7.70/4.23	YES
9.54/4.74	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.54/4.74	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.54/4.74	
9.54/4.74	
9.54/4.74	H-Termination with start terms of the given HASKELL could be proven:
9.54/4.74	
9.54/4.74	(0) HASKELL
9.54/4.74	(1) BR [EQUIVALENT, 0 ms]
9.54/4.74	(2) HASKELL
9.54/4.74	(3) COR [EQUIVALENT, 0 ms]
9.54/4.74	(4) HASKELL
9.54/4.74	(5) Narrow [EQUIVALENT, 31 ms]
9.54/4.74	(6) YES
9.54/4.74	
9.54/4.74	
9.54/4.74	----------------------------------------
9.54/4.74	
9.54/4.74	(0)
9.54/4.74	Obligation:
9.54/4.74	mainModule Main 
9.54/4.74	module Main where {
9.54/4.74	import qualified Prelude;
9.54/4.74	data List a = Cons a (List a)  | Nil ;
9.54/4.74	
9.54/4.74	data MyBool = MyTrue  | MyFalse ;
9.54/4.74	
9.54/4.74	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.54/4.74	
9.54/4.74	data Main.Nat = Succ Main.Nat  | Zero ;
9.54/4.74	
9.54/4.74	data Tup0 = Tup0 ;
9.54/4.74	
9.54/4.74	data Tup2 a b = Tup2 a b ;
9.54/4.74	
9.54/4.74	indexTup0 :: Tup2 Tup0 Tup0  ->  Tup0  ->  MyInt;
9.54/4.74	indexTup0 (Tup2 Tup0 Tup0) Tup0 = Main.Pos Main.Zero;
9.54/4.74	
9.54/4.74	null :: List a  ->  MyBool;
9.54/4.74	null Nil = MyTrue;
9.54/4.74	null (Cons vx vy) = MyFalse;
9.54/4.74	
9.54/4.74	otherwise :: MyBool;
9.54/4.74	otherwise = MyTrue;
9.54/4.74	
9.54/4.74	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
9.54/4.74	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
9.54/4.74	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
9.54/4.74	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
9.54/4.74	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
9.54/4.74	
9.54/4.74	primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
9.54/4.74	primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
9.54/4.74	primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
9.54/4.74	primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
9.54/4.74	primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);
9.54/4.74	
9.54/4.74	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.54/4.74	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.54/4.74	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.54/4.74	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.54/4.74	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.54/4.74	
9.54/4.74	psMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.54/4.74	psMyInt = primPlusInt;
9.54/4.74	
9.54/4.74	rangeSize0 vv vw MyTrue = psMyInt (indexTup0 (Tup2 vv vw) vw) (Main.Pos (Main.Succ Main.Zero));
9.54/4.74	
9.54/4.74	rangeSize1 vv vw MyTrue = Main.Pos Main.Zero;
9.54/4.74	rangeSize1 vv vw MyFalse = rangeSize0 vv vw otherwise;
9.54/4.75	
9.54/4.75	rangeSize2 (Tup2 vv vw) = rangeSize1 vv vw (null (rangeTup0 (Tup2 vv vw)));
9.54/4.75	
9.54/4.75	rangeSizeTup0 :: Tup2 Tup0 Tup0  ->  MyInt;
9.54/4.75	rangeSizeTup0 (Tup2 vv vw) = rangeSize2 (Tup2 vv vw);
9.54/4.75	
9.54/4.75	rangeTup0 :: Tup2 Tup0 Tup0  ->  List Tup0;
9.54/4.75	rangeTup0 (Tup2 Tup0 Tup0) = Cons Tup0 Nil;
9.54/4.75	
9.54/4.75	}
9.54/4.75	
9.54/4.75	----------------------------------------
9.54/4.75	
9.54/4.75	(1) BR (EQUIVALENT)
9.54/4.75	Replaced joker patterns by fresh variables and removed binding patterns.
