8.98/3.96	YES
10.99/4.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.99/4.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.99/4.49	
10.99/4.49	
10.99/4.49	H-Termination with start terms of the given HASKELL could be proven:
10.99/4.49	
10.99/4.49	(0) HASKELL
10.99/4.49	(1) BR [EQUIVALENT, 0 ms]
10.99/4.49	(2) HASKELL
10.99/4.49	(3) COR [EQUIVALENT, 0 ms]
10.99/4.49	(4) HASKELL
10.99/4.49	(5) Narrow [SOUND, 0 ms]
10.99/4.49	(6) AND
10.99/4.49	    (7) QDP
10.99/4.49	        (8) DependencyGraphProof [EQUIVALENT, 0 ms]
10.99/4.49	        (9) AND
10.99/4.49	            (10) QDP
10.99/4.49	                (11) QDPSizeChangeProof [EQUIVALENT, 13 ms]
10.99/4.49	                (12) YES
10.99/4.49	            (13) QDP
10.99/4.49	                (14) QDPOrderProof [EQUIVALENT, 0 ms]
10.99/4.49	                (15) QDP
10.99/4.49	                (16) DependencyGraphProof [EQUIVALENT, 0 ms]
10.99/4.49	                (17) QDP
10.99/4.49	                (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.99/4.49	                (19) YES
10.99/4.49	    (20) QDP
10.99/4.49	        (21) DependencyGraphProof [EQUIVALENT, 0 ms]
10.99/4.49	        (22) AND
10.99/4.49	            (23) QDP
10.99/4.49	                (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.99/4.49	                (25) YES
10.99/4.49	            (26) QDP
10.99/4.49	                (27) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.99/4.49	                (28) YES
10.99/4.49	    (29) QDP
10.99/4.49	        (30) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.99/4.49	        (31) YES
10.99/4.49	
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(0)
10.99/4.49	Obligation:
10.99/4.49	mainModule Main 
10.99/4.49	module Main where {
10.99/4.49	import qualified Prelude;
10.99/4.49	data MyBool = MyTrue  | MyFalse ;
10.99/4.49	
10.99/4.49	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.99/4.49	
10.99/4.49	data Main.Nat = Succ Main.Nat  | Zero ;
10.99/4.49	
10.99/4.49	error :: a;
10.99/4.49	error = stop MyTrue;
10.99/4.49	
10.99/4.49	modMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.99/4.49	modMyInt = primModInt;
10.99/4.49	
10.99/4.49	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.99/4.49	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
10.99/4.49	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
10.99/4.49	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
10.99/4.49	primGEqNatS Main.Zero Main.Zero = MyTrue;
10.99/4.49	
10.99/4.49	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.99/4.49	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
10.99/4.49	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
10.99/4.49	primMinusNatS x Main.Zero = x;
10.99/4.49	
10.99/4.49	primModInt :: MyInt  ->  MyInt  ->  MyInt;
10.99/4.49	primModInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
10.99/4.49	primModInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatP x (Main.Succ y));
10.99/4.49	primModInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatP x (Main.Succ y));
10.99/4.49	primModInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
10.99/4.49	primModInt vv vw = Main.error;
10.99/4.49	
10.99/4.49	primModNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.99/4.49	primModNatP Main.Zero Main.Zero = Main.error;
10.99/4.49	primModNatP Main.Zero (Main.Succ x) = Main.Zero;
10.99/4.49	primModNatP (Main.Succ x) Main.Zero = Main.error;
10.99/4.49	primModNatP (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
10.99/4.49	primModNatP (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatP0 x y (primGEqNatS x y);
10.99/4.49	
10.99/4.49	primModNatP0 x y MyTrue = primModNatP (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
10.99/4.49	primModNatP0 x y MyFalse = primMinusNatS (Main.Succ y) x;
10.99/4.49	
10.99/4.49	primModNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.99/4.49	primModNatS Main.Zero Main.Zero = Main.error;
10.99/4.49	primModNatS Main.Zero (Main.Succ x) = Main.Zero;
10.99/4.49	primModNatS (Main.Succ x) Main.Zero = Main.error;
10.99/4.49	primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
10.99/4.49	primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y));
10.99/4.49	
10.99/4.49	primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
10.99/4.49	primModNatS0 x y MyFalse = Main.Succ x;
10.99/4.49	
10.99/4.49	stop :: MyBool  ->  a;
10.99/4.49	stop MyFalse = stop MyFalse;
10.99/4.49	
10.99/4.49	}
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(1) BR (EQUIVALENT)
10.99/4.49	Replaced joker patterns by fresh variables and removed binding patterns.
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(2)
10.99/4.49	Obligation:
10.99/4.49	mainModule Main 
10.99/4.49	module Main where {
10.99/4.49	import qualified Prelude;
10.99/4.49	data MyBool = MyTrue  | MyFalse ;
10.99/4.49	
10.99/4.49	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.99/4.49	
10.99/4.49	data Main.Nat = Succ Main.Nat  | Zero ;
10.99/4.49	
10.99/4.49	error :: a;
10.99/4.49	error = stop MyTrue;
10.99/4.49	
10.99/4.49	modMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.99/4.49	modMyInt = primModInt;
10.99/4.49	
10.99/4.49	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.99/4.49	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
10.99/4.49	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
10.99/4.49	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
10.99/4.49	primGEqNatS Main.Zero Main.Zero = MyTrue;
10.99/4.49	
10.99/4.49	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.99/4.49	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
10.99/4.49	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
10.99/4.49	primMinusNatS x Main.Zero = x;
10.99/4.49	
10.99/4.49	primModInt :: MyInt  ->  MyInt  ->  MyInt;
10.99/4.49	primModInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
10.99/4.49	primModInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatP x (Main.Succ y));
10.99/4.49	primModInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatP x (Main.Succ y));
10.99/4.49	primModInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
10.99/4.49	primModInt vv vw = Main.error;
10.99/4.49	
10.99/4.49	primModNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.99/4.49	primModNatP Main.Zero Main.Zero = Main.error;
10.99/4.49	primModNatP Main.Zero (Main.Succ x) = Main.Zero;
10.99/4.49	primModNatP (Main.Succ x) Main.Zero = Main.error;
10.99/4.49	primModNatP (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
10.99/4.49	primModNatP (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatP0 x y (primGEqNatS x y);
10.99/4.49	
10.99/4.49	primModNatP0 x y MyTrue = primModNatP (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
10.99/4.49	primModNatP0 x y MyFalse = primMinusNatS (Main.Succ y) x;
10.99/4.49	
10.99/4.49	primModNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.99/4.49	primModNatS Main.Zero Main.Zero = Main.error;
10.99/4.49	primModNatS Main.Zero (Main.Succ x) = Main.Zero;
10.99/4.49	primModNatS (Main.Succ x) Main.Zero = Main.error;
10.99/4.49	primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
10.99/4.49	primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y));
10.99/4.49	
10.99/4.49	primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
10.99/4.49	primModNatS0 x y MyFalse = Main.Succ x;
10.99/4.49	
10.99/4.49	stop :: MyBool  ->  a;
10.99/4.49	stop MyFalse = stop MyFalse;
10.99/4.49	
10.99/4.49	}
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(3) COR (EQUIVALENT)
10.99/4.49	Cond Reductions:
10.99/4.49	The following Function with conditions
10.99/4.49	"undefined |Falseundefined;
10.99/4.49	"
10.99/4.49	is transformed to
10.99/4.49	"undefined  = undefined1;
10.99/4.49	"
10.99/4.49	"undefined0 True = undefined;
10.99/4.49	"
10.99/4.49	"undefined1  = undefined0 False;
10.99/4.49	"
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(4)
10.99/4.49	Obligation:
10.99/4.49	mainModule Main 
10.99/4.49	module Main where {
10.99/4.49	import qualified Prelude;
10.99/4.49	data MyBool = MyTrue  | MyFalse ;
10.99/4.49	
10.99/4.49	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.99/4.49	
10.99/4.49	data Main.Nat = Succ Main.Nat  | Zero ;
10.99/4.49	
10.99/4.49	error :: a;
10.99/4.49	error = stop MyTrue;
10.99/4.49	
