8.19/3.59	YES
10.16/4.14	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.16/4.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.16/4.14	
10.16/4.14	
10.16/4.14	H-Termination with start terms of the given HASKELL could be proven:
10.16/4.14	
10.16/4.14	(0) HASKELL
10.16/4.14	(1) BR [EQUIVALENT, 0 ms]
10.16/4.14	(2) HASKELL
10.16/4.14	(3) COR [EQUIVALENT, 0 ms]
10.16/4.14	(4) HASKELL
10.16/4.14	(5) Narrow [SOUND, 0 ms]
10.16/4.14	(6) QDP
10.16/4.14	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.16/4.14	(8) YES
10.16/4.14	
10.16/4.14	
10.16/4.14	----------------------------------------
10.16/4.14	
10.16/4.14	(0)
10.16/4.14	Obligation:
10.16/4.14	mainModule Main 
10.16/4.14	module Main where {
10.16/4.14	import qualified Prelude;
10.16/4.14	data MyBool = MyTrue  | MyFalse ;
10.16/4.14	
10.16/4.14	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.16/4.14	
10.16/4.14	data Main.Nat = Succ Main.Nat  | Zero ;
10.16/4.14	
10.16/4.14	data Ordering = LT  | EQ  | GT ;
10.16/4.14	
10.16/4.14	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
10.16/4.14	compareMyInt = primCmpInt;
10.16/4.14	
10.16/4.14	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.16/4.14	esEsOrdering LT LT = MyTrue;
10.16/4.14	esEsOrdering LT EQ = MyFalse;
10.16/4.14	esEsOrdering LT GT = MyFalse;
10.16/4.14	esEsOrdering EQ LT = MyFalse;
10.16/4.14	esEsOrdering EQ EQ = MyTrue;
10.16/4.14	esEsOrdering EQ GT = MyFalse;
10.16/4.14	esEsOrdering GT LT = MyFalse;
10.16/4.14	esEsOrdering GT EQ = MyFalse;
10.16/4.14	esEsOrdering GT GT = MyTrue;
10.16/4.14	
10.16/4.14	ltMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.16/4.14	ltMyInt x y = esEsOrdering (compareMyInt x y) LT;
10.16/4.14	
10.16/4.14	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
10.16/4.14	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
10.16/4.14	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
10.16/4.14	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
10.16/4.14	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
10.16/4.14	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
10.16/4.14	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
10.16/4.14	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
10.16/4.14	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
10.16/4.14	
10.16/4.14	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
10.16/4.14	primCmpNat Main.Zero Main.Zero = EQ;
10.16/4.14	primCmpNat Main.Zero (Main.Succ y) = LT;
10.16/4.14	primCmpNat (Main.Succ x) Main.Zero = GT;
10.16/4.14	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
10.16/4.14	
10.16/4.14	}
10.16/4.14	
10.16/4.14	----------------------------------------
10.16/4.14	
10.16/4.14	(1) BR (EQUIVALENT)
10.16/4.14	Replaced joker patterns by fresh variables and removed binding patterns.
10.16/4.14	----------------------------------------
10.16/4.14	
10.16/4.14	(2)
10.16/4.14	Obligation:
10.16/4.14	mainModule Main 
10.16/4.14	module Main where {
10.16/4.14	import qualified Prelude;
10.16/4.14	data MyBool = MyTrue  | MyFalse ;
10.16/4.14	
10.16/4.14	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.16/4.14	
10.16/4.14	data Main.Nat = Succ Main.Nat  | Zero ;
10.16/4.14	
10.16/4.14	data Ordering = LT  | EQ  | GT ;
10.16/4.14	
10.16/4.14	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
10.16/4.14	compareMyInt = primCmpInt;
10.16/4.14	
10.16/4.14	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.16/4.14	esEsOrdering LT LT = MyTrue;
10.16/4.14	esEsOrdering LT EQ = MyFalse;
10.16/4.14	esEsOrdering LT GT = MyFalse;
10.16/4.14	esEsOrdering EQ LT = MyFalse;
10.16/4.14	esEsOrdering EQ EQ = MyTrue;
10.16/4.14	esEsOrdering EQ GT = MyFalse;
10.16/4.14	esEsOrdering GT LT = MyFalse;
10.16/4.14	esEsOrdering GT EQ = MyFalse;
10.16/4.14	esEsOrdering GT GT = MyTrue;
10.16/4.14	
10.16/4.14	ltMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.16/4.14	ltMyInt x y = esEsOrdering (compareMyInt x y) LT;
10.16/4.14	
10.16/4.14	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
10.16/4.14	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
10.16/4.14	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
10.16/4.14	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
10.16/4.14	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
10.16/4.14	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
10.16/4.14	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
10.16/4.14	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
10.16/4.14	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
10.16/4.14	
10.16/4.14	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
10.16/4.14	primCmpNat Main.Zero Main.Zero = EQ;
