8.93/3.84	YES
11.11/4.45	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.11/4.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.11/4.45	
11.11/4.45	
11.11/4.45	H-Termination with start terms of the given HASKELL could be proven:
11.11/4.45	
11.11/4.45	(0) HASKELL
11.11/4.45	(1) BR [EQUIVALENT, 0 ms]
11.11/4.45	(2) HASKELL
11.11/4.45	(3) COR [EQUIVALENT, 0 ms]
11.11/4.45	(4) HASKELL
11.11/4.45	(5) Narrow [SOUND, 0 ms]
11.11/4.45	(6) AND
11.11/4.45	    (7) QDP
11.11/4.45	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.11/4.45	        (9) YES
11.11/4.45	    (10) QDP
11.11/4.45	        (11) DependencyGraphProof [EQUIVALENT, 0 ms]
11.11/4.45	        (12) AND
11.11/4.45	            (13) QDP
11.11/4.45	                (14) QDPSizeChangeProof [EQUIVALENT, 32 ms]
11.11/4.45	                (15) YES
11.11/4.45	            (16) QDP
11.11/4.45	                (17) MRRProof [EQUIVALENT, 0 ms]
11.11/4.45	                (18) QDP
11.11/4.45	                (19) PisEmptyProof [EQUIVALENT, 0 ms]
11.11/4.45	                (20) YES
11.11/4.45	
11.11/4.45	
11.11/4.45	----------------------------------------
11.11/4.45	
11.11/4.45	(0)
11.11/4.45	Obligation:
11.11/4.45	mainModule Main 
11.11/4.45	module Main where {
11.11/4.45	import qualified Prelude;
11.11/4.45	data Integer = Integer MyInt ;
11.11/4.45	
11.11/4.45	data MyBool = MyTrue  | MyFalse ;
11.11/4.45	
11.11/4.45	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
11.11/4.45	
11.11/4.45	data Main.Nat = Succ Main.Nat  | Zero ;
11.11/4.45	
11.11/4.45	data Ratio a = CnPc a a ;
11.11/4.45	
11.11/4.45	data Tup2 a b = Tup2 a b ;
11.11/4.45	
11.11/4.45	error :: a;
11.11/4.45	error = stop MyTrue;
11.11/4.45	
11.11/4.45	fromIntegerMyInt :: Integer  ->  MyInt;
11.11/4.45	fromIntegerMyInt (Integer x) = x;
11.11/4.45	
11.11/4.45	fromIntegral = pt fromIntegerMyInt toIntegerMyInt;
11.11/4.45	
11.11/4.45	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.11/4.45	primDivNatS Main.Zero Main.Zero = Main.error;
11.11/4.45	primDivNatS (Main.Succ x) Main.Zero = Main.error;
11.11/4.45	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
11.11/4.45	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
11.11/4.45	
11.11/4.45	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
11.11/4.45	primDivNatS0 x y MyFalse = Main.Zero;
11.11/4.45	
11.11/4.45	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
11.11/4.45	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
11.11/4.45	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
11.11/4.45	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
11.11/4.45	primGEqNatS Main.Zero Main.Zero = MyTrue;
11.11/4.45	
11.11/4.45	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.11/4.45	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
11.11/4.45	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
11.11/4.45	primMinusNatS x Main.Zero = x;
11.11/4.45	
11.11/4.45	primModNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.11/4.45	primModNatS Main.Zero Main.Zero = Main.error;
11.11/4.45	primModNatS Main.Zero (Main.Succ x) = Main.Zero;
11.11/4.45	primModNatS (Main.Succ x) Main.Zero = Main.error;
11.11/4.45	primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
11.11/4.45	primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y));
11.11/4.45	
11.11/4.45	primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
11.11/4.45	primModNatS0 x y MyFalse = Main.Succ x;
11.11/4.45	
11.11/4.45	primQrmInt :: MyInt  ->  MyInt  ->  Tup2 MyInt MyInt;
11.11/4.45	primQrmInt x y = Tup2 (primQuotInt x y) (primRemInt x y);
11.11/4.45	
11.11/4.45	primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
11.11/4.45	primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.11/4.45	primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.11/4.45	primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.11/4.45	primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.11/4.45	primQuotInt ww wx = Main.error;
11.11/4.45	
11.11/4.45	primRemInt :: MyInt  ->  MyInt  ->  MyInt;
11.11/4.45	primRemInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
11.11/4.45	primRemInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
11.11/4.45	primRemInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
11.11/4.45	primRemInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
11.11/4.45	primRemInt vy vz = Main.error;
11.11/4.45	
11.11/4.45	properFractionQ xv xw = properFractionQ1 xv xw (properFractionVu30 xv xw);
11.11/4.45	
11.11/4.45	properFractionQ1 xv xw (Tup2 q vw) = q;
11.11/4.45	
11.11/4.45	properFractionR xv xw = properFractionR0 xv xw (properFractionVu30 xv xw);
11.11/4.45	
11.11/4.45	properFractionR0 xv xw (Tup2 vx r) = r;
11.11/4.45	
11.11/4.45	properFractionRatio :: Ratio MyInt  ->  Tup2 MyInt (Ratio MyInt);
