30.60/15.72	MAYBE
33.10/16.34	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
33.10/16.34	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.10/16.34	
33.10/16.34	
33.10/16.34	H-Termination with start terms of the given HASKELL could not be shown:
33.10/16.34	
33.10/16.34	(0) HASKELL
33.10/16.34	(1) BR [EQUIVALENT, 0 ms]
33.10/16.34	(2) HASKELL
33.10/16.34	(3) COR [EQUIVALENT, 0 ms]
33.10/16.34	(4) HASKELL
33.10/16.34	(5) Narrow [SOUND, 0 ms]
33.10/16.34	(6) AND
33.10/16.34	    (7) QDP
33.10/16.34	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
33.10/16.34	        (9) YES
33.10/16.34	    (10) QDP
33.10/16.34	        (11) QDPOrderProof [EQUIVALENT, 19 ms]
33.10/16.34	        (12) QDP
33.10/16.34	        (13) DependencyGraphProof [EQUIVALENT, 0 ms]
33.10/16.34	        (14) QDP
33.10/16.34	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
33.10/16.34	        (16) YES
33.10/16.34	    (17) QDP
33.10/16.34	        (18) DependencyGraphProof [EQUIVALENT, 0 ms]
33.10/16.34	        (19) QDP
33.10/16.34	        (20) QDPOrderProof [EQUIVALENT, 12 ms]
33.10/16.34	        (21) QDP
33.10/16.34	        (22) DependencyGraphProof [EQUIVALENT, 0 ms]
33.10/16.34	        (23) QDP
33.10/16.34	        (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
33.10/16.34	        (25) YES
33.10/16.34	    (26) QDP
33.10/16.34	        (27) DependencyGraphProof [EQUIVALENT, 0 ms]
33.10/16.34	        (28) QDP
33.10/16.34	        (29) TransformationProof [EQUIVALENT, 0 ms]
33.10/16.35	        (30) QDP
33.10/16.35	        (31) UsableRulesProof [EQUIVALENT, 0 ms]
33.10/16.35	        (32) QDP
33.10/16.35	        (33) QReductionProof [EQUIVALENT, 0 ms]
33.10/16.35	        (34) QDP
33.10/16.35	        (35) MNOCProof [EQUIVALENT, 0 ms]
33.10/16.35	        (36) QDP
33.10/16.35	        (37) InductionCalculusProof [EQUIVALENT, 0 ms]
33.10/16.35	        (38) QDP
33.10/16.35	        (39) TransformationProof [EQUIVALENT, 0 ms]
33.10/16.35	        (40) QDP
33.10/16.35	            (41) DependencyGraphProof [EQUIVALENT, 0 ms]
33.10/16.35	            (42) QDP
33.10/16.35	            (43) TransformationProof [EQUIVALENT, 0 ms]
33.10/16.35	            (44) QDP
33.10/16.35	            (45) DependencyGraphProof [EQUIVALENT, 0 ms]
33.10/16.35	            (46) QDP
33.10/16.35	            (47) TransformationProof [EQUIVALENT, 0 ms]
33.10/16.35	            (48) QDP
33.10/16.35	            (49) DependencyGraphProof [EQUIVALENT, 0 ms]
33.10/16.35	            (50) QDP
33.10/16.35	            (51) TransformationProof [EQUIVALENT, 0 ms]
33.10/16.35	            (52) QDP
33.10/16.35	            (53) DependencyGraphProof [EQUIVALENT, 0 ms]
33.10/16.35	            (54) QDP
33.10/16.35	            (55) MNOCProof [EQUIVALENT, 0 ms]
33.10/16.35	            (56) QDP
33.10/16.35	            (57) InductionCalculusProof [EQUIVALENT, 0 ms]
33.10/16.35	            (58) QDP
33.10/16.35	(59) Narrow [COMPLETE, 0 ms]
33.10/16.35	(60) QDP
33.10/16.35	    (61) DependencyGraphProof [EQUIVALENT, 0 ms]
33.10/16.35	    (62) TRUE
33.10/16.35	
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(0)
33.10/16.35	Obligation:
33.10/16.35	mainModule Main 
33.10/16.35	module Main where {
33.10/16.35	import qualified Prelude;
33.10/16.35	data Main.Char = Char MyInt ;
33.10/16.35	
33.10/16.35	data List a = Cons a (List a)  | Nil ;
33.10/16.35	
33.10/16.35	data MyBool = MyTrue  | MyFalse ;
33.10/16.35	
33.10/16.35	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
33.10/16.35	
33.10/16.35	data Main.Nat = Succ Main.Nat  | Zero ;
33.10/16.35	
33.10/16.35	divMyInt :: MyInt  ->  MyInt  ->  MyInt;
33.10/16.35	divMyInt = primDivInt;
33.10/16.35	
33.10/16.35	error :: a;
33.10/16.35	error = stop MyTrue;
33.10/16.35	
33.10/16.35	modMyInt :: MyInt  ->  MyInt  ->  MyInt;
33.10/16.35	modMyInt = primModInt;
33.10/16.35	
33.10/16.35	primDivInt :: MyInt  ->  MyInt  ->  MyInt;
33.10/16.35	primDivInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
33.10/16.35	primDivInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y));
33.10/16.35	primDivInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y));
33.10/16.35	primDivInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
33.10/16.35	primDivInt vv vw = Main.error;
33.10/16.35	
33.10/16.35	primDivNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primDivNatP Main.Zero Main.Zero = Main.error;
33.10/16.35	primDivNatP (Main.Succ x) Main.Zero = Main.error;
33.10/16.35	primDivNatP (Main.Succ x) (Main.Succ y) = Main.Succ (primDivNatP (primMinusNatS x y) (Main.Succ y));
33.10/16.35	primDivNatP Main.Zero (Main.Succ x) = Main.Zero;
33.10/16.35	
33.10/16.35	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primDivNatS Main.Zero Main.Zero = Main.error;
33.10/16.35	primDivNatS (Main.Succ x) Main.Zero = Main.error;
33.10/16.35	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
33.10/16.35	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
33.10/16.35	
33.10/16.35	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
33.10/16.35	primDivNatS0 x y MyFalse = Main.Zero;
33.10/16.35	
33.10/16.35	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
33.10/16.35	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
33.10/16.35	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
33.10/16.35	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
33.10/16.35	primGEqNatS Main.Zero Main.Zero = MyTrue;
33.10/16.35	
33.10/16.35	primIntToChar :: MyInt  ->  Main.Char;
33.10/16.35	primIntToChar x = Main.Char x;
33.10/16.35	
33.10/16.35	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
33.10/16.35	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
33.10/16.35	primMinusNatS x Main.Zero = x;
33.10/16.35	
33.10/16.35	primModInt :: MyInt  ->  MyInt  ->  MyInt;
33.10/16.35	primModInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
33.10/16.35	primModInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatP x (Main.Succ y));
33.10/16.35	primModInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatP x (Main.Succ y));
33.10/16.35	primModInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
33.10/16.35	primModInt vx vy = Main.error;
33.10/16.35	
33.10/16.35	primModNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primModNatP Main.Zero Main.Zero = Main.error;
33.10/16.35	primModNatP Main.Zero (Main.Succ x) = Main.Zero;
33.10/16.35	primModNatP (Main.Succ x) Main.Zero = Main.error;
33.10/16.35	primModNatP (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
33.10/16.35	primModNatP (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatP0 x y (primGEqNatS x y);
33.10/16.35	
33.10/16.35	primModNatP0 x y MyTrue = primModNatP (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
33.10/16.35	primModNatP0 x y MyFalse = primMinusNatS (Main.Succ y) x;
33.10/16.35	
33.10/16.35	primModNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primModNatS Main.Zero Main.Zero = Main.error;
33.10/16.35	primModNatS Main.Zero (Main.Succ x) = Main.Zero;
33.10/16.35	primModNatS (Main.Succ x) Main.Zero = Main.error;
33.10/16.35	primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
33.10/16.35	primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y));
33.10/16.35	
33.10/16.35	primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
33.10/16.35	primModNatS0 x y MyFalse = Main.Succ x;
33.10/16.35	
33.10/16.35	primShowInt :: MyInt  ->  List Main.Char;
33.10/16.35	primShowInt (Main.Pos Main.Zero) = Cons (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero)))))))))))))))))))))))))))))))))))))))))))))))))) Nil;
33.10/16.35	primShowInt (Main.Pos (Main.Succ x)) = psPs (primShowInt (divMyInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))) (Cons (toEnumChar (modMyInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))) Nil);
33.10/16.35	primShowInt (Main.Neg x) = Cons (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))))))))))))))))))))))))))))))))))))) (primShowInt (Main.Pos x));
33.10/16.35	
33.10/16.35	psPs :: List a  ->  List a  ->  List a;
33.10/16.35	psPs Nil ys = ys;
33.10/16.35	psPs (Cons x xs) ys = Cons x (psPs xs ys);
33.10/16.35	
33.10/16.35	showMyInt :: MyInt  ->  List Main.Char;
33.10/16.35	showMyInt = primShowInt;
33.10/16.35	
33.10/16.35	stop :: MyBool  ->  a;
33.10/16.35	stop MyFalse = stop MyFalse;
33.10/16.35	
33.10/16.35	toEnumChar :: MyInt  ->  Main.Char;
33.10/16.35	toEnumChar = primIntToChar;
33.10/16.35	
33.10/16.35	}
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(1) BR (EQUIVALENT)
33.10/16.35	Replaced joker patterns by fresh variables and removed binding patterns.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(2)
33.10/16.35	Obligation:
33.10/16.35	mainModule Main 
33.10/16.35	module Main where {
33.10/16.35	import qualified Prelude;
33.10/16.35	data Main.Char = Char MyInt ;
33.10/16.35	
33.10/16.35	data List a = Cons a (List a)  | Nil ;
33.10/16.35	
33.10/16.35	data MyBool = MyTrue  | MyFalse ;
33.10/16.35	
33.10/16.35	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
33.10/16.35	
33.10/16.35	data Main.Nat = Succ Main.Nat  | Zero ;
33.10/16.35	
33.10/16.35	divMyInt :: MyInt  ->  MyInt  ->  MyInt;
33.10/16.35	divMyInt = primDivInt;
33.10/16.35	
33.10/16.35	error :: a;
33.10/16.35	error = stop MyTrue;
33.10/16.35	
33.10/16.35	modMyInt :: MyInt  ->  MyInt  ->  MyInt;
33.10/16.35	modMyInt = primModInt;
33.10/16.35	
33.10/16.35	primDivInt :: MyInt  ->  MyInt  ->  MyInt;
33.10/16.35	primDivInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
33.10/16.35	primDivInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y));
33.10/16.35	primDivInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y));
33.10/16.35	primDivInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
33.10/16.35	primDivInt vv vw = Main.error;
33.10/16.35	
33.10/16.35	primDivNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primDivNatP Main.Zero Main.Zero = Main.error;
33.10/16.35	primDivNatP (Main.Succ x) Main.Zero = Main.error;
33.10/16.35	primDivNatP (Main.Succ x) (Main.Succ y) = Main.Succ (primDivNatP (primMinusNatS x y) (Main.Succ y));
33.10/16.35	primDivNatP Main.Zero (Main.Succ x) = Main.Zero;
33.10/16.35	
33.10/16.35	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primDivNatS Main.Zero Main.Zero = Main.error;
33.10/16.35	primDivNatS (Main.Succ x) Main.Zero = Main.error;
33.10/16.35	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
33.10/16.35	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
33.10/16.35	
33.10/16.35	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
33.10/16.35	primDivNatS0 x y MyFalse = Main.Zero;
33.10/16.35	
33.10/16.35	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
33.10/16.35	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
33.10/16.35	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
33.10/16.35	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
33.10/16.35	primGEqNatS Main.Zero Main.Zero = MyTrue;
33.10/16.35	
33.10/16.35	primIntToChar :: MyInt  ->  Main.Char;
33.10/16.35	primIntToChar x = Main.Char x;
33.10/16.35	
33.10/16.35	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
33.10/16.35	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
33.10/16.35	primMinusNatS x Main.Zero = x;
33.10/16.35	
33.10/16.35	primModInt :: MyInt  ->  MyInt  ->  MyInt;
33.10/16.35	primModInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
33.10/16.35	primModInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatP x (Main.Succ y));
33.10/16.35	primModInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatP x (Main.Succ y));
33.10/16.35	primModInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
33.10/16.35	primModInt vx vy = Main.error;
33.10/16.35	
33.10/16.35	primModNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primModNatP Main.Zero Main.Zero = Main.error;
33.10/16.35	primModNatP Main.Zero (Main.Succ x) = Main.Zero;
33.10/16.35	primModNatP (Main.Succ x) Main.Zero = Main.error;
33.10/16.35	primModNatP (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
33.10/16.35	primModNatP (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatP0 x y (primGEqNatS x y);
33.10/16.35	
33.10/16.35	primModNatP0 x y MyTrue = primModNatP (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
33.10/16.35	primModNatP0 x y MyFalse = primMinusNatS (Main.Succ y) x;
33.10/16.35	
33.10/16.35	primModNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primModNatS Main.Zero Main.Zero = Main.error;
33.10/16.35	primModNatS Main.Zero (Main.Succ x) = Main.Zero;
33.10/16.35	primModNatS (Main.Succ x) Main.Zero = Main.error;
33.10/16.35	primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
33.10/16.35	primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y));
33.10/16.35	
33.10/16.35	primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
33.10/16.35	primModNatS0 x y MyFalse = Main.Succ x;
33.10/16.35	
33.10/16.35	primShowInt :: MyInt  ->  List Main.Char;
33.10/16.35	primShowInt (Main.Pos Main.Zero) = Cons (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero)))))))))))))))))))))))))))))))))))))))))))))))))) Nil;
33.10/16.35	primShowInt (Main.Pos (Main.Succ x)) = psPs (primShowInt (divMyInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))) (Cons (toEnumChar (modMyInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))) Nil);
33.10/16.35	primShowInt (Main.Neg x) = Cons (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))))))))))))))))))))))))))))))))))))) (primShowInt (Main.Pos x));
33.10/16.35	
33.10/16.35	psPs :: List a  ->  List a  ->  List a;
33.10/16.35	psPs Nil ys = ys;
