8.64/3.80	YES
10.63/4.38	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.63/4.38	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.63/4.38	
10.63/4.38	
10.63/4.38	H-Termination with start terms of the given HASKELL could be proven:
10.63/4.38	
10.63/4.38	(0) HASKELL
10.63/4.38	(1) BR [EQUIVALENT, 0 ms]
10.63/4.38	(2) HASKELL
10.63/4.38	(3) COR [EQUIVALENT, 0 ms]
10.63/4.38	(4) HASKELL
10.63/4.38	(5) Narrow [SOUND, 0 ms]
10.63/4.38	(6) AND
10.63/4.38	    (7) QDP
10.63/4.38	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.63/4.38	        (9) YES
10.63/4.38	    (10) QDP
10.63/4.38	        (11) DependencyGraphProof [EQUIVALENT, 0 ms]
10.63/4.38	        (12) AND
10.63/4.38	            (13) QDP
10.63/4.38	                (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.63/4.38	                (15) YES
10.63/4.38	            (16) QDP
10.63/4.38	                (17) TransformationProof [EQUIVALENT, 0 ms]
10.63/4.38	                (18) QDP
10.63/4.38	                (19) TransformationProof [EQUIVALENT, 0 ms]
10.63/4.38	                (20) QDP
10.63/4.38	                (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.63/4.38	                (22) YES
10.63/4.38	    (23) QDP
10.63/4.38	        (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.63/4.38	        (25) YES
10.63/4.38	
10.63/4.38	
10.63/4.38	----------------------------------------
10.63/4.38	
10.63/4.38	(0)
10.63/4.38	Obligation:
10.63/4.38	mainModule Main 
10.63/4.38	module Main where {
10.63/4.38	import qualified Prelude;
10.63/4.38	data MyBool = MyTrue  | MyFalse ;
10.63/4.38	
10.63/4.38	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.63/4.38	
10.63/4.38	data Main.Nat = Succ Main.Nat  | Zero ;
10.63/4.38	
10.63/4.38	divMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.63/4.38	divMyInt = primDivInt;
10.63/4.38	
10.63/4.38	error :: a;
10.63/4.38	error = stop MyTrue;
10.63/4.38	
10.63/4.38	primDivInt :: MyInt  ->  MyInt  ->  MyInt;
10.63/4.38	primDivInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
10.63/4.38	primDivInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y));
10.63/4.38	primDivInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y));
10.63/4.38	primDivInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
10.63/4.38	primDivInt vv vw = Main.error;
10.63/4.38	
10.63/4.38	primDivNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.63/4.38	primDivNatP Main.Zero Main.Zero = Main.error;
10.63/4.38	primDivNatP (Main.Succ x) Main.Zero = Main.error;
10.63/4.38	primDivNatP (Main.Succ x) (Main.Succ y) = Main.Succ (primDivNatP (primMinusNatS x y) (Main.Succ y));
10.63/4.38	primDivNatP Main.Zero (Main.Succ x) = Main.Zero;
10.63/4.38	
10.63/4.38	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.63/4.38	primDivNatS Main.Zero Main.Zero = Main.error;
10.63/4.38	primDivNatS (Main.Succ x) Main.Zero = Main.error;
10.63/4.38	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
10.63/4.38	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
10.63/4.38	
10.63/4.38	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
10.63/4.38	primDivNatS0 x y MyFalse = Main.Zero;
10.63/4.38	
10.63/4.38	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.63/4.38	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
10.63/4.38	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
10.63/4.38	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
10.63/4.38	primGEqNatS Main.Zero Main.Zero = MyTrue;
10.63/4.38	
10.63/4.38	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.63/4.38	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
10.63/4.38	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
10.63/4.38	primMinusNatS x Main.Zero = x;
10.63/4.38	
10.63/4.38	stop :: MyBool  ->  a;
10.63/4.38	stop MyFalse = stop MyFalse;
10.63/4.38	
10.63/4.38	}
10.63/4.38	
10.63/4.38	----------------------------------------
10.63/4.38	
10.63/4.38	(1) BR (EQUIVALENT)
10.63/4.38	Replaced joker patterns by fresh variables and removed binding patterns.
