8.22/3.61	YES
10.28/4.16	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.28/4.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.28/4.16	
10.28/4.16	
10.28/4.16	H-Termination with start terms of the given HASKELL could be proven:
10.28/4.16	
10.28/4.16	(0) HASKELL
10.28/4.16	(1) BR [EQUIVALENT, 0 ms]
10.28/4.16	(2) HASKELL
10.28/4.16	(3) COR [EQUIVALENT, 0 ms]
10.28/4.16	(4) HASKELL
10.28/4.16	(5) Narrow [SOUND, 0 ms]
10.28/4.16	(6) AND
10.28/4.16	    (7) QDP
10.28/4.16	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.28/4.16	        (9) YES
10.28/4.16	    (10) QDP
10.28/4.16	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.28/4.16	        (12) YES
10.28/4.16	    (13) QDP
10.28/4.16	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.28/4.16	        (15) YES
10.28/4.16	
10.28/4.16	
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(0)
10.28/4.16	Obligation:
10.28/4.16	mainModule Main 
10.28/4.16	module Main where {
10.28/4.16	import qualified Prelude;
10.28/4.16	data Float = Float MyInt MyInt ;
10.28/4.16	
10.28/4.16	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.28/4.16	
10.28/4.16	data Main.Nat = Succ Main.Nat  | Zero ;
10.28/4.16	
10.28/4.16	flip :: (c  ->  b  ->  a)  ->  b  ->  c  ->  a;
10.28/4.16	flip f x y = f y x;
10.28/4.16	
10.28/4.16	msFloat :: Float  ->  Float  ->  Float;
10.28/4.16	msFloat = primMinusFloat;
10.28/4.16	
10.28/4.16	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.28/4.16	msMyInt = primMinusInt;
10.28/4.16	
10.28/4.16	primMinusFloat :: Float  ->  Float  ->  Float;
10.28/4.16	primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);
10.28/4.16	
10.28/4.16	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
10.28/4.16	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
10.28/4.16	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
10.28/4.16	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
10.28/4.16	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
10.28/4.16	
10.28/4.16	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.28/4.16	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.28/4.16	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.28/4.16	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.28/4.16	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.28/4.16	
10.28/4.16	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
10.28/4.16	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
10.28/4.16	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
10.28/4.16	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
10.28/4.16	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
10.28/4.16	
10.28/4.16	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.28/4.16	primMulNat Main.Zero Main.Zero = Main.Zero;
10.28/4.16	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
10.28/4.16	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
10.28/4.16	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
10.28/4.16	
10.28/4.16	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.28/4.16	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.28/4.16	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.28/4.16	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.28/4.16	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.28/4.16	
10.28/4.16	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.28/4.16	srMyInt = primMulInt;
10.28/4.16	
10.28/4.16	subtractFloat :: Float  ->  Float  ->  Float;
10.28/4.16	subtractFloat = flip msFloat;
10.28/4.16	
10.28/4.16	}
10.28/4.16	
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(1) BR (EQUIVALENT)
10.28/4.16	Replaced joker patterns by fresh variables and removed binding patterns.
