8.36/3.79	MAYBE
10.22/4.28	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
10.22/4.28	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.22/4.28	
10.22/4.28	
10.22/4.28	H-Termination with start terms of the given HASKELL could not be shown:
10.22/4.28	
10.22/4.28	(0) HASKELL
10.22/4.28	(1) BR [EQUIVALENT, 0 ms]
10.22/4.28	(2) HASKELL
10.22/4.28	(3) COR [EQUIVALENT, 0 ms]
10.22/4.28	(4) HASKELL
10.22/4.28	(5) Narrow [SOUND, 0 ms]
10.22/4.28	(6) QDP
10.22/4.28	    (7) MRRProof [EQUIVALENT, 38 ms]
10.22/4.28	    (8) QDP
10.22/4.28	    (9) NonTerminationLoopProof [COMPLETE, 0 ms]
10.22/4.28	    (10) NO
10.22/4.28	(11) Narrow [COMPLETE, 0 ms]
10.22/4.28	(12) TRUE
10.22/4.28	
10.22/4.28	
10.22/4.28	----------------------------------------
10.22/4.28	
10.22/4.28	(0)
10.22/4.28	Obligation:
10.22/4.28	mainModule Main 
10.22/4.28	module Main where {
10.22/4.28	import qualified Prelude;
10.22/4.28	data List a = Cons a (List a)  | Nil ;
10.22/4.28	
10.22/4.28	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.22/4.28	
10.22/4.28	data Main.Nat = Succ Main.Nat  | Zero ;
10.22/4.28	
10.22/4.28	data Main.WHNF a = WHNF a ;
10.22/4.28	
10.22/4.28	dsEm :: (b  ->  a)  ->  b  ->  a;
10.22/4.28	dsEm f x = Main.seq x (f x);
10.22/4.28	
10.22/4.28	enforceWHNF :: Main.WHNF b  ->  a  ->  a;
10.22/4.28	enforceWHNF (Main.WHNF x) y = y;
10.22/4.28	
10.22/4.28	enumFromMyInt :: MyInt  ->  List MyInt;
10.22/4.28	enumFromMyInt = numericEnumFrom;
10.22/4.28	
10.22/4.28	fromIntMyInt :: MyInt  ->  MyInt;
10.22/4.28	fromIntMyInt x = x;
10.22/4.28	
10.22/4.28	numericEnumFrom n = Cons n (dsEm numericEnumFrom (psMyInt n (fromIntMyInt (Main.Pos (Main.Succ Main.Zero)))));
10.22/4.28	
10.22/4.28	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.22/4.28	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.22/4.28	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.22/4.28	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.22/4.28	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.22/4.28	
10.22/4.28	primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
10.22/4.28	primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
10.22/4.28	primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
10.22/4.28	primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
10.22/4.28	primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);
10.22/4.28	
10.22/4.28	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.22/4.28	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.22/4.28	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.22/4.28	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.22/4.28	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.22/4.28	
10.22/4.28	psMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.22/4.28	psMyInt = primPlusInt;
10.22/4.28	
10.22/4.28	seq :: a  ->  b  ->  b;
10.22/4.28	seq x y = Main.enforceWHNF (Main.WHNF x) y;
10.22/4.28	
10.22/4.28	}
10.22/4.28	
10.22/4.28	----------------------------------------
10.22/4.28	
10.22/4.28	(1) BR (EQUIVALENT)
10.22/4.28	Replaced joker patterns by fresh variables and removed binding patterns.