10.63/4.38	----------------------------------------
10.63/4.38	
10.63/4.38	(2)
10.63/4.38	Obligation:
10.63/4.38	mainModule Main 
10.63/4.38	module Main where {
10.63/4.38	import qualified Prelude;
10.63/4.38	data MyBool = MyTrue  | MyFalse ;
10.63/4.38	
10.63/4.38	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.63/4.38	
10.63/4.38	data Main.Nat = Succ Main.Nat  | Zero ;
10.63/4.38	
10.63/4.38	divMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.63/4.38	divMyInt = primDivInt;
10.63/4.38	
10.63/4.38	error :: a;
10.63/4.38	error = stop MyTrue;
10.63/4.38	
10.63/4.38	primDivInt :: MyInt  ->  MyInt  ->  MyInt;
10.63/4.38	primDivInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
10.63/4.38	primDivInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y));
10.63/4.38	primDivInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y));
10.63/4.38	primDivInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
10.63/4.38	primDivInt vv vw = Main.error;
10.63/4.38	
10.63/4.38	primDivNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.63/4.38	primDivNatP Main.Zero Main.Zero = Main.error;
10.63/4.38	primDivNatP (Main.Succ x) Main.Zero = Main.error;
10.63/4.38	primDivNatP (Main.Succ x) (Main.Succ y) = Main.Succ (primDivNatP (primMinusNatS x y) (Main.Succ y));
10.63/4.38	primDivNatP Main.Zero (Main.Succ x) = Main.Zero;
10.63/4.38	
10.63/4.38	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.63/4.38	primDivNatS Main.Zero Main.Zero = Main.error;
10.63/4.38	primDivNatS (Main.Succ x) Main.Zero = Main.error;
10.63/4.38	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
10.63/4.38	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
10.63/4.38	
10.63/4.38	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
10.63/4.38	primDivNatS0 x y MyFalse = Main.Zero;
10.63/4.38	
10.63/4.38	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.63/4.38	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
10.63/4.38	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
10.63/4.38	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
10.63/4.38	primGEqNatS Main.Zero Main.Zero = MyTrue;
10.63/4.38	
10.63/4.38	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.63/4.38	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
10.63/4.38	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
10.63/4.38	primMinusNatS x Main.Zero = x;
10.63/4.38	
10.63/4.38	stop :: MyBool  ->  a;
10.63/4.38	stop MyFalse = stop MyFalse;
10.63/4.38	
10.63/4.38	}
10.63/4.38	
10.63/4.38	----------------------------------------
10.63/4.38	
10.63/4.38	(3) COR (EQUIVALENT)
10.63/4.38	Cond Reductions:
10.63/4.38	The following Function with conditions
10.63/4.38	"undefined |Falseundefined;
10.63/4.38	"
10.63/4.38	is transformed to
10.63/4.38	"undefined  = undefined1;
10.63/4.38	"
10.63/4.38	"undefined0 True = undefined;
10.63/4.38	"
10.63/4.38	"undefined1  = undefined0 False;
10.63/4.38	"
10.63/4.38	
10.63/4.38	----------------------------------------
10.63/4.38	
10.63/4.38	(4)
10.63/4.38	Obligation:
10.63/4.38	mainModule Main 
10.63/4.38	module Main where {
10.63/4.38	import qualified Prelude;
10.63/4.38	data MyBool = MyTrue  | MyFalse ;
10.63/4.38	
10.63/4.38	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.63/4.38	
10.63/4.38	data Main.Nat = Succ Main.Nat  | Zero ;
10.63/4.38	
10.63/4.38	divMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.63/4.38	divMyInt = primDivInt;
10.63/4.38	
10.63/4.38	error :: a;
10.63/4.38	error = stop MyTrue;
10.63/4.38	
10.63/4.38	primDivInt :: MyInt  ->  MyInt  ->  MyInt;
10.63/4.38	primDivInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
10.63/4.38	primDivInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y));
10.63/4.38	primDivInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y));
10.63/4.38	primDivInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
10.63/4.38	primDivInt vv vw = Main.error;
10.63/4.38	
10.63/4.38	primDivNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.83/4.38	primDivNatP Main.Zero Main.Zero = Main.error;
10.83/4.38	primDivNatP (Main.Succ x) Main.Zero = Main.error;
10.83/4.38	primDivNatP (Main.Succ x) (Main.Succ y) = Main.Succ (primDivNatP (primMinusNatS x y) (Main.Succ y));
10.83/4.38	primDivNatP Main.Zero (Main.Succ x) = Main.Zero;
10.83/4.38	
10.83/4.38	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.83/4.38	primDivNatS Main.Zero Main.Zero = Main.error;
10.83/4.38	primDivNatS (Main.Succ x) Main.Zero = Main.error;
10.83/4.38	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
10.83/4.38	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
10.83/4.38	
10.83/4.38	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
10.83/4.38	primDivNatS0 x y MyFalse = Main.Zero;
10.83/4.38	
10.83/4.38	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.83/4.38	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
10.83/4.38	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
10.83/4.38	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
10.83/4.38	primGEqNatS Main.Zero Main.Zero = MyTrue;
10.83/4.38	
10.83/4.38	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.83/4.38	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
10.83/4.38	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
10.83/4.38	primMinusNatS x Main.Zero = x;
10.83/4.38	
10.83/4.38	stop :: MyBool  ->  a;
