28.97/15.41	MAYBE
31.40/15.99	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
31.40/15.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
31.40/15.99	
31.40/15.99	
31.40/15.99	H-Termination with start terms of the given HASKELL could not be shown:
31.40/15.99	
31.40/15.99	(0) HASKELL
31.40/15.99	(1) IFR [EQUIVALENT, 0 ms]
31.40/15.99	(2) HASKELL
31.40/15.99	(3) BR [EQUIVALENT, 0 ms]
31.40/15.99	(4) HASKELL
31.40/15.99	(5) COR [EQUIVALENT, 0 ms]
31.40/15.99	(6) HASKELL
31.40/15.99	(7) NumRed [SOUND, 5 ms]
31.40/15.99	(8) HASKELL
31.40/15.99	(9) Narrow [SOUND, 0 ms]
31.40/15.99	(10) AND
31.40/15.99	    (11) QDP
31.40/15.99	        (12) DependencyGraphProof [EQUIVALENT, 0 ms]
31.40/15.99	        (13) QDP
31.40/15.99	        (14) QDPOrderProof [EQUIVALENT, 0 ms]
31.40/15.99	        (15) QDP
31.40/15.99	        (16) DependencyGraphProof [EQUIVALENT, 0 ms]
31.40/15.99	        (17) QDP
31.40/15.99	        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
31.40/15.99	        (19) YES
31.40/15.99	    (20) QDP
31.40/15.99	        (21) DependencyGraphProof [EQUIVALENT, 0 ms]
31.40/15.99	        (22) QDP
31.40/15.99	        (23) TransformationProof [EQUIVALENT, 0 ms]
31.40/15.99	        (24) QDP
31.40/15.99	        (25) UsableRulesProof [EQUIVALENT, 0 ms]
31.40/15.99	        (26) QDP
31.40/15.99	        (27) QReductionProof [EQUIVALENT, 0 ms]
31.40/15.99	        (28) QDP
31.40/15.99	        (29) MNOCProof [EQUIVALENT, 0 ms]
31.40/15.99	        (30) QDP
31.40/15.99	        (31) InductionCalculusProof [EQUIVALENT, 0 ms]
31.40/15.99	        (32) QDP
31.40/15.99	        (33) TransformationProof [EQUIVALENT, 0 ms]
31.40/15.99	        (34) QDP
31.40/15.99	            (35) DependencyGraphProof [EQUIVALENT, 0 ms]
31.40/15.99	            (36) QDP
31.40/15.99	            (37) TransformationProof [EQUIVALENT, 0 ms]
31.40/15.99	            (38) QDP
31.40/15.99	            (39) DependencyGraphProof [EQUIVALENT, 0 ms]
31.40/15.99	            (40) QDP
31.40/15.99	            (41) TransformationProof [EQUIVALENT, 0 ms]
31.40/15.99	            (42) QDP
31.40/15.99	            (43) DependencyGraphProof [EQUIVALENT, 0 ms]
31.40/15.99	            (44) QDP
31.40/15.99	            (45) TransformationProof [EQUIVALENT, 1 ms]
31.40/15.99	            (46) QDP
31.40/15.99	            (47) DependencyGraphProof [EQUIVALENT, 0 ms]
31.40/15.99	            (48) QDP
31.40/15.99	            (49) MNOCProof [EQUIVALENT, 0 ms]
31.40/15.99	            (50) QDP
31.40/15.99	            (51) InductionCalculusProof [EQUIVALENT, 0 ms]
31.40/15.99	            (52) QDP
31.40/15.99	    (53) QDP
31.40/15.99	        (54) QDPSizeChangeProof [EQUIVALENT, 0 ms]
31.40/15.99	        (55) YES
31.40/15.99	    (56) QDP
31.40/15.99	        (57) DependencyGraphProof [EQUIVALENT, 0 ms]
31.40/15.99	        (58) QDP
31.40/15.99	        (59) QDPOrderProof [EQUIVALENT, 0 ms]
31.40/15.99	        (60) QDP
31.40/15.99	        (61) DependencyGraphProof [EQUIVALENT, 0 ms]
31.40/15.99	        (62) QDP
31.40/15.99	        (63) QDPSizeChangeProof [EQUIVALENT, 0 ms]
31.40/15.99	        (64) YES
31.40/15.99	(65) Narrow [COMPLETE, 0 ms]
31.40/15.99	(66) TRUE
31.40/15.99	
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(0)
31.40/15.99	Obligation:
31.40/15.99	mainModule Main 
31.40/15.99	module Main where {
31.40/15.99	import qualified Prelude;
31.40/15.99	}
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(1) IFR (EQUIVALENT)
31.40/15.99	If Reductions:
31.40/15.99	The following If expression
31.40/15.99	"if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x"
31.40/15.99	is transformed to
31.40/15.99	"primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y);
31.40/15.99	primModNatS0 x y False = Succ x;
31.40/15.99	"
31.40/15.99	The following If expression
31.40/15.99	"if primGEqNatS x y then primModNatP (primMinusNatS x y) (Succ y) else primMinusNatS y x"
31.40/15.99	is transformed to
31.40/15.99	"primModNatP0 x y True = primModNatP (primMinusNatS x y) (Succ y);
31.40/15.99	primModNatP0 x y False = primMinusNatS y x;
31.40/15.99	"
31.40/15.99	The following If expression
31.40/15.99	"if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero"
31.40/15.99	is transformed to
31.40/15.99	"primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y));
31.40/15.99	primDivNatS0 x y False = Zero;
31.40/15.99	"
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(2)
31.40/15.99	Obligation:
31.40/15.99	mainModule Main 
31.40/15.99	module Main where {
31.40/15.99	import qualified Prelude;
31.40/15.99	}
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(3) BR (EQUIVALENT)
31.40/15.99	Replaced joker patterns by fresh variables and removed binding patterns.
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(4)
31.40/15.99	Obligation:
31.40/15.99	mainModule Main 
31.40/15.99	module Main where {
31.40/15.99	import qualified Prelude;
31.40/15.99	}
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(5) COR (EQUIVALENT)
31.40/15.99	Cond Reductions:
31.40/15.99	The following Function with conditions
31.40/15.99	"undefined |Falseundefined;
31.40/15.99	"
31.40/15.99	is transformed to
31.40/15.99	"undefined  = undefined1;
31.40/15.99	"
31.40/15.99	"undefined0 True = undefined;
31.40/15.99	"
31.40/15.99	"undefined1  = undefined0 False;
31.40/15.99	"
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(6)
31.40/15.99	Obligation:
31.40/15.99	mainModule Main 
31.40/15.99	module Main where {
31.40/15.99	import qualified Prelude;
31.40/15.99	}
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(7) NumRed (SOUND)
31.40/15.99	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(8)
31.40/15.99	Obligation:
31.40/15.99	mainModule Main 
31.40/15.99	module Main where {
31.40/15.99	import qualified Prelude;
31.40/15.99	}
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(9) Narrow (SOUND)
31.40/15.99	Haskell To QDPs
31.40/15.99	
31.40/15.99	digraph dp_graph {
31.40/15.99	node [outthreshold=100, inthreshold=100];1[label="showsPrec",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
31.40/15.99	3[label="showsPrec ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
31.40/15.99	4[label="showsPrec ww3 ww4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
31.40/15.99	5[label="showsPrec ww3 ww4 ww5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
31.40/15.99	6 -> 36[label="",style="dashed", color="red", weight=0];
31.40/15.99	6[label="show ww4 ++ ww5",fontsize=16,color="magenta"];6 -> 37[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	6 -> 38[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	37[label="ww5",fontsize=16,color="green",shape="box"];38[label="show ww4",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
31.40/15.99	36[label="ww22 ++ ww21",fontsize=16,color="burlywood",shape="triangle"];924[label="ww22/ww220 : ww221",fontsize=10,color="white",style="solid",shape="box"];36 -> 924[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	924 -> 53[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	925[label="ww22/[]",fontsize=10,color="white",style="solid",shape="box"];36 -> 925[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	925 -> 54[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	52[label="primShowInt ww4",fontsize=16,color="burlywood",shape="triangle"];926[label="ww4/Pos ww40",fontsize=10,color="white",style="solid",shape="box"];52 -> 926[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	926 -> 55[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	927[label="ww4/Neg ww40",fontsize=10,color="white",style="solid",shape="box"];52 -> 927[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	927 -> 56[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	53[label="(ww220 : ww221) ++ ww21",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
31.40/15.99	54[label="[] ++ ww21",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3];
31.40/15.99	55[label="primShowInt (Pos ww40)",fontsize=16,color="burlywood",shape="box"];928[label="ww40/Succ ww400",fontsize=10,color="white",style="solid",shape="box"];55 -> 928[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	928 -> 59[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	929[label="ww40/Zero",fontsize=10,color="white",style="solid",shape="box"];55 -> 929[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	929 -> 60[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	56[label="primShowInt (Neg ww40)",fontsize=16,color="black",shape="box"];56 -> 61[label="",style="solid", color="black", weight=3];
31.40/15.99	57[label="ww220 : ww221 ++ ww21",fontsize=16,color="green",shape="box"];57 -> 62[label="",style="dashed", color="green", weight=3];
31.40/15.99	58[label="ww21",fontsize=16,color="green",shape="box"];59[label="primShowInt (Pos (Succ ww400))",fontsize=16,color="black",shape="box"];59 -> 63[label="",style="solid", color="black", weight=3];
31.40/15.99	60[label="primShowInt (Pos Zero)",fontsize=16,color="black",shape="box"];60 -> 64[label="",style="solid", color="black", weight=3];
31.40/15.99	61[label="Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))) : primShowInt (Pos ww40)",fontsize=16,color="green",shape="box"];61 -> 65[label="",style="dashed", color="green", weight=3];
31.40/15.99	62 -> 36[label="",style="dashed", color="red", weight=0];
31.40/15.99	62[label="ww221 ++ ww21",fontsize=16,color="magenta"];62 -> 66[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	63 -> 36[label="",style="dashed", color="red", weight=0];
31.40/15.99	63[label="primShowInt (div Pos (Succ ww400) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))) ++ toEnum (mod Pos (Succ ww400) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))) : []",fontsize=16,color="magenta"];63 -> 67[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	63 -> 68[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	64[label="Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))))))))))))))))))))))))) : []",fontsize=16,color="green",shape="box"];65 -> 52[label="",style="dashed", color="red", weight=0];
31.40/15.99	65[label="primShowInt (Pos ww40)",fontsize=16,color="magenta"];65 -> 69[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	66[label="ww221",fontsize=16,color="green",shape="box"];67[label="toEnum (mod Pos (Succ ww400) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))) : []",fontsize=16,color="green",shape="box"];67 -> 70[label="",style="dashed", color="green", weight=3];
31.40/15.99	68 -> 52[label="",style="dashed", color="red", weight=0];
31.40/15.99	68[label="primShowInt (div Pos (Succ ww400) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))",fontsize=16,color="magenta"];68 -> 71[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	69[label="Pos ww40",fontsize=16,color="green",shape="box"];70[label="toEnum (mod Pos (Succ ww400) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))",fontsize=16,color="black",shape="box"];70 -> 88[label="",style="solid", color="black", weight=3];
31.40/15.99	71 -> 75[label="",style="dashed", color="red", weight=0];
31.40/15.99	71[label="div Pos (Succ ww400) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))",fontsize=16,color="magenta"];71 -> 76[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	71 -> 77[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	88 -> 99[label="",style="dashed", color="red", weight=0];
31.40/15.99	88[label="primIntToChar (mod Pos (Succ ww400) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))",fontsize=16,color="magenta"];88 -> 100[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	88 -> 101[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	76[label="ww400",fontsize=16,color="green",shape="box"];77[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];75[label="div Pos (Succ ww27) Pos (Succ ww28)",fontsize=16,color="black",shape="triangle"];75 -> 87[label="",style="solid", color="black", weight=3];