10.22/4.28	----------------------------------------
10.22/4.28	
10.22/4.28	(2)
10.22/4.28	Obligation:
10.22/4.28	mainModule Main 
10.22/4.28	module Main where {
10.22/4.28	import qualified Prelude;
10.22/4.28	data List a = Cons a (List a)  | Nil ;
10.22/4.28	
10.22/4.28	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.22/4.28	
10.22/4.28	data Main.Nat = Succ Main.Nat  | Zero ;
10.22/4.28	
10.22/4.28	data Main.WHNF a = WHNF a ;
10.22/4.28	
10.22/4.28	dsEm :: (b  ->  a)  ->  b  ->  a;
10.22/4.28	dsEm f x = Main.seq x (f x);
10.22/4.28	
10.22/4.28	enforceWHNF :: Main.WHNF a  ->  b  ->  b;
10.22/4.28	enforceWHNF (Main.WHNF x) y = y;
10.22/4.28	
10.22/4.28	enumFromMyInt :: MyInt  ->  List MyInt;
10.22/4.28	enumFromMyInt = numericEnumFrom;
10.22/4.28	
10.22/4.28	fromIntMyInt :: MyInt  ->  MyInt;
10.22/4.28	fromIntMyInt x = x;
10.22/4.28	
10.22/4.28	numericEnumFrom n = Cons n (dsEm numericEnumFrom (psMyInt n (fromIntMyInt (Main.Pos (Main.Succ Main.Zero)))));
10.22/4.28	
10.22/4.28	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.22/4.28	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.22/4.28	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.22/4.28	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.22/4.28	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.22/4.28	
10.22/4.28	primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
10.22/4.28	primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
10.22/4.28	primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
10.22/4.28	primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
10.22/4.28	primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);
10.22/4.28	
10.22/4.28	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.22/4.28	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.22/4.28	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.22/4.28	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.22/4.28	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.22/4.28	
10.22/4.28	psMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.22/4.28	psMyInt = primPlusInt;
10.22/4.28	
10.22/4.28	seq :: b  ->  a  ->  a;
10.22/4.28	seq x y = Main.enforceWHNF (Main.WHNF x) y;
10.22/4.28	
10.22/4.28	}
10.22/4.28	
10.22/4.28	----------------------------------------
10.22/4.28	
10.22/4.28	(3) COR (EQUIVALENT)
10.22/4.28	Cond Reductions:
10.22/4.28	The following Function with conditions
10.22/4.28	"undefined |Falseundefined;
10.22/4.28	"
10.22/4.28	is transformed to
10.22/4.28	"undefined  = undefined1;
10.22/4.28	"
10.22/4.28	"undefined0 True = undefined;
10.22/4.28	"
10.22/4.28	"undefined1  = undefined0 False;
10.22/4.28	"
10.22/4.28	
10.22/4.28	----------------------------------------
10.22/4.28	
10.22/4.28	(4)
10.22/4.28	Obligation:
10.22/4.28	mainModule Main 
10.22/4.28	module Main where {
10.22/4.28	import qualified Prelude;
10.22/4.28	data List a = Cons a (List a)  | Nil ;
10.22/4.28	
10.22/4.28	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.22/4.28	
10.22/4.28	data Main.Nat = Succ Main.Nat  | Zero ;
10.22/4.28	
10.22/4.28	data Main.WHNF a = WHNF a ;
10.22/4.28	
10.22/4.28	dsEm :: (b  ->  a)  ->  b  ->  a;
10.22/4.28	dsEm f x = Main.seq x (f x);
10.22/4.28	
10.22/4.28	enforceWHNF :: Main.WHNF a  ->  b  ->  b;
10.22/4.28	enforceWHNF (Main.WHNF x) y = y;
10.22/4.28	
10.22/4.28	enumFromMyInt :: MyInt  ->  List MyInt;
10.22/4.28	enumFromMyInt = numericEnumFrom;
10.22/4.28	
10.22/4.28	fromIntMyInt :: MyInt  ->  MyInt;
10.22/4.28	fromIntMyInt x = x;
10.22/4.28	
10.22/4.28	numericEnumFrom n = Cons n (dsEm numericEnumFrom (psMyInt n (fromIntMyInt (Main.Pos (Main.Succ Main.Zero)))));
10.22/4.28	
10.22/4.28	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.22/4.28	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.22/4.28	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.22/4.28	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.22/4.28	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.22/4.28	
10.22/4.28	primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
10.22/4.28	primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
10.22/4.28	primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
10.22/4.28	primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