10.16/4.14	primCmpNat Main.Zero (Main.Succ y) = LT;
10.16/4.14	primCmpNat (Main.Succ x) Main.Zero = GT;
10.16/4.14	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
10.16/4.14	
10.16/4.14	}
10.16/4.14	
10.16/4.14	----------------------------------------
10.16/4.14	
10.16/4.14	(3) COR (EQUIVALENT)
10.16/4.14	Cond Reductions:
10.16/4.14	The following Function with conditions
10.16/4.14	"undefined |Falseundefined;
10.16/4.14	"
10.16/4.14	is transformed to
10.16/4.14	"undefined  = undefined1;
10.16/4.14	"
10.16/4.14	"undefined0 True = undefined;
10.16/4.14	"
10.16/4.14	"undefined1  = undefined0 False;
10.16/4.14	"
10.16/4.14	
10.16/4.14	----------------------------------------
10.16/4.14	
10.16/4.14	(4)
10.16/4.14	Obligation:
10.16/4.14	mainModule Main 
10.16/4.14	module Main where {
10.16/4.14	import qualified Prelude;
10.16/4.14	data MyBool = MyTrue  | MyFalse ;
10.16/4.14	
10.16/4.14	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.16/4.14	
10.16/4.14	data Main.Nat = Succ Main.Nat  | Zero ;
10.16/4.14	
10.16/4.14	data Ordering = LT  | EQ  | GT ;
10.16/4.14	
10.16/4.14	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
10.16/4.14	compareMyInt = primCmpInt;
10.16/4.14	
10.16/4.14	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.16/4.14	esEsOrdering LT LT = MyTrue;
10.16/4.14	esEsOrdering LT EQ = MyFalse;
10.16/4.14	esEsOrdering LT GT = MyFalse;
10.16/4.14	esEsOrdering EQ LT = MyFalse;
10.16/4.14	esEsOrdering EQ EQ = MyTrue;
10.16/4.14	esEsOrdering EQ GT = MyFalse;
10.16/4.14	esEsOrdering GT LT = MyFalse;
10.16/4.14	esEsOrdering GT EQ = MyFalse;
10.16/4.14	esEsOrdering GT GT = MyTrue;
10.16/4.14	
10.16/4.14	ltMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.16/4.14	ltMyInt x y = esEsOrdering (compareMyInt x y) LT;
10.16/4.14	
10.16/4.14	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
10.16/4.14	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
10.16/4.14	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
10.16/4.14	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
10.16/4.14	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
10.16/4.14	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
10.16/4.14	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
10.16/4.14	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
10.16/4.14	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
10.16/4.14	
10.16/4.14	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
10.16/4.14	primCmpNat Main.Zero Main.Zero = EQ;
10.16/4.14	primCmpNat Main.Zero (Main.Succ y) = LT;
10.16/4.14	primCmpNat (Main.Succ x) Main.Zero = GT;
10.16/4.14	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
10.16/4.14	
10.16/4.14	}
10.16/4.14	
10.16/4.14	----------------------------------------
10.16/4.14	
10.16/4.14	(5) Narrow (SOUND)
10.16/4.14	Haskell To QDPs
10.16/4.14	
10.16/4.14	digraph dp_graph {
10.16/4.14	node [outthreshold=100, inthreshold=100];1[label="ltMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.16/4.14	3[label="ltMyInt vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.16/4.14	4[label="ltMyInt vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.16/4.14	5[label="esEsOrdering (compareMyInt vx3 vx4) LT",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.16/4.14	6[label="esEsOrdering (primCmpInt vx3 vx4) LT",fontsize=16,color="burlywood",shape="box"];70[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];6 -> 70[label="",style="solid", color="burlywood", weight=9];
10.16/4.14	70 -> 7[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	71[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];6 -> 71[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	71 -> 8[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	7[label="esEsOrdering (primCmpInt (Pos vx30) vx4) LT",fontsize=16,color="burlywood",shape="box"];72[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];7 -> 72[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	72 -> 9[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	73[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];7 -> 73[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	73 -> 10[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	8[label="esEsOrdering (primCmpInt (Neg vx30) vx4) LT",fontsize=16,color="burlywood",shape="box"];74[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];8 -> 74[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	74 -> 11[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	75[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 75[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	75 -> 12[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	