9.30/3.99	YES
11.83/4.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.83/4.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.83/4.60	
11.83/4.60	
11.83/4.60	H-Termination with start terms of the given HASKELL could be proven:
11.83/4.60	
11.83/4.60	(0) HASKELL
11.83/4.60	(1) LR [EQUIVALENT, 0 ms]
11.83/4.60	(2) HASKELL
11.83/4.60	(3) IFR [EQUIVALENT, 0 ms]
11.83/4.60	(4) HASKELL
11.83/4.60	(5) BR [EQUIVALENT, 0 ms]
11.83/4.60	(6) HASKELL
11.83/4.60	(7) COR [EQUIVALENT, 0 ms]
11.83/4.60	(8) HASKELL
11.83/4.60	(9) LetRed [EQUIVALENT, 0 ms]
11.83/4.60	(10) HASKELL
11.83/4.60	(11) Narrow [SOUND, 0 ms]
11.83/4.60	(12) AND
11.83/4.60	    (13) QDP
11.83/4.60	        (14) DependencyGraphProof [EQUIVALENT, 0 ms]
11.83/4.60	        (15) AND
11.83/4.60	            (16) QDP
11.83/4.60	                (17) MRRProof [EQUIVALENT, 0 ms]
11.83/4.60	                (18) QDP
11.83/4.60	                (19) PisEmptyProof [EQUIVALENT, 0 ms]
11.83/4.60	                (20) YES
11.83/4.60	            (21) QDP
11.83/4.60	                (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.83/4.60	                (23) YES
11.83/4.60	    (24) QDP
11.83/4.60	        (25) DependencyGraphProof [EQUIVALENT, 0 ms]
11.83/4.60	        (26) AND
11.83/4.60	            (27) QDP
11.83/4.60	                (28) MRRProof [EQUIVALENT, 0 ms]
11.83/4.60	                (29) QDP
11.83/4.60	                (30) PisEmptyProof [EQUIVALENT, 0 ms]
11.83/4.60	                (31) YES
11.83/4.60	            (32) QDP
11.83/4.60	                (33) QDPOrderProof [EQUIVALENT, 6 ms]
11.83/4.60	                (34) QDP
11.83/4.60	                (35) DependencyGraphProof [EQUIVALENT, 0 ms]
11.83/4.60	                (36) QDP
11.83/4.60	                (37) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.83/4.60	                (38) YES
11.83/4.60	    (39) QDP
11.83/4.60	        (40) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.83/4.60	        (41) YES
11.83/4.60	
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(0)
11.83/4.60	Obligation:
11.83/4.60	mainModule Main 
11.83/4.60	module Main where {
11.83/4.60	import qualified Prelude;
11.83/4.60	}
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(1) LR (EQUIVALENT)
11.83/4.60	Lambda Reductions:
11.83/4.60	The following Lambda expression
11.83/4.60	"\(q,_)->q"
11.83/4.60	is transformed to
11.83/4.60	"q1 (q,_) = q;
11.83/4.60	"
11.83/4.60	The following Lambda expression
11.83/4.60	"\(_,r)->r"
11.83/4.60	is transformed to
11.83/4.60	"r0 (_,r) = r;
11.83/4.60	"
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(2)
11.83/4.60	Obligation:
11.83/4.60	mainModule Main 
11.83/4.60	module Main where {
11.83/4.60	import qualified Prelude;
11.83/4.60	}
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(3) IFR (EQUIVALENT)
11.83/4.60	If Reductions:
11.83/4.60	The following If expression
11.83/4.60	"if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x"
11.83/4.60	is transformed to
11.83/4.60	"primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y);
11.83/4.60	primModNatS0 x y False = Succ x;
11.83/4.60	"
11.83/4.60	The following If expression
11.83/4.60	"if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero"
11.83/4.60	is transformed to
11.83/4.60	"primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y));
11.83/4.60	primDivNatS0 x y False = Zero;
11.83/4.60	"
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(4)
11.83/4.60	Obligation:
11.83/4.60	mainModule Main 
11.83/4.60	module Main where {
11.83/4.60	import qualified Prelude;
11.83/4.60	}
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(5) BR (EQUIVALENT)
11.83/4.60	Replaced joker patterns by fresh variables and removed binding patterns.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(6)
11.83/4.60	Obligation:
11.83/4.60	mainModule Main 
11.83/4.60	module Main where {
11.83/4.60	import qualified Prelude;
11.83/4.60	}
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(7) COR (EQUIVALENT)
11.83/4.60	Cond Reductions:
11.83/4.60	The following Function with conditions
11.83/4.60	"undefined |Falseundefined;
11.83/4.60	"
11.83/4.60	is transformed to
11.83/4.60	"undefined  = undefined1;
11.83/4.60	"
11.83/4.60	"undefined0 True = undefined;
11.83/4.60	"
11.83/4.60	"undefined1  = undefined0 False;
11.83/4.60	"
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(8)
11.83/4.60	Obligation:
11.83/4.60	mainModule Main 
11.83/4.60	module Main where {
11.83/4.60	import qualified Prelude;
11.83/4.60	}
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(9) LetRed (EQUIVALENT)
11.83/4.60	Let/Where Reductions:
11.83/4.60	The bindings of the following Let/Where expression
11.83/4.60	"(fromIntegral q,r :% y) where {
11.83/4.60	q  = q1 vu30;
11.83/4.60	;
11.83/4.60	q1 (q,vv) = q;
11.83/4.60	;
11.83/4.60	r  = r0 vu30;
11.83/4.60	;
11.83/4.60	r0 (vw,r) = r;
11.83/4.60	;
11.83/4.60	vu30  = quotRem x y;
11.83/4.60	} 
11.83/4.60	"
11.83/4.60	are unpacked to the following functions on top level
11.83/4.60	"properFractionR wx wy = properFractionR0 wx wy (properFractionVu30 wx wy);
11.83/4.60	"
11.83/4.60	"properFractionVu30 wx wy = quotRem wx wy;
11.83/4.60	"
11.83/4.60	"properFractionQ1 wx wy (q,vv) = q;
11.83/4.60	"
11.83/4.60	"properFractionQ wx wy = properFractionQ1 wx wy (properFractionVu30 wx wy);
11.83/4.60	"
11.83/4.60	"properFractionR0 wx wy (vw,r) = r;
11.83/4.60	"
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(10)
11.83/4.60	Obligation:
11.83/4.60	mainModule Main 
11.83/4.60	module Main where {
11.83/4.60	import qualified Prelude;
11.83/4.60	}
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(11) Narrow (SOUND)
11.83/4.60	Haskell To QDPs
11.83/4.60	
11.83/4.60	digraph dp_graph {
11.83/4.60	node [outthreshold=100, inthreshold=100];1[label="properFraction",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.83/4.60	3[label="properFraction wz3",fontsize=16,color="burlywood",shape="triangle"];561[label="wz3/wz30 :% wz31",fontsize=10,color="white",style="solid",shape="box"];3 -> 561[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	561 -> 4[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	4[label="properFraction (wz30 :% wz31)",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
11.83/4.60	5[label="(fromIntegral (properFractionQ wz30 wz31),properFractionR wz30 wz31 :% wz31)",fontsize=16,color="green",shape="box"];5 -> 6[label="",style="dashed", color="green", weight=3];
11.83/4.60	5 -> 7[label="",style="dashed", color="green", weight=3];
11.83/4.60	6[label="fromIntegral (properFractionQ wz30 wz31)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
11.83/4.60	7[label="properFractionR wz30 wz31",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
11.83/4.60	8[label="fromInteger . toInteger",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
11.83/4.60	9[label="properFractionR0 wz30 wz31 (properFractionVu30 wz30 wz31)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
11.83/4.60	10[label="fromInteger (toInteger (properFractionQ wz30 wz31))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11.83/4.60	11[label="properFractionR0 wz30 wz31 (quotRem wz30 wz31)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
11.83/4.60	12[label="fromInteger (Integer (properFractionQ wz30 wz31))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
11.83/4.60	13[label="properFractionR0 wz30 wz31 (primQrmInt wz30 wz31)",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
11.83/4.60	14[label="properFractionQ wz30 wz31",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
11.83/4.60	15[label="properFractionR0 wz30 wz31 (primQuotInt wz30 wz31,primRemInt wz30 wz31)",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
11.83/4.60	16[label="properFractionQ1 wz30 wz31 (properFractionVu30 wz30 wz31)",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
