8.96/3.93	YES
11.17/4.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.17/4.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.17/4.54	
11.17/4.54	
11.17/4.54	H-Termination with start terms of the given HASKELL could be proven:
11.17/4.54	
11.17/4.54	(0) HASKELL
11.17/4.54	(1) BR [EQUIVALENT, 0 ms]
11.17/4.54	(2) HASKELL
11.17/4.54	(3) COR [EQUIVALENT, 0 ms]
11.17/4.54	(4) HASKELL
11.17/4.54	(5) Narrow [SOUND, 0 ms]
11.17/4.54	(6) AND
11.17/4.54	    (7) QDP
11.17/4.54	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.17/4.54	        (9) YES
11.17/4.54	    (10) QDP
11.17/4.54	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.17/4.54	        (12) YES
11.17/4.54	    (13) QDP
11.17/4.54	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.17/4.54	        (15) YES
11.17/4.54	    (16) QDP
11.17/4.54	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.17/4.54	        (18) YES
11.17/4.54	    (19) QDP
11.17/4.54	        (20) DependencyGraphProof [EQUIVALENT, 0 ms]
11.17/4.54	        (21) AND
11.17/4.54	            (22) QDP
11.17/4.54	                (23) QDPSizeChangeProof [EQUIVALENT, 29 ms]
11.17/4.54	                (24) YES
11.17/4.54	            (25) QDP
11.17/4.54	                (26) MRRProof [EQUIVALENT, 0 ms]
11.17/4.54	                (27) QDP
11.17/4.54	                (28) PisEmptyProof [EQUIVALENT, 0 ms]
11.17/4.54	                (29) YES
11.17/4.54	
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(0)
11.17/4.54	Obligation:
11.17/4.54	mainModule Main 
11.17/4.54	module Main where {
11.17/4.54	import qualified Prelude;
11.17/4.54	data Float = Float MyInt MyInt ;
11.17/4.54	
11.17/4.54	data MyBool = MyTrue  | MyFalse ;
11.17/4.54	
11.17/4.54	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
11.17/4.54	
11.17/4.54	data Main.Nat = Succ Main.Nat  | Zero ;
11.17/4.54	
11.17/4.54	data Tup2 b a = Tup2 b a ;
11.17/4.54	
11.17/4.54	error :: a;
11.17/4.54	error = stop MyTrue;
11.17/4.54	
11.17/4.54	floatProperFractionFloat (Float vv vw) = Tup2 (quotMyInt vv vw) (msFloat (Float vv vw) (fromIntFloat (quotMyInt vv vw)));
11.17/4.54	
11.17/4.54	fromIntFloat :: MyInt  ->  Float;
11.17/4.54	fromIntFloat = primIntToFloat;
11.17/4.54	
11.17/4.54	msFloat :: Float  ->  Float  ->  Float;
11.17/4.54	msFloat = primMinusFloat;
11.17/4.54	
11.17/4.54	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	msMyInt = primMinusInt;
11.17/4.54	
11.17/4.54	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.17/4.54	primDivNatS Main.Zero Main.Zero = Main.error;
11.17/4.54	primDivNatS (Main.Succ x) Main.Zero = Main.error;
11.17/4.54	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
11.17/4.54	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
11.17/4.54	
11.17/4.54	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
11.17/4.54	primDivNatS0 x y MyFalse = Main.Zero;
11.17/4.54	
11.17/4.54	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
11.17/4.54	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
11.17/4.54	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
11.17/4.54	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
11.17/4.54	primGEqNatS Main.Zero Main.Zero = MyTrue;
11.17/4.54	
11.17/4.54	primIntToFloat :: MyInt  ->  Float;
11.17/4.54	primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));
11.17/4.54	
11.17/4.54	primMinusFloat :: Float  ->  Float  ->  Float;
11.17/4.54	primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);
11.17/4.54	
11.17/4.54	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
11.17/4.54	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
11.17/4.54	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
11.17/4.54	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
11.17/4.54	
11.17/4.54	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
11.17/4.54	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
11.17/4.54	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
11.17/4.54	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
11.17/4.54	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
11.17/4.54	
11.17/4.54	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.17/4.54	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
11.17/4.54	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
11.17/4.54	primMinusNatS x Main.Zero = x;
11.17/4.54	
11.17/4.54	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
11.17/4.54	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
11.17/4.54	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
11.17/4.54	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
11.17/4.54	
11.17/4.54	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.17/4.54	primMulNat Main.Zero Main.Zero = Main.Zero;
11.17/4.54	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
11.17/4.54	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
11.17/4.54	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
11.17/4.54	
11.17/4.54	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.17/4.54	primPlusNat Main.Zero Main.Zero = Main.Zero;
11.17/4.54	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
11.17/4.54	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
11.17/4.54	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
11.17/4.54	
11.17/4.54	primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.17/4.54	primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.17/4.54	primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.17/4.54	primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.17/4.54	primQuotInt vx vy = Main.error;
11.17/4.54	
11.17/4.54	properFractionFloat :: Float  ->  Tup2 MyInt Float;
11.17/4.54	properFractionFloat = floatProperFractionFloat;
11.17/4.54	
11.17/4.54	quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	quotMyInt = primQuotInt;
11.17/4.54	
11.17/4.54	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	srMyInt = primMulInt;
11.17/4.54	
11.17/4.54	stop :: MyBool  ->  a;
11.17/4.54	stop MyFalse = stop MyFalse;
11.17/4.54	
11.17/4.54	}
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(1) BR (EQUIVALENT)
11.17/4.54	Replaced joker patterns by fresh variables and removed binding patterns.