10.83/4.38	stop MyFalse = stop MyFalse;
10.83/4.38	
10.83/4.38	}
10.83/4.38	
10.83/4.38	----------------------------------------
10.83/4.38	
10.83/4.38	(5) Narrow (SOUND)
10.83/4.38	Haskell To QDPs
10.83/4.38	
10.83/4.38	digraph dp_graph {
10.83/4.38	node [outthreshold=100, inthreshold=100];1[label="divMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.83/4.38	3[label="divMyInt vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.83/4.38	4[label="divMyInt vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.83/4.38	5[label="primDivInt vz3 vz4",fontsize=16,color="burlywood",shape="box"];289[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 289[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	289 -> 6[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	290[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 290[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	290 -> 7[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	6[label="primDivInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];291[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 291[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	291 -> 8[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	292[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 292[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	292 -> 9[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	7[label="primDivInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];293[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 293[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	293 -> 10[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	294[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 294[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	294 -> 11[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	8[label="primDivInt (Pos vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];295[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 295[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	295 -> 12[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	296[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 296[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	296 -> 13[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	9[label="primDivInt (Pos vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];297[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 297[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	297 -> 14[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	298[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 298[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	298 -> 15[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	10[label="primDivInt (Neg vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];299[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 299[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	299 -> 16[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	300[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 300[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	300 -> 17[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	11[label="primDivInt (Neg vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];301[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 301[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	301 -> 18[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	302[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 302[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	302 -> 19[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	12[label="primDivInt (Pos vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
10.83/4.38	13[label="primDivInt (Pos vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
10.83/4.38	14[label="primDivInt (Pos vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
10.83/4.38	15[label="primDivInt (Pos vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
10.83/4.38	16[label="primDivInt (Neg vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
10.83/4.38	17[label="primDivInt (Neg vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
10.83/4.38	18[label="primDivInt (Neg vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
10.83/4.38	19[label="primDivInt (Neg vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
10.83/4.38	20[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3];
10.83/4.38	21[label="error",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3];
10.83/4.38	22[label="Neg (primDivNatP vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];22 -> 30[label="",style="dashed", color="green", weight=3];
10.83/4.38	23 -> 21[label="",style="dashed", color="red", weight=0];
10.83/4.38	23[label="error",fontsize=16,color="magenta"];24[label="Neg (primDivNatP vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];24 -> 31[label="",style="dashed", color="green", weight=3];
10.83/4.38	25 -> 21[label="",style="dashed", color="red", weight=0];
10.83/4.38	25[label="error",fontsize=16,color="magenta"];26[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];26 -> 32[label="",style="dashed", color="green", weight=3];
10.83/4.38	27 -> 21[label="",style="dashed", color="red", weight=0];
