7.78/3.59	YES
9.72/4.10	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.72/4.10	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.72/4.10	
9.72/4.10	
9.72/4.10	H-Termination with start terms of the given HASKELL could be proven:
9.72/4.10	
9.72/4.10	(0) HASKELL
9.72/4.10	(1) BR [EQUIVALENT, 0 ms]
9.72/4.10	(2) HASKELL
9.72/4.10	(3) COR [EQUIVALENT, 0 ms]
9.72/4.10	(4) HASKELL
9.72/4.10	(5) Narrow [SOUND, 0 ms]
9.72/4.10	(6) QDP
9.72/4.10	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.72/4.10	(8) YES
9.72/4.10	
9.72/4.10	
9.72/4.10	----------------------------------------
9.72/4.10	
9.72/4.10	(0)
9.72/4.10	Obligation:
9.72/4.10	mainModule Main 
9.72/4.10	module Main where {
9.72/4.10	import qualified Prelude;
9.72/4.10	data MyBool = MyTrue  | MyFalse ;
9.72/4.10	
9.72/4.10	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.72/4.10	
9.72/4.10	data Main.Nat = Succ Main.Nat  | Zero ;
9.72/4.10	
9.72/4.10	data Ordering = LT  | EQ  | GT ;
9.72/4.10	
9.72/4.10	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
9.72/4.10	compareMyInt = primCmpInt;
9.72/4.10	
9.72/4.10	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.72/4.10	esEsOrdering LT LT = MyTrue;
9.72/4.10	esEsOrdering LT EQ = MyFalse;
9.72/4.10	esEsOrdering LT GT = MyFalse;
9.72/4.10	esEsOrdering EQ LT = MyFalse;
9.72/4.10	esEsOrdering EQ EQ = MyTrue;
9.72/4.10	esEsOrdering EQ GT = MyFalse;
9.72/4.10	esEsOrdering GT LT = MyFalse;
9.72/4.10	esEsOrdering GT EQ = MyFalse;
9.72/4.10	esEsOrdering GT GT = MyTrue;
9.72/4.10	
9.72/4.10	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.72/4.10	fsEsOrdering x y = not (esEsOrdering x y);
9.72/4.10	
9.72/4.10	gtEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
9.72/4.10	gtEsMyInt x y = fsEsOrdering (compareMyInt x y) LT;
9.72/4.10	
9.72/4.10	not :: MyBool  ->  MyBool;
9.72/4.10	not MyTrue = MyFalse;
9.72/4.10	not MyFalse = MyTrue;
9.72/4.10	
9.72/4.10	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
9.72/4.10	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
9.72/4.10	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
9.72/4.10	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
9.72/4.10	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
9.72/4.10	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
9.72/4.10	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
9.72/4.10	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
9.72/4.10	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
9.72/4.10	
9.72/4.10	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
9.72/4.10	primCmpNat Main.Zero Main.Zero = EQ;
9.72/4.10	primCmpNat Main.Zero (Main.Succ y) = LT;
9.72/4.10	primCmpNat (Main.Succ x) Main.Zero = GT;
9.72/4.10	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
9.72/4.10	
9.72/4.10	}
9.72/4.10	
9.72/4.10	----------------------------------------
9.72/4.10	
9.72/4.10	(1) BR (EQUIVALENT)
9.72/4.10	Replaced joker patterns by fresh variables and removed binding patterns.
