8.74/3.90	YES
11.08/4.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.08/4.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.08/4.53	
11.08/4.53	
11.08/4.53	H-Termination with start terms of the given HASKELL could be proven:
11.08/4.53	
11.08/4.53	(0) HASKELL
11.08/4.53	(1) BR [EQUIVALENT, 0 ms]
11.08/4.53	(2) HASKELL
11.08/4.53	(3) COR [EQUIVALENT, 0 ms]
11.08/4.53	(4) HASKELL
11.08/4.53	(5) Narrow [SOUND, 0 ms]
11.08/4.53	(6) AND
11.08/4.53	    (7) QDP
11.08/4.53	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.08/4.53	        (9) YES
11.08/4.53	    (10) QDP
11.08/4.53	        (11) DependencyGraphProof [EQUIVALENT, 0 ms]
11.08/4.53	        (12) AND
11.08/4.53	            (13) QDP
11.08/4.53	                (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.08/4.53	                (15) YES
11.08/4.53	            (16) QDP
11.08/4.53	                (17) MRRProof [EQUIVALENT, 2 ms]
11.08/4.53	                (18) QDP
11.08/4.53	                (19) PisEmptyProof [EQUIVALENT, 0 ms]
11.08/4.53	                (20) YES
11.08/4.53	
11.08/4.53	
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(0)
11.08/4.53	Obligation:
11.08/4.53	mainModule Main 
11.08/4.53	module Main where {
11.08/4.53	import qualified Prelude;
11.08/4.53	data Float = Float MyInt MyInt ;
11.08/4.53	
11.08/4.53	data MyBool = MyTrue  | MyFalse ;
11.08/4.53	
11.08/4.53	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
11.08/4.53	
11.08/4.53	data Main.Nat = Succ Main.Nat  | Zero ;
11.08/4.53	
11.08/4.53	data Tup2 b a = Tup2 b a ;
11.08/4.53	
11.08/4.53	error :: a;
11.08/4.53	error = stop MyTrue;
11.08/4.53	
11.08/4.53	floatProperFractionFloat (Float wy wz) = Tup2 (fromIntMyInt (quotMyInt wy wz)) (msFloat (Float wy wz) (fromIntFloat (quotMyInt wy wz)));
11.08/4.53	
11.08/4.53	fromIntFloat :: MyInt  ->  Float;
11.08/4.53	fromIntFloat = primIntToFloat;
11.08/4.53	
11.08/4.53	fromIntMyInt :: MyInt  ->  MyInt;
11.08/4.53	fromIntMyInt x = x;
11.08/4.53	
11.08/4.53	msFloat :: Float  ->  Float  ->  Float;
11.08/4.53	msFloat = primMinusFloat;
11.08/4.53	
11.08/4.53	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	msMyInt = primMinusInt;
11.08/4.53	
11.08/4.53	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.08/4.53	primDivNatS Main.Zero Main.Zero = Main.error;
11.08/4.53	primDivNatS (Main.Succ x) Main.Zero = Main.error;
11.08/4.53	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
11.08/4.53	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
11.08/4.53	
11.08/4.53	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
11.08/4.53	primDivNatS0 x y MyFalse = Main.Zero;
11.08/4.53	
11.08/4.53	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
11.08/4.53	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
11.08/4.53	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
11.08/4.53	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
11.08/4.53	primGEqNatS Main.Zero Main.Zero = MyTrue;
11.08/4.53	
11.08/4.53	primIntToFloat :: MyInt  ->  Float;
11.08/4.53	primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));
11.08/4.53	
11.08/4.53	primMinusFloat :: Float  ->  Float  ->  Float;
11.08/4.53	primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);
11.08/4.53	
11.08/4.53	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
11.08/4.53	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
11.08/4.53	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
11.08/4.53	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
11.08/4.53	
11.08/4.53	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
11.08/4.53	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
11.08/4.53	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
11.08/4.53	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
11.08/4.53	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
11.08/4.53	
11.08/4.53	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.08/4.53	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
11.08/4.53	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
11.08/4.53	primMinusNatS x Main.Zero = x;
11.08/4.53	
11.08/4.53	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
11.08/4.53	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
11.08/4.53	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
11.08/4.53	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
11.08/4.53	
11.08/4.53	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.08/4.53	primMulNat Main.Zero Main.Zero = Main.Zero;
11.08/4.53	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
11.08/4.53	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
11.08/4.53	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
11.08/4.53	
11.08/4.53	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.08/4.53	primPlusNat Main.Zero Main.Zero = Main.Zero;
11.08/4.53	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
11.08/4.53	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
11.08/4.53	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
11.08/4.53	
11.08/4.53	primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.08/4.53	primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.08/4.53	primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.08/4.53	primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.08/4.53	primQuotInt ww wx = Main.error;
11.08/4.53	
11.08/4.53	properFractionFloat :: Float  ->  Tup2 MyInt Float;
11.08/4.53	properFractionFloat = floatProperFractionFloat;
11.08/4.53	
11.08/4.53	quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	quotMyInt = primQuotInt;
11.08/4.53	
11.08/4.53	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	srMyInt = primMulInt;
11.08/4.53	
11.08/4.53	stop :: MyBool  ->  a;
11.08/4.53	stop MyFalse = stop MyFalse;
11.08/4.53	
11.08/4.53	truncateFloat :: Float  ->  MyInt;
11.08/4.53	truncateFloat x = truncateM x;
11.08/4.53	
11.08/4.53	truncateM xu = truncateM0 xu (truncateVu6 xu);
11.08/4.53	
11.08/4.53	truncateM0 xu (Tup2 m vv) = m;
11.08/4.53	
11.08/4.53	truncateVu6 xu = properFractionFloat xu;
11.08/4.53	
11.08/4.53	}
11.08/4.53	
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(1) BR (EQUIVALENT)
11.08/4.53	Replaced joker patterns by fresh variables and removed binding patterns.