10.83/4.38	27[label="error",fontsize=16,color="magenta"];28[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];303[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 303[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	303 -> 33[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	304[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 304[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	304 -> 34[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	29[label="stop MyTrue",fontsize=16,color="black",shape="box"];29 -> 35[label="",style="solid", color="black", weight=3];
10.83/4.38	30[label="primDivNatP vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];305[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];30 -> 305[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	305 -> 36[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	306[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 306[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	306 -> 37[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	31 -> 30[label="",style="dashed", color="red", weight=0];
10.83/4.38	31[label="primDivNatP vz30 (Succ vz400)",fontsize=16,color="magenta"];31 -> 38[label="",style="dashed", color="magenta", weight=3];
10.83/4.38	31 -> 39[label="",style="dashed", color="magenta", weight=3];
10.83/4.38	32 -> 28[label="",style="dashed", color="red", weight=0];
10.83/4.38	32[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];32 -> 40[label="",style="dashed", color="magenta", weight=3];
10.83/4.38	32 -> 41[label="",style="dashed", color="magenta", weight=3];
10.83/4.38	33[label="primDivNatS (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3];
10.83/4.38	34[label="primDivNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 43[label="",style="solid", color="black", weight=3];
10.83/4.38	35[label="error []",fontsize=16,color="red",shape="box"];36[label="primDivNatP (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];36 -> 44[label="",style="solid", color="black", weight=3];
10.83/4.38	37[label="primDivNatP Zero (Succ vz400)",fontsize=16,color="black",shape="box"];37 -> 45[label="",style="solid", color="black", weight=3];
10.83/4.38	38[label="vz30",fontsize=16,color="green",shape="box"];39[label="vz400",fontsize=16,color="green",shape="box"];40[label="vz30",fontsize=16,color="green",shape="box"];41[label="vz400",fontsize=16,color="green",shape="box"];42[label="primDivNatS0 vz300 vz400 (primGEqNatS vz300 vz400)",fontsize=16,color="burlywood",shape="box"];307[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];42 -> 307[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	307 -> 46[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	308[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 308[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	308 -> 47[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	43[label="Zero",fontsize=16,color="green",shape="box"];44[label="Succ (primDivNatP (primMinusNatS vz300 vz400) (Succ vz400))",fontsize=16,color="green",shape="box"];44 -> 48[label="",style="dashed", color="green", weight=3];
10.83/4.38	45[label="Zero",fontsize=16,color="green",shape="box"];46[label="primDivNatS0 (Succ vz3000) vz400 (primGEqNatS (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];309[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];46 -> 309[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	309 -> 49[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	310[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 310[label="",style="solid", color="burlywood", weight=9];
10.83/4.38	310 -> 50[label="",style="solid", color="burlywood", weight=3];
10.83/4.38	47[label="primDivNatS0 Zero vz400 (primGEqNatS Zero vz400)",fontsize=16,color="burlywood",shape="box"];311[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];47 -> 311[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	311 -> 51[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	312[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 312[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	312 -> 52[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	48 -> 30[label="",style="dashed", color="red", weight=0];
10.83/4.39	48[label="primDivNatP (primMinusNatS vz300 vz400) (Succ vz400)",fontsize=16,color="magenta"];48 -> 53[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	49[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];49 -> 54[label="",style="solid", color="black", weight=3];
10.83/4.39	50[label="primDivNatS0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3];
10.83/4.39	51[label="primDivNatS0 Zero (Succ vz4000) (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];51 -> 56[label="",style="solid", color="black", weight=3];
10.83/4.39	52[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];52 -> 57[label="",style="solid", color="black", weight=3];
10.83/4.39	53[label="primMinusNatS vz300 vz400",fontsize=16,color="burlywood",shape="triangle"];313[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];53 -> 313[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	313 -> 58[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	314[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 314[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	314 -> 59[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	54 -> 237[label="",style="dashed", color="red", weight=0];
