8.17/3.69	YES
9.93/4.19	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.93/4.19	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.93/4.19	
9.93/4.19	
9.93/4.19	H-Termination with start terms of the given HASKELL could be proven:
9.93/4.19	
9.93/4.19	(0) HASKELL
9.93/4.19	(1) BR [EQUIVALENT, 0 ms]
9.93/4.19	(2) HASKELL
9.93/4.19	(3) COR [EQUIVALENT, 0 ms]
9.93/4.19	(4) HASKELL
9.93/4.19	(5) Narrow [SOUND, 0 ms]
9.93/4.19	(6) QDP
9.93/4.19	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.93/4.19	(8) YES
9.93/4.19	
9.93/4.19	
9.93/4.19	----------------------------------------
9.93/4.19	
9.93/4.19	(0)
9.93/4.19	Obligation:
9.93/4.19	mainModule Main 
9.93/4.19	module Main where {
9.93/4.19	import qualified Prelude;
9.93/4.19	data List a = Cons a (List a)  | Nil ;
9.93/4.19	
9.93/4.19	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.93/4.19	
9.93/4.19	data Main.Nat = Succ Main.Nat  | Zero ;
9.93/4.19	
9.93/4.19	data Main.WHNF a = WHNF a ;
9.93/4.19	
9.93/4.19	dsEm :: (a  ->  b)  ->  a  ->  b;
9.93/4.19	dsEm f x = Main.seq x (f x);
9.93/4.19	
9.93/4.19	enforceWHNF :: Main.WHNF a  ->  b  ->  b;
9.93/4.19	enforceWHNF (Main.WHNF x) y = y;
9.93/4.19	
9.93/4.19	foldl' :: (a  ->  b  ->  a)  ->  a  ->  List b  ->  a;
9.93/4.19	foldl' f a Nil = a;
9.93/4.19	foldl' f a (Cons x xs) = dsEm (foldl' f) (f a x) xs;
9.93/4.19	
9.93/4.19	length :: List a  ->  MyInt;
9.93/4.19	length = foldl' length0 (Main.Pos Main.Zero);
9.93/4.19	
9.93/4.19	length0 n vv = psMyInt n (Main.Pos (Main.Succ Main.Zero));
9.93/4.19	
9.93/4.19	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
9.93/4.19	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
9.93/4.19	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
9.93/4.19	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
9.93/4.19	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
9.93/4.19	
9.93/4.19	primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
9.93/4.19	primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
9.93/4.19	primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
9.93/4.19	primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
9.93/4.19	primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);
9.93/4.19	
9.93/4.19	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.93/4.19	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.93/4.19	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.93/4.19	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.93/4.19	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.93/4.19	
9.93/4.19	psMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.93/4.19	psMyInt = primPlusInt;
9.93/4.19	
9.93/4.19	seq :: a  ->  b  ->  b;
9.93/4.19	seq x y = Main.enforceWHNF (Main.WHNF x) y;
9.93/4.19	
9.93/4.19	}
9.93/4.19	
9.93/4.19	----------------------------------------
9.93/4.19	
9.93/4.19	(1) BR (EQUIVALENT)
9.93/4.19	Replaced joker patterns by fresh variables and removed binding patterns.
