8.22/3.63	YES
10.14/4.16	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.14/4.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.14/4.16	
10.14/4.16	
10.14/4.16	H-Termination with start terms of the given HASKELL could be proven:
10.14/4.16	
10.14/4.16	(0) HASKELL
10.14/4.16	(1) BR [EQUIVALENT, 0 ms]
10.14/4.16	(2) HASKELL
10.14/4.16	(3) COR [EQUIVALENT, 0 ms]
10.14/4.16	(4) HASKELL
10.14/4.16	(5) Narrow [EQUIVALENT, 26 ms]
10.14/4.16	(6) YES
10.14/4.16	
10.14/4.16	
10.14/4.16	----------------------------------------
10.14/4.16	
10.14/4.16	(0)
10.14/4.16	Obligation:
10.14/4.16	mainModule Main 
10.14/4.16	module Main where {
10.14/4.16	import qualified Prelude;
10.14/4.16	data MyBool = MyTrue  | MyFalse ;
10.14/4.16	
10.14/4.16	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.14/4.16	
10.14/4.16	data Main.Nat = Succ Main.Nat  | Zero ;
10.14/4.16	
10.14/4.16	data Tup0 = Tup0 ;
10.14/4.16	
10.14/4.16	esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.14/4.16	esEsMyInt = primEqInt;
10.14/4.16	
10.14/4.16	fromEnumTup0 :: Tup0  ->  MyInt;
10.14/4.16	fromEnumTup0 Tup0 = Main.Pos Main.Zero;
10.14/4.16	
10.14/4.16	primEqInt :: MyInt  ->  MyInt  ->  MyBool;
10.14/4.16	primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
10.14/4.16	primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
10.14/4.16	primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.14/4.16	primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.14/4.16	primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.14/4.16	primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.14/4.16	primEqInt vv vw = MyFalse;
10.14/4.16	
10.14/4.16	primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.14/4.16	primEqNat Main.Zero Main.Zero = MyTrue;
10.14/4.16	primEqNat Main.Zero (Main.Succ y) = MyFalse;
10.14/4.16	primEqNat (Main.Succ x) Main.Zero = MyFalse;
10.14/4.16	primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;
10.14/4.16	
10.14/4.16	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.14/4.16	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.14/4.16	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.14/4.16	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.14/4.16	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.14/4.16	
10.14/4.16	primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
10.14/4.16	primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
10.14/4.16	primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
10.14/4.16	primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
10.14/4.16	primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);
10.14/4.16	
10.14/4.16	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.14/4.16	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.14/4.16	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.14/4.16	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.14/4.16	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.14/4.16	
10.14/4.16	psMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.14/4.16	psMyInt = primPlusInt;
10.14/4.16	
10.14/4.16	pt :: (a  ->  c)  ->  (b  ->  a)  ->  b  ->  c;
10.14/4.16	pt f g x = f (g x);
10.14/4.16	
10.14/4.16	succTup0 :: Tup0  ->  Tup0;
10.14/4.16	succTup0 = pt toEnumTup0 (pt (psMyInt (Main.Pos (Main.Succ Main.Zero))) fromEnumTup0);
10.14/4.16	
10.14/4.16	toEnum0 MyTrue vx = Tup0;
10.14/4.16	
10.14/4.16	toEnum1 vx = toEnum0 (esEsMyInt vx (Main.Pos Main.Zero)) vx;
10.14/4.16	
10.14/4.16	toEnumTup0 :: MyInt  ->  Tup0;
10.14/4.16	toEnumTup0 vx = toEnum1 vx;
10.14/4.16	
10.14/4.16	}
10.14/4.16	
10.14/4.16	----------------------------------------
10.14/4.16	
10.14/4.16	(1) BR (EQUIVALENT)
10.14/4.16	Replaced joker patterns by fresh variables and removed binding patterns.
