8.98/3.90	YES
11.13/4.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.13/4.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.13/4.47	
11.13/4.47	
11.13/4.47	H-Termination with start terms of the given HASKELL could be proven:
11.13/4.47	
11.13/4.47	(0) HASKELL
11.13/4.47	(1) BR [EQUIVALENT, 0 ms]
11.13/4.47	(2) HASKELL
11.13/4.47	(3) COR [EQUIVALENT, 0 ms]
11.13/4.47	(4) HASKELL
11.13/4.47	(5) Narrow [SOUND, 0 ms]
11.13/4.47	(6) AND
11.13/4.47	    (7) QDP
11.13/4.47	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.13/4.47	        (9) YES
11.13/4.47	    (10) QDP
11.13/4.47	        (11) DependencyGraphProof [EQUIVALENT, 0 ms]
11.13/4.47	        (12) AND
11.13/4.47	            (13) QDP
11.13/4.47	                (14) QDPSizeChangeProof [EQUIVALENT, 23 ms]
11.13/4.47	                (15) YES
11.13/4.47	            (16) QDP
11.13/4.47	                (17) MRRProof [EQUIVALENT, 0 ms]
11.13/4.47	                (18) QDP
11.13/4.47	                (19) PisEmptyProof [EQUIVALENT, 0 ms]
11.13/4.47	                (20) YES
11.13/4.47	
11.13/4.47	
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(0)
11.13/4.47	Obligation:
11.13/4.47	mainModule Main 
11.13/4.47	module Main where {
11.13/4.47	import qualified Prelude;
11.13/4.47	data Float = Float MyInt MyInt ;
11.13/4.47	
11.13/4.47	data MyBool = MyTrue  | MyFalse ;
11.13/4.47	
11.13/4.47	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
11.13/4.47	
11.13/4.47	data Main.Nat = Succ Main.Nat  | Zero ;
11.13/4.47	
11.13/4.47	data Tup2 a b = Tup2 a b ;
11.13/4.47	
11.13/4.47	error :: a;
11.13/4.47	error = stop MyTrue;
11.13/4.47	
11.13/4.47	floatProperFractionFloat (Float wy wz) = Tup2 (fromIntMyInt (quotMyInt wy wz)) (msFloat (Float wy wz) (fromIntFloat (quotMyInt wy wz)));
11.13/4.47	
11.13/4.47	fromEnumFloat :: Float  ->  MyInt;
11.13/4.47	fromEnumFloat = truncateFloat;
11.13/4.47	
11.13/4.47	fromIntFloat :: MyInt  ->  Float;
11.13/4.47	fromIntFloat = primIntToFloat;
11.13/4.47	
11.13/4.47	fromIntMyInt :: MyInt  ->  MyInt;
11.13/4.47	fromIntMyInt x = x;
11.13/4.47	
11.13/4.47	msFloat :: Float  ->  Float  ->  Float;
11.13/4.47	msFloat = primMinusFloat;
11.13/4.47	
11.13/4.47	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	msMyInt = primMinusInt;
11.13/4.47	
11.13/4.47	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.13/4.47	primDivNatS Main.Zero Main.Zero = Main.error;
11.13/4.47	primDivNatS (Main.Succ x) Main.Zero = Main.error;
11.13/4.47	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
11.13/4.47	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
11.13/4.47	
11.13/4.47	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
11.13/4.47	primDivNatS0 x y MyFalse = Main.Zero;
11.13/4.47	
11.13/4.47	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
11.13/4.47	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
11.13/4.47	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
11.13/4.47	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
11.13/4.47	primGEqNatS Main.Zero Main.Zero = MyTrue;
11.13/4.47	
11.13/4.47	primIntToFloat :: MyInt  ->  Float;
11.13/4.47	primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));
11.13/4.47	
11.13/4.47	primMinusFloat :: Float  ->  Float  ->  Float;
11.13/4.47	primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);
11.13/4.47	
11.13/4.47	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
11.13/4.47	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
11.13/4.47	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
11.13/4.47	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
11.13/4.47	
11.13/4.47	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
11.13/4.47	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
11.13/4.47	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
11.13/4.47	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
11.13/4.47	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
11.13/4.47	
11.13/4.47	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.13/4.47	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
11.13/4.47	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
