9.13/3.87	YES
10.99/4.37	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
10.99/4.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.99/4.37	
10.99/4.37	
10.99/4.37	H-Termination with start terms of the given HASKELL could be proven:
10.99/4.37	
10.99/4.37	(0) HASKELL
10.99/4.37	(1) IFR [EQUIVALENT, 0 ms]
10.99/4.37	(2) HASKELL
10.99/4.37	(3) BR [EQUIVALENT, 0 ms]
10.99/4.37	(4) HASKELL
10.99/4.37	(5) COR [EQUIVALENT, 0 ms]
10.99/4.37	(6) HASKELL
10.99/4.37	(7) Narrow [SOUND, 0 ms]
10.99/4.37	(8) AND
10.99/4.37	    (9) QDP
10.99/4.37	        (10) DependencyGraphProof [EQUIVALENT, 0 ms]
10.99/4.37	        (11) AND
10.99/4.37	            (12) QDP
10.99/4.37	                (13) MRRProof [EQUIVALENT, 0 ms]
10.99/4.37	                (14) QDP
10.99/4.37	                (15) PisEmptyProof [EQUIVALENT, 0 ms]
10.99/4.37	                (16) YES
10.99/4.37	            (17) QDP
10.99/4.37	                (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.99/4.37	                (19) YES
10.99/4.37	    (20) QDP
10.99/4.37	        (21) DependencyGraphProof [EQUIVALENT, 0 ms]
10.99/4.37	        (22) AND
10.99/4.37	            (23) QDP
10.99/4.37	                (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.99/4.37	                (25) YES
10.99/4.37	            (26) QDP
10.99/4.37	                (27) MRRProof [EQUIVALENT, 0 ms]
10.99/4.37	                (28) QDP
10.99/4.37	                (29) PisEmptyProof [EQUIVALENT, 0 ms]
10.99/4.37	                (30) YES
10.99/4.37	    (31) QDP
10.99/4.37	        (32) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.99/4.37	        (33) YES
10.99/4.37	
10.99/4.37	
10.99/4.37	----------------------------------------
10.99/4.37	
10.99/4.37	(0)
10.99/4.37	Obligation:
10.99/4.37	mainModule Main 
10.99/4.37	module Main where {
10.99/4.37	import qualified Prelude;
10.99/4.37	}
10.99/4.37	
10.99/4.37	----------------------------------------
10.99/4.37	
10.99/4.37	(1) IFR (EQUIVALENT)
10.99/4.37	If Reductions:
10.99/4.37	The following If expression
10.99/4.37	"if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x"
10.99/4.37	is transformed to
10.99/4.37	"primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y);
10.99/4.37	primModNatS0 x y False = Succ x;
10.99/4.37	"
10.99/4.37	The following If expression
10.99/4.37	"if primGEqNatS x y then primModNatP (primMinusNatS x y) (Succ y) else primMinusNatS y x"
10.99/4.37	is transformed to
10.99/4.37	"primModNatP0 x y True = primModNatP (primMinusNatS x y) (Succ y);
10.99/4.37	primModNatP0 x y False = primMinusNatS y x;
10.99/4.37	"
10.99/4.37	
10.99/4.37	----------------------------------------
10.99/4.37	
10.99/4.37	(2)
10.99/4.37	Obligation:
10.99/4.37	mainModule Main 
10.99/4.37	module Main where {
10.99/4.37	import qualified Prelude;
10.99/4.37	}
10.99/4.37	
10.99/4.37	----------------------------------------
10.99/4.37	
10.99/4.37	(3) BR (EQUIVALENT)
10.99/4.37	Replaced joker patterns by fresh variables and removed binding patterns.
10.99/4.37	----------------------------------------
10.99/4.37	
10.99/4.37	(4)
10.99/4.37	Obligation:
10.99/4.37	mainModule Main 
10.99/4.37	module Main where {
10.99/4.37	import qualified Prelude;
10.99/4.37	}
10.99/4.37	
10.99/4.37	----------------------------------------
10.99/4.37	
10.99/4.37	(5) COR (EQUIVALENT)
10.99/4.37	Cond Reductions:
10.99/4.37	The following Function with conditions
10.99/4.37	"undefined |Falseundefined;
10.99/4.37	"
10.99/4.37	is transformed to
10.99/4.37	"undefined  = undefined1;
10.99/4.37	"
10.99/4.37	"undefined0 True = undefined;
10.99/4.37	"
10.99/4.37	"undefined1  = undefined0 False;
10.99/4.37	"
10.99/4.37	
10.99/4.37	----------------------------------------
10.99/4.37	
10.99/4.37	(6)
10.99/4.37	Obligation:
10.99/4.37	mainModule Main 
10.99/4.37	module Main where {
10.99/4.37	import qualified Prelude;
10.99/4.37	}
10.99/4.37	
10.99/4.37	----------------------------------------
10.99/4.37	
10.99/4.37	(7) Narrow (SOUND)
10.99/4.37	Haskell To QDPs
10.99/4.37	
10.99/4.37	digraph dp_graph {
10.99/4.37	node [outthreshold=100, inthreshold=100];1[label="mod",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.99/4.37	3[label="mod vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.99/4.37	4[label="mod vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.99/4.37	5[label="primModInt vz3 vz4",fontsize=16,color="burlywood",shape="box"];679[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 679[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	679 -> 6[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	680[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 680[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	680 -> 7[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	6[label="primModInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];681[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 681[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	681 -> 8[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	682[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 682[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	682 -> 9[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	7[label="primModInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];683[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 683[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	683 -> 10[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	684[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 684[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	684 -> 11[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	8[label="primModInt (Pos vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];685[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 685[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	685 -> 12[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	686[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 686[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	686 -> 13[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	9[label="primModInt (Pos vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];687[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 687[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	687 -> 14[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	688[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 688[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	688 -> 15[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	10[label="primModInt (Neg vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];689[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 689[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	689 -> 16[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	690[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 690[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	690 -> 17[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	11[label="primModInt (Neg vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];691[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 691[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	691 -> 18[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	692[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 692[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	692 -> 19[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	12[label="primModInt (Pos vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
