8.05/3.65	YES
9.90/4.21	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.90/4.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.90/4.21	
9.90/4.21	
9.90/4.21	H-Termination with start terms of the given HASKELL could be proven:
9.90/4.21	
9.90/4.21	(0) HASKELL
9.90/4.21	(1) BR [EQUIVALENT, 0 ms]
9.90/4.21	(2) HASKELL
9.90/4.21	(3) COR [EQUIVALENT, 0 ms]
9.90/4.21	(4) HASKELL
9.90/4.21	(5) Narrow [SOUND, 0 ms]
9.90/4.21	(6) QDP
9.90/4.21	(7) TransformationProof [EQUIVALENT, 0 ms]
9.90/4.21	(8) QDP
9.90/4.21	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
9.90/4.21	(10) QDP
9.90/4.21	(11) TransformationProof [EQUIVALENT, 0 ms]
9.90/4.21	(12) QDP
9.90/4.21	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
9.90/4.21	(14) QDP
9.90/4.21	(15) UsableRulesProof [EQUIVALENT, 0 ms]
9.90/4.21	(16) QDP
9.90/4.21	(17) QReductionProof [EQUIVALENT, 0 ms]
9.90/4.21	(18) QDP
9.90/4.21	(19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.90/4.21	(20) YES
9.90/4.21	
9.90/4.21	
9.90/4.21	----------------------------------------
9.90/4.21	
9.90/4.21	(0)
9.90/4.21	Obligation:
9.90/4.21	mainModule Main 
9.90/4.21	module Main where {
9.90/4.21	import qualified Prelude;
9.90/4.21	data List a = Cons a (List a)  | Nil ;
9.90/4.21	
9.90/4.21	data MyBool = MyTrue  | MyFalse ;
9.90/4.21	
9.90/4.21	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.90/4.21	
9.90/4.21	data Main.Nat = Succ Main.Nat  | Zero ;
9.90/4.21	
9.90/4.21	data Ordering = LT  | EQ  | GT ;
9.90/4.21	
9.90/4.21	data Tup2 a b = Tup2 a b ;
9.90/4.21	
9.90/4.21	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
9.90/4.21	compareMyInt = primCmpInt;
9.90/4.21	
9.90/4.21	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.90/4.21	esEsOrdering LT LT = MyTrue;
9.90/4.21	esEsOrdering LT EQ = MyFalse;
9.90/4.21	esEsOrdering LT GT = MyFalse;
9.90/4.21	esEsOrdering EQ LT = MyFalse;
9.90/4.21	esEsOrdering EQ EQ = MyTrue;
9.90/4.21	esEsOrdering EQ GT = MyFalse;
9.90/4.21	esEsOrdering GT LT = MyFalse;
9.90/4.21	esEsOrdering GT EQ = MyFalse;
9.90/4.21	esEsOrdering GT GT = MyTrue;
9.90/4.21	
9.90/4.21	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.90/4.21	fsEsOrdering x y = not (esEsOrdering x y);
9.90/4.21	
9.90/4.21	ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
9.90/4.21	ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;
9.90/4.21	
9.90/4.21	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.90/4.21	msMyInt = primMinusInt;
9.90/4.21	
9.90/4.21	not :: MyBool  ->  MyBool;
9.90/4.21	not MyTrue = MyFalse;
9.90/4.21	not MyFalse = MyTrue;
9.90/4.21	
9.90/4.21	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
9.90/4.21	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
9.90/4.21	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
9.90/4.21	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
9.90/4.21	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
9.90/4.21	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
9.90/4.21	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
9.90/4.21	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
9.90/4.21	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
9.90/4.21	
9.90/4.21	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
9.90/4.21	primCmpNat Main.Zero Main.Zero = EQ;
9.90/4.21	primCmpNat Main.Zero (Main.Succ y) = LT;
9.90/4.21	primCmpNat (Main.Succ x) Main.Zero = GT;
9.90/4.21	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
9.90/4.21	
9.90/4.21	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
9.90/4.21	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
9.90/4.21	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
9.90/4.21	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
9.90/4.21	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
9.90/4.21	
9.90/4.21	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
9.90/4.21	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
9.90/4.21	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
9.90/4.21	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
9.90/4.21	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
9.90/4.21	
9.90/4.21	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.90/4.21	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.90/4.21	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.90/4.21	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.90/4.21	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.90/4.21	
