	76[label="Zero",fontsize=16,color="green",shape="box"];257 -> 218[label="",style="dashed", color="red", weight=0];
11.66/4.44	257[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS xz220 xz230)",fontsize=16,color="magenta"];257 -> 261[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	257 -> 262[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	258[label="primDivNatS0 (Succ xz20) (Succ xz21) True",fontsize=16,color="black",shape="triangle"];258 -> 263[label="",style="solid", color="black", weight=3];
11.66/4.44	259[label="primDivNatS0 (Succ xz20) (Succ xz21) False",fontsize=16,color="black",shape="box"];259 -> 264[label="",style="solid", color="black", weight=3];
11.66/4.44	260 -> 258[label="",style="dashed", color="red", weight=0];
11.66/4.44	260[label="primDivNatS0 (Succ xz20) (Succ xz21) True",fontsize=16,color="magenta"];82[label="Succ xz30000",fontsize=16,color="green",shape="box"];83[label="Zero",fontsize=16,color="green",shape="box"];261[label="xz230",fontsize=16,color="green",shape="box"];262[label="xz220",fontsize=16,color="green",shape="box"];263[label="Succ (primDivNatS (primMinusNatS (Succ xz20) (Succ xz21)) (Succ (Succ xz21)))",fontsize=16,color="green",shape="box"];263 -> 265[label="",style="dashed", color="green", weight=3];
11.66/4.44	264[label="Zero",fontsize=16,color="green",shape="box"];265 -> 35[label="",style="dashed", color="red", weight=0];
11.66/4.44	265[label="primDivNatS (primMinusNatS (Succ xz20) (Succ xz21)) (Succ (Succ xz21))",fontsize=16,color="magenta"];265 -> 266[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	265 -> 267[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	266[label="primMinusNatS (Succ xz20) (Succ xz21)",fontsize=16,color="black",shape="box"];266 -> 268[label="",style="solid", color="black", weight=3];
11.66/4.44	267[label="Succ xz21",fontsize=16,color="green",shape="box"];268[label="primMinusNatS xz20 xz21",fontsize=16,color="burlywood",shape="triangle"];310[label="xz20/Succ xz200",fontsize=10,color="white",style="solid",shape="box"];268 -> 310[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	310 -> 269[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	311[label="xz20/Zero",fontsize=10,color="white",style="solid",shape="box"];268 -> 311[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	311 -> 270[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	269[label="primMinusNatS (Succ xz200) xz21",fontsize=16,color="burlywood",shape="box"];312[label="xz21/Succ xz210",fontsize=10,color="white",style="solid",shape="box"];269 -> 312[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	312 -> 271[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	313[label="xz21/Zero",fontsize=10,color="white",style="solid",shape="box"];269 -> 313[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	313 -> 272[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	270[label="primMinusNatS Zero xz21",fontsize=16,color="burlywood",shape="box"];314[label="xz21/Succ xz210",fontsize=10,color="white",style="solid",shape="box"];270 -> 314[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	314 -> 273[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	315[label="xz21/Zero",fontsize=10,color="white",style="solid",shape="box"];270 -> 315[label="",style="solid", color="burlywood", weight=9];
11.66/4.44	315 -> 274[label="",style="solid", color="burlywood", weight=3];
11.66/4.44	271[label="primMinusNatS (Succ xz200) (Succ xz210)",fontsize=16,color="black",shape="box"];271 -> 275[label="",style="solid", color="black", weight=3];
11.66/4.44	272[label="primMinusNatS (Succ xz200) Zero",fontsize=16,color="black",shape="box"];272 -> 276[label="",style="solid", color="black", weight=3];
11.66/4.44	273[label="primMinusNatS Zero (Succ xz210)",fontsize=16,color="black",shape="box"];273 -> 277[label="",style="solid", color="black", weight=3];
11.66/4.44	274[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];274 -> 278[label="",style="solid", color="black", weight=3];
11.66/4.44	275 -> 268[label="",style="dashed", color="red", weight=0];
11.66/4.44	275[label="primMinusNatS xz200 xz210",fontsize=16,color="magenta"];275 -> 279[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	275 -> 280[label="",style="dashed", color="magenta", weight=3];
11.66/4.44	276[label="Succ xz200",fontsize=16,color="green",shape="box"];277[label="Zero",fontsize=16,color="green",shape="box"];278[label="Zero",fontsize=16,color="green",shape="box"];279[label="xz210",fontsize=16,color="green",shape="box"];280[label="xz200",fontsize=16,color="green",shape="box"];}
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(14)
11.66/4.44	Complex Obligation (AND)
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(15)
11.66/4.44	Obligation:
11.66/4.44	Q DP problem:
11.66/4.44	The TRS P consists of the following rules:
11.66/4.44	
11.66/4.44	   new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
11.66/4.44	   new_primDivNatS0(xz20, xz21, Succ(xz220), Zero) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
11.66/4.44	   new_primDivNatS0(xz20, xz21, Zero, Zero) -> new_primDivNatS00(xz20, xz21)
11.66/4.44	   new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) -> new_primDivNatS0(xz20, xz21, xz220, xz230)
11.66/4.44	   new_primDivNatS(Succ(Zero), Zero) -> new_primDivNatS(new_primMinusNatS2, Zero)
11.66/4.44	   new_primDivNatS(Succ(Succ(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero)
11.66/4.44	   new_primDivNatS00(xz20, xz21) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
11.66/4.44	
11.66/4.44	The TRS R consists of the following rules:
11.66/4.44	
11.66/4.44	   new_primMinusNatS0(Zero, Succ(xz210)) -> Zero
11.66/4.44	   new_primMinusNatS0(Zero, Zero) -> Zero
11.66/4.44	   new_primMinusNatS1(xz30000) -> Succ(xz30000)
11.66/4.44	   new_primMinusNatS2 -> Zero
11.66/4.44	   new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210)
11.66/4.44	   new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200)
11.66/4.44	
11.66/4.44	The set Q consists of the following terms:
11.66/4.44	
11.66/4.44	   new_primMinusNatS0(Succ(x0), Succ(x1))
11.66/4.44	   new_primMinusNatS0(Zero, Zero)
11.66/4.44	   new_primMinusNatS2
11.66/4.44	   new_primMinusNatS0(Succ(x0), Zero)
11.66/4.44	   new_primMinusNatS1(x0)
11.66/4.44	   new_primMinusNatS0(Zero, Succ(x0))
11.66/4.44	
11.66/4.44	We have to consider all minimal (P,Q,R)-chains.