11.13/4.47	primMinusNatS x Main.Zero = x;
11.13/4.47	
11.13/4.47	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
11.13/4.47	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
11.13/4.47	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
11.13/4.47	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
11.13/4.47	
11.13/4.47	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.13/4.47	primMulNat Main.Zero Main.Zero = Main.Zero;
11.13/4.47	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
11.13/4.47	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
11.13/4.47	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
11.13/4.47	
11.13/4.47	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.13/4.47	primPlusNat Main.Zero Main.Zero = Main.Zero;
11.13/4.47	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
11.13/4.47	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
11.13/4.47	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
11.13/4.47	
11.13/4.47	primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.13/4.47	primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.13/4.47	primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.13/4.47	primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.13/4.47	primQuotInt ww wx = Main.error;
11.13/4.47	
11.13/4.47	properFractionFloat :: Float  ->  Tup2 MyInt Float;
11.13/4.47	properFractionFloat = floatProperFractionFloat;
11.13/4.47	
11.13/4.47	quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	quotMyInt = primQuotInt;
11.13/4.47	
11.13/4.47	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	srMyInt = primMulInt;
11.13/4.47	
11.13/4.47	stop :: MyBool  ->  a;
11.13/4.47	stop MyFalse = stop MyFalse;
11.13/4.47	
11.13/4.47	truncateFloat :: Float  ->  MyInt;
11.13/4.47	truncateFloat x = truncateM x;
11.13/4.47	
11.13/4.47	truncateM xu = truncateM0 xu (truncateVu6 xu);
11.13/4.47	
11.13/4.47	truncateM0 xu (Tup2 m vv) = m;
11.13/4.47	
11.13/4.47	truncateVu6 xu = properFractionFloat xu;
11.13/4.47	
11.13/4.47	}
11.13/4.47	
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(1) BR (EQUIVALENT)
11.13/4.47	Replaced joker patterns by fresh variables and removed binding patterns.
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(2)
11.13/4.47	Obligation:
11.13/4.47	mainModule Main 
11.13/4.47	module Main where {
11.13/4.47	import qualified Prelude;
11.13/4.47	data Float = Float MyInt MyInt ;
11.13/4.47	
11.13/4.47	data MyBool = MyTrue  | MyFalse ;
11.13/4.47	
11.13/4.47	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
11.13/4.47	
11.13/4.47	data Main.Nat = Succ Main.Nat  | Zero ;
11.13/4.47	
11.13/4.47	data Tup2 a b = Tup2 a b ;
11.13/4.47	
11.13/4.47	error :: a;
11.13/4.47	error = stop MyTrue;
11.13/4.47	
11.13/4.47	floatProperFractionFloat (Float wy wz) = Tup2 (fromIntMyInt (quotMyInt wy wz)) (msFloat (Float wy wz) (fromIntFloat (quotMyInt wy wz)));
11.13/4.47	
11.13/4.47	fromEnumFloat :: Float  ->  MyInt;
11.13/4.47	fromEnumFloat = truncateFloat;
11.13/4.47	
11.13/4.47	fromIntFloat :: MyInt  ->  Float;
11.13/4.47	fromIntFloat = primIntToFloat;
11.13/4.47	
11.13/4.47	fromIntMyInt :: MyInt  ->  MyInt;
11.13/4.47	fromIntMyInt x = x;
11.13/4.47	
11.13/4.47	msFloat :: Float  ->  Float  ->  Float;
11.13/4.47	msFloat = primMinusFloat;
11.13/4.47	
11.13/4.47	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	msMyInt = primMinusInt;
11.13/4.47	
11.13/4.47	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.13/4.47	primDivNatS Main.Zero Main.Zero = Main.error;
11.13/4.47	primDivNatS (Main.Succ x) Main.Zero = Main.error;
11.13/4.47	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
11.13/4.47	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
11.13/4.47	
11.13/4.47	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
11.13/4.47	primDivNatS0 x y MyFalse = Main.Zero;
11.13/4.47	
11.13/4.47	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
11.13/4.47	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
11.13/4.47	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