11.83/4.60	17[label="primRemInt wz30 wz31",fontsize=16,color="burlywood",shape="triangle"];562[label="wz30/Pos wz300",fontsize=10,color="white",style="solid",shape="box"];17 -> 562[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	562 -> 19[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	563[label="wz30/Neg wz300",fontsize=10,color="white",style="solid",shape="box"];17 -> 563[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	563 -> 20[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	18[label="properFractionQ1 wz30 wz31 (quotRem wz30 wz31)",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
11.83/4.60	19[label="primRemInt (Pos wz300) wz31",fontsize=16,color="burlywood",shape="box"];564[label="wz31/Pos wz310",fontsize=10,color="white",style="solid",shape="box"];19 -> 564[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	564 -> 22[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	565[label="wz31/Neg wz310",fontsize=10,color="white",style="solid",shape="box"];19 -> 565[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	565 -> 23[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	20[label="primRemInt (Neg wz300) wz31",fontsize=16,color="burlywood",shape="box"];566[label="wz31/Pos wz310",fontsize=10,color="white",style="solid",shape="box"];20 -> 566[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	566 -> 24[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	567[label="wz31/Neg wz310",fontsize=10,color="white",style="solid",shape="box"];20 -> 567[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	567 -> 25[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	21[label="properFractionQ1 wz30 wz31 (primQrmInt wz30 wz31)",fontsize=16,color="black",shape="box"];21 -> 26[label="",style="solid", color="black", weight=3];
11.83/4.60	22[label="primRemInt (Pos wz300) (Pos wz310)",fontsize=16,color="burlywood",shape="box"];568[label="wz310/Succ wz3100",fontsize=10,color="white",style="solid",shape="box"];22 -> 568[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	568 -> 27[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	569[label="wz310/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 569[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	569 -> 28[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	23[label="primRemInt (Pos wz300) (Neg wz310)",fontsize=16,color="burlywood",shape="box"];570[label="wz310/Succ wz3100",fontsize=10,color="white",style="solid",shape="box"];23 -> 570[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	570 -> 29[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	571[label="wz310/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 571[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	571 -> 30[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	24[label="primRemInt (Neg wz300) (Pos wz310)",fontsize=16,color="burlywood",shape="box"];572[label="wz310/Succ wz3100",fontsize=10,color="white",style="solid",shape="box"];24 -> 572[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	572 -> 31[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	573[label="wz310/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 573[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	573 -> 32[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	25[label="primRemInt (Neg wz300) (Neg wz310)",fontsize=16,color="burlywood",shape="box"];574[label="wz310/Succ wz3100",fontsize=10,color="white",style="solid",shape="box"];25 -> 574[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	574 -> 33[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	575[label="wz310/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 575[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	575 -> 34[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	26 -> 35[label="",style="dashed", color="red", weight=0];
11.83/4.60	26[label="properFractionQ1 wz30 wz31 (primQuotInt wz30 wz31,primRemInt wz30 wz31)",fontsize=16,color="magenta"];26 -> 36[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	27[label="primRemInt (Pos wz300) (Pos (Succ wz3100))",fontsize=16,color="black",shape="box"];27 -> 37[label="",style="solid", color="black", weight=3];
11.83/4.60	28[label="primRemInt (Pos wz300) (Pos Zero)",fontsize=16,color="black",shape="box"];28 -> 38[label="",style="solid", color="black", weight=3];
11.83/4.60	29[label="primRemInt (Pos wz300) (Neg (Succ wz3100))",fontsize=16,color="black",shape="box"];29 -> 39[label="",style="solid", color="black", weight=3];
11.83/4.60	30[label="primRemInt (Pos wz300) (Neg Zero)",fontsize=16,color="black",shape="box"];30 -> 40[label="",style="solid", color="black", weight=3];
11.83/4.60	31[label="primRemInt (Neg wz300) (Pos (Succ wz3100))",fontsize=16,color="black",shape="box"];31 -> 41[label="",style="solid", color="black", weight=3];
11.83/4.60	32[label="primRemInt (Neg wz300) (Pos Zero)",fontsize=16,color="black",shape="box"];32 -> 42[label="",style="solid", color="black", weight=3];
11.83/4.60	33[label="primRemInt (Neg wz300) (Neg (Succ wz3100))",fontsize=16,color="black",shape="box"];33 -> 43[label="",style="solid", color="black", weight=3];
11.83/4.60	34[label="primRemInt (Neg wz300) (Neg Zero)",fontsize=16,color="black",shape="box"];34 -> 44[label="",style="solid", color="black", weight=3];
11.83/4.60	36 -> 17[label="",style="dashed", color="red", weight=0];
11.83/4.60	36[label="primRemInt wz30 wz31",fontsize=16,color="magenta"];35[label="properFractionQ1 wz30 wz31 (primQuotInt wz30 wz31,wz4)",fontsize=16,color="black",shape="triangle"];35 -> 45[label="",style="solid", color="black", weight=3];
11.83/4.60	37[label="Pos (primModNatS wz300 (Succ wz3100))",fontsize=16,color="green",shape="box"];37 -> 46[label="",style="dashed", color="green", weight=3];
11.83/4.60	38[label="error []",fontsize=16,color="black",shape="triangle"];38 -> 47[label="",style="solid", color="black", weight=3];
11.83/4.60	39[label="Pos (primModNatS wz300 (Succ wz3100))",fontsize=16,color="green",shape="box"];39 -> 48[label="",style="dashed", color="green", weight=3];
11.83/4.60	40 -> 38[label="",style="dashed", color="red", weight=0];
11.83/4.60	40[label="error []",fontsize=16,color="magenta"];41[label="Neg (primModNatS wz300 (Succ wz3100))",fontsize=16,color="green",shape="box"];41 -> 49[label="",style="dashed", color="green", weight=3];
11.83/4.60	42 -> 38[label="",style="dashed", color="red", weight=0];
11.83/4.60	42[label="error []",fontsize=16,color="magenta"];43[label="Neg (primModNatS wz300 (Succ wz3100))",fontsize=16,color="green",shape="box"];43 -> 50[label="",style="dashed", color="green", weight=3];
11.83/4.60	44 -> 38[label="",style="dashed", color="red", weight=0];
11.83/4.60	44[label="error []",fontsize=16,color="magenta"];45[label="primQuotInt wz30 wz31",fontsize=16,color="burlywood",shape="box"];576[label="wz30/Pos wz300",fontsize=10,color="white",style="solid",shape="box"];45 -> 576[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	576 -> 51[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	577[label="wz30/Neg wz300",fontsize=10,color="white",style="solid",shape="box"];45 -> 577[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	577 -> 52[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	46[label="primModNatS wz300 (Succ wz3100)",fontsize=16,color="burlywood",shape="triangle"];578[label="wz300/Succ wz3000",fontsize=10,color="white",style="solid",shape="box"];46 -> 578[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	578 -> 53[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	579[label="wz300/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 579[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	579 -> 54[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	47[label="error []",fontsize=16,color="red",shape="box"];48 -> 46[label="",style="dashed", color="red", weight=0];
11.83/4.60	48[label="primModNatS wz300 (Succ wz3100)",fontsize=16,color="magenta"];48 -> 55[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	49 -> 46[label="",style="dashed", color="red", weight=0];
11.83/4.60	49[label="primModNatS wz300 (Succ wz3100)",fontsize=16,color="magenta"];49 -> 56[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	50 -> 46[label="",style="dashed", color="red", weight=0];