11.11/4.45	properFractionRatio (CnPc x y) = Tup2 (fromIntegral (properFractionQ x y)) (CnPc (properFractionR x y) y);
11.11/4.45	
11.11/4.45	properFractionVu30 xv xw = quotRemMyInt xv xw;
11.11/4.45	
11.11/4.45	pt :: (c  ->  a)  ->  (b  ->  c)  ->  b  ->  a;
11.11/4.45	pt f g x = f (g x);
11.11/4.45	
11.11/4.45	quotRemMyInt :: MyInt  ->  MyInt  ->  Tup2 MyInt MyInt;
11.11/4.45	quotRemMyInt = primQrmInt;
11.11/4.45	
11.11/4.45	stop :: MyBool  ->  a;
11.11/4.45	stop MyFalse = stop MyFalse;
11.11/4.45	
11.11/4.45	toIntegerMyInt :: MyInt  ->  Integer;
11.11/4.45	toIntegerMyInt x = Integer x;
11.11/4.45	
11.11/4.45	truncateM xu = truncateM0 xu (truncateVu6 xu);
11.11/4.45	
11.11/4.45	truncateM0 xu (Tup2 m vv) = m;
11.11/4.45	
11.11/4.45	truncateRatio :: Ratio MyInt  ->  MyInt;
11.11/4.45	truncateRatio x = truncateM x;
11.11/4.45	
11.11/4.45	truncateVu6 xu = properFractionRatio xu;
11.11/4.45	
11.11/4.45	}
11.11/4.45	
11.11/4.45	----------------------------------------
11.11/4.45	
11.11/4.45	(1) BR (EQUIVALENT)
11.11/4.45	Replaced joker patterns by fresh variables and removed binding patterns.
11.11/4.45	----------------------------------------
11.11/4.45	
11.11/4.45	(2)
11.11/4.45	Obligation:
11.11/4.45	mainModule Main 
11.11/4.45	module Main where {
11.11/4.45	import qualified Prelude;
11.11/4.45	data Integer = Integer MyInt ;
11.11/4.45	
11.11/4.45	data MyBool = MyTrue  | MyFalse ;
11.11/4.45	
11.11/4.45	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
11.11/4.45	
11.11/4.45	data Main.Nat = Succ Main.Nat  | Zero ;
11.11/4.45	
11.11/4.45	data Ratio a = CnPc a a ;
11.11/4.45	
11.11/4.45	data Tup2 b a = Tup2 b a ;
11.11/4.45	
11.11/4.45	error :: a;
11.11/4.45	error = stop MyTrue;
11.11/4.45	
11.11/4.45	fromIntegerMyInt :: Integer  ->  MyInt;
11.11/4.45	fromIntegerMyInt (Integer x) = x;
11.11/4.46	
11.11/4.46	fromIntegral = pt fromIntegerMyInt toIntegerMyInt;
11.11/4.46	
11.11/4.46	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.11/4.46	primDivNatS Main.Zero Main.Zero = Main.error;
11.11/4.46	primDivNatS (Main.Succ x) Main.Zero = Main.error;
11.11/4.46	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
11.11/4.46	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
11.11/4.46	
11.11/4.46	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
11.11/4.46	primDivNatS0 x y MyFalse = Main.Zero;
11.11/4.46	
11.11/4.46	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
11.11/4.46	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
11.11/4.46	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
11.11/4.46	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
11.11/4.46	primGEqNatS Main.Zero Main.Zero = MyTrue;
11.11/4.46	
11.11/4.46	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.11/4.46	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
11.11/4.46	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
11.11/4.46	primMinusNatS x Main.Zero = x;
11.11/4.46	
11.11/4.46	primModNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.11/4.46	primModNatS Main.Zero Main.Zero = Main.error;
11.11/4.46	primModNatS Main.Zero (Main.Succ x) = Main.Zero;
11.11/4.46	primModNatS (Main.Succ x) Main.Zero = Main.error;
11.11/4.46	primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
11.11/4.46	primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y));
11.11/4.46	
11.11/4.46	primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
11.11/4.46	primModNatS0 x y MyFalse = Main.Succ x;
11.11/4.46	
11.11/4.46	primQrmInt :: MyInt  ->  MyInt  ->  Tup2 MyInt MyInt;
11.11/4.46	primQrmInt x y = Tup2 (primQuotInt x y) (primRemInt x y);
11.11/4.46	
11.11/4.46	primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
11.11/4.46	primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.11/4.46	primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.11/4.46	primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.11/4.46	primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.11/4.46	primQuotInt ww wx = Main.error;
11.11/4.46	
11.11/4.46	primRemInt :: MyInt  ->  MyInt  ->  MyInt;
11.11/4.46	primRemInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
11.11/4.46	primRemInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
11.11/4.46	primRemInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
11.11/4.46	primRemInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
11.11/4.46	primRemInt vy vz = Main.error;
11.11/4.46	
11.11/4.46	properFractionQ xv xw = properFractionQ1 xv xw (properFractionVu30 xv xw);
11.11/4.46	
11.11/4.46	properFractionQ1 xv xw (Tup2 q vw) = q;
11.11/4.46	
11.11/4.46	properFractionR xv xw = properFractionR0 xv xw (properFractionVu30 xv xw);
11.11/4.46	
11.11/4.46	properFractionR0 xv xw (Tup2 vx r) = r;
11.11/4.46	
