7.85/3.51	YES
9.78/4.05	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.78/4.05	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.78/4.05	
9.78/4.05	
9.78/4.05	H-Termination with start terms of the given HASKELL could be proven:
9.78/4.05	
9.78/4.05	(0) HASKELL
9.78/4.05	(1) BR [EQUIVALENT, 0 ms]
9.78/4.05	(2) HASKELL
9.78/4.05	(3) COR [EQUIVALENT, 0 ms]
9.78/4.05	(4) HASKELL
9.78/4.05	(5) Narrow [SOUND, 0 ms]
9.78/4.05	(6) AND
9.78/4.05	    (7) QDP
9.78/4.05	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.78/4.05	        (9) YES
9.78/4.05	    (10) QDP
9.78/4.05	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.78/4.05	        (12) YES
9.78/4.05	
9.78/4.05	
9.78/4.05	----------------------------------------
9.78/4.05	
9.78/4.05	(0)
9.78/4.05	Obligation:
9.78/4.05	mainModule Main 
9.78/4.05	module Main where {
9.78/4.05	import qualified Prelude;
9.78/4.05	data Float = Float MyInt MyInt ;
9.78/4.05	
9.78/4.05	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.78/4.05	
9.78/4.05	data Main.Nat = Succ Main.Nat  | Zero ;
9.78/4.05	
9.78/4.05	primMulFloat :: Float  ->  Float  ->  Float;
9.78/4.05	primMulFloat (Float x1 x2) (Float y1 y2) = Float (srMyInt x1 y1) (srMyInt x2 y2);
9.78/4.05	
9.78/4.05	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
9.78/4.05	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
9.78/4.05	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
9.78/4.05	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
9.78/4.05	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
9.78/4.05	
9.78/4.05	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.78/4.05	primMulNat Main.Zero Main.Zero = Main.Zero;
9.78/4.05	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
9.78/4.05	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
9.78/4.05	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
9.78/4.05	
9.78/4.05	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.78/4.05	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.78/4.05	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.78/4.05	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.78/4.05	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.78/4.05	
9.78/4.05	srFloat :: Float  ->  Float  ->  Float;
9.78/4.05	srFloat = primMulFloat;
9.78/4.05	
9.78/4.05	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.78/4.05	srMyInt = primMulInt;
9.78/4.05	
9.78/4.05	}
9.78/4.05	
9.78/4.05	----------------------------------------
9.78/4.05	
9.78/4.05	(1) BR (EQUIVALENT)
9.78/4.05	Replaced joker patterns by fresh variables and removed binding patterns.
9.78/4.05	----------------------------------------
9.78/4.05	
9.78/4.05	(2)
9.78/4.05	Obligation:
9.78/4.05	mainModule Main 
9.78/4.05	module Main where {
9.78/4.05	import qualified Prelude;
9.78/4.05	data Float = Float MyInt MyInt ;
9.78/4.05	
9.78/4.05	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.78/4.05	
9.78/4.05	data Main.Nat = Succ Main.Nat  | Zero ;
9.78/4.05	
9.78/4.05	primMulFloat :: Float  ->  Float  ->  Float;
9.78/4.05	primMulFloat (Float x1 x2) (Float y1 y2) = Float (srMyInt x1 y1) (srMyInt x2 y2);
9.78/4.05	
9.78/4.05	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
9.78/4.05	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
9.78/4.05	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
9.78/4.05	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
9.78/4.05	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
9.78/4.05	
9.78/4.05	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.78/4.05	primMulNat Main.Zero Main.Zero = Main.Zero;
9.78/4.05	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
9.78/4.05	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
9.78/4.05	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
9.78/4.05	
9.78/4.05	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.78/4.05	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.78/4.05	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.78/4.05	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.78/4.05	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.78/4.05	
9.78/4.05	srFloat :: Float  ->  Float  ->  Float;
9.78/4.05	srFloat = primMulFloat;
9.78/4.05	
9.78/4.05	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.78/4.05	srMyInt = primMulInt;
9.78/4.05	
9.78/4.05	}
9.78/4.05	
9.78/4.05	----------------------------------------
9.78/4.05	
9.78/4.05	(3) COR (EQUIVALENT)
9.78/4.05	Cond Reductions:
9.78/4.05	The following Function with conditions
9.78/4.05	"undefined |Falseundefined;
9.78/4.05	"
9.78/4.05	is transformed to
9.78/4.05	"undefined  = undefined1;
9.78/4.05	"
9.78/4.05	"undefined0 True = undefined;
9.78/4.05	"
9.78/4.05	"undefined1  = undefined0 False;
9.78/4.05	"
9.78/4.05	
9.78/4.05	----------------------------------------
9.78/4.05	
9.78/4.05	(4)
9.78/4.05	Obligation:
9.78/4.05	mainModule Main 
9.78/4.05	module Main where {