9.93/4.19	----------------------------------------
9.93/4.19	
9.93/4.19	(2)
9.93/4.19	Obligation:
9.93/4.19	mainModule Main 
9.93/4.19	module Main where {
9.93/4.19	import qualified Prelude;
9.93/4.19	data List a = Cons a (List a)  | Nil ;
9.93/4.19	
9.93/4.19	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.93/4.19	
9.93/4.19	data Main.Nat = Succ Main.Nat  | Zero ;
9.93/4.19	
9.93/4.19	data Main.WHNF a = WHNF a ;
9.93/4.19	
9.93/4.19	dsEm :: (b  ->  a)  ->  b  ->  a;
9.93/4.19	dsEm f x = Main.seq x (f x);
9.93/4.19	
9.93/4.19	enforceWHNF :: Main.WHNF a  ->  b  ->  b;
9.93/4.19	enforceWHNF (Main.WHNF x) y = y;
9.93/4.19	
9.93/4.19	foldl' :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
9.93/4.19	foldl' f a Nil = a;
9.93/4.19	foldl' f a (Cons x xs) = dsEm (foldl' f) (f a x) xs;
9.93/4.19	
9.93/4.19	length :: List a  ->  MyInt;
9.93/4.19	length = foldl' length0 (Main.Pos Main.Zero);
9.93/4.19	
9.93/4.19	length0 n vv = psMyInt n (Main.Pos (Main.Succ Main.Zero));
9.93/4.19	
9.93/4.19	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
9.93/4.19	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
9.93/4.19	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
9.93/4.19	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
9.93/4.19	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
9.93/4.19	
9.93/4.19	primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
9.93/4.19	primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
9.93/4.19	primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
9.93/4.19	primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
9.93/4.19	primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);
9.93/4.19	
9.93/4.19	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.93/4.19	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.93/4.19	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.93/4.19	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.93/4.19	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.93/4.19	
9.93/4.19	psMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.93/4.19	psMyInt = primPlusInt;
9.93/4.19	
9.93/4.19	seq :: a  ->  b  ->  b;
9.93/4.19	seq x y = Main.enforceWHNF (Main.WHNF x) y;
9.93/4.19	
9.93/4.19	}
9.93/4.19	
9.93/4.19	----------------------------------------
9.93/4.19	
9.93/4.19	(3) COR (EQUIVALENT)
9.93/4.19	Cond Reductions:
9.93/4.19	The following Function with conditions
9.93/4.19	"undefined |Falseundefined;
9.93/4.19	"
9.93/4.19	is transformed to
9.93/4.19	"undefined  = undefined1;
9.93/4.19	"
9.93/4.19	"undefined0 True = undefined;
9.93/4.19	"
9.93/4.19	"undefined1  = undefined0 False;
9.93/4.19	"
9.93/4.19	
9.93/4.19	----------------------------------------
9.93/4.19	
9.93/4.19	(4)
9.93/4.19	Obligation:
9.93/4.19	mainModule Main 
9.93/4.19	module Main where {
9.93/4.19	import qualified Prelude;
9.93/4.19	data List a = Cons a (List a)  | Nil ;
9.93/4.19	
9.93/4.19	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.93/4.19	
9.93/4.19	data Main.Nat = Succ Main.Nat  | Zero ;
9.93/4.19	
9.93/4.19	data Main.WHNF a = WHNF a ;
9.93/4.19	
9.93/4.19	dsEm :: (a  ->  b)  ->  a  ->  b;
9.93/4.19	dsEm f x = Main.seq x (f x);
9.93/4.19	
9.93/4.19	enforceWHNF :: Main.WHNF b  ->  a  ->  a;
9.93/4.19	enforceWHNF (Main.WHNF x) y = y;
9.93/4.19	
9.93/4.19	foldl' :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
9.93/4.19	foldl' f a Nil = a;
9.93/4.19	foldl' f a (Cons x xs) = dsEm (foldl' f) (f a x) xs;
9.93/4.19	
9.93/4.19	length :: List a  ->  MyInt;
9.93/4.19	length = foldl' length0 (Main.Pos Main.Zero);
9.93/4.19	
9.93/4.19	length0 n vv = psMyInt n (Main.Pos (Main.Succ Main.Zero));
9.93/4.19	
9.93/4.19	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
9.93/4.19	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
9.93/4.19	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
9.93/4.19	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
9.93/4.19	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