11.83/4.60	50[label="primModNatS wz300 (Succ wz3100)",fontsize=16,color="magenta"];50 -> 57[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	50 -> 58[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	51[label="primQuotInt (Pos wz300) wz31",fontsize=16,color="burlywood",shape="box"];580[label="wz31/Pos wz310",fontsize=10,color="white",style="solid",shape="box"];51 -> 580[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	580 -> 59[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	581[label="wz31/Neg wz310",fontsize=10,color="white",style="solid",shape="box"];51 -> 581[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	581 -> 60[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	52[label="primQuotInt (Neg wz300) wz31",fontsize=16,color="burlywood",shape="box"];582[label="wz31/Pos wz310",fontsize=10,color="white",style="solid",shape="box"];52 -> 582[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	582 -> 61[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	583[label="wz31/Neg wz310",fontsize=10,color="white",style="solid",shape="box"];52 -> 583[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	583 -> 62[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	53[label="primModNatS (Succ wz3000) (Succ wz3100)",fontsize=16,color="black",shape="box"];53 -> 63[label="",style="solid", color="black", weight=3];
11.83/4.60	54[label="primModNatS Zero (Succ wz3100)",fontsize=16,color="black",shape="box"];54 -> 64[label="",style="solid", color="black", weight=3];
11.83/4.60	55[label="wz3100",fontsize=16,color="green",shape="box"];56[label="wz300",fontsize=16,color="green",shape="box"];57[label="wz300",fontsize=16,color="green",shape="box"];58[label="wz3100",fontsize=16,color="green",shape="box"];59[label="primQuotInt (Pos wz300) (Pos wz310)",fontsize=16,color="burlywood",shape="box"];584[label="wz310/Succ wz3100",fontsize=10,color="white",style="solid",shape="box"];59 -> 584[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	584 -> 65[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	585[label="wz310/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 585[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	585 -> 66[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	60[label="primQuotInt (Pos wz300) (Neg wz310)",fontsize=16,color="burlywood",shape="box"];586[label="wz310/Succ wz3100",fontsize=10,color="white",style="solid",shape="box"];60 -> 586[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	586 -> 67[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	587[label="wz310/Zero",fontsize=10,color="white",style="solid",shape="box"];60 -> 587[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	587 -> 68[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	61[label="primQuotInt (Neg wz300) (Pos wz310)",fontsize=16,color="burlywood",shape="box"];588[label="wz310/Succ wz3100",fontsize=10,color="white",style="solid",shape="box"];61 -> 588[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	588 -> 69[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	589[label="wz310/Zero",fontsize=10,color="white",style="solid",shape="box"];61 -> 589[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	589 -> 70[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	62[label="primQuotInt (Neg wz300) (Neg wz310)",fontsize=16,color="burlywood",shape="box"];590[label="wz310/Succ wz3100",fontsize=10,color="white",style="solid",shape="box"];62 -> 590[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	590 -> 71[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	591[label="wz310/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 591[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	591 -> 72[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	63[label="primModNatS0 wz3000 wz3100 (primGEqNatS wz3000 wz3100)",fontsize=16,color="burlywood",shape="box"];592[label="wz3000/Succ wz30000",fontsize=10,color="white",style="solid",shape="box"];63 -> 592[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	592 -> 73[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	593[label="wz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];63 -> 593[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	593 -> 74[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	64[label="Zero",fontsize=16,color="green",shape="box"];65[label="primQuotInt (Pos wz300) (Pos (Succ wz3100))",fontsize=16,color="black",shape="box"];65 -> 75[label="",style="solid", color="black", weight=3];
11.83/4.60	66[label="primQuotInt (Pos wz300) (Pos Zero)",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3];
11.83/4.60	67[label="primQuotInt (Pos wz300) (Neg (Succ wz3100))",fontsize=16,color="black",shape="box"];67 -> 77[label="",style="solid", color="black", weight=3];
11.83/4.60	68[label="primQuotInt (Pos wz300) (Neg Zero)",fontsize=16,color="black",shape="box"];68 -> 78[label="",style="solid", color="black", weight=3];
11.83/4.60	69[label="primQuotInt (Neg wz300) (Pos (Succ wz3100))",fontsize=16,color="black",shape="box"];69 -> 79[label="",style="solid", color="black", weight=3];
11.83/4.60	70[label="primQuotInt (Neg wz300) (Pos Zero)",fontsize=16,color="black",shape="box"];70 -> 80[label="",style="solid", color="black", weight=3];
11.83/4.60	71[label="primQuotInt (Neg wz300) (Neg (Succ wz3100))",fontsize=16,color="black",shape="box"];71 -> 81[label="",style="solid", color="black", weight=3];
11.83/4.60	72[label="primQuotInt (Neg wz300) (Neg Zero)",fontsize=16,color="black",shape="box"];72 -> 82[label="",style="solid", color="black", weight=3];
11.83/4.60	73[label="primModNatS0 (Succ wz30000) wz3100 (primGEqNatS (Succ wz30000) wz3100)",fontsize=16,color="burlywood",shape="box"];594[label="wz3100/Succ wz31000",fontsize=10,color="white",style="solid",shape="box"];73 -> 594[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	594 -> 83[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	595[label="wz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];73 -> 595[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	595 -> 84[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	74[label="primModNatS0 Zero wz3100 (primGEqNatS Zero wz3100)",fontsize=16,color="burlywood",shape="box"];596[label="wz3100/Succ wz31000",fontsize=10,color="white",style="solid",shape="box"];74 -> 596[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	596 -> 85[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	597[label="wz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];74 -> 597[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	597 -> 86[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	75[label="Pos (primDivNatS wz300 (Succ wz3100))",fontsize=16,color="green",shape="box"];75 -> 87[label="",style="dashed", color="green", weight=3];
11.83/4.60	76 -> 38[label="",style="dashed", color="red", weight=0];
11.83/4.60	76[label="error []",fontsize=16,color="magenta"];77[label="Neg (primDivNatS wz300 (Succ wz3100))",fontsize=16,color="green",shape="box"];77 -> 88[label="",style="dashed", color="green", weight=3];
11.83/4.60	78 -> 38[label="",style="dashed", color="red", weight=0];
11.83/4.60	78[label="error []",fontsize=16,color="magenta"];79[label="Neg (primDivNatS wz300 (Succ wz3100))",fontsize=16,color="green",shape="box"];79 -> 89[label="",style="dashed", color="green", weight=3];
11.83/4.60	80 -> 38[label="",style="dashed", color="red", weight=0];
11.83/4.60	80[label="error []",fontsize=16,color="magenta"];81[label="Pos (primDivNatS wz300 (Succ wz3100))",fontsize=16,color="green",shape="box"];81 -> 90[label="",style="dashed", color="green", weight=3];
11.83/4.60	82 -> 38[label="",style="dashed", color="red", weight=0];
11.83/4.60	82[label="error []",fontsize=16,color="magenta"];83[label="primModNatS0 (Succ wz30000) (Succ wz31000) (primGEqNatS (Succ wz30000) (Succ wz31000))",fontsize=16,color="black",shape="box"];83 -> 91[label="",style="solid", color="black", weight=3];
11.83/4.60	84[label="primModNatS0 (Succ wz30000) Zero (primGEqNatS (Succ wz30000) Zero)",fontsize=16,color="black",shape="box"];84 -> 92[label="",style="solid", color="black", weight=3];
11.83/4.60	85[label="primModNatS0 Zero (Succ wz31000) (primGEqNatS Zero (Succ wz31000))",fontsize=16,color="black",shape="box"];85 -> 93[label="",style="solid", color="black", weight=3];