9.78/4.05	import qualified Prelude;
9.78/4.05	data Float = Float MyInt MyInt ;
9.78/4.05	
9.78/4.05	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.78/4.05	
9.78/4.05	data Main.Nat = Succ Main.Nat  | Zero ;
9.78/4.05	
9.78/4.05	primMulFloat :: Float  ->  Float  ->  Float;
9.78/4.05	primMulFloat (Float x1 x2) (Float y1 y2) = Float (srMyInt x1 y1) (srMyInt x2 y2);
9.78/4.05	
9.78/4.05	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
9.78/4.05	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
9.78/4.05	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
9.78/4.05	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
9.78/4.05	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
9.78/4.05	
9.78/4.05	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.78/4.05	primMulNat Main.Zero Main.Zero = Main.Zero;
9.78/4.05	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
9.78/4.05	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
9.78/4.05	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
9.78/4.05	
9.78/4.05	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.78/4.05	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.78/4.05	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.78/4.05	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.78/4.05	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.78/4.05	
9.78/4.05	srFloat :: Float  ->  Float  ->  Float;
9.78/4.05	srFloat = primMulFloat;
9.78/4.05	
9.78/4.05	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.78/4.05	srMyInt = primMulInt;
9.78/4.05	
9.78/4.05	}
9.78/4.05	
9.78/4.05	----------------------------------------
9.78/4.05	
9.78/4.05	(5) Narrow (SOUND)
9.78/4.05	Haskell To QDPs
9.78/4.05	
9.78/4.05	digraph dp_graph {
9.78/4.05	node [outthreshold=100, inthreshold=100];1[label="srFloat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.78/4.05	3[label="srFloat vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.78/4.05	4[label="srFloat vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.78/4.05	5[label="primMulFloat vx3 vx4",fontsize=16,color="burlywood",shape="box"];64[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];5 -> 64[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	64 -> 6[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	6[label="primMulFloat (Float vx30 vx31) vx4",fontsize=16,color="burlywood",shape="box"];65[label="vx4/Float vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];6 -> 65[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	65 -> 7[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	7[label="primMulFloat (Float vx30 vx31) (Float vx40 vx41)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.78/4.05	8[label="Float (srMyInt vx30 vx40) (srMyInt vx31 vx41)",fontsize=16,color="green",shape="box"];8 -> 9[label="",style="dashed", color="green", weight=3];
9.78/4.05	8 -> 10[label="",style="dashed", color="green", weight=3];
9.78/4.05	9[label="srMyInt vx30 vx40",fontsize=16,color="black",shape="triangle"];9 -> 11[label="",style="solid", color="black", weight=3];
9.78/4.05	10 -> 9[label="",style="dashed", color="red", weight=0];
9.78/4.05	10[label="srMyInt vx31 vx41",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
9.78/4.05	10 -> 13[label="",style="dashed", color="magenta", weight=3];
9.78/4.05	11[label="primMulInt vx30 vx40",fontsize=16,color="burlywood",shape="box"];66[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];11 -> 66[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	66 -> 14[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	67[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];11 -> 67[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	67 -> 15[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	12[label="vx31",fontsize=16,color="green",shape="box"];13[label="vx41",fontsize=16,color="green",shape="box"];14[label="primMulInt (Pos vx300) vx40",fontsize=16,color="burlywood",shape="box"];68[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 68[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	68 -> 16[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	69[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 69[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	69 -> 17[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	15[label="primMulInt (Neg vx300) vx40",fontsize=16,color="burlywood",shape="box"];70[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];15 -> 70[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	70 -> 18[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	71[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];15 -> 71[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	71 -> 19[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	16[label="primMulInt (Pos vx300) (Pos vx400)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
9.78/4.05	17[label="primMulInt (Pos vx300) (Neg vx400)",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