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(2)
11.17/4.54	Obligation:
11.17/4.54	mainModule Main 
11.17/4.54	module Main where {
11.17/4.54	import qualified Prelude;
11.17/4.54	data Float = Float MyInt MyInt ;
11.17/4.54	
11.17/4.54	data MyBool = MyTrue  | MyFalse ;
11.17/4.54	
11.17/4.54	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
11.17/4.54	
11.17/4.54	data Main.Nat = Succ Main.Nat  | Zero ;
11.17/4.54	
11.17/4.54	data Tup2 a b = Tup2 a b ;
11.17/4.54	
11.17/4.54	error :: a;
11.17/4.54	error = stop MyTrue;
11.17/4.54	
11.17/4.54	floatProperFractionFloat (Float vv vw) = Tup2 (quotMyInt vv vw) (msFloat (Float vv vw) (fromIntFloat (quotMyInt vv vw)));
11.17/4.54	
11.17/4.54	fromIntFloat :: MyInt  ->  Float;
11.17/4.54	fromIntFloat = primIntToFloat;
11.17/4.54	
11.17/4.54	msFloat :: Float  ->  Float  ->  Float;
11.17/4.54	msFloat = primMinusFloat;
11.17/4.54	
11.17/4.54	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	msMyInt = primMinusInt;
11.17/4.54	
11.17/4.54	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.17/4.54	primDivNatS Main.Zero Main.Zero = Main.error;
11.17/4.54	primDivNatS (Main.Succ x) Main.Zero = Main.error;
11.17/4.54	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
11.17/4.54	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
11.17/4.54	
11.17/4.54	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
11.17/4.54	primDivNatS0 x y MyFalse = Main.Zero;
11.17/4.54	
11.17/4.54	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
11.17/4.54	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
11.17/4.54	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
11.17/4.54	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
11.17/4.54	primGEqNatS Main.Zero Main.Zero = MyTrue;
11.17/4.54	
11.17/4.54	primIntToFloat :: MyInt  ->  Float;
11.17/4.54	primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));
11.17/4.54	
11.17/4.54	primMinusFloat :: Float  ->  Float  ->  Float;
11.17/4.54	primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);
11.17/4.54	
11.17/4.54	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
11.17/4.54	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
11.17/4.54	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
11.17/4.54	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
11.17/4.54	
11.17/4.54	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
11.17/4.54	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
11.17/4.54	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
11.17/4.54	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
11.17/4.54	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
11.17/4.54	
11.17/4.54	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.17/4.54	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
11.17/4.54	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
11.17/4.54	primMinusNatS x Main.Zero = x;
11.17/4.54	
11.17/4.54	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
11.17/4.54	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
11.17/4.54	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
11.17/4.54	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
11.17/4.54	
11.17/4.54	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.17/4.54	primMulNat Main.Zero Main.Zero = Main.Zero;
11.17/4.54	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
11.17/4.54	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
11.17/4.54	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
11.17/4.54	
11.17/4.54	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.17/4.54	primPlusNat Main.Zero Main.Zero = Main.Zero;
11.17/4.54	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
11.17/4.54	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
11.17/4.54	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
11.17/4.54	
11.17/4.54	primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.17/4.54	primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.17/4.54	primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.17/4.54	primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.17/4.54	primQuotInt vx vy = Main.error;
11.17/4.54	
11.17/4.54	properFractionFloat :: Float  ->  Tup2 MyInt Float;
11.17/4.54	properFractionFloat = floatProperFractionFloat;
11.17/4.54	
11.17/4.54	quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	quotMyInt = primQuotInt;
11.17/4.54	
11.17/4.54	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	srMyInt = primMulInt;
11.17/4.54	
11.17/4.54	stop :: MyBool  ->  a;
11.17/4.54	stop MyFalse = stop MyFalse;
11.17/4.54	
11.17/4.54	}
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(3) COR (EQUIVALENT)
11.17/4.54	Cond Reductions:
11.17/4.54	The following Function with conditions
11.17/4.54	"undefined |Falseundefined;
11.17/4.54	"
11.17/4.54	is transformed to
11.17/4.54	"undefined  = undefined1;
11.17/4.54	"
11.17/4.54	"undefined0 True = undefined;
11.17/4.54	"
11.17/4.54	"undefined1  = undefined0 False;
11.17/4.54	"
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(4)
11.17/4.54	Obligation:
11.17/4.54	mainModule Main 
11.17/4.54	module Main where {
11.17/4.54	import qualified Prelude;
11.17/4.54	data Float = Float MyInt MyInt ;
11.17/4.54	
11.17/4.54	data MyBool = MyTrue  | MyFalse ;
11.17/4.54	
11.17/4.54	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
11.17/4.54	
11.17/4.54	data Main.Nat = Succ Main.Nat  | Zero ;
11.17/4.54	
11.17/4.54	data Tup2 b a = Tup2 b a ;
11.17/4.54	
11.17/4.54	error :: a;
11.17/4.54	error = stop MyTrue;
11.17/4.54	
11.17/4.54	floatProperFractionFloat (Float vv vw) = Tup2 (quotMyInt vv vw) (msFloat (Float vv vw) (fromIntFloat (quotMyInt vv vw)));
11.17/4.54	
11.17/4.54	fromIntFloat :: MyInt  ->  Float;
11.17/4.54	fromIntFloat = primIntToFloat;
11.17/4.54	
11.17/4.54	msFloat :: Float  ->  Float  ->  Float;
11.17/4.54	msFloat = primMinusFloat;
11.17/4.54	
11.17/4.54	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	msMyInt = primMinusInt;
11.17/4.54	
11.17/4.54	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.17/4.54	primDivNatS Main.Zero Main.Zero = Main.error;
11.17/4.54	primDivNatS (Main.Succ x) Main.Zero = Main.error;
11.17/4.54	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
11.17/4.54	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
11.17/4.54	
11.17/4.54	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
11.17/4.54	primDivNatS0 x y MyFalse = Main.Zero;
11.17/4.54	
11.17/4.54	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
11.17/4.54	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
11.17/4.54	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
11.17/4.54	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
11.17/4.54	primGEqNatS Main.Zero Main.Zero = MyTrue;
11.17/4.54	
11.17/4.54	primIntToFloat :: MyInt  ->  Float;
11.17/4.54	primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));
11.17/4.54	