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(2)
10.28/4.16	Obligation:
10.28/4.16	mainModule Main 
10.28/4.16	module Main where {
10.28/4.16	import qualified Prelude;
10.28/4.16	data Float = Float MyInt MyInt ;
10.28/4.16	
10.28/4.16	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.28/4.16	
10.28/4.16	data Main.Nat = Succ Main.Nat  | Zero ;
10.28/4.16	
10.28/4.16	flip :: (c  ->  b  ->  a)  ->  b  ->  c  ->  a;
10.28/4.16	flip f x y = f y x;
10.28/4.16	
10.28/4.16	msFloat :: Float  ->  Float  ->  Float;
10.28/4.16	msFloat = primMinusFloat;
10.28/4.16	
10.28/4.16	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.28/4.16	msMyInt = primMinusInt;
10.28/4.16	
10.28/4.16	primMinusFloat :: Float  ->  Float  ->  Float;
10.28/4.16	primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);
10.28/4.16	
10.28/4.16	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
10.28/4.16	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
10.28/4.16	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
10.28/4.16	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
10.28/4.16	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
10.28/4.16	
10.28/4.16	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.28/4.16	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.28/4.16	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.28/4.16	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.28/4.16	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.28/4.16	
10.28/4.16	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
10.28/4.16	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
10.28/4.16	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
10.28/4.16	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
10.28/4.16	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
10.28/4.16	
10.28/4.16	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.28/4.16	primMulNat Main.Zero Main.Zero = Main.Zero;
10.28/4.16	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
10.28/4.16	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
10.28/4.16	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
10.28/4.16	
10.28/4.16	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.28/4.16	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.28/4.16	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.28/4.16	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.28/4.16	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.28/4.16	
10.28/4.16	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.28/4.16	srMyInt = primMulInt;
10.28/4.16	
10.28/4.16	subtractFloat :: Float  ->  Float  ->  Float;
10.28/4.16	subtractFloat = flip msFloat;
10.28/4.16	
10.28/4.16	}
10.28/4.16	
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(3) COR (EQUIVALENT)
10.28/4.16	Cond Reductions:
10.28/4.16	The following Function with conditions
10.28/4.16	"undefined |Falseundefined;
10.28/4.16	"
10.28/4.16	is transformed to
10.28/4.16	"undefined  = undefined1;
10.28/4.16	"
10.28/4.16	"undefined0 True = undefined;
10.28/4.16	"
10.28/4.16	"undefined1  = undefined0 False;
10.28/4.16	"
10.28/4.16	
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(4)
10.28/4.16	Obligation:
10.28/4.16	mainModule Main 
10.28/4.16	module Main where {
10.28/4.16	import qualified Prelude;
10.28/4.16	data Float = Float MyInt MyInt ;
10.28/4.16	
10.28/4.16	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.28/4.16	
10.28/4.16	data Main.Nat = Succ Main.Nat  | Zero ;
10.28/4.16	
10.28/4.16	flip :: (a  ->  c  ->  b)  ->  c  ->  a  ->  b;
10.28/4.16	flip f x y = f y x;
10.28/4.16	
10.28/4.16	msFloat :: Float  ->  Float  ->  Float;
10.28/4.16	msFloat = primMinusFloat;
10.28/4.16	
10.28/4.16	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.28/4.16	msMyInt = primMinusInt;
10.28/4.16	
10.28/4.16	primMinusFloat :: Float  ->  Float  ->  Float;
10.28/4.16	primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);
10.28/4.16	
10.28/4.16	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
10.28/4.16	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
10.28/4.16	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
10.28/4.16	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
10.28/4.16	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
10.28/4.16	
10.28/4.16	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.28/4.16	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.28/4.16	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.28/4.16	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.28/4.16	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.28/4.16	
10.28/4.16	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
10.28/4.16	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
10.28/4.16	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
10.28/4.16	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
10.28/4.16	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
10.28/4.16	
10.28/4.16	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.28/4.16	primMulNat Main.Zero Main.Zero = Main.Zero;
10.28/4.16	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
10.28/4.16	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
10.28/4.16	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
10.28/4.16	
10.28/4.16	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.28/4.16	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.28/4.16	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.28/4.16	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.28/4.16	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.28/4.16	