9[label="esEsOrdering (primCmpInt (Pos (Succ vx300)) vx4) LT",fontsize=16,color="burlywood",shape="box"];76[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];9 -> 76[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	76 -> 13[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	77[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];9 -> 77[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	77 -> 14[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	10[label="esEsOrdering (primCmpInt (Pos Zero) vx4) LT",fontsize=16,color="burlywood",shape="box"];78[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 78[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	78 -> 15[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	79[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 79[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	79 -> 16[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	11[label="esEsOrdering (primCmpInt (Neg (Succ vx300)) vx4) LT",fontsize=16,color="burlywood",shape="box"];80[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];11 -> 80[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	80 -> 17[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	81[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];11 -> 81[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	81 -> 18[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	12[label="esEsOrdering (primCmpInt (Neg Zero) vx4) LT",fontsize=16,color="burlywood",shape="box"];82[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 82[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	82 -> 19[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	83[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 83[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	83 -> 20[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	13[label="esEsOrdering (primCmpInt (Pos (Succ vx300)) (Pos vx40)) LT",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
10.16/4.15	14[label="esEsOrdering (primCmpInt (Pos (Succ vx300)) (Neg vx40)) LT",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
10.16/4.15	15[label="esEsOrdering (primCmpInt (Pos Zero) (Pos vx40)) LT",fontsize=16,color="burlywood",shape="box"];84[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];15 -> 84[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	84 -> 23[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	85[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 85[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	85 -> 24[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	16[label="esEsOrdering (primCmpInt (Pos Zero) (Neg vx40)) LT",fontsize=16,color="burlywood",shape="box"];86[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];16 -> 86[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	86 -> 25[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	87[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 87[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	87 -> 26[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	17[label="esEsOrdering (primCmpInt (Neg (Succ vx300)) (Pos vx40)) LT",fontsize=16,color="black",shape="box"];17 -> 27[label="",style="solid", color="black", weight=3];
10.16/4.15	18[label="esEsOrdering (primCmpInt (Neg (Succ vx300)) (Neg vx40)) LT",fontsize=16,color="black",shape="box"];18 -> 28[label="",style="solid", color="black", weight=3];
10.16/4.15	19[label="esEsOrdering (primCmpInt (Neg Zero) (Pos vx40)) LT",fontsize=16,color="burlywood",shape="box"];88[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];19 -> 88[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	88 -> 29[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	89[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 89[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	89 -> 30[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	20[label="esEsOrdering (primCmpInt (Neg Zero) (Neg vx40)) LT",fontsize=16,color="burlywood",shape="box"];90[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];20 -> 90[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	90 -> 31[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	91[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 91[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	91 -> 32[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	21[label="esEsOrdering (primCmpNat (Succ vx300) vx40) LT",fontsize=16,color="burlywood",shape="triangle"];92[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];21 -> 92[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	92 -> 33[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	93[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 93[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	93 -> 34[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	22[label="esEsOrdering GT LT",fontsize=16,color="black",shape="triangle"];22 -> 35[label="",style="solid", color="black", weight=3];