9.54/4.75	----------------------------------------
9.54/4.75	
9.54/4.75	(2)
9.54/4.75	Obligation:
9.54/4.75	mainModule Main 
9.54/4.75	module Main where {
9.54/4.75	import qualified Prelude;
9.54/4.75	data List a = Cons a (List a)  | Nil ;
9.54/4.75	
9.54/4.75	data MyBool = MyTrue  | MyFalse ;
9.54/4.75	
9.54/4.75	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.54/4.75	
9.54/4.75	data Main.Nat = Succ Main.Nat  | Zero ;
9.54/4.75	
9.54/4.75	data Tup0 = Tup0 ;
9.54/4.75	
9.54/4.75	data Tup2 b a = Tup2 b a ;
9.54/4.75	
9.54/4.75	indexTup0 :: Tup2 Tup0 Tup0  ->  Tup0  ->  MyInt;
9.54/4.75	indexTup0 (Tup2 Tup0 Tup0) Tup0 = Main.Pos Main.Zero;
9.54/4.75	
9.54/4.75	null :: List a  ->  MyBool;
9.54/4.75	null Nil = MyTrue;
9.54/4.75	null (Cons vx vy) = MyFalse;
9.54/4.75	
9.54/4.75	otherwise :: MyBool;
9.54/4.75	otherwise = MyTrue;
9.54/4.75	
9.54/4.75	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
9.54/4.75	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
9.54/4.75	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
9.54/4.75	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
9.54/4.75	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
9.54/4.75	
9.54/4.75	primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
9.54/4.75	primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
9.54/4.75	primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
9.54/4.75	primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
9.54/4.75	primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);
9.54/4.75	
9.54/4.75	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.54/4.75	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.54/4.75	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.54/4.75	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.54/4.75	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.54/4.75	
9.54/4.75	psMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.54/4.75	psMyInt = primPlusInt;
9.54/4.75	
9.54/4.75	rangeSize0 vv vw MyTrue = psMyInt (indexTup0 (Tup2 vv vw) vw) (Main.Pos (Main.Succ Main.Zero));
9.54/4.75	
9.54/4.75	rangeSize1 vv vw MyTrue = Main.Pos Main.Zero;
9.54/4.75	rangeSize1 vv vw MyFalse = rangeSize0 vv vw otherwise;
9.54/4.75	
9.54/4.75	rangeSize2 (Tup2 vv vw) = rangeSize1 vv vw (null (rangeTup0 (Tup2 vv vw)));
9.54/4.75	
9.54/4.75	rangeSizeTup0 :: Tup2 Tup0 Tup0  ->  MyInt;
9.54/4.75	rangeSizeTup0 (Tup2 vv vw) = rangeSize2 (Tup2 vv vw);
9.54/4.75	
9.54/4.75	rangeTup0 :: Tup2 Tup0 Tup0  ->  List Tup0;
9.54/4.75	rangeTup0 (Tup2 Tup0 Tup0) = Cons Tup0 Nil;
9.54/4.75	
9.54/4.75	}
9.54/4.75	
9.54/4.75	----------------------------------------
9.54/4.75	
9.54/4.75	(3) COR (EQUIVALENT)
9.54/4.75	Cond Reductions:
9.54/4.75	The following Function with conditions
9.54/4.75	"undefined |Falseundefined;
9.54/4.75	"
9.54/4.75	is transformed to
9.54/4.75	"undefined  = undefined1;
9.54/4.75	"
9.54/4.75	"undefined0 True = undefined;
9.54/4.75	"
9.54/4.75	"undefined1  = undefined0 False;
9.54/4.75	"
9.54/4.75	
9.54/4.75	----------------------------------------
9.54/4.75	
9.54/4.75	(4)
9.54/4.75	Obligation:
9.54/4.75	mainModule Main 
9.54/4.75	module Main where {
9.54/4.75	import qualified Prelude;
9.54/4.75	data List a = Cons a (List a)  | Nil ;
9.54/4.75	
9.54/4.75	data MyBool = MyTrue  | MyFalse ;
9.54/4.75	
9.54/4.75	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.54/4.75	
9.54/4.75	data Main.Nat = Succ Main.Nat  | Zero ;
9.54/4.75	
9.54/4.75	data Tup0 = Tup0 ;
9.54/4.75	
9.54/4.75	data Tup2 a b = Tup2 a b ;
9.54/4.75	
9.54/4.75	indexTup0 :: Tup2 Tup0 Tup0  ->  Tup0  ->  MyInt;
9.54/4.75	indexTup0 (Tup2 Tup0 Tup0) Tup0 = Main.Pos Main.Zero;
9.54/4.75	
9.54/4.75	null :: List a  ->  MyBool;
9.54/4.75	null Nil = MyTrue;
9.54/4.75	null (Cons vx vy) = MyFalse;
9.54/4.75	
9.54/4.75	otherwise :: MyBool;
9.54/4.75	otherwise = MyTrue;
9.54/4.75	
9.54/4.75	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
9.54/4.75	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
9.54/4.75	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
9.54/4.75	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
9.54/4.75	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