11.17/4.54	primMinusFloat :: Float  ->  Float  ->  Float;
11.17/4.54	primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);
11.17/4.54	
11.17/4.54	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
11.17/4.54	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
11.17/4.54	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
11.17/4.54	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
11.17/4.54	
11.17/4.54	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
11.17/4.54	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
11.17/4.54	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
11.17/4.54	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
11.17/4.54	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
11.17/4.54	
11.17/4.54	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.17/4.54	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
11.17/4.54	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
11.17/4.54	primMinusNatS x Main.Zero = x;
11.17/4.54	
11.17/4.54	primMulInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
11.17/4.54	primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
11.17/4.54	primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
11.17/4.54	primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);
11.17/4.54	
11.17/4.54	primMulNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.17/4.54	primMulNat Main.Zero Main.Zero = Main.Zero;
11.17/4.54	primMulNat Main.Zero (Main.Succ y) = Main.Zero;
11.17/4.54	primMulNat (Main.Succ x) Main.Zero = Main.Zero;
11.17/4.54	primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y);
11.17/4.54	
11.17/4.54	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
11.17/4.54	primPlusNat Main.Zero Main.Zero = Main.Zero;
11.17/4.54	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
11.17/4.54	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
11.17/4.54	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
11.17/4.54	
11.17/4.54	primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.17/4.54	primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.17/4.54	primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
11.17/4.54	primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
11.17/4.54	primQuotInt vx vy = Main.error;
11.17/4.54	
11.17/4.54	properFractionFloat :: Float  ->  Tup2 MyInt Float;
11.17/4.54	properFractionFloat = floatProperFractionFloat;
11.17/4.54	
11.17/4.54	quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	quotMyInt = primQuotInt;
11.17/4.54	
11.17/4.54	srMyInt :: MyInt  ->  MyInt  ->  MyInt;
11.17/4.54	srMyInt = primMulInt;
11.17/4.54	
11.17/4.54	stop :: MyBool  ->  a;
11.17/4.54	stop MyFalse = stop MyFalse;
11.17/4.54	
11.17/4.54	}
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(5) Narrow (SOUND)
11.17/4.54	Haskell To QDPs
11.17/4.54	
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11.17/4.54	349 -> 17[label="",style="solid", color="burlywood", weight=3];
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11.17/4.54	350 -> 18[label="",style="solid", color="burlywood", weight=3];
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11.17/4.54	23[label="primQuotInt (Pos wv300) (Neg Zero)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
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11.17/4.54	25[label="primQuotInt (Neg wv300) (Pos Zero)",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
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11.17/4.54	27[label="primQuotInt (Neg wv300) (Neg Zero)",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
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11.17/4.54	29[label="Pos (primDivNatS wv300 (Succ wv3100))",fontsize=16,color="green",shape="box"];29 -> 38[label="",style="dashed", color="green", weight=3];
11.17/4.54	30[label="error",fontsize=16,color="black",shape="triangle"];30 -> 39[label="",style="solid", color="black", weight=3];
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11.17/4.54	32 -> 30[label="",style="dashed", color="red", weight=0];
11.17/4.54	32[label="error",fontsize=16,color="magenta"];33[label="Neg (primDivNatS wv300 (Succ wv3100))",fontsize=16,color="green",shape="box"];33 -> 41[label="",style="dashed", color="green", weight=3];
11.17/4.54	34 -> 30[label="",style="dashed", color="red", weight=0];
11.17/4.54	34[label="error",fontsize=16,color="magenta"];35[label="Pos (primDivNatS wv300 (Succ wv3100))",fontsize=16,color="green",shape="box"];35 -> 42[label="",style="dashed", color="green", weight=3];
11.17/4.54	36 -> 30[label="",style="dashed", color="red", weight=0];
11.17/4.54	36[label="error",fontsize=16,color="magenta"];37[label="Float (msMyInt wv30 wv4) (srMyInt wv31 (Pos (Succ Zero)))",fontsize=16,color="green",shape="box"];37 -> 43[label="",style="dashed", color="green", weight=3];
11.17/4.54	37 -> 44[label="",style="dashed", color="green", weight=3];
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11.17/4.54	359 -> 45[label="",style="solid", color="burlywood", weight=3];
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11.17/4.54	360 -> 46[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	39[label="stop MyTrue",fontsize=16,color="black",shape="box"];39 -> 47[label="",style="solid", color="black", weight=3];
11.17/4.54	40 -> 38[label="",style="dashed", color="red", weight=0];
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11.17/4.54	42 -> 38[label="",style="dashed", color="red", weight=0];
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11.17/4.54	42 -> 51[label="",style="dashed", color="magenta", weight=3];
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11.17/4.54	44[label="srMyInt wv31 (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];44 -> 53[label="",style="solid", color="black", weight=3];
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11.17/4.54	46[label="primDivNatS Zero (Succ wv3100)",fontsize=16,color="black",shape="box"];46 -> 55[label="",style="solid", color="black", weight=3];