11.13/4.47	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
11.13/4.47	primGEqNatS Main.Zero Main.Zero = MyTrue;
11.13/4.47	
11.13/4.47	primIntToFloat :: MyInt  ->  Float;
11.13/4.47	primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));
11.13/4.47	
11.13/4.47	primMinusFloat :: Float  ->  Float  ->  Float;
11.13/4.47	primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);
11.13/4.47	
11.13/4.47	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
11.13/4.47	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
11.13/4.47	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
11.13/4.47	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
11.13/4.47	
11.13/4.47	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
11.13/4.47	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
11.13/4.47	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
11.13/4.47	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
11.13/4.47	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
11.13/4.47	
11.13/4.47	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.13/4.47	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
11.13/4.47	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
11.13/4.47	primMinusNatS x Main.Zero = x;
11.13/4.47	
11.13/4.47	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
11.13/4.47	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
11.13/4.47	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
11.13/4.47	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
11.13/4.47	
11.13/4.47	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.13/4.47	primMulNat Main.Zero Main.Zero = Main.Zero;
11.13/4.47	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
11.13/4.47	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
11.13/4.47	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
11.13/4.47	
11.13/4.47	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.13/4.47	primPlusNat Main.Zero Main.Zero = Main.Zero;
11.13/4.47	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
11.13/4.47	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
11.13/4.47	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
11.13/4.47	
11.13/4.47	primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.13/4.47	primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.13/4.47	primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.13/4.47	primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.13/4.47	primQuotInt ww wx = Main.error;
11.13/4.47	
11.13/4.47	properFractionFloat :: Float  ->  Tup2 MyInt Float;
11.13/4.47	properFractionFloat = floatProperFractionFloat;
11.13/4.47	
11.13/4.47	quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	quotMyInt = primQuotInt;
11.13/4.47	
11.13/4.47	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	srMyInt = primMulInt;
11.13/4.47	
11.13/4.47	stop :: MyBool  ->  a;
11.13/4.47	stop MyFalse = stop MyFalse;
11.13/4.47	
11.13/4.47	truncateFloat :: Float  ->  MyInt;
11.13/4.47	truncateFloat x = truncateM x;
11.13/4.47	
11.13/4.47	truncateM xu = truncateM0 xu (truncateVu6 xu);
11.13/4.47	
11.13/4.47	truncateM0 xu (Tup2 m vv) = m;
11.13/4.47	
11.13/4.47	truncateVu6 xu = properFractionFloat xu;
11.13/4.47	
11.13/4.47	}
11.13/4.47	
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(3) COR (EQUIVALENT)
11.13/4.47	Cond Reductions:
11.13/4.47	The following Function with conditions
11.13/4.47	"undefined |Falseundefined;
11.13/4.47	"
11.13/4.47	is transformed to
11.13/4.47	"undefined  = undefined1;
11.13/4.47	"
11.13/4.47	"undefined0 True = undefined;
11.13/4.47	"
11.13/4.47	"undefined1  = undefined0 False;
11.13/4.47	"
11.13/4.47	
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(4)
11.13/4.47	Obligation:
11.13/4.47	mainModule Main 
11.13/4.47	module Main where {
11.13/4.47	import qualified Prelude;
11.13/4.47	data Float = Float MyInt MyInt ;
11.13/4.47	
11.13/4.47	data MyBool = MyTrue  | MyFalse ;
11.13/4.47	
11.13/4.47	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
11.13/4.47	
11.13/4.47	data Main.Nat = Succ Main.Nat  | Zero ;
11.13/4.47	
11.13/4.47	data Tup2 b a = Tup2 b a ;
