8.81/3.79	YES
11.26/4.50	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
11.26/4.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.26/4.50	
11.26/4.50	
11.26/4.50	H-Termination with start terms of the given HASKELL could be proven:
11.26/4.50	
11.26/4.50	(0) HASKELL
11.26/4.50	(1) LR [EQUIVALENT, 0 ms]
11.26/4.50	(2) HASKELL
11.26/4.50	(3) IFR [EQUIVALENT, 0 ms]
11.26/4.50	(4) HASKELL
11.26/4.50	(5) BR [EQUIVALENT, 0 ms]
11.26/4.50	(6) HASKELL
11.26/4.50	(7) COR [EQUIVALENT, 0 ms]
11.26/4.50	(8) HASKELL
11.26/4.50	(9) LetRed [EQUIVALENT, 3 ms]
11.26/4.50	(10) HASKELL
11.26/4.50	(11) NumRed [SOUND, 0 ms]
11.26/4.50	(12) HASKELL
11.26/4.50	(13) Narrow [SOUND, 0 ms]
11.26/4.50	(14) AND
11.26/4.50	    (15) QDP
11.26/4.50	        (16) DependencyGraphProof [EQUIVALENT, 0 ms]
11.26/4.50	        (17) AND
11.26/4.50	            (18) QDP
11.26/4.50	                (19) MRRProof [EQUIVALENT, 13 ms]
11.26/4.50	                (20) QDP
11.26/4.50	                (21) PisEmptyProof [EQUIVALENT, 0 ms]
11.26/4.50	                (22) YES
11.26/4.50	            (23) QDP
11.26/4.50	                (24) QDPSizeChangeProof [EQUIVALENT, 18 ms]
11.26/4.50	                (25) YES
11.26/4.50	    (26) QDP
11.26/4.50	        (27) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.26/4.50	        (28) YES
11.26/4.50	
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(0)
11.26/4.50	Obligation:
11.26/4.50	mainModule Main 
11.26/4.50	module Main where {
11.26/4.50	import qualified Prelude;
11.26/4.50	}
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(1) LR (EQUIVALENT)
11.26/4.50	Lambda Reductions:
11.26/4.50	The following Lambda expression
11.26/4.50	"\(m,_)->m"
11.26/4.50	is transformed to
11.26/4.50	"m0 (m,_) = m;
11.26/4.50	"
11.26/4.50	The following Lambda expression
11.26/4.50	"\(q,_)->q"
11.26/4.50	is transformed to
11.26/4.50	"q1 (q,_) = q;
11.26/4.50	"
11.26/4.50	The following Lambda expression
11.26/4.50	"\(_,r)->r"
11.26/4.50	is transformed to
11.26/4.50	"r0 (_,r) = r;
11.26/4.50	"
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(2)
11.26/4.50	Obligation:
11.26/4.50	mainModule Main 
11.26/4.50	module Main where {
11.26/4.50	import qualified Prelude;
11.26/4.50	}
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(3) IFR (EQUIVALENT)
11.26/4.50	If Reductions:
11.26/4.50	The following If expression
11.26/4.50	"if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero"
11.26/4.50	is transformed to
11.26/4.50	"primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y));
11.26/4.50	primDivNatS0 x y False = Zero;
11.26/4.50	"
11.26/4.50	The following If expression
11.26/4.50	"if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x"
11.26/4.50	is transformed to
11.26/4.50	"primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y);
11.26/4.50	primModNatS0 x y False = Succ x;
11.26/4.50	"
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(4)
11.26/4.50	Obligation:
11.26/4.50	mainModule Main 
11.26/4.50	module Main where {
11.26/4.50	import qualified Prelude;
11.26/4.50	}
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(5) BR (EQUIVALENT)
11.26/4.50	Replaced joker patterns by fresh variables and removed binding patterns.