9.54/4.75	
9.54/4.75	primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
9.54/4.75	primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
9.54/4.75	primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
9.54/4.75	primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
9.54/4.75	primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);
9.54/4.75	
9.54/4.75	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.54/4.75	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.54/4.75	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.54/4.75	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.54/4.75	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.54/4.75	
9.54/4.75	psMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.54/4.75	psMyInt = primPlusInt;
9.54/4.75	
9.54/4.75	rangeSize0 vv vw MyTrue = psMyInt (indexTup0 (Tup2 vv vw) vw) (Main.Pos (Main.Succ Main.Zero));
9.54/4.75	
9.54/4.75	rangeSize1 vv vw MyTrue = Main.Pos Main.Zero;
9.54/4.75	rangeSize1 vv vw MyFalse = rangeSize0 vv vw otherwise;
9.54/4.75	
9.54/4.75	rangeSize2 (Tup2 vv vw) = rangeSize1 vv vw (null (rangeTup0 (Tup2 vv vw)));
9.54/4.75	
9.54/4.75	rangeSizeTup0 :: Tup2 Tup0 Tup0  ->  MyInt;
9.54/4.75	rangeSizeTup0 (Tup2 vv vw) = rangeSize2 (Tup2 vv vw);
9.54/4.75	
9.54/4.75	rangeTup0 :: Tup2 Tup0 Tup0  ->  List Tup0;
9.54/4.75	rangeTup0 (Tup2 Tup0 Tup0) = Cons Tup0 Nil;
9.54/4.75	
9.54/4.75	}
9.54/4.75	
9.54/4.75	----------------------------------------
9.54/4.75	
9.54/4.75	(5) Narrow (EQUIVALENT)
9.54/4.75	Haskell To QDPs
9.54/4.75	
9.54/4.75	digraph dp_graph {
9.54/4.75	node [outthreshold=100, inthreshold=100];1[label="rangeSizeTup0",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.54/4.75	3[label="rangeSizeTup0 wv3",fontsize=16,color="burlywood",shape="triangle"];19[label="wv3/Tup2 wv30 wv31",fontsize=10,color="white",style="solid",shape="box"];3 -> 19[label="",style="solid", color="burlywood", weight=9];
9.54/4.75	19 -> 4[label="",style="solid", color="burlywood", weight=3];
9.54/4.75	4[label="rangeSizeTup0 (Tup2 wv30 wv31)",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
9.54/4.75	5[label="rangeSize2 (Tup2 wv30 wv31)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.54/4.75	6[label="rangeSize1 wv30 wv31 (null (rangeTup0 (Tup2 wv30 wv31)))",fontsize=16,color="burlywood",shape="box"];20[label="wv30/Tup0",fontsize=10,color="white",style="solid",shape="box"];6 -> 20[label="",style="solid", color="burlywood", weight=9];
9.54/4.75	20 -> 7[label="",style="solid", color="burlywood", weight=3];
9.54/4.75	7[label="rangeSize1 Tup0 wv31 (null (rangeTup0 (Tup2 Tup0 wv31)))",fontsize=16,color="burlywood",shape="box"];21[label="wv31/Tup0",fontsize=10,color="white",style="solid",shape="box"];7 -> 21[label="",style="solid", color="burlywood", weight=9];
9.54/4.75	21 -> 8[label="",style="solid", color="burlywood", weight=3];
9.54/4.75	8[label="rangeSize1 Tup0 Tup0 (null (rangeTup0 (Tup2 Tup0 Tup0)))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.54/4.75	9[label="rangeSize1 Tup0 Tup0 (null (Cons Tup0 Nil))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
9.54/4.75	10[label="rangeSize1 Tup0 Tup0 MyFalse",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
9.54/4.75	11[label="rangeSize0 Tup0 Tup0 otherwise",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
9.54/4.75	12[label="rangeSize0 Tup0 Tup0 MyTrue",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
9.54/4.75	13[label="psMyInt (indexTup0 (Tup2 Tup0 Tup0) Tup0) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
9.54/4.75	14[label="primPlusInt (indexTup0 (Tup2 Tup0 Tup0) Tup0) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
9.54/4.75	15[label="primPlusInt (Pos Zero) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
9.54/4.75	16[label="Pos (primPlusNat Zero (Succ Zero))",fontsize=16,color="green",shape="box"];16 -> 17[label="",style="dashed", color="green", weight=3];
9.54/4.75	17[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
9.54/4.75	18[label="Succ Zero",fontsize=16,color="green",shape="box"];}
9.54/4.75	
9.54/4.75	----------------------------------------
9.54/4.75	
9.54/4.75	(6)
9.54/4.75	YES
9.69/4.79	EOF