8.19/3.61	YES
10.05/4.13	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.05/4.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.05/4.13	
10.05/4.13	
10.05/4.13	H-Termination with start terms of the given HASKELL could be proven:
10.05/4.13	
10.05/4.13	(0) HASKELL
10.05/4.13	(1) BR [EQUIVALENT, 0 ms]
10.05/4.13	(2) HASKELL
10.05/4.13	(3) COR [EQUIVALENT, 0 ms]
10.05/4.13	(4) HASKELL
10.05/4.13	(5) Narrow [SOUND, 0 ms]
10.05/4.13	(6) AND
10.05/4.13	    (7) QDP
10.05/4.13	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.05/4.13	        (9) YES
10.05/4.13	    (10) QDP
10.05/4.13	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.05/4.13	        (12) YES
10.05/4.13	    (13) QDP
10.05/4.13	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.05/4.13	        (15) YES
10.05/4.13	
10.05/4.13	
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(0)
10.05/4.13	Obligation:
10.05/4.13	mainModule Main 
10.05/4.13	module Main where {
10.05/4.13	import qualified Prelude;
10.05/4.13	data List a = Cons a (List a)  | Nil ;
10.05/4.13	
10.05/4.13	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.05/4.13	
10.05/4.13	data Main.Nat = Succ Main.Nat  | Zero ;
10.05/4.13	
10.05/4.13	data Main.WHNF a = WHNF a ;
10.05/4.13	
10.05/4.13	dsEm :: (a  ->  b)  ->  a  ->  b;
10.05/4.13	dsEm f x = Main.seq x (f x);
10.05/4.13	
10.05/4.13	enforceWHNF :: Main.WHNF b  ->  a  ->  a;
10.05/4.13	enforceWHNF (Main.WHNF x) y = y;
10.05/4.13	
10.05/4.13	foldl' :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
10.05/4.13	foldl' f a Nil = a;
10.05/4.13	foldl' f a (Cons x xs) = dsEm (foldl' f) (f a x) xs;
10.05/4.13	
10.05/4.13	fromIntMyInt :: MyInt  ->  MyInt;
10.05/4.13	fromIntMyInt x = x;
10.05/4.13	
10.05/4.13	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
10.05/4.13	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
10.05/4.13	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
10.05/4.13	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
10.05/4.13	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
10.05/4.13	
10.05/4.13	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.05/4.13	primMulNat Main.Zero Main.Zero = Main.Zero;
10.05/4.13	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
10.05/4.13	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
10.05/4.13	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
10.05/4.13	
10.05/4.13	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.05/4.13	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.05/4.13	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.05/4.13	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.05/4.13	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.05/4.13	
10.05/4.13	productMyInt :: List MyInt  ->  MyInt;
10.05/4.13	productMyInt = foldl' srMyInt (fromIntMyInt (Main.Pos (Main.Succ Main.Zero)));
10.05/4.13	
10.05/4.13	seq :: a  ->  b  ->  b;
10.05/4.13	seq x y = Main.enforceWHNF (Main.WHNF x) y;
10.05/4.13	
10.05/4.13	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.05/4.13	srMyInt = primMulInt;
10.05/4.13	
10.05/4.13	}
10.05/4.13	
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(1) BR (EQUIVALENT)
10.05/4.13	Replaced joker patterns by fresh variables and removed binding patterns.