11.26/4.50	
11.26/4.50	Binding Reductions:
11.26/4.50	The bind variable of the following binding Pattern
11.26/4.50	"frac@(Float vz wu)"
11.26/4.50	is replaced by the following term
11.26/4.50	"Float vz wu"
11.26/4.50	The bind variable of the following binding Pattern
11.26/4.50	"frac@(Double xu xv)"
11.26/4.50	is replaced by the following term
11.26/4.50	"Double xu xv"
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(6)
11.26/4.50	Obligation:
11.26/4.50	mainModule Main 
11.26/4.50	module Main where {
11.26/4.50	import qualified Prelude;
11.26/4.50	}
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(7) COR (EQUIVALENT)
11.26/4.50	Cond Reductions:
11.26/4.50	The following Function with conditions
11.26/4.50	"undefined |Falseundefined;
11.26/4.50	"
11.26/4.50	is transformed to
11.26/4.50	"undefined  = undefined1;
11.26/4.50	"
11.26/4.50	"undefined0 True = undefined;
11.26/4.50	"
11.26/4.50	"undefined1  = undefined0 False;
11.26/4.50	"
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(8)
11.26/4.50	Obligation:
11.26/4.50	mainModule Main 
11.26/4.50	module Main where {
11.26/4.50	import qualified Prelude;
11.26/4.50	}
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(9) LetRed (EQUIVALENT)
11.26/4.50	Let/Where Reductions:
11.26/4.50	The bindings of the following Let/Where expression
11.26/4.50	"m where {
11.26/4.50	m  = m0 vu6;
11.26/4.50	;
11.26/4.50	m0 (m,vv) = m;
11.26/4.50	;
11.26/4.50	vu6  = properFraction x;
11.26/4.50	} 
11.26/4.50	"
11.26/4.50	are unpacked to the following functions on top level
11.26/4.50	"truncateVu6 xw = properFraction xw;
11.26/4.50	"
11.26/4.50	"truncateM xw = truncateM0 xw (truncateVu6 xw);
11.26/4.50	"
11.26/4.50	"truncateM0 xw (m,vv) = m;
11.26/4.50	"
11.26/4.50	The bindings of the following Let/Where expression
11.26/4.50	"(fromIntegral q,r :% y) where {
11.26/4.50	q  = q1 vu30;
11.26/4.50	;
11.26/4.50	q1 (q,vw) = q;
11.26/4.50	;
11.26/4.50	r  = r0 vu30;
11.26/4.50	;
11.26/4.50	r0 (vx,r) = r;
11.26/4.50	;
11.26/4.50	vu30  = quotRem x y;
11.26/4.50	} 
11.26/4.50	"
11.26/4.50	are unpacked to the following functions on top level
11.26/4.50	"properFractionVu30 xx xy = quotRem xx xy;
11.26/4.50	"
11.26/4.50	"properFractionQ1 xx xy (q,vw) = q;
11.26/4.50	"
11.26/4.50	"properFractionR0 xx xy (vx,r) = r;
11.26/4.50	"
11.26/4.50	"properFractionR xx xy = properFractionR0 xx xy (properFractionVu30 xx xy);
11.26/4.50	"
11.26/4.50	"properFractionQ xx xy = properFractionQ1 xx xy (properFractionVu30 xx xy);
11.26/4.50	"
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(10)
11.26/4.50	Obligation:
11.26/4.50	mainModule Main 
11.26/4.50	module Main where {
11.26/4.50	import qualified Prelude;
11.26/4.50	}
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(11) NumRed (SOUND)
11.26/4.50	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(12)
11.26/4.50	Obligation:
11.26/4.50	mainModule Main 
11.26/4.50	module Main where {
11.26/4.50	import qualified Prelude;
11.26/4.50	}
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(13) Narrow (SOUND)
11.26/4.50	Haskell To QDPs
11.26/4.50	
11.26/4.50	digraph dp_graph {
11.26/4.50	node [outthreshold=100, inthreshold=100];1[label="fromEnum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.26/4.50	3[label="fromEnum xz3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
11.26/4.50	4[label="truncate xz3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
11.26/4.50	5[label="truncateM xz3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
11.26/4.50	6[label="truncateM0 xz3 (truncateVu6 xz3)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
11.26/4.50	7[label="truncateM0 xz3 (properFraction xz3)",fontsize=16,color="burlywood",shape="box"];288[label="xz3/xz30 :% xz31",fontsize=10,color="white",style="solid",shape="box"];7 -> 288[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	288 -> 8[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	8[label="truncateM0 (xz30 :% xz31) (properFraction (xz30 :% xz31))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
11.26/4.50	9[label="truncateM0 (xz30 :% xz31) (fromIntegral (properFractionQ xz30 xz31),properFractionR xz30 xz31 :% xz31)",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
11.26/4.50	10[label="fromIntegral (properFractionQ xz30 xz31)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11.26/4.50	11[label="fromInteger . toInteger",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
11.26/4.50	12[label="fromInteger (toInteger (properFractionQ xz30 xz31))",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
11.26/4.50	13[label="fromInteger (Integer (properFractionQ xz30 xz31))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
11.26/4.50	14[label="properFractionQ xz30 xz31",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
11.26/4.50	15[label="properFractionQ1 xz30 xz31 (properFractionVu30 xz30 xz31)",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
11.26/4.50	16[label="properFractionQ1 xz30 xz31 (quotRem xz30 xz31)",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
11.26/4.50	17[label="properFractionQ1 xz30 xz31 (primQrmInt xz30 xz31)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