9.72/4.10	----------------------------------------
9.72/4.10	
9.72/4.10	(2)
9.72/4.10	Obligation:
9.72/4.10	mainModule Main 
9.72/4.10	module Main where {
9.72/4.10	import qualified Prelude;
9.72/4.10	data MyBool = MyTrue  | MyFalse ;
9.72/4.10	
9.72/4.10	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.72/4.10	
9.72/4.10	data Main.Nat = Succ Main.Nat  | Zero ;
9.72/4.10	
9.72/4.10	data Ordering = LT  | EQ  | GT ;
9.72/4.10	
9.72/4.10	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
9.72/4.10	compareMyInt = primCmpInt;
9.72/4.10	
9.72/4.10	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.72/4.10	esEsOrdering LT LT = MyTrue;
9.72/4.10	esEsOrdering LT EQ = MyFalse;
9.72/4.10	esEsOrdering LT GT = MyFalse;
9.72/4.10	esEsOrdering EQ LT = MyFalse;
9.72/4.10	esEsOrdering EQ EQ = MyTrue;
9.72/4.10	esEsOrdering EQ GT = MyFalse;
9.72/4.10	esEsOrdering GT LT = MyFalse;
9.72/4.10	esEsOrdering GT EQ = MyFalse;
9.72/4.10	esEsOrdering GT GT = MyTrue;
9.72/4.10	
9.72/4.10	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.72/4.10	fsEsOrdering x y = not (esEsOrdering x y);
9.72/4.10	
9.72/4.10	gtEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
9.72/4.10	gtEsMyInt x y = fsEsOrdering (compareMyInt x y) LT;
9.72/4.10	
9.72/4.10	not :: MyBool  ->  MyBool;
9.72/4.10	not MyTrue = MyFalse;
9.72/4.10	not MyFalse = MyTrue;
9.72/4.10	
9.72/4.10	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
9.72/4.10	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
9.72/4.10	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
9.72/4.10	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
9.72/4.10	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
9.72/4.10	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
9.72/4.10	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
9.72/4.10	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
9.72/4.10	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
9.72/4.10	
9.72/4.10	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
9.72/4.10	primCmpNat Main.Zero Main.Zero = EQ;
9.72/4.10	primCmpNat Main.Zero (Main.Succ y) = LT;
9.72/4.10	primCmpNat (Main.Succ x) Main.Zero = GT;
9.72/4.10	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
9.72/4.10	
9.72/4.10	}
9.72/4.10	
9.72/4.10	----------------------------------------
9.72/4.10	
9.72/4.10	(3) COR (EQUIVALENT)
9.72/4.10	Cond Reductions:
9.72/4.10	The following Function with conditions
9.72/4.10	"undefined |Falseundefined;
9.72/4.10	"
9.72/4.10	is transformed to
9.72/4.10	"undefined  = undefined1;
9.72/4.10	"
9.72/4.10	"undefined0 True = undefined;
9.72/4.10	"
9.72/4.10	"undefined1  = undefined0 False;
9.72/4.10	"
9.72/4.10	
9.72/4.10	----------------------------------------
9.72/4.10	
9.72/4.10	(4)
9.72/4.10	Obligation:
9.72/4.10	mainModule Main 
9.72/4.10	module Main where {
9.72/4.10	import qualified Prelude;
9.72/4.10	data MyBool = MyTrue  | MyFalse ;
9.72/4.10	
9.72/4.10	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.72/4.10	
9.72/4.10	data Main.Nat = Succ Main.Nat  | Zero ;
9.72/4.10	
9.72/4.10	data Ordering = LT  | EQ  | GT ;
9.72/4.10	
9.72/4.10	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
9.72/4.10	compareMyInt = primCmpInt;
9.72/4.10	
9.72/4.10	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.72/4.10	esEsOrdering LT LT = MyTrue;
9.72/4.10	esEsOrdering LT EQ = MyFalse;
9.72/4.10	esEsOrdering LT GT = MyFalse;
9.72/4.10	esEsOrdering EQ LT = MyFalse;
9.72/4.10	esEsOrdering EQ EQ = MyTrue;
9.72/4.10	esEsOrdering EQ GT = MyFalse;
9.72/4.10	esEsOrdering GT LT = MyFalse;
9.72/4.10	esEsOrdering GT EQ = MyFalse;
9.72/4.10	esEsOrdering GT GT = MyTrue;
9.72/4.10	
9.72/4.10	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.72/4.10	fsEsOrdering x y = not (esEsOrdering x y);
9.72/4.10	
9.72/4.10	gtEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
9.72/4.10	gtEsMyInt x y = fsEsOrdering (compareMyInt x y) LT;
9.72/4.10	
9.72/4.10	not :: MyBool  ->  MyBool;
9.72/4.10	not MyTrue = MyFalse;
9.72/4.10	not MyFalse = MyTrue;
9.72/4.10	
9.72/4.10	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
9.72/4.10	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
9.72/4.10	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
9.72/4.10	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
