8.35/3.60	YES
10.35/4.11	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.35/4.11	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.35/4.11	
10.35/4.11	
10.35/4.11	H-Termination with start terms of the given HASKELL could be proven:
10.35/4.11	
10.35/4.11	(0) HASKELL
10.35/4.11	(1) BR [EQUIVALENT, 0 ms]
10.35/4.11	(2) HASKELL
10.35/4.11	(3) COR [EQUIVALENT, 0 ms]
10.35/4.11	(4) HASKELL
10.35/4.11	(5) Narrow [SOUND, 0 ms]
10.35/4.11	(6) QDP
10.35/4.11	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.35/4.11	(8) YES
10.35/4.11	
10.35/4.11	
10.35/4.11	----------------------------------------
10.35/4.11	
10.35/4.11	(0)
10.35/4.11	Obligation:
10.35/4.11	mainModule Main 
10.35/4.11	module Main where {
10.35/4.11	import qualified Prelude;
10.35/4.11	data MyBool = MyTrue  | MyFalse ;
10.35/4.11	
10.35/4.11	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.35/4.11	
10.35/4.11	data Main.Nat = Succ Main.Nat  | Zero ;
10.35/4.11	
10.35/4.11	data Ordering = LT  | EQ  | GT ;
10.35/4.11	
10.35/4.11	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
10.35/4.11	compareMyInt = primCmpInt;
10.35/4.11	
10.35/4.11	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.35/4.11	esEsOrdering LT LT = MyTrue;
10.35/4.11	esEsOrdering LT EQ = MyFalse;
10.35/4.11	esEsOrdering LT GT = MyFalse;
10.35/4.11	esEsOrdering EQ LT = MyFalse;
10.35/4.11	esEsOrdering EQ EQ = MyTrue;
10.35/4.11	esEsOrdering EQ GT = MyFalse;
10.35/4.11	esEsOrdering GT LT = MyFalse;
10.35/4.11	esEsOrdering GT EQ = MyFalse;
10.35/4.11	esEsOrdering GT GT = MyTrue;
10.35/4.11	
10.35/4.11	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.35/4.11	fsEsOrdering x y = not (esEsOrdering x y);
10.35/4.11	
10.35/4.11	ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.35/4.11	ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;
10.35/4.11	
10.35/4.11	not :: MyBool  ->  MyBool;
10.35/4.11	not MyTrue = MyFalse;
10.35/4.11	not MyFalse = MyTrue;
10.35/4.11	
10.35/4.11	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
10.35/4.11	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
10.35/4.11	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
10.35/4.11	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
10.35/4.11	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
10.35/4.11	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
10.35/4.11	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
10.35/4.11	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
10.35/4.11	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
10.35/4.11	
10.35/4.11	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
10.35/4.11	primCmpNat Main.Zero Main.Zero = EQ;
10.35/4.11	primCmpNat Main.Zero (Main.Succ y) = LT;
10.35/4.11	primCmpNat (Main.Succ x) Main.Zero = GT;
10.35/4.11	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
10.35/4.11	
10.35/4.11	}
10.35/4.11	
10.35/4.11	----------------------------------------
10.35/4.11	
10.35/4.11	(1) BR (EQUIVALENT)
10.35/4.11	Replaced joker patterns by fresh variables and removed binding patterns.