11.26/4.50	18[label="properFractionQ1 xz30 xz31 (primQuotInt xz30 xz31,primRemInt xz30 xz31)",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
11.26/4.50	19[label="primQuotInt xz30 xz31",fontsize=16,color="burlywood",shape="box"];289[label="xz30/Pos xz300",fontsize=10,color="white",style="solid",shape="box"];19 -> 289[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	289 -> 20[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	290[label="xz30/Neg xz300",fontsize=10,color="white",style="solid",shape="box"];19 -> 290[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	290 -> 21[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	20[label="primQuotInt (Pos xz300) xz31",fontsize=16,color="burlywood",shape="box"];291[label="xz31/Pos xz310",fontsize=10,color="white",style="solid",shape="box"];20 -> 291[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	291 -> 22[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	292[label="xz31/Neg xz310",fontsize=10,color="white",style="solid",shape="box"];20 -> 292[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	292 -> 23[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	21[label="primQuotInt (Neg xz300) xz31",fontsize=16,color="burlywood",shape="box"];293[label="xz31/Pos xz310",fontsize=10,color="white",style="solid",shape="box"];21 -> 293[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	293 -> 24[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	294[label="xz31/Neg xz310",fontsize=10,color="white",style="solid",shape="box"];21 -> 294[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	294 -> 25[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	22[label="primQuotInt (Pos xz300) (Pos xz310)",fontsize=16,color="burlywood",shape="box"];295[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];22 -> 295[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	295 -> 26[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	296[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 296[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	296 -> 27[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	23[label="primQuotInt (Pos xz300) (Neg xz310)",fontsize=16,color="burlywood",shape="box"];297[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];23 -> 297[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	297 -> 28[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	298[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 298[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	298 -> 29[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	24[label="primQuotInt (Neg xz300) (Pos xz310)",fontsize=16,color="burlywood",shape="box"];299[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];24 -> 299[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	299 -> 30[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	300[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 300[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	300 -> 31[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	25[label="primQuotInt (Neg xz300) (Neg xz310)",fontsize=16,color="burlywood",shape="box"];301[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];25 -> 301[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	301 -> 32[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	302[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 302[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	302 -> 33[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	26[label="primQuotInt (Pos xz300) (Pos (Succ xz3100))",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
11.26/4.50	27[label="primQuotInt (Pos xz300) (Pos Zero)",fontsize=16,color="black",shape="box"];27 -> 35[label="",style="solid", color="black", weight=3];
11.26/4.50	28[label="primQuotInt (Pos xz300) (Neg (Succ xz3100))",fontsize=16,color="black",shape="box"];28 -> 36[label="",style="solid", color="black", weight=3];
11.26/4.50	29[label="primQuotInt (Pos xz300) (Neg Zero)",fontsize=16,color="black",shape="box"];29 -> 37[label="",style="solid", color="black", weight=3];
11.26/4.50	30[label="primQuotInt (Neg xz300) (Pos (Succ xz3100))",fontsize=16,color="black",shape="box"];30 -> 38[label="",style="solid", color="black", weight=3];
11.26/4.50	31[label="primQuotInt (Neg xz300) (Pos Zero)",fontsize=16,color="black",shape="box"];31 -> 39[label="",style="solid", color="black", weight=3];
11.26/4.50	32[label="primQuotInt (Neg xz300) (Neg (Succ xz3100))",fontsize=16,color="black",shape="box"];32 -> 40[label="",style="solid", color="black", weight=3];
11.26/4.50	33[label="primQuotInt (Neg xz300) (Neg Zero)",fontsize=16,color="black",shape="box"];33 -> 41[label="",style="solid", color="black", weight=3];
11.26/4.50	34[label="Pos (primDivNatS xz300 (Succ xz3100))",fontsize=16,color="green",shape="box"];34 -> 42[label="",style="dashed", color="green", weight=3];
11.26/4.50	35[label="error []",fontsize=16,color="black",shape="triangle"];35 -> 43[label="",style="solid", color="black", weight=3];
11.26/4.50	36[label="Neg (primDivNatS xz300 (Succ xz3100))",fontsize=16,color="green",shape="box"];36 -> 44[label="",style="dashed", color="green", weight=3];
11.26/4.50	37 -> 35[label="",style="dashed", color="red", weight=0];