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(16) DependencyGraphProof (EQUIVALENT)
11.66/4.44	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(17)
11.66/4.44	Complex Obligation (AND)
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(18)
11.66/4.44	Obligation:
11.66/4.44	Q DP problem:
11.66/4.44	The TRS P consists of the following rules:
11.66/4.44	
11.66/4.44	   new_primDivNatS(Succ(Succ(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero)
11.66/4.44	
11.66/4.44	The TRS R consists of the following rules:
11.66/4.44	
11.66/4.44	   new_primMinusNatS0(Zero, Succ(xz210)) -> Zero
11.66/4.44	   new_primMinusNatS0(Zero, Zero) -> Zero
11.66/4.44	   new_primMinusNatS1(xz30000) -> Succ(xz30000)
11.66/4.44	   new_primMinusNatS2 -> Zero
11.66/4.44	   new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210)
11.66/4.44	   new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200)
11.66/4.44	
11.66/4.44	The set Q consists of the following terms:
11.66/4.44	
11.66/4.44	   new_primMinusNatS0(Succ(x0), Succ(x1))
11.66/4.44	   new_primMinusNatS0(Zero, Zero)
11.66/4.44	   new_primMinusNatS2
11.66/4.44	   new_primMinusNatS0(Succ(x0), Zero)
11.66/4.44	   new_primMinusNatS1(x0)
11.66/4.44	   new_primMinusNatS0(Zero, Succ(x0))
11.66/4.44	
11.66/4.44	We have to consider all minimal (P,Q,R)-chains.
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(19) MRRProof (EQUIVALENT)
11.66/4.44	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.
11.66/4.44	
11.66/4.44	Strictly oriented dependency pairs:
11.66/4.44	
11.66/4.44	   new_primDivNatS(Succ(Succ(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero)
11.66/4.44	
11.66/4.44	Strictly oriented rules of the TRS R:
11.66/4.44	
11.66/4.44	   new_primMinusNatS0(Zero, Succ(xz210)) -> Zero
11.66/4.44	   new_primMinusNatS0(Zero, Zero) -> Zero
11.66/4.44	   new_primMinusNatS1(xz30000) -> Succ(xz30000)
11.66/4.44	   new_primMinusNatS2 -> Zero
11.66/4.44	   new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210)
11.66/4.44	   new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200)
11.66/4.44	
11.66/4.44	Used ordering: Polynomial interpretation [POLO]:
11.66/4.44	
11.66/4.44	   POL(Succ(x_1)) = 1 + 2*x_1
11.66/4.44	   POL(Zero) = 1
11.66/4.44	   POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2
11.66/4.44	   POL(new_primMinusNatS0(x_1, x_2)) = x_1 + x_2
11.66/4.44	   POL(new_primMinusNatS1(x_1)) = 2 + 2*x_1
11.66/4.44	   POL(new_primMinusNatS2) = 2
11.66/4.44	
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(20)
11.66/4.44	Obligation:
11.66/4.44	Q DP problem:
11.66/4.44	P is empty.
11.66/4.44	R is empty.
11.66/4.44	The set Q consists of the following terms:
11.66/4.44	
11.66/4.44	   new_primMinusNatS0(Succ(x0), Succ(x1))
11.66/4.44	   new_primMinusNatS0(Zero, Zero)
11.66/4.44	   new_primMinusNatS2
11.66/4.44	   new_primMinusNatS0(Succ(x0), Zero)
11.66/4.44	   new_primMinusNatS1(x0)
11.66/4.44	   new_primMinusNatS0(Zero, Succ(x0))
11.66/4.44	
11.66/4.44	We have to consider all minimal (P,Q,R)-chains.