33.10/16.35	psPs (Cons x xs) ys = Cons x (psPs xs ys);
33.10/16.35	
33.10/16.35	showMyInt :: MyInt  ->  List Main.Char;
33.10/16.35	showMyInt = primShowInt;
33.10/16.35	
33.10/16.35	stop :: MyBool  ->  a;
33.10/16.35	stop MyFalse = stop MyFalse;
33.10/16.35	
33.10/16.35	toEnumChar :: MyInt  ->  Main.Char;
33.10/16.35	toEnumChar = primIntToChar;
33.10/16.35	
33.10/16.35	}
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(3) COR (EQUIVALENT)
33.10/16.35	Cond Reductions:
33.10/16.35	The following Function with conditions
33.10/16.35	"undefined |Falseundefined;
33.10/16.35	"
33.10/16.35	is transformed to
33.10/16.35	"undefined  = undefined1;
33.10/16.35	"
33.10/16.35	"undefined0 True = undefined;
33.10/16.35	"
33.10/16.35	"undefined1  = undefined0 False;
33.10/16.35	"
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(4)
33.10/16.35	Obligation:
33.10/16.35	mainModule Main 
33.10/16.35	module Main where {
33.10/16.35	import qualified Prelude;
33.10/16.35	data Main.Char = Char MyInt ;
33.10/16.35	
33.10/16.35	data List a = Cons a (List a)  | Nil ;
33.10/16.35	
33.10/16.35	data MyBool = MyTrue  | MyFalse ;
33.10/16.35	
33.10/16.35	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
33.10/16.35	
33.10/16.35	data Main.Nat = Succ Main.Nat  | Zero ;
33.10/16.35	
33.10/16.35	divMyInt :: MyInt  ->  MyInt  ->  MyInt;
33.10/16.35	divMyInt = primDivInt;
33.10/16.35	
33.10/16.35	error :: a;
33.10/16.35	error = stop MyTrue;
33.10/16.35	
33.10/16.35	modMyInt :: MyInt  ->  MyInt  ->  MyInt;
33.10/16.35	modMyInt = primModInt;
33.10/16.35	
33.10/16.35	primDivInt :: MyInt  ->  MyInt  ->  MyInt;
33.10/16.35	primDivInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
33.10/16.35	primDivInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y));
33.10/16.35	primDivInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y));
33.10/16.35	primDivInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
33.10/16.35	primDivInt vv vw = Main.error;
33.10/16.35	
33.10/16.35	primDivNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primDivNatP Main.Zero Main.Zero = Main.error;
33.10/16.35	primDivNatP (Main.Succ x) Main.Zero = Main.error;
33.10/16.35	primDivNatP (Main.Succ x) (Main.Succ y) = Main.Succ (primDivNatP (primMinusNatS x y) (Main.Succ y));
33.10/16.35	primDivNatP Main.Zero (Main.Succ x) = Main.Zero;
33.10/16.35	
33.10/16.35	primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primDivNatS Main.Zero Main.Zero = Main.error;
33.10/16.35	primDivNatS (Main.Succ x) Main.Zero = Main.error;
33.10/16.35	primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
33.10/16.35	primDivNatS Main.Zero (Main.Succ x) = Main.Zero;
33.10/16.35	
33.10/16.35	primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
33.10/16.35	primDivNatS0 x y MyFalse = Main.Zero;
33.10/16.35	
33.10/16.35	primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
33.10/16.35	primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
33.10/16.35	primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
33.10/16.35	primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
33.10/16.35	primGEqNatS Main.Zero Main.Zero = MyTrue;
33.10/16.35	
33.10/16.35	primIntToChar :: MyInt  ->  Main.Char;
33.10/16.35	primIntToChar x = Main.Char x;
33.10/16.35	
33.10/16.35	primMinusNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y;
33.10/16.35	primMinusNatS Main.Zero (Main.Succ y) = Main.Zero;
33.10/16.35	primMinusNatS x Main.Zero = x;
33.10/16.35	
33.10/16.35	primModInt :: MyInt  ->  MyInt  ->  MyInt;
33.10/16.35	primModInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y));
33.10/16.35	primModInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatP x (Main.Succ y));
33.10/16.35	primModInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatP x (Main.Succ y));
33.10/16.35	primModInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y));
33.10/16.35	primModInt vx vy = Main.error;
33.10/16.35	
33.10/16.35	primModNatP :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primModNatP Main.Zero Main.Zero = Main.error;
33.10/16.35	primModNatP Main.Zero (Main.Succ x) = Main.Zero;
33.10/16.35	primModNatP (Main.Succ x) Main.Zero = Main.error;
33.10/16.35	primModNatP (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
33.10/16.35	primModNatP (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatP0 x y (primGEqNatS x y);
33.10/16.35	
33.10/16.35	primModNatP0 x y MyTrue = primModNatP (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
33.10/16.35	primModNatP0 x y MyFalse = primMinusNatS (Main.Succ y) x;
33.10/16.35	
33.10/16.35	primModNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
33.10/16.35	primModNatS Main.Zero Main.Zero = Main.error;
33.10/16.35	primModNatS Main.Zero (Main.Succ x) = Main.Zero;
33.10/16.35	primModNatS (Main.Succ x) Main.Zero = Main.error;
33.10/16.35	primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero;
33.10/16.35	primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y));
33.10/16.35	
33.10/16.35	primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y));
33.10/16.35	primModNatS0 x y MyFalse = Main.Succ x;
33.10/16.35	
33.10/16.35	primShowInt :: MyInt  ->  List Main.Char;
33.10/16.35	primShowInt (Main.Pos Main.Zero) = Cons (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero)))))))))))))))))))))))))))))))))))))))))))))))))) Nil;
33.10/16.35	primShowInt (Main.Pos (Main.Succ x)) = psPs (primShowInt (divMyInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))) (Cons (toEnumChar (modMyInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))) Nil);
33.10/16.35	primShowInt (Main.Neg x) = Cons (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))))))))))))))))))))))))))))))))))))) (primShowInt (Main.Pos x));
33.10/16.35	
33.10/16.35	psPs :: List a  ->  List a  ->  List a;
33.10/16.35	psPs Nil ys = ys;
33.10/16.35	psPs (Cons x xs) ys = Cons x (psPs xs ys);
33.10/16.35	
33.10/16.35	showMyInt :: MyInt  ->  List Main.Char;
33.10/16.35	showMyInt = primShowInt;
33.10/16.35	
33.10/16.35	stop :: MyBool  ->  a;
33.10/16.35	stop MyFalse = stop MyFalse;
33.10/16.35	
33.10/16.35	toEnumChar :: MyInt  ->  Main.Char;
33.10/16.35	toEnumChar = primIntToChar;
33.10/16.35	
33.10/16.35	}
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(5) Narrow (SOUND)
33.10/16.35	Haskell To QDPs
33.10/16.35	
33.10/16.35	digraph dp_graph {
33.10/16.35	node [outthreshold=100, inthreshold=100];1[label="showMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
33.10/16.35	3[label="showMyInt wv3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
33.10/16.35	4[label="primShowInt wv3",fontsize=16,color="burlywood",shape="triangle"];721[label="wv3/Pos wv30",fontsize=10,color="white",style="solid",shape="box"];4 -> 721[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	721 -> 5[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	722[label="wv3/Neg wv30",fontsize=10,color="white",style="solid",shape="box"];4 -> 722[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	722 -> 6[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	5[label="primShowInt (Pos wv30)",fontsize=16,color="burlywood",shape="box"];723[label="wv30/Succ wv300",fontsize=10,color="white",style="solid",shape="box"];5 -> 723[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	723 -> 7[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	724[label="wv30/Zero",fontsize=10,color="white",style="solid",shape="box"];5 -> 724[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	724 -> 8[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	6[label="primShowInt (Neg wv30)",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
33.10/16.35	7[label="primShowInt (Pos (Succ wv300))",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
33.10/16.35	8[label="primShowInt (Pos Zero)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
33.10/16.35	9[label="Cons (Char (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))) (primShowInt (Pos wv30))",fontsize=16,color="green",shape="box"];9 -> 12[label="",style="dashed", color="green", weight=3];
33.10/16.35	10 -> 24[label="",style="dashed", color="red", weight=0];
33.10/16.35	10[label="psPs (primShowInt (divMyInt (Pos (Succ wv300)) (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))) (Cons (toEnumChar (modMyInt (Pos (Succ wv300)) (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))) Nil)",fontsize=16,color="magenta"];10 -> 25[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	10 -> 26[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	10 -> 27[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	11[label="Cons (Char (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))))))))))))))))))))))))))) Nil",fontsize=16,color="green",shape="box"];12 -> 4[label="",style="dashed", color="red", weight=0];
33.10/16.35	12[label="primShowInt (Pos wv30)",fontsize=16,color="magenta"];12 -> 16[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	25[label="wv300",fontsize=16,color="green",shape="box"];26 -> 4[label="",style="dashed", color="red", weight=0];
33.10/16.35	26[label="primShowInt (divMyInt (Pos (Succ wv300)) (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))",fontsize=16,color="magenta"];26 -> 29[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	27[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];24[label="psPs wv11 (Cons (toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))) Nil)",fontsize=16,color="burlywood",shape="triangle"];725[label="wv11/Cons wv110 wv111",fontsize=10,color="white",style="solid",shape="box"];24 -> 725[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	725 -> 30[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	726[label="wv11/Nil",fontsize=10,color="white",style="solid",shape="box"];24 -> 726[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	726 -> 31[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	16[label="Pos wv30",fontsize=16,color="green",shape="box"];29 -> 32[label="",style="dashed", color="red", weight=0];
33.10/16.35	29[label="divMyInt (Pos (Succ wv300)) (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))",fontsize=16,color="magenta"];29 -> 33[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	29 -> 34[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	30[label="psPs (Cons wv110 wv111) (Cons (toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))) Nil)",fontsize=16,color="black",shape="box"];30 -> 35[label="",style="solid", color="black", weight=3];
33.10/16.35	31[label="psPs Nil (Cons (toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))) Nil)",fontsize=16,color="black",shape="box"];31 -> 36[label="",style="solid", color="black", weight=3];
33.10/16.35	33[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];34[label="wv300",fontsize=16,color="green",shape="box"];32[label="divMyInt (Pos (Succ wv13)) (Pos (Succ wv14))",fontsize=16,color="black",shape="triangle"];32 -> 37[label="",style="solid", color="black", weight=3];
33.10/16.35	35[label="Cons wv110 (psPs wv111 (Cons (toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))) Nil))",fontsize=16,color="green",shape="box"];35 -> 38[label="",style="dashed", color="green", weight=3];
33.10/16.35	36[label="Cons (toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))) Nil",fontsize=16,color="green",shape="box"];36 -> 39[label="",style="dashed", color="green", weight=3];
33.10/16.35	37[label="primDivInt (Pos (Succ wv13)) (Pos (Succ wv14))",fontsize=16,color="black",shape="box"];37 -> 40[label="",style="solid", color="black", weight=3];
33.10/16.35	38 -> 24[label="",style="dashed", color="red", weight=0];
33.10/16.35	38[label="psPs wv111 (Cons (toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))) Nil)",fontsize=16,color="magenta"];38 -> 41[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	39[label="toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))",fontsize=16,color="black",shape="box"];39 -> 42[label="",style="solid", color="black", weight=3];
33.10/16.35	40[label="Pos (primDivNatS (Succ wv13) (Succ wv14))",fontsize=16,color="green",shape="box"];40 -> 43[label="",style="dashed", color="green", weight=3];
33.10/16.35	41[label="wv111",fontsize=16,color="green",shape="box"];42[label="primIntToChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
33.10/16.35	43[label="primDivNatS (Succ wv13) (Succ wv14)",fontsize=16,color="black",shape="triangle"];43 -> 45[label="",style="solid", color="black", weight=3];
33.10/16.35	44[label="Char (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))",fontsize=16,color="green",shape="box"];44 -> 46[label="",style="dashed", color="green", weight=3];
33.10/16.35	45[label="primDivNatS0 wv13 wv14 (primGEqNatS wv13 wv14)",fontsize=16,color="burlywood",shape="box"];727[label="wv13/Succ wv130",fontsize=10,color="white",style="solid",shape="box"];45 -> 727[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	727 -> 47[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	728[label="wv13/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 728[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	728 -> 48[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	46[label="modMyInt (Pos (Succ wv8)) (Pos (Succ wv10))",fontsize=16,color="black",shape="box"];46 -> 49[label="",style="solid", color="black", weight=3];