10.14/4.16	----------------------------------------
10.14/4.16	
10.14/4.16	(2)
10.14/4.16	Obligation:
10.14/4.16	mainModule Main 
10.14/4.16	module Main where {
10.14/4.16	import qualified Prelude;
10.14/4.16	data MyBool = MyTrue  | MyFalse ;
10.14/4.16	
10.14/4.16	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.14/4.16	
10.14/4.16	data Main.Nat = Succ Main.Nat  | Zero ;
10.14/4.16	
10.14/4.16	data Tup0 = Tup0 ;
10.14/4.16	
10.14/4.16	esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.14/4.16	esEsMyInt = primEqInt;
10.14/4.16	
10.14/4.16	fromEnumTup0 :: Tup0  ->  MyInt;
10.14/4.16	fromEnumTup0 Tup0 = Main.Pos Main.Zero;
10.14/4.16	
10.14/4.16	primEqInt :: MyInt  ->  MyInt  ->  MyBool;
10.14/4.16	primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
10.14/4.16	primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
10.14/4.16	primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.14/4.16	primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.14/4.16	primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.14/4.16	primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.14/4.16	primEqInt vv vw = MyFalse;
10.14/4.16	
10.14/4.16	primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.14/4.16	primEqNat Main.Zero Main.Zero = MyTrue;
10.14/4.16	primEqNat Main.Zero (Main.Succ y) = MyFalse;
10.14/4.16	primEqNat (Main.Succ x) Main.Zero = MyFalse;
10.14/4.16	primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;
10.14/4.16	
10.14/4.16	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.14/4.16	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.14/4.16	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.14/4.16	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.14/4.16	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.14/4.16	
10.14/4.16	primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
10.14/4.16	primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
10.14/4.16	primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
10.14/4.16	primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
10.14/4.16	primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);
10.14/4.16	
10.14/4.16	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.14/4.16	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.14/4.16	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.14/4.16	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.14/4.16	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.14/4.16	
10.14/4.16	psMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.14/4.16	psMyInt = primPlusInt;
10.14/4.16	
10.14/4.16	pt :: (c  ->  a)  ->  (b  ->  c)  ->  b  ->  a;
10.14/4.16	pt f g x = f (g x);
10.14/4.16	
10.14/4.16	succTup0 :: Tup0  ->  Tup0;
10.14/4.16	succTup0 = pt toEnumTup0 (pt (psMyInt (Main.Pos (Main.Succ Main.Zero))) fromEnumTup0);
10.14/4.16	
10.14/4.16	toEnum0 MyTrue vx = Tup0;
10.14/4.16	
10.14/4.16	toEnum1 vx = toEnum0 (esEsMyInt vx (Main.Pos Main.Zero)) vx;
10.14/4.16	
10.14/4.16	toEnumTup0 :: MyInt  ->  Tup0;
10.14/4.16	toEnumTup0 vx = toEnum1 vx;
10.14/4.16	
10.14/4.16	}
10.14/4.16	
10.14/4.16	----------------------------------------
10.14/4.16	
10.14/4.16	(3) COR (EQUIVALENT)
10.14/4.16	Cond Reductions:
10.14/4.16	The following Function with conditions
10.14/4.16	"undefined |Falseundefined;
10.14/4.16	"
10.14/4.16	is transformed to
10.14/4.16	"undefined  = undefined1;
10.14/4.16	"
10.14/4.16	"undefined0 True = undefined;
10.14/4.16	"
10.14/4.16	"undefined1  = undefined0 False;
10.14/4.16	"
10.14/4.16	
10.14/4.16	----------------------------------------
10.14/4.16	
10.14/4.16	(4)
10.14/4.16	Obligation:
10.14/4.16	mainModule Main 
10.14/4.16	module Main where {
10.14/4.16	import qualified Prelude;
10.14/4.16	data MyBool = MyTrue  | MyFalse ;
10.14/4.16	
10.14/4.16	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.14/4.16	
10.14/4.16	data Main.Nat = Succ Main.Nat  | Zero ;
10.14/4.16	
10.14/4.16	data Tup0 = Tup0 ;
10.14/4.16	
10.14/4.16	esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.14/4.16	esEsMyInt = primEqInt;
10.14/4.16	
10.14/4.16	fromEnumTup0 :: Tup0  ->  MyInt;
10.14/4.16	fromEnumTup0 Tup0 = Main.Pos Main.Zero;
10.14/4.16	
10.14/4.16	primEqInt :: MyInt  ->  MyInt  ->  MyBool;
10.14/4.16	primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
10.14/4.16	primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
10.14/4.16	primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.14/4.16	primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.14/4.16	primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.14/4.16	primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.14/4.16	primEqInt vv vw = MyFalse;
10.14/4.16	