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(2)
10.05/4.13	Obligation:
10.05/4.13	mainModule Main 
10.05/4.13	module Main where {
10.05/4.13	import qualified Prelude;
10.05/4.13	data List a = Cons a (List a)  | Nil ;
10.05/4.13	
10.05/4.13	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.05/4.13	
10.05/4.13	data Main.Nat = Succ Main.Nat  | Zero ;
10.05/4.13	
10.05/4.13	data Main.WHNF a = WHNF a ;
10.05/4.13	
10.05/4.13	dsEm :: (a  ->  b)  ->  a  ->  b;
10.05/4.13	dsEm f x = Main.seq x (f x);
10.05/4.13	
10.05/4.13	enforceWHNF :: Main.WHNF b  ->  a  ->  a;
10.05/4.13	enforceWHNF (Main.WHNF x) y = y;
10.05/4.13	
10.05/4.13	foldl' :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
10.05/4.13	foldl' f a Nil = a;
10.05/4.13	foldl' f a (Cons x xs) = dsEm (foldl' f) (f a x) xs;
10.05/4.13	
10.05/4.13	fromIntMyInt :: MyInt  ->  MyInt;
10.05/4.13	fromIntMyInt x = x;
10.05/4.13	
10.05/4.13	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
10.05/4.13	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
10.05/4.13	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
10.05/4.13	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
10.05/4.13	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
10.05/4.13	
10.05/4.13	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.05/4.13	primMulNat Main.Zero Main.Zero = Main.Zero;
10.05/4.13	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
10.05/4.13	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
10.05/4.13	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
10.05/4.13	
10.05/4.13	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.05/4.13	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.05/4.13	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.05/4.13	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.05/4.13	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.05/4.13	
10.05/4.13	productMyInt :: List MyInt  ->  MyInt;
10.05/4.13	productMyInt = foldl' srMyInt (fromIntMyInt (Main.Pos (Main.Succ Main.Zero)));
10.05/4.13	
10.05/4.13	seq :: b  ->  a  ->  a;
10.05/4.13	seq x y = Main.enforceWHNF (Main.WHNF x) y;
10.05/4.13	
10.05/4.13	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.05/4.13	srMyInt = primMulInt;
10.05/4.13	
10.05/4.13	}
10.05/4.13	
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(3) COR (EQUIVALENT)
10.05/4.13	Cond Reductions:
10.05/4.13	The following Function with conditions
10.05/4.13	"undefined |Falseundefined;
10.05/4.13	"
10.05/4.13	is transformed to
10.05/4.13	"undefined  = undefined1;
10.05/4.13	"
10.05/4.13	"undefined0 True = undefined;
10.05/4.13	"
10.05/4.13	"undefined1  = undefined0 False;
10.05/4.13	"
10.05/4.13	
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(4)
10.05/4.13	Obligation:
10.05/4.13	mainModule Main 
10.05/4.13	module Main where {
10.05/4.13	import qualified Prelude;
10.05/4.13	data List a = Cons a (List a)  | Nil ;
10.05/4.13	
10.05/4.13	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.05/4.13	
10.05/4.13	data Main.Nat = Succ Main.Nat  | Zero ;
10.05/4.13	
10.05/4.13	data Main.WHNF a = WHNF a ;
10.05/4.13	
10.05/4.13	dsEm :: (b  ->  a)  ->  b  ->  a;
10.05/4.13	dsEm f x = Main.seq x (f x);
10.05/4.13	
10.05/4.13	enforceWHNF :: Main.WHNF b  ->  a  ->  a;
10.05/4.13	enforceWHNF (Main.WHNF x) y = y;
10.05/4.13	
10.05/4.13	foldl' :: (a  ->  b  ->  a)  ->  a  ->  List b  ->  a;
10.05/4.13	foldl' f a Nil = a;
10.05/4.13	foldl' f a (Cons x xs) = dsEm (foldl' f) (f a x) xs;
10.05/4.13	
10.05/4.13	fromIntMyInt :: MyInt  ->  MyInt;
10.05/4.13	fromIntMyInt x = x;
10.05/4.13	
10.05/4.13	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
10.05/4.13	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
10.05/4.13	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
10.05/4.13	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
10.05/4.13	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
10.05/4.13	
10.05/4.13	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.05/4.13	primMulNat Main.Zero Main.Zero = Main.Zero;
10.05/4.13	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
10.05/4.13	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
10.05/4.13	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
10.05/4.13	