11.17/4.54	47[label="error []",fontsize=16,color="red",shape="box"];48[label="wv3100",fontsize=16,color="green",shape="box"];49[label="wv300",fontsize=16,color="green",shape="box"];50[label="wv300",fontsize=16,color="green",shape="box"];51[label="wv3100",fontsize=16,color="green",shape="box"];52[label="primMinusInt wv30 wv4",fontsize=16,color="burlywood",shape="box"];361[label="wv30/Pos wv300",fontsize=10,color="white",style="solid",shape="box"];52 -> 361[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	361 -> 56[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	362[label="wv30/Neg wv300",fontsize=10,color="white",style="solid",shape="box"];52 -> 362[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	362 -> 57[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	53[label="primMulInt wv31 (Pos (Succ Zero))",fontsize=16,color="burlywood",shape="box"];363[label="wv31/Pos wv310",fontsize=10,color="white",style="solid",shape="box"];53 -> 363[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	363 -> 58[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	364[label="wv31/Neg wv310",fontsize=10,color="white",style="solid",shape="box"];53 -> 364[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	364 -> 59[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	54[label="primDivNatS0 wv3000 wv3100 (primGEqNatS wv3000 wv3100)",fontsize=16,color="burlywood",shape="box"];365[label="wv3000/Succ wv30000",fontsize=10,color="white",style="solid",shape="box"];54 -> 365[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	365 -> 60[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	366[label="wv3000/Zero",fontsize=10,color="white",style="solid",shape="box"];54 -> 366[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	366 -> 61[label="",style="solid", color="burlywood", weight=3];
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11.17/4.54	367 -> 62[label="",style="solid", color="burlywood", weight=3];
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11.17/4.54	369 -> 64[label="",style="solid", color="burlywood", weight=3];
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11.17/4.54	70[label="primDivNatS0 Zero (Succ wv31000) (primGEqNatS Zero (Succ wv31000))",fontsize=16,color="black",shape="box"];70 -> 80[label="",style="solid", color="black", weight=3];
11.17/4.54	71[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];71 -> 81[label="",style="solid", color="black", weight=3];
11.17/4.54	72[label="primMinusNat wv300 wv40",fontsize=16,color="burlywood",shape="triangle"];375[label="wv300/Succ wv3000",fontsize=10,color="white",style="solid",shape="box"];72 -> 375[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	375 -> 82[label="",style="solid", color="burlywood", weight=3];
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11.17/4.54	376 -> 83[label="",style="solid", color="burlywood", weight=3];
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11.17/4.54	75 -> 72[label="",style="dashed", color="red", weight=0];
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11.17/4.54	75 -> 87[label="",style="dashed", color="magenta", weight=3];
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11.17/4.54	377 -> 88[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	378[label="wv310/Zero",fontsize=10,color="white",style="solid",shape="box"];76 -> 378[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	378 -> 89[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	77 -> 76[label="",style="dashed", color="red", weight=0];
11.17/4.54	77[label="primMulNat wv310 (Succ Zero)",fontsize=16,color="magenta"];77 -> 90[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	78 -> 281[label="",style="dashed", color="red", weight=0];
11.17/4.54	78[label="primDivNatS0 (Succ wv30000) (Succ wv31000) (primGEqNatS wv30000 wv31000)",fontsize=16,color="magenta"];78 -> 282[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	78 -> 283[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	78 -> 284[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	78 -> 285[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	79[label="primDivNatS0 (Succ wv30000) Zero MyTrue",fontsize=16,color="black",shape="box"];79 -> 93[label="",style="solid", color="black", weight=3];
11.17/4.54	80[label="primDivNatS0 Zero (Succ wv31000) MyFalse",fontsize=16,color="black",shape="box"];80 -> 94[label="",style="solid", color="black", weight=3];
11.17/4.54	81[label="primDivNatS0 Zero Zero MyTrue",fontsize=16,color="black",shape="box"];81 -> 95[label="",style="solid", color="black", weight=3];
11.17/4.54	82[label="primMinusNat (Succ wv3000) wv40",fontsize=16,color="burlywood",shape="box"];379[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];82 -> 379[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	379 -> 96[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	380[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];82 -> 380[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	380 -> 97[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	83[label="primMinusNat Zero wv40",fontsize=16,color="burlywood",shape="box"];381[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];83 -> 381[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	381 -> 98[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	382[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];83 -> 382[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	382 -> 99[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	84[label="primPlusNat wv300 wv40",fontsize=16,color="burlywood",shape="triangle"];383[label="wv300/Succ wv3000",fontsize=10,color="white",style="solid",shape="box"];84 -> 383[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	383 -> 100[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	384[label="wv300/Zero",fontsize=10,color="white",style="solid",shape="box"];84 -> 384[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	384 -> 101[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	85 -> 84[label="",style="dashed", color="red", weight=0];
11.17/4.54	85[label="primPlusNat wv300 wv40",fontsize=16,color="magenta"];85 -> 102[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	85 -> 103[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	86[label="wv300",fontsize=16,color="green",shape="box"];87[label="wv40",fontsize=16,color="green",shape="box"];88[label="primMulNat (Succ wv3100) (Succ Zero)",fontsize=16,color="black",shape="box"];88 -> 104[label="",style="solid", color="black", weight=3];
11.17/4.54	89[label="primMulNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];89 -> 105[label="",style="solid", color="black", weight=3];
11.17/4.54	90[label="wv310",fontsize=16,color="green",shape="box"];282[label="wv31000",fontsize=16,color="green",shape="box"];283[label="wv30000",fontsize=16,color="green",shape="box"];284[label="wv30000",fontsize=16,color="green",shape="box"];285[label="wv31000",fontsize=16,color="green",shape="box"];281[label="primDivNatS0 (Succ wv21) (Succ wv22) (primGEqNatS wv23 wv24)",fontsize=16,color="burlywood",shape="triangle"];385[label="wv23/Succ wv230",fontsize=10,color="white",style="solid",shape="box"];281 -> 385[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	385 -> 314[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	386[label="wv23/Zero",fontsize=10,color="white",style="solid",shape="box"];281 -> 386[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	386 -> 315[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	93[label="Succ (primDivNatS (primMinusNatS (Succ wv30000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];93 -> 110[label="",style="dashed", color="green", weight=3];
11.17/4.54	94[label="Zero",fontsize=16,color="green",shape="box"];95[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];95 -> 111[label="",style="dashed", color="green", weight=3];