11.83/4.60	86[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];86 -> 94[label="",style="solid", color="black", weight=3];
11.83/4.60	87[label="primDivNatS wz300 (Succ wz3100)",fontsize=16,color="burlywood",shape="triangle"];598[label="wz300/Succ wz3000",fontsize=10,color="white",style="solid",shape="box"];87 -> 598[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	598 -> 95[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	599[label="wz300/Zero",fontsize=10,color="white",style="solid",shape="box"];87 -> 599[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	599 -> 96[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	88 -> 87[label="",style="dashed", color="red", weight=0];
11.83/4.60	88[label="primDivNatS wz300 (Succ wz3100)",fontsize=16,color="magenta"];88 -> 97[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	89 -> 87[label="",style="dashed", color="red", weight=0];
11.83/4.60	89[label="primDivNatS wz300 (Succ wz3100)",fontsize=16,color="magenta"];89 -> 98[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	90 -> 87[label="",style="dashed", color="red", weight=0];
11.83/4.60	90[label="primDivNatS wz300 (Succ wz3100)",fontsize=16,color="magenta"];90 -> 99[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	90 -> 100[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	91 -> 362[label="",style="dashed", color="red", weight=0];
11.83/4.60	91[label="primModNatS0 (Succ wz30000) (Succ wz31000) (primGEqNatS wz30000 wz31000)",fontsize=16,color="magenta"];91 -> 363[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	91 -> 364[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	91 -> 365[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	91 -> 366[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	92[label="primModNatS0 (Succ wz30000) Zero True",fontsize=16,color="black",shape="box"];92 -> 103[label="",style="solid", color="black", weight=3];
11.83/4.60	93[label="primModNatS0 Zero (Succ wz31000) False",fontsize=16,color="black",shape="box"];93 -> 104[label="",style="solid", color="black", weight=3];
11.83/4.60	94[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];94 -> 105[label="",style="solid", color="black", weight=3];
11.83/4.60	95[label="primDivNatS (Succ wz3000) (Succ wz3100)",fontsize=16,color="black",shape="box"];95 -> 106[label="",style="solid", color="black", weight=3];
11.83/4.60	96[label="primDivNatS Zero (Succ wz3100)",fontsize=16,color="black",shape="box"];96 -> 107[label="",style="solid", color="black", weight=3];
11.83/4.60	97[label="wz3100",fontsize=16,color="green",shape="box"];98[label="wz300",fontsize=16,color="green",shape="box"];99[label="wz300",fontsize=16,color="green",shape="box"];100[label="wz3100",fontsize=16,color="green",shape="box"];363[label="wz31000",fontsize=16,color="green",shape="box"];364[label="wz30000",fontsize=16,color="green",shape="box"];365[label="wz31000",fontsize=16,color="green",shape="box"];366[label="wz30000",fontsize=16,color="green",shape="box"];362[label="primModNatS0 (Succ wz21) (Succ wz22) (primGEqNatS wz23 wz24)",fontsize=16,color="burlywood",shape="triangle"];600[label="wz23/Succ wz230",fontsize=10,color="white",style="solid",shape="box"];362 -> 600[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	600 -> 395[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	601[label="wz23/Zero",fontsize=10,color="white",style="solid",shape="box"];362 -> 601[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	601 -> 396[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	103 -> 46[label="",style="dashed", color="red", weight=0];
11.83/4.60	103[label="primModNatS (primMinusNatS (Succ wz30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];103 -> 112[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	103 -> 113[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	104[label="Succ Zero",fontsize=16,color="green",shape="box"];105 -> 46[label="",style="dashed", color="red", weight=0];
11.83/4.60	105[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];105 -> 114[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	105 -> 115[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	106[label="primDivNatS0 wz3000 wz3100 (primGEqNatS wz3000 wz3100)",fontsize=16,color="burlywood",shape="box"];602[label="wz3000/Succ wz30000",fontsize=10,color="white",style="solid",shape="box"];106 -> 602[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	602 -> 116[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	603[label="wz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];106 -> 603[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	603 -> 117[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	107[label="Zero",fontsize=16,color="green",shape="box"];395[label="primModNatS0 (Succ wz21) (Succ wz22) (primGEqNatS (Succ wz230) wz24)",fontsize=16,color="burlywood",shape="box"];604[label="wz24/Succ wz240",fontsize=10,color="white",style="solid",shape="box"];395 -> 604[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	604 -> 402[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	605[label="wz24/Zero",fontsize=10,color="white",style="solid",shape="box"];395 -> 605[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	605 -> 403[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	396[label="primModNatS0 (Succ wz21) (Succ wz22) (primGEqNatS Zero wz24)",fontsize=16,color="burlywood",shape="box"];606[label="wz24/Succ wz240",fontsize=10,color="white",style="solid",shape="box"];396 -> 606[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	606 -> 404[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	607[label="wz24/Zero",fontsize=10,color="white",style="solid",shape="box"];396 -> 607[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	607 -> 405[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	112[label="primMinusNatS (Succ wz30000) Zero",fontsize=16,color="black",shape="triangle"];112 -> 122[label="",style="solid", color="black", weight=3];
11.83/4.60	113[label="Zero",fontsize=16,color="green",shape="box"];114[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];114 -> 123[label="",style="solid", color="black", weight=3];
11.83/4.60	115[label="Zero",fontsize=16,color="green",shape="box"];116[label="primDivNatS0 (Succ wz30000) wz3100 (primGEqNatS (Succ wz30000) wz3100)",fontsize=16,color="burlywood",shape="box"];608[label="wz3100/Succ wz31000",fontsize=10,color="white",style="solid",shape="box"];116 -> 608[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	608 -> 124[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	609[label="wz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];116 -> 609[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	609 -> 125[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	117[label="primDivNatS0 Zero wz3100 (primGEqNatS Zero wz3100)",fontsize=16,color="burlywood",shape="box"];610[label="wz3100/Succ wz31000",fontsize=10,color="white",style="solid",shape="box"];117 -> 610[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	610 -> 126[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	611[label="wz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];117 -> 611[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	611 -> 127[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	402[label="primModNatS0 (Succ wz21) (Succ wz22) (primGEqNatS (Succ wz230) (Succ wz240))",fontsize=16,color="black",shape="box"];402 -> 408[label="",style="solid", color="black", weight=3];
11.83/4.60	403[label="primModNatS0 (Succ wz21) (Succ wz22) (primGEqNatS (Succ wz230) Zero)",fontsize=16,color="black",shape="box"];403 -> 409[label="",style="solid", color="black", weight=3];
11.83/4.60	404[label="primModNatS0 (Succ wz21) (Succ wz22) (primGEqNatS Zero (Succ wz240))",fontsize=16,color="black",shape="box"];404 -> 410[label="",style="solid", color="black", weight=3];
11.83/4.60	405[label="primModNatS0 (Succ wz21) (Succ wz22) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];405 -> 411[label="",style="solid", color="black", weight=3];