10.05/4.13	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.05/4.13	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.05/4.13	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.05/4.13	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.05/4.13	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.05/4.13	
10.05/4.13	productMyInt :: List MyInt  ->  MyInt;
10.05/4.13	productMyInt = foldl' srMyInt (fromIntMyInt (Main.Pos (Main.Succ Main.Zero)));
10.05/4.13	
10.05/4.13	seq :: b  ->  a  ->  a;
10.05/4.13	seq x y = Main.enforceWHNF (Main.WHNF x) y;
10.05/4.13	
10.05/4.13	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.05/4.13	srMyInt = primMulInt;
10.05/4.13	
10.05/4.13	}
10.05/4.13	
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(5) Narrow (SOUND)
10.05/4.13	Haskell To QDPs
10.05/4.13	
10.05/4.13	digraph dp_graph {
10.05/4.13	node [outthreshold=100, inthreshold=100];1[label="productMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.05/4.13	3[label="productMyInt vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.05/4.13	4[label="foldl' srMyInt (fromIntMyInt (Pos (Succ Zero))) vx3",fontsize=16,color="burlywood",shape="box"];76[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 76[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	76 -> 5[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	77[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 77[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	77 -> 6[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	5[label="foldl' srMyInt (fromIntMyInt (Pos (Succ Zero))) (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
10.05/4.13	6[label="foldl' srMyInt (fromIntMyInt (Pos (Succ Zero))) Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.05/4.13	7[label="dsEm (foldl' srMyInt) (srMyInt (fromIntMyInt (Pos (Succ Zero))) vx30) vx31",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
10.05/4.13	8[label="fromIntMyInt (Pos (Succ Zero))",fontsize=16,color="black",shape="triangle"];8 -> 10[label="",style="solid", color="black", weight=3];
10.05/4.13	9 -> 11[label="",style="dashed", color="red", weight=0];
10.05/4.13	9[label="seq (srMyInt (fromIntMyInt (Pos (Succ Zero))) vx30) (foldl' srMyInt (srMyInt (fromIntMyInt (Pos (Succ Zero))) vx30)) vx31",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	9 -> 13[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	10[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];12 -> 8[label="",style="dashed", color="red", weight=0];
10.05/4.13	12[label="fromIntMyInt (Pos (Succ Zero))",fontsize=16,color="magenta"];13 -> 8[label="",style="dashed", color="red", weight=0];
10.05/4.13	13[label="fromIntMyInt (Pos (Succ Zero))",fontsize=16,color="magenta"];11[label="seq (srMyInt vx4 vx30) (foldl' srMyInt (srMyInt vx5 vx30)) vx31",fontsize=16,color="black",shape="triangle"];11 -> 14[label="",style="solid", color="black", weight=3];
10.05/4.13	14[label="enforceWHNF (WHNF (srMyInt vx4 vx30)) (foldl' srMyInt (srMyInt vx5 vx30)) vx31",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
10.05/4.13	15[label="foldl' srMyInt (srMyInt vx5 vx30) vx31",fontsize=16,color="burlywood",shape="box"];78[label="vx31/Cons vx310 vx311",fontsize=10,color="white",style="solid",shape="box"];15 -> 78[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	78 -> 16[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	79[label="vx31/Nil",fontsize=10,color="white",style="solid",shape="box"];15 -> 79[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	79 -> 17[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	16[label="foldl' srMyInt (srMyInt vx5 vx30) (Cons vx310 vx311)",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
10.05/4.13	17[label="foldl' srMyInt (srMyInt vx5 vx30) Nil",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
10.05/4.13	18[label="dsEm (foldl' srMyInt) (srMyInt (srMyInt vx5 vx30) vx310) vx311",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
10.05/4.13	19[label="srMyInt vx5 vx30",fontsize=16,color="black",shape="triangle"];19 -> 21[label="",style="solid", color="black", weight=3];
10.05/4.13	20 -> 11[label="",style="dashed", color="red", weight=0];