10.83/4.39	54[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)",fontsize=16,color="magenta"];54 -> 238[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	54 -> 239[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	54 -> 240[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	54 -> 241[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	55[label="primDivNatS0 (Succ vz3000) Zero MyTrue",fontsize=16,color="black",shape="box"];55 -> 62[label="",style="solid", color="black", weight=3];
10.83/4.39	56[label="primDivNatS0 Zero (Succ vz4000) MyFalse",fontsize=16,color="black",shape="box"];56 -> 63[label="",style="solid", color="black", weight=3];
10.83/4.39	57[label="primDivNatS0 Zero Zero MyTrue",fontsize=16,color="black",shape="box"];57 -> 64[label="",style="solid", color="black", weight=3];
10.83/4.39	58[label="primMinusNatS (Succ vz3000) vz400",fontsize=16,color="burlywood",shape="box"];315[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];58 -> 315[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	315 -> 65[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	316[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 316[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	316 -> 66[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	59[label="primMinusNatS Zero vz400",fontsize=16,color="burlywood",shape="box"];317[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];59 -> 317[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	317 -> 67[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	318[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 318[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	318 -> 68[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	238[label="vz4000",fontsize=16,color="green",shape="box"];239[label="vz4000",fontsize=16,color="green",shape="box"];240[label="vz3000",fontsize=16,color="green",shape="box"];241[label="vz3000",fontsize=16,color="green",shape="box"];237[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz23 vz24)",fontsize=16,color="burlywood",shape="triangle"];319[label="vz23/Succ vz230",fontsize=10,color="white",style="solid",shape="box"];237 -> 319[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	319 -> 270[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	320[label="vz23/Zero",fontsize=10,color="white",style="solid",shape="box"];237 -> 320[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	320 -> 271[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	62[label="Succ (primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];62 -> 73[label="",style="dashed", color="green", weight=3];
10.83/4.39	63[label="Zero",fontsize=16,color="green",shape="box"];64[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];64 -> 74[label="",style="dashed", color="green", weight=3];
10.83/4.39	65[label="primMinusNatS (Succ vz3000) (Succ vz4000)",fontsize=16,color="black",shape="box"];65 -> 75[label="",style="solid", color="black", weight=3];
10.83/4.39	66[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3];
10.83/4.39	67[label="primMinusNatS Zero (Succ vz4000)",fontsize=16,color="black",shape="box"];67 -> 77[label="",style="solid", color="black", weight=3];
10.83/4.39	68[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];68 -> 78[label="",style="solid", color="black", weight=3];
10.83/4.39	270[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) vz24)",fontsize=16,color="burlywood",shape="box"];321[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];270 -> 321[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	321 -> 272[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	322[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];270 -> 322[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	322 -> 273[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	271[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero vz24)",fontsize=16,color="burlywood",shape="box"];323[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];271 -> 323[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	323 -> 274[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	324[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];271 -> 324[label="",style="solid", color="burlywood", weight=9];
10.83/4.39	324 -> 275[label="",style="solid", color="burlywood", weight=3];
10.83/4.39	73 -> 28[label="",style="dashed", color="red", weight=0];
10.83/4.39	73[label="primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];73 -> 83[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	73 -> 84[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	74 -> 28[label="",style="dashed", color="red", weight=0];
10.83/4.39	74[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];74 -> 85[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	74 -> 86[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	75 -> 53[label="",style="dashed", color="red", weight=0];
10.83/4.39	75[label="primMinusNatS vz3000 vz4000",fontsize=16,color="magenta"];75 -> 87[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	75 -> 88[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	76[label="Succ vz3000",fontsize=16,color="green",shape="box"];77[label="Zero",fontsize=16,color="green",shape="box"];78[label="Zero",fontsize=16,color="green",shape="box"];272[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) (Succ vz240))",fontsize=16,color="black",shape="box"];272 -> 276[label="",style="solid", color="black", weight=3];