31.40/15.99	100[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];101[label="ww400",fontsize=16,color="green",shape="box"];99[label="primIntToChar (mod Pos (Succ ww30) Pos (Succ ww31))",fontsize=16,color="black",shape="triangle"];99 -> 102[label="",style="solid", color="black", weight=3];
31.40/15.99	87[label="primDivInt (Pos (Succ ww27)) (Pos (Succ ww28))",fontsize=16,color="black",shape="box"];87 -> 98[label="",style="solid", color="black", weight=3];
31.40/15.99	102[label="primIntToChar (primModInt (Pos (Succ ww30)) (Pos (Succ ww31)))",fontsize=16,color="black",shape="box"];102 -> 104[label="",style="solid", color="black", weight=3];
31.40/15.99	98[label="Pos (primDivNatS (Succ ww27) (Succ ww28))",fontsize=16,color="green",shape="box"];98 -> 103[label="",style="dashed", color="green", weight=3];
31.40/15.99	104[label="primIntToChar (Pos (primModNatS (Succ ww30) (Succ ww31)))",fontsize=16,color="black",shape="box"];104 -> 106[label="",style="solid", color="black", weight=3];
31.40/15.99	103[label="primDivNatS (Succ ww27) (Succ ww28)",fontsize=16,color="black",shape="triangle"];103 -> 105[label="",style="solid", color="black", weight=3];
31.40/15.99	106[label="Char (primModNatS (Succ ww30) (Succ ww31))",fontsize=16,color="green",shape="box"];106 -> 109[label="",style="dashed", color="green", weight=3];
31.40/15.99	105[label="primDivNatS0 ww27 ww28 (primGEqNatS ww27 ww28)",fontsize=16,color="burlywood",shape="box"];930[label="ww27/Succ ww270",fontsize=10,color="white",style="solid",shape="box"];105 -> 930[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	930 -> 107[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	931[label="ww27/Zero",fontsize=10,color="white",style="solid",shape="box"];105 -> 931[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	931 -> 108[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	109[label="primModNatS (Succ ww30) (Succ ww31)",fontsize=16,color="black",shape="triangle"];109 -> 114[label="",style="solid", color="black", weight=3];
31.40/15.99	107[label="primDivNatS0 (Succ ww270) ww28 (primGEqNatS (Succ ww270) ww28)",fontsize=16,color="burlywood",shape="box"];932[label="ww28/Succ ww280",fontsize=10,color="white",style="solid",shape="box"];107 -> 932[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	932 -> 110[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	933[label="ww28/Zero",fontsize=10,color="white",style="solid",shape="box"];107 -> 933[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	933 -> 111[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	108[label="primDivNatS0 Zero ww28 (primGEqNatS Zero ww28)",fontsize=16,color="burlywood",shape="box"];934[label="ww28/Succ ww280",fontsize=10,color="white",style="solid",shape="box"];108 -> 934[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	934 -> 112[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	935[label="ww28/Zero",fontsize=10,color="white",style="solid",shape="box"];108 -> 935[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	935 -> 113[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	114[label="primModNatS0 ww30 ww31 (primGEqNatS ww30 ww31)",fontsize=16,color="burlywood",shape="box"];936[label="ww30/Succ ww300",fontsize=10,color="white",style="solid",shape="box"];114 -> 936[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	936 -> 119[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	937[label="ww30/Zero",fontsize=10,color="white",style="solid",shape="box"];114 -> 937[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	937 -> 120[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	110[label="primDivNatS0 (Succ ww270) (Succ ww280) (primGEqNatS (Succ ww270) (Succ ww280))",fontsize=16,color="black",shape="box"];110 -> 115[label="",style="solid", color="black", weight=3];
31.40/15.99	111[label="primDivNatS0 (Succ ww270) Zero (primGEqNatS (Succ ww270) Zero)",fontsize=16,color="black",shape="box"];111 -> 116[label="",style="solid", color="black", weight=3];
31.40/15.99	112[label="primDivNatS0 Zero (Succ ww280) (primGEqNatS Zero (Succ ww280))",fontsize=16,color="black",shape="box"];112 -> 117[label="",style="solid", color="black", weight=3];
31.40/15.99	113[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];113 -> 118[label="",style="solid", color="black", weight=3];
31.40/15.99	119[label="primModNatS0 (Succ ww300) ww31 (primGEqNatS (Succ ww300) ww31)",fontsize=16,color="burlywood",shape="box"];938[label="ww31/Succ ww310",fontsize=10,color="white",style="solid",shape="box"];119 -> 938[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	938 -> 126[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	939[label="ww31/Zero",fontsize=10,color="white",style="solid",shape="box"];119 -> 939[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	939 -> 127[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	120[label="primModNatS0 Zero ww31 (primGEqNatS Zero ww31)",fontsize=16,color="burlywood",shape="box"];940[label="ww31/Succ ww310",fontsize=10,color="white",style="solid",shape="box"];120 -> 940[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	940 -> 128[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	941[label="ww31/Zero",fontsize=10,color="white",style="solid",shape="box"];120 -> 941[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	941 -> 129[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	115 -> 640[label="",style="dashed", color="red", weight=0];
31.40/15.99	115[label="primDivNatS0 (Succ ww270) (Succ ww280) (primGEqNatS ww270 ww280)",fontsize=16,color="magenta"];115 -> 641[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	115 -> 642[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	115 -> 643[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	115 -> 644[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	116[label="primDivNatS0 (Succ ww270) Zero True",fontsize=16,color="black",shape="box"];116 -> 123[label="",style="solid", color="black", weight=3];
31.40/15.99	117[label="primDivNatS0 Zero (Succ ww280) False",fontsize=16,color="black",shape="box"];117 -> 124[label="",style="solid", color="black", weight=3];
31.40/15.99	118[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];118 -> 125[label="",style="solid", color="black", weight=3];
31.40/15.99	126[label="primModNatS0 (Succ ww300) (Succ ww310) (primGEqNatS (Succ ww300) (Succ ww310))",fontsize=16,color="black",shape="box"];126 -> 136[label="",style="solid", color="black", weight=3];
31.40/15.99	127[label="primModNatS0 (Succ ww300) Zero (primGEqNatS (Succ ww300) Zero)",fontsize=16,color="black",shape="box"];127 -> 137[label="",style="solid", color="black", weight=3];
31.40/15.99	128[label="primModNatS0 Zero (Succ ww310) (primGEqNatS Zero (Succ ww310))",fontsize=16,color="black",shape="box"];128 -> 138[label="",style="solid", color="black", weight=3];
31.40/15.99	129[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];129 -> 139[label="",style="solid", color="black", weight=3];
31.40/15.99	641[label="ww270",fontsize=16,color="green",shape="box"];642[label="ww280",fontsize=16,color="green",shape="box"];643[label="ww280",fontsize=16,color="green",shape="box"];644[label="ww270",fontsize=16,color="green",shape="box"];640[label="primDivNatS0 (Succ ww74) (Succ ww75) (primGEqNatS ww76 ww77)",fontsize=16,color="burlywood",shape="triangle"];942[label="ww76/Succ ww760",fontsize=10,color="white",style="solid",shape="box"];640 -> 942[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	942 -> 681[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	943[label="ww76/Zero",fontsize=10,color="white",style="solid",shape="box"];640 -> 943[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	943 -> 682[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	123[label="Succ (primDivNatS (primMinusNatS (Succ ww270) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];123 -> 134[label="",style="dashed", color="green", weight=3];
31.40/15.99	124[label="Zero",fontsize=16,color="green",shape="box"];125[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];125 -> 135[label="",style="dashed", color="green", weight=3];
31.40/15.99	136 -> 701[label="",style="dashed", color="red", weight=0];
31.40/15.99	136[label="primModNatS0 (Succ ww300) (Succ ww310) (primGEqNatS ww300 ww310)",fontsize=16,color="magenta"];136 -> 702[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	136 -> 703[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	136 -> 704[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	136 -> 705[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	137[label="primModNatS0 (Succ ww300) Zero True",fontsize=16,color="black",shape="box"];137 -> 148[label="",style="solid", color="black", weight=3];
31.40/15.99	138[label="primModNatS0 Zero (Succ ww310) False",fontsize=16,color="black",shape="box"];138 -> 149[label="",style="solid", color="black", weight=3];
31.40/15.99	139[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];139 -> 150[label="",style="solid", color="black", weight=3];
31.40/15.99	681[label="primDivNatS0 (Succ ww74) (Succ ww75) (primGEqNatS (Succ ww760) ww77)",fontsize=16,color="burlywood",shape="box"];944[label="ww77/Succ ww770",fontsize=10,color="white",style="solid",shape="box"];681 -> 944[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	944 -> 693[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	945[label="ww77/Zero",fontsize=10,color="white",style="solid",shape="box"];681 -> 945[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	945 -> 694[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	682[label="primDivNatS0 (Succ ww74) (Succ ww75) (primGEqNatS Zero ww77)",fontsize=16,color="burlywood",shape="box"];946[label="ww77/Succ ww770",fontsize=10,color="white",style="solid",shape="box"];682 -> 946[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	946 -> 695[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	947[label="ww77/Zero",fontsize=10,color="white",style="solid",shape="box"];682 -> 947[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	947 -> 696[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	134 -> 884[label="",style="dashed", color="red", weight=0];
31.40/15.99	134[label="primDivNatS (primMinusNatS (Succ ww270) Zero) (Succ Zero)",fontsize=16,color="magenta"];134 -> 885[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	134 -> 886[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	134 -> 887[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	135 -> 884[label="",style="dashed", color="red", weight=0];
31.40/15.99	135[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];135 -> 888[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	135 -> 889[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	135 -> 890[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	702[label="ww310",fontsize=16,color="green",shape="box"];703[label="ww300",fontsize=16,color="green",shape="box"];704[label="ww310",fontsize=16,color="green",shape="box"];705[label="ww300",fontsize=16,color="green",shape="box"];701[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS ww81 ww82)",fontsize=16,color="burlywood",shape="triangle"];948[label="ww81/Succ ww810",fontsize=10,color="white",style="solid",shape="box"];701 -> 948[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	948 -> 742[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	949[label="ww81/Zero",fontsize=10,color="white",style="solid",shape="box"];701 -> 949[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	949 -> 743[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	148 -> 788[label="",style="dashed", color="red", weight=0];
31.40/15.99	148[label="primModNatS (primMinusNatS (Succ ww300) Zero) (Succ Zero)",fontsize=16,color="magenta"];148 -> 789[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	148 -> 790[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	148 -> 791[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	149[label="Succ Zero",fontsize=16,color="green",shape="box"];150 -> 788[label="",style="dashed", color="red", weight=0];