9.90/4.21	splitAt :: MyInt  ->  List a  ->  Tup2 (List a) (List a);
9.90/4.21	splitAt n xs = splitAt3 n xs;
9.90/4.21	splitAt vv Nil = splitAt1 vv Nil;
9.90/4.21	splitAt n (Cons x xs) = splitAt0 n (Cons x xs);
9.90/4.21	
9.90/4.21	splitAt0 n (Cons x xs) = Tup2 (Cons x (splitAt0Xs' n xs)) (splitAt0Xs'' n xs);
9.90/4.21	
9.90/4.21	splitAt0Vu42 wy wz = splitAt (msMyInt wy (Main.Pos (Main.Succ Main.Zero))) wz;
9.90/4.21	
9.90/4.21	splitAt0Xs' wy wz = splitAt0Xs'0 wy wz (splitAt0Vu42 wy wz);
9.90/4.21	
9.90/4.21	splitAt0Xs'' wy wz = splitAt0Xs''0 wy wz (splitAt0Vu42 wy wz);
9.90/4.21	
9.90/4.21	splitAt0Xs''0 wy wz (Tup2 vw xs'') = xs'';
9.90/4.21	
9.90/4.21	splitAt0Xs'0 wy wz (Tup2 xs' vx) = xs';
9.90/4.21	
9.90/4.21	splitAt1 vv Nil = Tup2 Nil Nil;
9.90/4.21	splitAt1 wu wv = splitAt0 wu wv;
9.90/4.21	
9.90/4.21	splitAt2 n xs MyTrue = Tup2 Nil xs;
9.90/4.21	splitAt2 n xs MyFalse = splitAt1 n xs;
9.90/4.21	
9.90/4.21	splitAt3 n xs = splitAt2 n xs (ltEsMyInt n (Main.Pos Main.Zero));
9.90/4.21	splitAt3 ww wx = splitAt1 ww wx;
9.90/4.21	
9.90/4.21	}
9.90/4.21	
9.90/4.21	----------------------------------------
9.90/4.21	
9.90/4.21	(1) BR (EQUIVALENT)
9.90/4.21	Replaced joker patterns by fresh variables and removed binding patterns.
9.90/4.21	----------------------------------------
9.90/4.21	
9.90/4.21	(2)
9.90/4.21	Obligation:
9.90/4.21	mainModule Main 
9.90/4.21	module Main where {
9.90/4.21	import qualified Prelude;
9.90/4.21	data List a = Cons a (List a)  | Nil ;
9.90/4.21	
9.90/4.21	data MyBool = MyTrue  | MyFalse ;
9.90/4.21	
9.90/4.21	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.90/4.21	
9.90/4.21	data Main.Nat = Succ Main.Nat  | Zero ;
9.90/4.21	
9.90/4.21	data Ordering = LT  | EQ  | GT ;
9.90/4.21	
9.90/4.21	data Tup2 b a = Tup2 b a ;
9.90/4.21	
9.90/4.21	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
9.90/4.21	compareMyInt = primCmpInt;
9.90/4.21	
9.90/4.21	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.90/4.21	esEsOrdering LT LT = MyTrue;
9.90/4.21	esEsOrdering LT EQ = MyFalse;
9.90/4.21	esEsOrdering LT GT = MyFalse;
9.90/4.21	esEsOrdering EQ LT = MyFalse;
9.90/4.21	esEsOrdering EQ EQ = MyTrue;
9.90/4.21	esEsOrdering EQ GT = MyFalse;
9.90/4.21	esEsOrdering GT LT = MyFalse;
9.90/4.21	esEsOrdering GT EQ = MyFalse;
9.90/4.21	esEsOrdering GT GT = MyTrue;
9.90/4.21	
9.90/4.21	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.90/4.21	fsEsOrdering x y = not (esEsOrdering x y);
9.90/4.22	
9.90/4.22	ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
9.90/4.22	ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;
9.90/4.22	
9.90/4.22	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.90/4.22	msMyInt = primMinusInt;
9.90/4.22	
9.90/4.22	not :: MyBool  ->  MyBool;
9.90/4.22	not MyTrue = MyFalse;
9.90/4.22	not MyFalse = MyTrue;
9.90/4.22	
9.90/4.22	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
9.90/4.22	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
9.90/4.22	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
9.90/4.22	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
9.90/4.22	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
9.90/4.22	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
9.90/4.22	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
9.90/4.22	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
9.90/4.22	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
9.90/4.22	
9.90/4.22	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
9.90/4.22	primCmpNat Main.Zero Main.Zero = EQ;
9.90/4.22	primCmpNat Main.Zero (Main.Succ y) = LT;
9.90/4.22	primCmpNat (Main.Succ x) Main.Zero = GT;
9.90/4.22	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
9.90/4.22	
9.90/4.22	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
9.90/4.22	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
9.90/4.22	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
9.90/4.22	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
9.90/4.22	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
9.90/4.22	
9.90/4.22	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
9.90/4.22	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
9.90/4.22	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