10.83/4.39	273[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) Zero)",fontsize=16,color="black",shape="box"];273 -> 277[label="",style="solid", color="black", weight=3];
10.83/4.39	274[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero (Succ vz240))",fontsize=16,color="black",shape="box"];274 -> 278[label="",style="solid", color="black", weight=3];
10.83/4.39	275[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];275 -> 279[label="",style="solid", color="black", weight=3];
10.83/4.39	83 -> 53[label="",style="dashed", color="red", weight=0];
10.83/4.39	83[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="magenta"];83 -> 94[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	83 -> 95[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	84[label="Zero",fontsize=16,color="green",shape="box"];85 -> 53[label="",style="dashed", color="red", weight=0];
10.83/4.39	85[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];85 -> 96[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	85 -> 97[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	86[label="Zero",fontsize=16,color="green",shape="box"];87[label="vz4000",fontsize=16,color="green",shape="box"];88[label="vz3000",fontsize=16,color="green",shape="box"];276 -> 237[label="",style="dashed", color="red", weight=0];
10.83/4.39	276[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz230 vz240)",fontsize=16,color="magenta"];276 -> 280[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	276 -> 281[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	277[label="primDivNatS0 (Succ vz21) (Succ vz22) MyTrue",fontsize=16,color="black",shape="triangle"];277 -> 282[label="",style="solid", color="black", weight=3];
10.83/4.39	278[label="primDivNatS0 (Succ vz21) (Succ vz22) MyFalse",fontsize=16,color="black",shape="box"];278 -> 283[label="",style="solid", color="black", weight=3];
10.83/4.39	279 -> 277[label="",style="dashed", color="red", weight=0];
10.83/4.39	279[label="primDivNatS0 (Succ vz21) (Succ vz22) MyTrue",fontsize=16,color="magenta"];94[label="Zero",fontsize=16,color="green",shape="box"];95[label="Succ vz3000",fontsize=16,color="green",shape="box"];96[label="Zero",fontsize=16,color="green",shape="box"];97[label="Zero",fontsize=16,color="green",shape="box"];280[label="vz240",fontsize=16,color="green",shape="box"];281[label="vz230",fontsize=16,color="green",shape="box"];282[label="Succ (primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22)))",fontsize=16,color="green",shape="box"];282 -> 284[label="",style="dashed", color="green", weight=3];
10.83/4.39	283[label="Zero",fontsize=16,color="green",shape="box"];284 -> 28[label="",style="dashed", color="red", weight=0];
10.83/4.39	284[label="primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22))",fontsize=16,color="magenta"];284 -> 285[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	284 -> 286[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	285 -> 53[label="",style="dashed", color="red", weight=0];
10.83/4.39	285[label="primMinusNatS (Succ vz21) (Succ vz22)",fontsize=16,color="magenta"];285 -> 287[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	285 -> 288[label="",style="dashed", color="magenta", weight=3];
10.83/4.39	286[label="Succ vz22",fontsize=16,color="green",shape="box"];287[label="Succ vz22",fontsize=16,color="green",shape="box"];288[label="Succ vz21",fontsize=16,color="green",shape="box"];}
10.83/4.39	
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(6)
10.83/4.39	Complex Obligation (AND)
10.83/4.39	
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(7)
10.83/4.39	Obligation:
10.83/4.39	Q DP problem:
10.83/4.39	The TRS P consists of the following rules:
10.83/4.39	
10.83/4.39	   new_primDivNatP(Main.Succ(vz300), vz400) -> new_primDivNatP(new_primMinusNatS0(vz300, vz400), vz400)
10.83/4.39	
10.83/4.39	The TRS R consists of the following rules:
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000)
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero
10.83/4.39	
10.83/4.39	The set Q consists of the following terms:
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero)
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
10.83/4.39	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
10.83/4.39	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
10.83/4.39	
10.83/4.39	We have to consider all minimal (P,Q,R)-chains.
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(8) QDPSizeChangeProof (EQUIVALENT)
10.83/4.39	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
10.83/4.39	
10.83/4.39	Order:Polynomial interpretation [POLO]:
10.83/4.39	
10.83/4.39	   POL(Main.Succ(x_1)) = 1 + x_1
10.83/4.39	   POL(Main.Zero) = 0
10.83/4.39	   POL(new_primMinusNatS0(x_1, x_2)) = x_1
10.83/4.39	
10.83/4.39	
10.83/4.39	
10.83/4.39	
10.83/4.39	From the DPs we obtained the following set of size-change graphs:
10.83/4.39	*new_primDivNatP(Main.Succ(vz300), vz400) -> new_primDivNatP(new_primMinusNatS0(vz300, vz400), vz400) (allowed arguments on rhs = {1, 2})
10.83/4.39	The graph contains the following edges 1 > 1, 2 >= 2
10.83/4.39	
10.83/4.39	
10.83/4.39	
10.83/4.39	We oriented the following set of usable rules [AAECC05,FROCOS05].