10.28/4.16	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.28/4.16	srMyInt = primMulInt;
10.28/4.16	
10.28/4.16	subtractFloat :: Float  ->  Float  ->  Float;
10.28/4.16	subtractFloat = flip msFloat;
10.28/4.16	
10.28/4.16	}
10.28/4.16	
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(5) Narrow (SOUND)
10.28/4.16	Haskell To QDPs
10.28/4.16	
10.28/4.16	digraph dp_graph {
10.28/4.16	node [outthreshold=100, inthreshold=100];1[label="subtractFloat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.28/4.16	3[label="subtractFloat vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.28/4.16	4[label="subtractFloat vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.28/4.16	5[label="flip msFloat vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.28/4.16	6[label="msFloat vx4 vx3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.28/4.16	7[label="primMinusFloat vx4 vx3",fontsize=16,color="burlywood",shape="box"];88[label="vx4/Float vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];7 -> 88[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	88 -> 8[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	8[label="primMinusFloat (Float vx40 vx41) vx3",fontsize=16,color="burlywood",shape="box"];89[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];8 -> 89[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	89 -> 9[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	9[label="primMinusFloat (Float vx40 vx41) (Float vx30 vx31)",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10.28/4.16	10[label="Float (msMyInt vx40 vx30) (srMyInt vx41 vx31)",fontsize=16,color="green",shape="box"];10 -> 11[label="",style="dashed", color="green", weight=3];
10.28/4.16	10 -> 12[label="",style="dashed", color="green", weight=3];
10.28/4.16	11[label="msMyInt vx40 vx30",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
10.28/4.16	12[label="srMyInt vx41 vx31",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
10.28/4.16	13[label="primMinusInt vx40 vx30",fontsize=16,color="burlywood",shape="box"];90[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];13 -> 90[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	90 -> 15[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	91[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];13 -> 91[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	91 -> 16[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	14[label="primMulInt vx41 vx31",fontsize=16,color="burlywood",shape="box"];92[label="vx41/Pos vx410",fontsize=10,color="white",style="solid",shape="box"];14 -> 92[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	92 -> 17[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	93[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];14 -> 93[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	93 -> 18[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	15[label="primMinusInt (Pos vx400) vx30",fontsize=16,color="burlywood",shape="box"];94[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];15 -> 94[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	94 -> 19[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	95[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];15 -> 95[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	95 -> 20[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	16[label="primMinusInt (Neg vx400) vx30",fontsize=16,color="burlywood",shape="box"];96[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 96[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	96 -> 21[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	97[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 97[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	97 -> 22[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	17[label="primMulInt (Pos vx410) vx31",fontsize=16,color="burlywood",shape="box"];98[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];17 -> 98[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	98 -> 23[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	99[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];17 -> 99[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	99 -> 24[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	18[label="primMulInt (Neg vx410) vx31",fontsize=16,color="burlywood",shape="box"];100[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];18 -> 100[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	100 -> 25[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	101[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];18 -> 101[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	101 -> 26[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	19[label="primMinusInt (Pos vx400) (Pos vx300)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
10.28/4.16	20[label="primMinusInt (Pos vx400) (Neg vx300)",fontsize=16,color="black",shape="box"];20 -> 28[label="",style="solid", color="black", weight=3];
10.28/4.16	21[label="primMinusInt (Neg vx400) (Pos vx300)",fontsize=16,color="black",shape="box"];21 -> 29[label="",style="solid", color="black", weight=3];
10.28/4.16	22[label="primMinusInt (Neg vx400) (Neg vx300)",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3];
10.28/4.16	23[label="primMulInt (Pos vx410) (Pos vx310)",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
10.28/4.16	24[label="primMulInt (Pos vx410) (Neg vx310)",fontsize=16,color="black",shape="box"];24 -> 32[label="",style="solid", color="black", weight=3];
10.28/4.16	25[label="primMulInt (Neg vx410) (Pos vx310)",fontsize=16,color="black",shape="box"];25 -> 33[label="",style="solid", color="black", weight=3];