11.11/4.46	properFractionRatio :: Ratio MyInt  ->  Tup2 MyInt (Ratio MyInt);
11.11/4.46	properFractionRatio (CnPc x y) = Tup2 (fromIntegral (properFractionQ x y)) (CnPc (properFractionR x y) y);
11.11/4.46	
11.11/4.46	properFractionVu30 xv xw = quotRemMyInt xv xw;
11.11/4.46	
11.11/4.46	pt :: (c  ->  b)  ->  (a  ->  c)  ->  a  ->  b;
11.11/4.46	pt f g x = f (g x);
11.11/4.46	
11.11/4.46	quotRemMyInt :: MyInt  ->  MyInt  ->  Tup2 MyInt MyInt;
11.11/4.46	quotRemMyInt = primQrmInt;
11.11/4.46	
11.11/4.46	stop :: MyBool  ->  a;
11.11/4.46	stop MyFalse = stop MyFalse;
11.11/4.46	
11.11/4.46	toIntegerMyInt :: MyInt  ->  Integer;
11.11/4.46	toIntegerMyInt x = Integer x;
11.11/4.46	
11.11/4.46	truncateM xu = truncateM0 xu (truncateVu6 xu);
11.11/4.46	
11.11/4.46	truncateM0 xu (Tup2 m vv) = m;
11.11/4.46	
11.11/4.46	truncateRatio :: Ratio MyInt  ->  MyInt;
11.11/4.46	truncateRatio x = truncateM x;
11.11/4.46	
11.11/4.46	truncateVu6 xu = properFractionRatio xu;
11.11/4.46	
11.11/4.46	}
11.11/4.46	
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(3) COR (EQUIVALENT)
11.11/4.46	Cond Reductions:
11.11/4.46	The following Function with conditions
11.11/4.46	"undefined |Falseundefined;
11.11/4.46	"
11.11/4.46	is transformed to
11.11/4.46	"undefined  = undefined1;
11.11/4.46	"
11.11/4.46	"undefined0 True = undefined;
11.11/4.46	"
11.11/4.46	"undefined1  = undefined0 False;
11.11/4.46	"
11.11/4.46	
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(4)
11.11/4.46	Obligation:
11.11/4.46	mainModule Main 
11.11/4.46	module Main where {
11.11/4.46	import qualified Prelude;
11.11/4.46	data Integer = Integer MyInt ;
11.11/4.46	
11.11/4.46	data MyBool = MyTrue  | MyFalse ;
11.11/4.46	
11.11/4.46	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
11.11/4.46	
11.11/4.46	data Main.Nat = Succ Main.Nat  | Zero ;
11.11/4.46	
11.11/4.46	data Ratio a = CnPc a a ;
11.11/4.46	
11.11/4.46	data Tup2 b a = Tup2 b a ;
11.11/4.46	
11.11/4.46	error :: a;
11.11/4.46	error = stop MyTrue;
11.11/4.46	
11.11/4.46	fromIntegerMyInt :: Integer  ->  MyInt;
11.11/4.46	fromIntegerMyInt (Integer x) = x;
11.11/4.46	
11.11/4.46	fromIntegral = pt fromIntegerMyInt toIntegerMyInt;
11.11/4.46	
11.11/4.46	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.11/4.46	primDivNatS Main.Zero Main.Zero = Main.error;
11.11/4.46	primDivNatS (Main.Succ x) Main.Zero = Main.error;
11.11/4.46	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
11.11/4.46	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
11.11/4.46	
11.11/4.46	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
11.11/4.46	primDivNatS0 x y MyFalse = Main.Zero;
11.11/4.46	
11.11/4.46	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
11.11/4.46	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
11.11/4.46	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
11.11/4.46	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
11.11/4.46	primGEqNatS Main.Zero Main.Zero = MyTrue;
11.11/4.46	
11.11/4.46	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.11/4.46	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
11.11/4.46	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
11.11/4.46	primMinusNatS x Main.Zero = x;
11.11/4.46	
11.11/4.46	primModNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.11/4.46	primModNatS Main.Zero Main.Zero = Main.error;
11.11/4.46	primModNatS Main.Zero (Main.Succ x) = Main.Zero;
11.11/4.46	primModNatS (Main.Succ x) Main.Zero = Main.error;
11.11/4.46	primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
11.11/4.46	primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y));
11.11/4.46	
11.11/4.46	primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
11.11/4.46	primModNatS0 x y MyFalse = Main.Succ x;
11.11/4.46	
11.11/4.46	primQrmInt :: MyInt  ->  MyInt  ->  Tup2 MyInt MyInt;
11.11/4.46	primQrmInt x y = Tup2 (primQuotInt x y) (primRemInt x y);
11.11/4.46	
11.11/4.46	primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
11.11/4.46	primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.11/4.46	primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.11/4.46	primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.11/4.46	primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.11/4.46	primQuotInt ww wx = Main.error;
11.11/4.46	
11.11/4.46	primRemInt :: MyInt  ->  MyInt  ->  MyInt;
11.11/4.46	primRemInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
11.11/4.46	primRemInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
11.11/4.46	primRemInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
11.11/4.46	primRemInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
11.11/4.46	primRemInt vy vz = Main.error;
11.11/4.46	
11.11/4.46	properFractionQ xv xw = properFractionQ1 xv xw (properFractionVu30 xv xw);
11.11/4.46	