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(2)
11.08/4.53	Obligation:
11.08/4.53	mainModule Main 
11.08/4.53	module Main where {
11.08/4.53	import qualified Prelude;
11.08/4.53	data Float = Float MyInt MyInt ;
11.08/4.53	
11.08/4.53	data MyBool = MyTrue  | MyFalse ;
11.08/4.53	
11.08/4.53	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
11.08/4.53	
11.08/4.53	data Main.Nat = Succ Main.Nat  | Zero ;
11.08/4.53	
11.08/4.53	data Tup2 a b = Tup2 a b ;
11.08/4.53	
11.08/4.53	error :: a;
11.08/4.53	error = stop MyTrue;
11.08/4.53	
11.08/4.53	floatProperFractionFloat (Float wy wz) = Tup2 (fromIntMyInt (quotMyInt wy wz)) (msFloat (Float wy wz) (fromIntFloat (quotMyInt wy wz)));
11.08/4.53	
11.08/4.53	fromIntFloat :: MyInt  ->  Float;
11.08/4.53	fromIntFloat = primIntToFloat;
11.08/4.53	
11.08/4.53	fromIntMyInt :: MyInt  ->  MyInt;
11.08/4.53	fromIntMyInt x = x;
11.08/4.53	
11.08/4.53	msFloat :: Float  ->  Float  ->  Float;
11.08/4.53	msFloat = primMinusFloat;
11.08/4.53	
11.08/4.53	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	msMyInt = primMinusInt;
11.08/4.53	
11.08/4.53	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.08/4.53	primDivNatS Main.Zero Main.Zero = Main.error;
11.08/4.53	primDivNatS (Main.Succ x) Main.Zero = Main.error;
11.08/4.53	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
11.08/4.53	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
11.08/4.53	
11.08/4.53	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
11.08/4.53	primDivNatS0 x y MyFalse = Main.Zero;
11.08/4.53	
11.08/4.53	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
11.08/4.53	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
11.08/4.53	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
11.08/4.53	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
11.08/4.53	primGEqNatS Main.Zero Main.Zero = MyTrue;
11.08/4.53	
11.08/4.53	primIntToFloat :: MyInt  ->  Float;
11.08/4.53	primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));
11.08/4.53	
11.08/4.53	primMinusFloat :: Float  ->  Float  ->  Float;
11.08/4.53	primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);
11.08/4.53	
11.08/4.53	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
11.08/4.53	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
11.08/4.53	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
11.08/4.53	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
11.08/4.53	
11.08/4.53	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
11.08/4.53	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
11.08/4.53	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
11.08/4.53	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
11.08/4.53	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
11.08/4.53	
11.08/4.53	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.08/4.53	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
11.08/4.53	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
11.08/4.53	primMinusNatS x Main.Zero = x;
11.08/4.53	
11.08/4.53	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
11.08/4.53	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
11.08/4.53	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
11.08/4.53	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
11.08/4.53	
11.08/4.53	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.08/4.53	primMulNat Main.Zero Main.Zero = Main.Zero;
11.08/4.53	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
11.08/4.53	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
11.08/4.53	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
11.08/4.53	
11.08/4.53	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.08/4.53	primPlusNat Main.Zero Main.Zero = Main.Zero;
11.08/4.53	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
11.08/4.53	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
11.08/4.53	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
11.08/4.53	
11.08/4.53	primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.08/4.53	primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.08/4.53	primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.08/4.53	primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.08/4.53	primQuotInt ww wx = Main.error;
11.08/4.53	
11.08/4.53	properFractionFloat :: Float  ->  Tup2 MyInt Float;
11.08/4.53	properFractionFloat = floatProperFractionFloat;
11.08/4.53	
11.08/4.53	quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	quotMyInt = primQuotInt;
11.08/4.53	
11.08/4.53	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	srMyInt = primMulInt;