31.40/15.99	150[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];150 -> 792[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	150 -> 793[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	150 -> 794[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	693[label="primDivNatS0 (Succ ww74) (Succ ww75) (primGEqNatS (Succ ww760) (Succ ww770))",fontsize=16,color="black",shape="box"];693 -> 744[label="",style="solid", color="black", weight=3];
31.40/15.99	694[label="primDivNatS0 (Succ ww74) (Succ ww75) (primGEqNatS (Succ ww760) Zero)",fontsize=16,color="black",shape="box"];694 -> 745[label="",style="solid", color="black", weight=3];
31.40/15.99	695[label="primDivNatS0 (Succ ww74) (Succ ww75) (primGEqNatS Zero (Succ ww770))",fontsize=16,color="black",shape="box"];695 -> 746[label="",style="solid", color="black", weight=3];
31.40/15.99	696[label="primDivNatS0 (Succ ww74) (Succ ww75) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];696 -> 747[label="",style="solid", color="black", weight=3];
31.40/15.99	885[label="Succ ww270",fontsize=16,color="green",shape="box"];886[label="Zero",fontsize=16,color="green",shape="box"];887[label="Zero",fontsize=16,color="green",shape="box"];884[label="primDivNatS (primMinusNatS ww88 ww89) (Succ ww90)",fontsize=16,color="burlywood",shape="triangle"];950[label="ww88/Succ ww880",fontsize=10,color="white",style="solid",shape="box"];884 -> 950[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	950 -> 909[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	951[label="ww88/Zero",fontsize=10,color="white",style="solid",shape="box"];884 -> 951[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	951 -> 910[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	888[label="Zero",fontsize=16,color="green",shape="box"];889[label="Zero",fontsize=16,color="green",shape="box"];890[label="Zero",fontsize=16,color="green",shape="box"];742[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS (Succ ww810) ww82)",fontsize=16,color="burlywood",shape="box"];952[label="ww82/Succ ww820",fontsize=10,color="white",style="solid",shape="box"];742 -> 952[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	952 -> 752[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	953[label="ww82/Zero",fontsize=10,color="white",style="solid",shape="box"];742 -> 953[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	953 -> 753[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	743[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS Zero ww82)",fontsize=16,color="burlywood",shape="box"];954[label="ww82/Succ ww820",fontsize=10,color="white",style="solid",shape="box"];743 -> 954[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	954 -> 754[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	955[label="ww82/Zero",fontsize=10,color="white",style="solid",shape="box"];743 -> 955[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	955 -> 755[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	789[label="Succ ww300",fontsize=16,color="green",shape="box"];790[label="Zero",fontsize=16,color="green",shape="box"];791[label="Zero",fontsize=16,color="green",shape="box"];788[label="primModNatS (primMinusNatS ww84 ww85) (Succ ww86)",fontsize=16,color="burlywood",shape="triangle"];956[label="ww84/Succ ww840",fontsize=10,color="white",style="solid",shape="box"];788 -> 956[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	956 -> 819[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	957[label="ww84/Zero",fontsize=10,color="white",style="solid",shape="box"];788 -> 957[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	957 -> 820[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	792[label="Zero",fontsize=16,color="green",shape="box"];793[label="Zero",fontsize=16,color="green",shape="box"];794[label="Zero",fontsize=16,color="green",shape="box"];744 -> 640[label="",style="dashed", color="red", weight=0];
31.40/15.99	744[label="primDivNatS0 (Succ ww74) (Succ ww75) (primGEqNatS ww760 ww770)",fontsize=16,color="magenta"];744 -> 756[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	744 -> 757[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	745[label="primDivNatS0 (Succ ww74) (Succ ww75) True",fontsize=16,color="black",shape="triangle"];745 -> 758[label="",style="solid", color="black", weight=3];
31.40/15.99	746[label="primDivNatS0 (Succ ww74) (Succ ww75) False",fontsize=16,color="black",shape="box"];746 -> 759[label="",style="solid", color="black", weight=3];
31.40/15.99	747 -> 745[label="",style="dashed", color="red", weight=0];
31.40/15.99	747[label="primDivNatS0 (Succ ww74) (Succ ww75) True",fontsize=16,color="magenta"];909[label="primDivNatS (primMinusNatS (Succ ww880) ww89) (Succ ww90)",fontsize=16,color="burlywood",shape="box"];958[label="ww89/Succ ww890",fontsize=10,color="white",style="solid",shape="box"];909 -> 958[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	958 -> 911[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	959[label="ww89/Zero",fontsize=10,color="white",style="solid",shape="box"];909 -> 959[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	959 -> 912[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	910[label="primDivNatS (primMinusNatS Zero ww89) (Succ ww90)",fontsize=16,color="burlywood",shape="box"];960[label="ww89/Succ ww890",fontsize=10,color="white",style="solid",shape="box"];910 -> 960[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	960 -> 913[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	961[label="ww89/Zero",fontsize=10,color="white",style="solid",shape="box"];910 -> 961[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	961 -> 914[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	752[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS (Succ ww810) (Succ ww820))",fontsize=16,color="black",shape="box"];752 -> 766[label="",style="solid", color="black", weight=3];
31.40/15.99	753[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS (Succ ww810) Zero)",fontsize=16,color="black",shape="box"];753 -> 767[label="",style="solid", color="black", weight=3];
31.40/15.99	754[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS Zero (Succ ww820))",fontsize=16,color="black",shape="box"];754 -> 768[label="",style="solid", color="black", weight=3];
31.40/15.99	755[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];755 -> 769[label="",style="solid", color="black", weight=3];
31.40/15.99	819[label="primModNatS (primMinusNatS (Succ ww840) ww85) (Succ ww86)",fontsize=16,color="burlywood",shape="box"];962[label="ww85/Succ ww850",fontsize=10,color="white",style="solid",shape="box"];819 -> 962[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	962 -> 825[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	963[label="ww85/Zero",fontsize=10,color="white",style="solid",shape="box"];819 -> 963[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	963 -> 826[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	820[label="primModNatS (primMinusNatS Zero ww85) (Succ ww86)",fontsize=16,color="burlywood",shape="box"];964[label="ww85/Succ ww850",fontsize=10,color="white",style="solid",shape="box"];820 -> 964[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	964 -> 827[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	965[label="ww85/Zero",fontsize=10,color="white",style="solid",shape="box"];820 -> 965[label="",style="solid", color="burlywood", weight=9];
31.40/15.99	965 -> 828[label="",style="solid", color="burlywood", weight=3];
31.40/15.99	756[label="ww760",fontsize=16,color="green",shape="box"];757[label="ww770",fontsize=16,color="green",shape="box"];758[label="Succ (primDivNatS (primMinusNatS (Succ ww74) (Succ ww75)) (Succ (Succ ww75)))",fontsize=16,color="green",shape="box"];758 -> 770[label="",style="dashed", color="green", weight=3];
31.40/15.99	759[label="Zero",fontsize=16,color="green",shape="box"];911[label="primDivNatS (primMinusNatS (Succ ww880) (Succ ww890)) (Succ ww90)",fontsize=16,color="black",shape="box"];911 -> 915[label="",style="solid", color="black", weight=3];
31.40/15.99	912[label="primDivNatS (primMinusNatS (Succ ww880) Zero) (Succ ww90)",fontsize=16,color="black",shape="box"];912 -> 916[label="",style="solid", color="black", weight=3];
31.40/15.99	913[label="primDivNatS (primMinusNatS Zero (Succ ww890)) (Succ ww90)",fontsize=16,color="black",shape="box"];913 -> 917[label="",style="solid", color="black", weight=3];
31.40/15.99	914[label="primDivNatS (primMinusNatS Zero Zero) (Succ ww90)",fontsize=16,color="black",shape="box"];914 -> 918[label="",style="solid", color="black", weight=3];
31.40/15.99	766 -> 701[label="",style="dashed", color="red", weight=0];
31.40/15.99	766[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS ww810 ww820)",fontsize=16,color="magenta"];766 -> 775[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	766 -> 776[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	767[label="primModNatS0 (Succ ww79) (Succ ww80) True",fontsize=16,color="black",shape="triangle"];767 -> 777[label="",style="solid", color="black", weight=3];
31.40/15.99	768[label="primModNatS0 (Succ ww79) (Succ ww80) False",fontsize=16,color="black",shape="box"];768 -> 778[label="",style="solid", color="black", weight=3];
31.40/15.99	769 -> 767[label="",style="dashed", color="red", weight=0];
31.40/15.99	769[label="primModNatS0 (Succ ww79) (Succ ww80) True",fontsize=16,color="magenta"];825[label="primModNatS (primMinusNatS (Succ ww840) (Succ ww850)) (Succ ww86)",fontsize=16,color="black",shape="box"];825 -> 833[label="",style="solid", color="black", weight=3];
31.40/15.99	826[label="primModNatS (primMinusNatS (Succ ww840) Zero) (Succ ww86)",fontsize=16,color="black",shape="box"];826 -> 834[label="",style="solid", color="black", weight=3];
31.40/15.99	827[label="primModNatS (primMinusNatS Zero (Succ ww850)) (Succ ww86)",fontsize=16,color="black",shape="box"];827 -> 835[label="",style="solid", color="black", weight=3];
31.40/15.99	828[label="primModNatS (primMinusNatS Zero Zero) (Succ ww86)",fontsize=16,color="black",shape="box"];828 -> 836[label="",style="solid", color="black", weight=3];
31.40/15.99	770 -> 884[label="",style="dashed", color="red", weight=0];
31.40/15.99	770[label="primDivNatS (primMinusNatS (Succ ww74) (Succ ww75)) (Succ (Succ ww75))",fontsize=16,color="magenta"];770 -> 891[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	770 -> 892[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	770 -> 893[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	915 -> 884[label="",style="dashed", color="red", weight=0];
31.40/15.99	915[label="primDivNatS (primMinusNatS ww880 ww890) (Succ ww90)",fontsize=16,color="magenta"];915 -> 919[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	915 -> 920[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	916 -> 103[label="",style="dashed", color="red", weight=0];
31.40/15.99	916[label="primDivNatS (Succ ww880) (Succ ww90)",fontsize=16,color="magenta"];916 -> 921[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	916 -> 922[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	917[label="primDivNatS Zero (Succ ww90)",fontsize=16,color="black",shape="triangle"];917 -> 923[label="",style="solid", color="black", weight=3];
31.40/15.99	918 -> 917[label="",style="dashed", color="red", weight=0];
31.40/15.99	918[label="primDivNatS Zero (Succ ww90)",fontsize=16,color="magenta"];775[label="ww810",fontsize=16,color="green",shape="box"];776[label="ww820",fontsize=16,color="green",shape="box"];777 -> 788[label="",style="dashed", color="red", weight=0];
31.40/15.99	777[label="primModNatS (primMinusNatS (Succ ww79) (Succ ww80)) (Succ (Succ ww80))",fontsize=16,color="magenta"];777 -> 801[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	777 -> 802[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	777 -> 803[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	778[label="Succ (Succ ww79)",fontsize=16,color="green",shape="box"];833 -> 788[label="",style="dashed", color="red", weight=0];