11.26/4.50	37[label="error []",fontsize=16,color="magenta"];38[label="Neg (primDivNatS xz300 (Succ xz3100))",fontsize=16,color="green",shape="box"];38 -> 45[label="",style="dashed", color="green", weight=3];
11.26/4.50	39 -> 35[label="",style="dashed", color="red", weight=0];
11.26/4.50	39[label="error []",fontsize=16,color="magenta"];40[label="Pos (primDivNatS xz300 (Succ xz3100))",fontsize=16,color="green",shape="box"];40 -> 46[label="",style="dashed", color="green", weight=3];
11.26/4.50	41 -> 35[label="",style="dashed", color="red", weight=0];
11.26/4.50	41[label="error []",fontsize=16,color="magenta"];42[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="burlywood",shape="triangle"];303[label="xz300/Succ xz3000",fontsize=10,color="white",style="solid",shape="box"];42 -> 303[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	303 -> 47[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	304[label="xz300/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 304[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	304 -> 48[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	43[label="error []",fontsize=16,color="red",shape="box"];44 -> 42[label="",style="dashed", color="red", weight=0];
11.26/4.50	44[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="magenta"];44 -> 49[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	45 -> 42[label="",style="dashed", color="red", weight=0];
11.26/4.50	45[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="magenta"];45 -> 50[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	46 -> 42[label="",style="dashed", color="red", weight=0];
11.26/4.50	46[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="magenta"];46 -> 51[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	46 -> 52[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	47[label="primDivNatS (Succ xz3000) (Succ xz3100)",fontsize=16,color="black",shape="box"];47 -> 53[label="",style="solid", color="black", weight=3];
11.26/4.50	48[label="primDivNatS Zero (Succ xz3100)",fontsize=16,color="black",shape="box"];48 -> 54[label="",style="solid", color="black", weight=3];
11.26/4.50	49[label="xz3100",fontsize=16,color="green",shape="box"];50[label="xz300",fontsize=16,color="green",shape="box"];51[label="xz3100",fontsize=16,color="green",shape="box"];52[label="xz300",fontsize=16,color="green",shape="box"];53[label="primDivNatS0 xz3000 xz3100 (primGEqNatS xz3000 xz3100)",fontsize=16,color="burlywood",shape="box"];305[label="xz3000/Succ xz30000",fontsize=10,color="white",style="solid",shape="box"];53 -> 305[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	305 -> 55[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	306[label="xz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 306[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	306 -> 56[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	54[label="Zero",fontsize=16,color="green",shape="box"];55[label="primDivNatS0 (Succ xz30000) xz3100 (primGEqNatS (Succ xz30000) xz3100)",fontsize=16,color="burlywood",shape="box"];307[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];55 -> 307[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	307 -> 57[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	308[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];55 -> 308[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	308 -> 58[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	56[label="primDivNatS0 Zero xz3100 (primGEqNatS Zero xz3100)",fontsize=16,color="burlywood",shape="box"];309[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];56 -> 309[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	309 -> 59[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	310[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];56 -> 310[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	310 -> 60[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	57[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS (Succ xz30000) (Succ xz31000))",fontsize=16,color="black",shape="box"];57 -> 61[label="",style="solid", color="black", weight=3];
11.26/4.50	58[label="primDivNatS0 (Succ xz30000) Zero (primGEqNatS (Succ xz30000) Zero)",fontsize=16,color="black",shape="box"];58 -> 62[label="",style="solid", color="black", weight=3];
11.26/4.50	59[label="primDivNatS0 Zero (Succ xz31000) (primGEqNatS Zero (Succ xz31000))",fontsize=16,color="black",shape="box"];59 -> 63[label="",style="solid", color="black", weight=3];
11.26/4.50	60[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];60 -> 64[label="",style="solid", color="black", weight=3];
11.26/4.50	61 -> 225[label="",style="dashed", color="red", weight=0];
11.26/4.50	61[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS xz30000 xz31000)",fontsize=16,color="magenta"];61 -> 226[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	61 -> 227[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	61 -> 228[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	61 -> 229[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	62[label="primDivNatS0 (Succ xz30000) Zero True",fontsize=16,color="black",shape="box"];62 -> 67[label="",style="solid", color="black", weight=3];