10.05/4.13	20[label="seq (srMyInt (srMyInt vx5 vx30) vx310) (foldl' srMyInt (srMyInt (srMyInt vx5 vx30) vx310)) vx311",fontsize=16,color="magenta"];20 -> 22[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	20 -> 23[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	20 -> 24[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	20 -> 25[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	21[label="primMulInt vx5 vx30",fontsize=16,color="burlywood",shape="box"];80[label="vx5/Pos vx50",fontsize=10,color="white",style="solid",shape="box"];21 -> 80[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	80 -> 26[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	81[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];21 -> 81[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	81 -> 27[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	22 -> 19[label="",style="dashed", color="red", weight=0];
10.05/4.13	22[label="srMyInt vx5 vx30",fontsize=16,color="magenta"];23[label="vx311",fontsize=16,color="green",shape="box"];24[label="vx310",fontsize=16,color="green",shape="box"];25 -> 19[label="",style="dashed", color="red", weight=0];
10.05/4.13	25[label="srMyInt vx5 vx30",fontsize=16,color="magenta"];26[label="primMulInt (Pos vx50) vx30",fontsize=16,color="burlywood",shape="box"];82[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];26 -> 82[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	82 -> 28[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	83[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];26 -> 83[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	83 -> 29[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	27[label="primMulInt (Neg vx50) vx30",fontsize=16,color="burlywood",shape="box"];84[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];27 -> 84[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	84 -> 30[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	85[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];27 -> 85[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	85 -> 31[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	28[label="primMulInt (Pos vx50) (Pos vx300)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
10.05/4.13	29[label="primMulInt (Pos vx50) (Neg vx300)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
10.05/4.13	30[label="primMulInt (Neg vx50) (Pos vx300)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
10.05/4.13	31[label="primMulInt (Neg vx50) (Neg vx300)",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
10.05/4.13	32[label="Pos (primMulNat vx50 vx300)",fontsize=16,color="green",shape="box"];32 -> 36[label="",style="dashed", color="green", weight=3];
10.05/4.13	33[label="Neg (primMulNat vx50 vx300)",fontsize=16,color="green",shape="box"];33 -> 37[label="",style="dashed", color="green", weight=3];
10.05/4.13	34[label="Neg (primMulNat vx50 vx300)",fontsize=16,color="green",shape="box"];34 -> 38[label="",style="dashed", color="green", weight=3];
10.05/4.13	35[label="Pos (primMulNat vx50 vx300)",fontsize=16,color="green",shape="box"];35 -> 39[label="",style="dashed", color="green", weight=3];
10.05/4.13	36[label="primMulNat vx50 vx300",fontsize=16,color="burlywood",shape="triangle"];86[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];36 -> 86[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	86 -> 40[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	87[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 87[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	87 -> 41[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	37 -> 36[label="",style="dashed", color="red", weight=0];
10.05/4.13	37[label="primMulNat vx50 vx300",fontsize=16,color="magenta"];37 -> 42[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	38 -> 36[label="",style="dashed", color="red", weight=0];
10.05/4.13	38[label="primMulNat vx50 vx300",fontsize=16,color="magenta"];38 -> 43[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	39 -> 36[label="",style="dashed", color="red", weight=0];