31.40/15.99	833[label="primModNatS (primMinusNatS ww840 ww850) (Succ ww86)",fontsize=16,color="magenta"];833 -> 843[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	833 -> 844[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	834 -> 109[label="",style="dashed", color="red", weight=0];
31.40/15.99	834[label="primModNatS (Succ ww840) (Succ ww86)",fontsize=16,color="magenta"];834 -> 845[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	834 -> 846[label="",style="dashed", color="magenta", weight=3];
31.40/15.99	835[label="primModNatS Zero (Succ ww86)",fontsize=16,color="black",shape="triangle"];835 -> 847[label="",style="solid", color="black", weight=3];
31.40/15.99	836 -> 835[label="",style="dashed", color="red", weight=0];
31.40/15.99	836[label="primModNatS Zero (Succ ww86)",fontsize=16,color="magenta"];891[label="Succ ww74",fontsize=16,color="green",shape="box"];892[label="Succ ww75",fontsize=16,color="green",shape="box"];893[label="Succ ww75",fontsize=16,color="green",shape="box"];919[label="ww880",fontsize=16,color="green",shape="box"];920[label="ww890",fontsize=16,color="green",shape="box"];921[label="ww880",fontsize=16,color="green",shape="box"];922[label="ww90",fontsize=16,color="green",shape="box"];923[label="Zero",fontsize=16,color="green",shape="box"];801[label="Succ ww79",fontsize=16,color="green",shape="box"];802[label="Succ ww80",fontsize=16,color="green",shape="box"];803[label="Succ ww80",fontsize=16,color="green",shape="box"];843[label="ww840",fontsize=16,color="green",shape="box"];844[label="ww850",fontsize=16,color="green",shape="box"];845[label="ww86",fontsize=16,color="green",shape="box"];846[label="ww840",fontsize=16,color="green",shape="box"];847[label="Zero",fontsize=16,color="green",shape="box"];}
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(10)
31.40/15.99	Complex Obligation (AND)
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(11)
31.40/15.99	Obligation:
31.40/15.99	Q DP problem:
31.40/15.99	The TRS P consists of the following rules:
31.40/15.99	
31.40/15.99	   new_primDivNatS0(ww74, ww75, Zero, Zero) -> new_primDivNatS00(ww74, ww75)
31.40/15.99	   new_primDivNatS00(ww74, ww75) -> new_primDivNatS(Succ(ww74), Succ(ww75), Succ(ww75))
31.40/15.99	   new_primDivNatS(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS(ww880, ww890, ww90)
31.40/15.99	   new_primDivNatS1(Succ(ww270), Zero) -> new_primDivNatS(Succ(ww270), Zero, Zero)
31.40/15.99	   new_primDivNatS0(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS0(ww74, ww75, ww760, ww770)
31.40/15.99	   new_primDivNatS0(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS(Succ(ww74), Succ(ww75), Succ(ww75))
31.40/15.99	   new_primDivNatS1(Succ(ww270), Succ(ww280)) -> new_primDivNatS0(ww270, ww280, ww270, ww280)
31.40/15.99	   new_primDivNatS1(Zero, Zero) -> new_primDivNatS(Zero, Zero, Zero)
31.40/15.99	   new_primDivNatS(Succ(ww880), Zero, ww90) -> new_primDivNatS1(ww880, ww90)
31.40/15.99	
31.40/15.99	R is empty.
31.40/15.99	Q is empty.
31.40/15.99	We have to consider all minimal (P,Q,R)-chains.
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(12) DependencyGraphProof (EQUIVALENT)
31.40/15.99	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(13)
31.40/15.99	Obligation:
31.40/15.99	Q DP problem:
31.40/15.99	The TRS P consists of the following rules:
31.40/15.99	
31.40/15.99	   new_primDivNatS00(ww74, ww75) -> new_primDivNatS(Succ(ww74), Succ(ww75), Succ(ww75))
31.40/15.99	   new_primDivNatS(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS(ww880, ww890, ww90)
31.40/15.99	   new_primDivNatS(Succ(ww880), Zero, ww90) -> new_primDivNatS1(ww880, ww90)
31.40/15.99	   new_primDivNatS1(Succ(ww270), Zero) -> new_primDivNatS(Succ(ww270), Zero, Zero)
31.40/15.99	   new_primDivNatS1(Succ(ww270), Succ(ww280)) -> new_primDivNatS0(ww270, ww280, ww270, ww280)
31.40/15.99	   new_primDivNatS0(ww74, ww75, Zero, Zero) -> new_primDivNatS00(ww74, ww75)
31.40/15.99	   new_primDivNatS0(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS0(ww74, ww75, ww760, ww770)
31.40/15.99	   new_primDivNatS0(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS(Succ(ww74), Succ(ww75), Succ(ww75))
31.40/15.99	
31.40/15.99	R is empty.
31.40/15.99	Q is empty.
31.40/15.99	We have to consider all minimal (P,Q,R)-chains.
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(14) QDPOrderProof (EQUIVALENT)
31.40/15.99	We use the reduction pair processor [LPAR04,JAR06].
31.40/15.99	
31.40/15.99	
31.40/15.99	The following pairs can be oriented strictly and are deleted.
31.40/15.99	
31.40/15.99	   new_primDivNatS(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS(ww880, ww890, ww90)
31.40/15.99	   new_primDivNatS1(Succ(ww270), Zero) -> new_primDivNatS(Succ(ww270), Zero, Zero)
31.40/15.99	   new_primDivNatS1(Succ(ww270), Succ(ww280)) -> new_primDivNatS0(ww270, ww280, ww270, ww280)
31.40/15.99	The remaining pairs can at least be oriented weakly.
31.40/15.99	Used ordering:  Polynomial interpretation [POLO]:
31.40/15.99	
31.40/15.99	   POL(Succ(x_1)) = 1 + x_1
31.40/15.99	   POL(Zero) = 0
31.40/15.99	   POL(new_primDivNatS(x_1, x_2, x_3)) = x_1
31.40/15.99	   POL(new_primDivNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1
31.40/15.99	   POL(new_primDivNatS00(x_1, x_2)) = 1 + x_1
31.40/15.99	   POL(new_primDivNatS1(x_1, x_2)) = 1 + x_1
31.40/15.99	
31.40/15.99	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.40/15.99	none
31.40/15.99	
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(15)
31.40/15.99	Obligation:
31.40/15.99	Q DP problem:
31.40/15.99	The TRS P consists of the following rules:
31.40/15.99	
31.40/15.99	   new_primDivNatS00(ww74, ww75) -> new_primDivNatS(Succ(ww74), Succ(ww75), Succ(ww75))
31.40/15.99	   new_primDivNatS(Succ(ww880), Zero, ww90) -> new_primDivNatS1(ww880, ww90)
31.40/15.99	   new_primDivNatS0(ww74, ww75, Zero, Zero) -> new_primDivNatS00(ww74, ww75)
31.40/15.99	   new_primDivNatS0(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS0(ww74, ww75, ww760, ww770)
31.40/15.99	   new_primDivNatS0(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS(Succ(ww74), Succ(ww75), Succ(ww75))
31.40/15.99	
31.40/15.99	R is empty.
31.40/15.99	Q is empty.
31.40/15.99	We have to consider all minimal (P,Q,R)-chains.
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(16) DependencyGraphProof (EQUIVALENT)
31.40/15.99	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(17)
31.40/15.99	Obligation:
31.40/15.99	Q DP problem:
31.40/15.99	The TRS P consists of the following rules:
31.40/15.99	
31.40/15.99	   new_primDivNatS0(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS0(ww74, ww75, ww760, ww770)
31.40/15.99	
31.40/15.99	R is empty.
31.40/15.99	Q is empty.
31.40/15.99	We have to consider all minimal (P,Q,R)-chains.
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(18) QDPSizeChangeProof (EQUIVALENT)
31.40/15.99	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. 
31.40/15.99	
31.40/15.99	From the DPs we obtained the following set of size-change graphs:
31.40/15.99	*new_primDivNatS0(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS0(ww74, ww75, ww760, ww770)
31.40/15.99	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
31.40/15.99	
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(19)
31.40/15.99	YES
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(20)
31.40/15.99	Obligation:
31.40/15.99	Q DP problem:
31.40/15.99	The TRS P consists of the following rules:
31.40/15.99	
31.40/15.99	   new_primShowInt(Neg(ww40)) -> new_primShowInt(Pos(ww40))
31.40/15.99	   new_primShowInt(Pos(Succ(ww400))) -> new_primShowInt(new_div(ww400, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))
31.40/15.99	
31.40/15.99	The TRS R consists of the following rules:
31.40/15.99	
31.40/15.99	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/15.99	   new_div(ww27, ww28) -> Pos(new_primDivNatS3(ww27, ww28))
31.40/15.99	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/15.99	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/15.99	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/15.99	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/15.99	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/15.99	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/15.99	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/15.99	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/15.99	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/15.99	   new_primDivNatS4(ww90) -> Zero
31.40/15.99	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/15.99	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/15.99	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/15.99	
31.40/15.99	The set Q consists of the following terms:
31.40/15.99	
31.40/15.99	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/15.99	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/15.99	   new_primDivNatS3(Succ(x0), Zero)
31.40/15.99	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/15.99	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/15.99	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/15.99	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/15.99	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/15.99	   new_primDivNatS3(Zero, Succ(x0))
31.40/15.99	   new_primDivNatS01(x0, x1)
31.40/15.99	   new_primDivNatS4(x0)
31.40/15.99	   new_div(x0, x1)
31.40/15.99	   new_primDivNatS2(Zero, Zero, x0)
31.40/15.99	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/15.99	   new_primDivNatS3(Zero, Zero)
31.40/15.99	
31.40/15.99	We have to consider all minimal (P,Q,R)-chains.
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(21) DependencyGraphProof (EQUIVALENT)
31.40/15.99	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(22)
31.40/15.99	Obligation:
31.40/15.99	Q DP problem:
31.40/15.99	The TRS P consists of the following rules:
31.40/15.99	
31.40/15.99	   new_primShowInt(Pos(Succ(ww400))) -> new_primShowInt(new_div(ww400, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))
31.40/15.99	
31.40/15.99	The TRS R consists of the following rules:
31.40/15.99	
31.40/15.99	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/15.99	   new_div(ww27, ww28) -> Pos(new_primDivNatS3(ww27, ww28))
31.40/15.99	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/15.99	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/15.99	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/15.99	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/15.99	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/15.99	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/15.99	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/15.99	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/15.99	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/15.99	   new_primDivNatS4(ww90) -> Zero
31.40/15.99	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/15.99	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/15.99	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/15.99	
31.40/15.99	The set Q consists of the following terms:
31.40/15.99	
31.40/15.99	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/15.99	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/15.99	   new_primDivNatS3(Succ(x0), Zero)
31.40/15.99	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/15.99	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/15.99	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/15.99	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/15.99	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/15.99	   new_primDivNatS3(Zero, Succ(x0))
31.40/15.99	   new_primDivNatS01(x0, x1)
31.40/15.99	   new_primDivNatS4(x0)
31.40/15.99	   new_div(x0, x1)
31.40/15.99	   new_primDivNatS2(Zero, Zero, x0)
31.40/15.99	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/15.99	   new_primDivNatS3(Zero, Zero)
31.40/15.99	
31.40/15.99	We have to consider all minimal (P,Q,R)-chains.