11.08/4.53	
11.08/4.53	stop :: MyBool  ->  a;
11.08/4.53	stop MyFalse = stop MyFalse;
11.08/4.53	
11.08/4.53	truncateFloat :: Float  ->  MyInt;
11.08/4.53	truncateFloat x = truncateM x;
11.08/4.53	
11.08/4.53	truncateM xu = truncateM0 xu (truncateVu6 xu);
11.08/4.53	
11.08/4.53	truncateM0 xu (Tup2 m vv) = m;
11.08/4.53	
11.08/4.53	truncateVu6 xu = properFractionFloat xu;
11.08/4.53	
11.08/4.53	}
11.08/4.53	
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(3) COR (EQUIVALENT)
11.08/4.53	Cond Reductions:
11.08/4.53	The following Function with conditions
11.08/4.53	"undefined |Falseundefined;
11.08/4.53	"
11.08/4.53	is transformed to
11.08/4.53	"undefined  = undefined1;
11.08/4.53	"
11.08/4.53	"undefined0 True = undefined;
11.08/4.53	"
11.08/4.53	"undefined1  = undefined0 False;
11.08/4.53	"
11.08/4.53	
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(4)
11.08/4.53	Obligation:
11.08/4.53	mainModule Main 
11.08/4.53	module Main where {
11.08/4.53	import qualified Prelude;
11.08/4.53	data Float = Float MyInt MyInt ;
11.08/4.53	
11.08/4.53	data MyBool = MyTrue  | MyFalse ;
11.08/4.53	
11.08/4.53	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
11.08/4.53	
11.08/4.53	data Main.Nat = Succ Main.Nat  | Zero ;
11.08/4.53	
11.08/4.53	data Tup2 b a = Tup2 b a ;
11.08/4.53	
11.08/4.53	error :: a;
11.08/4.53	error = stop MyTrue;
11.08/4.53	
11.08/4.53	floatProperFractionFloat (Float wy wz) = Tup2 (fromIntMyInt (quotMyInt wy wz)) (msFloat (Float wy wz) (fromIntFloat (quotMyInt wy wz)));
11.08/4.53	
11.08/4.53	fromIntFloat :: MyInt  ->  Float;
11.08/4.53	fromIntFloat = primIntToFloat;
11.08/4.53	
11.08/4.53	fromIntMyInt :: MyInt  ->  MyInt;
11.08/4.53	fromIntMyInt x = x;
11.08/4.53	
11.08/4.53	msFloat :: Float  ->  Float  ->  Float;
11.08/4.53	msFloat = primMinusFloat;
11.08/4.53	
11.08/4.53	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	msMyInt = primMinusInt;
11.08/4.53	
11.08/4.53	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.08/4.53	primDivNatS Main.Zero Main.Zero = Main.error;
11.08/4.53	primDivNatS (Main.Succ x) Main.Zero = Main.error;
11.08/4.53	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
11.08/4.53	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
11.08/4.53	
11.08/4.53	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
11.08/4.53	primDivNatS0 x y MyFalse = Main.Zero;
11.08/4.53	
11.08/4.53	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
11.08/4.53	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
11.08/4.53	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
11.08/4.53	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
11.08/4.53	primGEqNatS Main.Zero Main.Zero = MyTrue;
11.08/4.53	
11.08/4.53	primIntToFloat :: MyInt  ->  Float;
11.08/4.53	primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));
11.08/4.53	
11.08/4.53	primMinusFloat :: Float  ->  Float  ->  Float;
11.08/4.53	primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);
11.08/4.53	
11.08/4.53	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
11.08/4.53	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
11.08/4.53	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
11.08/4.53	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
11.08/4.53	
11.08/4.53	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
11.08/4.53	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
11.08/4.53	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
11.08/4.53	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
11.08/4.53	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
11.08/4.53	
11.08/4.53	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.08/4.53	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
11.08/4.53	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
11.08/4.53	primMinusNatS x Main.Zero = x;
11.08/4.53	
11.08/4.53	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
11.08/4.53	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
11.08/4.53	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
11.08/4.53	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
11.08/4.53	
11.08/4.53	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.08/4.53	primMulNat Main.Zero Main.Zero = Main.Zero;
11.08/4.53	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
11.08/4.53	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
11.08/4.53	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
11.08/4.53	
11.08/4.53	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.08/4.53	primPlusNat Main.Zero Main.Zero = Main.Zero;