10.99/4.49	modMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.99/4.49	modMyInt = primModInt;
10.99/4.49	
10.99/4.49	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.99/4.49	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
10.99/4.49	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
10.99/4.49	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
10.99/4.49	primGEqNatS Main.Zero Main.Zero = MyTrue;
10.99/4.49	
10.99/4.49	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.99/4.49	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
10.99/4.49	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
10.99/4.49	primMinusNatS x Main.Zero = x;
10.99/4.49	
10.99/4.49	primModInt :: MyInt  ->  MyInt  ->  MyInt;
10.99/4.49	primModInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
10.99/4.49	primModInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatP x (Main.Succ y));
10.99/4.49	primModInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatP x (Main.Succ y));
10.99/4.49	primModInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
10.99/4.49	primModInt vv vw = Main.error;
10.99/4.49	
10.99/4.49	primModNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.99/4.49	primModNatP Main.Zero Main.Zero = Main.error;
10.99/4.49	primModNatP Main.Zero (Main.Succ x) = Main.Zero;
10.99/4.49	primModNatP (Main.Succ x) Main.Zero = Main.error;
10.99/4.49	primModNatP (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
10.99/4.49	primModNatP (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatP0 x y (primGEqNatS x y);
10.99/4.49	
10.99/4.49	primModNatP0 x y MyTrue = primModNatP (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
10.99/4.49	primModNatP0 x y MyFalse = primMinusNatS (Main.Succ y) x;
10.99/4.49	
10.99/4.49	primModNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.99/4.49	primModNatS Main.Zero Main.Zero = Main.error;
10.99/4.49	primModNatS Main.Zero (Main.Succ x) = Main.Zero;
10.99/4.49	primModNatS (Main.Succ x) Main.Zero = Main.error;
10.99/4.49	primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
10.99/4.49	primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y));
10.99/4.49	
10.99/4.49	primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
10.99/4.49	primModNatS0 x y MyFalse = Main.Succ x;
10.99/4.49	
10.99/4.49	stop :: MyBool  ->  a;
10.99/4.49	stop MyFalse = stop MyFalse;
10.99/4.49	
10.99/4.49	}
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(5) Narrow (SOUND)
10.99/4.49	Haskell To QDPs
10.99/4.49	
10.99/4.49	digraph dp_graph {
10.99/4.49	node [outthreshold=100, inthreshold=100];1[label="modMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.99/4.49	3[label="modMyInt vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.99/4.49	4[label="modMyInt vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.99/4.49	5[label="primModInt vz3 vz4",fontsize=16,color="burlywood",shape="box"];626[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 626[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	626 -> 6[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	627[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 627[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	627 -> 7[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	6[label="primModInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];628[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 628[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	628 -> 8[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	629[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 629[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	629 -> 9[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	7[label="primModInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];630[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 630[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	630 -> 10[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	631[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 631[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	631 -> 11[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	8[label="primModInt (Pos vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];632[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 632[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	632 -> 12[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	633[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 633[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	633 -> 13[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	9[label="primModInt (Pos vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];634[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 634[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	634 -> 14[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	635[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 635[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	635 -> 15[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	10[label="primModInt (Neg vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];636[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 636[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	636 -> 16[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	637[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 637[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	637 -> 17[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	11[label="primModInt (Neg vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];638[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 638[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	638 -> 18[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	639[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 639[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	639 -> 19[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	12[label="primModInt (Pos vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
10.99/4.49	13[label="primModInt (Pos vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
10.99/4.49	14[label="primModInt (Pos vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
10.99/4.49	15[label="primModInt (Pos vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
10.99/4.49	16[label="primModInt (Neg vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
10.99/4.49	17[label="primModInt (Neg vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
10.99/4.49	18[label="primModInt (Neg vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
10.99/4.49	19[label="primModInt (Neg vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
10.99/4.49	20[label="Pos (primModNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3];
10.99/4.49	21[label="error",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3];
10.99/4.49	22[label="Neg (primModNatP vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];22 -> 30[label="",style="dashed", color="green", weight=3];
10.99/4.49	23 -> 21[label="",style="dashed", color="red", weight=0];
10.99/4.49	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];
10.99/4.49	25 -> 21[label="",style="dashed", color="red", weight=0];
10.99/4.49	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];
10.99/4.49	27 -> 21[label="",style="dashed", color="red", weight=0];
10.99/4.49	27[label="error",fontsize=16,color="magenta"];28[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];640[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 640[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	640 -> 33[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	641[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 641[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	641 -> 34[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	29[label="stop MyTrue",fontsize=16,color="black",shape="box"];29 -> 35[label="",style="solid", color="black", weight=3];
10.99/4.49	30[label="primModNatP vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];642[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];30 -> 642[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	642 -> 36[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	643[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 643[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	643 -> 37[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	31 -> 30[label="",style="dashed", color="red", weight=0];