10.99/4.37	13[label="primModInt (Pos vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
10.99/4.37	14[label="primModInt (Pos vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
10.99/4.37	15[label="primModInt (Pos vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
10.99/4.37	16[label="primModInt (Neg vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
10.99/4.37	17[label="primModInt (Neg vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
10.99/4.37	18[label="primModInt (Neg vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
10.99/4.37	19[label="primModInt (Neg vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
10.99/4.37	20[label="Pos (primModNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3];
10.99/4.37	21[label="error []",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3];
10.99/4.37	22[label="Neg (primModNatP vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];22 -> 30[label="",style="dashed", color="green", weight=3];
10.99/4.37	23 -> 21[label="",style="dashed", color="red", weight=0];
10.99/4.37	23[label="error []",fontsize=16,color="magenta"];24[label="Pos (primModNatP vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];24 -> 31[label="",style="dashed", color="green", weight=3];
10.99/4.37	25 -> 21[label="",style="dashed", color="red", weight=0];
10.99/4.37	25[label="error []",fontsize=16,color="magenta"];26[label="Neg (primModNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];26 -> 32[label="",style="dashed", color="green", weight=3];
10.99/4.37	27 -> 21[label="",style="dashed", color="red", weight=0];
10.99/4.37	27[label="error []",fontsize=16,color="magenta"];28[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];693[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 693[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	693 -> 33[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	694[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 694[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	694 -> 34[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	29[label="error []",fontsize=16,color="red",shape="box"];30[label="primModNatP vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];695[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];30 -> 695[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	695 -> 35[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	696[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 696[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	696 -> 36[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	31 -> 30[label="",style="dashed", color="red", weight=0];
10.99/4.37	31[label="primModNatP vz30 (Succ vz400)",fontsize=16,color="magenta"];31 -> 37[label="",style="dashed", color="magenta", weight=3];
10.99/4.37	31 -> 38[label="",style="dashed", color="magenta", weight=3];
10.99/4.37	32 -> 28[label="",style="dashed", color="red", weight=0];
10.99/4.37	32[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];32 -> 39[label="",style="dashed", color="magenta", weight=3];
10.99/4.37	32 -> 40[label="",style="dashed", color="magenta", weight=3];
10.99/4.37	33[label="primModNatS (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];33 -> 41[label="",style="solid", color="black", weight=3];
10.99/4.37	34[label="primModNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 42[label="",style="solid", color="black", weight=3];
10.99/4.37	35[label="primModNatP (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];35 -> 43[label="",style="solid", color="black", weight=3];
10.99/4.37	36[label="primModNatP Zero (Succ vz400)",fontsize=16,color="black",shape="box"];36 -> 44[label="",style="solid", color="black", weight=3];
10.99/4.37	37[label="vz30",fontsize=16,color="green",shape="box"];38[label="vz400",fontsize=16,color="green",shape="box"];39[label="vz30",fontsize=16,color="green",shape="box"];40[label="vz400",fontsize=16,color="green",shape="box"];41[label="primModNatS0 vz300 vz400 (primGEqNatS vz300 vz400)",fontsize=16,color="burlywood",shape="box"];697[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];41 -> 697[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	697 -> 45[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	698[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 698[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	698 -> 46[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	42[label="Zero",fontsize=16,color="green",shape="box"];43[label="primModNatP0 vz300 vz400 (primGEqNatS vz300 vz400)",fontsize=16,color="burlywood",shape="box"];699[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];43 -> 699[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	699 -> 47[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	700[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];43 -> 700[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	700 -> 48[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	44[label="Zero",fontsize=16,color="green",shape="box"];45[label="primModNatS0 (Succ vz3000) vz400 (primGEqNatS (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];701[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];45 -> 701[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	701 -> 49[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	702[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 702[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	702 -> 50[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	46[label="primModNatS0 Zero vz400 (primGEqNatS Zero vz400)",fontsize=16,color="burlywood",shape="box"];703[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];46 -> 703[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	703 -> 51[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	704[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 704[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	704 -> 52[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	47[label="primModNatP0 (Succ vz3000) vz400 (primGEqNatS (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];705[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];47 -> 705[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	705 -> 53[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	706[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 706[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	706 -> 54[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	48[label="primModNatP0 Zero vz400 (primGEqNatS Zero vz400)",fontsize=16,color="burlywood",shape="box"];707[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];48 -> 707[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	707 -> 55[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	708[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 708[label="",style="solid", color="burlywood", weight=9];