9.78/4.05	18[label="primMulInt (Neg vx300) (Pos vx400)",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
9.78/4.05	19[label="primMulInt (Neg vx300) (Neg vx400)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
9.78/4.05	20[label="Pos (primMulNat vx300 vx400)",fontsize=16,color="green",shape="box"];20 -> 24[label="",style="dashed", color="green", weight=3];
9.78/4.05	21[label="Neg (primMulNat vx300 vx400)",fontsize=16,color="green",shape="box"];21 -> 25[label="",style="dashed", color="green", weight=3];
9.78/4.05	22[label="Neg (primMulNat vx300 vx400)",fontsize=16,color="green",shape="box"];22 -> 26[label="",style="dashed", color="green", weight=3];
9.78/4.05	23[label="Pos (primMulNat vx300 vx400)",fontsize=16,color="green",shape="box"];23 -> 27[label="",style="dashed", color="green", weight=3];
9.78/4.05	24[label="primMulNat vx300 vx400",fontsize=16,color="burlywood",shape="triangle"];72[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];24 -> 72[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	72 -> 28[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	73[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 73[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	73 -> 29[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	25 -> 24[label="",style="dashed", color="red", weight=0];
9.78/4.05	25[label="primMulNat vx300 vx400",fontsize=16,color="magenta"];25 -> 30[label="",style="dashed", color="magenta", weight=3];
9.78/4.05	26 -> 24[label="",style="dashed", color="red", weight=0];
9.78/4.05	26[label="primMulNat vx300 vx400",fontsize=16,color="magenta"];26 -> 31[label="",style="dashed", color="magenta", weight=3];
9.78/4.05	27 -> 24[label="",style="dashed", color="red", weight=0];
9.78/4.05	27[label="primMulNat vx300 vx400",fontsize=16,color="magenta"];27 -> 32[label="",style="dashed", color="magenta", weight=3];
9.78/4.05	27 -> 33[label="",style="dashed", color="magenta", weight=3];
9.78/4.05	28[label="primMulNat (Succ vx3000) vx400",fontsize=16,color="burlywood",shape="box"];74[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 74[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	74 -> 34[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	75[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 75[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	75 -> 35[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	29[label="primMulNat Zero vx400",fontsize=16,color="burlywood",shape="box"];76[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 76[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	76 -> 36[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	77[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 77[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	77 -> 37[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	30[label="vx400",fontsize=16,color="green",shape="box"];31[label="vx300",fontsize=16,color="green",shape="box"];32[label="vx300",fontsize=16,color="green",shape="box"];33[label="vx400",fontsize=16,color="green",shape="box"];34[label="primMulNat (Succ vx3000) (Succ vx4000)",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
9.78/4.05	35[label="primMulNat (Succ vx3000) Zero",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
9.78/4.05	36[label="primMulNat Zero (Succ vx4000)",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
9.78/4.05	37[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
9.78/4.05	38 -> 42[label="",style="dashed", color="red", weight=0];
9.78/4.05	38[label="primPlusNat (primMulNat vx3000 (Succ vx4000)) (Succ vx4000)",fontsize=16,color="magenta"];38 -> 43[label="",style="dashed", color="magenta", weight=3];
9.78/4.05	39[label="Zero",fontsize=16,color="green",shape="box"];40[label="Zero",fontsize=16,color="green",shape="box"];41[label="Zero",fontsize=16,color="green",shape="box"];43 -> 24[label="",style="dashed", color="red", weight=0];
9.78/4.05	43[label="primMulNat vx3000 (Succ vx4000)",fontsize=16,color="magenta"];43 -> 44[label="",style="dashed", color="magenta", weight=3];
9.78/4.05	43 -> 45[label="",style="dashed", color="magenta", weight=3];
9.78/4.05	42[label="primPlusNat vx5 (Succ vx4000)",fontsize=16,color="burlywood",shape="triangle"];78[label="vx5/Succ vx50",fontsize=10,color="white",style="solid",shape="box"];42 -> 78[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	78 -> 46[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	79[label="vx5/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 79[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	79 -> 47[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	44[label="vx3000",fontsize=16,color="green",shape="box"];45[label="Succ vx4000",fontsize=16,color="green",shape="box"];46[label="primPlusNat (Succ vx50) (Succ vx4000)",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