11.26/4.50	63[label="primDivNatS0 Zero (Succ xz31000) False",fontsize=16,color="black",shape="box"];63 -> 68[label="",style="solid", color="black", weight=3];
11.26/4.50	64[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];64 -> 69[label="",style="solid", color="black", weight=3];
11.26/4.50	226[label="xz31000",fontsize=16,color="green",shape="box"];227[label="xz31000",fontsize=16,color="green",shape="box"];228[label="xz30000",fontsize=16,color="green",shape="box"];229[label="xz30000",fontsize=16,color="green",shape="box"];225[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS xz22 xz23)",fontsize=16,color="burlywood",shape="triangle"];311[label="xz22/Succ xz220",fontsize=10,color="white",style="solid",shape="box"];225 -> 311[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	311 -> 258[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	312[label="xz22/Zero",fontsize=10,color="white",style="solid",shape="box"];225 -> 312[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	312 -> 259[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	67[label="Succ (primDivNatS (primMinusNatS (Succ xz30000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];67 -> 74[label="",style="dashed", color="green", weight=3];
11.26/4.50	68[label="Zero",fontsize=16,color="green",shape="box"];69[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];69 -> 75[label="",style="dashed", color="green", weight=3];
11.26/4.50	258[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS (Succ xz220) xz23)",fontsize=16,color="burlywood",shape="box"];313[label="xz23/Succ xz230",fontsize=10,color="white",style="solid",shape="box"];258 -> 313[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	313 -> 260[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	314[label="xz23/Zero",fontsize=10,color="white",style="solid",shape="box"];258 -> 314[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	314 -> 261[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	259[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero xz23)",fontsize=16,color="burlywood",shape="box"];315[label="xz23/Succ xz230",fontsize=10,color="white",style="solid",shape="box"];259 -> 315[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	315 -> 262[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	316[label="xz23/Zero",fontsize=10,color="white",style="solid",shape="box"];259 -> 316[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	316 -> 263[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	74 -> 42[label="",style="dashed", color="red", weight=0];
11.26/4.50	74[label="primDivNatS (primMinusNatS (Succ xz30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];74 -> 80[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	74 -> 81[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	75 -> 42[label="",style="dashed", color="red", weight=0];
11.26/4.50	75[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];75 -> 82[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	75 -> 83[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	260[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS (Succ xz220) (Succ xz230))",fontsize=16,color="black",shape="box"];260 -> 264[label="",style="solid", color="black", weight=3];
11.26/4.50	261[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS (Succ xz220) Zero)",fontsize=16,color="black",shape="box"];261 -> 265[label="",style="solid", color="black", weight=3];
11.26/4.50	262[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero (Succ xz230))",fontsize=16,color="black",shape="box"];262 -> 266[label="",style="solid", color="black", weight=3];
11.26/4.50	263[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];263 -> 267[label="",style="solid", color="black", weight=3];
11.26/4.50	80[label="Zero",fontsize=16,color="green",shape="box"];81[label="primMinusNatS (Succ xz30000) Zero",fontsize=16,color="black",shape="triangle"];81 -> 89[label="",style="solid", color="black", weight=3];
11.26/4.50	82[label="Zero",fontsize=16,color="green",shape="box"];83[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];83 -> 90[label="",style="solid", color="black", weight=3];
11.26/4.50	264 -> 225[label="",style="dashed", color="red", weight=0];
11.26/4.50	264[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS xz220 xz230)",fontsize=16,color="magenta"];264 -> 268[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	264 -> 269[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	265[label="primDivNatS0 (Succ xz20) (Succ xz21) True",fontsize=16,color="black",shape="triangle"];265 -> 270[label="",style="solid", color="black", weight=3];
11.26/4.50	266[label="primDivNatS0 (Succ xz20) (Succ xz21) False",fontsize=16,color="black",shape="box"];266 -> 271[label="",style="solid", color="black", weight=3];
11.26/4.50	267 -> 265[label="",style="dashed", color="red", weight=0];
11.26/4.50	267[label="primDivNatS0 (Succ xz20) (Succ xz21) True",fontsize=16,color="magenta"];89[label="Succ xz30000",fontsize=16,color="green",shape="box"];90[label="Zero",fontsize=16,color="green",shape="box"];268[label="xz230",fontsize=16,color="green",shape="box"];269[label="xz220",fontsize=16,color="green",shape="box"];270[label="Succ (primDivNatS (primMinusNatS (Succ xz20) (Succ xz21)) (Succ (Succ xz21)))",fontsize=16,color="green",shape="box"];270 -> 272[label="",style="dashed", color="green", weight=3];