10.35/4.11	----------------------------------------
10.35/4.11	
10.35/4.11	(2)
10.35/4.11	Obligation:
10.35/4.11	mainModule Main 
10.35/4.11	module Main where {
10.35/4.11	import qualified Prelude;
10.35/4.11	data MyBool = MyTrue  | MyFalse ;
10.35/4.11	
10.35/4.11	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.35/4.11	
10.35/4.11	data Main.Nat = Succ Main.Nat  | Zero ;
10.35/4.11	
10.35/4.11	data Ordering = LT  | EQ  | GT ;
10.35/4.11	
10.35/4.11	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
10.35/4.11	compareMyInt = primCmpInt;
10.35/4.11	
10.35/4.11	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.35/4.11	esEsOrdering LT LT = MyTrue;
10.35/4.11	esEsOrdering LT EQ = MyFalse;
10.35/4.11	esEsOrdering LT GT = MyFalse;
10.35/4.11	esEsOrdering EQ LT = MyFalse;
10.35/4.11	esEsOrdering EQ EQ = MyTrue;
10.35/4.11	esEsOrdering EQ GT = MyFalse;
10.35/4.11	esEsOrdering GT LT = MyFalse;
10.35/4.11	esEsOrdering GT EQ = MyFalse;
10.35/4.11	esEsOrdering GT GT = MyTrue;
10.35/4.11	
10.35/4.11	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.35/4.11	fsEsOrdering x y = not (esEsOrdering x y);
10.35/4.11	
10.35/4.11	ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.35/4.11	ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;
10.35/4.11	
10.35/4.11	not :: MyBool  ->  MyBool;
10.35/4.11	not MyTrue = MyFalse;
10.35/4.11	not MyFalse = MyTrue;
10.35/4.11	
10.35/4.11	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
10.35/4.11	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
10.35/4.11	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
10.35/4.11	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
10.35/4.11	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
10.35/4.11	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
10.35/4.11	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
10.35/4.11	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
10.35/4.11	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
10.35/4.11	
10.35/4.11	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
10.35/4.11	primCmpNat Main.Zero Main.Zero = EQ;
10.35/4.11	primCmpNat Main.Zero (Main.Succ y) = LT;
10.35/4.11	primCmpNat (Main.Succ x) Main.Zero = GT;
10.35/4.11	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
10.35/4.11	
10.35/4.11	}
10.35/4.11	
10.35/4.11	----------------------------------------
10.35/4.11	
10.35/4.11	(3) COR (EQUIVALENT)
10.35/4.11	Cond Reductions:
10.35/4.11	The following Function with conditions
10.35/4.11	"undefined |Falseundefined;
10.35/4.11	"
10.35/4.11	is transformed to
10.35/4.11	"undefined  = undefined1;
10.35/4.11	"
10.35/4.11	"undefined0 True = undefined;
10.35/4.11	"
10.35/4.11	"undefined1  = undefined0 False;
10.35/4.11	"
10.35/4.11	
10.35/4.11	----------------------------------------
10.35/4.11	
10.35/4.11	(4)
10.35/4.11	Obligation:
10.35/4.11	mainModule Main 
10.35/4.11	module Main where {
10.35/4.11	import qualified Prelude;
10.35/4.11	data MyBool = MyTrue  | MyFalse ;
10.35/4.11	
10.35/4.11	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.35/4.11	
10.35/4.11	data Main.Nat = Succ Main.Nat  | Zero ;
10.35/4.11	
10.35/4.11	data Ordering = LT  | EQ  | GT ;
10.35/4.11	
10.35/4.11	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
10.35/4.11	compareMyInt = primCmpInt;
10.35/4.11	
10.35/4.11	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.35/4.11	esEsOrdering LT LT = MyTrue;
10.35/4.11	esEsOrdering LT EQ = MyFalse;
10.35/4.11	esEsOrdering LT GT = MyFalse;
10.35/4.11	esEsOrdering EQ LT = MyFalse;
10.35/4.11	esEsOrdering EQ EQ = MyTrue;
10.35/4.11	esEsOrdering EQ GT = MyFalse;
10.35/4.11	esEsOrdering GT LT = MyFalse;
10.35/4.11	esEsOrdering GT EQ = MyFalse;
10.35/4.11	esEsOrdering GT GT = MyTrue;
10.35/4.11	
10.35/4.11	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.35/4.11	fsEsOrdering x y = not (esEsOrdering x y);
10.35/4.11	
10.35/4.11	ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.35/4.11	ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;
10.35/4.11	
10.35/4.11	not :: MyBool  ->  MyBool;
10.35/4.11	not MyTrue = MyFalse;
10.35/4.11	not MyFalse = MyTrue;
10.35/4.11	
10.35/4.11	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
10.35/4.11	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
10.35/4.11	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
10.35/4.11	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
10.35/4.11	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
10.35/4.11	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
10.35/4.11	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
10.35/4.11	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
10.35/4.11	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
10.35/4.11	
10.35/4.11	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
10.35/4.11	primCmpNat Main.Zero Main.Zero = EQ;
10.35/4.11	primCmpNat Main.Zero (Main.Succ y) = LT;