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000)
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.83/4.39	
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(9)
10.83/4.39	YES
10.83/4.39	
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(10)
10.83/4.39	Obligation:
10.83/4.39	Q DP problem:
10.83/4.39	The TRS P consists of the following rules:
10.83/4.39	
10.83/4.39	   new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240)
10.83/4.39	   new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22))
10.83/4.39	   new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22))
10.83/4.39	   new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz3000), Main.Zero), Main.Zero)
10.83/4.39	   new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22)
10.83/4.39	   new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
10.83/4.39	   new_primDivNatS(Main.Succ(Main.Zero), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Zero, Main.Zero), Main.Zero)
10.83/4.39	
10.83/4.39	The TRS R consists of the following rules:
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000)
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero
10.83/4.39	
10.83/4.39	The set Q consists of the following terms:
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero)
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
10.83/4.39	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
10.83/4.39	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
10.83/4.39	
10.83/4.39	We have to consider all minimal (P,Q,R)-chains.
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(11) DependencyGraphProof (EQUIVALENT)
10.83/4.39	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(12)
10.83/4.39	Complex Obligation (AND)
10.83/4.39	
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(13)
10.83/4.39	Obligation:
10.83/4.39	Q DP problem:
10.83/4.39	The TRS P consists of the following rules:
10.83/4.39	
10.83/4.39	   new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz3000), Main.Zero), Main.Zero)
10.83/4.39	
10.83/4.39	The TRS R consists of the following rules:
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000)
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero
10.83/4.39	
10.83/4.39	The set Q consists of the following terms:
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero)
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
10.83/4.39	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
10.83/4.39	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
10.83/4.39	
10.83/4.39	We have to consider all minimal (P,Q,R)-chains.
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(14) QDPSizeChangeProof (EQUIVALENT)
10.83/4.39	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
10.83/4.39	
10.83/4.39	Order:Polynomial interpretation [POLO]:
10.83/4.39	
10.83/4.39	   POL(Main.Succ(x_1)) = 1 + x_1
10.83/4.39	   POL(Main.Zero) = 1
10.83/4.39	   POL(new_primMinusNatS0(x_1, x_2)) = x_1
10.83/4.39	
10.83/4.39	
10.83/4.39	
10.83/4.39	
10.83/4.39	From the DPs we obtained the following set of size-change graphs:
10.83/4.39	*new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz3000), Main.Zero), Main.Zero) (allowed arguments on rhs = {1, 2})
10.83/4.39	The graph contains the following edges 1 > 1, 2 >= 2
10.83/4.39	
10.83/4.39	
10.83/4.39	
10.83/4.39	We oriented the following set of usable rules [AAECC05,FROCOS05].
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000)
10.83/4.39	
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(15)
10.83/4.39	YES
10.83/4.39	
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(16)
10.83/4.39	Obligation:
10.83/4.39	Q DP problem:
10.83/4.39	The TRS P consists of the following rules:
10.83/4.39	
10.83/4.39	   new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22))
10.83/4.39	   new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
10.83/4.39	   new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240)
10.83/4.39	   new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22)
10.83/4.39	   new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22))
10.83/4.39	
10.83/4.39	The TRS R consists of the following rules:
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000)
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero
10.83/4.39	
10.83/4.39	The set Q consists of the following terms:
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero)
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
10.83/4.39	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
10.83/4.39	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
10.83/4.39	
10.83/4.39	We have to consider all minimal (P,Q,R)-chains.
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(17) TransformationProof (EQUIVALENT)
10.83/4.39	By rewriting [LPAR04] the rule new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22)) at position [0] we obtained the following new rules [LPAR04]:
10.83/4.39	
10.83/4.39	   (new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)),new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)))
10.83/4.39	
10.83/4.39	
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(18)
10.83/4.39	Obligation:
10.83/4.39	Q DP problem:
10.83/4.39	The TRS P consists of the following rules:
10.83/4.39	
10.83/4.39	   new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
10.83/4.39	   new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240)
10.83/4.39	   new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22)
10.83/4.39	   new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22))
10.83/4.39	   new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22))
10.83/4.39	
10.83/4.39	The TRS R consists of the following rules:
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000)
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero
10.83/4.39	
10.83/4.39	The set Q consists of the following terms:
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero)
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
10.83/4.39	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
10.83/4.39	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
10.83/4.39	
10.83/4.39	We have to consider all minimal (P,Q,R)-chains.