10.99/4.37	708 -> 56[label="",style="solid", color="burlywood", weight=3];
10.99/4.37	49[label="primModNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];49 -> 57[label="",style="solid", color="black", weight=3];
10.99/4.37	50[label="primModNatS0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];50 -> 58[label="",style="solid", color="black", weight=3];
10.99/4.37	51[label="primModNatS0 Zero (Succ vz4000) (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];51 -> 59[label="",style="solid", color="black", weight=3];
10.99/4.37	52[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];52 -> 60[label="",style="solid", color="black", weight=3];
10.99/4.37	53[label="primModNatP0 (Succ vz3000) (Succ vz4000) (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];53 -> 61[label="",style="solid", color="black", weight=3];
10.99/4.37	54[label="primModNatP0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];54 -> 62[label="",style="solid", color="black", weight=3];
10.99/4.38	55[label="primModNatP0 Zero (Succ vz4000) (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];55 -> 63[label="",style="solid", color="black", weight=3];
10.99/4.38	56[label="primModNatP0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];56 -> 64[label="",style="solid", color="black", weight=3];
10.99/4.38	57 -> 548[label="",style="dashed", color="red", weight=0];
10.99/4.38	57[label="primModNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)",fontsize=16,color="magenta"];57 -> 549[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	57 -> 550[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	57 -> 551[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	57 -> 552[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	58[label="primModNatS0 (Succ vz3000) Zero True",fontsize=16,color="black",shape="box"];58 -> 67[label="",style="solid", color="black", weight=3];
10.99/4.38	59[label="primModNatS0 Zero (Succ vz4000) False",fontsize=16,color="black",shape="box"];59 -> 68[label="",style="solid", color="black", weight=3];
10.99/4.38	60[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];60 -> 69[label="",style="solid", color="black", weight=3];
10.99/4.38	61 -> 591[label="",style="dashed", color="red", weight=0];
10.99/4.38	61[label="primModNatP0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)",fontsize=16,color="magenta"];61 -> 592[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	61 -> 593[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	61 -> 594[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	61 -> 595[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	62[label="primModNatP0 (Succ vz3000) Zero True",fontsize=16,color="black",shape="box"];62 -> 72[label="",style="solid", color="black", weight=3];
10.99/4.38	63[label="primModNatP0 Zero (Succ vz4000) False",fontsize=16,color="black",shape="box"];63 -> 73[label="",style="solid", color="black", weight=3];
10.99/4.38	64[label="primModNatP0 Zero Zero True",fontsize=16,color="black",shape="box"];64 -> 74[label="",style="solid", color="black", weight=3];
10.99/4.38	549[label="vz4000",fontsize=16,color="green",shape="box"];550[label="vz4000",fontsize=16,color="green",shape="box"];551[label="vz3000",fontsize=16,color="green",shape="box"];552[label="vz3000",fontsize=16,color="green",shape="box"];548[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS vz52 vz53)",fontsize=16,color="burlywood",shape="triangle"];709[label="vz52/Succ vz520",fontsize=10,color="white",style="solid",shape="box"];548 -> 709[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	709 -> 589[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	710[label="vz52/Zero",fontsize=10,color="white",style="solid",shape="box"];548 -> 710[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	710 -> 590[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	67 -> 28[label="",style="dashed", color="red", weight=0];
10.99/4.38	67[label="primModNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];67 -> 79[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	67 -> 80[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	68[label="Succ Zero",fontsize=16,color="green",shape="box"];69 -> 28[label="",style="dashed", color="red", weight=0];
10.99/4.38	69[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];69 -> 81[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	69 -> 82[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	592[label="vz4000",fontsize=16,color="green",shape="box"];593[label="vz3000",fontsize=16,color="green",shape="box"];594[label="vz3000",fontsize=16,color="green",shape="box"];595[label="vz4000",fontsize=16,color="green",shape="box"];591[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS vz57 vz58)",fontsize=16,color="burlywood",shape="triangle"];711[label="vz57/Succ vz570",fontsize=10,color="white",style="solid",shape="box"];591 -> 711[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	711 -> 632[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	712[label="vz57/Zero",fontsize=10,color="white",style="solid",shape="box"];591 -> 712[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	712 -> 633[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	72 -> 30[label="",style="dashed", color="red", weight=0];
10.99/4.38	72[label="primModNatP (primMinusNatS (Succ vz3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];72 -> 87[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	72 -> 88[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	73[label="primMinusNatS (Succ vz4000) Zero",fontsize=16,color="black",shape="triangle"];73 -> 89[label="",style="solid", color="black", weight=3];
10.99/4.38	74 -> 30[label="",style="dashed", color="red", weight=0];
10.99/4.38	74[label="primModNatP (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];74 -> 90[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	74 -> 91[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	589[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS (Succ vz520) vz53)",fontsize=16,color="burlywood",shape="box"];713[label="vz53/Succ vz530",fontsize=10,color="white",style="solid",shape="box"];589 -> 713[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	713 -> 634[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	714[label="vz53/Zero",fontsize=10,color="white",style="solid",shape="box"];589 -> 714[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	714 -> 635[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	590[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS Zero vz53)",fontsize=16,color="burlywood",shape="box"];715[label="vz53/Succ vz530",fontsize=10,color="white",style="solid",shape="box"];590 -> 715[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	715 -> 636[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	716[label="vz53/Zero",fontsize=10,color="white",style="solid",shape="box"];590 -> 716[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	716 -> 637[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	79 -> 73[label="",style="dashed", color="red", weight=0];