33.10/16.35	47[label="primDivNatS0 (Succ wv130) wv14 (primGEqNatS (Succ wv130) wv14)",fontsize=16,color="burlywood",shape="box"];729[label="wv14/Succ wv140",fontsize=10,color="white",style="solid",shape="box"];47 -> 729[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	729 -> 50[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	730[label="wv14/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 730[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	730 -> 51[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	48[label="primDivNatS0 Zero wv14 (primGEqNatS Zero wv14)",fontsize=16,color="burlywood",shape="box"];731[label="wv14/Succ wv140",fontsize=10,color="white",style="solid",shape="box"];48 -> 731[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	731 -> 52[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	732[label="wv14/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 732[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	732 -> 53[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	49[label="primModInt (Pos (Succ wv8)) (Pos (Succ wv10))",fontsize=16,color="black",shape="box"];49 -> 54[label="",style="solid", color="black", weight=3];
33.10/16.35	50[label="primDivNatS0 (Succ wv130) (Succ wv140) (primGEqNatS (Succ wv130) (Succ wv140))",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3];
33.10/16.35	51[label="primDivNatS0 (Succ wv130) Zero (primGEqNatS (Succ wv130) Zero)",fontsize=16,color="black",shape="box"];51 -> 56[label="",style="solid", color="black", weight=3];
33.10/16.35	52[label="primDivNatS0 Zero (Succ wv140) (primGEqNatS Zero (Succ wv140))",fontsize=16,color="black",shape="box"];52 -> 57[label="",style="solid", color="black", weight=3];
33.10/16.35	53[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];53 -> 58[label="",style="solid", color="black", weight=3];
33.10/16.35	54[label="Pos (primModNatS (Succ wv8) (Succ wv10))",fontsize=16,color="green",shape="box"];54 -> 59[label="",style="dashed", color="green", weight=3];
33.10/16.35	55 -> 324[label="",style="dashed", color="red", weight=0];
33.10/16.35	55[label="primDivNatS0 (Succ wv130) (Succ wv140) (primGEqNatS wv130 wv140)",fontsize=16,color="magenta"];55 -> 325[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	55 -> 326[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	55 -> 327[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	55 -> 328[label="",style="dashed", color="magenta", weight=3];
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33.10/16.35	57[label="primDivNatS0 Zero (Succ wv140) MyFalse",fontsize=16,color="black",shape="box"];57 -> 63[label="",style="solid", color="black", weight=3];
33.10/16.35	58[label="primDivNatS0 Zero Zero MyTrue",fontsize=16,color="black",shape="box"];58 -> 64[label="",style="solid", color="black", weight=3];
33.10/16.35	59[label="primModNatS (Succ wv8) (Succ wv10)",fontsize=16,color="burlywood",shape="triangle"];733[label="wv10/Succ wv100",fontsize=10,color="white",style="solid",shape="box"];59 -> 733[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	733 -> 65[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	734[label="wv10/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 734[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	734 -> 66[label="",style="solid", color="burlywood", weight=3];
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33.10/16.35	735 -> 357[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	736[label="wv33/Zero",fontsize=10,color="white",style="solid",shape="box"];324 -> 736[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	736 -> 358[label="",style="solid", color="burlywood", weight=3];
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33.10/16.35	63[label="Zero",fontsize=16,color="green",shape="box"];64[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];64 -> 72[label="",style="dashed", color="green", weight=3];
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33.10/16.35	66[label="primModNatS (Succ wv8) (Succ Zero)",fontsize=16,color="black",shape="box"];66 -> 74[label="",style="solid", color="black", weight=3];
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33.10/16.35	737 -> 364[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	738[label="wv34/Zero",fontsize=10,color="white",style="solid",shape="box"];357 -> 738[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	738 -> 365[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	358[label="primDivNatS0 (Succ wv31) (Succ wv32) (primGEqNatS Zero wv34)",fontsize=16,color="burlywood",shape="box"];739[label="wv34/Succ wv340",fontsize=10,color="white",style="solid",shape="box"];358 -> 739[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	739 -> 366[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	740[label="wv34/Zero",fontsize=10,color="white",style="solid",shape="box"];358 -> 740[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	740 -> 367[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	71 -> 591[label="",style="dashed", color="red", weight=0];
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33.10/16.35	71 -> 593[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	71 -> 594[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	72 -> 591[label="",style="dashed", color="red", weight=0];
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33.10/16.35	72 -> 596[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	72 -> 597[label="",style="dashed", color="magenta", weight=3];
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33.10/16.35	741 -> 81[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	742[label="wv8/Zero",fontsize=10,color="white",style="solid",shape="box"];73 -> 742[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	742 -> 82[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	74[label="Zero",fontsize=16,color="green",shape="box"];364[label="primDivNatS0 (Succ wv31) (Succ wv32) (primGEqNatS (Succ wv330) (Succ wv340))",fontsize=16,color="black",shape="box"];364 -> 376[label="",style="solid", color="black", weight=3];
33.10/16.35	365[label="primDivNatS0 (Succ wv31) (Succ wv32) (primGEqNatS (Succ wv330) Zero)",fontsize=16,color="black",shape="box"];365 -> 377[label="",style="solid", color="black", weight=3];
33.10/16.35	366[label="primDivNatS0 (Succ wv31) (Succ wv32) (primGEqNatS Zero (Succ wv340))",fontsize=16,color="black",shape="box"];366 -> 378[label="",style="solid", color="black", weight=3];
33.10/16.35	367[label="primDivNatS0 (Succ wv31) (Succ wv32) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];367 -> 379[label="",style="solid", color="black", weight=3];
33.10/16.35	592[label="Succ wv130",fontsize=16,color="green",shape="box"];593[label="Zero",fontsize=16,color="green",shape="box"];594[label="Zero",fontsize=16,color="green",shape="box"];591[label="primDivNatS (primMinusNatS wv65 wv66) (Succ wv67)",fontsize=16,color="burlywood",shape="triangle"];743[label="wv65/Succ wv650",fontsize=10,color="white",style="solid",shape="box"];591 -> 743[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	743 -> 616[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	744[label="wv65/Zero",fontsize=10,color="white",style="solid",shape="box"];591 -> 744[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	744 -> 617[label="",style="solid", color="burlywood", weight=3];
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33.10/16.35	82[label="primModNatS0 Zero wv100 (primGEqNatS Zero (Succ wv100))",fontsize=16,color="black",shape="box"];82 -> 92[label="",style="solid", color="black", weight=3];
33.10/16.35	376 -> 324[label="",style="dashed", color="red", weight=0];
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33.10/16.35	376 -> 390[label="",style="dashed", color="magenta", weight=3];
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33.10/16.35	378[label="primDivNatS0 (Succ wv31) (Succ wv32) MyFalse",fontsize=16,color="black",shape="box"];378 -> 392[label="",style="solid", color="black", weight=3];
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33.10/16.35	745 -> 619[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	746[label="wv66/Zero",fontsize=10,color="white",style="solid",shape="box"];616 -> 746[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	746 -> 620[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	617[label="primDivNatS (primMinusNatS Zero wv66) (Succ wv67)",fontsize=16,color="burlywood",shape="box"];747[label="wv66/Succ wv660",fontsize=10,color="white",style="solid",shape="box"];617 -> 747[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	747 -> 621[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	748[label="wv66/Zero",fontsize=10,color="white",style="solid",shape="box"];617 -> 748[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	748 -> 622[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	91[label="primModNatS0 (Succ wv80) wv100 (primGEqNatS wv80 wv100)",fontsize=16,color="burlywood",shape="box"];749[label="wv80/Succ wv800",fontsize=10,color="white",style="solid",shape="box"];91 -> 749[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	749 -> 99[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	750[label="wv80/Zero",fontsize=10,color="white",style="solid",shape="box"];91 -> 750[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	750 -> 100[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	92[label="primModNatS0 Zero wv100 MyFalse",fontsize=16,color="black",shape="box"];92 -> 101[label="",style="solid", color="black", weight=3];
33.10/16.35	389[label="wv330",fontsize=16,color="green",shape="box"];390[label="wv340",fontsize=16,color="green",shape="box"];391[label="Succ (primDivNatS (primMinusNatS (Succ wv31) (Succ wv32)) (Succ (Succ wv32)))",fontsize=16,color="green",shape="box"];391 -> 400[label="",style="dashed", color="green", weight=3];
33.10/16.35	392[label="Zero",fontsize=16,color="green",shape="box"];619[label="primDivNatS (primMinusNatS (Succ wv650) (Succ wv660)) (Succ wv67)",fontsize=16,color="black",shape="box"];619 -> 625[label="",style="solid", color="black", weight=3];
33.10/16.35	620[label="primDivNatS (primMinusNatS (Succ wv650) Zero) (Succ wv67)",fontsize=16,color="black",shape="box"];620 -> 626[label="",style="solid", color="black", weight=3];
33.10/16.35	621[label="primDivNatS (primMinusNatS Zero (Succ wv660)) (Succ wv67)",fontsize=16,color="black",shape="box"];621 -> 627[label="",style="solid", color="black", weight=3];
33.10/16.35	622[label="primDivNatS (primMinusNatS Zero Zero) (Succ wv67)",fontsize=16,color="black",shape="box"];622 -> 628[label="",style="solid", color="black", weight=3];
33.10/16.35	99[label="primModNatS0 (Succ (Succ wv800)) wv100 (primGEqNatS (Succ wv800) wv100)",fontsize=16,color="burlywood",shape="box"];751[label="wv100/Succ wv1000",fontsize=10,color="white",style="solid",shape="box"];99 -> 751[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	751 -> 108[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	752[label="wv100/Zero",fontsize=10,color="white",style="solid",shape="box"];99 -> 752[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	752 -> 109[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	100[label="primModNatS0 (Succ Zero) wv100 (primGEqNatS Zero wv100)",fontsize=16,color="burlywood",shape="box"];753[label="wv100/Succ wv1000",fontsize=10,color="white",style="solid",shape="box"];100 -> 753[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	753 -> 110[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	754[label="wv100/Zero",fontsize=10,color="white",style="solid",shape="box"];100 -> 754[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	754 -> 111[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	101[label="Succ Zero",fontsize=16,color="green",shape="box"];400 -> 591[label="",style="dashed", color="red", weight=0];
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33.10/16.35	625 -> 591[label="",style="dashed", color="red", weight=0];
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33.10/16.35	625 -> 632[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	626 -> 43[label="",style="dashed", color="red", weight=0];
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33.10/16.35	109[label="primModNatS0 (Succ (Succ wv800)) Zero (primGEqNatS (Succ wv800) Zero)",fontsize=16,color="black",shape="box"];109 -> 120[label="",style="solid", color="black", weight=3];
33.10/16.35	110[label="primModNatS0 (Succ Zero) (Succ wv1000) (primGEqNatS Zero (Succ wv1000))",fontsize=16,color="black",shape="box"];110 -> 121[label="",style="solid", color="black", weight=3];
33.10/16.35	111[label="primModNatS0 (Succ Zero) Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];111 -> 122[label="",style="solid", color="black", weight=3];
33.10/16.35	598[label="Succ wv31",fontsize=16,color="green",shape="box"];599[label="Succ wv32",fontsize=16,color="green",shape="box"];600[label="Succ wv32",fontsize=16,color="green",shape="box"];631[label="wv650",fontsize=16,color="green",shape="box"];632[label="wv660",fontsize=16,color="green",shape="box"];633[label="wv67",fontsize=16,color="green",shape="box"];634[label="wv650",fontsize=16,color="green",shape="box"];635[label="Zero",fontsize=16,color="green",shape="box"];119 -> 526[label="",style="dashed", color="red", weight=0];
33.10/16.35	119[label="primModNatS0 (Succ (Succ wv800)) (Succ wv1000) (primGEqNatS wv800 wv1000)",fontsize=16,color="magenta"];119 -> 527[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	119 -> 528[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	119 -> 529[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	119 -> 530[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	120[label="primModNatS0 (Succ (Succ wv800)) Zero MyTrue",fontsize=16,color="black",shape="box"];120 -> 134[label="",style="solid", color="black", weight=3];