9.90/4.22	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
9.90/4.22	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
9.90/4.22	
9.90/4.22	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.90/4.22	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.90/4.22	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.90/4.22	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.90/4.22	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.90/4.22	
9.90/4.22	splitAt :: MyInt  ->  List a  ->  Tup2 (List a) (List a);
9.90/4.22	splitAt n xs = splitAt3 n xs;
9.90/4.22	splitAt vv Nil = splitAt1 vv Nil;
9.90/4.22	splitAt n (Cons x xs) = splitAt0 n (Cons x xs);
9.90/4.22	
9.90/4.22	splitAt0 n (Cons x xs) = Tup2 (Cons x (splitAt0Xs' n xs)) (splitAt0Xs'' n xs);
9.90/4.22	
9.90/4.22	splitAt0Vu42 wy wz = splitAt (msMyInt wy (Main.Pos (Main.Succ Main.Zero))) wz;
9.90/4.22	
9.90/4.22	splitAt0Xs' wy wz = splitAt0Xs'0 wy wz (splitAt0Vu42 wy wz);
9.90/4.22	
9.90/4.22	splitAt0Xs'' wy wz = splitAt0Xs''0 wy wz (splitAt0Vu42 wy wz);
9.90/4.22	
9.90/4.22	splitAt0Xs''0 wy wz (Tup2 vw xs'') = xs'';
9.90/4.22	
9.90/4.22	splitAt0Xs'0 wy wz (Tup2 xs' vx) = xs';
9.90/4.22	
9.90/4.22	splitAt1 vv Nil = Tup2 Nil Nil;
9.90/4.22	splitAt1 wu wv = splitAt0 wu wv;
9.90/4.22	
9.90/4.22	splitAt2 n xs MyTrue = Tup2 Nil xs;
9.90/4.22	splitAt2 n xs MyFalse = splitAt1 n xs;
9.90/4.22	
9.90/4.22	splitAt3 n xs = splitAt2 n xs (ltEsMyInt n (Main.Pos Main.Zero));
9.90/4.22	splitAt3 ww wx = splitAt1 ww wx;
9.90/4.22	
9.90/4.22	}
9.90/4.22	
9.90/4.22	----------------------------------------
9.90/4.22	
9.90/4.22	(3) COR (EQUIVALENT)
9.90/4.22	Cond Reductions:
9.90/4.22	The following Function with conditions
9.90/4.22	"undefined |Falseundefined;
9.90/4.22	"
9.90/4.22	is transformed to
9.90/4.22	"undefined  = undefined1;
9.90/4.22	"
9.90/4.22	"undefined0 True = undefined;
9.90/4.22	"
9.90/4.22	"undefined1  = undefined0 False;
9.90/4.22	"
9.90/4.22	
9.90/4.22	----------------------------------------
9.90/4.22	
9.90/4.22	(4)
9.90/4.22	Obligation:
9.90/4.22	mainModule Main 
9.90/4.22	module Main where {
9.90/4.22	import qualified Prelude;
9.90/4.22	data List a = Cons a (List a)  | Nil ;
9.90/4.22	
9.90/4.22	data MyBool = MyTrue  | MyFalse ;
9.90/4.22	
9.90/4.22	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.90/4.22	
9.90/4.22	data Main.Nat = Succ Main.Nat  | Zero ;
9.90/4.22	
9.90/4.22	data Ordering = LT  | EQ  | GT ;
9.90/4.22	
9.90/4.22	data Tup2 b a = Tup2 b a ;
9.90/4.22	
9.90/4.22	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
9.90/4.22	compareMyInt = primCmpInt;
9.90/4.22	
9.90/4.22	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.90/4.22	esEsOrdering LT LT = MyTrue;
9.90/4.22	esEsOrdering LT EQ = MyFalse;
9.90/4.22	esEsOrdering LT GT = MyFalse;
9.90/4.22	esEsOrdering EQ LT = MyFalse;
9.90/4.22	esEsOrdering EQ EQ = MyTrue;
9.90/4.22	esEsOrdering EQ GT = MyFalse;
9.90/4.22	esEsOrdering GT LT = MyFalse;
9.90/4.22	esEsOrdering GT EQ = MyFalse;
9.90/4.22	esEsOrdering GT GT = MyTrue;
9.90/4.22	
9.90/4.22	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.90/4.22	fsEsOrdering x y = not (esEsOrdering x y);
9.90/4.22	
9.90/4.22	ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
9.90/4.22	ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;
9.90/4.22	
9.90/4.22	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.90/4.22	msMyInt = primMinusInt;
9.90/4.22	
9.90/4.22	not :: MyBool  ->  MyBool;
9.90/4.22	not MyTrue = MyFalse;
9.90/4.22	not MyFalse = MyTrue;
9.90/4.22	
9.90/4.22	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
9.90/4.22	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
9.90/4.22	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
9.90/4.22	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
9.90/4.22	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
9.90/4.22	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
9.90/4.22	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
9.90/4.22	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
9.90/4.22	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
9.90/4.22	
9.90/4.22	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
9.90/4.22	primCmpNat Main.Zero Main.Zero = EQ;
9.90/4.22	primCmpNat Main.Zero (Main.Succ y) = LT;
9.90/4.22	primCmpNat (Main.Succ x) Main.Zero = GT;