11.08/4.53	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
11.08/4.53	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
11.08/4.53	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
11.08/4.53	
11.08/4.53	primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.08/4.53	primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.08/4.53	primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.08/4.53	primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.08/4.53	primQuotInt ww wx = Main.error;
11.08/4.53	
11.08/4.53	properFractionFloat :: Float  ->  Tup2 MyInt Float;
11.08/4.53	properFractionFloat = floatProperFractionFloat;
11.08/4.53	
11.08/4.53	quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	quotMyInt = primQuotInt;
11.08/4.53	
11.08/4.53	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.08/4.53	srMyInt = primMulInt;
11.08/4.53	
11.08/4.53	stop :: MyBool  ->  a;
11.08/4.53	stop MyFalse = stop MyFalse;
11.08/4.53	
11.08/4.53	truncateFloat :: Float  ->  MyInt;
11.08/4.53	truncateFloat x = truncateM x;
11.08/4.53	
11.08/4.53	truncateM xu = truncateM0 xu (truncateVu6 xu);
11.08/4.53	
11.08/4.53	truncateM0 xu (Tup2 m vv) = m;
11.08/4.53	
11.08/4.53	truncateVu6 xu = properFractionFloat xu;
11.08/4.53	
11.08/4.53	}
11.08/4.53	
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(5) Narrow (SOUND)
11.08/4.53	Haskell To QDPs
11.08/4.53	
11.08/4.53	digraph dp_graph {
11.08/4.53	node [outthreshold=100, inthreshold=100];1[label="truncateFloat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.08/4.53	3[label="truncateFloat vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
11.08/4.53	4[label="truncateM vy3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
11.08/4.53	5[label="truncateM0 vy3 (truncateVu6 vy3)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
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11.08/4.53	10[label="fromIntMyInt (quotMyInt vy30 vy31)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11.08/4.53	11[label="quotMyInt vy30 vy31",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
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11.08/4.53	285 -> 15[label="",style="solid", color="burlywood", weight=3];
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11.08/4.53	287 -> 17[label="",style="solid", color="burlywood", weight=3];
11.08/4.53	288[label="vy31/Neg vy310",fontsize=10,color="white",style="solid",shape="box"];14 -> 288[label="",style="solid", color="burlywood", weight=9];
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11.08/4.53	15[label="primQuotInt (Pos vy300) (Pos vy310)",fontsize=16,color="burlywood",shape="box"];289[label="vy310/Succ vy3100",fontsize=10,color="white",style="solid",shape="box"];15 -> 289[label="",style="solid", color="burlywood", weight=9];
11.08/4.53	289 -> 19[label="",style="solid", color="burlywood", weight=3];
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11.08/4.53	291 -> 21[label="",style="solid", color="burlywood", weight=3];
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11.08/4.53	18[label="primQuotInt (Neg vy300) (Neg vy310)",fontsize=16,color="burlywood",shape="box"];295[label="vy310/Succ vy3100",fontsize=10,color="white",style="solid",shape="box"];18 -> 295[label="",style="solid", color="burlywood", weight=9];
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11.08/4.53	20[label="primQuotInt (Pos vy300) (Pos Zero)",fontsize=16,color="black",shape="box"];20 -> 28[label="",style="solid", color="black", weight=3];
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11.08/4.53	22[label="primQuotInt (Pos vy300) (Neg Zero)",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3];
11.08/4.53	23[label="primQuotInt (Neg vy300) (Pos (Succ vy3100))",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
11.08/4.53	24[label="primQuotInt (Neg vy300) (Pos Zero)",fontsize=16,color="black",shape="box"];24 -> 32[label="",style="solid", color="black", weight=3];
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11.08/4.53	26[label="primQuotInt (Neg vy300) (Neg Zero)",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
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11.08/4.53	28[label="error",fontsize=16,color="black",shape="triangle"];28 -> 36[label="",style="solid", color="black", weight=3];
11.08/4.53	29[label="Neg (primDivNatS vy300 (Succ vy3100))",fontsize=16,color="green",shape="box"];29 -> 37[label="",style="dashed", color="green", weight=3];
11.08/4.53	30 -> 28[label="",style="dashed", color="red", weight=0];