9.72/4.10	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
9.72/4.10	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
9.72/4.10	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
9.72/4.10	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
9.72/4.10	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
9.72/4.10	
9.72/4.10	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
9.72/4.10	primCmpNat Main.Zero Main.Zero = EQ;
9.72/4.10	primCmpNat Main.Zero (Main.Succ y) = LT;
9.72/4.10	primCmpNat (Main.Succ x) Main.Zero = GT;
9.72/4.10	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
9.72/4.10	
9.72/4.10	}
9.72/4.10	
9.72/4.10	----------------------------------------
9.72/4.10	
9.72/4.10	(5) Narrow (SOUND)
9.72/4.10	Haskell To QDPs
9.72/4.10	
9.72/4.10	digraph dp_graph {
9.72/4.10	node [outthreshold=100, inthreshold=100];1[label="gtEsMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.72/4.10	3[label="gtEsMyInt vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.72/4.10	4[label="gtEsMyInt vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.72/4.10	5[label="fsEsOrdering (compareMyInt vx3 vx4) LT",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.72/4.10	6[label="not (esEsOrdering (compareMyInt vx3 vx4) LT)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.72/4.10	7[label="not (esEsOrdering (primCmpInt vx3 vx4) LT)",fontsize=16,color="burlywood",shape="box"];73[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];7 -> 73[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	73 -> 8[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	74[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];7 -> 74[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	74 -> 9[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	8[label="not (esEsOrdering (primCmpInt (Pos vx30) vx4) LT)",fontsize=16,color="burlywood",shape="box"];75[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];8 -> 75[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	75 -> 10[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	76[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 76[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	76 -> 11[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	9[label="not (esEsOrdering (primCmpInt (Neg vx30) vx4) LT)",fontsize=16,color="burlywood",shape="box"];77[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];9 -> 77[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	77 -> 12[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	78[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 78[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	78 -> 13[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	10[label="not (esEsOrdering (primCmpInt (Pos (Succ vx300)) vx4) LT)",fontsize=16,color="burlywood",shape="box"];79[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 79[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	79 -> 14[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	80[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 80[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	80 -> 15[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	11[label="not (esEsOrdering (primCmpInt (Pos Zero) vx4) LT)",fontsize=16,color="burlywood",shape="box"];81[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];11 -> 81[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	81 -> 16[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	82[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];11 -> 82[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	82 -> 17[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	12[label="not (esEsOrdering (primCmpInt (Neg (Succ vx300)) vx4) LT)",fontsize=16,color="burlywood",shape="box"];83[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 83[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	83 -> 18[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	84[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 84[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	84 -> 19[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	13[label="not (esEsOrdering (primCmpInt (Neg Zero) vx4) LT)",fontsize=16,color="burlywood",shape="box"];85[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 85[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	85 -> 20[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	86[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 86[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	86 -> 21[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	14[label="not (esEsOrdering (primCmpInt (Pos (Succ vx300)) (Pos vx40)) LT)",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