11.13/4.47	
11.13/4.47	error :: a;
11.13/4.47	error = stop MyTrue;
11.13/4.47	
11.13/4.47	floatProperFractionFloat (Float wy wz) = Tup2 (fromIntMyInt (quotMyInt wy wz)) (msFloat (Float wy wz) (fromIntFloat (quotMyInt wy wz)));
11.13/4.47	
11.13/4.47	fromEnumFloat :: Float  ->  MyInt;
11.13/4.47	fromEnumFloat = truncateFloat;
11.13/4.47	
11.13/4.47	fromIntFloat :: MyInt  ->  Float;
11.13/4.47	fromIntFloat = primIntToFloat;
11.13/4.47	
11.13/4.47	fromIntMyInt :: MyInt  ->  MyInt;
11.13/4.47	fromIntMyInt x = x;
11.13/4.47	
11.13/4.47	msFloat :: Float  ->  Float  ->  Float;
11.13/4.47	msFloat = primMinusFloat;
11.13/4.47	
11.13/4.47	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	msMyInt = primMinusInt;
11.13/4.47	
11.13/4.47	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.13/4.47	primDivNatS Main.Zero Main.Zero = Main.error;
11.13/4.47	primDivNatS (Main.Succ x) Main.Zero = Main.error;
11.13/4.47	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
11.13/4.47	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
11.13/4.47	
11.13/4.47	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
11.13/4.47	primDivNatS0 x y MyFalse = Main.Zero;
11.13/4.47	
11.13/4.47	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
11.13/4.47	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
11.13/4.47	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
11.13/4.47	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
11.13/4.47	primGEqNatS Main.Zero Main.Zero = MyTrue;
11.13/4.47	
11.13/4.47	primIntToFloat :: MyInt  ->  Float;
11.13/4.47	primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));
11.13/4.47	
11.13/4.47	primMinusFloat :: Float  ->  Float  ->  Float;
11.13/4.47	primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);
11.13/4.47	
11.13/4.47	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
11.13/4.47	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
11.13/4.47	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
11.13/4.47	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
11.13/4.47	
11.13/4.47	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
11.13/4.47	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
11.13/4.47	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
11.13/4.47	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
11.13/4.47	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
11.13/4.47	
11.13/4.47	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.13/4.47	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
11.13/4.47	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
11.13/4.47	primMinusNatS x Main.Zero = x;
11.13/4.47	
11.13/4.47	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
11.13/4.47	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
11.13/4.47	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
11.13/4.47	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
11.13/4.47	
11.13/4.47	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.13/4.47	primMulNat Main.Zero Main.Zero = Main.Zero;
11.13/4.47	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
11.13/4.47	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
11.13/4.47	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
11.13/4.47	
11.13/4.47	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.13/4.47	primPlusNat Main.Zero Main.Zero = Main.Zero;
11.13/4.47	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
11.13/4.47	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
11.13/4.47	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
11.13/4.47	
11.13/4.47	primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.13/4.47	primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.13/4.47	primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.13/4.47	primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.13/4.47	primQuotInt ww wx = Main.error;
11.13/4.47	
11.13/4.47	properFractionFloat :: Float  ->  Tup2 MyInt Float;
11.13/4.47	properFractionFloat = floatProperFractionFloat;
11.13/4.47	
11.13/4.47	quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	quotMyInt = primQuotInt;