11.26/4.50	271[label="Zero",fontsize=16,color="green",shape="box"];272 -> 42[label="",style="dashed", color="red", weight=0];
11.26/4.50	272[label="primDivNatS (primMinusNatS (Succ xz20) (Succ xz21)) (Succ (Succ xz21))",fontsize=16,color="magenta"];272 -> 273[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	272 -> 274[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	273[label="Succ xz21",fontsize=16,color="green",shape="box"];274[label="primMinusNatS (Succ xz20) (Succ xz21)",fontsize=16,color="black",shape="box"];274 -> 275[label="",style="solid", color="black", weight=3];
11.26/4.50	275[label="primMinusNatS xz20 xz21",fontsize=16,color="burlywood",shape="triangle"];317[label="xz20/Succ xz200",fontsize=10,color="white",style="solid",shape="box"];275 -> 317[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	317 -> 276[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	318[label="xz20/Zero",fontsize=10,color="white",style="solid",shape="box"];275 -> 318[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	318 -> 277[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	276[label="primMinusNatS (Succ xz200) xz21",fontsize=16,color="burlywood",shape="box"];319[label="xz21/Succ xz210",fontsize=10,color="white",style="solid",shape="box"];276 -> 319[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	319 -> 278[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	320[label="xz21/Zero",fontsize=10,color="white",style="solid",shape="box"];276 -> 320[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	320 -> 279[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	277[label="primMinusNatS Zero xz21",fontsize=16,color="burlywood",shape="box"];321[label="xz21/Succ xz210",fontsize=10,color="white",style="solid",shape="box"];277 -> 321[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	321 -> 280[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	322[label="xz21/Zero",fontsize=10,color="white",style="solid",shape="box"];277 -> 322[label="",style="solid", color="burlywood", weight=9];
11.26/4.50	322 -> 281[label="",style="solid", color="burlywood", weight=3];
11.26/4.50	278[label="primMinusNatS (Succ xz200) (Succ xz210)",fontsize=16,color="black",shape="box"];278 -> 282[label="",style="solid", color="black", weight=3];
11.26/4.50	279[label="primMinusNatS (Succ xz200) Zero",fontsize=16,color="black",shape="box"];279 -> 283[label="",style="solid", color="black", weight=3];
11.26/4.50	280[label="primMinusNatS Zero (Succ xz210)",fontsize=16,color="black",shape="box"];280 -> 284[label="",style="solid", color="black", weight=3];
11.26/4.50	281[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];281 -> 285[label="",style="solid", color="black", weight=3];
11.26/4.50	282 -> 275[label="",style="dashed", color="red", weight=0];
11.26/4.50	282[label="primMinusNatS xz200 xz210",fontsize=16,color="magenta"];282 -> 286[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	282 -> 287[label="",style="dashed", color="magenta", weight=3];
11.26/4.50	283[label="Succ xz200",fontsize=16,color="green",shape="box"];284[label="Zero",fontsize=16,color="green",shape="box"];285[label="Zero",fontsize=16,color="green",shape="box"];286[label="xz210",fontsize=16,color="green",shape="box"];287[label="xz200",fontsize=16,color="green",shape="box"];}
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(14)
11.26/4.50	Complex Obligation (AND)
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(15)
11.26/4.50	Obligation:
11.26/4.50	Q DP problem:
11.26/4.50	The TRS P consists of the following rules:
11.26/4.50	
11.26/4.50	   new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
11.26/4.50	   new_primDivNatS0(xz20, xz21, Succ(xz220), Zero) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
11.26/4.50	   new_primDivNatS0(xz20, xz21, Zero, Zero) -> new_primDivNatS00(xz20, xz21)
11.26/4.50	   new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) -> new_primDivNatS0(xz20, xz21, xz220, xz230)
11.26/4.50	   new_primDivNatS(Succ(Zero), Zero) -> new_primDivNatS(new_primMinusNatS2, Zero)
11.26/4.50	   new_primDivNatS(Succ(Succ(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero)
11.26/4.50	   new_primDivNatS00(xz20, xz21) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
11.26/4.50	
11.26/4.50	The TRS R consists of the following rules:
11.26/4.50	
11.26/4.50	   new_primMinusNatS0(Zero, Succ(xz210)) -> Zero
11.26/4.50	   new_primMinusNatS0(Zero, Zero) -> Zero
11.26/4.50	   new_primMinusNatS1(xz30000) -> Succ(xz30000)
11.26/4.50	   new_primMinusNatS2 -> Zero
11.26/4.50	   new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210)
11.26/4.50	   new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200)
11.26/4.50	
11.26/4.50	The set Q consists of the following terms:
11.26/4.50	
11.26/4.50	   new_primMinusNatS0(Succ(x0), Succ(x1))
11.26/4.50	   new_primMinusNatS0(Zero, Zero)
11.26/4.50	   new_primMinusNatS2
11.26/4.50	   new_primMinusNatS0(Succ(x0), Zero)
11.26/4.50	   new_primMinusNatS1(x0)
11.26/4.50	   new_primMinusNatS0(Zero, Succ(x0))
11.26/4.50	
11.26/4.50	We have to consider all minimal (P,Q,R)-chains.