11.11/4.46	properFractionQ1 xv xw (Tup2 q vw) = q;
11.11/4.46	
11.11/4.46	properFractionR xv xw = properFractionR0 xv xw (properFractionVu30 xv xw);
11.11/4.46	
11.11/4.46	properFractionR0 xv xw (Tup2 vx r) = r;
11.11/4.46	
11.11/4.46	properFractionRatio :: Ratio MyInt  ->  Tup2 MyInt (Ratio MyInt);
11.11/4.46	properFractionRatio (CnPc x y) = Tup2 (fromIntegral (properFractionQ x y)) (CnPc (properFractionR x y) y);
11.11/4.46	
11.11/4.46	properFractionVu30 xv xw = quotRemMyInt xv xw;
11.11/4.46	
11.11/4.46	pt :: (b  ->  c)  ->  (a  ->  b)  ->  a  ->  c;
11.11/4.46	pt f g x = f (g x);
11.11/4.46	
11.11/4.46	quotRemMyInt :: MyInt  ->  MyInt  ->  Tup2 MyInt MyInt;
11.11/4.46	quotRemMyInt = primQrmInt;
11.11/4.46	
11.11/4.46	stop :: MyBool  ->  a;
11.11/4.46	stop MyFalse = stop MyFalse;
11.11/4.46	
11.11/4.46	toIntegerMyInt :: MyInt  ->  Integer;
11.11/4.46	toIntegerMyInt x = Integer x;
11.11/4.46	
11.11/4.46	truncateM xu = truncateM0 xu (truncateVu6 xu);
11.11/4.46	
11.11/4.46	truncateM0 xu (Tup2 m vv) = m;
11.11/4.46	
11.11/4.46	truncateRatio :: Ratio MyInt  ->  MyInt;
11.11/4.46	truncateRatio x = truncateM x;
11.11/4.46	
11.11/4.46	truncateVu6 xu = properFractionRatio xu;
11.11/4.46	
11.11/4.46	}
11.11/4.46	
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(5) Narrow (SOUND)
11.11/4.46	Haskell To QDPs
11.11/4.46	
11.11/4.46	digraph dp_graph {
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11.11/4.46	9[label="fromIntegral (properFractionQ wy30 wy31)",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
11.11/4.46	10[label="pt fromIntegerMyInt toIntegerMyInt (properFractionQ wy30 wy31)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
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11.11/4.46	13[label="properFractionQ wy30 wy31",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
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11.11/4.46	17[label="properFractionQ1 wy30 wy31 (Tup2 (primQuotInt wy30 wy31) (primRemInt wy30 wy31))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
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11.11/4.46	289 -> 19[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	290[label="wy30/Neg wy300",fontsize=10,color="white",style="solid",shape="box"];18 -> 290[label="",style="solid", color="burlywood", weight=9];
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11.11/4.46	291 -> 21[label="",style="solid", color="burlywood", weight=3];
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11.11/4.46	294[label="wy31/Neg wy310",fontsize=10,color="white",style="solid",shape="box"];20 -> 294[label="",style="solid", color="burlywood", weight=9];
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11.11/4.46	23[label="primQuotInt (Neg wy300) (Pos wy310)",fontsize=16,color="burlywood",shape="box"];299[label="wy310/Succ wy3100",fontsize=10,color="white",style="solid",shape="box"];23 -> 299[label="",style="solid", color="burlywood", weight=9];
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11.11/4.46	300 -> 30[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	24[label="primQuotInt (Neg wy300) (Neg wy310)",fontsize=16,color="burlywood",shape="box"];301[label="wy310/Succ wy3100",fontsize=10,color="white",style="solid",shape="box"];24 -> 301[label="",style="solid", color="burlywood", weight=9];
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11.11/4.46	302[label="wy310/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 302[label="",style="solid", color="burlywood", weight=9];
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11.11/4.46	26[label="primQuotInt (Pos wy300) (Pos Zero)",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
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11.11/4.46	28[label="primQuotInt (Pos wy300) (Neg Zero)",fontsize=16,color="black",shape="box"];28 -> 36[label="",style="solid", color="black", weight=3];
11.11/4.46	29[label="primQuotInt (Neg wy300) (Pos (Succ wy3100))",fontsize=16,color="black",shape="box"];29 -> 37[label="",style="solid", color="black", weight=3];
11.11/4.46	30[label="primQuotInt (Neg wy300) (Pos Zero)",fontsize=16,color="black",shape="box"];30 -> 38[label="",style="solid", color="black", weight=3];
11.11/4.46	31[label="primQuotInt (Neg wy300) (Neg (Succ wy3100))",fontsize=16,color="black",shape="box"];31 -> 39[label="",style="solid", color="black", weight=3];
11.11/4.46	32[label="primQuotInt (Neg wy300) (Neg Zero)",fontsize=16,color="black",shape="box"];32 -> 40[label="",style="solid", color="black", weight=3];
11.11/4.46	33[label="Pos (primDivNatS wy300 (Succ wy3100))",fontsize=16,color="green",shape="box"];33 -> 41[label="",style="dashed", color="green", weight=3];
11.11/4.46	34[label="error",fontsize=16,color="black",shape="triangle"];34 -> 42[label="",style="solid", color="black", weight=3];