11.83/4.60	122[label="Succ wz30000",fontsize=16,color="green",shape="box"];123[label="Zero",fontsize=16,color="green",shape="box"];124[label="primDivNatS0 (Succ wz30000) (Succ wz31000) (primGEqNatS (Succ wz30000) (Succ wz31000))",fontsize=16,color="black",shape="box"];124 -> 133[label="",style="solid", color="black", weight=3];
11.83/4.60	125[label="primDivNatS0 (Succ wz30000) Zero (primGEqNatS (Succ wz30000) Zero)",fontsize=16,color="black",shape="box"];125 -> 134[label="",style="solid", color="black", weight=3];
11.83/4.60	126[label="primDivNatS0 Zero (Succ wz31000) (primGEqNatS Zero (Succ wz31000))",fontsize=16,color="black",shape="box"];126 -> 135[label="",style="solid", color="black", weight=3];
11.83/4.60	127[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];127 -> 136[label="",style="solid", color="black", weight=3];
11.83/4.60	408 -> 362[label="",style="dashed", color="red", weight=0];
11.83/4.60	408[label="primModNatS0 (Succ wz21) (Succ wz22) (primGEqNatS wz230 wz240)",fontsize=16,color="magenta"];408 -> 420[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	408 -> 421[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	409[label="primModNatS0 (Succ wz21) (Succ wz22) True",fontsize=16,color="black",shape="triangle"];409 -> 422[label="",style="solid", color="black", weight=3];
11.83/4.60	410[label="primModNatS0 (Succ wz21) (Succ wz22) False",fontsize=16,color="black",shape="box"];410 -> 423[label="",style="solid", color="black", weight=3];
11.83/4.60	411 -> 409[label="",style="dashed", color="red", weight=0];
11.83/4.60	411[label="primModNatS0 (Succ wz21) (Succ wz22) True",fontsize=16,color="magenta"];133 -> 505[label="",style="dashed", color="red", weight=0];
11.83/4.60	133[label="primDivNatS0 (Succ wz30000) (Succ wz31000) (primGEqNatS wz30000 wz31000)",fontsize=16,color="magenta"];133 -> 506[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	133 -> 507[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	133 -> 508[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	133 -> 509[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	134[label="primDivNatS0 (Succ wz30000) Zero True",fontsize=16,color="black",shape="box"];134 -> 147[label="",style="solid", color="black", weight=3];
11.83/4.60	135[label="primDivNatS0 Zero (Succ wz31000) False",fontsize=16,color="black",shape="box"];135 -> 148[label="",style="solid", color="black", weight=3];
11.83/4.60	136[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];136 -> 149[label="",style="solid", color="black", weight=3];
11.83/4.60	420[label="wz240",fontsize=16,color="green",shape="box"];421[label="wz230",fontsize=16,color="green",shape="box"];422 -> 46[label="",style="dashed", color="red", weight=0];
11.83/4.60	422[label="primModNatS (primMinusNatS (Succ wz21) (Succ wz22)) (Succ (Succ wz22))",fontsize=16,color="magenta"];422 -> 430[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	422 -> 431[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	423[label="Succ (Succ wz21)",fontsize=16,color="green",shape="box"];506[label="wz31000",fontsize=16,color="green",shape="box"];507[label="wz30000",fontsize=16,color="green",shape="box"];508[label="wz30000",fontsize=16,color="green",shape="box"];509[label="wz31000",fontsize=16,color="green",shape="box"];505[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS wz47 wz48)",fontsize=16,color="burlywood",shape="triangle"];612[label="wz47/Succ wz470",fontsize=10,color="white",style="solid",shape="box"];505 -> 612[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	612 -> 542[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	613[label="wz47/Zero",fontsize=10,color="white",style="solid",shape="box"];505 -> 613[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	613 -> 543[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	147[label="Succ (primDivNatS (primMinusNatS (Succ wz30000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];147 -> 160[label="",style="dashed", color="green", weight=3];
11.83/4.60	148[label="Zero",fontsize=16,color="green",shape="box"];149[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];149 -> 161[label="",style="dashed", color="green", weight=3];
11.83/4.60	430[label="primMinusNatS (Succ wz21) (Succ wz22)",fontsize=16,color="black",shape="box"];430 -> 436[label="",style="solid", color="black", weight=3];
11.83/4.60	431[label="Succ wz22",fontsize=16,color="green",shape="box"];542[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS (Succ wz470) wz48)",fontsize=16,color="burlywood",shape="box"];614[label="wz48/Succ wz480",fontsize=10,color="white",style="solid",shape="box"];542 -> 614[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	614 -> 544[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	615[label="wz48/Zero",fontsize=10,color="white",style="solid",shape="box"];542 -> 615[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	615 -> 545[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	543[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS Zero wz48)",fontsize=16,color="burlywood",shape="box"];616[label="wz48/Succ wz480",fontsize=10,color="white",style="solid",shape="box"];543 -> 616[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	616 -> 546[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	617[label="wz48/Zero",fontsize=10,color="white",style="solid",shape="box"];543 -> 617[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	617 -> 547[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	160 -> 87[label="",style="dashed", color="red", weight=0];
11.83/4.60	160[label="primDivNatS (primMinusNatS (Succ wz30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];160 -> 172[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	160 -> 173[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	161 -> 87[label="",style="dashed", color="red", weight=0];
11.83/4.60	161[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];161 -> 174[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	161 -> 175[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	436[label="primMinusNatS wz21 wz22",fontsize=16,color="burlywood",shape="triangle"];618[label="wz21/Succ wz210",fontsize=10,color="white",style="solid",shape="box"];436 -> 618[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	618 -> 440[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	619[label="wz21/Zero",fontsize=10,color="white",style="solid",shape="box"];436 -> 619[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	619 -> 441[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	544[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS (Succ wz470) (Succ wz480))",fontsize=16,color="black",shape="box"];544 -> 548[label="",style="solid", color="black", weight=3];
11.83/4.60	545[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS (Succ wz470) Zero)",fontsize=16,color="black",shape="box"];545 -> 549[label="",style="solid", color="black", weight=3];
11.83/4.60	546[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS Zero (Succ wz480))",fontsize=16,color="black",shape="box"];546 -> 550[label="",style="solid", color="black", weight=3];
11.83/4.60	547[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];547 -> 551[label="",style="solid", color="black", weight=3];
11.83/4.60	172 -> 112[label="",style="dashed", color="red", weight=0];
11.83/4.60	172[label="primMinusNatS (Succ wz30000) Zero",fontsize=16,color="magenta"];173[label="Zero",fontsize=16,color="green",shape="box"];174 -> 114[label="",style="dashed", color="red", weight=0];
11.83/4.60	174[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];175[label="Zero",fontsize=16,color="green",shape="box"];440[label="primMinusNatS (Succ wz210) wz22",fontsize=16,color="burlywood",shape="box"];620[label="wz22/Succ wz220",fontsize=10,color="white",style="solid",shape="box"];440 -> 620[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	620 -> 445[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	621[label="wz22/Zero",fontsize=10,color="white",style="solid",shape="box"];440 -> 621[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	621 -> 446[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	441[label="primMinusNatS Zero wz22",fontsize=16,color="burlywood",shape="box"];622[label="wz22/Succ wz220",fontsize=10,color="white",style="solid",shape="box"];441 -> 622[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	622 -> 447[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	623[label="wz22/Zero",fontsize=10,color="white",style="solid",shape="box"];441 -> 623[label="",style="solid", color="burlywood", weight=9];