10.99/4.49	31[label="primModNatP vz30 (Succ vz400)",fontsize=16,color="magenta"];31 -> 38[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	31 -> 39[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	32 -> 28[label="",style="dashed", color="red", weight=0];
10.99/4.49	32[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];32 -> 40[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	32 -> 41[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	33[label="primModNatS (Succ vz300) (Succ vz400)",fontsize=16,color="burlywood",shape="box"];644[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];33 -> 644[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	644 -> 42[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	645[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 645[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	645 -> 43[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	34[label="primModNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 44[label="",style="solid", color="black", weight=3];
10.99/4.49	35[label="error []",fontsize=16,color="red",shape="box"];36[label="primModNatP (Succ vz300) (Succ vz400)",fontsize=16,color="burlywood",shape="box"];646[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];36 -> 646[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	646 -> 45[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	647[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 647[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	647 -> 46[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	37[label="primModNatP Zero (Succ vz400)",fontsize=16,color="black",shape="box"];37 -> 47[label="",style="solid", color="black", weight=3];
10.99/4.49	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="vz30",fontsize=16,color="green",shape="box"];42[label="primModNatS (Succ vz300) (Succ (Succ vz4000))",fontsize=16,color="black",shape="box"];42 -> 48[label="",style="solid", color="black", weight=3];
10.99/4.49	43[label="primModNatS (Succ vz300) (Succ Zero)",fontsize=16,color="black",shape="box"];43 -> 49[label="",style="solid", color="black", weight=3];
10.99/4.49	44[label="Zero",fontsize=16,color="green",shape="box"];45[label="primModNatP (Succ vz300) (Succ (Succ vz4000))",fontsize=16,color="black",shape="box"];45 -> 50[label="",style="solid", color="black", weight=3];
10.99/4.49	46[label="primModNatP (Succ vz300) (Succ Zero)",fontsize=16,color="black",shape="box"];46 -> 51[label="",style="solid", color="black", weight=3];
10.99/4.49	47[label="Zero",fontsize=16,color="green",shape="box"];48[label="primModNatS0 vz300 vz4000 (primGEqNatS vz300 (Succ vz4000))",fontsize=16,color="burlywood",shape="box"];648[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];48 -> 648[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	648 -> 52[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	649[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 649[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	649 -> 53[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	49[label="Zero",fontsize=16,color="green",shape="box"];50[label="primModNatP0 vz300 vz4000 (primGEqNatS vz300 vz4000)",fontsize=16,color="burlywood",shape="box"];650[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];50 -> 650[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	650 -> 54[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	651[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 651[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	651 -> 55[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	51[label="Zero",fontsize=16,color="green",shape="box"];52[label="primModNatS0 (Succ vz3000) vz4000 (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
10.99/4.49	53[label="primModNatS0 Zero vz4000 (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
10.99/4.49	54[label="primModNatP0 (Succ vz3000) vz4000 (primGEqNatS (Succ vz3000) vz4000)",fontsize=16,color="burlywood",shape="box"];652[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];54 -> 652[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	652 -> 58[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	653[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];54 -> 653[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	653 -> 59[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	55[label="primModNatP0 Zero vz4000 (primGEqNatS Zero vz4000)",fontsize=16,color="burlywood",shape="box"];654[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];55 -> 654[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	654 -> 60[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	655[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];55 -> 655[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	655 -> 61[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	56[label="primModNatS0 (Succ vz3000) vz4000 (primGEqNatS vz3000 vz4000)",fontsize=16,color="burlywood",shape="box"];656[label="vz3000/Succ vz30000",fontsize=10,color="white",style="solid",shape="box"];56 -> 656[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	656 -> 62[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	657[label="vz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];56 -> 657[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	657 -> 63[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	57[label="primModNatS0 Zero vz4000 MyFalse",fontsize=16,color="black",shape="box"];57 -> 64[label="",style="solid", color="black", weight=3];
10.99/4.49	58[label="primModNatP0 (Succ vz3000) (Succ vz40000) (primGEqNatS (Succ vz3000) (Succ vz40000))",fontsize=16,color="black",shape="box"];58 -> 65[label="",style="solid", color="black", weight=3];
10.99/4.49	59[label="primModNatP0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];59 -> 66[label="",style="solid", color="black", weight=3];
10.99/4.49	60[label="primModNatP0 Zero (Succ vz40000) (primGEqNatS Zero (Succ vz40000))",fontsize=16,color="black",shape="box"];60 -> 67[label="",style="solid", color="black", weight=3];
10.99/4.49	61[label="primModNatP0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];61 -> 68[label="",style="solid", color="black", weight=3];
10.99/4.49	62[label="primModNatS0 (Succ (Succ vz30000)) vz4000 (primGEqNatS (Succ vz30000) vz4000)",fontsize=16,color="burlywood",shape="box"];658[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];62 -> 658[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	658 -> 69[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	659[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 659[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	659 -> 70[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	63[label="primModNatS0 (Succ Zero) vz4000 (primGEqNatS Zero vz4000)",fontsize=16,color="burlywood",shape="box"];660[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];63 -> 660[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	660 -> 71[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	661[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];63 -> 661[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	661 -> 72[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	64[label="Succ Zero",fontsize=16,color="green",shape="box"];65 -> 482[label="",style="dashed", color="red", weight=0];
10.99/4.49	65[label="primModNatP0 (Succ vz3000) (Succ vz40000) (primGEqNatS vz3000 vz40000)",fontsize=16,color="magenta"];65 -> 483[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	65 -> 484[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	65 -> 485[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	65 -> 486[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	66[label="primModNatP0 (Succ vz3000) Zero MyTrue",fontsize=16,color="black",shape="box"];66 -> 75[label="",style="solid", color="black", weight=3];
10.99/4.49	67[label="primModNatP0 Zero (Succ vz40000) MyFalse",fontsize=16,color="black",shape="box"];67 -> 76[label="",style="solid", color="black", weight=3];
10.99/4.49	68[label="primModNatP0 Zero Zero MyTrue",fontsize=16,color="black",shape="box"];68 -> 77[label="",style="solid", color="black", weight=3];