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(21) PisEmptyProof (EQUIVALENT)
11.66/4.44	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(22)
11.66/4.44	YES
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(23)
11.66/4.44	Obligation:
11.66/4.44	Q DP problem:
11.66/4.44	The TRS P consists of the following rules:
11.66/4.44	
11.66/4.44	   new_primDivNatS0(xz20, xz21, Succ(xz220), Zero) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
11.66/4.44	   new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
11.66/4.44	   new_primDivNatS0(xz20, xz21, Zero, Zero) -> new_primDivNatS00(xz20, xz21)
11.66/4.44	   new_primDivNatS00(xz20, xz21) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
11.66/4.44	   new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) -> new_primDivNatS0(xz20, xz21, xz220, xz230)
11.66/4.44	
11.66/4.44	The TRS R consists of the following rules:
11.66/4.44	
11.66/4.44	   new_primMinusNatS0(Zero, Succ(xz210)) -> Zero
11.66/4.44	   new_primMinusNatS0(Zero, Zero) -> Zero
11.66/4.44	   new_primMinusNatS1(xz30000) -> Succ(xz30000)
11.66/4.44	   new_primMinusNatS2 -> Zero
11.66/4.44	   new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210)
11.66/4.44	   new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200)
11.66/4.44	
11.66/4.44	The set Q consists of the following terms:
11.66/4.44	
11.66/4.44	   new_primMinusNatS0(Succ(x0), Succ(x1))
11.66/4.44	   new_primMinusNatS0(Zero, Zero)
11.66/4.44	   new_primMinusNatS2
11.66/4.44	   new_primMinusNatS0(Succ(x0), Zero)
11.66/4.44	   new_primMinusNatS1(x0)
11.66/4.44	   new_primMinusNatS0(Zero, Succ(x0))
11.66/4.44	
11.66/4.44	We have to consider all minimal (P,Q,R)-chains.
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(24) QDPSizeChangeProof (EQUIVALENT)
11.66/4.44	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
11.66/4.44	
11.66/4.44	Order:Polynomial interpretation [POLO]:
11.66/4.44	
11.66/4.44	   POL(Succ(x_1)) = 1 + x_1
11.66/4.44	   POL(Zero) = 1
11.66/4.44	   POL(new_primMinusNatS0(x_1, x_2)) = x_1
11.66/4.44	
11.66/4.44	
11.66/4.44	
11.66/4.44	
11.66/4.44	From the DPs we obtained the following set of size-change graphs:
11.66/4.44	*new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000) (allowed arguments on rhs = {1, 2, 3, 4})
11.66/4.44	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
11.66/4.44	
11.66/4.44	
11.66/4.44	*new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) -> new_primDivNatS0(xz20, xz21, xz220, xz230) (allowed arguments on rhs = {1, 2, 3, 4})
11.66/4.44	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
11.66/4.44	
11.66/4.44	
11.66/4.44	*new_primDivNatS0(xz20, xz21, Succ(xz220), Zero) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) (allowed arguments on rhs = {1, 2})
11.66/4.44	The graph contains the following edges 1 >= 1
11.66/4.44	
11.66/4.44	
11.66/4.44	*new_primDivNatS0(xz20, xz21, Zero, Zero) -> new_primDivNatS00(xz20, xz21) (allowed arguments on rhs = {1, 2})
11.66/4.44	The graph contains the following edges 1 >= 1, 2 >= 2
11.66/4.44	
11.66/4.44	
11.66/4.44	*new_primDivNatS00(xz20, xz21) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) (allowed arguments on rhs = {1, 2})
11.66/4.44	The graph contains the following edges 1 >= 1
11.66/4.44	
11.66/4.44	
11.66/4.44	
11.66/4.44	We oriented the following set of usable rules [AAECC05,FROCOS05].
11.66/4.44	
11.66/4.44	   new_primMinusNatS0(Zero, Zero) -> Zero
11.66/4.44	   new_primMinusNatS0(Zero, Succ(xz210)) -> Zero
11.66/4.44	   new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200)
11.66/4.44	   new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210)
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(25)
11.66/4.44	YES
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(26)
11.66/4.44	Obligation:
11.66/4.44	Q DP problem:
11.66/4.44	The TRS P consists of the following rules:
11.66/4.44	
11.66/4.44	   new_primMinusNatS(Succ(xz200), Succ(xz210)) -> new_primMinusNatS(xz200, xz210)
11.66/4.44	
11.66/4.44	R is empty.
11.66/4.44	Q is empty.
11.66/4.44	We have to consider all minimal (P,Q,R)-chains.
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(27) QDPSizeChangeProof (EQUIVALENT)
11.66/4.44	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. 
11.66/4.44	
11.66/4.44	From the DPs we obtained the following set of size-change graphs:
11.66/4.44	*new_primMinusNatS(Succ(xz200), Succ(xz210)) -> new_primMinusNatS(xz200, xz210)
11.66/4.44	The graph contains the following edges 1 > 1, 2 > 2
11.66/4.44	
11.66/4.44	
11.66/4.44	----------------------------------------
11.66/4.44	
11.66/4.44	(28)
11.66/4.44	YES
11.70/4.48	EOF