10.05/4.13	39[label="primMulNat vx50 vx300",fontsize=16,color="magenta"];39 -> 44[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	39 -> 45[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	40[label="primMulNat (Succ vx500) vx300",fontsize=16,color="burlywood",shape="box"];88[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];40 -> 88[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	88 -> 46[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	89[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];40 -> 89[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	89 -> 47[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	41[label="primMulNat Zero vx300",fontsize=16,color="burlywood",shape="box"];90[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];41 -> 90[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	90 -> 48[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	91[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 91[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	91 -> 49[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	42[label="vx300",fontsize=16,color="green",shape="box"];43[label="vx50",fontsize=16,color="green",shape="box"];44[label="vx50",fontsize=16,color="green",shape="box"];45[label="vx300",fontsize=16,color="green",shape="box"];46[label="primMulNat (Succ vx500) (Succ vx3000)",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3];
10.05/4.13	47[label="primMulNat (Succ vx500) Zero",fontsize=16,color="black",shape="box"];47 -> 51[label="",style="solid", color="black", weight=3];
10.05/4.13	48[label="primMulNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];48 -> 52[label="",style="solid", color="black", weight=3];
10.05/4.13	49[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];49 -> 53[label="",style="solid", color="black", weight=3];
10.05/4.13	50 -> 54[label="",style="dashed", color="red", weight=0];
10.05/4.13	50[label="primPlusNat (primMulNat vx500 (Succ vx3000)) (Succ vx3000)",fontsize=16,color="magenta"];50 -> 55[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	51[label="Zero",fontsize=16,color="green",shape="box"];52[label="Zero",fontsize=16,color="green",shape="box"];53[label="Zero",fontsize=16,color="green",shape="box"];55 -> 36[label="",style="dashed", color="red", weight=0];
10.05/4.13	55[label="primMulNat vx500 (Succ vx3000)",fontsize=16,color="magenta"];55 -> 56[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	55 -> 57[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	54[label="primPlusNat vx6 (Succ vx3000)",fontsize=16,color="burlywood",shape="triangle"];92[label="vx6/Succ vx60",fontsize=10,color="white",style="solid",shape="box"];54 -> 92[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	92 -> 58[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	93[label="vx6/Zero",fontsize=10,color="white",style="solid",shape="box"];54 -> 93[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	93 -> 59[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	56[label="vx500",fontsize=16,color="green",shape="box"];57[label="Succ vx3000",fontsize=16,color="green",shape="box"];58[label="primPlusNat (Succ vx60) (Succ vx3000)",fontsize=16,color="black",shape="box"];58 -> 60[label="",style="solid", color="black", weight=3];
10.05/4.13	59[label="primPlusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];59 -> 61[label="",style="solid", color="black", weight=3];
10.05/4.13	60[label="Succ (Succ (primPlusNat vx60 vx3000))",fontsize=16,color="green",shape="box"];60 -> 62[label="",style="dashed", color="green", weight=3];
10.05/4.13	61[label="Succ vx3000",fontsize=16,color="green",shape="box"];62[label="primPlusNat vx60 vx3000",fontsize=16,color="burlywood",shape="triangle"];94[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];62 -> 94[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	94 -> 63[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	95[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 95[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	95 -> 64[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	63[label="primPlusNat (Succ vx600) vx3000",fontsize=16,color="burlywood",shape="box"];96[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];63 -> 96[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	96 -> 65[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	97[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];63 -> 97[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	97 -> 66[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	64[label="primPlusNat Zero vx3000",fontsize=16,color="burlywood",shape="box"];98[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];64 -> 98[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	98 -> 67[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	99[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];64 -> 99[label="",style="solid", color="burlywood", weight=9];