10.35/4.11	primCmpNat (Main.Succ x) Main.Zero = GT;
10.35/4.11	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
10.35/4.11	
10.35/4.11	}
10.35/4.11	
10.35/4.11	----------------------------------------
10.35/4.11	
10.35/4.11	(5) Narrow (SOUND)
10.35/4.11	Haskell To QDPs
10.35/4.11	
10.35/4.11	digraph dp_graph {
10.35/4.11	node [outthreshold=100, inthreshold=100];1[label="ltEsMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.35/4.11	3[label="ltEsMyInt vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.35/4.11	4[label="ltEsMyInt vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.35/4.11	5[label="fsEsOrdering (compareMyInt vx3 vx4) GT",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.35/4.11	6[label="not (esEsOrdering (compareMyInt vx3 vx4) GT)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.35/4.11	7[label="not (esEsOrdering (primCmpInt vx3 vx4) GT)",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];
10.35/4.11	73 -> 8[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	74[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];7 -> 74[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	74 -> 9[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	8[label="not (esEsOrdering (primCmpInt (Pos vx30) vx4) GT)",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];
10.35/4.11	75 -> 10[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	76[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 76[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	76 -> 11[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	9[label="not (esEsOrdering (primCmpInt (Neg vx30) vx4) GT)",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];
10.35/4.11	77 -> 12[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	78[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 78[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	78 -> 13[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	10[label="not (esEsOrdering (primCmpInt (Pos (Succ vx300)) vx4) GT)",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];
10.35/4.11	79 -> 14[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	80[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 80[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	80 -> 15[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	11[label="not (esEsOrdering (primCmpInt (Pos Zero) vx4) GT)",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];
10.35/4.11	81 -> 16[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	82[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];11 -> 82[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	82 -> 17[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	12[label="not (esEsOrdering (primCmpInt (Neg (Succ vx300)) vx4) GT)",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];
10.35/4.11	83 -> 18[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	84[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 84[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	84 -> 19[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	13[label="not (esEsOrdering (primCmpInt (Neg Zero) vx4) GT)",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];
10.35/4.11	85 -> 20[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	86[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 86[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	86 -> 21[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	14[label="not (esEsOrdering (primCmpInt (Pos (Succ vx300)) (Pos vx40)) GT)",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
10.35/4.11	15[label="not (esEsOrdering (primCmpInt (Pos (Succ vx300)) (Neg vx40)) GT)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
10.35/4.11	16[label="not (esEsOrdering (primCmpInt (Pos Zero) (Pos vx40)) GT)",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];
10.35/4.11	87 -> 24[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	88[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 88[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	88 -> 25[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	17[label="not (esEsOrdering (primCmpInt (Pos Zero) (Neg vx40)) GT)",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];
10.35/4.11	89 -> 26[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	90[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 90[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	90 -> 27[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	18[label="not (esEsOrdering (primCmpInt (Neg (Succ vx300)) (Pos vx40)) GT)",fontsize=16,color="black",shape="box"];18 -> 28[label="",style="solid", color="black", weight=3];
10.35/4.11	19[label="not (esEsOrdering (primCmpInt (Neg (Succ vx300)) (Neg vx40)) GT)",fontsize=16,color="black",shape="box"];19 -> 29[label="",style="solid", color="black", weight=3];
10.35/4.11	20[label="not (esEsOrdering (primCmpInt (Neg Zero) (Pos vx40)) GT)",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];