33.10/16.35	121[label="primModNatS0 (Succ Zero) (Succ wv1000) MyFalse",fontsize=16,color="black",shape="box"];121 -> 135[label="",style="solid", color="black", weight=3];
33.10/16.35	122[label="primModNatS0 (Succ Zero) Zero MyTrue",fontsize=16,color="black",shape="box"];122 -> 136[label="",style="solid", color="black", weight=3];
33.10/16.35	527[label="Succ wv800",fontsize=16,color="green",shape="box"];528[label="wv1000",fontsize=16,color="green",shape="box"];529[label="wv800",fontsize=16,color="green",shape="box"];530[label="wv1000",fontsize=16,color="green",shape="box"];526[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS wv62 wv63)",fontsize=16,color="burlywood",shape="triangle"];755[label="wv62/Succ wv620",fontsize=10,color="white",style="solid",shape="box"];526 -> 755[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	755 -> 563[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	756[label="wv62/Zero",fontsize=10,color="white",style="solid",shape="box"];526 -> 756[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	756 -> 564[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	134 -> 675[label="",style="dashed", color="red", weight=0];
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33.10/16.35	134 -> 677[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	134 -> 678[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	135[label="Succ (Succ Zero)",fontsize=16,color="green",shape="box"];136 -> 675[label="",style="dashed", color="red", weight=0];
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33.10/16.35	136 -> 680[label="",style="dashed", color="magenta", weight=3];
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33.10/16.35	563[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS (Succ wv620) wv63)",fontsize=16,color="burlywood",shape="box"];757[label="wv63/Succ wv630",fontsize=10,color="white",style="solid",shape="box"];563 -> 757[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	757 -> 571[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	758[label="wv63/Zero",fontsize=10,color="white",style="solid",shape="box"];563 -> 758[label="",style="solid", color="burlywood", weight=9];
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33.10/16.35	564[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS Zero wv63)",fontsize=16,color="burlywood",shape="box"];759[label="wv63/Succ wv630",fontsize=10,color="white",style="solid",shape="box"];564 -> 759[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	759 -> 573[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	760[label="wv63/Zero",fontsize=10,color="white",style="solid",shape="box"];564 -> 760[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	760 -> 574[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	676[label="Succ Zero",fontsize=16,color="green",shape="box"];677[label="Succ Zero",fontsize=16,color="green",shape="box"];678[label="Succ (Succ wv800)",fontsize=16,color="green",shape="box"];675[label="primModNatS (primMinusNatS wv69 wv70) (Succ wv71)",fontsize=16,color="burlywood",shape="triangle"];761[label="wv69/Succ wv690",fontsize=10,color="white",style="solid",shape="box"];675 -> 761[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	761 -> 706[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	762[label="wv69/Zero",fontsize=10,color="white",style="solid",shape="box"];675 -> 762[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	762 -> 707[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	679[label="Succ Zero",fontsize=16,color="green",shape="box"];680[label="Succ Zero",fontsize=16,color="green",shape="box"];681[label="Succ Zero",fontsize=16,color="green",shape="box"];571[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS (Succ wv620) (Succ wv630))",fontsize=16,color="black",shape="box"];571 -> 579[label="",style="solid", color="black", weight=3];
33.10/16.35	572[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS (Succ wv620) Zero)",fontsize=16,color="black",shape="box"];572 -> 580[label="",style="solid", color="black", weight=3];
33.10/16.35	573[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS Zero (Succ wv630))",fontsize=16,color="black",shape="box"];573 -> 581[label="",style="solid", color="black", weight=3];
33.10/16.35	574[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];574 -> 582[label="",style="solid", color="black", weight=3];
33.10/16.35	706[label="primModNatS (primMinusNatS (Succ wv690) wv70) (Succ wv71)",fontsize=16,color="burlywood",shape="box"];763[label="wv70/Succ wv700",fontsize=10,color="white",style="solid",shape="box"];706 -> 763[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	763 -> 708[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	764[label="wv70/Zero",fontsize=10,color="white",style="solid",shape="box"];706 -> 764[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	764 -> 709[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	707[label="primModNatS (primMinusNatS Zero wv70) (Succ wv71)",fontsize=16,color="burlywood",shape="box"];765[label="wv70/Succ wv700",fontsize=10,color="white",style="solid",shape="box"];707 -> 765[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	765 -> 710[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	766[label="wv70/Zero",fontsize=10,color="white",style="solid",shape="box"];707 -> 766[label="",style="solid", color="burlywood", weight=9];
33.10/16.35	766 -> 711[label="",style="solid", color="burlywood", weight=3];
33.10/16.35	579 -> 526[label="",style="dashed", color="red", weight=0];
33.10/16.35	579[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS wv620 wv630)",fontsize=16,color="magenta"];579 -> 587[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	579 -> 588[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	580[label="primModNatS0 (Succ wv60) (Succ wv61) MyTrue",fontsize=16,color="black",shape="triangle"];580 -> 589[label="",style="solid", color="black", weight=3];
33.10/16.35	581[label="primModNatS0 (Succ wv60) (Succ wv61) MyFalse",fontsize=16,color="black",shape="box"];581 -> 590[label="",style="solid", color="black", weight=3];
33.10/16.35	582 -> 580[label="",style="dashed", color="red", weight=0];
33.10/16.35	582[label="primModNatS0 (Succ wv60) (Succ wv61) MyTrue",fontsize=16,color="magenta"];708[label="primModNatS (primMinusNatS (Succ wv690) (Succ wv700)) (Succ wv71)",fontsize=16,color="black",shape="box"];708 -> 712[label="",style="solid", color="black", weight=3];
33.10/16.35	709[label="primModNatS (primMinusNatS (Succ wv690) Zero) (Succ wv71)",fontsize=16,color="black",shape="box"];709 -> 713[label="",style="solid", color="black", weight=3];
33.10/16.35	710[label="primModNatS (primMinusNatS Zero (Succ wv700)) (Succ wv71)",fontsize=16,color="black",shape="box"];710 -> 714[label="",style="solid", color="black", weight=3];
33.10/16.35	711[label="primModNatS (primMinusNatS Zero Zero) (Succ wv71)",fontsize=16,color="black",shape="box"];711 -> 715[label="",style="solid", color="black", weight=3];
33.10/16.35	587[label="wv630",fontsize=16,color="green",shape="box"];588[label="wv620",fontsize=16,color="green",shape="box"];589 -> 675[label="",style="dashed", color="red", weight=0];
33.10/16.35	589[label="primModNatS (primMinusNatS (Succ wv60) (Succ (Succ wv61))) (Succ (Succ (Succ wv61)))",fontsize=16,color="magenta"];589 -> 688[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	589 -> 689[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	589 -> 690[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	590[label="Succ (Succ wv60)",fontsize=16,color="green",shape="box"];712 -> 675[label="",style="dashed", color="red", weight=0];
33.10/16.35	712[label="primModNatS (primMinusNatS wv690 wv700) (Succ wv71)",fontsize=16,color="magenta"];712 -> 716[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	712 -> 717[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	713 -> 59[label="",style="dashed", color="red", weight=0];
33.10/16.35	713[label="primModNatS (Succ wv690) (Succ wv71)",fontsize=16,color="magenta"];713 -> 718[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	713 -> 719[label="",style="dashed", color="magenta", weight=3];
33.10/16.35	714[label="primModNatS Zero (Succ wv71)",fontsize=16,color="black",shape="triangle"];714 -> 720[label="",style="solid", color="black", weight=3];
33.10/16.35	715 -> 714[label="",style="dashed", color="red", weight=0];
33.10/16.35	715[label="primModNatS Zero (Succ wv71)",fontsize=16,color="magenta"];688[label="Succ (Succ wv61)",fontsize=16,color="green",shape="box"];689[label="Succ (Succ wv61)",fontsize=16,color="green",shape="box"];690[label="Succ wv60",fontsize=16,color="green",shape="box"];716[label="wv700",fontsize=16,color="green",shape="box"];717[label="wv690",fontsize=16,color="green",shape="box"];718[label="wv690",fontsize=16,color="green",shape="box"];719[label="wv71",fontsize=16,color="green",shape="box"];720[label="Zero",fontsize=16,color="green",shape="box"];}
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(6)
33.10/16.35	Complex Obligation (AND)
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(7)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_psPs(Cons(wv110, wv111), wv8, wv10) -> new_psPs(wv111, wv8, wv10)
33.10/16.35	
33.10/16.35	R is empty.
33.10/16.35	Q is empty.
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(8) QDPSizeChangeProof (EQUIVALENT)
33.10/16.35	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. 
33.10/16.35	
33.10/16.35	From the DPs we obtained the following set of size-change graphs:
33.10/16.35	*new_psPs(Cons(wv110, wv111), wv8, wv10) -> new_psPs(wv111, wv8, wv10)
33.10/16.35	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
33.10/16.35	
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(9)
33.10/16.35	YES
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(10)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primModNatS0(wv60, wv61, Main.Zero, Main.Zero) -> new_primModNatS00(wv60, wv61)
33.10/16.35	   new_primModNatS1(Main.Succ(Main.Succ(wv800)), Main.Succ(Main.Succ(wv1000))) -> new_primModNatS0(Main.Succ(wv800), wv1000, wv800, wv1000)
33.10/16.35	   new_primModNatS1(Main.Succ(Main.Zero), Main.Succ(Main.Zero)) -> new_primModNatS(Main.Succ(Main.Zero), Main.Succ(Main.Zero), Main.Succ(Main.Zero))
33.10/16.35	   new_primModNatS0(wv60, wv61, Main.Succ(wv620), Main.Succ(wv630)) -> new_primModNatS0(wv60, wv61, wv620, wv630)
33.10/16.35	   new_primModNatS(Main.Succ(wv690), Main.Succ(wv700), wv71) -> new_primModNatS(wv690, wv700, wv71)
33.10/16.35	   new_primModNatS(Main.Succ(wv690), Main.Zero, wv71) -> new_primModNatS1(wv690, wv71)
33.10/16.35	   new_primModNatS0(wv60, wv61, Main.Succ(wv620), Main.Zero) -> new_primModNatS(Main.Succ(wv60), Main.Succ(Main.Succ(wv61)), Main.Succ(Main.Succ(wv61)))
33.10/16.35	   new_primModNatS1(Main.Succ(Main.Succ(wv800)), Main.Succ(Main.Zero)) -> new_primModNatS(Main.Succ(Main.Succ(wv800)), Main.Succ(Main.Zero), Main.Succ(Main.Zero))
33.10/16.35	   new_primModNatS00(wv60, wv61) -> new_primModNatS(Main.Succ(wv60), Main.Succ(Main.Succ(wv61)), Main.Succ(Main.Succ(wv61)))
33.10/16.35	
33.10/16.35	R is empty.
33.10/16.35	Q is empty.
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(11) QDPOrderProof (EQUIVALENT)
33.10/16.35	We use the reduction pair processor [LPAR04,JAR06].
33.10/16.35	
33.10/16.35	
33.10/16.35	The following pairs can be oriented strictly and are deleted.
33.10/16.35	
33.10/16.35	   new_primModNatS(Main.Succ(wv690), Main.Succ(wv700), wv71) -> new_primModNatS(wv690, wv700, wv71)
33.10/16.35	   new_primModNatS(Main.Succ(wv690), Main.Zero, wv71) -> new_primModNatS1(wv690, wv71)
33.10/16.35	The remaining pairs can at least be oriented weakly.
33.10/16.35	Used ordering:  Polynomial interpretation [POLO]:
33.10/16.35	
33.10/16.35	   POL(Main.Succ(x_1)) = 1 + x_1
33.10/16.35	   POL(Main.Zero) = 0
33.10/16.35	   POL(new_primModNatS(x_1, x_2, x_3)) = x_1
33.10/16.35	   POL(new_primModNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1
33.10/16.35	   POL(new_primModNatS00(x_1, x_2)) = 1 + x_1
33.10/16.35	   POL(new_primModNatS1(x_1, x_2)) = x_1
33.10/16.35	
33.10/16.35	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.10/16.35	none
33.10/16.35	
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(12)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primModNatS0(wv60, wv61, Main.Zero, Main.Zero) -> new_primModNatS00(wv60, wv61)
33.10/16.35	   new_primModNatS1(Main.Succ(Main.Succ(wv800)), Main.Succ(Main.Succ(wv1000))) -> new_primModNatS0(Main.Succ(wv800), wv1000, wv800, wv1000)
33.10/16.35	   new_primModNatS1(Main.Succ(Main.Zero), Main.Succ(Main.Zero)) -> new_primModNatS(Main.Succ(Main.Zero), Main.Succ(Main.Zero), Main.Succ(Main.Zero))
33.10/16.35	   new_primModNatS0(wv60, wv61, Main.Succ(wv620), Main.Succ(wv630)) -> new_primModNatS0(wv60, wv61, wv620, wv630)
33.10/16.35	   new_primModNatS0(wv60, wv61, Main.Succ(wv620), Main.Zero) -> new_primModNatS(Main.Succ(wv60), Main.Succ(Main.Succ(wv61)), Main.Succ(Main.Succ(wv61)))
33.10/16.35	   new_primModNatS1(Main.Succ(Main.Succ(wv800)), Main.Succ(Main.Zero)) -> new_primModNatS(Main.Succ(Main.Succ(wv800)), Main.Succ(Main.Zero), Main.Succ(Main.Zero))
33.10/16.35	   new_primModNatS00(wv60, wv61) -> new_primModNatS(Main.Succ(wv60), Main.Succ(Main.Succ(wv61)), Main.Succ(Main.Succ(wv61)))
33.10/16.35	
33.10/16.35	R is empty.
33.10/16.35	Q is empty.