10.99/4.49	69[label="primModNatS0 (Succ (Succ vz30000)) (Succ vz40000) (primGEqNatS (Succ vz30000) (Succ vz40000))",fontsize=16,color="black",shape="box"];69 -> 78[label="",style="solid", color="black", weight=3];
10.99/4.49	70[label="primModNatS0 (Succ (Succ vz30000)) Zero (primGEqNatS (Succ vz30000) Zero)",fontsize=16,color="black",shape="box"];70 -> 79[label="",style="solid", color="black", weight=3];
10.99/4.49	71[label="primModNatS0 (Succ Zero) (Succ vz40000) (primGEqNatS Zero (Succ vz40000))",fontsize=16,color="black",shape="box"];71 -> 80[label="",style="solid", color="black", weight=3];
10.99/4.49	72[label="primModNatS0 (Succ Zero) Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];72 -> 81[label="",style="solid", color="black", weight=3];
10.99/4.49	483[label="vz40000",fontsize=16,color="green",shape="box"];484[label="vz40000",fontsize=16,color="green",shape="box"];485[label="vz3000",fontsize=16,color="green",shape="box"];486[label="vz3000",fontsize=16,color="green",shape="box"];482[label="primModNatP0 (Succ vz39) (Succ vz40) (primGEqNatS vz41 vz42)",fontsize=16,color="burlywood",shape="triangle"];662[label="vz41/Succ vz410",fontsize=10,color="white",style="solid",shape="box"];482 -> 662[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	662 -> 523[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	663[label="vz41/Zero",fontsize=10,color="white",style="solid",shape="box"];482 -> 663[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	663 -> 524[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	75 -> 30[label="",style="dashed", color="red", weight=0];
10.99/4.49	75[label="primModNatP (primMinusNatS (Succ vz3000) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];75 -> 86[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	75 -> 87[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	76[label="primMinusNatS (Succ (Succ vz40000)) Zero",fontsize=16,color="black",shape="box"];76 -> 88[label="",style="solid", color="black", weight=3];
10.99/4.49	77 -> 30[label="",style="dashed", color="red", weight=0];
10.99/4.49	77[label="primModNatP (primMinusNatS Zero (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];77 -> 89[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	77 -> 90[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	78 -> 533[label="",style="dashed", color="red", weight=0];
10.99/4.49	78[label="primModNatS0 (Succ (Succ vz30000)) (Succ vz40000) (primGEqNatS vz30000 vz40000)",fontsize=16,color="magenta"];78 -> 534[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	78 -> 535[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	78 -> 536[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	78 -> 537[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	79[label="primModNatS0 (Succ (Succ vz30000)) Zero MyTrue",fontsize=16,color="black",shape="box"];79 -> 93[label="",style="solid", color="black", weight=3];
10.99/4.49	80[label="primModNatS0 (Succ Zero) (Succ vz40000) MyFalse",fontsize=16,color="black",shape="box"];80 -> 94[label="",style="solid", color="black", weight=3];
10.99/4.49	81[label="primModNatS0 (Succ Zero) Zero MyTrue",fontsize=16,color="black",shape="box"];81 -> 95[label="",style="solid", color="black", weight=3];
10.99/4.49	523[label="primModNatP0 (Succ vz39) (Succ vz40) (primGEqNatS (Succ vz410) vz42)",fontsize=16,color="burlywood",shape="box"];664[label="vz42/Succ vz420",fontsize=10,color="white",style="solid",shape="box"];523 -> 664[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	664 -> 529[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	665[label="vz42/Zero",fontsize=10,color="white",style="solid",shape="box"];523 -> 665[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	665 -> 530[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	524[label="primModNatP0 (Succ vz39) (Succ vz40) (primGEqNatS Zero vz42)",fontsize=16,color="burlywood",shape="box"];666[label="vz42/Succ vz420",fontsize=10,color="white",style="solid",shape="box"];524 -> 666[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	666 -> 531[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	667[label="vz42/Zero",fontsize=10,color="white",style="solid",shape="box"];524 -> 667[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	667 -> 532[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	86[label="Succ Zero",fontsize=16,color="green",shape="box"];87[label="primMinusNatS (Succ vz3000) (Succ Zero)",fontsize=16,color="black",shape="triangle"];87 -> 100[label="",style="solid", color="black", weight=3];
10.99/4.49	88[label="Succ (Succ vz40000)",fontsize=16,color="green",shape="box"];89[label="Succ Zero",fontsize=16,color="green",shape="box"];90[label="primMinusNatS Zero (Succ Zero)",fontsize=16,color="black",shape="triangle"];90 -> 101[label="",style="solid", color="black", weight=3];
10.99/4.49	534[label="vz40000",fontsize=16,color="green",shape="box"];535[label="vz30000",fontsize=16,color="green",shape="box"];536[label="vz40000",fontsize=16,color="green",shape="box"];537[label="Succ vz30000",fontsize=16,color="green",shape="box"];533[label="primModNatS0 (Succ vz44) (Succ vz45) (primGEqNatS vz46 vz47)",fontsize=16,color="burlywood",shape="triangle"];668[label="vz46/Succ vz460",fontsize=10,color="white",style="solid",shape="box"];533 -> 668[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	668 -> 574[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	669[label="vz46/Zero",fontsize=10,color="white",style="solid",shape="box"];533 -> 669[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	669 -> 575[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	93 -> 28[label="",style="dashed", color="red", weight=0];
10.99/4.49	93[label="primModNatS (primMinusNatS (Succ (Succ vz30000)) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];93 -> 106[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	93 -> 107[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	94[label="Succ (Succ Zero)",fontsize=16,color="green",shape="box"];95 -> 28[label="",style="dashed", color="red", weight=0];
10.99/4.49	95[label="primModNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];95 -> 108[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	95 -> 109[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	529[label="primModNatP0 (Succ vz39) (Succ vz40) (primGEqNatS (Succ vz410) (Succ vz420))",fontsize=16,color="black",shape="box"];529 -> 576[label="",style="solid", color="black", weight=3];
10.99/4.49	530[label="primModNatP0 (Succ vz39) (Succ vz40) (primGEqNatS (Succ vz410) Zero)",fontsize=16,color="black",shape="box"];530 -> 577[label="",style="solid", color="black", weight=3];
10.99/4.49	531[label="primModNatP0 (Succ vz39) (Succ vz40) (primGEqNatS Zero (Succ vz420))",fontsize=16,color="black",shape="box"];531 -> 578[label="",style="solid", color="black", weight=3];
10.99/4.49	532[label="primModNatP0 (Succ vz39) (Succ vz40) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];532 -> 579[label="",style="solid", color="black", weight=3];
10.99/4.49	100[label="primMinusNatS vz3000 Zero",fontsize=16,color="burlywood",shape="box"];670[label="vz3000/Succ vz30000",fontsize=10,color="white",style="solid",shape="box"];100 -> 670[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	670 -> 115[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	671[label="vz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];100 -> 671[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	671 -> 116[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	101[label="Zero",fontsize=16,color="green",shape="box"];574[label="primModNatS0 (Succ vz44) (Succ vz45) (primGEqNatS (Succ vz460) vz47)",fontsize=16,color="burlywood",shape="box"];672[label="vz47/Succ vz470",fontsize=10,color="white",style="solid",shape="box"];574 -> 672[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	672 -> 580[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	673[label="vz47/Zero",fontsize=10,color="white",style="solid",shape="box"];574 -> 673[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	673 -> 581[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	575[label="primModNatS0 (Succ vz44) (Succ vz45) (primGEqNatS Zero vz47)",fontsize=16,color="burlywood",shape="box"];674[label="vz47/Succ vz470",fontsize=10,color="white",style="solid",shape="box"];575 -> 674[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	674 -> 582[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	675[label="vz47/Zero",fontsize=10,color="white",style="solid",shape="box"];575 -> 675[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	675 -> 583[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	106[label="Succ Zero",fontsize=16,color="green",shape="box"];107 -> 87[label="",style="dashed", color="red", weight=0];