10.22/4.28	primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);
10.22/4.28	
10.22/4.28	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.22/4.28	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.22/4.28	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.22/4.28	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.22/4.28	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.22/4.28	
10.22/4.28	psMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.22/4.28	psMyInt = primPlusInt;
10.22/4.28	
10.22/4.28	seq :: a  ->  b  ->  b;
10.22/4.28	seq x y = Main.enforceWHNF (Main.WHNF x) y;
10.22/4.28	
10.22/4.28	}
10.22/4.28	
10.22/4.28	----------------------------------------
10.22/4.28	
10.22/4.28	(5) Narrow (SOUND)
10.22/4.28	Haskell To QDPs
10.22/4.28	
10.22/4.28	digraph dp_graph {
10.22/4.28	node [outthreshold=100, inthreshold=100];1[label="enumFromMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.22/4.28	3[label="enumFromMyInt vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.22/4.28	4[label="numericEnumFrom vx3",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.22/4.28	5[label="Cons vx3 (dsEm numericEnumFrom (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="green",shape="box"];5 -> 6[label="",style="dashed", color="green", weight=3];
10.22/4.28	6[label="dsEm numericEnumFrom (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero))))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.22/4.28	7 -> 8[label="",style="dashed", color="red", weight=0];
10.22/4.28	7[label="seq (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];7 -> 9[label="",style="dashed", color="magenta", weight=3];
10.22/4.28	9 -> 4[label="",style="dashed", color="red", weight=0];
10.22/4.28	9[label="numericEnumFrom (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero))))",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
10.22/4.28	8[label="seq (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero)))) vx4",fontsize=16,color="black",shape="triangle"];8 -> 11[label="",style="solid", color="black", weight=3];
10.22/4.28	10[label="psMyInt vx3 (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3];
10.22/4.28	11 -> 13[label="",style="dashed", color="red", weight=0];
10.22/4.28	11[label="enforceWHNF (WHNF (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero))))) vx4",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
10.22/4.28	12[label="primPlusInt vx3 (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="burlywood",shape="box"];40[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];12 -> 40[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	40 -> 15[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	41[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];12 -> 41[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	41 -> 16[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	14 -> 10[label="",style="dashed", color="red", weight=0];
10.22/4.28	14[label="psMyInt vx3 (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];13[label="enforceWHNF (WHNF vx5) vx4",fontsize=16,color="black",shape="triangle"];13 -> 17[label="",style="solid", color="black", weight=3];
10.22/4.28	15[label="primPlusInt (Pos vx30) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
10.22/4.28	16[label="primPlusInt (Neg vx30) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];16 -> 19[label="",style="solid", color="black", weight=3];
10.22/4.28	17[label="vx4",fontsize=16,color="green",shape="box"];18[label="primPlusInt (Pos vx30) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
10.22/4.28	19[label="primPlusInt (Neg vx30) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
10.22/4.28	20[label="Pos (primPlusNat vx30 (Succ Zero))",fontsize=16,color="green",shape="box"];20 -> 22[label="",style="dashed", color="green", weight=3];