10.05/4.13	99 -> 68[label="",style="solid", color="burlywood", weight=3];
10.05/4.13	65[label="primPlusNat (Succ vx600) (Succ vx30000)",fontsize=16,color="black",shape="box"];65 -> 69[label="",style="solid", color="black", weight=3];
10.05/4.13	66[label="primPlusNat (Succ vx600) Zero",fontsize=16,color="black",shape="box"];66 -> 70[label="",style="solid", color="black", weight=3];
10.05/4.13	67[label="primPlusNat Zero (Succ vx30000)",fontsize=16,color="black",shape="box"];67 -> 71[label="",style="solid", color="black", weight=3];
10.05/4.13	68[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3];
10.05/4.13	69[label="Succ (Succ (primPlusNat vx600 vx30000))",fontsize=16,color="green",shape="box"];69 -> 73[label="",style="dashed", color="green", weight=3];
10.05/4.13	70[label="Succ vx600",fontsize=16,color="green",shape="box"];71[label="Succ vx30000",fontsize=16,color="green",shape="box"];72[label="Zero",fontsize=16,color="green",shape="box"];73 -> 62[label="",style="dashed", color="red", weight=0];
10.05/4.13	73[label="primPlusNat vx600 vx30000",fontsize=16,color="magenta"];73 -> 74[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	73 -> 75[label="",style="dashed", color="magenta", weight=3];
10.05/4.13	74[label="vx600",fontsize=16,color="green",shape="box"];75[label="vx30000",fontsize=16,color="green",shape="box"];}
10.05/4.13	
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(6)
10.05/4.13	Complex Obligation (AND)
10.05/4.13	
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(7)
10.05/4.13	Obligation:
10.05/4.13	Q DP problem:
10.05/4.13	The TRS P consists of the following rules:
10.05/4.13	
10.05/4.13	   new_seq(vx4, vx30, vx5, Cons(vx310, vx311)) -> new_seq(new_srMyInt(vx5, vx30), vx310, new_srMyInt(vx5, vx30), vx311)
10.05/4.13	
10.05/4.13	The TRS R consists of the following rules:
10.05/4.13	
10.05/4.13	   new_primPlusNat1(Main.Succ(vx600), Main.Zero) -> Main.Succ(vx600)
10.05/4.13	   new_primPlusNat1(Main.Zero, Main.Succ(vx30000)) -> Main.Succ(vx30000)
10.05/4.13	   new_primMulNat0(Main.Zero, Main.Zero) -> Main.Zero
10.05/4.13	   new_srMyInt(Main.Pos(vx50), Main.Pos(vx300)) -> Main.Pos(new_primMulNat0(vx50, vx300))
10.05/4.13	   new_primPlusNat0(Main.Succ(vx60), vx3000) -> Main.Succ(Main.Succ(new_primPlusNat1(vx60, vx3000)))
10.05/4.13	   new_primMulNat0(Main.Succ(vx500), Main.Zero) -> Main.Zero
10.05/4.13	   new_primMulNat0(Main.Zero, Main.Succ(vx3000)) -> Main.Zero
10.05/4.13	   new_primPlusNat0(Main.Zero, vx3000) -> Main.Succ(vx3000)
10.05/4.13	   new_primPlusNat1(Main.Zero, Main.Zero) -> Main.Zero
10.05/4.13	   new_primMulNat0(Main.Succ(vx500), Main.Succ(vx3000)) -> new_primPlusNat0(new_primMulNat0(vx500, Main.Succ(vx3000)), vx3000)
10.05/4.13	   new_primPlusNat1(Main.Succ(vx600), Main.Succ(vx30000)) -> Main.Succ(Main.Succ(new_primPlusNat1(vx600, vx30000)))
10.05/4.13	   new_srMyInt(Main.Neg(vx50), Main.Neg(vx300)) -> Main.Pos(new_primMulNat0(vx50, vx300))
10.05/4.13	   new_srMyInt(Main.Pos(vx50), Main.Neg(vx300)) -> Main.Neg(new_primMulNat0(vx50, vx300))
10.05/4.13	   new_srMyInt(Main.Neg(vx50), Main.Pos(vx300)) -> Main.Neg(new_primMulNat0(vx50, vx300))
10.05/4.13	
10.05/4.13	The set Q consists of the following terms:
10.05/4.13	
10.05/4.13	   new_primPlusNat0(Main.Zero, x0)
10.05/4.13	   new_primMulNat0(Main.Succ(x0), Main.Succ(x1))
10.05/4.13	   new_primPlusNat1(Main.Succ(x0), Main.Zero)
10.05/4.13	   new_primPlusNat0(Main.Succ(x0), x1)
10.05/4.13	   new_primPlusNat1(Main.Succ(x0), Main.Succ(x1))
10.05/4.13	   new_srMyInt(Main.Neg(x0), Main.Neg(x1))
10.05/4.13	   new_primMulNat0(Main.Succ(x0), Main.Zero)
10.05/4.13	   new_primPlusNat1(Main.Zero, Main.Zero)
10.05/4.13	   new_primPlusNat1(Main.Zero, Main.Succ(x0))
10.05/4.13	   new_srMyInt(Main.Pos(x0), Main.Pos(x1))
10.05/4.13	   new_primMulNat0(Main.Zero, Main.Zero)
10.05/4.13	   new_primMulNat0(Main.Zero, Main.Succ(x0))
10.05/4.13	   new_srMyInt(Main.Pos(x0), Main.Neg(x1))
10.05/4.13	   new_srMyInt(Main.Neg(x0), Main.Pos(x1))
10.05/4.13	
10.05/4.13	We have to consider all minimal (P,Q,R)-chains.