10.99/4.49	107[label="primMinusNatS (Succ (Succ vz30000)) (Succ Zero)",fontsize=16,color="magenta"];107 -> 121[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	108[label="Succ Zero",fontsize=16,color="green",shape="box"];109 -> 87[label="",style="dashed", color="red", weight=0];
10.99/4.49	109[label="primMinusNatS (Succ Zero) (Succ Zero)",fontsize=16,color="magenta"];109 -> 122[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	576 -> 482[label="",style="dashed", color="red", weight=0];
10.99/4.49	576[label="primModNatP0 (Succ vz39) (Succ vz40) (primGEqNatS vz410 vz420)",fontsize=16,color="magenta"];576 -> 584[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	576 -> 585[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	577[label="primModNatP0 (Succ vz39) (Succ vz40) MyTrue",fontsize=16,color="black",shape="triangle"];577 -> 586[label="",style="solid", color="black", weight=3];
10.99/4.49	578[label="primModNatP0 (Succ vz39) (Succ vz40) MyFalse",fontsize=16,color="black",shape="box"];578 -> 587[label="",style="solid", color="black", weight=3];
10.99/4.49	579 -> 577[label="",style="dashed", color="red", weight=0];
10.99/4.49	579[label="primModNatP0 (Succ vz39) (Succ vz40) MyTrue",fontsize=16,color="magenta"];115[label="primMinusNatS (Succ vz30000) Zero",fontsize=16,color="black",shape="box"];115 -> 132[label="",style="solid", color="black", weight=3];
10.99/4.49	116[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];116 -> 133[label="",style="solid", color="black", weight=3];
10.99/4.49	580[label="primModNatS0 (Succ vz44) (Succ vz45) (primGEqNatS (Succ vz460) (Succ vz470))",fontsize=16,color="black",shape="box"];580 -> 588[label="",style="solid", color="black", weight=3];
10.99/4.49	581[label="primModNatS0 (Succ vz44) (Succ vz45) (primGEqNatS (Succ vz460) Zero)",fontsize=16,color="black",shape="box"];581 -> 589[label="",style="solid", color="black", weight=3];
10.99/4.49	582[label="primModNatS0 (Succ vz44) (Succ vz45) (primGEqNatS Zero (Succ vz470))",fontsize=16,color="black",shape="box"];582 -> 590[label="",style="solid", color="black", weight=3];
10.99/4.49	583[label="primModNatS0 (Succ vz44) (Succ vz45) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];583 -> 591[label="",style="solid", color="black", weight=3];
10.99/4.49	121[label="Succ vz30000",fontsize=16,color="green",shape="box"];122[label="Zero",fontsize=16,color="green",shape="box"];584[label="vz420",fontsize=16,color="green",shape="box"];585[label="vz410",fontsize=16,color="green",shape="box"];586 -> 30[label="",style="dashed", color="red", weight=0];
10.99/4.49	586[label="primModNatP (primMinusNatS (Succ vz39) (Succ (Succ vz40))) (Succ (Succ (Succ vz40)))",fontsize=16,color="magenta"];586 -> 592[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	586 -> 593[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	587[label="primMinusNatS (Succ (Succ vz40)) (Succ vz39)",fontsize=16,color="black",shape="box"];587 -> 594[label="",style="solid", color="black", weight=3];
10.99/4.49	132[label="Succ vz30000",fontsize=16,color="green",shape="box"];133[label="Zero",fontsize=16,color="green",shape="box"];588 -> 533[label="",style="dashed", color="red", weight=0];
10.99/4.49	588[label="primModNatS0 (Succ vz44) (Succ vz45) (primGEqNatS vz460 vz470)",fontsize=16,color="magenta"];588 -> 595[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	588 -> 596[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	589[label="primModNatS0 (Succ vz44) (Succ vz45) MyTrue",fontsize=16,color="black",shape="triangle"];589 -> 597[label="",style="solid", color="black", weight=3];
10.99/4.49	590[label="primModNatS0 (Succ vz44) (Succ vz45) MyFalse",fontsize=16,color="black",shape="box"];590 -> 598[label="",style="solid", color="black", weight=3];
10.99/4.49	591 -> 589[label="",style="dashed", color="red", weight=0];
10.99/4.49	591[label="primModNatS0 (Succ vz44) (Succ vz45) MyTrue",fontsize=16,color="magenta"];592[label="Succ (Succ vz40)",fontsize=16,color="green",shape="box"];593[label="primMinusNatS (Succ vz39) (Succ (Succ vz40))",fontsize=16,color="black",shape="box"];593 -> 599[label="",style="solid", color="black", weight=3];
10.99/4.49	594[label="primMinusNatS (Succ vz40) vz39",fontsize=16,color="burlywood",shape="box"];676[label="vz39/Succ vz390",fontsize=10,color="white",style="solid",shape="box"];594 -> 676[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	676 -> 600[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	677[label="vz39/Zero",fontsize=10,color="white",style="solid",shape="box"];594 -> 677[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	677 -> 601[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	595[label="vz470",fontsize=16,color="green",shape="box"];596[label="vz460",fontsize=16,color="green",shape="box"];597 -> 28[label="",style="dashed", color="red", weight=0];
10.99/4.49	597[label="primModNatS (primMinusNatS (Succ vz44) (Succ (Succ vz45))) (Succ (Succ (Succ vz45)))",fontsize=16,color="magenta"];597 -> 602[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	597 -> 603[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	598[label="Succ (Succ vz44)",fontsize=16,color="green",shape="box"];599[label="primMinusNatS vz39 (Succ vz40)",fontsize=16,color="burlywood",shape="triangle"];678[label="vz39/Succ vz390",fontsize=10,color="white",style="solid",shape="box"];599 -> 678[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	678 -> 604[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	679[label="vz39/Zero",fontsize=10,color="white",style="solid",shape="box"];599 -> 679[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	679 -> 605[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	600[label="primMinusNatS (Succ vz40) (Succ vz390)",fontsize=16,color="black",shape="box"];600 -> 606[label="",style="solid", color="black", weight=3];
10.99/4.49	601[label="primMinusNatS (Succ vz40) Zero",fontsize=16,color="black",shape="box"];601 -> 607[label="",style="solid", color="black", weight=3];
10.99/4.49	602[label="Succ (Succ vz45)",fontsize=16,color="green",shape="box"];603 -> 599[label="",style="dashed", color="red", weight=0];
10.99/4.49	603[label="primMinusNatS (Succ vz44) (Succ (Succ vz45))",fontsize=16,color="magenta"];603 -> 608[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	603 -> 609[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	604[label="primMinusNatS (Succ vz390) (Succ vz40)",fontsize=16,color="black",shape="box"];604 -> 610[label="",style="solid", color="black", weight=3];
10.99/4.49	605[label="primMinusNatS Zero (Succ vz40)",fontsize=16,color="black",shape="box"];605 -> 611[label="",style="solid", color="black", weight=3];
10.99/4.49	606[label="primMinusNatS vz40 vz390",fontsize=16,color="burlywood",shape="triangle"];680[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];606 -> 680[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	680 -> 612[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	681[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];606 -> 681[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	681 -> 613[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	607[label="Succ vz40",fontsize=16,color="green",shape="box"];608[label="Succ vz45",fontsize=16,color="green",shape="box"];609[label="Succ vz44",fontsize=16,color="green",shape="box"];610 -> 606[label="",style="dashed", color="red", weight=0];