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(13) DependencyGraphProof (EQUIVALENT)
33.10/16.35	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(14)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primModNatS0(wv60, wv61, Main.Succ(wv620), Main.Succ(wv630)) -> new_primModNatS0(wv60, wv61, wv620, wv630)
33.10/16.35	
33.10/16.35	R is empty.
33.10/16.35	Q is empty.
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(15) QDPSizeChangeProof (EQUIVALENT)
33.10/16.35	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. 
33.10/16.35	
33.10/16.35	From the DPs we obtained the following set of size-change graphs:
33.10/16.35	*new_primModNatS0(wv60, wv61, Main.Succ(wv620), Main.Succ(wv630)) -> new_primModNatS0(wv60, wv61, wv620, wv630)
33.10/16.35	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
33.10/16.35	
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(16)
33.10/16.35	YES
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(17)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS1(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS0(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS0(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS0(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS1(Main.Zero, Main.Zero) -> new_primDivNatS(Main.Zero, Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS00(wv31, wv32) -> new_primDivNatS(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32))
33.10/16.35	   new_primDivNatS(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS1(wv650, wv67)
33.10/16.35	   new_primDivNatS1(Main.Succ(wv130), Main.Zero) -> new_primDivNatS(Main.Succ(wv130), Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS0(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32))
33.10/16.35	   new_primDivNatS0(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS00(wv31, wv32)
33.10/16.35	
33.10/16.35	R is empty.
33.10/16.35	Q is empty.
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(18) DependencyGraphProof (EQUIVALENT)
33.10/16.35	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(19)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS1(wv650, wv67)
33.10/16.35	   new_primDivNatS1(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS0(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS0(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS0(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS0(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32))
33.10/16.35	   new_primDivNatS(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS0(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS00(wv31, wv32)
33.10/16.35	   new_primDivNatS00(wv31, wv32) -> new_primDivNatS(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32))
33.10/16.35	   new_primDivNatS1(Main.Succ(wv130), Main.Zero) -> new_primDivNatS(Main.Succ(wv130), Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	R is empty.
33.10/16.35	Q is empty.
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(20) QDPOrderProof (EQUIVALENT)
33.10/16.35	We use the reduction pair processor [LPAR04,JAR06].
33.10/16.35	
33.10/16.35	
33.10/16.35	The following pairs can be oriented strictly and are deleted.
33.10/16.35	
33.10/16.35	   new_primDivNatS(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS1(wv650, wv67)
33.10/16.35	   new_primDivNatS(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS(wv650, wv660, wv67)
33.10/16.35	The remaining pairs can at least be oriented weakly.
33.10/16.35	Used ordering:  Polynomial interpretation [POLO]:
33.10/16.35	
33.10/16.35	   POL(Main.Succ(x_1)) = 1 + x_1
33.10/16.35	   POL(Main.Zero) = 0
33.10/16.35	   POL(new_primDivNatS(x_1, x_2, x_3)) = x_1
33.10/16.35	   POL(new_primDivNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1
33.10/16.35	   POL(new_primDivNatS00(x_1, x_2)) = 1 + x_1
33.10/16.35	   POL(new_primDivNatS1(x_1, x_2)) = x_1
33.10/16.35	
33.10/16.35	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.10/16.35	none
33.10/16.35	
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(21)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS1(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS0(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS0(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS0(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS0(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32))
33.10/16.35	   new_primDivNatS0(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS00(wv31, wv32)
33.10/16.35	   new_primDivNatS00(wv31, wv32) -> new_primDivNatS(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32))
33.10/16.35	   new_primDivNatS1(Main.Succ(wv130), Main.Zero) -> new_primDivNatS(Main.Succ(wv130), Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	R is empty.
33.10/16.35	Q is empty.
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(22) DependencyGraphProof (EQUIVALENT)
33.10/16.35	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(23)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS0(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS0(wv31, wv32, wv330, wv340)
33.10/16.35	
33.10/16.35	R is empty.
33.10/16.35	Q is empty.
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(24) QDPSizeChangeProof (EQUIVALENT)
33.10/16.35	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. 
33.10/16.35	
33.10/16.35	From the DPs we obtained the following set of size-change graphs:
33.10/16.35	*new_primDivNatS0(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS0(wv31, wv32, wv330, wv340)
33.10/16.35	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
33.10/16.35	
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(25)
33.10/16.35	YES
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(26)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Neg(wv30)) -> new_primShowInt(Main.Pos(wv30))
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(wv300))) -> new_primShowInt(new_divMyInt(wv300, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_divMyInt(wv13, wv14) -> Main.Pos(new_primDivNatS2(wv13, wv14))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_divMyInt(x0, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(27) DependencyGraphProof (EQUIVALENT)
33.10/16.35	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(28)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(wv300))) -> new_primShowInt(new_divMyInt(wv300, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_divMyInt(wv13, wv14) -> Main.Pos(new_primDivNatS2(wv13, wv14))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_divMyInt(x0, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(29) TransformationProof (EQUIVALENT)
33.10/16.35	By rewriting [LPAR04] the rule new_primShowInt(Main.Pos(Main.Succ(wv300))) -> new_primShowInt(new_divMyInt(wv300, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))) at position [0] we obtained the following new rules [LPAR04]:
33.10/16.35	
33.10/16.35	   (new_primShowInt(Main.Pos(Main.Succ(wv300))) -> new_primShowInt(Main.Pos(new_primDivNatS2(wv300, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))),new_primShowInt(Main.Pos(Main.Succ(wv300))) -> new_primShowInt(Main.Pos(new_primDivNatS2(wv300, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(30)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(wv300))) -> new_primShowInt(Main.Pos(new_primDivNatS2(wv300, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_divMyInt(wv13, wv14) -> Main.Pos(new_primDivNatS2(wv13, wv14))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_divMyInt(x0, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(31) UsableRulesProof (EQUIVALENT)
33.10/16.35	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(32)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(wv300))) -> new_primShowInt(Main.Pos(new_primDivNatS2(wv300, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_divMyInt(x0, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(33) QReductionProof (EQUIVALENT)
33.10/16.35	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
33.10/16.35	
33.10/16.35	   new_divMyInt(x0, x1)
33.10/16.35	
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(34)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(wv300))) -> new_primShowInt(Main.Pos(new_primDivNatS2(wv300, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(35) MNOCProof (EQUIVALENT)
33.10/16.35	We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(36)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(wv300))) -> new_primShowInt(Main.Pos(new_primDivNatS2(wv300, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	Q is empty.
33.10/16.35	We have to consider all (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(37) InductionCalculusProof (EQUIVALENT)
33.10/16.35	Note that final constraints are written in bold face.
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	For Pair new_primShowInt(Main.Pos(Main.Succ(wv300))) -> new_primShowInt(Main.Pos(new_primDivNatS2(wv300, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))) the following chains were created:
33.10/16.35	*We consider the chain new_primShowInt(Main.Pos(Main.Succ(x0))) -> new_primShowInt(Main.Pos(new_primDivNatS2(x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))), new_primShowInt(Main.Pos(Main.Succ(x1))) -> new_primShowInt(Main.Pos(new_primDivNatS2(x1, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))) which results in the following constraint:
33.10/16.35	
33.10/16.35	(1)    (new_primShowInt(Main.Pos(new_primDivNatS2(x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))))=new_primShowInt(Main.Pos(Main.Succ(x1)))  ==>  new_primShowInt(Main.Pos(Main.Succ(x0)))_>=_new_primShowInt(Main.Pos(new_primDivNatS2(x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	We simplified constraint (1) using rules (I), (II), (VII) which results in the following new constraint:
33.10/16.35	
33.10/16.35	(2)    (Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))=x2 & new_primDivNatS2(x0, x2)=Main.Succ(x1)  ==>  new_primShowInt(Main.Pos(Main.Succ(x0)))_>=_new_primShowInt(Main.Pos(new_primDivNatS2(x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS2(x0, x2)=Main.Succ(x1) which results in the following new constraints:
33.10/16.35	
33.10/16.35	(3)    (new_primDivNatS02(x4, x3, x4, x3)=Main.Succ(x1) & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))=Main.Succ(x3)  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(x4))))_>=_new_primShowInt(Main.Pos(new_primDivNatS2(Main.Succ(x4), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))))
33.10/16.35	
33.10/16.35	(4)    (Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))=Main.Succ(x1) & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))=Main.Zero  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Zero)))_>=_new_primShowInt(Main.Pos(new_primDivNatS2(Main.Zero, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))))
33.10/16.35	
33.10/16.35	(5)    (Main.Succ(new_primDivNatS3(Main.Succ(x6), Main.Zero, Main.Zero))=Main.Succ(x1) & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))=Main.Zero  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(x6))))_>=_new_primShowInt(Main.Pos(new_primDivNatS2(Main.Succ(x6), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	We simplified constraint (3) using rules (I), (II), (VII) which results in the following new constraint:
33.10/16.35	
33.10/16.35	(6)    (x4=x7 & x3=x8 & new_primDivNatS02(x4, x3, x7, x8)=Main.Succ(x1) & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x3  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(x4))))_>=_new_primShowInt(Main.Pos(new_primDivNatS2(Main.Succ(x4), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS02(x4, x3, x7, x8)=Main.Succ(x1) which results in the following new constraints:
33.10/16.35	
33.10/16.35	(7)    (new_primDivNatS02(x12, x11, x10, x9)=Main.Succ(x1) & x12=Main.Succ(x10) & x11=Main.Succ(x9) & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x11 & (\/x13:new_primDivNatS02(x12, x11, x10, x9)=Main.Succ(x13) & x12=x10 & x11=x9 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x11  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(x12))))_>=_new_primShowInt(Main.Pos(new_primDivNatS2(Main.Succ(x12), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))))  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(x12))))_>=_new_primShowInt(Main.Pos(new_primDivNatS2(Main.Succ(x12), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))))
33.10/16.35	
33.10/16.35	(8)    (new_primDivNatS01(x15, x14)=Main.Succ(x1) & x15=Main.Zero & x14=Main.Zero & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x14  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(x15))))_>=_new_primShowInt(Main.Pos(new_primDivNatS2(Main.Succ(x15), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))))
33.10/16.35	
33.10/16.35	(9)    (new_primDivNatS01(x18, x17)=Main.Succ(x1) & x18=Main.Succ(x16) & x17=Main.Zero & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x17  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(x18))))_>=_new_primShowInt(Main.Pos(new_primDivNatS2(Main.Succ(x18), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	We simplified constraint (7) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
33.10/16.35	
33.10/16.35	(10)    (new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x10)))))_>=_new_primShowInt(Main.Pos(new_primDivNatS2(Main.Succ(Main.Succ(x10)), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	We solved constraint (8) using rules (I), (II), (III).We solved constraint (9) using rules (I), (II), (III).
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	To summarize, we get the following constraints P__>=_ for the following pairs.
33.10/16.35	
33.10/16.35	*new_primShowInt(Main.Pos(Main.Succ(wv300))) -> new_primShowInt(Main.Pos(new_primDivNatS2(wv300, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))))
33.10/16.35	
33.10/16.35	*(new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x10)))))_>=_new_primShowInt(Main.Pos(new_primDivNatS2(Main.Succ(Main.Succ(x10)), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(38)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(wv300))) -> new_primShowInt(Main.Pos(new_primDivNatS2(wv300, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(39) TransformationProof (EQUIVALENT)
33.10/16.35	By narrowing [LPAR04] the rule new_primShowInt(Main.Pos(Main.Succ(wv300))) -> new_primShowInt(Main.Pos(new_primDivNatS2(wv300, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))) at position [0,0] we obtained the following new rules [LPAR04]:
33.10/16.35	
33.10/16.35	   (new_primShowInt(Main.Pos(Main.Succ(Main.Succ(x0)))) -> new_primShowInt(Main.Pos(new_primDivNatS02(x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))),new_primShowInt(Main.Pos(Main.Succ(Main.Succ(x0)))) -> new_primShowInt(Main.Pos(new_primDivNatS02(x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))))
33.10/16.35	   (new_primShowInt(Main.Pos(Main.Succ(Main.Zero))) -> new_primShowInt(Main.Pos(Main.Zero)),new_primShowInt(Main.Pos(Main.Succ(Main.Zero))) -> new_primShowInt(Main.Pos(Main.Zero)))
33.10/16.35	
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(40)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Succ(x0)))) -> new_primShowInt(Main.Pos(new_primDivNatS02(x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Zero))) -> new_primShowInt(Main.Pos(Main.Zero))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(41) DependencyGraphProof (EQUIVALENT)
33.10/16.35	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(42)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Succ(x0)))) -> new_primShowInt(Main.Pos(new_primDivNatS02(x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(43) TransformationProof (EQUIVALENT)
33.10/16.35	By narrowing [LPAR04] the rule new_primShowInt(Main.Pos(Main.Succ(Main.Succ(x0)))) -> new_primShowInt(Main.Pos(new_primDivNatS02(x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))) at position [0,0] we obtained the following new rules [LPAR04]:
33.10/16.35	
33.10/16.35	   (new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x2))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(x2), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))),new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x2))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(x2), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))))
33.10/16.35	   (new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Zero)))) -> new_primShowInt(Main.Pos(Main.Zero)),new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Zero)))) -> new_primShowInt(Main.Pos(Main.Zero)))
33.10/16.35	
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(44)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x2))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(x2), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Zero)))) -> new_primShowInt(Main.Pos(Main.Zero))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(45) DependencyGraphProof (EQUIVALENT)
33.10/16.35	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(46)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x2))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(x2), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(47) TransformationProof (EQUIVALENT)
33.10/16.35	By narrowing [LPAR04] the rule new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x2))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(x2), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))) at position [0,0] we obtained the following new rules [LPAR04]:
33.10/16.35	
33.10/16.35	   (new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x2)))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(x2)), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))),new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x2)))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(x2)), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))))
33.10/16.35	   (new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))) -> new_primShowInt(Main.Pos(Main.Zero)),new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))) -> new_primShowInt(Main.Pos(Main.Zero)))
33.10/16.35	
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(48)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x2)))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(x2)), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))) -> new_primShowInt(Main.Pos(Main.Zero))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(49) DependencyGraphProof (EQUIVALENT)
33.10/16.35	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(50)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x2)))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(x2)), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(51) TransformationProof (EQUIVALENT)
33.10/16.35	By narrowing [LPAR04] the rule new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x2)))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(x2)), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))) at position [0,0] we obtained the following new rules [LPAR04]:
33.10/16.35	
33.10/16.35	   (new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x2))))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x2))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))),new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x2))))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x2))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	   (new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))) -> new_primShowInt(Main.Pos(Main.Zero)),new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))) -> new_primShowInt(Main.Pos(Main.Zero)))
33.10/16.35	
33.10/16.35	
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(52)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x2))))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x2))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))) -> new_primShowInt(Main.Pos(Main.Zero))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(53) DependencyGraphProof (EQUIVALENT)
33.10/16.35	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(54)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x2))))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x2))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(55) MNOCProof (EQUIVALENT)
33.10/16.35	We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(56)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x2))))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x2))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	Q is empty.