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(23) TransformationProof (EQUIVALENT)
31.40/15.99	By rewriting [LPAR04] the rule new_primShowInt(Pos(Succ(ww400))) -> new_primShowInt(new_div(ww400, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) at position [0] we obtained the following new rules [LPAR04]:
31.40/15.99	
31.40/15.99	   (new_primShowInt(Pos(Succ(ww400))) -> new_primShowInt(Pos(new_primDivNatS3(ww400, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))),new_primShowInt(Pos(Succ(ww400))) -> new_primShowInt(Pos(new_primDivNatS3(ww400, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))
31.40/15.99	
31.40/15.99	
31.40/15.99	----------------------------------------
31.40/15.99	
31.40/15.99	(24)
31.40/15.99	Obligation:
31.40/15.99	Q DP problem:
31.40/15.99	The TRS P consists of the following rules:
31.40/15.99	
31.40/15.99	   new_primShowInt(Pos(Succ(ww400))) -> new_primShowInt(Pos(new_primDivNatS3(ww400, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))
31.40/15.99	
31.40/15.99	The TRS R consists of the following rules:
31.40/15.99	
31.40/15.99	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/15.99	   new_div(ww27, ww28) -> Pos(new_primDivNatS3(ww27, ww28))
31.40/15.99	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/15.99	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/15.99	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/15.99	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/15.99	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/15.99	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/15.99	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/15.99	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/15.99	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/15.99	   new_primDivNatS4(ww90) -> Zero
31.40/15.99	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/15.99	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/15.99	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/15.99	
31.40/15.99	The set Q consists of the following terms:
31.40/15.99	
31.40/15.99	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/15.99	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/15.99	   new_primDivNatS3(Succ(x0), Zero)
31.40/15.99	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/15.99	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/15.99	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/15.99	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/15.99	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/15.99	   new_primDivNatS3(Zero, Succ(x0))
31.40/15.99	   new_primDivNatS01(x0, x1)
31.40/15.99	   new_primDivNatS4(x0)
31.40/15.99	   new_div(x0, x1)
31.40/15.99	   new_primDivNatS2(Zero, Zero, x0)
31.40/15.99	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/15.99	   new_primDivNatS3(Zero, Zero)
31.40/15.99	
31.40/15.99	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(25) UsableRulesProof (EQUIVALENT)
31.40/16.00	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(26)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(ww400))) -> new_primShowInt(Pos(new_primDivNatS3(ww400, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	The set Q consists of the following terms:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/16.00	   new_primDivNatS3(Succ(x0), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/16.00	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/16.00	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/16.00	   new_primDivNatS3(Zero, Succ(x0))
31.40/16.00	   new_primDivNatS01(x0, x1)
31.40/16.00	   new_primDivNatS4(x0)
31.40/16.00	   new_div(x0, x1)
31.40/16.00	   new_primDivNatS2(Zero, Zero, x0)
31.40/16.00	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/16.00	   new_primDivNatS3(Zero, Zero)
31.40/16.00	
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(27) QReductionProof (EQUIVALENT)
31.40/16.00	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
31.40/16.00	
31.40/16.00	   new_div(x0, x1)
31.40/16.00	
31.40/16.00	
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(28)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(ww400))) -> new_primShowInt(Pos(new_primDivNatS3(ww400, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	The set Q consists of the following terms:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/16.00	   new_primDivNatS3(Succ(x0), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/16.00	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/16.00	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/16.00	   new_primDivNatS3(Zero, Succ(x0))
31.40/16.00	   new_primDivNatS01(x0, x1)
31.40/16.00	   new_primDivNatS4(x0)
31.40/16.00	   new_primDivNatS2(Zero, Zero, x0)
31.40/16.00	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/16.00	   new_primDivNatS3(Zero, Zero)
31.40/16.00	
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(29) MNOCProof (EQUIVALENT)
31.40/16.00	We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(30)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(ww400))) -> new_primShowInt(Pos(new_primDivNatS3(ww400, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	Q is empty.
31.40/16.00	We have to consider all (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(31) InductionCalculusProof (EQUIVALENT)
31.40/16.00	Note that final constraints are written in bold face.
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	For Pair new_primShowInt(Pos(Succ(ww400))) -> new_primShowInt(Pos(new_primDivNatS3(ww400, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) the following chains were created:
31.40/16.00	*We consider the chain new_primShowInt(Pos(Succ(x0))) -> new_primShowInt(Pos(new_primDivNatS3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))), new_primShowInt(Pos(Succ(x1))) -> new_primShowInt(Pos(new_primDivNatS3(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) which results in the following constraint:
31.40/16.00	
31.40/16.00	(1)    (new_primShowInt(Pos(new_primDivNatS3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))=new_primShowInt(Pos(Succ(x1)))  ==>  new_primShowInt(Pos(Succ(x0)))_>=_new_primShowInt(Pos(new_primDivNatS3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	We simplified constraint (1) using rules (I), (II), (VII) which results in the following new constraint:
31.40/16.00	
31.40/16.00	(2)    (Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=x2 & new_primDivNatS3(x0, x2)=Succ(x1)  ==>  new_primShowInt(Pos(Succ(x0)))_>=_new_primShowInt(Pos(new_primDivNatS3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS3(x0, x2)=Succ(x1) which results in the following new constraints:
31.40/16.00	
31.40/16.00	(3)    (new_primDivNatS02(x4, x3, x4, x3)=Succ(x1) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Succ(x3)  ==>  new_primShowInt(Pos(Succ(Succ(x4))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x4), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))
31.40/16.00	
31.40/16.00	(4)    (Succ(new_primDivNatS2(Succ(x6), Zero, Zero))=Succ(x1) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Zero  ==>  new_primShowInt(Pos(Succ(Succ(x6))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x6), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))
31.40/16.00	
31.40/16.00	(5)    (Succ(new_primDivNatS2(Zero, Zero, Zero))=Succ(x1) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Zero  ==>  new_primShowInt(Pos(Succ(Zero)))_>=_new_primShowInt(Pos(new_primDivNatS3(Zero, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	We simplified constraint (3) using rules (I), (II), (VII) which results in the following new constraint:
31.40/16.00	
31.40/16.00	(6)    (x4=x7 & x3=x8 & new_primDivNatS02(x4, x3, x7, x8)=Succ(x1) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x3  ==>  new_primShowInt(Pos(Succ(Succ(x4))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x4), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS02(x4, x3, x7, x8)=Succ(x1) which results in the following new constraints:
31.40/16.00	
31.40/16.00	(7)    (new_primDivNatS02(x12, x11, x10, x9)=Succ(x1) & x12=Succ(x10) & x11=Succ(x9) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x11 & (\/x13:new_primDivNatS02(x12, x11, x10, x9)=Succ(x13) & x12=x10 & x11=x9 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x11  ==>  new_primShowInt(Pos(Succ(Succ(x12))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x12), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))  ==>  new_primShowInt(Pos(Succ(Succ(x12))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x12), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))
31.40/16.00	
31.40/16.00	(8)    (new_primDivNatS01(x16, x15)=Succ(x1) & x16=Succ(x14) & x15=Zero & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x15  ==>  new_primShowInt(Pos(Succ(Succ(x16))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x16), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))
31.40/16.00	
31.40/16.00	(9)    (new_primDivNatS01(x18, x17)=Succ(x1) & x18=Zero & x17=Zero & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x17  ==>  new_primShowInt(Pos(Succ(Succ(x18))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x18), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	We simplified constraint (7) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
31.40/16.00	
31.40/16.00	(10)    (new_primShowInt(Pos(Succ(Succ(Succ(x10)))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(Succ(x10)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	We solved constraint (8) using rules (I), (II), (III).We solved constraint (9) using rules (I), (II), (III).
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	To summarize, we get the following constraints P__>=_ for the following pairs.