10.16/4.15	23[label="esEsOrdering (primCmpInt (Pos Zero) (Pos (Succ vx400))) LT",fontsize=16,color="black",shape="box"];23 -> 36[label="",style="solid", color="black", weight=3];
10.16/4.15	24[label="esEsOrdering (primCmpInt (Pos Zero) (Pos Zero)) LT",fontsize=16,color="black",shape="box"];24 -> 37[label="",style="solid", color="black", weight=3];
10.16/4.15	25[label="esEsOrdering (primCmpInt (Pos Zero) (Neg (Succ vx400))) LT",fontsize=16,color="black",shape="box"];25 -> 38[label="",style="solid", color="black", weight=3];
10.16/4.15	26[label="esEsOrdering (primCmpInt (Pos Zero) (Neg Zero)) LT",fontsize=16,color="black",shape="box"];26 -> 39[label="",style="solid", color="black", weight=3];
10.16/4.15	27[label="esEsOrdering LT LT",fontsize=16,color="black",shape="triangle"];27 -> 40[label="",style="solid", color="black", weight=3];
10.16/4.15	28[label="esEsOrdering (primCmpNat vx40 (Succ vx300)) LT",fontsize=16,color="burlywood",shape="triangle"];94[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];28 -> 94[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	94 -> 41[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	95[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 95[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	95 -> 42[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	29[label="esEsOrdering (primCmpInt (Neg Zero) (Pos (Succ vx400))) LT",fontsize=16,color="black",shape="box"];29 -> 43[label="",style="solid", color="black", weight=3];
10.16/4.15	30[label="esEsOrdering (primCmpInt (Neg Zero) (Pos Zero)) LT",fontsize=16,color="black",shape="box"];30 -> 44[label="",style="solid", color="black", weight=3];
10.16/4.15	31[label="esEsOrdering (primCmpInt (Neg Zero) (Neg (Succ vx400))) LT",fontsize=16,color="black",shape="box"];31 -> 45[label="",style="solid", color="black", weight=3];
10.16/4.15	32[label="esEsOrdering (primCmpInt (Neg Zero) (Neg Zero)) LT",fontsize=16,color="black",shape="box"];32 -> 46[label="",style="solid", color="black", weight=3];
10.16/4.15	33[label="esEsOrdering (primCmpNat (Succ vx300) (Succ vx400)) LT",fontsize=16,color="black",shape="box"];33 -> 47[label="",style="solid", color="black", weight=3];
10.16/4.15	34[label="esEsOrdering (primCmpNat (Succ vx300) Zero) LT",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
10.16/4.15	35[label="MyFalse",fontsize=16,color="green",shape="box"];36 -> 28[label="",style="dashed", color="red", weight=0];
10.16/4.15	36[label="esEsOrdering (primCmpNat Zero (Succ vx400)) LT",fontsize=16,color="magenta"];36 -> 49[label="",style="dashed", color="magenta", weight=3];
10.16/4.15	36 -> 50[label="",style="dashed", color="magenta", weight=3];
10.16/4.15	37[label="esEsOrdering EQ LT",fontsize=16,color="black",shape="triangle"];37 -> 51[label="",style="solid", color="black", weight=3];
10.16/4.15	38 -> 22[label="",style="dashed", color="red", weight=0];
10.16/4.15	38[label="esEsOrdering GT LT",fontsize=16,color="magenta"];39 -> 37[label="",style="dashed", color="red", weight=0];
10.16/4.15	39[label="esEsOrdering EQ LT",fontsize=16,color="magenta"];40[label="MyTrue",fontsize=16,color="green",shape="box"];41[label="esEsOrdering (primCmpNat (Succ vx400) (Succ vx300)) LT",fontsize=16,color="black",shape="box"];41 -> 52[label="",style="solid", color="black", weight=3];
10.16/4.15	42[label="esEsOrdering (primCmpNat Zero (Succ vx300)) LT",fontsize=16,color="black",shape="box"];42 -> 53[label="",style="solid", color="black", weight=3];
10.16/4.15	43 -> 27[label="",style="dashed", color="red", weight=0];
10.16/4.15	43[label="esEsOrdering LT LT",fontsize=16,color="magenta"];44 -> 37[label="",style="dashed", color="red", weight=0];
10.16/4.15	44[label="esEsOrdering EQ LT",fontsize=16,color="magenta"];45 -> 21[label="",style="dashed", color="red", weight=0];
10.16/4.15	45[label="esEsOrdering (primCmpNat (Succ vx400) Zero) LT",fontsize=16,color="magenta"];45 -> 54[label="",style="dashed", color="magenta", weight=3];
10.16/4.15	45 -> 55[label="",style="dashed", color="magenta", weight=3];
10.16/4.15	46 -> 37[label="",style="dashed", color="red", weight=0];