9.90/4.22	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
9.90/4.22	
9.90/4.22	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
9.90/4.22	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
9.90/4.22	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
9.90/4.22	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
9.90/4.22	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
9.90/4.22	
9.90/4.22	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
9.90/4.22	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
9.90/4.22	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
9.90/4.22	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
9.90/4.22	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
9.90/4.22	
9.90/4.22	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.90/4.22	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.90/4.22	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.90/4.22	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.90/4.22	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.90/4.22	
9.90/4.22	splitAt :: MyInt  ->  List a  ->  Tup2 (List a) (List a);
9.90/4.22	splitAt n xs = splitAt3 n xs;
9.90/4.22	splitAt vv Nil = splitAt1 vv Nil;
9.90/4.22	splitAt n (Cons x xs) = splitAt0 n (Cons x xs);
9.90/4.22	
9.90/4.22	splitAt0 n (Cons x xs) = Tup2 (Cons x (splitAt0Xs' n xs)) (splitAt0Xs'' n xs);
9.90/4.22	
9.90/4.22	splitAt0Vu42 wy wz = splitAt (msMyInt wy (Main.Pos (Main.Succ Main.Zero))) wz;
9.90/4.22	
9.90/4.22	splitAt0Xs' wy wz = splitAt0Xs'0 wy wz (splitAt0Vu42 wy wz);
9.90/4.22	
9.90/4.22	splitAt0Xs'' wy wz = splitAt0Xs''0 wy wz (splitAt0Vu42 wy wz);
9.90/4.22	
9.90/4.22	splitAt0Xs''0 wy wz (Tup2 vw xs'') = xs'';
9.90/4.22	
9.90/4.22	splitAt0Xs'0 wy wz (Tup2 xs' vx) = xs';
9.90/4.22	
9.90/4.22	splitAt1 vv Nil = Tup2 Nil Nil;
9.90/4.22	splitAt1 wu wv = splitAt0 wu wv;
9.90/4.22	
9.90/4.22	splitAt2 n xs MyTrue = Tup2 Nil xs;
9.90/4.22	splitAt2 n xs MyFalse = splitAt1 n xs;
9.90/4.22	
9.90/4.22	splitAt3 n xs = splitAt2 n xs (ltEsMyInt n (Main.Pos Main.Zero));
9.90/4.22	splitAt3 ww wx = splitAt1 ww wx;
9.90/4.22	
9.90/4.22	}
9.90/4.22	
9.90/4.22	----------------------------------------
9.90/4.22	
9.90/4.22	(5) Narrow (SOUND)
9.90/4.22	Haskell To QDPs
9.90/4.22	
9.90/4.22	digraph dp_graph {
9.90/4.22	node [outthreshold=100, inthreshold=100];1[label="splitAt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.90/4.22	3[label="splitAt xu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.90/4.22	4[label="splitAt xu3 xu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.90/4.22	5[label="splitAt3 xu3 xu4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.90/4.22	6[label="splitAt2 xu3 xu4 (ltEsMyInt xu3 (Pos Zero))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.90/4.22	7[label="splitAt2 xu3 xu4 (fsEsOrdering (compareMyInt xu3 (Pos Zero)) GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.90/4.22	8[label="splitAt2 xu3 xu4 (not (esEsOrdering (compareMyInt xu3 (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.90/4.22	9[label="splitAt2 xu3 xu4 (not (esEsOrdering (primCmpInt xu3 (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];64[label="xu3/Pos xu30",fontsize=10,color="white",style="solid",shape="box"];9 -> 64[label="",style="solid", color="burlywood", weight=9];
9.90/4.22	64 -> 10[label="",style="solid", color="burlywood", weight=3];
9.90/4.22	65[label="xu3/Neg xu30",fontsize=10,color="white",style="solid",shape="box"];9 -> 65[label="",style="solid", color="burlywood", weight=9];
9.90/4.22	65 -> 11[label="",style="solid", color="burlywood", weight=3];
9.90/4.22	10[label="splitAt2 (Pos xu30) xu4 (not (esEsOrdering (primCmpInt (Pos xu30) (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];66[label="xu30/Succ xu300",fontsize=10,color="white",style="solid",shape="box"];10 -> 66[label="",style="solid", color="burlywood", weight=9];
9.90/4.22	66 -> 12[label="",style="solid", color="burlywood", weight=3];
9.90/4.22	67[label="xu30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 67[label="",style="solid", color="burlywood", weight=9];
9.90/4.22	67 -> 13[label="",style="solid", color="burlywood", weight=3];
9.90/4.22	11[label="splitAt2 (Neg xu30) xu4 (not (esEsOrdering (primCmpInt (Neg xu30) (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];68[label="xu30/Succ xu300",fontsize=10,color="white",style="solid",shape="box"];11 -> 68[label="",style="solid", color="burlywood", weight=9];