10.22/4.28	21[label="primMinusNat (Succ Zero) vx30",fontsize=16,color="burlywood",shape="box"];42[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];21 -> 42[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	42 -> 23[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	43[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 43[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	43 -> 24[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	22[label="primPlusNat vx30 (Succ Zero)",fontsize=16,color="burlywood",shape="box"];44[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];22 -> 44[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	44 -> 25[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	45[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 45[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	45 -> 26[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	23[label="primMinusNat (Succ Zero) (Succ vx300)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
10.22/4.28	24[label="primMinusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
10.22/4.28	25[label="primPlusNat (Succ vx300) (Succ Zero)",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
10.22/4.28	26[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
10.22/4.28	27[label="primMinusNat Zero vx300",fontsize=16,color="burlywood",shape="box"];46[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];27 -> 46[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	46 -> 31[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	47[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 47[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	47 -> 32[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	28[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];29[label="Succ (Succ (primPlusNat vx300 Zero))",fontsize=16,color="green",shape="box"];29 -> 33[label="",style="dashed", color="green", weight=3];
10.22/4.28	30[label="Succ Zero",fontsize=16,color="green",shape="box"];31[label="primMinusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];31 -> 34[label="",style="solid", color="black", weight=3];
10.22/4.28	32[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3];
10.22/4.28	33[label="primPlusNat vx300 Zero",fontsize=16,color="burlywood",shape="box"];48[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];33 -> 48[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	48 -> 36[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	49[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 49[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	49 -> 37[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	34[label="Neg (Succ vx3000)",fontsize=16,color="green",shape="box"];35[label="Pos Zero",fontsize=16,color="green",shape="box"];36[label="primPlusNat (Succ vx3000) Zero",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
10.22/4.28	37[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
10.22/4.28	38[label="Succ vx3000",fontsize=16,color="green",shape="box"];39[label="Zero",fontsize=16,color="green",shape="box"];}
10.22/4.28	
10.22/4.28	----------------------------------------
10.22/4.28	
10.22/4.28	(6)
10.22/4.28	Obligation:
10.22/4.28	Q DP problem:
10.22/4.28	The TRS P consists of the following rules:
10.22/4.28	
10.22/4.28	   new_numericEnumFrom(vx3) -> new_numericEnumFrom(new_psMyInt(vx3))
10.22/4.28	
10.22/4.28	The TRS R consists of the following rules:
10.22/4.28	
10.22/4.28	   new_psMyInt(Main.Neg(Main.Zero)) -> Main.Pos(Main.Succ(Main.Zero))
10.22/4.28	   new_psMyInt(Main.Neg(Main.Succ(Main.Zero))) -> Main.Pos(Main.Zero)
10.22/4.28	   new_primPlusNat(Main.Zero) -> Main.Succ(Main.Zero)
10.22/4.28	   new_primPlusNat(Main.Succ(vx300)) -> Main.Succ(Main.Succ(new_primPlusNat0(vx300)))
10.22/4.28	   new_psMyInt(Main.Pos(vx30)) -> Main.Pos(new_primPlusNat(vx30))
10.22/4.28	   new_psMyInt(Main.Neg(Main.Succ(Main.Succ(vx3000)))) -> Main.Neg(Main.Succ(vx3000))
10.22/4.28	   new_primPlusNat0(Main.Succ(vx3000)) -> Main.Succ(vx3000)
10.22/4.28	   new_primPlusNat0(Main.Zero) -> Main.Zero
10.22/4.28	
10.22/4.28	The set Q consists of the following terms:
10.22/4.28	
10.22/4.28	   new_psMyInt(Main.Neg(Main.Succ(Main.Zero)))
10.22/4.28	   new_primPlusNat0(Main.Zero)
10.22/4.28	   new_psMyInt(Main.Pos(x0))
10.22/4.28	   new_psMyInt(Main.Neg(Main.Succ(Main.Succ(x0))))
10.22/4.28	   new_psMyInt(Main.Neg(Main.Zero))
10.22/4.28	   new_primPlusNat(Main.Zero)
10.22/4.28	   new_primPlusNat0(Main.Succ(x0))
10.22/4.28	   new_primPlusNat(Main.Succ(x0))
10.22/4.28	
10.22/4.28	We have to consider all minimal (P,Q,R)-chains.