10.28/4.16	26[label="primMulInt (Neg vx410) (Neg vx310)",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
10.28/4.16	27[label="primMinusNat vx400 vx300",fontsize=16,color="burlywood",shape="triangle"];102[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];27 -> 102[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	102 -> 35[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	103[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 103[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	103 -> 36[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	28[label="Pos (primPlusNat vx400 vx300)",fontsize=16,color="green",shape="box"];28 -> 37[label="",style="dashed", color="green", weight=3];
10.28/4.16	29[label="Neg (primPlusNat vx400 vx300)",fontsize=16,color="green",shape="box"];29 -> 38[label="",style="dashed", color="green", weight=3];
10.28/4.16	30 -> 27[label="",style="dashed", color="red", weight=0];
10.28/4.16	30[label="primMinusNat vx300 vx400",fontsize=16,color="magenta"];30 -> 39[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	30 -> 40[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	31[label="Pos (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];31 -> 41[label="",style="dashed", color="green", weight=3];
10.28/4.16	32[label="Neg (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];32 -> 42[label="",style="dashed", color="green", weight=3];
10.28/4.16	33[label="Neg (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];33 -> 43[label="",style="dashed", color="green", weight=3];
10.28/4.16	34[label="Pos (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];34 -> 44[label="",style="dashed", color="green", weight=3];
10.28/4.16	35[label="primMinusNat (Succ vx4000) vx300",fontsize=16,color="burlywood",shape="box"];104[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 104[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	104 -> 45[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	105[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 105[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	105 -> 46[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	36[label="primMinusNat Zero vx300",fontsize=16,color="burlywood",shape="box"];106[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];36 -> 106[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	106 -> 47[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	107[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 107[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	107 -> 48[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	37[label="primPlusNat vx400 vx300",fontsize=16,color="burlywood",shape="triangle"];108[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];37 -> 108[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	108 -> 49[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	109[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];37 -> 109[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	109 -> 50[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	38 -> 37[label="",style="dashed", color="red", weight=0];
10.28/4.16	38[label="primPlusNat vx400 vx300",fontsize=16,color="magenta"];38 -> 51[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	38 -> 52[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	39[label="vx400",fontsize=16,color="green",shape="box"];40[label="vx300",fontsize=16,color="green",shape="box"];41[label="primMulNat vx410 vx310",fontsize=16,color="burlywood",shape="triangle"];110[label="vx410/Succ vx4100",fontsize=10,color="white",style="solid",shape="box"];41 -> 110[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	110 -> 53[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	111[label="vx410/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 111[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	111 -> 54[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	42 -> 41[label="",style="dashed", color="red", weight=0];
10.28/4.16	42[label="primMulNat vx410 vx310",fontsize=16,color="magenta"];42 -> 55[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	43 -> 41[label="",style="dashed", color="red", weight=0];
10.28/4.16	43[label="primMulNat vx410 vx310",fontsize=16,color="magenta"];43 -> 56[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	44 -> 41[label="",style="dashed", color="red", weight=0];
10.28/4.16	44[label="primMulNat vx410 vx310",fontsize=16,color="magenta"];44 -> 57[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	44 -> 58[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	45[label="primMinusNat (Succ vx4000) (Succ vx3000)",fontsize=16,color="black",shape="box"];45 -> 59[label="",style="solid", color="black", weight=3];
10.28/4.16	46[label="primMinusNat (Succ vx4000) Zero",fontsize=16,color="black",shape="box"];46 -> 60[label="",style="solid", color="black", weight=3];
10.28/4.16	47[label="primMinusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];47 -> 61[label="",style="solid", color="black", weight=3];
10.28/4.16	48[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];48 -> 62[label="",style="solid", color="black", weight=3];