31.40/16.00	
31.40/16.00	*new_primShowInt(Pos(Succ(ww400))) -> new_primShowInt(Pos(new_primDivNatS3(ww400, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))
31.40/16.00	
31.40/16.00	*(new_primShowInt(Pos(Succ(Succ(Succ(x10)))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(Succ(x10)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(32)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(ww400))) -> new_primShowInt(Pos(new_primDivNatS3(ww400, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	The set Q consists of the following terms:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/16.00	   new_primDivNatS3(Succ(x0), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/16.00	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/16.00	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/16.00	   new_primDivNatS3(Zero, Succ(x0))
31.40/16.00	   new_primDivNatS01(x0, x1)
31.40/16.00	   new_primDivNatS4(x0)
31.40/16.00	   new_primDivNatS2(Zero, Zero, x0)
31.40/16.00	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/16.00	   new_primDivNatS3(Zero, Zero)
31.40/16.00	
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(33) TransformationProof (EQUIVALENT)
31.40/16.00	By narrowing [LPAR04] the rule new_primShowInt(Pos(Succ(ww400))) -> new_primShowInt(Pos(new_primDivNatS3(ww400, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) at position [0,0] we obtained the following new rules [LPAR04]:
31.40/16.00	
31.40/16.00	   (new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS02(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))),new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS02(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))
31.40/16.00	   (new_primShowInt(Pos(Succ(Zero))) -> new_primShowInt(Pos(Zero)),new_primShowInt(Pos(Succ(Zero))) -> new_primShowInt(Pos(Zero)))
31.40/16.00	
31.40/16.00	
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(34)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS02(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))
31.40/16.00	   new_primShowInt(Pos(Succ(Zero))) -> new_primShowInt(Pos(Zero))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	The set Q consists of the following terms:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/16.00	   new_primDivNatS3(Succ(x0), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/16.00	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/16.00	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/16.00	   new_primDivNatS3(Zero, Succ(x0))
31.40/16.00	   new_primDivNatS01(x0, x1)
31.40/16.00	   new_primDivNatS4(x0)
31.40/16.00	   new_primDivNatS2(Zero, Zero, x0)
31.40/16.00	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/16.00	   new_primDivNatS3(Zero, Zero)
31.40/16.00	
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(35) DependencyGraphProof (EQUIVALENT)
31.40/16.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(36)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS02(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	The set Q consists of the following terms:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/16.00	   new_primDivNatS3(Succ(x0), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/16.00	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/16.00	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/16.00	   new_primDivNatS3(Zero, Succ(x0))
31.40/16.00	   new_primDivNatS01(x0, x1)
31.40/16.00	   new_primDivNatS4(x0)
31.40/16.00	   new_primDivNatS2(Zero, Zero, x0)
31.40/16.00	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/16.00	   new_primDivNatS3(Zero, Zero)
31.40/16.00	
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(37) TransformationProof (EQUIVALENT)
31.40/16.00	By narrowing [LPAR04] the rule new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS02(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) at position [0,0] we obtained the following new rules [LPAR04]:
31.40/16.00	
31.40/16.00	   (new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))),new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))
31.40/16.00	   (new_primShowInt(Pos(Succ(Succ(Zero)))) -> new_primShowInt(Pos(Zero)),new_primShowInt(Pos(Succ(Succ(Zero)))) -> new_primShowInt(Pos(Zero)))
31.40/16.00	
31.40/16.00	
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(38)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))
31.40/16.00	   new_primShowInt(Pos(Succ(Succ(Zero)))) -> new_primShowInt(Pos(Zero))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	The set Q consists of the following terms:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/16.00	   new_primDivNatS3(Succ(x0), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/16.00	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/16.00	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/16.00	   new_primDivNatS3(Zero, Succ(x0))
31.40/16.00	   new_primDivNatS01(x0, x1)
31.40/16.00	   new_primDivNatS4(x0)
31.40/16.00	   new_primDivNatS2(Zero, Zero, x0)
31.40/16.00	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/16.00	   new_primDivNatS3(Zero, Zero)
31.40/16.00	
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(39) DependencyGraphProof (EQUIVALENT)
31.40/16.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(40)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	The set Q consists of the following terms:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/16.00	   new_primDivNatS3(Succ(x0), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/16.00	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/16.00	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/16.00	   new_primDivNatS3(Zero, Succ(x0))
31.40/16.00	   new_primDivNatS01(x0, x1)
31.40/16.00	   new_primDivNatS4(x0)
31.40/16.00	   new_primDivNatS2(Zero, Zero, x0)
31.40/16.00	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/16.00	   new_primDivNatS3(Zero, Zero)
31.40/16.00	
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(41) TransformationProof (EQUIVALENT)
31.40/16.00	By narrowing [LPAR04] the rule new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) at position [0,0] we obtained the following new rules [LPAR04]:
31.40/16.00	
31.40/16.00	   (new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))),new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))
31.40/16.00	   (new_primShowInt(Pos(Succ(Succ(Succ(Zero))))) -> new_primShowInt(Pos(Zero)),new_primShowInt(Pos(Succ(Succ(Succ(Zero))))) -> new_primShowInt(Pos(Zero)))
31.40/16.00	
31.40/16.00	
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(42)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	   new_primShowInt(Pos(Succ(Succ(Succ(Zero))))) -> new_primShowInt(Pos(Zero))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	The set Q consists of the following terms:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/16.00	   new_primDivNatS3(Succ(x0), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/16.00	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/16.00	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/16.00	   new_primDivNatS3(Zero, Succ(x0))
31.40/16.00	   new_primDivNatS01(x0, x1)
31.40/16.00	   new_primDivNatS4(x0)
31.40/16.00	   new_primDivNatS2(Zero, Zero, x0)
31.40/16.00	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/16.00	   new_primDivNatS3(Zero, Zero)
31.40/16.00	
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(43) DependencyGraphProof (EQUIVALENT)
31.40/16.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(44)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	The set Q consists of the following terms:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/16.00	   new_primDivNatS3(Succ(x0), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/16.00	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/16.00	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/16.00	   new_primDivNatS3(Zero, Succ(x0))
31.40/16.00	   new_primDivNatS01(x0, x1)
31.40/16.00	   new_primDivNatS4(x0)
31.40/16.00	   new_primDivNatS2(Zero, Zero, x0)
31.40/16.00	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/16.00	   new_primDivNatS3(Zero, Zero)
31.40/16.00	
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(45) TransformationProof (EQUIVALENT)
31.40/16.00	By narrowing [LPAR04] the rule new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) at position [0,0] we obtained the following new rules [LPAR04]:
31.40/16.00	
31.40/16.00	   (new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero)))))))),new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	   (new_primShowInt(Pos(Succ(Succ(Succ(Succ(Zero)))))) -> new_primShowInt(Pos(Zero)),new_primShowInt(Pos(Succ(Succ(Succ(Succ(Zero)))))) -> new_primShowInt(Pos(Zero)))
31.40/16.00	
31.40/16.00	
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(46)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))))
31.40/16.00	   new_primShowInt(Pos(Succ(Succ(Succ(Succ(Zero)))))) -> new_primShowInt(Pos(Zero))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	The set Q consists of the following terms:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/16.00	   new_primDivNatS3(Succ(x0), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/16.00	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/16.00	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/16.00	   new_primDivNatS3(Zero, Succ(x0))
31.40/16.00	   new_primDivNatS01(x0, x1)
31.40/16.00	   new_primDivNatS4(x0)
31.40/16.00	   new_primDivNatS2(Zero, Zero, x0)
31.40/16.00	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/16.00	   new_primDivNatS3(Zero, Zero)
31.40/16.00	
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(47) DependencyGraphProof (EQUIVALENT)
31.40/16.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(48)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	The set Q consists of the following terms:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/16.00	   new_primDivNatS3(Succ(x0), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/16.00	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/16.00	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/16.00	   new_primDivNatS3(Zero, Succ(x0))
31.40/16.00	   new_primDivNatS01(x0, x1)
31.40/16.00	   new_primDivNatS4(x0)
31.40/16.00	   new_primDivNatS2(Zero, Zero, x0)
31.40/16.00	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/16.00	   new_primDivNatS3(Zero, Zero)
31.40/16.00	
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(49) MNOCProof (EQUIVALENT)
31.40/16.00	We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(50)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	Q is empty.
31.40/16.00	We have to consider all (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(51) InductionCalculusProof (EQUIVALENT)
31.40/16.00	Note that final constraints are written in bold face.
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	For Pair new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero)))))))) the following chains were created:
31.40/16.00	*We consider the chain new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x0))))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x1))))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x1))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Zero)))))))) which results in the following constraint:
31.40/16.00	
31.40/16.00	(1)    (new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Zero))))))))=new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x1)))))))  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x0)))))))_>=_new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	We simplified constraint (1) using rules (I), (II), (VII) which results in the following new constraint:
31.40/16.00	
31.40/16.00	(2)    (Succ(Succ(Succ(x0)))=x2 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x3 & Succ(Succ(Succ(Succ(Succ(Zero)))))=x4 & new_primDivNatS02(x2, x3, x0, x4)=Succ(Succ(Succ(Succ(Succ(x1)))))  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x0)))))))_>=_new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS02(x2, x3, x0, x4)=Succ(Succ(Succ(Succ(Succ(x1))))) which results in the following new constraints:
31.40/16.00	
31.40/16.00	(3)    (new_primDivNatS02(x8, x7, x6, x5)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(x6))))=x8 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x7 & Succ(Succ(Succ(Succ(Succ(Zero)))))=Succ(x5) & (\/x9:new_primDivNatS02(x8, x7, x6, x5)=Succ(Succ(Succ(Succ(Succ(x9))))) & Succ(Succ(Succ(x6)))=x8 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x7 & Succ(Succ(Succ(Succ(Succ(Zero)))))=x5  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x6)))))))_>=_new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x6))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x6, Succ(Succ(Succ(Succ(Succ(Zero)))))))))  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x6))))))))_>=_new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(Succ(x6)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x6), Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	
31.40/16.00	(4)    (new_primDivNatS01(x12, x11)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(x10))))=x12 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x11 & Succ(Succ(Succ(Succ(Succ(Zero)))))=Zero  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x10))))))))_>=_new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(Succ(x10)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x10), Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	
31.40/16.00	(5)    (new_primDivNatS01(x14, x13)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Zero)))=x14 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x13 & Succ(Succ(Succ(Succ(Succ(Zero)))))=Zero  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint:
31.40/16.00	
31.40/16.00	(6)    (new_primDivNatS02(x8, x7, x6, x5)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(x6))))=x8 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x7 & Succ(Succ(Succ(Succ(Zero))))=x5  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x6))))))))_>=_new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(Succ(x6)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x6), Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS02(x8, x7, x6, x5)=Succ(Succ(Succ(Succ(Succ(x1))))) which results in the following new constraints:
31.40/16.00	
31.40/16.00	(7)    (new_primDivNatS02(x21, x20, x19, x18)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(Succ(x19)))))=x21 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x20 & Succ(Succ(Succ(Succ(Zero))))=Succ(x18) & (\/x22:new_primDivNatS02(x21, x20, x19, x18)=Succ(Succ(Succ(Succ(Succ(x22))))) & Succ(Succ(Succ(Succ(x19))))=x21 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x20 & Succ(Succ(Succ(Succ(Zero))))=x18  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x19))))))))_>=_new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(Succ(x19)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x19), Succ(Succ(Succ(Succ(Succ(Zero)))))))))  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x19)))))))))_>=_new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(Succ(Succ(x19))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x19)), Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	
31.40/16.00	(8)    (new_primDivNatS01(x25, x24)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(Succ(x23)))))=x25 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x24 & Succ(Succ(Succ(Succ(Zero))))=Zero  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x23)))))))))_>=_new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(Succ(Succ(x23))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x23)), Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	
31.40/16.00	(9)    (new_primDivNatS01(x27, x26)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(Zero))))=x27 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x26 & Succ(Succ(Succ(Succ(Zero))))=Zero  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))_>=_new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Zero), Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint:
31.40/16.00	
31.40/16.00	(10)    (new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x19)))))))))_>=_new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(Succ(Succ(x19))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x19)), Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	We solved constraint (8) using rules (I), (II).We solved constraint (9) using rules (I), (II).
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	To summarize, we get the following constraints P__>=_ for the following pairs.
31.40/16.00	
31.40/16.00	*new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))))
31.40/16.00	
31.40/16.00	*(new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x19)))))))))_>=_new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(Succ(Succ(x19))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x19)), Succ(Succ(Succ(Succ(Succ(Zero)))))))))
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	
31.40/16.00	The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(52)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS02(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))))
31.40/16.00	
31.40/16.00	The TRS R consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(ww270), Succ(ww280)) -> new_primDivNatS02(ww270, ww280, ww270, ww280)
31.40/16.00	   new_primDivNatS3(Zero, Succ(ww280)) -> Zero
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Succ(ww770)) -> new_primDivNatS02(ww74, ww75, ww760, ww770)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Succ(ww760), Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Zero) -> new_primDivNatS01(ww74, ww75)
31.40/16.00	   new_primDivNatS02(ww74, ww75, Zero, Succ(ww770)) -> Zero
31.40/16.00	   new_primDivNatS01(ww74, ww75) -> Succ(new_primDivNatS2(Succ(ww74), Succ(ww75), Succ(ww75)))
31.40/16.00	   new_primDivNatS2(Succ(ww880), Succ(ww890), ww90) -> new_primDivNatS2(ww880, ww890, ww90)
31.40/16.00	   new_primDivNatS2(Succ(ww880), Zero, ww90) -> new_primDivNatS3(ww880, ww90)
31.40/16.00	   new_primDivNatS2(Zero, Zero, ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS2(Zero, Succ(ww890), ww90) -> new_primDivNatS4(ww90)
31.40/16.00	   new_primDivNatS4(ww90) -> Zero
31.40/16.00	   new_primDivNatS3(Succ(ww270), Zero) -> Succ(new_primDivNatS2(Succ(ww270), Zero, Zero))
31.40/16.00	   new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero))
31.40/16.00	
31.40/16.00	The set Q consists of the following terms:
31.40/16.00	
31.40/16.00	   new_primDivNatS3(Succ(x0), Succ(x1))
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Succ(x3))
31.40/16.00	   new_primDivNatS3(Succ(x0), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Succ(x2), Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Zero)
31.40/16.00	   new_primDivNatS02(x0, x1, Zero, Succ(x2))
31.40/16.00	   new_primDivNatS2(Succ(x0), Succ(x1), x2)
31.40/16.00	   new_primDivNatS2(Succ(x0), Zero, x1)
31.40/16.00	   new_primDivNatS3(Zero, Succ(x0))
31.40/16.00	   new_primDivNatS01(x0, x1)
31.40/16.00	   new_primDivNatS4(x0)
31.40/16.00	   new_primDivNatS2(Zero, Zero, x0)
31.40/16.00	   new_primDivNatS2(Zero, Succ(x0), x1)
31.40/16.00	   new_primDivNatS3(Zero, Zero)
31.40/16.00	
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(53)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_psPs(:(ww220, ww221), ww21) -> new_psPs(ww221, ww21)
31.40/16.00	
31.40/16.00	R is empty.