10.22/4.28	----------------------------------------
10.22/4.28	
10.22/4.28	(7) MRRProof (EQUIVALENT)
10.22/4.28	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
10.22/4.28	
10.22/4.28	
10.22/4.28	Strictly oriented rules of the TRS R:
10.22/4.28	
10.22/4.28	   new_psMyInt(Main.Neg(Main.Zero)) -> Main.Pos(Main.Succ(Main.Zero))
10.22/4.28	   new_psMyInt(Main.Neg(Main.Succ(Main.Zero))) -> Main.Pos(Main.Zero)
10.22/4.28	
10.22/4.28	Used ordering: Polynomial interpretation [POLO]:
10.22/4.28	
10.22/4.28	   POL(Main.Neg(x_1)) = 2 + 2*x_1
10.22/4.28	   POL(Main.Pos(x_1)) = 2 + x_1
10.22/4.28	   POL(Main.Succ(x_1)) = x_1
10.22/4.28	   POL(Main.Zero) = 2
10.22/4.28	   POL(new_numericEnumFrom(x_1)) = x_1
10.22/4.28	   POL(new_primPlusNat(x_1)) = x_1
10.22/4.28	   POL(new_primPlusNat0(x_1)) = x_1
10.22/4.28	   POL(new_psMyInt(x_1)) = x_1
10.22/4.28	
10.22/4.28	
10.22/4.28	----------------------------------------
10.22/4.28	
10.22/4.28	(8)
10.22/4.28	Obligation:
10.22/4.28	Q DP problem:
10.22/4.28	The TRS P consists of the following rules:
10.22/4.28	
10.22/4.28	   new_numericEnumFrom(vx3) -> new_numericEnumFrom(new_psMyInt(vx3))
10.22/4.28	
10.22/4.28	The TRS R consists of the following rules:
10.22/4.28	
10.22/4.28	   new_primPlusNat(Main.Zero) -> Main.Succ(Main.Zero)
10.22/4.28	   new_primPlusNat(Main.Succ(vx300)) -> Main.Succ(Main.Succ(new_primPlusNat0(vx300)))
10.22/4.28	   new_psMyInt(Main.Pos(vx30)) -> Main.Pos(new_primPlusNat(vx30))
10.22/4.28	   new_psMyInt(Main.Neg(Main.Succ(Main.Succ(vx3000)))) -> Main.Neg(Main.Succ(vx3000))
10.22/4.28	   new_primPlusNat0(Main.Succ(vx3000)) -> Main.Succ(vx3000)
10.22/4.28	   new_primPlusNat0(Main.Zero) -> Main.Zero
10.22/4.28	
10.22/4.28	The set Q consists of the following terms:
10.22/4.28	
10.22/4.28	   new_psMyInt(Main.Neg(Main.Succ(Main.Zero)))
10.22/4.28	   new_primPlusNat0(Main.Zero)
10.22/4.28	   new_psMyInt(Main.Pos(x0))
10.22/4.28	   new_psMyInt(Main.Neg(Main.Succ(Main.Succ(x0))))
10.22/4.28	   new_psMyInt(Main.Neg(Main.Zero))
10.22/4.28	   new_primPlusNat(Main.Zero)
10.22/4.28	   new_primPlusNat0(Main.Succ(x0))
10.22/4.28	   new_primPlusNat(Main.Succ(x0))
10.22/4.28	
10.22/4.28	We have to consider all minimal (P,Q,R)-chains.
10.22/4.28	----------------------------------------
10.22/4.28	
10.22/4.28	(9) NonTerminationLoopProof (COMPLETE)
10.22/4.28	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
10.22/4.28	Found a loop by semiunifying a rule from P directly.
10.22/4.28	
10.22/4.28	s = new_numericEnumFrom(vx3) evaluates to  t =new_numericEnumFrom(new_psMyInt(vx3))
10.22/4.28	
10.22/4.28	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
10.22/4.28	* Matcher: [vx3 / new_psMyInt(vx3)]
10.22/4.28	* Semiunifier: [ ]
10.22/4.28	
10.22/4.28	--------------------------------------------------------------------------------
10.22/4.28	Rewriting sequence
10.22/4.28	
10.22/4.28	The DP semiunifies directly so there is only one rewrite step from new_numericEnumFrom(vx3) to new_numericEnumFrom(new_psMyInt(vx3)).