9.78/4.05	47[label="primPlusNat Zero (Succ vx4000)",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
9.78/4.05	48[label="Succ (Succ (primPlusNat vx50 vx4000))",fontsize=16,color="green",shape="box"];48 -> 50[label="",style="dashed", color="green", weight=3];
9.78/4.05	49[label="Succ vx4000",fontsize=16,color="green",shape="box"];50[label="primPlusNat vx50 vx4000",fontsize=16,color="burlywood",shape="triangle"];80[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];50 -> 80[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	80 -> 51[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	81[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 81[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	81 -> 52[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	51[label="primPlusNat (Succ vx500) vx4000",fontsize=16,color="burlywood",shape="box"];82[label="vx4000/Succ vx40000",fontsize=10,color="white",style="solid",shape="box"];51 -> 82[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	82 -> 53[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	83[label="vx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];51 -> 83[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	83 -> 54[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	52[label="primPlusNat Zero vx4000",fontsize=16,color="burlywood",shape="box"];84[label="vx4000/Succ vx40000",fontsize=10,color="white",style="solid",shape="box"];52 -> 84[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	84 -> 55[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	85[label="vx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 85[label="",style="solid", color="burlywood", weight=9];
9.78/4.05	85 -> 56[label="",style="solid", color="burlywood", weight=3];
9.78/4.05	53[label="primPlusNat (Succ vx500) (Succ vx40000)",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
9.78/4.05	54[label="primPlusNat (Succ vx500) Zero",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3];
9.78/4.05	55[label="primPlusNat Zero (Succ vx40000)",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3];
9.78/4.05	56[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3];
9.78/4.05	57[label="Succ (Succ (primPlusNat vx500 vx40000))",fontsize=16,color="green",shape="box"];57 -> 61[label="",style="dashed", color="green", weight=3];
9.78/4.05	58[label="Succ vx500",fontsize=16,color="green",shape="box"];59[label="Succ vx40000",fontsize=16,color="green",shape="box"];60[label="Zero",fontsize=16,color="green",shape="box"];61 -> 50[label="",style="dashed", color="red", weight=0];
9.78/4.05	61[label="primPlusNat vx500 vx40000",fontsize=16,color="magenta"];61 -> 62[label="",style="dashed", color="magenta", weight=3];
9.78/4.05	61 -> 63[label="",style="dashed", color="magenta", weight=3];
9.78/4.05	62[label="vx500",fontsize=16,color="green",shape="box"];63[label="vx40000",fontsize=16,color="green",shape="box"];}
9.78/4.05	
9.78/4.05	----------------------------------------
9.78/4.05	
9.78/4.05	(6)
9.78/4.05	Complex Obligation (AND)
9.78/4.05	
9.78/4.05	----------------------------------------
9.78/4.05	
9.78/4.05	(7)
9.78/4.05	Obligation:
9.78/4.05	Q DP problem:
9.78/4.05	The TRS P consists of the following rules:
9.78/4.05	
9.78/4.05	   new_primMulNat(Main.Succ(vx3000), Main.Succ(vx4000)) -> new_primMulNat(vx3000, Main.Succ(vx4000))
9.78/4.05	
9.78/4.05	R is empty.
9.78/4.05	Q is empty.
9.78/4.05	We have to consider all minimal (P,Q,R)-chains.
9.78/4.05	----------------------------------------
9.78/4.05	
9.78/4.05	(8) QDPSizeChangeProof (EQUIVALENT)
9.78/4.05	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. 
9.78/4.05	
9.78/4.05	From the DPs we obtained the following set of size-change graphs:
9.78/4.05	*new_primMulNat(Main.Succ(vx3000), Main.Succ(vx4000)) -> new_primMulNat(vx3000, Main.Succ(vx4000))
9.78/4.05	The graph contains the following edges 1 > 1, 2 >= 2
9.78/4.05	
9.78/4.05	
9.78/4.05	----------------------------------------
9.78/4.05	
9.78/4.05	(9)
9.78/4.05	YES
9.78/4.05	
9.78/4.05	----------------------------------------
9.78/4.05	
9.78/4.05	(10)
9.78/4.05	Obligation:
9.78/4.05	Q DP problem:
9.78/4.05	The TRS P consists of the following rules:
9.78/4.05	
9.78/4.05	   new_primPlusNat(Main.Succ(vx500), Main.Succ(vx40000)) -> new_primPlusNat(vx500, vx40000)
9.78/4.05	
9.78/4.05	R is empty.
9.78/4.05	Q is empty.
9.78/4.05	We have to consider all minimal (P,Q,R)-chains.
9.78/4.05	----------------------------------------
9.78/4.05	
9.78/4.05	(11) QDPSizeChangeProof (EQUIVALENT)
9.78/4.05	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. 
9.78/4.05	
9.78/4.05	From the DPs we obtained the following set of size-change graphs:
9.78/4.05	*new_primPlusNat(Main.Succ(vx500), Main.Succ(vx40000)) -> new_primPlusNat(vx500, vx40000)
9.78/4.05	The graph contains the following edges 1 > 1, 2 > 2
9.78/4.05	
9.78/4.05	
9.78/4.05	----------------------------------------
9.78/4.05	
9.78/4.05	(12)
9.78/4.05	YES
9.96/4.11	EOF