11.13/4.47	
11.13/4.47	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.13/4.47	srMyInt = primMulInt;
11.13/4.47	
11.13/4.47	stop :: MyBool  ->  a;
11.13/4.47	stop MyFalse = stop MyFalse;
11.13/4.47	
11.13/4.47	truncateFloat :: Float  ->  MyInt;
11.13/4.47	truncateFloat x = truncateM x;
11.13/4.47	
11.13/4.47	truncateM xu = truncateM0 xu (truncateVu6 xu);
11.13/4.47	
11.13/4.47	truncateM0 xu (Tup2 m vv) = m;
11.13/4.47	
11.13/4.47	truncateVu6 xu = properFractionFloat xu;
11.13/4.47	
11.13/4.47	}
11.13/4.47	
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(5) Narrow (SOUND)
11.13/4.47	Haskell To QDPs
11.13/4.47	
11.13/4.47	digraph dp_graph {
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11.13/4.47	4[label="truncateFloat vy3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
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11.13/4.47	31 -> 29[label="",style="dashed", color="red", weight=0];
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11.13/4.47	33 -> 29[label="",style="dashed", color="red", weight=0];
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11.13/4.47	298 -> 41[label="",style="solid", color="burlywood", weight=3];
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11.13/4.47	43[label="error []",fontsize=16,color="red",shape="box"];44[label="vy3100",fontsize=16,color="green",shape="box"];45[label="vy300",fontsize=16,color="green",shape="box"];46[label="vy300",fontsize=16,color="green",shape="box"];47[label="vy3100",fontsize=16,color="green",shape="box"];48[label="primDivNatS0 vy3000 vy3100 (primGEqNatS vy3000 vy3100)",fontsize=16,color="burlywood",shape="box"];300[label="vy3000/Succ vy30000",fontsize=10,color="white",style="solid",shape="box"];48 -> 300[label="",style="solid", color="burlywood", weight=9];
11.13/4.47	300 -> 50[label="",style="solid", color="burlywood", weight=3];
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11.13/4.47	302 -> 52[label="",style="solid", color="burlywood", weight=3];
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11.13/4.47	257[label="primDivNatS0 (Succ vy20) (Succ vy21) (primGEqNatS Zero (Succ vy230))",fontsize=16,color="black",shape="box"];257 -> 261[label="",style="solid", color="black", weight=3];
11.13/4.47	258[label="primDivNatS0 (Succ vy20) (Succ vy21) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];258 -> 262[label="",style="solid", color="black", weight=3];
11.13/4.47	75[label="primMinusNatS (Succ vy30000) Zero",fontsize=16,color="black",shape="triangle"];75 -> 84[label="",style="solid", color="black", weight=3];
11.13/4.47	76[label="Zero",fontsize=16,color="green",shape="box"];77[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];77 -> 85[label="",style="solid", color="black", weight=3];
11.13/4.47	78[label="Zero",fontsize=16,color="green",shape="box"];259 -> 220[label="",style="dashed", color="red", weight=0];
11.13/4.47	259[label="primDivNatS0 (Succ vy20) (Succ vy21) (primGEqNatS vy220 vy230)",fontsize=16,color="magenta"];259 -> 263[label="",style="dashed", color="magenta", weight=3];
11.13/4.47	259 -> 264[label="",style="dashed", color="magenta", weight=3];
11.13/4.47	260[label="primDivNatS0 (Succ vy20) (Succ vy21) MyTrue",fontsize=16,color="black",shape="triangle"];260 -> 265[label="",style="solid", color="black", weight=3];
11.13/4.47	261[label="primDivNatS0 (Succ vy20) (Succ vy21) MyFalse",fontsize=16,color="black",shape="box"];261 -> 266[label="",style="solid", color="black", weight=3];
11.13/4.47	262 -> 260[label="",style="dashed", color="red", weight=0];
11.13/4.47	262[label="primDivNatS0 (Succ vy20) (Succ vy21) MyTrue",fontsize=16,color="magenta"];84[label="Succ vy30000",fontsize=16,color="green",shape="box"];85[label="Zero",fontsize=16,color="green",shape="box"];263[label="vy220",fontsize=16,color="green",shape="box"];264[label="vy230",fontsize=16,color="green",shape="box"];265[label="Succ (primDivNatS (primMinusNatS (Succ vy20) (Succ vy21)) (Succ (Succ vy21)))",fontsize=16,color="green",shape="box"];265 -> 267[label="",style="dashed", color="green", weight=3];