10.28/4.16	49[label="primPlusNat (Succ vx4000) vx300",fontsize=16,color="burlywood",shape="box"];112[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];49 -> 112[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	112 -> 63[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	113[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 113[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	113 -> 64[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	50[label="primPlusNat Zero vx300",fontsize=16,color="burlywood",shape="box"];114[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];50 -> 114[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	114 -> 65[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	115[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 115[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	115 -> 66[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	51[label="vx400",fontsize=16,color="green",shape="box"];52[label="vx300",fontsize=16,color="green",shape="box"];53[label="primMulNat (Succ vx4100) vx310",fontsize=16,color="burlywood",shape="box"];116[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];53 -> 116[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	116 -> 67[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	117[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 117[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	117 -> 68[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	54[label="primMulNat Zero vx310",fontsize=16,color="burlywood",shape="box"];118[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];54 -> 118[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	118 -> 69[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	119[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];54 -> 119[label="",style="solid", color="burlywood", weight=9];
10.28/4.16	119 -> 70[label="",style="solid", color="burlywood", weight=3];
10.28/4.16	55[label="vx310",fontsize=16,color="green",shape="box"];56[label="vx410",fontsize=16,color="green",shape="box"];57[label="vx410",fontsize=16,color="green",shape="box"];58[label="vx310",fontsize=16,color="green",shape="box"];59 -> 27[label="",style="dashed", color="red", weight=0];
10.28/4.16	59[label="primMinusNat vx4000 vx3000",fontsize=16,color="magenta"];59 -> 71[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	59 -> 72[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	60[label="Pos (Succ vx4000)",fontsize=16,color="green",shape="box"];61[label="Neg (Succ vx3000)",fontsize=16,color="green",shape="box"];62[label="Pos Zero",fontsize=16,color="green",shape="box"];63[label="primPlusNat (Succ vx4000) (Succ vx3000)",fontsize=16,color="black",shape="box"];63 -> 73[label="",style="solid", color="black", weight=3];
10.28/4.16	64[label="primPlusNat (Succ vx4000) Zero",fontsize=16,color="black",shape="box"];64 -> 74[label="",style="solid", color="black", weight=3];
10.28/4.16	65[label="primPlusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];65 -> 75[label="",style="solid", color="black", weight=3];
10.28/4.16	66[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3];
10.28/4.16	67[label="primMulNat (Succ vx4100) (Succ vx3100)",fontsize=16,color="black",shape="box"];67 -> 77[label="",style="solid", color="black", weight=3];
10.28/4.16	68[label="primMulNat (Succ vx4100) Zero",fontsize=16,color="black",shape="box"];68 -> 78[label="",style="solid", color="black", weight=3];
10.28/4.16	69[label="primMulNat Zero (Succ vx3100)",fontsize=16,color="black",shape="box"];69 -> 79[label="",style="solid", color="black", weight=3];
10.28/4.16	70[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];70 -> 80[label="",style="solid", color="black", weight=3];
10.28/4.16	71[label="vx3000",fontsize=16,color="green",shape="box"];72[label="vx4000",fontsize=16,color="green",shape="box"];73[label="Succ (Succ (primPlusNat vx4000 vx3000))",fontsize=16,color="green",shape="box"];73 -> 81[label="",style="dashed", color="green", weight=3];
10.28/4.16	74[label="Succ vx4000",fontsize=16,color="green",shape="box"];75[label="Succ vx3000",fontsize=16,color="green",shape="box"];76[label="Zero",fontsize=16,color="green",shape="box"];77 -> 37[label="",style="dashed", color="red", weight=0];
10.28/4.16	77[label="primPlusNat (primMulNat vx4100 (Succ vx3100)) (Succ vx3100)",fontsize=16,color="magenta"];77 -> 82[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	77 -> 83[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	78[label="Zero",fontsize=16,color="green",shape="box"];79[label="Zero",fontsize=16,color="green",shape="box"];80[label="Zero",fontsize=16,color="green",shape="box"];81 -> 37[label="",style="dashed", color="red", weight=0];
10.28/4.16	81[label="primPlusNat vx4000 vx3000",fontsize=16,color="magenta"];81 -> 84[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	81 -> 85[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	82 -> 41[label="",style="dashed", color="red", weight=0];
10.28/4.16	82[label="primMulNat vx4100 (Succ vx3100)",fontsize=16,color="magenta"];82 -> 86[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	82 -> 87[label="",style="dashed", color="magenta", weight=3];
10.28/4.16	83[label="Succ vx3100",fontsize=16,color="green",shape="box"];84[label="vx4000",fontsize=16,color="green",shape="box"];85[label="vx3000",fontsize=16,color="green",shape="box"];86[label="vx4100",fontsize=16,color="green",shape="box"];87[label="Succ vx3100",fontsize=16,color="green",shape="box"];}
10.28/4.16	
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(6)
10.28/4.16	Complex Obligation (AND)
10.28/4.16	
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(7)
10.28/4.16	Obligation:
10.28/4.16	Q DP problem:
10.28/4.16	The TRS P consists of the following rules:
10.28/4.16	
10.28/4.16	   new_primMulNat(Main.Succ(vx4100), Main.Succ(vx3100)) -> new_primMulNat(vx4100, Main.Succ(vx3100))
10.28/4.16	
10.28/4.16	R is empty.