11.83/4.60	623 -> 448[label="",style="solid", color="burlywood", weight=3];
11.83/4.60	548 -> 505[label="",style="dashed", color="red", weight=0];
11.83/4.60	548[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS wz470 wz480)",fontsize=16,color="magenta"];548 -> 552[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	548 -> 553[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	549[label="primDivNatS0 (Succ wz45) (Succ wz46) True",fontsize=16,color="black",shape="triangle"];549 -> 554[label="",style="solid", color="black", weight=3];
11.83/4.60	550[label="primDivNatS0 (Succ wz45) (Succ wz46) False",fontsize=16,color="black",shape="box"];550 -> 555[label="",style="solid", color="black", weight=3];
11.83/4.60	551 -> 549[label="",style="dashed", color="red", weight=0];
11.83/4.60	551[label="primDivNatS0 (Succ wz45) (Succ wz46) True",fontsize=16,color="magenta"];445[label="primMinusNatS (Succ wz210) (Succ wz220)",fontsize=16,color="black",shape="box"];445 -> 451[label="",style="solid", color="black", weight=3];
11.83/4.60	446[label="primMinusNatS (Succ wz210) Zero",fontsize=16,color="black",shape="box"];446 -> 452[label="",style="solid", color="black", weight=3];
11.83/4.60	447[label="primMinusNatS Zero (Succ wz220)",fontsize=16,color="black",shape="box"];447 -> 453[label="",style="solid", color="black", weight=3];
11.83/4.60	448[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];448 -> 454[label="",style="solid", color="black", weight=3];
11.83/4.60	552[label="wz470",fontsize=16,color="green",shape="box"];553[label="wz480",fontsize=16,color="green",shape="box"];554[label="Succ (primDivNatS (primMinusNatS (Succ wz45) (Succ wz46)) (Succ (Succ wz46)))",fontsize=16,color="green",shape="box"];554 -> 556[label="",style="dashed", color="green", weight=3];
11.83/4.60	555[label="Zero",fontsize=16,color="green",shape="box"];451 -> 436[label="",style="dashed", color="red", weight=0];
11.83/4.60	451[label="primMinusNatS wz210 wz220",fontsize=16,color="magenta"];451 -> 491[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	451 -> 492[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	452[label="Succ wz210",fontsize=16,color="green",shape="box"];453[label="Zero",fontsize=16,color="green",shape="box"];454[label="Zero",fontsize=16,color="green",shape="box"];556 -> 87[label="",style="dashed", color="red", weight=0];
11.83/4.60	556[label="primDivNatS (primMinusNatS (Succ wz45) (Succ wz46)) (Succ (Succ wz46))",fontsize=16,color="magenta"];556 -> 557[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	556 -> 558[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	491[label="wz220",fontsize=16,color="green",shape="box"];492[label="wz210",fontsize=16,color="green",shape="box"];557 -> 436[label="",style="dashed", color="red", weight=0];
11.83/4.60	557[label="primMinusNatS (Succ wz45) (Succ wz46)",fontsize=16,color="magenta"];557 -> 559[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	557 -> 560[label="",style="dashed", color="magenta", weight=3];
11.83/4.60	558[label="Succ wz46",fontsize=16,color="green",shape="box"];559[label="Succ wz46",fontsize=16,color="green",shape="box"];560[label="Succ wz45",fontsize=16,color="green",shape="box"];}
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(12)
11.83/4.60	Complex Obligation (AND)
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(13)
11.83/4.60	Obligation:
11.83/4.60	Q DP problem:
11.83/4.60	The TRS P consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primModNatS00(wz21, wz22) -> new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
11.83/4.60	   new_primModNatS(Succ(Succ(wz30000)), Zero) -> new_primModNatS(new_primMinusNatS0(wz30000), Zero)
11.83/4.60	   new_primModNatS(Succ(Succ(wz30000)), Succ(wz31000)) -> new_primModNatS0(wz30000, wz31000, wz30000, wz31000)
11.83/4.60	   new_primModNatS(Succ(Zero), Zero) -> new_primModNatS(new_primMinusNatS1, Zero)
11.83/4.60	   new_primModNatS0(wz21, wz22, Succ(wz230), Succ(wz240)) -> new_primModNatS0(wz21, wz22, wz230, wz240)
11.83/4.60	   new_primModNatS0(wz21, wz22, Succ(wz230), Zero) -> new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
11.83/4.60	   new_primModNatS0(wz21, wz22, Zero, Zero) -> new_primModNatS00(wz21, wz22)
11.83/4.60	
11.83/4.60	The TRS R consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primMinusNatS1 -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Zero) -> Succ(wz210)
11.83/4.60	   new_primMinusNatS2(Zero, Zero) -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Succ(wz220)) -> new_primMinusNatS2(wz210, wz220)
11.83/4.60	   new_primMinusNatS2(Zero, Succ(wz220)) -> Zero
11.83/4.60	   new_primMinusNatS0(wz30000) -> Succ(wz30000)
11.83/4.60	
11.83/4.60	The set Q consists of the following terms:
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Zero, Succ(x0))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Succ(x1))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Zero)
11.83/4.60	   new_primMinusNatS2(Zero, Zero)
11.83/4.60	   new_primMinusNatS1
11.83/4.60	   new_primMinusNatS0(x0)
11.83/4.60	
11.83/4.60	We have to consider all minimal (P,Q,R)-chains.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(14) DependencyGraphProof (EQUIVALENT)
11.83/4.60	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(15)
11.83/4.60	Complex Obligation (AND)
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(16)
11.83/4.60	Obligation:
11.83/4.60	Q DP problem:
11.83/4.60	The TRS P consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primModNatS(Succ(Succ(wz30000)), Zero) -> new_primModNatS(new_primMinusNatS0(wz30000), Zero)
11.83/4.60	
11.83/4.60	The TRS R consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primMinusNatS1 -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Zero) -> Succ(wz210)
11.83/4.60	   new_primMinusNatS2(Zero, Zero) -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Succ(wz220)) -> new_primMinusNatS2(wz210, wz220)
11.83/4.60	   new_primMinusNatS2(Zero, Succ(wz220)) -> Zero
11.83/4.60	   new_primMinusNatS0(wz30000) -> Succ(wz30000)
11.83/4.60	
11.83/4.60	The set Q consists of the following terms:
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Zero, Succ(x0))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Succ(x1))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Zero)
11.83/4.60	   new_primMinusNatS2(Zero, Zero)
11.83/4.60	   new_primMinusNatS1
11.83/4.60	   new_primMinusNatS0(x0)
11.83/4.60	
11.83/4.60	We have to consider all minimal (P,Q,R)-chains.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(17) MRRProof (EQUIVALENT)
11.83/4.60	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.
11.83/4.60	
11.83/4.60	Strictly oriented dependency pairs:
11.83/4.60	
11.83/4.60	   new_primModNatS(Succ(Succ(wz30000)), Zero) -> new_primModNatS(new_primMinusNatS0(wz30000), Zero)
11.83/4.60	
11.83/4.60	Strictly oriented rules of the TRS R:
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Zero) -> Succ(wz210)
11.83/4.60	   new_primMinusNatS2(Zero, Zero) -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Succ(wz220)) -> new_primMinusNatS2(wz210, wz220)
11.83/4.60	   new_primMinusNatS2(Zero, Succ(wz220)) -> Zero
11.83/4.60	
11.83/4.60	Used ordering: Polynomial interpretation [POLO]:
11.83/4.60	
11.83/4.60	   POL(Succ(x_1)) = 1 + x_1
11.83/4.60	   POL(Zero) = 2
11.83/4.60	   POL(new_primMinusNatS0(x_1)) = 1 + x_1
11.83/4.60	   POL(new_primMinusNatS1) = 2
11.83/4.60	   POL(new_primMinusNatS2(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
11.83/4.60	   POL(new_primModNatS(x_1, x_2)) = x_1 + x_2
11.83/4.60	
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(18)
11.83/4.60	Obligation:
11.83/4.60	Q DP problem:
11.83/4.60	P is empty.