9.72/4.10	15[label="not (esEsOrdering (primCmpInt (Pos (Succ vx300)) (Neg vx40)) LT)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
9.72/4.10	16[label="not (esEsOrdering (primCmpInt (Pos Zero) (Pos vx40)) LT)",fontsize=16,color="burlywood",shape="box"];87[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];16 -> 87[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	87 -> 24[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	88[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 88[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	88 -> 25[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	17[label="not (esEsOrdering (primCmpInt (Pos Zero) (Neg vx40)) LT)",fontsize=16,color="burlywood",shape="box"];89[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];17 -> 89[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	89 -> 26[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	90[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 90[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	90 -> 27[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	18[label="not (esEsOrdering (primCmpInt (Neg (Succ vx300)) (Pos vx40)) LT)",fontsize=16,color="black",shape="box"];18 -> 28[label="",style="solid", color="black", weight=3];
9.72/4.10	19[label="not (esEsOrdering (primCmpInt (Neg (Succ vx300)) (Neg vx40)) LT)",fontsize=16,color="black",shape="box"];19 -> 29[label="",style="solid", color="black", weight=3];
9.72/4.10	20[label="not (esEsOrdering (primCmpInt (Neg Zero) (Pos vx40)) LT)",fontsize=16,color="burlywood",shape="box"];91[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];20 -> 91[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	91 -> 30[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	92[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 92[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	92 -> 31[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	21[label="not (esEsOrdering (primCmpInt (Neg Zero) (Neg vx40)) LT)",fontsize=16,color="burlywood",shape="box"];93[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];21 -> 93[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	93 -> 32[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	94[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 94[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	94 -> 33[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	22[label="not (esEsOrdering (primCmpNat (Succ vx300) vx40) LT)",fontsize=16,color="burlywood",shape="triangle"];95[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];22 -> 95[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	95 -> 34[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	96[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 96[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	96 -> 35[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	23[label="not (esEsOrdering GT LT)",fontsize=16,color="black",shape="triangle"];23 -> 36[label="",style="solid", color="black", weight=3];
9.72/4.10	24[label="not (esEsOrdering (primCmpInt (Pos Zero) (Pos (Succ vx400))) LT)",fontsize=16,color="black",shape="box"];24 -> 37[label="",style="solid", color="black", weight=3];
9.72/4.10	25[label="not (esEsOrdering (primCmpInt (Pos Zero) (Pos Zero)) LT)",fontsize=16,color="black",shape="box"];25 -> 38[label="",style="solid", color="black", weight=3];
9.72/4.10	26[label="not (esEsOrdering (primCmpInt (Pos Zero) (Neg (Succ vx400))) LT)",fontsize=16,color="black",shape="box"];26 -> 39[label="",style="solid", color="black", weight=3];
9.72/4.10	27[label="not (esEsOrdering (primCmpInt (Pos Zero) (Neg Zero)) LT)",fontsize=16,color="black",shape="box"];27 -> 40[label="",style="solid", color="black", weight=3];
9.72/4.10	28[label="not (esEsOrdering LT LT)",fontsize=16,color="black",shape="triangle"];28 -> 41[label="",style="solid", color="black", weight=3];