11.17/4.54	96[label="primMinusNat (Succ wv3000) (Succ wv400)",fontsize=16,color="black",shape="box"];96 -> 112[label="",style="solid", color="black", weight=3];
11.17/4.54	97[label="primMinusNat (Succ wv3000) Zero",fontsize=16,color="black",shape="box"];97 -> 113[label="",style="solid", color="black", weight=3];
11.17/4.54	98[label="primMinusNat Zero (Succ wv400)",fontsize=16,color="black",shape="box"];98 -> 114[label="",style="solid", color="black", weight=3];
11.17/4.54	99[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];99 -> 115[label="",style="solid", color="black", weight=3];
11.17/4.54	100[label="primPlusNat (Succ wv3000) wv40",fontsize=16,color="burlywood",shape="box"];387[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];100 -> 387[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	387 -> 116[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	388[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];100 -> 388[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	388 -> 117[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	101[label="primPlusNat Zero wv40",fontsize=16,color="burlywood",shape="box"];389[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];101 -> 389[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	389 -> 118[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	390[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];101 -> 390[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	390 -> 119[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	102[label="wv40",fontsize=16,color="green",shape="box"];103[label="wv300",fontsize=16,color="green",shape="box"];104 -> 84[label="",style="dashed", color="red", weight=0];
11.17/4.54	104[label="primPlusNat (primMulNat wv3100 (Succ Zero)) (Succ Zero)",fontsize=16,color="magenta"];104 -> 120[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	104 -> 121[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	105[label="Zero",fontsize=16,color="green",shape="box"];314[label="primDivNatS0 (Succ wv21) (Succ wv22) (primGEqNatS (Succ wv230) wv24)",fontsize=16,color="burlywood",shape="box"];391[label="wv24/Succ wv240",fontsize=10,color="white",style="solid",shape="box"];314 -> 391[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	391 -> 316[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	392[label="wv24/Zero",fontsize=10,color="white",style="solid",shape="box"];314 -> 392[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	392 -> 317[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	315[label="primDivNatS0 (Succ wv21) (Succ wv22) (primGEqNatS Zero wv24)",fontsize=16,color="burlywood",shape="box"];393[label="wv24/Succ wv240",fontsize=10,color="white",style="solid",shape="box"];315 -> 393[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	393 -> 318[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	394[label="wv24/Zero",fontsize=10,color="white",style="solid",shape="box"];315 -> 394[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	394 -> 319[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	110 -> 38[label="",style="dashed", color="red", weight=0];
11.17/4.54	110[label="primDivNatS (primMinusNatS (Succ wv30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];110 -> 126[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	110 -> 127[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	111 -> 38[label="",style="dashed", color="red", weight=0];
11.17/4.54	111[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];111 -> 128[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	111 -> 129[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	112 -> 72[label="",style="dashed", color="red", weight=0];
11.17/4.54	112[label="primMinusNat wv3000 wv400",fontsize=16,color="magenta"];112 -> 130[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	112 -> 131[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	113[label="Pos (Succ wv3000)",fontsize=16,color="green",shape="box"];114[label="Neg (Succ wv400)",fontsize=16,color="green",shape="box"];115[label="Pos Zero",fontsize=16,color="green",shape="box"];116[label="primPlusNat (Succ wv3000) (Succ wv400)",fontsize=16,color="black",shape="box"];116 -> 132[label="",style="solid", color="black", weight=3];
11.17/4.54	117[label="primPlusNat (Succ wv3000) Zero",fontsize=16,color="black",shape="box"];117 -> 133[label="",style="solid", color="black", weight=3];
11.17/4.54	118[label="primPlusNat Zero (Succ wv400)",fontsize=16,color="black",shape="box"];118 -> 134[label="",style="solid", color="black", weight=3];
11.17/4.54	119[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];119 -> 135[label="",style="solid", color="black", weight=3];
11.17/4.54	120[label="Succ Zero",fontsize=16,color="green",shape="box"];121 -> 76[label="",style="dashed", color="red", weight=0];
11.17/4.54	121[label="primMulNat wv3100 (Succ Zero)",fontsize=16,color="magenta"];121 -> 136[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	316[label="primDivNatS0 (Succ wv21) (Succ wv22) (primGEqNatS (Succ wv230) (Succ wv240))",fontsize=16,color="black",shape="box"];316 -> 320[label="",style="solid", color="black", weight=3];
11.17/4.54	317[label="primDivNatS0 (Succ wv21) (Succ wv22) (primGEqNatS (Succ wv230) Zero)",fontsize=16,color="black",shape="box"];317 -> 321[label="",style="solid", color="black", weight=3];
11.17/4.54	318[label="primDivNatS0 (Succ wv21) (Succ wv22) (primGEqNatS Zero (Succ wv240))",fontsize=16,color="black",shape="box"];318 -> 322[label="",style="solid", color="black", weight=3];
11.17/4.54	319[label="primDivNatS0 (Succ wv21) (Succ wv22) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];319 -> 323[label="",style="solid", color="black", weight=3];
11.17/4.54	126[label="primMinusNatS (Succ wv30000) Zero",fontsize=16,color="black",shape="triangle"];126 -> 142[label="",style="solid", color="black", weight=3];
11.17/4.54	127[label="Zero",fontsize=16,color="green",shape="box"];128[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];128 -> 143[label="",style="solid", color="black", weight=3];
11.17/4.54	129[label="Zero",fontsize=16,color="green",shape="box"];130[label="wv400",fontsize=16,color="green",shape="box"];131[label="wv3000",fontsize=16,color="green",shape="box"];132[label="Succ (Succ (primPlusNat wv3000 wv400))",fontsize=16,color="green",shape="box"];132 -> 144[label="",style="dashed", color="green", weight=3];
11.17/4.54	133[label="Succ wv3000",fontsize=16,color="green",shape="box"];134[label="Succ wv400",fontsize=16,color="green",shape="box"];135[label="Zero",fontsize=16,color="green",shape="box"];136[label="wv3100",fontsize=16,color="green",shape="box"];320 -> 281[label="",style="dashed", color="red", weight=0];