11.08/4.53	30[label="error",fontsize=16,color="magenta"];31[label="Neg (primDivNatS vy300 (Succ vy3100))",fontsize=16,color="green",shape="box"];31 -> 38[label="",style="dashed", color="green", weight=3];
11.08/4.53	32 -> 28[label="",style="dashed", color="red", weight=0];
11.08/4.53	32[label="error",fontsize=16,color="magenta"];33[label="Pos (primDivNatS vy300 (Succ vy3100))",fontsize=16,color="green",shape="box"];33 -> 39[label="",style="dashed", color="green", weight=3];
11.08/4.53	34 -> 28[label="",style="dashed", color="red", weight=0];
11.08/4.53	34[label="error",fontsize=16,color="magenta"];35[label="primDivNatS vy300 (Succ vy3100)",fontsize=16,color="burlywood",shape="triangle"];297[label="vy300/Succ vy3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 297[label="",style="solid", color="burlywood", weight=9];
11.08/4.53	297 -> 40[label="",style="solid", color="burlywood", weight=3];
11.08/4.53	298[label="vy300/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 298[label="",style="solid", color="burlywood", weight=9];
11.08/4.53	298 -> 41[label="",style="solid", color="burlywood", weight=3];
11.08/4.53	36[label="stop MyTrue",fontsize=16,color="black",shape="box"];36 -> 42[label="",style="solid", color="black", weight=3];
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11.08/4.53	41[label="primDivNatS Zero (Succ vy3100)",fontsize=16,color="black",shape="box"];41 -> 48[label="",style="solid", color="black", weight=3];
11.08/4.53	42[label="error []",fontsize=16,color="red",shape="box"];43[label="vy3100",fontsize=16,color="green",shape="box"];44[label="vy300",fontsize=16,color="green",shape="box"];45[label="vy3100",fontsize=16,color="green",shape="box"];46[label="vy300",fontsize=16,color="green",shape="box"];47[label="primDivNatS0 vy3000 vy3100 (primGEqNatS vy3000 vy3100)",fontsize=16,color="burlywood",shape="box"];299[label="vy3000/Succ vy30000",fontsize=10,color="white",style="solid",shape="box"];47 -> 299[label="",style="solid", color="burlywood", weight=9];
11.08/4.53	299 -> 49[label="",style="solid", color="burlywood", weight=3];
11.08/4.53	300[label="vy3000/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 300[label="",style="solid", color="burlywood", weight=9];
11.08/4.53	300 -> 50[label="",style="solid", color="burlywood", weight=3];
11.08/4.53	48[label="Zero",fontsize=16,color="green",shape="box"];49[label="primDivNatS0 (Succ vy30000) vy3100 (primGEqNatS (Succ vy30000) vy3100)",fontsize=16,color="burlywood",shape="box"];301[label="vy3100/Succ vy31000",fontsize=10,color="white",style="solid",shape="box"];49 -> 301[label="",style="solid", color="burlywood", weight=9];
11.08/4.53	301 -> 51[label="",style="solid", color="burlywood", weight=3];
11.08/4.53	302[label="vy3100/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 302[label="",style="solid", color="burlywood", weight=9];
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11.08/4.53	50[label="primDivNatS0 Zero vy3100 (primGEqNatS Zero vy3100)",fontsize=16,color="burlywood",shape="box"];303[label="vy3100/Succ vy31000",fontsize=10,color="white",style="solid",shape="box"];50 -> 303[label="",style="solid", color="burlywood", weight=9];
11.08/4.53	303 -> 53[label="",style="solid", color="burlywood", weight=3];
11.08/4.53	304[label="vy3100/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 304[label="",style="solid", color="burlywood", weight=9];
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11.08/4.53	53[label="primDivNatS0 Zero (Succ vy31000) (primGEqNatS Zero (Succ vy31000))",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
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11.08/4.53	58[label="primDivNatS0 Zero Zero MyTrue",fontsize=16,color="black",shape="box"];58 -> 63[label="",style="solid", color="black", weight=3];
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11.08/4.53	76[label="Zero",fontsize=16,color="green",shape="box"];77[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];77 -> 84[label="",style="solid", color="black", weight=3];
11.08/4.53	258 -> 219[label="",style="dashed", color="red", weight=0];
11.08/4.53	258[label="primDivNatS0 (Succ vy20) (Succ vy21) (primGEqNatS vy220 vy230)",fontsize=16,color="magenta"];258 -> 262[label="",style="dashed", color="magenta", weight=3];
11.08/4.53	258 -> 263[label="",style="dashed", color="magenta", weight=3];
11.08/4.53	259[label="primDivNatS0 (Succ vy20) (Succ vy21) MyTrue",fontsize=16,color="black",shape="triangle"];259 -> 264[label="",style="solid", color="black", weight=3];