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(16) DependencyGraphProof (EQUIVALENT)
11.26/4.50	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(17)
11.26/4.50	Complex Obligation (AND)
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(18)
11.26/4.50	Obligation:
11.26/4.50	Q DP problem:
11.26/4.50	The TRS P consists of the following rules:
11.26/4.50	
11.26/4.50	   new_primDivNatS(Succ(Succ(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero)
11.26/4.50	
11.26/4.50	The TRS R consists of the following rules:
11.26/4.50	
11.26/4.50	   new_primMinusNatS0(Zero, Succ(xz210)) -> Zero
11.26/4.50	   new_primMinusNatS0(Zero, Zero) -> Zero
11.26/4.50	   new_primMinusNatS1(xz30000) -> Succ(xz30000)
11.26/4.50	   new_primMinusNatS2 -> Zero
11.26/4.50	   new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210)
11.26/4.50	   new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200)
11.26/4.50	
11.26/4.50	The set Q consists of the following terms:
11.26/4.50	
11.26/4.50	   new_primMinusNatS0(Succ(x0), Succ(x1))
11.26/4.50	   new_primMinusNatS0(Zero, Zero)
11.26/4.50	   new_primMinusNatS2
11.26/4.50	   new_primMinusNatS0(Succ(x0), Zero)
11.26/4.50	   new_primMinusNatS1(x0)
11.26/4.50	   new_primMinusNatS0(Zero, Succ(x0))
11.26/4.50	
11.26/4.50	We have to consider all minimal (P,Q,R)-chains.
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(19) MRRProof (EQUIVALENT)
11.26/4.50	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.
11.26/4.50	
11.26/4.50	Strictly oriented dependency pairs:
11.26/4.50	
11.26/4.50	   new_primDivNatS(Succ(Succ(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero)
11.26/4.50	
11.26/4.50	Strictly oriented rules of the TRS R:
11.26/4.50	
11.26/4.50	   new_primMinusNatS0(Zero, Succ(xz210)) -> Zero
11.26/4.50	   new_primMinusNatS0(Zero, Zero) -> Zero
11.26/4.50	   new_primMinusNatS1(xz30000) -> Succ(xz30000)
11.26/4.50	   new_primMinusNatS2 -> Zero
11.26/4.50	   new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210)
11.26/4.50	   new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200)
11.26/4.50	
11.26/4.50	Used ordering: Polynomial interpretation [POLO]:
11.26/4.50	
11.26/4.50	   POL(Succ(x_1)) = 1 + 2*x_1
11.26/4.50	   POL(Zero) = 1
11.26/4.50	   POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2
11.26/4.50	   POL(new_primMinusNatS0(x_1, x_2)) = x_1 + x_2
11.26/4.50	   POL(new_primMinusNatS1(x_1)) = 2 + 2*x_1
11.26/4.50	   POL(new_primMinusNatS2) = 2
11.26/4.50	
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(20)
11.26/4.50	Obligation:
11.26/4.50	Q DP problem:
11.26/4.50	P is empty.
11.26/4.50	R is empty.
11.26/4.50	The set Q consists of the following terms:
11.26/4.50	
11.26/4.50	   new_primMinusNatS0(Succ(x0), Succ(x1))
11.26/4.50	   new_primMinusNatS0(Zero, Zero)
11.26/4.50	   new_primMinusNatS2
11.26/4.50	   new_primMinusNatS0(Succ(x0), Zero)
11.26/4.50	   new_primMinusNatS1(x0)
11.26/4.50	   new_primMinusNatS0(Zero, Succ(x0))
11.26/4.50	
11.26/4.50	We have to consider all minimal (P,Q,R)-chains.
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(21) PisEmptyProof (EQUIVALENT)
11.26/4.50	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(22)
11.26/4.50	YES
11.26/4.50	
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(23)
11.26/4.50	Obligation:
11.26/4.50	Q DP problem:
11.26/4.50	The TRS P consists of the following rules:
11.26/4.50	
11.26/4.50	   new_primDivNatS0(xz20, xz21, Succ(xz220), Zero) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
11.26/4.50	   new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
11.26/4.50	   new_primDivNatS0(xz20, xz21, Zero, Zero) -> new_primDivNatS00(xz20, xz21)
11.26/4.50	   new_primDivNatS00(xz20, xz21) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
11.26/4.50	   new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) -> new_primDivNatS0(xz20, xz21, xz220, xz230)
11.26/4.50	
11.26/4.50	The TRS R consists of the following rules:
11.26/4.50	
11.26/4.50	   new_primMinusNatS0(Zero, Succ(xz210)) -> Zero
11.26/4.50	   new_primMinusNatS0(Zero, Zero) -> Zero
11.26/4.50	   new_primMinusNatS1(xz30000) -> Succ(xz30000)
11.26/4.50	   new_primMinusNatS2 -> Zero
11.26/4.50	   new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210)
11.26/4.50	   new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200)
11.26/4.50	
11.26/4.50	The set Q consists of the following terms:
11.26/4.50	
11.26/4.50	   new_primMinusNatS0(Succ(x0), Succ(x1))
11.26/4.50	   new_primMinusNatS0(Zero, Zero)
11.26/4.50	   new_primMinusNatS2
11.26/4.50	   new_primMinusNatS0(Succ(x0), Zero)
11.26/4.50	   new_primMinusNatS1(x0)
11.26/4.50	   new_primMinusNatS0(Zero, Succ(x0))
11.26/4.50	
11.26/4.50	We have to consider all minimal (P,Q,R)-chains.