11.11/4.46	35[label="Neg (primDivNatS wy300 (Succ wy3100))",fontsize=16,color="green",shape="box"];35 -> 43[label="",style="dashed", color="green", weight=3];
11.11/4.46	36 -> 34[label="",style="dashed", color="red", weight=0];
11.11/4.46	36[label="error",fontsize=16,color="magenta"];37[label="Neg (primDivNatS wy300 (Succ wy3100))",fontsize=16,color="green",shape="box"];37 -> 44[label="",style="dashed", color="green", weight=3];
11.11/4.46	38 -> 34[label="",style="dashed", color="red", weight=0];
11.11/4.46	38[label="error",fontsize=16,color="magenta"];39[label="Pos (primDivNatS wy300 (Succ wy3100))",fontsize=16,color="green",shape="box"];39 -> 45[label="",style="dashed", color="green", weight=3];
11.11/4.46	40 -> 34[label="",style="dashed", color="red", weight=0];
11.11/4.46	40[label="error",fontsize=16,color="magenta"];41[label="primDivNatS wy300 (Succ wy3100)",fontsize=16,color="burlywood",shape="triangle"];303[label="wy300/Succ wy3000",fontsize=10,color="white",style="solid",shape="box"];41 -> 303[label="",style="solid", color="burlywood", weight=9];
11.11/4.46	303 -> 46[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	304[label="wy300/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 304[label="",style="solid", color="burlywood", weight=9];
11.11/4.46	304 -> 47[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	42[label="stop MyTrue",fontsize=16,color="black",shape="box"];42 -> 48[label="",style="solid", color="black", weight=3];
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11.11/4.46	47[label="primDivNatS Zero (Succ wy3100)",fontsize=16,color="black",shape="box"];47 -> 54[label="",style="solid", color="black", weight=3];
11.11/4.46	48[label="error []",fontsize=16,color="red",shape="box"];49[label="wy3100",fontsize=16,color="green",shape="box"];50[label="wy300",fontsize=16,color="green",shape="box"];51[label="wy3100",fontsize=16,color="green",shape="box"];52[label="wy300",fontsize=16,color="green",shape="box"];53[label="primDivNatS0 wy3000 wy3100 (primGEqNatS wy3000 wy3100)",fontsize=16,color="burlywood",shape="box"];305[label="wy3000/Succ wy30000",fontsize=10,color="white",style="solid",shape="box"];53 -> 305[label="",style="solid", color="burlywood", weight=9];
11.11/4.46	305 -> 55[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	306[label="wy3000/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 306[label="",style="solid", color="burlywood", weight=9];
11.11/4.46	306 -> 56[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	54[label="Zero",fontsize=16,color="green",shape="box"];55[label="primDivNatS0 (Succ wy30000) wy3100 (primGEqNatS (Succ wy30000) wy3100)",fontsize=16,color="burlywood",shape="box"];307[label="wy3100/Succ wy31000",fontsize=10,color="white",style="solid",shape="box"];55 -> 307[label="",style="solid", color="burlywood", weight=9];
11.11/4.46	307 -> 57[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	308[label="wy3100/Zero",fontsize=10,color="white",style="solid",shape="box"];55 -> 308[label="",style="solid", color="burlywood", weight=9];
11.11/4.46	308 -> 58[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	56[label="primDivNatS0 Zero wy3100 (primGEqNatS Zero wy3100)",fontsize=16,color="burlywood",shape="box"];309[label="wy3100/Succ wy31000",fontsize=10,color="white",style="solid",shape="box"];56 -> 309[label="",style="solid", color="burlywood", weight=9];
11.11/4.46	309 -> 59[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	310[label="wy3100/Zero",fontsize=10,color="white",style="solid",shape="box"];56 -> 310[label="",style="solid", color="burlywood", weight=9];
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11.11/4.46	262[label="primDivNatS0 (Succ wy20) (Succ wy21) (primGEqNatS Zero (Succ wy230))",fontsize=16,color="black",shape="box"];262 -> 266[label="",style="solid", color="black", weight=3];
11.11/4.46	263[label="primDivNatS0 (Succ wy20) (Succ wy21) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];263 -> 267[label="",style="solid", color="black", weight=3];
11.11/4.46	80[label="Zero",fontsize=16,color="green",shape="box"];81[label="primMinusNatS (Succ wy30000) Zero",fontsize=16,color="black",shape="triangle"];81 -> 89[label="",style="solid", color="black", weight=3];
11.11/4.46	82[label="Zero",fontsize=16,color="green",shape="box"];83[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];83 -> 90[label="",style="solid", color="black", weight=3];
11.11/4.46	264 -> 225[label="",style="dashed", color="red", weight=0];
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11.11/4.46	264 -> 269[label="",style="dashed", color="magenta", weight=3];