9.90/4.22	68 -> 14[label="",style="solid", color="burlywood", weight=3];
9.90/4.22	69[label="xu30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 69[label="",style="solid", color="burlywood", weight=9];
9.90/4.22	69 -> 15[label="",style="solid", color="burlywood", weight=3];
9.90/4.22	12[label="splitAt2 (Pos (Succ xu300)) xu4 (not (esEsOrdering (primCmpInt (Pos (Succ xu300)) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
9.90/4.22	13[label="splitAt2 (Pos Zero) xu4 (not (esEsOrdering (primCmpInt (Pos Zero) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
9.90/4.22	14[label="splitAt2 (Neg (Succ xu300)) xu4 (not (esEsOrdering (primCmpInt (Neg (Succ xu300)) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
9.90/4.22	15[label="splitAt2 (Neg Zero) xu4 (not (esEsOrdering (primCmpInt (Neg Zero) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
9.90/4.22	16[label="splitAt2 (Pos (Succ xu300)) xu4 (not (esEsOrdering (primCmpNat (Succ xu300) Zero) GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
9.90/4.22	17[label="splitAt2 (Pos Zero) xu4 (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
9.90/4.22	18[label="splitAt2 (Neg (Succ xu300)) xu4 (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
9.90/4.22	19[label="splitAt2 (Neg Zero) xu4 (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
9.90/4.22	20[label="splitAt2 (Pos (Succ xu300)) xu4 (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
9.90/4.22	21[label="splitAt2 (Pos Zero) xu4 (not MyFalse)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
9.90/4.22	22[label="splitAt2 (Neg (Succ xu300)) xu4 (not MyFalse)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
9.90/4.22	23[label="splitAt2 (Neg Zero) xu4 (not MyFalse)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
9.90/4.22	24[label="splitAt2 (Pos (Succ xu300)) xu4 (not MyTrue)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
9.90/4.22	25[label="splitAt2 (Pos Zero) xu4 MyTrue",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
9.90/4.22	26[label="splitAt2 (Neg (Succ xu300)) xu4 MyTrue",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
9.90/4.22	27[label="splitAt2 (Neg Zero) xu4 MyTrue",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
9.90/4.22	28[label="splitAt2 (Pos (Succ xu300)) xu4 MyFalse",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
9.90/4.22	29[label="Tup2 Nil xu4",fontsize=16,color="green",shape="box"];30[label="Tup2 Nil xu4",fontsize=16,color="green",shape="box"];31[label="Tup2 Nil xu4",fontsize=16,color="green",shape="box"];32[label="splitAt1 (Pos (Succ xu300)) xu4",fontsize=16,color="burlywood",shape="box"];70[label="xu4/Cons xu40 xu41",fontsize=10,color="white",style="solid",shape="box"];32 -> 70[label="",style="solid", color="burlywood", weight=9];
9.90/4.22	70 -> 33[label="",style="solid", color="burlywood", weight=3];
9.90/4.22	71[label="xu4/Nil",fontsize=10,color="white",style="solid",shape="box"];32 -> 71[label="",style="solid", color="burlywood", weight=9];
9.90/4.22	71 -> 34[label="",style="solid", color="burlywood", weight=3];
9.90/4.22	33[label="splitAt1 (Pos (Succ xu300)) (Cons xu40 xu41)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
9.90/4.22	34[label="splitAt1 (Pos (Succ xu300)) Nil",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
9.90/4.22	35[label="splitAt0 (Pos (Succ xu300)) (Cons xu40 xu41)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
9.90/4.22	36[label="Tup2 Nil Nil",fontsize=16,color="green",shape="box"];37[label="Tup2 (Cons xu40 (splitAt0Xs' (Pos (Succ xu300)) xu41)) (splitAt0Xs'' (Pos (Succ xu300)) xu41)",fontsize=16,color="green",shape="box"];37 -> 38[label="",style="dashed", color="green", weight=3];
9.90/4.22	37 -> 39[label="",style="dashed", color="green", weight=3];
9.90/4.22	38[label="splitAt0Xs' (Pos (Succ xu300)) xu41",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
9.90/4.22	39[label="splitAt0Xs'' (Pos (Succ xu300)) xu41",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
9.90/4.22	40 -> 44[label="",style="dashed", color="red", weight=0];
9.90/4.22	40[label="splitAt0Xs'0 (Pos (Succ xu300)) xu41 (splitAt0Vu42 (Pos (Succ xu300)) xu41)",fontsize=16,color="magenta"];40 -> 45[label="",style="dashed", color="magenta", weight=3];