11.17/4.54	320[label="primDivNatS0 (Succ wv21) (Succ wv22) (primGEqNatS wv230 wv240)",fontsize=16,color="magenta"];320 -> 324[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	320 -> 325[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	321[label="primDivNatS0 (Succ wv21) (Succ wv22) MyTrue",fontsize=16,color="black",shape="triangle"];321 -> 326[label="",style="solid", color="black", weight=3];
11.17/4.54	322[label="primDivNatS0 (Succ wv21) (Succ wv22) MyFalse",fontsize=16,color="black",shape="box"];322 -> 327[label="",style="solid", color="black", weight=3];
11.17/4.54	323 -> 321[label="",style="dashed", color="red", weight=0];
11.17/4.54	323[label="primDivNatS0 (Succ wv21) (Succ wv22) MyTrue",fontsize=16,color="magenta"];142[label="Succ wv30000",fontsize=16,color="green",shape="box"];143[label="Zero",fontsize=16,color="green",shape="box"];144 -> 84[label="",style="dashed", color="red", weight=0];
11.17/4.54	144[label="primPlusNat wv3000 wv400",fontsize=16,color="magenta"];144 -> 151[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	144 -> 152[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	324[label="wv230",fontsize=16,color="green",shape="box"];325[label="wv240",fontsize=16,color="green",shape="box"];326[label="Succ (primDivNatS (primMinusNatS (Succ wv21) (Succ wv22)) (Succ (Succ wv22)))",fontsize=16,color="green",shape="box"];326 -> 328[label="",style="dashed", color="green", weight=3];
11.17/4.54	327[label="Zero",fontsize=16,color="green",shape="box"];151[label="wv400",fontsize=16,color="green",shape="box"];152[label="wv3000",fontsize=16,color="green",shape="box"];328 -> 38[label="",style="dashed", color="red", weight=0];
11.17/4.54	328[label="primDivNatS (primMinusNatS (Succ wv21) (Succ wv22)) (Succ (Succ wv22))",fontsize=16,color="magenta"];328 -> 329[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	328 -> 330[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	329[label="primMinusNatS (Succ wv21) (Succ wv22)",fontsize=16,color="black",shape="box"];329 -> 331[label="",style="solid", color="black", weight=3];
11.17/4.54	330[label="Succ wv22",fontsize=16,color="green",shape="box"];331[label="primMinusNatS wv21 wv22",fontsize=16,color="burlywood",shape="triangle"];395[label="wv21/Succ wv210",fontsize=10,color="white",style="solid",shape="box"];331 -> 395[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	395 -> 332[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	396[label="wv21/Zero",fontsize=10,color="white",style="solid",shape="box"];331 -> 396[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	396 -> 333[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	332[label="primMinusNatS (Succ wv210) wv22",fontsize=16,color="burlywood",shape="box"];397[label="wv22/Succ wv220",fontsize=10,color="white",style="solid",shape="box"];332 -> 397[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	397 -> 334[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	398[label="wv22/Zero",fontsize=10,color="white",style="solid",shape="box"];332 -> 398[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	398 -> 335[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	333[label="primMinusNatS Zero wv22",fontsize=16,color="burlywood",shape="box"];399[label="wv22/Succ wv220",fontsize=10,color="white",style="solid",shape="box"];333 -> 399[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	399 -> 336[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	400[label="wv22/Zero",fontsize=10,color="white",style="solid",shape="box"];333 -> 400[label="",style="solid", color="burlywood", weight=9];
11.17/4.54	400 -> 337[label="",style="solid", color="burlywood", weight=3];
11.17/4.54	334[label="primMinusNatS (Succ wv210) (Succ wv220)",fontsize=16,color="black",shape="box"];334 -> 338[label="",style="solid", color="black", weight=3];
11.17/4.54	335[label="primMinusNatS (Succ wv210) Zero",fontsize=16,color="black",shape="box"];335 -> 339[label="",style="solid", color="black", weight=3];
11.17/4.54	336[label="primMinusNatS Zero (Succ wv220)",fontsize=16,color="black",shape="box"];336 -> 340[label="",style="solid", color="black", weight=3];
11.17/4.54	337[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];337 -> 341[label="",style="solid", color="black", weight=3];
11.17/4.54	338 -> 331[label="",style="dashed", color="red", weight=0];
11.17/4.54	338[label="primMinusNatS wv210 wv220",fontsize=16,color="magenta"];338 -> 342[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	338 -> 343[label="",style="dashed", color="magenta", weight=3];
11.17/4.54	339[label="Succ wv210",fontsize=16,color="green",shape="box"];340[label="Zero",fontsize=16,color="green",shape="box"];341[label="Zero",fontsize=16,color="green",shape="box"];342[label="wv220",fontsize=16,color="green",shape="box"];343[label="wv210",fontsize=16,color="green",shape="box"];}
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(6)
11.17/4.54	Complex Obligation (AND)
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(7)
11.17/4.54	Obligation:
11.17/4.54	Q DP problem:
11.17/4.54	The TRS P consists of the following rules:
11.17/4.54	
11.17/4.54	   new_primMulNat(Main.Succ(wv3100)) -> new_primMulNat(wv3100)
11.17/4.54	
11.17/4.54	R is empty.
11.17/4.54	Q is empty.
11.17/4.54	We have to consider all minimal (P,Q,R)-chains.
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(8) QDPSizeChangeProof (EQUIVALENT)
11.17/4.54	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. 
11.17/4.54	
11.17/4.54	From the DPs we obtained the following set of size-change graphs:
11.17/4.54	*new_primMulNat(Main.Succ(wv3100)) -> new_primMulNat(wv3100)
11.17/4.54	The graph contains the following edges 1 > 1
11.17/4.54	
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(9)
11.17/4.54	YES
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(10)
11.17/4.54	Obligation:
11.17/4.54	Q DP problem:
11.17/4.54	The TRS P consists of the following rules:
11.17/4.54	
11.17/4.54	   new_primPlusNat(Main.Succ(wv3000), Main.Succ(wv400)) -> new_primPlusNat(wv3000, wv400)
11.17/4.54	
11.17/4.54	R is empty.
11.17/4.54	Q is empty.
11.17/4.54	We have to consider all minimal (P,Q,R)-chains.
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(11) QDPSizeChangeProof (EQUIVALENT)
11.17/4.54	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. 
11.17/4.54	
11.17/4.54	From the DPs we obtained the following set of size-change graphs:
11.17/4.54	*new_primPlusNat(Main.Succ(wv3000), Main.Succ(wv400)) -> new_primPlusNat(wv3000, wv400)
11.17/4.54	The graph contains the following edges 1 > 1, 2 > 2
11.17/4.54	
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(12)
11.17/4.54	YES
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(13)
11.17/4.54	Obligation:
11.17/4.54	Q DP problem:
11.17/4.54	The TRS P consists of the following rules:
11.17/4.54	
11.17/4.54	   new_primMinusNat(Main.Succ(wv3000), Main.Succ(wv400)) -> new_primMinusNat(wv3000, wv400)
11.17/4.54	
11.17/4.54	R is empty.