10.99/4.49	610[label="primMinusNatS vz390 vz40",fontsize=16,color="magenta"];610 -> 614[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	610 -> 615[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	611[label="Zero",fontsize=16,color="green",shape="box"];612[label="primMinusNatS (Succ vz400) vz390",fontsize=16,color="burlywood",shape="box"];682[label="vz390/Succ vz3900",fontsize=10,color="white",style="solid",shape="box"];612 -> 682[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	682 -> 616[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	683[label="vz390/Zero",fontsize=10,color="white",style="solid",shape="box"];612 -> 683[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	683 -> 617[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	613[label="primMinusNatS Zero vz390",fontsize=16,color="burlywood",shape="box"];684[label="vz390/Succ vz3900",fontsize=10,color="white",style="solid",shape="box"];613 -> 684[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	684 -> 618[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	685[label="vz390/Zero",fontsize=10,color="white",style="solid",shape="box"];613 -> 685[label="",style="solid", color="burlywood", weight=9];
10.99/4.49	685 -> 619[label="",style="solid", color="burlywood", weight=3];
10.99/4.49	614[label="vz390",fontsize=16,color="green",shape="box"];615[label="vz40",fontsize=16,color="green",shape="box"];616[label="primMinusNatS (Succ vz400) (Succ vz3900)",fontsize=16,color="black",shape="box"];616 -> 620[label="",style="solid", color="black", weight=3];
10.99/4.49	617[label="primMinusNatS (Succ vz400) Zero",fontsize=16,color="black",shape="box"];617 -> 621[label="",style="solid", color="black", weight=3];
10.99/4.49	618[label="primMinusNatS Zero (Succ vz3900)",fontsize=16,color="black",shape="box"];618 -> 622[label="",style="solid", color="black", weight=3];
10.99/4.49	619[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];619 -> 623[label="",style="solid", color="black", weight=3];
10.99/4.49	620 -> 606[label="",style="dashed", color="red", weight=0];
10.99/4.49	620[label="primMinusNatS vz400 vz3900",fontsize=16,color="magenta"];620 -> 624[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	620 -> 625[label="",style="dashed", color="magenta", weight=3];
10.99/4.49	621[label="Succ vz400",fontsize=16,color="green",shape="box"];622[label="Zero",fontsize=16,color="green",shape="box"];623[label="Zero",fontsize=16,color="green",shape="box"];624[label="vz400",fontsize=16,color="green",shape="box"];625[label="vz3900",fontsize=16,color="green",shape="box"];}
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(6)
10.99/4.49	Complex Obligation (AND)
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(7)
10.99/4.49	Obligation:
10.99/4.49	Q DP problem:
10.99/4.49	The TRS P consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primModNatS0(vz44, vz45, Main.Zero, Main.Zero) -> new_primModNatS00(vz44, vz45)
10.99/4.49	   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Succ(vz470)) -> new_primModNatS0(vz44, vz45, vz460, vz470)
10.99/4.49	   new_primModNatS00(vz44, vz45) -> new_primModNatS(new_primMinusNatS2(Main.Succ(vz44), Main.Succ(vz45)), Main.Succ(Main.Succ(vz45)))
10.99/4.49	   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS0(Main.Succ(vz30000)), Main.Succ(Main.Zero))
10.99/4.49	   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Succ(vz40000))) -> new_primModNatS0(Main.Succ(vz30000), vz40000, vz30000, vz40000)
10.99/4.49	   new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS0(Main.Zero), Main.Succ(Main.Zero))
10.99/4.49	   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Zero) -> new_primModNatS(new_primMinusNatS2(Main.Succ(vz44), Main.Succ(vz45)), Main.Succ(Main.Succ(vz45)))
10.99/4.49	
10.99/4.49	The TRS R consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
10.99/4.49	   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
10.99/4.49	   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
10.99/4.49	   new_primMinusNatS0(Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
10.99/4.49	
10.99/4.49	The set Q consists of the following terms:
10.99/4.49	
10.99/4.49	   new_primMinusNatS0(Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Zero, x0)
10.99/4.49	   new_primMinusNatS0(Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Succ(x0), x1)
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero)
10.99/4.49	
10.99/4.49	We have to consider all minimal (P,Q,R)-chains.
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(8) DependencyGraphProof (EQUIVALENT)
10.99/4.49	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(9)
10.99/4.49	Complex Obligation (AND)
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(10)
10.99/4.49	Obligation:
10.99/4.49	Q DP problem:
10.99/4.49	The TRS P consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS0(Main.Succ(vz30000)), Main.Succ(Main.Zero))
10.99/4.49	
10.99/4.49	The TRS R consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
10.99/4.49	   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
10.99/4.49	   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
10.99/4.49	   new_primMinusNatS0(Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
10.99/4.49	
10.99/4.49	The set Q consists of the following terms:
10.99/4.49	
10.99/4.49	   new_primMinusNatS0(Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Zero, x0)
10.99/4.49	   new_primMinusNatS0(Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Succ(x0), x1)
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero)
10.99/4.49	
10.99/4.49	We have to consider all minimal (P,Q,R)-chains.
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(11) QDPSizeChangeProof (EQUIVALENT)
10.99/4.49	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
10.99/4.49	
10.99/4.49	Order:Polynomial interpretation [POLO]:
10.99/4.49	
10.99/4.49	   POL(Main.Succ(x_1)) = 1 + x_1
10.99/4.49	   POL(Main.Zero) = 1
10.99/4.49	   POL(new_primMinusNatS0(x_1)) = x_1
10.99/4.49	
10.99/4.49	
10.99/4.49	
10.99/4.49	
10.99/4.49	From the DPs we obtained the following set of size-change graphs:
10.99/4.49	*new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS0(Main.Succ(vz30000)), Main.Succ(Main.Zero)) (allowed arguments on rhs = {1, 2})
10.99/4.49	The graph contains the following edges 1 > 1, 2 >= 2
10.99/4.49	
10.99/4.49	
10.99/4.49	
10.99/4.49	We oriented the following set of usable rules [AAECC05,FROCOS05].
10.99/4.49	
10.99/4.49	   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(12)
10.99/4.49	YES
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(13)
10.99/4.49	Obligation:
10.99/4.49	Q DP problem:
10.99/4.49	The TRS P consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primModNatS00(vz44, vz45) -> new_primModNatS(new_primMinusNatS2(Main.Succ(vz44), Main.Succ(vz45)), Main.Succ(Main.Succ(vz45)))
10.99/4.49	   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Succ(vz40000))) -> new_primModNatS0(Main.Succ(vz30000), vz40000, vz30000, vz40000)
10.99/4.49	   new_primModNatS0(vz44, vz45, Main.Zero, Main.Zero) -> new_primModNatS00(vz44, vz45)
10.99/4.49	   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Succ(vz470)) -> new_primModNatS0(vz44, vz45, vz460, vz470)
10.99/4.49	   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Zero) -> new_primModNatS(new_primMinusNatS2(Main.Succ(vz44), Main.Succ(vz45)), Main.Succ(Main.Succ(vz45)))
10.99/4.49	
10.99/4.49	The TRS R consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
10.99/4.49	   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
10.99/4.49	   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
10.99/4.49	   new_primMinusNatS0(Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
10.99/4.49	
10.99/4.49	The set Q consists of the following terms:
10.99/4.49	
10.99/4.49	   new_primMinusNatS0(Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Zero, x0)
10.99/4.49	   new_primMinusNatS0(Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Succ(x0), x1)
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero)
10.99/4.49	
10.99/4.49	We have to consider all minimal (P,Q,R)-chains.
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(14) QDPOrderProof (EQUIVALENT)
10.99/4.49	We use the reduction pair processor [LPAR04,JAR06].
10.99/4.49	
10.99/4.49	
10.99/4.49	The following pairs can be oriented strictly and are deleted.
10.99/4.49	
10.99/4.49	   new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Succ(vz40000))) -> new_primModNatS0(Main.Succ(vz30000), vz40000, vz30000, vz40000)
10.99/4.49	The remaining pairs can at least be oriented weakly.