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(19) TransformationProof (EQUIVALENT)
10.83/4.39	By rewriting [LPAR04] the rule new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22)) at position [0] we obtained the following new rules [LPAR04]:
10.83/4.39	
10.83/4.39	   (new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)),new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)))
10.83/4.39	
10.83/4.39	
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(20)
10.83/4.39	Obligation:
10.83/4.39	Q DP problem:
10.83/4.39	The TRS P consists of the following rules:
10.83/4.39	
10.83/4.39	   new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
10.83/4.39	   new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240)
10.83/4.39	   new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22)
10.83/4.39	   new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22))
10.83/4.39	   new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22))
10.83/4.39	
10.83/4.39	The TRS R consists of the following rules:
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000)
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero
10.83/4.39	
10.83/4.39	The set Q consists of the following terms:
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero)
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
10.83/4.39	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
10.83/4.39	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
10.83/4.39	
10.83/4.39	We have to consider all minimal (P,Q,R)-chains.
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(21) QDPSizeChangeProof (EQUIVALENT)
10.83/4.39	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
10.83/4.39	
10.83/4.39	Order:Polynomial interpretation [POLO]:
10.83/4.39	
10.83/4.39	   POL(Main.Succ(x_1)) = 1 + x_1
10.83/4.39	   POL(Main.Zero) = 1
10.83/4.39	   POL(new_primMinusNatS0(x_1, x_2)) = x_1
10.83/4.39	
10.83/4.39	
10.83/4.39	
10.83/4.39	
10.83/4.39	From the DPs we obtained the following set of size-change graphs:
10.83/4.39	*new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) (allowed arguments on rhs = {1, 2})
10.83/4.39	The graph contains the following edges 1 >= 1
10.83/4.39	
10.83/4.39	
10.83/4.39	*new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) (allowed arguments on rhs = {1, 2})
10.83/4.39	The graph contains the following edges 1 >= 1
10.83/4.39	
10.83/4.39	
10.83/4.39	*new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) (allowed arguments on rhs = {1, 2, 3, 4})
10.83/4.39	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
10.83/4.39	
10.83/4.39	
10.83/4.39	*new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) (allowed arguments on rhs = {1, 2, 3, 4})
10.83/4.39	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
10.83/4.39	
10.83/4.39	
10.83/4.39	*new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22) (allowed arguments on rhs = {1, 2})
10.83/4.39	The graph contains the following edges 1 >= 1, 2 >= 2
10.83/4.39	
10.83/4.39	
10.83/4.39	
10.83/4.39	We oriented the following set of usable rules [AAECC05,FROCOS05].
10.83/4.39	
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
10.83/4.39	   new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000)
10.83/4.39	   new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000)
10.83/4.39	
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(22)
10.83/4.39	YES
10.83/4.39	
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(23)
10.83/4.39	Obligation:
10.83/4.39	Q DP problem:
10.83/4.39	The TRS P consists of the following rules:
10.83/4.39	
10.83/4.39	   new_primMinusNatS(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS(vz3000, vz4000)
10.83/4.39	
10.83/4.39	R is empty.
10.83/4.39	Q is empty.
10.83/4.39	We have to consider all minimal (P,Q,R)-chains.
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(24) QDPSizeChangeProof (EQUIVALENT)
10.83/4.39	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. 
10.83/4.39	
10.83/4.39	From the DPs we obtained the following set of size-change graphs:
10.83/4.39	*new_primMinusNatS(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS(vz3000, vz4000)
10.83/4.39	The graph contains the following edges 1 > 1, 2 > 2
10.83/4.39	
10.83/4.39	
10.83/4.39	----------------------------------------
10.83/4.39	
10.83/4.39	(25)
10.83/4.39	YES
10.83/4.44	EOF