10.99/4.38	79[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="magenta"];79 -> 96[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	80[label="Zero",fontsize=16,color="green",shape="box"];81[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];81 -> 97[label="",style="solid", color="black", weight=3];
10.99/4.38	82[label="Zero",fontsize=16,color="green",shape="box"];632[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS (Succ vz570) vz58)",fontsize=16,color="burlywood",shape="box"];717[label="vz58/Succ vz580",fontsize=10,color="white",style="solid",shape="box"];632 -> 717[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	717 -> 638[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	718[label="vz58/Zero",fontsize=10,color="white",style="solid",shape="box"];632 -> 718[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	718 -> 639[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	633[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS Zero vz58)",fontsize=16,color="burlywood",shape="box"];719[label="vz58/Succ vz580",fontsize=10,color="white",style="solid",shape="box"];633 -> 719[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	719 -> 640[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	720[label="vz58/Zero",fontsize=10,color="white",style="solid",shape="box"];633 -> 720[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	720 -> 641[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	87 -> 73[label="",style="dashed", color="red", weight=0];
10.99/4.38	87[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="magenta"];87 -> 102[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	88[label="Zero",fontsize=16,color="green",shape="box"];89[label="Succ vz4000",fontsize=16,color="green",shape="box"];90 -> 81[label="",style="dashed", color="red", weight=0];
10.99/4.38	90[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];91[label="Zero",fontsize=16,color="green",shape="box"];634[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS (Succ vz520) (Succ vz530))",fontsize=16,color="black",shape="box"];634 -> 642[label="",style="solid", color="black", weight=3];
10.99/4.38	635[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS (Succ vz520) Zero)",fontsize=16,color="black",shape="box"];635 -> 643[label="",style="solid", color="black", weight=3];
10.99/4.38	636[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS Zero (Succ vz530))",fontsize=16,color="black",shape="box"];636 -> 644[label="",style="solid", color="black", weight=3];
10.99/4.38	637[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];637 -> 645[label="",style="solid", color="black", weight=3];
10.99/4.38	96[label="vz3000",fontsize=16,color="green",shape="box"];97[label="Zero",fontsize=16,color="green",shape="box"];638[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS (Succ vz570) (Succ vz580))",fontsize=16,color="black",shape="box"];638 -> 646[label="",style="solid", color="black", weight=3];
10.99/4.38	639[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS (Succ vz570) Zero)",fontsize=16,color="black",shape="box"];639 -> 647[label="",style="solid", color="black", weight=3];
10.99/4.38	640[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS Zero (Succ vz580))",fontsize=16,color="black",shape="box"];640 -> 648[label="",style="solid", color="black", weight=3];
10.99/4.38	641[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];641 -> 649[label="",style="solid", color="black", weight=3];
10.99/4.38	102[label="vz3000",fontsize=16,color="green",shape="box"];642 -> 548[label="",style="dashed", color="red", weight=0];
10.99/4.38	642[label="primModNatS0 (Succ vz50) (Succ vz51) (primGEqNatS vz520 vz530)",fontsize=16,color="magenta"];642 -> 650[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	642 -> 651[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	643[label="primModNatS0 (Succ vz50) (Succ vz51) True",fontsize=16,color="black",shape="triangle"];643 -> 652[label="",style="solid", color="black", weight=3];
10.99/4.38	644[label="primModNatS0 (Succ vz50) (Succ vz51) False",fontsize=16,color="black",shape="box"];644 -> 653[label="",style="solid", color="black", weight=3];
10.99/4.38	645 -> 643[label="",style="dashed", color="red", weight=0];
10.99/4.38	645[label="primModNatS0 (Succ vz50) (Succ vz51) True",fontsize=16,color="magenta"];646 -> 591[label="",style="dashed", color="red", weight=0];
10.99/4.38	646[label="primModNatP0 (Succ vz55) (Succ vz56) (primGEqNatS vz570 vz580)",fontsize=16,color="magenta"];646 -> 654[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	646 -> 655[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	647[label="primModNatP0 (Succ vz55) (Succ vz56) True",fontsize=16,color="black",shape="triangle"];647 -> 656[label="",style="solid", color="black", weight=3];
10.99/4.38	648[label="primModNatP0 (Succ vz55) (Succ vz56) False",fontsize=16,color="black",shape="box"];648 -> 657[label="",style="solid", color="black", weight=3];
10.99/4.38	649 -> 647[label="",style="dashed", color="red", weight=0];
10.99/4.38	649[label="primModNatP0 (Succ vz55) (Succ vz56) True",fontsize=16,color="magenta"];650[label="vz530",fontsize=16,color="green",shape="box"];651[label="vz520",fontsize=16,color="green",shape="box"];652 -> 28[label="",style="dashed", color="red", weight=0];
10.99/4.38	652[label="primModNatS (primMinusNatS (Succ vz50) (Succ vz51)) (Succ (Succ vz51))",fontsize=16,color="magenta"];652 -> 658[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	652 -> 659[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	653[label="Succ (Succ vz50)",fontsize=16,color="green",shape="box"];654[label="vz570",fontsize=16,color="green",shape="box"];655[label="vz580",fontsize=16,color="green",shape="box"];656 -> 30[label="",style="dashed", color="red", weight=0];
10.99/4.38	656[label="primModNatP (primMinusNatS (Succ vz55) (Succ vz56)) (Succ (Succ vz56))",fontsize=16,color="magenta"];656 -> 660[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	656 -> 661[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	657[label="primMinusNatS (Succ vz56) (Succ vz55)",fontsize=16,color="black",shape="triangle"];657 -> 662[label="",style="solid", color="black", weight=3];
10.99/4.38	658 -> 657[label="",style="dashed", color="red", weight=0];