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11.11/4.46	266[label="primDivNatS0 (Succ wy20) (Succ wy21) MyFalse",fontsize=16,color="black",shape="box"];266 -> 271[label="",style="solid", color="black", weight=3];
11.11/4.46	267 -> 265[label="",style="dashed", color="red", weight=0];
11.11/4.46	267[label="primDivNatS0 (Succ wy20) (Succ wy21) MyTrue",fontsize=16,color="magenta"];89[label="Succ wy30000",fontsize=16,color="green",shape="box"];90[label="Zero",fontsize=16,color="green",shape="box"];268[label="wy230",fontsize=16,color="green",shape="box"];269[label="wy220",fontsize=16,color="green",shape="box"];270[label="Succ (primDivNatS (primMinusNatS (Succ wy20) (Succ wy21)) (Succ (Succ wy21)))",fontsize=16,color="green",shape="box"];270 -> 272[label="",style="dashed", color="green", weight=3];
11.11/4.46	271[label="Zero",fontsize=16,color="green",shape="box"];272 -> 41[label="",style="dashed", color="red", weight=0];
11.11/4.46	272[label="primDivNatS (primMinusNatS (Succ wy20) (Succ wy21)) (Succ (Succ wy21))",fontsize=16,color="magenta"];272 -> 273[label="",style="dashed", color="magenta", weight=3];
11.11/4.46	272 -> 274[label="",style="dashed", color="magenta", weight=3];
11.11/4.46	273[label="Succ wy21",fontsize=16,color="green",shape="box"];274[label="primMinusNatS (Succ wy20) (Succ wy21)",fontsize=16,color="black",shape="box"];274 -> 275[label="",style="solid", color="black", weight=3];
11.11/4.46	275[label="primMinusNatS wy20 wy21",fontsize=16,color="burlywood",shape="triangle"];317[label="wy20/Succ wy200",fontsize=10,color="white",style="solid",shape="box"];275 -> 317[label="",style="solid", color="burlywood", weight=9];
11.11/4.46	317 -> 276[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	318[label="wy20/Zero",fontsize=10,color="white",style="solid",shape="box"];275 -> 318[label="",style="solid", color="burlywood", weight=9];
11.11/4.46	318 -> 277[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	276[label="primMinusNatS (Succ wy200) wy21",fontsize=16,color="burlywood",shape="box"];319[label="wy21/Succ wy210",fontsize=10,color="white",style="solid",shape="box"];276 -> 319[label="",style="solid", color="burlywood", weight=9];
11.11/4.46	319 -> 278[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	320[label="wy21/Zero",fontsize=10,color="white",style="solid",shape="box"];276 -> 320[label="",style="solid", color="burlywood", weight=9];
11.11/4.46	320 -> 279[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	277[label="primMinusNatS Zero wy21",fontsize=16,color="burlywood",shape="box"];321[label="wy21/Succ wy210",fontsize=10,color="white",style="solid",shape="box"];277 -> 321[label="",style="solid", color="burlywood", weight=9];
11.11/4.46	321 -> 280[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	322[label="wy21/Zero",fontsize=10,color="white",style="solid",shape="box"];277 -> 322[label="",style="solid", color="burlywood", weight=9];
11.11/4.46	322 -> 281[label="",style="solid", color="burlywood", weight=3];
11.11/4.46	278[label="primMinusNatS (Succ wy200) (Succ wy210)",fontsize=16,color="black",shape="box"];278 -> 282[label="",style="solid", color="black", weight=3];
11.11/4.46	279[label="primMinusNatS (Succ wy200) Zero",fontsize=16,color="black",shape="box"];279 -> 283[label="",style="solid", color="black", weight=3];
11.11/4.46	280[label="primMinusNatS Zero (Succ wy210)",fontsize=16,color="black",shape="box"];280 -> 284[label="",style="solid", color="black", weight=3];
11.11/4.46	281[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];281 -> 285[label="",style="solid", color="black", weight=3];
11.11/4.46	282 -> 275[label="",style="dashed", color="red", weight=0];
11.11/4.46	282[label="primMinusNatS wy200 wy210",fontsize=16,color="magenta"];282 -> 286[label="",style="dashed", color="magenta", weight=3];
11.11/4.46	282 -> 287[label="",style="dashed", color="magenta", weight=3];
11.11/4.46	283[label="Succ wy200",fontsize=16,color="green",shape="box"];284[label="Zero",fontsize=16,color="green",shape="box"];285[label="Zero",fontsize=16,color="green",shape="box"];286[label="wy200",fontsize=16,color="green",shape="box"];287[label="wy210",fontsize=16,color="green",shape="box"];}
11.11/4.46	
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(6)
11.11/4.46	Complex Obligation (AND)
11.11/4.46	
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(7)
11.11/4.46	Obligation:
11.11/4.46	Q DP problem:
11.11/4.46	The TRS P consists of the following rules:
11.11/4.46	
11.11/4.46	   new_primMinusNatS(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS(wy200, wy210)
11.11/4.46	
11.11/4.46	R is empty.
11.11/4.46	Q is empty.
11.11/4.46	We have to consider all minimal (P,Q,R)-chains.
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(8) QDPSizeChangeProof (EQUIVALENT)
11.11/4.46	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. 