10.35/4.11	91 -> 30[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	92[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 92[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	92 -> 31[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	21[label="not (esEsOrdering (primCmpInt (Neg Zero) (Neg vx40)) GT)",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];
10.35/4.11	93 -> 32[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	94[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 94[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	94 -> 33[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	22[label="not (esEsOrdering (primCmpNat (Succ vx300) vx40) GT)",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];
10.35/4.11	95 -> 34[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	96[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 96[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	96 -> 35[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	23[label="not (esEsOrdering GT GT)",fontsize=16,color="black",shape="triangle"];23 -> 36[label="",style="solid", color="black", weight=3];
10.35/4.11	24[label="not (esEsOrdering (primCmpInt (Pos Zero) (Pos (Succ vx400))) GT)",fontsize=16,color="black",shape="box"];24 -> 37[label="",style="solid", color="black", weight=3];
10.35/4.11	25[label="not (esEsOrdering (primCmpInt (Pos Zero) (Pos Zero)) GT)",fontsize=16,color="black",shape="box"];25 -> 38[label="",style="solid", color="black", weight=3];
10.35/4.11	26[label="not (esEsOrdering (primCmpInt (Pos Zero) (Neg (Succ vx400))) GT)",fontsize=16,color="black",shape="box"];26 -> 39[label="",style="solid", color="black", weight=3];
10.35/4.11	27[label="not (esEsOrdering (primCmpInt (Pos Zero) (Neg Zero)) GT)",fontsize=16,color="black",shape="box"];27 -> 40[label="",style="solid", color="black", weight=3];
10.35/4.11	28[label="not (esEsOrdering LT GT)",fontsize=16,color="black",shape="triangle"];28 -> 41[label="",style="solid", color="black", weight=3];
10.35/4.11	29[label="not (esEsOrdering (primCmpNat vx40 (Succ vx300)) GT)",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];
10.35/4.11	97 -> 42[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	98[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 98[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	98 -> 43[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	30[label="not (esEsOrdering (primCmpInt (Neg Zero) (Pos (Succ vx400))) GT)",fontsize=16,color="black",shape="box"];30 -> 44[label="",style="solid", color="black", weight=3];
10.35/4.11	31[label="not (esEsOrdering (primCmpInt (Neg Zero) (Pos Zero)) GT)",fontsize=16,color="black",shape="box"];31 -> 45[label="",style="solid", color="black", weight=3];
10.35/4.11	32[label="not (esEsOrdering (primCmpInt (Neg Zero) (Neg (Succ vx400))) GT)",fontsize=16,color="black",shape="box"];32 -> 46[label="",style="solid", color="black", weight=3];
10.35/4.11	33[label="not (esEsOrdering (primCmpInt (Neg Zero) (Neg Zero)) GT)",fontsize=16,color="black",shape="box"];33 -> 47[label="",style="solid", color="black", weight=3];
10.35/4.11	34[label="not (esEsOrdering (primCmpNat (Succ vx300) (Succ vx400)) GT)",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
10.35/4.11	35[label="not (esEsOrdering (primCmpNat (Succ vx300) Zero) GT)",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
10.35/4.11	36[label="not MyTrue",fontsize=16,color="black",shape="box"];36 -> 50[label="",style="solid", color="black", weight=3];
10.35/4.11	37 -> 29[label="",style="dashed", color="red", weight=0];
10.35/4.11	37[label="not (esEsOrdering (primCmpNat Zero (Succ vx400)) GT)",fontsize=16,color="magenta"];37 -> 51[label="",style="dashed", color="magenta", weight=3];
10.35/4.11	37 -> 52[label="",style="dashed", color="magenta", weight=3];
10.35/4.11	38[label="not (esEsOrdering EQ GT)",fontsize=16,color="black",shape="triangle"];38 -> 53[label="",style="solid", color="black", weight=3];
10.35/4.11	39 -> 23[label="",style="dashed", color="red", weight=0];
10.35/4.11	39[label="not (esEsOrdering GT GT)",fontsize=16,color="magenta"];40 -> 38[label="",style="dashed", color="red", weight=0];
10.35/4.11	40[label="not (esEsOrdering EQ GT)",fontsize=16,color="magenta"];41[label="not MyFalse",fontsize=16,color="black",shape="triangle"];41 -> 54[label="",style="solid", color="black", weight=3];
10.35/4.11	42[label="not (esEsOrdering (primCmpNat (Succ vx400) (Succ vx300)) GT)",fontsize=16,color="black",shape="box"];42 -> 55[label="",style="solid", color="black", weight=3];
10.35/4.11	43[label="not (esEsOrdering (primCmpNat Zero (Succ vx300)) GT)",fontsize=16,color="black",shape="box"];43 -> 56[label="",style="solid", color="black", weight=3];
10.35/4.11	44 -> 28[label="",style="dashed", color="red", weight=0];
10.35/4.11	44[label="not (esEsOrdering LT GT)",fontsize=16,color="magenta"];45 -> 38[label="",style="dashed", color="red", weight=0];