10.99/4.38	658[label="primMinusNatS (Succ vz50) (Succ vz51)",fontsize=16,color="magenta"];658 -> 663[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	658 -> 664[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	659[label="Succ vz51",fontsize=16,color="green",shape="box"];660 -> 657[label="",style="dashed", color="red", weight=0];
10.99/4.38	660[label="primMinusNatS (Succ vz55) (Succ vz56)",fontsize=16,color="magenta"];660 -> 665[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	660 -> 666[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	661[label="Succ vz56",fontsize=16,color="green",shape="box"];662[label="primMinusNatS vz56 vz55",fontsize=16,color="burlywood",shape="triangle"];721[label="vz56/Succ vz560",fontsize=10,color="white",style="solid",shape="box"];662 -> 721[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	721 -> 667[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	722[label="vz56/Zero",fontsize=10,color="white",style="solid",shape="box"];662 -> 722[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	722 -> 668[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	663[label="vz50",fontsize=16,color="green",shape="box"];664[label="vz51",fontsize=16,color="green",shape="box"];665[label="vz55",fontsize=16,color="green",shape="box"];666[label="vz56",fontsize=16,color="green",shape="box"];667[label="primMinusNatS (Succ vz560) vz55",fontsize=16,color="burlywood",shape="box"];723[label="vz55/Succ vz550",fontsize=10,color="white",style="solid",shape="box"];667 -> 723[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	723 -> 669[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	724[label="vz55/Zero",fontsize=10,color="white",style="solid",shape="box"];667 -> 724[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	724 -> 670[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	668[label="primMinusNatS Zero vz55",fontsize=16,color="burlywood",shape="box"];725[label="vz55/Succ vz550",fontsize=10,color="white",style="solid",shape="box"];668 -> 725[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	725 -> 671[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	726[label="vz55/Zero",fontsize=10,color="white",style="solid",shape="box"];668 -> 726[label="",style="solid", color="burlywood", weight=9];
10.99/4.38	726 -> 672[label="",style="solid", color="burlywood", weight=3];
10.99/4.38	669[label="primMinusNatS (Succ vz560) (Succ vz550)",fontsize=16,color="black",shape="box"];669 -> 673[label="",style="solid", color="black", weight=3];
10.99/4.38	670[label="primMinusNatS (Succ vz560) Zero",fontsize=16,color="black",shape="box"];670 -> 674[label="",style="solid", color="black", weight=3];
10.99/4.38	671[label="primMinusNatS Zero (Succ vz550)",fontsize=16,color="black",shape="box"];671 -> 675[label="",style="solid", color="black", weight=3];
10.99/4.38	672[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];672 -> 676[label="",style="solid", color="black", weight=3];
10.99/4.38	673 -> 662[label="",style="dashed", color="red", weight=0];
10.99/4.38	673[label="primMinusNatS vz560 vz550",fontsize=16,color="magenta"];673 -> 677[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	673 -> 678[label="",style="dashed", color="magenta", weight=3];
10.99/4.38	674[label="Succ vz560",fontsize=16,color="green",shape="box"];675[label="Zero",fontsize=16,color="green",shape="box"];676[label="Zero",fontsize=16,color="green",shape="box"];677[label="vz560",fontsize=16,color="green",shape="box"];678[label="vz550",fontsize=16,color="green",shape="box"];}
10.99/4.38	
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(8)
10.99/4.38	Complex Obligation (AND)
10.99/4.38	
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(9)
10.99/4.38	Obligation:
10.99/4.38	Q DP problem:
10.99/4.38	The TRS P consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primModNatS00(vz50, vz51) -> new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
10.99/4.38	   new_primModNatS(Succ(Succ(vz3000)), Zero) -> new_primModNatS(new_primMinusNatS0(vz3000), Zero)
10.99/4.38	   new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
10.99/4.38	   new_primModNatS(Succ(Zero), Zero) -> new_primModNatS(new_primMinusNatS1, Zero)
10.99/4.38	   new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) -> new_primModNatS0(vz50, vz51, vz520, vz530)
10.99/4.38	   new_primModNatS0(vz50, vz51, Succ(vz520), Zero) -> new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
10.99/4.38	   new_primModNatS0(vz50, vz51, Zero, Zero) -> new_primModNatS00(vz50, vz51)
10.99/4.38	
10.99/4.38	The TRS R consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primMinusNatS1 -> Zero
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
10.99/4.38	   new_primMinusNatS3(Zero, Zero) -> Zero
10.99/4.38	   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
10.99/4.38	   new_primMinusNatS0(vz4000) -> Succ(vz4000)
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
10.99/4.38	   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero
10.99/4.38	
10.99/4.38	The set Q consists of the following terms:
10.99/4.38	
10.99/4.38	   new_primMinusNatS3(Zero, Succ(x0))
10.99/4.38	   new_primMinusNatS3(Succ(x0), Zero)
10.99/4.38	   new_primMinusNatS3(Zero, Zero)
10.99/4.38	   new_primMinusNatS0(x0)
10.99/4.38	   new_primMinusNatS3(Succ(x0), Succ(x1))
10.99/4.38	   new_primMinusNatS1
10.99/4.38	   new_primMinusNatS2(x0, x1)
10.99/4.38	
10.99/4.38	We have to consider all minimal (P,Q,R)-chains.
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(10) DependencyGraphProof (EQUIVALENT)
10.99/4.38	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(11)
10.99/4.38	Complex Obligation (AND)
10.99/4.38	
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(12)
10.99/4.38	Obligation:
10.99/4.38	Q DP problem:
10.99/4.38	The TRS P consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primModNatS(Succ(Succ(vz3000)), Zero) -> new_primModNatS(new_primMinusNatS0(vz3000), Zero)
10.99/4.38	
10.99/4.38	The TRS R consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primMinusNatS1 -> Zero
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
10.99/4.38	   new_primMinusNatS3(Zero, Zero) -> Zero
10.99/4.38	   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
10.99/4.38	   new_primMinusNatS0(vz4000) -> Succ(vz4000)
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
10.99/4.38	   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero
10.99/4.38	
10.99/4.38	The set Q consists of the following terms:
10.99/4.38	
10.99/4.38	   new_primMinusNatS3(Zero, Succ(x0))
10.99/4.38	   new_primMinusNatS3(Succ(x0), Zero)
10.99/4.38	   new_primMinusNatS3(Zero, Zero)
10.99/4.38	   new_primMinusNatS0(x0)
10.99/4.38	   new_primMinusNatS3(Succ(x0), Succ(x1))
10.99/4.38	   new_primMinusNatS1
10.99/4.38	   new_primMinusNatS2(x0, x1)
10.99/4.38	
10.99/4.38	We have to consider all minimal (P,Q,R)-chains.