10.99/4.49	Used ordering:  Polynomial interpretation [POLO]:
10.99/4.49	
10.99/4.49	   POL(Main.Succ(x_1)) = 1 + x_1
10.99/4.49	   POL(Main.Zero) = 0
10.99/4.49	   POL(new_primMinusNatS2(x_1, x_2)) = x_1
10.99/4.49	   POL(new_primMinusNatS3(x_1, x_2)) = 1 + x_1
10.99/4.49	   POL(new_primModNatS(x_1, x_2)) = x_1
10.99/4.49	   POL(new_primModNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1
10.99/4.49	   POL(new_primModNatS00(x_1, x_2)) = 1 + x_1
10.99/4.49	
10.99/4.49	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
10.99/4.49	
10.99/4.49	   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
10.99/4.49	
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(15)
10.99/4.49	Obligation:
10.99/4.49	Q DP problem:
10.99/4.49	The TRS P consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primModNatS00(vz44, vz45) -> new_primModNatS(new_primMinusNatS2(Main.Succ(vz44), Main.Succ(vz45)), Main.Succ(Main.Succ(vz45)))
10.99/4.49	   new_primModNatS0(vz44, vz45, Main.Zero, Main.Zero) -> new_primModNatS00(vz44, vz45)
10.99/4.49	   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Succ(vz470)) -> new_primModNatS0(vz44, vz45, vz460, vz470)
10.99/4.49	   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Zero) -> new_primModNatS(new_primMinusNatS2(Main.Succ(vz44), Main.Succ(vz45)), Main.Succ(Main.Succ(vz45)))
10.99/4.49	
10.99/4.49	The TRS R consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
10.99/4.49	   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
10.99/4.49	   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
10.99/4.49	   new_primMinusNatS0(Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
10.99/4.49	
10.99/4.49	The set Q consists of the following terms:
10.99/4.49	
10.99/4.49	   new_primMinusNatS0(Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Zero, x0)
10.99/4.49	   new_primMinusNatS0(Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Succ(x0), x1)
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero)
10.99/4.49	
10.99/4.49	We have to consider all minimal (P,Q,R)-chains.
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(16) DependencyGraphProof (EQUIVALENT)
10.99/4.49	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(17)
10.99/4.49	Obligation:
10.99/4.49	Q DP problem:
10.99/4.49	The TRS P consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Succ(vz470)) -> new_primModNatS0(vz44, vz45, vz460, vz470)
10.99/4.49	
10.99/4.49	The TRS R consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
10.99/4.49	   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
10.99/4.49	   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
10.99/4.49	   new_primMinusNatS0(Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
10.99/4.49	
10.99/4.49	The set Q consists of the following terms:
10.99/4.49	
10.99/4.49	   new_primMinusNatS0(Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Zero, x0)
10.99/4.49	   new_primMinusNatS0(Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Succ(x0), x1)
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero)
10.99/4.49	
10.99/4.49	We have to consider all minimal (P,Q,R)-chains.
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(18) QDPSizeChangeProof (EQUIVALENT)
10.99/4.49	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. 
10.99/4.49	
10.99/4.49	From the DPs we obtained the following set of size-change graphs:
10.99/4.49	*new_primModNatS0(vz44, vz45, Main.Succ(vz460), Main.Succ(vz470)) -> new_primModNatS0(vz44, vz45, vz460, vz470)
10.99/4.49	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
10.99/4.49	
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(19)
10.99/4.49	YES
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(20)
10.99/4.49	Obligation:
10.99/4.49	Q DP problem:
10.99/4.49	The TRS P consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primModNatP(Main.Succ(Main.Zero), Main.Succ(Main.Zero)) -> new_primModNatP(new_primMinusNatS1, Main.Succ(Main.Zero))
10.99/4.49	   new_primModNatP00(vz39, vz40) -> new_primModNatP(new_primMinusNatS2(vz39, vz40), Main.Succ(Main.Succ(vz40)))
10.99/4.49	   new_primModNatP0(vz39, vz40, Main.Zero, Main.Zero) -> new_primModNatP00(vz39, vz40)
10.99/4.49	   new_primModNatP0(vz39, vz40, Main.Succ(vz410), Main.Succ(vz420)) -> new_primModNatP0(vz39, vz40, vz410, vz420)
10.99/4.49	   new_primModNatP0(vz39, vz40, Main.Succ(vz410), Main.Zero) -> new_primModNatP(new_primMinusNatS2(vz39, vz40), Main.Succ(Main.Succ(vz40)))
10.99/4.49	   new_primModNatP(Main.Succ(Main.Succ(vz3000)), Main.Succ(Main.Zero)) -> new_primModNatP(new_primMinusNatS0(vz3000), Main.Succ(Main.Zero))
10.99/4.49	   new_primModNatP(Main.Succ(Main.Succ(vz3000)), Main.Succ(Main.Succ(vz40000))) -> new_primModNatP0(vz3000, vz40000, vz3000, vz40000)
10.99/4.49	
10.99/4.49	The TRS R consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
10.99/4.49	   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
10.99/4.49	   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
10.99/4.49	   new_primMinusNatS0(Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
10.99/4.49	   new_primMinusNatS1 -> Main.Zero
10.99/4.49	
10.99/4.49	The set Q consists of the following terms:
10.99/4.49	
10.99/4.49	   new_primMinusNatS0(Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Zero, x0)
10.99/4.49	   new_primMinusNatS0(Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Succ(x0), x1)
10.99/4.49	   new_primMinusNatS1
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero)
10.99/4.49	
10.99/4.49	We have to consider all minimal (P,Q,R)-chains.
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(21) DependencyGraphProof (EQUIVALENT)
10.99/4.49	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(22)
10.99/4.49	Complex Obligation (AND)
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(23)
10.99/4.49	Obligation:
10.99/4.49	Q DP problem:
10.99/4.49	The TRS P consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primModNatP(Main.Succ(Main.Succ(vz3000)), Main.Succ(Main.Succ(vz40000))) -> new_primModNatP0(vz3000, vz40000, vz3000, vz40000)
10.99/4.49	   new_primModNatP0(vz39, vz40, Main.Zero, Main.Zero) -> new_primModNatP00(vz39, vz40)
10.99/4.49	   new_primModNatP00(vz39, vz40) -> new_primModNatP(new_primMinusNatS2(vz39, vz40), Main.Succ(Main.Succ(vz40)))
10.99/4.49	   new_primModNatP0(vz39, vz40, Main.Succ(vz410), Main.Succ(vz420)) -> new_primModNatP0(vz39, vz40, vz410, vz420)
10.99/4.49	   new_primModNatP0(vz39, vz40, Main.Succ(vz410), Main.Zero) -> new_primModNatP(new_primMinusNatS2(vz39, vz40), Main.Succ(Main.Succ(vz40)))
10.99/4.49	
10.99/4.49	The TRS R consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
10.99/4.49	   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
10.99/4.49	   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
10.99/4.49	   new_primMinusNatS0(Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
10.99/4.49	   new_primMinusNatS1 -> Main.Zero
10.99/4.49	
10.99/4.49	The set Q consists of the following terms:
10.99/4.49	
10.99/4.49	   new_primMinusNatS0(Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Zero, x0)
10.99/4.49	   new_primMinusNatS0(Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Succ(x0), x1)
10.99/4.49	   new_primMinusNatS1
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero)
10.99/4.49	
10.99/4.49	We have to consider all minimal (P,Q,R)-chains.