33.10/16.35	We have to consider all (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(57) InductionCalculusProof (EQUIVALENT)
33.10/16.35	Note that final constraints are written in bold face.
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	For Pair new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x2))))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x2))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))) the following chains were created:
33.10/16.35	*We consider the chain new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x0))))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x0))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x1))))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x1))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x1, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))) which results in the following constraint:
33.10/16.35	
33.10/16.35	(1)    (new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x0))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x1)))))))  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x0)))))))_>=_new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x0))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	We simplified constraint (1) using rules (I), (II), (VII) which results in the following new constraint:
33.10/16.35	
33.10/16.35	(2)    (Main.Succ(Main.Succ(Main.Succ(x0)))=x2 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x3 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))=x4 & new_primDivNatS02(x2, x3, x0, x4)=Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x1)))))  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x0)))))))_>=_new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x0))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x0, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS02(x2, x3, x0, x4)=Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x1))))) which results in the following new constraints:
33.10/16.35	
33.10/16.35	(3)    (new_primDivNatS02(x8, x7, x6, x5)=Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x1))))) & Main.Succ(Main.Succ(Main.Succ(Main.Succ(x6))))=x8 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x7 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))=Main.Succ(x5) & (\/x9:new_primDivNatS02(x8, x7, x6, x5)=Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x9))))) & Main.Succ(Main.Succ(Main.Succ(x6)))=x8 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x7 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))=x5  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x6)))))))_>=_new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x6))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x6, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x6))))))))_>=_new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x6)))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), Main.Succ(x6), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	
33.10/16.35	(4)    (new_primDivNatS01(x11, x10)=Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x1))))) & Main.Succ(Main.Succ(Main.Succ(Main.Zero)))=x11 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x10 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))=Main.Zero  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))_>=_new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(Main.Zero))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), Main.Zero, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	
33.10/16.35	(5)    (new_primDivNatS01(x14, x13)=Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x1))))) & Main.Succ(Main.Succ(Main.Succ(Main.Succ(x12))))=x14 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x13 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))=Main.Zero  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x12))))))))_>=_new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x12)))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), Main.Succ(x12), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint:
33.10/16.35	
33.10/16.35	(6)    (new_primDivNatS02(x8, x7, x6, x5)=Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x1))))) & Main.Succ(Main.Succ(Main.Succ(Main.Succ(x6))))=x8 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x7 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))=x5  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x6))))))))_>=_new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x6)))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), Main.Succ(x6), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS02(x8, x7, x6, x5)=Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x1))))) which results in the following new constraints:
33.10/16.35	
33.10/16.35	(7)    (new_primDivNatS02(x21, x20, x19, x18)=Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x1))))) & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x19)))))=x21 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x20 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))=Main.Succ(x18) & (\/x22:new_primDivNatS02(x21, x20, x19, x18)=Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x22))))) & Main.Succ(Main.Succ(Main.Succ(Main.Succ(x19))))=x21 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x20 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))=x18  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x19))))))))_>=_new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x19)))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), Main.Succ(x19), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x19)))))))))_>=_new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x19))))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), Main.Succ(Main.Succ(x19)), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	
33.10/16.35	(8)    (new_primDivNatS01(x24, x23)=Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x1))))) & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))=x24 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x23 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))=Main.Zero  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))_>=_new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), Main.Succ(Main.Zero), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	
33.10/16.35	(9)    (new_primDivNatS01(x27, x26)=Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x1))))) & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x25)))))=x27 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))=x26 & Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))=Main.Zero  ==>  new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x25)))))))))_>=_new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x25))))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), Main.Succ(Main.Succ(x25)), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint:
33.10/16.35	
33.10/16.35	(10)    (new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x19)))))))))_>=_new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x19))))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), Main.Succ(Main.Succ(x19)), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	We solved constraint (8) using rules (I), (II).We solved constraint (9) using rules (I), (II).
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	To summarize, we get the following constraints P__>=_ for the following pairs.
33.10/16.35	
33.10/16.35	*new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x2))))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x2))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))
33.10/16.35	
33.10/16.35	*(new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x19)))))))))_>=_new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x19))))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), Main.Succ(Main.Succ(x19)), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))))
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	
33.10/16.35	The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(58)
33.10/16.35	Obligation:
33.10/16.35	Q DP problem:
33.10/16.35	The TRS P consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primShowInt(Main.Pos(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(x2))))))) -> new_primShowInt(Main.Pos(new_primDivNatS02(Main.Succ(Main.Succ(Main.Succ(x2))), Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero)))))))), x2, Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Succ(Main.Zero))))))))
33.10/16.35	
33.10/16.35	The TRS R consists of the following rules:
33.10/16.35	
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Succ(wv140)) -> new_primDivNatS02(wv130, wv140, wv130, wv140)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(wv140)) -> Main.Zero
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Succ(wv340)) -> new_primDivNatS02(wv31, wv32, wv330, wv340)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Succ(wv330), Main.Zero) -> new_primDivNatS01(wv31, wv32)
33.10/16.35	   new_primDivNatS02(wv31, wv32, Main.Zero, Main.Succ(wv340)) -> Main.Zero
33.10/16.35	   new_primDivNatS01(wv31, wv32) -> Main.Succ(new_primDivNatS3(Main.Succ(wv31), Main.Succ(wv32), Main.Succ(wv32)))
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Succ(wv660), wv67) -> new_primDivNatS3(wv650, wv660, wv67)
33.10/16.35	   new_primDivNatS3(Main.Succ(wv650), Main.Zero, wv67) -> new_primDivNatS2(wv650, wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(wv660), wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, wv67) -> new_primDivNatS4(wv67)
33.10/16.35	   new_primDivNatS4(wv67) -> Main.Zero
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Zero, Main.Zero, Main.Zero))
33.10/16.35	   new_primDivNatS2(Main.Succ(wv130), Main.Zero) -> Main.Succ(new_primDivNatS3(Main.Succ(wv130), Main.Zero, Main.Zero))
33.10/16.35	
33.10/16.35	The set Q consists of the following terms:
33.10/16.35	
33.10/16.35	   new_primDivNatS4(x0)
33.10/16.35	   new_primDivNatS01(x0, x1)
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Succ(x0), x1)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Succ(x2))
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Succ(x1))
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Zero, x1)
33.10/16.35	   new_primDivNatS2(Main.Succ(x0), Main.Zero)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Zero)
33.10/16.35	   new_primDivNatS3(Main.Succ(x0), Main.Succ(x1), x2)
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Succ(x2), Main.Succ(x3))
33.10/16.35	   new_primDivNatS3(Main.Zero, Main.Zero, x0)
33.10/16.35	   new_primDivNatS2(Main.Zero, Main.Succ(x0))
33.10/16.35	   new_primDivNatS02(x0, x1, Main.Zero, Main.Zero)
33.10/16.35	
33.10/16.35	We have to consider all minimal (P,Q,R)-chains.
33.10/16.35	----------------------------------------
33.10/16.35	
33.10/16.35	(59) Narrow (COMPLETE)
33.10/16.35	Haskell To QDPs
33.10/16.35	
33.10/16.35	digraph dp_graph {
33.10/16.35	node [outthreshold=100, inthreshold=100];1[label="showMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
33.10/16.35	3[label="showMyInt wv3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
33.10/16.35	4[label="primShowInt wv3",fontsize=16,color="burlywood",shape="triangle"];721[label="wv3/Pos wv30",fontsize=10,color="white",style="solid",shape="box"];4 -> 721[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	721 -> 5[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	722[label="wv3/Neg wv30",fontsize=10,color="white",style="solid",shape="box"];4 -> 722[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	722 -> 6[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	5[label="primShowInt (Pos wv30)",fontsize=16,color="burlywood",shape="box"];723[label="wv30/Succ wv300",fontsize=10,color="white",style="solid",shape="box"];5 -> 723[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	723 -> 7[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	724[label="wv30/Zero",fontsize=10,color="white",style="solid",shape="box"];5 -> 724[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	724 -> 8[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	6[label="primShowInt (Neg wv30)",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
33.10/16.36	7[label="primShowInt (Pos (Succ wv300))",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
33.10/16.36	8[label="primShowInt (Pos Zero)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
33.10/16.36	9[label="Cons (Char (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))) (primShowInt (Pos wv30))",fontsize=16,color="green",shape="box"];9 -> 12[label="",style="dashed", color="green", weight=3];
33.10/16.36	10 -> 24[label="",style="dashed", color="red", weight=0];
33.10/16.36	10[label="psPs (primShowInt (divMyInt (Pos (Succ wv300)) (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))) (Cons (toEnumChar (modMyInt (Pos (Succ wv300)) (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))) Nil)",fontsize=16,color="magenta"];10 -> 25[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	10 -> 26[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	10 -> 27[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	11[label="Cons (Char (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))))))))))))))))))))))))))) Nil",fontsize=16,color="green",shape="box"];12 -> 4[label="",style="dashed", color="red", weight=0];
33.10/16.36	12[label="primShowInt (Pos wv30)",fontsize=16,color="magenta"];12 -> 16[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	25[label="wv300",fontsize=16,color="green",shape="box"];26 -> 4[label="",style="dashed", color="red", weight=0];
33.10/16.36	26[label="primShowInt (divMyInt (Pos (Succ wv300)) (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))",fontsize=16,color="magenta"];26 -> 29[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	27[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];24[label="psPs wv11 (Cons (toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))) Nil)",fontsize=16,color="burlywood",shape="triangle"];725[label="wv11/Cons wv110 wv111",fontsize=10,color="white",style="solid",shape="box"];24 -> 725[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	725 -> 30[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	726[label="wv11/Nil",fontsize=10,color="white",style="solid",shape="box"];24 -> 726[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	726 -> 31[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	16[label="Pos wv30",fontsize=16,color="green",shape="box"];29 -> 32[label="",style="dashed", color="red", weight=0];
33.10/16.36	29[label="divMyInt (Pos (Succ wv300)) (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))",fontsize=16,color="magenta"];29 -> 33[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	29 -> 34[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	30[label="psPs (Cons wv110 wv111) (Cons (toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))) Nil)",fontsize=16,color="black",shape="box"];30 -> 35[label="",style="solid", color="black", weight=3];