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(13) MRRProof (EQUIVALENT)
10.99/4.38	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
10.99/4.38	
10.99/4.38	Strictly oriented dependency pairs:
10.99/4.38	
10.99/4.38	   new_primModNatS(Succ(Succ(vz3000)), Zero) -> new_primModNatS(new_primMinusNatS0(vz3000), Zero)
10.99/4.38	
10.99/4.38	Strictly oriented rules of the TRS R:
10.99/4.38	
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
10.99/4.38	   new_primMinusNatS3(Zero, Zero) -> Zero
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
10.99/4.38	   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero
10.99/4.38	
10.99/4.38	Used ordering: Polynomial interpretation [POLO]:
10.99/4.38	
10.99/4.38	   POL(Succ(x_1)) = 1 + x_1
10.99/4.38	   POL(Zero) = 2
10.99/4.38	   POL(new_primMinusNatS0(x_1)) = 1 + x_1
10.99/4.38	   POL(new_primMinusNatS1) = 2
10.99/4.38	   POL(new_primMinusNatS2(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
10.99/4.38	   POL(new_primMinusNatS3(x_1, x_2)) = 1 + 2*x_1 + x_2
10.99/4.38	   POL(new_primModNatS(x_1, x_2)) = x_1 + x_2
10.99/4.38	
10.99/4.38	
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(14)
10.99/4.38	Obligation:
10.99/4.38	Q DP problem:
10.99/4.38	P is empty.
10.99/4.38	The TRS R consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primMinusNatS1 -> Zero
10.99/4.38	   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
10.99/4.38	   new_primMinusNatS0(vz4000) -> Succ(vz4000)
10.99/4.38	
10.99/4.38	The set Q consists of the following terms:
10.99/4.38	
10.99/4.38	   new_primMinusNatS3(Zero, Succ(x0))
10.99/4.38	   new_primMinusNatS3(Succ(x0), Zero)
10.99/4.38	   new_primMinusNatS3(Zero, Zero)
10.99/4.38	   new_primMinusNatS0(x0)
10.99/4.38	   new_primMinusNatS3(Succ(x0), Succ(x1))
10.99/4.38	   new_primMinusNatS1
10.99/4.38	   new_primMinusNatS2(x0, x1)
10.99/4.38	
10.99/4.38	We have to consider all minimal (P,Q,R)-chains.
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(15) PisEmptyProof (EQUIVALENT)
10.99/4.38	The TRS P is empty. Hence, there is no (P,Q,R) chain.
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(16)
10.99/4.38	YES
10.99/4.38	
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(17)
10.99/4.38	Obligation:
10.99/4.38	Q DP problem:
10.99/4.38	The TRS P consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
10.99/4.38	   new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) -> new_primModNatS0(vz50, vz51, vz520, vz530)
10.99/4.38	   new_primModNatS0(vz50, vz51, Succ(vz520), Zero) -> new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
10.99/4.38	   new_primModNatS0(vz50, vz51, Zero, Zero) -> new_primModNatS00(vz50, vz51)
10.99/4.38	   new_primModNatS00(vz50, vz51) -> new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
10.99/4.38	
10.99/4.38	The TRS R consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primMinusNatS1 -> Zero
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
10.99/4.38	   new_primMinusNatS3(Zero, Zero) -> Zero
10.99/4.38	   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
10.99/4.38	   new_primMinusNatS0(vz4000) -> Succ(vz4000)
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
10.99/4.38	   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero
10.99/4.38	
10.99/4.38	The set Q consists of the following terms:
10.99/4.38	
10.99/4.38	   new_primMinusNatS3(Zero, Succ(x0))
10.99/4.38	   new_primMinusNatS3(Succ(x0), Zero)
10.99/4.38	   new_primMinusNatS3(Zero, Zero)
10.99/4.38	   new_primMinusNatS0(x0)
10.99/4.38	   new_primMinusNatS3(Succ(x0), Succ(x1))
10.99/4.38	   new_primMinusNatS1
10.99/4.38	   new_primMinusNatS2(x0, x1)
10.99/4.38	
10.99/4.38	We have to consider all minimal (P,Q,R)-chains.
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(18) QDPSizeChangeProof (EQUIVALENT)
10.99/4.38	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
10.99/4.38	
10.99/4.38	Order:Polynomial interpretation [POLO]:
10.99/4.38	
10.99/4.38	   POL(Succ(x_1)) = 1 + x_1
10.99/4.38	   POL(Zero) = 1
10.99/4.38	   POL(new_primMinusNatS2(x_1, x_2)) = x_1
10.99/4.38	   POL(new_primMinusNatS3(x_1, x_2)) = x_1
10.99/4.38	
10.99/4.38	
10.99/4.38	
10.99/4.38	
10.99/4.38	From the DPs we obtained the following set of size-change graphs:
10.99/4.38	*new_primModNatS0(vz50, vz51, Succ(vz520), Zero) -> new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51)) (allowed arguments on rhs = {1, 2})
10.99/4.38	The graph contains the following edges 1 >= 1
10.99/4.38	
10.99/4.38	
10.99/4.38	*new_primModNatS00(vz50, vz51) -> new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51)) (allowed arguments on rhs = {1, 2})
10.99/4.38	The graph contains the following edges 1 >= 1
10.99/4.38	
10.99/4.38	
10.99/4.38	*new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatS0(vz3000, vz4000, vz3000, vz4000) (allowed arguments on rhs = {1, 2, 3, 4})
10.99/4.38	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
10.99/4.38	
10.99/4.38	
10.99/4.38	*new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) -> new_primModNatS0(vz50, vz51, vz520, vz530) (allowed arguments on rhs = {1, 2, 3, 4})
10.99/4.38	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
10.99/4.38	
10.99/4.38	
10.99/4.38	*new_primModNatS0(vz50, vz51, Zero, Zero) -> new_primModNatS00(vz50, vz51) (allowed arguments on rhs = {1, 2})
10.99/4.38	The graph contains the following edges 1 >= 1, 2 >= 2
10.99/4.38	
10.99/4.38	
10.99/4.38	
10.99/4.38	We oriented the following set of usable rules [AAECC05,FROCOS05].