11.11/4.46	
11.11/4.46	From the DPs we obtained the following set of size-change graphs:
11.11/4.46	*new_primMinusNatS(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS(wy200, wy210)
11.11/4.46	The graph contains the following edges 1 > 1, 2 > 2
11.11/4.46	
11.11/4.46	
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(9)
11.11/4.46	YES
11.11/4.46	
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(10)
11.11/4.46	Obligation:
11.11/4.46	Q DP problem:
11.11/4.46	The TRS P consists of the following rules:
11.11/4.46	
11.11/4.46	   new_primDivNatS(Main.Succ(Main.Succ(wy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(wy30000), Main.Zero)
11.11/4.46	   new_primDivNatS0(wy20, wy21, Main.Succ(wy220), Main.Succ(wy230)) -> new_primDivNatS0(wy20, wy21, wy220, wy230)
11.11/4.46	   new_primDivNatS00(wy20, wy21) -> new_primDivNatS(new_primMinusNatS0(wy20, wy21), Main.Succ(wy21))
11.11/4.46	   new_primDivNatS0(wy20, wy21, Main.Zero, Main.Zero) -> new_primDivNatS00(wy20, wy21)
11.11/4.46	   new_primDivNatS(Main.Succ(Main.Succ(wy30000)), Main.Succ(wy31000)) -> new_primDivNatS0(wy30000, wy31000, wy30000, wy31000)
11.11/4.46	   new_primDivNatS0(wy20, wy21, Main.Succ(wy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(wy20, wy21), Main.Succ(wy21))
11.11/4.46	   new_primDivNatS(Main.Succ(Main.Zero), Main.Zero) -> new_primDivNatS(new_primMinusNatS2, Main.Zero)
11.11/4.46	
11.11/4.46	The TRS R consists of the following rules:
11.11/4.46	
11.11/4.46	   new_primMinusNatS0(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS0(wy200, wy210)
11.11/4.46	   new_primMinusNatS0(Main.Succ(wy200), Main.Zero) -> Main.Succ(wy200)
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Succ(wy210)) -> Main.Zero
11.11/4.46	   new_primMinusNatS2 -> Main.Zero
11.11/4.46	   new_primMinusNatS1(wy30000) -> Main.Succ(wy30000)
11.11/4.46	
11.11/4.46	The set Q consists of the following terms:
11.11/4.46	
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.11/4.46	   new_primMinusNatS2
11.11/4.46	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.11/4.46	   new_primMinusNatS1(x0)
11.11/4.46	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.11/4.46	
11.11/4.46	We have to consider all minimal (P,Q,R)-chains.
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(11) DependencyGraphProof (EQUIVALENT)
11.11/4.46	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(12)
11.11/4.46	Complex Obligation (AND)
11.11/4.46	
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(13)
11.11/4.46	Obligation:
11.11/4.46	Q DP problem:
11.11/4.46	The TRS P consists of the following rules:
11.11/4.46	
11.11/4.46	   new_primDivNatS0(wy20, wy21, Main.Zero, Main.Zero) -> new_primDivNatS00(wy20, wy21)
11.11/4.46	   new_primDivNatS00(wy20, wy21) -> new_primDivNatS(new_primMinusNatS0(wy20, wy21), Main.Succ(wy21))
11.11/4.46	   new_primDivNatS(Main.Succ(Main.Succ(wy30000)), Main.Succ(wy31000)) -> new_primDivNatS0(wy30000, wy31000, wy30000, wy31000)
11.11/4.46	   new_primDivNatS0(wy20, wy21, Main.Succ(wy220), Main.Succ(wy230)) -> new_primDivNatS0(wy20, wy21, wy220, wy230)
11.11/4.46	   new_primDivNatS0(wy20, wy21, Main.Succ(wy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(wy20, wy21), Main.Succ(wy21))
11.11/4.46	
11.11/4.46	The TRS R consists of the following rules:
11.11/4.46	
11.11/4.46	   new_primMinusNatS0(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS0(wy200, wy210)
11.11/4.46	   new_primMinusNatS0(Main.Succ(wy200), Main.Zero) -> Main.Succ(wy200)
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Succ(wy210)) -> Main.Zero
11.11/4.46	   new_primMinusNatS2 -> Main.Zero
11.11/4.46	   new_primMinusNatS1(wy30000) -> Main.Succ(wy30000)
11.11/4.46	
11.11/4.46	The set Q consists of the following terms:
11.11/4.46	
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.11/4.46	   new_primMinusNatS2
11.11/4.46	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.11/4.46	   new_primMinusNatS1(x0)
11.11/4.46	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.11/4.46	
11.11/4.46	We have to consider all minimal (P,Q,R)-chains.