33.10/16.36	31[label="psPs Nil (Cons (toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))) Nil)",fontsize=16,color="black",shape="box"];31 -> 36[label="",style="solid", color="black", weight=3];
33.10/16.36	33[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];34[label="wv300",fontsize=16,color="green",shape="box"];32[label="divMyInt (Pos (Succ wv13)) (Pos (Succ wv14))",fontsize=16,color="black",shape="triangle"];32 -> 37[label="",style="solid", color="black", weight=3];
33.10/16.36	35[label="Cons wv110 (psPs wv111 (Cons (toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))) Nil))",fontsize=16,color="green",shape="box"];35 -> 38[label="",style="dashed", color="green", weight=3];
33.10/16.36	36[label="Cons (toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))) Nil",fontsize=16,color="green",shape="box"];36 -> 39[label="",style="dashed", color="green", weight=3];
33.10/16.36	37[label="primDivInt (Pos (Succ wv13)) (Pos (Succ wv14))",fontsize=16,color="black",shape="box"];37 -> 40[label="",style="solid", color="black", weight=3];
33.10/16.36	38 -> 24[label="",style="dashed", color="red", weight=0];
33.10/16.36	38[label="psPs wv111 (Cons (toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))) Nil)",fontsize=16,color="magenta"];38 -> 41[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	39[label="toEnumChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))",fontsize=16,color="black",shape="box"];39 -> 42[label="",style="solid", color="black", weight=3];
33.10/16.36	40[label="Pos (primDivNatS (Succ wv13) (Succ wv14))",fontsize=16,color="green",shape="box"];40 -> 43[label="",style="dashed", color="green", weight=3];
33.10/16.36	41[label="wv111",fontsize=16,color="green",shape="box"];42[label="primIntToChar (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
33.10/16.36	43[label="primDivNatS (Succ wv13) (Succ wv14)",fontsize=16,color="black",shape="triangle"];43 -> 45[label="",style="solid", color="black", weight=3];
33.10/16.36	44[label="Char (modMyInt (Pos (Succ wv8)) (Pos (Succ wv10)))",fontsize=16,color="green",shape="box"];44 -> 46[label="",style="dashed", color="green", weight=3];
33.10/16.36	45[label="primDivNatS0 wv13 wv14 (primGEqNatS wv13 wv14)",fontsize=16,color="burlywood",shape="box"];727[label="wv13/Succ wv130",fontsize=10,color="white",style="solid",shape="box"];45 -> 727[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	727 -> 47[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	728[label="wv13/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 728[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	728 -> 48[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	46[label="modMyInt (Pos (Succ wv8)) (Pos (Succ wv10))",fontsize=16,color="black",shape="box"];46 -> 49[label="",style="solid", color="black", weight=3];
33.10/16.36	47[label="primDivNatS0 (Succ wv130) wv14 (primGEqNatS (Succ wv130) wv14)",fontsize=16,color="burlywood",shape="box"];729[label="wv14/Succ wv140",fontsize=10,color="white",style="solid",shape="box"];47 -> 729[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	729 -> 50[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	730[label="wv14/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 730[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	730 -> 51[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	48[label="primDivNatS0 Zero wv14 (primGEqNatS Zero wv14)",fontsize=16,color="burlywood",shape="box"];731[label="wv14/Succ wv140",fontsize=10,color="white",style="solid",shape="box"];48 -> 731[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	731 -> 52[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	732[label="wv14/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 732[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	732 -> 53[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	49[label="primModInt (Pos (Succ wv8)) (Pos (Succ wv10))",fontsize=16,color="black",shape="box"];49 -> 54[label="",style="solid", color="black", weight=3];
33.10/16.36	50[label="primDivNatS0 (Succ wv130) (Succ wv140) (primGEqNatS (Succ wv130) (Succ wv140))",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3];
33.10/16.36	51[label="primDivNatS0 (Succ wv130) Zero (primGEqNatS (Succ wv130) Zero)",fontsize=16,color="black",shape="box"];51 -> 56[label="",style="solid", color="black", weight=3];
33.10/16.36	52[label="primDivNatS0 Zero (Succ wv140) (primGEqNatS Zero (Succ wv140))",fontsize=16,color="black",shape="box"];52 -> 57[label="",style="solid", color="black", weight=3];
33.10/16.36	53[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];53 -> 58[label="",style="solid", color="black", weight=3];
33.10/16.36	54[label="Pos (primModNatS (Succ wv8) (Succ wv10))",fontsize=16,color="green",shape="box"];54 -> 59[label="",style="dashed", color="green", weight=3];
33.10/16.36	55 -> 324[label="",style="dashed", color="red", weight=0];
33.10/16.36	55[label="primDivNatS0 (Succ wv130) (Succ wv140) (primGEqNatS wv130 wv140)",fontsize=16,color="magenta"];55 -> 325[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	55 -> 326[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	55 -> 327[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	55 -> 328[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	56[label="primDivNatS0 (Succ wv130) Zero MyTrue",fontsize=16,color="black",shape="box"];56 -> 62[label="",style="solid", color="black", weight=3];
33.10/16.36	57[label="primDivNatS0 Zero (Succ wv140) MyFalse",fontsize=16,color="black",shape="box"];57 -> 63[label="",style="solid", color="black", weight=3];
33.10/16.36	58[label="primDivNatS0 Zero Zero MyTrue",fontsize=16,color="black",shape="box"];58 -> 64[label="",style="solid", color="black", weight=3];
33.10/16.36	59[label="primModNatS (Succ wv8) (Succ wv10)",fontsize=16,color="burlywood",shape="triangle"];733[label="wv10/Succ wv100",fontsize=10,color="white",style="solid",shape="box"];59 -> 733[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	733 -> 65[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	734[label="wv10/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 734[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	734 -> 66[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	325[label="wv130",fontsize=16,color="green",shape="box"];326[label="wv130",fontsize=16,color="green",shape="box"];327[label="wv140",fontsize=16,color="green",shape="box"];328[label="wv140",fontsize=16,color="green",shape="box"];324[label="primDivNatS0 (Succ wv31) (Succ wv32) (primGEqNatS wv33 wv34)",fontsize=16,color="burlywood",shape="triangle"];735[label="wv33/Succ wv330",fontsize=10,color="white",style="solid",shape="box"];324 -> 735[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	735 -> 357[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	736[label="wv33/Zero",fontsize=10,color="white",style="solid",shape="box"];324 -> 736[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	736 -> 358[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	62[label="Succ (primDivNatS (primMinusNatS (Succ wv130) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];62 -> 71[label="",style="dashed", color="green", weight=3];
33.10/16.36	63[label="Zero",fontsize=16,color="green",shape="box"];64[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];64 -> 72[label="",style="dashed", color="green", weight=3];
33.10/16.36	65[label="primModNatS (Succ wv8) (Succ (Succ wv100))",fontsize=16,color="black",shape="box"];65 -> 73[label="",style="solid", color="black", weight=3];
33.10/16.36	66[label="primModNatS (Succ wv8) (Succ Zero)",fontsize=16,color="black",shape="box"];66 -> 74[label="",style="solid", color="black", weight=3];
33.10/16.36	357[label="primDivNatS0 (Succ wv31) (Succ wv32) (primGEqNatS (Succ wv330) wv34)",fontsize=16,color="burlywood",shape="box"];737[label="wv34/Succ wv340",fontsize=10,color="white",style="solid",shape="box"];357 -> 737[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	737 -> 364[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	738[label="wv34/Zero",fontsize=10,color="white",style="solid",shape="box"];357 -> 738[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	738 -> 365[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	358[label="primDivNatS0 (Succ wv31) (Succ wv32) (primGEqNatS Zero wv34)",fontsize=16,color="burlywood",shape="box"];739[label="wv34/Succ wv340",fontsize=10,color="white",style="solid",shape="box"];358 -> 739[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	739 -> 366[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	740[label="wv34/Zero",fontsize=10,color="white",style="solid",shape="box"];358 -> 740[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	740 -> 367[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	71 -> 591[label="",style="dashed", color="red", weight=0];
33.10/16.36	71[label="primDivNatS (primMinusNatS (Succ wv130) Zero) (Succ Zero)",fontsize=16,color="magenta"];71 -> 592[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	71 -> 593[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	71 -> 594[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	72 -> 591[label="",style="dashed", color="red", weight=0];
33.10/16.36	72[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];72 -> 595[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	72 -> 596[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	72 -> 597[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	73[label="primModNatS0 wv8 wv100 (primGEqNatS wv8 (Succ wv100))",fontsize=16,color="burlywood",shape="box"];741[label="wv8/Succ wv80",fontsize=10,color="white",style="solid",shape="box"];73 -> 741[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	741 -> 81[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	742[label="wv8/Zero",fontsize=10,color="white",style="solid",shape="box"];73 -> 742[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	742 -> 82[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	74[label="Zero",fontsize=16,color="green",shape="box"];364[label="primDivNatS0 (Succ wv31) (Succ wv32) (primGEqNatS (Succ wv330) (Succ wv340))",fontsize=16,color="black",shape="box"];364 -> 376[label="",style="solid", color="black", weight=3];
33.10/16.36	365[label="primDivNatS0 (Succ wv31) (Succ wv32) (primGEqNatS (Succ wv330) Zero)",fontsize=16,color="black",shape="box"];365 -> 377[label="",style="solid", color="black", weight=3];
33.10/16.36	366[label="primDivNatS0 (Succ wv31) (Succ wv32) (primGEqNatS Zero (Succ wv340))",fontsize=16,color="black",shape="box"];366 -> 378[label="",style="solid", color="black", weight=3];
33.10/16.36	367[label="primDivNatS0 (Succ wv31) (Succ wv32) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];367 -> 379[label="",style="solid", color="black", weight=3];
33.10/16.36	592[label="Succ wv130",fontsize=16,color="green",shape="box"];593[label="Zero",fontsize=16,color="green",shape="box"];594[label="Zero",fontsize=16,color="green",shape="box"];591[label="primDivNatS (primMinusNatS wv65 wv66) (Succ wv67)",fontsize=16,color="burlywood",shape="triangle"];743[label="wv65/Succ wv650",fontsize=10,color="white",style="solid",shape="box"];591 -> 743[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	743 -> 616[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	744[label="wv65/Zero",fontsize=10,color="white",style="solid",shape="box"];591 -> 744[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	744 -> 617[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	595[label="Zero",fontsize=16,color="green",shape="box"];596[label="Zero",fontsize=16,color="green",shape="box"];597[label="Zero",fontsize=16,color="green",shape="box"];81[label="primModNatS0 (Succ wv80) wv100 (primGEqNatS (Succ wv80) (Succ wv100))",fontsize=16,color="black",shape="box"];81 -> 91[label="",style="solid", color="black", weight=3];
33.10/16.36	82[label="primModNatS0 Zero wv100 (primGEqNatS Zero (Succ wv100))",fontsize=16,color="black",shape="box"];82 -> 92[label="",style="solid", color="black", weight=3];
33.10/16.36	376 -> 324[label="",style="dashed", color="red", weight=0];
33.10/16.36	376[label="primDivNatS0 (Succ wv31) (Succ wv32) (primGEqNatS wv330 wv340)",fontsize=16,color="magenta"];376 -> 389[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	376 -> 390[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	377[label="primDivNatS0 (Succ wv31) (Succ wv32) MyTrue",fontsize=16,color="black",shape="triangle"];377 -> 391[label="",style="solid", color="black", weight=3];
33.10/16.36	378[label="primDivNatS0 (Succ wv31) (Succ wv32) MyFalse",fontsize=16,color="black",shape="box"];378 -> 392[label="",style="solid", color="black", weight=3];
33.10/16.36	379 -> 377[label="",style="dashed", color="red", weight=0];
33.10/16.36	379[label="primDivNatS0 (Succ wv31) (Succ wv32) MyTrue",fontsize=16,color="magenta"];616[label="primDivNatS (primMinusNatS (Succ wv650) wv66) (Succ wv67)",fontsize=16,color="burlywood",shape="box"];745[label="wv66/Succ wv660",fontsize=10,color="white",style="solid",shape="box"];616 -> 745[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	745 -> 619[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	746[label="wv66/Zero",fontsize=10,color="white",style="solid",shape="box"];616 -> 746[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	746 -> 620[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	617[label="primDivNatS (primMinusNatS Zero wv66) (Succ wv67)",fontsize=16,color="burlywood",shape="box"];747[label="wv66/Succ wv660",fontsize=10,color="white",style="solid",shape="box"];617 -> 747[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	747 -> 621[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	748[label="wv66/Zero",fontsize=10,color="white",style="solid",shape="box"];617 -> 748[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	748 -> 622[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	91[label="primModNatS0 (Succ wv80) wv100 (primGEqNatS wv80 wv100)",fontsize=16,color="burlywood",shape="box"];749[label="wv80/Succ wv800",fontsize=10,color="white",style="solid",shape="box"];91 -> 749[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	749 -> 99[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	750[label="wv80/Zero",fontsize=10,color="white",style="solid",shape="box"];91 -> 750[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	750 -> 100[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	92[label="primModNatS0 Zero wv100 MyFalse",fontsize=16,color="black",shape="box"];92 -> 101[label="",style="solid", color="black", weight=3];
33.10/16.36	389[label="wv330",fontsize=16,color="green",shape="box"];390[label="wv340",fontsize=16,color="green",shape="box"];391[label="Succ (primDivNatS (primMinusNatS (Succ wv31) (Succ wv32)) (Succ (Succ wv32)))",fontsize=16,color="green",shape="box"];391 -> 400[label="",style="dashed", color="green", weight=3];