11.13/4.47	266[label="Zero",fontsize=16,color="green",shape="box"];267 -> 36[label="",style="dashed", color="red", weight=0];
11.13/4.47	267[label="primDivNatS (primMinusNatS (Succ vy20) (Succ vy21)) (Succ (Succ vy21))",fontsize=16,color="magenta"];267 -> 268[label="",style="dashed", color="magenta", weight=3];
11.13/4.47	267 -> 269[label="",style="dashed", color="magenta", weight=3];
11.13/4.47	268[label="primMinusNatS (Succ vy20) (Succ vy21)",fontsize=16,color="black",shape="box"];268 -> 270[label="",style="solid", color="black", weight=3];
11.13/4.47	269[label="Succ vy21",fontsize=16,color="green",shape="box"];270[label="primMinusNatS vy20 vy21",fontsize=16,color="burlywood",shape="triangle"];312[label="vy20/Succ vy200",fontsize=10,color="white",style="solid",shape="box"];270 -> 312[label="",style="solid", color="burlywood", weight=9];
11.13/4.47	312 -> 271[label="",style="solid", color="burlywood", weight=3];
11.13/4.47	313[label="vy20/Zero",fontsize=10,color="white",style="solid",shape="box"];270 -> 313[label="",style="solid", color="burlywood", weight=9];
11.13/4.47	313 -> 272[label="",style="solid", color="burlywood", weight=3];
11.13/4.47	271[label="primMinusNatS (Succ vy200) vy21",fontsize=16,color="burlywood",shape="box"];314[label="vy21/Succ vy210",fontsize=10,color="white",style="solid",shape="box"];271 -> 314[label="",style="solid", color="burlywood", weight=9];
11.13/4.47	314 -> 273[label="",style="solid", color="burlywood", weight=3];
11.13/4.47	315[label="vy21/Zero",fontsize=10,color="white",style="solid",shape="box"];271 -> 315[label="",style="solid", color="burlywood", weight=9];
11.13/4.47	315 -> 274[label="",style="solid", color="burlywood", weight=3];
11.13/4.47	272[label="primMinusNatS Zero vy21",fontsize=16,color="burlywood",shape="box"];316[label="vy21/Succ vy210",fontsize=10,color="white",style="solid",shape="box"];272 -> 316[label="",style="solid", color="burlywood", weight=9];
11.13/4.47	316 -> 275[label="",style="solid", color="burlywood", weight=3];
11.13/4.47	317[label="vy21/Zero",fontsize=10,color="white",style="solid",shape="box"];272 -> 317[label="",style="solid", color="burlywood", weight=9];
11.13/4.47	317 -> 276[label="",style="solid", color="burlywood", weight=3];
11.13/4.47	273[label="primMinusNatS (Succ vy200) (Succ vy210)",fontsize=16,color="black",shape="box"];273 -> 277[label="",style="solid", color="black", weight=3];
11.13/4.47	274[label="primMinusNatS (Succ vy200) Zero",fontsize=16,color="black",shape="box"];274 -> 278[label="",style="solid", color="black", weight=3];
11.13/4.47	275[label="primMinusNatS Zero (Succ vy210)",fontsize=16,color="black",shape="box"];275 -> 279[label="",style="solid", color="black", weight=3];
11.13/4.47	276[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];276 -> 280[label="",style="solid", color="black", weight=3];
11.13/4.47	277 -> 270[label="",style="dashed", color="red", weight=0];
11.13/4.47	277[label="primMinusNatS vy200 vy210",fontsize=16,color="magenta"];277 -> 281[label="",style="dashed", color="magenta", weight=3];
11.13/4.47	277 -> 282[label="",style="dashed", color="magenta", weight=3];
11.13/4.47	278[label="Succ vy200",fontsize=16,color="green",shape="box"];279[label="Zero",fontsize=16,color="green",shape="box"];280[label="Zero",fontsize=16,color="green",shape="box"];281[label="vy200",fontsize=16,color="green",shape="box"];282[label="vy210",fontsize=16,color="green",shape="box"];}
11.13/4.47	
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(6)
11.13/4.47	Complex Obligation (AND)
11.13/4.47	
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(7)
11.13/4.47	Obligation:
11.13/4.47	Q DP problem:
11.13/4.47	The TRS P consists of the following rules:
11.13/4.47	
11.13/4.47	   new_primMinusNatS(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS(vy200, vy210)
11.13/4.47	
11.13/4.47	R is empty.
11.13/4.47	Q is empty.
11.13/4.47	We have to consider all minimal (P,Q,R)-chains.
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(8) QDPSizeChangeProof (EQUIVALENT)
11.13/4.47	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. 