10.28/4.16	Q is empty.
10.28/4.16	We have to consider all minimal (P,Q,R)-chains.
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(8) QDPSizeChangeProof (EQUIVALENT)
10.28/4.16	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. 
10.28/4.16	
10.28/4.16	From the DPs we obtained the following set of size-change graphs:
10.28/4.16	*new_primMulNat(Main.Succ(vx4100), Main.Succ(vx3100)) -> new_primMulNat(vx4100, Main.Succ(vx3100))
10.28/4.16	The graph contains the following edges 1 > 1, 2 >= 2
10.28/4.16	
10.28/4.16	
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(9)
10.28/4.16	YES
10.28/4.16	
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(10)
10.28/4.16	Obligation:
10.28/4.16	Q DP problem:
10.28/4.16	The TRS P consists of the following rules:
10.28/4.16	
10.28/4.16	   new_primPlusNat(Main.Succ(vx4000), Main.Succ(vx3000)) -> new_primPlusNat(vx4000, vx3000)
10.28/4.16	
10.28/4.16	R is empty.
10.28/4.16	Q is empty.
10.28/4.16	We have to consider all minimal (P,Q,R)-chains.
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(11) QDPSizeChangeProof (EQUIVALENT)
10.28/4.16	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. 
10.28/4.16	
10.28/4.16	From the DPs we obtained the following set of size-change graphs:
10.28/4.16	*new_primPlusNat(Main.Succ(vx4000), Main.Succ(vx3000)) -> new_primPlusNat(vx4000, vx3000)
10.28/4.16	The graph contains the following edges 1 > 1, 2 > 2
10.28/4.16	
10.28/4.16	
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(12)
10.28/4.16	YES
10.28/4.16	
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(13)
10.28/4.16	Obligation:
10.28/4.16	Q DP problem:
10.28/4.16	The TRS P consists of the following rules:
10.28/4.16	
10.28/4.16	   new_primMinusNat(Main.Succ(vx4000), Main.Succ(vx3000)) -> new_primMinusNat(vx4000, vx3000)
10.28/4.16	
10.28/4.16	R is empty.
10.28/4.16	Q is empty.
10.28/4.16	We have to consider all minimal (P,Q,R)-chains.
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(14) QDPSizeChangeProof (EQUIVALENT)
10.28/4.16	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. 
10.28/4.16	
10.28/4.16	From the DPs we obtained the following set of size-change graphs:
10.28/4.16	*new_primMinusNat(Main.Succ(vx4000), Main.Succ(vx3000)) -> new_primMinusNat(vx4000, vx3000)
10.28/4.16	The graph contains the following edges 1 > 1, 2 > 2
10.28/4.16	
10.28/4.16	
10.28/4.16	----------------------------------------
10.28/4.16	
10.28/4.16	(15)
10.28/4.16	YES
10.43/4.20	EOF