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(14) QDPSizeChangeProof (EQUIVALENT)
11.11/4.46	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
11.11/4.46	
11.11/4.46	Order:Polynomial interpretation [POLO]:
11.11/4.46	
11.11/4.46	   POL(Main.Succ(x_1)) = 1 + x_1
11.11/4.46	   POL(Main.Zero) = 1
11.11/4.46	   POL(new_primMinusNatS0(x_1, x_2)) = x_1
11.11/4.46	
11.11/4.46	
11.11/4.46	
11.11/4.46	
11.11/4.46	From the DPs we obtained the following set of size-change graphs:
11.11/4.46	*new_primDivNatS00(wy20, wy21) -> new_primDivNatS(new_primMinusNatS0(wy20, wy21), Main.Succ(wy21)) (allowed arguments on rhs = {1, 2})
11.11/4.46	The graph contains the following edges 1 >= 1
11.11/4.46	
11.11/4.46	
11.11/4.46	*new_primDivNatS(Main.Succ(Main.Succ(wy30000)), Main.Succ(wy31000)) -> new_primDivNatS0(wy30000, wy31000, wy30000, wy31000) (allowed arguments on rhs = {1, 2, 3, 4})
11.11/4.46	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
11.11/4.46	
11.11/4.46	
11.11/4.46	*new_primDivNatS0(wy20, wy21, Main.Succ(wy220), Main.Succ(wy230)) -> new_primDivNatS0(wy20, wy21, wy220, wy230) (allowed arguments on rhs = {1, 2, 3, 4})
11.11/4.46	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
11.11/4.46	
11.11/4.46	
11.11/4.46	*new_primDivNatS0(wy20, wy21, Main.Zero, Main.Zero) -> new_primDivNatS00(wy20, wy21) (allowed arguments on rhs = {1, 2})
11.11/4.46	The graph contains the following edges 1 >= 1, 2 >= 2
11.11/4.46	
11.11/4.46	
11.11/4.46	*new_primDivNatS0(wy20, wy21, Main.Succ(wy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(wy20, wy21), Main.Succ(wy21)) (allowed arguments on rhs = {1, 2})
11.11/4.46	The graph contains the following edges 1 >= 1
11.11/4.46	
11.11/4.46	
11.11/4.46	
11.11/4.46	We oriented the following set of usable rules [AAECC05,FROCOS05].
11.11/4.46	
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Succ(wy210)) -> Main.Zero
11.11/4.46	   new_primMinusNatS0(Main.Succ(wy200), Main.Zero) -> Main.Succ(wy200)
11.11/4.46	   new_primMinusNatS0(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS0(wy200, wy210)
11.11/4.46	
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(15)
11.11/4.46	YES
11.11/4.46	
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(16)
11.11/4.46	Obligation:
11.11/4.46	Q DP problem:
11.11/4.46	The TRS P consists of the following rules:
11.11/4.46	
11.11/4.46	   new_primDivNatS(Main.Succ(Main.Succ(wy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(wy30000), Main.Zero)
11.11/4.46	
11.11/4.46	The TRS R consists of the following rules:
11.11/4.46	
11.11/4.46	   new_primMinusNatS0(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS0(wy200, wy210)
11.11/4.46	   new_primMinusNatS0(Main.Succ(wy200), Main.Zero) -> Main.Succ(wy200)
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Succ(wy210)) -> Main.Zero
11.11/4.46	   new_primMinusNatS2 -> Main.Zero
11.11/4.46	   new_primMinusNatS1(wy30000) -> Main.Succ(wy30000)
11.11/4.46	
11.11/4.46	The set Q consists of the following terms:
11.11/4.46	
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.11/4.46	   new_primMinusNatS2
11.11/4.46	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.11/4.46	   new_primMinusNatS1(x0)
11.11/4.46	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.11/4.46	
11.11/4.46	We have to consider all minimal (P,Q,R)-chains.
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(17) MRRProof (EQUIVALENT)
11.11/4.46	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.
11.11/4.46	
11.11/4.46	Strictly oriented dependency pairs:
11.11/4.46	
11.11/4.46	   new_primDivNatS(Main.Succ(Main.Succ(wy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(wy30000), Main.Zero)
11.11/4.46	
11.11/4.46	Strictly oriented rules of the TRS R:
11.11/4.46	
11.11/4.46	   new_primMinusNatS0(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS0(wy200, wy210)
11.11/4.46	   new_primMinusNatS0(Main.Succ(wy200), Main.Zero) -> Main.Succ(wy200)
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Succ(wy210)) -> Main.Zero
11.11/4.46	
11.11/4.46	Used ordering: Polynomial interpretation [POLO]:
11.11/4.46	
11.11/4.46	   POL(Main.Succ(x_1)) = 1 + x_1
11.11/4.46	   POL(Main.Zero) = 2
11.11/4.46	   POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2
11.11/4.46	   POL(new_primMinusNatS0(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
11.11/4.46	   POL(new_primMinusNatS1(x_1)) = 1 + x_1
11.11/4.46	   POL(new_primMinusNatS2) = 2
11.11/4.46	
11.11/4.46	
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(18)
11.11/4.46	Obligation:
11.11/4.46	Q DP problem:
11.11/4.46	P is empty.
11.11/4.46	The TRS R consists of the following rules:
11.11/4.46	
11.11/4.46	   new_primMinusNatS2 -> Main.Zero
11.11/4.46	   new_primMinusNatS1(wy30000) -> Main.Succ(wy30000)
11.11/4.46	
11.11/4.46	The set Q consists of the following terms:
11.11/4.46	
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.11/4.46	   new_primMinusNatS2
11.11/4.46	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.11/4.46	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.11/4.46	   new_primMinusNatS1(x0)
11.11/4.46	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.11/4.46	
11.11/4.46	We have to consider all minimal (P,Q,R)-chains.
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(19) PisEmptyProof (EQUIVALENT)
11.11/4.46	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.11/4.46	----------------------------------------
11.11/4.46	
11.11/4.46	(20)
11.11/4.46	YES
11.32/4.51	EOF