9.72/4.10	29[label="not (esEsOrdering (primCmpNat vx40 (Succ vx300)) LT)",fontsize=16,color="burlywood",shape="triangle"];97[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];29 -> 97[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	97 -> 42[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	98[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 98[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	98 -> 43[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	30[label="not (esEsOrdering (primCmpInt (Neg Zero) (Pos (Succ vx400))) LT)",fontsize=16,color="black",shape="box"];30 -> 44[label="",style="solid", color="black", weight=3];
9.72/4.10	31[label="not (esEsOrdering (primCmpInt (Neg Zero) (Pos Zero)) LT)",fontsize=16,color="black",shape="box"];31 -> 45[label="",style="solid", color="black", weight=3];
9.72/4.10	32[label="not (esEsOrdering (primCmpInt (Neg Zero) (Neg (Succ vx400))) LT)",fontsize=16,color="black",shape="box"];32 -> 46[label="",style="solid", color="black", weight=3];
9.72/4.10	33[label="not (esEsOrdering (primCmpInt (Neg Zero) (Neg Zero)) LT)",fontsize=16,color="black",shape="box"];33 -> 47[label="",style="solid", color="black", weight=3];
9.72/4.10	34[label="not (esEsOrdering (primCmpNat (Succ vx300) (Succ vx400)) LT)",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
9.72/4.10	35[label="not (esEsOrdering (primCmpNat (Succ vx300) Zero) LT)",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
9.72/4.10	36[label="not MyFalse",fontsize=16,color="black",shape="triangle"];36 -> 50[label="",style="solid", color="black", weight=3];
9.72/4.10	37 -> 29[label="",style="dashed", color="red", weight=0];
9.72/4.10	37[label="not (esEsOrdering (primCmpNat Zero (Succ vx400)) LT)",fontsize=16,color="magenta"];37 -> 51[label="",style="dashed", color="magenta", weight=3];
9.72/4.10	37 -> 52[label="",style="dashed", color="magenta", weight=3];
9.72/4.10	38[label="not (esEsOrdering EQ LT)",fontsize=16,color="black",shape="triangle"];38 -> 53[label="",style="solid", color="black", weight=3];
9.72/4.10	39 -> 23[label="",style="dashed", color="red", weight=0];
9.72/4.10	39[label="not (esEsOrdering GT LT)",fontsize=16,color="magenta"];40 -> 38[label="",style="dashed", color="red", weight=0];
9.72/4.10	40[label="not (esEsOrdering EQ LT)",fontsize=16,color="magenta"];41[label="not MyTrue",fontsize=16,color="black",shape="box"];41 -> 54[label="",style="solid", color="black", weight=3];
9.72/4.10	42[label="not (esEsOrdering (primCmpNat (Succ vx400) (Succ vx300)) LT)",fontsize=16,color="black",shape="box"];42 -> 55[label="",style="solid", color="black", weight=3];
9.72/4.10	43[label="not (esEsOrdering (primCmpNat Zero (Succ vx300)) LT)",fontsize=16,color="black",shape="box"];43 -> 56[label="",style="solid", color="black", weight=3];
9.72/4.10	44 -> 28[label="",style="dashed", color="red", weight=0];
9.72/4.10	44[label="not (esEsOrdering LT LT)",fontsize=16,color="magenta"];45 -> 38[label="",style="dashed", color="red", weight=0];
9.72/4.10	45[label="not (esEsOrdering EQ LT)",fontsize=16,color="magenta"];46 -> 22[label="",style="dashed", color="red", weight=0];
9.72/4.10	46[label="not (esEsOrdering (primCmpNat (Succ vx400) Zero) LT)",fontsize=16,color="magenta"];46 -> 57[label="",style="dashed", color="magenta", weight=3];
9.72/4.10	46 -> 58[label="",style="dashed", color="magenta", weight=3];
9.72/4.10	47 -> 38[label="",style="dashed", color="red", weight=0];
9.72/4.10	47[label="not (esEsOrdering EQ LT)",fontsize=16,color="magenta"];48[label="not (esEsOrdering (primCmpNat vx300 vx400) LT)",fontsize=16,color="burlywood",shape="triangle"];99[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];48 -> 99[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	99 -> 59[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	100[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 100[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	100 -> 60[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	49 -> 23[label="",style="dashed", color="red", weight=0];
9.72/4.10	49[label="not (esEsOrdering GT LT)",fontsize=16,color="magenta"];50[label="MyTrue",fontsize=16,color="green",shape="box"];51[label="Zero",fontsize=16,color="green",shape="box"];52[label="vx400",fontsize=16,color="green",shape="box"];53 -> 36[label="",style="dashed", color="red", weight=0];