10.16/4.15	46[label="esEsOrdering EQ LT",fontsize=16,color="magenta"];47[label="esEsOrdering (primCmpNat vx300 vx400) LT",fontsize=16,color="burlywood",shape="triangle"];96[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];47 -> 96[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	96 -> 56[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	97[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 97[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	97 -> 57[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	48 -> 22[label="",style="dashed", color="red", weight=0];
10.16/4.15	48[label="esEsOrdering GT LT",fontsize=16,color="magenta"];49[label="Zero",fontsize=16,color="green",shape="box"];50[label="vx400",fontsize=16,color="green",shape="box"];51[label="MyFalse",fontsize=16,color="green",shape="box"];52 -> 47[label="",style="dashed", color="red", weight=0];
10.16/4.15	52[label="esEsOrdering (primCmpNat vx400 vx300) LT",fontsize=16,color="magenta"];52 -> 58[label="",style="dashed", color="magenta", weight=3];
10.16/4.15	52 -> 59[label="",style="dashed", color="magenta", weight=3];
10.16/4.15	53 -> 27[label="",style="dashed", color="red", weight=0];
10.16/4.15	53[label="esEsOrdering LT LT",fontsize=16,color="magenta"];54[label="Zero",fontsize=16,color="green",shape="box"];55[label="vx400",fontsize=16,color="green",shape="box"];56[label="esEsOrdering (primCmpNat (Succ vx3000) vx400) LT",fontsize=16,color="burlywood",shape="box"];98[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];56 -> 98[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	98 -> 60[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	99[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];56 -> 99[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	99 -> 61[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	57[label="esEsOrdering (primCmpNat Zero vx400) LT",fontsize=16,color="burlywood",shape="box"];100[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];57 -> 100[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	100 -> 62[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	101[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];57 -> 101[label="",style="solid", color="burlywood", weight=9];
10.16/4.15	101 -> 63[label="",style="solid", color="burlywood", weight=3];
10.16/4.15	58[label="vx400",fontsize=16,color="green",shape="box"];59[label="vx300",fontsize=16,color="green",shape="box"];60[label="esEsOrdering (primCmpNat (Succ vx3000) (Succ vx4000)) LT",fontsize=16,color="black",shape="box"];60 -> 64[label="",style="solid", color="black", weight=3];
10.16/4.15	61[label="esEsOrdering (primCmpNat (Succ vx3000) Zero) LT",fontsize=16,color="black",shape="box"];61 -> 65[label="",style="solid", color="black", weight=3];
10.16/4.15	62[label="esEsOrdering (primCmpNat Zero (Succ vx4000)) LT",fontsize=16,color="black",shape="box"];62 -> 66[label="",style="solid", color="black", weight=3];
10.16/4.15	63[label="esEsOrdering (primCmpNat Zero Zero) LT",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3];
10.16/4.15	64 -> 47[label="",style="dashed", color="red", weight=0];
10.16/4.15	64[label="esEsOrdering (primCmpNat vx3000 vx4000) LT",fontsize=16,color="magenta"];64 -> 68[label="",style="dashed", color="magenta", weight=3];
10.16/4.15	64 -> 69[label="",style="dashed", color="magenta", weight=3];
10.16/4.15	65 -> 22[label="",style="dashed", color="red", weight=0];
10.16/4.15	65[label="esEsOrdering GT LT",fontsize=16,color="magenta"];66 -> 27[label="",style="dashed", color="red", weight=0];
10.16/4.15	66[label="esEsOrdering LT LT",fontsize=16,color="magenta"];67 -> 37[label="",style="dashed", color="red", weight=0];
10.16/4.15	67[label="esEsOrdering EQ LT",fontsize=16,color="magenta"];68[label="vx3000",fontsize=16,color="green",shape="box"];69[label="vx4000",fontsize=16,color="green",shape="box"];}
10.16/4.15	
10.16/4.15	----------------------------------------
10.16/4.15	
10.16/4.15	(6)
10.16/4.15	Obligation:
10.16/4.15	Q DP problem:
10.16/4.15	The TRS P consists of the following rules:
10.16/4.15	
10.16/4.15	   new_esEsOrdering(Main.Succ(vx3000), Main.Succ(vx4000)) -> new_esEsOrdering(vx3000, vx4000)
10.16/4.15	
10.16/4.15	R is empty.
10.16/4.15	Q is empty.
10.16/4.15	We have to consider all minimal (P,Q,R)-chains.
10.16/4.15	----------------------------------------
10.16/4.15	
10.16/4.15	(7) QDPSizeChangeProof (EQUIVALENT)
10.16/4.15	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. 
10.16/4.15	
10.16/4.15	From the DPs we obtained the following set of size-change graphs:
10.16/4.15	*new_esEsOrdering(Main.Succ(vx3000), Main.Succ(vx4000)) -> new_esEsOrdering(vx3000, vx4000)
10.16/4.15	The graph contains the following edges 1 > 1, 2 > 2
10.16/4.15	
10.16/4.15	
10.16/4.15	----------------------------------------
10.16/4.15	
10.16/4.15	(8)
10.16/4.15	YES
10.35/4.18	EOF