10.99/4.38	
10.99/4.38	   new_primMinusNatS3(Zero, Zero) -> Zero
10.99/4.38	   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
10.99/4.38	   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
10.99/4.38	
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(19)
10.99/4.38	YES
10.99/4.38	
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(20)
10.99/4.38	Obligation:
10.99/4.38	Q DP problem:
10.99/4.38	The TRS P consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primModNatP(Succ(Succ(vz3000)), Zero) -> new_primModNatP(new_primMinusNatS0(vz3000), Zero)
10.99/4.38	   new_primModNatP00(vz55, vz56) -> new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
10.99/4.38	   new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatP0(vz3000, vz4000, vz3000, vz4000)
10.99/4.38	   new_primModNatP0(vz55, vz56, Succ(vz570), Zero) -> new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
10.99/4.38	   new_primModNatP(Succ(Zero), Zero) -> new_primModNatP(new_primMinusNatS1, Zero)
10.99/4.38	   new_primModNatP0(vz55, vz56, Zero, Zero) -> new_primModNatP00(vz55, vz56)
10.99/4.38	   new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) -> new_primModNatP0(vz55, vz56, vz570, vz580)
10.99/4.38	
10.99/4.38	The TRS R consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primMinusNatS1 -> Zero
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
10.99/4.38	   new_primMinusNatS3(Zero, Zero) -> Zero
10.99/4.38	   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
10.99/4.38	   new_primMinusNatS0(vz4000) -> Succ(vz4000)
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
10.99/4.38	   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero
10.99/4.38	
10.99/4.38	The set Q consists of the following terms:
10.99/4.38	
10.99/4.38	   new_primMinusNatS3(Zero, Succ(x0))
10.99/4.38	   new_primMinusNatS3(Succ(x0), Zero)
10.99/4.38	   new_primMinusNatS3(Zero, Zero)
10.99/4.38	   new_primMinusNatS0(x0)
10.99/4.38	   new_primMinusNatS3(Succ(x0), Succ(x1))
10.99/4.38	   new_primMinusNatS1
10.99/4.38	   new_primMinusNatS2(x0, x1)
10.99/4.38	
10.99/4.38	We have to consider all minimal (P,Q,R)-chains.
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(21) DependencyGraphProof (EQUIVALENT)
10.99/4.38	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(22)
10.99/4.38	Complex Obligation (AND)
10.99/4.38	
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(23)
10.99/4.38	Obligation:
10.99/4.38	Q DP problem:
10.99/4.38	The TRS P consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatP0(vz3000, vz4000, vz3000, vz4000)
10.99/4.38	   new_primModNatP0(vz55, vz56, Succ(vz570), Zero) -> new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
10.99/4.38	   new_primModNatP0(vz55, vz56, Zero, Zero) -> new_primModNatP00(vz55, vz56)
10.99/4.38	   new_primModNatP00(vz55, vz56) -> new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
10.99/4.38	   new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) -> new_primModNatP0(vz55, vz56, vz570, vz580)
10.99/4.38	
10.99/4.38	The TRS R consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primMinusNatS1 -> Zero
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
10.99/4.38	   new_primMinusNatS3(Zero, Zero) -> Zero
10.99/4.38	   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
10.99/4.38	   new_primMinusNatS0(vz4000) -> Succ(vz4000)
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
10.99/4.38	   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero
10.99/4.38	
10.99/4.38	The set Q consists of the following terms:
10.99/4.38	
10.99/4.38	   new_primMinusNatS3(Zero, Succ(x0))
10.99/4.38	   new_primMinusNatS3(Succ(x0), Zero)
10.99/4.38	   new_primMinusNatS3(Zero, Zero)
10.99/4.38	   new_primMinusNatS0(x0)
10.99/4.38	   new_primMinusNatS3(Succ(x0), Succ(x1))
10.99/4.38	   new_primMinusNatS1
10.99/4.38	   new_primMinusNatS2(x0, x1)
10.99/4.38	
10.99/4.38	We have to consider all minimal (P,Q,R)-chains.
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(24) QDPSizeChangeProof (EQUIVALENT)
10.99/4.38	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
10.99/4.38	
10.99/4.38	Order:Polynomial interpretation [POLO]:
10.99/4.38	
10.99/4.38	   POL(Succ(x_1)) = 1 + x_1
10.99/4.38	   POL(Zero) = 1
10.99/4.38	   POL(new_primMinusNatS2(x_1, x_2)) = x_1
10.99/4.38	   POL(new_primMinusNatS3(x_1, x_2)) = x_1
10.99/4.38	
10.99/4.38	
10.99/4.38	
10.99/4.38	
10.99/4.38	From the DPs we obtained the following set of size-change graphs:
10.99/4.38	*new_primModNatP0(vz55, vz56, Succ(vz570), Zero) -> new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56)) (allowed arguments on rhs = {1, 2})
10.99/4.38	The graph contains the following edges 1 >= 1
10.99/4.38	
10.99/4.38	
10.99/4.38	*new_primModNatP00(vz55, vz56) -> new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56)) (allowed arguments on rhs = {1, 2})
10.99/4.38	The graph contains the following edges 1 >= 1
10.99/4.38	
10.99/4.38	
10.99/4.38	*new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primModNatP0(vz3000, vz4000, vz3000, vz4000) (allowed arguments on rhs = {1, 2, 3, 4})
10.99/4.38	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
10.99/4.38	
10.99/4.38	
10.99/4.38	*new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) -> new_primModNatP0(vz55, vz56, vz570, vz580) (allowed arguments on rhs = {1, 2, 3, 4})
10.99/4.38	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
10.99/4.38	
10.99/4.38	
10.99/4.38	*new_primModNatP0(vz55, vz56, Zero, Zero) -> new_primModNatP00(vz55, vz56) (allowed arguments on rhs = {1, 2})
10.99/4.38	The graph contains the following edges 1 >= 1, 2 >= 2
10.99/4.38	
10.99/4.38	
10.99/4.38	
10.99/4.38	We oriented the following set of usable rules [AAECC05,FROCOS05].