31.40/16.00	Q is empty.
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(54) QDPSizeChangeProof (EQUIVALENT)
31.40/16.00	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. 
31.40/16.00	
31.40/16.00	From the DPs we obtained the following set of size-change graphs:
31.40/16.00	*new_psPs(:(ww220, ww221), ww21) -> new_psPs(ww221, ww21)
31.40/16.00	The graph contains the following edges 1 > 1, 2 >= 2
31.40/16.00	
31.40/16.00	
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(55)
31.40/16.00	YES
31.40/16.00	
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(56)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primModNatS(Succ(ww840), Zero, ww86) -> new_primModNatS1(ww840, ww86)
31.40/16.00	   new_primModNatS1(Zero, Zero) -> new_primModNatS(Zero, Zero, Zero)
31.40/16.00	   new_primModNatS00(ww79, ww80) -> new_primModNatS(Succ(ww79), Succ(ww80), Succ(ww80))
31.40/16.00	   new_primModNatS0(ww79, ww80, Succ(ww810), Zero) -> new_primModNatS(Succ(ww79), Succ(ww80), Succ(ww80))
31.40/16.00	   new_primModNatS(Succ(ww840), Succ(ww850), ww86) -> new_primModNatS(ww840, ww850, ww86)
31.40/16.00	   new_primModNatS0(ww79, ww80, Succ(ww810), Succ(ww820)) -> new_primModNatS0(ww79, ww80, ww810, ww820)
31.40/16.00	   new_primModNatS1(Succ(ww300), Succ(ww310)) -> new_primModNatS0(ww300, ww310, ww300, ww310)
31.40/16.00	   new_primModNatS0(ww79, ww80, Zero, Zero) -> new_primModNatS00(ww79, ww80)
31.40/16.00	   new_primModNatS1(Succ(ww300), Zero) -> new_primModNatS(Succ(ww300), Zero, Zero)
31.40/16.00	
31.40/16.00	R is empty.
31.40/16.00	Q is empty.
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(57) DependencyGraphProof (EQUIVALENT)
31.40/16.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(58)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primModNatS1(Succ(ww300), Succ(ww310)) -> new_primModNatS0(ww300, ww310, ww300, ww310)
31.40/16.00	   new_primModNatS0(ww79, ww80, Succ(ww810), Zero) -> new_primModNatS(Succ(ww79), Succ(ww80), Succ(ww80))
31.40/16.00	   new_primModNatS(Succ(ww840), Succ(ww850), ww86) -> new_primModNatS(ww840, ww850, ww86)
31.40/16.00	   new_primModNatS(Succ(ww840), Zero, ww86) -> new_primModNatS1(ww840, ww86)
31.40/16.00	   new_primModNatS1(Succ(ww300), Zero) -> new_primModNatS(Succ(ww300), Zero, Zero)
31.40/16.00	   new_primModNatS0(ww79, ww80, Succ(ww810), Succ(ww820)) -> new_primModNatS0(ww79, ww80, ww810, ww820)
31.40/16.00	   new_primModNatS0(ww79, ww80, Zero, Zero) -> new_primModNatS00(ww79, ww80)
31.40/16.00	   new_primModNatS00(ww79, ww80) -> new_primModNatS(Succ(ww79), Succ(ww80), Succ(ww80))
31.40/16.00	
31.40/16.00	R is empty.
31.40/16.00	Q is empty.
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(59) QDPOrderProof (EQUIVALENT)
31.40/16.00	We use the reduction pair processor [LPAR04,JAR06].
31.40/16.00	
31.40/16.00	
31.40/16.00	The following pairs can be oriented strictly and are deleted.
31.40/16.00	
31.40/16.00	   new_primModNatS1(Succ(ww300), Succ(ww310)) -> new_primModNatS0(ww300, ww310, ww300, ww310)
31.40/16.00	   new_primModNatS(Succ(ww840), Succ(ww850), ww86) -> new_primModNatS(ww840, ww850, ww86)
31.40/16.00	   new_primModNatS1(Succ(ww300), Zero) -> new_primModNatS(Succ(ww300), Zero, Zero)
31.40/16.00	The remaining pairs can at least be oriented weakly.
31.40/16.00	Used ordering:  Polynomial interpretation [POLO]:
31.40/16.00	
31.40/16.00	   POL(Succ(x_1)) = 1 + x_1
31.40/16.00	   POL(Zero) = 0
31.40/16.00	   POL(new_primModNatS(x_1, x_2, x_3)) = x_1
31.40/16.00	   POL(new_primModNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1
31.40/16.00	   POL(new_primModNatS00(x_1, x_2)) = 1 + x_1
31.40/16.00	   POL(new_primModNatS1(x_1, x_2)) = 1 + x_1
31.40/16.00	
31.40/16.00	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.40/16.00	none
31.40/16.00	
31.40/16.00	
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(60)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primModNatS0(ww79, ww80, Succ(ww810), Zero) -> new_primModNatS(Succ(ww79), Succ(ww80), Succ(ww80))
31.40/16.00	   new_primModNatS(Succ(ww840), Zero, ww86) -> new_primModNatS1(ww840, ww86)
31.40/16.00	   new_primModNatS0(ww79, ww80, Succ(ww810), Succ(ww820)) -> new_primModNatS0(ww79, ww80, ww810, ww820)
31.40/16.00	   new_primModNatS0(ww79, ww80, Zero, Zero) -> new_primModNatS00(ww79, ww80)
31.40/16.00	   new_primModNatS00(ww79, ww80) -> new_primModNatS(Succ(ww79), Succ(ww80), Succ(ww80))
31.40/16.00	
31.40/16.00	R is empty.
31.40/16.00	Q is empty.
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(61) DependencyGraphProof (EQUIVALENT)
31.40/16.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(62)
31.40/16.00	Obligation:
31.40/16.00	Q DP problem:
31.40/16.00	The TRS P consists of the following rules:
31.40/16.00	
31.40/16.00	   new_primModNatS0(ww79, ww80, Succ(ww810), Succ(ww820)) -> new_primModNatS0(ww79, ww80, ww810, ww820)
31.40/16.00	
31.40/16.00	R is empty.
31.40/16.00	Q is empty.
31.40/16.00	We have to consider all minimal (P,Q,R)-chains.
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(63) QDPSizeChangeProof (EQUIVALENT)
31.40/16.00	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. 
31.40/16.00	
31.40/16.00	From the DPs we obtained the following set of size-change graphs:
31.40/16.00	*new_primModNatS0(ww79, ww80, Succ(ww810), Succ(ww820)) -> new_primModNatS0(ww79, ww80, ww810, ww820)
31.40/16.00	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
31.40/16.00	
31.40/16.00	
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(64)
31.40/16.00	YES
31.40/16.00	
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(65) Narrow (COMPLETE)
31.40/16.00	Haskell To QDPs
31.40/16.00	
31.40/16.00	digraph dp_graph {
31.40/16.00	node [outthreshold=100, inthreshold=100];1[label="showsPrec",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
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31.40/16.00	938 -> 126[label="",style="solid", color="burlywood", weight=3];
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31.40/16.00	138[label="primModNatS0 Zero (Succ ww310) False",fontsize=16,color="black",shape="box"];138 -> 149[label="",style="solid", color="black", weight=3];
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31.40/16.00	946 -> 695[label="",style="solid", color="burlywood", weight=3];
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31.40/16.00	947 -> 696[label="",style="solid", color="burlywood", weight=3];
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31.40/16.00	948 -> 742[label="",style="solid", color="burlywood", weight=3];
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31.40/16.00	148 -> 791[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	149[label="Succ Zero",fontsize=16,color="green",shape="box"];150 -> 788[label="",style="dashed", color="red", weight=0];
31.40/16.00	150[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];150 -> 792[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	150 -> 793[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	150 -> 794[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	693[label="primDivNatS0 (Succ ww74) (Succ ww75) (primGEqNatS (Succ ww760) (Succ ww770))",fontsize=16,color="black",shape="box"];693 -> 744[label="",style="solid", color="black", weight=3];
31.40/16.00	694[label="primDivNatS0 (Succ ww74) (Succ ww75) (primGEqNatS (Succ ww760) Zero)",fontsize=16,color="black",shape="box"];694 -> 745[label="",style="solid", color="black", weight=3];
31.40/16.00	695[label="primDivNatS0 (Succ ww74) (Succ ww75) (primGEqNatS Zero (Succ ww770))",fontsize=16,color="black",shape="box"];695 -> 746[label="",style="solid", color="black", weight=3];
31.40/16.00	696[label="primDivNatS0 (Succ ww74) (Succ ww75) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];696 -> 747[label="",style="solid", color="black", weight=3];
31.40/16.00	885[label="Succ ww270",fontsize=16,color="green",shape="box"];886[label="Zero",fontsize=16,color="green",shape="box"];887[label="Zero",fontsize=16,color="green",shape="box"];884[label="primDivNatS (primMinusNatS ww88 ww89) (Succ ww90)",fontsize=16,color="burlywood",shape="triangle"];950[label="ww88/Succ ww880",fontsize=10,color="white",style="solid",shape="box"];884 -> 950[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	950 -> 909[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	951[label="ww88/Zero",fontsize=10,color="white",style="solid",shape="box"];884 -> 951[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	951 -> 910[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	888[label="Zero",fontsize=16,color="green",shape="box"];889[label="Zero",fontsize=16,color="green",shape="box"];890[label="Zero",fontsize=16,color="green",shape="box"];742[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS (Succ ww810) ww82)",fontsize=16,color="burlywood",shape="box"];952[label="ww82/Succ ww820",fontsize=10,color="white",style="solid",shape="box"];742 -> 952[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	952 -> 752[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	953[label="ww82/Zero",fontsize=10,color="white",style="solid",shape="box"];742 -> 953[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	953 -> 753[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	743[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS Zero ww82)",fontsize=16,color="burlywood",shape="box"];954[label="ww82/Succ ww820",fontsize=10,color="white",style="solid",shape="box"];743 -> 954[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	954 -> 754[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	955[label="ww82/Zero",fontsize=10,color="white",style="solid",shape="box"];743 -> 955[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	955 -> 755[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	789[label="Succ ww300",fontsize=16,color="green",shape="box"];790[label="Zero",fontsize=16,color="green",shape="box"];791[label="Zero",fontsize=16,color="green",shape="box"];788[label="primModNatS (primMinusNatS ww84 ww85) (Succ ww86)",fontsize=16,color="burlywood",shape="triangle"];956[label="ww84/Succ ww840",fontsize=10,color="white",style="solid",shape="box"];788 -> 956[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	956 -> 819[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	957[label="ww84/Zero",fontsize=10,color="white",style="solid",shape="box"];788 -> 957[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	957 -> 820[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	792[label="Zero",fontsize=16,color="green",shape="box"];793[label="Zero",fontsize=16,color="green",shape="box"];794[label="Zero",fontsize=16,color="green",shape="box"];744 -> 640[label="",style="dashed", color="red", weight=0];