11.08/4.53	260[label="primDivNatS0 (Succ vy20) (Succ vy21) MyFalse",fontsize=16,color="black",shape="box"];260 -> 265[label="",style="solid", color="black", weight=3];
11.08/4.53	261 -> 259[label="",style="dashed", color="red", weight=0];
11.08/4.53	261[label="primDivNatS0 (Succ vy20) (Succ vy21) MyTrue",fontsize=16,color="magenta"];83[label="Succ vy30000",fontsize=16,color="green",shape="box"];84[label="Zero",fontsize=16,color="green",shape="box"];262[label="vy230",fontsize=16,color="green",shape="box"];263[label="vy220",fontsize=16,color="green",shape="box"];264[label="Succ (primDivNatS (primMinusNatS (Succ vy20) (Succ vy21)) (Succ (Succ vy21)))",fontsize=16,color="green",shape="box"];264 -> 266[label="",style="dashed", color="green", weight=3];
11.08/4.53	265[label="Zero",fontsize=16,color="green",shape="box"];266 -> 35[label="",style="dashed", color="red", weight=0];
11.08/4.53	266[label="primDivNatS (primMinusNatS (Succ vy20) (Succ vy21)) (Succ (Succ vy21))",fontsize=16,color="magenta"];266 -> 267[label="",style="dashed", color="magenta", weight=3];
11.08/4.53	266 -> 268[label="",style="dashed", color="magenta", weight=3];
11.08/4.53	267[label="Succ vy21",fontsize=16,color="green",shape="box"];268[label="primMinusNatS (Succ vy20) (Succ vy21)",fontsize=16,color="black",shape="box"];268 -> 269[label="",style="solid", color="black", weight=3];
11.08/4.53	269[label="primMinusNatS vy20 vy21",fontsize=16,color="burlywood",shape="triangle"];311[label="vy20/Succ vy200",fontsize=10,color="white",style="solid",shape="box"];269 -> 311[label="",style="solid", color="burlywood", weight=9];
11.08/4.53	311 -> 270[label="",style="solid", color="burlywood", weight=3];
11.08/4.53	312[label="vy20/Zero",fontsize=10,color="white",style="solid",shape="box"];269 -> 312[label="",style="solid", color="burlywood", weight=9];
11.08/4.53	312 -> 271[label="",style="solid", color="burlywood", weight=3];
11.08/4.53	270[label="primMinusNatS (Succ vy200) vy21",fontsize=16,color="burlywood",shape="box"];313[label="vy21/Succ vy210",fontsize=10,color="white",style="solid",shape="box"];270 -> 313[label="",style="solid", color="burlywood", weight=9];
11.08/4.53	313 -> 272[label="",style="solid", color="burlywood", weight=3];
11.08/4.53	314[label="vy21/Zero",fontsize=10,color="white",style="solid",shape="box"];270 -> 314[label="",style="solid", color="burlywood", weight=9];
11.08/4.53	314 -> 273[label="",style="solid", color="burlywood", weight=3];
11.08/4.53	271[label="primMinusNatS Zero vy21",fontsize=16,color="burlywood",shape="box"];315[label="vy21/Succ vy210",fontsize=10,color="white",style="solid",shape="box"];271 -> 315[label="",style="solid", color="burlywood", weight=9];
11.08/4.53	315 -> 274[label="",style="solid", color="burlywood", weight=3];
11.08/4.53	316[label="vy21/Zero",fontsize=10,color="white",style="solid",shape="box"];271 -> 316[label="",style="solid", color="burlywood", weight=9];
11.08/4.53	316 -> 275[label="",style="solid", color="burlywood", weight=3];
11.08/4.53	272[label="primMinusNatS (Succ vy200) (Succ vy210)",fontsize=16,color="black",shape="box"];272 -> 276[label="",style="solid", color="black", weight=3];
11.08/4.53	273[label="primMinusNatS (Succ vy200) Zero",fontsize=16,color="black",shape="box"];273 -> 277[label="",style="solid", color="black", weight=3];
11.08/4.53	274[label="primMinusNatS Zero (Succ vy210)",fontsize=16,color="black",shape="box"];274 -> 278[label="",style="solid", color="black", weight=3];
11.08/4.53	275[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];275 -> 279[label="",style="solid", color="black", weight=3];
11.08/4.53	276 -> 269[label="",style="dashed", color="red", weight=0];
11.08/4.53	276[label="primMinusNatS vy200 vy210",fontsize=16,color="magenta"];276 -> 280[label="",style="dashed", color="magenta", weight=3];
11.08/4.53	276 -> 281[label="",style="dashed", color="magenta", weight=3];
11.08/4.53	277[label="Succ vy200",fontsize=16,color="green",shape="box"];278[label="Zero",fontsize=16,color="green",shape="box"];279[label="Zero",fontsize=16,color="green",shape="box"];280[label="vy200",fontsize=16,color="green",shape="box"];281[label="vy210",fontsize=16,color="green",shape="box"];}
11.08/4.53	
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(6)
11.08/4.53	Complex Obligation (AND)
11.08/4.53	
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(7)
11.08/4.53	Obligation:
11.08/4.53	Q DP problem:
11.08/4.53	The TRS P consists of the following rules:
11.08/4.53	
11.08/4.53	   new_primMinusNatS(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS(vy200, vy210)
11.08/4.53	
11.08/4.53	R is empty.