11.26/4.50	----------------------------------------
11.26/4.50	
11.26/4.50	(24) QDPSizeChangeProof (EQUIVALENT)
11.26/4.50	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
11.26/4.51	
11.26/4.51	Order:Polynomial interpretation [POLO]:
11.26/4.51	
11.26/4.51	   POL(Succ(x_1)) = 1 + x_1
11.26/4.51	   POL(Zero) = 1
11.26/4.51	   POL(new_primMinusNatS0(x_1, x_2)) = x_1
11.26/4.51	
11.26/4.51	
11.26/4.51	
11.26/4.51	
11.26/4.51	From the DPs we obtained the following set of size-change graphs:
11.26/4.51	*new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000) (allowed arguments on rhs = {1, 2, 3, 4})
11.26/4.51	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
11.26/4.51	
11.26/4.51	
11.26/4.51	*new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) -> new_primDivNatS0(xz20, xz21, xz220, xz230) (allowed arguments on rhs = {1, 2, 3, 4})
11.26/4.51	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
11.26/4.51	
11.26/4.51	
11.26/4.51	*new_primDivNatS0(xz20, xz21, Succ(xz220), Zero) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) (allowed arguments on rhs = {1, 2})
11.26/4.51	The graph contains the following edges 1 >= 1
11.26/4.51	
11.26/4.51	
11.26/4.51	*new_primDivNatS0(xz20, xz21, Zero, Zero) -> new_primDivNatS00(xz20, xz21) (allowed arguments on rhs = {1, 2})
11.26/4.51	The graph contains the following edges 1 >= 1, 2 >= 2
11.26/4.51	
11.26/4.51	
11.26/4.51	*new_primDivNatS00(xz20, xz21) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) (allowed arguments on rhs = {1, 2})
11.26/4.51	The graph contains the following edges 1 >= 1
11.26/4.51	
11.26/4.51	
11.26/4.51	
11.26/4.51	We oriented the following set of usable rules [AAECC05,FROCOS05].
11.26/4.51	
11.26/4.51	   new_primMinusNatS0(Zero, Zero) -> Zero
11.26/4.51	   new_primMinusNatS0(Zero, Succ(xz210)) -> Zero
11.26/4.51	   new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200)
11.26/4.51	   new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210)
11.26/4.51	
11.26/4.51	----------------------------------------
11.26/4.51	
11.26/4.51	(25)
11.26/4.51	YES
11.26/4.51	
11.26/4.51	----------------------------------------
11.26/4.51	
11.26/4.51	(26)
11.26/4.51	Obligation:
11.26/4.51	Q DP problem:
11.26/4.51	The TRS P consists of the following rules:
11.26/4.51	
11.26/4.51	   new_primMinusNatS(Succ(xz200), Succ(xz210)) -> new_primMinusNatS(xz200, xz210)
11.26/4.51	
11.26/4.51	R is empty.
11.26/4.51	Q is empty.
11.26/4.51	We have to consider all minimal (P,Q,R)-chains.
11.26/4.51	----------------------------------------
11.26/4.51	
11.26/4.51	(27) QDPSizeChangeProof (EQUIVALENT)
11.26/4.51	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. 
11.26/4.51	
11.26/4.51	From the DPs we obtained the following set of size-change graphs:
11.26/4.51	*new_primMinusNatS(Succ(xz200), Succ(xz210)) -> new_primMinusNatS(xz200, xz210)
11.26/4.51	The graph contains the following edges 1 > 1, 2 > 2
11.26/4.51	
11.26/4.51	
11.26/4.51	----------------------------------------
11.26/4.51	
11.26/4.51	(28)
11.26/4.51	YES
11.54/4.54	EOF