9.93/4.19	
9.93/4.19	primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
9.93/4.19	primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
9.93/4.19	primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
9.93/4.19	primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
9.93/4.19	primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);
9.93/4.19	
9.93/4.19	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.93/4.19	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.93/4.19	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.93/4.19	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.93/4.19	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.93/4.19	
9.93/4.19	psMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.93/4.19	psMyInt = primPlusInt;
9.93/4.19	
9.93/4.19	seq :: b  ->  a  ->  a;
9.93/4.19	seq x y = Main.enforceWHNF (Main.WHNF x) y;
9.93/4.19	
9.93/4.19	}
9.93/4.19	
9.93/4.19	----------------------------------------
9.93/4.19	
9.93/4.19	(5) Narrow (SOUND)
9.93/4.19	Haskell To QDPs
9.93/4.19	
9.93/4.19	digraph dp_graph {
9.93/4.19	node [outthreshold=100, inthreshold=100];1[label="length",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.93/4.19	3[label="length vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.93/4.19	4[label="foldl' length0 (Pos Zero) vy3",fontsize=16,color="burlywood",shape="box"];60[label="vy3/Cons vy30 vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 60[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	60 -> 5[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	61[label="vy3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 61[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	61 -> 6[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	5[label="foldl' length0 (Pos Zero) (Cons vy30 vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.93/4.19	6[label="foldl' length0 (Pos Zero) Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.93/4.19	7[label="dsEm (foldl' length0) (length0 (Pos Zero) vy30) vy31",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.93/4.19	8[label="Pos Zero",fontsize=16,color="green",shape="box"];9 -> 18[label="",style="dashed", color="red", weight=0];
9.93/4.19	9[label="seq (length0 (Pos Zero) vy30) (foldl' length0 (length0 (Pos Zero) vy30)) vy31",fontsize=16,color="magenta"];9 -> 19[label="",style="dashed", color="magenta", weight=3];
9.93/4.19	9 -> 20[label="",style="dashed", color="magenta", weight=3];
9.93/4.19	9 -> 21[label="",style="dashed", color="magenta", weight=3];
9.93/4.19	9 -> 22[label="",style="dashed", color="magenta", weight=3];
9.93/4.19	19[label="Pos Zero",fontsize=16,color="green",shape="box"];20[label="vy31",fontsize=16,color="green",shape="box"];21[label="vy30",fontsize=16,color="green",shape="box"];22[label="Pos Zero",fontsize=16,color="green",shape="box"];18[label="seq (length0 vy4 vy310) (foldl' length0 (length0 vy5 vy310)) vy311",fontsize=16,color="black",shape="triangle"];18 -> 25[label="",style="solid", color="black", weight=3];
9.93/4.19	25[label="enforceWHNF (WHNF (length0 vy4 vy310)) (foldl' length0 (length0 vy5 vy310)) vy311",fontsize=16,color="black",shape="box"];25 -> 26[label="",style="solid", color="black", weight=3];
9.93/4.19	26[label="foldl' length0 (length0 vy5 vy310) vy311",fontsize=16,color="burlywood",shape="box"];62[label="vy311/Cons vy3110 vy3111",fontsize=10,color="white",style="solid",shape="box"];26 -> 62[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	62 -> 27[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	63[label="vy311/Nil",fontsize=10,color="white",style="solid",shape="box"];26 -> 63[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	63 -> 28[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	27[label="foldl' length0 (length0 vy5 vy310) (Cons vy3110 vy3111)",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