11.83/4.60	The TRS R consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primMinusNatS1 -> Zero
11.83/4.60	   new_primMinusNatS0(wz30000) -> Succ(wz30000)
11.83/4.60	
11.83/4.60	The set Q consists of the following terms:
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Zero, Succ(x0))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Succ(x1))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Zero)
11.83/4.60	   new_primMinusNatS2(Zero, Zero)
11.83/4.60	   new_primMinusNatS1
11.83/4.60	   new_primMinusNatS0(x0)
11.83/4.60	
11.83/4.60	We have to consider all minimal (P,Q,R)-chains.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(19) PisEmptyProof (EQUIVALENT)
11.83/4.60	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(20)
11.83/4.60	YES
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(21)
11.83/4.60	Obligation:
11.83/4.60	Q DP problem:
11.83/4.60	The TRS P consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primModNatS(Succ(Succ(wz30000)), Succ(wz31000)) -> new_primModNatS0(wz30000, wz31000, wz30000, wz31000)
11.83/4.60	   new_primModNatS0(wz21, wz22, Succ(wz230), Succ(wz240)) -> new_primModNatS0(wz21, wz22, wz230, wz240)
11.83/4.60	   new_primModNatS0(wz21, wz22, Succ(wz230), Zero) -> new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
11.83/4.60	   new_primModNatS0(wz21, wz22, Zero, Zero) -> new_primModNatS00(wz21, wz22)
11.83/4.60	   new_primModNatS00(wz21, wz22) -> new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
11.83/4.60	
11.83/4.60	The TRS R consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primMinusNatS1 -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Zero) -> Succ(wz210)
11.83/4.60	   new_primMinusNatS2(Zero, Zero) -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Succ(wz220)) -> new_primMinusNatS2(wz210, wz220)
11.83/4.60	   new_primMinusNatS2(Zero, Succ(wz220)) -> Zero
11.83/4.60	   new_primMinusNatS0(wz30000) -> Succ(wz30000)
11.83/4.60	
11.83/4.60	The set Q consists of the following terms:
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Zero, Succ(x0))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Succ(x1))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Zero)
11.83/4.60	   new_primMinusNatS2(Zero, Zero)
11.83/4.60	   new_primMinusNatS1
11.83/4.60	   new_primMinusNatS0(x0)
11.83/4.60	
11.83/4.60	We have to consider all minimal (P,Q,R)-chains.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(22) QDPSizeChangeProof (EQUIVALENT)
11.83/4.60	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
11.83/4.60	
11.83/4.60	Order:Polynomial interpretation [POLO]:
11.83/4.60	
11.83/4.60	   POL(Succ(x_1)) = 1 + x_1
11.83/4.60	   POL(Zero) = 1
11.83/4.60	   POL(new_primMinusNatS2(x_1, x_2)) = x_1
11.83/4.60	
11.83/4.60	
11.83/4.60	
11.83/4.60	
11.83/4.60	From the DPs we obtained the following set of size-change graphs:
11.83/4.60	*new_primModNatS0(wz21, wz22, Succ(wz230), Zero) -> new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22)) (allowed arguments on rhs = {1, 2})
11.83/4.60	The graph contains the following edges 1 >= 1
11.83/4.60	
11.83/4.60	
11.83/4.60	*new_primModNatS00(wz21, wz22) -> new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22)) (allowed arguments on rhs = {1, 2})
11.83/4.60	The graph contains the following edges 1 >= 1
11.83/4.60	
11.83/4.60	
11.83/4.60	*new_primModNatS(Succ(Succ(wz30000)), Succ(wz31000)) -> new_primModNatS0(wz30000, wz31000, wz30000, wz31000) (allowed arguments on rhs = {1, 2, 3, 4})
11.83/4.60	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
11.83/4.60	
11.83/4.60	
11.83/4.60	*new_primModNatS0(wz21, wz22, Succ(wz230), Succ(wz240)) -> new_primModNatS0(wz21, wz22, wz230, wz240) (allowed arguments on rhs = {1, 2, 3, 4})
11.83/4.60	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
11.83/4.60	
11.83/4.60	
11.83/4.60	*new_primModNatS0(wz21, wz22, Zero, Zero) -> new_primModNatS00(wz21, wz22) (allowed arguments on rhs = {1, 2})
11.83/4.60	The graph contains the following edges 1 >= 1, 2 >= 2
11.83/4.60	
11.83/4.60	
11.83/4.60	
11.83/4.60	We oriented the following set of usable rules [AAECC05,FROCOS05].
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Zero, Zero) -> Zero
11.83/4.60	   new_primMinusNatS2(Zero, Succ(wz220)) -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Zero) -> Succ(wz210)
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Succ(wz220)) -> new_primMinusNatS2(wz210, wz220)
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(23)
11.83/4.60	YES
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(24)
11.83/4.60	Obligation:
11.83/4.60	Q DP problem:
11.83/4.60	The TRS P consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primDivNatS(Succ(Succ(wz30000)), Succ(wz31000)) -> new_primDivNatS0(wz30000, wz31000, wz30000, wz31000)
11.83/4.60	   new_primDivNatS0(wz45, wz46, Zero, Zero) -> new_primDivNatS00(wz45, wz46)
11.83/4.60	   new_primDivNatS(Succ(Succ(wz30000)), Zero) -> new_primDivNatS(new_primMinusNatS0(wz30000), Zero)
11.83/4.60	   new_primDivNatS00(wz45, wz46) -> new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))
11.83/4.60	   new_primDivNatS0(wz45, wz46, Succ(wz470), Succ(wz480)) -> new_primDivNatS0(wz45, wz46, wz470, wz480)
11.83/4.60	   new_primDivNatS0(wz45, wz46, Succ(wz470), Zero) -> new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))
11.83/4.60	   new_primDivNatS(Succ(Zero), Zero) -> new_primDivNatS(new_primMinusNatS1, Zero)
11.83/4.60	
11.83/4.60	The TRS R consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primMinusNatS1 -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Zero) -> Succ(wz210)
11.83/4.60	   new_primMinusNatS2(Zero, Zero) -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Succ(wz220)) -> new_primMinusNatS2(wz210, wz220)
11.83/4.60	   new_primMinusNatS2(Zero, Succ(wz220)) -> Zero
11.83/4.60	   new_primMinusNatS0(wz30000) -> Succ(wz30000)
11.83/4.60	
11.83/4.60	The set Q consists of the following terms:
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Zero, Succ(x0))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Succ(x1))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Zero)
11.83/4.60	   new_primMinusNatS2(Zero, Zero)
11.83/4.60	   new_primMinusNatS1
11.83/4.60	   new_primMinusNatS0(x0)
11.83/4.60	
11.83/4.60	We have to consider all minimal (P,Q,R)-chains.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(25) DependencyGraphProof (EQUIVALENT)
11.83/4.60	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(26)
11.83/4.60	Complex Obligation (AND)
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(27)
11.83/4.60	Obligation:
11.83/4.60	Q DP problem:
11.83/4.60	The TRS P consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primDivNatS(Succ(Succ(wz30000)), Zero) -> new_primDivNatS(new_primMinusNatS0(wz30000), Zero)
11.83/4.60	
11.83/4.60	The TRS R consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primMinusNatS1 -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Zero) -> Succ(wz210)
11.83/4.60	   new_primMinusNatS2(Zero, Zero) -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Succ(wz220)) -> new_primMinusNatS2(wz210, wz220)
11.83/4.60	   new_primMinusNatS2(Zero, Succ(wz220)) -> Zero
11.83/4.60	   new_primMinusNatS0(wz30000) -> Succ(wz30000)
11.83/4.60	
11.83/4.60	The set Q consists of the following terms:
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Zero, Succ(x0))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Succ(x1))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Zero)
11.83/4.60	   new_primMinusNatS2(Zero, Zero)
11.83/4.60	   new_primMinusNatS1
11.83/4.60	   new_primMinusNatS0(x0)
11.83/4.60	
11.83/4.60	We have to consider all minimal (P,Q,R)-chains.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(28) MRRProof (EQUIVALENT)
11.83/4.60	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.
11.83/4.60	
11.83/4.60	Strictly oriented dependency pairs:
11.83/4.60	
11.83/4.60	   new_primDivNatS(Succ(Succ(wz30000)), Zero) -> new_primDivNatS(new_primMinusNatS0(wz30000), Zero)
11.83/4.60	
11.83/4.60	Strictly oriented rules of the TRS R:
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Zero) -> Succ(wz210)
11.83/4.60	   new_primMinusNatS2(Zero, Zero) -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Succ(wz220)) -> new_primMinusNatS2(wz210, wz220)
11.83/4.60	   new_primMinusNatS2(Zero, Succ(wz220)) -> Zero
11.83/4.60	
11.83/4.60	Used ordering: Polynomial interpretation [POLO]:
11.83/4.60	
11.83/4.60	   POL(Succ(x_1)) = 1 + x_1
11.83/4.60	   POL(Zero) = 2
11.83/4.60	   POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2
11.83/4.60	   POL(new_primMinusNatS0(x_1)) = 1 + x_1
11.83/4.60	   POL(new_primMinusNatS1) = 2
11.83/4.60	   POL(new_primMinusNatS2(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
11.83/4.60	
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(29)
11.83/4.60	Obligation:
11.83/4.60	Q DP problem:
11.83/4.60	P is empty.