11.13/4.47	
11.13/4.47	From the DPs we obtained the following set of size-change graphs:
11.13/4.47	*new_primMinusNatS(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS(vy200, vy210)
11.13/4.47	The graph contains the following edges 1 > 1, 2 > 2
11.13/4.47	
11.13/4.47	
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(9)
11.13/4.47	YES
11.13/4.47	
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(10)
11.13/4.47	Obligation:
11.13/4.47	Q DP problem:
11.13/4.47	The TRS P consists of the following rules:
11.13/4.47	
11.13/4.47	   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vy30000), Main.Zero)
11.13/4.47	   new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Succ(vy230)) -> new_primDivNatS0(vy20, vy21, vy220, vy230)
11.13/4.47	   new_primDivNatS00(vy20, vy21) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21))
11.13/4.47	   new_primDivNatS0(vy20, vy21, Main.Zero, Main.Zero) -> new_primDivNatS00(vy20, vy21)
11.13/4.47	   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Succ(vy31000)) -> new_primDivNatS0(vy30000, vy31000, vy30000, vy31000)
11.13/4.47	   new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21))
11.13/4.47	   new_primDivNatS(Main.Succ(Main.Zero), Main.Zero) -> new_primDivNatS(new_primMinusNatS2, Main.Zero)
11.13/4.47	
11.13/4.47	The TRS R consists of the following rules:
11.13/4.47	
11.13/4.47	   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
11.13/4.47	   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
11.13/4.47	   new_primMinusNatS2 -> Main.Zero
11.13/4.47	   new_primMinusNatS1(vy30000) -> Main.Succ(vy30000)
11.13/4.47	
11.13/4.47	The set Q consists of the following terms:
11.13/4.47	
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.13/4.47	   new_primMinusNatS2
11.13/4.47	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.13/4.47	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.13/4.47	   new_primMinusNatS1(x0)
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.13/4.47	
11.13/4.47	We have to consider all minimal (P,Q,R)-chains.
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(11) DependencyGraphProof (EQUIVALENT)
11.13/4.47	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(12)
11.13/4.47	Complex Obligation (AND)
11.13/4.47	
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(13)
11.13/4.47	Obligation:
11.13/4.47	Q DP problem:
11.13/4.47	The TRS P consists of the following rules:
11.13/4.47	
11.13/4.47	   new_primDivNatS0(vy20, vy21, Main.Zero, Main.Zero) -> new_primDivNatS00(vy20, vy21)
11.13/4.47	   new_primDivNatS00(vy20, vy21) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21))
11.13/4.47	   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Succ(vy31000)) -> new_primDivNatS0(vy30000, vy31000, vy30000, vy31000)
11.13/4.47	   new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Succ(vy230)) -> new_primDivNatS0(vy20, vy21, vy220, vy230)
11.13/4.47	   new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21))
11.13/4.47	
11.13/4.47	The TRS R consists of the following rules:
11.13/4.47	
11.13/4.47	   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
11.13/4.47	   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
11.13/4.47	   new_primMinusNatS2 -> Main.Zero
11.13/4.47	   new_primMinusNatS1(vy30000) -> Main.Succ(vy30000)
11.13/4.47	
11.13/4.47	The set Q consists of the following terms:
11.13/4.47	
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.13/4.47	   new_primMinusNatS2
11.13/4.47	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.13/4.47	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.13/4.47	   new_primMinusNatS1(x0)
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.13/4.47	
11.13/4.47	We have to consider all minimal (P,Q,R)-chains.