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(24) QDPSizeChangeProof (EQUIVALENT)
10.99/4.49	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
10.99/4.49	
10.99/4.49	Order:Polynomial interpretation [POLO]:
10.99/4.49	
10.99/4.49	   POL(Main.Succ(x_1)) = 1 + x_1
10.99/4.49	   POL(Main.Zero) = 1
10.99/4.49	   POL(new_primMinusNatS2(x_1, x_2)) = x_1
10.99/4.49	   POL(new_primMinusNatS3(x_1, x_2)) = x_1
10.99/4.49	
10.99/4.49	
10.99/4.49	
10.99/4.49	
10.99/4.49	From the DPs we obtained the following set of size-change graphs:
10.99/4.49	*new_primModNatP0(vz39, vz40, Main.Succ(vz410), Main.Zero) -> new_primModNatP(new_primMinusNatS2(vz39, vz40), Main.Succ(Main.Succ(vz40))) (allowed arguments on rhs = {1, 2})
10.99/4.49	The graph contains the following edges 1 >= 1
10.99/4.49	
10.99/4.49	
10.99/4.49	*new_primModNatP00(vz39, vz40) -> new_primModNatP(new_primMinusNatS2(vz39, vz40), Main.Succ(Main.Succ(vz40))) (allowed arguments on rhs = {1, 2})
10.99/4.49	The graph contains the following edges 1 >= 1
10.99/4.49	
10.99/4.49	
10.99/4.49	*new_primModNatP(Main.Succ(Main.Succ(vz3000)), Main.Succ(Main.Succ(vz40000))) -> new_primModNatP0(vz3000, vz40000, vz3000, vz40000) (allowed arguments on rhs = {1, 2, 3, 4})
10.99/4.49	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
10.99/4.49	
10.99/4.49	
10.99/4.49	*new_primModNatP0(vz39, vz40, Main.Succ(vz410), Main.Succ(vz420)) -> new_primModNatP0(vz39, vz40, vz410, vz420) (allowed arguments on rhs = {1, 2, 3, 4})
10.99/4.49	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
10.99/4.49	
10.99/4.49	
10.99/4.49	*new_primModNatP0(vz39, vz40, Main.Zero, Main.Zero) -> new_primModNatP00(vz39, vz40) (allowed arguments on rhs = {1, 2})
10.99/4.49	The graph contains the following edges 1 >= 1, 2 >= 2
10.99/4.49	
10.99/4.49	
10.99/4.49	
10.99/4.49	We oriented the following set of usable rules [AAECC05,FROCOS05].
10.99/4.49	
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
10.99/4.49	   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
10.99/4.49	   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(25)
10.99/4.49	YES
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(26)
10.99/4.49	Obligation:
10.99/4.49	Q DP problem:
10.99/4.49	The TRS P consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primModNatP(Main.Succ(Main.Succ(vz3000)), Main.Succ(Main.Zero)) -> new_primModNatP(new_primMinusNatS0(vz3000), Main.Succ(Main.Zero))
10.99/4.49	
10.99/4.49	The TRS R consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(vz3900)) -> Main.Zero
10.99/4.49	   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS3(vz400, vz3900)
10.99/4.49	   new_primMinusNatS2(Main.Zero, vz40) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS3(Main.Succ(vz400), Main.Zero) -> Main.Succ(vz400)
10.99/4.49	   new_primMinusNatS0(Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS2(Main.Succ(vz390), vz40) -> new_primMinusNatS3(vz390, vz40)
10.99/4.49	   new_primMinusNatS1 -> Main.Zero
10.99/4.49	
10.99/4.49	The set Q consists of the following terms:
10.99/4.49	
10.99/4.49	   new_primMinusNatS0(Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Succ(x0))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1))
10.99/4.49	   new_primMinusNatS3(Main.Succ(x0), Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Zero, x0)
10.99/4.49	   new_primMinusNatS0(Main.Zero)
10.99/4.49	   new_primMinusNatS2(Main.Succ(x0), x1)
10.99/4.49	   new_primMinusNatS1
10.99/4.49	   new_primMinusNatS3(Main.Zero, Main.Zero)
10.99/4.49	
10.99/4.49	We have to consider all minimal (P,Q,R)-chains.
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(27) QDPSizeChangeProof (EQUIVALENT)
10.99/4.49	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
10.99/4.49	
10.99/4.49	Order:Polynomial interpretation [POLO]:
10.99/4.49	
10.99/4.49	   POL(Main.Succ(x_1)) = 1 + x_1
10.99/4.49	   POL(Main.Zero) = 1
10.99/4.49	   POL(new_primMinusNatS0(x_1)) = x_1
10.99/4.49	
10.99/4.49	
10.99/4.49	
10.99/4.49	
10.99/4.49	From the DPs we obtained the following set of size-change graphs:
10.99/4.49	*new_primModNatP(Main.Succ(Main.Succ(vz3000)), Main.Succ(Main.Zero)) -> new_primModNatP(new_primMinusNatS0(vz3000), Main.Succ(Main.Zero)) (allowed arguments on rhs = {1, 2})
10.99/4.49	The graph contains the following edges 1 > 1, 2 >= 2
10.99/4.49	
10.99/4.49	
10.99/4.49	
10.99/4.49	We oriented the following set of usable rules [AAECC05,FROCOS05].
10.99/4.49	
10.99/4.49	   new_primMinusNatS0(Main.Zero) -> Main.Zero
10.99/4.49	   new_primMinusNatS0(Main.Succ(vz30000)) -> Main.Succ(vz30000)
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(28)
10.99/4.49	YES
10.99/4.49	
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(29)
10.99/4.49	Obligation:
10.99/4.49	Q DP problem:
10.99/4.49	The TRS P consists of the following rules:
10.99/4.49	
10.99/4.49	   new_primMinusNatS(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS(vz400, vz3900)
10.99/4.49	
10.99/4.49	R is empty.
10.99/4.49	Q is empty.
10.99/4.49	We have to consider all minimal (P,Q,R)-chains.
10.99/4.49	----------------------------------------
10.99/4.49	
10.99/4.49	(30) QDPSizeChangeProof (EQUIVALENT)
10.99/4.49	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. 
10.99/4.49	
10.99/4.49	From the DPs we obtained the following set of size-change graphs:
10.99/4.49	*new_primMinusNatS(Main.Succ(vz400), Main.Succ(vz3900)) -> new_primMinusNatS(vz400, vz3900)
10.99/4.49	The graph contains the following edges 1 > 1, 2 > 2
10.99/4.49	
10.99/4.49	
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10.99/4.49	
10.99/4.49	(31)
10.99/4.49	YES
11.06/7.76	EOF