10.35/4.11	45[label="not (esEsOrdering EQ GT)",fontsize=16,color="magenta"];46 -> 22[label="",style="dashed", color="red", weight=0];
10.35/4.11	46[label="not (esEsOrdering (primCmpNat (Succ vx400) Zero) GT)",fontsize=16,color="magenta"];46 -> 57[label="",style="dashed", color="magenta", weight=3];
10.35/4.11	46 -> 58[label="",style="dashed", color="magenta", weight=3];
10.35/4.11	47 -> 38[label="",style="dashed", color="red", weight=0];
10.35/4.11	47[label="not (esEsOrdering EQ GT)",fontsize=16,color="magenta"];48[label="not (esEsOrdering (primCmpNat vx300 vx400) GT)",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];
10.35/4.11	99 -> 59[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	100[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 100[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	100 -> 60[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	49 -> 23[label="",style="dashed", color="red", weight=0];
10.35/4.11	49[label="not (esEsOrdering GT GT)",fontsize=16,color="magenta"];50[label="MyFalse",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 -> 41[label="",style="dashed", color="red", weight=0];
10.35/4.11	53[label="not MyFalse",fontsize=16,color="magenta"];54[label="MyTrue",fontsize=16,color="green",shape="box"];55 -> 48[label="",style="dashed", color="red", weight=0];
10.35/4.11	55[label="not (esEsOrdering (primCmpNat vx400 vx300) GT)",fontsize=16,color="magenta"];55 -> 61[label="",style="dashed", color="magenta", weight=3];
10.35/4.11	55 -> 62[label="",style="dashed", color="magenta", weight=3];
10.35/4.11	56 -> 28[label="",style="dashed", color="red", weight=0];
10.35/4.11	56[label="not (esEsOrdering LT GT)",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) GT)",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];
10.35/4.11	101 -> 63[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	102[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 102[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	102 -> 64[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	60[label="not (esEsOrdering (primCmpNat Zero vx400) GT)",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];
10.35/4.11	103 -> 65[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	104[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];60 -> 104[label="",style="solid", color="burlywood", weight=9];
10.35/4.11	104 -> 66[label="",style="solid", color="burlywood", weight=3];
10.35/4.11	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)) GT)",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3];
10.35/4.11	64[label="not (esEsOrdering (primCmpNat (Succ vx3000) Zero) GT)",fontsize=16,color="black",shape="box"];64 -> 68[label="",style="solid", color="black", weight=3];
10.35/4.11	65[label="not (esEsOrdering (primCmpNat Zero (Succ vx4000)) GT)",fontsize=16,color="black",shape="box"];65 -> 69[label="",style="solid", color="black", weight=3];
10.35/4.11	66[label="not (esEsOrdering (primCmpNat Zero Zero) GT)",fontsize=16,color="black",shape="box"];66 -> 70[label="",style="solid", color="black", weight=3];
10.35/4.11	67 -> 48[label="",style="dashed", color="red", weight=0];
10.35/4.11	67[label="not (esEsOrdering (primCmpNat vx3000 vx4000) GT)",fontsize=16,color="magenta"];67 -> 71[label="",style="dashed", color="magenta", weight=3];
10.35/4.11	67 -> 72[label="",style="dashed", color="magenta", weight=3];
10.35/4.11	68 -> 23[label="",style="dashed", color="red", weight=0];
10.35/4.11	68[label="not (esEsOrdering GT GT)",fontsize=16,color="magenta"];69 -> 28[label="",style="dashed", color="red", weight=0];
10.35/4.11	69[label="not (esEsOrdering LT GT)",fontsize=16,color="magenta"];70 -> 38[label="",style="dashed", color="red", weight=0];
10.35/4.11	70[label="not (esEsOrdering EQ GT)",fontsize=16,color="magenta"];71[label="vx3000",fontsize=16,color="green",shape="box"];72[label="vx4000",fontsize=16,color="green",shape="box"];}
10.35/4.11	
10.35/4.11	----------------------------------------
10.35/4.11	
10.35/4.11	(6)
10.35/4.11	Obligation:
10.35/4.11	Q DP problem:
10.35/4.11	The TRS P consists of the following rules:
10.35/4.11	
10.35/4.11	   new_not(Main.Succ(vx3000), Main.Succ(vx4000)) -> new_not(vx3000, vx4000)
10.35/4.11	
10.35/4.11	R is empty.
10.35/4.11	Q is empty.
10.35/4.11	We have to consider all minimal (P,Q,R)-chains.
10.35/4.11	----------------------------------------
10.35/4.11	
10.35/4.11	(7) QDPSizeChangeProof (EQUIVALENT)
10.35/4.11	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. 
10.35/4.11	
10.35/4.11	From the DPs we obtained the following set of size-change graphs:
10.35/4.11	*new_not(Main.Succ(vx3000), Main.Succ(vx4000)) -> new_not(vx3000, vx4000)
10.35/4.11	The graph contains the following edges 1 > 1, 2 > 2
10.35/4.11	
10.35/4.11	
10.35/4.11	----------------------------------------
10.35/4.11	
10.35/4.11	(8)
10.35/4.11	YES
10.44/4.16	EOF