11.17/4.54	Q is empty.
11.17/4.54	We have to consider all minimal (P,Q,R)-chains.
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(14) QDPSizeChangeProof (EQUIVALENT)
11.17/4.54	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. 
11.17/4.54	
11.17/4.54	From the DPs we obtained the following set of size-change graphs:
11.17/4.54	*new_primMinusNat(Main.Succ(wv3000), Main.Succ(wv400)) -> new_primMinusNat(wv3000, wv400)
11.17/4.54	The graph contains the following edges 1 > 1, 2 > 2
11.17/4.54	
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(15)
11.17/4.54	YES
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(16)
11.17/4.54	Obligation:
11.17/4.54	Q DP problem:
11.17/4.54	The TRS P consists of the following rules:
11.17/4.54	
11.17/4.54	   new_primMinusNatS(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS(wv210, wv220)
11.17/4.54	
11.17/4.54	R is empty.
11.17/4.54	Q is empty.
11.17/4.54	We have to consider all minimal (P,Q,R)-chains.
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(17) QDPSizeChangeProof (EQUIVALENT)
11.17/4.54	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. 
11.17/4.54	
11.17/4.54	From the DPs we obtained the following set of size-change graphs:
11.17/4.54	*new_primMinusNatS(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS(wv210, wv220)
11.17/4.54	The graph contains the following edges 1 > 1, 2 > 2
11.17/4.54	
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(18)
11.17/4.54	YES
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(19)
11.17/4.54	Obligation:
11.17/4.54	Q DP problem:
11.17/4.54	The TRS P consists of the following rules:
11.17/4.54	
11.17/4.54	   new_primDivNatS(Main.Succ(Main.Succ(wv30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(wv30000), Main.Zero)
11.17/4.54	   new_primDivNatS0(wv21, wv22, Main.Succ(wv230), Main.Succ(wv240)) -> new_primDivNatS0(wv21, wv22, wv230, wv240)
11.17/4.54	   new_primDivNatS00(wv21, wv22) -> new_primDivNatS(new_primMinusNatS0(wv21, wv22), Main.Succ(wv22))
11.17/4.54	   new_primDivNatS0(wv21, wv22, Main.Zero, Main.Zero) -> new_primDivNatS00(wv21, wv22)
11.17/4.54	   new_primDivNatS(Main.Succ(Main.Succ(wv30000)), Main.Succ(wv31000)) -> new_primDivNatS0(wv30000, wv31000, wv30000, wv31000)
11.17/4.54	   new_primDivNatS0(wv21, wv22, Main.Succ(wv230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(wv21, wv22), Main.Succ(wv22))
11.17/4.54	   new_primDivNatS(Main.Succ(Main.Zero), Main.Zero) -> new_primDivNatS(new_primMinusNatS2, Main.Zero)
11.17/4.54	
11.17/4.54	The TRS R consists of the following rules:
11.17/4.54	
11.17/4.54	   new_primMinusNatS0(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS0(wv210, wv220)
11.17/4.54	   new_primMinusNatS0(Main.Succ(wv210), Main.Zero) -> Main.Succ(wv210)
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Succ(wv220)) -> Main.Zero
11.17/4.54	   new_primMinusNatS2 -> Main.Zero
11.17/4.54	   new_primMinusNatS1(wv30000) -> Main.Succ(wv30000)
11.17/4.54	
11.17/4.54	The set Q consists of the following terms:
11.17/4.54	
11.17/4.54	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.17/4.54	   new_primMinusNatS2
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.17/4.54	   new_primMinusNatS1(x0)
11.17/4.54	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.17/4.54	
11.17/4.54	We have to consider all minimal (P,Q,R)-chains.
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(20) DependencyGraphProof (EQUIVALENT)
11.17/4.54	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(21)
11.17/4.54	Complex Obligation (AND)
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(22)
11.17/4.54	Obligation:
11.17/4.54	Q DP problem:
11.17/4.54	The TRS P consists of the following rules:
11.17/4.54	
11.17/4.54	   new_primDivNatS0(wv21, wv22, Main.Zero, Main.Zero) -> new_primDivNatS00(wv21, wv22)
11.17/4.54	   new_primDivNatS00(wv21, wv22) -> new_primDivNatS(new_primMinusNatS0(wv21, wv22), Main.Succ(wv22))
11.17/4.54	   new_primDivNatS(Main.Succ(Main.Succ(wv30000)), Main.Succ(wv31000)) -> new_primDivNatS0(wv30000, wv31000, wv30000, wv31000)
11.17/4.54	   new_primDivNatS0(wv21, wv22, Main.Succ(wv230), Main.Succ(wv240)) -> new_primDivNatS0(wv21, wv22, wv230, wv240)
11.17/4.54	   new_primDivNatS0(wv21, wv22, Main.Succ(wv230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(wv21, wv22), Main.Succ(wv22))
11.17/4.54	
11.17/4.54	The TRS R consists of the following rules:
11.17/4.54	
11.17/4.54	   new_primMinusNatS0(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS0(wv210, wv220)
11.17/4.54	   new_primMinusNatS0(Main.Succ(wv210), Main.Zero) -> Main.Succ(wv210)
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Succ(wv220)) -> Main.Zero
11.17/4.54	   new_primMinusNatS2 -> Main.Zero
11.17/4.54	   new_primMinusNatS1(wv30000) -> Main.Succ(wv30000)
11.17/4.54	
11.17/4.54	The set Q consists of the following terms:
11.17/4.54	
11.17/4.54	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.17/4.54	   new_primMinusNatS2
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.17/4.54	   new_primMinusNatS1(x0)
11.17/4.54	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.17/4.54	
11.17/4.54	We have to consider all minimal (P,Q,R)-chains.