10.22/4.28	
10.22/4.28	
10.22/4.28	
10.22/4.28	
10.22/4.28	----------------------------------------
10.22/4.28	
10.22/4.28	(10)
10.22/4.28	NO
10.22/4.28	
10.22/4.28	----------------------------------------
10.22/4.28	
10.22/4.28	(11) Narrow (COMPLETE)
10.22/4.28	Haskell To QDPs
10.22/4.28	
10.22/4.28	digraph dp_graph {
10.22/4.28	node [outthreshold=100, inthreshold=100];1[label="enumFromMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.22/4.28	3[label="enumFromMyInt vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.22/4.28	4[label="numericEnumFrom vx3",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.22/4.28	5[label="Cons vx3 (dsEm numericEnumFrom (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="green",shape="box"];5 -> 6[label="",style="dashed", color="green", weight=3];
10.22/4.28	6[label="dsEm numericEnumFrom (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero))))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.22/4.28	7 -> 8[label="",style="dashed", color="red", weight=0];
10.22/4.28	7[label="seq (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];7 -> 9[label="",style="dashed", color="magenta", weight=3];
10.22/4.28	9 -> 4[label="",style="dashed", color="red", weight=0];
10.22/4.28	9[label="numericEnumFrom (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero))))",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
10.22/4.28	8[label="seq (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero)))) vx4",fontsize=16,color="black",shape="triangle"];8 -> 11[label="",style="solid", color="black", weight=3];
10.22/4.28	10[label="psMyInt vx3 (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3];
10.22/4.28	11 -> 13[label="",style="dashed", color="red", weight=0];
10.22/4.28	11[label="enforceWHNF (WHNF (psMyInt vx3 (fromIntMyInt (Pos (Succ Zero))))) vx4",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
10.22/4.28	12[label="primPlusInt vx3 (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="burlywood",shape="box"];40[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];12 -> 40[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	40 -> 15[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	41[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];12 -> 41[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	41 -> 16[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	14 -> 10[label="",style="dashed", color="red", weight=0];
10.22/4.28	14[label="psMyInt vx3 (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];13[label="enforceWHNF (WHNF vx5) vx4",fontsize=16,color="black",shape="triangle"];13 -> 17[label="",style="solid", color="black", weight=3];
10.22/4.28	15[label="primPlusInt (Pos vx30) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
10.22/4.28	16[label="primPlusInt (Neg vx30) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];16 -> 19[label="",style="solid", color="black", weight=3];
10.22/4.28	17[label="vx4",fontsize=16,color="green",shape="box"];18[label="primPlusInt (Pos vx30) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
10.22/4.28	19[label="primPlusInt (Neg vx30) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
10.22/4.28	20[label="Pos (primPlusNat vx30 (Succ Zero))",fontsize=16,color="green",shape="box"];20 -> 22[label="",style="dashed", color="green", weight=3];
10.22/4.28	21[label="primMinusNat (Succ Zero) vx30",fontsize=16,color="burlywood",shape="box"];42[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];21 -> 42[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	42 -> 23[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	43[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 43[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	43 -> 24[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	22[label="primPlusNat vx30 (Succ Zero)",fontsize=16,color="burlywood",shape="box"];44[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];22 -> 44[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	44 -> 25[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	45[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 45[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	45 -> 26[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	23[label="primMinusNat (Succ Zero) (Succ vx300)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
10.22/4.28	24[label="primMinusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
10.22/4.28	25[label="primPlusNat (Succ vx300) (Succ Zero)",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
10.22/4.28	26[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
10.22/4.28	27[label="primMinusNat Zero vx300",fontsize=16,color="burlywood",shape="box"];46[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];27 -> 46[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	46 -> 31[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	47[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 47[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	47 -> 32[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	28[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];29[label="Succ (Succ (primPlusNat vx300 Zero))",fontsize=16,color="green",shape="box"];29 -> 33[label="",style="dashed", color="green", weight=3];
10.22/4.28	30[label="Succ Zero",fontsize=16,color="green",shape="box"];31[label="primMinusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];31 -> 34[label="",style="solid", color="black", weight=3];
10.22/4.28	32[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3];
10.22/4.28	33[label="primPlusNat vx300 Zero",fontsize=16,color="burlywood",shape="box"];48[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];33 -> 48[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	48 -> 36[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	49[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 49[label="",style="solid", color="burlywood", weight=9];
10.22/4.28	49 -> 37[label="",style="solid", color="burlywood", weight=3];
10.22/4.28	34[label="Neg (Succ vx3000)",fontsize=16,color="green",shape="box"];35[label="Pos Zero",fontsize=16,color="green",shape="box"];36[label="primPlusNat (Succ vx3000) Zero",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
10.22/4.28	37[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
10.22/4.28	38[label="Succ vx3000",fontsize=16,color="green",shape="box"];39[label="Zero",fontsize=16,color="green",shape="box"];}
10.22/4.28	
10.22/4.28	----------------------------------------
10.22/4.28	
10.22/4.28	(12)
10.22/4.28	TRUE
10.33/6.88	EOF