9.90/4.22	41 -> 49[label="",style="dashed", color="red", weight=0];
9.90/4.22	41[label="splitAt0Xs''0 (Pos (Succ xu300)) xu41 (splitAt0Vu42 (Pos (Succ xu300)) xu41)",fontsize=16,color="magenta"];41 -> 50[label="",style="dashed", color="magenta", weight=3];
9.90/4.22	45[label="splitAt0Vu42 (Pos (Succ xu300)) xu41",fontsize=16,color="black",shape="triangle"];45 -> 47[label="",style="solid", color="black", weight=3];
9.90/4.22	44[label="splitAt0Xs'0 (Pos (Succ xu300)) xu41 xu5",fontsize=16,color="burlywood",shape="triangle"];72[label="xu5/Tup2 xu50 xu51",fontsize=10,color="white",style="solid",shape="box"];44 -> 72[label="",style="solid", color="burlywood", weight=9];
9.90/4.22	72 -> 48[label="",style="solid", color="burlywood", weight=3];
9.90/4.22	50 -> 45[label="",style="dashed", color="red", weight=0];
9.90/4.22	50[label="splitAt0Vu42 (Pos (Succ xu300)) xu41",fontsize=16,color="magenta"];49[label="splitAt0Xs''0 (Pos (Succ xu300)) xu41 xu6",fontsize=16,color="burlywood",shape="triangle"];73[label="xu6/Tup2 xu60 xu61",fontsize=10,color="white",style="solid",shape="box"];49 -> 73[label="",style="solid", color="burlywood", weight=9];
9.90/4.22	73 -> 52[label="",style="solid", color="burlywood", weight=3];
9.90/4.22	47 -> 4[label="",style="dashed", color="red", weight=0];
9.90/4.22	47[label="splitAt (msMyInt (Pos (Succ xu300)) (Pos (Succ Zero))) xu41",fontsize=16,color="magenta"];47 -> 53[label="",style="dashed", color="magenta", weight=3];
9.90/4.22	47 -> 54[label="",style="dashed", color="magenta", weight=3];
9.90/4.22	48[label="splitAt0Xs'0 (Pos (Succ xu300)) xu41 (Tup2 xu50 xu51)",fontsize=16,color="black",shape="box"];48 -> 55[label="",style="solid", color="black", weight=3];
9.90/4.22	52[label="splitAt0Xs''0 (Pos (Succ xu300)) xu41 (Tup2 xu60 xu61)",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
9.90/4.22	53[label="msMyInt (Pos (Succ xu300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
9.90/4.22	54[label="xu41",fontsize=16,color="green",shape="box"];55[label="xu50",fontsize=16,color="green",shape="box"];56[label="xu61",fontsize=16,color="green",shape="box"];57[label="primMinusInt (Pos (Succ xu300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];57 -> 58[label="",style="solid", color="black", weight=3];
9.90/4.22	58[label="primMinusNat (Succ xu300) (Succ Zero)",fontsize=16,color="black",shape="box"];58 -> 59[label="",style="solid", color="black", weight=3];
9.90/4.22	59[label="primMinusNat xu300 Zero",fontsize=16,color="burlywood",shape="box"];74[label="xu300/Succ xu3000",fontsize=10,color="white",style="solid",shape="box"];59 -> 74[label="",style="solid", color="burlywood", weight=9];
9.90/4.22	74 -> 60[label="",style="solid", color="burlywood", weight=3];
9.90/4.22	75[label="xu300/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 75[label="",style="solid", color="burlywood", weight=9];
9.90/4.22	75 -> 61[label="",style="solid", color="burlywood", weight=3];
9.90/4.22	60[label="primMinusNat (Succ xu3000) Zero",fontsize=16,color="black",shape="box"];60 -> 62[label="",style="solid", color="black", weight=3];
9.90/4.22	61[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];61 -> 63[label="",style="solid", color="black", weight=3];
9.90/4.22	62[label="Pos (Succ xu3000)",fontsize=16,color="green",shape="box"];63[label="Pos Zero",fontsize=16,color="green",shape="box"];}
9.90/4.22	
9.90/4.22	----------------------------------------
9.90/4.22	
9.90/4.22	(6)
9.90/4.22	Obligation:
9.90/4.22	Q DP problem:
9.90/4.22	The TRS P consists of the following rules:
9.90/4.22	
9.90/4.22	   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
9.90/4.22	   new_splitAt0Vu42(xu300, xu41, h) -> new_splitAt(new_primMinusNat(xu300), xu41, h)
9.90/4.22	   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt(new_primMinusNat(xu300), xu41, h)
9.90/4.22	
9.90/4.22	The TRS R consists of the following rules:
9.90/4.22	
9.90/4.22	   new_primMinusNat(Main.Succ(xu3000)) -> Main.Pos(Main.Succ(xu3000))
9.90/4.22	   new_primMinusNat(Main.Zero) -> Main.Pos(Main.Zero)
9.90/4.22	
9.90/4.22	The set Q consists of the following terms:
9.90/4.22	
9.90/4.22	   new_primMinusNat(Main.Zero)
9.90/4.22	   new_primMinusNat(Main.Succ(x0))
9.90/4.22	
9.90/4.22	We have to consider all minimal (P,Q,R)-chains.