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(23) QDPSizeChangeProof (EQUIVALENT)
11.17/4.54	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
11.17/4.54	
11.17/4.54	Order:Polynomial interpretation [POLO]:
11.17/4.54	
11.17/4.54	   POL(Main.Succ(x_1)) = 1 + x_1
11.17/4.54	   POL(Main.Zero) = 1
11.17/4.54	   POL(new_primMinusNatS0(x_1, x_2)) = x_1
11.17/4.54	
11.17/4.54	
11.17/4.54	
11.17/4.54	
11.17/4.54	From the DPs we obtained the following set of size-change graphs:
11.17/4.54	*new_primDivNatS00(wv21, wv22) -> new_primDivNatS(new_primMinusNatS0(wv21, wv22), Main.Succ(wv22)) (allowed arguments on rhs = {1, 2})
11.17/4.54	The graph contains the following edges 1 >= 1
11.17/4.54	
11.17/4.54	
11.17/4.54	*new_primDivNatS(Main.Succ(Main.Succ(wv30000)), Main.Succ(wv31000)) -> new_primDivNatS0(wv30000, wv31000, wv30000, wv31000) (allowed arguments on rhs = {1, 2, 3, 4})
11.17/4.54	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
11.17/4.54	
11.17/4.54	
11.17/4.54	*new_primDivNatS0(wv21, wv22, Main.Succ(wv230), Main.Succ(wv240)) -> new_primDivNatS0(wv21, wv22, wv230, wv240) (allowed arguments on rhs = {1, 2, 3, 4})
11.17/4.54	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
11.17/4.54	
11.17/4.54	
11.17/4.54	*new_primDivNatS0(wv21, wv22, Main.Zero, Main.Zero) -> new_primDivNatS00(wv21, wv22) (allowed arguments on rhs = {1, 2})
11.17/4.54	The graph contains the following edges 1 >= 1, 2 >= 2
11.17/4.54	
11.17/4.54	
11.17/4.54	*new_primDivNatS0(wv21, wv22, Main.Succ(wv230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(wv21, wv22), Main.Succ(wv22)) (allowed arguments on rhs = {1, 2})
11.17/4.54	The graph contains the following edges 1 >= 1
11.17/4.54	
11.17/4.54	
11.17/4.54	
11.17/4.54	We oriented the following set of usable rules [AAECC05,FROCOS05].
11.17/4.54	
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Succ(wv220)) -> Main.Zero
11.17/4.54	   new_primMinusNatS0(Main.Succ(wv210), Main.Zero) -> Main.Succ(wv210)
11.17/4.54	   new_primMinusNatS0(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS0(wv210, wv220)
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(24)
11.17/4.54	YES
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(25)
11.17/4.54	Obligation:
11.17/4.54	Q DP problem:
11.17/4.54	The TRS P consists of the following rules:
11.17/4.54	
11.17/4.54	   new_primDivNatS(Main.Succ(Main.Succ(wv30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(wv30000), Main.Zero)
11.17/4.54	
11.17/4.54	The TRS R consists of the following rules:
11.17/4.54	
11.17/4.54	   new_primMinusNatS0(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS0(wv210, wv220)
11.17/4.54	   new_primMinusNatS0(Main.Succ(wv210), Main.Zero) -> Main.Succ(wv210)
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Succ(wv220)) -> Main.Zero
11.17/4.54	   new_primMinusNatS2 -> Main.Zero
11.17/4.54	   new_primMinusNatS1(wv30000) -> Main.Succ(wv30000)
11.17/4.54	
11.17/4.54	The set Q consists of the following terms:
11.17/4.54	
11.17/4.54	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.17/4.54	   new_primMinusNatS2
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.17/4.54	   new_primMinusNatS1(x0)
11.17/4.54	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.17/4.54	
11.17/4.54	We have to consider all minimal (P,Q,R)-chains.
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(26) MRRProof (EQUIVALENT)
11.17/4.54	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.
11.17/4.54	
11.17/4.54	Strictly oriented dependency pairs:
11.17/4.54	
11.17/4.54	   new_primDivNatS(Main.Succ(Main.Succ(wv30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(wv30000), Main.Zero)
11.17/4.54	
11.17/4.54	Strictly oriented rules of the TRS R:
11.17/4.54	
11.17/4.54	   new_primMinusNatS0(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS0(wv210, wv220)
11.17/4.54	   new_primMinusNatS0(Main.Succ(wv210), Main.Zero) -> Main.Succ(wv210)
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Succ(wv220)) -> Main.Zero
11.17/4.54	
11.17/4.54	Used ordering: Polynomial interpretation [POLO]:
11.17/4.54	
11.17/4.54	   POL(Main.Succ(x_1)) = 1 + x_1
11.17/4.54	   POL(Main.Zero) = 2
11.17/4.54	   POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2
11.17/4.54	   POL(new_primMinusNatS0(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
11.17/4.54	   POL(new_primMinusNatS1(x_1)) = 1 + x_1
11.17/4.54	   POL(new_primMinusNatS2) = 2
11.17/4.54	
11.17/4.54	
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(27)
11.17/4.54	Obligation:
11.17/4.54	Q DP problem:
11.17/4.54	P is empty.
11.17/4.54	The TRS R consists of the following rules:
11.17/4.54	
11.17/4.54	   new_primMinusNatS2 -> Main.Zero
11.17/4.54	   new_primMinusNatS1(wv30000) -> Main.Succ(wv30000)
11.17/4.54	
11.17/4.54	The set Q consists of the following terms:
11.17/4.54	
11.17/4.54	   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Zero)
11.17/4.54	   new_primMinusNatS2
11.17/4.54	   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
11.17/4.54	   new_primMinusNatS1(x0)
11.17/4.54	   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
11.17/4.54	
11.17/4.54	We have to consider all minimal (P,Q,R)-chains.
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(28) PisEmptyProof (EQUIVALENT)
11.17/4.54	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.17/4.54	----------------------------------------
11.17/4.54	
11.17/4.54	(29)
11.17/4.54	YES
11.17/4.57	EOF