10.99/4.38	
10.99/4.38	   new_primMinusNatS3(Zero, Zero) -> Zero
10.99/4.38	   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
10.99/4.38	   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
10.99/4.38	
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(25)
10.99/4.38	YES
10.99/4.38	
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(26)
10.99/4.38	Obligation:
10.99/4.38	Q DP problem:
10.99/4.38	The TRS P consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primModNatP(Succ(Succ(vz3000)), Zero) -> new_primModNatP(new_primMinusNatS0(vz3000), Zero)
10.99/4.38	
10.99/4.38	The TRS R consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primMinusNatS1 -> Zero
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
10.99/4.38	   new_primMinusNatS3(Zero, Zero) -> Zero
10.99/4.38	   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
10.99/4.38	   new_primMinusNatS0(vz4000) -> Succ(vz4000)
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
10.99/4.38	   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero
10.99/4.38	
10.99/4.38	The set Q consists of the following terms:
10.99/4.38	
10.99/4.38	   new_primMinusNatS3(Zero, Succ(x0))
10.99/4.38	   new_primMinusNatS3(Succ(x0), Zero)
10.99/4.38	   new_primMinusNatS3(Zero, Zero)
10.99/4.38	   new_primMinusNatS0(x0)
10.99/4.38	   new_primMinusNatS3(Succ(x0), Succ(x1))
10.99/4.38	   new_primMinusNatS1
10.99/4.38	   new_primMinusNatS2(x0, x1)
10.99/4.38	
10.99/4.38	We have to consider all minimal (P,Q,R)-chains.
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(27) MRRProof (EQUIVALENT)
10.99/4.38	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
10.99/4.38	
10.99/4.38	Strictly oriented dependency pairs:
10.99/4.38	
10.99/4.38	   new_primModNatP(Succ(Succ(vz3000)), Zero) -> new_primModNatP(new_primMinusNatS0(vz3000), Zero)
10.99/4.38	
10.99/4.38	Strictly oriented rules of the TRS R:
10.99/4.38	
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Zero) -> Succ(vz560)
10.99/4.38	   new_primMinusNatS3(Zero, Zero) -> Zero
10.99/4.38	   new_primMinusNatS3(Succ(vz560), Succ(vz550)) -> new_primMinusNatS3(vz560, vz550)
10.99/4.38	   new_primMinusNatS3(Zero, Succ(vz550)) -> Zero
10.99/4.38	
10.99/4.38	Used ordering: Polynomial interpretation [POLO]:
10.99/4.38	
10.99/4.38	   POL(Succ(x_1)) = 1 + x_1
10.99/4.38	   POL(Zero) = 2
10.99/4.38	   POL(new_primMinusNatS0(x_1)) = 1 + x_1
10.99/4.38	   POL(new_primMinusNatS1) = 2
10.99/4.38	   POL(new_primMinusNatS2(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
10.99/4.38	   POL(new_primMinusNatS3(x_1, x_2)) = 1 + 2*x_1 + x_2
10.99/4.38	   POL(new_primModNatP(x_1, x_2)) = x_1 + x_2
10.99/4.38	
10.99/4.38	
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(28)
10.99/4.38	Obligation:
10.99/4.38	Q DP problem:
10.99/4.38	P is empty.
10.99/4.38	The TRS R consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primMinusNatS1 -> Zero
10.99/4.38	   new_primMinusNatS2(vz56, vz55) -> new_primMinusNatS3(vz56, vz55)
10.99/4.38	   new_primMinusNatS0(vz4000) -> Succ(vz4000)
10.99/4.38	
10.99/4.38	The set Q consists of the following terms:
10.99/4.38	
10.99/4.38	   new_primMinusNatS3(Zero, Succ(x0))
10.99/4.38	   new_primMinusNatS3(Succ(x0), Zero)
10.99/4.38	   new_primMinusNatS3(Zero, Zero)
10.99/4.38	   new_primMinusNatS0(x0)
10.99/4.38	   new_primMinusNatS3(Succ(x0), Succ(x1))
10.99/4.38	   new_primMinusNatS1
10.99/4.38	   new_primMinusNatS2(x0, x1)
10.99/4.38	
10.99/4.38	We have to consider all minimal (P,Q,R)-chains.
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10.99/4.38	
10.99/4.38	(29) PisEmptyProof (EQUIVALENT)
10.99/4.38	The TRS P is empty. Hence, there is no (P,Q,R) chain.
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(30)
10.99/4.38	YES
10.99/4.38	
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(31)
10.99/4.38	Obligation:
10.99/4.38	Q DP problem:
10.99/4.38	The TRS P consists of the following rules:
10.99/4.38	
10.99/4.38	   new_primMinusNatS(Succ(vz560), Succ(vz550)) -> new_primMinusNatS(vz560, vz550)
10.99/4.38	
10.99/4.38	R is empty.
10.99/4.38	Q is empty.
10.99/4.38	We have to consider all minimal (P,Q,R)-chains.
10.99/4.38	----------------------------------------
10.99/4.38	
10.99/4.38	(32) QDPSizeChangeProof (EQUIVALENT)
10.99/4.38	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
10.99/4.38	
10.99/4.38	From the DPs we obtained the following set of size-change graphs:
10.99/4.38	*new_primMinusNatS(Succ(vz560), Succ(vz550)) -> new_primMinusNatS(vz560, vz550)
10.99/4.38	The graph contains the following edges 1 > 1, 2 > 2
10.99/4.38	
10.99/4.38	
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10.99/4.38	
10.99/4.38	(33)
10.99/4.38	YES
10.99/4.43	EOF