9.90/4.22	----------------------------------------
9.90/4.22	
9.90/4.22	(7) TransformationProof (EQUIVALENT)
9.90/4.22	By narrowing [LPAR04] the rule new_splitAt0Vu42(xu300, xu41, h) -> new_splitAt(new_primMinusNat(xu300), xu41, h) at position [0] we obtained the following new rules [LPAR04]:
9.90/4.22	
9.90/4.22	   (new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2),new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2))
9.90/4.22	   (new_splitAt0Vu42(Main.Zero, y1, y2) -> new_splitAt(Main.Pos(Main.Zero), y1, y2),new_splitAt0Vu42(Main.Zero, y1, y2) -> new_splitAt(Main.Pos(Main.Zero), y1, y2))
9.90/4.22	
9.90/4.22	
9.90/4.22	----------------------------------------
9.90/4.22	
9.90/4.22	(8)
9.90/4.22	Obligation:
9.90/4.22	Q DP problem:
9.90/4.22	The TRS P consists of the following rules:
9.90/4.22	
9.90/4.22	   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
9.90/4.22	   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt(new_primMinusNat(xu300), xu41, h)
9.90/4.22	   new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
9.90/4.22	   new_splitAt0Vu42(Main.Zero, y1, y2) -> new_splitAt(Main.Pos(Main.Zero), y1, y2)
9.90/4.22	
9.90/4.22	The TRS R consists of the following rules:
9.90/4.22	
9.90/4.22	   new_primMinusNat(Main.Succ(xu3000)) -> Main.Pos(Main.Succ(xu3000))
9.90/4.22	   new_primMinusNat(Main.Zero) -> Main.Pos(Main.Zero)
9.90/4.22	
9.90/4.22	The set Q consists of the following terms:
9.90/4.22	
9.90/4.22	   new_primMinusNat(Main.Zero)
9.90/4.22	   new_primMinusNat(Main.Succ(x0))
9.90/4.22	
9.90/4.22	We have to consider all minimal (P,Q,R)-chains.
9.90/4.22	----------------------------------------
9.90/4.22	
9.90/4.22	(9) DependencyGraphProof (EQUIVALENT)
9.90/4.22	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
9.90/4.22	----------------------------------------
9.90/4.22	
9.90/4.22	(10)
9.90/4.22	Obligation:
9.90/4.22	Q DP problem:
9.90/4.22	The TRS P consists of the following rules:
9.90/4.22	
9.90/4.22	   new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
9.90/4.22	   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
9.90/4.22	   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt(new_primMinusNat(xu300), xu41, h)
9.90/4.22	
9.90/4.22	The TRS R consists of the following rules:
9.90/4.22	
9.90/4.22	   new_primMinusNat(Main.Succ(xu3000)) -> Main.Pos(Main.Succ(xu3000))
9.90/4.22	   new_primMinusNat(Main.Zero) -> Main.Pos(Main.Zero)
9.90/4.22	
9.90/4.22	The set Q consists of the following terms:
9.90/4.22	
9.90/4.22	   new_primMinusNat(Main.Zero)
9.90/4.22	   new_primMinusNat(Main.Succ(x0))
9.90/4.22	
9.90/4.22	We have to consider all minimal (P,Q,R)-chains.