11.83/4.60	The TRS R consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primMinusNatS1 -> Zero
11.83/4.60	   new_primMinusNatS0(wz30000) -> Succ(wz30000)
11.83/4.60	
11.83/4.60	The set Q consists of the following terms:
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Zero, Succ(x0))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Succ(x1))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Zero)
11.83/4.60	   new_primMinusNatS2(Zero, Zero)
11.83/4.60	   new_primMinusNatS1
11.83/4.60	   new_primMinusNatS0(x0)
11.83/4.60	
11.83/4.60	We have to consider all minimal (P,Q,R)-chains.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(30) PisEmptyProof (EQUIVALENT)
11.83/4.60	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(31)
11.83/4.60	YES
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(32)
11.83/4.60	Obligation:
11.83/4.60	Q DP problem:
11.83/4.60	The TRS P consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primDivNatS0(wz45, wz46, Zero, Zero) -> new_primDivNatS00(wz45, wz46)
11.83/4.60	   new_primDivNatS00(wz45, wz46) -> new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))
11.83/4.60	   new_primDivNatS(Succ(Succ(wz30000)), Succ(wz31000)) -> new_primDivNatS0(wz30000, wz31000, wz30000, wz31000)
11.83/4.60	   new_primDivNatS0(wz45, wz46, Succ(wz470), Succ(wz480)) -> new_primDivNatS0(wz45, wz46, wz470, wz480)
11.83/4.60	   new_primDivNatS0(wz45, wz46, Succ(wz470), Zero) -> new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))
11.83/4.60	
11.83/4.60	The TRS R consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primMinusNatS1 -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Zero) -> Succ(wz210)
11.83/4.60	   new_primMinusNatS2(Zero, Zero) -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Succ(wz220)) -> new_primMinusNatS2(wz210, wz220)
11.83/4.60	   new_primMinusNatS2(Zero, Succ(wz220)) -> Zero
11.83/4.60	   new_primMinusNatS0(wz30000) -> Succ(wz30000)
11.83/4.60	
11.83/4.60	The set Q consists of the following terms:
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Zero, Succ(x0))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Succ(x1))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Zero)
11.83/4.60	   new_primMinusNatS2(Zero, Zero)
11.83/4.60	   new_primMinusNatS1
11.83/4.60	   new_primMinusNatS0(x0)
11.83/4.60	
11.83/4.60	We have to consider all minimal (P,Q,R)-chains.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(33) QDPOrderProof (EQUIVALENT)
11.83/4.60	We use the reduction pair processor [LPAR04,JAR06].
11.83/4.60	
11.83/4.60	
11.83/4.60	The following pairs can be oriented strictly and are deleted.
11.83/4.60	
11.83/4.60	   new_primDivNatS(Succ(Succ(wz30000)), Succ(wz31000)) -> new_primDivNatS0(wz30000, wz31000, wz30000, wz31000)
11.83/4.60	The remaining pairs can at least be oriented weakly.
11.83/4.60	Used ordering:  Polynomial interpretation [POLO]:
11.83/4.60	
11.83/4.60	   POL(Succ(x_1)) = 1 + x_1
11.83/4.60	   POL(Zero) = 1
11.83/4.60	   POL(new_primDivNatS(x_1, x_2)) = x_1
11.83/4.60	   POL(new_primDivNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1
11.83/4.60	   POL(new_primDivNatS00(x_1, x_2)) = 1 + x_1
11.83/4.60	   POL(new_primMinusNatS2(x_1, x_2)) = x_1
11.83/4.60	
11.83/4.60	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Succ(wz220)) -> new_primMinusNatS2(wz210, wz220)
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Zero) -> Succ(wz210)
11.83/4.60	   new_primMinusNatS2(Zero, Zero) -> Zero
11.83/4.60	   new_primMinusNatS2(Zero, Succ(wz220)) -> Zero
11.83/4.60	
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(34)
11.83/4.60	Obligation:
11.83/4.60	Q DP problem:
11.83/4.60	The TRS P consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primDivNatS0(wz45, wz46, Zero, Zero) -> new_primDivNatS00(wz45, wz46)
11.83/4.60	   new_primDivNatS00(wz45, wz46) -> new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))
11.83/4.60	   new_primDivNatS0(wz45, wz46, Succ(wz470), Succ(wz480)) -> new_primDivNatS0(wz45, wz46, wz470, wz480)
11.83/4.60	   new_primDivNatS0(wz45, wz46, Succ(wz470), Zero) -> new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))
11.83/4.60	
11.83/4.60	The TRS R consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primMinusNatS1 -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Zero) -> Succ(wz210)
11.83/4.60	   new_primMinusNatS2(Zero, Zero) -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Succ(wz220)) -> new_primMinusNatS2(wz210, wz220)
11.83/4.60	   new_primMinusNatS2(Zero, Succ(wz220)) -> Zero
11.83/4.60	   new_primMinusNatS0(wz30000) -> Succ(wz30000)
11.83/4.60	
11.83/4.60	The set Q consists of the following terms:
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Zero, Succ(x0))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Succ(x1))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Zero)
11.83/4.60	   new_primMinusNatS2(Zero, Zero)
11.83/4.60	   new_primMinusNatS1
11.83/4.60	   new_primMinusNatS0(x0)
11.83/4.60	
11.83/4.60	We have to consider all minimal (P,Q,R)-chains.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(35) DependencyGraphProof (EQUIVALENT)
11.83/4.60	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(36)
11.83/4.60	Obligation:
11.83/4.60	Q DP problem:
11.83/4.60	The TRS P consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primDivNatS0(wz45, wz46, Succ(wz470), Succ(wz480)) -> new_primDivNatS0(wz45, wz46, wz470, wz480)
11.83/4.60	
11.83/4.60	The TRS R consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primMinusNatS1 -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Zero) -> Succ(wz210)
11.83/4.60	   new_primMinusNatS2(Zero, Zero) -> Zero
11.83/4.60	   new_primMinusNatS2(Succ(wz210), Succ(wz220)) -> new_primMinusNatS2(wz210, wz220)
11.83/4.60	   new_primMinusNatS2(Zero, Succ(wz220)) -> Zero
11.83/4.60	   new_primMinusNatS0(wz30000) -> Succ(wz30000)
11.83/4.60	
11.83/4.60	The set Q consists of the following terms:
11.83/4.60	
11.83/4.60	   new_primMinusNatS2(Zero, Succ(x0))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Succ(x1))
11.83/4.60	   new_primMinusNatS2(Succ(x0), Zero)
11.83/4.60	   new_primMinusNatS2(Zero, Zero)
11.83/4.60	   new_primMinusNatS1
11.83/4.60	   new_primMinusNatS0(x0)
11.83/4.60	
11.83/4.60	We have to consider all minimal (P,Q,R)-chains.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(37) QDPSizeChangeProof (EQUIVALENT)
11.83/4.60	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. 
11.83/4.60	
11.83/4.60	From the DPs we obtained the following set of size-change graphs:
11.83/4.60	*new_primDivNatS0(wz45, wz46, Succ(wz470), Succ(wz480)) -> new_primDivNatS0(wz45, wz46, wz470, wz480)
11.83/4.60	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
11.83/4.60	
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(38)
11.83/4.60	YES
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(39)
11.83/4.60	Obligation:
11.83/4.60	Q DP problem:
11.83/4.60	The TRS P consists of the following rules:
11.83/4.60	
11.83/4.60	   new_primMinusNatS(Succ(wz210), Succ(wz220)) -> new_primMinusNatS(wz210, wz220)
11.83/4.60	
11.83/4.60	R is empty.
11.83/4.60	Q is empty.
11.83/4.60	We have to consider all minimal (P,Q,R)-chains.
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(40) QDPSizeChangeProof (EQUIVALENT)
11.83/4.60	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. 
11.83/4.60	
11.83/4.60	From the DPs we obtained the following set of size-change graphs:
11.83/4.60	*new_primMinusNatS(Succ(wz210), Succ(wz220)) -> new_primMinusNatS(wz210, wz220)
11.83/4.60	The graph contains the following edges 1 > 1, 2 > 2
11.83/4.60	
11.83/4.60	
11.83/4.60	----------------------------------------
11.83/4.60	
11.83/4.60	(41)
11.83/4.60	YES
11.89/4.64	EOF