9.93/4.19	28[label="foldl' length0 (length0 vy5 vy310) Nil",fontsize=16,color="black",shape="box"];28 -> 30[label="",style="solid", color="black", weight=3];
9.93/4.19	29[label="dsEm (foldl' length0) (length0 (length0 vy5 vy310) vy3110) vy3111",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
9.93/4.19	30[label="length0 vy5 vy310",fontsize=16,color="black",shape="triangle"];30 -> 32[label="",style="solid", color="black", weight=3];
9.93/4.19	31 -> 18[label="",style="dashed", color="red", weight=0];
9.93/4.19	31[label="seq (length0 (length0 vy5 vy310) vy3110) (foldl' length0 (length0 (length0 vy5 vy310) vy3110)) vy3111",fontsize=16,color="magenta"];31 -> 33[label="",style="dashed", color="magenta", weight=3];
9.93/4.19	31 -> 34[label="",style="dashed", color="magenta", weight=3];
9.93/4.19	31 -> 35[label="",style="dashed", color="magenta", weight=3];
9.93/4.19	31 -> 36[label="",style="dashed", color="magenta", weight=3];
9.93/4.19	32[label="psMyInt vy5 (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];32 -> 37[label="",style="solid", color="black", weight=3];
9.93/4.19	33 -> 30[label="",style="dashed", color="red", weight=0];
9.93/4.19	33[label="length0 vy5 vy310",fontsize=16,color="magenta"];34[label="vy3111",fontsize=16,color="green",shape="box"];35[label="vy3110",fontsize=16,color="green",shape="box"];36 -> 30[label="",style="dashed", color="red", weight=0];
9.93/4.19	36[label="length0 vy5 vy310",fontsize=16,color="magenta"];37[label="primPlusInt vy5 (Pos (Succ Zero))",fontsize=16,color="burlywood",shape="box"];64[label="vy5/Pos vy50",fontsize=10,color="white",style="solid",shape="box"];37 -> 64[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	64 -> 38[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	65[label="vy5/Neg vy50",fontsize=10,color="white",style="solid",shape="box"];37 -> 65[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	65 -> 39[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	38[label="primPlusInt (Pos vy50) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
9.93/4.19	39[label="primPlusInt (Neg vy50) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
9.93/4.19	40[label="Pos (primPlusNat vy50 (Succ Zero))",fontsize=16,color="green",shape="box"];40 -> 42[label="",style="dashed", color="green", weight=3];
9.93/4.19	41[label="primMinusNat (Succ Zero) vy50",fontsize=16,color="burlywood",shape="box"];66[label="vy50/Succ vy500",fontsize=10,color="white",style="solid",shape="box"];41 -> 66[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	66 -> 43[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	67[label="vy50/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 67[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	67 -> 44[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	42[label="primPlusNat vy50 (Succ Zero)",fontsize=16,color="burlywood",shape="box"];68[label="vy50/Succ vy500",fontsize=10,color="white",style="solid",shape="box"];42 -> 68[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	68 -> 45[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	69[label="vy50/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 69[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	69 -> 46[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	43[label="primMinusNat (Succ Zero) (Succ vy500)",fontsize=16,color="black",shape="box"];43 -> 47[label="",style="solid", color="black", weight=3];
9.93/4.19	44[label="primMinusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3];
9.93/4.19	45[label="primPlusNat (Succ vy500) (Succ Zero)",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3];
9.93/4.19	46[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3];