11.08/4.53	Q is empty.
11.08/4.53	We have to consider all minimal (P,Q,R)-chains.
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(8) QDPSizeChangeProof (EQUIVALENT)
11.08/4.53	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. 
11.08/4.53	
11.08/4.53	From the DPs we obtained the following set of size-change graphs:
11.08/4.53	*new_primMinusNatS(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS(vy200, vy210)
11.08/4.53	The graph contains the following edges 1 > 1, 2 > 2
11.08/4.53	
11.08/4.53	
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(9)
11.08/4.53	YES
11.08/4.53	
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(10)
11.08/4.53	Obligation:
11.08/4.53	Q DP problem:
11.08/4.53	The TRS P consists of the following rules:
11.08/4.53	
11.08/4.53	   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vy30000), Main.Zero)
11.08/4.53	   new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Succ(vy230)) -> new_primDivNatS0(vy20, vy21, vy220, vy230)
11.08/4.53	   new_primDivNatS00(vy20, vy21) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21))
11.08/4.53	   new_primDivNatS0(vy20, vy21, Main.Zero, Main.Zero) -> new_primDivNatS00(vy20, vy21)
11.08/4.53	   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Succ(vy31000)) -> new_primDivNatS0(vy30000, vy31000, vy30000, vy31000)
11.08/4.53	   new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21))
11.08/4.53	   new_primDivNatS(Main.Succ(Main.Zero), Main.Zero) -> new_primDivNatS(new_primMinusNatS2, Main.Zero)
11.08/4.53	
11.08/4.53	The TRS R consists of the following rules:
11.08/4.53	
11.08/4.53	   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
11.08/4.53	   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
11.08/4.53	   new_primMinusNatS1(vy30000) -> Main.Succ(vy30000)
11.08/4.53	   new_primMinusNatS2 -> Main.Zero
11.08/4.53	
11.08/4.53	The set Q consists of the following terms:
11.08/4.53	
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.08/4.53	   new_primMinusNatS2
11.08/4.53	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.08/4.53	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.08/4.53	   new_primMinusNatS1(x0)
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.08/4.53	
11.08/4.53	We have to consider all minimal (P,Q,R)-chains.
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(11) DependencyGraphProof (EQUIVALENT)
11.08/4.53	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(12)
11.08/4.53	Complex Obligation (AND)
11.08/4.53	
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(13)
11.08/4.53	Obligation:
11.08/4.53	Q DP problem:
11.08/4.53	The TRS P consists of the following rules:
11.08/4.53	
11.08/4.53	   new_primDivNatS0(vy20, vy21, Main.Zero, Main.Zero) -> new_primDivNatS00(vy20, vy21)
11.08/4.53	   new_primDivNatS00(vy20, vy21) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21))
11.08/4.53	   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Succ(vy31000)) -> new_primDivNatS0(vy30000, vy31000, vy30000, vy31000)
11.08/4.53	   new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Succ(vy230)) -> new_primDivNatS0(vy20, vy21, vy220, vy230)
11.08/4.53	   new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21))
11.08/4.53	
11.08/4.53	The TRS R consists of the following rules:
11.08/4.53	
11.08/4.53	   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
11.08/4.53	   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
11.08/4.53	   new_primMinusNatS1(vy30000) -> Main.Succ(vy30000)
11.08/4.53	   new_primMinusNatS2 -> Main.Zero
11.08/4.53	
11.08/4.53	The set Q consists of the following terms:
11.08/4.53	
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.08/4.53	   new_primMinusNatS2
11.08/4.53	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.08/4.53	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.08/4.53	   new_primMinusNatS1(x0)
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.08/4.53	
11.08/4.53	We have to consider all minimal (P,Q,R)-chains.