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(14) QDPSizeChangeProof (EQUIVALENT)
11.13/4.47	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
11.13/4.47	
11.13/4.47	Order:Polynomial interpretation [POLO]:
11.13/4.47	
11.13/4.47	   POL(Main.Succ(x_1)) = 1 + x_1
11.13/4.47	   POL(Main.Zero) = 1
11.13/4.47	   POL(new_primMinusNatS0(x_1, x_2)) = x_1
11.13/4.47	
11.13/4.47	
11.13/4.47	
11.13/4.47	
11.13/4.47	From the DPs we obtained the following set of size-change graphs:
11.13/4.47	*new_primDivNatS00(vy20, vy21) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21)) (allowed arguments on rhs = {1, 2})
11.13/4.47	The graph contains the following edges 1 >= 1
11.13/4.47	
11.13/4.47	
11.13/4.47	*new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Succ(vy31000)) -> new_primDivNatS0(vy30000, vy31000, vy30000, vy31000) (allowed arguments on rhs = {1, 2, 3, 4})
11.13/4.47	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
11.13/4.47	
11.13/4.47	
11.13/4.47	*new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Succ(vy230)) -> new_primDivNatS0(vy20, vy21, vy220, vy230) (allowed arguments on rhs = {1, 2, 3, 4})
11.13/4.47	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
11.13/4.47	
11.13/4.47	
11.13/4.47	*new_primDivNatS0(vy20, vy21, Main.Zero, Main.Zero) -> new_primDivNatS00(vy20, vy21) (allowed arguments on rhs = {1, 2})
11.13/4.47	The graph contains the following edges 1 >= 1, 2 >= 2
11.13/4.47	
11.13/4.47	
11.13/4.47	*new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21)) (allowed arguments on rhs = {1, 2})
11.13/4.47	The graph contains the following edges 1 >= 1
11.13/4.47	
11.13/4.47	
11.13/4.47	
11.13/4.47	We oriented the following set of usable rules [AAECC05,FROCOS05].
11.13/4.47	
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
11.13/4.47	   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
11.13/4.47	   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
11.13/4.47	
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(15)
11.13/4.47	YES
11.13/4.47	
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(16)
11.13/4.47	Obligation:
11.13/4.47	Q DP problem:
11.13/4.47	The TRS P consists of the following rules:
11.13/4.47	
11.13/4.47	   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vy30000), Main.Zero)
11.13/4.47	
11.13/4.47	The TRS R consists of the following rules:
11.13/4.47	
11.13/4.47	   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
11.13/4.47	   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
11.13/4.47	   new_primMinusNatS2 -> Main.Zero
11.13/4.47	   new_primMinusNatS1(vy30000) -> Main.Succ(vy30000)
11.13/4.47	
11.13/4.47	The set Q consists of the following terms:
11.13/4.47	
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.13/4.47	   new_primMinusNatS2
11.13/4.47	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.13/4.47	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.13/4.47	   new_primMinusNatS1(x0)
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.13/4.47	
11.13/4.47	We have to consider all minimal (P,Q,R)-chains.
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(17) MRRProof (EQUIVALENT)
11.13/4.47	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
11.13/4.47	
11.13/4.47	Strictly oriented dependency pairs:
11.13/4.47	
11.13/4.47	   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vy30000), Main.Zero)
11.13/4.47	
11.13/4.47	Strictly oriented rules of the TRS R:
11.13/4.47	
11.13/4.47	   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
11.13/4.47	   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
11.13/4.47	
11.13/4.47	Used ordering: Polynomial interpretation [POLO]:
11.13/4.47	
11.13/4.47	   POL(Main.Succ(x_1)) = 1 + x_1
11.13/4.47	   POL(Main.Zero) = 2
11.13/4.47	   POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2
11.13/4.47	   POL(new_primMinusNatS0(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
11.13/4.47	   POL(new_primMinusNatS1(x_1)) = 1 + x_1
11.13/4.47	   POL(new_primMinusNatS2) = 2
11.13/4.47	
11.13/4.47	
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(18)
11.13/4.47	Obligation:
11.13/4.47	Q DP problem:
11.13/4.47	P is empty.
11.13/4.47	The TRS R consists of the following rules:
11.13/4.47	
11.13/4.47	   new_primMinusNatS2 -> Main.Zero
11.13/4.47	   new_primMinusNatS1(vy30000) -> Main.Succ(vy30000)
11.13/4.47	
11.13/4.47	The set Q consists of the following terms:
11.13/4.47	
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.13/4.47	   new_primMinusNatS2
11.13/4.47	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.13/4.47	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.13/4.47	   new_primMinusNatS1(x0)
11.13/4.47	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.13/4.47	
11.13/4.47	We have to consider all minimal (P,Q,R)-chains.
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(19) PisEmptyProof (EQUIVALENT)
11.13/4.47	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.13/4.47	----------------------------------------
11.13/4.47	
11.13/4.47	(20)
11.13/4.47	YES
11.27/5.90	EOF