9.93/4.19	47[label="primMinusNat Zero vy500",fontsize=16,color="burlywood",shape="box"];70[label="vy500/Succ vy5000",fontsize=10,color="white",style="solid",shape="box"];47 -> 70[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	70 -> 51[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	71[label="vy500/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 71[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	71 -> 52[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	48[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];49[label="Succ (Succ (primPlusNat vy500 Zero))",fontsize=16,color="green",shape="box"];49 -> 53[label="",style="dashed", color="green", weight=3];
9.93/4.19	50[label="Succ Zero",fontsize=16,color="green",shape="box"];51[label="primMinusNat Zero (Succ vy5000)",fontsize=16,color="black",shape="box"];51 -> 54[label="",style="solid", color="black", weight=3];
9.93/4.19	52[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];52 -> 55[label="",style="solid", color="black", weight=3];
9.93/4.19	53[label="primPlusNat vy500 Zero",fontsize=16,color="burlywood",shape="box"];72[label="vy500/Succ vy5000",fontsize=10,color="white",style="solid",shape="box"];53 -> 72[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	72 -> 56[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	73[label="vy500/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 73[label="",style="solid", color="burlywood", weight=9];
9.93/4.19	73 -> 57[label="",style="solid", color="burlywood", weight=3];
9.93/4.19	54[label="Neg (Succ vy5000)",fontsize=16,color="green",shape="box"];55[label="Pos Zero",fontsize=16,color="green",shape="box"];56[label="primPlusNat (Succ vy5000) Zero",fontsize=16,color="black",shape="box"];56 -> 58[label="",style="solid", color="black", weight=3];
9.93/4.19	57[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];57 -> 59[label="",style="solid", color="black", weight=3];
9.93/4.19	58[label="Succ vy5000",fontsize=16,color="green",shape="box"];59[label="Zero",fontsize=16,color="green",shape="box"];}
9.93/4.19	
9.93/4.19	----------------------------------------
9.93/4.19	
9.93/4.19	(6)
9.93/4.19	Obligation:
9.93/4.19	Q DP problem:
9.93/4.19	The TRS P consists of the following rules:
9.93/4.19	
9.93/4.19	   new_seq(vy4, vy310, vy5, Cons(vy3110, vy3111), h) -> new_seq(new_length0(vy5, vy310, h), vy3110, new_length0(vy5, vy310, h), vy3111, h)
9.93/4.19	
9.93/4.19	The TRS R consists of the following rules:
9.93/4.19	
9.93/4.19	   new_primPlusNat(Main.Zero) -> Main.Succ(Main.Zero)
9.93/4.19	   new_length0(Main.Neg(Main.Zero), vy310, h) -> Main.Pos(Main.Succ(Main.Zero))
9.93/4.19	   new_primPlusNat(Main.Succ(vy500)) -> Main.Succ(Main.Succ(new_primPlusNat0(vy500)))
9.93/4.19	   new_length0(Main.Neg(Main.Succ(Main.Zero)), vy310, h) -> Main.Pos(Main.Zero)
9.93/4.19	   new_length0(Main.Neg(Main.Succ(Main.Succ(vy5000))), vy310, h) -> Main.Neg(Main.Succ(vy5000))
9.93/4.19	   new_primPlusNat0(Main.Succ(vy5000)) -> Main.Succ(vy5000)
9.93/4.19	   new_length0(Main.Pos(vy50), vy310, h) -> Main.Pos(new_primPlusNat(vy50))
9.93/4.19	   new_primPlusNat0(Main.Zero) -> Main.Zero
9.93/4.19	
9.93/4.19	The set Q consists of the following terms:
9.93/4.19	
9.93/4.19	   new_primPlusNat0(Main.Succ(x0))
9.93/4.19	   new_primPlusNat0(Main.Zero)
9.93/4.19	   new_length0(Main.Neg(Main.Zero), x0, x1)
9.93/4.19	   new_length0(Main.Neg(Main.Succ(Main.Succ(x0))), x1, x2)
9.93/4.19	   new_length0(Main.Pos(x0), x1, x2)
9.93/4.19	   new_primPlusNat(Main.Succ(x0))
9.93/4.19	   new_primPlusNat(Main.Zero)
9.93/4.19	   new_length0(Main.Neg(Main.Succ(Main.Zero)), x0, x1)
9.93/4.19	
9.93/4.19	We have to consider all minimal (P,Q,R)-chains.
9.93/4.19	----------------------------------------
9.93/4.19	
9.93/4.19	(7) QDPSizeChangeProof (EQUIVALENT)
9.93/4.19	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. 
9.93/4.19	
9.93/4.19	From the DPs we obtained the following set of size-change graphs:
9.93/4.19	*new_seq(vy4, vy310, vy5, Cons(vy3110, vy3111), h) -> new_seq(new_length0(vy5, vy310, h), vy3110, new_length0(vy5, vy310, h), vy3111, h)
9.93/4.19	The graph contains the following edges 4 > 2, 4 > 4, 5 >= 5
9.93/4.19	
9.93/4.19	
9.93/4.19	----------------------------------------
9.93/4.19	
9.93/4.19	(8)
9.93/4.19	YES
9.98/4.27	EOF