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(8) QDPSizeChangeProof (EQUIVALENT)
10.05/4.13	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. 
10.05/4.13	
10.05/4.13	From the DPs we obtained the following set of size-change graphs:
10.05/4.13	*new_seq(vx4, vx30, vx5, Cons(vx310, vx311)) -> new_seq(new_srMyInt(vx5, vx30), vx310, new_srMyInt(vx5, vx30), vx311)
10.05/4.13	The graph contains the following edges 4 > 2, 4 > 4
10.05/4.13	
10.05/4.13	
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(9)
10.05/4.13	YES
10.05/4.13	
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(10)
10.05/4.13	Obligation:
10.05/4.13	Q DP problem:
10.05/4.13	The TRS P consists of the following rules:
10.05/4.13	
10.05/4.13	   new_primMulNat(Main.Succ(vx500), Main.Succ(vx3000)) -> new_primMulNat(vx500, Main.Succ(vx3000))
10.05/4.13	
10.05/4.13	R is empty.
10.05/4.13	Q is empty.
10.05/4.13	We have to consider all minimal (P,Q,R)-chains.
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(11) QDPSizeChangeProof (EQUIVALENT)
10.05/4.13	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. 
10.05/4.13	
10.05/4.13	From the DPs we obtained the following set of size-change graphs:
10.05/4.13	*new_primMulNat(Main.Succ(vx500), Main.Succ(vx3000)) -> new_primMulNat(vx500, Main.Succ(vx3000))
10.05/4.13	The graph contains the following edges 1 > 1, 2 >= 2
10.05/4.13	
10.05/4.13	
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(12)
10.05/4.13	YES
10.05/4.13	
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(13)
10.05/4.13	Obligation:
10.05/4.13	Q DP problem:
10.05/4.13	The TRS P consists of the following rules:
10.05/4.13	
10.05/4.13	   new_primPlusNat(Main.Succ(vx600), Main.Succ(vx30000)) -> new_primPlusNat(vx600, vx30000)
10.05/4.13	
10.05/4.13	R is empty.
10.05/4.13	Q is empty.
10.05/4.13	We have to consider all minimal (P,Q,R)-chains.
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(14) QDPSizeChangeProof (EQUIVALENT)
10.05/4.13	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. 
10.05/4.13	
10.05/4.13	From the DPs we obtained the following set of size-change graphs:
10.05/4.13	*new_primPlusNat(Main.Succ(vx600), Main.Succ(vx30000)) -> new_primPlusNat(vx600, vx30000)
10.05/4.13	The graph contains the following edges 1 > 1, 2 > 2
10.05/4.13	
10.05/4.13	
10.05/4.13	----------------------------------------
10.05/4.13	
10.05/4.13	(15)
10.05/4.13	YES
10.05/4.16	EOF