31.40/16.00	744[label="primDivNatS0 (Succ ww74) (Succ ww75) (primGEqNatS ww760 ww770)",fontsize=16,color="magenta"];744 -> 756[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	744 -> 757[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	745[label="primDivNatS0 (Succ ww74) (Succ ww75) True",fontsize=16,color="black",shape="triangle"];745 -> 758[label="",style="solid", color="black", weight=3];
31.40/16.00	746[label="primDivNatS0 (Succ ww74) (Succ ww75) False",fontsize=16,color="black",shape="box"];746 -> 759[label="",style="solid", color="black", weight=3];
31.40/16.00	747 -> 745[label="",style="dashed", color="red", weight=0];
31.40/16.00	747[label="primDivNatS0 (Succ ww74) (Succ ww75) True",fontsize=16,color="magenta"];909[label="primDivNatS (primMinusNatS (Succ ww880) ww89) (Succ ww90)",fontsize=16,color="burlywood",shape="box"];958[label="ww89/Succ ww890",fontsize=10,color="white",style="solid",shape="box"];909 -> 958[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	958 -> 911[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	959[label="ww89/Zero",fontsize=10,color="white",style="solid",shape="box"];909 -> 959[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	959 -> 912[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	910[label="primDivNatS (primMinusNatS Zero ww89) (Succ ww90)",fontsize=16,color="burlywood",shape="box"];960[label="ww89/Succ ww890",fontsize=10,color="white",style="solid",shape="box"];910 -> 960[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	960 -> 913[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	961[label="ww89/Zero",fontsize=10,color="white",style="solid",shape="box"];910 -> 961[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	961 -> 914[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	752[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS (Succ ww810) (Succ ww820))",fontsize=16,color="black",shape="box"];752 -> 766[label="",style="solid", color="black", weight=3];
31.40/16.00	753[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS (Succ ww810) Zero)",fontsize=16,color="black",shape="box"];753 -> 767[label="",style="solid", color="black", weight=3];
31.40/16.00	754[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS Zero (Succ ww820))",fontsize=16,color="black",shape="box"];754 -> 768[label="",style="solid", color="black", weight=3];
31.40/16.00	755[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];755 -> 769[label="",style="solid", color="black", weight=3];
31.40/16.00	819[label="primModNatS (primMinusNatS (Succ ww840) ww85) (Succ ww86)",fontsize=16,color="burlywood",shape="box"];962[label="ww85/Succ ww850",fontsize=10,color="white",style="solid",shape="box"];819 -> 962[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	962 -> 825[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	963[label="ww85/Zero",fontsize=10,color="white",style="solid",shape="box"];819 -> 963[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	963 -> 826[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	820[label="primModNatS (primMinusNatS Zero ww85) (Succ ww86)",fontsize=16,color="burlywood",shape="box"];964[label="ww85/Succ ww850",fontsize=10,color="white",style="solid",shape="box"];820 -> 964[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	964 -> 827[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	965[label="ww85/Zero",fontsize=10,color="white",style="solid",shape="box"];820 -> 965[label="",style="solid", color="burlywood", weight=9];
31.40/16.00	965 -> 828[label="",style="solid", color="burlywood", weight=3];
31.40/16.00	756[label="ww760",fontsize=16,color="green",shape="box"];757[label="ww770",fontsize=16,color="green",shape="box"];758[label="Succ (primDivNatS (primMinusNatS (Succ ww74) (Succ ww75)) (Succ (Succ ww75)))",fontsize=16,color="green",shape="box"];758 -> 770[label="",style="dashed", color="green", weight=3];
31.40/16.00	759[label="Zero",fontsize=16,color="green",shape="box"];911[label="primDivNatS (primMinusNatS (Succ ww880) (Succ ww890)) (Succ ww90)",fontsize=16,color="black",shape="box"];911 -> 915[label="",style="solid", color="black", weight=3];
31.40/16.00	912[label="primDivNatS (primMinusNatS (Succ ww880) Zero) (Succ ww90)",fontsize=16,color="black",shape="box"];912 -> 916[label="",style="solid", color="black", weight=3];
31.40/16.00	913[label="primDivNatS (primMinusNatS Zero (Succ ww890)) (Succ ww90)",fontsize=16,color="black",shape="box"];913 -> 917[label="",style="solid", color="black", weight=3];
31.40/16.00	914[label="primDivNatS (primMinusNatS Zero Zero) (Succ ww90)",fontsize=16,color="black",shape="box"];914 -> 918[label="",style="solid", color="black", weight=3];
31.40/16.00	766 -> 701[label="",style="dashed", color="red", weight=0];
31.40/16.00	766[label="primModNatS0 (Succ ww79) (Succ ww80) (primGEqNatS ww810 ww820)",fontsize=16,color="magenta"];766 -> 775[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	766 -> 776[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	767[label="primModNatS0 (Succ ww79) (Succ ww80) True",fontsize=16,color="black",shape="triangle"];767 -> 777[label="",style="solid", color="black", weight=3];
31.40/16.00	768[label="primModNatS0 (Succ ww79) (Succ ww80) False",fontsize=16,color="black",shape="box"];768 -> 778[label="",style="solid", color="black", weight=3];
31.40/16.00	769 -> 767[label="",style="dashed", color="red", weight=0];
31.40/16.00	769[label="primModNatS0 (Succ ww79) (Succ ww80) True",fontsize=16,color="magenta"];825[label="primModNatS (primMinusNatS (Succ ww840) (Succ ww850)) (Succ ww86)",fontsize=16,color="black",shape="box"];825 -> 833[label="",style="solid", color="black", weight=3];
31.40/16.00	826[label="primModNatS (primMinusNatS (Succ ww840) Zero) (Succ ww86)",fontsize=16,color="black",shape="box"];826 -> 834[label="",style="solid", color="black", weight=3];
31.40/16.00	827[label="primModNatS (primMinusNatS Zero (Succ ww850)) (Succ ww86)",fontsize=16,color="black",shape="box"];827 -> 835[label="",style="solid", color="black", weight=3];
31.40/16.00	828[label="primModNatS (primMinusNatS Zero Zero) (Succ ww86)",fontsize=16,color="black",shape="box"];828 -> 836[label="",style="solid", color="black", weight=3];
31.40/16.00	770 -> 884[label="",style="dashed", color="red", weight=0];
31.40/16.00	770[label="primDivNatS (primMinusNatS (Succ ww74) (Succ ww75)) (Succ (Succ ww75))",fontsize=16,color="magenta"];770 -> 891[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	770 -> 892[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	770 -> 893[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	915 -> 884[label="",style="dashed", color="red", weight=0];
31.40/16.00	915[label="primDivNatS (primMinusNatS ww880 ww890) (Succ ww90)",fontsize=16,color="magenta"];915 -> 919[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	915 -> 920[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	916 -> 103[label="",style="dashed", color="red", weight=0];
31.40/16.00	916[label="primDivNatS (Succ ww880) (Succ ww90)",fontsize=16,color="magenta"];916 -> 921[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	916 -> 922[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	917[label="primDivNatS Zero (Succ ww90)",fontsize=16,color="black",shape="triangle"];917 -> 923[label="",style="solid", color="black", weight=3];
31.40/16.00	918 -> 917[label="",style="dashed", color="red", weight=0];
31.40/16.00	918[label="primDivNatS Zero (Succ ww90)",fontsize=16,color="magenta"];775[label="ww810",fontsize=16,color="green",shape="box"];776[label="ww820",fontsize=16,color="green",shape="box"];777 -> 788[label="",style="dashed", color="red", weight=0];
31.40/16.00	777[label="primModNatS (primMinusNatS (Succ ww79) (Succ ww80)) (Succ (Succ ww80))",fontsize=16,color="magenta"];777 -> 801[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	777 -> 802[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	777 -> 803[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	778[label="Succ (Succ ww79)",fontsize=16,color="green",shape="box"];833 -> 788[label="",style="dashed", color="red", weight=0];
31.40/16.00	833[label="primModNatS (primMinusNatS ww840 ww850) (Succ ww86)",fontsize=16,color="magenta"];833 -> 843[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	833 -> 844[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	834 -> 109[label="",style="dashed", color="red", weight=0];
31.40/16.00	834[label="primModNatS (Succ ww840) (Succ ww86)",fontsize=16,color="magenta"];834 -> 845[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	834 -> 846[label="",style="dashed", color="magenta", weight=3];
31.40/16.00	835[label="primModNatS Zero (Succ ww86)",fontsize=16,color="black",shape="triangle"];835 -> 847[label="",style="solid", color="black", weight=3];
31.40/16.00	836 -> 835[label="",style="dashed", color="red", weight=0];
31.40/16.00	836[label="primModNatS Zero (Succ ww86)",fontsize=16,color="magenta"];891[label="Succ ww74",fontsize=16,color="green",shape="box"];892[label="Succ ww75",fontsize=16,color="green",shape="box"];893[label="Succ ww75",fontsize=16,color="green",shape="box"];919[label="ww880",fontsize=16,color="green",shape="box"];920[label="ww890",fontsize=16,color="green",shape="box"];921[label="ww880",fontsize=16,color="green",shape="box"];922[label="ww90",fontsize=16,color="green",shape="box"];923[label="Zero",fontsize=16,color="green",shape="box"];801[label="Succ ww79",fontsize=16,color="green",shape="box"];802[label="Succ ww80",fontsize=16,color="green",shape="box"];803[label="Succ ww80",fontsize=16,color="green",shape="box"];843[label="ww840",fontsize=16,color="green",shape="box"];844[label="ww850",fontsize=16,color="green",shape="box"];845[label="ww86",fontsize=16,color="green",shape="box"];846[label="ww840",fontsize=16,color="green",shape="box"];847[label="Zero",fontsize=16,color="green",shape="box"];}
31.40/16.00	
31.40/16.00	----------------------------------------
31.40/16.00	
31.40/16.00	(66)
31.40/16.00	TRUE
31.49/16.10	EOF