9.90/4.22	----------------------------------------
9.90/4.22	
9.90/4.22	(11) TransformationProof (EQUIVALENT)
9.90/4.22	By narrowing [LPAR04] the rule new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt(new_primMinusNat(xu300), xu41, h) at position [0] we obtained the following new rules [LPAR04]:
9.90/4.22	
9.90/4.22	   (new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3),new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3))
9.90/4.22	   (new_splitAt(Main.Pos(Main.Succ(Main.Zero)), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Zero), y2, y3),new_splitAt(Main.Pos(Main.Succ(Main.Zero)), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Zero), y2, y3))
9.90/4.22	
9.90/4.22	
9.90/4.22	----------------------------------------
9.90/4.22	
9.90/4.22	(12)
9.90/4.22	Obligation:
9.90/4.22	Q DP problem:
9.90/4.22	The TRS P consists of the following rules:
9.90/4.22	
9.90/4.22	   new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
9.90/4.22	   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
9.90/4.22	   new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3)
9.90/4.22	   new_splitAt(Main.Pos(Main.Succ(Main.Zero)), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Zero), y2, y3)
10.28/4.22	
10.28/4.22	The TRS R consists of the following rules:
10.28/4.22	
10.28/4.22	   new_primMinusNat(Main.Succ(xu3000)) -> Main.Pos(Main.Succ(xu3000))
10.28/4.22	   new_primMinusNat(Main.Zero) -> Main.Pos(Main.Zero)
10.28/4.22	
10.28/4.22	The set Q consists of the following terms:
10.28/4.22	
10.28/4.22	   new_primMinusNat(Main.Zero)
10.28/4.22	   new_primMinusNat(Main.Succ(x0))
10.28/4.22	
10.28/4.22	We have to consider all minimal (P,Q,R)-chains.
10.28/4.22	----------------------------------------
10.28/4.22	
10.28/4.22	(13) DependencyGraphProof (EQUIVALENT)
10.28/4.22	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
10.28/4.22	----------------------------------------
10.28/4.22	
10.28/4.22	(14)
10.28/4.22	Obligation:
10.28/4.22	Q DP problem:
10.28/4.22	The TRS P consists of the following rules:
10.28/4.22	
10.28/4.22	   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
10.28/4.22	   new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
10.28/4.22	   new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3)
10.28/4.22	
10.28/4.22	The TRS R consists of the following rules:
10.28/4.22	
10.28/4.22	   new_primMinusNat(Main.Succ(xu3000)) -> Main.Pos(Main.Succ(xu3000))
10.28/4.22	   new_primMinusNat(Main.Zero) -> Main.Pos(Main.Zero)
10.28/4.22	
10.28/4.22	The set Q consists of the following terms:
10.28/4.22	
10.28/4.22	   new_primMinusNat(Main.Zero)
10.28/4.22	   new_primMinusNat(Main.Succ(x0))
10.28/4.22	
10.28/4.22	We have to consider all minimal (P,Q,R)-chains.
10.28/4.22	----------------------------------------
10.28/4.22	
10.28/4.22	(15) UsableRulesProof (EQUIVALENT)
10.28/4.22	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
10.28/4.22	----------------------------------------
10.28/4.22	
10.28/4.22	(16)
10.28/4.22	Obligation:
10.28/4.22	Q DP problem:
10.28/4.22	The TRS P consists of the following rules:
10.28/4.22	
10.28/4.22	   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
10.28/4.22	   new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
10.28/4.22	   new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3)
10.28/4.22	
10.28/4.22	R is empty.
10.28/4.22	The set Q consists of the following terms:
10.28/4.22	
10.28/4.22	   new_primMinusNat(Main.Zero)
10.28/4.22	   new_primMinusNat(Main.Succ(x0))
10.28/4.22	
10.28/4.22	We have to consider all minimal (P,Q,R)-chains.
10.28/4.22	----------------------------------------
10.28/4.22	
10.28/4.22	(17) QReductionProof (EQUIVALENT)
10.28/4.22	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
10.28/4.22	
10.28/4.22	   new_primMinusNat(Main.Zero)
10.28/4.22	   new_primMinusNat(Main.Succ(x0))
10.28/4.22	
10.28/4.22	
10.28/4.22	----------------------------------------
10.28/4.22	
10.28/4.22	(18)
10.28/4.22	Obligation:
10.28/4.22	Q DP problem:
10.28/4.22	The TRS P consists of the following rules:
10.28/4.22	
10.28/4.22	   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
10.28/4.22	   new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
10.28/4.22	   new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3)
10.28/4.22	
10.28/4.22	R is empty.
10.28/4.22	Q is empty.
10.28/4.22	We have to consider all minimal (P,Q,R)-chains.
10.28/4.22	----------------------------------------
10.28/4.22	
10.28/4.22	(19) QDPSizeChangeProof (EQUIVALENT)
10.28/4.22	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. 
10.28/4.22	
10.28/4.22	From the DPs we obtained the following set of size-change graphs:
10.28/4.22	*new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
10.28/4.22	The graph contains the following edges 2 >= 2, 3 >= 3
10.28/4.22	
10.28/4.22	
10.28/4.22	*new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3)
10.28/4.22	The graph contains the following edges 2 > 2, 3 >= 3
10.28/4.22	
10.28/4.22	
10.28/4.22	*new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
10.28/4.22	The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
10.28/4.22	
10.28/4.22	
10.28/4.22	----------------------------------------
10.28/4.22	
10.28/4.22	(20)
10.28/4.22	YES
10.28/4.26	EOF