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(14) QDPSizeChangeProof (EQUIVALENT)
11.08/4.53	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
11.08/4.53	
11.08/4.53	Order:Polynomial interpretation [POLO]:
11.08/4.53	
11.08/4.53	   POL(Main.Succ(x_1)) = 1 + x_1
11.08/4.53	   POL(Main.Zero) = 1
11.08/4.53	   POL(new_primMinusNatS0(x_1, x_2)) = x_1
11.08/4.53	
11.08/4.53	
11.08/4.53	
11.08/4.53	
11.08/4.53	From the DPs we obtained the following set of size-change graphs:
11.08/4.53	*new_primDivNatS00(vy20, vy21) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21)) (allowed arguments on rhs = {1, 2})
11.08/4.53	The graph contains the following edges 1 >= 1
11.08/4.53	
11.08/4.53	
11.08/4.53	*new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Succ(vy31000)) -> new_primDivNatS0(vy30000, vy31000, vy30000, vy31000) (allowed arguments on rhs = {1, 2, 3, 4})
11.08/4.53	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
11.08/4.53	
11.08/4.53	
11.08/4.53	*new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Succ(vy230)) -> new_primDivNatS0(vy20, vy21, vy220, vy230) (allowed arguments on rhs = {1, 2, 3, 4})
11.08/4.53	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
11.08/4.53	
11.08/4.53	
11.08/4.53	*new_primDivNatS0(vy20, vy21, Main.Zero, Main.Zero) -> new_primDivNatS00(vy20, vy21) (allowed arguments on rhs = {1, 2})
11.08/4.53	The graph contains the following edges 1 >= 1, 2 >= 2
11.08/4.53	
11.08/4.53	
11.08/4.53	*new_primDivNatS0(vy20, vy21, Main.Succ(vy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vy20, vy21), Main.Succ(vy21)) (allowed arguments on rhs = {1, 2})
11.08/4.53	The graph contains the following edges 1 >= 1
11.08/4.53	
11.08/4.53	
11.08/4.53	
11.08/4.53	We oriented the following set of usable rules [AAECC05,FROCOS05].
11.08/4.53	
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
11.08/4.53	   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
11.08/4.53	   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
11.08/4.53	
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(15)
11.08/4.53	YES
11.08/4.53	
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(16)
11.08/4.53	Obligation:
11.08/4.53	Q DP problem:
11.08/4.53	The TRS P consists of the following rules:
11.08/4.53	
11.08/4.53	   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vy30000), Main.Zero)
11.08/4.53	
11.08/4.53	The TRS R consists of the following rules:
11.08/4.53	
11.08/4.53	   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
11.08/4.53	   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
11.08/4.53	   new_primMinusNatS1(vy30000) -> Main.Succ(vy30000)
11.08/4.53	   new_primMinusNatS2 -> Main.Zero
11.08/4.53	
11.08/4.53	The set Q consists of the following terms:
11.08/4.53	
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.08/4.53	   new_primMinusNatS2
11.08/4.53	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.08/4.53	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.08/4.53	   new_primMinusNatS1(x0)
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.08/4.53	
11.08/4.53	We have to consider all minimal (P,Q,R)-chains.
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(17) MRRProof (EQUIVALENT)
11.08/4.53	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.
11.08/4.53	
11.08/4.53	Strictly oriented dependency pairs:
11.08/4.53	
11.08/4.53	   new_primDivNatS(Main.Succ(Main.Succ(vy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vy30000), Main.Zero)
11.08/4.53	
11.08/4.53	Strictly oriented rules of the TRS R:
11.08/4.53	
11.08/4.53	   new_primMinusNatS0(Main.Succ(vy200), Main.Succ(vy210)) -> new_primMinusNatS0(vy200, vy210)
11.08/4.53	   new_primMinusNatS0(Main.Succ(vy200), Main.Zero) -> Main.Succ(vy200)
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Succ(vy210)) -> Main.Zero
11.08/4.53	
11.08/4.53	Used ordering: Polynomial interpretation [POLO]:
11.08/4.53	
11.08/4.53	   POL(Main.Succ(x_1)) = 1 + x_1
11.08/4.53	   POL(Main.Zero) = 2
11.08/4.53	   POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2
11.08/4.53	   POL(new_primMinusNatS0(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
11.08/4.53	   POL(new_primMinusNatS1(x_1)) = 1 + x_1
11.08/4.53	   POL(new_primMinusNatS2) = 2
11.08/4.53	
11.08/4.53	
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(18)
11.08/4.53	Obligation:
11.08/4.53	Q DP problem:
11.08/4.53	P is empty.
11.08/4.53	The TRS R consists of the following rules:
11.08/4.53	
11.08/4.53	   new_primMinusNatS1(vy30000) -> Main.Succ(vy30000)
11.08/4.53	   new_primMinusNatS2 -> Main.Zero
11.08/4.53	
11.08/4.53	The set Q consists of the following terms:
11.08/4.53	
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.08/4.53	   new_primMinusNatS2
11.08/4.53	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.08/4.53	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.08/4.53	   new_primMinusNatS1(x0)
11.08/4.53	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.08/4.53	
11.08/4.53	We have to consider all minimal (P,Q,R)-chains.
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(19) PisEmptyProof (EQUIVALENT)
11.08/4.53	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.08/4.53	----------------------------------------
11.08/4.53	
11.08/4.53	(20)
11.08/4.53	YES
11.25/6.07	EOF