33.10/16.36	392[label="Zero",fontsize=16,color="green",shape="box"];619[label="primDivNatS (primMinusNatS (Succ wv650) (Succ wv660)) (Succ wv67)",fontsize=16,color="black",shape="box"];619 -> 625[label="",style="solid", color="black", weight=3];
33.10/16.36	620[label="primDivNatS (primMinusNatS (Succ wv650) Zero) (Succ wv67)",fontsize=16,color="black",shape="box"];620 -> 626[label="",style="solid", color="black", weight=3];
33.10/16.36	621[label="primDivNatS (primMinusNatS Zero (Succ wv660)) (Succ wv67)",fontsize=16,color="black",shape="box"];621 -> 627[label="",style="solid", color="black", weight=3];
33.10/16.36	622[label="primDivNatS (primMinusNatS Zero Zero) (Succ wv67)",fontsize=16,color="black",shape="box"];622 -> 628[label="",style="solid", color="black", weight=3];
33.10/16.36	99[label="primModNatS0 (Succ (Succ wv800)) wv100 (primGEqNatS (Succ wv800) wv100)",fontsize=16,color="burlywood",shape="box"];751[label="wv100/Succ wv1000",fontsize=10,color="white",style="solid",shape="box"];99 -> 751[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	751 -> 108[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	752[label="wv100/Zero",fontsize=10,color="white",style="solid",shape="box"];99 -> 752[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	752 -> 109[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	100[label="primModNatS0 (Succ Zero) wv100 (primGEqNatS Zero wv100)",fontsize=16,color="burlywood",shape="box"];753[label="wv100/Succ wv1000",fontsize=10,color="white",style="solid",shape="box"];100 -> 753[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	753 -> 110[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	754[label="wv100/Zero",fontsize=10,color="white",style="solid",shape="box"];100 -> 754[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	754 -> 111[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	101[label="Succ Zero",fontsize=16,color="green",shape="box"];400 -> 591[label="",style="dashed", color="red", weight=0];
33.10/16.36	400[label="primDivNatS (primMinusNatS (Succ wv31) (Succ wv32)) (Succ (Succ wv32))",fontsize=16,color="magenta"];400 -> 598[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	400 -> 599[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	400 -> 600[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	625 -> 591[label="",style="dashed", color="red", weight=0];
33.10/16.36	625[label="primDivNatS (primMinusNatS wv650 wv660) (Succ wv67)",fontsize=16,color="magenta"];625 -> 631[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	625 -> 632[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	626 -> 43[label="",style="dashed", color="red", weight=0];
33.10/16.36	626[label="primDivNatS (Succ wv650) (Succ wv67)",fontsize=16,color="magenta"];626 -> 633[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	626 -> 634[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	627[label="primDivNatS Zero (Succ wv67)",fontsize=16,color="black",shape="triangle"];627 -> 635[label="",style="solid", color="black", weight=3];
33.10/16.36	628 -> 627[label="",style="dashed", color="red", weight=0];
33.10/16.36	628[label="primDivNatS Zero (Succ wv67)",fontsize=16,color="magenta"];108[label="primModNatS0 (Succ (Succ wv800)) (Succ wv1000) (primGEqNatS (Succ wv800) (Succ wv1000))",fontsize=16,color="black",shape="box"];108 -> 119[label="",style="solid", color="black", weight=3];
33.10/16.36	109[label="primModNatS0 (Succ (Succ wv800)) Zero (primGEqNatS (Succ wv800) Zero)",fontsize=16,color="black",shape="box"];109 -> 120[label="",style="solid", color="black", weight=3];
33.10/16.36	110[label="primModNatS0 (Succ Zero) (Succ wv1000) (primGEqNatS Zero (Succ wv1000))",fontsize=16,color="black",shape="box"];110 -> 121[label="",style="solid", color="black", weight=3];
33.10/16.36	111[label="primModNatS0 (Succ Zero) Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];111 -> 122[label="",style="solid", color="black", weight=3];
33.10/16.36	598[label="Succ wv31",fontsize=16,color="green",shape="box"];599[label="Succ wv32",fontsize=16,color="green",shape="box"];600[label="Succ wv32",fontsize=16,color="green",shape="box"];631[label="wv650",fontsize=16,color="green",shape="box"];632[label="wv660",fontsize=16,color="green",shape="box"];633[label="wv67",fontsize=16,color="green",shape="box"];634[label="wv650",fontsize=16,color="green",shape="box"];635[label="Zero",fontsize=16,color="green",shape="box"];119 -> 526[label="",style="dashed", color="red", weight=0];
33.10/16.36	119[label="primModNatS0 (Succ (Succ wv800)) (Succ wv1000) (primGEqNatS wv800 wv1000)",fontsize=16,color="magenta"];119 -> 527[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	119 -> 528[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	119 -> 529[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	119 -> 530[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	120[label="primModNatS0 (Succ (Succ wv800)) Zero MyTrue",fontsize=16,color="black",shape="box"];120 -> 134[label="",style="solid", color="black", weight=3];
33.10/16.36	121[label="primModNatS0 (Succ Zero) (Succ wv1000) MyFalse",fontsize=16,color="black",shape="box"];121 -> 135[label="",style="solid", color="black", weight=3];
33.10/16.36	122[label="primModNatS0 (Succ Zero) Zero MyTrue",fontsize=16,color="black",shape="box"];122 -> 136[label="",style="solid", color="black", weight=3];
33.10/16.36	527[label="Succ wv800",fontsize=16,color="green",shape="box"];528[label="wv1000",fontsize=16,color="green",shape="box"];529[label="wv800",fontsize=16,color="green",shape="box"];530[label="wv1000",fontsize=16,color="green",shape="box"];526[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS wv62 wv63)",fontsize=16,color="burlywood",shape="triangle"];755[label="wv62/Succ wv620",fontsize=10,color="white",style="solid",shape="box"];526 -> 755[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	755 -> 563[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	756[label="wv62/Zero",fontsize=10,color="white",style="solid",shape="box"];526 -> 756[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	756 -> 564[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	134 -> 675[label="",style="dashed", color="red", weight=0];
33.10/16.36	134[label="primModNatS (primMinusNatS (Succ (Succ wv800)) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];134 -> 676[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	134 -> 677[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	134 -> 678[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	135[label="Succ (Succ Zero)",fontsize=16,color="green",shape="box"];136 -> 675[label="",style="dashed", color="red", weight=0];
33.10/16.36	136[label="primModNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];136 -> 679[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	136 -> 680[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	136 -> 681[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	563[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS (Succ wv620) wv63)",fontsize=16,color="burlywood",shape="box"];757[label="wv63/Succ wv630",fontsize=10,color="white",style="solid",shape="box"];563 -> 757[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	757 -> 571[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	758[label="wv63/Zero",fontsize=10,color="white",style="solid",shape="box"];563 -> 758[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	758 -> 572[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	564[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS Zero wv63)",fontsize=16,color="burlywood",shape="box"];759[label="wv63/Succ wv630",fontsize=10,color="white",style="solid",shape="box"];564 -> 759[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	759 -> 573[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	760[label="wv63/Zero",fontsize=10,color="white",style="solid",shape="box"];564 -> 760[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	760 -> 574[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	676[label="Succ Zero",fontsize=16,color="green",shape="box"];677[label="Succ Zero",fontsize=16,color="green",shape="box"];678[label="Succ (Succ wv800)",fontsize=16,color="green",shape="box"];675[label="primModNatS (primMinusNatS wv69 wv70) (Succ wv71)",fontsize=16,color="burlywood",shape="triangle"];761[label="wv69/Succ wv690",fontsize=10,color="white",style="solid",shape="box"];675 -> 761[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	761 -> 706[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	762[label="wv69/Zero",fontsize=10,color="white",style="solid",shape="box"];675 -> 762[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	762 -> 707[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	679[label="Succ Zero",fontsize=16,color="green",shape="box"];680[label="Succ Zero",fontsize=16,color="green",shape="box"];681[label="Succ Zero",fontsize=16,color="green",shape="box"];571[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS (Succ wv620) (Succ wv630))",fontsize=16,color="black",shape="box"];571 -> 579[label="",style="solid", color="black", weight=3];
33.10/16.36	572[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS (Succ wv620) Zero)",fontsize=16,color="black",shape="box"];572 -> 580[label="",style="solid", color="black", weight=3];
33.10/16.36	573[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS Zero (Succ wv630))",fontsize=16,color="black",shape="box"];573 -> 581[label="",style="solid", color="black", weight=3];
33.10/16.36	574[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];574 -> 582[label="",style="solid", color="black", weight=3];
33.10/16.36	706[label="primModNatS (primMinusNatS (Succ wv690) wv70) (Succ wv71)",fontsize=16,color="burlywood",shape="box"];763[label="wv70/Succ wv700",fontsize=10,color="white",style="solid",shape="box"];706 -> 763[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	763 -> 708[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	764[label="wv70/Zero",fontsize=10,color="white",style="solid",shape="box"];706 -> 764[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	764 -> 709[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	707[label="primModNatS (primMinusNatS Zero wv70) (Succ wv71)",fontsize=16,color="burlywood",shape="box"];765[label="wv70/Succ wv700",fontsize=10,color="white",style="solid",shape="box"];707 -> 765[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	765 -> 710[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	766[label="wv70/Zero",fontsize=10,color="white",style="solid",shape="box"];707 -> 766[label="",style="solid", color="burlywood", weight=9];
33.10/16.36	766 -> 711[label="",style="solid", color="burlywood", weight=3];
33.10/16.36	579 -> 526[label="",style="dashed", color="red", weight=0];
33.10/16.36	579[label="primModNatS0 (Succ wv60) (Succ wv61) (primGEqNatS wv620 wv630)",fontsize=16,color="magenta"];579 -> 587[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	579 -> 588[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	580[label="primModNatS0 (Succ wv60) (Succ wv61) MyTrue",fontsize=16,color="black",shape="triangle"];580 -> 589[label="",style="solid", color="black", weight=3];
33.10/16.36	581[label="primModNatS0 (Succ wv60) (Succ wv61) MyFalse",fontsize=16,color="black",shape="box"];581 -> 590[label="",style="solid", color="black", weight=3];
33.10/16.36	582 -> 580[label="",style="dashed", color="red", weight=0];
33.10/16.36	582[label="primModNatS0 (Succ wv60) (Succ wv61) MyTrue",fontsize=16,color="magenta"];708[label="primModNatS (primMinusNatS (Succ wv690) (Succ wv700)) (Succ wv71)",fontsize=16,color="black",shape="box"];708 -> 712[label="",style="solid", color="black", weight=3];
33.10/16.36	709[label="primModNatS (primMinusNatS (Succ wv690) Zero) (Succ wv71)",fontsize=16,color="black",shape="box"];709 -> 713[label="",style="solid", color="black", weight=3];
33.10/16.36	710[label="primModNatS (primMinusNatS Zero (Succ wv700)) (Succ wv71)",fontsize=16,color="black",shape="box"];710 -> 714[label="",style="solid", color="black", weight=3];
33.10/16.36	711[label="primModNatS (primMinusNatS Zero Zero) (Succ wv71)",fontsize=16,color="black",shape="box"];711 -> 715[label="",style="solid", color="black", weight=3];
33.10/16.36	587[label="wv630",fontsize=16,color="green",shape="box"];588[label="wv620",fontsize=16,color="green",shape="box"];589 -> 675[label="",style="dashed", color="red", weight=0];
33.10/16.36	589[label="primModNatS (primMinusNatS (Succ wv60) (Succ (Succ wv61))) (Succ (Succ (Succ wv61)))",fontsize=16,color="magenta"];589 -> 688[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	589 -> 689[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	589 -> 690[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	590[label="Succ (Succ wv60)",fontsize=16,color="green",shape="box"];712 -> 675[label="",style="dashed", color="red", weight=0];
33.10/16.36	712[label="primModNatS (primMinusNatS wv690 wv700) (Succ wv71)",fontsize=16,color="magenta"];712 -> 716[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	712 -> 717[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	713 -> 59[label="",style="dashed", color="red", weight=0];
33.10/16.36	713[label="primModNatS (Succ wv690) (Succ wv71)",fontsize=16,color="magenta"];713 -> 718[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	713 -> 719[label="",style="dashed", color="magenta", weight=3];
33.10/16.36	714[label="primModNatS Zero (Succ wv71)",fontsize=16,color="black",shape="triangle"];714 -> 720[label="",style="solid", color="black", weight=3];
33.10/16.36	715 -> 714[label="",style="dashed", color="red", weight=0];
33.10/16.36	715[label="primModNatS Zero (Succ wv71)",fontsize=16,color="magenta"];688[label="Succ (Succ wv61)",fontsize=16,color="green",shape="box"];689[label="Succ (Succ wv61)",fontsize=16,color="green",shape="box"];690[label="Succ wv60",fontsize=16,color="green",shape="box"];716[label="wv700",fontsize=16,color="green",shape="box"];717[label="wv690",fontsize=16,color="green",shape="box"];718[label="wv690",fontsize=16,color="green",shape="box"];719[label="wv71",fontsize=16,color="green",shape="box"];720[label="Zero",fontsize=16,color="green",shape="box"];}
33.10/16.36	
33.10/16.36	----------------------------------------
33.10/16.36	
33.10/16.36	(60)
33.10/16.36	Obligation:
33.10/16.36	Q DP problem:
33.10/16.36	The TRS P consists of the following rules:
33.10/16.36	
33.10/16.36	   new_primShowInt(Main.Neg(wv30), []) -> new_primShowInt(Main.Pos(wv30), [])
33.10/16.36	
33.10/16.36	R is empty.
33.10/16.36	Q is empty.
33.10/16.36	We have to consider all (P,Q,R)-chains.
33.10/16.36	----------------------------------------
33.10/16.36	
33.10/16.36	(61) DependencyGraphProof (EQUIVALENT)
33.10/16.36	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
33.10/16.36	----------------------------------------
33.10/16.36	
33.10/16.36	(62)
33.10/16.36	TRUE
33.21/16.40	EOF