9.72/4.10	53[label="not MyFalse",fontsize=16,color="magenta"];54[label="MyFalse",fontsize=16,color="green",shape="box"];55 -> 48[label="",style="dashed", color="red", weight=0];
9.72/4.10	55[label="not (esEsOrdering (primCmpNat vx400 vx300) LT)",fontsize=16,color="magenta"];55 -> 61[label="",style="dashed", color="magenta", weight=3];
9.72/4.10	55 -> 62[label="",style="dashed", color="magenta", weight=3];
9.72/4.10	56 -> 28[label="",style="dashed", color="red", weight=0];
9.72/4.10	56[label="not (esEsOrdering LT LT)",fontsize=16,color="magenta"];57[label="vx400",fontsize=16,color="green",shape="box"];58[label="Zero",fontsize=16,color="green",shape="box"];59[label="not (esEsOrdering (primCmpNat (Succ vx3000) vx400) LT)",fontsize=16,color="burlywood",shape="box"];101[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];59 -> 101[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	101 -> 63[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	102[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 102[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	102 -> 64[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	60[label="not (esEsOrdering (primCmpNat Zero vx400) LT)",fontsize=16,color="burlywood",shape="box"];103[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];60 -> 103[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	103 -> 65[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	104[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];60 -> 104[label="",style="solid", color="burlywood", weight=9];
9.72/4.10	104 -> 66[label="",style="solid", color="burlywood", weight=3];
9.72/4.10	61[label="vx400",fontsize=16,color="green",shape="box"];62[label="vx300",fontsize=16,color="green",shape="box"];63[label="not (esEsOrdering (primCmpNat (Succ vx3000) (Succ vx4000)) LT)",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3];
9.72/4.10	64[label="not (esEsOrdering (primCmpNat (Succ vx3000) Zero) LT)",fontsize=16,color="black",shape="box"];64 -> 68[label="",style="solid", color="black", weight=3];
9.72/4.10	65[label="not (esEsOrdering (primCmpNat Zero (Succ vx4000)) LT)",fontsize=16,color="black",shape="box"];65 -> 69[label="",style="solid", color="black", weight=3];
9.72/4.10	66[label="not (esEsOrdering (primCmpNat Zero Zero) LT)",fontsize=16,color="black",shape="box"];66 -> 70[label="",style="solid", color="black", weight=3];
9.72/4.10	67 -> 48[label="",style="dashed", color="red", weight=0];
9.72/4.10	67[label="not (esEsOrdering (primCmpNat vx3000 vx4000) LT)",fontsize=16,color="magenta"];67 -> 71[label="",style="dashed", color="magenta", weight=3];
9.72/4.10	67 -> 72[label="",style="dashed", color="magenta", weight=3];
9.72/4.10	68 -> 23[label="",style="dashed", color="red", weight=0];
9.72/4.10	68[label="not (esEsOrdering GT LT)",fontsize=16,color="magenta"];69 -> 28[label="",style="dashed", color="red", weight=0];
9.72/4.10	69[label="not (esEsOrdering LT LT)",fontsize=16,color="magenta"];70 -> 38[label="",style="dashed", color="red", weight=0];
9.72/4.10	70[label="not (esEsOrdering EQ LT)",fontsize=16,color="magenta"];71[label="vx3000",fontsize=16,color="green",shape="box"];72[label="vx4000",fontsize=16,color="green",shape="box"];}
9.72/4.10	
9.72/4.10	----------------------------------------
9.72/4.10	
9.72/4.10	(6)
9.72/4.10	Obligation:
9.72/4.10	Q DP problem:
9.72/4.10	The TRS P consists of the following rules:
9.72/4.10	
9.72/4.10	   new_not(Main.Succ(vx3000), Main.Succ(vx4000)) -> new_not(vx3000, vx4000)
9.72/4.10	
9.72/4.10	R is empty.
9.72/4.10	Q is empty.
9.72/4.10	We have to consider all minimal (P,Q,R)-chains.
9.72/4.10	----------------------------------------
9.72/4.10	
9.72/4.10	(7) QDPSizeChangeProof (EQUIVALENT)
9.72/4.10	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. 
9.72/4.10	
9.72/4.10	From the DPs we obtained the following set of size-change graphs:
9.72/4.10	*new_not(Main.Succ(vx3000), Main.Succ(vx4000)) -> new_not(vx3000, vx4000)
9.72/4.10	The graph contains the following edges 1 > 1, 2 > 2
9.72/4.10	
9.72/4.10	
9.72/4.10	----------------------------------------
9.72/4.10	
9.72/4.10	(8)
9.72/4.10	YES
9.90/4.22	EOF