10.14/4.16	primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.14/4.16	primEqNat Main.Zero Main.Zero = MyTrue;
10.14/4.16	primEqNat Main.Zero (Main.Succ y) = MyFalse;
10.14/4.16	primEqNat (Main.Succ x) Main.Zero = MyFalse;
10.14/4.16	primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;
10.14/4.16	
10.14/4.16	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.14/4.16	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.14/4.16	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.14/4.16	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.14/4.16	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.14/4.16	
10.14/4.16	primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
10.14/4.16	primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
10.14/4.16	primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
10.14/4.16	primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
10.14/4.16	primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);
10.14/4.16	
10.14/4.16	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.14/4.16	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.14/4.16	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.14/4.16	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.14/4.16	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.14/4.16	
10.14/4.16	psMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.14/4.16	psMyInt = primPlusInt;
10.14/4.16	
10.14/4.16	pt :: (b  ->  a)  ->  (c  ->  b)  ->  c  ->  a;
10.14/4.16	pt f g x = f (g x);
10.14/4.16	
10.14/4.16	succTup0 :: Tup0  ->  Tup0;
10.14/4.16	succTup0 = pt toEnumTup0 (pt (psMyInt (Main.Pos (Main.Succ Main.Zero))) fromEnumTup0);
10.14/4.16	
10.14/4.16	toEnum0 MyTrue vx = Tup0;
10.14/4.16	
10.14/4.16	toEnum1 vx = toEnum0 (esEsMyInt vx (Main.Pos Main.Zero)) vx;
10.14/4.16	
10.14/4.16	toEnumTup0 :: MyInt  ->  Tup0;
10.14/4.16	toEnumTup0 vx = toEnum1 vx;
10.14/4.16	
10.14/4.16	}
10.14/4.16	
10.14/4.16	----------------------------------------
10.14/4.16	
10.14/4.16	(5) Narrow (EQUIVALENT)
10.14/4.16	Haskell To QDPs
10.14/4.16	
10.14/4.16	digraph dp_graph {
10.14/4.16	node [outthreshold=100, inthreshold=100];1[label="succTup0",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.14/4.16	3[label="succTup0 wu3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.14/4.16	4[label="pt toEnumTup0 (pt (psMyInt (Pos (Succ Zero))) fromEnumTup0) wu3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
10.14/4.16	5[label="toEnumTup0 (pt (psMyInt (Pos (Succ Zero))) fromEnumTup0 wu3)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.14/4.16	6[label="toEnum1 (pt (psMyInt (Pos (Succ Zero))) fromEnumTup0 wu3)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.14/4.16	7[label="toEnum0 (esEsMyInt (pt (psMyInt (Pos (Succ Zero))) fromEnumTup0 wu3) (Pos Zero)) (pt (psMyInt (Pos (Succ Zero))) fromEnumTup0 wu3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
10.14/4.16	8[label="toEnum0 (primEqInt (pt (psMyInt (Pos (Succ Zero))) fromEnumTup0 wu3) (Pos Zero)) (pt (psMyInt (Pos (Succ Zero))) fromEnumTup0 wu3)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
10.14/4.16	9[label="toEnum0 (primEqInt (psMyInt (Pos (Succ Zero)) (fromEnumTup0 wu3)) (Pos Zero)) (psMyInt (Pos (Succ Zero)) (fromEnumTup0 wu3))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10.14/4.16	10[label="toEnum0 (primEqInt (primPlusInt (Pos (Succ Zero)) (fromEnumTup0 wu3)) (Pos Zero)) (primPlusInt (Pos (Succ Zero)) (fromEnumTup0 wu3))",fontsize=16,color="burlywood",shape="box"];17[label="wu3/Tup0",fontsize=10,color="white",style="solid",shape="box"];10 -> 17[label="",style="solid", color="burlywood", weight=9];
10.14/4.16	17 -> 11[label="",style="solid", color="burlywood", weight=3];
10.14/4.16	11[label="toEnum0 (primEqInt (primPlusInt (Pos (Succ Zero)) (fromEnumTup0 Tup0)) (Pos Zero)) (primPlusInt (Pos (Succ Zero)) (fromEnumTup0 Tup0))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
10.14/4.16	12[label="toEnum0 (primEqInt (primPlusInt (Pos (Succ Zero)) (Pos Zero)) (Pos Zero)) (primPlusInt (Pos (Succ Zero)) (Pos Zero))",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
10.14/4.16	13[label="toEnum0 (primEqInt (Pos (primPlusNat (Succ Zero) Zero)) (Pos Zero)) (Pos (primPlusNat (Succ Zero) Zero))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
10.14/4.16	14[label="toEnum0 (primEqInt (Pos (Succ Zero)) (Pos Zero)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
10.14/4.16	15[label="toEnum0 MyFalse (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
10.14/4.16	16[label="error []",fontsize=16,color="red",shape="box"];}
10.14/4.16	
10.14/4.16	----------------------------------------
10.14/4.16	
10.14/4.16	(6)
10.14/4.16	YES
10.14/4.21	EOF